# Letter to Edmund Halley on changes to the Principia

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.

My thanks for your note of

{De la Hire}

## 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 ♐.

< text from f 58ar resumes >