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

Hadly Suffolk. Aug. 22d.

Amasia Cappadociæ65.5243.20
Amida Mesopotamiæ78.1539.30
Amisus Cappadociæ65.5044.15
Anararbus Ciliciæ64.2038.50
Anemurium Ciliciæ65.1036.50
Antiochia Syriæ68.1036.20
Antiochia ad Taurum68.4039.20
Antiochia Ciliciæ62.3038.30
{Arania} Mesopotamiæ79.5034.20
Arbela Assyriæ89.035.52
Arbua Persidis92.1530.0
Aria Ariæ106.4036.45
Arsatia Mediæ91.0031.30
Armusa Persiæ sinûe95.3023.30
Arserate Armeniæ maj.79.3043 30
Artaxata Armeniæ major78.042.40
Artemita Armen maj78 4040 30
Aziris Armeniæ major72.0042 30
Babylon Babyloniæ79.00.35.00
Berrhæa Syriæ71.{3}0.36.00
Cæsaria Cappadociæ seu Mazaca64.40.41 40
Carrhæ Mesopotamiæ73.20.36.10
Chaboras Mesopotamiæ78 055 30
Chorsa Armeniæ maj74 4042 30
Chotena Armen min67 3040 40
Cyropolis Mediæ83 12.44 00
Cyrrhum Syriæ70 1036 0
Damascus Syriæ69 033 0
Ecbatana Mediæ88 041 10
Edessa Mesopotamiæ72 3037 30
Emesa Syriæ69 4034 0
Gabala Syriæ68 2034 56
Hecatompilos Parthiæ96 037 50
Heliopolis Syriæ68 4033 40
Heraclea Syriæ68 2035 10
Hierapolis Syriæ70 3038 0
Iban Armeniæ majoris87 0040 5
Iconium Cappadociæ63 4539 49
Marde Mespopotamiæ76 0.38 15
Melita Melitane Arm. min71 0.40 32
Nicephorium Mesopotamiæ73 6.35 20
Nicopolis Armeniæ69 0.42 25
Nisibis Ariæ111.0.35.3
Nisibis Mesopotamiæ75.10.37.30
Nyssa & Nysa Arm. min66 30.40 20
Orchoe Babyloniæ78 30.32 40.
Palmyra Syriæ71 30.34 0
Persepolis Persidis91 033 20
Samosata Syriæ71 3037 36
Samunis Mediæ79 046 40
Sebaste Cappadociæ
{Seleucia} magna Mesopotamiæ79 20.{illeg}
Seleucia Syriæ{68 36}{illeg}
S{illeg}bra Armen. major{illeg}{illeg}
Susa Susianæ{illeg}{illeg}
Tabresium Mediæ89.{illeg}{illeg}
Teredon Babyloniæ84.1031 37
Thospia Armen maj76 4041 17
Trapezus Cappadociæ{illeg}644 {illeg}

For his worthy freind Mr Is: Newton

Math: Prof.


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