Review of the Principia of Newton. 243 
val work, for, of them, if he was not the sole, he was cer- 
tainly the principal inventor; of his philosophy, nothing can 
be said, more or less, than that it is entirely his own, and by 
the power of truth, has subverted the Aristotelian, Cartesian, 
and all other systems. 
he 11 preparatory Lemmas in our opinion, are the most 
concise, perspicuous, and complete demonstrations of the 
vanishing ratios of variable magnitudes, which have ever 
appeared. The methods of Archimedes and of the ancients 
have always been considered as elegant and conclusive ; but 
they dwindle very much when compared with those of New- 
ton; and I know not whether any of the boasted methods of 
the modern analysts would not suffer even more in a com- 
parison of that, which in the mathematics should ever be 
considered as the most important object, and as comveehiia 
its universal value, logical evidence. 
The 7th, 9th, 10th, and 11th Lemmas, are the foundation of 
many of those intricate theories of curvilinear ratios, and o 
the variation of curvature which, since the time of N ewton, 
— pe be have been spun outinto volumes. These ulti- - 
mat el amie are not the ratios of 
varie = Sete dependent on e another, while 
actually have an augmentation or anh inution, for unless 
they have always to one another, a esti ratio, that ratio 
must vary from the true ratio of variation at the very point 
from whence the variation commenced; neither is it the 
imagined ratio of the quantities, when their movements have 
actually vanished, called by Berkley, the ghosts of tw mie 
quantities ; but it is the ratio under which they or 
These ratios 
m 
other ratio between them and their limits, can be taken. 
The Fine, therefore, as approximating to them by less than 
any finite scenines are the proper, determinate, and only 
fixed ratios in questi 
he modern iahytieal methods, which consist in assuming 
the second term of a developed function of an increment as 
that ultimate ratio, is either a mere assumption, or it must be 
founded on the same logical principles of reasoning. In some 
parts of this work, and more particular in the quadrature 
of curves, we have the substance of this theory of derivative 
functions, as will appear in the se uel of our review. The 
2d section, or the first which Sites to the general: sub- 
