8 Oct. 15, '58. " P. S.–To-morrow will be the fifteenth of such a line (when used as a factor or negative unity, we may interpret the effect birthday of the Quaternions. They started into life, or light, full grown, on ) the 16th of Octo- operator) as a left (or right) handed rotation ber 1843, as I was walking with Lady Hamil- through one right angle about its direction. ton to Dublin, and came up to Brougham The effect of a repetition of the operation is Bridge, which my boys have since called the a rotation through two right angles, or a Quaternion Bridge. That is to say, I then and simple inversion. Thus, if we operate with there felt the galvanic circuit of thought close; i on j we turn the northward line left-handand the sparks which fell from it were the edly through a right angle about an eastward such as I have used them ever since." ' I pulled axis, i.e., we raise it to a direction vertically out, on the spot, a pocket-book which still upward. Thus we see that ij=k. But, if exists, and made an entry, on which, at the very we perform the operation again, we see that moment, I felt that it might be worth my while ik or ij is now the southward line or -j. to expend the labour of at least ten (or it might Thus i=-1. And similarly with the squares be fifteen) years to come. But then, it is fair to of j and k. It is to be noticed that in all say that this was because I felt a problem to these cases the operating line is supposed to have been at that moment solved, -an intellectaal want relieved, which had haunted me for be perpendicular to the operand. Also that at least fifteen years before. we have taken for granted (what is easily “ Less than an hour elasped, before I had proved), that i, j, k may stand indifferently asked and obtained leave, of the Council of the for the unit line themselves or for the operaRoyal Irish Academy, of which Society I was, tion of turning through a right angle. Thus, at that time, the President,—to read at the the equation ij=k may either mean (as next general Meeting, a Paper on Quaternions; above) that i acting on the line j turns it which I accordingly did, on November 13, into the line k; or that the rectangular 1843. “ Some of those early communications of rotation j, succeeded by i, is equivalent to mine to the Academy may still have some the single rotation k. We may easily verify interest for a person like you, who has since so the last assertion by taking i as the operand. well studied my Volume, which was not pub- j changes it to -k, and i changes this to j. lished for ten years afterwards. But k turns i into j at once. “In the meantime, will you not do honour Even these simple ideas lead us at once to to the birthday, to-morrow, in an extra cup of one of the most remarkable properties of -ink? for it may be obsolete now to propose quarternions. When turning the northward XXX,—or even XYZ.” line (j) about the eastward line (i), we wrote We must now endeavour to explain, in as the operator first,—thus ij=k. Now, the popular a manner as possible, the nature of order of multiplication is not indifferent, for the new Calculus. In order to do so, let us ji is not equal to k. ji, in fact, expresses recur to the suggestion of Wallis, before that i (the eastward line) has been made to described, and endeavour to ascertain the rotate left-handedly through a right angle exact nature of its defects. We easily see about į (the northward line). This obthat one great defect is want of symmetry. viously brings it to the downward direction, As before stated, if we take an eastward or we have ji= -k. Similar expressions line, of proper length, to represent positive hold for the other products two and two of unity, an equal westward line represents the three symbols. Thus we have the laws negative unity; but all lines perpendicular of their multiplication complete. And on or inclined to these are represented by so this basis the whole theory may be erected. called imaginary quantities. Hamilton's Now any line whatever may be resolved (as great step was the attainment of the desired velocities and forces are) into so much eastsymmetry by making all lines alike express. ward (or westward), so much northward (or ible by so-called imaginary quantities. Thus, southward), and so much upward (or downinstead of 1 for the eastward line, and the ward). Hence every line may be expressed square root of negative unity for a north- as the sum of three parts, numerical mulward line, he represents every line in space tiples of i, j, and k respectively. Call these whose length is unity by a distinct square numbers x, y, and 2, then the line may be root of negative unity. All are thus equally expressed by si+yj+zk. If we square this imaginary, or rather equally real. The i, j, we find -(x2 +y2 +z2); for the other terms k mentioned in the extract just given, are occur in pairs like xix yj and yj xxi, and three such quantities; which (merely for destroy each other; since we have, as above, illustration, because any other set of three ij+ji=0, with similar results for the other mutually rectangular directions will do as pairs of the three rectangular unit-lines. well) we may take as representing unit lines Now co+ya +22 is the square of the length drawn respectively eastwards, northwards, of the line (by a double application of and upwards. The square of each being Euclid 1. xlvii.) Hence the square of every cess line of unit length is negative unity. And lowing passage is extracted from a letter to herein consists the grand symmetry and con a mathematical friend, who was at the time sequent simplicity of the method; for we engaged in studying the new calculus :may now write a single symbol such as a (Greek letters are usually employed by " Whatever may be the future sucHamilton in this sense), instead of the much of Quaternions as an Instrument more cumbrous and not more expressive form of Investigation, they furnish already, to those who have