tion, the one different from the other. This is the first idea we want our reader to get a firm hold of; that multiplication is not necessarily addition, but an operation self-contained, self-interpretablé springing originally out of addition; but, when full-grown, existing apart from its parent.

The second idea we want our reader to fix his mind on is this, that when a science has been extended into a new form, certain limitations, which appeared to be of the nature of essential truths in the old science, are found to be utterly untenable; that it is, in fact, by throwing these limitations aside that room is made for the growth of the new science. We have instanced Algebra as a growth out of Arithmetic by the removal of the restriction that subtraction shall require something to subtract from. The word "subtraction" may indeed be inappropriate, as the word multiplication appeared to be to Record's scholar, who failed to see how the multiplication of a thing could make it less. In the advance of the sciences the old terminology often becomes inappropriate; but if the mind can extract the right idea from the sound or sight of a word, it is the part of wisdom to retain it. And so all the old words have been retained in the science of Quaternions to which we are now to advance.

The fundamental idea on which the science is based is that of motion of transference. Real motion is indeed not needed, any more than real superposition is needed in Euclid's Geometry. An appeal is made to mental transference in the one science, to mental superposition in the other.

We are then to consider how it is possible to frame a new science which shall spring out of Arithmetic, Algebra, and Geometry, and shall add to them the idea of motion-of transference. It must be confessed the project we entertain is not a project due to the nineteenth century. The Geometry of Des Cartes was based on something very much resembling the idea of motion, and so far the mere introduction of the idea of transference was not of much value. The real advance was due to the thought of severing multiplication from addition, so that the one might be the representative of a kind of motion absolutely different from that which was represented by

the other, yet capable of being combined with it. What the nineteenth century has done, then, is to divorce addition from multiplication in the new form in which the two are presented, and to cause the one, in this new character, to signify motion forwards and backwards, the other motion round and round.

We do not purpose to give a history of the science, and shall accordingly content ourselves with saying, that the notion of separating addition from multiplication-attributing to the one, motion from a point, to the other motion about a point-had been floating in the minds of mathematicians for half a century, without producing many results worth recording, when the subject fell into the hands of a giant, Sir William Rowan Hamilton, who early found that his road was obstructed he knew not by what obstacle-so that many points which seemed within his reach were really inaccessible. He had done a considerable amount of good work, obstructed as he was, when, about the year 1843, he perceived clearly the obstruction to his progress in the shape of an old law which, prior to that time, had appeared like a law of common sense. The law in question is known as the commutative law of multiplication. Presented in its simplest form it is nothing more than this, "five times three is the same as three times five;" more generally, it appears under the form of "abba whatever a and b may represent." When it came distinctly into the mind of Hamilton that this law is not a necessity, with the extended signification of multiplication, he saw his way clear, and gave up the law. The barrier being removed, he entered on the new science as a warrior enters a besieged city through a practicable breach. The reader will find it easy to enter after him.

CHAPTER II.

VECTOR ADDITION AND SUBTRACTION.

1. Definition of a Vector. A vector is the representative of transference through a given distance, in a given direction. Thus if AB be a straight line, the idea to be attached to "vector AB" is that of transference from A to B.

For the sake of definiteness we shall frequently abbreviate the phrase "vector AB" by a Greek letter, retaining in the meantime (with one exception to be noted in the next chapter) the English letters to denote ordinary numerical quantities.

If we now start from B and advance to C in the same direction, BC being equal to AB, we may, as in ordinary geometry, designate "vector BC" by the same symbol, which we adopted to designate "vector AB."

Further, if we start from any other point 0 in space, and advance from that point by the distance OX equal to and in the same direction as AB, we are at liberty to designate "vector OX” by the same symbol as that which represents AB.

Other circumstances will determine the starting point, and individualize the line to which a specific vector corresponds. Our definition is therefore subject to the following condition :—All lines which are equal and drawn in the same direction are represented by the same vector symbol.

We have purposely employed the phrase "drawn in the same direction" instead of "parallel," because we wish to guard the student against confounding "vector AB" with "vector BA."

2. In order to apply algebra to geometry, it is necessary to impose on geometry the condition that when a line measured in one direction is represented by a positive symbol, the same line measured in the opposite direction must be represented by the corresponding negative symbol.

In the science before us the same condition is equally requisite, and indeed the reason for it is even more manifest. For if a transference from A to B be represented by +a, the transference which neutralizes this, and brings us back again to A, cannot be conceived to be represented by anything but -a, provided the symbols + and - are to retain any of their old algebraic meaning. The vector AB, then, being represented by + a, the vector BA will be represented by -α.

3. Further it is abundantly evident that so far as addition and subtraction of parallel vectors are concerned, all the laws of Algebra must be applicable. Thus (in Art. 1) AB+ BC or a + a produces the same result at AC which is twice as great as AB, and is therefore properly represented by 2a; and so on for all the rest. The distributive law of addition may then be assumed to hold in all its integrity so long at least as we deal with vectors which are parallel to one another. In fact there is no reason whatever, so far, why a should not be treated in every respect as if it were an ordinary algebraic quantity. It need scarcely be added that vectors in the same direction have the same proportion as the lines which correspond to them.

We have then advanced to the following—

LEMMA. All lines drawn in the same direction are, as vectors, to be represented by numerical multiples of one and the same symbol, to which the ordinary laws of Algebra, so far as their addition, subtraction, and numerical multiplication are concerned, may be unreservedly applied.

4. The converse is of course true, that if lines as vectors are represented by multiples of the same vector symbol, they are parallel.

It is only necessary to add to what has preceded, that if BC be

a line not in the same direction with

The vector symbol a must A

B

C

AB, then the vector BC cannot be represented by a or by any multiple of a. be limited to express transference in a certain direction, and cannot, at the same time, express transference in any other direction. To express "vector BC" then, another and quite independent symbol ẞ must be introduced. This symbol, being united to a by the signs and, the laws of algebra will, of course, apply to the combination.

5. If we now join AC, and thus form a triangle ABC, and if we denote vector AB by a, BC by B, AC by y, it is clear that we shall be presented with the equation a + B = y.

This equation appears at first sight to be a violation of Euclid I. 20: "Any two sides of a triangle are together greater than the third side." But it is not really so. The anomalous appearance arises from the fact that whilst we have extended the meaning of the symbol beyond its arithmetical signification, we have said nothing about that of a symbol. It is clearly necessary that the signification of this symbol shall be extended along with that of the other. It must now be held to designate, as it does perpetually in algebra, "equivalent to." This being premised, the equation above is freed from its anomalous appearance, and is perfectly consistent with everything in ordinary geometry. Expressed in words it reads thus: "A transference from A to B followed by a transference from B to C is equivalent to a transference from A to C."

6. AXIOм. If two vectors have not the same direction, it is impossible that the one can neutralize the other.

This is quite obvious, for when a transference has been effected from A to B, it is impossible to conceive that any amount of transference whatever along BC can bring the moving point back to A.

It follows as a consequence of this axiom, that if a, ß be different actual vectors, i. e. finite vectors not in the same direction, and if