1 Ad quod Systema Mundi fa­bricentur, & quibus Planetis accommodentur.

HAe Theoricae ad Hy­potheses Copernica­nas instituuntur, in quibus cum Sol Cen­trum Mundi possideat, hujus ap­parentes motus, realiter exi­stunt in terra. Unde haec loco Solis inter septem Planetas nu­meratur.

De quinque tantum ex his septem eorumque locis investi­gandis hic dicemus. Nam Lunae motus, & passiones quas conjun­ctim habet cum terra, quia plu­res reliquis admittit varietates non nisi per instrumentum par­ticulare commode absolvi ne­queunt, quare Lunam hic mis­sam facimus.

Rursus locus terrae in his Theoricis non tam sui ipsius quam aliorum Planetarum cau­sa requiritur; quorum loca in Zodiaco deprehendi nequeunt, [Page 4] nisi prius in qua mundi parte terra sit (hoc est nos ipsi simus) dignoscatur. Interim tamen ve­rus terrae locus respectu Ecclip­ticae, & per consequens appa­rens solis, modo requiratur, hic inveniri poterit. Uti postea in octavâ Propositione indicabi­tur.

2 Quomodo tempus omne cal­culo accommodetur.

UT tempus calculo accom­modetur haec sunt obser­vanda.

• 1 Omnes motus colligendi sunt ad tempora completa.
• 2 Dies inchoatur in suo me­ridie completur vero in meridie die sequentis. Ita quod,
• 3 Meridies primi diei Janua­rii est terminus communis ve­teris, & novi Anni: periodus (sc.) praecedentis, & principium Anni sequentis.

3 Quid sit locus Planetae, cum methodo colligendi aequales Anomalias.

HAe Theoricae, uti antea di­ctum est, praecipue institu­untur ad expeditam inventionē locorum Saturni, Iovis, Martis, Veneris, & Mercurii, a cujusque diei meridiem & in formâ quâ nunc sunt ad annum septingen­tesimum [Page 5] supra millesimum abs­que sensibili errore inser­vient.

Locus Planetae est ejusdem situs ad planum Eclipticae re­spectu longitudinis in illâ, lati­tudinisque ab eadem. Cui eti­am intervallum seu distantia Planetae á terra addi poterit.

Ad haec invenienda primo dignoscendum est quaenam tempori dato debeatur Ano­malia tam terrae, quam Plane­tae cujus locus inquiratur. Hae vero Anomaliae ex propriis Ta­bulis orbitae cujusque Planetae annexis excerpenda. Numeri­que Tabulares pro gradibus graduumque partibus centesi­mis aestimandi sunt.

His praemissis modus colligendi aequales Anomalias hujus­modi est.

• Primo, Exscribe Epocham anni proxim praecedentis.
• 2 Sub ista Epochâ, seu nu­mero scribe motus competen­tes tot annis, mensibus, & die­bus quot ab anno Epochae completis sint, hi ex propriis Tabulis sunt sigillatim sumen­di, & invicem ordinatim sub­jicendi: quod ut siat numero­rum dis unctio satis doceb it.
• [Page 6]3 Horum aggregatum da­bit Anomaliam quaesitam, sin vero excedat circulum seu 360 gr. integer circulus quoties po­terit rejiciendus est, & resi­duum sumendum pro Ano­malia.

Haec tam pro terra quam Planeta sigillatim facienda sunt. Qua de causa Anomaliae terre­stris Tabula bis repetitur, ut scilicet in quaque lamia semel in promptu sit, pro singulari in­strumenti faciebus quaecunque illarum in usum venerit, & sine qua nec Planetae locus, nec passiones aliquot quibus subji­citur inveniri possunt.

Sequitur jam

• 1 Longitudinem Planetae in Ecliptica investigare,
• 2 Latitudinem ab Ecliptica investigare.

Huc rei centro instrumenti, hoc est centro Solis filum ap­pendendum est. Insuper com­paranda est tenuis e metallo re­gula cum linea fiduciali ejus­dem (aut circiter) longitudinis cujus est diametrus instrumen­ti. Quae solute sit oportet & mobilis nullo modo alligata, sed datis duobus quibuslibet instrumenti punctis applicabi­lis.

4 Cujuslibet e quinque Plane­tis longitudinem invenire.

1 COllige Anomalias tam terrae, quam Planetae cujus Longitudo inquiritur ex propriis Tabulis, uti antea praeceptum est.

2 Numera Anomaliam Pla­netae in Orbita ipsius, Ano­maliam terrae super illam ter­rae Orbitam quae in eadem in­strumenti facie, qua etiam est Planetae Theorica describitur. Haec duo puncta observa nam in illis erit & Planetae & terrae locus pro dato tempore.

3 His punctis lineam regulae fiducialem ita applicabis ut eadem regulae linea, & Solem respiciat, & limbum seu Zo­diacum secet, vel praetergre­diatur prout ratio postulet, & disponatur major ejus portio á terra versus Planetam, sae­pius enim ad operationes se­quentes illud requiretur.

4 Per circinum cape mini­mam distantiam inter Cen­trum Solis, & lineam regulae fi­ducialem, & invariatâ aperturâ fige pedem unam super ali­quem Zodiaci exterioris sive limbi gradum in eodem regulae latere in quo erat Solis Cen­trum, & versus eam Zodiaci [Page 8] plagam quae à terra versus Planetam respicit. Quae om­nia ita dirigenda sunt ut alter pes circini lineam regulae fi­ducialem tangat. Tunc enim pes iste super Zodiacam posi­tus ostendet Planetae Longi­tudinem in signis & partibus ejus.

Videas exempla post praecep­tum sequens.

5 Cujuslibet è 5 Planetis La­titudinem investigare.

1 COgnitis Anomaliis tam terrae quam Planetae, ap­plica filum Centro affixum Anomaliae Planetae in suâ Or­bitâ numeratae, & immoto filo cape minimam distantiam in­ter illud & istum Planetae cha­racterem (cujus locum inqui­ris) filo magis commodum, nam uterque aptus non erit: Et observa utrum filum Bo­realem an Australem inclina­tionem secuerit.

2 Metire istam distantiam in Scala pro inclinationibus Pla­netae, facta & ei circinus in­clinationem ostendet (plaga vero antea detecta est.)

[Page 9]3 Restant adhuc duae di­stantiae mensurandae. Prima, est distantia Planetae a terrâ, hoc est à punctis Anomalia­rum quae sunt loca eorum in ipsorum Orbitis. Secunda, est Planetae à sole. Quae fiunt ap­plicando distantias in circino captas Scalae huic rei factae Scalae (sc.) Decimali quae in singulis Theoricis grad. 360 sive exterioris Planetae pun­ctum Aphelium secat. Hoc pacto distantias ipsas, vel sal­tem earum proportionem dig­nosces.

4 Adi Scalam in partes 120 aequales divisam cum arcu gra­duationum sibi appendente, & super istum arcum numera Planetae inclinationem prius inventam cui filum applica. Deinde super eandem Scalam numera Planetae distantiam a Sole, & minimum abinde ad filum spatium per circinum cape, & serva. Denuo in ea­dem Scalâ Planetae à terra di­stantiam nota, & circini pe­dem alteram istic fige. Filum verum ita move ut pes circini alter conversus invariata aper­tura filum exacte tangat. Sic demum filum super arcum ap­pendentem ostendet Planetae latitudinem quaesitam. Quae semper ejusdem erit denomi­nationis [Page 10] cujus est inclinatio prius inventa.

Duo plenissima Exempla hic sequuntur. Longitudi­nis, Latitudinis, Distantiae­que terrae reliquorumque 5 Planetarum. Unum ad quar­tum Octobris 1649 in Meri­die. Alterum ad 19 Feb. 1651 in Meridie.

6 Quot Semidiametris terrae Planeta quispiam distabit à Sole, vel Terra dignosce­re.

MEnsuratis prius distantiis Planetae à Terrâ, & Sole in Scalis propriis ut ante prae­ceptum est

Pro

• ♄
• ♃
• ♂
• ♀
• ☿

Duc distan­tias in

• 400
• 200
• 100
• 50
• 50

Factum erit inter vallum quaesitum in Semidiame­tris Terrae.

In acquirendâ distantia Terrae & Sole majori opus est cautelâ: attamen eodem pa­riter modo investigatur.

Theoricae huic rei magis idoneae sunt istae Veneris, Mer­curii, aut Martis, si distantia Terrae à Sole mensuretur in Theorica Veneris, aut Mercu­rii, numerus inventus per Sca­lam istius laminis ducendus est in 50 numerum (scil.) Ve­neris, & Mercurii, sin vero in Theorica Martis ducatur in 100 Marti propriam.

[Page 12]

7 Ex Planetae Longitudine & Latitudine datis rectam as­censionem & declinationem invenire.

COmmodissimè haec fiunt per Astrolabia, aut instru­menta istiusmodi Spherica. Ad supplendum autem hunc defectum Scalas addidi quibus licet majori cum molestiâ, ista perficiantur. Huic rei deli­neationes in Theoricis Saturni & Iovis bis repetitae inservi­unt, ut unaquaeque lamina suam habeat Scalam istis The­oricis quae super illâ ducuntur paratam.

¶ Primo, igitur inquirenda est ascensio recta istius puncti [Page 13] Eclipticae quod longitudini Planetae respondet, quasi La­titudis esset expers. Quod per­ficitur in scala ascensionum re­ctarum partium Eclipticae. Quae ex inspectione tituli di­gnosci potest.

Numera igitur in Zodiaco Elliptico Planetae Longitudi­nem, id est, signum & gradum ubi per quartum praecedens inventus fuerit, & ibi appli­cato filo centrali observa ubi arcum secuerit notatum 1, 2, 3. Qui in gradibus graduumque partibus aestimatus ostendit differentiam Longitudinis ab ascensione recta, & proinde appellari potest Longitudinis aequatio. Haec aequatio Lon­gitudini antea inventae vel ad­denda est, vel subtrahenda prout filum ostenderit cadens in titulos Additivos, vel Sub­tractivos pone hunc differen­tialem arcum scriptos. Hoc cite facto prout oportet, sum­ma vel differentia inventa erit ascensio recta merae Lon­gitudinis Planetae. Quod pri­mum erat requisitum.

