VOL. XL.] VHILOSOPHICAL TRANSACTIONS. JQJ 



And hence it is, there are so many species of mathematical quantity, as there 

 are forms of composite numbers, or ways in their composition ; among which 

 there are two, more eminent for their simplicity and universality, than the rest: 

 one is the geometrical power formed from a constant root ; and the other, 

 though well known, yet wanting a name as well as a notation, may be called the 

 arithmetical power; or the power of a root uniformly increasing or diminishing; 

 the one is only for the form of the quantity itself, the other is for the constitu- 

 tion of it from its elements. 



Now from the properties of either of these, it would be easy to show how 

 the quadratures of simple figures are deducible from the areas of their limiting 

 polygons. Mr. M. just points out the method from the arithmetical power, as 

 being the shortest and readiest at hand. 



Let z, 2, z, &c. or z, z, z, &c be quantities in arithmetical progression, 

 diminishing or increasing by the common difference z ; and let, as before ex- 



plained, J'" signify the arithmetical power of z, denominated by the potential 



index m, namely, z X z X z, &c. whose first root is z, and last z — m — 1 X z; 



which being supposed, the element of the arithmetical power will be mz X i; 

 that is, the product made from the multiplication of the two indices, and the 

 next inferior power of the next root in order. For the first arithmetical 



power z is = z.z , and the next z is = z X z — mz, therefore the 

 difference will be as is explained. 



And consequently, since the sum of these elements or differences, taken in 

 order from the first to the last, make up the quantity according to its termini ; 

 hence, if z be the absciss of a curvilinear figure, whose ordinate y is equal to 

 wjz*""', a demonstration might easily be made, that the [form of the quantity 

 for] the area will be z"; that is, the same multiple of the next superior power of 

 z divided by the index of that power. 



For since the arithmetical powers do both unite and become the same with 

 the geometrical power, when the differential index z is supposed to be nothing; 

 the magnitude of the geometrical figure will be implied from the magnitudes of 

 the two polygons made up of rectangles, one from the increasing arithmetical 

 power, the other from the diminishing, though it be true, that the elements 

 of the polygons cannot be summed up, when z, the measure of the absciss z, 

 is supposed to be nothing. 



In like manner, in any other case where z and z are two abscisses, whose 