learned to read them, (pwvavra ouvewi+yi +zk. We have seen that the product, and con TOLOL,) a powerful ORGAN OF EXPRESSION, especially in geometrical science, and in all that sequently the quotient, of two lines at right widening field of physical inquiry, to which angles to each other, is a third line perpen relations of space (not always easy to express dicular to both; and that the product or with clearness by the Cartesian Method) are quotient of two parallel lines is a number: subsidiary, or rather are indispensable." what is the product, or the quotient, of two lines not at right angles, and not parallel, to To follow up the illustration we began each other? It is a QUATERNION. This is with, it may be well merely to mention here very easily seen thus. Take the case of the that a quaternion may in all cases be reprequotient of one line by another, and suppose sented as a power of a line. When a line them drawn from the same point: the first is raised to the first, third, or any odd in. line may, by letting fall a perpendicular tegral power, it represents a right-angled from its extremity upon the second, be quaternion, or one which contains no pure resolved into two parts, one parallel, the numerical part; when to the second, fourth, other perpendicular, to the second line. The or other even integral power, it degenerates quotient of the two parallel lines is a mere into a mere number; for all other powers it number, that of the two perpendicular lines contains four distinct terms. Compare this is a line, and can therefore be expressed as with the illustration already given, lead. the sum of multiples of i, j, and k. Hence, ing to De Moivre's theorem ; and we see w representing the numeral quotient of the what a grand step Hamilton supplied parallel portions, the quotient of any two by assigning in every case a definite direcIines may be written as w+ci+yj+xk; tion to the axis about which the rotation and, in this form, is seen to depend essentially takes place. upon FOUR perfectly distinct numbers; There are many other ways in which we whence its name. In actually working with can exhibit the essential dependence of the quaternions, however, this cumbrous form is product or quotient of two directed lines not necessary; we may express it as B-a, ß (or Vectors as Hamilton calls them), on four and a being the two lines of which it is the numbers (or Scalars), and the consequent quotient; and in various other equivalent fitness of the name "Quaternion. It may algebraic forms; or we may at once sub. interest the reader to see another of them. stitute for it a single letter (Hamilton uses Let us now regard the quaternion as the the early letters, a, b, c, etc.) The amount factor or operator required to change one of condensation, and consequent shortening, side of a triangle into another; and let us of the work of any particular problem which suppose the process to be performed by turn. is thus attained, though of immense im- ing one of the sides round till it coincides portance, is not by any means the only or in direction with the other, and then stretcheven the greatest advantage possessed by ing or shortening it till they coincide in quaternions over other methods of treating length also. For the first operation we must analytical geometry. They render us en know the axis about which the rotation is to tirely independent of special lines, axes of take place, and the angle or amount of rotaco-ordinates, etc., devised for the application tion. Now the direction of the axis depends of other methods, and take their reference on two numbers (in Astronomy they may be lines in every case from the particular Altitude and Azimuth, Right Ascension and problem to which they are applied. They Declination, or Latitude and Longitude); have thus what Hamilton calls an “internal the amount of rotation is a third number; character of their own; and give us, without and the amount of stretching or shortening trouble, an insight into each special question, in the final operation is the fourth. which other methods only yield to a com Among the many curious results of the bination of great acuteness with patient invention of quaternions, must be noticed labour. In fact, before their invention, no the revival of Auxions, or, at all events, a process was known for treating problems in mode of treating differentials closely allied tridimensional space in a thoroughly natural to that originally introduced by Newton. and inartificial manner. The really useful, but over-praised differenBut let him speak for himself. The fol- tial coefficients, have, as a rule, no meaning in quaternions; so that, except when dealing must be perpendicular to each of the factors. with scalar variables (which are simply It is easy to carry this further, but enough degraded quaternions), we must employ in has been said to show the character of the differentiation fluxions or differentials. And reasoning. the reader may easily understand the cause It is characteristic of Hamilton that he of this. It lies in the fact that quaternion fancied he saw in the quaternion, with its multiplication is not commutative; so that, scalar and vector elements, the one merely in differentiating a product, for instance, numerical, the other having reference to each factor must be differentiated where it position in space, a realization of the Pythastands; and thus the differential of such a gorean Tetractys product is not generally a mere algebraic multiple of the differential of the inde παγάν αενάου φύσεως ριζώματ’ έχουσαν, pendent quaternion-variable. It is thus that the whirligig of time brings its revenges. as it is called in the Carmen Aureum. The shameless theft which Leibnitz com Of course, so far as mere derivation goes, mitted, and which he sought to disguise by it is hard to see any difference between the altering the appearance of the stolen goods, Tetractys and the Quaternion. But we are must soon