552 fl« the Methods of detecting [May, 



Thus another objection of Lagranee is removed. This objection 

 is, that what in the fluxional or difterential calculus are called 

 limits, are not so, in the strict ancient meaning of the word limit. 



Whatever be the magnitude of each of the quantities E F and 



F G = X and i, the fraction -^ is of the same value at the point 



E of the curve ; the fluxions or differentials, therefore, are not 

 zeroes, though Euler and others have endeavoured to show that 

 they are zeroes only ; but they are indeterminate quantities with a 

 determinate ratio, ' They arc, therefore, common algebraic quan- 

 tities ; and may be sometimes zeroes, sometimes infinite. 



I have taken, as a function of x, y = x". But I might have 

 taken another function of x. Perhaps it would have been better, 

 because more general, if I had taken y ^fx; y^ =/(-r + I) 

 T= fx + pi + (] i^ + r i' + &c., ? being an indeterminate quan- 

 tity ; the reasoning, however, would, with certain conditions, have 

 been the same, but probably less intelligil»le to a learner. The 

 conditions are, that from x \o x + i the curve turns its concavity 

 to the same side, and that the ordinates continue to increase. 



From what has been said, it is easy to teach the learner the 

 formulas for tangents, quadratures, cubatures, complaiiHtions, and 

 rectifications of lines, whether of single or of double curvature. 



It is likewise so easy to teach him to calculate a fe^' logarithms, 

 by the series derived from the asymptotic spaces of the hyperbola, 

 that I think the calculation and demonstration should always pre- 

 cede trigonometry. All this he may do without hearing the hyper- 

 bola even named. 



From the 47th proposition of the first book, or from the I4th 

 pro])Osition of the second book, of Luclid, he may derive an 

 algebraic equation to the circle ; and from that equation he may, 

 without a previous know ledge of the conic sections, obtain equations 

 to the ellipse, hyperbola, and parabola ; he may, indeed, obtain an 

 equation to the parabola without any knowledge of an equation to 

 the circle. Thus he would have various subjects to which he might 

 apply the fiuxional calculus. 



Might i!0t those authors wj;io in algebra treat of variable quan- 

 tities, subjoin the rate of variation, and thus make fluxions a part 

 of common algebra ? 



Article III. 



On the Methods of dctecliug Carbureted Hydrogen Gas in Coal- 

 Miiies without setting it on fire. By John B. Longmire. 



It is not generally known that miners are in possession of means 

 which, if properly used, will detect the presence of carbureted 



