Front Page Titles (by Subject) APPENDIX.—No. VIII. - The Works of Jeremy Bentham, vol. 8 (Chrestomathia, Essays on Logic and Grammar, Tracts on Poor Laws, Tracts on Spanish Affairs)
Return to Title Page for The Works of Jeremy Bentham, vol. 8 (Chrestomathia, Essays on Logic and Grammar, Tracts on Poor Laws, Tracts on Spanish Affairs)
The Online Library of Liberty
A project of Liberty Fund, Inc.
Search this Title:
APPENDIX.—No. VIII. - Jeremy Bentham, The Works of Jeremy Bentham, vol. 8 (Chrestomathia, Essays on Logic and Grammar, Tracts on Poor Laws, Tracts on Spanish Affairs) 
The Works of Jeremy Bentham, published under the Superintendence of his Executor, John Bowring (Edinburgh: William Tait, 1838-1843). In 11 vols. Volume 8.
About Liberty Fund:
Liberty Fund, Inc. is a private, educational foundation established to encourage the study of the ideal of a society of free and responsible individuals.
The text is in the public domain.
Fair use statement:
This material is put online to further the educational goals of Liberty Fund, Inc. Unless otherwise stated in the Copyright Information section above, this material may be used freely for educational and academic purposes. It may not be used in any way for profit.
New Principles of Instruction, proposed as applicable to Geometry and Algebra, principally for the purpose of supplying to those Superior Branches of Learning, the Exercises already applied with so much success to the Elementary Branches.
The following principles not having any particular connexion with the New System, nor having been included in the attestation given in favour of that system by extensive experience, could not present a sufficient title to be included in the Table. In the character of candidates for examination, they are, however, submitted to the consideration of the competent authorities.
It will, at the same time, be a question for learners and adepts in the science to answer to themselves, whether, in this same method, additional promptitude may not be found, as well as positive facilities, for the arranging of geometrical ideas in their minds, and aiding the communication of them, upon occasion, to others, whether in the character of learners or adepts.
Principles, with correspondent Exercises, applying specially or exclusively to Geometry.
5.Key-Presenting, or Contrivance-Indicating Principle.*
Among the five above-mentioned principles, of the four that apply to Geometry, between the first and the fourth an intimate relation will, at first glance, be seen to have place, they being in fact the converse of each other. But, of the fourth, neither the use, nor consequently the nature, can be fully explained till that of the third, to which it is subservient, has been brought to view. In the character of exercises or modes of learning, the utility of them, has, in both instances, received, though in respect of the number of the learners, as yet but on a narrow scale, the attestation of experience.
Geometrical-Relation-Verbally-Expressing, or Purely-Verbal-Expression-Maximizing, or Diagram-Occasionally-Discarding Principle.
Of this principle, the great use is, to serve as a test, and, by that means, an instrument of, and security for, intellection.—See above, Exercise No. 9, Princip. No. 24. (pp. 44, 51.)
Mode of Performing this Exercise.
Without the aid of any diagram, and, consequently, without the use of any of those signs, such as the letters of the alphabet, which, as often as for bringing to mind the figure in question, a diagram or delineation of it is employed, are necessary for designating and distinguishing from each other the parts of the figure, a proposition (in the geometrical sense of the word) and, consequently, the figure which is the subject of it, is expressed in words alone; which words will, of course, be such, that every proposition (using the word proposition in the logical and grammatical sense) which they serve to constitute, will be what is called a general proposition, having for its subject not merely an individual object, but a class, genus, or sort, of objects.
Without the aid of any diagram, any such description, can it then be given in such manner as to be intelligible? From his own experience,—from experiments made at his suggestion, in the instance of three persons, at two widely distant points of time,—the writer of these pages is enabled to answer in the affirmative. In the instance of two of them, the experiment began with Euclid’s Elements, and went no farther than the first six books. In the instance of the other person, it began with one of the most copious, and, at that time, best approved institutional works on Conic Sections,* and was continued, if he misrecollects not, to the very end. In both instances the papers in which the descriptions in question are contained, are in his possession, though at this moment not accessible. In all three instances the learners were of a self-directing age. What was done, was done purely for the satisfaction as well as instruction of the operators themselves, and not in the way of exercise for the satisfaction of a teacher; for, except the learners themselves, in no one of the three instances was there any teacher in the case. In the case of the Conic Sections, though he himself neither did at that time read, nor since then has ever read, so much as a page of what was written, yet, so it was, that the whole of it was written in his presence: and well does he remember the tokens of self-satisfaction, as well as promptitude and velocity, with which the performance of the self-imposed task, continued as it was during a course of some months, was accompanied,—symptoms which, for such a length of time, nothing but the full sense of continual success could assuredly have produced.
But, without a perfect conception, or, at any rate, without the supposed consciousness of such perfect conception, a task of this kind and of this length never could have been performed. From first to last no diagram having been employed, consequently, no reference to any actually drawn diagram made, it is only by words—by words of a purely general nature, that the several relations borne by the several parts of the figures in question to each other, that the ideas in question, could have been expressed. But, in this way, the ideas in question having actually been expressed, how much superior, in the character of an intellectual test, this species of exercise cannot but have been, in comparison with any other, will, it is hoped, without entering into any diagrammatical exemplifications, be found sufficiently intelligible.
For this purpose a particular mode of designation, applicable to the several parts of a geometrical figure, required to be devised, and was devised and settled accordingly. For example, in order that such words of designation as right, left, top, bottom, and the like, might be capable of being employed, it was necessary that, of the figure of which a description was to be given, the position should be determined. But, once for all, care was taken to declare and record, that it was merely for the purpose of description and exemplification that, so far as concerned position, this declaration was made; and that, in whatsoever position the figure were placed, the species it belonged to and the properties it possessed would be the same.
I. Enunciation or enunciative part,—enunciative, viz. of the proposition to be demonstrated.†
II. Demonstration or demonstrative part. In every portion of discourse, to the whole of which the term proposition—a proposition—is customarily applied by geometricians, these two parts at least will be found. To these will in most instances be found added what may be termed the direction—directive part, or preparatively directive part, viz. the part by which direction is given for the operation to be performed, and for necessary additions to be made to the originally exhibited or conceived figure, for the purpose of preparing the ground for the demonstration.
In Euclid’s Elements without exception, and for a considerable extent, if not for the most part, in other books, in the higher branches of geometry, in giving expression to the enunciation as above, the mode of purely verbal designation here proposed for all the above-mentioned parts is actually employed. But, so far as this practice is pursued, the propositions (taking the word proposition in the logical sense) are all general; the ideas conveyed by them are all general ideas; and in this original state it is, and without need of extension, that, in so far as they have place in the mind, they lie there ready to be applied, upon occasion, to all such individual figures as are respectively comprehended within their import.
But now, instead of being thus general, suppose the mode of designating them such, as to confine the application of them to the individual figure, exhibited by the diagram that accompanies them. For example, instead of saying, a square having for its side the longest boundary of a right angled triangle is exactly equal to both the squares taken together, which have for their sides respectively the two other boundaries of the same triangle, suppose the proposition worded thus—in the triangle in question (describing it by letters,) the square having for its side such boundary (describing it again by letters,) is exactly equal to both the squares taken together, which have for their sides respectively the two other boundary lines, (describing them also by letters.)
By such a mode of expression or designation, if it be supposed that no other more general mode is ever added or substituted to it, what general ideas—what practically applicable instruction would be conveyed? Answer;—Surely not any. For rendering the proposition susceptible of conveying any such instruction, what would be the course necessary to be pursued? Answer;—To substitute to this diagrammatical and individualizing mode of expression or designation, the purely verbal, and thence general, mode of expression or designation here in the first place brought to view. Here, then, before any real acquisition in the way of science can be made, there is an additional operation that must be performed,—an additional operation requiring much greater exertion of mind to perform it, as well as a much greater strength and maturity of mind to be able to perform it, than the original one.
This general mode of expression or designation, which, to the purpose of useful and practically applicable intellection, will, in the case of the enunciation, as above explained, be acknowledged to be, at least to a very considerable extent, absolutely necessary, will, it is hoped, in the case of the other two parts of the proposition, be acknowledged to be at least useful; useful, viz. on the supposition of its being practicable: and that it is practicable hath, as above, been already proved by repeated experience, without any contrary experience to oppose to it.
Diagram occasionally discarding principle: by the word occasionally, thus inserted in the composition of this, one of the names to be employed for the designation of this principle, intimation is given, that upon the diagrammatical, i. e. the ordinary mode of designation, no permanent exclusion is proposed to be put: that it is in aid, and not in lieu, of that ordinary mode, that the one proposed—the purely verbal mode—is proposed to be employed. So far is any such constant exclusion of the diagrammatical mode from being intended, that by the principle mentioned in the fourth place, this diagrammatical mode is to some purposes, by means of a set of adapted signs, proposed to be employed by itself: by itself, and thereby to the occasional and temporary exclusion of the verbal mode.
That, under the burthen imposed by the labour of forming, by means of a description given in the purely verbal mode, a conception of the figure meant to be presented to the mind, considerable relief will very frequently be afforded by a glance at the figure, cannot admit of doubt. For facilitating conception, in the first instance, the verbal mode and the diagrammatical mode will thus be employed in conjunction in conjunction, and so far, perhaps, with not very unequal advantage.
In comparison with the diagrammatical mode, no mean advantages will, it is believed, be found attendant on the purely verbal mode.
1. One is—the giving to the general ideas, the presence of which in the mind is, in every instance, necessary to intellection, a sort of perpetual and uniform fixation, by means of a determinate set of words,—thoroughly considered, apposite, and thereby, sooner or later, perfectly adequate words,—instead of leaving these general ideas to be, on each individual occasion, in a hasty, and, therefore, frequently in an inadequate manner, caught up in the way of abstraction: caught up without words for the fixation of them; and therefore, in case of error, without possibility of correction, there being no permanent or determinate object to which correction can apply.
2. The other advantage is—the saving that will frequently be made of the expense of time and labour, necessarily attached to the making out the several parts of the figure, by means of the letters employed in the designation; and, moreover, of the perplexity, and, as it were, mental stammering, with which the operation of ringing the changes upon these letters is, especially in unpractised minds, so apt to be attended. Sometimes, it is true, it may happen that, in addition to the general glance taken of the figure, recurrence to these letters may, for the purpose of forming a conception of this or that part of it, be found necessary. But at other times it may happen that no such recurrence will be found necessary: the need of it having been effectually superseded by the purely verbal description, by means of the general words contained in it.
A question here presents itself, as one which, by any learner in geometry, might not unaptly be put to the author of any institutional work, by means of which he was occupied in teaching himself. The directions and reasonings, the only use of which is to convey so many general ideas, why is it that for giving expression to them you have not (while in the case of the enunciation made of the proposition to be demonstrated you actually have) employed the correspondent general words? These general words, did you know where or how to find them? Then, why is it that you have not found them and produced them? With all its load of unavoidable and immoveable difficulty, is not the task heavy enough for us? Must this additional, this moveable difficulty, be left pressing on us? These same general and only adequate words, is it then that you have not been able to find them? You, to whom, by so many years of study, and so often continually repeated applications to practice, the subject has been rendered so perfectly familiar,—with what degree of consistency can you entertain any such expectation as that we, to all of whom the subject is perfectly new, and many of whom are, in various degrees, dull or inattentive, or both, should be able to accomplish at the moment, and at every moment, a work, which our master has not been able to perform in so many years?*
Thus much will not, it is believed, be found open to dispute. The only idea which, in any case, is conveyed by the individual figure in question, as delineated in the diagram in question,—the individual figure, of which the parts are designated by the letters of reference,—is an individual one. But, except in as far as by abstraction from these individual ideas, general or specific ideas are formed, from no number of such individual diagrams, can any general ideas, applicable to any practical purpose, be deduced. This process of generalization, the learner in question, is he competent to the performance of it? If he is, then proportioned and equal to the number of these acts of generalization that he is competent to, and performs accordingly, is the stock of mathematical science which he actually lays up, at any rate, for the time. But, in any given instance, suppose a general idea thus formed, and for the moment laid up, note well the great disadvantage under which the operation is performed. No precise form of general words has the learner before him, by which this idea of his stands expressed, and by which, were he provided with it, the idea might, as it were, be anchored in his mind. If the occasion of making application of it recur with a certain degree of frequency, he will, notwithstanding the want of apt words for the expression of it, retain it in a state fit for use. But let it, for a certain length of time, be unemployed, the words which should have held it fast being wanting, the consequence is, it drops out of his mind, and as well might it never have been lodged there.
Whatsoever form of words is necessary and sufficient to the giving expression to the general idea, which the individual diagram with the letters which all along apply to it, are intended to convey,—now, suppose it, as in the case of Euclid’s proposition as above-mentioned, ready provided, and extended not only to the propositions, but also to the demonstrations, and the directions by which the preparatory additions to the figure are described. Things being in this state, the idea from the very first presents itself to his mind, in all its generality: in the only garb and condition in which it is capable of being applied to use. If then so it is, that, from the proposition in question, demonstrations, as above included, he has succeeded in deducing any idea at all, that idea is a general one, an idea fit for use, it is not a mere individual idea, having for its necessary support the individual figure. In that case, employing the general words in question, or others that are equivalent to them, he will, in addressing himself either to a teacher for the purpose of proving, or to a learner for the purpose of communicating, his proficiency, find himself, on the occasion of any line, for instance, which, for the purpose of the demonstration, requires to be drawn, in a condition able to describe it by words designative of the relation which, when drawn, it will bear to the other parts of the figure: he will not say, draw A B, or draw A C, leaving it for the party addressed to make discovery of the place which the line, when drawn, will occupy; a discovery which, otherwise than by seeing the diagram, and thereupon copying that part of the diagram, he will, for want of the general words in question, find it impossible to make.
True it is, that without actually having given, either by word of month, or in writing, any such purely verbal description of it, to have framed and entertained a clear, correct, and complete conception of the proposition in question, be it what it may, is altogether possible; if it were not, scarcely perhaps would so much as a single person be found by whom, in relation to any such proposition, any such conception had ever been entertained. But not the less true is it, that by one who, upon being required, were to find himself ultimately unable to give, in relation to it, that sort of purely verbal description, no such clear, correct, and complete conception of it could really be entertained.
Of the propositions themselves (considered as distinct from the demonstrations and the introductory steps, as above) by Euclid a description of the sort in question—a purely verbal description—has, as in every instance, been actually given. But, when he comes to the introductory steps, (preparatory additions,) then it is, that, as if to save the trouble of finding for his conceptions an adequate assistance of general expressions, having given his diagram, it is to the component parts of that individual diagram, as indicated by the letters of the alphabet, that he refers us. Draw the line A B, or draw B C, says the direction that he gives us. But on what account was it that he required us to draw this line? Plainly on this account, and no other, viz. on account of a certain relation which the line so drawn would, when drawn, be found to bear to the other parts of the figure; it is only in virtue of some such relation that the lines, when drawn, can be applicable to the purpose. But, by the letters A B, or B C, is this relation in any degree expressed? Not it, indeed. That same instructive, that same intellection-proving, and, at the same time, intellection-conveying mode of expression which he uniformly applied to his propositions,—i. e. the mere grammatical sentence, enunciative, in each instance, of the geometrical relation, the existence of which is thereby undertaken to be demonstrated—how happened it that he did not continue the application of it to his demonstrations, and the directions given for the preparatory steps? Had the question been put to him; for despatch, would probably have been his answer. But, for want of knowing very well how, would not improbably have been the more correct answer; and, at any rate, what should be not only a correct answer, but, moreover, an addition to such effect as would have been necessary to the forming a complete one. For the composition of a book of instruction upon that plan, the human mind had not, in his time, made sufficient advance. The mathematician is one sort of person; the logician is another. It is by generalization that all inventions are accomplished; most discoveries made. But generalization by wholesale, generalization upon an all-comprehensive scale, is the work of the logician: it is, by the same process, performed upon a comparatively small scale,—performed, as it were, by driblets,—that the particular discoveries in Mechanical Philosophy, in Chemical Philosophy, and even in Mathematics, have been made. But it is one thing to make progress in a certain track; another thing to be able to give a description, a clear, and correct, and complete, and easily apprehensible, description of the progress so made in that same track.
