Schott, Gaspar Mechanica Hydraulico-pneumatica. Pars I. Mechanicæ Hydraulico-pnevmaticæ Theoriam continet. 1657 la schot_mecha_051_la_1657.xml 051.xml

A. R.

P. GASPARIS SCHOTTI

SOCIETATIS JESU

MECHANICA HIDRAULICO -PNEVMATICA

Cum

FIGURIS ÆNÊIS, ET PRIVILEGIO SACRÆ

Ce&longs;aræ Maje&longs;tatis.

ANNO M DC LVIII.

MECHANICA HYDRAVLICOPNEVMATICA

Ad Eminenti&longs;s: S.R.I. Principem Ioannem Philippum Electorem Mogunt: Auctore.

P. GASPARE SCHOTTO.

Soci Iesu

P. GASPARIS SCHOTTI

REGISCURIANI, E SOCIE­TATE JESU,

Olim in Panormitana Siciliæ, nunc in Herbipo­litana Franconiæ eju&longs;dem SOCIETATIS Academia Mathe&longs;eos Profe&longs;&longs;oris,

MECHANICA HYDRAULICO-PNEV­MATICA,

Qua Præterquàm quòd Aquei elementi natura, proprietas, vis motrix, atque occultus cum aëre conflictus, à primis fundamentis de­mon&longs;tratur; omnis quoque generis Experimenta Hydraulico-pnevmatica recluduntur; & ab&longs;oluta Machinarum aquâ & aere animandarum ratio ac methodus præ&longs;cribitur.

OPUS BIPARTITUM,

Cujus

Pars I. Mechanicæ Hydraulico-pnevmaticæ Theoriam continet.

Pars II. Eju&longs;dem Praxin exhibet, Machinasque Aquarias innumeras, uti & Organa, ali­aque In&longs;trumenta, in motum ac &longs;onum concitat; nec non varia techna&longs;inata, quæ motum perpetuum vi aquæ &longs;pondent,exponit.

ACCESSIT

Experimentum novum Magdeburgicum, quo vacuum alij &longs;ta­bilire, alij evertere conantur.

Sumptu Heredum JOANNIS GODEFRIDI Schôn wetteri,

Bibliopol: Francofurten&longs;.

Excudebat HENRICUS PIGRIN Typographus Herbipoli,ANNO M. DC. LVII.

DEDICATIO.

Eminenti&longs;&longs;imo, & Reverendi&longs;&longs;imo Principi ac Domino,

D. JOANNI PHILIPPO,

SACRÆ SEDIS MOGUNTINÆ

Archiepi&longs;copo, S. Rom. Imp. per Germani­am Archi-Cancellario, & Principi Electori; Epi&longs;copo Herbipolen&longs;i, Franciæ Orientalis Duci, Domino meo clementi&longs;&longs;imo.

EMINENTISSIME PRIN­CEPS. Fontes quos Natura producit ad hominum utilitatem, Mare petunt, ut fluant. Omnia quippe flumina, ac proinde & fontes flumi- num origines, intrant in Mare, & Mare non redundat; ad locum, unde exeunt flumina, revertun­tur, vt iterum fluant. Fontes quos hoc in Opere produco ex Hydrotech­nica Arte ad Principum delecta­tionem, ut &longs;aliant, FONTEM petunt PULCHRUM, Te inquam, EMI­NENTISSIME DOMINE, Tuumque favorem ac benevolentiam. Ni&longs;i enim ope Tua, qui FONS es PUL­CHERRIMUS, foveantur, & Tua augeantur ditenturque co­pia; exare&longs;cant nece&longs;&longs;e e&longs;t, ma­gno licet labore, nec minori Arte, in fluxum ac &longs;altum àme animati. Pateat igitur Fontibus meis FONS PULCHER; deriventur in eos Humanitatis Tuæ aquæ, ut gloriari liceat, ac dicere: FONS PULCHER SUFFICIT UNDAS. Dixi quod volebain, & hîc finio, nè peccare cogar peccatum quo Scriptorum plerique reos &longs;e con­&longs;tituunt, dum in Mecænatum &longs;uorum laudes ip&longs;o in Librorum ve&longs;tibulo quàm pro loci oppor­tunitate longiùs excurrunt. Ma­jor e&longs;t virtutum Tuarum &longs;plen­dor, majora in Eccle&longs;iam & Im­perium Romanum merita, quàm ut exili calamo meo, præ&longs;ertim hoc loco, exarentur.

Vale PRINCEPS EMINEN­TISSIME, Imperii columen, & Eccle&longs;iæ decus; ae bono utrius­que quàm diutis&longs;imè vive. Ita optat Herbipoli Die VIII. Julij, Anno MDCL VII.

EMINENTISSIMÆ TUÆ

CELSITUDINIS

Humillimus Cliens

Ca&longs;parus Schottè So­cietate JESU.

IN FRONTEM LIBRI

ACROAMA EXEGETICUM.

FONS PULCHER SUFFICIT UNDAS.

Obmute&longs;ce Antiquitas; Jam dudum exaruit lympha, Quam propinavit Pincerna vatum

PEGASUS.

Riget in&longs;tar pumicis, qui Parna&longs;&longs;um irrigavit: Qui infantium etiam linguas fecerat di&longs;ertas, Dudum, prô pudor, e&longs;t de&longs;ertus. Pone cri&longs;tas Heliconis Nympha;

HIPPOCRENE FUIT!

Unde, aut ubi fluxerit olim, quis &longs;ciat? Ca&longs;talides ip&longs;æ nefciunt, Ex quo toties potæ cecinerunt.

Et quid mirum Helicona nu&longs;quam apparere, Si ip&longs;a Græcia di&longs;paruit, & migravit In EUKOPAM reliquam;

In quâ non unas numeres Athenas, Quando pæne innumera vides Athenæa, Suo quodque formo&longs;um Apolline, Suo quodque irriguum fonte, E quo bibunt &longs;apientes Europæi, Et entheantur.

De Europa univer&longs;a periculum facin &longs;ola

GERMANIA:

De Germania verò pronuncia, Si, quæ Rhenus, Mœnusve alluit, invi&longs;as

ATHENÆA, MOGANUM ET ARTAUNIUM.

Quæ, qualésve ea incolant Mu&longs;æ, Di&longs;ce ex latice, qui eas reficit.

FONS

Medio bullit è Nobilitatis Rhenanæ jugo, Non &longs;onipedis Pega&longs;i extu&longs;us ungulâ; Sed jubati LEONIS effo&longs;&longs;us ungue,

CALLOCRENE.

Hoc uno ambæ libant & vivunt de fonte, Cælo divi&longs;æ, non zelo,

MOGANIDES MUSÆ ET FRANCONIDES.

His Seu &longs;itim arentis gutturis levent, Seu barbita pulvere glorio&longs;o &longs;ordida lavent, Seu in carmina animent Vates, Seu fugientes Vatibus venas revocent, Seu in pr&ecedil;mia lauros & palmas rigent, Seu fe&longs;tivè cum Nymphis ludant Apollinis honori,

FONS PULCHER SUFFICIT UNDAS.

Verùm quid Mu&longs;as, Nymphásque memoro? Ip&longs;a adeò Natura hoc fonte haurit, Ut parturientem juvet Artem:

Hinc Ars bibit, ut languentem fulciat Naturam, Experti di&longs;cimus: Quod non &longs;ola Natura audet. Nec Ars &longs;ola pote&longs;t, Et pote&longs;t cum Arte Natura & audet. Verùm deficerent, ni&longs;i &longs;ufficeret,

FONS PULCHER

Undas, Animos, Vires. Hoc manante animatur emortuus Æolus, Et &longs;pirat:

Hoc &longs;tillante lique&longs;cunt, licet ære fu&longs;æ, Phaëtontiades, Et lacrymantur:

Hoc illabente alterni folles inflantur & efflantur, Et organa re&longs;onant:

Hoc &longs;aliente Deorum &longs;imulacra irrorantur, Et tripudiant.

Verbo, Paucis multa dicam:

CRESCIT QUODCUNQUE RIGAT, VIGET QUODCUNQUE LAVAT,

FONS PULCHER.

Hinc e&longs;t quod &longs;tupent ob&longs;tipi; Et ægrè vident Invidi,

HOC FONTE,

Virere lauros, Musásque vigere.

FONS PULCHER.

Feras cicurat, mon&longs;tra domat: Hercule felicior!

Quæ is exanimabat, hic animat;

Quæ i&longs;te clavâ fregit, ille regit Suaviter influendo.

Hoc &longs;tratagemate, Rotas Moganas firmavit, Ut bene verterentur, Quæ à procellis, quia rapiebantur, Pæne evertebantur.

Pluris &longs;cilicet &longs;æpe e&longs;t e&longs;&longs;e humanum quàm fortem, Ubi gloria e&longs;t major, Vincere amoribus quàm viribus. In fontibus reliquis eximium hoc habet

FONS PULCHER,

Quòd non aquam &longs;olùm fundat, Quâ &longs;uam effæta reparet Juventam

AQUILA;

Sed & oleum &longs;tillet, Quo repullulantes pennas ad robur Ungat.

Nec Aquilæ modò hoc Fonte lotæ Juvene&longs;cunt:

Sed, quia cum Igne quid commune habet

FONS PVLCHER,

Hoc etiam in FONTE emortuus,

Novo prodigio Novus Revivi&longs;cit

PHOENIX.

Unde infelici&longs;&longs;imorum temporum Hæc una e&longs;t felicitas, Quod mergantur in FONTE PULCHRO, In quo ni&longs;i tota &longs;ubmergantur, Nunquam emergent Feliciora.

Et Gloria hæc &longs;umma erit

FONTIS PVLCHRI,

In ea incidi&longs;&longs;e tempora, In quæ ni&longs;i incidi&longs;&longs;et, Forent tempora omnino Profligata.

De FONTE PULCHRO quid ampliùs dicam? Fontes alij aut æ&longs;tu pro&longs;pero ex&longs;iccantur, Aut algore &longs;ini&longs;tro congelant, Ut fluxum &longs;i&longs;tant:

FONS PVLCHER.

FLUET DONEC INFLUAT IN FONTEM QUOCUM SALIET IN VITAM ÆTERNAM.

Quod vovent & precantur mecum

OMNES BONÆ MUSÆ A FONTE PULCHRO LOTÆ, RIGATÆ, POTÆ.

P. NICOLAUS MOHR è Socie­tate JESU.

Facultas admodum R. P.

PRÆPOSITI GENERALIS SOCIETATIS JESU.

GOSWINUS NICKEL SOCIETA­TIS JESU

PRÆPOSITUS GENERALIS.

CUm Opus, quod in&longs;cribitur Mechanica Hydraulico-pnevmatica, à P. GASPARE SCHOTT noftræ Societatis Sacerdote compo&longs;i­tum, aliquot eju&longs;dem Societatis Religio&longs;i recognove­rint, & in lucem edi po&longs;&longs;e probaverint; facultatem concedimus vt typis mandetur, &longs;i ijs ad quos pertinet, ita videbitur. In cujus rei te&longs;timonium has litteras manu no&longs;trâ &longs;ub&longs;criptas, &longs;igillo&queacute; no&longs;tro munitas da­mus Romæ, 23. Januarij 1655.

GOSWINUS NICKEL.

FACULTAS

R. P. PROVINCIALIS SOCIETATIS IESV

Per Rheni Superioris Provinciam Bi­bliopolæ facta.

CUm ex Mandato Sacræ Cæ&longs;areæ Maje&longs;tatis omnibus & &longs;ingulis Typographis, Bibliopolis, ac aliis quamcun­que librariam negotiationem exercentibus, &longs;eriò firmiterque in­hibeatur, nè qui&longs;quam libros ullos à SOCIETATIS no&longs;træ Pa­tribus hactenus editos, aut impo&longs;terum edendos intra S.R. Im­perij, Regnorum, & Dominiorum Suæ Cæ&longs;areæ Maje&longs;tatis hæ­reditariorum fines, &longs;imili aliovè charactere aut formâ, &longs;ive in toto, &longs;ive in parte recudere, vel aliò recudendos mittere, aut ali­bi etiam impre&longs;&longs;os adducere, vendere, & di&longs;trahere, clàm &longs;eu palàm, citrà &longs;upradictorum Patrum con&longs;en&longs;um ac te&longs;timonium, audeat, vel præ&longs;umat: Ego NITHARDUS BIBERUS, SOCIETATIS JESU per Rheni &longs;uperioris Provinciam Pro­vincialis, concedo Joanni Godefrido Schôn wettero fa­cultatem, &longs;uis &longs;umptibus excudendi P. GASPARIS SCHOTTI è SOCIEIATE JESU Mechanicam Hydraulico-pnev­maticam. In quorum fidem hoc ei te&longs;timonium manu no&longs;trâ &longs;ub&longs;criptum, & Sigillo munitum dare voluimus. Herbipoli21. Fanuarij 1656.

NITHARDUS BIBERUS.

L. S.

ELENCHUS

ELENCHUS TITU­LORUM,

Sive SYNOPSIS OPERIS.

Præloquium ad Lectorem, de Operis Occa&longs;ione, Divi&longs;ione, In&longs;criptione; deque Hydraulicorum & Pnevmaticorum Scriptoribus. pag. 1.

PARS I.

THEORETICA.

De Machinarum Hydraulico-Pnevmaticarum principiis &longs;eu fundamentis. 15

PROTHEORIA I.

De vi Attractiva corporum, ad vacuum vel replendum, vel fugiendum: &longs;eu de primo Machinarum Hydro-pnevmaticarum principio. 18

§. I. Hero Alexandrinus vacuum in aëre & aqua di&longs;&longs;eminatum agno&longs;cit. 19

§. II. Experimenta quibus Hero vacuum in corporibus di&longs;&longs;eminatum pro­bat. 21

§. III. Experimenta quibus vacuum nullum e&longs;&longs;e probatur. 25

§. IV. Vis attractiva ob vacui metum, primum Machinarum Hydro-pnevma­ticarum principium, quæ, & qualis, quibusvis in&longs;it corporibus. 28

§. V. Experimentum, quo o&longs;tenditur vis attractiva ad aquas in altum evchen­das, vacui vitandi causâ, per Machinas Hydro-pnevmaticas. 30

§. VI. Notantur nonnulla circa prædictum Experimentum, &longs;imulque &longs;tabili­tur, in attractione aquæ metu vacui habendam e&longs;&longs;e rationem perpendiculi aquarum. 33

§. VII. Heronis Alexandrini error in elevanda aqua vi attractiva. 36

§. VIII. Alius Heronis error in elevanda aqua vi attractiva. 37

§. IX. In aquarum elevatione vi attractivâ, habenda e&longs;t ratio perpendiculi &longs;olùm illorum tuborum, per quos aqua, non per quos aër vehitur. 39

TITULORUM.

§. X. Viattractivâ elevari pote&longs;t aqua ultra perpendiculi altitudinem, aëre me­diante. 40

PROTHEORIA II.

De vi Expul&longs;iva propter corporum impenetrabilitatem; &longs;eu de &longs;ecundo Machi­narum Hydro-pnevmaticarum principio. 43

§. I. Omnia univer&longs;i corpora &longs;unt inter &longs;e contigua. ibid.

§. II. Corpora &longs;e mutuò expellunt. 45

§. III. Experimentum, quo o&longs;tenditur vis expul&longs;iva, propter corporum im­penetrabilitatem, ad aquas elevandas in altum. 46

§. IV. Notantur nonnulla circa prædictum Experimentum. 48

§. V. Heronis error in elevando oleo vi expul&longs;ivâ in lucerna. 50

§. VI. Attractione & expul&longs;ione &longs;imul aquam elevare po&longs;&longs;umus. 53

§. VII. Salomon Caus meritò &longs;ugillat Heronem. 55

§. VIII. Idem perpendiculum aquæ cadentis non pote&longs;t &longs;ervire &longs;imul attra­ctioni & expul&longs;ioni. 57

§. IX. Error apud Joannem Bapti&longs;tam Portam indicatur. 58

§. X. Alia ratio aquam elevandi in altum propter corporum impenetrabili­tatem. 59

PROTHEORIA III.

De vi Rarefactiva; &longs;eu de tertio Hydro-pnevmaticarum Machinarum prin­cipio. 61

§. I. Rarefactionis vis, & effectus mirabilis. ibid.

§. II. Alia rarefactionis exempla. 62

PROTHEORIA IV.

De fluxu aquæ naturali; &longs;eu de quarto Hydro-pnevmaticarum Machinarum principio. 64

CAPUT I. De Proprietatibus aquæ fluentis liberè. 65

PROPRIETAS

I. Aqua tendit ad loca decliviora. 67

II. Aquæ &longs;uperficies &longs;uperior, ce&longs;&longs;ante fluxu, &longs;phærica e&longs;t; reliquæ conforman­tur va&longs;is & receptaculis. ibid.

III. Aqua minùs pre&longs;&longs;a expellitur à magis pre&longs;&longs;a. 68

IV. Aquæ, & humidi cuiuscunque, pars unaquæque premitur humido &longs;upra ip&longs;am exi&longs;tente ad perpendiculum, &longs;i humidum &longs;it de&longs;cendens in aliquo, aut abalio aliquo pre&longs;&longs;um. ibid.

V. Aquâ in &longs;itu naturali con&longs;i&longs;tente, partes &longs;uperiores non premunt inferio­res. 70

VI. Aquæ in &longs;itu naturali po&longs;itæ una pars non expellit alteram. 71

VII. Non omnes æquæ æquales magnitudine, &longs;unt æquales pondere. ibid.

VIII. Aqua naturaliter non a&longs;cendit ad locum altiorem &longs;uâ origine. ibid.

CAPUT II. De Proprietatibus aquæ fluentis per &longs;iphones. 72

PROPOSITIO

I. Siphonis varias divi&longs;iones a&longs;&longs;ignare. ibid.

II. Siphonis erecti proprietates a&longs;&longs;ignare. 74

III. Siphonis inver&longs;i proprietates a&longs;&longs;ignare. 77

IV. Siphonis mixti inæqualium crurium proprietates a&longs;&longs;ignare. 81

V. Siphonis mixti æqualium crurium proprietates a&longs;&longs;ignare. 84

VI. Inver&longs;i &longs;iphonis, va&longs;i uno crure impo&longs;ito, proprietates a&longs;&longs;ignare. ibid.

VII. Cau&longs;am ad&longs;ignare, cur quando o&longs;culum externi cruris &longs;iphonis inver&longs;i e&longs;t altius, aut æquè altum ac aquæ &longs;uperficies &longs;uprema in va&longs;e, aqua non fluat; quando verò e&longs;t demi&longs;&longs;ius, fluat. 88

VIII. Siphonem ad libram &longs;eu vectem revocare, & filtrum ad &longs;iphonem. 93

IX. Diabetem &longs;piritalem de&longs;cribere, eiu&longs;que proprietates ad&longs;ignare. 94

X. Siphonis inver&longs;i fluxum æqualem reddere. 96

XI. Siphone inver&longs;o aquam ex uno montis latere in alterum, per verticem, de­ducere. 98

XII. Siphone inver&longs;o aquam ex montis radice ad ejus verticem elevare. 102

XIII. Cau&longs;am a&longs;&longs;ignare, cur in &longs;iphone inver&longs;o interrupto perpendiculum aquæ de&longs;cendentis debeat &longs;uperare perpendiculum aquæ a&longs;cendentis. 104

XIV. Cau&longs;am a&longs;&longs;ignare, cur in Fonte Heronis perpendiculum aquæ de&longs;cen­dentis debeat e&longs;&longs;e longius perpendiculo aquæ a&longs;cendentis. 107

CAPUT III. De Proprietatibus aquæ fluentis per tubos. 110

PROPOSITIO

I. Aqua decurrit per tubi verticalis foramen ba&longs;is, in&longs;tar columnæ aqueæ, cuius ba&longs;is æqualis foramini, altitudo perpendiculares à ba&longs;i erectæ; &longs;ive tubi &longs;int &longs;emper pleni, &longs;ive non. 111

II. Per tubos tam &longs;emper, quàm non &longs;emper plenos æqualis altitudinis, & æqua­lium foraminum, effluit æqualis aquæ copia, eodem vel æquali tempore, cujuscunque capacitatis & formæ &longs;int tubi. 113

III. Per tubos tam &longs;emper, quàm non &longs;emper plenos æqualium luminum, &longs;ed inæqualium altitudinum, effluit eodem, vel æquali tempore, inæqualis aquæ copia. 114

IV. Per tubos &longs;emper, & non &longs;emper plenos inæqualium luminum, &longs;ed æquali­um altitudinum, effluit eodem, vel æquali tempore, inæqualis aquæ copia. ibid.

V. Per tubos æquè altos, & æqualium luminum, non &longs;emper plenos, fluit eodem tempore æqualis aquæ copia; &longs;ed tantò fluit unus diutiùs altero, quantò plus aquæ continet unus quàm alter. 115

VI. Per tubos non &longs;emper plenos, & non æquè altos, æqualium tamen lumi­num, eodem vel æquali tempore non fluit æqualis aquæ copia. ibid.

VII. Tubus altitudine quadrupedalis, cui pro ba&longs;is diametro pedis Pari&longs;ien&longs;is uncia, aquâ &longs;emper, plenus, effundit per lumen lineare in ba&longs;i &longs;itum, aquæ libram &longs;patio tredecim minutorum &longs;ecundorum temporis. 116

VIII. Aquæ fluentes ex tubis tam &longs;emper quàm non &longs;emper plenis æqualium foraminum, &longs;ed inæqualium altitudinum, habent rationem &longs;ubduplicatam altitudinum tuborum; habentque dicti tubi duplicatam rationem aqua­rum, quas fundunt. 117

IX. Aqua naturali motu de&longs;cendens & effluens per tubos, imitatur leges alio­rum gravium naturali motu de&longs;cendentium. 120

X. Velocitates motus aquæ de&longs;cendentis & effluentis per tubos æqualium fo­raminum, &longs;ed inæqualium altitudinum, habent &longs;ubduplicatam rationem altitudinum. 125

XI. Cau&longs;am a&longs;&longs;ignare, cur aquæ fluentes per tubos æqualium luminum, &longs;ed in­æqualium altitudinum, habeant rationem &longs;ubduplicatam altitudinum tu­borum. 126

XII. Tempora quibus æqualis aquæ quantitas è tubis æqualium luminum, &longs;ed inæqualium altitudinum effluit, habent &longs;ubduplicatam rationem tubo­rum. ibid.

XIII. Si tubi, &longs;ive &longs;emper pleni, &longs;ive non &longs;emper pleni, &longs;int ejusdem altitudinis, &longs;ed inæqualium foraminum, e&longs;t eadem ratio aquæ ad aquam, quæ foraminis ad foramen, phy&longs;icè &longs;eu ad &longs;en&longs;um. 127

XIV. Tubi non &longs;emper pleni æquè alti, & æqualium foraminum; &longs;ed inæquali­um ba&longs;ium, evacuantur inæqualibus temporibus, e&longs;tque eadem ratio tem­porum, quæ ba&longs;ium. 129

XV. Tempora quibus deplentur tubi non &longs;emper pleni, æquèlati, &longs;ed non æquè alti, per æqualia foramina, &longs;unt in altitudinum ratione &longs;ubduplicata. 131

XVI. Tempora quibus evacuantur tubi non &longs;emper pleni &longs;imiles, & æquales quoad altitudines & ba&longs;es, per lumina &longs;imilia inæqualia, &longs;unt reciprocè, ut lumina. ibid.

