Methodus Directa & Geometrica, Cujus Ope Investigantur Aphelia, Eccentricitates, Proportionesque Orbium Planetarum Primariorum, Absque Supposita Aequalitate Anguli Motus, ad Alterum Ellipsews Focum, ab Astronomis Hactenus Usurpati. Anth. Edmundo Hally Jun. e Collegio Reginae Oxon

Satellites that come to pass conformable to the calculus of Cassini, and differ days and hours from the calculus and predictions made upon the hypotheses of Galilei: Besides that there should happen a great many which do not happen according to the system of Cassini. E.g. according to the hypothesis of Galilei, the fourth of the Satellites should have more than 96 Eclipses in a year, of the duration of three or four hours; but according to the system of Cassini, the same Satellite will be three or four years without suffering any Eclipse. Which proceeds from nothing but the false situation of the Orbs supposed by Galilei; as the great difference of the time of the Eclipses that happen depends from this, that neither Galileo nor the other Astronomers do separate from the proper motion of the Satellites the appearances which do befall it by that of Jupiter about the Sun. And therefore 'tis, that they have taken for a simple and equal motion a motion compounded of an equal and unequal; whence they have slipped into an error about the Mean motions, which in progress of time hath so increased, that the Configurations drawn from their hypotheses for that time have almost no likeness at all with those that are observed. These old hypotheses were therefore far off from serving to find the Longitudes, as their Authors intended them; since it was impossible for them nor only to observe the Eclipses of the Satellites for some years to the nearness of an hour, but even to make us know and distinguish at this time one Satellite from another, whereas by the System of Signor Cassini one may predict for many years to come the Eclipses of the Satellites with as much preciseness, as those of the Sun and Moon by the Astronomical Tables. Methodus directa & Geometrica, cujus ope investigantur Aphelia, Eccentricitates, Proportionesque orbium Planetarum primariorum, absque supposita æqualitate anguli motûs, ad alterum Ellipses focum, ab Astronomis hactenus usurpatâ. Auth. Edmundo Hally Jun. è Collegio Reginae Oxon. Motus Terra annus per Eclipticam, opticam inaequalitatem inducit motibus cæterorum planetarum, Astronomis Copernicani nomine Parallaxeos orbis notissimam; quam quidem inaequalitatem, ex observationibus non multâ operâ datam, methodi sequentis basin firmissimam constituâ; ubi praeter observata nihil alius supponitur, quàm quod orbis Planetarum sint Ellipses, quodque Sol in foco omnium orbibus communis, sit constitutus, & denique, quod tempora periodica singulorum singulorum ita innoteascant, ut non sentiatur error aliquis, saltem in duabus vel tribus revolutionibus: His concessis, motus Terra, pro ceteris Planetis necessario requisitus, primò aggregariendus est. Sit S Sol; ABCDE, orbis Terra; P, Planeta Mars (qui in hanc rem plurimis de causis longè praeferendius est;) & primo observetur verum tempus & locum, quo Mars opponitur Soli; tunc enim Sol & Terra coincident in linam rectam cum Marte; vel, quod fere semper accidit si habuerit latitudinem, cum puncto ubi perpendicularis à Marte demissa in planum Eclipticae incidit. Sic in Schemate, S, A, & P sunt in linea recta; deinde post 687 dies, Mars revertitur ad idem punctum P, ubi in priori observatione Soli opponebatur; Terra vero, cum non revertatur ad A, nisi post 730½ dies, in B, Solem recipit in linea SB, Martem vero in linea BP, & observatis longitudinibus Solis & Martis, omnes anguli Trianguli PBS dantur, & supposita PS 100000, in iisdem partibus invenitur longitudo linea SB; pari ratione post alteram Martis periodum, Terra existente in C invenitur linea SC, nec assimiliter lineae SD, SE, SF; differentiæque observatarum locorum Solis, sunt anguli ad Solem ASB, BSC, CSD, DSE: Sic tandem verum est ad hoc problema Geometricum: Datis tribus lineis, in uno Ellipsoide foco coeuntibus, tam longitudine quam positione, invenire longitudinem transversæ diametrii, cum distantia focorum: Cujus resolutio extenditur etiam ad reliquos planetas, si, post Theoriam motus Terra cognitam, scrutemur (secundum methodum propositam Reverendiss. Episcopo Sarisburiensi in Astronomia ejus Geometrica lib. 2. part. 2. cap. 5.) tres distantias planetarum aliquid a Sole in positionibus suis. Quoniam vero Rev. Episcopus supponit planetam ita ferri in orbe suo, ut aequalibus temporibus aequales angulos ad focum alterum Ellipsoide absolutum, & ei calculum suum superstitum, non incongruum videtur, ostendere, quomodo id ipsum fieri possit absque ista suppositione, quam observatio nos rejiciendam monet. Sit S, Sol; ALBK, orbis Terra; P, Planeta, vel Punctum in plano Eclipticae, ubi perpendicularis à planeta