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“ 2 Oct. 15, '58.
“P. S.–To-morrow will be the fifteenth birthday of the Quaternions. They started into life, or light, full grown, on D the 16th of October 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge, which my boys have since called the Quatermion Bridge. That is to say, I then and there felt the galvanic circuit of thought close; and the sparks which fell from it were the fundamental equations between i, j, k, eractly such as I have used them ever since. I pulled out, on the spot, a pocket-book which still exists, and made an entry, on which, at the very moment, I felt that it might be worth my while to expend the labour of at least ten (or it might be fifteen) years to come. But then, it is fair to say that this was because I felt a problem to have been at that moment solved,—an intellectual want relieved,—which had haunted me for at least fifteen years before.
“Less than an hour elasped, before I had asked and obtained leave, of the Council of the Royal Irish Academy, of which Society I was, at that time, the President, to read at the merz general Meeting, a Paper on Quaternions; which I accordingly did, on November 13, 1843.
“Some of those early communications of mine to the Academy may still have some interest for a person like you, who has since so well studied my Volume, which was not published for ten years afterwards.
“In the meantime, will you not do honour to the birthday, to-morrow, in an extra cup of —ink? for it may be obsolete now to propose XXX,+or even XYZ.”
We must now endeavour to explain, in as popular a manner as possible, the nature of the new Calculus. In order to do so, let us recur to the suggestion of Wallis, before described, and endeavour to ascertain the exact nature of its defects. We easily see that one great defect is want of symmetry. As before stated, if we take an eastward line, of proper length, to represent positive unity, an equal westward line represents negative unity; but all lines perpendicular or inclined to these are represented by socalled imaginary quantities. Hamilton's great step was the attainment of the desired symmetry by making all lines alike expressible by so-called imaginary quantities. Thus, instead of 1 for the eastward line, and the square root of negative unity for a northward line, he represents every line in space whose length is unity by a distinct square root of negative unity. All are thus equally imaginary, or rather equally real. The i,j, k mentioned in the extract just given, are three such quantities; which (merely for illustration, because any other set of three mutually rectangular directions will do as well) we may take as representing unit lines drawn respectively eastwards, northwards, and upwards. The square of each being
negative unity, we may interpret the effect
of such a line (when used as a factor or operator) as a left (or right) handed rotation through one right angle about its direction. The effect of a repetition of the operation is a rotation through two right angles, or a simple inversion. Thus, if we operate with i on f we turn the northward line left-handedly through a right angle about an eastward axis, i.e., we raise it to a direction vertically upward. Thus we see that j=k. But, if we perform the operation again, we see that ik or is is now the southward line or -j. Thus "E-1. And similarly with the squares of j and k. It is to be noticed that in all these cases the operating line is supposed to be perpendicular to the operand. Also that we have taken for granted (what is easily proved), that i, j, k may stand indifferently for the unit line themselves or for the operation of turning through a right angle. Thus, the equation is =k may either mean (as above) that i acting on the line j turns it into the line k; or that the rectangular rotation f, succeeded by i, is equivalent to the single rotation k. We may easily verify the last assertion by taking i as the operand. j changes it to -k, and i changes this to j. But k turns i into 7 at once.
Even these ... ideas lead us at once to one of the most remarkable properties of quarternions. When turning the northward line (j) about the eastward line (i), we wrote the operator first-thus ij=k. Now, the order of multiplication is not indifferent, for ji is not equal to k. ji, in fact, expresses that i (the eastward line) has been made to rotate left-handedly through a right angle about j (the northward line). This obviously brings it to the downward direction, or we have ji= -k, Similar expressions hold for the other products two and two of the three symbols. Thus we have the laws of their multiplication complete. And on this basis the whole theory may be erected. Now any line whatever may be resolved (as velocities and forces are) into so much eastward (or westward), so much northward (or southward), and so much upward (or downward). Hence every line may be expressed as the sum of three parts, numerical multiples of i, j, and k respectively. Call these numbers 2, y, and 2, then the line may be expressed by zi+yj + zk. If we square this we find -(z” +y^+2*); for the other terms occur in phirs like zi xuj and yj xzi, and destroy each other; since we have, as above, ij-i-ji=0, with similar results for the other pairs of the three rectangular unit-lines. Now z*-ī-y?--z" is the square of the length of the line (by a double application of Euclid I. xlvii.) Hence the square of every
line of unit length is negative unity. And herein consists the grand symmetry and consequent simplicity of the method; for we may now write a single symbol such as o. (Greek letters are usually employed by Hamilton in this sense), instead of the much more cumbrous and not more expressive form zi-Hyj + zk. We have seen that the product, and consequently the quotient, of two lines at right angles to each other, is a third line perpendicular to both; and that the product or quotient of two parallel lines is a number: what is the product, or the quotient, of two lines not at right angles, and not parallel, to each other ? It is a QUATERNION. This is very easily seen thus. Take the case of the quotient of one line by another, and suppose them drawn from the same point: the first line may, by letting fall a perpendicular from its extremity upon the second, be resolved into two parts, one parallel, the other perpendicular, to the second line. The quotient of the two parallel lines is a mere number, that of the two perpendicular lines is a line, and can therefore be expressed as the sum of multiples of i, j, and k. Hence, w representing the