between certain determinate limits. The above conclusion is therefore not perfectly correct, and the equation | = e* for n=500 cannot be admitted as a general (formal) equation. But if we consider % as no longer insignificant, but as either actual or imaginary, and therefore of the form pcos + -1.p.sin o, then s" = p. cos ro+ - 1.p. sin rø, and then I (both in the series found for (1 + )" when n=+o, and in the series for e*) would approach zero the more nearly, the greater r were taken, i.e. the further the terms were taken on towards infinity. And hence at the moment that the assertion that nrl-1 -=1 for n=+00 no ceases to be correct, the terms which are influenced by this incorrect assertion, may be considered as equal to zero, and hence these two discrepancies correct one another. Hence although the equation (2) (1+)* = e for n == is not correct as a general (formal) equation, it is yet not incorrect for any actual or imaginary value of z as a numeric equation, and that, because both series, that for ), and also that for et are convergent for any such value of z.* Finally if the terms of these series had not these large denominators r! (that is, 1.2.3.4.5.6...r) which cause the series to converge for any value of z, the abovementioned equation would have only held for those values of s for which the said series would have been convergent. Hence while the equation (No. 1) holds with perfect generality, when x is considered as perfectly insignificant, being a mere supporter of the operations, and when therefore we cannot speak of the convergence or divergence of the series, we perceive in (No. 2) an equation which is only correct as a numeric equation, i. e. which is only correct when the series are convergent. * Hence if we wish to calculate the value of ex approximately for any positive or negative values of , we have only to calculate the value of for any (disregarding the sign) very large positive or negative, whole or broken, rational or irrational n, and the approximation will be the closer the greater n is taken. For example, we can in this manner calculate the number e itself. A definite Integral always presupposes numeric values; consequently equations in which definite integrals occur are seldom or never correct as general (formal) equations, but can only be admitted as numeric equations; consequently the convergence of any infinite series which may occur in them is an indispensable condition, whereas the condition of convergence with respect to a general series in general investigations, such as must be necessarily first established as the foundation of the possibility of any calculation, is quite as absurd, as if, upon an investigation of the capabilities of a living and yet powerful man, the condition were premised as indispensable,—that he should be already dead. But if equations can occur which no longer hold generally but only upon the particular hypothesis that they are numeric equations, the theory of such numeric equations must be established for itself in a second Essay, and the “ Theory of Definite Integrals” (in which numeric equations first occur in considerable numbers,) will have determinately and distinctly to solve the problem of discovering a method of calculating with such equations. The Author believes that he has now shewn the Reader in the clearest possible manner, what he will have to expect from a second Essay. While namely this first Essay treats of perfectly general forms, as the first and most necessary foundation of all calculation, the second Essay will have to continue these general investigations, but at the same time to bestow more attention upon the passing of general into particular and numeric forms, which passing takes place when the former are considered upon determinate and particular hypotheses. Among the practical results, which have been the consequences of the view here established, and which are to be found at once in the present first Essay, the Author thinks that he ought to distinguish the following: (a) A fully assured method of calculating with roots in general, and with imaginary expressions in particular ; (6) The establishment of those formula, which must take the place of the usual rules a”. a* = q*+* ; a" : a*= a*-*; (am)* = q*s; and so on, in order that we may calculate with general powers and logarithms in perfect safety, inasmuch as the above formulæ which are those usually employed, are only partially correct; (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 Mathe matics," 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. (Lehr buch der Elementar-Mathematik), which has been chiefly written for beginners; most incompletely in the “Shorter Instruction-Book in the whole of elementary Mathe matics,” 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. TABLE OF CONTENTS. ELEMENTARY ALGEBRA. FIRST CHAPTER, pp. 19–24. 2. Three simplest “ Equations for whole numbers”. 3. General “Sum”; general “Difference"; general “Equation ” ...... 4. Propositions concerning general equations....... 5. The Laws of Addition and Subtraction......... 6. General Idea of "Calculation"...... 7. General Idea of “Zero”................................................ 8. General Idea of +b and -b ......... 9. General Idea of Algebraical Sum 10. Idea of Positive and Negative Whole Numbers .............. Tlallah SECOND CHAPTER, pp. 24–33. 11. General Ideas of Multiplication and Division... 12. Idea of Whole Product ..... 13. Idea of Difference-Product........................... 14. Idea of General Product ................................................ 15. Idea of Difference-Quotient ......... 16. It is never allowable to divide by zero ............ 17. Laws of these Quotients ..... Idea of General Quotient ............. Means of ascertaining the equality of expressions ............. How equations are formed from one another ........ Multiplication and Division by zero, by -b and by +b; by Algebraical Sums .................................. 22. Idea of Broken Number ; of Positive and Negative Broken Number ; 23. Idea of greater and less Actual Number, as also of the continuous increase (and diminution) of all actual numbers from – co to too. 32 28. Idea of Actual Power........................... 29. Idea of Actual Logarithm ..................... 30. Five Formulæ for these Actual Logarithms........ 31. Idea of Numerical Number. Common Cyphering, as the first appli. cation of the above doctrines .............. 32. Idea of Decimal Fraction. Method of Calculation with the same ... Idea of Determining Equation, in opposition to Identical Equation Solution of Algebraical Equations of the first Degree with one or several unknown expressions ....... Idea of General Square Root; Imaginary Root; Precautionary rules for calculating with general (and therefore also with imaginary) square roots ................. Solution of the General Quadratic Equation..... Idea of General-Numerical Number; calculation with the same...... Higher Algebraical Equation of the nth Degree......... Idea of general nth root. Formulæ for such roots................ Binomial Theorem for whole exponents. Transformation of the Bino. mial Series for (1+b) into a series proceeding according to whole 43. Idea of Infinite Series proceeding according to whole powers of x ... 44. Identity of the Coefficients when the series are identical................ 45. Addition, Subtraction, Multiplication, and Division of two such in- 46. Potentiation and Radication of infinite series ...... Idea of Summation of infinite series .. Convergence and Divergence of numerical series. Idea of the Value 486. Practical rules for calculating with infinite series ............ 49. Idea of the Natural Power e", where x is any actual or imaginary number. Three Formulæ for such Powers. cosines and sines ........... on........................ |