The most striking instance in point which history affords (though not an example of an experimental science rendered deductive, but of an unparalleled extension given to the deductive process in a science which was deductive already) is the revolution in geometry which originated with Descartes and was completed by Clairaut. These great mathematicians pointed out the importance of the phenomenon better known. Thus the covered that variations of quality in science of sound, which previously any class of phenomena correspond stood in the lowest rank of merely regularly to variations of quantity experimental science, became deduc- either in those same or in some other tive when it was proved by experiment phenomena; every formula of mathethat every variety of sound was con- matics applicable to quantities which sequent on, and therefore a mark of, vary in that particular manner bea distinct and definable variety of comes a mark of a corresponding oscillatory motion among the particles general truth respecting the variaof the transmitting medium. When tions in quality which accompany this was ascertained it followed that them; and the science of quantity every relation of succession or co-ex- being (as far as any science can be) istence which obtained between phe- altogether deductive, the theory of nomena of the more known class, that particular kind of qualities beobtained also between the phenomena comes, to this extent, deductive likewhich correspond to them in the wise. other class. Every sound, being a mark of a particular oscillatory motion, became a mark of everything which, by the laws of dynamics, was known to be inferrible from that motion; and everything which by those same laws was a mark of any oscillatory motion among the particles of an elastic medium became a mark of the corresponding sound. And thus many truths, not before suspected, concerning sound become deducible fact, that to every variety of position from the known laws of the propagation of motion through an elastic medium; while facts already empirically known respecting sound become an indication of corresponding properties of vibrating bodies, previously undiscovered. But the grand agent for trans-ordinates vary relatively to one anforming experimental into deductive other, every other geometrical prosciences is the science of number. The properties of number, alone among all known phenomena, are, in the most rigorous sense, properties of all things whatever. All things are not coloured, or ponderable, or even extended; but all things are numerable. And if we consider this science in its whole extent, from common arithmetic up to the calculus of variations, the truths already ascertained seem all but infinite, and admit of indefinite extension. in points, direction in lines, or form in curves or surfaces, (all of which are Qualities,) there corresponds a peculiar relation of quantity between either two or three rectilineal co-ordinates; insomuch that if the law were known according to which those co perty of the line or surface in question, whether relating to quantity or quality, would be capable of being inferred. Hence it followed that every geometrical question could be solved, if the corresponding algebraical one could; and geometry received an accession (actual or potential) of new truths, corresponding to every property of numbers which the progress of the calculus had brought, or might in future bring, to light. In the same general manner, mechanics, These truths, though affirmable of astronomy, and in a less degree every all things whatever, of course apply branch of natural philosophy comto them only in respect of their monly so called, have been made quantity. But if it comes to be dis-algebraical. The varieties of physi cal phenomena with which those same subject of inquiry, and drawing sciences are conversant have been a case within one induction by means found to answer to determinable of another; wherein lies the pecuvarieties in the quantity of some liar certainty always ascribed to the circumstance or other; or at least to sciences which are entirely, or almost varieties of form or position for which entirely, deductive? Why are they corresponding equations of quantity called the Exact Sciences? Why had already been, or were susceptible are mathematical certainty, and the of being, discovered by geometers. evidence of demonstration, common phrases to express the very highest degree of assurance attainable by reason? Why are mathematics by almost all philosophers, and (by some) even those branches of natural philosophy which, through the medium of mathematics, have been converted into deductive sciences, considered to be independent of the evidence of experience and observation, and characterised as systems of Necessary Truth? The answer I conceive to be, that this character of necessity ascribed to the truths of mathematics, and even (with some reservations to be hereafter In these various transformations, the propositions of the science of number do but fulfil the function proper to all propositions forming a train of reasoning, viz. that of enabling us to arrive in an indirect method, by marks of marks, at such of the properties of objects as we cannot directly ascertain (or not so conveniently) by experiment. We travel from a given visible or tangible fact through the truths of numbers to the facts sought. The given fact is a mark that a certain relation subsists between the quantities of some of the elements concerned; while the fact sought pre-made) the peculiar certainty attributed supposes a certain relation between the quantities of some other elements. Now, if these last quantities are dependent in some known manner upon the former, or vice versa, we can argue from the numerical relation between the one set of quantities to determine that which subsists between the other set; the theorems of the calculus affording the intermediate links. And thus one of the two physical facts becomes a mark of the other, by being a mark of a mark of a mark of it. to them, is an illusion; in order to sustain which, it is necessary to suppose that those truths relate to, and express the properties of purely ima ginary objects. It is acknowledged that the conclusions of geometry are deduced, partly at least, from the socalled Definitions, and that those definitions are assumed to be correct representations, as far as they go, of the objects with which geometry is