B 1 N 



..a.-r^'=«'[,-^(i)-i(i)'-ji-. 



And, in the fame manner, if I, divided by the cube root 

 of the fquare of j ± *, be converttd into a fcrics, we (hull 



2-S-7 / * 



B I N 



con.lroAing both figurate numbers and the coefficients of 

 the terms of the various powers of a binomiiil, which, lince 

 his time, has been often uled for thefe and other piirpofes ; 

 and, more than a century after, was, by Pafcal, otherwile 

 called the arithmetical triangk, and of which he has com- 

 moniy been called the inventor, though he only mentioned 

 fome of its additional propeities. 



+ 



•S-7 / * \ ' 



5:c. 



iiut thtfc feries are only commodious in calculation, in 

 proportion to tlieir degree of convcrgcncy. For ii N be 

 made to repreftnt the rank wliich any term holds in the fe- 

 ries anfing from llie binomial a—b being raifed to the mth 

 power, then that term will be to the following one as I to 



t 



-N+i 



N 



from which it is evident, that for the 



terms of the fcrics to go on dccreafnig, bxm — 'ii+l, taken 

 pofitively, mull be always Itfs than <7N. 



With refpeft to the hiftory of thij theorem, the prevail- 

 ing opinion, till within thefe few years, has been, that it 

 was not only invented by Newton, but full given by him in 

 that ftate of perfcftion, jn which the terms of the feries, 

 for any affigntd power whatever, can be found, indepen- 

 dently of the terms of the preceding powers ; viz. the 

 fccond term from the fnft, the third from the fecond, 

 the fourth from the third, and fo on, by a general rule. 

 But it has Gnce been found, that in the cafe of integral 

 powers, the theorem had been defcribed by Briggs, in his 

 " Trigonometrica Britannica," long before Newton was 

 born i and that, by the general law of the terms, independ- 

 ently of thofe of the preceding powers. For, as far as re- 

 gards the generation of the coefficients of the teims of one 

 power from thofe of the former ones, fucceffively one after 

 another, it was remarked by Vieta, Oughtred, and many 

 others ; and was not unknown to much more early writers 

 on arithmetic and algebra, as will be manifeft by a flight in- 

 fpeftion of their works, as well as the gradual advance the 

 property made, both in extent and perfpicuity, under the 

 hands of the latter authors, mofl of whom added fomethiug 

 more towards its pcrftdtion. 



The knowledge, indeed, of this property of the coeffi- 

 cients of the terms of the integral powers of a binomial, is, 

 at lead, as old as the praftice of the extraftion of roots, of 

 which it is both the foundation and principle. And as the 

 writers on arithmetic became acquainted with the nature of 

 the coefficients in the higher powers, they extended the ex- 

 traftiou of roots accordingly, ftill making ufe of this pro- 

 perty. At 6rll, they appear to have been only acquainted 

 with the nature of the iquare, the coefficients of which are 

 the three terms, i, 2, 1 ; and, by their means, extrafted 

 the fquare roots of numbers, but went no farther. The na- 

 ture of the cube next prtfented itfelf, which confifts of the 

 coefficients, i, 3, 3, i ; and, by means of thefe, they ex- 

 trafted tlie cube roots of numbers, in the fame way as is 

 praftifed at prefent. And this was the extent of tlw;ir ex- 

 traftions, in the time of Lucas de Burgo, who, from 1470 

 to 1500, wrote feveral tra£ls on arithmetic, containing the 

 fubllance of what was then known of this fcience. 



It was not long, however, before the nature of the coeffi- 

 cients of all the higher powers became known, and tables 

 formed for conftrufting tliem indefinitely. For, in the year 

 1543, Michael Stifeliuj, a German, publitlied an excellent 

 work on arithmetic and algebra, under the title of Arlth- 

 Diiika Integra, in which he gives the following table, for 



15 

 21 

 28 



l^ 

 45 

 55 

 66 

 78 



9' 

 105 



120 

 136 



10 

 20 



35 

 56 



8+ 

 120 



35 

 70 

 126 

 210 



i65'33o 



495 



715 



1 00 1 



5365; 



220 

 286 

 364 

 455 

 560 

 680238016188 



126 

 252 

 462 



792 

 1287 

 2002 

 3303 



- 1 

 1820:4368 



462 



924 



1716 



3003 



5005 



[1716 

 3432 

 16435 



, _ 6435 



S008 11144012^70 



123761944.8,24310124310! 



In this table Stifelius obferves, that the horizontal lines 

 furnidi the coefficients of the terms of the correfpondent 

 powers of a binomial ; and teaches how to ufe them in ex- 

 trafting the roots of all powers whatever. The fame table 

 was alfo ufed, for a fimilar purpofe, by Cardan, Stevin, and 

 other writers on arithmetic ; and it is highly probable that 

 it was known much earlier than the time of Stifelius, at 

 leaft as far as regards the progreffions of figurate numbers, 

 which had been amply treated of by Nicomachus, who lived, 

 according to fome, before Euclid, but not till long after 

 him, according to others ; and whofe work on arithmetic 

 was publiflied at Paris iu 1538, and is fuppofed to have been 

 chiefly copied in the treatife on the fame fubjedt by Boethius. 



The contemplation of this table has alfo, probably, been 

 attended wth tiie invention and extenfion of fome of our 

 mod curious difcoveries in mathematics, both with refpeft 

 to the powers of a binomial, the confequent extraftion of 

 roots, the dodlrine of angular feftions by Vieta, and the 

 differential method of Briggs, and others. For a few of the 

 powers or feftions being once known, the table would be 

 of the greatell ufe in difcovering and conftrudling the reft ; 

 and accordingly it appears to have been ufed, on many oc- 

 cafions of this kind, by Stifslius, Cardan, Stevin, Vieta, 

 Briggs, Oughtred, Mercator, Pafcal, &c. 



But although the nature and conftruftion of this table 

 were thus early known, and employed in raifing powers and 

 extrafting roots, it was yet only by railing tlie numbers 

 from one another, by continual additions, and taking them 

 from the table, for ufe, when wanted ; till Briggs firft 

 pointed out the way of raifing any horizontal line in the 

 table, by itfelf, without any of the preceding lines ; and 

 thus teaching to raife the terms of any integral powers of a 

 binomial, independently of any other powers ; which was, 

 in faft, giving the fubllance of the binomial theorem in 

 words, but wanting the notation in fymbols. 



It may, however, be fairly queflioncd, whether Briggs 

 knew how, even in the cafe of an integral exponent, to 

 exhibit the law of the formation of the coefficients, under 



the form "'('"-■)• C^'- j ) -. (''->-+i ) f ^^^^^ 



i . 2 . 3 n ' 6 



his method of forming the fucccflivc coefficients amounts to 



nearly 



