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.
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