(c) A fully assured method of calculating with such infinite series as are yet perfectly general, and precisely for that reason not convergent. In the "Tracts upon some parts of higher Mathematics" (Aufsätze aus dem Gebiete der höheren Mathematik, Berlin, 1823), will be found some applications of these elementary, but assured methods of calculation, especially in the last of those Tracts, to which the Author thus expressly refers the Reader, because the object of the present Essay prescribes the greatest possible brevity, and such application could consequently not be presented in this place. Those Readers, finally, who desire to see these views developed at length, but in such a manner as is required for pædagogical objects, will find their wishes gratified in the "Attempt at a perfectly consequential System of Mathematics," 7 vols. (Versuch eines vollkommen consequenten Systems der Mathematik, Berlin), especially in the two first volumes (2nd edition); further, but less fully, in the first vol. (2nd edition) of the "Instruction-Book in Elementary Mathematics," 3 vols. (Lehrbuch der Elementar-Mathematik), which has been chiefly written for beginners; most incompletely in the "Shorter Instruction-Book in the whole of elementary Mathematics," 3rd edition, Leipzig, 1842 (Lehrbuch für den gesammten mathematischen Elementar-Unterricht), which is intended as a guide for the very earliest beginners; on the other hand more fully and fundamentally in the "Instruction-Book in the whole of higher Mathematics," in two volumes, Leipzig, 1839. BERLIN, January, 1842. M. OHM. SECOND CHAPTER, pp. 24-33. 2 Idea of Difference-Product.... 15. Idea of Difference-Quotient Multiplication and Division by zero, by -6 and by +b; by Algebraical 22. Idea of Broken Number; of Positive and Negative Broken Number; 24. Idea of Whole Power, with 5 Formulæ ; Idea of Difference-Power 181818 32. Idea of Decimal Fraction. Method of Calculation with the same... 38 33. Further resolution of calculation with cyphers........... 34. Idea and resolution of calculation with letters................. 35. Idea of Determining Equation, in opposition to Identical Equation 36. Solution of Algebraical Equations of the first Degree with one or 37. Idea of General Square Root; Imaginary Root; Precautionary rules for calculating with general (and therefore also with imaginary) 38. Solution of the General Quadratic Equation....... 39. Idea of General-Numerical Number; calculation with the same...... 40. Higher Algebraical Equation of the nth Degree......... 41. Idea of general nth root. Formulæ for such roots.. 42. Binomial Theorem for whole exponents. Transformation of the Bino- mial Series for (1+b) into a series proceeding according to whole SECOND PART. 43. Idea of Infinite Series proceeding according to whole powers of a ... 44. Identity of the Coefficients when the series are identical.... 45. Addition, Subtraction, Multiplication, and Division of two such in- 49. Idea of the_Natural Power e, where x is any actual or imaginary 50. Idea of Natural Logarithm, and of General Cosine and Sine......... 66 51. Formulæ for these general cosines and sines Causes for supposing that there is a periodical return in those values 68 Decisive Proof of the correctness of this supposition...................... 55. Evaluation of the sine and cosine with either actual or imaginary 56. 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.. 57. Simplest value of the same 59. Formulæ for the simplest values of natural logarithms Formulæ for infinitely multiple-meaning natural logarithms 61. Definition of General Power, and its Simplest Value 62. Properties of the (single-meaning) Simplest Values of general powers 78 ............... 64. Properties of the infinitely multiple-meaning general powers.. Remarks concerning the Formula for Powers with broken exponents 86 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 (Œuvres 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. 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 I |