One geometrical interpretation of the negative sign of algebra was early seen to be mere reversal of direction along a line. Thus, when an image is formed by a plane mirror, the distance of any point in it from the mirror is simply the negative of that of the corresponding point of the object. Or if motion in one direction along a line be treated as positive, motion in the opposite direction along the same - line is negative. In the case of time, measured from the Christian era, this distinction is at once given by the letters a.d. or b.c., prefixed to the date. And to find the position, in time, of one event relatively to another, we have only to subtract the date of the second (taking account of its sign) from that of the first. Thus to find the interval between the battles of Marathon (490 b.c. ) and Waterloo (1815 a.d.) we have

+1815 - ( - 490) = 2305 years.

And it is obvious that the same process applies in all cases in which we deal with quantities which may be regarded as of one directed dimension only, such as distances along a line, rotations about an axis, &c. But it

is essential to notice that this is by no means necessarily true of operators. To turn a line through a certain angle in a given plane, a certain operator is required; but when we wish to turn it through an equal negative angle we must not, in general, employ the negative of the former operator. For the negative of the operator which turns a line through a given angle in a given plane will in all cases produce the negative of the original result, which is not the result of the reverse operator, unless the angle involved be an odd multiple of a right angle. This is, of course, on the usual assumption that the sign of a product is changed when that of any one of its factors is changed, —which merely means that -1 is commutative with all other quantities.

The celebrated Wallis seems to have been the first to push this idea further. In his Treatise of Algebra (1685) he distinctly proposes to construct the imaginary roots of a quadratic equation by going out of the line on which the roots, if real, would have been constructed.

In 1804 the Abbé Buée, apparently without any knowledge of Wallis’s work, developed this idea so far as to make it useful in geometrical applications. He gave, in fact, the theory of what in Hamilton’s system is called Composition of Vectors in one plane—i.e., the combination, by + and —, of complanar directed lines. His constructions are based on the idea that the imaginaries ± f - 1 represent a unit line, and its reverse, perpendicular to the line on which the real units ± 1 are measured. In this sense the imaginary expression a + b f - 1 is constructed by measuring a length a along the fundamental line (for real quantities), and from its extremity a line of length b in some direction perpendicular to the fundamental line. But he did not attack the question of the representation of products or quotients of directed lines. The step he took is really nothing more than the kinematical principle of the composition of linear velocities, but expressed in terms of the algebraic imaginary.

In 1806 (the year of publication of Buée’s paper) Argand published a pamphlet in which precisely the same ideas are developed, but to a considerably greater extent. For an interpretation is assigned to the product of two directed lines in one plane, when each is expressed as the sum of a real and an imaginary part. This product is interpreted as another directed line, forming the fourth term of a proportion, of which the first term is the real (positive) unit-line, and the other two are the factor-lines. Argand’s work remained unnoticed until the question was again raised in Gergonne's Annales, 1813, by Français. This writer stated that he had found the germ of his remarks among the papers of his deceased brother, and that they had come from Legendre, who had himself received them from some one unnamed. This led to a letter from Argand, in which he stated his communications with Legendre, and gave a résumé of the contents of his pamphlet. In a further communication to the Annales, Argand pushed on the applications of his theory. He has given by means of it a simple proof of the existence of n roots, and no more, in every rational algebraic equation of the nth order with real coefficients. About 1828 Warren in England, and Mourey in France, independently of one another and of Argand, reinvented these modes of interpretation; and still later, in the writings of

Cauchy, Gauss, and others, the properties of the expression a + b f - 1 were developed into the immense and most important subject now called the theory of complex numbers (see Numbers, Theory of). From the more purely symbolical view it was developed by Peacock, De Morgan, <fcc., as double algebra.

Argand’s method may be put, for reference, in the following form. The directed line whose length is a, and which makes an angle 0 with the real (positive) unit line, is expressed by α(cos0 + τsin0), where i is regarded as + √ - 1. The sum of two such lines (formed by adding together the real and the imaginary parts of two such expressions) can, of course, be expressed as a third directed line— the diagonal of the parallelogram of which they are conterminous sides. The product, P, of two such lines is, as we have seen, given by 1: α(cos0 + isin0) :: α'(cos0 ’ + fsin0'): P, or P=αα ’ {cos(0 + 0') + isin(0 + 0')}.

Its length is, therefore, the product of the lengths of the factors, and its inclination to the real unit is the sum of those of the factors. If we write the expressions for the two lines in the form A + Bi, A' + B'i, the product is AA' - BB' + i(AB' + BA'); and the fact that the length of the product line is the product of those of the factors is seen in the form

(A 2 + B 2 )(A' 2 + B' 2 ) = (AA' - BB , ) 2 + (AB , + BA ’) 2 .

In the modern theory of complex numbers this is expressed by saying that the Norm of a product is equal to the product of the norms of the factors.

