analogy would lead us to make p= V-1=KB. it, which can only be so long as the plane of Thus the expression for BKP will be (V-1)-1, the angle + B-1 bisects the angle of the We will not further enlarge on these sugges- planes + and -B. Now these planes cointions, and observe only in closing that the cide; therefore the plane of BV-1 is perpenexpressions a, b, abe, which designate lines dicular to the plane ay. Conversely, since considered in reference to one, two and three every plane perpendicular to ry bisects the dimensions, are only the first terms of a series angles between the planes of the positive and which can be indefinitely extended. negative angles, every angle B, in such a plane, may be considered a mean proportional in magnitude and position between + and -6; hence its value, in respect to both magni If the above ideas are admissable, the ques tion so often raised, as to whether every function can be reduced to the form p+q-1, would be answered in the negative; and KP tude and position, is ±√-1. (−1)(-1) would offer the simplest example of a quantity irreducible to this form, and as heterogeneous with respect to V-1 as is the latter with respect to +1. It is true there are demonstrations going to show that the form (a+b-1m+nV-1 can always be reduced to the form p+qV-1; but we may be permitted to remark that those which make use of series are not conclusive so long as it is not proved that p and q are finite. Indeed it often happens in analysis that a series, which, from its very nature can only be true for real quantities, assumes an infinite value, or rather form, when it is made to represent an imaginary quantity; and in like manner it is presumable that a series composed of terms of the form p+q V-1 or a can become infinite if it is to express a quantity of the order ab. As for those demonstrations which employ logarithms, they also seem somewhat obscure, because we have as yet no definite conceptions of imaginary logarithms. It is also necessary to ascertain whether the same logarithm may not belong at the same time to several quantities of different orders; a, a, ap ̧· Moreover the several values resulting from the radicals of the proposed expression is another source of ambiguity, so that one may succeed in rigorously reducing (a+b −1)m+nV-1 to the form p+qV-1 without its being necessarily true that this expression has no other values of the order an irreducible to this form. Before this second paper of Argand's had come to the notice of Francais, the latter also had endeavored to extend the new theory of imaginaries to tri-dimensional space. In the fourth Vol. of Gergonne's Annales a letter appeared from Francais, from which the following is an Extract: According to my previous definition, positive and negative angles are taken in the same plane, which for brevity I shall designate as the plane ry. It would then seem natural to suppose that imaginary angles are situated in planes perpendicular to y, and this supposition would be justified by analogy alone; but its legitimacy may be shown as follows: the angle BV-1 is a mean proportional, both as to magnitude and position, between + and -ẞ; it is therefore situated with respect to the angle + as is the angle -ẞ with respect to From the above, and my 2nd and 3d theorems, it follows that 1 BV-1 2п =12 cos(BV-1)+V-1 sin(-1). Lambert's hyperbolic sine and cosine are thus reduced to the theory of circular arcs, Naperian logarithms, and roots of unity. 1 It further follows that a· 18V-1 =ea =cosa cos(-1)+ V-1 sinacos(B = 1) +1.61 sin(3 ♦ − 1). Whence a √-1 sin (ẞ v−1). =a cosa cos(ẞv=1)+ √−1 a sin acos(-1)+ √—1. aea Hence the projections of a on the three coordinate axes, or rather its three components, will be a cosa cos (B1), V-1.a sin acos(B1 = 1), V-1.aa sin( 1 −1). reached; but I confess I am not yet satisfied These, Monsieur, are the results I have with them. I desire to suppress wholly the old imaginary notation, as I have done for geometry of two dimensions; that is, for the latter I have reduced oblique lines of the form A+B -1 to that of da, where a denotes the absolute length of the line, and a the angle it makes with the axis of reference. In tri-diposition of any line by da A, a denoting the mensional geometry, I desire to express the absolute length a the above angle and A the angle made by the plane of a with ay; but as yet all my efforts in that direction have proved unsuccessful. I trust some one more skillful than myself may succeed in filling up this true method of extending our theory of imaggap. At all events, I am confident that the inaries to tri-dimensional geometry consists in the