638 Notices respecting New Books. 



Chap. I, to meet the objections there suggested, of which the 

 author seems to go in fear. 



The integration of the circular functions, ?/ = sin x, cos a?, can 

 be shown in a similar picture, equating the differential element 

 •of the curve to a corresponding element associated with the 

 circle. No need then for the long-winded summation penultimate 

 .to the integral, with its gritty approximation so indigestible to 

 the careful young Berkleian thinker. 



The author goes in fear of this young critic, apt to conclude 

 that the Calculus is after all only approximately true ; hence 

 much of the lengthy explanations in Chapter I, only to strew his 

 path with difficulty and trip him up. 



But if the young heretic w T ill not be convinced in bis heresy, 

 agree with him then in his error, and allow that there is an 

 outstanding error, but that the error is finally proved zero. The 

 young engineer is inclined to these heresies, but Perry knew 

 how to interest him. 



Perry states the result of a differentiation of sin mx is to 

 increase the phase angle mx by a lead Jtt, and to multiply by m ; 



~ 8in (W + e)=m sin {mx + e+±TT), 

 ax cos v J cos 



without changing jironi sine to cosine. Conversely an integration 

 reverses these steps ; it subtracts a lag of £71-, and divides by m, 



sin / , \ , 1 sin / , 7 . A 



-and no confused thinking is required of any change of sign. 



Similar results are easily remembered for a combination of 

 circular and exponential or hyperbolic functions, expressed in a 

 change of phase and a multiplier or divisor. 



The result of the operation of A -7- +B on B * n (mx + e) is to 

 give a lead tan" 1 jt and a multiplier V(m 2 A 2 -\-B 2 ) ; the in vers 



operation solves the differential equation Aj- + By = (mx + e). 



The differential equation (D.E.) is not included in the scope of 

 this treatise, and the name even is kept out of sight; so that a 

 third volume looms in the future, to replace the treatise prepared 

 by Bertrand, destroyed in the siege of Paris. 



But the novice should become accustomed at the outset to 

 the name and the simple operations. To cure his fright, the 

 integration of f(x) should be proposed as the solution of the D.E. 



( ^-=f(x) , and then the interpretation of the arbitrary constant, 



singular solution, node-, tac-, and cusp-locus, can be illustrated in 

 a graph as it occurs. 



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