This is called the Law of Commutation.

The second law says that in forming the sum of a, b, and c, we may either add b to a and then c to their sum, or we may first form the sum of b and c and add this to a; in symbols

a+b+c= a + (b + c).

This is called the Law of Association.

Law II.

The brackets mean that the operation inside is supposed performed. The last equation says we may put brackets in or leave them out.

From these two laws it follows that in addition we may change the order in which the terms are added in any manner we please.

Proof. The sum a+b+c+d+ to their sum c and so on, hence

...

means to a add b,

a+b+c+d=(a+b)+c+d.

This is the sum of 3 terms, hence by Law II we can put c+d in brackets and by Law I can write d+c for c+d; in symbols

a+b+c+d= (a + b) + (c+d)

Hence we can interchange any two consecutive terms. But by the interchange of two consecutive terms we can bring any arrangement into any other and therefore we may change the order of terms in any manner we please.

11. Multiplication. In multiplication we have the two corresponding laws of Commutation and Association.

The Law of Commutation is:-In forming the product of a and b we may either multiply a by b or b by a; in symbols

Besides these, there are the Laws of Distribution (a + b) c = ac+bc... (V1)

a (b+c)=ab + ac... (V2)

Law v.

Law v is written in two forms according as the first or the second factor is a sum. If Law III holds for a product, then both forms say the same thing, but if we have an Algebra where Law III does not hold, then the two forms are essentially different. It will be seen in Chap. IV that the Law of Commutation does not always hold for products of Vectors.

12. As a sum can always be written as the sum of two terms for a+b+c+d means (a+b+c)+d}, the Law of Distribution holds for any number of terms. Example. From these laws we get

If c =

(a + b) (c + d) = a (c + d) + b (c + d) by V1

= (ac + ad) + (bc+bd) by V2

= ac+aa+bc + bd by II ..............(i).

= a and d = b, we get

(a + b) (a + b) = aa + ab + ba + bb

=ad+2ab+bb by III,

or writing a2 for aa etc.

(a+b)2= a2+2ab+b2

.(ii).

It is thus seen that the laws stated are sufficient to obtain the formulae (i) and (ii).

13. The last law which concerns us is, A product can

only vanish if one factor vanishes.

In Algebra we have also the law of indices

am × an = am+n,

Law VI.

but as we shall not require it for Vectors, no more need be said about it.

14. The above laws are proved in arithmetic where a, b, c, ... denote numbers.

Algebra may be defined as the art or science of combining symbols a, b, c, ... according to certain laws laid

down. If we take the laws above given, we get the common Algebra; if we change the laws we get another Algebra. A variety of such Algebras have been worked out.

15. It should here be noticed that the formulae given above are true whenever the laws on which they are based are true. An example will make this clear. Let a and b denote not numbers, but lengths, then by a+b we mean a length obtained by placing the given lengths end to end, and it will be seen at once that for this Addition Laws I and II hold.

In Arithmetic a product has a clear meaning only if one factor is a number. Hence it is not clear what we have to understand by the product of two lengths. For this we require a new definition. The product of two lengths a and b is defined as meaning the area of a rectangle whose base is a, and whose altitude is b. The area being obtained by placing the lengths perpendicular to each other and then moving one along the other.

ac

b.c

a

b

Fig. 4.

As

in a rectangle either side may
be taken as base, we see that
the Commutative law holds,
viz. ab = ba.

Again consider Law v.
(a+b) c = ac+bc.

This is Euc. II, i.

This means that the area of the rectangle of base a+b and altitude c is the sum of the rectangles whose bases

are a and b respectively and whose common altitude is c.

Hence the law of Distribution holds. It will also be seen at once that the two laws of Addition hold for areas. Hence we get the formula

(a+b)2= a2+2ab+b2

as true in the present case.

It says, the square on a line is equal to the sum of the squares on the two parts together with twice the rectangle contained by the parts. This is of course Euc. II, iv.

16. It will be seen that all the theorems in the Second Book of Euclid can be proved in the same

manner.

The two problems contained in it are (if x denotes the length to be found) expressed by the equations

x2= ab

x2 + ax = α2

II, xiv.
II, xi.

Euclid in solving these problems, from our point of view, solves two quadratic equations. The whole book may thus be algebraically treated.

17. Subtraction is best defined as the operation inverse to Addition; a-b means that number which added to b gives a, or

a-b=c if b+c=a.

By introducing negative quantities it is always possible to transform a difference into a sum and it is therefore unnecessary to state special laws for subtraction.

Of the negative in Geometry, something more will be said presently.

18. Division is defined as the operation inverse to multiplication: the quotient a/b means that quantity c which when multiplied by b gives a, or

a/b = c if bc = a.

By allowing the symbols to denote fractions we can replace division by multiplication.

As the operation of division will not be applied to vectors, nothing more need be said on this subject.

SECTION (III). THE NEGATIVE IN GEOMETRY.

It

19. Steps in a line. If a point moves along a line, it may be either moving towards or away from any particular point in that line, this may be also expressed by saying that the point moves in one or in the opposite sense is convenient to call one sense positive and the other negative, and to denote them by the algebraical signs + and -. In a figure the positive sense is indicated by an arrowhead.

A

Fig. 5.

B

If A and B are two points, then AB shall denote the length of the line between the points as generated by a point moving from A to B.

Definition. This motion of a point from A to B is called the step AB.

The steps AB and BA are those of the same magnitude but of opposite sense, and we write.

AB-BA.

Definition. If two steps AB and CD in the same straight line are in the same sense and of equal magnitude, we say that they are equal and write

20.

AB= CD.

Addition of Steps. If a point moves from A to B, and then from В to C, it has made the two steps AB and BC, but these are equivalent to the one step AC, hence we write

and this defines the addition of steps in a line.

A

A

Fig. 6.

It will be seen that the formula is true whatever the position of the points B in the line may be, the two steps AB and BC always bring the moving point from A to C.