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It is strange, indeed, that the one particu- | again form a cone. Thus the prediction was, lar result of this theory, which, perhaps more that in a plate formed of a biaxal crystal, a than anything else that Hamilton has done, single ray, incident in a certain direction, has rendered his name known beyond the little would emerge as a hollow cylinder of light; world of true philosophers, should have been and that light, forced to pass through such a easily within the reach of Fresnel and others plate in a certain direction, would enter and for many years before ; and in no way requir. emerge as a hollow cone. ed Hamilton's new conceptions or methods, al. These two phenomena are deducible at though it was by them that he was led to once from the form of the Wave Surface its discovery. This singular result is still (as it is called) in biaxal crystals, long beknown by the name Conical Refraction, fore assigned by Fresnel ; but no one seems which he proposed for it when he first pre- to have anticipated Hamilton in closely dicted its existence in the third Supplement studying the form of that surface from its to his Systems of Rays, read in 1832. To equation, certainly not in recognizing the give the reader an idea of its nature, let us fact that it possesses four conical cusps, and, suppose light from a brilliant point to fall also, that it has four tangent planes, each of on a plate of glass, or other singly refracting which touches it, not in one point, but in an body, the side next the light being covered infinite number of points forming a circle. by a plate of metal with a very small hole The reader may get a rough idea of such in it. A single ray will thus be admitted properties by thinking of the portion of an into the glass, will be refracted in the ordi- apple which is nearest to the stalk. nary way, and will escape from the plate as But, besides these very remarkable results a single ray parallel to the direction of in- which Hamilton showed must be obtained by cidence. Try the same experiment with a proper experimental methods, he predicted slice of Iceland-spar, or other doubly re-others of, perhaps, still more decisive charfracting crystal. În general, the single acter, with reference to the polarization of incident ray will be split into two, which the light of the cone and cylinder above de. will pursue separate paths in the crystal, but scribed. All these results of theory were will emerge parallel to each other and to the experimentally verified, at Hamilton's reincident ray. But if a plate of a biaxal quest, in 1833, by Dr. Lloyd, the substance crystal be used, Hamilton showed that there employed being a plate of arragonite. are two directions in which if the incident The step from Optics to Dynamics, in the ray fall it will be divided in the crystal, not application of the method of Varying Acinto two, but into an infinite number of rays, tion, was made in 1827, and communicated forming a hollow cone. Each of these rays to the Royal Society, in whose Philosophiemerges parallel to the incident ray, so that cal Transactions for 1834 aud 1835 there they form on emergence a hollow cylinder are two papers on the subject. These disof light.
play, like the “Systems of Rays," a mas. But, further, suppose the same three sub- tery over symbols, and a flow of mathematistances to be experimented on as follows: cal language (if the expression can be used) place on each side of the plate a leaf of tin- almost unequalled. But they contain, what foil, in which a very small hole is pierced, is far more valuable still, the greatest addiand expose the whole to light, proceeding, tion which Dynamical Science has received not from a point, but from a large surface. since the grand strides made by Newton and The particular ray which passes in glass, Lagrange. Jacobi and other mathematiand other singly refracting bodies, from hole cians have developed to a great extent, and to hole through the plate, comes from one as a question of pure mathematics only, definite point of the luminous body and Hamilton's processes, and have thus made emerges from the second hole as a single extensive additions to our knowledge of Difray. In uniaxal crystals, and generally in ferential Equations. But there can be litbiaxal crystals, two definite and distinct rays | tle doubt that we have as yet obtained only from the luminary are so refracted as to pass a mere glimpse of the vast physical results from hole to hole; and therefore, at emer- of which they contain the germ. And gence, as each passes out parallel to its di- though this, of course, is by far the more rection at incidence, we have two emergent valuable aspect in which any such contriburays. But Hamilton showed that there are tion to science can be looked at, the other two directions in every biaxal crystal, such must not be despised. It is characteristic that if the line between the holes be made of most of Hamilton's, as of nearly all great to coincide with either, the light which passes discoveries, that even their indirect consefrom hole to hole will belong to an infinite quences are of high value. number of different incident rays, forming a After the remarks we have made on the cone. On emergence, they will of course Optical Paper, we may dismiss the Dynami
cal ones very briefly; for the reader who has trically imaginable, but dynamically impossible followed the illustration we gave of an ele- motion, be made to differ infinitely little from mentary case of the former, will easily un
the actual manner of motion of the system, bederstand its bearing on the latter; and, if
tween the given extreme positions, then the
varied value of the definite integral called acthe Optical example be not understood, we