Hoc modo absque ulterio­ri [Page 14] labore acquiruntur ascensio­nes rectae vel Solis, vel Terrae, quia latitud. expertes semper versentur in plano Eclipticae.

¶ Secundo haec ascensio recta corrigenda est juxta La­titudinem Planetae ab Eclipti­ca modo aliquam (quod fre­quentissime accidit) habuerit. Et huic rei maxima pars alte­rius Systematis Scalarum in­servit. Hoc modo.

Super duodecim signis juxta ordinem quo in Ellipsi inscri­buntur (quae signis in exte­riori Zodiaco respondent licet characteres aliter signentur) & super gradus exterioris Zo­diaci (cujus gr. 30 antedictis signis per integram Scalam respondent) numera Planetae Longitudinem, & filum ap­plica. Deinde in Scalâ lineae mediae quae Centrum petit, Planetae latitudinemnu mera. A quo puncto ad filum cape per circinum minimam distan­tiam; haec minima distantia applicata Scalae lineae mediae a Centro exteriùs, aequatio­nem exhibebit in gradibus & minutis. Sit haec Latitudinis aequatio. Quae ascensioni prius inventae addi vel ab eadem subtrahi debet juxta titulos in Ellipsi notato, Haec summa aut differentia sic ultimo in­venta e [...]it exacta ascensio [Page 15] recta Planetae pro Longitudi­ne, & Latitudine datis.

¶ Ad declinationem Planetae acquirendam Zodiaco tantum utimur exteriori cum arcu cir­culari utrinque ad 25 gr. nu­merato. Hoc modo.

Numera Planetae latitu­dinem in arcu 25 grad. la­titudini Planetae pro eo tem­pore quoad plagam congruo, & illuc filum porrige. Deinde in Zodiaco exteriori (juxta ordinem signorum & gradu­um illic numeratorum) nume­ra longitudinem Planetae: in quo puncto fige circini pedem alterum; altero vero cape mi­nimam distantiam a filo: illud observans utrum in hâc ope­ratione circinus supra vel in­fra filum steterit. Minima haec distantia applicetur lineae re­ctae 35 partium ab initio Sca­lae procedendo & ostendet de­clinationem quaesiram. Pla­gam vero Septent. vel Au­stral. situs circini infra vel su­pra filum ostender. Nam su­perior situs Borealem inferior plagam Meridionalem deno­tat. Et ut haec directio sem­per [Page 16] presto sit utrisque exterio­ris Zodiaci terminis inscribi­tur.

Terrae sive Solis declinatio nullâ molestiâ invenitur ap­plicando Scalae 35 longitudini ab Ariete vel Libra in ex­teriori Zodiaco recto.

Sequitur Exemplum As­censionis rectae, & Declina­tionis Terrae reliquorumque Planetarum juxta Longitu­dines Latitudinesque in prio­ribus Exemplis inventas, & ad Meridiem quarti diei Octo­bris 1649 computatum.

8 Invenire locum Solis vel Terrae in Eclipticâ.

HOc faciliùs fit pro Terra quàm pro reliquis 5 Pla­netis, quia Terra & Latitudi­nis & commutationis est ex­pers, & ad inveniendum ve­rum locum Terrae in Eclipticâ commodius utemur majori Theoricâ: illâ (sc.) quae com­prehendit Venerum & Mercu­rium unâ parte, vel illâ alterâ quae comprehenditur à Marte ex altera instrumenti facie.

In Orbitâ Terrae numera Anomaliam ad datum tempus inventam, & ad hunc termi­num filum extende quod in exteriori Zodiaco locum terrae designabit, cujus oppositum est locus Solis.

Sic habes in duobus priori­bus exemplis locum Terrae ad datum tempus, viz. Aries 21 gr. 45 m. & Virgo 11 gr. 30 m. quorum oppositam sunt 5 loca Solis viz. Libra 21 gr. 45 m. & Pisces 11 gr. 30 m.

9 De praecipuis nonnullis Pla­netarum passionibus.

PRincipium harum Theorica­rum officium est ut per illas inveniantur loca Planetarum quoad longitudinem & latitu­dinem: quod quia jam antea tractavimus operae praetium erit de praecipuis eorum passioni­bus pauca addere. Quarum tria praecipuè sunt capita.

• 1 Planetae (ob motum lon­gitudinis quem faciunt in E­clipticâ) non nunquam viden­tur secundum seriem signo­rum procedere (hoc est) 1 Di­recti sunt in Motu. Aliquando videntur retrocedere (i.e.) sunt 2 Retrogradi. Et in illorum transmutationibus inter u­trunque horum motuum ne­cessario videbuntur stare hoc est sunt 3 Stationarii.
• 2 Loca Planetarum con­siderantur vel quoad distan­tiam à Sole, vel ab invicem; unde varios habent aspectus. Quorum 1 conjunctio dicitur quando duo quilibet Planetae sunt in eodem gradu longitu­dinis. 2 Opposite quando sunt in opposita longitudine. 3 Trinus quando ⅓ circuli [Page 19] vel quatuor signis, 4 Quarti­lis quando 3 signis vel circuli quadrante, 5 Sextilis quan­do sextâ parte circuli vel duo­bus signis ab invicem dista­bunt. Venus, & Mercurius nunquam hos aspectus praeter conjunctionem habent ad So­lem nec inter se invicem ul­lum faciunt praeter sextilem quo saepius distant.
• 3 Locis eorum ad Solem comparatis, vel sunt sub ra­diis, & dicuntur combusti. Vel post ortum Solis interdiu oriuntur, & vocantur Orien­tales: aut post Solis occasum seu noctu occidunt, & sunt Occidentales: vel Soli sunt oppositi, & dicuntur Acro­nychi. Venus & Mercurius nunquam sunt Acronychi, quia Venus nunquam à Sole ultrà 48 gr. Mercurius ultrà 29 gr. recedit.

10 De Directione, Retrogra­datione, & Statione.

CUm inventio justi tempo­ris harum mutationum in Planetarum cursibus res sit per se difficilis; per has Theoricas vix accuratè detegentur. Mo­dus optimus est (cognitis prius locis ad diem certum) pro 5 aut decimo post die eorum [Page 20] longitudines inquirere. Prae­sertim in Saturno Iove & Marte quia verò motus Vene­ris & Mercurii velociores sunt sufficiet eorum longitudines ad secundum aut quartum post diem investigare. Quo pacto exploratis eorum longitudini­bus ad duo tempora diversa quem curiam teneant ratione progressioni, regressionis, aut stationis facile perceperis.

Sic si ad priùs Exemplum loca ad aliquot sequentes diei examinaveris, erunt om­nium motus juxta seriem sig­norum directi, in posteriori omnes excepto Iove retrogra­di, cujus etiam locus invenie­tur parum distans à priori in praecedentia tunc primam in­tracturus stationem.

Nam illud semper est no­tandum quod si Planeta dire­ctio transiverit ad stationem ista dicitur prima statio: quan­do vero à retrogrado motu, ista statio secunda nuncupa­tur.

11 De Latitudine ascen­dente & descendente.

INventis sic prius latitudini­nibus ad rectum tempus ex­aminentur de novo ad 2, 3, 5, vel 10 diem sequentem, & u­trum sint ascendentes, vel des­cendentes dignosces. Hoc modo.

Si post secundam inquisi­tionem inventi fuerint in eâ­dem plagâ (viz. vel Septen­trionali vel Meridionali) quâ antea, tum si sit cujusque latitudo ad utrumque tempus, vel Meridionalis decrescens, vel à Meridie ad Boream mu­tata, & crescens, dicuntur ascendentes.

Sin verò ad utrunque tem­pus latitudo fuerit Septentrio­nalis decrescens, vel mutata à Boreâ ad Meridiem, & tum crescens, vocantur descenden­tes.

Denique si ad utrumque tempus consistant: sunt in puncto variationis. viz. si in Boreâ latitudine constiterint ab ascendente vergunt ad de­scendentem; si in Meridiona­li à descendente ad ascenden­tem.

12 De Planetarum Aspectibus.

COmpara duorum quorum­libet loca ad datum tem­pus & deprehendes Aspe­ctus juxta regulas noni prae­cepti.

Exempli gratiâ in primo praecedentium Exemplorum Sol & Iupiter sunt propemo­dum in conjunctione. Sol & Saturnus prope Trinum. Satur­nus & Iupiter non procul à Tri­no. Saturnus & Mercurius pro­pe Trinum. Venus & Mercurius non procul à Sextilo. Et pa­riter de reliquis.

Attamen illud obiter no­tandum, quod licet Iupiter & Sol tendant ad conjunctio­nem, & nobis terricolis re­vera appareant conjuncti, ta­men per sextam praecedens di­stant ab invicem 18700 semi­diametris Terrae.

13 Utrum Planetae sunt Com­busti, Acronychi, Orientales, vel Occidentales.

PLanetae dicuntur Orienta­les quorum loca distabunt à terra minus semicirculo juxta seriem signorum numerato. Occidentales è contra. Si sint [Page 23] in loco Terrae sunt Acronychi, sin loco Terrae oppositi vocan­tur combusti.

Sic in praecedentium exem­plorum primo Saturnus erit Orientalis quia à 21 Arietis ad primum Cancri juxta s. s. non completur semicirculus Iupiter combustus, Mars Occidenta­lis, quia à 21 Arietis loco (sc.) Terrae ad quartum Sagittarii locum Martis intercipiuntur plus 180 gradibus. Venus Orientalis, Mercurius Occi­dentalis. Nullus hic Acrony­chus quia eorum loca multum distant à terra.

