completely fill the area ACB D, and whose heights are all equal to P P'. 'D' D FIG. 51. We obtain an oblique cylinder from the above right cylinder by moving the face A'C' B'D' parallel to itself anywhere in its own plane. But such a motion will only shear the elementary right six-faces, such as PP', and so not change their volume. Hence the volume of an oblique cylinder is equal to the product of its base, and the perpendicular distance between its faces. § 14. On the Measurement of Angles. Hitherto we have been concerned with quantities of area and quantities of volume; we must now turn to quantities of angle. In our chapter on Space (p. 66) we have noted one method of measuring angles; but that was a merely relative method, and did not lead us to fix upon an absolute unit. We might, in fact, have taken any opening of the compasses for unit angle, and determined the magnitude of any other angle by its ratio to this angle. But there is an absolute unit which naturally suggests itself in our measurement of angles, and one which we must consider here, as we shall frequently have to make use of it in our chapter on Position. Let A O B be any angle, and let a circle of radius a be described about o as centre to meet the sides of this Б FIG. 52. angle in A and B. Then if we were to double the angle A O B, we should double the arc A B; if we were to treble it, we should treble the arc; shortly, if we were to take any multiple of the angle, we should take the same multiple of the arc. We may thus state that angles at the centre of a circle vary as the arcs on which they stand. Hence if 0 and ' be two angles, which are subtended by arcs s and s' respectively, the ratio of 0 to e' will be the same as that of s to s'. Now suppose ' to represent four right angles; then s' will be the entire circumference, or, in our previous notation, 2 π u. We have thus Now it is extremely convenient to choose a unit angle which shall be independent of the circle upon which we measure our arcs. We should obtain such an independent unit if we took the arc subtended by it equal to the radius of the circle or if we took s = a. In this case our unit equals of four right angles, = 1 П 2 π of two right angles, = 636 of a right angle approximately. Thus we see that the angle subtended at the centre of any circle by an arc equal to the radius is a constant fraction of a right angle. If this angle be chosen as the unit, we deduce from the proportion is to 'as s is to s', that unity as s is to the radius a; or:— s = a 0. must be to Thus, if we choose the above angle as our unit of angle, the measure of any other angle will be the ratio of the arc it subtends from the centre to the radius ; but we have seen (p. 125) that the arcs subtended from the centre in different circles by equal angles are in the ratio of the radii of the respective circles. Hence the above measurement of angle is independent of the radius of the circle upon which we base our measurement. This is the primary property of the socalled circular measurement of angles, and it is this which renders it of such great value. The circular measure of any angle is thus the ratio of the arc it subtends from the centre of any circle to the radius of the circle. It follows that the circular measure of four right angles is the ratio of the whole circumference to the radius, or equals that is, equals 2πα α ; 2π. The circular measure of two right angles will then be π, of one right angle of three right π 2' 15. On Fractional Powers. Before we leave the subject of quantity it will be necessary to refer once more to the subject of powers which we touched upon in our chapter on Number (p. 16). We there used a" as a symbol signifying the result of multiplying a by itself n times. From this definition we easily deduce the following identity : an × a1 × a2 × aˆ = a2+p+q+r. For the left hand side denotes that we are first to multiply a by itself n times, and then multiply this by a2, or a multiplied by itself p times, and so on. Hence we may write the left hand side αχαχαχα to n factors) × (a xaxaxa.. to p factors) x (axaxaxa.. to r factors). But this is obviously equal to (αχαχαχαχ n+p+q+r factors), or to a" + p + q + r ̧ If b be such a quantity that bra, b is termed an nth root of a, and this is written symbolically b = Thus, since 8=23, 2 is a 3rd, or cube root of 8. Or, again, since 243=35, 3 is termed a 5th root of 243. Now we have seen at the conclusion of our first chapter that we can often learn a very great deal by extending the meaning of our terms. Let us now see if we cannot extend the meaning of the symbol a". Does it cease to have a meaning when n is a fraction or negative? Obviously we cannot multiply a quantity by itself a fractional number of times, nor can we do Hence the old mean so a negative number of times. ing of a", where n is a positive integer, becomes sheer nonsense when we try to adapt it to the case of n being fractional or negative. Is then a" in this latter case meaningless? In an instance like this we are thrown back upon the results of our definition, and we endeavour to give to our symbol such a meaning that it will satisfy these results. Now the fundamental result of our theory of integer powers is that— This will obviously be true however many quantities, n, p, q, r, we take. Now let us suppose we wish to interpret a where is a fraction. We begin by as m suming it satisfies the above relation, and in order to arrive at its meaning we suppose that n = = p = q and that there are m such quantities. เ = (a). Thus am must be such a quantity that, multiplied by itself m times, it equals a'. But we have defined above (p. 144) an mth root of a' to be such a quantity that, multiplied m times by itself, it equals a'. Hence we am say that am is equal to an mth root of a'; or, as it is written for shortness, |