demissa, incidit; AB linea Apsidum orbis Terra: Observentur primò Planetæ in P, longitudo & latitudine, simulque Solis Longitudo à Terra in K; & post periodum ejusdem planetæ, Terra existente in L, observentur denno positiones Planetæ Suisque, ut prius: Jam ex observatis longitudinibus Solis & Aphelii Terræ, anguli ASK, ASL dantur, & consequenter latera SK, SL: (Nam si angulus Anomalie coaequatus sit acutus, proportio est ut differentia distantiae media & Cosinus anguli in Eccentricitatem ducti, ad distantiam Apheliam, ita Perihelia distantia, ad distantiam Planetæ à Sole in data Anomalia; quæ si angulus fuerit obtusus, primus terminus proportionis onis est summa duarum partium, quarum in priori analogia fuit differen- sia: Hujus Theorematis demonstrationem neminem Analytice modice peritum latere posse arbitror, & idcirco ei supersedco:) Jam in Tri- angulo KSL dantur latera KS, LS, & angulus KSL, quaeruntur Latus KL, & anguli SKL, SLK: Deinde in Triangulo KLP, dantur KL, KLP, differentia observatarum Longitudinum planetæ, & PKL diffe- rentia angularum SKL ultimo inventi, & SKP Elongationis Planetæ à Sole in prima observatione, quaeritur LP: Tum in Triangulo LSP, latera LS, LP, & angulus PLS elongatio Planetæ à Sole in secunda ob- servazione, dantur latus SP & angulus LSP requiruntur, quibus inven- tis, ut SP ad LP, ita Tangens Latitudinis observatae ex L, ad Tangentem Inclinationis seu Latitudinis ad Solem; & ut Co-sinus Inclinationis ad Radium, ita SP curta distantiæ, ad veram distantiam planetæ à Sole: Sic tandem invenimus positionem & longitudinem desideratam. Jam restat ut ostendam, quomodo ex datis tribus distantias à Sole cum angulis interceptis, invenienda sit media distantia cum Eccentricitate Ellipso. Sit S Sol, & SA, SB, SC tres distantia in debita positio, ductisque AE, BC, sit AB distantia focorum Hyperbola, & SA-SB=EH trans- versa diameter; quibus positis, describatur linea ista Hyperbolica, cujus focus interior est punctum A, extremitas lineæ longioris SA: Parì modo sint B, C, foci alterius Hyperbolæ, cujus diameter SB-SC=KL; ex quibus describatur linea Hyperbolica focum habens interiorem in pun- cto B: Dico has duas Hyperbolas sic descriptas se se intersecare in pun- cto F, qui est alter Ellipso quæstæ focus, ductaque linea FA, FB, vel FC, SA+FA, SB+FB vel SC+FC aquabitur transverse diametro, & SF est distantia focorum: quibus positis descriptio Ellipsoe facillima est. Cum vero hujus constructionis ratio non omnibus ita facile percipiatur, non abs re erit, illustrationem ejus aliquam asserre; Ideò dico, quod ex notissima Ellipsoe proprietate SB+FE=SA+FA, & transpositis equati- onis partibus FB-FA=SA-SB, itant etiam si FB & FA nos lateant, earum tamen differentia aequalis sit SA-SB, hoc est, EH, cujque sit ex natura Hyperbola, ut habeat quasvis duas lineas à suis foci ad quodvis punctum in sua curva constanter differentes quantitate trans- versæ diametri; constat, punctum F esse aliquid in curva Hyperbola, cujus diameter transversæ æquatur SA-SB, & Foci A, B: Parì modo demonstrari potest punctum F esse in Hyperbola cujus diameter est SB-SC, & foci B, C. Ergo necesse est, ut sit in intersectione dua- rum istarum Hyperbolarum, que, cum se se intersectent in unico solum puncto, clarè ostendunt ubi sit Focus alter Ellipsoe quæstæ. Jam ut id ipsum Analyticè expeditatur, puta factum, siique FF=1, SA-SB=FB-FA=b, AB=c, SB-SC=FC-FB=d, BC=f, siique Sinus anguli ABC=S, Co-sinus ejusdem=s. Tum ut c ad b, ita 2a-b ad 2ab-bb & \frac{2a-b}{c} = BD per 36. 3 Eucl. & ut f add, ita 2a+d ad 2ad+fd & \frac{2a+d}{f} = BG per eandem, & ut mi- nuatur labor calculi sit \frac{cc-hb}{2c} = g & \frac{b}{c} = h, similiter sit \frac{11-ad}{2i} = k Per 12 & 1 lib. 2. Euclid. 3 \frac{c}{i} = l, tunc DE = g+ha, & BC = k-la; & quoniam in omni Triangulo Obtusangulo Quadratum basis æquatur Summa Acutangulo Quadratorum laterum, & dupli rectanguli laterum in Co- sinum anguli comprehensi ducti, erit gg + 2gha + hha + kk - 2kla + llaa + 2gks - 2glsa + 2khsa - 2hlsa aequalis quadrato DG: Sed DG aequalis est sinui Anguli DFG vel DBG in a, id est FB ducto, (est enim qua- drilaterum FBDG circulo, cujus diameter est FB, inscriptum;) ideo SSaa = gg + 2gha + hha + kk - 2kla + llaa + 2gks - 2glsa + 2khsa - 2hlsa; qua equatio facile resolvitur, cum non excedat quadraticam affectam, semperque componitur ex istis Quadratis & Rectangulis; signa tamen + & -- ob diversam trium linearum constitutionem multâ cautione sunt rectangulis adhibenda. Nostram equationem aptavimus figu- ra IV, sed in alio quovis casu non erit difficilis attendenti, ex his quae superius tradidimus, similem constituere. Sic tandem absolvi propositionem meam, & ostendi, quomodo ex tribus locis Heliocentricis Planete, & distantia à Sole observatis, describi posse Orbita istius Planete; quoâ non nisi quinque talibus observationibus haec tenus effe- ctum vidimus.