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It is characteristic of Hamilton that he fancied he saw in the quaternion, with its scalar and vector elements, the one merely numerical, the other having reference to position in space, a realization of the Pythagorean Tetractys

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 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 com

mitted, 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 Hamil

ton.

One very remarkable speculation of Hamilton's is that in which he deduces, by a species of metaphysical or á 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

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παγὰν ἀενάου φύσεως ῥιζώματ ̓ ἔχουσαν,

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 AbraYet there is no cadabra of later times. 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

Studies, even in books, you may judge from my "Little as I have pursued such [physical] 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 out offence to me, consider that I abused the superficial, course of reading. You might, withlicense 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 particular, 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 à priori principles of mine, the Law of the Four Scales, and the Conabout the multiplication of vectors-including ception 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 very à priori (and poetical) sort of man himself, discovery of OERSTED; who, by the way, was a 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

THE TETRACTYS.

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 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

Or high Mathesis, with her charm severe,
Of line and number, was our theme; and we
Sought to behold her unborn progeny,
And thrones reserved in Truth's celestial sphere:
While views, before attained, became more
clear;

And how the One of Time, of Space the Three,
Might, in the Chain of Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echoes of an ancient strain,

seems to have carefully

Some shadowy outlines of old thoughts sub- studied only one particular system, which

lime,

Gently he smiled to see, revived again,
In later age, and occidental clime,
A dimly traced Pythagorean lore,
A westward floating, mystic dream of FOUR.

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 mathematician 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

depends mainly upon two distinct and non-
commutative fifth roots of positive unity,
which, for ease of reference, we will call,
with their inventor, A and μ. 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 dodecahe-
dron (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 count-
ing two for each edge. That is, there are
twenty corners. Now, in that one of Hamil-
ton's systems which he most fully worked
out, the operators A and , 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
in any chosen order. And, as the reader
may easily convince himself by trial, such a
group of twenty operations as this, consist-
ing of the series μ, A, u, y fly fly fly dy λ, λ,
taken twice, brings us back to the edge we
started from, after passing through each

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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) HXWRSTVJKLMNPQZ. If LTSRQ, 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 м, 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 VWRST. 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 μ of this new calculus, either of which produces precisely the same effect whatever edge of whatever face of the dodecahedron it be 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

formerly to have been assumed, an approximate one. But, unless the earth's orbit were exactly circular the true place of the star will not be the centre of the hodograph. To enter into further details on this subject we should require geometrical diagrams or analytical symbols.

of his Lectures are almost painful to the eye.

Hamilton had, at one time, serious intentions of entering the Church, and was, more than once, offered ordination. The following letter, written to the Editor of the Irish Ecclesiastical Journal, and published in that work, contains a very singular attempt to elucidate one of the grandest questions connected with the Christian religion.

"ON THE ASCENSION OF OUR BLESSED LORD. "Whitsun Eve, 1842.

sacred season, turn naturally on that seeming "SIR,-The meditations of a Christian, at this pause in the operations of divine Providence, when, as at this time, the disciples who had seen their Lord parted from them, and taken up into heaven, were waiting at Jerusalem for the promised coming of the Comforter. You will judge whether the following remarks, in part confessedly conjectural, but offered (it is hoped) in no presumptuous spirit, may properly

occupy any portion of your columns, in connexion with the events which the Church at this season commemorates.

The discoveries we have already described, and the papers and treatises we have mentioned, might well have formed the whole work of a long and laborious life. But, not to speak of the enormous collection of Ms. books, full to overflowing with new and origi nal matter, left by Hamilton, which have been handed over to Trinity College, Dublin, and of whose contents we hope a large portion at least may soon be published, the works we have already called attention to barely form the greater portion of what he has published. His extraordinary investigations connected with the solution of algebraic equations of the Fifth Degree, and his exami. nation of the results arrived at by Abel, Jerrard, and Badano, in their researches on this subject, form another grand contribution to science. There is also his great paper on "It may be assumed that your readers are Fluctuating Functions, a subject which, disposed to adopt, in its simplicity, the teaching since the time of Fourier, has been of im- of the 4th article, that Christ did truly rise mense and ever increasing value in physical again from death, and took again his body, with applications of mathematics. Of his extensive flesh, bones, and all things appertaining to the investigations into the solution (especially by ascended into heaven, and there sitteth, until he perfection of Man's nature; wherewith he numerical approximation) of certain classes return to judge all Men at the last day.' They of differential equations, which constantly will not be inclined to explain away the dococcur in the treatment of physical questions, trine of the Ascension of the Lord's Humanity, only a few items have been published, at in- into what some have sought to substitute for tervals, in the Philosophical Magazine. Be-it,--a ceasing of the Godhead to be manifested sides all this, Hamilton was a voluminous correspondent. Often a single letter of his occupied from fifty to a hundred or more closely written pages, all devoted to the minute consideration of every feature of some particular problem; for it was one of the peculiar characteristics of his mind, never to be satisfied with a general understanding of a question, he pursued it until he knew it in all its details. He was ever courteous and kind in answering any applications for assistance in the study of his works, even when his compliance must have cost him much valuable time. He was excessively precise and hard to please, with reference to the final polish of his own works for publication; and it was probably for this reason that he published so little, compared with the extent of his investigations. His peculiar use of capitals, italics, and other typographical artifices for the purpose of imitating in writing and type, as closely as possible, the effects of emphasis and pause in a vivá voce lecture, will be evident from almost any of the extracts we have made from his works. To such an extent did he carry this, that some pages

in the person of Christ. Far rather will they be ready to believe that the 'glorious' Ascension was the epoch of a more bright manifes tation of God in Christ, than any which had been vouchsafed before though perhaps rather to angelic than to human beings; and that no merely figurative, though in part a spiritual sense, is to be assigned to those passages of Holy highly exalted, and seated at the right hand of Writ, which speak of Jesus as having been God. As God, indeed, we know that Heaven, and the Heaven of Heavens, cannot contain him; yet it is also declared that Heaven is His Throne, and Earth is His Footstool: and Scripture and the Church seem to attest alike, that the risen and glorified Humanity of Christ is now in Heaven, as in some holiest place, where worshipped; his power, his name, and his presGod is eminently manifested, and eminently ence dwelling there.