consideration of imaginary angles. In a postscript to this letter, Francais acknowledges the receipt of Argand's memoir, and that to the latter belongs the credit of the discovery of the geometrical representation of imaginaries. He then adds: In starting from the same principle we have M. Francais concludes: reached different results. I have said above natural extension of the ordinary definithat I have not succeeded in reducing the ex- tion of Algebra.' pression for the position of any right line in The reasons for my aa aA failure are these: I attempted to make, from analogy space to the form ¶ ̧=a, ea1=1=a(cosA+ √−1 sinA) whence = in writing 2-1, should introduce a new unit, I do not quite see why M. Argand (No. 12), rendering, it seems to me, the rest of his paper obscure. Finally, I should be loth to admit the correctness of his assertion that (cosa +1 sin a) cos A+1sin A is irreducible to the form A+B-1. fact, we have which, when a=17, A=‡TM, gives which agrees with the result of M. Argand. But, developing the general case, we have 1a A = ( ea √ —I) eA √ =1_ ̧ (a.eA 1 = 1) -1= (a cos A+-1 a sin A) V-1__ e cV=1=e log (cv — 1_, log c+log =1 therefore, In [cos (dlog c) + 1-1 sin (d log c)], which is certainly of the form A+B 1-1. I, 1 therefore, think myself correct in regarding the expression (c-1)=1, which he assigns to the third dimension, as simply a conjecture [cos(V=1.asinA)+1=1sin(=1.asinA)]=open to serious objection. =[cos(a cos A)+ V-1 sin(a cos A)] × cos(acosA)cos(-1.asinA)+ 1—1 sin(acosA)cos(-1.asinA)+ On Nov. 13th, 1813, M. Servois addressed a letter to Gergonne, which is 1.e-1.acosa.sin(-1.asinA), especially interesting as bearing upon an expression which, on account of its doubly the extension of Argand's theory to space transcendental character, would seem inadmissable. On comparing it with of three dimensions. He objected first to Francais' proof of his first theorem. 1λ+μ-1=cosλcos(u V-1)+ V-1sin cos (1)+1.e1sin(-1) This proposition, that av-1 is a mean proportional in magnitude and position between a and-a, he claimed to consist of two, one of which, viz: that av-1, was a mean proportional as to position is not evident, and is indeed precisely what is to be proved. To this criticism Gergonne replied that, although Servois thinks it evident that av-1 is a mean as regards magnitude, between +a anda, it seemed to him difficult to see how such an expression, which, with its signs, is anegation of magnitude, could be a mean between two reals; that as regards magnitude, the mean could 1.only be a; but, taking position into account, the mean must also be conceived under this new aspect, and is for this very reason a mean in position as well as magnitude, so that the interpretation of 1 sin a cos(A-1)] × sin a sin(A-1) 1-sin asiu (A 1-1) From this it seems to me clear that aA cannot be determined as da was, and that the supposed analogy between angles and lines does not exist. You must have remarked, Monsieur, that M. Argand does not prove my proposition aa=a cosa+1 sin a), and that this fundamental equality is, with him, simply a supposition justified only by a few examples. Servois objected, secondly, that the On this remark M. Gergonne very new theory was not only founded merely justly observes that no demonstration on analogy, but was not even justified a was needed, inasmuch as Argand had posteriori by its applications. Emphadefined the sum of directed lines as a sizing Argand's remark that it consisted certain composition of motions, "a very in the use of a special notation, he char av-1 is reduced to the selection of a line which is situated with reference to + a as a is to it. acterized it as 66 a sort of geometric mask, But 3°, does it follow from the proportion superadded to analytic forms whose (a,n) _o(c,y) direct use was more simple and expedi-(b,) ̄ ̄¢ (d‚§)' as M. Francais says, that we tious." For example, he says: take Argand's first application, e a must have a с = d and a-By-d? I do not where he proposes to develope sin (a+b) and see that this necessarily follows from the concos (a+b). From the general formula -1 ception of 9, The very meaning of this ratio =cos a +1 sin a, I obtain e (a+b) 1 ́ −1. = cos (a+b) + √' -- 1 sin (a+b), and thence (a+b) 119(b,) is quite obscure. What indeed is meant =eav_ī by doubling, trebling, etc, a directed line? (a,a) e =(cos a+ 1 sin a) (cos b+ priori this is not intelligible. M. Francais seems = ♦ — 1 sin b), or e (a+b) 1 -1 (cos a cos b-sin a sin b) +1 (sin a cos b+ cos a sin b); equating these two values of e(a+b)-1, and subsequently the real and imaginary parts separately, we have cos (a+b)=cos a cos b-sin