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.