Thus it is, that in this as in so many other parts of the field of science, infancy, under the preposterous name of antiquity,—infancy continues to set the law to maturity; inexperience to experience.
In regard to this gap in the mass of requisite instruction, ask for the reason of its existence; if, by the word reason, be meant a productive cause, having its root in the essential nature of the subject, no such reason will be found. But if, by the word reason, be meant a cause having its root in the nature of the human mind, there is nothing in it but what, in every part of the field of thought and action, lies constantly under our eyes.
Authority and habit.—In these two words, in as far as sinister interest is out of the question, may be seen the cause of all deficiencies in the system of instruction which (time for the operation not having been wanting) continue unsupplied. Authority,—the authority of great names: habit,—the habit of continuing to travel without reflection, in the track in which, with or without reflection, men have begun, or continued to travel already.
In the use of general terms for giving expression to the correspondent general relations between the correspondent sorts of figures and parts of figures, Euclid, the father of Geometry, went not beyond the collection of words expressive of the purely enunciative part of the discourse called a proposition; for the demonstrative part and the preparatory part he left it to the learner to deduce the general ideas from the individual objects, presented by the individual diagram, in company with the words, of which, by the reference made to it, the import was in like manner individualized. Can there be any need of doing, or so much as use in doing, that which, in the eyes of the father of the science, was not fit, or at least not necessary, to be done?
The papers in question, in and by which application was, so long ago made, of the purely-verbal-expression-maximizing principle to a large portion of Euclid’s Elements, not being immediately accessible, an exemplification of it applied to the first proposition of these Elements, has, by the writer of these pages, been hastily formed for the purpose, and will be found in the Appendix.* To save recurrence to books, along with it is given a reprint of the same proposition as exhibited in the customary form in Mr Professor Playfair’s Elements of that science.
Whether in any, and if in any, in what degree, the conception of the subject is facilitated by the mode here proposed, is a question, to the answering of which, an understanding matured, and in other respects not ill furnished, but by which little or no attention has happened to have been bestowed upon this branch of science, will be in a particular degree well adapted.
Mode of making the experiment, to try the utility of the proposed mode, so far as concerns facility of conception.
1. Try whether the purely verbal mode of designation is intelligible without a diagram. For this purpose, the diagram, as given without the letters of reference, and the diagram, as given with the letters of reference, should both be covered.
2. If it be not perfectly or readily intelligible without a diagram, uncover that diagram which has not any letter of reference.
3. If it be not perfectly or readily intelligible even then, uncover now the diagram which has the letters of reference.
As to the giving facilities to conception, by this advantage, should it in any way be found included among the effects of the proposed mode, not only in the instance of each scholar would the labour be alleviated, and expenditure of time diminished, but in a greater degree than antecedently to experience would perhaps be expected, the number of the scholars reaping from this part of the instruction substantial benefit would be increased.
Even in the grammar school, under the old and still subsisting mode, large according to an eminent and most amply experienced master,* is the proportion of scholars by whom, at the end of a long series of years, no efficient learning is obtained. Larger, again, by far, among those by whom, after years spent in the endeavour, on one part, to infuse learning in this shape, on the other to imbibe it, [is the proportion by whom,] no efficient stock of it is obtained.
Under the name of the Ass’s Bridge, the 5th proposition, in the very first book of Euclid, is the known stumbling-block, the ne plus ultra to many a labouring mind.† Why? Because, to the purpose of clear conception, to the purpose of efficient instruction, the method traced out by Euclid, and followed blindfold for so many ages, is lamentably incompetent. In the Chrestomathic School, it may be presumed with some confidence, there will be no Ass’s Bridge.
The Ass’s Bridge having thus presented itself to view, the temptation of exhibiting this additional test of the utility of the purely verbal-expression-maximizing principle was too strong to be resisted. To the labour of giving expression in this mode to Euclid’s first proposition, has, accordingly, been added in the Appendix, the corresponding-like labour applied to the 5th proposition, called the Ass’s Bridge.‡
To what length in the field of mathematics this substitution of ordinary and unabbreviated language, to scientific and abbreviated, is in the nature of the case capable of being carried with advantage, can scarcely be determined antecedently to experiment. What is certain is, that in the details, in the actual performance of algebraical operations, i. e. on any other occasion, or for any other purpose than that of explanation, practised in the way of instruction, it cannot be carried over the whole. For in as far as pursued in detail, the system of abbreviation is essentially necessary to the performance of the operations themselves, when taken in the aggregate. But for this assistance, a long life might be consumed before more than a small part of those which have actually been performed, could be perused and understood, after their being respectively invented, not to speak of the labour expended in the course of the invention.
But while the uses of ordinary language were confined to the giving expression to principles, i. e. to propositions of so general and extensive a nature, as that by each of them large bundles of details, bundles more or less large and copious [might be embraced,] whether a degree of progress, considerable enough to be productive of sensible advantage, might not thus be made, is a matter to which experiment may be looked to for a determinate answer; and in the meantime the conjectures, in anticipative views taken of the subject by the learned, for a provisional one.
In proportion as in the character of principles, a number of these propositions, all expressed in ordinary language, are brought to view,—and laid before the reader all of them in one view,—such point of conformity and disconformity will, it may be expected, be found to have place among them, as will enable the mind to bind a number of them together into bundles, capable of being each of them designated by a term of more extensive import, these bundles into still smaller bundles, and so on: at each step of this abstractive process, the number of the bundles thus diminishing, and the extent of each thus receiving increase. To what length the nature of the case would suffer this process to be carried on, the greatest adept would scarcely venture to predict. But, that the further it were carried on, the more clear and complete would be the view thus rendered obtainable, will hardly be regarded as matter of dispute.
That, for this purpose, changes would require to be made in the stock of expression afforded by ordinary language, seems scarcely to admit of doubt: some terms might require to be added, others substituted, to that part of the ordinary language which is applicable to the purpose. But it is in the way of definition that the whole of this business might be despatched. In these definitions, in as far as the word had been already employed in different senses, the object and effect of the operation would be to fix the import: in as far as it was new, to give to it, for the first time, an import applicable to the subject. In all these cases, in the first instance, the defined word alone would be the word which would be foreign to the stock of the ordinary language: to the ordinary language would belong all the words employed in the explanation of it. True it is that, when once a word in itself new, and thence foreign to the ordinary language, had thus received its explanation, viz. in ordinary language, it then, without inconvenience, might be employed, and of necessity would be employed, in the explanations given of other such new words.
But in comparison with the perplexity produced by the introduction of an extensive system of new characters, the utmost perplexity that would be produced by the introduction of new words, supposing them to be, in a moderate degree, expressive, and at the same time elucidated, by explanations expressed in ordinary language, would be inconsiderable indeed, especially if the number of them was so insignificant as to admit of their being, in the form of a synoptic table, spread under the eye all together at one time.
Practical-use-indication maximizing, or practical-application maximizing principle.
Signal would be the service rendered to mankind, if, by some competent hand, a line were to be drawn between those parts in the field of Mathematics, the contents of which are, and those the contents of which are not, susceptible of practically useful application.
1. In some instances the whole contents of the field are of this useful kind, and, in respect of right practice, absolutely necessary. Such is the case, for example, with the doctrine of probabilities, so far at least as the application of it is confined to such events as, besides being actually exemplified, or liable to be exemplified, are of a nature interesting to, that is, liable to be productive of pain or pleasure to, mankind. In these instances, figure has no place. To the field of Arithmetic, deloporic or adeloporic—simple, or algebraical,—(manifestly expressive or non-manifestly expressive)—this class of instances is confined. Such, again, is the quantity added to any mass of money, or money’s worth, by allowance paid for it, whether in the shape of interest or discount.
2. Another class of instances there is, in which the whole contents of the field are of this useful, and, at the same, necessary kind. The field is the field of uranological geography or topography: the field of astronomy, in as far as the mass of art and science belonging to it is applicable to the ascertainment of the extent of portions, or the relative position of single places or spots, on the earth’s surface.
In this class of instances not only number but figure is a necessary object of regard. The field to which they belong lies therefore within that portion of the field of mathematics, which is common to geometry and arithmetic.
3. In another class of instances the contents of the field are, beyond question, occasionally useful, but without being constantly and in every part of it necessary. This field is the field of Mechanics, taken in the largest sense in which that appellation is employed.
In this field, the most general and intelligible use consists in the saving of what may be called fumbling: viz. experiment—first experiments or observations employed to ascertain some general matter of fact, which, by calculation alone—calculation grounded on existing experiments and observations, might, without the aid of fresh ones made on purpose, have sufficed.
How great a quantity of labour, and thereby of the matter of wealth, and of time,—and thereby of the matter of life, which might have been saved by mathematical calculation, has been wasted in fumbling, may be more easily imagined than ascertained.
In this case too the field belongs to that portion of the field of mathematics, which is common to algebra and geometry.
Between what is susceptible of practically useful application, and what is not susceptible of practically useful application, why is it that this line ought to be drawn? What is it that calls upon professional men engaged in the teaching of this branch of art and science, to take this task upon themselves?
Answer;—That persons who either cannot afford, or on any other account are not willing to bestow, any part of their time upon any parts of the field, from which no practical use can be reaped, may not, by ignorance of this distinction, be drawn into any such misapplication of time and labour. A moral transgression, though unpunishable, an injury analogous to the crime called fraudulent obtainment, or obtainment of money on false pretences, would be the act of that teacher, who, knowing that the purpose of the pupil was not to go beyond the productive part of the field, should, for want of the land-mark or warning-post in question, here called for, lead him upon the irremediably barren part of the field.
Of a proposition which, in any shape, has, as above, a physical use, the use will be found exemplified either in some branch or branches of physical art and science, i. e. of Natural Philosophy, as it is so commonly, though unaptly, called, or in the doctrine of probabilities. Of these branches, see a list, though not exactly a complete one, in Table I.
Without having any immediate application to any branch of physics, as above, and therefore without having any immediate use, a proposition may still have a practical use. If it has, this use may, in this latter case, be termed a preparatory use.
A proposition belonging to geometry, suppose it to be itself not susceptible of application to any branch of physics, but suppose it, at the same time, necessary to the demonstration of another which is susceptible of such application. Immediate use it has none; but it has a preparatory use.
Such preparatory use may, by any number of degrees, be removed from the immediate use. A proposition is of no use but in respect of its being necessary to the demonstration of another; that other is of no use but in respect of its being necessary to the demonstration of a third: let a series of this sort be of any length, if at the end of it we come to a proposition which has an immediate use, every proposition in the series has its use, for every one of them has a preparatory use.
In the Chrestomathic school, time will not allow of the giving admission to more than a comparatively small part of these mathematical propositions, which are not only practically true, but practically useful: much less of the giving admission to any that possess not this essential requisite.
In so far as practicable, it will, therefore, be highly useful that selection should be made.
For making the selection a principle of distinction, has already just been pointed out; and for the making application of it a process, mainly mechanical, is altogether obvious.
In relation to each of the several branches of natural science, as above, look over some work or works the most correct, and upon the whole the most complete that can be found, in which, to any part of the physical subject is question, application has been made of mathematical, and, in particular, of geometrical propositions: in as far as this has been done, the work is a work of what is called mixed mathematics. In each of these works, note under the occasions in which, and the places in which, use has been made of any proposition, beginning at least, if not ending with, those, for example, of Euclid. From them make out a list or table, headed with the names of these several propositions.
This done, in any new edition published of that elementary work [Euclid,] under the head of each proposition, make reference, if not to the several instances, at any rate to some of the most eminently useful of the instances, in which application has thus been made of it; ranging them under the head of the branch of physical science, to which they respectively belong, and referring to the work in which they have been found. So in the case of those whose use is of the preparatory kind. For labour, whether of body or mind, there exists not any more effectual sweetener than the indication of use. That branch of useful art or science is scarcely to be found, in which, for the acquisition of the instruction it affords, labour of mind so intense, or in itself so irksome, is necessary as in Mathematics.
In the existing mode, the manner of administering the instruction is pregnant with perplexity to the learner, and no such indication as above, is employed to sweeten it. In the now proposed mode, the manner in which the instruction is administered will be found much less perplexing; and, in the addition of the practical use, the labour will find its natural edulceration, the indication of the reward naturally attached to it.
* By the humble and sincere desire of rendering himself useful to mankind, by contribution made to an association which has for its object the giving extent, in every sense of the word, to useful instruction, the writer of these pages finds, and that not without very serious and unfeigned regret, that he has fallen into a sort of system which, at Edinburgh, and probably in many other seats of learning, is deemed heretical; for true it is, that such is his fortune, and, in this respect, his misfortune, that he belongs to that school to which, in 1793, the late Dr Beddoes, in 1811, the present Mr Professor Leslie, not to speak of Mr Locke, have been found to belong. To this same school it was, moreover, his good or ill fortune to belong, as from what is above stated may be suspected, many years before the work of Dr Beddoes, on this subject, was published, and perhaps before that ingenious philosopher belonged, or had even been sent, to any school.
To him it is, not a matter of exultation but of regret, not a pleasureable reflection, but a painful one, that if this his view of the matter should be found correct and useful; if, by means of institutional books, composed upon the purely-verbal-expression-maximizing principle, geometry, for example, should be found to be learned at the same time, either more easily or more thoroughly than in the present mode, all the institutional books at present existing on this subject, would be found comparatively useless, and cease to be the subjects of purchase.
That without regret, or even without displeasure, such a state of things should be contemplated by persons interested, either in respect of pecuniary matters or in respect of reputation, in the existing stock of writers on this subject, is not consistent with human nature; and if, in this instance, that line of conduct should, on the part of persons so circumstanced, be pursued, which, in all other instances, has been pursued, the object of general research will be, by what means the reputation of the idea, and thence of him by whom it was advanced, may most effectually be depressed.
But if, by considerations of this sort, men, to whom it seemed that they had anything new and useful to offer, had been induced to suppress them, no improvement would ever have been made in any part of the field of art and science. And, in the present instance, a circumstance fortunate to the heretic is, that in no case could the resentment of orthodoxy fall lighter than in his.
Of this school, in as far as concerns Mathematics, the principle or principles may thus briefly be brought to view.
Otherwise than in so far as it is applicable to physics, Mathematics (except for amusement, as chess is useful) is neither useful nor so much as true. 1. That, except as excepted, it is not useful, is a proposition which, when clearly understood, will be seen to be identical: a proposition disaffirming it would be a self-contradictory one. 2. That it is not so much as true, will, it is believed, be found, upon calm and careful reflection, to be little if anything different from an identical, proposition; a proposition contradicting it, little if anything different from a self-contradictory one.
A proposition in Mathematics, [Geometry excepted] what is it? A proposition, in which physical existences, i. e. bodies and portions of space are considered in respect of their quantities, and nothing else.
