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1. If upon BM be erected infinitely narrow parallelograms BGhb and M Nom and their distance Mb and altitudes MN, BG be
Mm + Bb given, and the semi sum of their bases
be also given and
Mm - Bb called s and their semi difference
be called x: and if the
2 lines BG, bh, MN, mo, butt upon the curve nNgG in the points n, N, g, and G, and the infinitely little lines on and hg be equal to one another and called c, and the figure mnNgGB be turned about its axis BM to generate a solid, and this solid move uniformly in water from M to B according to the direction of its axis BM :
MN x ġ
the summ of the resistances of the two surfaces generated by the infinitely little lines Gg, Nn shall be least when gG!! is to nN!! as BGx Bb to MN x Mm. For the resistances of the surfaces generated by the revolution
MN of Gg and Nn are
that is, if Ggquad Ggquad
BG MN and Nnquad be called p and q, as and and their summ
9 BG MN is least when the fluxion thereof
is р 9
99 nothing, or
99 Now p=Ggquad = Bbquad + ghquad = 88 – 2sx + 2x + cc and therefore p=– 28€ + 2x, and by the same argument ♡ 2sic + 2.cic and BG x 2sic 2xcić MN x 2sc + 2acic
2 MNx8+2 therefore pp 99
PP and thence pp is to 99 as BG x 8 to MN x 8 + x, that is, gG! to nN!! as BG x Bb to MN x Mm.
2. If the curve line DnNgG be such that the surface of the solid generated by its revolution feels the least resistance of any solid with the same top and bottom BG and CD, then the resistance of the two narrow annular surfaces generated by the revolution of the [infinitely little lines nN] and Gg is less then if the intermediate solid bgNM be removed (along CB without altering Mb, until bg comes [to BG], supposing as before that on is equal to hg,) and by consequence it is the least that can be, and therefore gG99 is to nN19 as BG ~ Bb [is to MNR Mm].
*[Also if] gh be equal to hG so that the angle [gGh is 45degr] then will 4B699 be [to nN" as BG ~ Bb is to] MN * Mm, and by consequence 4BG!! is to GR" as BGA is to MN BR or 4BG! X BR is to GRcub (as GR to MN].
Whence the proposition to be demonstrated easily follows.
But its to be noted that in the booke pag 327 lin. 7 instead of Quod si figura DN FB it should be written Quod si figura DNFGB, and that DNFG is an uniform curve meeting with the right line GB in G in an angle of 135degr.
I have not yet made any experiments about the resistance of the air and water nor am resolved to see Oxford this
But perhaps the next
may. I had answered your letter sooner but that I wanted time to examin this Theorem and the Lem. 1 in the 3d Book. I do not see how to derive the resistance of the air from the ascent of water. The reasoning which must be about it seems too complicate to come under an exact calculus, and what allowance must be made for the retardation of the water by the contact of the pipe or hole at its going out of the vessel is hard to know.
II. LIST OF PROPOSITIONS APPARENTLY INTENDED TO BE INSERTED
IN A 2ND EDITION OF THE PRINCIPIA.
In Theoria Lunae tractentur hae Propositiones.
8 PROP. XXV. PROB. V. PAGE 434, PRINCIP. Orbem Lunae ad aequilibrium reducere.
* If the altitude of the frustum of the cone spoken of in the preceding paragraph be infinitely small, the semi-angle of the cone becomes equal to 45°. Hence when the total resistance is a minimum, the curve meets the extreme ordinate GB at an angle of 45°.
5 PROP. XXVI. Aream orbis totius Lunaris in plano immobili descriptam mensi synodico proportionalem esse.
6 PROP. XXVII. Invenire distantiam mediam Lunae a Terra.
7 PROP. XXVIII. Invenire motum medium Lunae.
1 PROP. XXIX. In mediocri distantia Terrae a Sole invenire vires solis tam ad perturbandos motus Lunae quam ad mare movendum.
2 PROP. Invenire vires Lunae ad mare movendum.
3 PROP. XXX. Invenire incrementum horarium areae quam Luna in orbe non excentrico revolvens radio ad terram ducto in plano immobili describit.
4 PROP. XXXI. Ex motu horario Lunae invenire distantiam ejus a terra.
10 PROP. Invenire formam orbis Lunaris non excentrici.
Invenire variationem Lunae in orbe non excentrico.
9 PROP. Invenire aequationem parallacticam.
Invenire formam orbis Lunaris excentrici.
13 PROP. Invenire incrementum horarium areae quam Luna in orbe excentrico revolvens radio ad terram ducto in plano immobili describit.
Invenire parallaxim solis.
PROP. Invenire motum horarium Apogaei Lunaris in Quadraturis consistentis.
PROP. Invenire motum horarium Apogaei Lunaris in conjunctione et oppositione consistentis.
Invenire locum solis.
Ex Solis motu medio et prostaphaeresi dabitur locus centri gravitatis Terrae et Lunae deinde ex hoc loco et parallaxi menstrua (quae in quadraturis Lunae est 20" vel 30" circiter) dabitur locus terrae cum loco opposito solis.
PROP. Invenire motum Apheliorum.
Invenire motum nodorum.
Nodus orbium Jovis et Saturni movetur in plano immobili quod transit per nodum illum & secat angulum orbium in ratione corporum in distantias ductorum inverse, id est in ratione equalitatis circiter, existente angulo quem hoc planum continet cum angulo orbis Jovis minore quam angulo altero quem continet cum orbe Saturni. Serventur forte inclinationes orbium omnium, ad hoc planum, & quaerantur motus intersectionum quas orbes cum ipso faciunt et habebuntur motus planorum orbium respectu fixarum.
PROP. Invenire perturbationes Orbis Saturni ab ejus gravitate in Jovem oriundas.
PROP. Invenire perturbationes Orbis Jovis ab ejus gravitate in Saturnum oriundas.
In systemate Planetarum invenire planum immobile.
A centro solis per orbes Planetarum ducatur linea recta sic ut si Planetae singuli in minimas suas ab hac linea distantias ducantur, summa contentorum ad unam lineae partem aequetur summa contentorum ad alteram; et haec linea jacebit in plano immobili quam proxime.
Vel sic accuratius:
Per solem et orbes Planetarum et commune centrum gravitatis eorum omnium ducatur linea recta sic ut si sol et semisses Planetarum in minimis orbium ab hac linea distantiis ad utramque solis partem siti augeantur vel minuantur in ratione distantiarum verarum a centro solis ad distantias mediocres ab eodem centro, deinde ducantur in distantias suas ab hac linea: summa productorum ab una rectæ parte et ab una etiam parte communis centri gravitatis, conjuncta cum summa productorum ex altera utriusque parte aequetur sun productorum reliquorum : jacebit haec recta in plano immobili, et hujusmodi rectae duae planum illud determinabunt.
III. ON THE MOTION OF THE APOGEE IN AN ELLIPTIC ORBIT OF
VERY SMALL ECCENTRICITY.
From a somewhat mutilated MS. which seems to have been prepared
for the press.
Si Luna P in orbe elliptico QPR axem QR, umbilicos S, F habente, revolvatur circa Terram S et interea vi aliqua V a pondere suo in Terram diversa continuò impellatur versus Terram; sit autem umbilicorum distantia SF infinitè parva: erit motus Apogaei ab impulsibus illis oriundus ad motum medium Lunae circa Terram in ratione composita ex ratione duplae vis V ad Lunae pondus mediocre