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