A proposition in Geometry, what is it? A proposition in which physical existences, as above, are considered in respect of their figure, and thereby in respect of their quantity, but in no other respect.
A proposition, having for its subject the geometrical figure called a sphere, is a proposition having for its subject all such bodies as can with propriety be termed spherical bodies, as likewise all such individual portions of space, as can with propriety be termed spherical spaces; and so in the case of a cone, a cube, and so forth.
In as far as any such individual portions of matter and space are actually in existence, the proposition is actually true. In as far as any such portions of matter or space may be considered as likely to come into existence, or as capable of coming into existence, it may be considered as having a sort of potential truth, which, as soon as any such portions of matter or space come into existence, would be converted into actual truth.
In point of fact, no portion, either of matter or space, such as agrees exactly with the description given by Mathematicians of the sort of figure called a sphere, ever has come into existence, (there seems reason to believe.) But, by this circumstance, though in a strict sense,—that is, to the mere purpose of absolutely correct expression,—the truth of all propositions concerning the sort of figure called a sphere is destroyed; yet, in no degree is the utility of any of them either destroyed, or so much as lessened; in no degree is the truth of them destroyed or lessened with reference to any useful purpose, with reference to any purpose, or in any sense, other than a perfectly useless one.
A general proposition which has no individual object to which it is truly applicable, is not a true one. It is no more a true proposition than an army which has no soldier in it is a true army; a fagot which has no stick in it, a true fagot.
A Mathematical proposition which has no individual portion of matter or space to which it is truly applicable, is a general proposition which has no individual object to which it is truly applicable.
Among the sorts of things which are the subjects of mathematical propositions, there is not one which contains any individual objects which, with strict truth, can be said to belong to it.
There are, however, many which, without any error attended with any practical inconvenience, may be considered as belonging to it. These then may, without practical disadvantage, and, at the same time, with great practical advantage, be considered as having individuals belonging to them; be considered, in a word, as true.
Take any body—a billiard ball, for example—that is intended to be spherical, assuredly it is not exactly spherical. Of all the geometrical propositions which have the sphere for their subject, there is not one of them that is exactly true when applied to it; but it is so near to the being spherical, that all these propositions may, without any material error, be applied to it.
Among a number of billiard balls all perfectly capable of being applied to the use for which they were designed, some will come nearer to an exactly spherical figure than others. The nearer any one comes to this figure, the nearer, in that instance, will these several propositions come to the being exactly true.
From the list of the applications, and thereby of the uses made of the several propositions of pure mathematics, the order of invention will follow as a sort of corollary. Amongst other things it may, on that occasion, be seen how, in point of fact, mathematical ideas—how all mathematical ideas—have their root in physical ones—in physical observations. The actual applications thus made to practice,—the indications thus afforded, will be pregnant with immediate practical uses. The general observations deduced as above, in the way of inference, from those observations of detail, will be but matter of curiosity and theory. Curious as it may be it will not be very easy to find the class of persons to whom it will be acceptable. To the non-mathematician it will be neither very interesting nor comprehensible. To the mathematician it will not be very acceptable. That, before any such surface as a circular one had any existence, all its radii were equal, is, in his creed, as in Montesquieu’s, a fundamental article. That fluxions and equations should have had their origin in so impure a source as matter, is, to an ardent-minded mathematician, an idea no more to be endured than, by certain religionists it is, that moral evil should have no other source than physical; or, by the sentimental poet, the sentimental orator, or the hypocritical politician, it is that sympathy (whether for the individual or the particular class of the community-political body he belongs to, the nation at large, or the human race) should have so unhonoured a parent, or so despicable an antagonist, as self-regard, either in his own pure bosom, or that of any of his friends.
In the construction of the sort of Genealogical Tables here brought to view, the difference between the order of invention and the order of demonstration, must not be out of view. It is by observation made of the practical applications of which the several propositions have been found susceptible, that the order of invention in as far as it is capable of being determined, will be determined; and, for the benefit of posterity, the secrets of inventive genius brought to light. The path of genius in the intellectual world has been like that of a comet in the physical world. To the eye of the ordinary observer few marks by which it can be discovered are visible. In the spreading of this veil, love of ease concurs with love of fame, or what, in dyslogistic language,—(language, with the addition of disapprobation attached to the practice)—is the same thing, pride and vanity concur with indolence. In these circumstances may, perhaps, be found the causes of that obscurity in which, from Euclid, through Newton, down to the present time, the works of mathematicians have been so generally involved. To display to the wondering, and not unenvious, eyes of the adept, inventions and discoveries of a man’s own, in all their freshness, is an operation, not only more pleasant, but less tedious than that of endeavouring to facilitate, to the vulgar mind, the conception of discoveries that, whether they were or were not his, are already become stale. As in the order of time, so in the order of dignity and reputation, communication is preceded by invention. But, to communicate in the promptest, easiest, and most effectual manner, what has already been invented and discovered, is itself the work of inventive genius and the matter of an art;—it is a branch of logic, that commanding art, of which invention, to whatever subject applied, constitutes one branch, and no more than one.
Genealogical-Table employing, or Synoptic-Filiation indicating principle.
Viz. Of the sort of relation of which the propositions in Geometry are susceptible, in respect of use.
Immediate or preparatory; to one or other, or both, of these denominations, will be referable the use of any proposition in mathematics that has any use.
In as far as in either way, it has a use, how to point out, and, in the most satisfactory, not to say the only satisfactory, way, afford a demonstration of that use, was shown under the last head.
In as far as the use is not only preparatory but mathematical,—and, between any two propositions, of the last of which the use is ultimate, while, of the first of them, the use is, with reference to the last, preparatory, others, connected with one another in a series or chain, are interposed, each being in like manner preparatory with reference to that which stands next to it,—a chain or tree of this sort (or whatever be the sensible image employed for elucidation) will bear some resemblance to the chains or trees of which a genealogical table is composed.
The business is nothing more than to propose for consideration the composition of a table, or set of tables, in and by which these several relations may all of them stand exhibited at one view.
Of this sort of matter, what quantity will be capable of being, in a commodious manner, brought together, so as to be presented in one view, remains to be determined by experiment.
Something will depend on the application which may be found capable of being made with advantage of the principle next mentioned.
For the giving connexion to these several elementary units, use—practical use, in its several modifications, as above explained, will show itself the strongest possible cementing principle. A rope of sand is the emblem of a cluster of propositions, for none of which, be it ever so copious, use in any shape is discernible.
How to construct a Geometrical Genealogical-Filiation Table.
Of this sort of Table, the one essential property is—that the more advanced the proposition is, and thence the greater the number by which it is expressed, the greater the number of the propositions on which the demonstration of it may depend.
Thus, in the case of proposition the first, no proposition on which it has any dependence can have existence. Definitions and axioms are the only materials of which the foundation of it can be composed. In the case of proposition second, there exists one proposition, but no more than one, on which, besides definitions and axioms, it is possible for it to have dependence. In the case of proposition third, there may be two such supports, and so on throughout.
The higher the proposition in question stands in the geometrical scale thus described, the more numerous the list or string is capable of being, the list or string of propositions on which it depends.
In any tabular or synoptic exhibition, the demonstrative part, or the corresponding diagram of the proposition in question, being included in a graphical compartment of correspondent bulk and convenient form, a circle, an oval, a square, or a long square, for example;—a circle, an oval, or a pear-shaped figure, may be considered as the body of the sort of plaything by means of which Franklin drew thunder from the sky, called a kite; of this kite, the string of numbers which, one below another, give indication of the several sources or foundation-stones of the proposition, as above, naturally may be so disposed as to represent the tail of this kite.
The higher the place of the proposition is in this scale of filiation (the word descent cannot, without a sort of verbal contradiction, be employed,) the longer will naturally be this tail. If, therefore, in this Table, the propositions are ranged in horizontal rows, one above another, according to their places in the scale, the higher the proposition or kite stands, the greater is the quantity of room which, in a vertical direction will naturally be requisite to give lodgment to its tail.
In a tail of this sort, over and above the series of propositions, the axioms and definitions will require to be designated. For the designation of the propositions, convenience will require the employing of the Arabic numerals. If then, for the designation of the axioms, Roman numerals in an upright form be employed, and, for the designation of the definitions, the same numerals in a leaning form,—upon this plan the function of designation will be performed in the most simple, and, at the same time, on the most familiar plan.
An explanation of the purpose to which these numerals are respectively applied, might constitute part of the contents of a border, with which a Table of this sort might and should be garnished.
As to the postulates, being but three in number, and these of perpetual recurrence, it seems questionable whether, after the first use, any repetition need be made of them; and thence, whether any particular numerals, or other instruments of designation for them need be provided.
For the composition of the border other ingredients are—a list of the definitions and another of the axioms employed in the demonstration of the several propositions included in the Table.
In the case of the definitions and the axioms, what seems to render this concomitant exhibition necessary (but not to the exclusion of the propositions) is, that in the case of the definitions and the axioms, there exist no such means of elucidation as have place in the case of the propositions, viz. by means of the reciprocal exercises afforded by the purely verbal mode of designation, in the one case, and the purely diagrammatical in the other.
In some instances the same proposition will be susceptible of demonstration, from two or more different sources. Wheresoever this multiplicity has place, the kite will have the corresponding number of tails.
As to the border, the string of axioms will be comparatively a short one: a dozen, or some such matter. For the whole number of propositions contained in the geometrical scale, be it ever so ample, this small number will suffice.
Much longer will be the number of definitions. At every considerable step it will necessarily receive increase.
The same border might and should be inserted in both of the two corresponding Filiation Tables, viz. the verbally expressed and the diagrammatically expressed one.
The degree of closeness as between proposition and proposition in the several rows, consequently the number capable of being inserted with convenience in each row, and the inequalities, if any, in the distances between proposition and proposition in each row, i. e. between kite and kite, (tail or tails included,) will depend upon the room, if any, necessary to be left in each inferior row for the tails belonging to the several kites, ranged in the several superior rows. For the construction of such a Table, the most convenient course, it is believed, that could be taken, would be—having settled the scale of magnitude, as determined i. e. by the size of the type, form the several kites separately, and then having ready a sheet of paper of the proposed size and dimensions, attach them to it in order: the mark of attachment temporary till everything is finally settled.
In respect of its contents, a Table of this sort, shall it be confined to the propositions contained in Euclid’s Elements?—to the propositions contained in Euclid’s works at large?—to the propositions contained in the sum of the works of the Grecian geometers?—or shall it, as far as it goes, comprise all such geometrical propositions, as in any way present themselves as susceptible of practical use? To all these questions, surely the last suggests the only natural answer, viz. that which is implicitly contained in the last of them.
By a very simple expedient in the verbally expressed Table, a distinction might be made, by a particular type, between those of modern and those of ancient date. In the elementary branch, in which no curve but the circle is introduced, let Euclid’s propositions, for example, as constituting the main part of the work, be in the ordinary Roman type: propositions found in the works of other ancients might be either in the same Roman type with Euclid’s, or in another Roman type of different, suppose of inferior size: if the type could not conveniently be diminished, the black letter might answer the purpose.
Another part of the above-mentioned border might be composed of references to the original works, in which the several propositions, denoted by the number by which they are designated in the Table, have been found.
In this case, as in every other, the application made of the exercises, with the place-capturing principle for their support,* will be determined by the nature of the particular object to be accomplished. Having for his guides a corresponding pair of Tables, viz. one containing the propositions (the enunciative parts) verbally expressed; the other with the same diagrammatically expressed; both of them without any of the references by which the filiation is indicated, the exercise is performed either by the extempore pronunciation, or by the extempore writing, of the references. Briefly thus: given the kites, required the tails.
By a system of exercitation thus conducted, the object to the attainment of which the process of demonstration in form is directed, would, it is believed, be not only attained, but attained in a much more perfect degree. By the form of demonstration, what is brought to view is the connexion between that individual proposition, and those on which it depends more immediately—that and nothing more. But by this system of genealogy, what is brought to view is the connexion between each such proposition and every other. In the one case, you have first one part by itself, then another part by itself, and so on; in the other case, all the parts are knit together into one connected whole.
At the outset, at any rate, an enunciative part, the preparatory part, and the demonstrative part, being distinguished as above, in the demonstrative the forms of demonstration might and should be strictly observed; in the preparative as well as the demonstrative part, each distinguishable step being carefully distinguished from every other, and for that purpose formed into a distinct paragraph. But, the mode of reasoning being once thoroughly understood, sooner or later the former, by which so much room is occupied, might, it is supposed, without prejudice to intellection, be discarded.
Scarcely in the compass of a single Table thus constructed, could any very considerable part of the field of geometry be exhibited. A number of such Tables, standing in succession, would be found requisite, any two or more of which might, upon occasion, by so simple an operation as juxtaposition, be made into one.*
Special-visible-sign-employment-maximizing—Purely-diagrammatic-expression occacasionally-employing—Verbal-expression ocsionally-discarding principle.
Special sign, special in contradistinction to ordinary: special in contradistinction to the ordinary signs of which language is composed.
Arbitrary, in contradistinction to imitative, are, moreover, the signs to be understood to be in both cases.
By any of these special and arbitrary signs, imitation being out of the question, nothing can be intended to be expressed, which is not capable of being expressed by the ordinary signs; to the expression of which the signs of which ordinary language is composed, are not capable of being applied.
But in this case, as in every other, the labour necessary to the faculty of making use of the ordinary signs of which language is composed, has already been undergone, and the faculty acquired.
Whatsoever may be the special signs in question, in the acquisition of the faculty of making use of them, whatsoever labour requires to be employed, is so much extra labour added to that which has been expended in the acquisition of the faculty of employing the ordinary signs.
In as far as any use is made of special signs, here there is an account of profit and loss: or say rather of loss and profit: cost, the labour necessarily expended in acquiring the faculty of making use of these signs: profit, the advantage, whatever it be, derived from the application made of these signs, in lieu of, or in addition to, the ordinary signs, to the purpose in question. First in order of consideration comes the article of profit, that being the final cause, but for which the expenditure would not be made.
Profit derivable from the employing of special signs: or uses of special signs in Mathematics.
I. Exemplification, viz. employing individual signs, or assemblages of signs, to serve as examples of the general propositions which compose the matter of mathematical language, and, by that means, the more clearly and promptly to convey the general ideas of which they are intended to be the expression.
In as far, however, as it is to this use, and no other, that the assemblage of special signs in question is applied, the epithet of unanalogous does not belong to them. On the contrary, they are imitative. Thus, geometrical diagrams are a species of drawing: and as, in the case of a square table, the draught of the whole table, in proportion or otherwise, is an imitation of the whole table, so the diagram of a square is an imitation of the principal part of it.
II. To the head of Abbreviation, or say Condensation, will be found referable whatsoever useful effect is producible by this means.
Ordinary language is the sort of vehicle, and the only sort of vehicle, which is in possession of the employment of conveying ideas to the mind. In as far as any other sign, or set of signs, shares in this employment,—in as far as this function is performed by any special set of signs,—it is only through the medium of those ordinary signs: those ordinary signs, not the ideas themselves which they are employed to denote, are the objects immediately presented to the mind by any fresh special signs.
Unless they present spoken words, i. e. the sounds in question in a shorter compass than the shortest in which they can, with an equal degree of conspicuousness, be presented by the ordinary signs or characters of which written language is composed, the effect, if any, of special signs, must necessarily be to retard, not to accelerate, conception; for, first, they have to bring to view the ordinary signs, and, when they have so done, then it is that they are, in respect of promptitude, upon a par, and no more than upon a par, with those ordinary signs.