numeral quotient of the parallel portions, the quotient of any two lines, may be written as w-Hzi+yj +zk; and, in this form, is seen to depend essentially upon Four perfectly distinct numbers; whence its name. In actually working with quaternions, however, this cumbrous form is not necessary; we may express it as B-i-a, B and a being the two lines of which it is the quotient; and in various other equivalent algebraic forms; or we may at once substitute for it a single letter (Hamilton uses the early letters, a, b, c, etc.) The amount of condensation, and consequent shortening, of the work of any particular problem which is thus attained, though of immense importance, is not by any means the only or even the greatest advantage possessed by quaternions over other methods of treating analytical geometry. They render us entirely independent of special lines, axes of co-ordinates, etc., devised for the application of other methods, and take their reference lines, in every case from the particular Fo to which they are applied. They ave thus what Hamilton calls an “internal” character of their own; and give us, without trouble, an insight into each special question, which other methods only yield to a combination of great acuteness with patient labour. In fact, before their invention, no process was known for treating problems in tridimensional space in a thoroughly natural and inartificial manner. But let him speak for himself. The fol
lowing passage is extracted from a letter to a mathematical friend, who was at the time engaged in studying the new calculus:—
- “Whatever may be the future success of Quaternions as an Instrument of Investigation, they furnish already, to those who have learned to read them, (povávra aruverotorw,) a powerful ORGAN of ExPREssion, especially in geometrical science, and in all that widening field of physical inquiry, to which relations of space (not always easy to express with clearness by the Cartesian Method) are subsidiary, or rather are indispensable.”
To follow up the illustration we began with, it may be well merely to mention here that a quaternion may in all cases be represented as a power of a line. When a line is raised to the first, third, or any odd integral power, it represents a right-angled quaternion, or one which contains no pure numerical part; when to the second, fourth, or other even integral power, it degenerates into a mere number; for all other powers it contains four distinct terms. Compare this with the illustration already given, leading to De Moivre's theorem; and we see what a grand step Hamilton supplied by assigning in every case a definite direction to the axis about which the rotation takes place.
There are many other ways in which we can exhibit the essential dependence of the product or quotient of two directed lines (or Vectors as Hamilton calls them), on four numbers (or Scalars), and the consequent fitness of the name Quaternion. It may interest the reader to see another of them. Let us now regard the quaternion as the factor or operator required to change one side of a triangle into another; and let us suppose the process to be performed by turning one of the sides round till it coincides in direction with the other, and then stretching or shortening it till they coincide in length also. For the first operation we must know the axis about which the rotation is to take place, and the angle or amount of rotation. Now the direction of the axis depends on two numbers (in Astronomy they may be Altitude and Azimuth, Right Ascension and Declination, or Latitude and Longitude); the amount of rotation is a third number; and the amount of stretching or shortening in the final operation is the fourth.
Among the many curious results of the invention of quaternions, must be noticed the revival of fluzions, or, at all events, a mode of treating differentials closely allied to that originally introduced by Newton. The really useful, but over-praised differential coefficients, have, as a rule, no meaning in quaternions; so that, except when dealing with scalar variables (which are simply degraded quaternions), we must employ in differentiation fluxions or differentials. And the reader may easily understand the cause of this. It lies in the fact that quaternion multiplication is not commutative; so that, in differentiating a product, for instance, each factor must be differentiated where it stands; and thus the differential of such a product is not generally a mere algebraic multiple of the differential of the independent quaternion-variable. It is thus that the whirligig of time brings its revenges. The shameless theft which Leibnitz committed, and which he sought to disguise by altering the appearance of the stolen goods, must soon be obvious, even to his warmest partisans. They can no longer pretend to regard Leibnitz as even a second inventor when they find that his only possible claim, that of devising an improvement in notation, merely unfits Newton's method of fluxions for application to the simple and symmetrical, yet massive, space-geometry of Hamilton. One very remarkable speculation of Hamilton's is that in which he deduces, by a species of metaphysical or a priori reasoning, the results previously mentioned, viz., that the product (or quotient) of two parallel vectors must be a number, and that of two mutually perpendicular vectors a third perpendicular to both. We cannot give his reasoning at full length, but will try to make part of it easily intelligible. Suppose that there is no direction in space pre-eminent, and that the product of two vectors is something which has quantity, so as to vary in amount if the factors are changed, and to have its sign changed if that of one of them is reversed; if the vectors be parallel, their product cannot be, in whole or in part, a vector inclined to them, for there is nothing to determine the direction in which it must lie. It cannot be a vector parallel to them; for by changing the sign of both factors the product is unchanged, whereas, as the whole system has been reversed, the product vector ought to have been reversed. Hence it must be a number. Again, the product of two perpendicular vectors cannot be wholly or partly a number, because on inverting one of them the sign of that number ought to change; but inverting one of them is simply equivalent to a rotation through two right angles about the other, and from the symmetry of space ought to leave the number unchanged. Hence the product of two perpendicular vectors must be a vector, and an easy extension of the same reasoning shows that it WOL. XLV.
