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• Errata—vol. VIII.
• Chrestomathia: Being a Collection of Papers, Explanatory of the Design of an Institution, Proposed to Be Set On Foot Under the Name of the Chrestomathic Day School, Or Chrestomathic School, For the Extension of the New System of Instruction to the High
• Introduction By the Editor.
• First Preface to the First Edition.
• Second Preface to the First Edition.
• Chrestomathic (a) Instruction Tables. Table I.
• Chrestomathic Instruction Tables. Table II.
• Appendix. No. I.
• Appendix.—no. II.
• Appendix.—no. III.
• Appendix.—no. IV.: Essay On Nomenclature and Classification. *
• Appendix.—no. V.: Sources of Motion.
• Appendix.—no. VI.
• Appendix.—no. VII.
• Appendix.—no. VIII.
• Appendix No. IX.
• A Fragment On Ontology; Now First Published, From the Manuscripts of Jeremy Bentham.
• Note By the Editor.
• Introduction.
• Chapter I.: Classification of Entities.
• Chapter II.: Fictitious Entities Classified.
• Essay On Logic: Now First Published, From the Manuscripts of Jeremy Bentham.
• Note By the Editor.
• Introduction.
• Chapter I.: Logic—what ?
• Chapter II.: Logic, Its Characteristics.
• Chapter III.: Præcognita: Or, Preliminary and General Indications Concerning Logic, According to the Aristotelians.
• Chapter IV.: Of Aristotle’s Predicaments and Postpredicaments.
• Chapter V. *: Mode of Discussion.
• Chapter VI.: Relation of Logic to the Business of Human Life In General, and Therein to Arts and Sciences, I. E. to Disciplines.
• Chapter VII.: Clearness In Discourse, How to Produce It? and Hence of Exposition.
• Chapter VIII.: Of Division.
• Chapter IX.: Of Methodization, Otherwise Termed Arrangement. † ‡
• Chapter X.: Of the Art of Invention.
• Appendix.—a.: Phenomena of the Human Mind.
• Appendix B. Division of Art and Science. †
• Essay On Language; Now First Published, From the Manuscripts of Jeremy Bentham.
• Note By the Editor.
• Introduction.
• Chapter I.: Modes Or Forms of Which Discourse Or Language Has Been Found Susceptible, Viz. Audible, Visible, and Their Respective Substitutes.
• Chapter II.: Uses of Language.
• Chapter III.: Operations Which, In the Character of an Art, Are Performable In Relation to Discourse, Or Language In General.
• Chapter IV.: Properties Desirable In a Language.
• Chapter V.: Of Improvement Considered As Applicable to Language, Or the Means By Which, In So Far As the Particular Language Employed By an Individual Admits of the Possession of Them, the Properties Desirable In Language May, On Each Occasion, Be Secured
• Chapter VI.: Analytical View of the Matter of Thought and Internal Action; Correspondent View of the Matter of Language.
• Fragments On Universal Grammar; Now First Published, From the Manuscripts of Jeremy Bentham.
• Introduction.
• Chapter I.: Definitions.
• Chapter II.: Uses of Universal Grammar.
• Chapter III.: Of the Parts of Speech.
• Chapter IV.: Of the Noun-substantive.
• Chapter V.: Of the Adjective.
• Chapter VI.: Of Pronouns. *
• Chapter VII.: Of Verbs.
• Chapter VIII.: Of Government and Concord.
• Chapter IX.: Of Prepositions, Adverbs, and Conjunctions.
• Chapter X.: Of Interjections.
• Tracts On Poor Laws and Pauper Management.
• Note By the Editor On the Tracts On Poor Laws.
• Situation and Relief of the Poor.
• Observations On the Pauper Population Table Hereunto Annexed.
• Outline of the Non-adult Value Table.
• Outline of a Work Entitled Pauper Management Improved.
• Book I.: Political Arrangements.
• Book II.: Plan of Management.
• Book III.: Collateral Benefits.
• Book IV.: Pauper Comforts.
• Observations On the Poor Bill. Introduced By the Rt. Hon. William Pitt (feb, 1797). : Chapter I. Introduction
• Chapter II.: 1. Under-ability, Or Supplemental-wages Clause.
• Chapter III.: 2. Family-relief, Or Extra-children Clause.
• Chapter IV.: 3. Cow-money Clause.
• Chapter V.: 4. Relief-extension, Or Opulence-relief Clause.
• Chapter VI.: 5. Apprenticeship Clause.
• Three Tracts Relative to Spanish and Portuguese Affairs; With a Continual Eye to English Ones.
• Advertisement For Tract the First and Second; of This Second * Publication, Namely, On the Then Proposed Spanish House of Lords. ( Anno 1820.)
• Tract, No. I.: Letter to the Spanish Nation On a Then ( Anno 1820) Proposed House of Lords.
• Advertisement to Tract the Second.
• Tract, No. II.: Observations On Judge Advocate Hermosa’s Panegyric On Judicial Delays; On the Occasion of the Impunity As Yet Given By Him to the Loyal Authors of the Cadiz Massacre, a Counterpart to the Manchester Massacre; Explaining, Moreover, the Effe
• Advertisement to Tract the Third.
• Tract, No. III.: Letter to the Portuguese Nation, On Antiquated Constitutions; On the Spanish Constitution Considered As a Whole, and On Certain Defects Observable In It; In Particular, the Immutability-enacting, Or Infallibility-assuming, the Non-re-elig
• Letters to Count Toreno, On the Proposed Penal Code, Delivered In By the Legislation Committee of the Spanish Cortes, April 25th, 1821.
• Advertisement.
• Letter I.
• Letter II.: On the Course Taken By the Legislative Committee, to Prevent, Otherwise Than By Punishment Eo Nomine, the Free Examination of Their Proposed Penal Code.
• Letter III.
• Letter IV.
• Letter V.
• Letter VI.
• Letter VII.
• Supplemental Advertisement.
• Securities Against Misrule, Adapted to a Mahommedan State, and Prepared With Particular Reference to Tripoli In Barbary.
• Note By the Editor.
• Chapter I.: Preliminary Explanations.
• Chapter II.: Public Opinion the Sole Remedy—parallel Between the Public-opinion Tribunal and the Official Judicatories.
• Chapter III.: Notification and Publication In Reference to Securities.
• Chap. IV.: The Securities In Detail.
• Part I.: Securities In Favour of the Nation Considered In the Aggregate.
• Part II.: Securities In Favour of Individuals.
• Chapter V.: Hopes of Success For Any Project Having Such Securities For Its End.

