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Extra info for (2s1) Designs withs intersection numbers
Not only for those formed from simple parabolas but also for those from all higher parabolas and from their eomplements, on which so far there has been 5 Fermat had done this, but Wallis was ignorant of it , see Introduction p. xiv and note 15. 4 The Arithmetic of Infinitesimals complete silence from everyone, nor has anyone (as far as I know) anywhere attempted it . But also I saw here that it was possible to derive as a direct consequence an almost complete teaching of spirals; and indeed I have taught the comparison with a circ1e, not only for the space contained within the usual spirals (as Archimedes did), but also for that contained within other spirals.
Further, he could integrate and differentiate such functions by operating on the series term by term. 83 In those letters he was explicit about his debt to Wallis ,84 and Wallis was not slow to respond. By 1676 Wallis had completed a large part, possibly the first seventy-two chapters, of A treatise 01 algebra. It was probably Newton's Epistola posterior that prompted hirn to add a furt her twenty-five chapters in 80 81 82 83 84 Newton 1665; see also Whiteside 1961, Dennis and Confrey 1996, Steda1l2002, 175-180.
An d t herefore a lso t he lines M T , M T, are prop ort ion al to t he ares A 0 , A 0, as is dear. Then , having eonstrueted any number of straight lines MT, MT, ete. making a eo nt inuo us sequenee of ang les AMT, TM T, ete. equal to eac h other (a nd t herefore [the lines M T a re] in arit hmetic pr opor tion ), we may suppose (su perimposed on these a ngles) t he same number of sim ilar sectors (or rather , one fewer beeause a sector may not be inseribed in t he first spaee) inseribing t he figure'" M TM (bo unded by t he t rue spiral line M T a nd t he straight line TM ).