also relate to realities, not independent realities, but realities of the same nature as those with which we started in our original definitions. Thus, whatever conclusions are arrived at in regard to lines, or circles, or ellipses, will apply to all objects, so far as we consider them as having length, or a circular or elliptic form. We find, in fact, that the conclusions reached in mathematics do hold true of all bodies in earth or sky, so far as we find them occupying space, or having numerical relations.

If this view be correct, we see how inadequate is the representation of those who, like D. Stewart and Mr. J. S. Mill, represent mathematical definitions as merely hypothetical, and represent the whole consistency and necessity as being between a supposition and the consequences drawn from it.1 This is to overlook the concrete cognitions or beliefs from which the definition is derived. It is likewise to overlook the fact that these refer to objects, and the further fact that the abstractions from the concretes also imply a reality. This theory also fails to account for the circumstance that the conclusions reached in mathematics admit of an application to the settlement of so many questions in astronomy, and in other departments of natural philosophy. Thus, what was demonstrated of the conic sections by Apollonius, is found true in the orbits of the planets and comets, as revealed by modern discovery. All this can at once be explained if we suppose that the mind starts with cognitions and beliefs, that it abstracts from these, and discovers relations among the things thus abstracted: the reality that was in the original conviction goes on to the farthest conclusion.

I am inclined to look on the primitive cognitions as constituting, properly speaking, the foundation of mathematics. The mind, looking at the things under the clear and distinct aspects in which they are set before it by abstraction, discovers relations between them, and can draw deductions from the combination. In this process the mind proceeds spontaneously, without thinking of the general principle involved in the reasoning. It finds that A is equal to B, and B to C, and it at once concludes that A is equal to C. It does not feel that in order to reach this conclusion it 1 Stewart's Elem. Vol. п. chap. ii. Mill's Logic, I. v. 1.

needs any generalized maxim, such as that "Things which are equal to the same things are equal to one another." The reasoning appears clear anterior to the general principle being announced; and when the principle is announced, it does not seem to add to the force of the ratiocination. It does not, in fact, add to the cogency of the argument; it is merely the expression of the general principle on which it proceeds. Still, it serves many important scientific purposes, as Locke and Stewart admit, to have this general principle expressed in the form of an axiom.1 It allows the reflective mind to dwell on the general principle underlying the spontaneous conviction; by its clearness it enables us to test the ratiocination; and it shows what those must be prepared to disprove who would dispute or deny the conclusion. If this view be correct, the abstracted cognitions or beliefs in the definitions constitute the proper foundation of mathematical demonstration, while the axioms being the generalizations of our primitive judgments pronounced on looking at the things defined, are the links which bind together the parts of the superstructure added.

1 Locke's Essay, iv. vii. 11. Stewart's Elem. I. chap. i.

2 There is truth, then, in a statement of D. Stewart: “The doctrine which I have been attempting to establish, so far from degrading axioms from that rank which Dr. Reid would assign them, tends to identify them still more than he has done, with the exercise of our reasoning powers; inasmuch as, instead of comparing them with the data, on the accuracy of which that of our conclusion necessarily depends, it considers them as the vincula which give coherence to all the particular links of the chain; or (to vary the metaphor) as component elements, without which the faculty of reasoning is inconceivable and impossible" (Elem. Vol. п. chap. i.)

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CHAPTER IV.

INTUITIVE PRINCIPLES INVOLVED IN THE PHYSICAL SCIENCES.

These sciences must ever be conducted in the method of induction, with sense and artificial instruments as the agents of observation. But nearly the whole of them do at times go down to first principles, and the inquirer is obliged, in the last resort, to appeal to what the mind sees to be true. At the same time, it is not the special business of these sciences to inquire into the nature or guarantee of ultimate truths; this it leaves very properly to metaphysicians, who should be prepared to announce laws of intuition, which the physicist might probably employ to suit his purposes. They might be more profitably employed in such a work which lies exclusively within their own province, than in pursuing speculative ends which can never be attained by human

reason.