14 De Ortu & Occasu Poëtico.

A Pud Poëtas dicuntur Pla­netae oriri, & occidere Cosmicè, Acronycè, & Helia­cè; harum passionum detectio (utpote etiam occultationum, & emersionum) in his Theori­cis expectari non debet. Res [Page 24] est per se ardua praesertim in Planetis ob eorum continuum motum & tum Longitudinis, tum Latitudinis variationem. Praeterea ad elevationes Poli, & Horizontes particulares re­feruntur; quapropter Astro­labiis, atque istiusmodi pro­jectionibus Spherae, non Theoricis conveniunt. Exa­ctè ex Tabulis Astronomicis, & Calculo Trigonometrico deducuntur. Qui curiosiùs in haec inquirunt exinde satisfa­ctionem petant. Haec quae scrip­simus pro introductione in­serviant ad magis praecisas o­perationes, vel saltem ad sup­plendos eorum defectus quo­rum peritiâ, vel desiderium eousque non attingit, & quo­rum gratia haec praecipuè in­tendimus.

1 Quomodo quaevis Theorica commodissimè disponatur.

OPtimè describuntur super duas laminas ut cujusvis Planetae orbita, seu Eccen­tricus majoris sit Diametri.

Methodus quâ incedo, in genere, concordat cum Syste­mate mundi Copernicano, in specie cum istâ ejusdem dispo­sitione quàm introduxit Ke­plerus in suis Tabulis Rudol­phinis cùm hâc tantùm diffe­rentiâ. Keplerus orbitas Pla­netarum facit Ellipses, quòd verò proprius, Ego perfectos Circulos facilitatis gratià fa­cio. Defectus ex hoc discri­mine procedens non erit ma­gni momenti in Instrumentis non nimium magis amplis.

Ad majorem concinnitatem Saturnum & Martem in oppo­sitis [Page 26] faciebus ejusdem laminae disposui. In alterius laminae facie è quidem altera Iovem alterâ terram cum Venere & Mercurio: interiùs compre­hensis, locavi. Scalas etiam aliàs vacuis locis ad alios usus addidi. Insuper, necessitate id requirente, orbita terrae qua­ter repetitur, viz. in utrâque laminâ utrinque cum propor­tione ad exigentiam cujusque Planetae requisitâ.

2 De Planetarum & Terrae eccentricis.

PRimò in singulis laminarum faciebus describatur. Cir­culus qui priùs in 360 gr. di­visus, ulterius in duodecem partes cum 12 Zodiaci signis notatas distinguatur. Nume­retur quodlibet signum 10, 20, 30. Itaque hi Circuli Zodia­cum ad colligendas Planeta­rum Longitudines necessari­um designabunt. In Centro pingatur Solaris essigies mon­strans Solem in Centro Mun­di locum habere.

[Page 27]2 Hoc facto, sic perge (sit pro exemplo Saturnus.) Ex Tabula C, excerpe Aphelium in columna directè sub Satur­ni charactere (nempe, Sagitta­rius 27 gr. 30 m.) A Centro ad 27 gr. 30 min. Sagitarii in Zodiaco, duc Semidiametrum, in quâ paululum distans à limbo versus Centrum assume punctum, quòd pro Saturni A­phelio, habeatur. Distantia verò abindè ad Centrum, divi­di concipiatur in 100000 par­tes aequales quae instar Scalae decimalis ad reliquum opus peragendum inserviat.

In hâc Scalâ 100000 suma­tur Saturni eccentricitas, ex Tabulâ A, nempe 05387 & super eadem lineâ à Centro Solis versus punctum Aphe­lium transferatur. Istud inter­vallum vocetur Saturni eccen­triticas, vel si malueris cape nu­merum 94631 ex eadem Ta­bula A, qui super Scalâ eadem, â puncto Aphelio versus So­lem translatus, dabit ideme c­centricitatis punctum, quod ita inventum erit Centrum orbitae Saturni.

[Page 28]Si igitur, ab hoc Centro ad punctum Aphelii, ut Semidia­metro describatur circulus or­bitum Saturni descripseris.

3 Denuo regulâ ad Cen­trum Solis applicatâ juxta sig­na & numeros in Tabula C sub charactre Saturni notatos, decimum quemque Anoma­liae sive divisionis orbitae Sa­turni gradum transferas; & tandem sub divisis his parti­bus majoribus in decem mi­nores aequales (nam aequales sufficient licet rigidè sumptae inaequales esse debent) habe­bis 360 gradus Anomalos pro Saturni orbitâ. Hi â puncto Aphelio per 10, 20, 30, ad 360 & secundum seriem singulo­rum numerentur.

4 Orbita terrae circa So­lem ad orbitam Saturni justè proportionata nunc venit in­ferenda. Ad quod faciendum inspiciatur secundo Tabula C cujus numerus primus sub sig­no terrae. Ostendit Apheli­um terrae in Capricorni 7 gr. 00 m. applicatâ igitur regulâ à centro ad septimum Capri­corni gr. ducatur linea delebi­lis quae lineam terrae Apheliam representabit.

[Page 29]Deinde consule Tabulam A, ubi deprehendes punctum Aphelium Terrae à centro Solis distare 10128 partibus prioris Scalae lineae sc. Saturni in 100000 partes divisae. Per has partes ex scalâ desumptas punctum terrae Aphelium in debitâ distantia transferas. Consulo rursus praedictam ta­bula A. Et videbis terrae ec­centricitatem esse 00179 par­tium prioris scalae decimalis quae ex scalâ praedictâ de­sumptae in lineam terrae A­pheliam à centro Solis trans­ferendae sunt. Punctum trans­latum erit Eccentrici terrae centrum. Vel si distantia ista sit nimis brevis in eâdem ta­bulâ invenias distantiam A­phelii terrae à centro Eccen­trici ejusdem esse 09949 par­tium quae ex priori scalâ de­cerptae & à puncto Aphelii terrae super lineâ terrae Aphe­liâ versus Solis centrum trans­missae centrum eccentrici terrae monstrabunt. Super hoc cen­tro ad intervallum puncti ter­rae Aphelii scribe circulum qui orbitam terrae repraesentabit ad magnum Saturni orbem justè proportionatam.

5 Minor hic circulus seu­terrae [Page 30] orbita in debitas partes anomalias dividenda est, qua­rum decima quaelibet numeris Tabularibus sub charactere Terrae in tabula A inscribi potest: regulâ (scilicet) ad centrum Solis fixâ, & ad gra­dus & signorum Zodiaci mi­nuta in praedictâ Tabulâ datis applicatâ. Hae partes denuo bisecentur ut quaelibet pars quinque gradus signisicet, vel in Instrumentis majoribus in quinque partes aequales possint dividi quarum quaeli­bet duos gradus Anomaliae de­notabit. Hae partes à puncto terrae Aphelio per 10, 20, 30, &c. ad 360 numerandae sunt. Atque hoc modo Eccentrici Saturni & Terrae debite pro­portionati disponuntur, & di­viduntur.

Eodem pariter modo in Theoricis Martis & Iovis operandum est, usurpando co­lumnas Marti & Iovi desti­natas in Tab. A, unâ cum co­lumnâ terrae & quales nume­ri pro Saturno ex Tabula A tales pro Marte & Iove ex Tabula E & D desumendi sunt.

Similiter per Terrâ, Marte, & Mercurio: qui tres ex una la­minarum facie collocandi sunt. Linea terrae Aphelia à centro [Page 31] Solis ad punctum Terrae aphe­lium extensa & in 100000 di­visa inservit pro decimali scalâ ad inserendos omnes nu­meros eccentricos horum tri­um Planetarum. Ex hâc scalâ numeri proportionandis ec­centricis Terrae, Veneris & Mer­curii in tabulis B, F & G, de­sumantur. Quorum lineae A­pheliae & divisiones graduum Anomalorum disponuntur, & determinantur per columnas tabulae C, istis Planetis respon­dentibus: regulâ ut antea ad centrum fixâ, & ad signa, & gradus Zodiaci super has The­oricas ducendos applicata.

Minores istae Tabulae nu­merales pro colligendis Ano­maliis Terrae reliquorumque Planetarum eodem modo cui­que orbitae inscribantur, prout in scematibus appareat. Et ii­dem sunt numeri posteà in Anomaliarum Tabulis tran­scripti.

Tabulae numerales pro Terra bis repetuntur in utrâ­que laminâ semel. viz. in Theorica Martis, & in illis Vene­ris & Mercurii eo fine ut utra­que lamina cursum terrae te­neret absque alterius ope. Et istic loci disponuntur quia non datur alius magis conveniens. [Page 32] Circuli enim terrae in Theori­cis Saturni & Iovis nimis sunt parvi ad eas commodè tenen­das.

3 De scalis Distantia­rum.

IN singulis Instrumenti fa­ciebus scalae partium aequa­lium describuntur ad metien­das distantias Planetae tam à Sole quàm à Terrâ inscribun­tur in lineis Apheliis exterio­ris Planetae, viz. in Apheliis Saturni, Iovis, Martis & Terrae Determinantur ex tabulâ H, & ratio hujus limitationis est ut ejusdem proximè essent ad invicem magnitudinis, & in­terim numeros admitterent ad semidiametros sine magno la­bore reducibiles.

Modus consiciendi videatur in exemplo Saturni. Numerus Saturni in tabulâ H est 85 63/100 si igitur (ope Sectoris aut ali­ter) hujus Planetae lineam A­pheliam (ex Theoricâ) à Solis centro ad Saturni Aphelium sumpseris, & Sectoris crura ad hanc longitudinem in termi­nis 85 63/100 in lineâ partium [Page 33] aequalium aperueris habebis numeros quos volueris rotun­dos utpote 80, 70, &c. pro hujus scalae divisionibus. Qui à sectore ad lineam Apheliam à puncto Saturni Aphelio translati dabunt longitudinem 80, 70, &c. partium in scalâ aequalium quas denuo dividas & prout in schemate continues in Saturno, & Marte, ad 100 in Iove et Terra ad 120. In­tegra scala non necessario di­viditur in plures 10 partibus largioribus quarum supremae in 10 minores subdivisae (prout moris est) numeri apponantur ut in schematibus videre est.