CHAPTER VIII.

OF DIVISION.

Section I.

Of the different Modes of Division.

Division is either systematical or unsystematical.* Systematical division is the indication of the species, without the assignment of their reciprocal differentiæ.

Division may be complete or incomplete. The following is an example of complete and exhaustive division:—Vitale est vel sensitivum, i. e. animal, vel non sensitivum, i. e. planta; sensitivum est vel rationale, i. e. homo, vel irrationale, i. e. brutum.

Strict division is bipartite; loose division is multipartite.

Division imports separation; the separation has not been performed if the parcels are not distinct. They are not distinct if any object which is included in the one is included in the other.

Physical and Psychical—under one or other of these two epithets may every possible mode of division be comprised; physical, where the subject matter to be divided—say the dividend—is a physical body or aggregate; psychical, where it is a psychical, or say an ideal aggregate, viz., any aggregate of objects individually assignable or unassignable, for the designation of which a common name, or say appellation, has been provided.

Of these modes, the first—the physical—is the only original and proper mode. It is the archetype of the other. The ideal aggregate is feigned to be—is considered as being a body, a mass of matter; any number of lesser aggregates into which, they being contained in it, it is considered as being capable of being resolved, are considered as so many parts into which it is considered as capable of being divided.

Of these two modes, the psychical is the only one that belongs to the present design; the only one employed in the exercise of the art of logic. In an institute of that art, the physical would not have had any title to a place, had it not been for the light which it may serve to throw upon, the explanation which it may serve to give of the psychical, which has been deduced from it.

