t%6 Carnot on the Theory of 



of a very great number of ficlcs. If, indeed, we conceive a 

 regular polygon to be inferibed in a circle, it is evident that 

 thofe figures, though they be always different, and can never 

 become identical, yet refemble each other more and more as 

 the fides of the polygon increafe in number. The circum- 

 ferences of thofe figures, their furfaccs, the folids formed by 

 their revolutions about a given axis, the analogous lines 

 drawn without and within them, the angles formed by thofe 

 lines, Sec. are, if not respectively equal, at lead the nearer to 

 equality the more the number of fides augments. Hence, 

 by fuppofing the number of fides to be very great, we may, 

 without any fenfible error, attribute to the circumfcribing 

 circle, the properties which we have difcovered to belong to 

 the inferibed polygon. 



Farther, the fides of this polygon evidently diminifh in 

 magnitude in proportion as their number is augmented ; and 

 confequently, if we fuppofe the polygon to be really compofed 

 of a very great number of fides, we may alfo affirm that each 

 of thofe fides is really very fmall. 



This being underftood, if, in the courfe of calculation, a 

 eafe happen to occur in which many operations would be 

 much Amplified by neglecting, for example, one of thofe 

 fides, which is fmall in comparifon with a given line; that is, 

 by employing in the calculation that given line inftead of a- 

 quantity equal to the fum of that given line and the little fide 

 in quedion, it is clear that this may be done without incon- 

 venience; for the refulting error muff be extremely fmall, 

 and it will not be worth while to inquire into its value. 



3. For example, if it were propofed to draw a tangent to 

 a oiven point M, of the circumference of the circle MBD 



(fig. i>* 



Let C be the centre of the circle, and DCB the axis. Sup- 

 pofe the abfeifs DP = x, the correfponding ordinate MP =jv 

 5nd let TP reprefent the fnbtangent required. 



In order to find that fubtangent, let us confider the circle 

 as a polygon of a very great number of fides. Let MNbe 

 one of thofe fides, and be produced till it meet the axis. It 

 W-ill then evidently be the tangent in qucfiion, becaufe 

 ItJ&lls not within the polygon. Moreover, let fall the per- 

 pendicular 