Sic in Iove dividendum est spatium ab Aphelio ad So­lis centrum in 92 87/100 & ita de reliquis juxta numeros Tabu­lae H.

4 De Nodis & scalis inclina­tionum.

USus Tabulae M est ad in­serviendos nodos quinque Planetarum nam Terra nullum [Page 34] habet. Methodus videatur in exemplo Saturni. Nodus Sa­turni ascendens est 22 grad. 27 min. Cancri. Positâ igitur regulâ à centro Solis ad 22 gr. 27 min. Cancri: in limbo de­lebilem ducas lineam quae erit communis sectio plani ec­centrici Planetae, & Eclipticae. In hâc lineâ duo quaelibet puncta opposita aequalis utrin­que à centro distantiae assu­mas ut in schemate ad chara­cteres ♄ ♄, ob planum in quo cursus Saturni describitur. Per haec duo puncta du­citur ellipsis punctis dister­minata (vel aliâ circularis quaelibet ad libitum figura) in cujus altera medietate (ista scilicet) quae à 22 grad. ½ Caneri, juxta seriem signorum procedit) scribatur SATUR­NI Inclinatio Borea. In reli­quâ SATURNI Inclinatio Austrina.

Minor scala ad metiendas Saturni inclinationes terminos habet et suos limites in hunc modum. Inspicè Tabulam N, ubi invenies maximam Satur­ni inclinationem 2 gr. 32. m. Cape igitur distantiam alter­utrius [Page 35] puncti (notari ♄, ♄) à centro Solis, & ad hanc di­stantiam aperiantur crura se­ctoris in lineâ partium aequa­lium à terminis 2 32/60.

Ex sectore sic aperto capi­as distantiam in terminis 3, 3, in lineâ partium sectoris aequa­lium tres partes ex quae lon­gitudinem dabit scalae notatae 1, 2, 3, ad mensurandas Satur­ni inclinationes. Quae in tres partes, significantes tres gra­dus, quarum singula in qua­tuor aliàs aequales dividatur. Hoc modo opus harum linea­rum in Theoricis Saturni per­agitur.

Similiter faciendum est pro reliquis Planetis usurpando numeros illis pertinentes & in Tabulis M & N expressos. Ampliore igitur non opus erit directione.

5 De Scalis Latitudi­num.

IN utrâque laminâ, & super istam faciem ubi Theoricae Martis & Veneris ducuntur una istiusmodi scala describi­tur, ut neutra alterius indi­geat. Linea à Solis Centro ducta est partium 120 aequa­lium. Arcus seu scala curvilinea [Page 36] super priorem pendens in 10 grad. dispescitur Martis Ta­bula Q. Veneris Tabulâ nota­tâ R, quod varietatis tantùm causâ sit nam aliter Tabula Q sola utrique satisfecisset. Sed haec cautio observata digna est, quod scilicet recta à Cen­tro Solis ad peripheriam ten­dens, justum aliquem Zodiaci gradum secet. Quia gradus isti Tabulares (per quos in­aequales scalarum partes ex­penduntur) ex limbi gradibus sumi debent, & proptereà commodiùs, & ad faciliorem numerationem lineâ praedictâ in aequalem gradum cadat.

Atque hoc modo Theoricae scalis satis commodis ad inveniendas tam Longitudines quàm Latitudines quin­que Planetarum instruuntur. Reliquae de quibus dicendum restat accomo­dantur ad convertendas Longitudines, & Latitudines in Declinationes, & As­censiones Rectas.

6 De Scalis Ascensionum Re­ctarum.

SCalae Ascensionum Recta­rum, & Declinationum in Planis Saturni & Iovis de­scribantur, quia magis amplum [Page 37] est in illis spatium ad eas com­modè tenendas.

1 (In loco convenienti) ducenda est lineâ rectâ, & à Centro Solis arcus descri­bendus commodae attamen ar­bitrariae distantiae cum nume­ris 1, 2, 3, ex utrâque parte lineae rectae adfixis. Gradus isti 1, 2, 3, sunt etiam arbitra­rii, interim quantitatis aptae recipiendis Ellipticae figurae di­visionibus adeò amplis ut di­stinctè in quatuor equales partes possint dividi.

2 Ex utrâque parte lineae rectae mediae in scalâ Circulari sic divisâ numera 2 gr. 29 min. per quorum terminos à Cen­tro Solis duc duas lineas dele­biles.

3 Intra lineas obscuras duc cujusvis formae Ellipsim ita tamen ut ejus extremitates justè tangant praedictas lineas delebiles per grad. 2. 29 min. ductas.

4 Huic figurae ovali inscri­bantur graduationes ope Ta­bellae W, quintus aut deci­mus quilibet gradus inseri potest reliquis tantum aequa­liter divisis. Ordo characte­rum, numerationis, & divisio­nis modus videatur in schema­tibus. Atque haec pro ratione conficiendi has scalas.

7 De scalis Declinatio­num.

HAE super iisdem Theorica­rum planis quibus scalae A rectarum insistunt.

1 A Centro Solis ducatur rectâ lineâ. Cujus extremitas Soli proximâ dividatur in 10 partes aequales, quarum quaelibet quadri secetur [sin ulterius procedere in animo sit inaequaliter instar tan­gentium dividenda est] haec scalâ etiam est arbitrariae mo­do, recipiendis minoribus di­visionibus, commodae sit lon­gitudinis.

2 A Centro Solis & super istâ lineâ describitur arcus Circuli continentis ex utrâ­que parte lineae rectae 25 gr. istiusmodi quales integer Cir­culus contineret 360 numeris utrinque ad fixis 00, 5, 10, 15, 20, 25, &c.

3 Ultra hunc arcum Cir­culi, ducitur lineâ rectâ in­finite protensa quae priori du­ctae insistit ad rectos, & posteà terminatur regulâ à Centro Solis utrinque per gradus [Page 39] Circuli 23 grad. ½ dimissâ: Atque ita lineae ductae per 23 grad. ½ ad Cancrem & Capricor­num justos hujus perpendiculi limites distinguent. Dividitur verò haec linea utrinque per Canonem sinuum: quilibet quintus decimusque gradus à caeteris distinguitur, & trige­simus quisque duplici chara­ctere signi alicujus insignitur, prout in schemate videre licet.

4 Quartò, In loco commo­do describenda est altera si­gura ad libitum Elliptica. At eâ conditione, ut ejus extremi­tates directè tangant delebi­les istas lineas prius per gradus arcus circularis 23 ½ ductas.

Divisiones imponuntur ope Zodiaci recti linei prius descri­pti applicando regulam ad ini­tium cujusque signi, & in hanc ovalem transferendo. Inscri­ptio initiorum sufficiet, nam gradus ex Zodiaco rectilineo desumendi sunt. Et ista ova­lis divisio non sit alio sine nisi ad commodius transferendos gradus Zodiaci prioris, nam in hoc novo signa contrario stant ordine quam in priori Cancro cum Capricorn in me­dio Aries & Libra ad extre­mitates.

[Page 40]5 Remanet adhuc Scala altera finuum rectorum ad gradus circiter 35, ubicunque volueris inserenda quae sic de­terminabitur. Cape longitu­dinem Zodiaci rectilinei ab Aricte ad Canceri vel Capri­corni, ad quam aperiatur Sector (commodissimè enim perficitur per illud instrumen­tum) in lineis sinuum & in terminis 23 ½. Deindè trans­ferantur sinus 35 grad. in hanc lineam rectam & sic in partes debitas dividetur. Exemplar omnium videas in schemati­bus.

Hucusque progressus sum in declaratione Methodi quâ hae Theoricae cum omni earum apparatu, construendae sunt sequuntur Tabulae anteà sae­piùs nominatae, ad plurima tam inserenda quàm determinanda necessariae.

[Page 41]

Quomodo Tabula praecedens té­pori futuro accommodetur.

IN 100 annis Aphelia & Nodi Planetarum progrediuntur, ut in adjunctâ Tabellâ.

[Page 42]Per hos numeros Tabulae praecedentes (ad annum 1673 completum constructae) ad alium quemlibet adaptari pos­sunt. Tabulae istae notatae C (quas solummodò intelligo) prout nunc sunt ad annum 1700 inservient. Post perio­dum istam adimpletam ad an­num 1730 ad 30 (scilicet) an­nos sequentes accommodari possunt, & tunc ad 1760 foe­liciter inservient. Nam in 30 annis Nodi progressum faci­unt adjunctae tabulae, qui in eruendis Lati­tudinibus non causa­bit errorem plus ⅛ gr. in ipsis Marte & Ve­nere ubi error erit maximus.

Repeto igitur has Tabulas notatas C, factas esse ad 1763 completum quas si desideras rectificare ad annum 1730 completum. Primo sume diffe­rentiam horum annorum (sc.) 57, & in hunc numerum duc progressus Aphelios Tabulae K. Abscissis quinque dextimis figuris residuum erit gradus. Fractio decimales graduum partes, quae in sexagesimas facile converti possunt. Et deinde numeri sic inventi ad­dendi sunt numeris Planeta­rum respectivis in Tabula C, atque ita ad annum 1730 rectificantur.

[Page 43]Eodem modo rectificabis Nodorum loca multiplicando per 57 motum eorum in Ta­bula K, ut antè correctio de­indè cuique Planetae respecti­ve est addenda juxta motum in Tabulâ M expressum.

Aphelia, & Nodii (rigidè sumpti) non sunt fixi sed con­tinuo moventur minimò spa­tio. Interim quia motus est tardissimus (quòd ad hoc In­strumentum) absque notabili errore per aliquot annorum spatium fixâ imaginemur.

Error enim oriens ex Nodis fixis in annis 30, non excedit 8 min. scrupula prima in ipsis Marte & Venere, ut anteà monstratum. Error etiam ex fixis Apheliis in 30 annorum cursu erit circiter 31 min. in Terra vel Sole, 38 min. in Sa­turno, 24 min. in Iove, 33 m. in Marte, 39 min. in Venere, 52 min. in Mercurio. Error sanè in his Instrumentis satis tolerabilis.

