SrSir

In yethe Scholium you write of, the words vel Hyperbolæ in yethe 3d line are to be struck out, & in yethe 5t or 6t line the words quæ sit ad $\underset{\_}{\mathrm{GK}}$ should be quæ sit ad $\frac{1}{2}\mathrm{GK}$. I send you inclosed yethe beginning of this Scholium wthwith yethe 63d figure as I would have them printed. I thank you heartily for giving me notice that it was amiss. The ground of yethe transmutation of a trapezium into a paralleglogram I lay down pag 87 in these words: "Nam rectæ quævis convergentes transmutantur in parallelas, adhibendo pro radio ordinato primo $\mathrm{AO}$ lineam quamvis rectam 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 yethe figure pag 86 conceive yethe curve $\mathrm{HGI}$ to be produced both ways till it meet & intersect it self any where in yethe radius ordinatus primus $\mathrm{AO}$: & when yethe point $\mathrm{G}$ moving up & down in yethe curve $\mathrm{HI}$ arrives at ytthat intersection point, I say yethe point $\mathrm{g}$ moving in like manner up & down in yethe curve $\mathrm{hi}$ will become infinitely distant. For yethe point $\mathrm{G}$ falling upon yethe line $\mathrm{OA}$, yethe line $\mathrm{OD}$ point $\mathrm{D}$ will fall upon yethe point $\mathrm{A}$ & yethe line $\mathrm{OD}$ upon yethe line $\mathrm{OA}$ & so becoming parallel to $\mathrm{AB}$ their intersection point $\mathrm{d}$ will become infinitely distant & consequently yethe line $\mathrm{dg}$ will become infinitlely distant & so will its point $\mathrm{g}$. Q.E.D. So then if any two lines of yethe primary figure $\mathrm{HGID}$ intersect in yethe radius ordinatus primus $\mathrm{AO}$ their intersection in yethe new figure $\mathrm{hgid}$ 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 parallel. Therefore if in yethe primary figure there be any Trapezium whose opposite sides converge to points in yethe radius ordinatus primus $\mathrm{OA}$ those sides in yethe new figure shal become parallel & so yethe trapezium be converted into a parallelogram.