a sin b, sin (a+b)=sin a cos b + cos a sin b. All the other geometrical applications are easily made in the same manner. They may be found in various works, and especially in "A Purely Algebraic Theory of Imaginary Quantities," by M. Suremain-de-Missery (Paris, 1801). The single application to algebra (close of Argand's treatise) seems to me quite unsatisfactory. I do not think it sufficient to find values for x which render the polynomial of less and less value; it is necessary, besides this, that the law of decrease should necessarily render it zero; and that it should be such that zero is not, so to speak, the asymptote of the polynomial. After citing Euler's proof that in reply to Argand's assertion that this expression was irreducible to the form p+qv-1, he raises two other objections, which are important and given in full. Accustomed to designate the position of a point in a plane by an angle and radius vector, geometers have certainly not been ignorant of the consequences of M. Francais' definition. But, content with distinguishing between the magnitude and position of a right line in a plane, they had not yet formed, from these two simple ideas, a single complex one, or rather they had not yet created a new etre geometrique, uniting at once both the ideas of magnitude and position. The length of a right line and its position, i. e. the angle it makes with a fixed axis, are two quantities, which we may term homogeneous; now, how can they be so combined as to form this new entity called a directed line? It seems to me this problem is not yet satisfactorily solved. If a is the length of the line and a the arc of the unit circle which measures the angle it makes with a fixed axis, undoubtedly we may in general represent the line by (a, a), and the function must be determined by the essential condition it is to satisfy. Thus 1°, evidently (a,a)=+a must correspond to a=o, α=2nя, and (a,a)=—a to a=π, a=(2n+1); 2°, also, evidently from_p(a,a)=(b,B) we must have a=b, a=, a=2π, а=3π, difficulty, inasmuch as he speaks of the sum of directed lines only as a consequence of his first two theorems. Still, I do not object to admitting this condition as an essential characteristic of o; but in that case the complete definition of a directed line will be a definition nominis non rei, or, in other words, directed line will be the name of a certain analytic function of the length and direction of a right line. From this it unfortunately follows that we are no longer constructing imaginaries, but simply reducing them to the same analytic form. However, let us see what this function is. It is, in the first place, clear that the expression (a,a)=a.ea V-1 satisfies the three foregoing conditions. In fact, we have 1°(a,o)=a.eo=1=a, (a,)=a.e11_a(cos + √−1 sin~)=—a; 2° the equation (a,a)=(b,B) becomes a.e-1. b.eß-1, or, passing to logarithms, equating, and returning to numbers, a=b, a=p; 3° the above proportion, by similar transformations becomes to have been aware of this a b с d and a = -B=y-d. But is this form a.ea-1 the only one which satisfies these three conditions? I think not, and it seems to me evident that they will be equally true if we substitute an arbitrary coefficient for the imaginary -1. So that the form a.ea-1 will, in my opinion, only be a special case of the analytic expression for a directed line, in its conventional signification. Are there any other conditions which follow from this signification? To this question no answer is made, nor do I either see any. Again, 4 the table of double argument which you (Gergonne) propose, as applied to a plane supposed to be so divided into points or infinitesimal squares that each square corresponds to a number which would be its index, would very properly indicate the length and position of the radii vectores which revolve about the point or central square corresponding to ±0; and it is quite remarkable that if we designate the length of a radius vector by a, and the angle it makes with the real line...,-1, ±0,+1,.... by a, the rectangular co-ordinates of its extremity remote from the origin by x, y, the real line being the axis of x, the point would be determined by x+y V-1, and consequently, since x = a cosa, y=asin by a, a.ea-1. Thus we have a new geometrical interpretation of the function a. ea1 which, it seems to me, is of more value than that of MM. Argand and Francais; but certainly we should not thereby conclude that' this was a new method of constructing, geometrically, imaginary quantities, for the above indices presuppose them. However this may be, it is clear that your ingenious tabular arrangement of numerical magnitudes