INTRODUCTORY EXPLANATION. 17 



it is not possible to explain either the principle, or the 

 reason of its utility., to any but a mathematician. We 

 can only explain its mere construction, as follows : 



Let A B be one (inch, for example) ; and take an inde- 

 finitely extended line A X, perpendicular to A B : from 

 A towards X let a curve be conceived to be described, 

 so that every ordinate N P shall be connected with its 

 abscissa A N, by the following law. Measure A N in 

 inches and parts of inches; and multiply the result by 

 itself; and the product by 4342945. Find the num- 

 ber to which this product is the common logarithm, 

 and divide ! 12 84 by the result. The quotient is the 

 fraction of an inch in N P, and in the table we find, 

 not N P, but the area A N P B expressed as a fraction of 

 a square inch. The curve itself is what is called an 

 asymptote to A X, continually approaching, but never 

 reaching, AX : and the whole area, AX being continued 

 for ever, is one square inch. To this table I shall have 

 continual occasion to refer : into it, in fact, is condensed 

 almost the whole use I shall have to make of the higher 

 mathematics. 



I have thus drawn the distinction between the prin- 

 ciples of the subject, as derived from very obvious 

 results of self-knowledge, and the principles of mathe- 

 matics, applied merely to the abbreviation of the tedious 

 operations which large numbers require. I now pro- 

 ceed to the several assertions which have been made 

 upon the nature and tendency of the subject. 



I. That it is not true. The whole weight of this 

 assertion, and of all arguments in its favour, falls 

 entirely upon the method of measurement in page 11., 

 and ultimately upon the second axiom, in page 9. 

 Again, as we are most unquestionably justified in say- 

 ing that it is more probable we shall draw one of the 

 c 



