To find all the values of a which make K,μ and S,=v 56. Most general idea of calculation with cyphers....... 57. Calculation of all the values of a th root.... 63'. Simplest Values of general logarithms; formulæ for these simplest 64. Properties of the infinitely multiple-meaning general powers.... 65. Remarks concerning the Formula for Powers with broken exponents 67. Infinite series for logarithms 68. The perfectly General Binomial Theorem..... 70. Formulæ for general calculation with general (actual or imaginary) ERRATA. PAGE 28 line 7, for sect. 15, read sect. 16. INTRODUCTION. It is a remarkable fact that complaints of the want of clearness and rigour in that part of Mathematics which respects calculation, whether it be called Arithmetic, Universal Arithmetic, Mathematical Analysis, or aught else,-recur from time to time, now uttered by subordinate writers, now repeated by the most distinguished of the learned. One finds contradictions in the theory of "opposed magnitudes;"-another is merely disquieted by "imaginary quantities;"-a third finally meets with difficulties in "infinite series," either because Euler and other distinguished mathematicians have applied them with success in a divergent form, while the complainant thinks himself convinced that their convergence is a fundamental condition,- or because in general investigations general series occur, which, precisely because they are general, can be neither accounted divergent nor convergent. These considerations force themselves continually upon the Author of these sheets, and forced themselves more especially upon him lately when reading a letter of Abel, (who died so unfortunately soon for Mathematics,) which will be found in his Complete Works (Euvres complètes de N. H. Abel. Christiania, 1839) and in which he writes thus: Divergent series are in general very mischievous affairs, and it is shameful that any one should have founded a demonstration upon them. You can demonstrate any thing you please by employing them, and it is they who have caused so much misfortune, and given birth to so many paradoxes. Can any thing be conceived more horrible than to declare that 0=1-2"+ 3′′ – 4′′ + 5′′ - &c. &c. when n is a whole positive number?-At last my eyes have been opened in a most striking manner, for, with the exception of the simplest cases, as for example the geometric series, there can scarcely be found in the whole of mathematics a single infinite series, whose sum has been rigorously determined; that is to say, the most important part of mathematics is without foundation. The greater part of the results are correct, that is true, but that is a most extraordinary circumstance. I am engaged in discovering the reason of this, -a most interesting problem. I do not think that you could propose to me more than a very small number of problems or theorems containing infinite series, without my being able to make wellfounded objections to their demonstration. Do so, and I will answer you. Not even the Binomial Theorem has as yet been rigorously demonstrated. I have found that for all values of x which are less than 1. When x+1 the same formula holds, but only provided that m is >-1; and when x=-1, the formula only holds for positive values of m. For all other values of m the series 1+mx+... is divergent. Taylor's Theorem, the foundation of the whole infinitesimal calculus, has no better foundation. I have only found one single rigorous demonstration of it, and that is the one given by M. Cauchy in his "Abstract of Lectures upon the Infinitesimal Calculus" (Resumé des leçons sur le calcul infinitésimal) where he has demonstrated that we shall have as long as the series is convergent; but it is usually employed without ceremony in all cases. "The Theory of infinite series in general rests at present upon a very bad foundation. All operations are applied to them as if they were finite; but is this permissible? I think not. Where is it demonstrated that the differential of an infinite series is found by taking the differential of each term? Nothing is easier than to give examples where this rule is not correct; for example, cosx-cos 2x + cos 3x- &c. an utterly false result, for this series is divergent. "The same remark holds for the multiplication and division of infinite series. I have begun to examine the most important rules which are (at present) esteemed to hold good in this respect, and to shew in what cases they are correct, and in what not so. This work proceeds tolerably well, and interests me infinitely." Thus laments Abel. But d'Alembert (in many places of his Opuscules"), Carnot, even Laplace make similar complaints, although more briefly, and not always precisely concerning infinite series. Kramp in his " Analysis of astronomical and terrestrial refractions, 1799, Chap. III. Analysis of numerical faculties." (Analyse des réfractions astronomique et terrestres, 1799, Chap. III. Analyse des facultés numériques), encounters a formula, of which he himself says that it is incorrect, adding: "I acknowledge frankly that all the trouble which I have given myself to find out the reason of this paralogism has been hitherto ineffectual. I should be infinitely obliged to any mathematician who would point it out to me. It seems to me to belong essentially to our