XVII. Datis altitudine & foramine tubi &longs;emper pleni, invenire quantitatem aquæ quam dato tempore effundat; vel, datis iisdem, invenire magnitudi­nem ci&longs;ternæ quæ dato tempore repleatur. 132

XVIII. Datis altitudine & lumine tubi &longs;emper pleni, invenire tempus quo da­tam aquæ quantitatem effundat, &longs;ive quo datam ci&longs;ternam impleat. 135

XIX. Datis tempore, quantitate aquæ, &longs;eu ci&longs;terna, & lumine tubi, invenire alti­t&udot;dinem tubi, qui &longs;emper plenus ci&longs;ternam dato tempore repleat. 136

XX. Dato va&longs;e, & foramine per quod effluit aqua, invenire tempus quo eva­cuatur. 137

XXI. Dato va&longs;e, & tempore, invenire foramen per quod evacuetur tempore dato. 139

XXII. Altitudinem &longs;caturiginis dati fontis per tubos fluentis invenire. 140

XXIII. Data alicujus tubi, aut va&longs;is erogatorii altitudine, ac tempore quo deter­minatam aquæ quantitatem è &longs;uo lumine effundit, invenire altitudinem ejusdem aut alterius tubi, qui æquali tempore, per æquale lumen, aliam de­terminatam aquæ quantitatem effundat. 140

XXIV. In tubo &longs;eu va&longs;e &longs;emper pleno determinare &longs;patia, quæ temporibus æqualibus &longs;ibi &longs;uccedentibus evacuantur; uti & menfuram &longs;eu pondus a­quæ quæ effluit. 141

PARERGUM CAPITIS III.

De inventione mediæ, ac tertiæ proportionalis quantitatis, in numeris, & lineis. 144.

PROPOSITIO

I. Inter duos numeros medium proportionalem invenire. 145

II. Datis duobus numeris, tertium continuè proportionalem invenire. ibid.

III. Inter duas rectas lineas datas invenire tertiam proportionalem. ibid.

IV. Datis duabus rectis, invenire tertiam proportionalem. 146

V. Aliter invenire tertiam proportionalem. 147

VI. Adhuc aliter invenire tertiam proportionalem. ibid.

VII. Datis tribus numeris invenire quartum proportionalem. 148

VIII. Datis tribus rectis lineis, quartam proportionalem invenire. ibid.

CAPUT IV.

De proprietatibus aquæ &longs;alientis ex tubis. 149

PROPOSITIO

I. Salientium &longs;ive ecdromorum horizontalium, & mediorum, &longs;uper eodem ho­rizonte, longitudines &longs;unt in ratione &longs;ubduplicata tuborum, ex quibus exi­liunt. 151

III. Salientes horizontales & mediæ, ejusdem tubi, eò &longs;unt longiores, quò lu­men tubi fuerit altius &longs;upra horizontem. 153

III. Salientes horizontales & mediæ, eandem longitudinis rationem &longs;ervant, quam altitudines tuborum &longs;uper eundem horizontem. 154

IV. Salientium verticalium in quacunque elevatione tubi &longs;upra horizontem &longs;emper eadem e&longs;t altitudo. 155

V. Saliens verticalis nunquam adæquat aluitudinem originis. 156

VI. Saliens verticalis tubi quadrupedalis proximè æquat quinque &longs;extas tubi &longs;ui partes. ibid.

VII. Salientes verticales eò &longs;unt longiores, quò tubi &longs;unt longiores; &longs;ed non eâ­dem proportione illæ ac hi cre&longs;cunt. ibid.

VIII. Data tubi altitudine, & &longs;upra horizontem elevatione, invenire longitudi­nem &longs;alientis horizontalis, & mediæ. 157

IX. Data longitudine &longs;alientis horizontalis, aut mediæ, invenire altitudinem tu­bi, cognitâ ejus elevatione &longs;upra horizontem. ibid.

X. Ex nota &longs;caturiginis altitudine fontis unius ex tubo exilientis horizontaliter, altitudinem &longs;caturiginis cujuscunque alterius æqualiter &longs;upra horizontem elevati invenire. 158

CAPUT V.

De fluxu aquæ per diver&longs;a ejusdem va&longs;is aut tubi foramina. 158

PROPOSITIO

I. Per foramina æqualia, æquè à &longs;ummo tubi di&longs;tantia, &longs;ive in ba&longs;e, &longs;ive in latere, æquali tempore æquales fluunt aquarum quantitates. 160

II. Aquæ è foraminibus æqualiter à &longs;ummo tubi di&longs;tantibus decurrentes, &longs;unt in­ter &longs;e ut foramina. ibid.

III. Aqua per foramina va&longs;is eo impetu &longs;eu velocitate decurrit, quo per tubos æqualium foraminum & altitudinum. 161

IV. Velocitates aquæ decurrentis per foramina æqualia ejusdem va&longs;is, inæqua­liter di&longs;tantia à &longs;ummo va&longs;is, &longs;unt in &longs;ubduplicata ratione di&longs;tantiæ. 162

V. Aquæ per æqualia foramina inæqualiter à &longs;ummitate va&longs;is di&longs;tantia fluentes, &longs;unt in &longs;ubduplicata ratione di&longs;tantiarum. ibid.

VI. Secto foramine laterali va&longs;is in partes æquales, à rectis horizontalibus, in­venire rationes aquarum ex eis fluentium. 163

VII. Secto foramine laterali va&longs;is in partes inæquales, à rectis horizontalibus, reperire rationes aquarum effluentium ex ip&longs;is. 164

VIII. Datis foraminibus inæqualibus &longs;uper eadem horizontali, venari rationes aquarum. ibid.

IX. Datis foraminibus ejusdem va&longs;is, quorum unum &longs;uperius, alterum inferius, inter easdem parallelas perpendiculares, reperire rationes aquarum. 165

X. Datis foraminibus ejusdem va&longs;is, quorum unum &longs;uperius, alterum inferius; non inter easdem parallelas, reperire rationes aquarum. ibid.

XI. Dato foramine, & linea horizontali, in aliquo va&longs;e, con&longs;tituere &longs;uper illa fo­ramen, è quo æqualis aqua fluat eodem tempore. 166

XII. Dato foramine, & latere alterius in eodem va&longs;e, reperire foramen, è quo æqualis aqua effluat. ibid.

XIII. Dato foramine, reperire aliud æquale in eodem va&longs;e, è quo fluat aqua in ratione data. 167

XIV. Dato foramine, aptare in eodem va&longs;e aliud datum &longs;imile, magnitudinis di­ver&longs;æ, à quo aqua fluens cum fluente à primo, habeat rationem datam. 168

CAPUT VI.

De aëris gravitate, rarefactione, & conden&longs;atione. 169

PROPOSITIO

I. Aëris gravitatem invenire. 169

II. Quantum conden&longs;ari aër po&longs;&longs;it, invenire. 170

III. Quantum rarefieri aër po&longs;&longs;it, invenire. 171

PARS II.

PRACTICA.

De fabrica Machinarum Hydro-pnevmaticarum ex traditis principiis. 173

CLASSIS PRIMA.

De variis Hydraulicis atque Pnevmaticis Machinis. 176

CAPUT I.

De Machinis quæ fiunt vi attractiva. ibid.

MACHINA

I. Fonticulus phialæ vitreæ inclu&longs;us. 177

II. Siphon inver&longs;us interruptus, aquam in determinatam altitudinem evchens. 179.

III. Fons Cæ&longs;areus. 181

IV. Scyphus plenus per fundum effundens liquorem, non plenus retinens. 183

V. Cancer vomitor. 185

VI. Sphæra vitrea Aqui&longs;uga. 186

VII. Siphon inver&longs;us interruptus, elevans aquam in quamvis altitudinem. ibid.

VIII. Navis Horologa. 188

IX. Siphon inver&longs;us horologus. 189

X. Bina va&longs;a, quorum uni &longs;i aqua infundatur, alterum reddit vinum. 190

CAPUT II.

De Machinis quæ fiunt vi expul&longs;iva. 191

I. Fons Heronis in va&longs;is immediatis. 192

Item Fonticulus &longs;imilis Fonti Heronis. 194

II. Clep&longs;ydra Heroniana. 195

III. Fons Heronis in va&longs;is mediatis: item &longs;cyphus diver&longs;os ejiciens liquores. 197

IV. Fons novus Poly&longs;iphonius. 201

V. Fons perennis, alto in loco aquam è puteo profundo &longs;ubmini&longs;trans. 203

VI. Antlia, &longs;eu Cte&longs;ibia Machina. 205

VII. Speculator cornu inflans. 207

VIII. Fonticulus compre&longs;&longs;ione aquam &longs;pargens in altum: Baculus item viato­rius aquivomus, & Catellus mingens. 208

IX. Phiala vitrea, compre&longs;&longs;ione aquam projiciens in altum. 211

X. Infundibulum pnevmatico-hydraulicum, aquam in determinatam altitudi­nem attollens. 211

XI. Infundibulum alterum pnevmatico-hydraulicum, aquam in quamlibet al­titudinem extollens. 213

XII. Fons eâdem fi&longs;tulâ di&longs;colores ejiciens liquores. 214

XIII. Sclopetum Æolium &longs;eu pnevmaticum. 216

XIV. Aliud &longs;clopetum Pnevmaticum. 217

XV. Sedes Aquivoma. 219

XVI. Heronis va&longs;a quæ vinum pro aquainfu&longs;a reddunt: &longs;eu Hydriæ Canæ Ga­lilææ. 220

XVII. Hydraconti&longs;terium antiquum. 222

XVIII. Hydraconti&longs;terium novum. 223

CAPUT III.

De Machinis quæ fiunt rarefactione. 226

MACHINA

I. Pyrobolus fons, incale&longs;centis per ignem aëris vi aquam expellens. 226

II. Fons alius pyrobolus, projectum liquorem convertens in aërem, autignem 226 Item Meteorologicæ impre&longs;&longs;iones igneæ. 229

III. Thermo&longs;copium progno&longs;ticum Hybernum. 229 & Æ&longs;tivum. 231

IV. In&longs;trumentum novum, gradus humidi & &longs;icci indicans. 232

V. Cacabus ejiciens, & retrahens eandem aquam. 236

VI. Pilæ Æoliæ. 237

VII. Ci&longs;ta Æolia. 238

VIII. Æolus ventum efflans. 239

IX. Memnonia &longs;tatua, citharæ, & humanæ vocis &longs;onum ad orientem &longs;olem edens. 240

X. Memnoniæ aves, voce & motu animatæ. 242

XI. Ara a&longs;pide adornata, in qua igne po&longs;ito I&longs;is & O&longs;iris vinum & lac &longs;acrificant a&longs;pis verò &longs;ibilando applaudit. 244

XII. Valvæ &longs;acelli &longs;uccen&longs;o &longs;acrificii igne &longs;ponte aperibiles, & extincto clau&longs;æ. 246

XIII. Ara Deorum imagines tripudiantes exhibens. 247

XIV. Hydrologium horarum A&longs;tronomicarum, &longs;eu æqualium. 248

XV. Fonticulus horarius. 249

XVI. Præco horarius, &longs;ingulis horis cornu inflans. 252

CAPUT IV.

De Machinis quæ fiunt naturali lap&longs;u aquæ. 254

MACHINA

I. Clep&longs;ydra Kircheriana, quæ fontis in&longs;tar ejaculatur aquam, & inver&longs;a ite­rum fluit. 254

II. Multimammia Deorum mater, lac ex uberibus promens. 256

III. Rota ver&longs;atilis, aquam lu&longs;trationi nece&longs;&longs;ariam fundens. 257

IV. Cycocephalus Ægyptius ex veretillo aquam fundens, quæ horas æquino­ctiales in &longs;ubjecto va&longs;e mon&longs;trat. 259

V. Hydrologium horarum antiquarum, &longs;eu inæqualium. 261

VI. Hydrologium Bettinianum. 264

VII. Hercules clavâ Draconem percutiens. 265

VIII. Aquila horodictica. 269

IX. Chorea &longs;erpentum aquivomorum. 270

X. Coluber volitans. 271

XI. Nauta Hydro-horologus. 272

XII. Libra hydraulica horodictica. 274

XIII. Hydraulicum horolabium facillimum. 275

XIV. Incubus Hydro-horologus. 276

XV. Hydrologium Magneticum. 277

CAPUT V.

De Machinis quæ habent principium mixtum. 278

MACHINA

I. Avis ex&longs;putam à &longs;erpente aquam &longs;orbens è cratere. 279

II. Vas hydro-pnevmaticum, omnis generis jocos exhibens. 281

III. Triton buccinâ inflatâ cur&longs;um fluminum fi&longs;tens. 285

IV. Atlas cælum humeris impo&longs;itum torquens in gyrum. 289

V. Lucerna Grunbergeriana. 290

VI. Hydrotechnicus tubus, varia ludentis Naturæ &longs;pectacula exhibens. 291

CAPUT VI.

De Machinis hydraulicis variis. 299

MACHINA

I. Sphæra in aëre fu&longs;pen&longs;a, ac circa &longs;uum centrum mota. 300

II. Cribrum Ve&longs;talis virginis. 303

III. Tubus vitreus, aquam perpetuò incorruptam con&longs;ervans. 304

IV. Phœnix hydro-botanica. ibid.

V. Phiala tabaci fumum refrigerans. 305

VI. Torricelli & Berti tubus vacuo vacuus. 306

VII. Hydropota varij coloris ac &longs;aporis aquam ex&longs;puens. 310

VIII. Palingene&longs;ia Cæ&longs;area. 313

IX. Vas eodem ore vinum & aquam, calidam & frigidam, hauriens ac fundens. 315

X. Prochyta Heronis Alexandrini malè con&longs;tructa. 317

XI. Libra hydro&longs;tatica mirabilis. 318

XII. Canopus Ægyptiacus. 320

XIII. Lamina plumbea plana, aquis innatans. 321

XIV. Anemo&longs;copium commune. ibid.

XV. Anemo&longs;copium Kircherianum. 323

Epilogus Cla&longs;&longs;is primæ, ubi nonnulla de fontibus Romanis, ac Tu&longs;culanis, aliisque. 325

CLASSIS SECUNDA

De Machinis motum perpetuum æmulantibus. 329

PROLUSIO. De motus perpetui arte procurati impo&longs;&longs;ibilitate. ibid.

MACHINA

I. Cochlea Archimedea motum perpetuum æmulans. 336

II. Tubus &longs;piralis in plano, aquam attollens, & motum perpetuum æmulans. 341

III. Horarium hydraulicum, motum perpetuum adumbrans. 346

IV. Kircheriana Machinula, motus perpetui rarefactionis & conden&longs;ationis &longs;pecimen exhibens. 351

V. Fonticulus motum perpetuum vi rarefactionis & conden&longs;ationis aëris e­mentitus. 353

VI. Horo&longs;copium Hydropnevmato-Magneticum, quo &longs;phæra Magnetica in medio liquorum librata perpetuò circumducitur, ad mon&longs;trandas ho­ras, circulos cœle&longs;tes, totiusque A&longs;trolabij my&longs;teria. 354

VII. Machinamentum aliud hydro-pnevmatico-magneticum, perpetuans mo­tum per ventum. 357

VIII. Cte&longs;ibica Machina perpetui motus æmula. 360

IX. Catona perpetuò mobilis in gyrum hydro&longs;taticâ arte, ut putabatur. 362

X. Situlæ automatæ, &longs;pontaneo atque perpetuo motu aquam haurientes. 365

XI. Funigiana indu&longs;tria perpetuo motu &longs;itulæ unicæ è puteis aquam hauriens. 368

XII. Perennis fontium ac fluviorum in Terraqua circulatio. 371

XIII. Gnomon Scheinerianus in centro mundi. 374

XIV. Mobile perpetuum Chymico-hydraulicum. 376

ANNOTATIO II. De Motu perpetuo quem Drebellius & alij adumbrarunt. 377

ANNOTATIO III. De motu perpetuo quem Boëklerus promittit. 379

ANNOTATIO IV. De motu perpetuo quem D. Har&longs;torfferus excogitavit. 381

CLASSIS TERTIA.

De Organis hydraulicis, aliisque in&longs;trumentis harmonicis hydropnevma­ticis. 383

MACHINA I. Organum hydraulicum automatum & autophonum. 384

PRAGMATIA I. Cameras Æolias fabricari, ad ventum organo hydraulico &longs;ubmini&longs;trandum. ibid.

PRAGMATIA II. Secundus modus Æolias cameras fabricandi. 386

PRAGMATIA III. Tertius modus Æolias cameras con&longs;truendi. 387

PRAGMATIA IV. Ventum per folles perpetuum producere in ordine ad Organa hydraulica. 388

PRAGMATIA V. Folles aliter inflare ad in&longs;trumenta chordophôna &longs;ive fidicina incitanda. 389

PRAGMATIA VI. Cylindrum phonotacticum con&longs;truere. 390

§. I. Cylindrum phonotacticum harmonicè delineare. ibid.

§. II. Cantilenam in Cylindrum phonotacticum harmonicè delineatum transferre. 397

§. III. Dentibus Cylindrum phonotacticum in&longs;truere. 399

§. IV. Plures cantilenas in eundem cylindrum transferre. 400

§. V. Cantilenas &longs;yncopatas, & b molli &longs;ignatas, ut & &longs;emitonia, in cylin­drum phonotacticum transferre. 401

§. VI. Minimi valoris notas cylindro phonotactico in&longs;erere. 406

PRAGMATIA VII. Cylindrum phonotacticum in gyrum movere. 407

PRAGMATIA VIII. Organum hydraulicum automatum atque antophô­num fabricare. 408

MACHINA II. Organum hydraulicum Vitruvianum. 409

MACHINA III. Cuculus cantans, atque tripudians. 414

MACHINA IV. Gallus cantans & alas quatiens. 414

MACHINA V. Diver&longs;æ volucres garrientes, ac &longs;e moventes. 417

MACHINA VI. Pan Octaulum inflans, Nympha Echo reflans. 418

MACHINA VII. Cyclopes automati Mu&longs;icam Pythagoricam exhibentes. 420.

MACHINA VIII. Clavicymbalum automatum, omnis generis in&longs;tru­mentorum fidibus in&longs;tructorum &longs;ymphoniam exhibens. 432

MACHINA IX. Automatum Kircherianum, omnis generis in&longs;trumento­rum &longs;ymphoniam exhibens. 436

MACHINA X. Ci&longs;ta pnevmato-harmonica antophóna, &longs;olo vento harmo­nio&longs;um &longs;onum edens. 438

EXPERIMENTUM NOVUM MAGDEBURGICUM.

§. I. Experimentum quomodo fiat. 445

§. II. Experimento jam invento quænam de novo addita Herbipoli. 447

§. III. Argumenta ad &longs;tabiliendum vacuum ex hoc Experimento &longs;umpta. 449

§. IV. Argumenta ad evertendum vacuum ex eodem Experimento deprom­pta. 450

§. V. P. Athana&longs;ii Kircheri de novo Experimento judicium. 452

§. VI. Litteræ Auctoris Experimenti, ejusdemque ad varia quæ&longs;ita re&longs;pon&longs;io. 453

§. VII. Ejusdem Authoris re&longs;pon&longs;io ad no&longs;trum quæ&longs;itum. 450

§. VIII. P. Nicolai Zucchii è Societate Je&longs;u de novo Experimento judicium. 463

§. IX. P. Melchioris Cornæi è Societate Je&longs;u de eodem Experimento judi­cium. 465

InMECHANICAM HYDRAU­LICAM

Odarion Encomia&longs;ticon.CEdat &longs;uperbas Amphitryonides Palmas Minervæ. Vis jacet: Artium Tropæa clarent. Cuncta nutus Jam Dominæ venerantur Artis. Non ille no&longs;tri temporis Hercules, Qui frangit armis cornua fluminum, Dum cana &longs;eclorum &longs;enectus Ingeniis animis que floret. Quid? ergò Reges nè furor incitet Vincire Pontum. Nil ni&longs;i fabulam Seris propinabit nepotum Fa&longs;tus Achæmenidæ cachinnis. Risêre &longs;tultas æquora compedes Manare ritu certa &longs;uo, licet Ringatur & flagro rebelles Mulctet aquæ male&longs;anus iras. In&longs;anientis quid Domini minæ Tangunt proteruum marmor? Herus jubet? Surde&longs;cit, & plaudente fluctu Ludificat pelagus Tyrannum. At non & Artem vis Acheloia Impunè ludet. Capta &longs;uis Aqua Technis tenetur, liberosque Victa jocos, patiturque ri&longs;us. Et quas domando non potuit mari Sevùm fremi&longs;cens ardor adoreas, Has lenè &longs;olers Machinator Præripuit meliore ni&longs;u. Spectare ge&longs;tin? en tibi panditur Hoc &longs;cena libro, quâ &longs;trepitu procul, Quæ dulce &longs;it &longs;pecta&longs;&longs;e, &longs;pectes, Quæque juvet didici&longs;&longs;e, di&longs;cas, Quàm fœderatum non violabilis Lex nectat orbem; &longs;i qua pericula Vexent, ut imis &longs;umma, raris Den&longs;a, levi grave det &longs;alutem. Quò pondus vnd vergat, & aëris; Quod pellat; aut quod corpora jus trahat; Cur i&longs;te gurges ructet vndas, Quas avido bibit ales ore; Quid mille lu&longs;us, mille jocos aquæ Fundet, docebit SCHOTTVS Hydraulicæ Con&longs;ultus artis, doctas idem Ingenium fluitantis auræ. Non Ille nomen fluctibus apprimat, Quod turbet Au&longs;ter, dignus aheneis Scribi columnis, quæ nec ævi Dente, nec invidiæ terantur.

L. G. S. J.

MONITIO AD LECTOREM.

Opu&longs;culum hoc Appendice exceptâ Romæ con&longs;criptum, acprælo præparatum fuit; ideo in eo pa&longs;sim ita loquitur Auctor, qua&longs;i Romæ adhuc degeret. Vale, & nè quid te moretur, errores paucos in calce Librinotatos, priùs corrige.

MECHANICA HYDRAVLICO-PNEVMATICA.

PRÆLOQVIVM AD LECTOREM;

De Operis Occa&longs;ione, Divi&longs;ione, In&longs;criptione: deque Hydraulicorum, & Pnevmaticorum Scripto­ribus.

OCCASIO OPERIS.

