C58 PHILOSOPHICAL TRANSACTIONS. [aNNO 1702. 



same man, and at the same time, but successively, by the joint advice of se- 

 veral inquisitive men, and in a considerable tract of time; yet all perhaps of the 

 same nation, and probably the English. But whoever gave the first hint of 

 this invention, certain it is, that the great improvements of the magnetic doc- 

 trine are due to the English ; and chiefly to those about London, and Gresham- 

 Coilege: and it is fit the memory of it should be preserved. 



The case is much the same with that of printing, which we cannot reason- 

 ably suppose to be invented all at once ; nor perhaps all by the same man ; 

 but rather by the concurrent advice of several, and in a considerable tract of 

 time, before it came to that degree of perfection, which we now call printing. 



A Method of squaring some Kinds of Curves, or of reducing ihem to Simpler 

 ones. By A. Devioivre. N° 278, p. 1 1 13. Translated fiom the Latin. 

 Theorem I. Let a denote the area of a curve, whose absciss is x, and or- 

 dinate a:"' V'^/x — XX. Let B denote the area of another curve, having the same 

 absciss as the former, but its ordinate .T"'~"v'rfar — xx. Put "^ dx — xx = y, 

 then will the area be A := 



.r"""' ?/ = — Q 



OT + 2 -^ 



d 2m+I „, „ 3 



m+l 2m + 4 -^ 



(!'■ Im+l 2wi-l „, o 3 



m l7n + 4, Im + l ' -^ 



rf3 ^ 1m+l ^ 2m- 1 ^ 1m-3 ,„ . ;j 

 m-1 Im + A, 2TO + 2 2m -^ 



&C. 



Where it is to be observed, 1st. That n is supposed to be a positive integer 

 number; 2d. That the quantity d" b, in the series denoted by p, must be mul- 

 tiplied into as many terms as there are units in n ; 3d. That there must be taken 

 as many of the following series, denoted by — q, — r, — s, — t, &c. as there 

 are units in n. So that, to make this plain by an example or two, \( n = 1, 



then the area will be A = f/" B x ^ —„ x'"-h/^. And if ?i = 2, then 



2m + 4 wi + 2 -^ 

 2m +1 2m— 2 1 ,,,-13 ^^ v ^"' "^ ^ "' " ^ 



4thly. That if ?/ be put = ^ dx — xx, then will Abe = Q — R-|-s— t &c. ± p. 

 Corol. 1 . If m be put equal to any term of this scries, — -i^, ; , j, y, ^, -g , &c; 

 then the quadrature of the curve whose ordinate is x^W dx — xx, or 

 "» \/dx -^ XX, becomes finite, and will be exhibited by our series. For example, 

 to find the area of the curve whose ordinate is x-^V dx — xx; suppose this 



