A History of Mathematics From Mesopotamia to Modernity

166 A History ofMathematics Modern scholars (see particularly Hall 1980) are agreed that Newton’s communication was both too lat ...

TheCalculus 167 A y C R O B b a Fig. 1Indian calculation of the arc. Ifyis the arc AC,a=AB is the ‘Sine’ of the angle AOC, BO th ...

168 A History ofMathematics Here ‘Sine’ and ‘Cosine’ mean the lines AB, BO in the diagram (Fig. 1.)—sine and cosine multiplied b ...

TheCalculus 169 the Indian work as, precisely,nota version of the (later) European ‘method of infinite series’, let alone that b ...

170 A History ofMathematics separately. This is how Newton’s first private draft of his ideas begins, if with no explanation of ...

TheCalculus 171 T qo po AA' BCB' x y Fig. 3A and A′are infinitely close on the curve (a parabola, represented as a polygon with ...

172 A History ofMathematics integration—finding the fluent—is established as the central problem; so is the fact that you can fi ...

TheCalculus 173 was in no hurry to publish—it took nine years from his discovery of the method in 1675 to his first paper, which ...

174 A History ofMathematics However, it is only after three pages of this exposition of ‘rules’ for the calculus, which tells yo ...

TheCalculus 175 we have claiming pride in the discovery, such as the one which opens this section, come in the main from some ti ...

176 A History ofMathematics Leibniz’s manuscript notes). However, none of this matters in terms of the impact such a solution wo ...

TheCalculus 177 are complicated. As has been pointed out for example, by Hall, the work would have been doubly unfamiliar to its ...

178 A History ofMathematics Fig. 5Newton’s picture forPrincipiaI, proposition 1. This argument does not use Newton’s version of ...

TheCalculus 179 Leibniz was in Italy and did not reply to this crawling letter (there is more in the same vein). However, the br ...

180 A History ofMathematics showed that it worked because two errors in calculation happened to cancel. (It just happened that o ...

TheCalculus 181 –0.5 0.5 2 4 6 8 2 –1.5 –1 1 1.5 A C B T W T 0 c Fig. 6The catenaryy=acosh(x/a)=(a/ 2 )(ex/a+e−x/a). [Jakob Bern ...

182 A History ofMathematics And it is now time to return to a version—if possible updated—of Boris Hessen’s thesis: that the who ...

TheCalculus 183 du 5 –5 55 r Fig. 7A cardioid,r= 2 ( 1 +cosθ); and an element of area. which (you can either believe this or w ...

184 A History ofMathematics T ABb E D d c Fig. 8Newton’s picture of the tangent to a curve. T F D BLA M E G e C Fig. 9Newton’s ‘ ...

TheCalculus 185 In consequence, sinceBDis equal toy,BTwill bez−x−(by+yy)/z. This is−BT=AL+(BD× GM/BL). Here the sign−prefixed to ...