LOGARITHMS. 



0,000000000542813 ; and if the logarithm of the geometrical mean, viz, 

 4.301051/09302416 be added to the quotient, the sum will be 



4.301051709845230 = the logarithm of 20001. 



Wherefore it is manifest, that to have the 

 logarithm to 14 places of figures, there is 

 no necessity of continuing out the quoti- 

 ent beyond 6 places of figures. But if 

 you have a mind to have the logarithm to 

 10 places of figures only, the two first fi- 

 gures are enough. And if the logarithms 

 of the numbers above 20000 are to be 

 found by this way, the labour of doing 

 them will mostly consist in setting down 

 the numbers. This series is easily de- 

 duced from the consideration of the hy- 

 perbolic spaces aforesaid. The first figure 

 of every logarithm towards the left hand, 

 which is separated from the rest by a 

 point, is called the index of that loga- 

 rithm ; because it points out the highest 

 or remotest place of that number from 

 the place of unity in the infinite scale of 

 proportionals towards the left hand : thus, 

 if the index of the logarithm be 1, it 

 shows that its highest place towards the 

 left hand is the tenth place from unity ; 

 and therefore all logarithms which have 

 1 for their index, will be found between 

 the tenth and hundredth place in the 

 order of numbers. And for the same 

 reason all logarithms which have 2 for 

 their index, will be found between the 

 hundredth and thousandth place in the 

 order of numbers, Sec. Whence univer- 

 sally the index or characteristic of any 

 logarithm is always less by one than the 

 number of figures in whole numbers, 

 which answer to the given logarithm ; 

 and, in decimals, the index is negative. 



As all systems of logarithms whatever 

 are composed of similar quantities, it will 

 be easy to form, from any system of loga- 

 rithms, another system in any given ra- 

 tio ; and consequently to reduce one table 

 of logarithms into another of any given 

 form. For as any one logarithm in the given 

 form is to its correspondent logarithm in 

 another form, so is any other logarithm in 

 the given form to its correspondent loga- 

 rithm in the required form ; and hence 

 we may reduce the logarithms of Lord 

 Neper into the form of Briggs's, and 

 contrarywise. For as 2.302585092, &c. 

 Lord Neper's logarithm of 10, is to 

 1.0000000000, Mr. Briggs's logarithm of 

 10 ; so is any other logarithm in Lord 

 Neper's form to the correspondent tabu- 

 lar logarithm in Mr. Briggs's form : and 

 because the two first numbers constantly 

 remain the sEune^ if Lord Neper's loga- 



rithm of any one number be divided by 

 2.302585, &c. or multiplied by .4342944, 

 &c. the ratio of 1.0000, &c. to 2.30258, 

 &c. as is found by dividing 1.0000 j, &c. 

 by 2.30258, &c. the quotient in the for- 

 mer, and the product in the latter, will 

 give the correspondent logarithm in 

 Briggs's form, and the contrary. And, 

 after the same manner, the ratio of natu- 

 ral logarithms to that of Briggs's will be 

 found = 868588963806. 



The use and application of LOGARITHMS. 

 It is evident, from what has been said of 

 the construction of logarithms, that addi- 

 tion of logarithms must be the same thing 

 as multiplication in common arithmetic ; 

 and subtraction in logarithms the same as 

 division : therefore, in multiplication by 

 logarithms, add the logarithms of the mul- 

 tiplicand and multiplier together, their 

 sum is the logarithm of the product. 



num. logarithms, 



Example. Multiplicand.. 8.5 0.9294189 

 Multiplier 10 1.0000000 



Product 85 1.9294189 



1 



And in division, subtract the logarithm of 

 the divisor from the logarithm of the divi- 

 dend, the remainder is the logarithm of 

 the quotient. 



num. logarithms. 



Example. Dividend... 9712.8 3.9873444 

 Divisor 456 2.6589648 



Quotient... 21.3 1.3283796 



LOGARITHM, to fold the complement of a. 

 Begin at the left hand, and write down 

 what each figure wants of 9, only what 

 the last significant figure wants of 10 ; so 

 the complement of the logarithm of 456, 

 viz. 2.6589648, is 7.3410352. 



In the rule of three. Add the loga- 

 rithms of the second and third terms 

 together, and from the sum subtract the 

 logarithm of the first, the remainder is 

 the logarithm of the fourth. Or, in- 

 stead of subtracting a logarithm, add its 

 complement, and the result will be the 

 same. 



LOGARITHMS, to raise powers by. Mul- 

 tiply the logarithm of the number given 

 by the index of the power required, the 