MAgiam Naturalem con&longs;cribere ani-

mus mihi est, benevole Lector, è Viri Docti&longs;&longs;imi, toto&queacute; orbe noti&longs;&longs;imi Atha­na&longs;ij Kircheri Libris, quà typo excu&longs;is, quà exar atis manu collectam; at&que; ex omnibus eius­dem Adver&longs;ariis ac Schediasmatis, quæ penes me &longs;unt; nec non ex alijs probatorum Auctorum monu­mentis, nostris&queacute; inventis, fide &longs;ummâ, pari&queacute; &longs;tudio concinnatam, propriis&queacute; & aliorum experimentis &longs;tabilitam: illam inquam Magiam Naturalem, cuius in Catalogo Librorum posthàc à prædicto Docti&longs;&longs;imo Auctore edendorum Magneticæ Arti denuò hîc Ro­mæ nuper editæ appo&longs;ito feci mentionem. Spartam hanc ab ip&longs;o Auctore mihi commi&longs;&longs;am præ alijs a&longs;&longs;um­p&longs;i excolendam, quoniam præterquàm quòd omnium ab ip&longs;o con&longs;criptorum librorum, &longs;eu prælo commi&longs;&longs;orum, &longs;eu in Adver&longs;aria coniectorum, copiam habeo, quo­tidianâ eiusdem vtor con&longs;uetudine, vtpote inre litte­rariâ &longs;ocius; cuius proinde &longs;ententiam aut minùs intellectam, aut breviùs &longs;ubinde explicatam, exquire­re nullo negotio po&longs;&longs;um. Opus erit ingens, multi&queacute; la­boris ac &longs;tudij, nec minùs reconditâ rerum cùm natu­ralium, tùm arte factarum refertum &longs;cientiâ. Habe­bis in eo varia, curio&longs;a, exotica admir andorum effe­ctuum &longs;pectacula, reconditarúmque inventionum mi­racula, & quæ meritò cen&longs;ebuntur magica; ab omni tamen imposturâ, & illicitæ artis &longs;u&longs;picione aliena. Habebis Magiam Arithmeticam; at non vulgarem: Habebis Geometricam; at paucis perviam: Habebis Astronomicam; at penitùs abstru&longs;am Invenies Ma­giam Magneticam, Gnomonicam, Staticam, Opti­cam, Dioptricam, Catoptricam, Hydraulicam, Pnev- maticam, Pyrobolicam, Harmonicam, Phonocampticam Anacampticam, Anaclasticam, Phy&longs;iognomicam, Mecha­nicam, Sympathicam, Steganologicam, Cryptographi­cam, Divinatoriam, Cabalicam, Hieroglyphicam, Sacram. Miraberis in eodem Opere &longs;ecreta Ani­malium, Herbarum, Plantarum, Lapidum, aliarúm­que rerum per variam activorum cum pa&longs;&longs;ivis appli­cationem, virtutum occultarum combinationem, Na­turæ & Artis connubium, Paranymphâ Experientiâ. Jdeam aliquam in fine huius Operis invenies.

Magia Na­turalis Au­ctoris.

Quoniam verò Opus, vt dixi, vastum erit, mul­ti&queacute; laboris ac &longs;tudij, nec ni&longs;i &longs;ubci&longs;ivis horis, quas ordinariis occupationibus &longs;ubtr ahere licebit, perfici­endum; operæ pretium me facturum, gratúmque Rei­publicæ Litterariæ existimavi, &longs;iquæ paulatim elabo­ra&longs;&longs;em, per partes protruderem. Ab Hydraulicis igi­tur atque Pnevmaticis initium &longs;umere decrevi, id&queacute; non tàm meâ, quàm aliorum voluntate. Scribendi occa&longs;io hæc fuit. Est in &longs;upradicti Docti&longs;&longs;imi Auctoris Mu&longs;eo &longs;anè celeberrimo, frequentati&longs;&longs;imóque (quod brevi typis evulgabimus) non exigua Hydraulicarum ac Pnevmaticarum Machinarum copia, quas &longs;um- mâ animi voluptate &longs;pectant atque mirantur ij, qui ex omnibus Vrbis & Orbis partibus ad ip&longs;um vi&longs;en­dum accurrunt Viri Principes ac Litter ati, avidé­que &longs;cire de&longs;iderant, & Machinarum constructa­rum rationes, & machinalium motionum cau&longs;as. Ho­rum de&longs;iderìo vt &longs;atisfacerem, omnium dicti Mu&longs;ej Machinarum fabricam & qua&longs;i anatomiam edoce­re, aut alicubi jam ab ip&longs;o Auctore edoctam enarrare, brevi opu&longs;culo aggre&longs;&longs;us &longs;um. Quod dum faciebam, tam multa ejusdem argumenti animo calamóque occur­rebant, quæ alibi videram, legeram, excogitaveram ip&longs;e, nec vulgaria, nec injucunda, vt difficiliùs mihi fuerit &longs;istere, quàm pro&longs;equi inceptum cur&longs;um; nec mi­nor in non &longs;eribendo, quàm in &longs;cribendo labor &longs;ubeun­dus: multóque plurafuerunt omittenda, quàm char­tæ committenda, vt &longs;umptibus & tempori parceretur.

Mu&longs;eum Kircheria­num.

OPERIS DIVISIO.

Cùm verò vt in aliis &longs;cientiis & artibus, ita in Hydrauli­cis atque Pnevmaticis, vel maximè Praxis & Theoria conjunctionem ita ambiant, vt vna &longs;i alteram de&longs;tituat, meri­tò vtraque cadat; & nec &longs;ola Praxis &longs;uam &longs;ine Theoria obtine­at exi&longs;timationem, nec &longs;ine Praxi&longs;ola &longs;uam Theoria; vtramque; coniungendam putavi; præ&longs;ertim cùm vix vllum id hactenus factita&longs;&longs;e videam, tuque mecum per&longs;picies, Lector, &longs;ievolves, quotquot de Aquaticis ac Spiritalibus quoquo modo tractârunt Auctores, quorum Elenchum po&longs;tea dabimus. Plerique enim nudam duntaxat tradunt Praxin, non per&longs;criptis, ac nè indi­catis quidem operandi principijs &longs;eu fundamentis, quæ veligno­rarunt ip&longs;i, vel alijs inviderunt: quò fit, vt alijs &longs;æpe errandi præbeant occa&longs;ionem, & ip&longs;i non rarò pudendos committant erro­res, vel tunc etiam, cùm aliorum errores detegere atque corri­gere præ&longs;umptuo&longs;iùs attentant. In duas igitur partes Opus to­tum dividimus. In prima theoriam damus ad omnis generis Machinas Hydraulicas, Pnevmaticas, & Mixtas ex Vtris­que con&longs;truendas facilitate &longs;ummâ, &longs;ucce&longs;&longs;u infallibili. In &longs;e­cunda fabricam docemus, & praxim exhibemus innumera­rum Machinarum, &longs;eu purè Hydraulicarum, &longs;eu purè Pnevmaticarum, &longs;eu Hydro-pnevmaticarum, hoc e&longs;t, ex v­tris que Mixtarum, ex traditis principijs con&longs;tructarum, con­&longs;truendarumvè, ad hortorum delicias, domorumque vtilita­tes, commoditates, ornamenta, virorum præ&longs;ertim Principum, qui magis oculorum inde, animique oblectamentum, quàm rei familiaris quæ&longs;tum ex&longs;pectant. Nec oculos tantum modò pa­&longs;ere &longs;atagimus; auribus etiam &longs;ua paramus delinimenta, va­riaque Organa atque Jn&longs;trumenta automata, & autophona, &longs;olo aquarum lap&longs;u, aëris que allap&longs;u, in motum ac &longs;onum con­citamus, non minori facilitate, quàm arte. Et quoniam in- credibilis e&longs;t hodie, &longs;i unquam aliàs, ne&longs;cio an manuum, an ve­rò ingeniorum pruritus, motum, quem vocant perpetuum (quem tot modis ac vijs non pauci hactenus quæ&longs;iverunt, &longs;ed fru&longs;trà, per­ennis quietis potiùs quàm motionis Architecti) reap&longs;e exhibendi, & Mundo iam dudum eius exæ&longs;tuanti de&longs;iderio obtrudendi; multique putant, non alijs &longs;aciliùs Machinis quàm hydrauli­cis in opus deduci id po&longs;&longs;e; in vnum collegi refutavique, quot­quot videre, audire, legere potui aliorum artificia, qui eâ in re &longs;e aliquid præ&longs;titi&longs;&longs;e putarunt, aut præ&longs;tari po&longs;&longs;e &longs;ibi per&longs;ua&longs;e­runt; vt visâ alieni conatus vanitate, curio&longs;us Lector judi­cium ferre de alijs queat, & à &longs;imili vano labore &longs;ibi temperet. Tres igitur Cla&longs;&longs;es habebit Pars &longs;ecunda Operis.

Divi&longs;io O­peris præ­&longs;entis bi­partita.

Motus per­petui effi­ciendi pru­ritus mul­torum.

OPERIS INSCRIPTIO.

In&longs;criptio operis pr&ecedil;­&longs;entis.

MEchanicam porrò Hydraulico-pnevmaticam vocamus hoc Opus; & Machinas Hydro pnevmaticas, &longs;eu Hydrau­lico-pnevmaticas, quas in eodem producimus, appellamus, quòd pleræque tales &longs;int, taliumque tradantur in prima Operis parte principia ac regulæ; Hydraulico-pnevmaticas, inquam, qua&longs;i dicas, Aquatico-Spiritales, hoc e&longs;t, aqua & &longs;piritu &longs;eu aëre

animatas. Aquaticæ etenim Machinæ, hauriendis è profun­do, deducendis per planitiem, educendis in altum per fi&longs;tulas ac tubos (a)ul\s Græci appellant) aquis excogitatæ, triplicis &longs;unt generis. Aliæ iumentorum, hominum, ponderum ope, rotis variè inter &longs;e&longs;e implicatis, moventur; cuiusmodi &longs;unt tympa­na, antliæ, cochleæ, tollenones, &longs;imiliaque, quæ Vitruvius pro­ducit libro 10. Capite 9, 10, 11, 13, & 14. alijque Auctores multi ad nau&longs;eam vsque confarcinatis libris inculcant. Aliæ &longs;olo aqua­rum lap&longs;u fontes exhibent amœni&longs;&longs;imos, & aquas per &longs;iphones tubosque variè configuratos protru&longs;as nunc expandunt in &longs;ub­tili&longs;&longs;imum lucidi&longs;&longs;imum que velum, nunc diffundunt in radios, figurant in &longs;tellas, effingunt in flores, extendunt in iacula, cri­&longs;pant in pluvias, conglobant in grandines, &longs;ummâ oculorum voluptate, vt pa&longs;&longs;im in vrbanis ac &longs;uburbanis Romæ vide­mus hortis.

Aquaticæ Machinæ triphcis &longs;unt generis.

Tractoriæ.

Hydrauli­cæ.

Aliæ denique inclu&longs;i intru&longs;ive äeris violentiâ eiaculan- tur in altum aquas, inflant tibias, tubasque, volucrum ani­maliumque aliorum imitantur motum ac Cantum, aliaque præ­&longs;tant tam mira atque exotica, vix vt ab humano ingenio præ­&longs;tari po&longs;&longs;e videantur. Eius generis fuit apud Cornelium Seve­rum in Ætna Triton, qui aquarum a&longs;&longs;ultu auram concipiebat, ac per buccinam ori admotam vocem edebat, &longs;ive vt ait Poëta, irriguum Carmen. Tales etiam &longs;unt, quas magno numero, nec minori ingenio prodit Hero in &longs;uo de Spiritalibus libello Machi­nas, ab antiquis (quos Ægyptios interpretatur Kircherus in Mechanica Ægyptiaca tom. 2. Oedipi parte 2.) excogitatas: cuiusmodi &longs;unt &longs;tatuæ ad aras vinum & lac &longs;acrificantes, va­&longs;a melancoryphi vocem edentia, Syringæ mentientes vocem a­vicularum multarum per ordinem di&longs;po&longs;itarum, quibus a&longs;&longs;idet noctua, quæ &longs;ua &longs;ponte ad aviculas conver&longs;a eas terret& à cantu ab&longs;terret; aliæque multæ. Tales denique, quas toto hoc Opere magno numero producimus.

Hydrauli­co-pnev­maticæ.

Primi generis Machinas appellamus nos cum alijs Tracto­rias: & de illis non agimus hoc Opere; pertinent enim ad illam Mechanicæ partem, quæ viponderum atque rotarum &longs;uas perfi­cit operationes, & &longs;ine ip&longs;ius principijs nec tradi, nec ab alijs percipi po&longs;&longs;unt: quare Magiæ no&longs;træ Mechanicæ eas re&longs;erva­mus. Secundi generis Machinas appellamus Hydraulicas. Tertij denique generis Hydraulico-pnevmaticas. Quoniam igitur pleræque Machinæ, quas producimus, huiu&longs;modi &longs;unt, hoc e&longs;t, aquæ & aëris &longs;ub&longs;idio con&longs;tructæ; ideo Hydraulico­pnevmaticas, vt dicebam, appellamus hoc Opere traditas Machinas. Dico, plerasque aquæ & aëris &longs;ub&longs;idio e&longs;&longs;e con­&longs;tructas; non omnes: &longs;unt enim & purè hydraulicæ multæ, & purè pnevmaticæ non paucæ; imò nonnullæ, quæ licet aquæ au­xilio fiant, hydraulicæ tamen propriè ac rigorosè, juxta &longs;en&longs;um expo&longs;itum, cen&longs;eri non po&longs;&longs;unt, vt videbis in Operis decur&longs;u, & ex Machinarum indice apparebit.

Machinæ huius Ope ris pleræque &longs;unt hydrau­lico-pnev­maticæ.

SCRIPTORES HYDRAVLICORVM, ET PNEVMATICORVM.

SCriptores quod attinet, qui de Hydraulicis, Pnevmaticis­que quoquo modo &longs;crip&longs;erunt, & ad manus meas pervene­runt, &longs;unt &longs;equentes, Alphabeti ordine. Archimedes, Atha­na&longs;ius Kircherus, Augu&longs;tinus Ramellus, Benedictus Ca­&longs;tellus, Daniel Lip&longs;torpius, Daniel Schvventerus, Evan­geli&longs;ta Torricellus, Galilæus Galilæi, Ga&longs;par Ens, Ge­orgius Agricola, Georgius Philippus Har&longs;torfferus, Ge­orgius Valla, Hero Alexandrinus, Hieronymus Carda­nus, Iacobus Be&longs;&longs;on, Ioannes Bapti&longs;ta Porta, Ioannes Ba­pti&longs;ta Balianus, Ioannes Bapti&longs;ta Aleottus, Ioannes Iaco­bus Weckerus, Ioannes Leurechon, Io&longs;ephus Ceredus, Ioannes Branca, Marinus Ghetaldus, Marinus Mer&longs;en­nus, Marius Bettinus, Nicolaus Cabæus, Petrus Herigo­nius, Raphaël Magiottus, Robertus Valturius, Rober- tus à Fluctibus, Salomon Caus, Simon Stevinus, Theo­dorus Muretus, Va&longs;&longs;or Arabs, Vitruvius, eiusque Com­mentatores, vt Philander, Barbarus, Cæ&longs;arinus, alijque. Hæc Auctorum nomina.

Ex his Archimedes duos per&longs;crip&longs;it libros peri\ tw=n o)xoume/nwn, hoc e&longs;t, de In&longs;identibus in humido, ex quibus pauca no&longs;tro in­&longs;erviunt propo&longs;ito. Athana&longs;ius Kircherus Societ. Je&longs;u multa pa&longs;&longs;im hydraulica atque pnevmatica machinamenta in&longs;erit &longs;uis docti&longs;&longs;imis libris, præ&longs;ertim de Arte Magnetica, de Lumine & Vmbra, de Con&longs;ono & Di&longs;&longs;ono, ac novi&longs;&longs;imè &longs;uo incompara­bili de Ægyptiorum hieroglyphicis Operi quem Oedipum Ægy­ptiacuin in&longs;crip&longs;it: multa quoque inter eius Manu&longs;cripta re­peri, multaque oretenus ab eodem percepi. Augu&longs;tinus Ra- mellus ingens &longs;crip&longs;it Opus de Machinis Italico & Gallico idio­mate, interque illas ingentem habet copiam Antliarum diver&longs;is modis con&longs;tructarum, aliasque nonnullas praxes ad elevandas aquas, &longs;ed omnes tractoriæ &longs;unt Machinæ ad primum pertinen­tes genus. Benedictus Castellus, olim Mathematicus Pon- tificius, libellum compo&longs;uit Italico idiomate de aqua cur­rente, &longs;ed per alveos, aquæductus, & tubos inclinatos; quæ proinde ad nos non pertinent. Daniel Lip&longs;torpius Lubecen&longs;is edidit Specimina Philo&longs;ophiæ Carte&longs;ianæ, in cuius parte 3. c. 3. multas adducit Machinas hydraulico-pnevmaticas, easque explicat ex Carte&longs;ij principijs. Daniel Schvventerus, & Ge- orgius Philippus Harstorfferus Norimbergen&longs;is Patritius, in Delicijs Mathematicis multas afferunt, easque iucundi&longs;&longs;imas Machinas hydraulicas, & pnevmaticas. Multas etiam in Thaumaturgo &longs;uo Mathematico adducit ex alijs Ga&longs;par Ens; vt & Weckerus in &longs;uis Secretis nonnullas ex Cardano de&longs;um- ptas, quas dictus Cardanus in&longs;erit &longs;uis de &longs;ubtilitate libris. E- vangelista Torricellus, Magni Hetruriæ Ducis Mathemati­cus, & Benedicti Ca&longs;telli di&longs;cipulus, &longs;crip&longs;it libros 3. de Motu proiectorum; quorum &longs;ecundo inter&longs;erit nonnulla de motu a­quarum per tubos, &longs;ed quàm &longs;ubtilia, tàm parum ad praxin v- tilia, vt ip&longs;emet fatetur. Galilæus Galilæi edidit Archimedem redivivum, quem ex latino Italicum fecit Ioannes Baptista Hodierna Siculus, Vir rerum Astronomicarum apprimè gnarus, & &longs;ide­rum, adeoque Naturæ totius &longs;crutator indefe&longs;&longs;us, amicusque &longs;inceri&longs;&longs;imus. Opu&longs;culum e&longs;t ad Hydrostaticam pertinens; quale etiam e&longs;t Archimedes promotus Marini Ghetaldi; vt & quæ habet Simon Stevinus; ideoque nobis nullam hîc &longs;uppetunt materiam. Georgius Agricola libro 6. de rebus metallicis in- &longs;erit nonnullas Machinas hydraulicas, nonnullas Robertus Valturius &longs;uis de Militia libris, nonnullas Iacobus Be&longs;&longs;on Del- phinas &longs;uo de In&longs;trumentis Mathematicis ac Mechanicis tracta­tui; &longs;ed omnes &longs;unt tractoriæ, & alienæ à no&longs;tro instituto. Ge- orgius Valla Placentinus libro 15. Expetendorum & fugiendo­rum, qui e&longs;t &longs;extus Geometriæ, &longs;eu Spiritalium vnicus, multas habet Machinas Spiritales, &longs;ed ad vnam omnes ex Herone Ale­xandrino decerptas, nè nominato quidem Herone. Habebat vir bonus, vt Io&longs;ephus Ceredus te&longs;tatur, Heronis librum græ­cum de Spiritalibus, nondum tunc à Federico Commandino, aliovè Latinitati donatum, putabatque &longs;ecurè atque impunè &longs;e po&longs;&longs;e in alienam me&longs;&longs;em mittere falcem &longs;uam.

Archime­des.

Kircherus.

Ramellus.

Ca&longs;tellus.

Lip&longs;torpi'.

Schvven­terus.

Har&longs;torffe­rus.

Ga&longs;par Ens

Wecker'.

Cardanus.

Torricell9.

Galilæus.

Ghetaldus.

Stevinus.

Agricola.

Valturius. Be&longs;&longs;on.

Valla.

Hero Alexandrinus Auctor antiquus, græcè edidit libel- lum de Spiritalibus, in quo quæ ab antiquis (quos Ægyptios interpretatur Kircherus, vt &longs;uprà dicebam) tradita in illo ge­nere fuêre, in ordinem redegit; & quæ ip&longs;emet invenit, expo­&longs;uit: quæ quidem omnia pulcherrima &longs;unt, & ingenio plena. Idem Auctor ait &longs;e quatuor libros con&longs;crip&longs;i&longs;&longs;e de aquaticis horo­&longs;copijs; quos tamen temporum iniquitas nobis invidit. De- &longs;cribit autem Hero nudam praxim, nec vllibi præcepta & regu­las tradit, ni&longs;i quòd in principio de vacuo agat; ideoque & ip&longs;e &longs;ubinde errat, (&longs;i tamen ip&longs;ius, & non Interpretis &longs;unt er­rores) & alios in errorem inducit; nec omnia quæ promittit, effectum &longs;ortiuntur. E&longs;t nihilominùs ip&longs;e de Republica litteraria egregiè meritus, & vnicus qui ex antiquis de rebus hydraulicis & pnevmaticis &longs;ua ad nos monumenta transmi&longs;it. Reperies in hoc Opere no&longs;tro nonnullas ex ip&longs;o de&longs;umptas Machinas; quas ideo adducimus, quòd aliquæ indigent correctione, vtpote fal&longs;æ: aliquæ explicatione, vtpote ob&longs;curiùs traditæ; aliquæ excu&longs;a­tione ac defen&longs;ione, vtpote &longs;ini&longs;trâ ab alijs cen&longs;urâ notatæ; aliæ commendatione, vtpote no&longs;trarum ectypa; aliæ denique ad­miratione, vtpote ingenio pleni&longs;&longs;imæ. Heronem Latinum fe­cit Federicus Commandinus Vrbinas; Italicum Joannes Ba- pti&longs;ta Aleottus, qui & in fine addidit quinque Problemata no­va, &longs;anè jucundi&longs;&longs;ima, pariterque ingenio&longs;a.

Hero.

Hero Ale­xandrinus nudam pra­xin hydrau lico-pnev­maticarum Machina­rum tradit.

Eius Ma­chinæ non­nullæ qua­re in hoc Opere ad­ducantur.

Aleottus.

Ioannes Bapti&longs;ta Porta Neapolitanus tres latinè &longs;crip&longs;it egregios libellos de Spiritalibus, qui tamen pe&longs;&longs;imum nacti &longs;unt Interpretem Joannem E&longs;crivanum Hi&longs;panum, qui eos Italicè at que Hi&longs;panicè vertit; &longs;ed cùm Italicam linguam non calleret, mirum e&longs;t, quàm barbarum vbique, quàm ob&longs;curum, quàm mutilum reddiderit optimum Auctorem. Addidit nonnulla de &longs;uo, oretenus à Porta, &longs;ed malè intellecta, & peiùs per&longs;cri­pta, itáque alienis permixta, vt di&longs;cerni nequeant, ni&longs;i cum genuino comparentur partu. Tradit Porta Spiritalium prin­cipia, tradit machinamenta multa; vtraque longis experi­mentis, nec &longs;ine expen&longs;is, confirmata. Huic ego primas inter omnes Spiritalium Scriptores dandas cen&longs;eo. Latinum exem­plar Neapoli impre&longs;&longs;um Anno 1601. innumeris &longs;catet erroribus typographicis, qui &longs;en&longs;um &longs;æpe pror&longs;us pervertunt. Idem Au­ctor lib. 18. & 19. Magiæ Naturalis multas alias habet Machi­nas Hydraulicas & Pnevmaticas.

