greater than they really are; and can with equal ease suppose the force to disappear altogether.

To this I reply, that certainly we may conceive the weight of bodies to vary in intensity and direction, and by an additional effort of imagination, may conceive the weight to vanish: but that in all these suppositions, even in the extreme one, we must suppose the rule to be universal. If any bodies have weight, all bodies must have weight. If the direction of weight be different in different points, this direction must still vary according to the law of continuity; and the same is true of the intensity of the weight. For if this were not so, the rest and motion, the velocity and direction, the permanence and change of bodies, as to their mechanical condition, would be arbitrary and incoherent: they would not be subject to mechanical ideas; that is, not to ideas at all: and hence these conditions of objects would in fact be inconceivable. In order that the universe may be possible, that is, may fall under the conditions of intelligible conceptions, we must be able to conceive a body at rest. But the rest of bodies (except in the absolute negation of all force) implies the equilibrium of opposite forces. And one of these opposite forces must be a general force, as weight, in order that the universe may be governed by general conditions. And this general force, by the conception of force, may produce motion, as well as equilibrium; and this motion again must be determined, and determined by general conditions; which cannot be, except the communication of motion be regulated by an inertia proportional to the weight.

But it will be asked, Is it then pretended that Newton's experiment, by which it was intended to prove inertia proportional to weight, does really prove nothing but what may be demonstrated à priori? Could we know, without experiment, that all bodies,-gold, iron, wood, cork,— have inertia proportional to their weight? And to this we reply, that experiment holds the same place in the establishment of this, as of the other fundamental doctrines of mechanics. Intercourse with the external world is requisite for developing our ideas; measurement of phenomena is needed to fix our conceptions and to render them precise: but the result of our experimental studies is, that we reach a position in which our convictions do not rest upon experiment. We learn by observation

truths of which we afterwards see the necessity. This is the case with the laws of motion, as I have repeatedly endeavoured to shew. The same will appear to be the case with the proposition, that bodies of different kinds have their inertia proportional to their weight.

For bodies of the same kind have their inertia proportional to their weight, both quantities being proportional to the quantity of matter. And if we compress the same quantity of matter into half the space, neither the weight nor the inertia is altered, because these depend on the quantity of matter alone. But in this way we obtain a body of twice the density; and in the same manner we obtain a body of any other density. Therefore whatever be the density, the inertia is proportional to the quantity of matter. But the mechanical relations of bodies cannot depend upon any difference of kind, except a difference of density. For if we suppose any fundamental difference of mechanical nature in the particles or component elements of bodies, we are led to the same conclusion, of arbitrary, and therefore impossible, results, which we deduced from this supposition with regard to weight. Therefore all bodies of different density, and hence, all bodies whatever, must have their inertia proportional to their weight.

Hence we see, that the propositions, that all bodies are heavy, and that inertia is proportional to weight, necessarily follow from those fundamental ideas which we unavoidably employ in all attempts to reason concerning the mechanical relations of bodies. This conclusion may perhaps appear the more startling to many, because they have been accustomed to expect that fundamental ideas and their relations should be self-evident at our first contemplation of them. This, however, is far from being the case, as I have already shewn. It is not the first, but the most complete and developed condition of our conceptions which enables us to see what are axiomatic truths in each province of human speculation. Our fundamental ideas are necessary conditions of knowledge, universal forms of intuition, inherent types of mental developement; they may even be termed, if any one chooses, results of connate intellectual tendencies; but we cannot term them innate ideas, without calling up a large array of false opinions. For innate ideas were considered as capable of composition, but by no means of simplification: as most perfect in their original condition; as to

be found, if any where, in the most uneducated and most uncultivated minds; as the same in all ages, nations, and stages of intellectual culture; as capable of being referred to at once, and made the basis of our reasonings, without any special acuteness or effort: in all which circumstances the Fundamental Ideas of which we have spoken, are opposed to Innate Ideas so understood.

I shall not, however, here prosecute this subject. I will only remark, that Fundamental Ideas, as we view them, are not only not innate, in any usual or useful sense, but they are not necessarily ultimate elements of our knowledge. They are the results of our analysis so far as we have yet prosecuted it; but they may themselves subsequently be analysed. It may hereafter appear, that what we have treated as different Fundamental Ideas have, in fact, a connexion, at some point below the structure which we erect upon them. For instance, we treat of the mechanical ideas of force, matter, and the like, as distinct from the idea of substance. Yet the principle of measuring the quantity of matter by its weight, which we have deduced from mechanical ideas, is applied to determine the substances which enter into the composition of bodies. The idea of substance supplies the axiom, that the whole quantity of matter of a compound body is equal to the sum of the quantities of matter of its elements. The mechanical ideas of force and matter lead us to infer that the quantity both of the whole and its parts must be measured by their weights. Substance may, for some purposes, be described as that to which properties belong; matter in like manner may be described as that which resists force. The former involves the Idea of permanent Being; the latter, the Idea of Causation. There may be some elevated point of view from which these ideas may be seen to run together. But even if this be so, it will by no means affect the validity of reasonings founded upon these notions, when duly determined and developed. If we once adopt a view of the nature of knowledge which makes necessary truth possible at all, we need be little embarrassed by finding how closely connected different necessary truths are; and how often, in exploring towards their roots, different branches appear to spring from the same stem.

GRANGE,

Aug. 31, 1840.

W. WHEWELL.

XIII. On the Position of the Axes of Optical Elasticity in Crystals belonging to the Oblique-Prismatic System. By W. H. MILLer, M.A. F.R.S. Fellow and Tutor of St. John's College, and Professor of Mineralogy in the University of Cambridge.

[Read March 21, 1836.]

IN a Memoir printed in the 5th Volume of the Cambridge Transactions it is stated, that in crystals belonging to the Oblique-Prismatic System one of the three rectangular axes of optical elasticity was always found to coincide with that crystallographic axis (Y, Y') which, in crystals of this system, is perpendicular to the other two: but that the positions of the other axes of optical elasticity (E', ') had no known relation to the form of the crystal. In some oblique-prismatic crystals, however, it was found that one of the axes of optical elasticity

', ' was also the axis of a principal zone. In the crystals which I have examined since the publication of the paper already alluded to, by the same method, this coincidence is found to occur less frequently. Upon the whole, however, there seems to be no reason for supposing it accidental in the instances (five or six out of twenty) in which it has been observed; but rather that it is a particular case of some general law connecting the form and optical properties of crystals, in the discovery of which it is hoped the observations here recorded may be in some degree instrumental.

The crystals selected for examination are taken principally from among those which have been described by Mr Brooke in the Annals of Philosophy for 1823 and 1824. The mutual inclination of two faces is expressed by the angle between their normals, or the angular distance of their "poles." An explanation of the notation in which the symbols VOL. VII. PART II.

DD