As to the first named of these uses, what is certain is, that, for a length of time, more or less considerable, it cannot take place, or so much as begin to take place. Every new sign of this kind is part and parcel of a new language: and of no new language can any part or parcel be ever learned, without a proportionable expense in the article of time. All this is so much loss. When once the portion in question of the new language has been learned, i. e. when between the thing meant to be signified and the new sign an association has been sufficiently formed, then, and not till then, if there be a profit, comes the profit.*
In the instance of each such sign, taken by itself, if between the thing signified and the sign there be any analogy, the closer the analogy the less will be the cost: the more frequently the occasion occurs for putting the sign to use, the greater will be the profit.
Thence, taking the whole number of the signs together, the aggregate number of the occasions in which they can be employed being given, the profit will be the greater the less the number of the signs.
In algebra, in contradistinction to, and almost to the exclusion of, geometry, has the employment thus given to this principle been most copious. Of the signs of which this language is composed, the number even absolutely taken is very small. The number of the occasions on which they are employed, being, even in a work of a very moderate scope, immense, relatively taken, its smallness is still more conspicuous.
It is, however, to the second head, to speak shortly in the way of abridgment, that, in algebra, any part of the advantages derived, from the use therein made of peculiar signs, can be referred. The effect produced by them is neither more nor less than the presenting, in a smaller compass, the same ideas as those which are produced by the corresponding portion of ordinary language. By the cross employed to signify addition, the effect is neither more nor less than that which would be produced by the word, addition, together with such other words as may be necessary to complete the sentence—the grammatical or logical proposition, for which this one simple sign is capable of being employed, and is commonly made to serve as a substitute.
Of this sort of calculation, the importance, as well as the nature, may be not uninstructively illustrated by an instance in which, by a scientific person of no mean note, ingenuity, labour, time, and expense, (typographic expense,) in no small quantity, were actually thrown away. On the publication of the then new system of chemistry, which bears the name of Lavoisier, the business was divided among three hands. The contrivance of a new set of characters, termed chemical characters, adapted to the new theory, being at that time regarded as constituting the subject of a necessary part of that business, was announced as having fallen exclusively to the lot of one of these three hands. Since that time, so different in many parts, as well as so much more extensive is the culture received by the field of chemistry, that even had the principle of the contrivance been good, the application given to it could no longer have continued useful, without having undergone, in every shape, such alteration as would have rendered it hardly recognisable. But it was bad in principle. The new signs were characters or signs to which every imaginable exertion was made to give what analogy could be given to them to the things signified. But had these exertions been even much more successful than they were, these special and newly published characters would never have presented to the mind, especially to the mind of a learner, the ideas of the respective chemical substances, with the same perfection, much less with the like certainty, as that with which they come presented by the corresponding set of names, as expressed by those already and commonly adopted general characters, of which ordinary written language is composed.
In the way of facility afforded to conception, whatsover effect they were productive of was wholly on the side of disadvantage.
In respect of abbreviation or condensation, it was not productive of any advantage. For giving lodgment to each one of these signs, a receptacle of the same form for each was, as in the case of a Genealogical Table, it is believed, or, at any rate, for illustration, may be conceived to have been, provided. But within every such receptacle, the name of the substance in question, expressed in ordinary letter-press, might have been included, and in such form and size as to be altogether as conspicuous, as readily apprehensible, as the new sign, for the giving lodgment to which it was employed.
Of the notion of this mode of expression, what was the source? Imitation: imitation, without sufficient thought.
In the infancy of chemistry, when as yet she was little better than a slave to the impostor alchemy, a set of special signs were employed, for the designation of such of the metals as were then known; together with some others of the simple, or supposed simple, substances then known, or supposed to be known. But the design, in pursuance of which these characters were framed, was of a mixed character, made up of the opposite ingredients divulgation and concealment; and entertained by minds in which, in sharers of power, perpetually varying and perpetually unascertainable, credulity and imposture maintained a conjunct sway. By an effort of economy, as whimsical as it was elaborate, the same set of seven signs served for a set of chemical substances, namely, metals, and the same number of heavenly bodies at the same time; that the use might be the more profound, and the adepts, including or not including the inventor himself, the more effectually deluded.
At the same time that, by the pair of self-teaching learners, application, as above,* was made of the purely verbal expression maximizing principle, by the same persons was application made of the principle which there corresponds to and contrasts with it, viz. this same verbal expression occasionally discarding principle, or purely diagrammatical expression employing principle.
What the signs had for their immediate purpose, was to convey to the mind, by these means alone, without the use of words, a conception, in the first place, of the enunciative part of the proposition; in the next place, of the several operations which, in the preparatory part, were required to be performed; and, lastly, of the several assertions contained in so many distinct steps of the demonstrative part.
What, in relation to this head, is recollected of them, is as follows:—1. The signs employed were, or at least were endeavoured to be, made analogous, i. e. naturally expressive. If, for example, on the occasion of the first step in the preparatory part—on the occasion of the first operation required by it to be performed,—a line of a certain description was, at a certain part of the figure, exhibited in conformity to the enunciative part, or representation of the subject of it, required to be drawn,—in this case, immediately after this original figure or diagram, came another, in which it was copied, with the addition of the thus prescribed line, and so on for every fresh step a fresh figure.
So, again, when, on the occasion of the demonstrative part, expression came to be given to the first step, a set of marks, of which a small number was found sufficient, were employed for distinguishing those parts, whether lines or angles, which were the subjects of that part of the demonstration, from the succeeding ones; and so on, as above.
Another condition necessary to usefulness is that, taken together, the collection of signs employed should not be too bulky for use,—should not occupy so great a quantity of space as not to be capable, in a number sufficient for instruction, of being brought together into one table.
Neither was this condition, it is believed, altogether unfulfilled. In the ordinary mode of designation, a circumstance which necessitates the allotting to each figure a larger space than would otherwise be necessary, is the affording room enough for the letters of reference: these letters large enough to be clearly distinguishable, and so placed as that no doubt should exist in regard to the part which, in each instance, they were employed to designate. But in the proposed plan, these arbitrary and naturally inexpressive marks would have no place.
In some such way would the matter stand in regard to the several propositions separately taken.
In regard to the Genealogical Tables above-mentioned. On this occasion, each proposition, taken by itself, being supposed to be already understood, having, by the means already mentioned, been rendered intelligible, in a Table of this sort, all that could require to be exhibited, would be the diagrams or figures representative of the enunciative parts of the several propositions. For showing, in relation to each subsequent proposition, what were the preceding propositions on which it is grounded, and which in the demonstrative part were accordingly referred to, nothing more would be necessary than a cypher, or cyphers, expressive of the numbers by which, in the same Table, those propositions stand respectively designated. The diagrams expressive of the several propositions being included in similar compartments, circular suppose or quadrangular, and those compartments ranged in lines descending from the top to the bottom of the Table, an equal number on each line, the eye would thus be conducted to them with instantaneous rapidity. For this purpose, the order of the numbers should, from first to last, in the whole series of the propositions, be the order of the names upon the Table. Whether in each proposition (the order of the propositions being the same as in Euclid,) to the number expressive of its place in the series, should or should not be added the two sets of numbers expressive of the book to which it belonged in Euclid, and the place of it in that book, experiment would soon determine.
In the case when the same proposition is capable of being demonstrated from any one of several sets of antecedent propositions, sets of cyphers, expressive of them, might be inserted: each set being distinguished from every other by the word or, or by a simple line of separation.
In respect of promptitude of conception, could any additional facility be afforded by a set of lines drawn issuing from the succeeding proposition, to the several antecedent ones, by means of which it has been or might be demonstrated? The negative seems most probable: confusion rather than elucidation presenting itself as the most probable result of a tissue or piece of network, thus irregular and thus complicated.
To the propositions that are in Euclid, shall not all such others be added, by which equally useful instruction, relative to the same class of figures, promises to be afforded, and this, too, in the same Table? Yes, unless propagation of superstitious and delusive errors be preferred to propagation of useful knowledge. But in the character of a certificate of acknowledged truth, the authority of Euclid being naturally more extensively received than any other, propositions derived from other sources might be distinguished from those of Euclid by some mark common to them all, and immediately discernible; suppose, for example, by different colours, (or what would be much less expensive,) by being included in somewhat smaller compartments.
That, in the instance of the pair of self-teachers above-mentioned, after a few general hints received from their distantly situated adviser, the carrying into effect these little devices was a matter of no small instruction as well as amusement, is perfectly remembered.
That, in the instance of other learners, by whom no part in the pleasure of invention would be shared, any real profit, either in the way of amusement or of instruction, would be reaped, does not absolutely follow.
One consideration, however, does present itself as promising to turn the scale in favour of the affirmative side. This is the applicability of the two correspondent and opposite modes of expression to the purpose of affording a test of intellection, and such a test as admits of the application of the place-capturing principle.—(Table II., No. 10.) The correspondent exercises will consist of two correspondent and opposite translations: one the recitative, the other the organic exercise.
In the case of a proposition taken by itself, the scholar having before him the process expressed in the purely diagrammatic mode, repeats, by the help of it, the same process in its several steps, as expressed in the purely verbal mode. In this way is performed one of the two (the simple recitative) exercises. At another time, having before him the process expressed in the purely verbal mode, he delineates on the spot the same process as expressed in the purely diagrammatic mode. In this way is performed the Organic Exercise.
In a similar manner might the corresponding pair of reciprocal translation exercises be grounded on a pair of Genealogical Geometrical Tables.
Suppose one of these Tables expressed in the purely verbal, the other in the purely diagrammatic, mode. In this case the same correspondent exercises might be performed, as have been just described.
Another exercise might have either of these Tables for its ground. The figures of reference (arithmetical numbers) by which the genealogy of the proposition is, in each instance, expressed, being suppressed or concealed for the occasion, the exercise consists in the giving an indication of that analogy, viz. either by the mere naming or writing of the numbers, by the pronouncing or writing the lines or purport of the proposition as expressed in the purely verbal mode, or by delineating it as expressed in the purely diagrammatic mode.
Key-presenting, or special contrivance-indicating principle.
Key, viz. to the expedient by which the demonstration is effected, and by which, accordingly, in many instances, the entire proposition, whether theorem or problem was first suggested.
This principle will be found applicable as well to Algebra as to Geometry.
Of the sort of intellectual instrument here in view, as applied to Geometry, the Appendix presents two specimens; one applied to Euclid’s first proposition, which is a problem, the other to his fifth proposition, which is a theorem. In both instances, this part, termed the key, forms the second of the four points exemplified in these two propositions, as expressed upon the purely verbal expression-maximizing principle.*
Of the use of this sort of instrument, the effect, it is believed, will be found to be the letting the learner into the secret, as it were, of the invention; by showing him what, on the occasion of the invention passed in the inventor’s mind.
In these two instances each individual proposition has its own key; the key which belongs to the one, will not be found to apply exactly to the other.
But should all the propositions delivered by Euclid, together with such others as it might be found practicable and useful to add to them, come to have been exhibited upon this same proposed principle, some circumstances common to a number of them, will probably be brought to view, by means of which they will be found distinguishable, with advantage, into so many classes: and, in that case, what will probably be found is, that in addition to, or in lieu of, the keys belonging to the individual propositions, a key will be found applicable to the whole class. Out of these classes may, perhaps, be found compoundable other more extensive classes—say, perhaps, of the second order;—each such class with its key, as before.
Of the sort of instrument of elucidation, for the designation of which the word key is here ventured to be employed, happily for the science and the learners, examples, even now, are not altogether wanting in the works of Mathematicians; and, as far as concerns the purpose of instruction at least, howsoever it may be in regard to further discovery and advancement, it will scarcely be denied that the greater the number of these keys, supposing them equally well constructed, that the work affords, the better adapted it is to the purpose.
One example which, of itself, is worth a multitude, is afforded by Montucla, in his Histoire des Mathématiques, tom i., lib. iii., note B., pp. 197-201.
In it the several peculiar figures, three in number, capable of being produced by the cutting of a cone, (or rather a pair of cones,) are brought together, are confronted with each other, and their principal characteristic properties, viz. those in which they agree with, and those in which they differ from, each other, are placed together in one view,—all in the compass of no more than four, though it must be acknowledged, closely printed quarto pages.
A circumstance which renders this example the better adapted to the present purpose is, that, on this occasion, nothing more is given than the enunciative parts of the several propositions, preceded by such definitions, no more than six in number, as were judged necessary. Total number of propositions, according to the numerical figures, no more than 21; though, if it be considered that, in most of them, the three species of conic sections in question are comprised, that number may, in that respect, be required to be nearly tripled.
In this explanation, use, it is true, as could not but be expected, is made of diagrams, for reference to which alphabetical letters are in the usual way employed: consequently, neither the purely diagrammatic mode in any part, nor the purely verbal mode of expression, except here or there are or can be employed. But to no inconsiderable extent upon the whole, sometimes for five or six lines together, the purely verbal mode is employed.
Taken together, therefore, in the hands of a liberal minded and unprejudiced institutionalist, out of these four pages, upon the plan here proposed, might be made an admirable and most instructive set of exercises, for the geometrical section of the proposed Chrestomathic School.
Few, perhaps, if any institutional books are in use, in which keys of this sort, in greater or less abundance, may not be found. In particular, wherever anything is seen in form of a note, search may be made for an implement of this kind, with considerable probability of success.
To the natural aridity of the subject, more or less of humectation may be expected to be afforded from the springs of criticism.
Neither in the case of Algebra (as above announced will this same principle, it is believed, be found inapplicable.
In the branch of mathematics called Algebra,—viz. in such problems and such only as have no direct relation to figure—in which figure is not as such taken into the account; two sorts of operations, in themselves perfectly distinct, may be distinguished: viz. the mode of designation or expression, and the contrivance or species of investigation employed in the resolution of problems: the system of abbreviation, and the system of contrivance for the purpose of performing the several particular operations, for the facilitation of which the same system of abbreviation is throughout employed. Between-these two the relation is that between the means and the end: the mode of expression the means; the resolution of problems the end.
As to the mode of designation, the object which it has in view, the advantage which in comparison with common arithmetic it affords, may be expressed in a word, abbreviation; room, labour, and time, all these precious objects are saved by it. It is a particular species of short-hand, differing only from the sort commonly designated by that name in two particulars. 1. In its application it is confined to that sort of discourse which has quantity for its subject. 2. Within its field of action the degree of power which it exercises is much greater than any that is exercised by ordinary short-hand. All that short-hand does, is the employing, for the giving expression to each word, strokes in less number, or more easily and quickly described, than those which are employed in ordinary hand. The mode pursued in writing before the invention of printing, and in printing itself for some time afterwards,—in a word, the system of contractions was a species of short-hand.
Multifarious as well as great are the savings made by the mode of notation employed in algebra. As far as it goes, the following may serve as a specimen.
1. In the room of a number of single words, being those of most frequent occurrence, such as those of addition, subtraction, &c., it employs so many marks in a great degree more simple.
2. Of an assemblage of figures, i. e. the common Arabic characters expressive of the names of numbers,—characters which of themselves constitute a species of short-hand,—of an assemblage of this sort, however long and complicated, it performs the office, by a single letter of the alphabet.
3. Where the assemblage of these abridgments of abridgments present themselves as susceptible of ulterior abridgment, of a line of any length composed of letters, with or without figures, it performs the office, it expresses the import, by means of a single letter; and so toties quoties.
From this function of algebra, the other, the efficient it may be termed, which consists in the solution of problems, in the performance of tasks proposed, in the rendering of services requested or demanded, is, as has been shown above, altogether different. To the last mentioned the former bears the relation of a means to an end.