It is by arrangement in a line of subalternation in this mode, and no other, that the operation of division, understand of psychical division, can be performed. In the character of a dividend, a name constituting a receptacle of a comparatively larger content, is assumed. Its contents, the articles contained in it, are lodged in two or more other receptacles, so constituted, in respect of extent, as to contain all of them together the exact amount of the contents of the dividend or greater receptacle; the aggregate contained in the greater receptacle being considered as divided, the component articles or units of the aggregate mass are considered as distributed among the compartments which, by such division, have been created.

Section II.

Of bipartite, dichotomous, or perfect Division.†

Be the subject of discourse, and the purpose for which the subject is taken in hand, what they may, correctness, and, with reference to the end, completeness,—these are the two qualities which every man, in proportion as it is his desire that the expression given to his thoughts should be at once true and useful, would be desirous should be found to appertain to it.

For correctness at large, not much instruction can be given, without special reference all along made to the particular nature of the subject, and the purpose in view, i. e. the portion of the field of thought and action operated upon. But, be the purpose and the subject what they may, correctness will, in a considerable degree, depend upon clearness; correctness of conception, on the one part, upon clearness of expression on the other; and, in so far, some instruction on the subject of correctness has already been endeavoured to be administered.

But a case may be brought to view, nor that a narrow one, in which correctness altogether depends upon completeness; insomuch, that if the discourse be incomplete, it is certain, to the exact amount of the degree of incompleteness, to be likewise incorrect. Such, it is evident, is the case, in so far as it has happened to a man to undertake, whether for his own sake, or for the sake of others, that his view of the subject, and accordingly the expression which he gives as the result of that view, shall prove complete.

Towards the accomplishment of an object at once so desirable, and at the same time so much above the reach of human power, logic may perhaps be seen to afford a sort of instrument or engine of greater power than might readily have been imagined. This engine is the exhaustive mode of division.

To answer its intention in the completest manner, an analysis or division must always be throughout dichotomous; “the condividents,” says the logical compend, “ought to be distinct and opposite.”*

Every division is a distribution of individuals:—an assignment of distinct names, simple or compound, univocal or multivocal, under which, as if in ideal compartments, these individuals may be found. These compartments were marked out in the mind of him who distinguished them, marked out by certain properties or qualities in such sort, that an individual possessing such a property, was deemed to belong to such a compartment, an individual possessing such another property, to such another compartment.

For the purposes of discourse, these respective groups of individuals are distinguished by certain names corresponding to these properties;—one name denoting that the individuals it is applied to, the individuals comprisable within the compartment it denominates, possess all of them such a property, or set of properties; another name that the individuals it is applied to possess another property, or set of properties.

Now the use of different names is to distinguish different individuals,—to distinguish the individuals possessing one property, or set of properties, from the individuals possessing another property, or set of properties;—in so much, that if on any occasion a man wish to be understood to say of any one individual, or set of individuals, what he wishes not to be understood to say of another, he may have the means of making himself understood accordingly.

Conceive, then, a group of individuals which are known apart, and distinguished from other individuals by a certain name. Of a part of this whole number of individuals, is it wished to say something which it is not wished to say of the remaining part; in what way, then, is this to be done? There is but one way, that is, by dividing in imagination the whole set, into a certain number of lesser sets,—in the present instance into two; in other words, by distributing the whole assemblage of individuals into two compartments, in one of which shall be contained all the individuals to which it is wished to apply the proposition; in the other, all those to which it is not wished to apply it.—And these compartments that they may be known wherever there is occasion to bring them to view, must be characterized each by a peculiar name.

A division, to answer this purpose, must be exhaustive, must comprise the whole of the subject.

Call the parcel to be divided A: instead of dividing it into two parcels only, such as B and C, divide it at once into three parcels, B, C, and D. Is this division as satisfactory as it might be? No; it will probably be perceived that it is not, though the reason may not be immediately perceived: what then is the reason? It is as follows:—It exhibits all the discongruencies of the three parts or members, but it does not exhibit all their congruencies. Let them be properly distributed and named, that is, so distributed and named, that to no article to which one of the names is applied, could either of the other two names be applied; then, no A that is a B, is either a C or a D; no A that is a C is either a B or a D; and no A that is a D, is either a B or a C. The discongruencies between these several sets are sufficiently expressed,—expressed by the circumstance of their being condivident members of the whole in question, according to a plan of partition which is announced to be an exhaustive one; but on the other hand, there is a congruency between them which is not expressed.—Every A which is a B, agrees with a C, in this that is not a D.—Every A which is a C, agrees with a D in this that it is not a B.—And every A which is a D, agrees with a B in this, that it is not a C.

