BOOK V.

S 47. The fifth book of the _Elements_ is not exclusively geometrical. It contains the theory of ratios and proportion of quantities in general. The treatment, as here given, is admirable, and in every respect superior to the algebraical method by which Euclid's theory is now generally replaced. We shall treat the subject in order to show why the usual algebraical treatment of proportion is not really sound. We begin by quoting those definitions at the beginning of Book V. which are most important. These definitions have given rise to much discussion.

The only definitions which are essential for the fifth book are Defs. 1, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more than useless, and probably not Euclid's, but additions of later editors, of whom Theon of Alexandria was the most prominent. Defs. 10 and 11 belong rather to the sixth book, whilst all the others are merely nominal. The really important ones are 4, 5, 6 and 7.

S 48. To define a magnitude is not attempted by Euclid. The first two definitions state what is meant by a "part," that is, a submultiple or measure, and by a "multiple" of a given magnitude. The meaning of Def. 4 is that two given quantities can have a ratio to one another only in case that they are comparable as to their magnitude, that is, if they are of the same kind.

Def. 3, which is probably due to Theon, professes to define a ratio, but is as meaningless as it is uncalled for, for all that is wanted is given in Defs. 5 and 7.

In Def. 5 it is explained what is meant by saying that two magnitudes have the same ratio to one another as two other magnitudes, and in Def. 7 what we have to understand by a greater or a less ratio. The 6th definition is only nominal, explaining the meaning of the word _proportional_.

Euclid represents magnitudes by lines, and often denotes them either by single letters or, like lines, by two letters. We shall use only single letters for the purpose. If a and b denote two magnitudes of the same kind, their ratio will be denoted by a : b; if c and d are two other magnitudes of the same kind, but possibly of a different kind from a and b, then if c and d have the same ratio to one another as a and b, this will be expressed by writing--

a : b :: c : d.

Further, if m is a (whole) number, ma shall denote the multiple of a which is obtained by taking it m times.

S 49. The whole theory of ratios is based on Def. 5.

Def. 5. _The first of four magnitudes is said to have the same ratio to the second that the third has to the fourth when, any equimultiples whatever of the first and the third being taken, and any equimultiples whatever of the second and the fourth, if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; and if the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth; and if the multiple of the first is greater than that of the second, the multiple of the third is also greater than that of the fourth._

It will be well to show at once in an example how this definition can be used, by proving the first part of the first proposition in the sixth book. _Triangles of the same altitude are to one another as their bases_, or if a and b are the bases, and [alpha] and [beta] the areas, of two triangles which have the same altitude, then a : b :: [alpha] : [beta].

To prove this, we have, according to Definition 5, to show--

if ma > nb, then m[alpha] > n[beta], if ma = nb, then m[alpha] = n[beta], if ma < nb, then m[alpha] < n[beta].

That this is true is in our case easily seen. We may suppose that the triangles have a common vertex, and their bases in the same line. We set off the base a along the line containing the bases m times; we then join the different parts of division to the vertex, and get m triangles all equal to [alpha]. The triangle on ma as base equals, therefore, m[alpha]. If we proceed in the same manner with the base b, setting it off n times, we find that the area of the triangle on the base nb equals n[beta], the vertex of all triangles being the same. But if two triangles have the same altitude, then their areas are equal if the bases are equal; hence m[alpha] = n[beta] if ma = nb, and if their bases are unequal, then that has the greater area which is on the greater base; in other words, m[alpha] is greater than, equal to, or less than n[beta], according as ma is greater than, equal to, or less than nb, which was to be proved.

S 50. It will be seen that even in this example it does not become evident what a ratio really is. It is still an open question whether ratios are magnitudes which we can compare. We do not know whether the ratio of two lines is a magnitude of the same kind as the ratio of two areas. Though we might say that Def. 5 defines _equal _ratios, still we do not know whether they are equal in the sense of the axiom, that two things which are equal to a third are equal to one another. That this is the case requires a proof, and until this proof is given we shall use the :: instead of the sign = , which, however, we shall afterwards introduce.

As soon as it has been established that all ratios are like magnitudes, it becomes easy to show that, in some cases at least, they are numbers. This step was never made by Greek mathematicians. They distinguished always most carefully between continuous magnitudes and the discrete series of numbers. In modern times it has become the custom to ignore this difference.

If, in determining the ratio of two lines, a common measure can be found, which is contained m times in the first, and n times in the second, then the ratio of the two lines equals the ratio of the two numbers m : n. This is shown by Euclid in Prop. 5, X. But the ratio of two numbers is, as a rule, a fraction, and the Greeks did not, as we do, consider fractions as numbers. Far less had they any notion of introducing irrational numbers, which are neither whole nor fractional, as we are obliged to do if we wish to say that all ratios are numbers. The incommensurable numbers which are thus introduced as ratios of incommensurable quantities are nowadays as familiar to us as fractions; but a proof is generally omitted that we may apply to them the rules which have been established for rational numbers only. Euclid's treatment of ratios avoids this difficulty. His definitions hold for commensurable as well as for incommensurable quantities. Even the notion of incommensurable quantities is avoided in Book V. But he proves that the more elementary rules of algebra hold for ratios. We shall state all his propositions in that algebraical form to which we are now accustomed. This may, of course, be done without changing the character of Euclid's method.

