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this enquiry he began with the more fimple feries, firft confidering 

 arithmetical progrelfions ; then he proceeded to thofe whofe 

 terms were as the fquares, cubes, biquadrates, &c. or as the 

 fquare roots, cube roots, &c. of the terms of thofe arithmetical 

 progreffions. He afterwards confidered thofe progreffions whofe 

 terms were as any dimenfion whatfoevcr of the terms of the 

 arithmetical progreffions ; that is, the indices or exponents of 

 whofe dimenfions were as any numbers, integral, fradional, or 

 furd, whether pofitive or negative. He confidered thefe pro- 

 greffions as confifting of an infinite number of terms, the laf^ 

 term, which reprefented the lafl ordinate of the curve, being 

 ftill finite ; and the intermediate terms from o to the laft, being 

 infinite in number, reprefented ordinates applied to the axis, 

 at infinitely fmall and equal diflances, between the vertex and 

 laft ordinate. Or perhaps thefe terms reprefented any other lines 

 right or curve ; or any plain or curve furfaces, in the cafe 

 of folids, which were proportional to them. At length, by an 

 induQion of particulars, he came to this general theorem, which 

 is the 64th of his Arithmetica Infinitorum, " In any infinite 

 " feries of quantities beginning from o, and continually increaf- 

 " ing according to any power whatfoever, whether fimple or 

 " compounded of fuch as are fimple, the ratio of all the terms 

 " of fuch a feries is to as many times the greatefl, as unity to the 

 " index of that power increafed by unity." And this is the fame 

 in fubftance with Sir I. Newton's firfl rule for the quadrature 

 of fimple curves, in his Anal, per Equat. t. n. infin. which was 

 inveftigated, in the manner juft now mentioned, by an induc- 

 tion of particulars by Wallis, but which Newton demonftrated 

 univerfally by an indefinite index, as was his manner, compre- 

 hending, in one general propofition, all thofe particular cafes which 



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