On a new Species of Imaginary Quantities connected with a theory of Quaternions

By Sir William R. Hamilton

[Proceedings of the Royal Irish Academy, Nov. 13, 1843, vol. 2, 424-434]

It is known to all students of algebra that an imaginary equation of the form tex2html_wrap_inline1310 has been employed so as to conduct to very varied and important results. Sir Wm. Hamilton proposes to consider some of the consequences which result from the following system of imaginary equations, or equations between a system of three different imaginary quantities:




no linear relation between i, j, k being supposed to exist, so that the equation


in which


and w, x, y, z, w', x', y', z' are real, is equivalent to the four separate equations


Sir W. Hamilton calls an expression of the form tex2html_wrap_inline1334 a quaternion; and the four real quantities w, x, y, z he calls the constituents thereof. Quaternions are added or subtracted by adding or subtracting their constituents, so that


Their multiplication is, in virtue of the definitions (A) (B) (C), effected by the formulae



which give


and therefore


if we call the positive quantity


the modulus of the quaternion tex2html_wrap_inline1334 . The modulus of the product of any two quaternions is therefore equal to the product of the moduli. Let


then, because the equations (D) give


we have


Consider x, y, z as the rectangular coordinates of a point of space, and let tex2html_wrap_inline1352 be the point where the radius vector of x, y, z (prolonged if necessary) intersects the spheric surface described about the origin with a radius equal to unity; call tex2html_wrap_inline1352 the representative point of the quaternion  tex2html_wrap_inline1334 , and let the polar coordinates tex2html_wrap_inline1364 and tex2html_wrap_inline1366 , which determine tex2html_wrap_inline1352 upon the sphere, be called the co-latitude and the longitude of the representative point  tex2html_wrap_inline1352 , or of the quaternion tex2html_wrap_inline1334 itself; let also the other angle  tex2html_wrap_inline1374 be called the amplitude of the quaternion; so that a quaternion is completely determined by its modulus, amplitude, co-latitude and longitude. Construct the representative points tex2html_wrap_inline1376 and tex2html_wrap_inline1378 , of the other factor  tex2html_wrap_inline1380 , and of the product  tex2html_wrap_inline1382 ; and complete the spherical triangle tex2html_wrap_inline1384 by drawing the arcs tex2html_wrap_inline1386 , tex2html_wrap_inline1388 , tex2html_wrap_inline1390 . Then, the equations (G) become


and consequently shew that the angles of the triangle tex2html_wrap_inline1384 are


these angles are therefore respectively equal to the amplitudes of the factors, and the supplement (to two right angles) of the amplitude of the product. The equations (D) show, further, that the product-point  tex2html_wrap_inline1378 is to the right or left of the multiplicand-point  tex2html_wrap_inline1376 , with respect to the multipler-point  tex2html_wrap_inline1352 , according as the semiaxis of +z (or its intersection with the spheric surface) is to the right or left of the semiaxis of +y, with respect to the semiaxis of +x: that is, according as the positive direction of rotation in longitude is towards the right or left. A change in the order of the two quaternion-factors would throw the product-point tex2html_wrap_inline1378 from the right to the left, or from the left to the right of tex2html_wrap_inline1386 .

It results from these principles, that if tex2html_wrap_inline1384 be any spherical triangle; if, also, tex2html_wrap_inline1412 be the rectangular coordinates of tex2html_wrap_inline1352 , tex2html_wrap_inline1416 those of tex2html_wrap_inline1376 , and tex2html_wrap_inline1420 of tex2html_wrap_inline1378 , the centre of the sphere being origin, and the radius being unity; and if the rotation round +x from +y to +z be of the same (right-handed or left-handed) character as that round tex2html_wrap_inline1352 from tex2html_wrap_inline1376 to tex2html_wrap_inline1378 ; then the following formula of multiplication, according to the rules of quaternions, will hold good:



Developing and decomposing this imaginary or symbolic formula (I), we find that it is equivalent to the system of the four following real equations, or equations between real quantities:


