157 



This propofition is aflumed in the third cor. of the 7th, and in the cor. of the 8th lemma, 

 I ft Book of the Principia. 



III. From the properties of limits, the three following may be felefted, as frequently 

 ufed; of which the fecond, as being more connefled with the above theorem, is particularly 

 conCdered. The others eafdy follow from the definition of a limit. 



1. Quantities which are the fame limits of the fame variable quantity, are equal to one 



another. 



2. Prop. B. If the limiting ratio of the correfponding increments of two magnitudes 



Commencing together, be always a ratio of equality ; the magnitudes thcmfelves are in a ra- 

 tio of equality. 



For : Let M and N be the two magnitudes, and let M be divided into any number 

 n of equal parts p, and let alfo N be divided into the lame number of correfponding parts 

 g, r, s, t, &c. Then the parts of N are all equal to each other, or fome of them are 

 unequal. 



If they are always all equal M : N : ^ : g, and the limiting ra«o of the increments mud 

 be the fame as the conftant ratio of the increments themfelves, and therefore M : N is the 

 ratio of equality. 



If the parts of N are not equal to each other, one of them muft be greateft and one of 

 them muft be leaft ; therefore, taking n any number, if q be greateft, and r leaft, nq is 

 greater than N, and nr lefs. If M : N be not the ratio of equality, let it differ from the 

 ratio of equality by the ratio e if. Now the ratio of M : N is between the ratios np : nq, 

 QT p : q and np : nr or p : r, confequently one of them muft differ from the ratio of equa 

 lity by a ratio greater than the ratio e : f; but by hypothefis the ratios/ : q, p : r, Sac^ 

 may be nearer the ratio of equality than by any affigned ratio. Hence the ratio M : N can- 

 not differ from a ratio of equality. ( Vid. cor. 4 Lemma Prin. Math.) 



3. If an equation exifts between the correfponding variable quantities L + ?, L -}- c, &c. 



where e, e &c. may be lefs than any afligned magnitude, however fmall, L L &c. being fixed, 



and therefore the limits of thefe quantities ; then, if in that equation L, L &c. be fubftitutcd 



ft 

 for L + f, L -f- e, &c. an equation will be had between the limits. 



IV. The following inftances will ferve for illuftrating the application of thefe principles. 



1 . In the demonftration of the above geometrical theorem, the limiting ratio of the incre- 

 ment of the difference of the elliptic arcs to the increment of the tangent is deduced by the 

 aplication of the principles of prop. A. From the limiting ratio of thefe increments, the 

 ratio of the difference of the arcs to the tangent is deduced by prop. B ; not from the ratio 

 of the evanefcent increments, but from the Hmiting ratio of the increments. 



2. The follov/ing method of finding the fine in terms of the arc, ihews the applications 

 in an algebraic procefs. 



Let j=the fine of arc a to rad. i , and let fine := — e, n reprefenting any affigned 



n n 



number, however great. Then by a well known theorem for the Cgn of a multiple are 