[Page 44]

Distantia Apheliorum di­videndae sunt per numeros cui­que Planetae in Tabula H ad­junctos, ultra Centrum in iis­dem partibus quousque opus fuerit continuandae. Sic distan­tiam Solis à Terrâ compara­veris in Semidiametris Terrae. Si primò, in propriâ cuique Pla­netae scalâ mensuraveris, & secundò, si Saturni distantiam multiplicaveris in 400, Iovis in 200, Martis in 100, Vene­ris, Mercurii, & Terrae in eâ­dem, cum illis Tabula per 50 numeros facile ob eorum pro­portionem subduplam in me­moriâ retinueris.

[Page 45]

[Page 46]

Hae Epochae uti nunc sunt durabunt ad 1700, & ulte­rius ab 8 in 8 annos continua­buntur hoc modo. Ab ultimâ Terrae Epochâ subducatur nu­merus Terrae affixus in Tabulâ adnexâ, viz. 0.077, in reli­quis Planetis ultimis eorum Epochis numeri affixi prout Tabula monstrabit sunt ad­dendi Tabulae motuum se­quentes nullâ indigent corre­ctione, correctis enim Epochis nihil amplius restat corrigen­dum.

[Page 47]

[Page 48]

Sic tandem absolvimus om­nes Tabulas his Theoricis ne­cessarias ad colligendas aequa­les sive medias Anomalias in cujusque diei Meridie. Quo­modo autem concinne inscri­bantur in Instrumentis, & unaquaeque affixa Orbitae, propriae Planetae convenien­tissimè disponatur ad usum, absque reliqui operis impedi­mento in schematibus videre est.

1 To what Systeme of the world these Theorics are framed & to what planets they serve.

THese Theorics are framed according to Copernicus his Hypo­thesis: in which the Sun is supposed to be in the Cen­ter of the World, and those moti­ons that are apparently in the Sun, to be really in the Earth. And so the earth, in the Suns roome comes to be numbred a­mong the 7 Planets.

Of these 7 we shall properly enquire after the places of five onely. For, the perfect absolution of the Moones motion, and pas­sions jointly with the Earth, be­ing of more varieties then the rest, will require an Instrument alone, and so the Moon is dismis­sed hence.

Again, the earths place is requi­red in these Theorics, not so much for it self, as for the other five Planets, whose places in the Zo­diac [Page 4] cannot be had in respect of us, unlesse we first know in what part or place of the World the earth (that is, our selves upon the earth) do stand. Yet the true place of the earth in respect of the E­cliptick, & consequently the ap­parent longitude of the Sun, may here likewise be found, when at any time it shall be required, as is shewed afterwards in the 8^th Proposition.

2 How all time is to be fitted for computation.

FOr the accomodation of time to calculation, we may ob­serve these things.

• 1 All motions are to be col­lected for complete times.
• 2 A day begins upon its own noon, and ends upon the noon of the next day. So that,
• 3 The noon of the first day of Ianuary is the common term of the old and new years, being the end of the former and the begin­ning of the latter.

3 What the place of a Planet is, with the manner of colle­cting the equal Anomalies.

THese Theorics (as is said be­fore) do especially concern the 5 Planets, Saturn, Jupiter, Mars, Venus, & Mercury, & are in­tended for the speedy finding out of their places for every day at noon. They will serve as they are [Page 5] now framed, till the year 1700 without any notable altera­tion.

The place of a Planet is the situation of it to the plain of the Ecliptick, in respect of longitude therein, and latitude therefrom. To which also may be added the interval or distance of it from the Earth.

To find these things, we must first know, what Anomaly is due, for the time assigned, both to the earth, and likewise to the Planet whose place is required. These are severally to be gathered out of their proper Tables, annexed to every Planets Orbit. And the numbers in those Tables are to be esteemed for degrees and cen­tesimal parts of degrees.

The manner of collecting the equal Anomalies is this.

• First, Exscribe the Epocha which belongs to that year, w^ch most neerly precedeth the year wherein you seeke the place of any Planet.
• 2 Vnder that Epocha or num­ber, write the motions belonging to so many years, moneths, and dayes, as are completely expired since the year of the Epocha. Each of these numbers must be taken out of their proper Tables, & set orderly one under another which the disjunction of the numbers will give direction enough to doe.
• [Page 6]3 All these numbers must be added into one, and their summe shall give the Anomaly for the time assigned. If the sum rise to be above a Circle or 360 d. you must then cast away the said number of 360 as oft as you may, and the remaining number must be taken for the Anomaly.

These thinges are to be done both in the Earth and Planet se­verally. And for that purpose the Table of the Earths Anoma­ly is twice set down upon each plate once; that which soever of the plates you are to use, you may have the earths Table at hand: without which neither the Pla­nets place, nor some of the pas­sions thereto belonging can be found. Now it follows to be shewed,

• 1 How to find the Longitude of a Planet in the Ecliptic.
• 2 How to find the Latitude of a Planet from the Ecliptic.

And for this purpose you must have a thread fixed to the Cen­ter of your plate, which is the Center of the Sun. And besides, there must be a thin plate-ruler, with a streight or fiducial edge, of such length as may be neer about the Diameters of the plates. It must not at all be fa­stened to them, but be separate and loose, that it may be ap­plyed to any two points pre­scribed upon the superficies of the plates.

4 How to find the longitude of any of the 5 Planets.

1 GAther the Anomalies of the Earth and of the Planet whose longitude is re­quired, each out of their own proper Tables: in such manner as was before shewed.

2 Count the Planets Ano­maly upon the Planets Orbit, & the Earths Anomaly upon that Orbit of the earth which is drawn upon the same side of the plate with the course of your Planet, and observe these two points, for in them are the places of the earth and Planet, for the time assigned.

3 To both these points, ap­ply the fiducial edge of your little plate-ruler, so, as that the same edge may look towards the Sun, and that it may also cut the limbe or Zodiac, and goe beyond it as occasion shall be: and let the greatest part of it lye from the earth towards the planet, for many times it will be requisite so to lay it, because of the work that next follows.

4 Measure with your Com­passes the least distance between the Center of the Sun and the fi­ducial edge of the same ruler: and set one foot of this distance upon any part on the exteriour limbe or Zodiac of the plate, & on the same side of the ruler that the Suns Center is, and on that [Page 8] part of the Zodiac which is from the Earth towards the Planet. All this must be done in such wise, that the other foot of the Compasses being turned about may justly touch the edge of the ruler. In this posture, that foot which standeth upon the Zodiac will there shew the signe and degrees of the Pla­nets longitude.

See examples after the next Precept.

5 How to find the Latitude of any of the 5 Planets.

1 HAving found the Ano­malies of the Earth and Planet, lay the threed that is fixed at the center upon the Planets Anomaly numbred in its proper Orbit. And to the threed so laid, take the least distance from that character of the Planet (whose place you seeke) that lyes fitted to the threed, for both will not: and observe whether the threed cut through the title of North or South inclination.

2 Measure the same least di­stance, upon the Scale which is made for the measure of the Planets inclination, and upon that Scale the Compasses will shew how much the inclination is: the coast or title of it being discovered before.

[Page 9]3 You are then to measure two distances more. The first, is from the Planet to the earth, that is, from the points of their Anomalyes, which are their places in their Orbits. The second, is from the Planet to the Sun. And these are done, by taking the said distances in your compasses, and applying those lengths to the Scale ap­pointed for that purpose [namely that Decimal Scale, which on every Theoric pas­seth through 360, or the Aphe­lial point of the exteriour Pla­net.] By this meanes you shall know their distances, or the proportion of them at least.

4 Next, goe to the equal Scale divided into 120, which hath an ark of graduations ap­pendent to it. And upon that ark, Count the inclination of the Planet, which you found before, and thereto lay the threed. Afterwards, upon the Scale of 120 count the number of the Planets distance from the Sun, and take the least ex­tent from that number to the threed, keeping it still in your compasses. Then again, upon the same Scale, count the di­stance of the Planet from the Earth, and there set one foot of the former extent, and apply the threed to the other foot, so, that the said other foot being turned about, may onely reach [Page 10] the threed neither going be­yond, nor falling short of it. So the threed, in this position, will shew upon the appendent arke the quantity of the Planets la­titude. And for the coast or de­nomination of the Latitude it must alwayes be the same that the Inclination was, whether North or South.

See two examples at large here following for the Longit. Latit. and Dist. of the earth and the other 5 Planets. One Example is for the 4^th of October at noon 1649. The other is for the 19^th of February at noon, 1651.

6 To know how many Semi­diameters of the Earth any Planet at any time is distant from the Earth, or from the Sun.

HAving measured the di­stances of the Planet from the Earth and from the Sun, upon its proper Scale, as was shewed before; Then

For

• ♄
• ♃
• ♂
• ♀
• ☿

Multi­ [...]ly the said di­stances by

• 400
• 200
• 100
• 50
• 50

And the product will be the re­quired interval in Semidiame­ters of the Earth.

The Earths distance also from the Sun may be had in the same manner, but with a little more caution. For the fittest Theories for this work are those of Venus, and Mer­cury, or else Mars. If you take the Earths distance from the Sun upon the plate of Venus, and Mercury, then you must multiply the number found by the Scale of that plate, by 50, which is the number given be­fore for Venus, and Mercury. But if you take it from the Theoric of Mars, then you must multiply the number there found, by 100, which is the multiplying number given be­fore for Mars.

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7 By the Longitude & Lati­tude of a Planet being known, how to find the right ascension & declinati­on thereto belonging.

THis work is most proper for Astrolabes, and other such Spherical instruments. Yet because these Theorics should not be altogether defe­ctive herein, I have added such Scales as will perform these things, though it be with more trouble. For this purpose those Delineations upon the two Theorics of Saturn & Jupiter are added; both which are the same thing done twice over, that each plate may have one ready at hand, for those Planets which are drawn upon it.

