TGL. XXIV.] PHILOSOPHICAL TRANSACTIONS. 215 



many others. But then so many terms of the series must be taken in at first, as 

 shall serve to the purposes of the intended approximation, otherwise as often as 

 it shall fall short of that, the operation must be begun de novo. Many like- 

 wise are the properties peculiar to this theorem, and great variety of problems 

 might be framed ; and I scruple not to say, many may occur in practice, which 

 are solvable by this, and no other method whatever. 



Hence may be found the number of words whereof the 24 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 may be produced from any number of figures given. 



Hence also the compass of a musical instrument being given, the time and 

 number of the bars, whereof each tune shall consist, the number of tunes may 

 be found which that instrument is capable of. 



To give an instance of the prodigious variety that there is in music, I have 

 calculated the number of tunes in common time, consisting of eight bars each, 

 which may be played on an instrument of one note compass only, and it is 

 this, viz. 27584.270157.013570.368536.999728.299176; whereas the changes 

 on 24 bells are only 620448.401733.239439.360000, which is but the 



444588 604583 P^*^^ ^^ ^^^ number of tunes ; and yet Dr. Wallis, in his algebra, 

 demonstrates, they could not be dispatched in 31557.600000.000000 years. If 

 then the instrument were of as many notes compass as any instrument now in 

 use, how prodigiously must the number of tunes be increased ; the calculation 

 of which (though much more intricate and operose) would be equally attainable 

 by our theorem. 



Of Ossifications or Petrifactions in the Coats of Arteries, particularly in the 

 Valves of the Great Artery. By William Cowper^ Surgeon^ and F. R. S. 

 N°299, p. 1970. 



How far anatomical inquiries inform us respecting the true seats and causes of 

 diseases, which have been ascribed to the want of spirits in some, and of radi- 

 cal moisture in aged people, &c. may be in some measure seen by two observa- 

 tions, among others, published in the Trans. N° 280 : the first there mentioned 

 is of a young gentlewoman, in whom the parietes, or membranes, that com- 

 pose the trunks of the arteries of the arm near the axilla, being very much 

 thickened, so that the diameter of its bore was lessened to more than a third 

 part of its natural size ; insomuch that a part of the trunk of the artery cut 

 transversely, very much resembled a bit of the stem of a tobacco-pipe, its sides 

 were so thick, and its bore consequently so much lessened : the other was of 



