





On the Fundamental Principles of Mathematics. 337 



Y. If a similar construction should exist downward, or in the 



opposite direction, the sum of the areas of both would still be 



finite; it being equivalent to 40,; but the limit of the surface 



along OY would be a straight line, (19.), specifically infinite. 



This last must be true, since S would still differ from 20,, (being 



an infinitesimal less than it,) if OY produced were only, (19.), 



relatively infinite : it will be equivalent to 20, only in case the 



border or limit OY really have no termination in the direction 

 of Y * 



Comparison and Contrast of a Finite Quantity with the Infi- 

 nite of its own Species. Relative Zero. 



(22,) The distinctions of the various infinites having now been 

 exhibited, we may be the better prepared for the comparison of a 

 finite quantity with an infinite of its own species. 



If a curve be drawn as in the figure, this curve will be the ordinary hyperbola, 

 and OY its asymptote. Now the Integral Cal >dus will indicate that tin rea bor- 

 dered by OP, OY, and the curve is not finite when the two latter are interminable 

 *n the directions in which the approach of the one to other takes place. Yet this 

 area is less than that of the other surface already described ; a portion of that other 

 surface being left out by the construction of the curve : i.e., the area bordered by the 

 curve is less than 2Q ; or it must be finite. Here, then, is a paradox. May it not be 

 true that in this case a concealed term exists in the constant which must be intro- 

 duced, in the integration; especially since the equation applicable to thi i( in- 

 tegrated according to the rule for the integration of differential quantities containing 

 j a power of the variable; will exhibit infinity in the result: it being in fact the - 

 j cepted case ; which however may be made to exhibit a finite result, when integrated 

 I V the aid of logarithms. 



The equation of the hyperbola, the asymptotes being the axes, is 



A2+B2 



xy= - =7. Hence, 



4 



x=— , and ax = — 



y V 



2 



—qdy - f 



.\ydx= — l '=-gy <*y\ 



"tfhich is the excepted case. 



M. IJAbbe Moigno (Lecons de Calcul Differential et de Calcul Integral,— Calctd 

 Integral, Ire Partie, 16,) disposes of the excepted caae, in the general, thus: Le 

 second membre de la for mule, 



8 emble devenir infini pour ■»— 1 ; mais comme on peut lecrire sou- la forme 



fx m dx= r- \-C ; 



ft devient reellement indeterniin6 ; on obtient sn rentable valeur en pi ttanl le rap- 

 port j^-f-i log. x — Qfn+i log. a, des d£riv£es du nurnerateur et du dtnomenat t 



7 faisant mv-l, ce qui donne 



dx 



f x m- ifa~~f ——\ ;. a--log. a-f C = log. ar-f-C, 



" X 



comme on le aait a priori. 

 Second Series, YoL VII, No. 21.— May, 1849. 43 