¶ The first thing to be done is, to get the right ascension of [Page 13] the meer longitude of the Pla­net, as if it were without all Latitude, or in that very point of the Ecliptic which answers to the Longitude. And this is performed upon that Systeme of Scales which is made for the finding out of the right ascen­sions of the parts of the Eclip­tic, as in the title thereof is ex­pressed, by which title it may also be known.

Count therefore upon the Elliptical Zodiac, the Planets Longitude, that is, the signe & degree, in which you found it by the 4^th precedent: and thereto applying the Center threed, ob­serve where the same threed cuts the ark noted with 1, 2, 3, the same ark being estimated in degrees & mi­nutes, is that which shews how much the Longitude differs from the right ascension, which may be called, the longitude E­quation. This Equation or dif­ference must either be added to, or subtracted from, the Longit. before found, according as the threed will intimate by falling upon the directions for additi­on or subtraction, written close­ly behind this differential ark. And this being accordingly done the sum or difference so found, shall be the right ascension of the Planets meer Longitude, which was the first thing re­quired.

And thus much alone doth [Page 14] get the true right ascension for the Earth or Sun, because they lye in the plaine of the Eclip­tic & have no latitude from it.

¶ The second thing to be done, is to correct this forego­ing right ascension, which cor­rection must alwayes be made when the Planet hath any La­titude from the Ecliptic, as most commonly it hath. And for the effecting of this, The greatest part of the other Sy­steme of Scales is to be used, and in this manner.

Vpon the 12 signes as they are ordered and inscribed into the Ellipsis (which signes do answer to those in the exteriour Zodiac, though the character­ing of them be different) and upon the degrees of the exteri­our Zodiac (30 of which deg. quite through that Scale do an­swer to these forementioned signes) count the Planets Lon­gitude, and thereto apply the threed. Then again, upon the Scale of the middle line that goes to the Center, count the Planets Latitude; & from that point to the threed, take the least distance with your Com­passes. This least distance ap­plyed to the same Scale of the middle line, from the Center outwards, will give the equati­on in degr. and min. This may be the latitude equation. And it must be either added or sub­tracted from that right ascen­sion [Page 15] that was found before, ac­cording as the Directions that are written upon the Ellipsis shall prescribe.

By which meanes, the last sum or difference thus found, shall be the perfect right ascen­sion of the Planet, agreeable to the Longit. and Latit. given. This for the right ascension.

¶ For the Planets declinati­on, you are to make use onely of the exteriour Zodiac, and the circular ark, numbred both wayes to 25 d. The way is this. Count the latitude of the Planet upon one of the arks of 25 deg. namely that w^ch is noted with the same kind of latitude that the Planet at that time hath, & thereto apply the threed. Then upon the exteriour Zodiac (ac­cording to the order of the signes and degr. as they are there set on) rekon the Planets longitude; & setting one foot of your com­passes in that point, with the o­ther foot take the least distance to the threed, observing whether your compasses in this work do stand above or below the threed. This least distance being so takē must be applyed to the right line of 35 parts, from the beginning forwards upon the Scale, where it will shew you the quantity of the Planets declinatiō. And for the coast of this Declination, whether it be North or South, the former observation of the [Page 16] standing of the compasses, either above or below the threed, will resolve. For if the compasses do stand above the threed, then the declination is North: if they stand below, then the declination is South. And this directiō also, that it might be alwayes neer at hand, is written at both ends of the exteriour Zodiac.

The Earth or Suns declin. is had, by taking the length from Aries or Libra in the exteriour streight Zodiac, and applying it to the Scale of 35, for it will there give the declination without more adoe.

Here follows an Example of the right ascensions & declina­tions of the Earth and the other 5 planets, according to the Long. & Latit. of them, found in the first of the two former Exam­ples computed for the fourth day of October at Noon, 1649.

8 How to find the place of the Earth or Sun in the E­cliptic.

THis is much more easie to be done for the Earth then it was for the other 5 Planets; because the earths place is free both from commutatiō & La­tit. And for the finding of the true place in the Ecliptic, it will be best to use the earths largest Theorics: namely, either that which comprehends Venus & Mercury upon one Table, or else that which is comprehended by Mars upon the other Table.

Having therefore found the earths Anomaly for the assigned time, Count the same upon the Orbit of the earth, and thereto lay the center-threed, which be­ing so laid, will give the place of the earth, in the degrees of the exteriour Zodiac. And the opposite thereto, is the place of the Sun.

In the two former examples you have the earths places (for those assigned times) ex­pressed by the signe and degree, wherein it then shall be: namely Aries 21 d. 45 m. and Virgo 11 d. 30 m. And the opposites to these are the pla­ces of the Sun at those times: that is, Libra 21 d. 45 min. and Pisces 11 d. 30 m.

9 Concerning some of the principal passions of the Planets.

THe finding out of the places of the 5 Planets in respect of Longit. and Latit. is the thing principally intended in these Theorics. Now this having been already declared, it shall not be amisse to adde somewhat of the principal passions belonging unto them: of which there are these 3 chief heads.

• 1 At some times these 5 Pla­nets (in respect of that motion which they make according to the longit. of the Ecliptic) doe appeare to goe forward, agree­ably to the order & succession of the signes, that is, they ap­peare to be 1 Direct in motion. Sometimes again the seeme to goe backward in motion, or to be 2 Retrograde. And in their changes from the one of these motions to the other, they must necessarily appeare to be stand­ing still, or to be 3 Stationary.
• 2 Their places being com­pared in respect of distance from the Sun, or one from the other, the Planets may have se­veral aspects: as 1 Conjunction, when they are (any two of them) in one place of longit. 2 Opposi­tion, when they are in opposite longit. 3 Trine, when they are ⅓ part of a circle or 4 signes [Page 19] distant from each other: 4 Quartile, when they are three signes or a quadrant of a circle distant: 5 Sextile, when they are ⅙ part of a circle or two signes distant. Venus and Mer­cury cannot make any of these Aspects with the Sun. And one of them with the other can make none but the Sextile, which often they doe.
• 3 Their places being com­pared with the Suns place, they are either under the Sun beames & are thē said to be 1 Combust: or else they rise after the Sun, rising when the Sun is up, and are called 2 Oriental: or they set after the Sun, while the Sun is down, and are called 3 Occi­dētal: or are opposite to the Sun; and are called 4 Acronychal. Venus and Mercury can never be Acronychal, because they ne­ver goe farre enough from the Sun: Venus onely 48 d. Mer­curius onely 29 degrees.

10 Of Direction, Retrogra­dation, and Station.

THese things will not well be discovered by these Theo­ries, it being a difficult business to set the just times of these changes in their courses. If you desire to know in which of these motions any Planet is, the best way will be (when you have [Page 20] found their places for any one day) to enquire their longitudes about 5 or 10 dayes after in Saturn, Jupiter and Mars, or about 2 or 4 dayes after for Venus and Mercurius, because the motions of these are much swifter then of the other. And so having found their places of longitude at two several times, you shall perceive what course they hold in respect of progresse or regresse of stand­ing still.

So if in the first Example the places were again examined for some other dayes after, they would all be found direct in their motions according to the succession of the 12 signes. But in the second Example, they would all be found Retrograde except Jupiter: which Planet also will be found to be very neer to his former place, yet a little more forward, and conse­quently neer to his first station, then going to enter into it.

For it must alwayes be no­ted, that, if a Planet passe from direct motion to station, then that standing is the first station. But if it passe from re­trograde motion, then is the sta­tion following to be taken for the second station.

11 Of latitudes ascendent or descendent

AFter the latitudes of the Planets are found for any assigned time, if they be again examined for 2, 3, 5, or 10 dayes after, you may know whe­ther they be ascendent, or des­cendent, in this manner.

If in the second enquiry they be found still in the same coast or denomination (of North or South latitude) that they were before, then

If the latitude at both times be either South and decreasing, or else changed from South to North, and then increasing, they are then said to be ascendent. But

If their latitude at both times of enquiry be either North de­creasing, or else change from North to South and then increa­sing afterwards, they are then said to be descent.

If at these two times of en­quiry they be found consistent, then are they upon their change, namely, if consistent and in North latitude, they are chang­ing from ascendent to descen­dent: but if consistent and in South latitude, then are they changing from descendent to ascendent.

12 Of the Planets Aspects.

COmpare the places of any two of the Planets together, & you shall have their Aspects for the time assigned, according to the former rules in the ninth precept.

Thus (rudely) in the first of the former Examples. The Sun and Jupiter are neer in Conjun­ction. The Sun and Saturn not farre from a Trine. Saturn & Jupiter not farre from a Trine. Saturn and Mercury neer to a Trine. Venus and Mercury not farre from a Sextile. In the same manner you may deale with the rest.

But by the way note this, that though Jupiter and the Sun are neer to a conjunction, and to us that are upon the earth doe ap­pear as if they were really toge­ther, yet by the precedent sixth Proposition, they are distant from each other 18700 semi­diameters of the Earth.

13 Whether the Planets be combust Acronychal, Ori­ental, or Occidental.

THose Planets are Orientall whose places being reckoned from the place of the Earth, ac­cording to the succession of the 12 signes, are distant from it [Page 23] lesse then a semicircle, or 6 signes. And they again are Oc­cidental whose places so count­ed, are distant from the Earths place more then a semicircle. If their places be the same with the Earths place, they are Acronychal, if opposite, they are Combust.

Thus in the first of the two former Examples; Saturn is Oriental, because from the 21 deg. of Aries to the 1 deg. of Cancer (which is according to the order of the signes) is lesse then a semicircle. Jupiter is combust. Mars is Occidental, because from the Earths place which is Aries 21 deg. to the place of Mars which is Sagit­tarius 4 deg. is more then a se­micircle or 6 signes. Venus is Oriental. Mercury is Occiden­tal. None of them are Acrony­chal, because their places are not neer to the place of the Earth, but much differing from it.