S. 51. Using the notation explained above we express the first propositions as follows:--

Prop. 1. If a = ma', b = mb', c = mc', then a + b + c = m(a' + b' + c').

Prop. 2. If a = mb, and c = md, e = nb, and f = nd,

then a + e is the same multiple of b as c + f is of d, viz.:--

a + e = (m + n)b, and c + f = (m + n)d.

Prop. 3. If a = mb, c = md, then is na the same multiple of b that nc is of d, viz. na = nmb, nc = nmd.

Prop. 4. If a : b :: c : d, then ma : nb :: mc : nd.

Prop. 5. If a = mb, and c = md, then a - c = m(b - d).

Prop. 6. If a = mb, c = md,

then are a - nb and c - nd either equal to, or equimultiples of, b and d, viz. a - nb = (m - n)b and c - nd = (m - n)d, where m - n may be unity.

All these propositions relate to _equimultiples_. Now follow propositions about ratios which are compared as to their magnitude.

S 52. Prop. 7. If a = b, then a : c :: b : c and c : a :: c : b.

The proof is simply this. As a = b we know that ma = mb; therefore

if ma > nc, then mb > nc, if ma = nc, then mb = nc, if ma < nc, then mb < nc,

therefore the first proportion holds by Definition 5.

Prop. 8. If a > b, then a : c > b : c, and c : a < c : b.

The proof depends on Definition 7.

Prop. 9 (converse to Prop. 7). If a : c :: b : c, or if c : a :: c : b, then a = b.

Prop. 10 (converse to Prop. 8). If a : c > b : c, then a > b, and if c : a < c : b, then a < b.

Prop. 11. If a : b :: c : d, and a : b :: e : f, then c : d :: e : f.

In words, _if too ratios are equal to a third, they are equal to one another_. After these propositions have been proved, we have a right to consider a ratio as a _magnitude_, for only now can we consider a ratio as something for which the axiom about magnitudes holds: things which are equal to a third are equal to one another.

We shall indicate this by writing in future the sign = instead of ::. The remaining propositions, which explain themselves, may then be stated as follows:

S 53. Prop. 12. If a : b = c : d = e : f, then a + c + e : b + d + f = a : b.

Prop. 13. If a : b = c : d and c : d > e : f, then a : b > e : f.

Prop. 14. If a : b = c : d, and a > c, then b > d.

Prop. 15. Magnitudes have the same ratio to one another that their equimultiples have--

ma : mb = a : b.

Prop. 16. If a, b, c, d are magnitudes of the same kind, and if a : b = c : d, then a : c = b : d.

Prop. 17. If a + b : b = c + d : d, then a : b = c : d.

Prop. 18 (converse to 17). If a : b = c : d then a + b : b = c + d : d.

Prop. 19. If a, b, c, d are quantities of the same kind, and if a : b = c : d, then a - c : b - d = a : b.

S 54. Prop. 20. _If there be three magnitudes, and another three, which have the same ratio, taken two and two, then if the first be greater than the third, the fourth shall be greater than the sixth: and if equal, equal; and if less, less._

If we understand by

a : b : c : d : e : ... = a' : b' : c' : d' : e' : ...

that the ratio of any two consecutive magnitudes on the first side equals that of the corresponding magnitudes on the second side, we may write this theorem in symbols, thus:--

If a, b, c be quantities of one, and d, e, f magnitudes of the same or any other kind, such that

a : b : c = d : e : f, and if a > c, then d > f, but if a = c, then d = f, and if a < c, then d < f.

Prop. 21. If a : b = e : f and b : c = d : e, or if a : b : c = 1/f : 1/e : 1/d, and if a > c, then d > f, but if a = c, then d = f, and if a < c, then d < f.

By aid of these two propositions the following two are proved.

S 55. Prop. 22. _If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last._

We may state it more generally, thus:

If a : b : c : d : e: ... = a' : b' : c' : d' : e' : ... ,

then not only have two consecutive, but any two magnitudes on the first side, the same ratio as the corresponding magnitudes on the other. For instance--

a : c = a' : c'; b : e = b' : e', &c.

Prop. 23 we state only in symbols, viz.:--

If a : b : c : d : e : ... = 1/a' : 1/b' : 1/c' : 1/d' : 1/e' ...,

then a : c = c' : a', b : e = e' : b',

and so on.

Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then

a + e : b = c + f : d.

Some of the proportions which are considered in the above propositions have special names. These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view.

S 56. The last proposition in the fifth book is of a different character.

Prop. 25. _If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together._ In symbols--

If a, b, c, d be magnitudes of the same kind, and if a : b = c : d, and if a is the greatest, hence d the least, then a + d > b + c.

S 57. We return once again to the question. What is a ratio? We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude. It will presently be shown that ratios of lines may be considered as _quotients_ of lines, so that a ratio appears as answer to the question, How often is one line contained in another? But the answer to this question is given by a number, at least in some cases, and in all cases if we admit incommensurable numbers. Considered from this point of view, we may say the fifth book of the _Elements_ shows that some of the simpler algebraical operations hold for incommensurable numbers. In the ordinary algebraical treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject.