1 82 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738. 



for arithmetic and geometry. A truth which, though certain in itself, may 

 perhaps seem doubtful from the nature and tendency of the present inquiries in 

 mathematics. For, among the moderns, some have thought it necessary, for 

 investigating the relations of quantities, to have recourse to very hard hypo- 

 theses ; such as that of number infinite and indeterminate ; and that of mag- 

 nitudes in statu fieri, existing in a potential manner, which are actually of no 

 magnitude. And others, whose names are truly to be reverenced on account 

 of their great and singular inventions, have thought it requisite to have recourse 

 even to principles foreign to mathematics, and have introduced the considera- 

 tion of efficient causes, and physical powers, for the production of mathemati- 

 cal quantities ; and have spoken of them, and used them, as if they were a 

 species of quantities by themselves. 



N. B. In the following proposition Mr. Machin has, for the sake of brevity, 

 made use of a peculiar notation for composite numbers, or such quantities as 

 are analogous to them, whose factors are in arithmetical progression. 



The quantity expressed by this notation has a double index: that at the head 

 of the root at the right-hand, but separated by a hook to distinguish it from the 

 common index, denotes the number of factors; and that above, within the 

 hook on the left hand, denotes the common difl^erence of the factors proceed- 

 ing, in a decreasing or increasing arithmetical progession. 



Thus the quantity — ^(""denotes, by its index m on the right hand, that 



it is a composite quantity, consisting of so many factors as there are units in 

 the number m; and the index a above, on the left, denotes the common differ- 

 ence of the factors, decreasing in an arithmetical progression, if it be positive; 

 or increasing, if it be negative; and so signifies, in the common notation, 

 the composite number or quantity, n -{■ a.n -\- a — a. n-f-a — 2a.. n-^ a — 3a. 

 and so on. 



2 



For example : — ^^if" is = 7z-j-5. n-f-3. n-|- l.n — \,n — 3. n — 5. con- 

 » + 5 



sisting of six factors whose common difference is 2. After the same manner 



2 

 ^(^ is = n -|- 4. n -|- 2. n. n — 2. n — 4, consisting of five factors. Accord- 

 it + 4 



ing to which method it will easily appear, that if a be any integer, then 

 2 

 j=l{^'' + ^ will be = nn — 1. nn — Q nn — 25, continued to such a num- 



n + 2a + X 



bar of double factors, as are expressed by a -|- 1, or half the index, which in 



2 



this case is an even number. So ^('•' + ' will be equal to 



K + 2a 



n.nn — 4.nn — \6. nn — 36, and so on, where there are to be as many double 