14 Of the Poetical risings and settings.

THe Poëtical kindes of ri­sing and setting are called Cosmical, Acronychal, and He­liacal. These and some other passions of the Planets (such as are the Emersions and Occultations) are not to be [Page 24] expected from these Theorics. They are difficult to be found, especially for the Planets, which are alwayes in motion, not resi­ding any long time in one Lon­gitude and Latitude. Besides, the same things have relation to the elevations of the Pole above several Horizons, which kind of conclusions are not pro­per for Theorics, but must be referred to Astrolabes and o­ther Spherical Instruments. The most exact practice this way is to be had in the Astro­nomical Tables, and Trigono­metrical Spheric works to be conjoyned therewith for such purposes. They therefore that would have more, must there seek help and wayes to satis­fie themselves. This that is here done, may serve for an in­troduction to more exact work­ings: at least it may supply the wants of such, whose skill and desires reach not so farre; for whose sakes it was principally intended.

1 How every particular Theo­ric is to be disposed for best convenience.

IT is best to make them upon two plates, that each Planets Or­bit or Eccentric may be of the larger extent.

The way that I goe is (in general) agreeable to Coper­nicus his frame of the World; and in particular, to that which Kepler useth in his Rudolphin Tables. Onely this difference there is: Kepler makes the Orbits of the Planets to be El­lipses, which is the better way; and I here doe make them per­fect Circles, which is the easier way. And though it be defe­ctive yet it makes no great dif­ference in these small Instru­ments.

For most convenience I have put Saturn and Mars upon one [Page 26] Table, each of them taking up one side. Vpon the other Table, on one side is set Jupiter, and upon the other side is the earth at large, with Venus and Mer­cury comprehended within it. Other Scales there are added (in spare places) for other uses. Likewise the orbit of the earth is placed upon each side of the two plates, that is, it is four times repeated, need re­quiring it should be so often iterated. It is also proportioned for the quantity of it, according to the exigence of each several Planet.

2 Concerning the Eccen­trics of the Planets and the Earth.

FIrst you are to make 4 limbes upon the 4 sides of your two plates, dividing each of them into 360 deg. and distinguish­ing the whole Circle into 12 signes, unto which their 12 names, or 12 characters, or both, must be annexed. Each signe is to be numbred by 10, 20, 30 deg. and so these Circles will (each of them) represent the Zodiac, in which the Long. & of the Planets must be found. In the Center you may draw the effigies of the Sun, signifying thereby, that the middle or Center of the World is his proper place.

[Page 27]2 Then for the other work (for instance suppose the Planet Saturn) you are first out of the Table C, to look where the place of his Aphelium is (which is shewed by the first number in the Table under the character of Saturn) namely Sagittarius 27 gr. 30 m. Wherefore from the center of the Sun, to the 27 g. of Sagittarius in the Zodiac, draw a Semidiameter: in which, a little within the Zodiac to­wards the Center, assume any point, which you must suppose to be the Aphelial point of Sa­turn: and the distance from that Aphelial point to the Cen­ter, must be supposed to be di­vided into 100000 equal parts, which must serve as a decimal Scale for the rest of the work.

Out of that Scale of 100000, take Saturns eccentricity, ac­cording to the quantity of it set down in the Table A, namely, 05387, and set it off upon the same line, from the Center of the Sun towards the Aphelial point. This distance is called Saturns eccentricity. Or you may take the number 94613 (which is also in the same Table A) out of the equal Scale, and set that distance from the Aphelial point towards the Center of the Sun, and it will give the same point of eccentricity. This point thus found, is the Center of Saturns orbit.

[Page 28]And therefore; if you set one foot of your compasses upon that Center, opening the other to the Aphelial point, & describe a Circle to that extent, and upon that Center, you shall then de­scribe the orbit of Saturn.

3 After this, By laying a ru­ler to the Center of the Sun, and by the numbers & signes in the Table C under the character of Saturn, you may inscribe each 10^th deg. of the Anomaly or di­vision of Saturns Orb. And a­gain, dividing each of those large parts into ten lesser equal parts (for, equal will well serve though in rigour they ought to be otherwise) you shall have the 360 Anomalar deg. of Saturns Orbit. These are to be numbred from the Aphelial point, by 10, 20, 30, to 360, ending in the same point: and the order of numeration must be according to the series of the 12 signes in the Zodiac.

4 The next thing to be done, is the setting in of the earth course about the sun, proportioned just­ly to this orbit of Saturn. And for this, look again in the Table C, the first number whereof un­der Earth shewes where the I­phelium of the Earth lyes, viz. in Capricorn 7. d. 00 m. There­fore laying a ruler from the cen­ter of the Sun to the 7^th deg. of Capricorn, draw an obscure line, which will be the Earths Aphelial line.

[Page 29]Then look into the Table A, where you shall find the Earths Aphelial point to be distant from the center of the Sun 10128 parts of the former de­cimal scale or 100000 equal parts of Saturns line. By which parts taken from that scale, you may set off the Earths Aphelial point in a true distance. Again, look into the Table A, and you shall there see the Earths eccen­tricity to be 00179, of the same parts of the former decimal scale, which you are to take and set from the center of the Sun, up on the earths Aphelial line, and that point shall be the Center of the earths eccentric. Or if that be too short a distance, you may in the same Table find the distance of the Aphelium (or Aphelial point) of the earth from the cen­ter of the Earths orbit or eccen­tric to be 09949: & this num­ber taken out of the former deci­mal scale, & one foot of it set in the Aphelial point of the earth, the other upon the Aphelial line of the Earth, towards the center of the Sun, will shew the same center of the earths eccentric. Vpon this center therefore, and to the extent of the Aphelial point of the earth from it, de­scribe a little circle, which is to resemble the earths orbit, being justly proportioned to the great orb of Saturn.

5 This little orbit or circle of [Page 30] the Earth, is to be divided in­to its just Anomalar parts. Each tenth of which may be inscribed by the numbers of the Table C, which are placed un­der the word of Earth, by a ruler laid to the Center of the Sun, and to such degrees and minutes of the signes in the Zo­diac, as shall be given out of the forementioned Table. And these 10^ths may be bisected, & so each division may signifie 5 deg. Or else each of them may be divided into 5 equal parts, every one of them signifying 2 deg. of Anomaly: this is to be done in larger Theorics. These Anomalor parts of the Earth are to be numbred from their Aphelial point, by 10, 20, 30, and to 360. Thus are the Ec­centrics of Saturn and the Earth to be proportioned, pla­ced, and divided.

In the same manner you are to work for the Theorics of Mars and Jupiter, if you use the columnes of Mars and Ju­piter in the Table C, together with the columne of the Earth: and what numbers were taken for Saturn out of the Table A, the like numbers must be taken out if the Tables E and D for Mars and Jupiter.

So also for the Earth, Ve­nus, and Mercury. These three are to be placed together upon one side of one of the plates. [Page 31] The decimal scale for all the nū ­bers of eccentricity for these 3 Planets, is the Aphelial line of the Earth, reaching from the Center of the Sun to the Aphe­lial point of the Earth, divided into 100000 equal parts. And out of that scale the numbers of the Earth, Venus and Mercury in the Tables B, F and G, must be taken for the proportioning of their eccentrics. And the right placing of their Aphelial lines, with the divisions of their Anomalar degrees, must be limited by the columns of the Table C, which answer to those Planets: a ruler being laid from the Center of the Sun to the signes and degrees of the Zodiacal limbes drawn upon the Theorical plates.

The little numeral Tables, for gathering the Anomalyes of the Earth and any Planet, may be written to each orbit, in such fashion as my draughts of these Theorics doe shew: & are the same numbers that are set down in the Tables of Ano­malyes hereafter specifyed.

The numeral Tables for the earth are twice written, upon each plate once; namely, in the Theoric of Mars, and in that of Venus and Mercurie; to the end that each table might have the earths motions upon it, without being beholden to the other. And they are there set, [Page 32] because in those two places onely is convenient roome for them. For, the Circles of the earth upon the Theorics of Sa­turn and Jupiter, are two little to hold them.

3 Concerning the scales of distance.

UPon every side of the two Plates, there are scales of equal parts to measure the di­stances of the Planet from the Sun and from the Earth. They are inscribed upon the Aphe­lial lines of the exteriour Pla­net: namely, upon the Aphe­lial lines Saturn, Mars, Jupi­ter, and Earth. The limiting of them is taken from the table H: and the reason of this limi­tation is, because they should be of somewhat neer an equal bigness one to another, and yet also that they might be of some such numbers that may be re­duced to semidiameters with­out any great trouble.

The manner of making them, may be seen in the example of Saturn. The number for Sa­turn (in the table H) is, 85 63/100 If therefore (by help of the Se­ctor, or otherwise) you take the Aphelial line of this Planet (out of the Theoric) from the center of the Sun to the Aphe­lial point of Saturn, and open [Page 33] the Sector to that extent, in the number 85 63/100 in the line of e­qual parts, you shall then have any even number or divisiō from the same scale of equal parts, as of 80, or 70, &c. which be­ing taken from the sector, and transferred to the Aphelial line, and being set thereon, from the Aphelial point of Saturn, you shall have the length of 80 or 70 of those equal parts. These you may divide and con­tinue as farre as they are in my Theorics: namely, in Saturn, and Mars, to 100, in Jupiter and the Earth to 120. You need not divide the whole scale any more then into 10 large parts, and the uppermost of them alone may be sub-divided into 10 lesser equal parts. After which they are to be numbred in such manner as is usual in such de­cimal scales, and as in those Theorics is to be seen.

So for Jupiter, you are to di­vide the space from his Aphe­lial to the center of the Sun, into 92 87/100, and so all the rest accor­dingly as their numbers, in the Table H, do require.