Argand’s attempts to extend his method to space generally were fruitless. The reasons will be obvious later; but we mention them just now because they called forth from Servois (Gergonne’s Annales, 1813) a very remarkable comment, in which was contained the only yet discovered trace of an anticipation of the method of Hamilton. Argand had been led to deny that such an expression as i i could be expressed in the form A + Bz,— although, as is well known, Euler showed that one of its values is a real quantity, the exponential function of - ττ∕2 . Servois says, with reference to the general representation of a directed line in space :—

“L’analogie semblerait exiger que le trinôme fût de la forme 7>cosα + <∕eos)3 + rcosγ5 a, β, -y étant les angles d’une droite avec trois axes rectangulaires; et qu’on eût

( 2> cos a + q cos 3 + rcosγ)( p'cos a + /cos β + r'cos γ)

= cos 2 α+cos 2 3 + cos 2 γ = l. Les valeurs de p, q, r, p', q', r' qui satisferaient à cette condition seraient absurdes ; mais seraient-elles imaginaires, réductibles à la forme générale Α + ΒχΖ-l? Voilà une question d’analyse fort singulière que je soumets a vos lumières. La simple proposition que je vous en fais suffit pour vous faire voir que je ne crois point que toute fonction analytique non réelle soit vraiment réductible à la forme A + B√- 1.”

As will be seen later, the fundamental i, j, k of quaternions, with their reciprocals, furnish a set of six quantities which satisfy the conditions imposed by Servois. And it is quite certain that they cannot be represented by ordinary imaginaries.

Something far more closely analogous to quaternions than anything in Argand’s work ought to have been suggested by De Moivre’s theorem j^1730). Instead of regarding, as Buée and Argand had done, the expression a (cos θ + i sin θ) as a directed line, let us suppose it to represent the operator which, when applied to any line in the plane in which θ is measured, turns it in that plane through the angle θ, and at the same time increases its length in the ratio a : 1. From the new point of view we see at once, as it were, why it is true that

(cos θ + i sin θ) m =cos mθ + i sin mθ.

For this equation merely states that m turnings of a line through successive equal angles, in one plane, give the same result as a single turning through m times the common angle. To make this process applicable to any plane in space, it is clear that we must have a special value of i for each such plane. In other words, a unit line, drawn in any direction whatever, must have - 1 for its square. In such a system there will be no line in space specially distinguished as the real unit line : all will be alike imaginary, or rather alike real. We may state, in passing, that every quaternion can be represented as α(cos0 + ≡rsin0),—where a is a real number, θ a real angle, and w a directed unit line whose square is - 1. Hamilton took this grand step, but, as we have already said, without any help from the previous work of De Moivrc. The course of his investigations is minutely described in the preface to his first great work on the subject. Hamilton, like most of the many inquirers who endeavoured to give a real interpretation to the imaginary of common algebra, found that at least two kinds, orders, or ranks of quantities were necessary for the purpose. But, instead of dealing with points on a line, and then wandering out at right angles to it, as Buée and Argand had done, he chose to look on algebra as the science of pure time ? and to investigate the properties of “sets” of time-steps. In its essential nature a set is a linear function of any number of distinct units of the same species. Hence the simplest form of a set is a couple; and it was to the possible laws of combination of couples that Hamilton first directed his attention. It is obvious that the way in which the two separate time-steps are involved in the couple will determine these laws of combination. But Hamilton’s special object required that these laws should be such as to lead to certain assumed results; and he therefore commenced by assuming these, and from the assumption determined how the separate time-steps must be involved in the couple. If we use Roman letters for mere numbers, capitals for instants of time, Greek letters for time-steps, and a parenthesis to denote a couple, the laws assumed by Hamilton as the basis of a system were as follows :—

(B 1 , B 2 )-(A 1 , A 2 ) = (B 1 -A 1 , B 2 ~ A 2 ) = (<x, 3);

(a, b)(α, ∕3) = (aa-bj8, ba + aβ).

To show how we give, by such assumptions, a real interpretation to the ordinary algebraic imaginary, take the simple case a = 0, b = l, and the second of the above formulae gives

(0, l)(α, 3) = (-3, a)·

Multiply once more by the number-couple (0,1), and we have

(0, l)(0, l)(α, 3) = (0, l)(-0, a) = (-a,~3) = (-l, 0)(a, 3)=-(a, 3)∙

Thus the number-couple (0,1), when twice applied to a step-couple, simply changes its sign. That we have here a perfectly real and intelligible interpretation of the ordinary algebraic imaginary is easily seen by an illustration, even if it be a somewhat extravagant one. Some Eastern potentate, possessed of absolute power, covets the vast possessions of his vizier and of his barber. He determines to rob them both (an operation which may be very satisfactorily expressed by - 1); but, being a wag, he chooses his own way of doing it. He degrades his vizier to the office of barber, taking all his goods in the process; and makes the barber his vizier. Next day he repeats the operation. Each of the victims has been restored to his former rank, but the operator - 1 has been applied to both.

