In ye Scholium you write of, the words vel Hyperbolæ in ye 3d line are to be struck out, & in ye 5t or 6t line the words quæ sit ad GK_ should be quæ sit ad 12GK. I send you inclosed ye beginning of this Scholium wth ye 63d figure as I would have them printed. I thank you heartily for giving me notice that it was amiss. The ground of ye 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 ye figure pag 86 conceive ye curve HGI to be produced both ways till it meet & intersect it self \any where/ in ye radius ordinatus primus AO: & when ye point G moving up & down in ye curve HI arrives at yt intersection point, I say ye point g moving in like manner up & down in ye curve hi will become infinitely distant. For ye point G falling upon ye line OA, ye line OD point D will fall upon ye point A & ye line OD upon ye line OA & so becoming parallel to AB their intersection point d will become infinitely distant & consequently ye line dg will become infinitl|e|ly distant & so will its point g. Q.E.D. So then if any two lines of ye primary figure HGID intersect in ye radius ordinatus primus AO their intersection in ye 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 ye primary figure there be any Trapezium wh{ose} opposite sides converge to points in ye radius ordinatus primus {OA} those sides in ye new figure shal become parallel & so ye trapezium be converted into a parallelogram.

The printed sheets I intend to looke over. Mr Paget in his stay here has noted these \errata/ of wch ye 3d is a fault in ye Copy.

p. 6, l. 27. velocitate. p. 8{,} l. 19. –tur. Sunt. p. 14{,} l. 30: reciproce ut DO_. p. 18{,} l. 1 recta.       I wish ye printer be carefull to mend all you note. Sr I am very sensible of ye great trouble you are at in this business, & ye great care you take about it. Pray take your own time. And if you meet wth any thing else wch you think need either correcting or further explaining be pleased to signify it to

Yor humble & Obliged Servt

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 >

Mr Newton
Octob. 18. 1686

< text from f 58v resumes > <58ar> < insertion from f 58r >

Errata sensum turbantia
sic Emenda.

pag. 14 lin. 30 lege|.| ut OK ad OD seu OL_. pag. 18 l. 1 recta. p. 61 l. 22 et p. 62 lin. 2 leg|pro| AC_ lege AB_. Symbol (upwards arrow with double strike) in text p. 95. l. 3 post majori adde{sic} & perpendicularia minori. pag. 96 l. 30 & 31 lege ABCdef_, et l. {illeg}|3|2. abcDEF p. 104 l. 16 pro GOq+HGPO_q lege HPq=GOq+POHG_q 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 SL×SI12 lege SL×SI32. pag 2 lin. 11 pro 2B2 seu 2Bcub_ lege 2B3_ et dele reciproce{.}

{p.} 416 l. 17 pro 132000 lege 13200{.} p. 494 L. 4 lege|pro| lege .

< text from f 58ar resumes >

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