I l 
Simplicity and Umverfality than the reft: One is the Geometrical 
Tower formed from a conftant Root ; and the other, though well known* 
yet wanting a Name as well as a Notation, may be called the Arith- 
rnetical Tower , or the Power of a Root uniformly increaling or dimi- 
nishing, and is that whofe Notation is defigned in Page 213 : The one 
is only for the Form of the Quantity itfelf, the other is for the Confti- 
tution of it from its Elements. 
Now from the Properties of either of thefe it would be eafy to lliew 
how the Quadratures of fimple Figures are deducible from the Areas 
of their limiting Polygons. I fhaU juft point out the Method from 
the Arithmetical Power, as being the Ihorteft and readieft at hand. 
Let z y zyz See, or z, z } z] 6 cc. be Quantities in Arithmetical Pro- 
greftion, diminiftiing or increafmg by the common Difference z } and 
2 
let, as before explained, dignify the Arithmetical Power of js, de- 
nominated by the potentia l Index m, namely, zXzx'z, &c. whole 
firft Root is a and laft z— w — 1 xz , which being fuppofed, the Ele- 
mentof the Arithmetical Power will be m'zxL that is, the Pro- 
duct made from the Multiplication of the two Indices, and the next 
inferior Power of the next Root in Order. For the firft Arithmetical 
2 2 2 2 
^®.». ytm Vi, _ 
Power z is as z. z , and the next z is=z z Xz — mz, 
wherefore the Difference will be as is explained. 
And confequently, fince the Sum of thefe Elements or Differences, 
taken in order from the firft to the laft, do make up the Quantity ac- 
cording to -its termini ; hence, it z be the Abfcifs of a curvilinear 
Figure whofe Ordinate y is equal to a Demonftration might 
eafily be made that the [Form of the Quantity for] the Area will be 
z m -, that is, the fame Multiple of the next fuperior Power of & divided 
by the Index of that Power. 
For fince the Arithmetical Powers do . both unite, and become the 
fame with the Geometrical Power, when the differential Index z 
is fuppofed to be nothing; the Magnitude of the Geometrical Figure* 
will be implied from the Magnitudes of the two Polygons made up of 
Re&angles, one from the increafmg Arithmetical Power, the other* 
from the diminiftiing, although it be true, that the Elements of the 
Polygons cannot be fummed up> when s, the Meafure of the Abfcifs - 
^ is.fiYppofed to be nothing 
