of time, then we understand what is meant by the ratio of b to a, and the ratio of d to c, and these ratios may very well be equal to one another; but there is no such thing as a ratio of c to a, or of d to b, because the quantities compared are not of the same kind. When, however, four quantities of the same kind are proportionals, they are also proportionals when taken alternately; that is to say, when the two middle ones are interchanged. § 7. Of Fractions. We have seen in § 3, page 101, that a ratio may be expressed in the form of a fraction. Thus, let a be represented by the fraction and b by the fraction r where p, q, r, s are numbers. Then the result on page 115 may be written Let us examine a little more closely into the meaning of either side of this equation. Suppose we were to take a rectangle oQTs, of which one side, o Q, contained q units of length, and another, os, s units. Then this rectangle could be obtained from the unit square by operating upon it with the two stretches q Its area would thus contain qs square units. and s. Now let us apply to this rectangle in succession the ? and. If we stretch the q two stretches denoted by rectangle in the direction of the side oq in the ratio of we divide the side o Q into q equal parts, and then take op equal p times one of those parts. But each of these parts will be equal to unity, hence oP contains P units. We thus convert our rectangle or into one OP', of which one side, o P, contains p and the other, os, 8 units. Now let us apply to this rectangle the stretch-parallel to the side os (as the figure is drawn r denotes a squeeze). We must divide os into s equal parts and take r such parts, or we must measure a length O R along o s equal to r units. Thus this second stretch converts the rectangle OP' into a rectangle OR', of which the side OP contains p and the side OR contains r units of length, or into a rectangle containing p r square units. Hence the two stretches P and applied in succession to the rectangle or con r S q vert it into the rectangle OR'. Now this may be written symbolically thus : 1 Now unit-rectangle may obviously be obtained from the rectangle or by squeezing it first in the ratio in q 1 - S the direction of o Q, and then in the ratio in the direction o s. Now this is simply saying that o T contains γ q8 unit-rectangles. Hence the operation? × applied 1 of the result of its to unit-rectangle must produce q8 That is: .pr unit-rectangle, pr. unit-rectangle. 98 is equal to the operation denoted by or to multi pr plying unity by p r and then dividing the result by q 8. This equivalence is termed the multiplication of frac tions. A special case of the multiplication of fractions arises when s equals r. We then have― r Р r pr X = q r gr But the operation denotes that we are to divide unity r into r equal parts, and then taker of them; in other words, we perform a null operation on unity. The symbol of operation may therefore be omitted, and we read This result is then expressed in words as follows: Given a fraction, we do not alter its value by multiplying the numerator and denominator by equal quantities. From this last result we can easily interpret the operation P Or, to apply first the operation to unity and then to до add to this the result of the operation is the same thing as dividing unity into qs parts, taking ps of those parts, and then adding to them qr more of the like parts. But this is the same thing as to take at once ps + qr of those parts. Thus we may write— This result is termed the addition of fractions. The reader will find no difficulty in interpreting addition graphically by a succession of stretches and squeezes of the unit-rectangle. We term division the operation by which we reverse the result of multiplication. Hence when we ask the meaning of dividing by the fraction we put the P q question: What is the operation which, following on the operation 2 just reverses its effect? or, to multiply unity by P, and then by 2, is to perform Ρ the operation of dividing unity into qp parts and then taking p q of them, or to leave unity unaltered. Hence the stretch completely reverses the stretch? ; p it is, in fact, a squeeze which just counteracts the preceding stretch. Thus multiplying by Ρ operation equivalent to dividing by P. 9 must be an Ρ Or, to divide by is the same thing as to multiply by. This result is termed the division of fractions. § 8. Of Areas; Shear. Ρ Hitherto we have been concerned with stretching or squeezing the sides of a rectangle. These operations alter its area, but leave it still of rectangular shape. We shall now describe an operation which changes its angles, but leaves its area unaltered. Let ABCD be a rectangle, and let A B EF be a parallelogram (or a four-sided figure whose opposite sides are equal), having the same side, A B, as the rectangle, but having the opposite side, EF (equal to A B, and |