It is plain, therefore, that A, instead, of being divided into three parcels, B, C, and D, might always, in the first instance, be divided into two, only B and C; for of the three parcels, any two may be consolidated into one, having this property in common, that no A that belongs to either of them, belongs to a third.

And this plan of division is the more simple of the two: first step,—Every A is either a B or a C; second step,—Every B is either a D or an E. In the first step, the attention gets repose: it has but two compartments to examine, in order to see that every A belongs to one or the other of them—which is shown by the circumstance of their having for names the name A, with an epithet, and that no A that belongs to the one, belongs to the other.

In the case of an aggregate of the physical kind, the greatest number of integral parts into which it is capable of being divided, is always a determinate number: in a bushel of apples, containing 400 apples,—400 is that number; in a bushel of wheat, containing 400,000 grains of wheat,—400,000 is that number: in a garden containing every species of plants, suppose 65,536 to be the number of each different species,—65,536 is that number.

These 65,536 plants, each of them of a species distinguishable from every other species; suppose it required so to divide into subordinate and lesser aggregates, the universal or all-comprehensive aggregate, of which, by the supposition, the word plant is the name;—to divide it in such sort, that by a series of successive divisions, from the descriptions given of the products of these several divisions, it should be made to appear in what points each of these 65,536 plants coincided with and in what points it disagreed with the description given of every other; the following is the only mode of proceeding by which the object can be accomplished:—Divide the whole aggregate into two equal parts, or say, divisions; divide each of these divisions into others, which call divisions of the second order,—calling the two first-mentioned divisions, divisions of the first order; each of these divisions, dividing always by two, divide into divisions of the third order,—the total number of divisions eight: and go on, dividing always by two, until the whole number of the component aggregates, thus formed, comes to be 65,536, the assumed number of different species of plants. This mode of division is termed from the Greek, dichotomous; from the Latin, bifurcate,—two-forked.*

This mode of division is subservient to the obtaining of the properties of clearness and correctness, in respect of the conceptions formed and entertained of the subject matter of consideration.

Assured, and altogether incontrovertible, is its all-comprehensive, or say exhaustive, property,—it has place at the very first step, or stage of division,—it has place at every other, be they ever so numerous.

At every step, one article of information it affords as equally incontestible;—it shows a point of agreement and a point of difference between the two results which it brings to view,—point of agreement, the properties belonging to the genus of which they are species,—point of difference, some property which the one has, the other has not.

Still, however, of all the distinguishable species contained in the highest genus, the genus generalissimum, scarcely are there any limits to the number of those which may still remain unincluded. At the same time, still do there, whatsoever be the number, remain means of reaching them by fresh divisions from new sources.

This points to another resource for aiding the mind in the performance of this task.

When, after a first division, the all-comprehensive process has proceeded on in a course of subdivision, till it have picked up as many of the objects belonging to the source as are found capable of being designated by it, if any remain unarrested and unsorted, look out for a fresh source of division, and go on as far as that will carry you.

If any still remain behind unenlisted, look out for another source of division, and so on.

When in a number more or less considerable, divers sources have thus been employed and exhausted, take in hand the sources themselves, apply to them the exhaustive mode of analysis, their eventual points of agreement and difference will, at any rate, be elicited; and if the articles that require to be taken up are not all of them enlisted, some fresh means of enlistment may perhaps be brought to view.

Section III.

Of the Aristotelian Rules of Division.

By the Aristotelians no division was recognised as legitimate, or at any rate as perfect, unless it were exhaustive.