Port

Balianus.

Joannes Bapti&longs;ta Balianus Patritius Genuen&longs;is, &longs;cri­p&longs;it libros &longs;ex de motu naturali gravium & liquidorum quorum tres vltimi &longs;unt de liquidorum fluxu per canales & foramina; è quorum vltimo nonnulla in Protheoriam no&longs;tram quartam

derivavimus. P. Ioannes Leurechon è Societ ate no&longs;tra, Lo­tharingus &longs;crip&longs;it gallico idiomate, tacito nomine &longs;uo, Recreatio­nes Mathematicas; quas deinde germanicas fecit Daniel Schvventer, latinas verò Ga&longs;par Ens. Io&longs;ephus Ceredus, Medicus Placentinus, &longs;crip&longs;it tres egregios di&longs;cur&longs;us de Cochleæ Archimedææ compo&longs;itione & vtilitate, ad elevandas magnâ quantitate ac facilitate aquas. Ioannes Branca civis Roma­nus, & Architectus Lauretanæ domus, librum &longs;crip&longs;it Italico ac Latino &longs;ermone de Machinis, quas inter multæ &longs;unt hydrau­licæ, & nonnullæ hydro-pnevmaticæ, at pleræque fal&longs;æ, & hy- dro-pnevmaticis principijs contrariæ. Marinus Mer&longs;ennus &longs;crip&longs;it Phœnomena hydraulica & pnevmatica, multasque ex­perientias à &longs;e in illa materia factas, ex quibu⋅ multa deducit docti&longs;&longs;imaque; Theoremata atque Problemata; & nos nonnulla ex ip&longs;o in&longs;eruimus quartæ no&longs;træ Protheoriæ. Nicolaus Cabæus Commentarijs &longs;uis in Ari&longs;totelis Meteora, nonnulla habet hy- draulica: nonnulla Marius Bettinus in Apiarijs Mathematicis Apiario 4. Progymnas. 1. Nonnulla denique Petrus Herigo­nius to. 3. Cur&longs;us Mathematici tract. de Mechanica. Raphael Magiottus, Mathematicus Excellenti&longs;&longs;imus, Opu&longs;culum &longs;cri- p&longs;it Italicum de aquæ Renitentia compre&longs;&longs;ioni; in quo tradit Experimentum novum, quod nos adducimus Part. 2. Cla&longs;&longs;e 1. cap. 5. Machina 6. Robertus à Fluctibus in &longs;ua Philo&longs;ophia Mo&longs;aica alijs&qtail; Operibus, multas pa&longs;&longs;im inter&longs;erit Machinas, a­liâ occa&longs;ione. Salomon à Caus &longs;crip&longs;it Gallico idiomate libros tres de hydraulicis quos ip&longs;e de viribus motricibus intitulat: Opus est egregium, in eoque theoria aliqualiter cum praxi conjungi- tur. Theodorus Muretus, è Societate IESV, in Pragen&longs;i Vni­ver&longs;itate Mathematicæ Profe&longs;&longs;or, edidit Problema Mathema- ticum de Fontibus. Vitruvius lib. 8. Architecturæ cap. 6 & 7. agit de perductionibus & librationibus aquarum, & in&longs;trumen­tis ad hunc v&longs;um; & libro 10. varia habet in&longs;trumenta hy­draulica, at pleraque tractoria. Va&longs;&longs;or Arabs habet varias praxes Spiritales ad aquas in altum educendas: Liber e&longs;t Ma­nu&longs;criptus, latinus, quem reperi inter libros P. Athana&longs;ij Kir­cheri, at mutilum, & &longs;ine figuris nece&longs;&longs;aijs: Su&longs;picor e&longs;&longs;e fragmentum Scriptoris cuiusdam Græci antiqui, à Ua&longs;&longs;ore in Arabicam, ab alio in Latinam linguam translatum. Edidi&longs;- &longs;em vnà cum hoc Opere, &longs;i operæ pretium me facturum judica&longs;­&longs;em.

Leurechon

Ceredus.

Branca.

Mer&longs;ennus

Cabæus.

Bettinus.

Herigoni'

Magiottus

Robertus a Fluctibus

Salomon à Caus.

Muretus.

Vitruvius.

Va&longs;&longs;or A­rabs.

Hi &longs;unt Auctores, qui de Hydraulicis atque pnevmati­cis &longs;crip&longs;erunt, &longs;eu ex profe&longs;&longs;o, &longs;eu incidenter. Ex quibus ta­men, &longs;i Kircherum, Portam, Mer&longs;ennum, aliosque nonnullos excipias, pauci no&longs;trum triverunt iter, & vix vllus theoriam cum praxi conjunxit; quod nos facimus. Aliorum Auctorum nomina, qui vnam aut alteram Machinam &longs;uis inter&longs;erue­runt Operibus, proferemus &longs;uis locis: Cuiusmodi &longs;unt Oron- tius Finæus, & Bonaventura Cavallerius, eximius vterque Mathematicus; quorum ille in fine tractatus de Horologijs vnicam habet hydraulicam Machinam; hic alteram in fine Exercitationis &longs;extæ Geometricæ; quarum vtram­que nos infrà Parte 2. afferimus.

Finæus.

Cavalleri'.

MECHANICÆ HYDRAVLI­CO-PNEVMATICÆ

PARS PRIMA THEORETICA,

De Machinarum Hydraulico­pnevmaticarum principijs, &longs;eu fundamentis.

PROœMIVM.

QUatuor reperio Machinarum Hydrau- lico-pnevmaticarum principia, quibus ab Arte animantur in motum, illósque præstant effectus, quos tantâ voluptate miramur quotidie, tantâ pa&longs;&longs;im experimur vtilita- te. Primum est vis Attractiva, ad vacuum vel fu­giendum, velreplendum; quâ vi Aqua, elementu&mtail; alioquingrave, iners, lentum, & humi repens, tracta accurrit non invita, vt videtur, &longs;uæ&queacute; oblita naturæ in &longs;ublime nititur, tripudianti &longs;imilis, vacui velreplen- di, vel impediendi causâ. Alterum est vis expul­&longs;iva, ad corporum penetrationem fugiendam; quâ vi eadem Aqua, dum aëri alterivé corpori validè in­cumbenticedere cogitur ho&longs;pitio, expul&longs;a fugam prori­pit eò celeriorem, excel&longs;iorémque, quò vehementiori- bus &longs;e&longs;timulis agitatam &longs;en&longs;erit. Tertium est vis ‘Ra­refactiva; quâ ob&longs;e&longs;&longs;um elementum in tantas &longs;ubin­de redigitur angustias, vt dum &longs;e&longs;e, quâ data porta, vel &longs;ubducit, vel ingerit, alas indui&longs;&longs;e videatur, abje­ctis gravitatis propriæ compedibus. Quartum de- ni&queacute; estgravitas Aquæ naturalis, ad decliviora loca tendentis, & æquilibrium affectantis; quæ gravitas non rarò (quod mirêre) in altum attollit ip&longs;am, quam paulò antè depre&longs;&longs;er at in profundum.

Principia Machinarum Hydrau­lico-pnev­maticarum quatuor.

I. Attracti­va vis ad vacuum e­vitandum.

II. Expul&longs;i­va vis ad corporum penetrati­onem fu­giendam:

III. Rarefa­ctio & con­den&longs;atio, aquam vel expellens, vel attra­hens.

IV. Gravi­tas aquæ æquilibri­um affe­ctantis.

Hæc quatuor principia qui applicare ritè nove­rit, quaslibet ingenio&longs;as Machinas, ad Naturæ ne&longs;cio an æmulationem, an invidiamfabricari poterit faci­litate &longs;ummâ, &longs;ucce&longs;&longs;u infallibili; cùm nulla Machina hydraulica, aut pnevmatica, aut ex vtrisque mixta ad&longs;ignari po&longs;&longs;it quæ non aliquâ harumfacultatum in- &longs;tituatur. Jmò multis par a&longs;angis Naturæ &longs;uper abit co­natus u(dro texni/ths. Natura etenim non ni&longs;i vno motus principio aquas terrâ eijcit, at&que; in fontes efformat, na­turali videlicet fluxu aquæ de&longs;cendentis primum, ac de­inde a&longs;cendentis: at verò Ars non vno, &longs;ed multiplici, vt dixi, &longs;cilicet &longs;uctu &longs;eu attractione, pre&longs;&longs;ione, dila­tatione, præcipitatione: queis artibus, vel potiùs &longs;tra­tagematis, in valles, in montes, in hortos, in domos de­ducit, non fontes tantummodò, &longs;ed machinamenta a­lia varia, jucunda, ac pror&longs;us qaumasiw/tata, aliorum vel nece&longs;&longs;itati, vel vtilitati, vel delectationi: Veri&longs;&longs;imum quippe est illud Antiphontis, te/xnh| ga/r xratou=men, w(=n fu/sei nixw/­meqa. At hæc ip&longs;a quatuor hydro-pnevmaticarum Machinarum principia &longs;eu fundamenta, antequam vlteriùs progrediamur, paulô fu&longs;iùs explicare lubet, vt &longs;ecuriùs deinde in praxi pro­cedamus.

Ars vincit Naturam infontibus producen­dis.

Pars I.

PROTHEORIA PRIMA

De vi attractiva Corporum, ad vacuum vel replendum, vel fugiendum: &longs;eu de primo Machi­narum hydro-pnevmaticarum principio.

Vacuum Phi­lo&longs;ophorum alij admit tunt, alij respuunt.

VAcuum appellat vulgus quodlibet vas, aut lo­cum, in quo nullum cernunt vi&longs;ibile corpus, etiam&longs;i aëre &longs;int plena. At Philo&longs;ophi Vacuum definiunt locum nullo repletum corpore. Hoc po&longs;terius, quod &longs;olum vacui nomen meretur propriè, non tam invi&longs;um Naturæ e&longs;t vi&longs;um De­mocrito, Leucippo, Epicuro, alijsque, vt eliminandum penitus cen&longs;uerint è rerum cen&longs;u. Alij verò plerique Philo&longs;ophi cum Stagyrita &longs;uo adeo Naturæ adver&longs;um, adeo pernicio&longs;um exi&longs;ti­mant Vacuum, vt ad evitandum ip&longs;um dicant, eandem Naturam nullum non movere lapidem, & omnia &longs;usque deque vertere corpora, ip&longs;aque permi&longs;cere elementa. Qui de hydraulicis, pnevma­ticisque tractant Machinis, vnanimi con&longs;en&longs;u fatentur omnes

(&longs;i Lip&longs;torpium cum Carte&longs;io &longs;uo excipias) vim attractivam, quam in Machinis nonnullis aperti&longs;&longs;im è elucere videmus, aliun­de non provenire, ni&longs;i quòd Aqua, & Aër, Naturæ ob&longs;ecunda­turi, accurrant tracti &longs;eu &longs;ponte, &longs;eu invitè, vel ad replendum va­cuum quod oderunt, vel ad impediendum quod timent. Vtra­que igitur &longs;ententia, & quæ vacuum recipit, & quæ rejicit, Hy­draulicorum ac Pnevmaticorum Artificibus favet; ac proinde ex vtrius que placitis vis attractiva prædicta e&longs;t explicanda.

Attractiva vis ex appetitu vacui vel replen­di, vel fu­giendi ori­tur in Ma­chinis hy­dro-pnev­maticis.

Protheoria I.

§. I.

Hero Alexandrinus Vacuum in aëre & aqua di&longs;seminatum agno&longs;cit.

HEro Alexandrinus, &longs;imul Spiritalium Magi&longs;trum agens, &longs;i­ mul Democritici, Epicureique Philo&longs;ophi per&longs;onam, prio­rem de Vacuo &longs;ententiam doctè non minùs quàm fusè explicat, atque tuetur, in Præfatione Libelli &longs;ui de Pnevmaticis &longs;eu Spi­ritalibus, vbi ait: Alij enim vnivërsè nullum omnino vacuum e&longs;&longs;e affirmant; alij coacervatum quidem naturâ nullum e&longs;&longs;e vacuum, &longs;ed &longs;ecundùm exiguas partes di&longs;&longs;eminatum in aëre, humido, & igne, alijsque corporibus; quibus maximè a&longs;&longs;entiri convenit. Vacuum coacerva­tum veteres vocabant a)n, di&longs;&longs;eminatum verò parespame/non, vt notat Petrus Ga&longs;lendus in Philo&longs;ophia Epicuri; à quo Epicuro Hero &longs;ontentiam &longs;uam mutuatus e&longs;t. *alla\ me\n, inquit Epicurus Epi&longs;tola ad Herodotum, kai\| to\ pa_n e)si\ ph_ me/n *sw_ma, ph_| de\ xeno\n vniver&longs;um autem partim quidem Corpus, partim verò inane e&longs;t.

Hero vacu­um admit­tit in aëre & aqua di&longs; &longs;eminatum

Epicuriu vacuum ad mittit.

Suam porrò de vacuo &longs;ententiam Hero tot probat rationi­bus & experientijs, vt demon&longs;tratam exi&longs;timet, easque dignas cen&longs;uit Marinus Mer&longs;ennus, quas Hydraulicis &longs;uis præmitteret Phœnomenis. Ac principio quidem o&longs;tendit Hero. Va&longs;a, quæ multis vacua videntur, eò quòd nullum in ijs vi&longs;ibile deprehen­dunt Corpus, non e&longs;&longs;e vacua, vt putant, &longs;ed aëre plena. Quod quidem patet ex eo, inquit Hero, quòd &longs;i vas, quod vacuum vide­tur, quis inver&longs;um in aquam depre&longs;&longs;erit, rectumque &longs;ervarit, aqua in ip&longs;um non ingredietur, quamvis totum occultatum fuerit. Vtique non alia de Cau&longs;a, ni&longs;i quia aër, cum Corpus &longs;it, & totum vas occupet, non permittit aquam ingredi, vt ip&longs;emet &longs;ubjungit. Idem patetin phialis vitreis, ollis, alijsque va&longs;is, &longs;i inver&longs;a aquis immergantur: non enim ni&longs;i difficulter admodum demergun­tur; & dum retrahuntur, latera interiora habent &longs;icca; non ob aliud profectò, ni&longs;i quia inclu&longs;us aër ingre&longs;&longs;um prohibuit aquæ.

Va&longs;a ina­nia aëre e&longs;&longs;e plena, vari­is probatur experimen­tis.

Quòd &longs;i prædictorum Va&longs;orum inver&longs;orum aquis immer­&longs;orum fundum quis perforet, aqua quidem per os demer&longs;um in ip&longs;a in­gredietur, aër verò per foramen exibit, adeo manife&longs;tè, vt non &longs;o- lùm manui admotæ &longs;en&longs;um inferat, &longs;ed etiam aures &longs;ono de­mulceat, & quidem harmonico, &longs;i foramini aptentur fi&longs;tulæ &longs;ono Harmonico edendo aptæ, prout ip&longs;emet Hero fieri jubet in multis, quas de&longs;cribit in citato Libello, hydraulicis at­que pnevmaticis Machinis. Quod etiam quotidie experimur in Hortis Romanis, Tu&longs;culanis, Tiburtinis, & in ip&longs;o etiam P. Athana&longs;ij Kircheri mu&longs;eo. Non igitur rbitrandum e&longs;t, infert Hero, in his quæ&longs;unt, vacui naturam quandam coacervatam ip&longs;am per­&longs;e ip&longs;am exi&longs;tere &longs;ed&longs;ecundùm exiguas partes in aëre, & humido, alijsque corporib di&longs;&longs;eminatam, etiam in ip&longs;o adamante, duri&longs;&longs;imo licet, den&longs;i&longs;&longs;oque, quippe qui neque igniri, neque rumpi pote&longs;t, & percu&longs;­ &longs;us in incudes & malleos totus ing reditur (vt cum Plio putat Hero; quod tamen, Mer&longs;enno te&longs;te, experientiæ repugnat, nam ictu mallei in varias partes di&longs;&longs;ilit, & ita pote&longs;t iguiri, vt pereat.) Hoc autem ip&longs;iaccidit, &longs;ubiungit Hero, non quòd vacuo careat, &longs;ed ob conti­nuatam den&longs;itatem. Corpora porro illa (aëris præ&longs;ertim) quæ partes &longs;uas habent inter &longs;e&longs;e cohærentes, non|tamen ex omni parte, &longs;ed interiecta habent inter valla quædam vacua, (quæ vacuola appel­lat Mer&longs;ennus) bellè comparat Hero arenarum cumulo, inter cuius grana multæ aëris particulæ intercedunt; aitque, arenæ par­ticulas particulis aëris &longs;imiles e&longs;&longs;e, aërem verò, qui inter arenas in­terjicitur, &longs;imilem vacuolis intra aërem contentis; quæ quidem vocuola tunc di&longs;inere putat in aëre (alijsque corporibus) cùm accedente vi quapiam conden&longs;atur aër; redire verò, cùm facta remi&longs;&longs;ione in pri&longs;tinum ordinem re&longs;tituitur aër ob naturalem contentionem, quâ, velut arcus inflexus, aut &longs;icca &longs;pongia vi pre&longs;&longs;a, molem ab ip&longs;o rerum conditu &longs;ibi tributam repetit, &longs;ta­ tim atque vis externa de&longs;init. At aëris corpora, inquit Hero, inter &longs;e&longs;e quidem cohærent, non tamen ex omni parte, &longs;edinteriecta habent intervalla quædam vacua, &longs;icut arena, quæ e&longs;t in littoribus. Itaque a­nimo concipiendum e&longs;t, arenæ particulas corporibus aëris &longs;imiles e&longs;&longs;e, aë­rem vero, qui inter particulas arenæ interiicitur, &longs;imilem vacuis intra aërem contentis. Quamobrem vi quadam accedente äerem den&longs;ari con­tingit, & in vacuorum loca re&longs;idere, corporibus præter naturam inter &longs;e&longs;e compre&longs;&longs;is: remi&longs;sione verò factârar&longs;us in cun dem ordinem re&longs;titui- tur, ob naturalem Corporum contentionem; quemadmodum & in cor­nuum ramentis, & in &longs;pongiis &longs;iccis, quæ&longs;i compre&longs;&longs;a remittantur, rur­&longs;us in cundem locum redeunt, eandem que accipiunt molem. Similiter &longs;i aliqua vi aëris particulæ à &longs;e in vicem di&longs;tractæ fuerint, & maior præ­ter naturam locus vacuus fiat, rur&longs;us ad &longs;e&longs;e recurrunt, per vacuum e­nim celerem corporum lationem fieri contingit, nullo ob&longs;tunte, aut re­pellente, quóusque corpora ad &longs;e&longs;e applicentur.

Adamas i­gni & mal­len cedit.

Vacuola He ronis inter aëris parti­culas de&longs;&longs;e­minata.

Amterie Ha habens vacua &longs;ecun­dum Hero­nem compa atur are­næ.

§. II.

Experimenta quibus Hero vacuum in Corpo­ribus di&longs;&longs;eminatum probat.

Argumeta Heronis pro vacuo.

HIs præmi&longs;&longs;is, multa, & pa&longs;&longs;im obvia adducit experimenta

Hero, quibus probat dari vacuum inter corporum, aëris præ­&longs;ertim, particulas di&longs;&longs;eminatum; & aquam, aliaque corpora ad illud replendum, &longs;i nimiùm di&longs;trahantur particulæ vi externâ, ac­currere attracta; atque adeo vim attractivam, quam primum hydro pnevmaticarum Machinarum principium e&longs;&longs;e diximus, tribuit Vacuo inter corpora di&longs;&longs;eminato.

Vacuum in corporibus di&longs;&longs;eminatum experimen­tis probat Hero.

Primò enim, &longs;i quis vas levi&longs;&longs;imum & angu&longs;ti oris a&longs;&longs;umens, atque ori admovens aërem ex&longs;uxerit, dimi&longs;erit&que;; ex labiis &longs;tatim vas appendetur, vacuo carnem attrahente, vt locus exinanitus repleatur.

Va&longs;a ex&longs;u­cto aëre la­biorum car­nes attra­hunt.

Secundò. Ova medica, quæ ex vitro con&longs;tant, & angu&longs;ti &longs;unt oris, quando volunt humido replere, ex&longs;ugentes ore aërem, qui in ip&longs;is continetur, o&longs;culumque digito comprehendentes, in humidum in ver­tunt, & digito remi&longs;&longs;o in locum exin anitum attrahitur aqua, atque id­ip&longs;um humidum præter naturam &longs;ur&longs;um fertur. Idem contingit in Siphonibus, qui ex&longs;ucto aëre humidum è va&longs;is hauriunt, vt quo­tidiana docet experientia, & dicemus fusè infrà Protheoria 4. Cap. 2. Idem in multis Machinis hydraulicis, aut pnevmaticis e­venit, vt &longs;uo loco videbimus.

Ova medi­ca quomodo repleantur aquá.

Tertò, Cucurbitæ à Chirurgis igni priùs admotæ, vt aêr in ip&longs;is contentus rarefiat, dum corpori applicantur, non &longs;olùm non excidunt, manife&longs;tam gravitatem habentes, &longs;ed adiacentem ma­teriam per corporis raritates attrahunt, eandem ob cau&longs;am, vt &longs;cilicet locus exinanitus repleatur: iniectus enim ignis, &longs;ubjungit Hero, corrumpit & extenuat aërem, qui in ip&longs;is continetur, ideoque relin­quitur in ip&longs;is vacuum; cùm verò cucurbitula re&longs;piraverit; aër in e­vacuatum locum &longs;uccedit, materiânon ampliùs attractâ, deciditque cucurbitula.

Cucurbitu­la cur car­nem attra­hant.

Sphæra con­cava, & aë­replena, cur plùs ris per inflatio­nem recipi­at &longs;ecundum Heronem.

