•icT. i.] DISSERTATION SECOND. 13 



thus two magnitudes which are not similar, are shown to 

 have equal areas. In like manner, if a triangle and a paral- 

 lelogram have the same base and altitude, the triangle is 

 shown to be half the parallelogram; because, if to the 

 triangle there be added another, similar and equal to it- 

 self, but in (he reverse position, the two together will com- 

 pose a parallelogram, having the same base and altitude 

 with (he given triangle. The same is true of the com- 

 parison of all other rectilineal figures ; and if the reasoning 

 be carefully analyzed, it will always be found to be redu- 

 cible to the primitive and original idea of equality, derived 

 from things that coincide or occupy the same space ; that 

 is to say, the areas which are proved equal, are always 

 such as, by the addition or subtraction of equal and simi- 

 lar parts, may be rendered capable of coinciding with one 

 another. 



. This principle, which is quite general with respect to 

 rectilineal figures, must fail, when we would compare cur- 

 vilineal and rectilineal spaces with one another, and make 

 the latter serve as measures of the former, because no ad- 

 dition or subtraction of rectilineal figures can ever produce 

 a figure which is curvilineal. It is possible, indeed, to com- 

 bine curvilineal figures, so as to produce one that is rectili- 

 neal ; but this principle is of very limited extent; it led 

 to the quadrature of the lunulae of Hippocrates, but has 

 hardly furnished any other result which can be considered 

 as valuable in science. 



In the difficulty to which geometers were thus reduced, 

 it might occur, that, by inscribing a rectilineal figure within 

 a curve, and circumscribing another round it. two limits 

 could be obtained, one greater and the other less than the 

 area required. It was also evident, that, by increasing the 

 number, and diminishing the sides of those figures, (he (wo 

 limits might be brought continually nearer to one another, 



