of the rectangle contained by the lines a and b. For aa, which means the square on the line a, we write a2.
§ 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 = a2 if b + c = a. |
| ” | 3. | a (a + b) = a2 + ab. |
| ” | 4. | (a + b)2 = a2 + 2ab + b2. |
| ” | 5. | (a + b)(a − b) + b2 = a2. |
| ” | 6. | (a + b)(a − b) + b2 = a2. |
| ” | 7. | a2 + (a − b)2 = 2a (a − b) + b2. |
| ” | 8. | 4(a + b)a + b2 = (2a + b)2. |
| ” | 9. | (a + b)2 + (a − b)2 = 2a2 + 2b2. |
| ” | 10. | (a + b)2 + (a − b)2 = 2a2 + 2b2. |
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.
§ 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 | |||
| and | (b + c)a = ba + ca | (6). |
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) the commutative law for multiplication; (2) and (4) the associative laws for addition; (6) the distributive law. |
§ 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:—
But
and
Therefore
| (a + b)2 | = aa + ab + (ab + bb) |
| = aa + (ab + ab) + bb | |
| = 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—
Prop. 7, which is an easy consequence of Prop. 4, may be transformed. If we denote by c the line a + b, so that
we get
| c2 + (c − b)2 | = 2c(c − b) + b2 |
| = 2c2 − 2bc + b2. |
Subtracting c2 from both sides, and writing a for c, we get
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.
§ 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.
§ 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) = x2. Prop. 11 contains, therefore, the solution of a quadratic equation, which we may write x2 + ax = a2. 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
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
where x denotes the unknown altitude.
Book III.
§ 26. The third book of the Elements relates exclusively to properties of the circle. A circle and its circumference have been defined in Book I., Def. 15. We restate it here in slightly different words:—
Definition.—The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane. This point is called the “centre” of the circle.
Of the new definitions, of which eleven are given at the beginning of the third book, a few only require special mention. The first, which says that circles with equal radii are equal, is in part a theorem, but easily proved by applying the one circle to the other. Or it may be considered proved by aid of Prop. 24, equal circles not being used till after this theorem.
In the second definition is explained what is meant by a line which “touches” a circle. Such a line is now generally called a tangent to the circle. The introduction of this name allows us to state many of Euclid’s propositions in a much shorter form.
For the same reason we shall call a straight line joining two points on the circumference of a circle a “chord.”
Definitions 4 and 5 may be replaced with a slight generalization by the following:—
Definition.—By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.
§ 27. From the definition of a circle it follows that every circle has a centre. Prop. 1 requires to find it when the circle is given, i.e. when its circumference is drawn.
To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected. Euclid then proves, first, that no point off this perpendicular can be the centre, hence that the centre must lie in this line; and, secondly, that of the points on the perpendicular one only can be the centre, viz. the one which bisects the parts of the perpendicular bounded by the circle. In the second part Euclid silently assumes that the perpendicular there used does cut the circumference in two, and only in two points. The proof therefore is incomplete. The proof of the first part, however, is exact. By drawing two non-parallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.
§ 28. In Prop. 2 it is proved that a chord of a circle lies altogether within the circle.
