S 20. The propositions in the second book are very different in
character from those in the first; they all relate to areas of
rectangles and squares. Their true significance is best seen by
stating them in an algebraic form. This is often done by expressing
the lengths of lines by aid of numbers, which tell how many times a
chosen unit is contained in the lines. If there is a unit to be found
which is contained an exact number of times in each side of a
rectangle, it is easily seen, and generally shown in the teaching of
arithmetic, that the rectangle contains a number of unit squares equal
to the product of the numbers which measure the sides, a unit square
being the square on the unit line. If, however, no such unit can be
found, this process requires that connexion between lines and numbers
which is only established by aid of ratios of lines, and which is
therefore at this stage altogether inadmissible. But there exists
another way of connecting these propositions with algebra, based on
modern notions which seem destined greatly to change and to simplify
mathematics. We shall introduce here as much of it as is required for
our present purpose.
At the beginning of the second book we find a definition according to
which "a rectangle is said to be 'contained' by the two sides which
contain one of its right angles"; in the text this phraseology is
extended by speaking of rectangles contained by any two straight
lines, meaning the rectangle which has two adjacent sides equal to the
two straight lines.
We shall denote a finite straight line by a single small letter, a, b,
c, ... x, and the area of the rectangle contained by two lines a and b
by ab, and this we shall call the product of the two lines a and b. It
will be understood that this definition has nothing to do with the
definition of a product of numbers.
We define as follows:--
The _sum_ of two straight lines a and b means a straight line c which
may be divided in two parts equal respectively to a and b. This sum is
denoted by a + b.
The _difference_ of two lines a and b (in symbols, a-b) means a line c
which when added to b gives a; that is,
a - b = c if b + c = a.
The _product_ of two lines a and b (in symbols, ab) means the area of
the rectangle contained by the lines a and b. For aa, which means the
square on the line a, we write a^2.
S 21. The first ten of the fourteen propositions of the second book
may then be written in the form of formulae as follows:--
Prop. 1. a(b + c + d + ... ) = ab + ac + ad + ...
" 2. ab + ac = a^2 if b + c = a.
" 3. a(a + b) = a^2 + ab.
" 4. (a + b)^2 = a^2 + 2ab + b^2.
" 5. (a + b)(a - b) + b^2 = a^2.
" 6. (a + b)(a - b) + b^2 = a^2.
" 7. a^2 + (a - b)^2 = 2a(a - b) + b^2.
" 8. 4(a + b)a + b^2 = (2a + b)^2.
" 9. (a + b)^2 + (a - b)^2 = 2a^2 + 2b^2.
" 10. (a + b)^2 + (a - b)^2 = 2a^2 + 2b^2.
It will be seen that 5 and 6, and also 9 and 10, are identical. In
Euclid's statement they do not look the same, the figures being
arranged differently.
If the letters a, b, c, ... denoted numbers, it follows from algebra
that each of these formulae is true. But this does not prove them in
our case, where the letters denote lines, and their products areas
without any reference to numbers. To prove them we have to discover
the laws which rule the operations introduced, viz. addition and
multiplication of segments. This we shall do now; and we shall find
that these laws are the same with those which hold in algebraical
addition and multiplication.
S 22. In a sum of numbers we may change the order in which the numbers
are added, and we may also add the numbers together in groups and then
add these groups. But this also holds for the sum of segments and for
the sum of rectangles, as a little consideration shows. That the sum
of rectangles has always a meaning follows from the Props. 43-45 in
the first book. These laws about addition are reducible to the two--
a + b = b + a (1),
a + (b + c) = a + b + c (2);
or, when expressed for rectangles,
ab + ed = ed + ab (3),
ab + (cd + ef) = ab + cd + ef (4).
The brackets mean that the terms in the bracket have been added
together before they are added to another term. The more general cases
for more terms may be deduced from the above.
