# Letter to Edmund Halley on work on ellipses and the descent of falling bodies

July 14. 16{illeg}|8|6.

S^{r}

I have considered yo^{r} proposal about wooden cuts & beleive it will be much convenienter for y^{e} Reader & may be sufficiently handsome but I leave it to your determination. If you go this way, then I desire you would divide y^{e} first figure into these two. I crouded y^{m} into one to save y^{e} trouble of altering y^{e} numbers in y^{e} schemes you have. I am very sensible of y^{e} great kindness of y^{e} Gentlemen of your Society to me, far beyond w^{t} I could ever expect or deserve & know how to distinguish between their favour & anothers humour. Now I understand he was in some respects misrepresented to me I wish I had spared y^{e} Postscript in my last. This is true, that his Letters occasioned my finding the method of determining Figures, w^{ch} when I had tried in y^{e} Ellipsis, I threw the calculation by being upon other studies & so it rested for about 5 yeares till upon your request I sought for y^{t} paper, & not finding it did it again & reduced it into y^{e} Propositions shewed you by M^{r} Paget: but for y^{e} duplicate proportion I can affirm y^{t} I gathered it from Keplers Theorem about 20 yeares ago. And so S^{r} Christopher Wren's examining y^{e} Ellipsis over ag^{t} y^{e} Focus shews y^{t} he knew it many yeares ago before he left of his enquiry after y^{e} figure by an imprest motion & a descent compounded together. There was another thing in M^{r} Hooks letters w^{ch} he will think I had from him. He told me y^{t} my proposed exp^{t} about y^{e} descent of falling bodies was not y^{e} only way to prove y^{e} motion ofy^{e} earth & so added y^{e} exp^{t} of yo^{r} Pendulum Clock at S^{t} Helle{na} as an argum^{t} of gravities being lessened at y^{e} equator by y^{e} diurnal moti{on.} The exp^{t} was new to me but not y^{e} notion. For in y^{t} very paper w^{ch} I told you was writ some time above 15 yeares ago & to y^{e} best of my memory was writ 18 or 19 years ago, I calculated y^{e} force of ascent at y^{e} Equator arising from y^{e} earth's diurnal motion in order to know what would be y^{e} diminution \of gravity/ thereby. But yet to do this business right is a thing of far greater difficulty then I was aware of. A third thing there was in his letters, w^{ch} was new to me & I shall acknowledge it if I make use of it. 'Twas y^{e} deflexion of falling f|b|odies to y^{e} south east in o^{r} Latitude. And now having sincerely told you y^{e} case between M^{r} Hook & me I hope I shall be free for y^{e} future from y^{e} prejudice of his Letters. I have considered how best to compose y^{e} present dispute & I think it may be done by y^{e} inclosed Scholium to y^{e} fourth Proposition. In {illeg} turning over some old papers I met with another demonstration of that Proposition, w^{ch} I have added at y^{e} end of this Scholium. Which is all at present from

Yo^{r} affectionate Friend & humble Servant

Is. Newton.

<56v> < insertion from the left of f 56v > < text from f 56v resumes > < insertion from the right of f 56v >Lres from M^{r} Newton

Aug. 20. July. 14. Octob. 18