may be regarded as a central slice (tranche centrale) of a table of triple argument representing points and lines in tri-dimensional space. You would doubtless give to each term a tri-nomial form; but what would be the co-efficient of the third term? For my part I cannot tell. Analogy would seem to indicate that the tri-nominal should be of the form pcos a+q cos B+r cosy, a, 3 and being the angles made by a right line with three rectangular axes, and that we should have (p cos a+q cos 3+rcos ) (p'cosa+q'cos +r'cosy)=cos2a+cos23+cos2)=1. The values of p‚q‚r, p',q',r' satisfying this condition would be absurd; but would they be imaginaries, reducible to the general form A+B V-1? Council of the Royal Irish Academy, of which society I was at that time president, to read, at the next general meeting, a paper on Quaternions, which I accordingly did on Nov. 13th, 1843." It is also proper here to add a disclaimer from Gergonne as to any thought of the extension of his table to tridimensional space, until after the appearance of Argand's and Francais papers; and that even then he saw no way by which to effect that result. The above letter from M. Servois called forth a reply from Francais (Annales, Vol. IV., p. 364–367), and a third paper from Argand (Annales, Vol. V., p 197– 209). In the former, Francais sustains Gergonne, who had already said that Servois asked too much of the new theory, demanding rigorous demonstrations of that which, as in the early history of negative quantities or the calculus, was of whose fundamental principles the perceived by a sort of instinct, the proofs earlier writers were not in a state to On this letter Hamilton remarks in his Lectures on Quaternions, (Preface, p. 57), “The six non-reals which thus Servois with remarkable sagacity foresaw, without being able to determine them, may now be identified with the then unknown symbols +i, +j, +k, produce. He then adds a few examples then unknown symbols +i, +j, +, -j, -k of the quaternion theory ;" and it -j, -k of the quaternion theory;"andit of the facility with which one might pass may here be interesting to quote (North from the proposed to the ordinary notaBritish Review, 1866), from a letter of Hamilton on the discovery of these symbols: -- Oct. 15, '58. tion. cides with the axis of reference is aa+b-8=C, The equation of a triangle whose base coin whence a cos a+bcos,3=c, and a sin a-b sin 3=0, or, taking the sum and difference of the squares a+b2+2ab cos(a+3 =c2, a2cos2a+b2cos2,3+2ab cos(a-3)=c2. The equation of the circle referred to the center is a=x+y V−1, whence "P. S.-To-morrow will be the fifteenth birthday of the Quaternions. They started into life, or light, full grown, on the 16th of Oct., 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham a coso=x, a sino y, x2+y=ao. Bridge, which my boys have since called The equation of a circle referred to a diamethe Quaternion Bridge. That is to say, ter is poб-4=2a, whence I then and there felt the galvanic circuit cos + sin p=2a, p sin —σ cos ¢=0, of thought to close; and the sparks p2=2ap cos, x2+y2=2ax. which fell from it were the fundamental The equation of an ellipse referred to the equations between i, j, k; exactly such as focus is po+(2a-p)=2e, whence I have used them ever since. I pulled pcos +(2a-p)cos ч=2e, out, on the spot, a pocketbook, which a2-es p sin +(2a-p)sin Y=o, p= ́a-e cos' The reply of Argand is appended. The new theory of imaginaries, already retwo distinct and independent objects; it seeks, ferred to several times in this publication, has first, to render intelligible certain expressions whose presence in analysis has been inevitable, but which have not yet been referred to any known evaluable quantity; and, second, it which employs geometric symbols concurrentpresents a method, or a particular notation ly with the ordinary algebraic signs. Hence, from this double point of view, two questions arise: Has it been rigorously shown that -1 represents a line perpendicular to those denoted by +1 and -1? Can the notation of directed lines furnish, in certain cases, demonstrations and solutions preferable either for their simplicity or brevity, etc., to those which they are intended to replace? The first of these will, perhaps, always be open to discussion so long as we seek to establish the meaning of -1 by analogy, from the commonly received ideas on positive and negative quantities and their ratios. Negative quantities have been and are still the subject of discussion; it will, therefore, be all the easier to raise objections to the new theory of imaginaries. But this difficulty will vanish if, with M. Francais, we define