A, it will meet A E on the side of E towards A; because the triangle which it forms with the lines AC and AE must have the same shape as a C E. So also any one of these lines which starts from A C on the side of c away from a will meet A E on the side of E away from A.

Looking then at the various fractions of A B which are now marked off, it is clear that, if one of them is less than a c, the corresponding fraction of A D is less than A E; and if greater, greater. It follows, therefore, that the line AE which is given by this construction satisfies, in the case of any fraction we choose, the condition which is necessary for the fourth proportional. Consequently, if the second assumption which we made about space be true, there always is a fourth proportional, and this process will enable us to find it.

There is, however, still one objection to be made against our definition of the fourth proportional, or rather one point in which we can make it a firmer ground-work for the study of ratios. For it assumes that quantities are continuous; that is, that any quantity can be divided into any number of equal parts, this being implied in the process of taking any numerical fraction of a quantity.

We say, for example, that if a, b, c, d, are proportionals, and if a is greater than three-fifths of b, c will be greater than three-fifths of d. Now the process of finding three-fifths of b is one or other of the following two processes. Either we divide b into five equal parts and take three of them, or we multiply b by three and divide the result into five equal parts. (We know of course that these two processes give us the same result.) But it is assumed in both cases that we can divide a given quantity into five equal parts.

Now in a definition it is desirable to assume as

little as possible; and accordingly the Greek geometers in defining proportion, or (which is really the same thing) in defining the fourth proportional of three given quantities, have tried to avoid this assumption.

Nor is it difficult to do this. For let us consider the same example. We say that if a is greater than three-fifths of b, c will be greater than the same fraction of d. Now let us multiply both the quantities a and b by five. Then for a to be greater than three-fifths of b, the quantity which a has now become must be greater than three-fifths of the quantity which b has become; that is, if the new b be divided into five equal parts the new a must be greater than three of them. But each of these five equal parts is the same as the original b; and so our statement as to the relative greatness of a and b is the same as this, that five times a is greater than three times b; and similarly for c and d.

Now every fraction involves two numbers. It is a compound process made up of multiplying by one number and dividing by another, and it is clear therefore that we may, not only in this particular case of three-fifths but in general, transform our rule for the fourth proportional into this new form. According as m times a is greater or less than n times b, so is m times c greater or less than n times d, where m and n are any whole numbers whatever.

This last form is the one in which the rule is given by the Greek geometers; and it is clear that it does not depend on the continuity of the quantities considered, for whether it be true or not that we can divide a number into any given number of equal parts, we can certainly take any multiple of it that we like.

These fundamental ideas, of ratio, of the equality of

ratios, and of the nature of the fourth proportional are now established generally, and with reference to quantities of any kind, not with regard to lengths alone; provided merely that it is always possible to take any given multiple of any given quantity.

§ 6. Of Areas; Stretch and Squeeze.

We shall now proceed to apply these ideas to areas, or quantities of surface, and in particular to plane areas. The simplest of these for the purposes of measurement is a rectangle. The finding of the area of a rectangle is in many cases the same process as numerical multiplication. For example, a rectangle which is 7 inches long and 5 inches broad will contain 35 square inches, and this follows from our fundamental ideas about the multiplication of numbers. But this process, the multiplication of numbers, is only applicable to the case in which we know how many times each side of the rectangle contains the unit of length, and it then tells us how many times the area of the rectangle contains the square described upon the unit of length. It remains to find a method which can always be used.

For this purpose we first of all observe that when one side of a rectangle is lengthened or shortened in any ratio, the other side being kept of a fixed length, the area of the rectangle will be increased or diminished in exactly the same ratio.

In order then to make any rectangle OPR Q out of a square O A C B, we have first of all to stretch the side o A until it becomes equal to o P, and thereby to stretch the whole square into the rectangle o D, which increases its area in the ratio of o A to o P. Then we must stretch the side O B of this figure until it is equal to o Q, and

thereby the figure OD becomes o R, and its area is increased in the ratio of OB to oq. Or we may, if we

like, first stretch oв to the length oq, whereby the square o c becomes o E, and then stretch o A to o P, by which o E becomes o R.

Thus the whole operation of turning the square o c into the rectangle OR is made up of two stretches; or, as we have agreed to call them, multiplications'; viz. the square has to be multiplied by the ratio of op to o A, and by the ratio of oq to OB; and we may find from the result that the order of these two processes is immaterial.

For let us represent the ratio of oP to OA by the letter a, and the ratio of oQ to oв by b. Then the ratio of the rectangle o D to the square o c is also a; in other words, a times o c is equal to o D. And the ratio of o R to O D is b, so that b times o D is equal to oR; that is, b times a times oc is equal to o R, or, as we write it, b a times o c is o R.1

And in the same way b times oc is equal to OE and a times b times oc is a times o E, which is O R.

It is a matter of convention which has grown up in consequence of our ordinary habit of reading from left to right, that we always read the symbols of a multiplication, or of any other operation, from right to left. Thus ab times any quantity x, means a times b times a; that is to say, we first multiply a by b, and then by a; that operation being first performed whose symbol comes last.

Consequently we have b a times oc giving the same result as ab times oc; or, as we write it

ba = ab,

which means that the effect of multiplying first by the ratio a and then by the ratio b is the same as that of multiplying first by the ratio b and then by the ratio a.

This proposition, that in multiplying by ratios we may take them in any order we please without affecting the result, can be put into another form.

Suppose that we have four quantities, a, b, c, d, then I can make a into d by two processes performed in succession; namely, by first multiplying by the ratio of b to a, which turns it into b, and then by the ratio of d to b. But I might have produced the same effect on a by first multiplying it by the ratio of c to a, which turns it into c, and then multiplying by the ratio of d to C. We are accustomed to write the ratio of b to a in shorthand in any of the four following ways:

-

and so the fact we have just stated may be written thus:

a

Now let us assume that the four quantities, a, b, c, d, are proportionals; that is, that the ratios / and 4. are equal to one another. It follows then that the ratios / and /, are equal to one another.

с

a

This proposition may be otherwise stated in this form; that if a, b, c, d are proportionals, then a, b, b, d will also be proportionals: provided always that this latter statement has any meaning, for it is quite possible that it should have no meaning at all. Suppose, for instance, that a and b are two lengths, c and d two intervals