[ 20 7 '] 
XXXII. Of the Theory of Circulating De- 
cimal Fra&ions. By John Robertfon, 
Lib . R. S. 
Read Nov. 24., 
1768. 
T HE great advantages arifing from 
the ule of Arithmetic, particularly 
in philofophy and commerce, is fufficiently known 5 
therefore, every ftep taken towards its perfec- 
tion, has always been countenanced by thole who 
were bed: acquainted with its nature and value. On 
thefe motives I have been induced to offer the annexed 
paper to this learned Society. 
Regiomontanus, it is faid, firft among Europeans* 
added to the then known arithmetic, an operation 
by decimal fractions; which he exemplified in his 
triangular table. Its utility was readily feen, and 
embraced in many nations, and particularly in this > 
where it appears to have been cultivated in its theory, 
and facile modes of operation, more than in other 
places. 
Many writers have remarked its excellency in nu- 
meral computation, and have pointed out compen^ 
diums to avoid the trouble of writing down fuper- 
fluous figures ; particularly in the operations with 
concrete numbers, or thole relative to money, weight, 
and meafure j where the gradations from one deno- 
mination to another do not proceed in an uniform 
progreffion. 
In finding the decimal values of the fractional 
parts of concrete, and other numbers, it often hap- 
D d 2 pens, 
