232 Involutions and Evolutions by Logarithms 



Art. XXL — General Rule for Involution and Evolution by 

 Logarithms ; by J. L. Riddell, M.D., Professor of Chemistry. 



Having recently had occasion to use decimal exponents, in the 

 involution of decimal quantities ; and not finding in any mathe- 

 matical works at my command, any directions or rules therefor, 

 sufficiently explicit, it became necessary to seek out de novo some 

 certain and convenient plau of operation. 



To the following method, among several, I give preference; 

 and I offer it for publication, (though I cannot flatter myself that 

 it will be found by mathematicians to possess novelty,) because I 

 am sure it may prove convenient and reliable to students and 

 others, especially in the management of decimal quantities with 

 decimal exponents. 



The truth of the method must, upon inspection, be so obvious 

 to mathematicians, as to render an accompanying demonstration 

 unnecessary. 



Precaution. — Let the numerical quantity to be involved, and 

 also if convenient the exponent, be expressed in integers or deci- 

 mals, or both if necessary. 



General Rule for Involution and Evolution. 



Take the decimal part of the logarithm of the numerical quan- 

 tity from the tables, and make the index equal the number less 

 by 1 than the number of places in the quantity from the first sig- 

 nificant figure or place on the left, to the last figure on the right. 

 Multiply this logarithm by the exponent, and from the product, 

 subtract the product of the exponent into the number of decimal 

 places in the quantity : the remainder will be the logarithm of the 

 root or power sought, with its appropriate index and sign. 

 In order to express this rule by symbols. 



Let Q,~ the numerical quantity to be involved or evolved. 

 Dec log Q, 3= the decimal part of the logarithm of Q. 



R s» the root or power sought. 

 M « the exponent. 



N = number diminished by unity of integral and deci- 

 mal places in Q,, from the first significant figure 

 on the left, to the last place on the right. 

 D= number of decimal places in Q,. 

 The formula then becomes 



M (N+dec log a)-MD=log R. 

 Example 1. — Let it be required to raise 01 11 to the OOlll 



pow 



o 



£ = •0457 Ml. M = 001ll. N = 2. D~3. 



0111x20457141 -00333= -1+8837782= log R, the 

 •responding number of which is 0-7652. 



