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 es sentially 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, yainly 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 imaginary 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 Irish 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 Hamil ton'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 Art, or in some other Science, to which Algebra do or to discover something else, in some other 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 Notawhen a Formula must be written with pretion, or the symmetrical structure of his Syntax; caution, 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 light, 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 improving 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 negative 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 expression harmonize, according to the Laws of Language, with other Algebraical expressions?'

"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

We have seen that the product, and consequently the quotient, of two lines at right angles to each other, is a third line perpendicular to both; and that the product or quotient of two parallel lines is a number: what is the product, or the quotient, of two lines not at right angles, and not parallel, to each other? It is a QUATERNION. This is very easily seen thus. Take the case of the quotient of one line by another, and suppose them drawn from the same point: the first line may, by letting fall a perpendicular from its extremity upon the second, be resolved into two parts, one parallel, the other perpendicular, to the second line. The quotient of the two parallel lines is a mere number, that of the two perpendicular lines is a line, and can therefore be expressed as the sum of multiples of i, j, and k. Hence, w representing the numeral quotient of the parallel portions, the quotient of any two lines may be written as w+zi+yj+xk; and, in this form, is seen to depend essentially upon FOUR perfectly distinct numbers; whence its name. In actually working with quaternions, however, this cumbrous form is not necessary; we may express it as B÷a, B and a being the two lines of which it is the quotient; and in various other equivalent algebraic forms; or we may at once substitute for it a single letter (Hamilton uses the early letters, a, b, c, etc.) The amount of condensation, and consequent shortening, of the work of any particular problem which is thus attained, though of immense importance, is not by any means the only or even the greatest advantage possessed by quaternions over other methods of treating analytical geometry. They render us entirely independent of special lines, axes of co-ordinates, etc., devised for the application of other methods, and take their reference lines in every case from the particular problem to which they are applied. They have thus what Hamilton calls an "internal" character of their own; and give us, without trouble, an insight into each special question, which other methods only yield to a combination of great acuteness with patient labour. In fact, before their invention, no process was known for treating problems in tridimensional space in a thoroughly natural and inartificial manner.

But let him speak for himself. The fol

cess

"Whatever may be the future suc

of Quaternions as an Instrument of Investigation, they furnish already, to those who have learned to read them, (pwvâvra σuveTOLOW,) a powerful ORGAN OF EXPRESSION, especially in geometrical science, and in all that widening field of physical inquiry, to which relations of space (not always easy to express with clearness by the Cartesian Method) are subsidiary, or rather are indispensable."

To follow up the illustration we began with, it may be well merely to mention here that a quaternion may in all cases be represented as a power of a line. When a line is raised to the first, third, or any odd integral power, it represents a right-angled quaternion, or one which contains no pure numerical part; when to the second, fourth, or other even integral power, it degenerates into a mere number; for all other powers it contains four distinct terms. Compare this with the illustration already given, leading to De Moivre's theorem; and we see what a grand step Hamilton supplied by assigning in every case a definite direction to the axis about which the rotation takes place.

There are many other ways in which we can exhibit the essential dependence of the product or quotient of two directed lines (or Vectors as Hamilton calls them), on four numbers (or Scalars), and the consequent fitness of the name Quaternion. It may interest the reader to see another of them. Let us now regard the quaternion as the factor or operator required to change one side of a triangle into another; and let us suppose the process to be performed by turning one of the sides round till it coincides in direction with the other, and then stretching or shortening it till they coincide in length also. For the first operation we must know the axis about which the rotation is to take place, and the angle or amount of rotation. Now the direction of the axis depends on two numbers (in Astronomy they may be Altitude and Azimuth, Right Ascension and Declination, or Latitude and Longitude); the amount of rotation is a third number; and the amount of stretching or shortening in the final operation is the fourth.

Among the many curious results of the invention of quaternions, must be noticed the revival of fluxions, or, at all events, a mode of treating differentials closely allied to that originally introduced by Newton. The really useful, but over-praised differential coefficients, have, as a rule, no meaning