Hamilton, still keeping prominently before him as his great object the invention of a method applicable to space of three dimensions, proceeded to study the properties of triplets of the form x + iy +jz, by which he proposed to represent the directed line in space whose projections on the coordinate axes are x, y, z. The composition of two such lines by the algebraic addition of their several projections agreed with the assumption of Buée and Argand for the case of coplanar lines. But, assuming the distributive principle, the product of two lines appeared to give the expression xx' - yy' - zz' + i{yx ’ + xyj +j{xz' + zxj + ijfiz , + zyj.

For the square of j, like that of i, was assumed to be negative unity. But the interpretation of ij presented a difficulty,—in fact the main difficulty of the whole investigation,—and it is specially interesting to see how Hamilton attacked it. He saw that he could get a hint from the simpler case, already thoroughly discussed, provided the two factor lines were in one plane through the real unit line. This requires merely that y. z ::y': z'; or yz'-zy ’ =0∙, but then the product should be of the same form as the separate factors. Thus, in this special case, the term in ij ought to vanish. But the numerical factor appears to be yz' + zy', while it is the quantity yz — zy which really vanishes. Hence Hamilton was at first inclined to think that ij must be treated as nil. But he soon saw that “a less harsh supposition” would suit the simple case. For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' axd ∖ ,jizy', the term would vanish of itself when the factor-lines are coplanar provided ij= -ji, for it would then assume the form ij (yz' - zyj. He had now the following expression for the product of any two directed lines xx' - yy' - zz' + ijjx' + xyj +j(xz , + ≈c') + ijjy≈' - zyj.

But his result had to be submitted to another test, the Law of the Norms. As soon as he found, by trial, that this law was satisfied, he took the final step. “This led me,” he says, “to conceive that perhaps, instead of seeking to confine ourselves to triplets, we ought to regard these as only imperfect forms of Quaternions, and that thus my old conception of sets might receive a new and useful application.” In a very short time he settled his fundamental assumptions. He had now three distinct space-units i, j, k; and the following conditions regulated their combination by multiplication :—

i 2 =y 2 =⅛ 2 = -1, tj'= -ji=k, jk≈ - kj=i, ki= -ik=j. It will be easy to see that, instead of the last three of these, we may write the single one ijk≈ - 1. i And noιc the product of two quaternions could be at once expressed as a third quaternion, thus—

(a + ib -{-jc 4- kdja' + ib' Λ^jc + kdj = A + z"B +J*C -I-Z-I~), where [table]

Hamilton at once found that the Law of the Norms holds,— not being aware that Euler had long before decomposed the product of two sums of four squares into this very set of four squares. And now a directed line in space came to be represented as ix+jy + kz, while the product of two lines is the quaternion - (xx! + yy ’ +zzj + i(yz' - zyj +j(za! - xzj + k(xy' -yxj.

To any one acquainted, even to a slight extent, with the elements of Cartesian geometry of three dimensions, a glance at the extremely suggestive constituents of this expression shows how justly Hamilton was entitled to say —“When the conception .... had been so far unfolded and fixed in my mind, I felt that the new instrument for applying calculation to geometry, for which I had so long

sought, was now, at least in part, attained.” The date ol this memorable discovery is October 16, 1843.

We can devote but a few lines to the consideration oi the expression above. Suppose, for simplicity, the factorlines to be each of unit length. Then x, y, z, x', y', z express their direction-cosines. Also, if 0 be the angle between them, and x", y", z" the direction-cosines of a line perpendicular to each of them, we have xx' + yy' + zz — cos θ, yz' — zy' = x" sinθ, &c., so that the product of two unit lines is now expressed as - cosØ + (ix" +jy" + ½z")sin0. Thus, when the factors are parallel, or 0 = 0, the product, which is now the square of any (unit) line, is—1. And when the two factor lines are at right angles to one another, or 0 = τr∕2, the product is simply ix" +jy"+kz'', the unit line perpendicular to both. Hence, and in this lies the main element of the symmetry and simplicity of the quaternion calculus, all systems of three mutually rectangular unit lines in space have the same properties as the fundamental system i, j, k. In other words, if the system (considered as rigid) be made to turn about till the first factor coincides with i and the second with j, the product will coincide with k. This fundamental system, therefore, becomes unnecessary; and the quaternion method, in every case, takes its reference lines solely from the problem to which it is applied. It has therefore, as it were, a unique internal character of its own.

Hamilton, having gone thus far, proceeded to evolve these results from a train of a priori or metaphysical reasoning, which is so interesting in itself, and so characteristic of the man, that we briefly sketch its nature.

Let it be supposed that the product of two directed lines is something which has quantity; i.e., it may be halved, or doubled, for instance. Also let us assume (a) space to have the same properties in all directions, and make the convention (ό) that to change the sign of any one factor changes the sign of a product. Then the product of two lines which have the same direction cannot be, even in part, a directed quantity. For, if the directed part...