tion, or the accumulated living force of the syscannot find a Dynamical one which can be tem' in the motion thus imagined, will differ presented with any more chance of being in- infinitely less from the actual value of that intelligible to him. We will merely quote | tegral. But when this well-known law of some of Hamilton's own remarks, inserting least, or as it might be called, of stationary (in square brackets), a few hints to help the
action, is applied to the determination of the
actual motion of a system, it serves only to reader :
form, by the rules of the calculus of variations,
the differential equations of motion of the se"In the solar system, when we consider only
cond order, which can always be otherwise the mutual attractions of the sun and of the
found. It seems, therefore, to be with reason ten known planets, the determination of the
that LAGRANGE, LAPLACE, and Poisson have motions of the latter about the former is reduced, by the usual methods, to the integration
spoken lightly of the utility of this principle
in the present state of dynamics. A different of a system of thirty ordinary differential equations of the second order, between the co-ordi
estimate, perhaps, will be formed of that other
principle which has been introduced in the nates and the time; or, by a transformation of LAGRANGE, to the integration of a system of
present paper, under the name of the law of
varying action, in which we pass from an acsixty ordinary differential equations of the first
tual motion to another motion dynamically posorder, between the time and the elliptic ele
sible, by varying the extreme positions of the ments; by which integrations, the thirty vary
system, and (in general) the quantity H, and ing co-ordinates, or the sixty varying elements, are to be found as functions of the time. In
which serves to express, by means of a single
| function, not the mere differential equations of the method of the present essay, this problem
motion, but their intermediate and their final is reduced to the search and differentiation of a 1 : single function, which satisfies two partial dif.
integrals." ferential equations of the first order and of the These extracts give a very good idea, not second degree : and every other dynamical pro- l only of the method itself, but of Hamilton's blem, respecting the motions of any system, however numerous, of attracting or repelling
own opinion of it, though certain phrases points, (even if we suppose those points re
employed may reasonably be objected to. stricted by any conditions of connexion consistent with the law of living force,) is reduced, To give the popular reader an idea of the in like manner, to the study of one central nature of the Quaternions, and the steps by function, of which the form marks out and which Hamilton was, during some fifteen characterises the properties of the moving sys
years, .gradually conducted to their inventem, and is to be determined by a pair of par
tion, it is necessary to refer to the history tial differential equations of the first order, combined with some simple considerations.
of a singular question in algebra and analy. The difficulty is therefore at least transferred | tical geometry, the representation or interfrom the integration of many equations of one pretation of negative and imaginary (or class to the integration of two of another; and impossible) quantities. even if it should be thought that no practical Descartes' analytical geometry and allied facility is gained, yet an intellectual pleasure methods easily gave the representation of a may result from the reduction of the most complex and, probably, of all researches respecting
negative quantity. For it was seen at once the forces and motions of body, to the study of
to be a useful convention, and consistent one characteristic function, the unfolding of one
with all the fundamental laws of the subject, central relation.
to interpret a negative quantity as a quan“ Although LAGRANGE and others, in treating tity measured in the opposite direction to of the motion of a system, have shown that the that in which positives of the same kind are variation of this definite integral [the Action measured. Thus a negative amount of elevaof the system) vanishes when the extreme co
tion is equivalent to depth, negative gain is ordinates and the constant H [the initial energy] ||
| loss, a negative push is a pull, and so on. are given, they appear to have deduced from this result only the well-known law of least | And no error; but rather great gain in como action ; namely, that if the points or bodies of pleteness and generality, results from the a system be imagined to move from a given set employment of this convention in algebra, of initial to a given set of final positions, not trigonometry, geometry, and dynamics. as they do nor even as they could move consis- But it is not precisely from this point of tently with the general dynamical laws or dif- 1 view that we can readily see our way to the ferential equations of motion, but so as not to
interpretation of impossible quantities. Such violato any supposed geometrical connexions, nor that one dynamical relation between velo
quantities arise thus: If a positive quantity cities and configurations which constitutes the be squared, the result is positive; and the law of living force; and if, besides, this geome- / same is true of a negative quantity. Hence,