Quartò, con&longs;truatur Sphæra cra&longs;&longs;itudinem habens laminæ, vt nè facilè rumpatur, continensque circiter cotylas octo, & vndique præclu­&longs;a; & per foramen in ip&longs;a factum immittamus Siphonem æreum, hoc e&longs;t, tubum gracilem, non contingentem locum, qui puncto perfora­to &longs;ecundùm diametrum opponitur, ita vt aqua fluere po&longs;&longs;it; alia autem ip&longs;ius pars extra &longs;phæram emineat circiter digitos tres, ip&longs;i&queacute; diligenter adferruminetur, ambitu foraminis circa tubum &longs;tanno præclu&longs;o, vt cùm volumus ore per &longs;iphonem inflare, &longs;piritus ex &longs;phærâ nullo modo excidat.His factis,, in&longs;piciamus quæ contingunt. Aëre enim in ip&longs;a exi&longs;tente(quemadmodum & in aliis va&longs;is quæ vacua appellantur) totum­que qui in ipsa locum replente, & per continuationem quandam ad eius ambitum applicato, nullo denique loco, ut arbitrantur, penitus exi&longs;ten­te vacuo; neque aquam immittere po&longs;&longs;emus, neque alium aërem, non excedente priori aëre qui in ipsa erat; & &longs;i multa vi immi&longs;&longs;ionem facia­mus, priùs di&longs;rumpetur vas, quàm aliquid &longs;u&longs;cipiat, cùm plenum &longs;it: ne­que enim aëris corpora in minorem magnitudinem contrahi po&longs;&longs;unt: quare nece&longs;&longs;e erit, vt in &longs;eip&longs;is habeant quædam intervalla, in quæ com­pre&longs;&longs;a minorem obtineant molem; hoc autem veri&longs;imile non e&longs;t, nullo penitus exi&longs;tente vacuo: & cùm corpora ad &longs;e&longs;e applicentur &longs;ecundùm omnes &longs;uperficies, & &longs;imiliter ad va&longs;is ambitum, non po&longs;&longs;unt expul&longs;a lo­cum alicui facere, non exi&longs;tente vacuo aliquo, quamobrem nullo modo in &longs;phæram immittetur aliquid eorum, quæ &longs;unt extra, ni&longs;i excedat ali­qua pars aëris priùs in ea contenti, &longs;iquidem totus locus con&longs;tipatus est, & continuatus, vt arbitrantur. At experientiâ con&longs;tat contrari­um: nam &longs;i quis &longs;iphonem ori admovens &longs;phæraminflet, multum im­mittit &longs;piritum, non excedente aëre, qui in ea e&longs;t. Quod cùm ita &longs;em­per contingat; manife&longs;tò o&longs;tenditur, contractionem fieri corporum in &longs;phæra exi&longs;tentium in vacua implicata: contractio autem præter naturam fit, ob immi&longs;&longs;ionis violentiam. Si igitur aliquis infla, & ad ip&longs;um os ma­num apponens, &longs;tatim digito &longs;iphonem obturet; manebit omni tempore con&longs;tipatus aër in &longs;phæra: quod&longs;i quis aperiat, rur&longs;us extra erumpt cum agno &longs;trepitu & clamore immi&longs;&longs;us aër, propterea quòd, vt propo&longs;uimus, expellitur à dilatatione prexi&longs;tentis aëris impetu quodam factâ.

Quintò, &longs;i quis velit aërem, qui in Sphæra e&longs;t, per &longs;iphonem at­

trahere, magna copia con&longs;equetur, nullâ aliâ&longs;ub&longs;tantiâ in &longs;phæram &longs;uc­cedente, quemadmodum in ovo antè dictum e&longs;t. Quare ex hoc per &longs;pi­cuè o&longs;tenditur, magnam vacui coacervationem in &longs;phæra factam e&longs;&longs;e: non enim aëris corpora, quærelinquuntur eo tempore, maiora fieri po&longs;­&longs;unt, ita vt expul&longs;orum corporum locum repleant: nam&longs;i augerentur, cùm nulla ip&longs;is &longs;ub&longs;tantia extrin&longs;ecus accedat, veri&longs;imile e&longs;&longs;et, augmen­tum fieri per rarefactionem; hoc autem e&longs;&longs;et implicatio &longs;ecundùm eva­cuationem; &longs;ed vacuum nullum e&longs;&longs;e dicunt; non igitur augebuntur Cor­pora; neque enim aliud augmentum ip&longs;is accidere mente concipi pote&longs;t. Ex quibus per &longs;picuum e&longs;t, in aëris corporibus di&longs;&longs;eminata e&longs;&longs;e quædam vacua intermedia, & vi quadam adveniente ea præter naturam in va­cua reclinare.

Sphæra con cava ex&longs;u­cto aëre, va cua conti­net inter­valla &longs;ecun­dum Hero­nem.

Sextò, vacuis non exi&longs;tentibus, neque per aquam, neque per aërem, neque per aliud corpus vllum po&longs;&longs;et lumen tranfire, aut calidit as, aut alia potentia &longs;eu qualitas corporea. Nam quomodo radij &longs;olares per aquam in fundam va&longs;is penetrarent? Si enim aqua non haberet poros, &longs;ed radij aquam vi &longs;cinderent, contingeret vasa plena &longs;uperfun­di; quod fieri non videmus: Ad hæc &longs;i aquam vi &longs;cinderent, non vti­que radiorum alij ad&longs;uperiorem locum frangerentur, alij deor&longs;um cade­rent: nunc autem quicunque aquæ particulis occurrunt, franguntur ad &longs;uperiorem locum; quicunque verò incidunt in aquæ vacua, paucis par­ticulis occurrentes in va&longs;is fundum pervadunt. Adde quòd &longs;i nulla e&longs;&longs;ent in corporibus vacua, nullum corpus per alia corpora tran­&longs;ire po&longs;&longs;et: quod tamen oleis contingit, quorum aliqua tam &longs;ub­tilia, &longs;unt vt omnia metalla, ip&longs;umque; etiam vitrum penetrent. Idem etiam fieri ab igne, nemo ne&longs;cit, qui ferrum, aurum, & vitrum candentia vidit. Idem præterea à virtute magnetica fieri, quo­tidie experimur.

Corpora & qualitates corporeæ, pe­netrant a­lia corpora, &longs;ecundùm Heronem, propter va­cua interie­cta.

Vinum mi­&longs;cetur aquæ ob inteer &longs;a vacua uxta He­ronem.

Septimò, Vinum in aquam iniectum videtur &longs;ecundùm fu&longs;io­nem quandam per totum aquæ locum per meare; quod non fieret, ni&longs;i va­cua aquæ ine&longs;&longs;ent. Lumen quoque alterum per alterum fertur: nam cùm plures lucernas quis accenderit, omnia magis illu&longs;tr abuntur, ijsdem quoquò ver&longs;us per &longs;e invicem penetrantibus. Sed & per æs, per ferrum, & per alia corpora omnia penetratio fit; quemadmodum & quod in tor­pedine marina contingit.

Argumen­ta aliorum o vaco.

Hæ &longs;unt rationes Heronis, quibus probat. mne corpus ex tenuibus con&longs;tare corporibus, inter quæ&longs;unt vacua di&longs;&longs;eminata particulis minora. Quibus addi pote&longs;t vlteriùs, quòd eadem vacuola vi­dentur admittenda inter aquæ partes frigore con&longs;trictas in gla­ciem: etenim aqua in congelatione ad maius intervallum occu­pandum extenditur; va&longs;a enim licet benè compacta, &longs;ià &longs;inu latiori in angu&longs;tum a&longs;&longs;urgant, & repleantur aquâ ad maximam il­lam latitudinem, vel vltra; in congelatione diffringuntur: non &longs;ic, &longs;i infra illam latitudinem repleantur, vtdilatationi &longs;patium de­tur. In maioribus etiam va&longs;is, &longs;ed recurvo intus labro, validius re&longs;i&longs;tentibus, ip&longs;a glacies curvatur, &longs;ur&longs;um &longs;ecundùm convexum elevata. Omnis etiam glacies aquæ iniecta &longs;uperna Dila­tatur ergo in congelatione aqua, & vacuitates parvas intus ha­bet.

Addi præterea in confirmationem dictorum pote&longs;t duplex Experimentum, vnum argenti vivi, alterum aquæ tubo inclusæ; de quibus agemus infrà Cla&longs;se 1. cap. 6. Machina 7. Item expe­rimentum Thermo&longs;copij, de quo ibidem cap. 3. Mach 3.

Vis attra­ctiva quo­modo ab He rone expli­cetur.

Ex his alijsque argumentis atque experimentis, quæ vide­ri po&longs;&longs;unt apud alios, præ&longs;ertim Neotericos nonnullos, con&longs;tare po­te&longs;t, quomodoex Heronis & eorum &longs;ententia, qui vacuum admittunt, Natura adnitatur trahere alia corpora, aquam præ&longs;ertim & aë­rem, ad vacuum replendum, quod vi extrin&longs;eca præter naturam e&longs;t alicubi coacervatum; & quomodo corporibus omnibus in­&longs;it vis quædam attractiva, vt repleant vacuum illud; ac denique quomodo vis attractiva corporum, aquæ inquam & aëris, cau&longs;a e&longs;&longs;e po&longs;&longs;it ac principium Machinarum hydro-pnev­maticarum. Lege Heronis Spiritalia.

§. III.

Experimenta quibus vacuum nullum e&longs;&longs;e probatur

HIs tamen non ob&longs;tantibus, communior & &longs;anior Philo&longs;o­

phorum Veterum ac Recentiorum &longs;en&longs;us e&longs;t, nullum in re­rum natura vacuum e&longs;&longs;e, neque coacervatum in vnum, neque di&longs;&longs;eminatum inter partes corporum di&longs;ci&longs;&longs;as; & neque &longs;pontè ortum, neque productum violenter; adeoque corporum tra­ctionem, quæ in allatis Experimentis elucet, non fieri ad replen­dum vacuum, quod nullibi &longs;t, neque e&longs;&longs;e pote&longs;t; &longs;ed ad impe­diendam, quam Natura tantopere abhorret, inanitatem. Sua­dent hæc tum experientia Philo&longs;ophiæ Magi&longs;tra, tum rationes. Experientias adducam paucas ex multis; rationes apud alios leges.

Argumen­ta contra vacuum.

Primò, Latera follium, &longs;i arctè con&longs;tricta &longs;int, & inter &longs;e contigua, nulla vi diduci po&longs;&longs;unt, etiam Angelica, &longs;i obturetur orificium, per quod aër inter iectum inter diducenda latera &longs;pa­tium occupaturus &longs;ubeat. Cur hoc, ni&longs;i quia vacuum dari non pote&longs;t? quod tamen dari deberet, &longs;i occlu&longs;o orificio aër, & quodcunque aliud corpus &longs;ubiturum excluderetur, & tamen la­tera diducerentur. Eâdem de cau&longs;a duæ tabulæ plani&longs;&longs;imæ, & &longs;ibi mutuò &longs;ecundùm planitiem cohærentes, divelli non po&longs;­&longs;unt, &longs;i ita tentetur divul&longs;io, vt aër aut aliud corpus inter &longs;uper­ficies &longs;ubintrare tempe&longs;tivè non po&longs;&longs;it. Iuveni lacertorum &longs;uorum robur iactanti (inquit P. Nicolaus Zucchius in &longs;ua de vacuo do­cti&longs;&longs;ima Diatribe) propo&longs;ita-&longs;emel e&longs;t lamina ærea per an&longs;am in me­dio extantem apprehen&longs;am elevanda è tabula marmorea, cui optimè con­gruebat: qui primò tanquam rem ludicram puero committendam con­temp&longs;it: tum in&longs;tantibus amicis manum vtramque admovens, cùm lu­ctatus diu hærentem non removi&longs;&longs;et, excu&longs;avit impotentiam, obiectâ peregrini & potenti&longs;&longs;imi glutinis interpo&longs;itione, quo forti&longs;&longs;imè copulan­te nequiret divelli; donec vidit ab alio per tabulam facillimè laminam deduci, & ad extrema productam, & actam in transver&longs;um, inde de­portari. Pa&longs;&longs;im etiam experimur, dum duo corpora &longs;ecundùm planas &longs;uperficies coniuncta &longs;unt, vt vnum illorum elevatur, &longs;e­qui etiam alterum, quantumvis grave, atque in aëre libero ve­luti &longs;u&longs;pen&longs;um hærere.

Latera fol­lium diauci non po&longs;&longs;unt, ore occlu&longs;e, ob metum vacus.

Tabulæ pla­næ metu va cui divelli non po&longs;&longs;unt.

Secundò, Si vas vitreum aquæ immergas atque impleas & deinde intra aquam invertas, vt patens os deor&longs;um vergat, tum æqualiter ita inver&longs;um eleves, & extrahere tentes; &longs;pectabis a­quam illo inclu&longs;am &longs;ur&longs;um intra illud &longs;imul attolli; & &longs;i vas illud ex parte iam extractum, & ex parte adhuc immer&longs;um, manu &longs;u­&longs;tineas, videbis etiam aquam inclu&longs;am in illo per&longs;i&longs;tere &longs;ic eleva­tam &longs;upra ambientis aquæ &longs;uperficiem æquilibratam, & &longs;uo et­iam pondere manum &longs;u&longs;tinentis aggravare. Cur hoc? Quia non pote&longs;t aër &longs;uccedere ad occupandum locum inter aquam de&longs;cendentem, & partes va&longs;is, quæ aquâ de&longs;cendente de&longs;ere­rentur. Idem continget in quolibet alio va&longs;e, &longs;ed in vitreo et­iam oculis patebit experimentum. Quòd &longs;i in fundo va&longs;is fora­men aliquod &longs;it clau&longs;um, vt po&longs;&longs;it aperiri; eo aperto, & aëre li­berè &longs;ubintrante, aqua intra vas &longs;u&longs;pen&longs;um de&longs;cendet, & de&longs;i­net gravare manum.

Vas inver­&longs;um difficul ter extrahi­tur ex aqua ob vacui metum.

Va&longs;a &longs;ubs pertu&longs;a, &longs;u­periùs occlu &longs;a, aquam retinent vuí metu.

Tertiò, In va&longs;is ad irrigandos hortos pa&longs;&longs;im adhiberi &longs;oli­tis (cuiu&longs;modi proponimus infrà Cla&longs;s. 1. cap. 6. Mach 2, & Cri­brum Ve&longs;talium vocamus) licet fundum ip&longs;orum &longs;it perforatum, & multis foraminibus pervium, dum aquâ plena, æqualiter ele­vantur in libero aëre, aqua non defluit, quàm diu digitus fora­mini illorum &longs;upernè in collo patenti adpre&longs;&longs;us non permittit in­gre&longs;&longs;um aëri ad replendum locum, qui, &longs;i deflueret aqua, relin­queretur inter &longs;uperiorem va&longs;is &longs;uperficiem, & defluentem a­quam. Et in tali con&longs;i&longs;tentia permanente aquâ intra vas, tan­tum gravat, & deor&longs;um nititur aqua, quantum &longs;i fundo integro & collo aperto, &longs;imul cum va&longs;e &longs;u&longs;tineretur. Remoto verò di­gito, & permi&longs;&longs;o ingre&longs;&longs;u aëri per &longs;uperius foramen, &longs;tatim de­fluit aqua ex fundi foraminibus. Quòd &longs;i in medio defluxu ite­rum appo&longs;ito digito foramen ob&longs;truatur, pendet iterum aqua, contra inclinacionem &longs;uæ gravitatis, tunc etiam manum gra­vantis, nec defluit in apertum aërem.

Quartò, Si fi&longs;tulæ vnam extremitatem aquæ immergas,

ex altera ori admota aërem ex&longs;ugas, elevatur confe&longs;tim aqua, &longs;ualicet gravitate reluctans, po&longs;t aërem extractum ad os ex&longs;u­gentis; vtique nè detur vacuum in fi&longs;tula ex&longs;ucto aêre.

Fi&longs;tula ex­&longs;ucto aëre a­quam recipit vacui metu

Quintò, Si phialam cupream firmis lateribus compactam, nè facilè rumpatur, igni admotam calefacias, vt aër intus con­tentus vehementer rarefiat, eamque deinde frigidæ immergas, vt aër phialæ à violenta rarefactione &longs;e recolligens conden&longs;etur, & minorem occupet locum; intus attrahitur aqua, dum alius aër in locum relinquendum &longs;uccedere non pote&longs;t, ne &longs;cilicet detur vacuum in phiala. Sic &longs;i ex fi&longs;tula, cuius vnum orificium occlu­ &longs;um, alterum apertum, aut ex clave fœminea benè compacta, ex&longs;ugas fortiter & continuato tractu aërem, & ce&longs;&longs;ante &longs;uctione &longs;tatim apponas fi&longs;tulæ orificio patenti digitum aut labrum, nè &longs;uccedat ambiens aër; attrahitur intra fi&longs;tulam, aut clavem, caro digiti aut labri, vt notabiliter vellicet, & de carne depende­at. Non alia de cau&longs;a, ni&longs;i quia durante ex&longs;uctione rarefit rema­nens intus aër, & violenter totum illud &longs;patium &longs;olus implet; ce&longs;­&longs;ante verò &longs;uctu recolligit &longs;e à violenta rarefactione, & minorem locum occupans advocat aliud, quod ip&longs;i proximum e&longs;t, ad lo­cum de&longs;ertum replendum. Eademque cau&longs;a e&longs;t, cur cucurbi­ tulæ, quæ Chirurgis in v&longs;u e&longs;&longs;e &longs;olent, carnem &longs;ur&longs;um attrahant; nimirum quia aêr, qui priùs igne incaluerat, & in raritatem abi­erat, extincta flamma frigiditatem & den&longs;itatem recipit, atque in minorem contractus molem inane &longs;patium relinqueret, ni&longs;i ad id replendum caro elevaretur.

Phiala cale facta, & a­quæ immer­&longs;a, attrahit ip&longs;am va­cui metu.

Fi&longs;tula ex­tracto aëre carnë ad­motam tra hunt, ne vacuum detur.

Cucurbitu­læ carnem attrahunt metu vacui

Sextò, Si &longs;clopeto æneo, aut maiori bombardæ, immit­ tas embolum interiori &longs;uperficiei benè congruentem, vt aêr in­ter ip&longs;am & embolum &longs;ubire non po&longs;&longs;it, & deinde igniarium fo­ramen occludas; non ni&longs;i difficulter extrahitur embolus, & vi aliquantisper extractus retroagitur intra fi&longs;tulam:

Embolus &longs;clopeto, foramini i­gniarioclau &longs;o, difficul­ter extrahi­tur, vacui metu.

Atque hæc &longs;ecunda &longs;ententia, quæ Naturæ vacuum tan­toperè fugientis partes tuendas &longs;u&longs;cipit, meritò à melioris notæ Philo&longs;ophis, vt olim, ita nunc quoque præfertur: quidquid ob- ganniant Neoterici quidam, qui ne&longs;cio quo novitatis &longs;tudio in­ citati, xenodoci/as vento inflati, nova quotidie dogmata cudere, aut ob&longs;oleta veterum commenta recoquere non verecundantur. Satiùs enim ac faciliùs e&longs;&longs;e exi&longs;timatur, admittere contra di&longs;po­&longs;itiones particulares corporum gravium elevationem, levium depre&longs;&longs;ionem, den&longs;iorum dilatationem, rariorum conden&longs;a­tionem, durorum inflectionem, & &longs;imilia, quàm vacuum, quod Naturam adeo abhorrere con&longs;picitur, vt nulla vi naturalis agen­tis, nullo Machinæ artificio dari po&longs;&longs;it. Huiu&longs;modi autem na­turæ pugnas & &longs;tratagemata multa &longs;pectabis manife&longs;tè infrà in multis Machinis, quibus hæc Naturæ inanitatem fugientis &longs;o­lertia &longs;ummam præbet commoditatem. Alia etiam paulò po&longs;t afferam, &longs;i priùs, quæuam &longs;it vis illa Attractiva corporum, quâ Natura vtitur ad vacuum impediendum, o&longs;tendero.

Vacuum da rinaturali­ter non po­te&longs;t.

§. IV.

Vis attractiva|ob vacui metum, primum Machinarum hydro-pnevmaticarum principium, quæ, & qua­lis, quibusvè in&longs;it corporibus.

HÆ &longs;unt binæ de vacuo &longs;ententiæ, quarum vtraque fontium artificialium, aliarumque hydro-pnevmaticarum Machina­rum con&longs;tructioni, & effectuum mirabilium patrationi viam a­perit & &longs;ubmini&longs;trat arma; quamvis evidentiùs &longs;ecunda, quàm prima, cui & illam idcirco prætulimus. Quæ&longs;tio tamen non le­ vis remanet inter &longs;ecundæ huius &longs;ententiæ fautores circa vim il­lam, quæ cogit corpora gravia accurrere ad vacuum impedi­endum, etiam contra propriam inclinationem; dubitaturque, quibusnam illa vis in&longs;it ceu &longs;ubiectis; trahentibusnè, an tractis corporibus?

Vis attra­ctiva aquæ quibus in&longs;it corporibus.

Alij enim putant, e&longs;&longs;e virtutem motivam ip&longs;is corporibus accurrentibus intrin&longs;ecam, quæ in bonum Vniver&longs;i corpora gra­via in altum, contra con&longs;uetum modum, ad vacuum impedien­dum, vrgeat atque impellat. At hoc experientia ip&longs;a &longs;atis con­vellit, & fal&longs;itatis arguit; &longs;iquidem, vt benè advertunt Cabæus & Zucchius, aqua a&longs;cendens intra tubum, ad vacuum impediendum (quod in &longs;equentibus Machinis frequenti&longs;&longs;imum e&longs;t) tantundem gravat, quantum gravaret in libero aëre &longs;u&longs;pen&longs;a. Quòd benè adver­terejubet Cabæus illos, qui &longs;pecio&longs;o nimis nomine &longs;e Ingenie­ros appellari volunt; & Zucchius ait, &longs;e huius rei experimento coëgi&longs;&longs;e, virum alioqui non ineruditum, vt fallaciam agno&longs;ceret in via, quâ motum perpetuum &longs;e inveni&longs;&longs;e putabat, & iam dicandum Ma­gno Principi &longs;chema in æs incidi curaverat, &longs;upponendo aquam ad impe­diendum vacuum &longs;ua vi a&longs;cendentem non egere virtute æquilibrante, ne­dum &longs;uperante pondus illius, qua eget cùm &longs;i&longs;tenda vel elevanda e&longs;t in aëre, liberè accurrente ad replendum &longs;patium inter eam, & alia corpora relinquendum.

Vis attra­ctiva non ine&longs;t corpo­ribus attra Hic.

Aqua in tu bis elevata adfugiendvacuum gravat.