By means of the relation which it bears to some quantity or quantities already known, to make known some quantity which as yet is unknown,—to this one problem may be referred all problems whatsoever, to which the name of algebraical can be applied.
For the accomplishment of this purpose on different occasions, different contrivances, over and above those which consist in nothing more than an abbreviated mode of expression, have suggested themselves to persons conversant with this art. In no instance, perhaps, certainly not in every instance, to the giving expression to these contrivances, are the modes of abridgment employed in algebra considered as a species of short-hand indispensably necessary.
As yet not even in algebraical, that abbreviated and technical language, has any mathematician, it is believed, unfolded, or so much as endeavoured to unfold, for the boy, what may accordingly still be called the secrets of his art.
Not even in abbreviated and technical language do we possess any such key constructed out of unabbreviated and ordinary language.
As to the abbreviative principle, algebra is not the only branch of the mathematics in which the abbreviative system or method is applicable with advantage. Though not in its whole extent, nor to anything near its whole extent, it is to a part of that extent applicable, and with like, if not altogether equal advantage to geometry. In Payne’s Geometry, not to look for others, application is accordingly made of it, and with very considerable advantage.
If abbreviation were the only use of the function here distinguished by the appellation of abbreviative, it would follow that in the performance of the essential function everything which at present is not only customarily but exclusively expressed by the exercise of the abbreviative function is capable of being expressed without it,—may be expressed in a word in ordinary language. To any such purpose as the practice of the art, what is plain enough is, that by no such substitution could any advantage be gained; on the contrary, it would by the amount of the whole of its effect be disadvantageous. Instruction is the only purpose to which it could be made serviceable; but that to this purpose it might be rendered eminently, in a very high degree, serviceable, seems sufficiently evident.
In this case the same substitution of signs immediately expressive of general ideas, to signs immediately expressive of none but individual ones, would be the result, as has been already shown to be the result in the case of geometry; and in respect of intellection, and command of the subject, that result would be attended with the same advantages.
In this case the whole method of the art might be explained and taught,—the whole secrets of the art laid open, to an intelligent mind, without its being subjected to any part of that hard labour which must so unavoidably be bestowed upon the subject, before the signs and modes of proceeding, by means of which the abbreviation is performed, have been learned.
But supposing this done, the number of persons more or less acquainted with the principles of this art might be increased,—increased by the whole number of those who at present are repelled from it, by the formidable apparatus of magical characters now employed, by means of which the abbreviative function of it is performed. And when the principle of each distinguishable contrivance was held up to view in ordinary language, each principle characterized and fixed by an appropriate name, with a definition annexed, even the adepts themselves might, in the clearness and expressive generality of the language, find facilities according to the nature of the case, either for the invention of new contrivances, or for showing if such were the case, and as soon as it came to be the case, that the nature of the case admitted not of any others.
An observation which, it is believed, will be found general among mathematicians, is, that by the use of different inventions, contrivances, and expedients, from the number of years which even in the case of an amateur of this branch of art and science, would be necessary to carry him over the whole field of it, several years have been struck off, principally by the ingenuity of the French mathematicians. These applications of inventive genius, what then are they? To this question—and the whole field of the science cannot present a more important one—an answer might, if what is said above be correct, be given in ordinary language.
In the case of Algebra, (Fluxions included,) elucidation, if so it may be termed, though the same in respect of its end, will, in respect of the description of the means requisite to be taken for the accomplishment of that end, be somewhat different from what it has been seen to be in the case of Geometry.
In the case of Geometry, the enunciative parts of the proposition excepted, nor even they throughout the whole of the field—the language is particular, being, by the want of general terms, confined, in respect of the subject, to the individual figures and parts of figures exhibited by the individual diagrams, and designated—not by any indication given of their intrinsic and permanent relations one to another, but—by the arbitrary and unexplanatory denomination given to them by means of so many combinations of the letters of the alphabet. In this case, one great instrument of elucidation, therefore, consists in the substitution of terms expressive of general ideas, being those of so many sorts of relation, to denominations thus individual and unexpressive. But in the case of Algebra, the terms employed, abbreviated, and, to those to whom the use of them is not familiar, obscure and perplexing, are as general as it would be in the power of words—of words at length and unabbreviated, to make them. For generalizing designation, in the character of a new and as yet unknown instrument of elucidation, no room is left in Algebra.
But though of the application of the purely verbal expression employing principle the effect is not in Algebra, to add in any respect to the generality of the language, that, even in Algebra, it is capable of being made to act, and with very considerable effect, in the character of an instrument of elucidation, seems scarcely to admit of doubt.*
It consists in simply forbearing to employ the algebraic formulæ or forms, while those explanations are going on, by which the rationale of the art and science is brought to view.
In the algebraic branch of mathematics, in idea at least, two sorts of operations, as above pointed out, may be distinguished—the abbreviative or condensative, and the effective or efficient. The abbreviative are but a species of short-hand: they perform, on the occasion of discourse applied to this particular subject, though with a degree of efficiency incomparaably superior, the sort of function which the characters of which short-hand is composed, in relation to discourse at large, perform. In as far as this is the case, it follows that, in the exercise of this art, every particular contrivance, which does not consist in the mere employment of this general system of abbreviation, may as effectually and intelligibly be expressed in ordinary characters, and without this particular species of short-hand, as any other subject of discourse may be expressed in these same ordinary characters, and without the use of that species of short-hand commonly called short-hand, the use of which is applicable to every subject of discourse.
In regard to these abbreviative contrivances, what may very well happen is, that some apply principally or exclusively to this or that subject; to the solution of this or that particular problem or group of problems; and in so far the invention of the mode of abbreviation is the invention of the mode of solving the problem, and thus the abbreviative part and the efficient part are in a manner confounded. But, at any rate, it is not in every instance that this sort of confusion has place; and, on the other hand, a number there are of these contrivances for condensation, which are employed on all occasions alike.
True it is that, on the explanation given of the several substitutions by which the condensation is performed, the characters, the instruments themselves by which it is performed, cannot but be brought to view. But, for this particular purpose, no one of them need be brought to view more than once, or some other small and limited number of times; and between this use of them for the mere purpose of explanation, and the constant use of them through the whole of every page, how great the difference cannot but be to the mind of a young scholar, is sufficiently obvious.
By one passage, or some other small number of passages, consisting of the abbreviative forms or characters, every contrivance that belongs to the head of abbreviation may be explained; and even without so much as one such assemblage of uncouth forms, every contrivance, which does not operate as an instrument of abbreviation, or in so far as it operates otherwise than as an instrument of abbreviation, may be explained.
Prodigious would be the relief thus afforded to the uninitiated juvenile learner’s mind, made by the indulgence thus afforded to his love of ease.
Under the head of Language-learning, the dark spot produced by every hard word, by every word which, being derived from a foreign language, has no relative belonging to it in the vernacular language, has already been brought to view. To an uninitiated eye, a page of algebra is a surface covered almost wholly with the like dark spots.
True it is that, for the explanation of the different contrivances, words in no small number that to the learner will be new, some of them already in use, others which it may be necessary to coin for the particular purpose here proposed, would be found requisite: and these new words will be so many hard words, so many dark spots.
But no sooner would one of these new words present itself, than a definition or explanation, composed either purely of common words, or partly of common words and partly of such peculiar words as had already, in this same way, received their explanation, would be subjoined. No sooner has the dark spot made its appearance, than the requisite light will have been thrown upon it: and how much more thickly darkened a portion of discourse is by unknown characters, than even by hard words expressed in familiar characters, few but must have experienced.
In the case of Geometry, the word key was confined in its application to such explanations as were annexed to particular propositions, or groups of propositions, over and above such explanations as, in the case of the demonstrative and preparatory parts of the several propositions, could not but result from the translation of the individualizing modes of designation employed, in so far as diagrams are employed with letters of reference, into the general expressions of which purely verbal discourse is composed.
In the case of Algebra, every paragraph in which the use of forms and characters were abstained from, would, in so far as it were instructive, operate as a key. For it would have as its object, either the explanation of the several contrivances of abbreviation, or of the several contrivances whereby these instruments of condensation were applied to practice and endeavoured to be put to use. Of no other sort of matter could it be composed; for, to the solution of the several problems, unless it be, in a few instances, as above, for illustration, the use of these forms would, of course, be necessary.
In this case, as in that of Geometry, an additional instrument of elucidation would be afforded by the application of the use indication-prescribing principle, by the indication of the use, the practical use, derivable from the solution of the several sorts of problems, for the solution of which the Algebraic language is wont to be employed.
On this occasion it is not by any application which may be, or that has been, made of them that, in the sense here in view, they could with propriety be said to be put to use. Only in so far as it had been, or was capable of being made, subservient, either to some security or comfort in the business of ordinary life, whether immediately, or through the medium of this or that spot in the field of art and science, is it that the application made could with propriety be termed a useful one.
Take, for instance, the collection of articles intituled Praxes, or Questions for Praxis, subjoined to the English translation of Euler’s Algebra. The number of them is 213. Of this number, a part more or less considerable, consist of a sort of jokes, named paradoxes, having the excitation of wonder manifestly for their effect, and perhaps for their only effect. In every one of them application is made of the Algebraic form, to the solution of some problem. But of these 213 problems, it is not from every one that, by any person, benefit in any shape, over and above the pleasure derivable from playing at this kind of game, seems capable of being received. The additional praxis, therefore, would be from this miscellaneous list to point out such as are in their nature applicable to beneficial use, and by indication of the occasion to show in what shape they are respectively capable of being put to use.
To answer the purpose of elucidation in the completest manner—understand always, with reference to the uninitiated—a key should not only have the effect of letting the reader into the heart (so to speak) of the contrivance, by which the proposed object is effected, the proposed advantage gained, but in the production of this effect the purely verbal mode of expression alone, unless it be with the sort of exception above hinted at, should be employed: the purely verbal mode; viz. in Geometry, to the exclusion of the diagrammatic, in Algebra to the exclusion of the Algebraic, characters and forms.
To what precise length it may be possible, with any degree of net advantage, to carry this principle of elucidation, which consists in the temporary exclusion of peculiar signs, is a question on which, antecedently to experience, it can never be within the reach of the most expert mathematician to pronounce. Thus much, however, may be asserted: viz. that the further the institutionalist can find means to carry on his system of instruction in this track, the greater will be the number of the learners whom he will carry with him.
To Geometry,—as it seems pretty well agreed among the learned,—to Geometry to the exclusion of, and in contradistinction to, Algebra, (including Fluxions,) is confined what may be called the tonic or invigorative use of Mathematics: the service done to mental health and strength by a sort of exercise by which the process of close reasoning is carried on, and to the performance of which close and unremitted attention is indispensable. It is in consideration of this use, that by some the Algebraic form is held in a sort of contempt, and that, in the immense class of occasions in that vast portion of the mathematical field which belongs to Geometry and Algebra in common, and on which the same conclusion may be arrived at by either track, the same problem effected in the algebraic mode is considered as done in the way of makeshift, and not productive of use or advantage in any shape, over and above what may happen to be attached to the solution of the particular problem for the solution of which it is employed.
This being admitted, although by the solution of a single problem in the algebraic mode, no such service could be rendered to the mental frame, as in manner above mentioned, may be rendered to it by the solution of the same single problem in the geometrical mode, yet by the indication of this or that particular contrivance, by means of which this or that class of problems may be solved in the algebraic mode, there seems little reason to doubt that, to the mental frame, a service might be rendered, though not exactly of the same sort, yet of a sort not to be absolutely neglected. In the Geometrical case, it is to the judgment and the attention, that the service would be rendered; in the algebraical case, it is to the conceptive and inventive faculty that the most immediate part of the service would be rendered.
The case of the uninitiated is here all along the only principal case in view. But, neither to the adepts does it seem that the mode of elucidation thus here proposed, would be altogether without its use. By the survey that would thus be made of the ground, in a point of view so new, it could scarcely happen but that in one way or other an increase of command would be acquired with reference to it, and new discoveries made in it such as otherwise, for a long time, if ever, might not have been made.
The sort of intellectual instrument, the key thus proposed, or rather the apparatus or collection of keys, would be very far from being complete, if in its purpose it did not include all the several fictions, which, in the framing of this branch of art and science, have been invented and employed.
For illustration, without looking any further, two may here be mentioned: viz., the conversion of the algebraical method into geometrical, and the contrivance, called by its first inventor Newton, and from him by British mathematicians the method of fluxions, and by its second but not less original inventor Leibnitz, and from him by the mathematicians of all other countries, the differential and integral calculus.
For the explanation of these fictions, and, indeed, for the justification of the use so copiously made of them, two operations would, it should seem, require to be performed. One is, the indication of the really exemplified state of things, to which the fiction is now wont to be applied, or is considered as applicable, the other is the indication of the advantage derived from the use of this the fictitious language, in contradistinction to the language by which the state of things in question would be expressed plainly and clearly without having recourse to fiction.
1. As to the conversion of the forms of Algebra into those of Geometry, or of the algebraic mode of expression into the geometrical. If in a case in which figure has no place,—as in a case where the quantity of money to be paid or received, or given under the name of interest for the use of money during a certain time, is the subject of investigation,—the geometrical forms should be employed, or the subject of investigation, thereby represented in the character of a portion of matter or space, exhibiting a certain figure, here a fiction, is employed: figure is said to have place in a case where it really has no place.
2. In cases where the geometrical form is the form in which the subject presents itself in the first instance, and the translation which is made is a translation from this geometrical form into the algebraical, here in this case no fiction has place: here what is done may be done, and is done, without any recourse to fiction; and as to the advantage looked for from this translation, an obvious one that presents itself is the abbreviation which constitutes an essential character of the algebraic form. In the opposite species of translation: viz. that from the algebraic form into the geometrical, fiction is inseparable. Why?—because when by the supposition figure does not form part of the case, figure is stated as forming part of the case. But when the translation is from the geometrical form into the algebraical, neither in this, nor in any other shape, has fiction any place. Why?—because, though in the case as first stated, figure has place, yet if reference to the figure be not necessary to the finding the answer which is sought, to the doing what is required or proposed to be done, the particular nature of the figure, is a circumstance which, without fiction, may be neglected, and left out of the account.
So in the case of the method of fluxions, which is but a particular species of algebra distinguished by that name.
Take some question for the solution of which this new method is wont to be employed. This question, could it be solved by ordinary algebra, or could it not? If it could, then why is it that this new method is employed? i. e. what is the advantage resulting from the employment of it? if it could not, then what is the expedient which is supplied by fluxions, and which could not be supplied by algebra?
In this method a fiction is employed: a point, or a line, or a surface, is said to have kept flowing where in truth there has been no flowing in the case. With this falsehood, how is it that mathematical truth, spoken of as truth by excellence, is compatible?
What is here meant is, not that no such fictions ought to be employed, but that to the purpose and on the occasion of instruction, whenever they are employed, the necessity or the use of them should be made known.
To say that, in discourse, fictitious language ought never, on any occasion, to be employed, would be as much as to say that no discourse in the subject of which the operations, or affections, or other phenomena of the mind are included, ought ever to be held: for no ideas being ever to be found in it which have not their origin in sense, matter is the only direct subject of any portion of verbal discourse; on the occasion and for the purpose of the discourse, the mind is all along considered and spoken of as if it were a mass of matter: and it is only in the way of fiction that when applied to any operation, or affection of the mind, anything that is said is either true or false.
