LOGARITHMS.] 



NAVIGATION. 



1037 



of two integers, however many decimals may follow, is 1 ; 

 of a number consisting of three integers, the index of 

 the log. is 2 ; of a number consisting of four integers, the 

 index ia 3 ; and, generally, of a number consisting of n 

 integers, the index of the log. ia n 1. 



We have, therefore, only to count the figures in the 

 integral part of a number, disregarding the decimals, 

 and then to write down, for the index of the log. of that 

 number, the figure denoting what was counted, diminished 

 by 1 : to this index the proper decimal part of the log. , 

 called the mantissa, which is furnished by the table, is to 

 be united, when the complete log. will be exhibited. 

 Thus, for the numbers 43, 58, 32-47, 67 '813, &c., the 

 index, in each case, is 1 ; for each of the numbers 246, 

 835, 647-29, 158-72, &c., the index is 2 ; and so on. 



If, however, the number proposed have no integral 

 figures that is, if it consist wholly of decimals then 

 the index of the log. will be negative. For instance, 

 suppose the number is "3247, we may write it thus 



:;-L't7 



... ; and, from the principles already explained, the 



lo o <. of the quotient thus indicated is log. 3 247 log. 

 10 ; that is, log. 3 -247 1. Consequently, the index of 

 the log. of -3247 is the index of log. 3-247 diminished by 

 1 : but as there is only one integer in the number 3-247, 

 the index of its log. is ; therefore the index of log. 

 3_' 47 is 1. Suppose the number is -03247 ; this may 



3247 

 be written -TTT = hence log. -03247= log. -3247 log. 



but the index of log. 



10 



10; that is, = log. "3247 1 



as just shown, is 1 ; hence the index of log. 

 03247 is 2. Similarly, the index of log. -003247 is- 

 3, that of log. "0003247 is 4, and so on. Consequently, 

 whatever be the number whose log. is required, we find 

 the index of that log. thus : 



Count how many places the leading figure of the 

 number is from the unit's place, and put down what U 

 counted, for index : the figure thus put down will be plus 

 if the counting is towards the right, and minus if towards 

 the left ; or the rule is expressed otherwise, thus : 

 " Place your pen between the first and second FIGURE 

 (NOT CIPHER), and count one for each figure or cipher, 

 until you come to the decimal point ; the number this 

 gives will be the index. If you count to the right the 

 index is positive, if to the left it is negative." (See 



522). 



The docimal part of the log. of a number is always 

 found against that number in the table, which is to be re- 

 ferred to so soon as the index is written down. Because 

 of this ready way of ascertaining what the index of the 

 log. of a number is, it is not necessary to insert more 

 than the mantissa, or decimal part of the log., in the 

 table : but it is to be observed that this decimal part is 

 always plus. For instance, having first written down 

 the proper index, we find, on referring to the table, that 

 log. -3247= -1 + -511482, log. -03247= -2 + 511432, 



log. -003247= -3+ -511482, &o. 



But a more compact and convenient way of writing these 

 logs. U this, namely : 



log. -3247=1-511482, log. -03247 = 2^511482, log. 



003247=3 511482, <tc. 



And in this form the logarithms of numbers less than 

 unity that is, of numbers of which all the figures are 



!s are always used in practice. 



Tlin logarithmic operation for finding a product from 



its factors suggests tfiat for finding any power of a known 



root, or any root of a known power : thus, since * merely 



tea the product of p factors, each equal to the 



1) 



number n, we have log. n* = p log. n ; and since if np= 

 r, we must have n=r', it follows that log. n=p log. r, 



and consequently that log. r= We thus derive 



the following practical rules for performing the more 

 troublesome operations of arithmetic in a short and easy 

 manner liy ln.'l]i of a t.-iMo of logarithms. 



1 Multiplication. From the table take the log. of each 

 factor ; add these logs, together: the sum will be the log. 



of the product of the factors ; and against this log. in 

 the tables, will be found the product sought. 



2. Division. Subtract the log. of the divisor from the 

 log. of the dividend : the remainder is the log. of the 

 quotient. 



3. Powers, and Soots. Multiply the log. of tha 

 number to which the exponent is attached, by that ex- 

 ponent, whether it be integral or fractional : the result 

 will be the log. of the power or root. 



As to the means which algebraists have devised for 

 constructing such a table as that here referred to, the 

 student may consult the Chapter on Logarithms and 

 Series in the Section on Mathematics : it has been 

 thought sufficient, in this introductory article, to show 

 the general principles on which tables of logs, are based, 

 without entering into details as to their actual for- 

 mation. These tables are given at page 525, et set}. 



Ox THE SINES, COSINES, TANGENTS, <fec., OF ANGLES. 

 In the Section on Mathematics (p. 613) the trigonomet- 

 rical terms, sine, cosine, tangent, <fcc., are defined in 

 two different ways. For the purposes of navigation, the 

 first of these ways will be found to be the more conve- 

 nient, as well as the more intelligible to the learner. 

 Taking the diagram at page 613, we may explain the 

 lines there figured as follows (Fig. 1) : 



Fig. 1. 



,S7u>. The sine of an angle A O B, or of the arc A B, 

 which measures that angle, is the straight line Bn drawn 

 from the extremity B of the arc, at right angle* to, and 

 terminating in, the radius O A, drawn to tlie commence- 

 ment of the arc. 



Cosine. And the cosine of the same angle is the por- 

 tion OH of the radius between the centre O and the tout 

 n of the sine. 



Tangent. The tangent of the same angle is the line A.t, 

 touching the measuring arc A B at its commencement A, 

 and terminating in the prolonged radius drawn through 

 the extremity B. 



'int. And this prolonged radius, that is, the line 

 Ot, is the secant of the angle A O B. 



As A is regarded as the origin or commencement of 

 the arc which measures the angle A O B, so C is regarded 

 as the commencement of the complement of that arc ; 

 that is, of the arc which, united to the former, makes up 

 a quadrant or 90. The sine, cosine, <tc., of this arc (or 

 of the angle which it measures) are the cosine, sine, <fcc. , 

 of the former, as is obvious ; and the tangent and secant 

 of the latter are called the cotangent and cosecant of the 

 former ; thus : 



Cotangent. The cotangent of the angle A OB, or of 

 the arc A B which measures it, is the line Cit, drawn to 

 touch the complement of that arc at C, and terminating 

 in the secant Ot, or in that secant prolonged. 



Cosecant. And Ou, drawn from the centre through 

 B, up to the cotangent, is the cosecant of the au"le 

 A O B, or of the arc A B. 



The lines here defined refer to circular arcs .and the 

 angles which they measure. The sides O A, O B of the 

 angle at O, are each equal to the radius of the circle 

 here considered : if these sides or radii had been of any 

 other length, no change would have taken place in the 

 angle at O ; neither would any change have taken place 

 in the number of degrees contained in the measuring arc ; 

 but the actual length of that arc would have varied, 

 being longer than AB in a circle of longer radius, an I 

 shorter than A B in a circle of shorter radius. On this 

 account the sine, cosine, &c., of an arc are not lines of 

 fixed lengths for a fixed number of degrees : a degree of 