4 Of the Nodes and scales of inclination.

THe Table M serves to put in the Ascendent Nodes of the 5 Planets; for the Earth [Page 34] hath none. The manner of it may be seen in the example of Saturn. Saturns Ascendent Node is in the 22 deg. 27 min. of Cancer. Therefore laying a ruler from the Center of the Sun to the 22 deg. 27 min. of Can­cer in the limbe, you may draw an obscure line at length: this line is the common section of the plain Planets eccentric with the plain of the Ecliptic. In this obscure line you may assume any 2 points, opposite one to the other, and of equal distance from the Suns Center on both sides, as is done in my Theorics at the characters of ♄ ♄, for the plain on which the course of Saturn is drawn. Through which two points is drawn a prickt ovall (which might have been of any other compassing form, as a Circle, or the like) in the one half of which (namely, that which goes from the 221/2 deg. of Cancer, accor­ding to the series of the 12 signes) is written SATURNI Inclinatio Borea; and on the other half is written SATUR­NI Inclinatio Austrina. So this particular is done.

Then for the little scale, which is to be the measure of Saturns inclinations, that is thus to be limited. Look in the Table N, where you shall see the greatest inclination of Sa­turn to be 2 deg. 32. min. Take [Page 35] then the length or distance of either of the fore-named two points (noted with ♄ ♄) from the Center of the Sun, and with that distance, open the sector in the line of equal parts from 2 22/60.

When the sector is so opened, you may take off 3 in the line of equal parts, and that shall give the length of that Scale which is to measure the incli­nations of Saturn, noted with 1, 2, 3. This scale may be divi­ded into 3 equal parts: first, which are to signifie 3 degrees: and these again may be quar­tered. This is the work to be done for these lines upon the Theoric of Saturn.

The like must be done for every other Planet, by making use of the numbers belonging to each of them, expressed in the Tables M and N. There will therefore here need no more di­rection.

5 Concerning those Scales that are to find the Lati­tudes.

THere is upon each of the two plates one of this sort of scales, that so one plate may have no need to seek help from the other. They are drawn upon those sides on which Mars and Venus are placed. The line drawn from the Center of the [Page 36] Sun is an equal scale divided into 120 parts. The arke or cur­ved scale which hangeth upon the former, is divided into 10 degrees; that upon Mars, by the Table noted with Q: that upon Venus, by the Table R. They might have been done both by one Table (as by that with Q) but onely for variety. This caution alone is here to be obser­ved, namely, that the streight line comming from the Center be made to but upon some iust degree of the Zodiac or limbe: because those degrees in the forementioned Tables (by which the un-equal parts of the an­nexed scales are limited out) are to be taken in the limbe. And therefore it will be most expedient for ease in account to let the line point upon some even degree.

Thus these Theorics are fitted with scales sufficient for the finding out of the Longitudes and Latitudes of the 5 Planets. The other scales that yet remain to be spo­ken of, are fitted to turn the Longitudes and Latitudes into Right Ascensions and Declinations.

6 Concerning the Scales for Right Ascension.

THese scales for Right As­censions with those of De­clinations, are set upon the planes of Saturn and Jupiter, [Page 37] because their is most roome to hold them.

1 There is first a right line drawn (in some convenient place) without any divisions upon it, and upon the Center of the Sun and ark described at any fit distance, numbred with 1, 2, 3, on both sides the right line. The degrees 1, 2, 3, are of any arbitrary length, so large that the oval figure may be of some quantity to receive a fit number of divisions, and that the same divisions may receive sub-divisions into large quar­ters. This is the first work.

2 Vpon the Circular scale so divided, count 2 deg. 29 m. on both sides the middle right line, and through these limits draw two obscure right lines from the Center of the Sun.

3 Within these two obscure lines, draw an oval figure of any forme, but so, as that the two extreme parts of it may justly touch the two former ob­scure lines drawn through 2 d. 29 minutes.

4 After this oval figure is drawn, it is also to be graduated by help of the Table W; you may put in onely every 5^th & 10^th d. & whē they are put in, the rest of the lesser parts may be inserted by equal subdivisions. The order of their characte ring & num­ration, and the manner of their division, may best be seen in my [Page 38] Theorics. This will serve for direction to make these scales.

7 Concerning the scales for Declinations.

THese stand upon the same plaines of the Theorics, with the other scales of right ascension.

1 Here is first drawn a streight line from the Center of the Sun. That part which is neerest to the Center is divided into 10 equal parts [but if they should goe further then 10, they must then be unequal as Tan­gents are] standing for de­grees: and each of them is cut into quarters. This scale of 10 degr. is not limited, but may be of any fit length for the sub­divisions.

2 From the Center of the Sun and upon this line, is de­scribed an ark of a Circle, which contains upon it (on each side of the streight line former­ly protracted) 25 true degrees (such as the whole circle should contain 360) which are accor­dingly numbred on both sides, from 00, to 5, 10, 15, 20, 25.

3 Without this Circular ark is set a line perpendicu­lar to that first drawn, and ex­tended at length on both sides, but afterwards it is to be li­mited, by laying a ruler from [Page 39] the Center of the Sun to 23½d. counted upon the Circular ark both wayes: so shall lines drawn through these 23½ deg. give just limits to this perpen­dicular line, at Cancer and Capricorn. The divisions of this line are nothing but a dou­ble scale of sines. Every 10^th and 5^th degree is to be distin­guished from the rest, and every 30^th degree is to be double cha­ractered with some or other of the 12 signes, as is to be seen in my Theorics.

4 Again, there must an oval be here described, it may be of any fashion, but must be set in place convenient, and in such manner, that it may lye justly between the two former obscure lines drawn through 23½ degrees touching them with its extremities.

The divisions of it are to be taken from the former streight charactered Zodiac, by laying a ruler from the Center, to the be­ginning of each of those signes, and so transferring them into this oval. This inscription of the onely beginnings of the 12 signes into the oval is sufficient: for the degrees of these 12 signes must be taken out of the former streight Zodiac, this new division being onely added for conveniency of new chara­ctering the degrees of the old Zodiac. For in this new one you [Page 40] Cancer and Capricorn to stand in the middle, and Aries and Libra in the two extream pla­ces, contrary to what they did in the former Zodiac.

5 One Scale yet more re­maines, containing the right sines of 35 degrees. It may stand any where, and is thus to be limited. Take the length from Aries to Cancer or Ca­pricorn, in the streight Zodiac, and with that length open the Sector (for it is soonest done by that instrument) in the line of sines from 23½ degrees thereon. Then from the Sector so opened, take the several sines of 35 degrees, and insert them into this line, so it shall be di­vided into its requisite parts. The pattern of these things may be seen in my Theorics.

Thus farre I have gone in declaring the manner how these Theorics are made in all their particulars. There now follow the Tables that are mentioned before, by which many things are to be divided and limited.

[Page 41]

How to make the praecedent Table serve for times to come.

IN 100 years, the Aphelia and Nodes of the Planets move forward thus much,

[Page 42]And by these numbers, the Tables precedent (which are made to the year 1673 complet) may be fitted to any year to come. For these said Tables (those noted with C, I onely speak of) as they now are, will serve till the year 1700. And afterwards they may be fitted to 1730; that is, for 30 years to come, after that period of time, and so they will serve in use till 1760 very well. For in 30 years the Nodes make this progresse onely, which in their latitudes will not erre above ⅛ of a degree, no not in Mars and Venus, in which two Planets this errour must be greatest.

I say these tables noted with C, are made for the year 1673 com­plete. And if you would rectifie them to the year 1730 complete, you are first to take the diffe­rence of these two years, 1673 and 1730, which will be 57: and by 57 multiply the Aphe­lial numbers or progresses at K, and from the product cut off the 5 last figures; the remainder shall be the degrees, and the fraction shall be the decimal parts of degrees, which will easily be turned into sexagesi­mal parts. And then the num­ber so found out for each Pla­net, must be added respectively to every number of his proper Planet in the precedent Table [Page 43] C: and so the numbers of that Table shall be rectified for the year 1730.

In the same manner you may rectifie the places of the Nodes by multiplying the former num­bers of the Nodes motion at K, into 57, &c. as before. Then the corrections must be added to each Planet respectively ac­cording as the places of their Nodes are expressed in the Ta­ble M.

The Aphelia, and Nodes ought not to stand still (in ri­gour) but to move continually some small quantity. Yet be­cause these motions are very slow, they may be permitted to stand still for some number of years without much prejudice to these Planetary Instruments.

The errour of Latitude wch ariseth from the immobility of the Nodes, is in 30 years (even in Mars and Venus) not above 8 minutes, as was shewed be­fore. And the errour in Longi­tude, which ariseth by reason of the immobility of the Aphe­lia, will in 30 years time be about 31 minutes in the Earth or Sun; 38 min. in Saturn; 24 min. in Jupiter; 33 m. in Mars, [Page 44] 39 min. in Venus; 52 min. in Mercury; which may well be endured in these mannuary Theorics.

Let the Aphelial distances be divided into these numbers here set to every Planet, and continued in the same parts be­yond the Center, so farre as is needfull. So shall their distan­ces from the Earth and the Sun be had in semidiameters of the Earth; If first they be measu­red upon their proper scales: and secondly, if Saturns di­stance be multiplyed by 400; Jupiters by 200, Mars his di­stance by 100; Venus, Mercu­ry and the Earth upon the same side with them by 50. Which numbers may be easily remem­bred, because they goe in a sub­duple proportion.

[Page 45]

[Page 46]

These Epochaes do endure till 1700. If it be required to continue them further for every 8 years, then from the last Epocha of the Earth must be subtracted the number here standing by the Earth, namely, 0. 077; and in all the other Planets the numbers here set down must be added to the last Epocha of each of them standing in the superiour Ta­ble of Epochaes. All the cor­ction that is requisite is to be done in the Epochaes, in the rest of the Tables of motions, which now follow, there will be no need of any such things.

[Page 47]

[Page 48]

These are all the Tables that are to be set upon the Theori­cal plates, whereby the equal or Mean Anomalyes may be ga­thered to any day at Noon. The manner how they are to stand upon the two Plates with such convenience that they may be ready for use, annexed each to the proper Orbit of its own Planet, without hindrance of the other work that is there drawn, may best be seen upon my Theorics.