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Sir

I have lately tryed to looke into Cartes 3d 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 69th. 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 − 9xx + 2bx − 29. But set you thus x 2 x 3 x 4 then the first æquation on each hand is xx = 6 the 2d æquation x3 = 29 Now heare all the midle species are lost, such as −9xx + 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 Mr Michell or Mr Yard of Kings & they will send it to your

humble & thankfull pupill

T. Horne

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 Long Lat Amasia Cappadociæ 65.52 43.20 Amida Mesopotamiæ 78.15 39.30 Amisus Cappadociæ 65.50 44.15 Anararbus Ciliciæ 64.20 38.50 Anemurium Ciliciæ 65.10 36.50 Antiochia Syriæ 68.10 36.20 Antiochia ad Taurum 68.40 39.20 Antiochia Ciliciæ 62.30 38.30 {Arania} Mesopotamiæ 79.50 34.20 Arbela Assyriæ 89.0 35.52 Arbua Persidis 92.15 30.0 Aria Ariæ 106.40 36.45 Arsatia Mediæ 91.00 31.30 Armusa Persiæ sinûe 95.30 23.30 Arserate Armeniæ maj. 79.30 43 30 Artaxata Armeniæ major 78.0 42.40 Artemita Armen maj 78 40 40 30 Aziris Armeniæ major 72.00 42 30 Babylon Babyloniæ 79.00. 35.00 Berrhæa Syriæ 71.{3}0. 36.00 Cæsaria Cappadociæ seu Mazaca 64.40. 41 40 Carrhæ Mesopotamiæ 73.20. 36.10 Chaboras Mesopotamiæ 78 0 55 30 Chorsa Armeniæ maj 74 40 42 30 Chotena Armen min 67 30 40 40 Cyropolis Mediæ 83 12. 44 00 Cyrrhum Syriæ 70 10 36 0 Damascus Syriæ 69 0 33 0 Ecbatana Mediæ 88 0 41 10 Edessa Mesopotamiæ 72 30 37 30 Emesa Syriæ 69 40 34 0 Gabala Syriæ 68 20 34 56 Hecatompilos Parthiæ 96 0 37 50 Heliopolis Syriæ 68 40 33 40 Heraclea Syriæ 68 20 35 10 Hierapolis Syriæ 70 30 38 0 Iban Armeniæ majoris 87 00 40 5 Iconium Cappadociæ 63 45 39 49 Marde Mespopotamiæ 76 0. 38 15 Melita Melitane Arm. min 71 0. 40 32 Nicephorium Mesopotamiæ 73 6. 35 20 Nicopolis Armeniæ 69 0. 42 25 Nisibis Ariæ 111.0. 35.3 Nisibis Mesopotamiæ 75.10. 37.30 Nyssa & Nysa Arm. min 66 30. 40 20 Orchoe Babyloniæ 78 30. 32 40. Palmyra Syriæ 71 30. 34 0 Persepolis Persidis 91 0 33 20 Samosata Syriæ 71 30 37 36 Samunis Mediæ 79 0 46 40 Sebaste Cappadociæ {Seleucia} magna Mesopotamiæ 79 20. {illeg} Seleucia Syriæ {68 36} {illeg} S{illeg}bra Armen. major {illeg} {illeg} Susa Susianæ {illeg} {illeg} Tabresium Mediæ 89.{illeg} {illeg} Teredon Babyloniæ 84.10 31 37 Thospia Armen maj 76 40 41 17 Trapezus Cappadociæ {illeg}6 44 {illeg}

For his worthy freind Mr Is: Newton

Math: Prof.

Cambridge.