be obvious, even to his warmest almost entirely ignorant of the meaning partisans. They can no longer pretend to Pythagoras attached to his mystic idea, and regard Leibnitz as even a second inventor it certainly must have been excessively when they find that his only possible claim, vague, if not quite so senseless as the Abrathat of devising an improvement in notation, cadabra of later times. Yet there is no merely unfits Newton's method of fluxions doubt that Hamilton was convinced that for application to the simple and symmetri- Quaternions, in virtue of some process cal, yet massive, space-geometry of Hamil- analogous to the quasi-metaphysical specuton. lation we have just sketched, are calculated One very remarkable speculation of Ham- to lead to important discoveries in physical ilton's is that in which he deduces, by a science; and, in fact, he writesspecies of metaphysical or á priori reasoning, the results previously mentioned, viz. Studies, even in books, you may judge from my “Little as I have pursued such [physical] that the product (or quotient) of two parallel Presidential Addresses, pronounced on the occavectors must be a number, and that of two sions of delivering Medals (long ago), from the mutually perpendicular vectors a third per- chair of the R.I.X., to Apjohn and to Kane, pendicular to both. We cannot give his that physical (as distinguished from mathereasoning at full length, but will try to make matical) investigations have not been wholly part of it easily intelligible. alien to my somewhat wide, but doubtless very Suppose that there is no direction in space out offence to me, consider that I abused the superficial, course of reading. You might, withpre-eminent, and that the product of two license of hope, which may be indulged to an vectors is something which has quantity, so inventor, if I were to confess that I expect the as to vary in amount if the factors are Quaternions to supply hereafter, not merely changed, and to have its sign changed if mathematical methods, but also physical sugthat of one of them is reversed; if the gestions. And, in particular, you are quite vectors be parallel, their product cannot be, welcome to smile, if I say that it does not seem in whole or in part, a vector inclined to extravagant to me to suppose that a full posthem, for there is nothing to determine the session of those à priori principles of mine, direction in which it must lie. It cannot be the Law of the Four Scales, and the Con about the multiplication of vectors—including a vector parallel to them; for by changing ception of the Extra-spatial Unit,—which have the sign of both factors the product is un as yet not been much more than hinted to the changed, whereas, as the whole system has public—might have led (I do not at all mean been reversed, the product vector ought to that in my hands they ever would have done so,) have been reversed. Hence it must be a to an ANTICIPATION of something like the grand number. discovery of OERSTED; who, by the way, was a pendicular vectors cannot be wholly or partly very à priori (and poetical) sort of man himself , a number, because on inverting one of them received from him some printed pamphlets, the sign of that number ought to change ; several years ago. It is impossible to estimate but inverting one of them is simply equiva- the chances given, or opened up, by any new lent to a rotation through two right angles way of looking at things; especially when that about the other, and from the symmetry of way admits of being intimately combined space ought to leave the number unchanged. with calculation of a most rigorous kind.” Hence the product of two perpendicular vectors must be a vector, and an easy ex. This idea is still further developed in the tension of the same reasoning shows that it following sonnet, which gives besides a good VOL. XLV. N-3 idea of his powers of poetical composition. quaternions, but more numerous, and gave It is understood to refer to Sir John Her- vaious applications of them. These groups schel, who had, at a meeting of the British have, generally, a direct connexion with the Association, compared the Quaternion Cal. “Sets” with which he was occupied just beculus to a Cornucopia, from which, turn it fore the invention of the quaternions : and as you will, something new and valuable it would be vain to attempt to explain their must escape. nature to the general reader. But we must say a few words about another, and most THE TETRACTYS. extraordinary, system which Hamilton seems to have invented about 1856, and which has Or high Mathesis, with her charm severe, no connexion whatever with any previous Of line and number, was our theme; and we Sought to behold her unborn progeny, group. Unfortunately, Hamilton has pubAnd thrones reserved in Truth's celestial sphere: | lished but a page or two with reference to While views, before attained, became more them, yet that little is enough to show the clear ; probability of their becoming, at some future And how the One of Time, of Space the Three, time, of great importance in the study of Might, in the Chain of Symbol, girdled be: crystals and and polyhedra in general. The And when my eager and reverted ear subject is capable of indefinite estension ; Caught some faint echoes of an ancient strain, but Hamilton seems to have carefully Some shadowy outlines of old thoughts sublime, studied only one particular system, which Gently he smiled to see, revived again, depends mainly upon two distinct and nonIn later age, and occidental clime, commutative fifth roots of positive unity, A dimly traced Pythagorean lore, which, for ease of reference, we will call, A westward floating, mystic dreard of Four. with their inventor, I and M. Although nothing more practical than an ingenious Whatever may be the future of Quaterni- “puzzle” has yet resulted from these inons, and it may possibly far surpass all that vestigations, their singular