1 when we come to perform the inverse opera- this proposition leads at once (but not by the tion, i. e., extract the square root, we do not route pursued by its discoverer) to what is at once see what is to be done when the called De Moivre's Theorem, one of the most quantity to be operated on is negative. valuable propositions in plane trigonometry. When it is positive, its square root may be Warren, Argand, Grassmann, and various either a negative or a positive number, as others, especially in the present century, we have just seen. If positive, it is to be vainly attempted to extend this process to measured off in some definite direction, if space of three dimensions. The discovery negative, in the opposite. But how shall was reserved for Hamilton, but was not atwe proceed to lay off the square root of a tained even by him till after fifteen or twenty negative quantity ? Wallis, in the end of years of arduous work. And it is a curious the sixteenth century, suggested that this fact that it was by speculations totally unmight be done by going out of the line connected with geometry that he was so preon which the result, when real, would have pared as to see, almost at the instant of been laid down; and his method is equiva. seizing it, the full value of his invention. lent to this :-Positive unity being repre- The frightful complexity of the results to sented by an eastward line, negative unity which Warren was led in endeavouring to will of course be represented by an equal express as lines the products and quotients westward line, and these are the two square of directed lines in one plane, seems to have roots of positive unity. According to Wallis' induced Hamilton to seek for a representasuggestion a northward and a southward tion of imaginary quantities altogether indeline may now be taken to represent the two pendent of geometry. The results of some square roots of negative unity, or the so- at least of his investigations are given in a called impossibles or imaginaries of algebra. very curious essay, Algebra as the Science But the defect of this is that we might have of pure Time, communicated to the Royal assumed with equal reason any other line Irish Academy in 1833, and published, along (perpendicular to the eastward one) as that with later developments, in the seventeenth on which the imaginary quantities are to be volume of their Transactions.
We quote represented. In fact Wallis' process is es considerable portions of the introductory resentially limited to plane problems, and has marks prefaced to this Essay, as they show, no application to tridimensional space. But, in a very distinct manner, the logical charimperfect as this step is, it led at once to acter and the comprehensive grasp of Hamilanother of great importance, the considera- ton's mind. tion of the length, and direction, of a line independently of one another.
And we now
“The Study of Algebra may be pursued in see that as the factor negative unity simply three very different schools, the Practical
, the reverses a line, while the square root of Philological, or the Theoretical
, according as negative unity (employed as a factor) turns Language, or a Contemplation; according as
Algebra itself is accounted an Instrument, or a it through a right angle, the one operation case of operation, or symmetry of expression, may be looked upon as in a certain sense a or clearness of thought, (the agere, the fari, or duplication of the other. In other words, the sapere,) is eminently prized and sought for. twice turning through a right angle, about The Practical person seeks a Rule which he the same axis, is equivalent to a reversal; may apply, the Philological person seeks a Foror, negative unity, being taken to imply re
mula which he may write, the Theoretical versal of direction, may be considered as
person seeks a Theorem on which he may medi
tate. The felt imperfections of Algebra are of rotation through two right angles, and its three answering kinds. The Practical Algesquare root (the ordinary imaginary or im- braist complains of imperfection when he finds possible quantity) may thus be represented his instrument limited in power; when a rule, as the agent which effects a certain quad- which he could happily apply to many cases, rantal rotation. But, as before remarked, can be hardly or not at all applied by him to the axis of this rotation is indeterminate; it some new case ; when it fails to enable him to
do or to discover something else, in some other may have any direction whatever
perpendicular to the positive unit line. If we fix with him was but subordinate, and for the sake
Art, or in some other Science, to which Algebra on a particular direction, everything becomes of which and not for its own sake, he studied definite, and we can on the same plan inter- Algebra. The Philological Algebraist compret the (imaginary) cube roots of negative plains of imperfection, when his Language preunity as factors or operators which turn a sents him with an Anomaly ; when he finds an line through an angle of sixty degrees posi- Exception disturb the simplicity of his Notatively or negatively. Similarly, any power when a Formula must be written with pre
tion, or the symmetrical structure of his Syntax; of negative unity, positive or negative, whole caution, and a Symbolism is not universal. The or fractional, obtains an immediate repre. Theoretical Algebraist complains of imperfecsentation. And the general statement of tion, when the clearness of his Contemplation
is obscured; when the Reasonings of his Science | as set forth by Euclid in his Elements, two seem anywhere to oppose each other, or become thousand years ago; though he may well dein any part too complex or too little valid for sire to see them treated in a clearer and better his belief to rest firmly upon them; or when, method. The doctrine involves no obscurity though trial may bave taught him that a rule is nor confusion of thought, and leaves in the useful, or that a forinula gives true results, he mind no reasonable ground for doubt, although cannot prove that rule, nor understand that ingenuity may usefully be exercised in imformula: when he cannot rise to intuition from proving the plan of the argument. But it reinduction, or cannot look beyond the signs to quires no peculiar scepticism to doubt, or even the things signified.
