504 PHILOSOPHICAL TRANSACTIONS. [aNNO iSSl. 



a, viz. bg = 1 ag; and consequently bg + (ig, or the whole line ab, is equal 

 to 3 ag: and therefore ag =^ -^ of ab, which was to be demonstrated. 



The like construction and demonstration may be used in taking away any other 

 part of any quantity, and the like part again of the first mentioned part, and so 

 in infinitum. The total sum of all the parts so taken, or supposed to be taken, 

 shall be equal to a certain quantity, or part, or fraction, whose denominator 

 shall be less by an unit than the denominator of the first mentioned part ; as 



X — JL J- _1_ -4- 1 &r JL — _J__ -1- 1 -I- 1 Sirn 



T 7 1^ 4 9 1^ 3 4 3) *-vv^. g , y -f- -j-o-gr -p 1000) ^^'~" 



And so, that which the incomparable Archimedes, in squaring the parabola, has 

 only demonstrated in one particular case, viz. -i. = ^ -|- _i^ -|- ^'_. -}- _^ -|- ^-jjVt, 

 &c. and that with a huge apparatus of preliminary propositions, amounting to a 

 whole book, is here universally demonstrated in all cases, which are infinite, and 

 by a very simple and easy method, in Des Cartes way. 



Now if each of the said ranks of powers, together with their respective roots, 

 be equal to the several numbers or fractions under A; as is demonstrated by the 

 lemma ; then is A the sum of them all, or equal to them all : that is, R + p = 

 A = R 4- 1 : for R is the same with A — -f , or l, or but {-^) one infinite part 

 larger. Wherefore p= 1. Q. E. D, viz. infinitely infinite fractions are equal 

 to a unit, viz. to the least integer root. 



Hence it appears; 1. That there is given a progression in infinitum. — 2. That 

 there is a progression not only to one infinite, but to several infinites, or rather 

 to infinite infinites. — 3. That this may be performed, viz. this calculation, by a 

 very limited and slight capacity. — 4. That this whole progression, or these infi- 

 nite progressions, may be numbered or summed up. — 5. And that into a sum, 

 not only not infinite, but so very small, as to be less than any number. — It ap- 

 pears further, that some infinites are equal, others unequal; and that one infi- 

 nite is equal to two, or three, or more, either finites or infinites. 

 For, 



1. The infinite powers of the first rank are = -i- = -j\-^, and also equal to all 

 the infinitely-infine powers of all the other ranks. 



1 8 



The infinite powers of the second rank are = 

 Those of the third are = 

 Those of the fourth are = -^ = -^i 

 Those of the fifth are = -gV = -rl 

 &c. in infinitum. 



viz. equal to the re- 

 spective mean pro- 

 portional numbers 

 between the square 

 numbers respective- 

 i^ly, ex. gr. 



4, 6, 9 



9, 12, i(j 



l6, 20, 25 



25, 30, 36 



&c. in infinitum. 



