Letter to Edmund Halley on changes to the Principia

<58r>

S^r

In y^e Scholium you write of, the words vel Hyperbolæ in y^e 3^d line are to be struck out, & in y^e 5^t or 6^t line the words quæ sit ad $\underset{\_}{\mathrm{GK}}$ should be quæ sit ad $\frac{1}{2}\mathrm{GK}$. I send you inclosed y^e beginning of this Scholium w^th y^e 63^d figure as I would have them printed. I thank you heartily for giving me notice that it was amiss. The ground of y^e transmutation of a trape{illeg}|z|ium into a paralleg|l|ogram I lay down pag 87 in these wo{rds:} "Nam rectæ quævis convergentes transmutantur in parallelas, adhibendo pro radio ordinato primo AO lineam quamvis rect{illeg}|am| quæ per concursum convergentium transit: id adeo quia concursus ille hoc pacto abit in infinitum, lineæ autem parallelæ sunt quæ ad punctum infinitè distans tendunt." In y^e figure pag 86 conceive y^e curve HGI to be produced both ways till it meet & intersect it self \any where/ in y^e radius ordinatus primus AO: & when y^e point G moving up & down in y^e curve HI arrives at y^t intersection point, I say y^e point g moving in like manner up & down in y^e curve hi will become infinitely distant. For y^e point G falling upon y^e line OA, y^e line OD point D will fall upon y^e point A & y^e line OD upon y^e line OA & so becoming parallel to AB their intersection point d will become infinitely distant & consequently y^e line dg will become infinitl|e|ly distant & so will its point g. Q.E.D. So then if any two lines of y^e primary figure HGID intersect in y^e radius ordinatus primus AO their intersection in y^e new figure hgid shall become infinitely distant & therefore if the two intersecting lines be right ones they shall become parallel. For right lines which tend to a point infinitely distant do not intersect one another \& diverge/ but are par{allel.} Therefore if in y^e primary figure there be any Trapezium wh{ose} opposite sides converge to points in y^e radius ordinatus primus {OA} those sides in y^e new figure shal become parallel & so y^e trapezium be converted into a parallelogram.

The printed sheets I intend to looke over. M^r Paget in his stay here has noted these \errata/ of w^ch y^e 3d is a fault in y^e Copy.

p. 6, l. 27. velocitate. p. 8{,} l. 19. –tur. Sunt. p. 14{,} l. 30: reciproce ut $\underset{\_}{\mathrm{DO}}$. p. 18{,} l. 1 recta.       I wish y^e printer be carefull to mend all you note. S^r I am very sensible of y^e great trouble you are at in this business, & y^e great care you take about it. Pray take your own time. And if you meet w^th any thing else w^ch you think need either correcting or further explaining be pleased to signify it to

Yo^r humble & Obliged Serv^t

Is. Newton.

Trin. Coll.
Octob. 18. 1686.

My thanks for your note of
{De la Hire}

<58v> < insertion from the left margin of f 58v >

M^r Newton
Octob. 18. 1686

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Errata sensum turbantia
sic Emenda.

pag. 14 lin. 30 lege|.| $\underset{\_}{\text{ut }\mathrm{OK}\text{ ad }\mathrm{OD}\text{ seu }\mathrm{OL}}$. pag. 18 l. 1 recta. p. 61 l. 22 et p. 62 lin. 2 leg|pro| $\underset{\_}{\mathrm{AC}}$ lege $\underset{\_}{\mathrm{AB}}$. p. 95. l. 3 post majori adde, {sic} & perpendicularia minori. pag. 96 l. 30 & 31 lege $\underset{\_}{\mathrm{ABCdef}}$, et l. {illeg}|3|2. abcDEF p. 104 l. 16 pro ${\mathrm{GO}}^q+\left.{\stackrel{\_}{\mathrm{HG}-\mathrm{PO}}}^q\right)$ lege ${\mathrm{HP}}^q={\mathrm{GO}}^q+\left.{\stackrel{\_}{\mathrm{PO}-\mathrm{HG}}}^q\right)$ pag. 105 lin 7 pro G scribe H. pa|.| 118 l. 17 pro CP lege PfB et lin 19 pro CP lege BP. p. 122 lin 28 pro L scribe M. {illeg} p. 125. l. 16. pro omnibus altitudinibus lege omnibus æqualibus altitudinibus ‡ p 209. l. 18. pro $\mathrm{SL}\times {\mathrm{SI}}^{\frac{1}{2}}$ lege $\mathrm{SL}\times {\mathrm{SI}}^{\frac{3}{2}}$. pag 2 lin. 11 pro $\underset{\_}{2{\mathrm{B}}^{-2}\text{ seu }\frac{2}{{\mathrm{B}}^{cub}}}$ lege $\underset{\_}{2{\mathrm{B}}^3}$ et dele reciproce{.}

{p.} 416 l. 17 pro $\frac{1}{32000}$ lege $\frac{1}{3200}${.} p. 494 L. 4 lege|pro| ♏ lege ♐.

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