I think I do understand Your thoughts as to that Proposition. The conclusion of it puts me in mind of an allowance which ought to be made in Prop. xxxviith on account of the Moons coming nearer to ye Earth in Syzygiis & going further from it in Quadraturis than in her mean distance at the Octants. But this allowance would increase the number 45 so much as to give some disturbance to the XXXIXth Proposition & the Scholium of the Ivth as they now stand, unless You think fit to ballance it some other way, for there is a latitude in that xxxvIIth Proposition.

I am, Sr, Your most Humble Servant

I have reconsidered the third Corollary of the vith Proposition. And for preventing the cavils of those who are ready to put two or more sorts of matter you may add these word{s} to the end of the Corollary. Vim inertiæ proportionalem esse gravitati corporis constitit per experimenta pendulorum. Vis inertiæ oritur a quantitate materiæ in corpore ideoq: est ut ejus massa. Corpus condensatur per contractionem pororum, & poris destitutum (ob impenitrabilitatem materia) non amplius condensari potest; ideoq: in spatiis plenis est ut magnitudo spatii. Et concessis hisce tribus Principiis Corollarium valet.

Your emendations of the XIXth Proposition may all of them stand.

In the emendation of the xxth Proposition pag 425 lin. 8 the word triplicata should be struck out as you

observe. The rest may stand unto the words [Et simili computo fit Tabula sequens] correcting only the numbers as you propose & putting the numbers 229 & 230 instead of 689 & 692. The Table is computed to ye excess of 17 miles rather then to that of 32 miles, because that of 17 is the least that can be & is certain upon a supposition that the earth is uniform, that of 32 is not yet sufficiently ascertained, & I suspect that it is too big.

After the last observations of Des Hayes ending wth these words [et quod in insula S. Dominici eadem esset ped. 3, lin. 7] add this Paragraph.

Deniq: anno 1704, P. Fuelleus invenit in Porto-belo in America longitudinem Penduli ad minuta secunda oscillantis esse pedum trium Parisiensium et linearum 57, id est tribus circiter lineis breviorem quam in Latitudine Lutetiæ Parisiorum; & subinde ad insulam Martinicam navigans invenit longitudinem Penduli isochroni esse pedum trium Parisiensium et linearum 55.

Latitudo autem Paraibæ est 68r 38′ in austrum et ea Porto-beli 9gr 33′ in boream, et Latitudines insularum &c. You may here put the Latitude of Goree 148 40'. I have not books by me to examin it.

...

Let the next Paragraph run thus. Observavit utiq: ad ignem calefacta evasit pedis unius cum quarta parte lineæ......... In priore casu calor major fuit quam in posteriore, in hoc vero major fuit quam calor externarum partium corporis humani. Nam metalla ad solem æstivum valde incalescunt......... sed excessu quartam partem lineæ unius vix superante......... differentia prodiit non minor quam 118 lines non multo major quam linearum 2. Et inter hos limites quantitas mediocris est 2. Propter

calores locorum in Zona torrida negligamus tres decimas partes lineæ et manebit differentia duarum linearum circiter.......... jam autus in ratione differentiarum fiet milli arium plus minus 32. Est igitur excessus ille non minor

quam milliarium arium 32.

17, non multo major quam milli

I think the words [excessu 56" fere] are right. ffor the Moons parallax 57′ 30′′ must have an increase in the proportion of 32 miles to the earths semidiameter, that is an increase of 28", wch doubled give 56" to be added to ye diameter of the earths shadow. ffor the Suns diameter & parallax remain without sensible alteration. And for yo same reason I take [excessu 2′ 45′′] to be right.

