Infinite Series • 
93 
If H= 
and /, u, j 3 ,y, &c. whole 
numbers and n = i ; then can the fum of any feries of the above« 
mentioned formula be found in finite algebraical fundions of x+, 
and the circular arcs and logarithms of them> 
n THE dodrine of proportional parts was probably very, 
early known in the asra of fcience; for when men could not 
find the exad value of a quantity, they were induced to find 
near approximations by trials, and* from thence, by proportion, 
an approximation ftill nearer : which method is commonly 
denominated the Rule of Falfe. 
This was often found to deviate considerably from the exad 
value ; and the fame operation was repeated, which frequently 
produced a nearer approximate value, and fc on. 
This method of approximations, the moft general yet 
known, has been ufed in refolving problems by feveral of the 
moft eminent mathematicians in different ages, and in this 
particularly by M. Euler. 
2. The following observation,.. T believe, was firft publiflied 
in the Meditationes , in the year 1770, viz. that the convef- 
gency of the approximate values, found by the rule of falfe 
find method of infinite feries, generally depended on this, vi% 0 . 
E" A R T. m 
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