to disbelieve, the doctrine of Negatives and “It is not here asserted that every or any Imaginaries, when set forth (as it has commonly Algebraist belongs exclusively to any one of these been) with principles like these : that a greater three schools, so as to be only Practical, or magnitude may be subtracted from a less, and only Philological, or only Theoretical. Lan- that the remainder is less than nothing ; that guage and Thought react, and Theory and Prac- two negative numbers, or numbers denoting tice help each other. No man can be so merely magnitudes, each less than nothing, may be practical as to use frequently the rules of Al- multiplied the one by the other, and that the gebra, and never to admire the beauty of the product will be a positive number, or a number language which expresses those rules, nor care noting a ma nitude greater than nothing; to know the reasoning which deduces them. and that although the square of a number, or No man can be so merely philological an Alge- the product obtained by multiplying that numbraist but that things or thoughts will at some ber by itself, is therefore always positive, times intrude upon signs; and occupied as he whether the number be positive or negative, may babitually be with the logical building up yet that numbers, called imaginary, can be of his expressions, he will feel sometimes a de- found or conceived or determined, and operated sire to know what they mean, or to apply them. on by all the rules of positive and negative And no man can be so merely Theoretical, or numbers, as if they were subject to those rules, so exclusively devoted to thoughts, and to the although they have negative squares, and must contemplation of theorems in Algebra, as not therefore be supposed to be themselves neither to feel an interest in its notation and language, positive nor negative, nor yet null numbers, so its symmetrical system of signs, and the logical that the magnitudes which they are supposed forms of their combinations; or not prize those to denote can neither be greater than nothing, practical aids, and especially those methods of nor less than nothing, nor even equal to nothresearch, which the discoveries and contem- ing. It must be hard to found a SCIENCE on plations of Algebra have given to other sciences. such grounds as these, though the forms of But, distinguishing without dividing, it is per- logic may build up from them a symmetrical syshaps correct to say that every Algebraical tem of expressions, and a practical art may be Student and every Algebraical Composition learned of rightly applying useful rules which may be referred upon the whole to one or other seem to depend upon them. of these three schools, according as one or other “So useful are those rules, so symmetrical of these three views habitually actuates the those expressions, and yet so unsatisfactory man, and eminently marks the work.
those principles from which they are supposed “These remarks have been premised, that to be derived, that a growing tendency may be the reader may more easily and distinctly per- perceived to the rejection of that view which ceive what the design of the following commu- regarded Algebra as a SCIENCE, in some sense nication is, and what the Author hopes or at analogous to Geometry, and to the adoption of least desires to accomplish. That design is one or other of those two different views, which Theoretical, in the sense already explained, as regard Algebra as an Art, or as a Language: as distinguished from what is Practical on the one a System of Rules, or else as a System of Exhand, and from what is Philological on the pressions, but not as a System of Truths, or other. The thing aimed at, is to improve the Results having any other validity than what they Science, not the Art, nor the Language of Al may derive from their practical usefuluess, or gebra. The imperfections sought to be re- their logical or philological coherence. Opinions moved, are confusions of thought, and ob- thus are tending to substitute for the Theoretical scurities or errors of reasoning; not difficulties question, — Is à Theorem of Algebra true?' of application of an instrument, nor failures of the Practical question,— Can it be applied as symmetry in expression. And that confusions an Instrument, to do or to discover something of thought and errors of reasoning, still darken else, in some research which is not Algebraical ?' the beginnings of Algebra, is the earnest and or else the Philological question,-'Does its exjust complaint of sober and thoughtful men, pression harmonize, according to the Laws of who in a spirit of love and honour have studied Language, with other Algebraical expressions ?' Algebraic Science, admiring, extending, and “Yet a natural regret might be felt, if such applying what has been already brought to were the destiny of Algebra; if a study, which light, and feeling all the beauty and consistence is continually engaging mathematicians more of many a remote deduction, from principles and more, and has almost superseded the Study which yet remain obscure, and doubtful. of Geometrical Science, were found at last to
"For it has not fared with the principles of be not, in any strict and proper sense, the Study Algebra as with the principles of Geometry. of a Science at all: and if, in thus exchanging No candid and intelligent person can doubt the the ancient for the modern Mathesis, there truth of the chief properties of Parallel Lines, were a gain only of Skill or Elegance, at the
expense of Contemplation and Intuition. Indul-real interpretation for the so-called imagin. gence, therefore, may be hoped for, by any one ary quantity, but it cannot be called simple, who would inquire, whether existing Algebra, nor is it at all adapted for elementary instrucin the state to which it has been already unfolded by the masters of its rules and of its tion. The reader will observe that Hamillanguage, offers indeed no rudiment which may ton, with his characteristic sagacity, has encourage a hope of developing a SCIENCE of chosen a form of interpretation which admits Algebra: a Science properly so called; strict, of no indeterminateness. Unlike Wallis and pure, and independent; deduced by valid reason- others, who strove to express ordinary algeings from its own intuitive principles; and thus braic imaginaries by directions in space, not less an object of priori contemplation than Hamilton gave his illustration by time or Geometry, nor less distinct, in its own essence, progression, which admits, so to speak, of from the Signs by which it may express its but one dimension. We may attempt to meaning.
