Figures with re-en ABC, ACD, ADE, and abc, acd, ade, similarly formed in each figure, are similar to each other: and the same conclusion will equally follow, whatever be the number of the sides of the figure: it follows, therefore, that all rectilinear figures are similar to each other, in which all the triangles similarly formed in each are similar to each other. 852. In the geometrical theory of rectilinear figures, where trant angles A represents an interior, and A' the corresponding exterior angle, we assume A and A', in the equation may come under the general equation of A+ A' =π figure and to be both of them positive, and also less than : but figures angles. may be conceived to be formed possessing re-entrant angles, (as we have already seen in the case of stellated figures), where one of these angles may exceed 180°, and where this equation cannot be satisfied unless the other be negative, and conversely: thus, if the movement of transfer, in passing from the line BA to AC, be from right to left in one case (Fig. 1), and from left to right in another (Fig. 2), the external angles (AC) generated will have different signs: and if these movements of transfer Fig. 1. Fig. 2. Fig. 3. Fig. 4. B be continued through more than 180°, the interior angle BAC All these relations of exterior and interior angles are exem- Fig. 2. D Fig. 3. Fig. 4. OLAM B E B E C. plified in the formation of the equilateral pentagonal figures (1), (2), (3) and (4), of which the first is the regular pentagon of Geometry: the second has the re-entrant and interior angle at A greater than 180°: and therefore A' negative: the third has the angle at A negative, and the fourth the angle at A positive and greater than 4 right angles. Such rectilinear figures, therefore, though not geometrical, satisfy the equations of figure and of angles, and they may be determined, like geometrical figures, by the aid of those equations, from the requisite data. CHAPTER XXXII. Definition of a logarithm. A system of logarithms. There are only two systems of ON LOGARITHMS AND LOGARITHMIC TABLES AND THEIR USE. 853. THE index or exponent x, in the equation a = n is called the logarithm of n to the base a: this definition includes the primary notion of a logarithm, in which the term originated, which is mentioned in Art. 273. It will be shewn in a subsequent Chapter, that if a and n are numbers {where the term number is used in its largest sense (Art. 169 and 416)} greater than 1, there is always an arithmetical value of x which satisfies the equation a* =n: or, in other words, there is always an arithmetical logarithm of n to the base a. 854. A system of logarithms is the series of indices of the same base, which correspond to the successive values of n: such a system is formed by the series of logarithms of the natural numbers from 1 to 100000, to the base 10, which constitute the logarithms registered in our ordinary tables, and which are therefore called tabular logarithms. 855. Though our general reasonings have reference to systems of logarithms calculated to any base whatever, there are logarithms two systems only which are commonly employed in analysis: the which are first is that of tabular logarithms above referred to, and which are commonly employed, exclusively used in numerical calculation: the other is the system and the whose base is 2.7182818..., which is called Napierian, from the Napierian. name of the great man to whom their invention is due, and which is almost exclusively used in analytical formulæ. the tabular General 856. The properties of logarithms are the properties of properties indices of powers of the same symbol, and may be stated as of logarithms. follows. Let x and x' be the logarithms of n and n' respectively to the base a, and therefore a*=n and ar=n': we then find (Art. 635) (1) a* × a* = a*+* = nn', where x + x' is the logarithm of nn': in other words, the logarithm of a product is the sum of the logarithms of its factors. other words, the logarithm of the quotient of two numbers is the logarithm of the dividend diminished by the logarithm of the divisor. (3) (a*)P = a2* = n', where px is the logarithm of no: in other words, the logarithm of the pth power of a number is P times the logarithm of the number. 1 x 1 (4) (a")}" = a" = n}, where is the logarithm of n2: in other words, the logarithm of the pth root of a number is th part of the logarithm of the number. 857. The logarithm of the base of a system is 1. The logarithm of the For a1 = a (Art. 42), and 1 is therefore the logarithm of a, base is 1. where a is the base of the system, whatever a may be. 858. The logarithm of 1, in all systems, is 0. therefore the logarithm of 1. 859. The logarithm of 0 is infinite and negative, or in all systems, where the base is greater than 1. For if a be greater than 1, then a-~ = The logarithm of 1, in all systems, is 0. ∞, The logarithm of 0, when the base is 1 1 = = 0: but if a a be less than 1, then a = 0; and consequently, under such circumstances, ∞ is the logarithm of 0. greater than 1, is -8. The loga 860. The logarithms of all numbers which are not integral rithms of powers of the base, involve a decimal part. all numbers which are not integral powers of the base involve a For if a be the base of the system, then the logarithms of a, a2, a3, a1, or of the integral powers of a are the indices of a, and therefore whole numbers: the logarithms of all numbers decimal part. between 1 and a, a and a2, a2 and a3, a3 and at have a decimal part: thus, in the tabular system, the logarithms of 10, 100, 1000, 10000, 100000, 1000000, &c., which are the articulate numbers of the decimal scale, are expressed by the series of natural numbers 1, 2, 3, 4, 5, 6, &c. being less by 1 than the number of places in each number: consequently, the tabular logarithms of all numbers The logarithms of numbers less than 1, where the base is and it may be observed that the integral part, or characteristic as it is called, of such logarithms will always be known from the number of integral places in the number, and therefore will not require to be tabulated. 861. The logarithms of numbers less than 1, in a system whose base is greater than 1, are negative. For, if a be greater than 1, and if x be positive, then a* is 1 greater than greater than 1, and a-*: is less than 1: the logarithms of 1, are negative. Classifica fundamen tions of = απ such numbers, therefore, are necessarily negative. 862. The fundamental operations of arithmetic are Addition tion of the and Subtraction, Multiplication and Division, Involution and tal opera- Evolution, and the order of succession in which they are thus Arithmetic arranged is likewise the order of the difficulty and labour of in the order performing them, the inverse operations of Subtraction, Division, sion and and Evolution being generally less simple, direct, and expeditious difficulty. than the corresponding operations of Addition, Multiplication, and Involution. of succes The order of the cor 863. If we compare the operations with numbers with the responding corresponding operations with their logarithms, it will be observed logarithmic that their order, with reference to the preceding classification, is operations is lower by reduced by two unities: thus the operation of Multiplication two unities. with numbers corresponds to that of Addition with logarithms, |