originality and its inventor ever dared to hope, there can be | (if we may use the word) oddity, and the but one opinion of the extraordinary genius, wonderful series of new transformations and the untiring energy of him who, unaided, | which they suggest to the mathematician, composed in so short a time two such enor render them well worthy of further study mous treatises as the Lectures (1853), and and development. Some idea of a small the Elements of Quaternions (1866). As a class of their properties may be derived from repertory of mathematical facts, and a tri- the consideration of a pentagonal dodecaheumph of analytical and geometrical power, dron (a solid enclosed by twelve faces, each they can be compared only with such im- of which has five sides). The number of perishable works as the Principia and the edges of this solid is thirty; as we may see Mécanique Analytique. They cannot be by remarking that, if we count five edges said to be adapted to the wants of elemen- for each of the twelve faces, each edge will tary teaching, but we are convinced that have been taken twice. Also, since three every one who has a real liking for mathe- edges meet in each corner, and since each matics, and who can get over the preliminary edge passes through two corners, we shall difficulties, will persevere till he finishes the get three times too many corners by countwork, whichever of the two it may be, he ing two for each edge. That is, there are has commenced. They have all that exqui- twenty corners. Now, in that one of Hamilsite charm of combined beauty, power, and ton's systems which he most fully worked originality which made Hamilton compare out, the operators 1 and M, applied to any Lagrange's great work to a “scientific edge of the pentagonal dodecahedron, change poem.” And they conduct the mathemati. it into one of the adjoining edges. Thus, cian to a boundless expanse of new territory going along an edge, to a corner, we meet of the richest promise, the cultivation of two new edges, that to the right is derived which cannot be said to have been more than from the first by the operator 1, that to the commenced, even by labour so unremitting, left by p. Every possible way of moving and genius so grand, as Hamilton brought | along successive edges of such a solid may to bear on it. therefore be symbolized by performing on the first edge the successive operations 1 and je The unit vectors of the quaternion calcu. in any chosen order. And, as the reader lus are not the only roots of unity which may easily convince himself by trial, such a Hamilton introduced into practical analysis. group of twenty operations as this, consistIn various articles in the Philosophical ing of the series u, l, m, n, M, M, M, d, n, , Magazino he developed the properties of taken twice, brings us back to the edge we groups of symbols analogous to the i, j, k of started from, after passing through each corner once, and only once. Such results as being given, finish with a given number of ysis employed, though very simple, is more 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 by projection (the eye being supposed to be give of this is a special case, the hodograph placed very near to the middle of one face), of the earth's motion in its orbit. In conin order to prevent any two of the lines sequence of the fact that light moves with a which represent the edges from crossing each finite, though very great, velocity, its appaother. The game is played by inserting rent direction when it reaches the eye varies pegs, numbered 1, 2, 3, ... 20, in success with the motion of the spectator. The posisive holes, which are cut at the points of the tion of a star in the heavens appears to be figure representing the corners of the dodec- nearer than it really is to the point towards ahedron; taking care to pass only along which the earth is moving; in fact, the star the lines which represent the edges. It is seems to be displaced in a direction parallel characteristic of Hamilton that he has se- to that in which the earth is moving, and lected the 20 consonants of our alphabet to through a space such as the earth would denote these holes. travel in the time occupied by light in comWhen five pegs are placed in any five suc- ing from the star. This is the phenomenon 'essive holes, it is always possible in two detected by Bradley, and known as the aber. ways, sometimes in four, to insert the whole ration of light. Thus the line joining the twenty, so as to form a continuous circuit. true place of the star with its apparent place Thus, let BCDFG be the given five, we may represents at every instant, by its length and complete the series by following the order direction, the velocity of the earth in its of the consonants; or we may take the fol- orbit. We are now prepared to give a genlowing order (after G) AXWRSTVJKLMNPQZ. eral definition. The hodograph correspondIf LTSRQ, or ZBCDM, be given there are four | ing to any case whatever of motion of a solutions. If fewer than five be fixed at point is formed by drawing at every instarting there are, of course, more solutions. I stant, from a fixed point, lines representing This is only the simplest case of the game. the velocity of the moving point in magniPuzzles without number, and of a far higher tude and direction. One of the most singuorder of difficulty, can be easily suggested | lar properties of the hodograph, discovered after a little practice, but even more readily by Hamilton, is that the hodograph of the by the proper mathematical processes. orbit of every planet and comet, however Thus, BCD may be given, the problem being excentric its path may be, is a circle. A to insert all the pegs in order, and end at a star, therefore, in consequence of aberration, given hole. If that hole be m, it is impossi- appears to describe an exact circle surroundble; if t, there is one solution; if J, two; ing its true place, in a plane parallel to the |