Yet in as far as any such fictions are employed, the necessity of them, if, as in the case just mentioned, necessary, or the use of them, if simply useful, should be made known. Why? In the first place, to prevent that perplexity which has place in the mind, in as far as truth and falsehood being confounded, that which is not true is supposed to be true; in the next place, by putting it as far as possible in the power of the learner to perceive and understand the use and value, as well as the nature of the instruction communicated to him, to lighten the burthen of the labour necessary to be employed in the acquisition of it.
When for purposes such as the above, a survey comes to be taken of the field of mathematics, another object or subject of inquiry may be, whether in mathematics in general, but more particularly in algebra, fluxions included, the language is, in every instance, as expressive as it ought to be. Antecedently to association, with a very few exceptions for the designation of anything which is to be signified, any one sign is as proper as another. But when associations have once been formed, this original indifference is at an end: for the designation of any object, some word or phrase should be looked out, which, in virtue of some meaning with which they have already been invested, serve in some measure to lead the mind to the conception of the thing meant to be designated, and in that respect are better adapted to the purpose than any words taken at random: than any words, in short, between which and the object which is to be designated, no such relation has place.
Thence it is, that, for the idea, be the object what it may, the choice of the words employed for the designation of it, is never a matter of indifference; nor will there perhaps ever exist the case in which a number of words or phrases may not be found, all of them possessing, in respect of the designation of the object in question, so many different degrees in the scale of aptitude.
In the practice of Mathematicians, propositions of the geometrical cast, and propositions of the algebraical cast, are, to an extent which seems not to have been as yet determined, considered as interconvertible: employed indifferently, the one or the other, and upon occasion translated into each other. When, in the particular subject to which they are respectively applied, figure, although it have place, may, without inconvenience in the shape of error, or any other shape, be laid out of consideration;—in this case, instead of geometry, which, in this case, seems the more apposite and natural form, Algebra, if employed, is employed without fiction, and may, therefore, be employed without production of obscurity, without inconvenience in that shape; and, in proportion as the sought for result is arrived at with less labour and more promptitude, with clear, and peculiar, and net advantage.
But if, in a case in which figure cannot have place, as in the case of calculation concerning degrees of probability, as expressed by numbers, if any proposition be clothed in the geometrical form, so far will fiction have been employed, and with it, its never-failing accompaniment—obscurity, have been induced.
In the mind of him by whom they are employed, when the natural and individual ideas in which they have their source, and the individual or other particular objects, from which those ideas were drawn, are once lost sight of, all extensive general expressions soon become empty sounds.
In the use made of Algebra, at any rate, on the occasion of instruction given in this art to learners, the particular application which, either at the time in question, was made, or at any future time, was proposed to be made of it, should never be out of sight.
It is for want of this test of intellection—it is for want of this check, that, in books on Algebra, so many propositions, that are self-contradictory, and thereby void of all real and intelligible import, are to be found. Quantities that are negative, which, being interpreted, means less than nothing: and by the multiplying one of these quantities by another, that is, by adding together a certain number of these quantities,—a number of quantities equal to the product, and each of them greater than nothing, generated.
Algebraical language, even where, in the use made of it no fiction is involved, is a sort of abbreviated or short-hand language. So far, and so far only, as the abbreviated expressions which it employs, are, by him who employs them, capable of being, upon occasion, translated into propositions delivered at length, and in the form of ordinary language; so far, and so far only, as in the room of every such fiction as it employs, expressions by which nothing but the plain truth is asserted,—expressions significative, in a direct way, of those ideas for the giving expression to which the fictitious language here employed—were capable of being substituted, and accordingly are substituted; so far, and so far only, are they in the mouth or pen of him by whom they are employed, of him by whom, or of him to whom, they are addressed, anything better than empty sounds.
It is for want of all regular recurrence to these sorts of intellection, it is for want of this undiscontinued reference to unabbreviated and unsophisticated language, that algebra is in so many minds a collection of signs, unaccompanied by the things signified, of words without import, and therefore without use.
Employed on a number of different occasions, in so many different senses, and without any clear indication of the difference, or enumeration attempted to be made of these different occasions, the tissue of fictions involved in the use made of the negative sign, fills with obscurity the field of quantity, as the fiction of a debt where there is no debt covers with obscurity the field of commercial arrangement and commercial intercourse. See Tab I., Stage V., Book-keeping (p. 39.)
It was by an abstract consideration of the nature of the case (i. e. by a metaphysical view of the subject, as some mathematicians would incline to say, or a logical, as it might be more correct to say,) that this notion of the natural distinctness between the contrivances for abbreviation on the one hand, and the contrivances for the actual solution of problems, though with the assistance afforded by those abbreviative contrivances on the other, were suggested to the writer of these pages. It was with no small satisfaction that, for this same idea, he found afterwards a confirmation, and a sort of sanction, in the writings of two first-rate mathematicians, viz. a passage in Euler, adopted and quoted with applause by Carnot.—Euler, Mémoires de l’Academie de Berlin, Année 1754; Reflexions sur la Metaphysique du Calcul infinitesimal. Paris, 1813, p. 202.
Persons there are, says he, in whose view of this matter, Geometry and Algebra (la géomètrie et l’analyse) do not require many reasonings (raisonnemens); in their view, the rules (les regles) which these sciences prescribe to us, include already the points of knowledge (les connoissances) necessary to conduct us to the solution, so that all that we have to do is to perform the operations in conformity to those rules, without troubling ourselves with the reasonings on which those rules are grounded. This opinion, if it were well-grounded, would be strongly in opposition to that almost general opinion, according to which Geometry and Algebra are regarded as the most appropriate instruments for cultivating the mental powers (l’esprit,) and giving exercise to the faculty of ratiocination (la faculté de raisonner.) Although the persons in question are not without a tincture of mathematical learning, yet surely they can have been but little habituated to the solution of problems in which any considerable degree of difficulty is involved; for, soon would they have perceived that the mere habit of making application of those prescribed rules, goes but a very little way towards enabling a man to resolve problems of this description; and that, before application is actually made of them, it is necessary to bestow a very serious examination upon the several particular circumstances of the problem, and on this ground to carry on reasonings of this sort in abundance (faire la-dessus quantité de raisonnemens,) before he is in a condition to apply to it those general rules, in which are comprised that class of reasonings, of which, even during the time that, occupied in the calculation, we are reaping the benefit of them, scarce any distinct perception has place in our minds. This preparation, necessary as it is that it should be before the operation of calculation is so much as begun,—this preparation it is, that requires very often a train of reasonings, longer, perhaps, than is ever requisite in any other branch of science: a train, in the carrying on of which a man has this great advantage, that he may all along make sure of their correctness, while in every other branch of science he finds himself under the frequent necessity of taking up with such reasonings as are very far from being conclusive. Moreover, the very process of calculation itself, notwithstanding that, by Algebra, the rules of it are ready made to his hands (quoique l’analyse en préserve les règles,) requires throughout to have for its support a solid body of reasoning (un raisonnement solide,) without which he is, at every turn, liable to fall into some mistakes. The algebraist, therefore, (le géomètre is the word, but it is in his algebraic, and not in his geometrical, capacity, that, on the present occasion, the mathematician is evidently meant to be brought to view); the algebraist, then, (concludes this Grand Master of the Order,) finds, on every part of the field, occasion to keep his mind in exercise by the formation of those reasonings by which alone, if the problem be a difficult one, he can be conducted to the solution of it.
Thus far this illustrious pair of mathematicians. Now these reasonings (raisonnemens) so often mentioned, and always as so many works or operations perfectly distinct from those which consist in the mere application of the algebraic formulæ, what are they? Plainly the very things for the designation of which the words, contrivances for the coming at the solution of the problem, or some such words, have all along been employed. Thus much, then, is directly asserted, viz. that the operations, which consist in the as it were mechanical application of this set of rules, which for all cases is the same, on the one hand; and, on the other hand, those which consist in the other more particular contrivances for solving the particular problem, or set of problems, in question, by the application of these same general rules, are two classes of operations perfectly distinct from each other. But, moreover, another thing which, if not directly asserted, seems all along to be implied, is, that, to one or other of these two heads, everything that is or can be done in the way of algebra is referable.
Of the descriptions given of these different contrivances and sets of contrivances, of this sort of materials it is, that, in as far as they apply to the algebraic (not to speak here of the geometric) method, all these keys and sets of keys, as employed by the hand of the mathematician, will have to be composed. But, these contrivances being in themselves thus distinct from the general formulæ, it follows that, for the explanation of them, language other than that in which these formulæ are delivered, may consequently be employed: other language, viz. (—for there is no other) that language which is in common use. And thus it is that not only to Geometry, but to Algebra, may the purely verbal mode of designation be applied, to give to the several quantities which have place in the problem, such a mode of expression, as by indicating the several relations they bear to each other, shall prepare them for being taken for the subjects of that sort of operation, which consists in the putting them in that point of view in which, by means of those relations, those quantities which at first were not known, but which it is desired to know, become known accordingly. This, when expressed in the most general terms of which it is susceptible, will, it is believed, be found to be a tolerably correct account of the sort of operation which, on each particular occasion, must proceed. No direct, and, as it were, mechanical application of the set of general rules. Of what, then, is it, that a sort of algebraic key, or set of keys, of the kind in question, must be composed? Of a system of abbreviations or directions by which it shall be shown in what manner, in the several cases to which it is applicable, this sort of preliminary tactical operation may be performed, and to the best advantage.
As these two intimately connected yet distinguishable operations, viz. the application of the use-indicating (No. II.) and that of the key-presenting principle, went on together—the order of invention, i. e. the order in which the several propositions, or groups of propositions, come to be invented, would, in conjunction with the order of demonstration, i. e. the order in which, for the purpose of demonstration, it is either necessary or most convenient that they should be presented, be brought to light.
But in proportion as the order of invention came thus to be detected and displayed, in that same proportion would it be rendered manifest that theory was formed, and in what manner it was so formed, by abstraction, out of positive ideas; more and more general out of particulars; and, in a word, originally out of individual ones.
Supposing the whole field of Geometry, or, in a word, of Mathematics, measured and delineated upon this plan, what would, in that case, be signified by the word understanding, in such phrases as these, viz. he understands plain elementary geometry, he understands conic sections, or, in general, he understands the subject, would be a state of mind considerably different from that which at present is indicated by these same phrases, and accordingly, in the signification of the words learning and teaching, as applied to the same subject, the correspondent changes would be undergone.
Field of Mathematics—need of a general revision of it, for the purpose of Chrestomathic instruction.
Should there be any person, in whose eyes any of the observations above hazarded afford a prospect of their being conducive, in any degree, to the wished for purpose, to that same person a general revision or survey of the whole field of the science, with a view to the same purpose, may, perhaps, present itself as a task neither altogether needless nor unpromising.
In this, as in every other track of art and science, invention and teaching what has already been invented, are very different operations; and, for the performance of them to the best advantage, talents, in some respects different, and, at any rate, different situations, will, in general, be found necessary.
To the removal of the difficulties by which, in the minds of the generality of learners, progress is most apt to be impeded, a strong and clear sense of them is at least useful, if not indispensably necessary: and the larger the possession a man has or that sort and strength of talent by which he is qualified for invention, the less strong will be the impression left by any such difficulties on his mind.
Placed on the threshold of the science, upon crossing the track of it, a little verbal inaccuracy, which, to the eyes and feet of an adept standing in the higher regions, will, like a thread of grossamer, be an object altogether imperceptible, will, in the eyes of many a learner, be, if not an insurmountable bar, a troublesome, and, for a long time, a disheartening, stumbling-block.
In this part, as in so many others of the field of art and science, dazzled, not to say blinded, by the splendour which encircles a great name, professors have scarce suffered their eyes to be opened to see anything like an imperfection in the object of their admiration; and hence it is that so long as it affects not the substance—the very vital part, of the art and science, inaccuracies by which, though imperceptible to proficients, learners are put to torture, might, if searched for by eyes wholly unprejudiced, be found, it is believed, in greater numbers than is commonly so much as suspected.
For illustration, and as far as they go, even for demonstration, the following examples, taken from each of the three great divisions of Mathematics, viz. Geometry, Algebra, and Fluxions, no one of them requiring, for the conception of it, any the smallest degree of proficiency in the science to which it belongs will, it is believed, be considered as neither irrelevant nor unsatisfactory.
Euclid, Euler, and Newton,—men of no less account than these, will each of them be seen to afford an example of the sort of relation, and hitherto imperceptible, but not less operative sort of imperfection here in view: Euclid in Geometry, Euler in Algebra, Newton in the world of his own creation, Fluxions. If in the greater number, or in all these instances, the seat of imperfection should appear to belong rather to Logic or Grammar, than to Mathematics, neither the inconvenience to the learner, nor, consequently, the demand for indication, will by this circumstance be at all diminished.
In regard to Geometry, on the occasion of the exemplifications, which have already been mentioned, and for which reference has been made to the Appendix,* three have already been brought to view.
But those which are seen are but three out of a much greater number of imperfections, real or supposed, which, in the course of the inquiry already mentioned, the pair of self-teaching learners detected or supposed themselves to have detected. Without an adequate motive no labour at all, much less any course of labour so persevering as that which was here necessary, was ever undertaken; and on this occasion, in the character of an adequate motive and efficient cause, none presented itself as being so analogous, or in all respects so promising, as the sort of triumph which, in every instance, would follow upon the supposition of success. Many of these supposed triumphs the then adviser remembers to have been occasionally reported by these two pupils, if, on the ground of a few general hints, furnished at the outset, pupils they could be called: and sometimes it was the Grecian sage, sometimes his disciple, Simpson; sometimes both the one and the other, that were thus dragged, in imagination, at the tail of the audacious stripling’s car. For one most lengthy and perplext proposition, viz., the enunciative part of it on the subject of proportions, Simpson, who, in his quality of modern, could be treated with the less ceremony, Simpson, it is perfectly remembered, was not only drawn and quartered, but gibbeted.
Next, as to Algebra.
A seeming paradox, not to say absurdity, in which many a mind, it is believed, contrives even now to be entangled, is the rule, according to which, the product of two negative quantities, multiplied by each other, each of them less than nothing, (for in that mystery this other is but included in part,) produce a positive quantity; yea, verily, and that altogether as great as if they had both been positive.
In the third chapter and thirty-third Article of his Algebra, Euler, when he has observed that, by the multiplication of a positive by a negative, or of a negative by a positive, quantity, the product is still negative; and therefore, if the product of two negative quantities were not positive, it would be the same with these, thinks he has made the matter sufficiently clear. That the conception remaining in the mind of this adept, after the utterance of these words, was abundantly clear, need not be doubted; and no less clear would it have been whatever other words it had on this same occasion happened to him to employ. But, as to a learner, taught by such a demonstration, the chances seem many to one that his tongue would be silenced; yet, the chances seem, at least, as many that his mind would be rather darkened than enlightened.
Fortunate it is, on this occasion, for the learner in Algebra, if, being an Englishman, it is through the medium of the translations that have been made into his own language, that he betakes himself for instruction to that celebrated work. At the end of the first volume are inserted a number of notes, some by a former translator of the work from German or Latin into French—some by the translator into English. In the second of these notes, should perseverance have carried him thus far, or fortune set him down at the place, the learner will find what light the subject admits of, thrown upon this the original darkness. Without employing the gloom of Algebraic characters to throw again their darkness upon this first light, a short passage or two, extracted from two pages, may suffice to afford to the intelligent though uninitiated, unmathematical reader, a clue which, if not immediately, will, it is believed, with the help of a little reflection, lead to a solution of the paradox.