theory of fractional powers and logarithms of negative numbers, and that the theorem log (-a) = log a + log (− 1), despite its appearance of extreme simplicity, is far from having been rigorously demonstrated." In another place of the same chapter where he meets with similar contradictions, he says further: "I should not be indisposed to believe in fact, that every application which we have hitherto made of our general theory of powers to roots and logarithms of negative quantities, were a conclusion à particulari ad universale, which ought not to be excusable in analysis." Further on he says: "As regards the differential of (-1)" there is not a single mathematician, who, conforming himself to received ideas, can tell us what it is. "But what shall we do with (-1)2? what will (-1) become for the infinite number of cases in which the exponents are irrational, exponential, circular, or finally transcendental quantities of any description ?" Thus Kramp. It is clear that he complains less of infinite series and more of powers and logarithms, and, thinking that he was operating correctly, he has obtained a great number of incorrect results, but has also happily remarked that they were incorrect. On the other hand the greatest analysts, as Euler and Lagrange, have sometimes exhibited such results in their writings, without having always remarked that they were incorrect. For example, all the results which are to be found in the XIth Lecture of the "Lectures upon the Calculus of Functions," (Leçons sur le Calcul des Fonctions, 1806.) by Lagrange, are incorrect, and yet these incorrect results had been previously given in the same form by Euler; their incorrectness for a general and not merely whole exponent surprises us in Lagrange's work the more, that the object of the above named Lecture was precisely to prove the general correctness of these formulæ with perfect rigour. But if on the one side such facts speak loudly enough, and on the other we hear complaints from men who stand on the highest height of science, and who have themselves more or less extended its bounds, we cannot but repeatedly inquire: 1. Are these complaints just or unjust, and how far so? 2. Are, as Abel appears to complain, the infinite series the only cause of all the paradoxes of calculation, or have we to seek for their sources elsewhere also, and where? 3. When we say in mathematical analysis: "this or that result is correct or is false," what do we mean by so saying?. or in other words; if two results contradict one another, what characteristic have we, which will help us to distinguish the correct from the false? 4. How may the paradoxes of calculation be avoided with certainty ? And so forth. When the Author of these sheets considers these questions from his own point of sight, it appears to him, in examining the first question, that the reproach made by Abel, that calculations or demonstrations are conducted with divergent series, only affects in all its generality the mathematicians of the last century, since all living mathematicians of note, as Gauss, Dirichlet, Jacobi, Bessel, Cauchy and the rest, do not employ them, while Poisson among others has spoken decidedly against their being employed. But that the series which are used, and from which deductions are drawn, ought to be always and necessarily convergent, is a circumstance of which the Author of this Essay has not been at all able to convince himself; on the contrary, it is his opinion that series, as long as they are GENERAL, so that we cannot speak of their convergence or divergence, must always, when properly treated, necessarily and unconditionally produce correct results. An examination of this opinion is one of the objects of the present pages. With regard to the second question, namely, where are the sources of the parodoxes of calculation to be looked for? the Author finds them especially: (a) in a onesided conception of the idea of zero; (b) in a disregard of the properties of manymeaning or infinitely multiple-meaning expressions; (c) in the circumstance, that through a want of proper attention in mathematical analysis, and algebra, general propositions are liable to be generally converted, which, as is well known, offends against the first rules of logic, and must necessarily lead to the most erroneous results; (d) in an imperfectly correct method of treating infinite series; and finally, also (e) in the fact, that results are considered and applied as generally correct, which have only been demonstrated for particular cases, and which, too, are only true in those particular cases. This last case namely occurs in the above cited work of Kramp, who has applied for all faculties (factorials) the law h". amIv = (ha)mIv viz. for those factorials also, in which the exponent m is a broken number, whereas it is precisely this formula, among those which are employed for calculating with faculties (factorials), which does not hold for broken factorials, as may be easily proved. This is exactly the same error as would be committed if we were to pronounce that the equation (-1)"= = COS Nπ, which is correct for any whole number n, would also hold for any broken n. For n=, for example, it would give (-1)=cos, that is√-1=0. But although even not long ago Tralles in two Essays of the |