as it is called in the Carmen Aureum. Of course, so far as mere derivation goes, it is hard to see any difference between the Tetractys and the Quaternion. But we are almost entirely ignorant of the meaning Pythagoras attached to his mystic idea, and it certainly must have been excessively vague, if not quite so senseless as the Abracadabra of later times. Yet there is no doubt that Hamilton was convinced that Quaternions, in virtue of some process analogous to the quasi-metaphysical speculation we have just sketched, are calculated to lead to important discoveries in physical science; and, in fact, he writes—
“Little as I have pursued such [physical] Studies, even in books, you may judge from my Presidential Addresses, pronounced on the occasions of delivering Medals (long ago), from the chair of the R.I.A., to Apjohn and to Kane, that physical (as distinguished from mathematical) investigations have not been wholly alien to my somewhat wide, but doubtless very superficial, course of reading. You might, with
" out offence to me, consider that I abused the
license of hope, which may be indulged to an inventor, if I were to confess that I expect the Quaternions to supply hereafter, not merely mathematical methods, but also physical suggestions. And, in Fo you are quite welcome to smile, if I say that it does not seem extravagant to me to suppose that a full possession of those d priori principles of mine, about the multiplication of vectors—including the Law of the Four Scales, and the Conception of the Extra-spatial Unit, which have as yet not been much more than hinted to the public—Might have led (I do not at all mean that in my hands they ever would have done so,) to an ANTICIPATION of something like the grand discovery of OERSTED: who, by the way, was a very a priori (and poetical) sort of man himself, as I know from having conversed with him, and received from him some printed pamphlets, several years ago. It is impossible to estimate the chances given, or opened up, by any new way of looking at things; especially when that way admits of being intimately combined with calculation of a most rigorous kind.”
This idea is still further developed in the following sonnet, which gives besides a good idea of his powers of poetical composition. It is understood to refer to Sir John Herschel, who had, at a meeting of the British Association, compared the Quaternion Calculus to a Cornucopia, from which, turn it as you will, something new and valuable must escape.
Or high Mathesis, with her charm severe,
Whatever may be the future of Quaternions, and it may possibly far surpass all that its inventor ever dared to hope, there can be but one opinion of the extraordinary genius, and the untiring energy of him who, unaided, composed in so short a time two such enormous treatises as the Lectures (1853), and the Elements of Quaternions (1866). As a repertory of mathematical facts, and a triumph of analytical and geometrical power, they can be compared only with such imperishable works as the Principia and the Mécanique Analytique. They cannot be said to be adapted to the wants of elementary teaching, but we are convinced that every one who has a real liking for mathematics, and who can get over the preliminary difficulties, will persevere till he finishes the work, whichever of the two it may be, he has commenced. They have all that exquisite charm of combined beauty, power, and originality which made Hamilton compare Lagrange's great work to a “scientific poem.” And they conduct the mathemati. cian to a boundless expanse of new territory of the richest promise, the cultivation of which cannot be said to have been more than commenced, even by labour so unremitting, and genius, so grand, as Hamilton brought to bear on it.