“The taking of a negative quantity negatively destroys” (says the intelligent annotator) “the very property of negation, and is the conversion of negative into positive numbers.” Of the non-conception or misconception, so apt to have place on this subject, he thus points out the cause. “Multiplication,” (says he,) “has been erroneously called a compendious method of performing addition:” (which it might without impropriety be called when the quantities are both positive,) “whereas,” (continues he,) “it is the taking or repeating of one given number as many times as the number by which it is to be multiplied contains units. Thus (any number multiplied by one-half) 9, for instance, multiplied by ½, means that it is to be taken half a time;” (i. e. that of that same number the half is to be taken instead of the whole.) “Hence,” (continues he, a little further on,) “it appears that numbers may be diminished by multiplication, as well as increased, in any given ratio, which is wholly inconsistent with the nature of addition.”
Happy as the young Algebraist may have reason to think himself, if perseverance has thus carried him to the end of the first and longest of the two stages into which the road is divided, it will have been still more fortunate for him, if at the very place at which, by the obscure exposition, he has at the very threshold of the science been, as above, tormented, it has by any means happened to him to be conducted to that other spot, at which light is let into the subject, and satisfaction substituted to perplexity. True it is that, drowned in a flood of Algebra, a figure of two, being the same which is prefixt to the note, may, after the flood has been dragged, upon a close inspection be found. But, in point of fact, how stands the matter of reference? It is by the note itself that the eye was conducted to the reference in the text. By that reference it was not, nor probably ever would, have been conducted to the note.
Here belongs a practice, begun, it is believed, as well as continued, in Scotland, and but too much copied in England,—the throwing the matter of elucidation to a distance from the matter to be elucidated. The consequence is, that many, at the suggestion of indolence, refuse from first to last to go a-hunting, time after time, in quest of the light thus proffered, but, at the same time, hidden under a bushel; while others, groaning under a toil thus causelessly imposed upon them, purchase or leave unpurchased, at the humour of the moment, the light with which, without any additional expense to the writer, they might have been accommodated, without being thus made to pay for it.
Lastly, as to fluxions: a modification of the algebraic form,—a mode of calculating invented under that name by Newton,—under the name of the differential and integral calculus, by Leibnitz, whose denomination is employed in every language but the English.
The original work of Newton is not at present within reach. But the word employed on this occasion in English, being in all English books the same, no such suspicion can arise, as that in the use of so elementary and radical an expression, any departure from the language of the great master can have had place.
In a logical and grammatical point of view, this word is not exactly the word which the object intended to be denoted required for the expression of it: instead of the clear idea meant to be conveyed, to an unpractised mind the idea presented is very apt to be a confused one: a confusion by which the very first steps taken on this ground are but too apt to be involved.
By a word or two of explanation, this confusion might have been effectually dispelled; but nowhere is any such explanation to be found.
The agent or operating instrument of action, and the product or result of it, in as far as the operation is effective; on every occasion both these entities are as necessary as they are distinct and distinguishable from each other. But owing to the poverty of the language, or to the want of clear discernment on the part of the generality of those who begun and of those who continue to use it, the two last of these objects are apt to be confounded under one name.
To the above examples, though in this case on particular great name, no individual mathematician can be brought to view, that no branch of mathematics may want its exemplification, may be added a source of confused conception, observable in the lowest field of mathematics, viz. arithmetic.
Square root, cube root: of the objects which these expressions are employed to signify, that in the head of many a student the ideas obtained remain from first to last in a state of confusion, is a proposition, the truth of which would, it is believed be, upon inquiry, but too abundantly exemplified.
Square-root, i. e. root of the square: just as we say, fountain-head, house-top. In a book of instruction, suppose an explanation to this effect were subjoined upon the first mention of this compound appellative, many a scholar’s mind, it is believed, would be saved from a load of perplexity and confusion under which at present it has to struggle.
Or without the explanation, short and simple as it is, suppose the hyphen and no more inserted, as above, between the two elements of this compounded appellative, this, if it had not of itself afforded a complete solution of the enigma, would, in many instances, have afforded a clue to it. Accordingly sometimes, though not constantly, this simple though of itself inadequate instrument of explanation is inserted.
For want of such an explanation of the two adjuncts, viz. square and cube, thus applied, what in many a mind is at present the effect?
Square root and cube root, two different roots belonging to the same imaginary plant. Square root, as being that one of the two which is of the most frequent occurrence, a root, such as that of the common radish, which runs out into length, made square, viz. as it might be by four strokes of a knife made in proper situations and directions.
Cube root, a root of another shape, such as that for instance of a turnip radish brought into the shape of a cube or die by four such strokes as the above, with the addition of two others, viz. at the top and bottom of the radish.
Matter is infinitely divisible, matter is not infinitely divisible—both these propositions cannot be true, one of them must be true: which of them is true it is scarce possible to prove. For the present purpose, let the latter be supposed to be true; true or not true, it is rather more distinctly conceivable than the other; and for the present purpose the only one that can serve. For the present purpose, then, let it be supposed true.
On this supposition, all matter is composed of atoms, and all of them of the same size.
These smallest existing atoms, suppose them, all or some of them, cubes—so many perfect dice. These dice may be conceived to be composed each of them of a determinate number of particles of the same form, which though never in fact separated, may as easily be conceived to be separable and separated as if they really were so. These component particles, call them points: and let the number of them be exactly 512. Ranged in a column regular, eight of these points make a line; the lines being all of them straight and ranged in appropriate order, one above another, eight of them, each containing eight points, make a surface—a surface of a square form, such as that exhibited by a chess-board; and ranged again in a correspondent order, eight of these chess-board surfaces compose the atomic cube or die.
The sixty-four points first mentioned, points which thus placed in the due and correspondent order—in the order adapted to the purpose, exhibit the superficial figure called a square; the square composed of these sixty-four stands upon, and placed in any direction, has for each of its sides, (of which the square placed in a certain position, may be called the base,) the line composed of eight of these points. The whole atom is composed of eight of these squares, piled one upon another, constituting a cube, having for its base the square first mentioned.* The number contained in the cube is then with relation to each of the lines of each of these squares, a cube, containing eight times as many of these points as any one of the squares contains; each such square, containing eight times as many points as any one of its component lines contains.
Eight, the number of the points in each of these lines, is the cube root of 512, the whole solid composed of 512 such points, the whole number of the points contained in the solid atom, the form of which is, by the supposition, that of a cube or die: eight, this same number, eight, is at the same time the square root of sixty-four, which is the number of the points contained in each of the surfaces by which that atom is bounded; the form of each of which is by the supposition the form of a square.
As often as in any institutional work in mathematics an explanation of these terms square root and cube root is undertaken to be given, the figure of a square at least is, it is believed, exhibited; and for the representation of it a number of points or lines are employed.
But nowhere, it is believed, is the explanation so full as above; nor in the giving it are the points put together in such a manner as to present the idea of a cube. Yet this cube being, of all the entities in question the only one which, in a separate state has, in the nature of things, its exemplification, the ideas of a surface, a line, and a point, having, respectively, been deduced from the idea of this solid in the way of abstraction, the consequence seems to be, that when images come to be exhibited, the image of a cube ought no more to have been omitted than the image of a square.
Neither is it very distinctly explained why or how one of the surfaces by which a cube or die is bounded, comes to be considered as constituting the root of it; nor why or how one of the lines by which one of these surfaces is bounded, comes to be considered as constituting the root of that surface.
Supposing these matters to admit of explanation, the explanation it is believed will be to some such effect as this: Take a die and set it down upon a table resting on any one of its faces or surfaces—suppose that which is marked with one spot—then suppose the die to be a plant, that surface may naturally enough be considered as representing the root of the plant. Of any figure approaching to that of a die, true it is that no plant has ever yet been found. But of a figure approaching very nearly to that of a hemisphere, such as that which might on all sides be contained exactly within the compass of a die, of correspondent dimensions, plants have actually been found, witness a species of the genus cactus.
In like manner, in a vertical position, at right angles to the table, set up a chess board, composed, as above, of the rows of squares of which it (this square figure) is composed; the lowest, i. e. that which is in contact with the table, represents that boundary which in geometrical language is frequently called the base of the square; and which in the language of arithmetic, as above, may be termed the root of it, bearing, as it does, the same relation to the number of lines contained in the whole surface, as the number of lines contained in the whole surface bears to the number of lines contained in the whole solid, termed, as above, a cube or die.
Simple as the above explanation is, and useful at least as it seems to be for the obviating confused conceptions and misconceptions, such as those of which the above exemplifications may serve as a sample, no such explanation will, it is believed, be as yet to be found in any institutional book.
Unfortunately, coupled as it is with the expressions used for the designating of the other objects that are so closely related to, and inseparably connected with it, the word root, considering the material image which it cannot fail to present, and which if it did not present, it would be altogether insignificant and inexpressive, seems not very happily suited to the purpose.
In correspondency with the word root, is employed the word power; root being, in a certain proposition, indicative of decrease; power, in the same proportion of increase. Here, with no other difference than that between decrease and increase, the objects themselves match exactly. But the symbols that are thus employed for the designation of those same objects, very badly do they match with each other.
1. No image correspondent in any way to that which is exhibited by the word root, is exhibited by the word power. With the correspondent idea, for the expression of which the word root is employed, it has no analogy; it does not match with it: of itself neither of them has any tendency to call up to mind the other.
2. On the other hand, power has the advantage, and an indispensable one it is, of carrying the increase to any number of degrees, and consequently the length, say also the height, to any extent that can be desired.
On the other hand, when for expressing decrease, and thus, in the scale of magnitude, descent, you employ the word root, at the first step in the line of descent you have the square root; at the next, the cube root; but there your stock of roots, of different species of roots, each less, and running down lower than the preceding one, is at an end.
In one point of view, and that the main one, power, it is true, is not ill adapted to present the ideas that belong to the subject. The idea of power includes in it the idea of the effect produced or producible by the operation or action of that power; and the greater the quantity of power, the greater will be expected to be the quantity of the effect. Whatsoever be the number in question, by the quantity expressed by the term the third power, of that same number, the effect producible, be it of what nature it will, will be greater than the effect producible by the quantity expressed by the term the second power of that same number; taken in this point of view, of two numbers employed for giving expression to two powers of different magnitude, the greater will therefore be expressive of the greater power.
But taken in another sense,—as resulting from another of the sorts of occasions on which it is wont to be employed,—another sense, and that to many minds a more familiar one, of any increase of the number attached to the word power, the result will be the idea not of increase but of decrease. Apply it, for example, to statistics. What is meant by the first power in Europe? Is it not that which is capable of producing the greatest effects? What is meant by the second power in Europe? Is it not that which is not capable of producing any effects but such as will be less than those producible by the first power? and so on, the greater the number the less the power indicated by it.
Though, as above, in itself and of itself, were no correspondent and apposite idea required to be expressed along with it, power might, have been not altogether ill adapted to the purpose; yet this incapacity of finding its match in any other word, is such an objection to it as seems insuperable and conclusive.
Retaining the word root for giving expression to decrease in quantity and descent in altitude, suppose that for giving expression to increase and ascent in the same proposition, the word branch were employed. Branches ascending in the sky, we might have as many as powers; descending, roots we might have as many as branches; roots,—not square roots and cube roots indeed,—after which our stock of roots would be exhausted; but first roots, and second roots, and third roots, and so on, down to the centre of the earth; exactly as many as branches; for every branch a root, wherever a root were wanted; for every root a branch, wherever a branch were wanted.
The plain and standard number, neither multiplied by itself nor divided, neither increased nor diminished, shall it be root or branch, or both, or neither? Keeping still to the same figure, shall it not be trunk? Second root will then be to trunk, what trunk will be to second branch. In this case, as in the case of logarithms, there are points which would require to be settled.
To the use of the word branch an objection not unanalogous to that which, as above applied to the word power, does, it must be confessed, present itself. In the ascending series of branches, the greater the number employed in giving expression to any term in the series—in a word, to any branch,—the greater should be the effect of any portion of matter taken in that number, repeated the number of times indicated by that numerical denomination: the effect producible by the third branch of the number should be greater than the effect producible by the second branch of the same number and so on. But, in the case of the class of material beings, from the sensible properties of which the image is deduced; in the case of a tree, (for example,) the higher the branch is, it is not the stronger the more powerful, but the weaker the less powerful; and it is by the greater number that the higher branch will be presented to view; and, in particular, no branch can fail to present itself as being in a greater or less degree weaker, instead of stronger than the trunk.
Here, then, applying to the word branch is an objection analogous to that which we have seen applying to the word power: analogous to it, and perhaps equal to it.
But, when the one objection is set against the other, there remains in favour of the word branch, the circumstance of its being analogous, to the word root—the word already in use to designate in corresponding propositions the correspondent and opposite effect.
What must be confessed is, that supposing the superior aptitude of the proposed new terms, when compared with the old established terms, were ever so unquestionable, the utility of any such undertaking as that of substituting in any institutional work, or scheme of oral instruction, the new to the old, would still be very questionable. It is in the terms now in use for the designating of the ideas in question, that all the existing works on the subject stand expressed: these works could, therefore, no further be understood, than in as far as the terms here in question are understood.
But how conclusive soever this consideration may be, in the character of an objection to any such attempt as that of substituting these new terms to the old established ones, it applies not in the character of an objection, to the adding, in a scheme of instruction, to an explanation of the old, an explanation of the new. If, therefore, the ideas presented by the proposed new terms should, in any instance, be found clearer than the ideas presented by the old, here will so much new light be thrown upon the subject, without any of the inconveniences so frequently, if not constantly, attached to change.
Nor would the preferable use of the new language be altogether incompatible with the reaping the instruction contained in the books in which the old terms are employed. All along, since the days of Newton and Leibnitz, while, in the English school, the terms fluent and fluxion, with their appendages, have been employed,—by the German and French schools, for the conveyance of the same ideas, the terms integral and differential, with their appendages, have been employed.
A principle of nomenclature so inadequate—a principle by which neither multiplication nor division could be carried on more than two stages, how came it to be adopted? To what cause shall it be ascribed? Obviously enough to this, viz. the continual conversion of the algebraical and the geometrical forms into each other. In geometry, when from your point you laid down a line, when from your line you had erected your square, and on your square you had erected your solid in the form of a cube, then you found yourself at a stand, no other ulterior dimensions did the nature of things afford. So much as to the scale of increase. So, on the other hand, in regard to roots. In the square you possessed a figure, of which the metaphorical root represented by any of its boundaries, might be found; in the cube you possessed another figure, for which a still deeper root, viz. the same by which the root of the square had been represented, might be found. But, the nature of things not affording anything more solid or substantial than a cube, there ended also the corresponding line of roots. So much as to the scale of decrease, for in the number called a square number, in other words, in the second power of that (the correspondent) number, or in the first branch of that same number, considered in the character of a branch, Geometry affords an image capable, in some sort, of representing it; so likewise in the number called a cube number. But, at that point the representation, and, consequently, the interconvertibility ends; at that next point you come to the third power, or the second branch of the number in question, and to that the stores of Geometry afford not any correspondent image.
Here, then, may perhaps be seen the cause of this obscure and imperfect portion of nomenclature. But, by the indication thus given of the cause of the imperfection, the inconvenience resulting from it is not by any means diminished; nor, therefore, the demand for the application of such remedy as the nature of the case admits of.
