<42r>

Jan: 22: 1675

Mr Dary

With my thanks for your problem, I should have sent you a Continuation of ye two series you desire but I have not any Computations of them by mee, & perceive them so tedious to Compute yt I am constrained to deferr them to another tyme: The series for the length of the Ellipsis I Computed when I sent it you, & sent you so much as I Computed of it. But at prsent in stead of these I have sent you an approximation for the length of the Ellipsis which you may send Sr Anthony Dean instead of that you propounded, if you please.

Suppose AB, AD, rectangular Conjugate semidiameters of the Ellipsis, BCD a quadrant of it, AD it {sic} chord bisected in H; Draw AH abutting upon the Ellipsis at C, Joyne BC and CD, Take $BE=BC+CD$, & $EF=\frac{1}{3}DE$: and BF shall be the length of the quadrant quamproximè: This is derived from Hugeniu's Quadrature of ye Circle, and I beli{illeg}|ev|{sic} approaches ye Ellipsis as near as his doth the Circle.

In like maner if you would know the length of any other arch; as BGC, bisect its Chord in K, Draw AKG, and ye Chords BG & GC: and make $\frac{4BG+4GC-BC}{3}$ ye length of the arch BGC.

Thus you may find BC and CD severally and the sue of them will give ye Quadrant BCD, exacter then before.

I: Newton

|Dr Barrow's Euclid I make no question but is safe enough now I know it was delivered.|

<42av>

|Approach for the Perimeter of an Ellipsis|

|Mr Isaac Newton's Letre to Mich: Dary Jan: 22: 1675|