cise our greatest powers of imagination, do with it what we please, we cannot make our senses indicate to us an increase or diminution in a given quantity of what we call matter. We find it so far amenable to our control that we can alter its arrangement, form, density, state of aggregation, temperature, etc.; nay, by so approximating it to other matter as to produce a chemical combination, we may entirely transform its appearance and properties,-all but one: its mass or quantity is completely beyond our control. Measure it by what process we please, by the "muscular sense,' by weight, anyhow, there it is, altogether independent of us, laughing our efforts to scorn! Can this be a mere mental idea which the mind that conceived it (or, at all events, in some way received the conception of it) is unable to destroy?

But there is one other argument on this point which must be mentioned. Not only do our own senses invariably indicate to us the impossibility of altering the quantity of matter, but the senses of all men alike point to the same quantity, quality, and collocation of matter in the earth and external to the earth. Whence this extraordinary agreement between the evidences of the senses in different men, when the minds are so different?

Our conviction then of the objective reality of matter is based upon the experimental truth that we can neither increase nor diminish its quantity, in fact on what we may conveniently for our present purpose call the Conservation of Matter.

95. Here let us pause for a moment to compare together this view of matter and the definition of the laws of the universe, which we have already given. The laws of the universe we defined (Art. 54) to be the laws according to

which the beings in the universe are trammelled by the Governor thereof as regards time, space, and sensation. Now, it may be asked, is this definition consistent with a belief in the objective reality of matter? We reply, that the two are in perfect accordance.

We do not here intend to enter into any metaphysical discussion. It is enough for us to say that our practical working certainty of the reality of matter means, firstly, that it offers resistance to our imagination and our will, and, secondly, that in particular it offers absolute resistance to all attempts to change its quantity. We shall soon see that both properties belong to something else.

96. Returning from this digression let us therefore assume that the objective reality of the external universe has been proved, and that this reality is strongly impressed upon us in virtue of that principle which we have called the conservation of matter.

But as soon as we grant this, we are obliged by our reason, however little our senses may incline us to it, or rather however much they may dispose us against it, to allow objective reality to whatever is found to be in the same sense conserved. (We have here italicised these four words for a reason which will afterwards appear.) This is a question which deserves and must get careful consideration.

97. In abstract dynamics several things are said and mathematically proved by deductions from experiment to be conserved, but one only of these in the strict sense in which we have spoken of the conservation of matter. We will examine them briefly, and our non-mathematical readers must pardon us if in this examination we make use of certain technical expressions belonging to the domain of mathematical physics.

(1.) Conservation of Momentum.-What is understood by this is a mere direct consequence of Newton's first interpretation of his third law of motion, viz., that Action and Reaction are equal and opposite. Stated in its simplest form it asserts that the momentum of a system of bodies, measured in any direction whatever, is not altered by their mutual action, whether that action be of the nature of traction, attraction, repulsion, or impact. And we see at once from this third law of motion that it must be so, because the change of momentum, in any direction, of any one part of the system, per unit of time, is the measure of the force acting on that part in that direction. Whatever momentum in this particular direction is gained by one member of the system must have been lost by other members, but not from their whole momentum, merely from the part of it in this direction. It thus appears that the (algebraic) sum of the momenta generated by the mutual actions of the system is zero.

These momenta are in fact directed magnitudes (like the forces of which they are the measure), and are therefore capable of cancelling one another. In this sense the conservation is of the same nature as that of the imagined electric or magnetic fluids, where no portion whatever of one kind can be produced without the simultaneous appearance of an equal quantity of the other, a quantity just capable of neutralizing it. This is obviously not in any sense analogous to the Conservation of Matter of which we have just spoken.

(2.) Conservation of Moment of Momentum.-Here we deal with quantities of the order of the moments of forces about an axis, i.g. couples in Poinsot's sense. These also are directed magnitudes depending for their conservation

upon the first interpretation of Newton's third law, and therefore the same remarks apply to them as to the preceding.

(3.) Conservation of Vis Viva.--Vis viva is the old name for energy or the power of doing work. We now deal with quantities which cannot possess direction, because they are essentially products of pairs of quantities similarly directed, and are therefore all to be treated as of the same algebraic sign, or rather (to adopt the language of Sir W. R. Hamilton) as signless quantities. With such there can of course be no cancelling.

To make our meaning clear, let us consider upon what vis viva depends. It depends upon and is proportional to the product of the mass into the square of the velocity. Now mass is of course a signless quantity; evidently we cannot have negative mass. Then with regard to the square of the velocity, this will be positive whether the velocity be positive or negative, whether it be in one direction or the opposite. Vis viva, therefore, or energy, is something which is not affected with the sign of direction, or, as we have already said, it is a signless quantity.

98. We have said that the energy which a body contains-its vis viva-its power of doing work, is independent of the direction in which it is moving; and, further, that while the mass is the same, it is proportional to the square of the velocity. For instance, we may measure the energy of a cannon ball or of an arrow by the distance it will carry itself up against the force of gravity, represented by its own weight, when shot vertically upwards, and we find that with a double velocity it will go Or we may point the cannon horizontally, and measure the energy of the same ball by the number of planks of oak wood which it can penetrate, and

four times as high.

we shall find that a ball with double the velocity will penetrate nearly four times as many as one with the single velocity. All these experiments concur together in convincing us that the energy of the ball is independent of the direction in which the cannon is pointed, and is proportional to the square of the velocity, so that a double velocity will give a fourfold energy.

99. We have just now spoken about a cannon ball fired into the air against the force of gravity. Such a ball, as it mounts, will each moment lose part of its velocity, until it finally comes to a standstill, after which it will begin to descend. When it is just turning it is perfectly harmless, and if we were standing on the top of a cliff to which it had just reached, we might without danger catch it in our arms and lodge it on the cliff. Its energy has apparently disappeared. Let us, however, see whether this is really true or not. It was fired up at us, let us say, by a foe at the bottom of the cliff, and the thought occurs to us to drop it down upon him again, which we do with great success, for he is smashed to pieces by the ball.

In truth, dynamics informs us that such a ball will again strike the ground with a velocity, and therefore with an energy precisely equal to that with which it was originally projected upwards. Now, when at the top of the cliff, if it had not the energy due to actual motion, it had nevertheless some sort of energy due to its elevated position, for it had obviously the power of doing work. We thus recognise two forms of energy which change into one another, the one due to actual motion and the other to position; the former of these is generally called kinetic, and the latter potential energy. All this appears to have been clearly perceived by Newton, who gave it as a second interpreta