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—
Further, if m is a (whole) number, ma shall denote the multiple of a which is obtained by taking it m times.
§ 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 α and β the areas, of two triangles which have the same altitude, then a : b :: α : β.
To prove this, we have, according to Definition 5, to show—
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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 α. The triangle on ma as base equals, therefore, mα. 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 nβ, 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 mα = nβ if ma = nb, and if their bases are unequal, then that has the greater area which is on the greater base; in other words, mα is greater than, equal to, or less than nβ, according as ma is greater than, equal to, or less than nb, which was to be proved.
§ 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.
§. 51. Using the notation explained above we express the first propositions as follows:—
Prop. 1. If
then
Prop. 2. If
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then a + e is the same multiple of b as c + f is of d, viz.:—
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
then
Prop. 5. If
then
Prop. 6. If
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.
§ 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
if
if
therefore the first proportion holds by Definition 5.
Prop. 8. If
and
The proof depends on Definition 7.
Prop. 9 (converse to Prop. 7). If
or if
Prop. 10 (converse to Prop. 8). If
and if
Prop. 11. If
and
then
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:
§ 53. Prop. 12. If
then
Prop. 13. If
then
Prop. 14. If
Prop. 15. Magnitudes have the same ratio to one another that their equimultiples have—
Prop. 16. If a, b, c, d are magnitudes of the same kind, and if
then
Prop. 17. If
then
Prop. 18 (converse to 17). If
then
Prop. 19. If a, b, c, d are quantities of the same kind, and if
then
§ 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
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
and if
but if
and if
Prop. 21. If
or if