conversant. Now we have pointed out that, from a definition as such, no proposition, unless it be one concerning the meaning of a word, can ever follow; and that what apparently follows from a definition, follows in reality from an implied assumption that there exists a real thing conformable thereto. This assumption in the case of the definitions of geo§ 1. IF, as laid down in the two metry, is not strictly true: there preceding chapters, the foundation of exist no real things exactly conformall sciences, even deductive or demon- able to the definitions. There exist strative sciences, is Induction; if every no points without magnitude; no step in the ratiocinations even of geo-lines without breadth, nor perfectly metry is an act of induction; and if straight; no circles with all their a train of reasoning is but bringing radii exactly equal, nor squares with many inductions to bear upon the all their angles perfectly right. It CHAPTER V. OF DEMONSTRATION AND NECESSARY TRUTHS. Since, then, neither in nature, nor in the human mind, do there exist any objects exactly corresponding to the definitions of geometry, while yet that science cannot be supposed to be conversant about non-entities; thing remains but to consider geo no will perhaps be said that the assump-matical line, thinks so from the evition does not extend to the actual, dence of his consciousness: I suspect but only to the possible existence of it is rather because he supposes that such things. I answer that, accord- unless such a conception were possible, ing to any test we have of possibility, mathematics could not exist as a they are not even possible. Their science: a supposition which there existence, so far as we can form any will be no difficulty in showing to be judgment, would seem to be incon- entirely groundless. sistent with the physical constitution of our planet at least, if not of the universe. To get rid of this difficulty, and at the same time to save the credit of the supposed system of necessary truth, it is customary to say that the points, lines, circles, and squares which are the subject of geometry as conversant with such lines, metry, exist in our conceptions merely, and are part of our minds; which minds, by working on their own materials, construct an à priori science, the evidence of which is purely mental, and has nothing whatever to do with outward experience. By howsoever high authorities this doctrine may have been sanctioned, it appears to me psychologically incorrect. The points, lines, circles, and squares which any one has in his mind, are (I apprehend) simply copies of the points, lines, circles, and squares which he has known in his experience. Our idea of a point I apprehend to be simply our idea of the minimum visibile, the smallest portion of sur face which we can see. A line as defined by geometers is wholly inconceivable. We can reason about a line as if it had no breadth; because we have a power, which is the foundation of all the control we can exercise over the operations of our minds; the power, when a perception is present to our senses or a conception to our intellects, of attending to a part only of that perception or conception, instead of the whole. But we cannot conceive a line without breadth; we can form no mental picture of such a So long, however, as there line; all the lines which we have in exists no practical necessity for atour minds are lines possessing breadth. tending to any of the properties of If any one doubts this, we may refer the object except its geometrical prohim to his own experience. I much perties, or to any of the natural irrequestion if any one who fancies that gularities in those, it is convenient to he can conceive what is called a mathe-neglect the consideration of the other angles, and figures as really exist; and the definitions, as they are called, must be regarded as some of our first and most obvious generalisations concerning those natural objects. The correctness of those generalisations, as generalisations, is without a flaw: the equality of all the radii of a circle is true of all circles, so far as it is true of any one: but it is not exactly true of any circle; it is only nearly true; so nearly that no error of any importance in practice will be incurred by feigning it to be exactly true. When we have occasion to extend these inductions, or their consequences, to cases in which the error would be appreciable-to lines of perceptible breadth or thickness, parallels which deviate sensibly from equidistance, and the like-we correct our conclusions by combining with them a fresh set of propositions relating to the aberration; just as we also take in propositions relating to the physical or chemical properties of the material, if those properties happen to introduce any modification into the result; which they easily may, even with respect to figure and magnitude, as in the case, for instance, of expansion by heat. properties and of the irregularities, and to reason as if these did not exist: accordingly, we formally announce in the definitions, that we intend to proceed on this plan. But it is an error to suppose, because we resolve to confine our attention to a certain number of the properties of an object, that we therefore conceive, or have an idea of, the object denuded of its other properties. We are thinking, all the time, of precisely such objects as we have seen and touched, and with all the properties which naturally belong to them; but, for scientific convenience, we feign them to be divested of all properties, except those which are material to our purpose, and in regard to which we design to consider them. The peculiar accuracy, supposed to be characteristic of the first principles of geometry thus appears to be fictitious. The assertions on which the reasonings of the science are founded do not, any more than in other sciences, exactly correspond with the fact, but we suppose that they do so for the sake of tracing the consequences which follow from the supposition. The opinion of Dugald Steward respecting the foundations of geometry, is, I conceive, substantially correct; that it is built on hypotheses; that it owes to this alone the peculiar certainty supposed to distinguish it; and that in any science whatever, by reasoning from a set of hypotheses, we may obtain a body of conclusions as certain as those of geometry, that is, as strictly in accordance with the hypotheses, and as irresistibly compelling assent, on condition that those hypotheses are true.