For the product of two numbers we have the law that it remains
unaltered if the factors be interchanged. This also holds for our
geometrical product. For if ab denotes the area of the rectangle which
has a as base and b as altitude, then ba will denote the area of the
rectangle which has b as base and a as altitude. But in a rectangle we
may take either of the two lines which contain it as base, and then
the other will be the altitude. This gives
ab = ba (5).
In order further to multiply a sum by a number, we have in algebra the
rule:--Multiply each term of the sum, and add the products thus
obtained. That this holds for our geometrical products is shown by
Euclid in his first proposition of the second book, where he proves
that the area of a rectangle whose base is the sum of a number of
segments is equal to the sum of rectangles which have these segments
separately as bases. In symbols this gives, in the simplest case,
a(b + c) = ab + ac \
> (6).
and (b + c)a = ba + ca /
To these laws, which have been investigated by Sir William Hamilton
and by Hermann Grassmann, the former has given special names. He calls
the laws expressed in
(1) and (3) the commutative law for addition;
(5) " " " multiplication;
(2) and (4) the associative laws for addition;
(6) the distributive law.
S 23. Having proved that these six laws hold, we can at once prove
every one of the above propositions in their algebraical form.
The first is proved geometrically, it being one of the fundamental
laws. The next two propositions are only special cases of the first.
Of the others we shall prove one, viz. the fourth:--
(a + b)^2 = (a + b)(a + b) = (a + b)a + (a + b)b by (6).
But (a + b)a = aa + ba by (6),
= aa + ab by (5);
and (a + b)b = ab + bb by (6).
Therefore (a + b)^2 = aa + ab + (ab + bb) \
= aa + (ab + ab) + bb > by (4).
= aa + 2ab + bb /
This gives the theorem in question.
In the same manner every one of the first ten propositions is proved.
It will be seen that the operations performed are exactly the same as
if the letters denoted numbers.
Props. 5 and 6 may also be written thus--
(a + b)(a - b) = a^2 - b^2.
Prop. 7, which is an easy consequence of Prop. 4, may be transformed.
If we denote by c the line a + b, so that
c = a + b, a = c - b,
we get
c^2 + (c - b)^2 = 2c(c - b) + b^2
= 2c^2 - 2bc + b^2.
Subtracting c^2 from both sides, and writing a for c, we get
(a - b)^2 = a^2 - 2ab + b^2.
In Euclid's _Elements_ this form of the theorem does not appear, all
propositions being so stated that the notion of subtraction does not
enter into them.
S 24. The remaining two theorems (Props. 12 and 13) connect the square
on one side of a triangle with the sum of the squares on the other
sides, in case that the angle between the latter is acute or obtuse.
They are important theorems in trigonometry, where it is possible to
include them in a single theorem.
S 25. There are in the second book two problems, Props. 11 and 14.
If written in the above symbolic language, the former requires to find
a line x such that a(a - x) = x^2. Prop. 11 contains, therefore, the
solution of a quadratic equation, which we may write x^2 + ax = a^2.
The solution is required later on in the construction of a regular
decagon.
More important is the problem in the last proposition (Prop. 14). It
requires the construction of a square equal in area to a given
rectangle, hence a solution of the equation
x^2 = ab.
In Book I., 42-45, it has been shown how a rectangle may be
constructed equal in area to a given figure bounded by straight lines.
By aid of the new proposition we may therefore now determine a line
such that the square on that line is equal in area to any given
rectilinear figure, or we can _square_ any such figure.
As of two squares that is the greater which has the greater side, it
follows that now the comparison of two areas has been reduced to the
comparison of two lines.
The problem of reducing other areas to squares is frequently met with
among Greek mathematicians. We need only mention the problem of
squaring the circle (see CIRCLE).
In the present day the comparison of areas is performed in a simpler
way by reducing all areas to rectangles having a common base. Their
altitudes give then a measure of their areas.
The construction of a rectangle having the base u, and being equal in
area to a given rectangle, depends upon Prop. 43, I. This therefore
gives a solution of the equation
ab = ux,
where x denotes the unknown altitude.