0,%% Carrot on the Theory of. 



investigating the value of TP, on the principle of the tangent 

 being perpeiulicular to the extremity of the radius; for it is 

 evidem that the fimilar triangle?, 



CPM, MPT, give CP : MP : : MP : TP; 



and therefore TP = == -=— , as before. 



CP a—x 



5. As a fecond example, fuppofe we arc to find the furface 

 of a given circle. Let us ftlll consider that curve as a regular 

 polvgon of a great number of fides. The area of any regular 

 polygon is equal to the product of its circumference into half 

 the perpendicular let fall from the centre on one of its fides. 

 The circle, therefore, being confidered as a regular polvgon 

 of a great number of fides, its furface ought to be equal to 

 the product of its circumference (that is, the fum of very 

 numerous fides) into half the radius; a proposition as exact 

 as the other. 



6. However vague and indeterminate the exprefhons very 

 great and very J mull } or others of the fame nature, may ap- 

 pear, we fee, by the two preceding examples, that they are 

 not unufefully employed in mathematical combinations, and 

 that '.he quantities fignilied by thefe expressions may afford 

 much help in facilitating the folution of divers questions 

 which may be propofed. For, the notion of fuch- quantities 

 being once rightly underfiood and admitted, all other curves, 

 as weil as the circle, may be confidered as polygons of a great 

 number of tides ; all surfaces may be divided into a great 

 number of fillets or zones ; all bodies may be refolvcd into 

 corpufcles ; and, in a word, all quantities may be deeompofed 

 into particles of the fame fpecies with themfelves. Hence 

 arife new relations and combinations; and we may eafily 

 judge, from the examples already adduced, of the refources 

 in calculation, which the introduction of fuch elementary 

 quantities must afford. 



7. But the advantages of admitting thefe quantities is ftill 

 more considerable than could, at first fight, have been ex- 

 pected. For we have feen, in the examples propofed, that 

 what was considered, at the outfet, as a fimple method of 

 approximation, conducts us, at leaft in certain cafes, to re- 

 fults perfectly accurate. It becomes interesting, therefore, 



to 



