<61r>

A Curve passing through the ends of The Rayes PA PC PD & which are supposed to be in continued Proportion and make equall angles at the Pole P I call the rumb Spirall it is desired to draw a touch line to the said Curve both by ayd of a Series and, a geometricall approach

and this tis presumed according to Dr Wallis in Libro de Cycloide cannot be done without streightening the Curve

Conversly if one Ray as PA and the angle the touch line makes therewith be given FAP, it is desired from these data to describe the Spirall

The Rayes PC or PD and PA may be conceived to be the tangents of halfe the Complements of the Latitudes of two Places numbred with the double arke or whole Complement, and the angle DPA to be the difference of Longitude, the drawing of a touch line to such Spirall gives the angle of the rumbe from one place to the other, a Problem of great use in Navigation and no Construction or good approach for it, yet in lines not knowne, vouchsafe therefore to consider this most usefull Probleme and impart the same as an Appendage to your Letters to Leibnitz which we hope you will consent to have printed in English

<61v>

About the Rumb Spirall to Mr Newton 12 Ocr 1678

$\begin{array}{ccc}\begin{array}{c}\begin{array}{c}216\\ 108\end{array}\\ 324\end{array}& \begin{array}{ccc}\begin{array}{l}\begin{array}{ll}1500& -\\ \phantom{0}225& \end{array}\\ \begin{array}{l}1275\end{array}\end{array}& —& \begin{array}{l}\begin{array}{l}150\\ \phantom{0}75& \end{array}\end{array}\end{array}& \begin{array}{c}\text{commanding}\\ \text{you}\end{array}\\ \begin{array}{cc}\begin{array}{c}20\end{array}& \begin{array}{rr}5& 1098\\ & 549|0\\ & 274\frac{1}{2}\\ & \begin{array}{cc}22& 10\frac{1}{2}\\ 1& 6\end{array}\end{array}\end{array}& \begin{array}{l}\begin{array}{rr}167& \\ 16& 14\\ 16& 14\\ 16& 14\end{array}\\ \begin{array}{rr}217& \phantom{0}2\end{array}\\ \begin{array}{rr}167& \\ 33& :8\end{array}\\ \begin{array}{rr}133& 12\\ 2& 3\end{array}\end{array}\end{array}$