Alij igitur, & meliùs, exi&longs;timant, omnibus corporibus contiguis hanc ine&longs;&longs;e vim, à Natura attributam, ad Vniver&longs;i bonum ac con&longs;ervationem, vt quotiescunque vnum illorum lo­co cedit, &longs;eu tractum ab extrin&longs;eco agente violenter, &longs;eu &longs;ponte & innatâ inclinatione recedens, pondere videlicet, autlevitate: trahat &longs;ecum alterum &longs;ibi contiguum, quantumvis reluctans, quoties inter ip&longs;um & contiguum non pote&longs;t &longs;ubintrare corpus aliud, quod partium Vniver&longs;i contiguitatem tueatur, & impedi­at inanitatem; adeoque trahentibus, non tractis corporibus vim illam motivam ine&longs;&longs;e. Et hanc vim appellant vim attracti­vam, e&longs;tque illa ip&longs;a quam nos primum Machinarum hydro-pnev­maticarum principium &longs;eu fundamentum dicimus, vnaque e&longs;t ex cau&longs;is vacui impeditivis. Dico, vna ex cau&longs;is, non enim v­nica e&longs;t via atque indu&longs;tria, quâ vacuum Natura evitare adni­titur. Modò enim vtitur prædicta vi attractiva: modò ad&longs;ci&longs;cit operam alteratricium qualitatum, quæ corpora rarefaciunt, & in ampliorem molem diffundunt: non nunquam magno impe­tu va&longs;a frangit, aut alia id genus media, prout occa&longs;io & nece&longs;­&longs;itas fert, accommodat: quin etiam haud rarò nullo motu, nulla actione, nulla denique vi effectrice, &longs;ed per negativam quandam ob&longs;i&longs;tentiam vacuo repugnat; vt cùm Iatera follium compre&longs;&longs;a clau&longs;o orificio, vt diximus &longs;uprà, divellinon &longs;init. Semper ta- men Natura in his, vtin cæteris rebus, compendio &longs;tudet, & quan­tum pote&longs;t, facilioribus ac &longs;implicioribus vtitur medijs; vt &longs;i ad fugam vacui corpus &longs;ur&longs;um attrahere, quàm den&longs;atione impedi­re promptius &longs;it, illud præ&longs;tat, non hoc; &longs;i contrà, hoc, non il­lud: item &longs;i per &longs;olum motum localem &longs;ine rarefactione corpus aliunde trahere &longs;ufficit, rarefactioni parcit; alioquin vtrumque mi&longs;cet.

Attractiva vis trahen­tibus corpo­ribus ine&longs;t, ad vacuum impediendum

Vacui cau­&longs;a impediti­va multi­plex.

Attractiva vis ad va­cuum im­pediendum omnibus corporibus ine&longs;t.

Sed hæc accuratiús Philo&longs;ophi; nobis &longs;ufficit varijs experi­entijs mon&longs;tra&longs;&longs;e, ine&longs;&longs;e corporibus Vniver&longs;um componentibus vim attractivam vel ad replendum, vel ad impediendum vacuum; quæ vis vt alia corpora, ita multò magis aërem & aquam attra­hat, & Machinarum Hydro-pnevmaticarum principium ac fun­damentum primum con&longs;tituat.

Nunc tempus e&longs;t, vt experimento vnico & veluti re ip&longs;a mon&longs;tremus, quomodo vis attractiva, quam hactenus corpori­bus Vniver&longs;um componentibus ine&longs;&longs;e o&longs;tendimus ad vacuum impediendum (vel vt Hero vult, replendum) cau&longs;a e&longs;&longs;e po&longs;&longs;it, atque principium Hydro-pnevmaticarum Machinarum.

§. V.

Experimentum, quo o&longs;tenditur vis attractiva ad aquas in altum evehendas, vacui vitandi causâ, per Machinas Hydro-pnevmaticas.

INfrà Cla&longs;se 1. cap. 1. Machina 11. Modum præ&longs;cribimus aquam

ad certam ac determinatam aliquam altitudinem evehendi, eumque appellamus Siphonem inver&longs;um interruptum: qui mo­dus cùm nitatur vi Attractivâ orta ex vacui metu, multaque im­plicet notatu digni&longs;&longs;ima, & ad vim Attractivam penitiùs intelli­gendam omninò nece&longs;&longs;aria, hîc eum paulò fu&longs;iùs explicandum cen&longs;ui; ibi enim breviter tantùm, & quantum ad praxin &longs;ufficit, illum proponimus. Sit itaque in ci&longs;terna, puteo, pi&longs;cina, aut va&longs;e quocunque B, aquâ pleno, elevanda ad pedes ex.g.centũ (di­co, exempli gratia, nam alioquin non pote&longs;t ad tantam altitudinem attolli per vim attractivam) aqua in vas KA, quod &longs;it con&longs;titu- tum in parte &longs;uperiore domus, habeatque epi&longs;tomium H, per quod, vbi repletum fuerit vas, depleri po&longs;&longs;it. Fiat vas C, cu­iuscunque capacitatis (quamvis meliùs &longs;it, vt &longs;it æqualis capa­citatis cum va&longs;e KA, aut &longs;altem non minoris) ex eoque derive­tur ad vas KA, &longs;iphon &longs;eu tubus OI, qui in va&longs;e C, incipiat im­mediatè infra operculum ip&longs;ius, in va&longs;e verò KA, de&longs;inat paulò infra eiu&longs;dem operculum, tantum ab illo di&longs;tans, quantum &longs;uf­ficit vt aër permeare po&longs;&longs;it. Deinde ex va&longs;e B, derivetur ad vas KA, tubus BK; qui in va&longs;e B, incipiat paulò &longs;upra fundum ip&longs;i- us, in va&longs;e verò KA, de&longs;inat paulò infra operculum; vbi etiam nonnihil in curvetur, vti in figura apparet. Tandem ex va&longs;e C, derivetur alius tubus EF, cum epi&longs;tomio E, habens longitudi­nem paulò majorem longitudine tubi BK, capacitatem verò &longs;eu amplitudinem eidem BK, omnino æqualem. Debent autem omnes tres tubi diligenti&longs;&longs;imè adferruninari &longs;eu coarctari plum­bo, aliavè materia, &longs;uis va&longs;is in loco tran&longs;itus; & tam vas C, quam vas KA, claudi obturarique vndiq, &longs;tudio&longs;i&longs;&longs;imè, nè aërem ali­unde, quàm per tubos, recipere aut ejicere valeant. His ita præparatis, impleatur vas C. aquâ per foramen O, immi&longs;&longs;o ip&longs;i foramini iufundibulo; & vbi repletum fuerit, claudatur vt di­ctum. Deinde clau&longs;o epi&longs;tomio H, aperiatur epi&longs;tomium E, tubi EF; defluetque aqua ex va&longs;e C, & in locum ip&longs;ius, nè va­cuum in va&longs;e admittatur, &longs;equetur per &longs;iphonem IO, aër in va­&longs;e KA, contentus; in locum verò aëris extracti è va&longs;e KA, &longs;eque­tur, propter vacui metum, aqua va&longs;is &longs;eu ci&longs;ternæ aut pi&longs;cinæ B, per &longs;iphonem BK; & tam diu durabit a&longs;cens aquæ per &longs;iphonem BK, in vas KA, quàm diu de&longs;cendet aqua per &longs;iphonem EF ex va­&longs;e C. Lege quæ dicimus infrà loco cit. En ergo, quomodo aq ia, ad vacuum impediendum, a&longs;cendat contra naturalem &longs;uam in­clinationem, & quomodo propter eandem cau&longs;am aër de&longs;cendat.

Experimen tum vim attractivam o&longs;tendens ad vacuum impedien­dum.

§. VI.

Notantur nonnulla circa prædictum experimentum, &longs;i­mulque; &longs;tabilitur, in attractione aquæ metu vacui habendam e&longs;&longs;e rationem perpendiculi aquarum.

DIximus, meliùs e&longs;&longs;e vt vas C, &longs;it æqlis capacitatis cum va&longs;e KA, aut&longs;altem non minoris: & præterea &longs;iphonem EF debe­re e&longs;&longs;e paulò longiorem &longs;iphone BK, capacitatis verò &longs;eu ampli­tudinis æqualis. Ratio quoad capacitatem va&longs;orum & &longs;ipho­num e&longs;t, quia tamdiu a&longs;cendit aqua per BK, quàm diu de&longs;cen­dit per EF, vt dixi (tam diu enim, & non ampliùs, duratvacui metus, ob quam aqua a&longs;cendit per tubum BK:) &longs;i ergo vas C mi­nus e&longs;&longs;et quàm vas KA, aut tubus EF capacior quàm tubus BK; ef­flueret tota aqua ex C, antequam repleretur KA. Accedit & hoc commodi ex æquali vtrorumque capacitate, quòd &longs;cire po&longs;­&longs;it apud C exi&longs;tens, quandonam plenum &longs;it vas KA; quando ni­mirum effluxerit tota aqua C. Ratio verò cur tubus EF, debeat e&longs;&longs;e longior tubo BK, patebit ex dicendis Protheoria IV. capite 2. Propo&longs;it. 13. interim hîc id confirmo hoc Experimento. Ac­cipe vas aquâ plenum, eique in&longs;ere &longs;iphonem inver&longs;um ABCD æqualium crurium. Accipe deinde tubum rectum DE aquâ plenum, eumque, obturato priùs ore E, in&longs;ere extremitati D, cruris CD, & diligenter coarcta in­ter &longs;e&longs;e, nè aër inter vtrumque tran&longs;ire po&longs;&longs;it. His factis aperios inferius. Et &longs;i quidem tubus rectus DE lon­gior fuerit quàm crus AB à &longs;upre­ma &longs;uperficie aquæ va&longs;is vsque ad B; effluet aqua ex E, a&longs;cendetque a­qua va&longs;is A, per crus AB, indeque per CD, crus de&longs;cendet, donec tota effluxerit ex ore E. Si aut de­urtetur tubus DE in F, fiatque æqualis cruri AB; a&longs;endet aqua va­&longs;is A &longs;olùm u&longs;que ad B, quàm diu vi­delicet aqua ex F effluit; quâ elapsâ, relabetur aqua cruris AB intra vas A, Si idem tubus DE decurtetur in G, aut H, fiatque brevior quàm crus AB; a&longs;cendet aqua per AB &longs;o­lùm aliquòu&longs;que, & deinde rela­betur. Experientiam quilibet fa­cilè poterit facere.

Experimë­tum aliud: o&longs;tendens vim attra­ctivam a­quæ in altum.

Nec &longs;ufficit vt tubus EF, &longs;upe­rioris figuræ &longs;it quacunque ratione longior tubo BK, &longs;ed nece&longs;&longs;ari­um e&longs;t, vt perpendiculum ip&longs;ius EF, &longs;it longius quàm perpendicu­lum BK, hoc e&longs;t, vt tubus EF perpendiculariter de&longs;cendat profundiùs infra vas C, quàm tubus BK a&longs;cendat perpendi­culariter &longs;upra vas B. Quare &longs;i tubus EF centum & vnius pe­ dum inclinaretur infra vas C, aut convolveretur in helicem, vt hîc apparet, ita vt perpendiculum ip&longs;ius &longs;olùm e&longs;&longs;et pedum v.g. 60. ab E v&longs;que ad D v.g. & non v&longs;que ad N; neutiquam &longs;ufficeret; &longs;ed omnino nece&longs;&longs;arium e&longs;t, vt perpendiculum &longs;it etiam ad minimùm centum pedum, & pertingat v&longs;que ad N. Et hæc perpendiculi men&longs;ura adeo e&longs;t nece&longs;&longs;aria in hydraulicis hi­&longs;ce operationibus, vt Natura falli nulla arte, nullo dolo, nullaque; experientia po&longs;&longs;it. Ioannes Bapti&longs;ta Porta lib. 2. Spirital. cap. 1. ait, &longs;e, cúm indigeret determi­nato perpendiculo infra vas de­&longs;cendente, nec locum haberet ad demittendum perpendicula­riter tubum, multis viis tenta&longs;&longs;e, vt defectum &longs;uppleret; præcipuè verò duabus memoratis, nimi­rum inclinando primùm, deinde in helicem convolvendo tubum requi&longs;itæ longitudinis; &longs;ed nun­quam de&longs;ideratum effectum fu­i&longs;&longs;e con&longs;ecutum, nec vnquam a&longs;cendi&longs;&longs;e aquam ad maiorem altitudinem quàm erat perpen­diculum tubi inclinati, aut tor­tuo&longs;i. Demi&longs;it etiam infra vas canalem capaciorem, licet bre­viorem illo, qui erat &longs;upra vas: præterea loco vnius longioris demi&longs;it multos breviores, qui &longs;i­mul &longs;uperabant longitudinem &longs;uperioris; &longs;ed vano &longs;emper labore. Adeo &longs;ui juris tenax e&longs;t Natura, & vt non fallit nos in &longs;uis operationibus, ita nec falli à nobis vult. Infrà tamen Cla&longs;se 1. cap. 1. Machina 7. Modum do­cebo elevandi aquam ad quamvis altitudinem perpendiculis bre­vibus multiplicatis, &longs;ed vtrisque, de&longs;cendentis nimirum & a&longs;cen­dentis aquæ. Notandum præterea e&longs;t hoc loco, Tubum OI, aut CI, po&longs;&longs;e e&longs;&longs;e quantævis longitudinis, vt infrà iterum dicam §. IX. Quomodo autem vi attractiva & expul&longs;iva &longs;imul elevari po&longs;&longs;it aqua ad maiorem altitudinem quàm &longs;it perpen­diculum aquæ de&longs;cendentis, dicemus infrà Protheoria II.

Perpendicu lum aquæ de&longs;cenden­tis debet e&longs;&longs;e longius, quàm a&longs;cen dentis.

§. VII.

Heronis Alexandrini error in elevanda a­qua vi attractivâ.

Error He­ronis.

HEro Alexandrinus cap. 53. libri de Spiritalibus modum præ­&longs;cribit elevandi aquam per vim attractivam ex vacui metu; &longs;ed vehementer hallucinatur, ob perpendiculi prædicti defectum; &longs;i tamen Heronis e&longs;t error ille, & non potiùs interpretis, qui &longs;che­ma appo&longs;uit, vt notavit etiam Ioannes Bapti&longs;ta Porta lib. 2. de Spirital. cap. 2. Affero verba vnà cum Schemate Heronis, ex ver­&longs;ione & editione Federici Commandini Vrbinatis, qui ex Græco in latinum eum tran&longs;tulit quem errorem repetit etiam Ioannes Bapti&longs;ta Aleottus in ver&longs;ione Italica.

E&longs;t etiam alia con&longs;tructio, inquit Hero, in qua humidum &longs;en­&longs;im fertur, & manet, ita vt &longs;emper a&longs;cendens videatur. Sit ba&longs;is AB vndique præclu&longs;a, quæ dia­phragma habeat CD, & vitreum oper­culum Cylindri forma EF &longs;imiliter vndi­que præclu&longs;um. In operculo autem EF &longs;it tubus GH, ab ip&longs;ius tecto parum di&longs;tans, & &longs;imul perforatus cum diaphragmate. A­lius etiam tubus KL &longs;imul perforetur cum tecto ba&longs;is, & à diaphragmate parum di&longs;tet. In&longs;it autem ba&longs;i extra vitreum operculum foramen M, per quod vas AB, impleatur: & ba&longs;is AB ad fundum canalem habeatN. Sit præterea alius tubus XO, &longs;imul perforatus cum diaphragmate, parumque à ba&longs;i di&longs;tans, per quem implebitur vas CB. Itaque ob&longs;tructo canali N, aër qui e&longs;t in CB, per tubos, & per fora­men M excedet extra. Et cùm vas CB repletum fuerit, implebi­mus AD per foramen M, aër enim qui in ip&longs;o e&longs;t, per idem foramen excedit. Si igitur canalem N fluere dimittamus, in locum CB exina­nitum, aër ex vitreo operculo procedet per tubum GH: in locum autem vitrei operculi exinanitum ex AD humidum a&longs;cendet per tubum KL, & ar per foramen M ingredietur. Atque hoc vsque eò fiet, quoad vitreum operculum repletum fuerit. Oportebit autem loca AD, CB, EF, inter &longs;e æqualia e&longs;&longs;e, vt ad invicem & aër, & humidum transferantur. Quan­do autem vas CB exinanitum fuertt, & con&longs;titerit aër is continuitas, rur&longs;us ex vitreo operculo aqua in vas AD deferetur, aëre per canalemN, & per GH tubum in vitreum operculum ingrediente, qui verò e&longs;t in vase AD aër per foramen M excedet.

Ita Hero; &longs;ed errat autip&longs;e, aut eius interpres: debete­nim canalis N de&longs;cendere infra vas perpendiculariter ad tantam profunditatem, quanta e&longs;t longitudo KL, vt attrahatur tota a­qua va&longs;is AD; alioquin po&longs;tquam effluxerit aliquantulum a­quæ ex canali N, traxeritque aliquantulum aëris ex cylindro EF, & aquæ ex va&longs;e AD, ce&longs;&longs;abit omnino aquæ fluxus, ob de­fectum perpendiculi æqualis perpendiculo KL, &longs;altem ab &longs;u­prema &longs;uperficie aquæ va&longs;is AD, v&longs;que ad orificium K tubi prædict KL Qui defectus &longs;i &longs;uppleatur, &longs;equetur effectus de&longs;ide­ratus. Porta tamen loco citato, occa&longs;ione huius Heronianæ indu&longs;triæ, alium excogita vit modum elevandi vi attractiva ex inferiori ad &longs;uperius vas aquam: quem legere poteris apud ipsum.

§. VIII.

Alius Heronis error in elevanda aqua vi attractiva.

ALium errorem committit Hero lib. cit. de Spiritalibus cap. 5.

&longs;i tamen & hic non e&longs;t ad&longs;cribendus interpreti. Docere vult Hero modum elevandi atque attrahendi humidum ex va&longs;e per &longs;iphonem inver&longs;um, cuius vnum crus &longs;it humido immer&longs;um, alterum extra vas promineat, non attrahendo prius per os no­&longs;trum aërem ex &longs;iphone, &longs;ed appendendo vas quodpiam aquâ plenum: Sic ergo di&longs;currit.

Error alius Heronis.

Sit &longs;merismation aliquod, cuius ma&longs;culum quidem apponatur extrin&longs;eco &longs;iphonis cruri, ita vt per ip&longs;um fluat, &longs;itque TV; faminave- rò TYV priùs adglutinata va&longs;iLZ, quod paulò plùs aquæ, quàm &longs;iphon capiat; habeat autem ad fundum effluxionem *w Quan­do igitur volumus per &longs;iphonem e­ducere aquam, quæ e&longs;t in va&longs;e AB, effluxionem va&longs;is LZ diito con­prehendentes, ip&longs;um aquâ imple­bimus. Po&longs;tea fœmineum &longs;meri­&longs;ma ma&longs;culo aptantes effluxionem*w dimittemus. Evacuato au­tem LZ va&longs;e, aër in &longs;iphone exi&longs;tens in evacuatum locum pro­cedet; quem con&longs;equetur humi­dum quod in va&longs;e AB, ita vt &longs;iphom impleatur. Deinde aufferentes vas LZ Siphonem fluere &longs;ine­mus. Oportet autem &longs;iphonem rectè de&longs;cendere, &longs;i officium &longs;uum præ­&longs;tare debeat; illud autem erit, quando ad va&longs;is AB labrum duas regu­las rectas compingemus, atque inter ip&longs;as intrin&longs;ecum &longs;iphonis crus con&longs;tituemus, ita vt vtrasque contingat; ad idem verò &longs;iphonis crus intrin&longs;ecum vtrimque clauvicuulam affigemus, regulas continentem parte interiori: ita enim neque obliquum, neque in nteriorem par­tem declinabit; rectò enim & exqui&longs;itè de&longs;cendet, claviculis regulas atterentibus.

Hæc Hero; quæ tamen fal&longs;a &longs;unt, & contraria experien­tiæ, rationi, & hydro-pnevmaticis principijs. Ni&longs;i enim va&longs;is altitudo ab V ad *w tanta &longs;it, quantum e&longs;t crus internum CD Siphonis; non a&longs;cendet aqua per crus CD, ni&longs;i ad tantam al­titudinem, quanta e&longs;t altitudo va&longs;is prædicta ab Y, v&longs;que ad *w. Et ratio e&longs;t, quia cùm perpendiculum Y *w minus &longs;it, quàm perpendiculum CD; non pote&longs;t aqua attolli ad altitudinem CD, &longs;ed po&longs;tquam a&longs;cenderit ad altitudinem æqualem perpen­diculo Y *w, relabetur intra vas AB. Ioannes Bapti&longs;ta Porta lib. 2. de &longs;pirital. cap. 3. ait. aquam non effluxuram è va&longs;e LZ, ab­ lato impedimento apud *w, propter defectum &longs;cilicet perpendi­culi prædicti. Sed ratio & experientia convincit, effluere aquam è va&longs;e; & a&longs;cendere aquam per crus internum ad altitudinem ex­plicatam; quod ip&longs;emet Porta alibi fatetur.

Error Ioan­nis Bapti&longs;tæ Porta.

Ut igitur ex va&longs;e AB extrahi po&longs;&longs;it aqua per &longs;iphonem ab&longs;­que eo quòd ore attrahatur aër, oportet loco Va&longs;is LZ adhibere tubum tantæ longitudinis, quantæ e&longs;t crus CD, imò paulò ma­joris: tunc enim effluente aquâ ex tubo appen&longs;o a&longs;cendet paula­tim aqua per &longs;iphonem, donec pervenerit u&longs;que ad &longs;ignum V, quod &longs;uppono e&longs;&longs;e &longs;emper demi&longs;&longs;ius quàm e&longs;t &longs;uprema &longs;uperfi­cies aquæ in Va&longs;e AB: tunc enim &longs;i aufferatur tubus appen&longs;us, ef­fluxus aquæ ex &longs;iphone tam diu durabit, donec tota aqua ex va&longs;e AB effluxerit, &longs;i os C ad fundum v&longs;que va&longs;is pertigerit. Vi­de Salomonem de Caus lib. 1. de viribus motricibus. Vide præ­terea quæ &longs;cribimus de fluxu aquæ per &longs;iphones infrà Protheo­ria IV. Cap. 2. præ&longs;ertim Propo&longs;it. XI. ubi in Annotatione 2. no­tamus errorem Marini Mer&longs;enni &longs;imilem prædicti Heronis errori.

Monuit hîc, & optimè, acuti&longs;&longs;imeque P. Paulus Ca&longs;atus Mathe&longs;eos in Romano no&longs;tro Collegio Profe&longs;&longs;or, quæ diximus contra Heronem hoc loco, tunc &longs;olùm haberelocum, quando &longs;uprema aquæ &longs;uperficies non e&longs;&longs;et multùm &longs;upra punctum C. Cæterùm &longs;i vas AB e&longs;t plenum, & perpendiculum *y*w &longs;uperat per­pendiculum à &longs;uprema aquæ &longs;uperficie u&longs;q ad D; poterit a&longs;cen­dere aqua, & per crus DY defluere quâ &longs;emel defluente, &longs;em­per deinde defluet, donec evacuetur vas vsque ad punctum C, quoniam perpendiculum DY &longs;emper manebit longius quàm perpendiculum CD.

§. IX.

In aquarum elevatione vi attractivâ, habenda e&longs;t ratio perpendiculi &longs;olùm illorum tuborum, per quos aqua, non per quos aër vehitur.

