( 19^9 > 
rais'd to any integer power. To raifc ah Infinite Series to 
an integer power, though of an interrupted Order, with- 
out introducing any thing immaterkl, or which muft af- 
terwards be expung'd 5^ and many others. But then fo 
many Terms of the Series muft be taken in at firft as^ (hall 
ferve to the purpofes of the intended approximation, 
otherwife as often as it fhall fall fhort of that, the Opera- 
tion muft be be'gan de novo. 
Manylikewife are the Properties peculiar to this Theorem, 
and great Variety of Problems might be framed 5 and I 
fcruple not to fay, many may occur in Pradice, which 
are folvable by this, and no other Method whatever. 
Hence may be found the number of Words whereof the 
34 Letters are capable, from one Letter in each Word, to 
any number of Letters given. 
Hence may be found the number of all Numbers, to 
any given number of Places, which maybe produced from 
^ny number of Figures given. 
Hence alfo the Compafs of a Mufical Inftrument being 
given, the Time and number of the Bars, whereof each 
Tune (hall confift, the number of Tunes may be found 
which that Inftrument is capable of. 
To give an inftance of the prodigious variety that there 
is in Muficlc, I have Calculated the number of Tunes in 
Common Time, confifting of eight Bars each, which may 
be pJay'd on an Inftrument of one Note Compafs only, 
and it is this, mz. 27584. 270157; 013570. 968586. 
999728. 299176. whereas the Changes on 24 Bells is but 
620448. 401733. 239439. 3doooo, which is but 
\ - g of the number of Tunes, and yet Dr 
444588. 604583 ^ 
Wdlif in his Algebra dcmonftrates, could not be difpatch'd 
in 31557. 600000. 000000 years. 
If then the Inftrument were of as many Notes Compafs 
as any Inftrument now in ufe, how prodigioufly muft ihe 
Bbbbbbbbbbbb num* 
