Copy of a letter from Newton to Michael Dary, dated 22 January 1675/6

<42r>

Jan: 22: 1675

M^r Dary

With my thanks for your problem, I should have sent you a Continuation of y^e two series you desire but I have not any Computations of them by mee, & perceive them so tedious to Compute y^t 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 p^rsent in stead of these I have sent you an approximation for the length of the Ellipsis which you may send S^r 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 $\mathrm{BE}=\mathrm{BC}+\mathrm{CD}$, & $\mathrm{EF}=\frac{1}{3}\mathrm{DE}$: and BF shall be the length of the quadrant quamproximè: This is derived from Hugeniu's Quadrature of y^e Circle, and I beli{illeg}|ev|v {sic} approaches y^e 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 y^e Chords BG & GC: and make $\frac{4\mathrm{BG}+4\mathrm{GC}-\mathrm{BC}}{3}$ y^e length of the arch BGC.

Thus you may find BC and CD severally and the sum̄e of them will give y^e Quadrant BCD, exacter then before.

Your Loving friend

I: Newton

|D^r 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|

|M^r Isaac Newton's Letre to Mich: Dary Jan: 22: 1675|