give a rough explanation of his process, for “The Author of this paper has been led to the reader who is not familiar with algebraic the belief, that the Intuition of Time is such a signs, in some such way as this :-If an rudiment. This belief involves the three follow- officer and a private be set upon by thieves, ing as components: First, that the notion of and both be plundered of all they have, this Time is connected with existing Algebra ; Second, that this notion or intuition of Time operation may be represented by negative
unity. And the imaginary quantity of may be unfolded into an independent Pure Science; and Third, that the science of Pure algebra, or the square root of negative unity, Time, thus unfolded, is co-extensive and identi- will then be represented by a process which cal with Algebra, só far as Algebra itself is a would rob the private only, but at the same Science. The first component judgment is the time exchange the ranks of the two soldiers. result of an induction; the second of a deduc- It is obvious that on a repetition of this tion; the third is a joint result of the deductive and inductive processes."
process both would be robbed, while they
would each be left with the same rank as at It would not be easy, in our limited space, first. But what is most essential for remark and without using algebraic symbols freely, here is that the operation corresponding to to give the reader more than a very vague the so-called imaginary of algebra is throughidea of the nature of this Essay. What we out regarded as perfectly real. are most concerned with at present is the In 1835 Hamilton seems to have extended bearing of its processes upon the interpreta- the above theory from Couples to Triplets, tion of imaginary quantities, and even on and even to a general theory of Sets, each that we can only say a few words. The step containing an assigned number of time-steps. in time from one definite moment to another Many of his results are extremely remarkdepends, as is easily seen, solely on the able, as may be gathered from the only pubrelative position in time of the two moments, lished account of them, a brief notice in the not on the absolute date of either. And, in Preface to his Lectures on Quaternions. comparing one such step with another, there After having alluded to them, he proceeds: can be difference only in duration and direc- “ There was, however, a motive which in. tion, --i.e., one step may be longer or shorter duced me then to attach a special importance than the other, and the two may be in the to the consideration of triplets. ... This same or in opposite directions, progressive was the desire to connect, in some new and or retrograde. Here numerical factors, posi- useful (or at least interesting) way, calculative and negative, come in. But to intro- tion with geometry, through some undisduce something analogous to the imaginary covered extension, to space of three dimenof algebra, Hamilton had to compare with sions, of a method of construction or repreeach other, not two, but two pairs or Couples sentation which had been employed with of, steps. Thus, a and b representing steps success by Mr. Warren (and indeed also by in time, (a, b) is called a Couple; and its other authors
, of whose writings I had not value depends on the order as well as the then heard), for operations on right lines in magnitude of its constituent steps. It is one plane : which method had given a species shown that (-a,-b) is the same couple taken of geometrical interpretation to the usual negatively. And the imaginary of common and well-known imaginary symbol of algealgebra is now represented by that operation bra." After many attempts, most of which on a step-couple which changes the sign (or launched him, like his predecessors and conorder of progression) of the second step of temporaries, into a maze of expressions of the couple, and makes the steps change fearful complexity, he suddenly lit upon a places. That is, it is the factor or operator system of extreme simplicity and elegance. which changes (a, b) into (-b,a): for a second The following remarkably interesting extract application will obviously produce the result from a letter gives his own account of the (-a,-6). There is, no doubt, here a perfectly discovery :