This Table was mentioned in the Note to p. xiv. of the Contents, as one
likely to facilitate reference. In fact, the references in the text of the Elements
are almost entirely to Articles (with their sub-articles), and not to pages.

Figure. Page. Figure. Page. Figure. Page. Figure. Page.

Figure. Page.

NOTE.-It appears by these Tables that the Author intended to have com-
pleted the work by the addition of Seven Articles, and Two Figures.-ED.

ELEMENTS OF QUATERNIONS.

BOOK I.

ON VECTORS, CONSIDERED WITHOUT REFERENCE TO ANGLES, OR TO ROTATIONS.

CHAPTER I.

FUNDAMENTAL PRINCIPLES RESPECTING VECTORS.

SECTION 1.- On the Conception of a Vector; and on Equality of Vectors.

ART. 1.-A right line AB, considered as having not only length, but also direction, is said to be a VECTOR. Its initial point a is said to be its origin; and its final point B is said to be its term. A vector AB is conceived to be (or to construct) the difference of its two extreme points; or, more fully, to be the result of the subtraction of its own origin from its own term; and, in conformity with this conception, it is also denoted by the symbol B A: a notation which will be found to be extensively useful, on account of the analogies which it serves to express between geometrical and algebraical operations. When the extreme points A and B are distinct, the vector AB or B A is said to be an actual (or an effective) vector; but when (as a limit) those two points are conceived to coincide, the vector AA or AA, which then results, is said to be null. Opposite vectors, such as AB and BA,

A

Vector.

B

B-A

A-B

A

B

Revector.

Fig. 1.

to be vector and provector: the line ac, or c-A, which is

B

drawn from the origin A of the first to the term c of the second,

being then said to be the transvector. At a later stage, we shall have to consider vector-arcs and vector-angles; but at present, our only vectors are (as above) right A lines.

B

Fig. 2.

2. Two vectors are said to be EQUAL to each other, or the equation AB = CD, or B-AD-C, is said to hold good, when (and only when) the origin and term of the one can be brought to coincide respectively with the corresponding points of the other, by transports (or by translations) without rotation. It follows that all null vectors are equal, and may therefore be denoted by a common symbol, such as that used for zero; so that we may write,

A - A=B-B = &c. = 0;

but that two actual vectors, AB and CD, are not (in the present full sense) equal to each other, unless they have not merely equal lengths, but also similar directions. If then they do not happen to be parts of one common line, they must be opposite sides of a parallelogram,

one circle (or sphere), can never be equal vectors; because their

directions differ.