Tubi per quos aër de­fertur in

NOtat optimè Ioannes Bapti&longs;ta Porta lib 2. de Spiritalibus. C. 6. & nos etiam &longs;uprà in fine §. VI. in&longs;inuavimus, in Machinis

vi attractiva elevan­tibus aquam, haben­dam e&longs;&longs;e rationem longitudinis ac bre­vitatis &longs;olùm illorum tuborum &longs;eu canali­um, qui conferunt ad aquarum perpen­diculum, & per quos tran&longs;ire debet aqua, atque adeo quorum aquæ &longs;e mutuò tra­here ac v de­bent, quoru&longs;olùbet. eli­ùs intelligatur, in­ciat figura porta &longs;uprà §. citato, in qua figura henda e&longs;t &longs;olùm ratio tuborum BK, & EF; hi enim duo continent aquas quarum perpendi­cula ad invicem pu­gnant, juxta dicenda infrà Protheoria IV. cap. 2. Propo&longs;it. 13. tubus verò OI pote&longs;t e&longs;&longs;e quantævis longitudinis aut brevita­tis, ac proinde vas C pote&longs;t e&longs;&longs;e & infra & &longs;upra vas B. Patet hoc experientia, vt dicemus loco citato; vbi etiam experientiæ cau&longs;am dabimus.

Machinis attractivis metu vacui po&longs;&longs;unt e&longs;&longs;e quantævis longitudinis

§. X.

Vi attractivâ elevari pote&longs;t aqua vltra perpen­culi altitudinem, aëre mediante.

Ingenio&longs;&longs;imum &longs;anè modum Naturam quodamodo decipiendi, & aquam vltra perpendiculi à Natura requi&longs;iti altitudinem e­levandi vi attractiva, affert Porta loco proximè citato, cap. 11. aitque &longs;e illum ca&longs;u reperi&longs;&longs;e inter contuas hydraulicas atque pnevmaticas experientias quas fecit. Fit hoc aëre mediante, aquæ immi&longs;to. Hac ratione, inquit Porta, perpendiculo 50. pedum elevari pote&longs;t aqua ad altitudinem pedum 80. Modus hic e&longs;t.

Sit ex va&longs;e AB portatili e­ levanda aqua ad vas D, ad alti­tudinem pedum 80. Fiat canalis vitreus GK (vt&longs;pectari, oculis­que percipi manife&longs;tè po&longs;&longs;itar­tificium) vtrimque apertus, & va&longs;i D optimè coarctatus a­pud K. Fiat deinde vas C, æ­qualis aut maioris capacitatis, quàm vas D; in&longs;eraturque v­trique canalis PQ vtrimque a­pertus immediatè &longs;ub operculis &longs;eu tectis va&longs;orum. Tandem ex va&longs;e C de&longs;cendat perpendi­culariter canalis EF quinqua­ginta pedum longitudinis, ha­bens epi&longs;tomium E, & &longs;it opti­mè adferruminatus fundo va&longs;is C. His factis, impleantur va&longs;a AB, & C, aquâ, admoveaturque vas AB tubo GK, ita vt o­&longs;culum G &longs;it aquæ immer&longs;um, & aperiatur epi&longs;tomium E; de­&longs;cendet aqua ex va&longs;e C, &longs;ecumque trahet aërem ex tubis QP, & KG, & ex va&longs;e D, &longs;imulque aquam ex va&longs;e AB. Po&longs;tquam a&longs;cenderit aqua intra canalem GK v&longs;que ad aliquod &longs;patium exiguum, v.g. v&longs;que ad H; remove aliquantulùm vas AB; & aqua canalis a&longs;cendet vlteriús, tracta &longs;cilicet ab aqua effluente per FE) & po&longs;t aquam &longs;equetur aër. Admove deinde iterum vas AB, & attrahetur alia aqua, eritque aër HI inter aquam GH, & IM. Iterum remove vas AB, & ingredietur per os G alius aër, &longs;equeturque aquam. Quo facto, &longs;i vas iterum admôris, di­ctamque operationem repetive­ris; a&longs;cendent tandem aqua & aër mi&longs;ti inter &longs;e, v&longs;que ad vas D, illudque replebunt, antequam ef­fluat tota aqua va&longs;is C, quod ob id minus e&longs;&longs;e debet quam vas D.

Experimen tum Porta ad aquam elevandam vltra perpen­diculi alti­tudinem.

Procedendum e&longs;t autem valde cautè in hoc negotio, nè nimium aéris ingrediatur per os G, & levitate &longs;ua &longs;ur&longs;um ten­dens pellat modicam aquam antea in tromiffam, repleatque totum canalem GK, &longs;icque de­&longs;truatur continuitas aquæ & aëris intra canalem: hoc enim &longs;i fieret, non repleretur vas D, ni&longs;i po&longs;t longi&longs;&longs;imum tempus, & multas operationes, deberetque &longs;æpius repleri vas C. Nun­quam præterea tantum aquæ &longs;imul continere debet canalis GK, vt ejus perpendiculum &longs;uperet perpendiculum aquæ ca­n alis EF. Ex hac praxi deducit Porta modum mirabilem, & omnibus antiquis ignotum, ut ip&longs;e ait, elevandi per &longs;iphonem inver&longs;um aquam è montis radice ad ejus verticem; quem mo­dum afferam infrà Protheoria IV. cap. 2. Propo&longs;it. 12. ex quo etiam loco meliùs intelligetur hoc præ&longs;ens Experimentum.

Protheoria II.

PROTHEORIA SECVNDA.

De vi Expul&longs;iva propter corporum impe­netrabilitatem; &longs;eu de &longs;ecundo Machinarum Hy­dropnevmaticarum principio.

§. I.

Omnia Vniver&longs;i corpora &longs;unt inter &longs;e contigua.

COrpora omnia, quæ DEVS Optimus Maximus in prima rerum conditione intra præ&longs;titutum in Vni­ver&longs;i ambitu conclu&longs;it locum, Cœlum dico, Elemen­ta, & Mixta, tali &longs;unt ordine di&longs;po&longs;ita, ut multi, nec vulgares Philo&longs;ophi, exi&longs;timent, motum localem in Mundo non futurum ullum, ni&longs;i aut concedatur di&longs;per&longs;um per omnia vacuum (quale vidimus invehere Heronem, alio&longs;que nonnullos) aut corporum mutua penetratio, aut rerum omnium commotio, atque permixtio. Audi Epicurum apud Diogenem Laêrtium lib. 10. in Epi&longs;tola ad Herodotum, ubi ait. *ei) de\ mh\ h)_n o( kai\ xeno\s, kai\ xw/an, kai\ a)naf fu/sin o)noma/comen, x a)/n ei)_xe ta\ sw/mata o(/pn de\ di) exinei_to, xaa/pe fa/inetw xinmena Ni&longs;i verò id e&longs;&longs;et quoque, quod & inane, &longs;eu vacuum, & regionem, &longs;eu &longs;patium, ac locum & intacti­lem &longs;ive in corpoream naturam vocamus; non haberent corpora ne­que ubi e&longs;&longs;ent, neque quà motus &longs;uos obirent; cùm moveri ea quidem manife&longs;tum &longs;it.

Corpora o­mnia Vni­ver&longs;i conti­gua inter &longs;e.

Hanc autem, quam petitam ex motu rationem in&longs;inuat &longs;olummodò Epicurus, deducunt manife&longs;tiùs Democritus atque Leucippus (quos &longs;ecutus e&longs;t Epicurus) apud Ari&longs;totelem lib. 4. Phy&longs;. tex. 6. (ubi etiam alteram de corporum penetratione ur­gent) his verbis ga\ a)\n doxo/ih ei)/nai ei) m xeno\n. to\ ga/ ples a)du/na­ton ei)/nai de/ca, &c. Non videtur quippe e&longs;&longs;e motus, ni&longs;i &longs;it inane. Quod enim plenum e&longs;t, admittere in &longs;e mobile non pote&longs;t. Nam &longs;i ad- mittat, & fuerint in eodem loco duo corpora; nihil repugnabit, quò mi­nùs quotquot libuerit, &longs;imul &longs;int; quia dici, cur id non valeat evenire, minimè pote&longs;t. Sin autem eveniat, tunc quod minimum, id admittet quod maximum; quando ip&longs;um magnum multa parva e&longs;t: adeo ut &longs;i multa æqualia in eodem e&longs;&longs;e contigerit, e&longs;&longs;e & in æqualia contingat. Ean­dem con&longs;ecutionem deducit Lucretius in &longs;ua ver&longs;ibus ligata Phi­ lo&longs;ophia; additque non &longs;olùm, &longs;i Inane &longs;eu Vacuum non admit­tatur, rem nullam exijs quæ &longs;unt, moveri, &longs;ed nec ullam denuò na&longs;ci po&longs;&longs;e in mundo.

Argumen­tum Epicuri Leucippi, ac Democriti pro vacuo.

Corporum penetratio non pote&longs;t fieri natu­raliter.

Vacuum non admittit Lucretius.

Quæ &longs;i non e&longs;&longs;et Inane,

Non tam &longs;olicito motu privata careret, Quàm genita omninò nulla ra­tione fui&longs;&longs;et, vndique materies quoniam &longs;tipata fui&longs;&longs;et. Cogita enim vniver&longs;um mundum (ait vir docti&longs;&longs;imus Petrus Ga&longs;&longs;endus in &longs;uis de Phy&longs;iologia Epicuri Animad ver&longs;ionibus, ex Epicuri men­te,) &longs;i nihil Inanis inter&longs;per&longs;um habeat, conferti&longs;&longs;imam e&longs;&longs;e mo­lem, adeo ut nè minimum quidem corpu&longs;culum valeat de novo &longs;u&longs;cipere (quod quidem a&longs;&longs;erendum nece&longs;&longs;ariò e&longs;t, &longs;i nullum in rebus Inane, &longs;i nihil non plenum, &longs;i locus nullus re&longs;tat complen­dus.) Quare aut corpus non admittetur denuò genitum, aut in illo loco collocabitur, in quo aliud jam &longs;itum e&longs;t; &longs;icque idem locus duo corpora &longs;e&longs;e undique penetrantia capiet. Quod &longs;anè per vires Naturæ fieri po&longs;&longs;e non dixeris, ut ut alij id a&longs;&longs;erant, in­ducti exemplo panis, &longs;pongiæque &longs;iccæ aquam, cui injiciuntur, in&longs;ugentis; & cinerum tantundem aquæ in con&longs;ortium &longs;uum ad­mittentium in va&longs;e, quantum &longs;ine cineribus vas recipere pote&longs;t.

Argumen­tum Epicu reorum ac Democriti torum pro vacuo.

Si ab&longs;onum videtur vacuum admittere, &longs;i a)du/naton corpo­rum dari penetrationem, & tamen motum admittimus localem; &longs;u&longs;que deque verti corpora omnia, quæ Univer&longs;o concluduntur, fateamur nece&longs;&longs;e e&longs;t toties, quoties eorum unum movebitur, di­cet Democriticus qui&longs;piam, aut Epicureus: moveatur enim quodcunque eorum, quæ &longs;unt, de &longs;uo quem occupat loco, alte­riu&longs;que in vadere tentet &longs;tationem; &longs;anè cùm locum plenum offendat, nece&longs;&longs;e erit, ex eo pellat, quod corpus illum occupat. Illud porrò quónam concedat, &longs;i omnia quidem plena &longs;unt? An non ip&longs;um rur&longs;us expellet aliud? At de eo par redibit difficultas: hoc enim, aliud expellere loco debebit; & illud aliud; &longs;icque con­tinuari nece&longs;&longs;e erit inchoatam &longs;omel corporum commotionem in ævum, & totum commoveri Vniver&longs;um vel ad mu&longs;cæ vola­tum. Si enim vel unum corpus cedere loco non valeat, aut non volet, omnis ce&longs;&longs;abit illicò motus, eritque unaquæque res mun­di &longs;ic in&longs;erta in aliam, ut magis è loco excedere nequeat, quàm mytuli (inquit Ga&longs;&longs;endus paulò antè citatus) &longs;eu cochleæ illæ, quæ intra &longs;ilices na&longs;cuntur. Neque dicas, e&longs;&longs;e in mundo rara, liquida, fluxaque corpora, quæ cedere magis quàm &longs;ilices po&longs;­&longs;int. Sint enim quantum vis rara, fluxa, & cedere nata; &longs;i admi­&longs;tum Inane non habent, &longs;i particulæ omnes illorum contiguita­tem undique, &longs;eu &longs;ecundùm totas &longs;uperficies habent; tale con­tinuum corpus facient, ut perinde &longs;it &longs;eu ex &longs;ilice, &longs;eu ex aqua, &longs;eu ex aëre illud dicas: neque enim aër vel paucioribus, vel minùs compactis &longs;ui partibus occupabit locum minorem, quàm &longs;ilex, cùm quotcunque loci particulas feceris, tam nulla earum &longs;it aë­ris, quàm &longs;ilicis particulæ expers. Hactenus Democritici, & Epicurei.

§. II.

Corpora &longs;e mutuò expellunt.

HIs quid reponendum, viderint Philo&longs;ophi. Ari&longs;toteles certe,

qui nec vacuum admittit, nec corporum penetrationem mutuam, nequaquam timet rerum omnium ad quemlibet mo­tum commotionem; quoniam liquida non &longs;olùm pelluntur, &longs;ed dividuntur etiam dum urgentur; & quæ antror&longs;um vergebant partes, latera lambentes claudunt terga. At quidquid &longs;it, tam evidens e&longs;t, corpus unum expellere alterum è &longs;uo, quem occu­pat, loco, ni &longs;ponte excedere velit, quàm e&longs;t manife&longs;tum, mo­ tum dari localem; idque verum e&longs;t non tantùm in duris &longs;oli­di&longs;vé, &longs;ed liquidis etiam ac fluidis corporibus, aëre dico, & aqua. videmus enim quotidie, aquam va&longs;i, tuboque aëre pleno illa­bentem, expellere aërem, &longs;i exeundi locum habet; & &longs;i non ha­bet, neque aquam po&longs;&longs;e, aut velle infundi. Videmus item aë- rem inflatum va&longs;i per os unum, expellere aquam per alterum, aut etiam per idem, &longs;i elabendi detur locus; qui &longs;i non datur, nec aë­ri patere ingre&longs;&longs;um. Videmus follibus compre&longs;&longs;is erumpere aërem, & aquam etiam, &longs;i aquâ erant repleti. Videmus, &longs;i va&longs;i aquâ pleno operculum in medio perforatum imponamus, de­primamu&longs;que, aquam pre&longs;&longs;am, &longs;i inter latera va&longs;is & opercu­lum erumpere non pote&longs;t, ex&longs;ilire per foramen magno impetu. Taceo centena exempla alia. Atque hæc mutua corporum ex­pul&longs;io alterum e&longs;t Machinarum Hydro-pnevmaticarum prin­cipium, quo innumera, & non minùs iucunda, quàm admiran­da exhibentur &longs;pectacula pa&longs;&longs;im, vt apparebit aperti&longs;&longs;imè ex &longs;e­quentibus Machinis. Interim&longs;equens accipe Experimentum.

Corpora ex­pellunt &longs;e mutuò.

Experimen­ta varia, quibus pro­batur cor­pora&longs;e mu­mutuò ex­pellere.

§. III.

Experimentum, quo o&longs;tenditur vis expul&longs;iva, propter corporum impenetrabilitatem, ad aquas elevan­das in altum.

FAcex&longs;tanno, cu­pro, creta, ligno, aut alia quacunque materia aquis re&longs;i­&longs;tente tria va&longs;cula A, F, & G, columnis inter&longs;e di&longs;tincta, vt mon&longs;trat figura; aut alia ratione, prout placuerit, aut com­modiùs fuerit: &longs;int­que hæc va&longs;cula vn­dique clau&longs;a quàm diligenti&longs;&longs;ime, nè aër po&longs;&longs;it aut ingre­di, aut elabi, &longs;altem ex A & G. Vas infe­rius G habeat epi­&longs;tomium I. per quod aqua effluere po&longs;&longs;it. Va&longs;a A & F habeant in operculis forami­na C & E, vt per ea infundi po&longs;&longs;it aqua, & iterum obturari. Exva&longs;e A egrediatur tubulus AB vtrimque, apertus qui apud A fundum va&longs;is non attingat, &longs;ed tantum ab illo di&longs;tet, vt aqua interfluere po&longs;&longs;it: apud B verò habeat o&longs;culum &longs;tricti&longs;&longs;i­mum. Habeat præterea epi&longs;tomium K. Ex eodem va&longs;e A de­&longs;cendat intra vas inferius G alius tubulus DN, vtrimque aper­tus, qui apud D non attingat operculum va&longs;is A, &longs;ed tantum ab illo di&longs;tet, vt po&longs;&longs;it egredi aër: apud H verò tran&longs;eat &longs;olum operculum va&longs;is G, & non extendatur vlteriùs.| Ex va&longs;e deni­que F de&longs;cendatintra vas G alius tubus FG vtrimque aper­tus, qui apud F tran&longs;eat &longs;olùm fundum va&longs;is, & non progredia­tur vlteriùs; apud G verò di&longs;tet tantum à fundo, vt aqua ef­fluere po&longs;&longs;it, habeatque vas G epi&longs;tominm L. Hi tres tubi debentita &longs;tanno alia vè materia coarctari, & adferruminari va­&longs;is in loco tran&longs;itus, vt nullus aër penetrare intra va&longs;a, aut elabi ex ijsdem po&longs;&longs;it. Cavendum præterea diligenter e&longs;t, nè tubu­lus AB &longs;it longior, aut æquè longus, ac tubulus FG; quare meliùs erit, &longs;i inter B & K decurtetur quantum pote&longs;t, & quàm minimum extra vas A protendatur.

His ita ritè præparatis, repleantur aqua duo va&longs;a A & F, per foramina C & E, & obturentur quàm diligenti&longs;&longs;imè, maximè foramen C (foramen enim E pote&longs;t manere apertum;) vas verò G maneatvacuum; & omnia epi&longs;tomia &longs;int clau&longs;a. De­inde aperiatur epi&longs;tomium L, vt aqua va&longs;is F defluere po&longs;&longs;it intra vas G: & po&longs;tquam defluxit aliquantulum aquæ, aperia­tur epi&longs;tomium K; erumpetque maximo impetu aqua va­&longs;is A per tubulum AB, pro&longs;ilietque in altum, vi expul&longs;iva, pro­pter corporum impetrabilitatem. Nàm aqua va&longs;is F de&longs;cen­dens intra vas G, expellit inde aërem per tubum HD intra vas A; qui aer expellit inde aquam per tubulum AB, quoniam ne­que in G, neque in A, aër & aqua &longs;imul in eodem loco manere po&longs;&longs;unt, &longs;ed nece&longs;&longs;ariò vnum corpus expellit alterum, propter eorum impenetrabilitatem. Vides igitur quomodo vis expul­&longs;iva propter corporum impenetrabilitatem &longs;it principium & cau&longs;a Machinarum Hydro-pnevmaticarum? Vbi de&longs;cenderit aqua va&longs;is F intra vas G, & ex&longs;iliuerit aqua va&longs;is A per tubum AB; depromi poterit aqua va&longs;is G per epi&longs;tomium I, & reple­ri iterum va&longs;a A & F, vt antea, & in&longs;titui idem lu&longs;us.

§. IV.

Notantur nonnulla circa prædictum Experimentum.

Fundame­tum omni­um Machi­narum Hy­dro-, nev­maticarum quæ fiunt vi expul&longs;iva.

HOc experimentum e&longs;t fundamentum omnium Hydro­pnevmaticarum Machinarum quæ fiunt vi expul&longs;iva ob cor­ porum impenetra­bilitatem; ideoque Tyrones diligenter procurare debent, vt illud intelligant: quare nonnulla cir­ca ipsum &longs;unt adver­tenda.

Primum e&longs;t, ex­pedire vt vas A non &longs;it minùs capax quam vas F, & vt tubus FG non &longs;it amplior tubo AB: &longs;ic enim fiet, vt quàmdiu a­qua ex va&longs;e F de&longs;cen­dit intra G, tamdiu &longs;aliat aqua ex vase A per tubum AB. Hoc tamen non e&longs;t omni­nò nece&longs;&longs;arium.

Secundum e&longs;t, Non e&longs;&longs;e nece&longs;&longs;arium, vt fiat vas F, &longs;ed fie­ri pote&longs;t&longs;olus canalis &longs;eu tubus FG, qui apud F habeat infun­dibulum, vt aqua infundi po&longs;&longs;it; dummodò &longs;it longior quàm tubus AB; vt mox dicam.

Tertium e&longs;t, omninò nece&longs;&longs;arium e&longs;&longs;e, vt tubus FG longi­or &longs;it quam tubus AB, quia, vt diximus Protheoria I. §. VI. & di­cemus iterum infrà Protheoria IV. cap. 2. Propo&longs;it. 14. hîc et­iam, vt vbiq, in hydraulicis, perpendiculis certatur, non verò a­quarum copiâ. Perpendiculun|autem tubi FG computatur ab F v&longs;que ad

&longs;upremam &longs;uperficiem aquæ de&longs;cendentis & quie&longs;centis in va&longs;e G. Quò verò longius fuerit perpendiculum FG &longs;upra perpendicu­lum AB eò altiùs &longs;alit aqua ex o&longs;culo B. Quare cùm in prin­cipio, quando incipit de&longs;cendere aqua intra vas G, longius &longs;it perpendiculum FG, quàm po&longs;tea (repleto paulatim va&longs;e G) & id &longs;emper magis ac magis minuatur; con&longs;equens e&longs;t, vtin prin­cipio altiùs &longs;aliat aqua ex B, quàm po&longs;t principium, & vt &longs;em­per minùs & minùs altè ex&longs;iliat. Vide etiam quæ dicimus in­frà Parte 2. Cla&longs;se I. cap. 2. Machina I. Annot. 2.

Perpendi­culis, non a­qudrum co­piâ certatur in hydrau­licis.

Quartum e&longs;t, Non &longs;olùm &longs;ervandam e&longs;&longs;e proportionem quoad longitudinem inter tubos AB, & GF, &longs;ed etiam quoad capacitatem, nè &longs;cilicet AB laxior &longs;it quàm GF; alioquin gra­vior erit aqua intra ip&longs;um, quàm vt elevari po&longs;&longs;it ab aëre modi­co intra vas A pul&longs;o à modica aqua cadente intra vas G. Si verò FG tubus laxior e&longs;&longs;e quàm AB, nimis citò repleretur vas G, & antequam tota aqua va&longs;is A expelleretur per AB.

Quintum e&longs;t, Nihil referre quantæ &longs;it altitudinis tubus HD, per quem aër defertur, quia nihil confert ad perpendiculum. Pro­curandum tamen e&longs;t, nè &longs;it nimis laxus, aliàs multùm aëris intra ip&longs;um ab&longs;orberetur, qui alioquin expelli deberet intra vas A.

Sextum e&longs;t, prædictum artificium elevandi aquam vi ex­pul&longs;iva propter corporum impenetra bilitatem, e&longs;&longs;e typum & exemplar omnium Machinarum Hydro-pnevmaticarum præ­dicta virtute animatarum; ac proinde &longs;emperante oculos in &longs;imilibus Machinis fabricandis e&longs;&longs;e habendum, vtin principio dicebam.