The object to be divided being termed the dividendum, the parts into which it is divided, and which constitute the result of the operation, the dividing, or condivident members;—follow, according to Sanderson, under the name of canons of perfect division, the following rules.†

1. Let the parts be such as that, by their union, the whole shall be recomposed. Membra absorbeant totum divisum.

2. The dividend is greater in extent than any one of its members—it cannot but be equal to all of them when put together, Divisum esto [say rather est] latius singulis suis membris, adæquatum universis.

3. Let the condivident members, in their import, be distinct from, and opposite to, each other;—in such sort, adds an explanation, that they be not liable either in any point to be coincident, or to be confounded. Membra condividentia sint contradistincta et opposita, ita ut confundi nequeant, vel coincidere.

4. The members which are produced by each division, let them be the nearest and immediate members, and let the number of them be as small as may be. Divisio fiat in membra proxima et immediata et (quam fieri commodè potest) paucissima.

From the nearest result thus formed (continues the text in the way of explanation) to the more remote and minute portions descend by subdivisions. A proximis porro ad remotiora et minutiora descendendum per subdivisiones.

The dichotomous mode of division (it goes on to say) is that which has been most approved of, where it can conveniently be employed. Dichotomiæ (dichotomies) sunt laudatissimæ, ubi commodè haberi possunt.

Nor yet (continues this explanation) ought it to be everywhere hunted after, too superstitiously and anxiously pursued, as it is by the Ramæans. Non tamen nimium superstitiosè et anxiè ubique venandæ, quod faciunt Ramæi.

An example, unfortunately not a very unfrequent one, of the conjunction of self-sufficiency and emptiness, may be seen in the account of Dichotomy, as above delivered.

Of the existence of a state of things in which dichotomy can be employed commodiously, intimation is given, and in that state of things, says the instructor, dichotomies are most praiseworthy things. What is that state of things? To any such question not so much as any, the smallest endeavour and attempt, is made to find an answer.

What renders the deficiency the more to be regretted, is the danger which it seems there is of dichotomies being too superstitiously and anxiously hunted after, a danger which, in the practice of the sort of persons here called Ramæans, (meaning, it should seem, the followers of a certain Peter Ramus,) had actually been realized.

Upon the whole, however, of two propositions relative to the matter in question, viz. the dichotomical, or bifurcate mode of division, intimation is hereby given, viz. 1, that there are certain cases in which this mode of division has its use,—2, that there are cases in which,—forasmuch as in those cases, it either has no use at all, or none but what is outweighed by inconvenience,—the practice of employing it may be considered as matter of abuse. Let us see whether some criterion may not be discernible whereby the one of these classes may be distinguished from the other.*

Section IV.

Relation of Synthesis to Analysis.

Psychical, or say logical, division supposes the antecedent existence of psychical aggregation. A bushel of apples, a bushel of wheat cannot be divided until it has been collected. Psychical division has no subject but the ideas commonly called general ideas. These general ideas are all aggregate, or say abstract, ideas, formed by aggregation and abstraction out of simple ones.

Of the aggregate thus formed, the extent is determined and measured by that of the import of the term, the appellation employed for the expression of it.

If of this extent the amplitude be, in a certain degree considerable, the aggregate idea, of which that appellative is the sign, will hardly have been formed, but that, antecedently to its formation, some other aggregate idea, or ideas, less ample in extent and in their whole, contained within the one in question, will also have been formed, formed, and by their respective appellatives designated and fixed.

Thus, in a country in which human society has, in the scale of civilisation, reached the pastoral state, an appellative correspondent to the word animal will scarcely have been formed till after the two appellations corresponding respectively to the words man and sheep have been formed and brought into use: by the observation of the properties which are possessed by all men and not possessed by any sheep, the aggregate idea expressed by the word man will have been formed; by the observation of those properties which are possessed by all sheep and not possessed by any man, the aggregate idea expressed by the word sheep will have been formed.

In the instance of man, the properties which are common to all men, can never have been presented to the senses of any man, but at the same time the properties by which the several different men that have come under observation, have differed from each other, have also, and at the same time, been presented to his senses.