Of these equations (K), the first is only an expression of the well-known theorem, already employed in these remarks, which serves to connect a side of any spherical triangle with the three angles thereof. The three other equations (K) are an expression of another theorem (which possibly is new), namely that a force tex2html_wrap_inline1436 , directed from the centre of the sphere to the point  tex2html_wrap_inline1378 , is statically equivalent to the system of three other forces, one directed to tex2html_wrap_inline1352 , and equal to tex2html_wrap_inline1442 , another directed to tex2html_wrap_inline1376 , and equal to tex2html_wrap_inline1446 , and the third equal to tex2html_wrap_inline1448 , and directed towards that pole of the arc tex2html_wrap_inline1386 , which lies at the same side of this arc as tex2html_wrap_inline1378 . It is not difficult to prove this theorem otherwise; but it may be regarded as interesting to see that the four equations (K) are included so simply in the one formula (I) of multiplication of quaternions, and are obtained so easily by developing and decomposing that formula, according to the fundamental definitions (A) (B) (C). A new sort of algorithm, or calculus, for spherical trigonometry, appears to be thus given, or indicated. And by supposing the three corners of the spherical triangle tex2html_wrap_inline1384 to tend indefinitely to close up in that one point which is the intersection of the spheric surface with the positive semiaxis of x, each coordinate tex2html_wrap_inline1458 will tend to become = 1, and each tex2html_wrap_inline1462 and tex2html_wrap_inline1464 to vanish, while the sum of the three angles will tend to become tex2html_wrap_inline1466 ; so that the following well known and important equations in the usual calculus of imaginaries, as connected with plane trigonometry, namely,


(in which tex2html_wrap_inline1310 ), is found to result, as a limiting case, from the more general formula (I).

In the ordinary theory there are only two different square roots of negative unity (+i and -i), and they differ only in their signs. In the present theory, in order that a quaternion, w + ix + jy + kz, should have its square = -1, it is necessary and sufficient that we should have


we are conducted, therefore, to the extended expression


which may be called an imaginary unit, because its modulus is = 1, and its square is negative unity. To distinguish one such imaginary unit from another, we may adopt the notation,


tex2html_wrap_inline1352 being still that point on the spheric surface which has tex2html_wrap_inline1458 , tex2html_wrap_inline1462 , tex2html_wrap_inline1464 (or tex2html_wrap_inline1490 , tex2html_wrap_inline1492 , tex2html_wrap_inline1494 ) for its rectangular coordinates; and then the formula of multiplication (I) becomes, for any spherical triangle, in which the rotation round tex2html_wrap_inline1352 , from tex2html_wrap_inline1376 to tex2html_wrap_inline1378 , is positive,


If tex2html_wrap_inline1504 be the positive pole of the arc tex2html_wrap_inline1386 , or the pole to which the least rotation from tex2html_wrap_inline1376 round tex2html_wrap_inline1352 is positive, then the product of the two imaginary units in the first member of this formula (which may be any two such units), is the following:


we have also, for the product of the same two factors, taken in the opposite order, the expression


which differs only in the sign of the imaginary part; and the product of these two products is unity, because, in general,


we have, therefore,


and the products tex2html_wrap_inline1512 and tex2html_wrap_inline1514 may be said to be reciprocals of each other.

In general, in virtue of the fundamental equations of definition, (A), (B), (C), although the distributive character of the multiplication of ordinary algebraic quantities (real or imaginary) extends to the operation of the same name in the theory of quaternions, so that


yet the commutative character is lost, and we cannot generally write for the new as for the old imaginaries,


since we have, for example, ji = - ij. However, in virtue of the same definitions, it will be found that another important property of the old multiplication is preserved, or extended to the new, namely, that which may be called the associative character of the operation, and which may have for its type the formula


thus we have, generally,



and so on for any number of factors; the notation tex2html_wrap_inline1520 being employed to express that one determined quaternion, which, in virtue of the theorem (Q), is obtained, whether we first multiply tex2html_wrap_inline1382 as a multiplicand by tex2html_wrap_inline1380 as a multiplier, and then multiply the product tex2html_wrap_inline1526 as a multiplicand by tex2html_wrap_inline1334 as a multiplier; or multiply first tex2html_wrap_inline1380 by tex2html_wrap_inline1334 , and then tex2html_wrap_inline1382 by tex2html_wrap_inline1536 . With the help of this principle, we might easily prove the equation (P), by observing that its first member tex2html_wrap_inline1538 .

In the same manner it is seen at once that


whatever n points upon the spheric surface may be denoted by tex2html_wrap_inline1352 , tex2html_wrap_inline1376 , tex2html_wrap_inline1378 , tex2html_wrap_inline1550 ,... tex2html_wrap_inline1552 : and by combining this principle with that expressed by (M), it is not difficult to prove that for any spherical polygon tex2html_wrap_inline1554 , the following formula holds good:



which includes the theorem (I') for the case of a spherical triangle, and in which the arrangement of the n points may be supposed, for simplicity, to be such that the rotations round tex2html_wrap_inline1352 from tex2html_wrap_inline1376 to tex2html_wrap_inline1378 , round tex2html_wrap_inline1376 from tex2html_wrap_inline1378 to tex2html_wrap_inline1550 , and so on, are all positive, and each less than two right angles, though it is easy to interpret the expression so as to include also the cases where any or all of these conditions are violated. When the polygon becomes infinitely small, and therefore plane, the imaginary units become all equal to each other, and may be denoted by the common symbol i; and the formula (R) agrees then with the known relation, that