what is meant by a ratio of magnitude and position between two lines. Indeed, the relation between two such lines may be conceived of with all necessary precision. Whether this relation be called ratio or something else, it may always be made the subject of exact reasoning, and its consequences, in analysis and geometry, of which M. Francais and myself have given some examples, may be traced. The only remaining question, then, is whether it is proper to designate this relation as a ratio or proportion, words which already possess, in analysis, a determinate and fixed meaning. Now, this is permissible, because the new meaning is an extension, not a contradict on, of the old one. The latter is so generalized that the ordinary meaning becomes, so to speak, a particular case of the new one. There is then, no question here of demonstration. 1 an Thus, for the analyst who first wrote a-n=- this equation was a definition of negative exponents, not a proposition proved or to be proved. All that it was incumbent upon him to show was that this definition was only a generalization of that of positive exponents, the only ones before known, and so for fractional, irrational and imaginary exponents. It has been said that Euler proved ( √ −1) √ — 1=e—1⁄2". The word prored may be exact if we mean that this equation is derived from ex√=1= = cos x + √ - 1 sin x, which is readily shown to be the case; but it is not so as regards this latter; for to show that a certain expression has a definite value, implies the previous definition of the expression. But is there any definition of imaginary exponents antedating the so called demonstration of Euler? It seems not. When Euler sought to evaluate ex-1, he naturally resorted to the 2 22 previously was defined as representing a quantity equal to cos.r+ V-1 sine, we should thereby bring both real and imaginary exponents under the same law. Here again, then, we have the extension of a principle, not the demonstration of a theorem. It is also by an extension of principles that I was led to regard (-1) (-1) as representing a perpendicular to the plane ±1, ± 1-1. The two results conflict, and I certainly have not insisted upon my own; I only wish to observe that MM. Francais and Servois have attacked it from considerations which are after all of the very nature of those on which I relied to establish it. But if the above perpendicular cannot be expressed by (−1) = 1, how then shall it be represented? Or, rather, can any expression tive of the perpendicular, shall bring all dibe found, whose adoption, as the representarected lines whatever under a common law, as is already the case for every line of the plane +1, -1? This is a question which must be of interest to geometers, at least to those who admit the new theory. To return to the original question, I observe that whether -1 does or does not represent the perpendicular on ±1 must depend upon the meaning of the word ratio; for it is agreed by all that So that M. Servois' objection to Francais' proof of his first theorem, viz: "That it is not proved that a V-1 is a mean, as to position, between +a and -a" is equivalent to the assertion that the word ratio has no reference to position. In its usual acceptation, this is true; and on the other hand, it may be said that, in the conception of a ratio between quantities with different signs, the signs must be regarded. In the new meaning, direction · and magnitude make up the idea of ratio. It is thus seen to be a question of words, decided by the exact definition given by Francais, which is an extension of the usual one. The second point under discussion is more important. Doubtless no truth is reached by the notation of directed lines which cannot be attained by ordinary methods; but which method is the simplest? This question is, I influence of methods and notations on the think, worthy of examination. It is to the progress of the science, that modern mathematics owes its superiority. So that when anything new of this kind appears, we may at least examine it in this respect. Since the publication of the new theory, M. Servois alone bas expressed an opinion on this point, theorem ez=1+ + + 1 1.2 demonstrated for all real values of 2. By mak- and his opinion is not favorable to the new ing z=x V-1 he found notation. Analytic formulæ seem to him more simple and expeditious. I would, however, claim for my method a more careful examination. I admit that it is novel, and that the mental operations it requires, although quite simple, demand some familiarity in order that they may be performed with the ease which follows practice in the ordinary operations of |