The respective aggregates, composed of the several simple ideas presented by each of these men respectively, may be termed individual aggregates; the aggregate composed of the several simple ideas, drawn alike from the contemplation of all these several men, and fixed and designated by the classical term man, may be termed a classical aggregate.†

In the formation of the aggregate idea corresponding to the term man, in the formation of the classical psychical aggregate termed man, the attention has turned itself aside from all the several simple ideas that have alike been presented by the above-mentioned individual aggregates, turning itself at the same time, and therefore confining itself, to such of those simple ideas as have been presented by every individual belonging to that class comprehended under that appellative; and those to which it has thus exclusively turned itself and confined itself, it may be said by so doing to have abstracted, i. e. drawn off from the rest. It is thus that to the process or operation by which, in this way, classical aggregates are formed, the term abstraction has been applied, and to the classical aggregates themselves the term abstract ideas as well as general ideas.

These explanations premised, the time may have come for observing, that where, of the name of a classical aggregate the extent is to a certain degree considerable, it will scarcely have been formed but by repeated exercises of the process of abstraction, a certain cluster of ideas having been first abstracted, or as it were, distilled from the cluster contained in the several individual, i.e. physical aggregates; and from the product of this first distillation, others, drawn off to compose what may be termed a classical aggregate of the second stage from the bottom; from this product of the second distillation others again drawn off to compose an aggregate of the third stage from the bottom, and so on.

By certain terms, which, in the description of this process, have sometimes been employed, (viz. synthesis and analysis,) it seems as if it had been taken for granted, that the two operations thus denominated were each of them the exact counterpart and converse of the other: that the stages passed over in the one process and in the other would, everywhere, and on all occasions, be exactly the same, consequently the number of those stages likewise; and that whatsoever had by synthesis been put together, the putting of that asunder—of all that, as far as they both went, and no more than that,—was the operation performed by analysis.

Wide indeed from the truth of the case would any such conception, however, be found. Small has probably been the number of the successive operations of the kind in question—viz. abstractions, by which,—correspondently small the number of the stages in passing in or through which, the idea of the most amply extensive classical aggregate of which the mind is capable of forming to itself an idea, has in this way been formed. Of this most extensive aggregate, termed by the logicians of antiquity, the genus generalissimum, being, or existence, or entity, is the name. Five, or at the utmost, six, is the number of successive distillations by which this most sublimated and refined of all abstract ideas has, as appears, been formed. Five, or, at the utmost, six, has accordingly been the number of steps successively taken by the mind in its ascent towards this most exalted pinnacle: five, or, at the utmost, six, the number of stages at which it has stopped. Of these abstractions, these distillations, these steps, these stages, the number corresponds to and is indicated by the number of the ramifications exhibited by the famous Porphyrian Tree,* and of these operations and their results, indication has been given, and at the same time recordation made, by the names respectively employed for the designation of the classical aggregates of different amplitudes which have been their respective products.

Far indeed from being thus limited is the number of aggregates of different orders capable of being formed by the decomposition of that all-embracing aggregate.

Division the first all-comprehensive.—Divide the aggregate of universal amplitude being or substance into its two aggregates immediately issuing from it, you have corporeal beings and incorporeal. For corporeal beings, say, in one word, bodies; as, on the other hand, for incorporeal beings, in one word, spirits.

Division the Second.—Divide the aggregate, corporeal beings, into its proximate component aggregates, living and not living; for its proximate component aggregates, you have those endued with life and those not endued with life, to which latter description belong mineral bodies.

Division the Third.—Divide the aggregate corporeal beings endued with life, into its proximate component aggregates, you have such as are endued with animal life, with sensation as well as life, say in one word animals; and such as are not endued with sensation as well as life, say in one word vegetables.

It was from the observation and contemplation of individual animals, and from the observation made of a quality which such of them as were most exposed to observation had in common with one another, and which was not observed or observable in vegetables, viz. the habitual act of respiration, that the common name, expressive of the faculty corresponding to the exercising of that act, viz. animal, was first formed. Here then are four stages which have place alike in the ascending and the descending line.

But in the descending line, between the psychical aggregate designated by the word animal, and the individuals from the observation of which the existence was deduced of that faculty from which the classical appellative of that aggregate was deduced as above, there has been framed a whole nest of physical aggregates, one within another, in a long chain or series of intus-susceptions or enclosures; and so in the case of vegetables another such series; and so in the case of minerals another such; each such box, with its companions, at the same distance from the all-enclosing box, being the result of a division which at that part had been made of the contents of that larger box, within the limits of which they had all of them been contained.

