Letter from Thomas Horne to Newton, with notes by Newton on Middle Eastern cities

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Sir

I have lately tryed to looke into Cartes 3^d booke of the nature of æquations, thinking to understand what goes before by the helpe of his rules delivered there: & I begin to hope I may by my owne strength, & j judge it is better to find one conclusion out than have 20 shewed me, which made me defer moving questions to you so long, & partly because j cannot move my many doubts in proper termes. But I know you are to good & wise to deride me. Some Rules I think j understand in that discourse of æquations, but I stick at the first hypothesis pag 69^th. for first though I may suppose that x is æquall to 2 or x − 2 æquall to nothing &c: –; yet methinks this does not reach the universall nature of æquations, though some may be imagined so to be made & then all things follow according to the subsequent rules. let x − 2 be 0 & x − 3 = 0 x − 4 = 0 the first magnitude that rises is xx − ex + 6 & the cube (if j may so call it) x³ − 9^xx + 2bx − 29. But set you thus x 2 x 3 x 4 then the first æquation on each hand is xx = 6 the 2^d æquation x³ = 29 Now heare all the midle species are lost, such as −9^xx + 26x. In the next place why must I alter the value of x & make it some times æquall to 2 & some times to 3 methinks this is more like a square x − 2 = 0 x − 2 = 0 & xx − 4x + 4 0. &c:

In pag. 74. I cannot understand the note H.

pag. 78. I cannot find out the order of dividing the noted æquation by yy − aa − cc. neither am j able to find the meaning of those words in the bottome of the page. Id quod monstrat radiem quæsitam esse aa + cc quemadmodum per multiplicationem probari potest

pag. 79 the rule + x4*.pxx.qx.r &c. I have practised upon some of his æquations but have not the reason of it nor {Basins} demonstration. p. 137.       Nor the rule p. 81 of making 2 æquations out of one.

I remember Cartes says if one try methodically to examine his rules, one shall find the demonstration of them, be pleased to shew me the methodicall examination in that aponation pag. 78 or some other.

pray sir if you can spare time to ease a doubter, deliver a note to M^r Michell or M^r Yard of Kings & they will send it to your

humble & thankfull pupill

T. Horne

Hadly Suffolk. Aug. 22^d.

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For his worthy freind Mr Is: Newton

Math: Prof.

Cambridge.