In the calculation of the Moons force (Prop. xxxvii) your scruple may be eased (I think) by relying more upon the observation of the tyde at Chepstow then on that at Plymouth, but I have mislaid my copy of the calculation. If the nearer access of the Moon to the earth in the Syzygies then in the Quadratures create any difficulty be pleased to send me a copy of the calculation & I will reconsider it. The Latitude of Paris should be 485 50', I am Sr

I have look'd over Your new addition to ye 3d Corollary of ye vith Proposition, but I am not yet satisfied as to the difficulty, unless You will be pleased to add, that it is true upon this concession that the Primigenial particles out of which the world may be supposed to have been fram'd (concerning which You discourse at large in

ye additions to Your Opticks pag. 343 & seqq.) were all of them created equally dense, that is, (as I would rather speak,) have all the same vis Inertiæ in respect of their real magnitude or extension in spatio pleno. I call this a concession, because I cannot see how it may be certainly proved either a priori by bare reasoning from the nature of the thing, or be inferrd from Experiments. I am not certain whether You do not Your self allow the contrary to be possible. Your words seem to mean so in pag: 347. lin: 5 Optic: [forte etiam & diversis densitatibus diversisq: viribus]

I do not clearly understand how You would have ye alteration settled in Prop: xxth, I mean that which begins with [Unde tale confit Theorema] & ends with [et simili computo confit Tabula sequens]. You may be pleased to send me a transcript of ye Context leaving void spaces for the Numbers. You may let me know at ye same time time whether You choose 57200 or 57230 Toises for the Measure of a degree between the Latitudes 48°. 49°. I suppose You retain 85 lines for ye length of ye Pendulum.

I am satisfied that 56′′ is the right increase of ye shadow of ye Earth, 'twas my oversight in making the figure of ye shadow to be similar to that of ye Earth.

As to the xxxvIIth Proposition, I take it that the Moons force must be augmented in her Syzygies & diminished in her Quadratures in the proportion of 47 to 46 nearly. Whence by my computation, if nothing else be altered in the Proposition, S will be to L nearly as 1 to 5. To make S to L as 1 to 47% or 4; instead of putting L + S to L-S as 7 to 4, it may be put 46, 6

47 L+ Sto L-S as 11 to 6. But this proportion

47,7

of 11 to 6 falls without ye Limits at Bristol & Plymouth. I shall therefore leave it to Your self to settle ye whole Proposition as You shall judge it may best be done. In

ye xxvIIth Proposition I shall hereafter take notice, that I find the proportion to be as 694 to 70 instead of 6810 to 6910. I think 69 to 70 may every where be used. Your Copy of yo xxxvirth Proposition is as follows*. {Vis Lunæ ad mare movendum colligenda est ex ejus proportione ad vim Solis, et hæc proportio colligenda est ex proportione motuum maris qui ab his viribus oriuntur. Ante ostium fluvii Avonæ ad lapidem tertium infra Bristoliam, tempore verno et autumnali totus aquæ ascensus in conjunctione et oppositione Luminarium, observante Samuele Sturmio, est pedum plus minus 45, in Quadraturis autem est pedum tantum 25. Altitudo prior ex summa virium posterior ex eorundem† differentia oritur. Solis igitur et Lunæ in Equatore versantium et mediocriter a Terra distantium sunto vires S et L, et erit L + S ad LS ut 45 ad 25 seu 9 ad 5.

In portu Plymuthi æstus maris (ex observatione Samuelis Colepressi) ad pedes plus minus sexdecim altitudine mediocri attollitur, ac tempore verno et autumnali altitudo æstus in syzygiis superare potest altitudinem ejus in quadraturis pedibus plus septem vel octo. Si maxima harum altitudinum differentia sit pedum octo, erit L + S ad L - S ut 20 ad 12 seu 5 ad 3. Donec aliquid certius ex phænomenis constiterit, assumamus L + S esse ad L S (proportione mediocri) ut 7 ad 4.

Cæterum ob aquarum reciprocos motus æstus maximi non incidunt in ipsas Luminarium syzygias sed sunt tertii a syzygiis ut dictum fuit, et incidunt in horam Lunarem plus minus tricesimam sextam a syzygiis, id est, in horam solarem tricesimam septimam circiter. Oritur hic æstus ab actione Lunæ in ejus præcedente appulsu ad meridianum

* I have transcribed the Proposition from Newton's MS. Nos. 193, 194, Cotes not having copied it into this draught of his letter. The heading is 'Invenire vim Lunæ ad Mare movendum."