In all the sciences which meet in their researches with regular forms, and correlated numbers, and constant or periodical motion, -such as mechanical science, statics and dynamics, and certain. departments of astronomy, optics, and thermotics, mathematics have an important part to act, and they come in with all their intuitive axioms and demonstrations. On these I need not dwell further. I leave them, to refer to those sciences in which intuition enters otherwise than in a mathematical form.

Most, if not all, of our intuitive convictions enter, in a tacit way, into physical investigation. Thus, the conviction as to the identity of being leads us to chase the substance through the

various forms it may assume, and constrains even those who are most opposed to hypotheses, to speak of ultimate molecules or atoms, which change not with changing circumstances. The intuition of whole and parts prompts us to seek for the missing part after we have found certain parts which have been separated by analysis, and it constrains us to look on the abstract as implying the concrete. Our intuitions as to space make the physicist certain, when he sees body now in one place and again in another, that it must have passed through the whole intermediate space. They should prevent him from ever giving in to the theory which represents matter as consisting merely of points of force; these points cannot, properly speaking, be unextended, and there must always be space between. Our conviction as to time assures us that there can be no break in it, and that when we fall in with the same object at two different times, it must have existed the whole intervening time. Our intuitions as to quantity, as to number and proportion, enter more or less formally into all natural investigation. Our intuition as to generalization insists that, in division, the sub-classes should make up the class. Our conviction as to substance and property prompts us, when we discover a new object, to look out for the exercise of its properties; and leads the physicist, when he meets with such agencies as electricity and galvanism, to declare that they must either be separate substances (which is very improbable), or properties or states of substances. Finally, the fundamental law of causality directs us to seek for a cause to every effect. The physical investigator, engrossed with external facts, and seeking to clear them up, will seldom so much as observe these fundamental principles, which are unconsciously guiding him; and only on rare occasions will he find it necessary to make a formal appeal to them. Still, there will be times when those most prejudiced against metaphysics will be tempted or compelled to fall back on them, when going down to the depths of a deep subject, or when hard pressed by an opponent. It often happens that, when they do so, their expression of the principle is sufficiently awkward and blundering; and I think they have reason to complain of the metaphysician that he has been wasting his ingenuity in unprofitable and unattainable pursuits, and has

done so little to aid physical investigation in a matter in which he might have lent it effectual aid.1

There is a class of sciences which proceed on our intuition as to the resemblances among objects and classes. These have been called the classificatory sciences by Whewell; they embrace zoology and botany, and mineralogy so far as it is not a branch of chemistry, and geology so far as it deals with organisms. In all these the mind is guided and guarded by our convictions regarding individuals, classes, genera, and species. Another class of sciences have underlying them our conviction as to substance and property; of this description is chemistry, and the sciences which treat of electricity and magnetism and the cognate agencies. A number of sciences proceed on the conviction as to causation; such are all departments of natural philosophy, as it seeks to determine the laws which regulate force; and such too is geology, so far as it strives to find the circumstances and agencies which have brought the earth's surface to its present state. In physiology, too, there is an inquiry after the properties, be they mechanical or chemical or vital, which have brought the organism into the state in which we find it.

The metaphysician should in no case pretend to be able to construct any department of natural science; but keeping within his own province, it is competent for him to furnish an expression of the fundamental principles of cognition, belief, and thought, and the physicist might then be able to use them under the forms which are best suited to his special purposes.

1 It has been shown by Dr. Whewell, in his great work on the Philosophy of the Inductive Sciences, more particularly in his History of Scientific Ideas, that each kind of science has its special fundamental idea at its basis, and he classifies the sciences according to the ideas which regulate them. The phrase “ideas” does not seem a good one to express the intuitive convictions of the mind, either in their spontaneous exercises or formal enunciation, and I think he is altogether wrong in supposing that these ideas “superinduce” on the facts something not in the facts. But he has in that work developed great truths, which physical investigators were almost universally overlooking. I have not in this chapter deemed it necessary to follow him in his elaborate exposition of the ideas and conceptions involved in the various sciences; I have contented myself with showing how certain intuitive principles enter into special sciences.