Again, let tex2html_wrap_inline1352 , tex2html_wrap_inline1376 , tex2html_wrap_inline1378 be, respectively, the representative points of any three quaternions tex2html_wrap_inline1334 , tex2html_wrap_inline1380 , tex2html_wrap_inline1382 , and let tex2html_wrap_inline1586 , tex2html_wrap_inline1588 , tex2html_wrap_inline1590 be the representative points of the three other quaternions, tex2html_wrap_inline1536 , tex2html_wrap_inline1526 , tex2html_wrap_inline1520 , derived by multiplication from the former; then the algebraical principle expressed by the formula (Q) may be geometrically enunciated by saying that the two points tex2html_wrap_inline1586 and tex2html_wrap_inline1588 are the foci of a spherical conic which touches the four sides of the spherical quadrilateral tex2html_wrap_inline1602 ; and analogous theorems respecting spherical pentagons and other polygons may be deduced, by constructing similarly the formulæ (Q'),&c.

In general, a quaternion tex2html_wrap_inline1334 , like an ordinary imaginary quantity, may be put under the form,


provided that we assign to tex2html_wrap_inline1606 , or tex2html_wrap_inline1608 , the extended meaning (L), which involves two arbitrary angles; and the same general quaternion tex2html_wrap_inline1334 may be considered as a root of a quadratic equation, with real coefficients, namely


which easily conducts to the following expression for a quotient, or formula for the division of quaternions,


if we define tex2html_wrap_inline1616 or tex2html_wrap_inline1618 to mean that quaternion tex2html_wrap_inline1380 which gives the product tex2html_wrap_inline1382 , when it is multiplied as a multiplicand by tex2html_wrap_inline1334 as a multiplier. The same general formula (S'') of division may easily be deduced from the equation (O), by writing that equation as follows,


or it may be obtained from the four general equations of multiplication (D), by treating the four constituents of the multiplicand, namely w', x', y', z', as the four sought quantities, while w, x, y, z, and w'', x'', y'', z'', are given; or from a construction of spherical trigonometry, on principles already laid down.

The general expression (S) for a quaternion may be raised to any power with a real exponent q, in the same manner as an ordinary imaginary expression, by treating the square root of -1 which it involves as an imaginary unit tex2html_wrap_inline1658 having (in general) a fixed direction; raising the modulus  tex2html_wrap_inline1660 to the proposed real power; and multiplying the amplitude tex2html_wrap_inline1374 , increased or diminished by any whole number of circumferences, by the exponent q: thus


if q be real, and if n be any whole number. For example, a quaternion has in general two, and only two, different square roots, and they differ only in their signs, being both included in the formula,


in which it is useless to assign to n any other values than 0 and 1; although, in the particular case where the original quaternion reduces itself to a real and negative quantity, so that tex2html_wrap_inline1678 , this formula (T') becomes


the direction of tex2html_wrap_inline1658 remaining here entirely undetermined; a result agreeing with the expression (L) or (L') for tex2html_wrap_inline1608 . In like manner the quaternions, which are cube roots of unity, are included in the expression


tex2html_wrap_inline1658 denoting here again an imaginary unit, with a direction altogether arbitrary.

If we make, for abridgment


the series here indicated will be always convergent, whatever quaternion tex2html_wrap_inline1334 may be; and we can always separate its real and imaginary parts by the formula


which gives, reciprocally, for the inverse function tex2html_wrap_inline1698 , the expression


u being any whole number, and tex2html_wrap_inline1704 being the natural, or Napierian, logarithm of tex2html_wrap_inline1660 , or, in other words, that real quantity, positive or negative, of which the function f is equal to the given real and positive modulus  tex2html_wrap_inline1660 . And although the ordinary property of exponential functions, namely


does not in general hold good, in the present theory, unless the two quaternions tex2html_wrap_inline1334 and tex2html_wrap_inline1380 be codirectional, yet we may raise the function f to any real power by the formula


which it is natural to extend, by definition, to the case where the exponent q becomes itself a quaternion. The general equation,


when put under the form


will then give


and thus the general expression for a quaternion q, which is one of the logarithms of a given quaternion tex2html_wrap_inline1728 to a given base tex2html_wrap_inline1730 , is found to involve two independent whole numbers n and n', as in the theories of Graves and Ohm, respecting the general logarithms of ordinary imaginary quantities to ordinary imaginary bases.

For other developments and applications of the new theory, it is necessary to refer to the original paper from which this abstract is taken, and which will probably appear in the twenty-first volume of the Transactions of the Academy.


D.R. Wilkins
School of Mathematics
Trinity College, Dublin