Though at a few of the highest stages, the steps taken in the ascending line, and the steps taken in the descending line, are coincident, agreeing with one another in number,—in the line of ascent, they were taken with the seven-leagued boots of fairy land, assisted by the wings of the genius Imagination; in the line of descent, they were taken by Observation, retarded at every step by Reflection and Discernment, and in several tracks by Experiment.

Taken in their original acceptation, synthesis and analysis, the synthetic method and the analytic, are packing and unpacking; the latter operation being the exact converse or reverse and counterpart of the former: the road the same, the stages or stepping places the same, the direction alone different, and that opposite.

But very different it has been seen from a course thus simple in description, is the course taken by the mind when occupied in working in the field of logical method.

Instead, therefore, of the number of integral parts contained in a logical aggregate being a limited, in a word, a given, a determinate, or, at any rate, a determinable number, as it would be were it not for the powers—the unlimited powers, of decomposition and recomposition possessed by the human mind,—of these powers, one effect is to exclude as fruitless every possible attempt at circumscribing within any limited extent the number of such parts into which a logical whole is capable of being divided.

In the case of physical aggregates, it may be done; but not so in the case of logical ones. Take a bushel of apples: the number of integrant parts of that aggregate, each apple constituting one of those integrant parts, will be the number of apples that were put into the bushel, neither more nor less.

Some years ago, the aggregate number of all the species of plants then known was estimated at 40,000. Suppose a garden, and in it a specimen of every one of these 40,000 species; 40,000, neither more nor less, is, in this case, the exact number of integrant parts into which the aggregate here in question is capable of being divided. But, upon this supposition, 40,000 is not equal to the number of integrant parts called species, into which the logical aggregate, designated by the names of plant and vegetable, is capable of being divided.

In this supposed case, for every species there is one individual, and no more; for every individual, one species, and no more.

But as, within the extent of one species, an indefinite number of individuals may be, and habitually are, contained, so from any one individual, much more from a greater number of individuals, an unlimited number of species may be deduced.

No new species, it is true, can be formed, except so far as in description it is capable of being rendered different from every species which had been described, before it had ever been described. But, in regard to any of the observable species of natural bodies, taken as they come out of the hands of Nature, this is a condition, of the failure of which no reasonable probability seems to present itself. Take, for instance, the 40,000 different species of plants, that having, some years ago, been said to be the number of those species already known to be in existence. Of these, there exists not any one which has not some property, or aggregate of properties, which is not to be found in any of the others, and which constitute that difference, or say differential character, whereby it stands distinguished from every other. Of these differences, the ideas were respectively formed in the mind by the process of abstraction. They were formed from the observations made of some individual plant or plants, which, at the time of observation, were respectively considered as belonging to those same species. On this occasion, in the formation of any such species, what was done was, not to take for the character, or essence of the species, every mark whereby the individual in question, the individual, or individuals, then and there observed, was seen to differ from all individuals that had ever been observed before, but only some one or other small number of these marks. For, in all the different species of plants that have thus been formed, take any one whatsoever: answerable to the description, how ample and particular soever of that one species, will be found individuals in a multitude absolutely inexhaustible, no two of them so perfectly similar but that, upon a simultaneous comparison, differences, perceptible and describable differences, between them might be found.

Hence it is, that the denomination given to the operation by which the fictitious aggregates created by the joint powers of observation and imagination, or by the imagination alone, is, abstraction. Out of an indefinite number of peculiar marks, by which the several really existing individuals lying open to observation are distinguishable, the mind fixes upon some one or other comparatively small number, and leaving the others unnoticed, and in this way separating them from these others, makes its own use of them, applies to the purpose in question the property, or properties, thus abstracted; establishing them in the character of so many marks, whereby the thus new-formed species, and as many individuals as will ever come to be included under it, i. e. be found to exhibit marks of the same description, are made to stand distinguished, as supposed, from all other species* and individuals that are, every have been, or ever will be, in existence.