In the case of this science, as in the case of so many other branches of art and science, the knowledge which the artist, or man of science possesses, in relation to the subject, is derived from the several particulars of detail which belong to the subject, from the acquaintance which continual practice has given him with these several particulars. The ideas which, from these several particulars it has happened to him to derive and store his mind with, are perhaps, without exception, clear ones. On the several occasions on which these particulars have been brought to view and spoken of, spoken of in language which, in the mind of him by whom it has been employed, has all along had clear ideas for its accompaniment—the language attached to the subject by usage has, of course, been all along employed. In the mind of this artist, or man of science, by whom this current language is employed, it is all along conjoined with clear ideas. The conclusion which very naturally, however erroneously he forms in his own mind,—forms all along, as a matter of course, and in such a manner as he would move his legs in walking, almost without thinking of it—is, that in the minds of other persons, in the minds of learners, ideas similarly, if not altogether equally, clear, will be attached to this same language.
But in the minds of persons in general, and of young scholars in particular, the phrases in question have no such accompaniment; with the particulars belonging to the science they have no such already formed acquaintance. When, therefore, without sufficient warning, perhaps without any warning at all, of the impropriety of the application thus made of them, the phrases in question are by the teachers in question (through which soever of the two modes of conveying his instruction, viz. discourse scriptitiously or discourse orally delivered,) employed in the delivery of instruction in relation to the art or science, the consequence is, that, instead of ascribing to them the latent and multifarious meaning which, by long practice and acquaintance with particulars, the teacher has learned to attach to them, the meaning which he, the learner, attaches to them, is no other than that which has been attached to them by the usage of ordinary language. When, with the assurance so naturally attached to the possessor of acknowledged and undisputed infallibility, he is told that every number which is brought to view, to which the sign called the negative sign is prefixed, is expressive of a quantity which is, and exactly by so much as the figure indicates, less than nothing, the belief of the existence of an infinite number of quantities, each of them less than nothing, is thus added to his creed.
When, again, after having been required to take one of these quantities that are by so much less than nothing, and multiply it by another of these quantities that are less than nothing,—two, for example, by three—the product is composed of a number of quantities, all of them greater than nothing, viz. six in number (being exactly the same number of quantities, all greater than nothing, that would have been the result, if, instead of quantities all of them less than nothing, an equal number of quantities, all of them greater than nothing, is employed;) what is the consequence? We remain astonished and confounded. But the more astonishing the matter of science thus imbibed, the greater the glory attached in the acquisition of it; and to comfort the learner under his confusion, is the use and benefit of this glory, the glory of having, by dint of hard labour, succeeded in treasuring up in his mind, under the name of science, this mass of palpable nonsense.
A determination (suppose) is taken to substitute, on this ground, the language of simple truth for the language of scientific falsehood, and thereby to substitute light for darkness. For the production of this effect what is the course that a man will have to take? On the occasion of every sort of transaction, operation, event, or state of things, in which this sort of fictitious language is in use to be employed, he will have to bring to view the nature of the transaction, operation, event, or state of things, and, at the same time, to bring to view the effect which the supposed existence of some supposed negative quantity is productive of. Of this transaction, operation, event, or state of things having given an indicative description, employing, in so far as susceptible of application to the subject, the terms of ordinary language, he will thereupon, in the like language, give an indication of the effect so produced by the negative quantity, as above. So far as this mode of explanation shall have been made to extend, so far, and no farther, will the science have been brought and put into that state in which it ought to be put for the instruction of the young beginner; into which it must be put before it can have been fitted for rendering more than a very small part of that quantity of service which, in its own nature, it is capable of rendering to mankind.
In what circumstances shall we look for the cause of so apparently extraordinary a phenomenon; such flagrant impropriety, inappositeness, falsity, and thence so thick a veil of factitious obscurity in the language of science? Of inappositeness, impropriety, falsity, in that science, of all others, which reckons infallibility in the number of its pretensions; of which infallibility is commonly regarded as the unquestionable and exclusive attribute?
In as far as language which, on ordinary occasions, is used in one sense, is, on the occasion of scientific instruction, used in another, an effect similar to that which, by the species of secret discourse called cypher, is produced in any mind which is not in possession of the key, is produced in the mind to which instruction in the science has lately begun to be communicated.
To him who is in possession of the key, the language of the cypher, obscure, mysterious, and perhaps nonsensical, (as to the conception of this very person it would be otherwise,) is clear, correct, and instructive. But does it ever happen to him to entertain any such expectation, that to any person who is not a possessor of that necessary instrument, it should present itself in that more satisfactory character? So soon as any such persuasion to any such effect were entertained by him, so soon would an assurance equally strong be possessed by him that the purpose for which alone the language had with so much pains been devised, was already defeated.
To the experienced instructor, the particulars which he has been accustomed to have in view on the occasion of which the technical and inapposite language in question, which, and which alone, in speaking of the subject, it has been usual to him to employ, and have employed, is the cypher: to this cypher the particulars which, on these same occasions, it has been usual for him to have in view, compose the key. What wonder if, among those to whom, while not yet in possession of the key, the cypher comes to be pored over, the number of those to whose minds the words of the cypher have imparted clear ideas, is comparatively so inconsiderable.
By a comparatively small number of privileged minds, to the constitution of which the subject happens to be in a peculiar degree adapted, at the end of a certain number of years thus employed, an acquaintance with the science—an acquaintance more or less clear, correct, and extensive—comes to have been attained. Attained! but how? by means of the cypher? by means of the inapposite, the ill-constructed, the fictitious language? No: but in spite of it. Instead of being left to be drawn by abstraction, like Truth out of her well, from the bottom of an ocean of perturbers, had the key been conveyed, in the first instance, and terms of compact texture constructed out of apposite, familiar, and unfictitious language, a small part of the time so unprofitably employed would have sufficed for extracting from the subject a set of conceptions much more clear, correct, and extensive, than those obtained by a process so full of perplexity and inquietude.
To no man can any truth, or set of truths, the effect of which is to prove and expose the vanity of any part of that treasure of science, real or pretended, in which he has been accustomed to place any part of his title to the expectation of distinction and respect, be naturally expected to be otherwise than unacceptable. In no better light than that of an enemy can the author of so unwelcome an importation be regarded. By the feeling of the uneasiness thus produced, the passion of anger directing itself toward the immediate author of the uneasiness, will be produced; and with it the appetite for vengeance. To the gratification of this appetite, the readiest instrument which the nature of the case will, generally speaking, be found to present, will be the imputation of ignorance: matter by which an imputation of this sort may be fixed will of course be looked out for, and never does it fail to be looked out for with a diligence correspondent to the provocation given, and the temper of the individual to whom it has been given. Ingenuity is set to work to devise by what means the dart cast at the mind may be rendered most sharp; the wound deepest and most afflictive.
But by no such artifices will the mind of a judicious reader be led astray from the view of the proffered benefit. By the means indicated, or by any other means, is the art of teaching, as applied to the branch of art and science in question, susceptible of being improved? If so, whether on the part of him by whom any useful and practically applicable means of improvement have been suggested, the marks of ignorance be more or less palpable, or more or less numerous, are questions not worth a thought.
To the man of science, in whose breast the predominant affection is not the self-regarding love of reputation and desire of intellectual fame, but the social affection of philanthropy, observations which have for their tendency, as well as their object, to put him on his guard against those different propensities which, with more or less power and effect, operate in every human breast, will be regarded, not as an injury but as a service; will be received, not with anger, at least not with any durable emotion of that kind, but rather with complacency and thankfulness. He will find himself thus put upon his guard against an intestine, against a latent and insidious, enemy.
On the occasion of propounding any extensive plan of useful instruction—in this, as in every other walk of useful art and science, the lover of mankind will propose to himself two main objects: the one to maximize the quantity of use capable of being derived from it, the other to maximize the facility, and thence the promptitude with which each given portion or degree of it may be rendered obtainable. Usefulness and facility, by these two words, may be expressed the main objects of his regard.
Long after these, the advancement, in as far as that is distinct from usefulness; long after these, though still not as an object to be neglected, will the mere extension of science, that science being but a speculative one, rank in his estimation and endeavours.
Interconversion of Geometry and Algebra.
In speaking of Geometry and Algebra—of Geometry in the first place, of Algebra in the next place—thus far it has been necessary to speak of these two objects, as if they were so many distinct branches of mathematical art and science; one of them, and that alone, applicable to one sort of subject or occasion; another, and that alone, to another. But, to his surprise not improbably, and to his no small annoyance certainly, the learner will sooner or later have occasion to observe, that, in point of practice, no such separation has place; and that, for the obtaining of one and the same result, for the solution of one and the same problem, for the finding an answer to one and the same question, for the demonstrating the truth of one and the same assertion, for instance, in the way the problem in question* has been solved, both these branches of mathematical art and science have been employed at once; and that, for the arriving at no more than one conclusion, he will have to feel his way through the two distinct sorts of labyrinths, the labyrinth constructed out of the capitals of the letterpress alphabet, or the field of geometry; and the labyrinth constructed out of the small letters of the same alphabet, in the field of Algebra, with dots put over some of them, in the upper quarter of it, if in the part occupied by the Newtonians; and d’s put before as many of them, if in the part occupied by the Leibnitzian corps. Accordingly when, after leaving out a swarm of other lines, he has learned that, for the designation of the line which, in the first place, he is in search of, two of these capital letters have been appointed, a supposition which he will naturally be led to make is, that now he has formed with it that sort of acquaintance which will be sufficient for the purpose. Not he, indeed; for too soon, whenever it is, for his peace, will he find it snatched out of his hands, and thrown into the algebraic mill, out of which it will not come without having stamped upon it a new name, made out of a single letter of the alphabet, and that a small one; and so with regard to all the other Geometric personages, for giving names to some of which, nothing less than three, or even more than three, of these letters a-piece, will suffice.
Of so troublesome a repetition of labour, especially on a branch of the field of art and science, of which, by means of the abbreviative and condensative forms, saving of labour is acknowledged to be the grand instrument, wherein consists the use? To what cause is the usage that has taken place in this matter to be ascribed? To find any answer to this question, the new search that has been made in the works of mathematicians has not been attended with success.
One effect seems inseparably to follow, of course, from the very nature of the two modes; and that is, that the mode of expression which, in the geometrical mode is, by the references to the individual diagram, confined to that individual diagram, and thus reduced down (narrowed, to the minimum or maximum shall we call it of narrowness) is, in the algebraic mode, and for the opposite reason, generalized, or, to use an expression more conformable to the language of logicians, universalized; and, to this circumstance, without our being always if ever fully aware of it, may frequently, perhaps, be found the cause, not less real, how imperfectly soever perceived, of the trouble taken to translate the preparatory and demonstrative part of the proposition out of the geometrical form into the algebraic.
Then why not translate it at once into the ordinary unabbreviated language? In answer to this question several reasons may be given, none of them unapt.
1. Of one of the most obvious of them, intimation is already conveyed by the word unabbreviated. Abbreviation is the main characteristic of the algebraic mode of notation, as distinguished from the simply arithmetical.
Applied in so many cases where it was in a prodigious degree, beneficent, habit would suffice to cause it to be applied to other cases in which the employment of it would not be attended with any such advantages.
2. Be it of what kind it may, an instrument which, after much trouble, a man has at length succeeded in rendering himself expert in the use of, he is naturally fond of playing with; love of power and love of admiration,—both these appetites find their gratification in it.
3. By this symbolical, in contradistinction to the purely verbal, mode of designation, much embarrassment and difficulty is saved; and, in lieu of a variable, an invariable mode of expression is employed. For framing the expression in the purely verbal mode, how much more effectually soever if framed by a masterly hand, instructive to the scholar at first entrance, a much fuller and clearer comprehension of the subject will commonly be necessary than every one is able to attain, as well as much more labour than every one is willing to bestow.
For the giving expression to the same matter in the purely verbal mode, it is impossible to say how many forms, all more or less different from one another, some more apt, some less apt, whether in respect of choice of words or arrangement, might be capable of being employed. Expressed in the symbolical mode, all these variants are reduced to one. To the reader, great, whatsoever be the subject, is the advantage derived in the shape of clearness of conception, from this unity and simplicity in the mode of expression; to the writer it, moreover, affords, though a different, a correspondent advantage. Reducing all styles to one, it places the most inexpert grammarian upon a level with the most expert.
[* ] Besides those here enumerated, two other subjects are discussed in this No. of Appendix, viz. Demand for Revision of the Field of Mathematics, and Interconvertibility of Geometry and Algebra.
[* ] Hamilton’s, as far as recollection serves.
[† ] [Demonstrated.] Viz., either that what is asserted, as in the case of the sort of proposition called a theorem, is correct and true; or that what has been undertaken to be done (as in the case of the sort of proposition called a problem,) has accordingly been done. When any such phrase as the demonstration of a proposition occurs, the sort of proposition which on any such occasion is most apt to present itself, is a theorem; and this, not only because theorems more frequently occur than problems, but because in the case of a problem, the term demonstration will not apply in such manner as to complete the sense, without the insertion of a number of words, of which, in the other case, there is no need.
[* ] It is noticed, as the result of experience, that such a question would not be likely to be put; and that learners generally avoid verbal modes.—Ed.
[* ] This “exemplification” has not been discovered among the MSS.: but the method proposed by Mr Bentham was (perhaps is) in use, to some extent, in the University of Dublin, and there is in existence a small tract, containing the proposition of the first book of Euclid demonstrated in words alone, (published in Dublin.)—Ed.
[* ] See Mr Pillans’ Letter, supra, p. 61.
[† ] In practice, this is not found to be the case.—Ed.
[‡ ] This, like the former exemplification, has not been found.—Ed.
[* ] The remarks which follow bear the title, “All true Mathematics is Physics; or, Beddoism defended against Edinburgh Review for July 1812.” Dr Thomas Beddoes published, in 1793, “Observations on the nature of demonstrative Evidence,” in which he maintained (but by no means to the same extent as the author) the physical origin of mathematical conceptions.—Ed.
[* ]Vide supra, p. 48.
[* ] In the six first Books of Euclid’s Elements, being all that relate to plane figures to the exclusion of solids, the sum of the propositions is 231. This number might, perhaps, not be too great to be conveniently included in one Table.
In one of the latest, and it is supposed, upon the whole, most instructive of the books professing to exhibit the elements of Geometry, one book has for its title, “The Comparison of Solids.” In this book, the number of propositions is twenty-one. Besides their respective mathematical and preparatory uses, many of these have their physical and immediate uses. Witness cylinders and spheres, and thereby and therein milk measures, ale measures, and oranges. Of these twenty-one, no one is to be found in the elementary worka of Euclid: every one, perhaps, is in some way or other, descended from its contents. Might not here be another occasion for marking the filiation of the branches contained in this useful supplement, and thereby affording indication and demonstration of the utility of the venerable original?
[a ] The first six Books.—Ed.
[* ] Except by means of the abbreviative and concentrative, it cannot facilitate conception more than ordinary language, of which it is the sign, does.
[* ] Vide p. 156.
[* ] This, like the other exemplifications already alluded to, has not been found among the MSS.—Ed.
[* ] There is here a frequent incidental repetition of views already discussed in, and properly belonging to, other departments of this Essay. It was written at considerable intervals of time, and the author sometimes overlooked the fact that he had already gone over the same ground.—Ed.
[* ] Vide supra, p. 159. note.
[* ] Of these conceivable ultimate particles, eight ranged in proper order exhibit the figure of a line; eight such lines, containing sixty-four such particles, the figure of a square; eight such squares, containing sixty-four such lines, and 512 such particles, the figure of a cube or die.
[* ] This seems to have been meant to follow some exemplification which has not been found among the MSS.—Ed.