The unit vectors of the quaternion calculus are not the only roots of unity which Hamilton introduced into practical analysis. In various articles in the Philosophical Magazine he developed the properties of "roups of symbols analogous to the i, j, k of
quaternions, but more numerous, and gave vaious applications of them. These groups have, generally, a direct connexion with the “Sets” with which he was occupied just be. fore the invention of the quaternions: and it would be vain to attempt to explain their nature to the general reader. But we must say a few words about another, and most extraordinary, system which Hamilton seems to have invented about 1856, and which has no connexion whatever with any previous group. Unfortunately, Hamilton has published but a page or two with reference to them, yet that little is enough to show the probability of their becoming, at some future time, of great importance in the study of crystals and and polyhedra in general. The subject is capable of indefinite extension; but Hamilton seems to have carefully studied only one particular system, which depends mainly upon two distinct and noncommutative fifth roots of positive unity, which, for ease of reference, we will call, with their inventor, A and ps. Although nothing more practical than an ingenious “puzzle” has yet resulted from these in: vestigations, their singular originality and (if we may use the word) oddity, and the wonderful series of new transformations which they suggest to the mathematician, render them well worthy of further study and development. Some idea of a small class of their properties may be derived from the consideration of a pentagonal dodecahedron (a solid enclosed by twelve faces, each of which has five sides). The number of edges of this solid is thirty; as we may see by remarking that, if we count five edges for each of the twelve faces, each edge will have been taken twice. Also, since three edges meet in each corner, and since each edge passes through two corners, we shall get three times too many corners by counting two for each edge. That is, there are twenty corners. Now, in that one of Hamilton's systems which he most fully worked out, the operators A and p, applied to any edge of the pentagonal dodecahedron, change it into one of the adjoining edges. Thus, going along an edge, to a corner, we meet two new edges, that to the right is derived from the first by the operator A, that to the left by p. Every possible way of moving along successive edges of such a solid may therefore be symbolized by performing on the first edge the successive operations A and po in any chosen order. And, as the reader may easily convince himself by trial, such a group of twenty operations as this, consisting of the series p, A, p., A, p, p, u, \, \, \, taken twice, brings us back to the edge we started from, after passing through each
by projection (the eye being supposed to be placed very near to the middle of one face), in order to prevent any two of the lines which represent the edges from crossing each other. The game is played by inserting pegs, numbered 1, 2, 3, . . . 20, in successive holes, which are cut at the points of the figure representing the corners of the dodecahedron; taking care to pass only along the lines which represent the edges. It is characteristic of Hamilton that he has selected the 20 consonants of our alphabet to denote these holes. When five pegs are placed in any five suc‘essive holes, it is always possible in two ways, sometimes in four, to insert the whole twenty, so as to form a continuous circuit. Thus, let BCDFG be the given five, we may complete the series by following the order of the consonants; or we may take the following order (after G) HxwnstvjKLMNPQz. If LTsko, or ZBCDM, be given there are four solutions. If fewer than five be fixed at starting there are, of course, more solutions. This is only the simplest case of the game. Puzzles without number, and of a far higher order of difficulty, can be easily suggested after a little practice, but even more readily by the proper mathematical processes. Thus, BCD may be given, the problem being to insert all the pegs in order, and end at a given hole. If that hole be M, it is impossible; if T, there is one solution; if J, two; and, if R, four. Again, certain initial points
being given, finish with a given number of
pegs. Thus, given KJV, finish with the eighth. The other five are TsNML, for when we have got to L no other peg can be inserted. If LKJ be given the others are wwRst. Similarly to finish with any additional number short of 18. We have been thus explicit on this apparently trivial matter, because we do not know of any other game of skill which is so closely allied to mathematics, and because the analysis employed, though very simple, is more startlingly novel than even that of the quaternions. The i, j, k of quaternions can, as we have seen, be represented by three definite unit lines at right angles to each other. How can we represent geometrically the A or the p of this new calculus, either of which produces precisely the same effect whatever edge of whatever face of the dodecahedron it i. applied to ? Another very elegant invention of Hamilton's, and one which appears to have been suggested to him by his quaternion investigations, is the Hodograph, which supplies a graphic representation of the velocity and acceleration in every case of motion of a particle. The easiest illustration we can give of this is a special case, the hodograph of the earth's motion in its orbit. In consequence of the fact that light moves with a finite, though very great, velocity, its apparent direction when it reaches the eye varies with the motion of the spectator. The position of a star in the heavens appears to be nearer than it really is to the point towards which the earth is moving; in fact, the star seems to be displaced in a direction parallel to that in which the earth is moving, and through a space such as the earth would travel in the time occupied by light in coming from the star. This is the phenomenon detected by Bradley, and known as the aberration of light. Thus the line joining the true place of the star with its apparent place represents at every instant, by its length and direction, the velocity of the earth in its orbit. We are now prepared to give a general definition. The hodograph corresponding to any case whatever of motion of a point is formed by drawing at every instant, from a fixed point, lines representing the velocity of the moving point in magnitude and direction. One of the most singular properties of the hodograph, discovered by Hamilton, is that the hodograph of the orbit of every planet and comet, however excentric its path may be, is a circle. A star, therefore, in consequence of aberration, appears to describe an exact circle surrounding its true place, in a plane parallel to the plane of the ecliptic; not merely, as seems