* It is justly remarked by Professor Bain (Logic, ii. 134) that the word Hypothesis is used here in a somewhat peculiar sense. An hypothesis, in science, usually means a supposition not proved to be true, but surmised to be so because if true it would account for certain known facts; and the final result of the speculation may be to prove its truth. The hypotheses spoken of in the text are of a different character; they are known not to be literally true, while as much of them as is true is not When, therefore, it is affirmed that the conclusions of geometry are necessary truths, the necessity consists in reality only in this, that they correctly follow from the suppositions from which they are deduced. Those suppositions are so far from being necessary, that they are not even true; they purposely depart, more or less widely, from the truth. The only sense in which necessity can be ascribed to the conclusions of any scientific investigation, is that of legitimately following from some assumption, which, by the conditions of the inquiry, is not to be questioned. In this relation, of course, the derivative truths of every deductive science must stand to the inductions, or assumptions, on which the science is founded, and which, whether true or untrue, certain or doubtful in themselves, are always supposed certain for the purposes of the particular science. And therefore the conclusions of all deductive sciences were said by the ancients to be necessary propositions. We have observed already that to be predicated necessarily was characteristic of the predicable Proprium, and that a proprium was any property of a thing which could be deduced from its essence, that is, from the properties included in its definition. § 2. The important doctrine of Dugald Stewart, which I have endeavoured to enforce, has been con hypothetical, but certain. The two cases, however, resemble in the circumstance that in both we reason, not from a truth, but from an assumption, and the truth therefore of the conclusions is conditional, not categorical. This suffices to justify, in point of logical propriety, Stewart's use of the term. It is of course needful to bear in mind that the hypothetical element in the definitions of geometry is the assumption that what is very nearly true is exactly so. This unreal exactitude might be called a fiction, as properly as an hypothesis; but that appellation, still more than the other, would fail to point out the close relation which exists between the fictitious point or line and the points and lines of which we have experience. tested by Dr. Whewell, both in the dissertation appended to his excellent Mechanical Euclid, and in his elaborate work on the Philosophy of the Inductive Sciences; in which last he also replies to an article in the Edinburgh Review, (ascribed to a writer of great scientific eminence,) in which Stewart's opinion was defended against his former strictures. The supposed refutation of Stewart consists in proving against him (as has also been done in this work) that the premises of geometry are not definitions, but assumptions of the real existence of things corresponding to those definitions. This, however, is doing little for Dr. Whewell's purpose; for it is these very assumptions which are asserted to be hypotheses, and which he, if he denies that geometry is founded on hypotheses, must show to be absolute truths. All he does, however, is to observe, that they, at any rate, are not arbitrary hypotheses; that we should not be at liberty to substitute other hypotheses for them; that not only "a definition, to be admissible, must necessarily refer to and agree with some conception which we can distinctly frame in our thoughts," but that the straight lines, for instance, which we define, must be "those by which angles are contained, those by which triangles are bounded, those of which parallelism may be predicated, and the like."* And this is true: but this has never been contradicted. Those who say that the premises of geometry are hypotheses, are not bound to maintain them to be hypotheses which have no relation whatever to fact. Since an hypothesis framed for the purpose of scientific inquiry must relate to something which has real existence, (for there can be no science respecting non-entities,) it follows that any hypothesis we make respect ing an object, to facilitate our study of it, must not involve anything which is distinctly false, and repugnant to * Mechanical Buclid, pp. 149 et seq. its real nature: we must not ascribe to the thing any property which it has not; our liberty extends only to slightly exaggerating some of those which it has, (by assuming it to be completely what it really is very nearly,) and suppressing others, under the indispensable obligation of restoring them whenever, and in as far as, their presence or absence would make any material difference in the truth of our conclusions. Of this nature, accordingly, are the first principles involved in the definitions of geometry. That the hypotheses should be of this particular character is, however, no further necessary, than inasmuch as no others could enable us to deduce conclusions which, with due corrections, would be true of real objects: and in fact, when our aim is only to illustrate truths, and not to investigate them, we are not under any such restriction. We might suppose an imaginary animal, and work out by deduction, from the known laws of physiology, its natural history; or an imaginary commonwealth, and from the elements composing it might argue what would be its fate. And the conclusions which we might thus draw from purely arbitrary hypotheses might form a highly useful intellectual exercise: but as they could only teach us what would be the properties of objects which do not really exist, they would not constitute any addition to our knowledge of nature while, on the contrary, if the hypothesis merely divests a real object of some portion of its properties, without clothing it in false ones, the conclusions will always express, under known liability to correction, actual truth. § 3. But though Dr. Whewell has not shaken Stewart's doctrine as to the hypothetical character of that portion of the first principles of geometry which are involved in the so-called definitions, he has, I conceive, greatly the advantage of Stewart on another important point in the |