"A local translation of Christ's Body being thus believed, it is natural to believe also that this change of place was accomplished in time, and not with that strict instantaneity which may be attributed to a purely spiritual operation. Accordingly we read that at least the which the Apostles were witnesses,-was gradfirst part of the act of Ascension,-the part of ual; their gaze could follow for a while their ascending Lord: nor was it instantly, though it

may have been soon, that a cloud received him out of their sight. And to suppose that the remainder of that wonderful translation was effected without occupying some additional time, seems almost as much against the truth of Christ's natural Body,' as that it should be at one time in more places than one,' which latter notion a rubric of our Book of Common Prayer rejects as error and absurdity. The Cloud which hovered over Bethany was surely not that Heaven where Jesus sitteth at the right hand of God; and to believe that his arrival, as Man, at the latter, was subsequent to his arrival at the former, seems to be a just as well as an obvious inference, from the Doctrine of the Ascension of His Body.

Jesus take his seat at the right hand of God, than the Spirit fell upon the Apostles. The finished work, of ascending up on high, may have been followed instantly by the receiving of gifts for men.

"Should this conjecture be admitted, of the Ascension not having been completed till the Day of Pentecost, although commenced ten days before, it might suggest much interesting meditation respecting the glory,' the 'great triumph,' with which our Saviour Christ was then exalted into God's Kingdom of Heaven. May not the transit from the Cloud to the Throne have been but one continued passage, in long triumphal pomp, through powers and principalities made subject? May not the Only Begotten Son' have then again been brought forth into the world,— not by a new Nativity, but (as it were) by Proclamation and Investiture,-while the Universe beheld its God, and all the Angels worshipped him? And would not such triumphal progress harmonize well with that Psalm, which has al

"But how long was it subsequent? We dare not, by mere reasoning, attempt to decide this question. That place to which the Saviour has been exalted, and which, although in one sense 'Heaven,' is in another sense declared to be 'far above all heavens,' may well be thought to be inconceivably remote from the whole astro-ways been referred in a special manner to the nomical universe; no eye, no telescope, we may suppose, has pierced the mighty interspace: light may not yet have been able to spread from thence to us, if such an effluence as light be suffered thence to radiate. And, on the other hand, it must be owned, that, vast beyond all thought of ours as the interval in space may be, Christ's glorious Body may have been trans-by ported over it, in any interval of time, however short.

"Reason is silent then: nor can we expect to find, on this point, a clear revelation in Scripture; but do we meet with no indications? Does Holy Writ leave us here entirely without light? I think that it does not: and shall submit to you a view, which it seems to me to suggest.

"First, it is clear from Scripture, that the Ascension of Christ had been entirely performed before the Descent of the Spirit on the Day of Pentecost. Thus, in a well-known verse of that sixty-eighth Psalm, which the Church has connected with the Service for Whitsunday, and which St. Paul has quoted in reference to the Ascension; in the first sermon of Peter to the Jews; and in other passages of the Bible: the obtaining of gifts for men,' the receiving from the Father the promise of the Holy Ghost, is spoken of as a result or consequence of Christ's having ascended up on high,--having been exalted by the right hand of God,-having ascended, as did not David, into the Heavens. The act of ascending occupied therefore no longer time than that from Holy Thursday to Whitsunday.

"But may it not have been allowed to occupy so long a time as this? No reason à priori can be given against the supposition; no passage of Scripture, no decision of the Church, so far as I know, is against it. The very close connexion announced, in the texts above alluded to, between the Ascension of Christ into Heaven, and the Descent of the Holy Ghost upon Earth, appears to me an indication in its favour. For the purely spiritual nature of the later descent prevents the necessity, almost the possibility, of our supposing it to have occupied time at all. No sooner, it may reasonably be thought, did

Ascension, and whieh speaks of the everlasting
Gates as lifting up their heads, that the King of
Glory might come in?

"Many other reflections occur to me, but I forbear. If anything unscriptural or uncatholic shall be detected by you in the foregoing remarks, or (in the event of you publishing them) your readers, the pointing it out will be received as an obligation by, Sir, your obedient servant,

says

"William] R[owan] H[amilton]."

his elder

son,

Like most men of great originalty, Hamilton generally matured his ideas before putting pen to paper. "He used to carry on," and arithmetical calculation in his mind, "long trains of algebraical during which he was unconscious of the earthly necessity of eating: we used to bring in a 'snack' and leave it in his study, but a brief nod of recognition of the intrusion of the chop or cutlet was often the only result, and his thoughts went on soaring upwards. I have been much with him in his periods of mathematical incubation, and would divide them into three, thus:-First, that of contemplation, above indicated. Second, that of construction. In this he committed to paper (or, if nothing else were at hand, as when in the garden, a few formula written on his finger-nails) the skeleton, afterwards results arrived at. Third, the didactic stage. to be clothed with flesh and blood, of the Having now completely satisfied himself of the correctness of the results (and sometimes having retraced and simplified the method of discovery) he proceeded to consider how to teach it, and this by experiment. I was so long with him in his periods of mathematical incubation that I knew, almost by the tones of his voice and the expression of his eyes, when the didactic period had arrived, and generally anticipated it by fetching the black

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