Section V.

Misapplication of the Terms Synthesis and Analysis to Geometry and Algebra.

Expressing the difference between Geometry and Algebra is another of the purposes to which the opposite terms Synthesis and Analysis, with the methods respectively denominated from them, viz. the Synthetic method, and the Analytic method, have been employed.

But, between these two branches of science, no such difference or distinction will be found as that of which intimation is given, by that pair of correspondent and opposite appellatives.

In Geometry, quantity is never considered but with relation to figure; in Algebra it never is considered with relation to figure: of the difference between these two branches of Mathematics, this account is at once true, short, and clear, and no other account that is in equal, if in any degree at all endowed with these qualities, will, it is believed, be found.*

In Algebra, as well as in Geometry,—in Geometry as well as in Algebra,—that which is unknown, or supposed to be unknown, is inferred from its relation to that which is known, or supposed to be known: in Algebra, unknown quantities, as expressed by letters, are made known by means of the relation they bear to known ones, as expressed by figures; in Geometry, unknown quantities, as expressed by figure, and supposed to exist as between figure and figure, or parts of the same figure, are made known by means of the relations they bear to known quantities, as expressed by figure.

In Geometry, true it is, that objects are put together; quantities known and unknown are put together; whereupon, of such as are unknown, a description is given, and a conception conveyed by means of the relation they bear to certain known ones.

Of Geometry this is true; nor is it less so when applied to Algebra.

A quantity is mentioned to me, of which I wish to know the amount, it being as yet unknown to me. By the amount, in this case, is always meant the amount in numbers; for, in truth, the subject of Algebra is number—numbers and nothing else. Suppose the number in question six;—in answer to my question, What is the number? the number six is not mentioned by that name; but I am told it is that number which is as great again as number three, or half as great exactly as number twelve. Simple as they are, either of these answers is already Algebra.

And it is thus that, by Algebra, the known and unknown quantities being put together, a description of such as are unknown is given, and a conception conveyed by means of the relation they bear to certain known ones.

[* ] Consolidation is the converse of Division. Division is per descensum. Definition per ascensum; Synonymation per æquum.

[† ] See farther on this subject, Appendix B.; and supra, p. 102, et seq.

[* ] Sanderson, lib. i. cap. xviii.

[* ] On this plan of division, between the number of operations performed, and the number of included aggregates to which the original all-embracing aggregate is reduced by those operations, there exists an established ratio.

[† ] Sanderson, lib. i. cap. xviii. De Divisione.

[* ] With these words the MS. of this section ends, and the subject does not appear to have been further pursued.—Ed.

[† ] To this purpose classical is preferred to general; class having no such determinate ratio as genus bears to species.

[* ] For an exemplification of the Porphyrian Tree, see Table IV. of Chrestomathia, and the correspondent notes, supra, p. 110.

[* ] Among natural philosophers, and more particularly among botanists, the word species has a particular and narrow import, whereby it stands distinguished from that of variety; for the composition of a species, those marks or combinations of matter alone are on this occasion taken and employed, of which it is supposed that on the part of all individuals descended, in the way of botanical descent, from an individual thus described, they will for ever continue to have existence; those which are regarded as being in the contrary case, being, for distinction sake, termed varieties; in such sort that by the term variety is expressed an aggregate subordinate to, and contained within, some one species, of which it is a variety. But of these attributions of everlasting immutability, in the one case, and mutability on the other, there can never be any ground stronger than conjecture; a conjecture which, though by experience found to be so far true as that the acting in conformity to the indication afforded by it, is found to be productive of practical benefit, yet is every now and then found to be erroneous. Accordingly, it is with suitable diffidence that the existence of the sort of distinction in question is commonly announced,—that such a collection of marks is indicative of a species, and not of a variety; that such another of a mere variety, and not of a species. Thus it is that species, in the botanical, or say phytological, sense, differs from a species in the logical sense.

[* ] In algebra, quantity alone, mere quantity, without regard to figure, is throughout the subject; in geometry, the subject of figure—of that other subject, is superadded; in fluxions, the idea of motion is introduced, in addition to those of quantity and figure.