§ V.

Heronis error in elevando oleo vi Expul&longs;ivâ in lucerna.

HEro Alexandrinus in libello de Spiritalibus cap. 72. proponit hanc Machinam. Lucernæ con&longs;tructio, vt ellychnio impo­&longs;ito, quando oleum deficiat, ex aure eius oleum in ellychnium infun­datur, quantum quis voluerit, nullo ip&longs;i appo&longs;ito va&longs;e, ex quo oleum in­ fluat. Qua quidem in con&longs;tructione errorem committit gravi&longs;­&longs;imum (&longs;i tamen ip&longs;ius e&longs;t error, & non potiùs Interpretis, qui figuras appo&longs;uit) quem hîc detegere oportet, ne alijs etiam &longs;it errandi occa&longs;io. Quod eò et­iam libentiùs facio, quòd nec Federicus Commandinus, qui ex Græco in latinum tran&longs;tulit Heronem; nec Ioannes Bapti­&longs;ta Aleottus, qui Commandini interpretationem red didit Itali­cam; nec Ioannes Bapti&longs;ta Porta, qui Heronis errorem etiam ob­&longs;ervavit, genuinam lucernæ de­lineationem dederunt. Eam verò hîc dare conabimur, allatis Heronis verbis ex Commandini translatione, addendo nonnulla verba majoris claritatis gratia, quæ a&longs;teri&longs;cis in margine no­tantur. Sic ergo di&longs;currit Hero. Con&longs;truatur lucerna, ba&longs;im habens concavam, & triangularem in&longs;tar pyramidis; &longs;it que ba&longs;is concava AB CD, & in ea diaphragma EF. Lu­cernæ autem caulis &longs;it GH, & ip&longs;e concavus; &longs;upra quem &longs;it concavus calathus KL, plurimum olei conti­nens. Ex diaphragmate EF pro- cedat MN tubus, &longs;imul cum eo perforatus, tantum di&longs;tans à calathi operculo KL (in quo ellychnium imponitur) quantum ad aër is exitum &longs;ufficiat. Alius autem tubulus XO demittatur per opercu­lm KL di&longs;tans à fundo calathi, quantum &longs;atis &longs;it ad aquæ fluxum & ex operculo paululum excedens. Exce&longs;&longs;ui verò aptetur alius tubulusP habens &longs;uperius o&longs;culum ob&longs;tructum; qui tubulus per fundum lychnij impul&longs;us coagmentetur lychno. At tubum autem P adglutine­tur alius tubulus exilis, procedens ad extremitatem auris, & &longs;imul com ea perforetur, vt in fluat in concavitatem ellychnij, foramen habens quemadmodum & alij. Et&longs;ub diaphragmate EF conglutinetur clavi­cula deferens in locum CDEF, ita, vt &longs;i aperiatur, aqua ex locoABEF, in ip&longs;um CDEF tran&longs;eat. Sit autem in operculo AB par­rum foramen, per quod locum ABEF implebimus aquâ& qui in ip&longs;o e&longs;t aër, per dictum foramen excedet. Itaque &longs;ublato ellych­nio, calathum oleo imple bimus per tubum XO, aëre per NM tubulum excedente, & adhuc per clavem apertam, quæ e&longs;t in fundo C.D, quando & quæ e&longs;t in CDEF aqua effluet. Impo&longs;ito igitur el­lychnio per &longs;meri&longs;ma P, quando opus &longs;it oleum infundere, aperiemus clavem, quæ in fundo CD; & aquâ ex loco ABEF in locumCDEF &longs;ecedente, aėr qui e&longs;t in ip&longs;o, per tubum MN ad cala­thum perveniens elidet oleum; quod quidem per tubulum XO, & per alium ip&longs;i cohærentem ad ellychnium procedet. Quando au­tem non amplùs fluere volumus, clavicula claudetur, & ce&longs;&longs;abit; & rur&longs;us quando opus &longs;it, idem faciemus.

Error He­ronis in lu­cernæ con­&longs;tructione.

Lucernæ Heronis ge­auina figu­ra.

Hæc Commandinus Heronis Interpres; quæ ego omni­ no mendo&longs;a e&longs;&longs;e exi&longs;timo: puto enim vbique lychnum pro el­lychnio ponendum e&longs;&longs;e, & alia pleraque legenda prout in mar­gine &longs;unt po&longs;ita a&longs;teri&longs;cis notata. Non potui in venire Græcum Heronis textum, vt vt diù quæ&longs;iverim. Tubuli etiam &longs;upra ca­lathi operculum di&longs;ponendi &longs;unt &longs;ine dubio, vt nos di&longs;po&longs;uimus, quoniam Heronis textus latinus apud ip&longs;umnet Commandinum id exigit. Nec mireris Lector, &longs;i &longs;u&longs;picer Commandini præ&longs;entem Latinum de lucerna textum non e&longs;&longs;e vndequaque corre&longs;pon­dentem græco textui; po&longs;thumum enim e&longs;t Opus Commandini, nec po&longs;tremam ip&longs;ius manum expertum, vt habetur initio libelli in præfatiuncula ad Lectorem. E&longs;t autem in Commandini te­xtu ac &longs;chemate in&longs;ignis error in eo, quòd clavicula R trans­mittit aquam quæ e&longs;t in ABEF intra ba&longs;im EFCD per aper­tum &longs;olùm foramen &longs;ine tubulo, qui paulò longior &longs;it quàm tu­bulus OX; quod tamen omnino requiritur, vt diximus paulò antè §. IV. & cau&longs;am dabimusintrà Protheoria IV. cap. 2. Pro­po&longs;it. 14. experientia &longs;iquidem con&longs;tat, cylindrum aquæ per fo­ramen de&longs;cendentem (&longs;i quandoque de&longs;cendit aëre &longs;uccedente) in aêre libero non habere eandem vim, ac de&longs;cendentem per tubum; tamet&longs;i vtrobique &longs;it eadem aquæ quantitas, cum eodem perpendiculo. Cau&longs;a e&longs;t, quia &longs;i liberè & &longs;ine tubo defluit a queus cylindrus, &longs;ignum e&longs;t aërem &longs;ubintrare; &longs;i aër &longs;ubintrat, ce&longs;&longs;at me­tus vacui, ac proinde aqua debet elevari vt laboranti Naturæ &longs;uc­currat. Quòd &longs;i tam &longs;trictum e&longs;t foramen, vt aër &longs;ubintrare non po&longs;&longs;it, nè gutta quidem aquæ effluet, vt experientia quotidiana docet, idque propter vacui metum. Hunc tamen errorem ego nequaquam Heroni ad&longs;cribendum putem, vt Porta facit, &longs;ed exi&longs;timo Heronem produxi&longs;&longs;e tubulum R ferè v&longs;que fundum CD; &longs;ic enim major erit quàm tubulus XO, & aqua per ip&longs;am R decideus habebit perpendiculum præponderans perpendi­culo aquæ &longs;eu olei a&longs;cendentis per XO.

Heronis lo­cus apud Comman­dinum men do&longs;.

Heronis er­ror excu&longs;a­tur.

Similis error reperitur apud Heronem cap. 36. vbi docet &longs;atyri&longs;cum con&longs;truere &longs;uper quadam ba&longs;i, vtrem in manibus te­nentem, cui adiacet labellum; vt infu&longs;o in labellum humore, ita vt repleatur, aqua per vtrem in idem labellum in fluat, & non &longs;u­perfundatur, quòusque tota per vtrem aqua evacuata fuerit. Ex qua quidem con&longs;tructione de&longs;umpta e&longs;t praxis con&longs;truendi fon­tem Heronis, de quo infrà Parte 2. Cla&longs;se I. cap. 2. Machi­na 1. 2. & 3. vbi errorem Heronis detegemus; &longs;i tamen Heronis e&longs;t, & non In­terpretis.

§. VI.

Attractione & expul&longs;ione &longs;imul aquam ele­vare po&longs;&longs;umus.

DIximus §. IV præcedente, perpen diculum aquæ cadentis & pellentis aërem, longius e&longs;&longs;e debere perpendiculo aquæ a&longs;cendentis per expul&longs;ionem. Diximus præterea Protheoria I. §. VI. perpendiculum aquæ cadentis & trahentis aërem, debe­re &longs;uperare longitudine perpendiculum aquæ attractæ. Infrâ Parte 2. Cla&longs;se I. cap. 1. Machina 7. & cap. 2. Machina 10. & 11. di­cemus, qua ratione per multiplicationem plurium perpem dicu­lorum brevium elevari po&longs;&longs;it aqua, tam per expul&longs;ionem, quàm per attractionem, ad quam vis altitudinem. Nunc &longs;ubjiciam modum attollendi aquam per attractionem & expul&longs;ionem &longs;i­ fimul ad duplam altitudinem aquæ cadentis, quoniam ingenio­fus e&longs;t, & ad multa pote&longs;t e&longs;&longs;e vtilis. Refert illum Porta lib. 2. Spiritalium cap. 2. eumque magnificè extollit, tanquam à &longs;e in­ventum; & ait &longs;uperare omnem humanum intellectum, nec vn­quam in mentem veni&longs;&longs;e antiquis, &longs;e verò po&longs;t expen&longs;as multas, & labores plurimos illum tandem reperi&longs;&longs;e. Modus hic e&longs;t.

Modus in­genio&longs;us e­levandi aquam in quam vis al titudinem per attracti onem & ex­pul&longs;ionem, &longs;imul.

Sit ex va&longs;e B elevanda aqua v&longs;que ad vas E, ad altitudinem centum pe­dum, mediante perpen­diculo 50. pedum aquæ cadentis. Fiant alia duo va&longs;a A, & C, in eodem plano horizontali cum va&longs;e B con&longs;tituta, eju&longs;dem capacitatis cum B; & in­fra ip&longs;a con&longs;tituatur vas D, cujus &longs;uprema oper­culi pars di&longs;tet à fundis di­ctorum va&longs;orum 50. pedi­bus. Ex va&longs;e A de&longs;cen­dat in vas D tubus GX, 50. pedum; qui tamen ab X ex­tendi ac prolongari pote&longs;t u&longs;que ad H, dummodò tantum di&longs;tet à fundo, ut aqua effluere po&longs;&longs;it. Ha­beat autem hic tubus epi­&longs;tomium V. Ex va&longs;e B de­&longs;cendat in idem vas D ali­us tubus KI, incipiens paulò infra operculum apud K, & de&longs;inens pau­lò infra operculum apud I. Ex va&longs;e C de&longs;cendat tubus OP 50. pedum, habens epi&longs;tomium T. Ex va&longs;e E de&longs;cendant in va­&longs;a B & C duo tubi: pri­mus NC, incipiens infra operculum apud N, & de&longs;inens in­fra operculum apud C: alter ML, incipiens immediatè infra operculum apud M, & de&longs;inens &longs;upra fundum apud L. Lon­gitudo verò hujus tubi ML ab operculo M u&longs;que ad punctum L &longs;it centum pedum. Tria va&longs;a A, B, C, habeant &longs;ua forami­na cum infundibulis F, per quæ aqua infundi po&longs;&longs;it intra va&longs;a, & obturari. Vas tamen A pote&longs;t manere apertum. Duo va&longs;a D & E habeant &longs;ua epi&longs;tomia H & S. Omnia præterea va­&longs;a, excepto A, debent e&longs;&longs;e undique ac diligenti&longs;&longs;imè clau&longs;a; ideoque tubi debent in loco tran&longs;itus accurati&longs;&longs;imè adferrumi­nari ip&longs;orum va&longs;orum operculis, & fundis. His ita con&longs;titutis, repleantur tria va&longs;a A, B, C, aquâ, & claudantur, clau&longs;is priùs omnibus epi&longs;tomijs omnium va&longs;orum ac tuborum: reliqua ve­rò duo va&longs;a, D & E, maneant vacua. Deinde aperiatur epi&longs;tomium T; & aqua va&longs;is C de&longs;cendet ver&longs;us P, &longs;ecum­que trahet, ad vacuum vitandum, aërem va&longs;is E, per tubum NC, &longs;imulque aquam va&longs;is B u&longs;que ad altitudinem 50. pedum ubi V, quantum nimirum e&longs;t perpendiculum tubl OP, per quem aqua cadit. Po&longs;tquam verò aqua de&longs;cenderit hinc u&longs;que ad P, inde u&longs;que ad V, hærebit utrimque velut in æquilibrio immobilis, propter perpendiculorum æqualitatem. Tum verò aperiatur epi&longs;tomium V, & aqua va&longs;is A de&longs;cendet intra vas D, indeque expellet aërem per tubum IK intra vas B; qui aër premet aquam, atque propellet &longs;ur&longs;um per tubum LM ver&longs;us M, &longs;icque elevabitur aqua ultra punctum V, effluetque aqua ex tubo OP, penitu&longs;que a&longs;cendet aqua va&longs;is B, per tubum LM, intra vas E; nec ce&longs;&longs;abit fluxus, donec tria va&longs;a A, B, C, &longs;imul fuerint evacuata, totaque aqua B a&longs;cenderitin E. Sijam aqua va&longs;orum D & E per &longs;ua epi&longs;tomia depromatur, & va&longs;a A, B, C, denuò repleantur; in&longs;titui poterit eadem circulatio, idque to­ties, quoties libuerit. Hæc e&longs;t praxis Portæ; ingenio&longs;us atque indu&longs;trius artifex addere, demere, mutare poterit, quæ libuerit, ornatûs aut commoditatis gratia, modò non alteretur rel &longs;ub­&longs;tantia, quam nos præcisè hîc dedimus.

Nota hîc, ut aperto epi&longs;tomio T de&longs;cendat aqua va&longs;is C ver&longs;us P, laxandum &longs;imul e&longs;&longs;e epi&longs;tomium V va&longs;is A; alio­quin nunquam de&longs;cendet aqua va&longs;is C, nè detur vacuum, ut con&longs;ideranti patebit.

Nota præterea, quod diximus de aqua a&longs;cendente ad alti­tudinem 50. pedum, tantùm gratia exempli fui&longs;&longs;e dictum: nam aqua non pote&longs;t per attractionem|elevari ad tantam altitudinem, ut alibi fu&longs;iùs dicemus.

§. VII.

Salomon Caus meritò &longs;ugillat Heronem.

HEro cap. 55. Spiritalium ait, quædam va&longs;a à principio humore injecto fluunt, intermi&longs;&longs;ione verò facta non ampliùs fluunt injecto humore, quou&longs;&queacute; ad dimidium repleta fuerint; tunc enim incipiunt flue­re: At intermi&longs;&longs;ione facta, non ampliùs fluunt, quou&longs;&que; impleantur.Hujus rei probandæ gratia proponit huju&longs;modi machinam. Sit enim vas AB, habens in &longs;eip&longs;tres &longs;iphones inflexos C, D, E, in ven­tre occultatos, quorum altera quidem crura &longs;int ad fundum va&longs;is, altera ve­rò extra ferantur in canales efforma­ta; atque eorum extremis apponantur va&longs;a F, H, G, quorum fundum di&longs;tet ab o&longs;culis, quantum ad aquæ fluxum &longs;a­tis &longs;it. Omnia verò comprehendantur alio quodam va&longs;e, tanquam ba&longs;i KLMN, quæ canalem X habeat. Et &longs;iphonis quidem C curvit as &longs;it ad fundum va­&longs;is; &longs;iphonis verò D curvitas ad di­midium altitudinis ejus & &longs;iphonis E ad ip&longs;um collum. Si igitur in vas AB aquam infundamus, à principio quidem fluet per &longs;iphonem C, quoniam cur­vitas ip&longs;ius e&longs;t ad fundum. Quòd &longs;i intermittamus, evacuabitur humidum infu&longs;um per canalem X, & vas F relin quetur aqua plenum, &longs;iphonis verò C reliqua pars aëre plena erit. Rur&longs;um cùm aquam infunda­mus, non procedet per &longs;iphonem C, propterea quòd aër in eo contine­tur inter aquam infu&longs;am, & eam quæ e&longs;t in va&longs;e F. A&longs;cendet ergo humidum u&longs;que ad &longs;iphonis D curvitatem, quæ e&longs;t ad dimidium alti­tudinis va&longs;is; & tunc fluere incipiet: &longs;ed inter mi&longs;&longs;ione facta, rur&longs;us idem continget quod in &longs;iphone C jam dictum e&longs;t. Eadem & in &longs;i­phone E intelligantur. Oportebit autem humidum &longs;en&longs;im infunde­re, nè aër, qui in &longs;iphonibus comprehen&longs;us e&longs;t, &longs;umma vi expellatur.Hæc Hero; Quæ tamen, & quidem optimè, atque ex Hydro­pnevmaticæ artis principijs, Salomon Caus in &longs;uis hydraulicis, &longs;eu lib. 1. de viribus motricibus, reprehendit, dicens effluere aquam per &longs;iphonem C, etiam po&longs;t &longs;ecundam infu&longs;ionem in­tra vas, eò quòd aër contentus intra &longs;iphonem C pellatur ab aqua extra inferius o&longs;culum, erumpatque extra aquam va&longs;is F Quod quidem veri&longs;&longs;imum e&longs;t, cùm aqua infu&longs;a intra vas AB per &longs;e­cundam infu&longs;ionem, facillimè a&longs;cendat per crus internum u&longs;que ad curvitatem C, non re&longs;i&longs;tente aëre; qui aër intra &longs;iphonem CF compre&longs;­&longs;us cedit aquæ a&longs;cendenti primùm u&longs;q, ad curvitatem C, indeque de&longs;cendenti pondere &longs;uo, pul&longs;o aëre, erumpente per aquam va­&longs;is F. Adde quòd &longs;i tam altum e&longs;&longs;et va&longs;culum F, ut aër per ejus aquam exitum non inveniret; a&longs;cenderet is per &longs;iphonem aquæ mixtus, prout fieri videmus in canali vitreo quantumvis gracili atque longo: &longs;i enim aquâ aliquòusque eum repleas, & ob­tùrato utroque orificio invertas, ut aqua &longs;uperiorem, aër inferio­rem occupat locum; &longs;tatim de&longs;cendit aqua, & a&longs;cenditaër, aper­tâ &longs;ibi vel per medias aquas viâ, ut te&longs;tantur bullæ frequentes in­ter aquas erumpentes; & qui paulò antè &longs;uccumbebat aquæ, ei­dem paulò po&longs;t incumbit aër.

Heronis Ma china defe­ctuo&longs;a.

Heronis r­ror.

§. VIII.

Idem perpendiculum aquæ cadentis non pote&longs;t &longs;ervire &longs;imul attractioni & expul&longs;ioni.

POrta loco &longs;æpe citato de &longs;piritalibus Cap. VIII. ait, &longs;e fre­quenter tenta&longs;&longs;e ut aquam eodem perpendiculo mediante elevaret ad majorem altitudinem, quàm perpendiculum erat, partim attractione, partim expul&longs;ione. Eum in finem fecit vas A, ex quo elevare volebat aquam u&longs;que ad vas B, ad alti­tudinem pedum centum verbi gratia. Con&longs;truxit igitur aliud æquale vas C in eodem plano, & aliud DF demi&longs;&longs;ius utroque pedibus 50. va&longs;a con&longs;tructa in&longs;truxit tubis & epi&longs;tomijs, prout figura mon&longs;trat. Hi&longs;ce peractis, replevit va&longs;a C & A aquis, & diligenter clau&longs;it utrumque, clau&longs;is etiam eadem diligentia va&longs;is B & DE. Deinde aperuit tubum ED, ut de&longs;cendente aqua va&longs;is C intra vas DE, traheretur aër ex tubo CK, va&longs;e B, & tubo IH, &longs;imulque attraheretur aqua va&longs;is A &longs;altem ad L, altitu­dinem pedum 50; quod & factum fuit. Sperabat|autem ut aër va&longs;is DF pul&longs;us per tubum EG, intra vas A, expelleret inde aliam aquam, & &longs;ublatam jam antea aquam v&longs;que ad L, attolleret vlteriùs v&longs;que ad vas B; at nunquam id obtinere potuit: nam po&longs;t­quam a&longs;cenderat aqua v&longs;­que ad L, hæ&longs;it &longs;u&longs;pen&longs;a, & aqua va&longs;is C non deflu­xit ampliùs per tubum ED. Collegit inde, idem per­pendiculum ED non po&longs;­&longs;e in&longs;ervire attractioni & expul&longs;ioni &longs;imul, &longs;ed vel &longs;oli attractioni, vel &longs;oli ex­pul&longs;ioni; quod benè notan­dum e&longs;t. Cau&longs;am dabi­mus infrà Protheoria IV. capite 2. Propo&longs;it. 13. & 14. quam tamen ignora&longs;&longs;e Por­ta videtur, quoniam nul­lam ad&longs;ignat.

§. IX.

Error apud Ioannem Bapti&longs;tam Portam, indicatur.

APud Ioannem Bapti&longs;tam Portam lib 2. Spiritalium cap. 10. traditur modus elevandi aquam per expul&longs;ionem ad &longs;tellas v&longs;que, vt ibi dicitur, mediante &longs;olo vnico perpendiculo quin­quaginta verbi gratia pedum altitudinis: qui tamen e&longs;t omnino fal&longs;us, & repugnat doctrinæ Portæ in præcedentibus traditæ; Vnde &longs;u&longs;picabar ab Hi&longs;pano traductore fuiffe additum: crede­re enim nulla ratione poteram, à Porta tam enormem errorem potui&longs;&longs;e committi. Reperi tamen deinde eundem modum in editione latina cap. 9. Modus hic e&longs;t. E&longs;to vas C, & &longs;upra ip&longs;um ad di&longs;tantiam pedum verbi gratia quinquaginta, va&longs;a D & K, cum &longs;uis infundibulis; & &longs;upra hæc ali­ud G; aliudque &longs;upra hoc; & &longs;ic deinceps ad quamvis altitudinem, etiam v&longs;que ad &longs;tellas. Conjun­gantur dicta va&longs;a tubis, vt mon­&longs;trat figura, habeatque tubus DB &longs;uum epi&longs;tomium, vt & vas C &longs;u­um; & va&longs;a D & K repleantur a­quâ. Deinde aperiatur epi&longs;tomi­um tubi DB, & aqua de&longs;cendens intra vas C expellet inde aërem per tubum FK intra vas K; qui aër inde expellet aquam per tu­bum FG in vas G, & ex G per tubum HI expelletur aqua in aliud vas, & &longs;ic vlteriùs. Sed hæc pror­ &longs;us &longs;unt erronea, & contra princi­pia hydraulica. Perpendiculum enim DB aquæ cadentis pedum quinquaginta, non pote&longs;t ele­vare aliam præter aquam va&longs;is K per tubum FG pedum etiam 50. ad &longs;ummum, v&longs;que ad vas G, vbi quie&longs;cet aqua. Dato dein­de quòd ex G po&longs;&longs;et elevari aqua eodem perpendiculo DB, de­beret tubus IH pertingere ferè v&longs;que ad fundum va&longs;is G, vt aqua pre&longs;&longs;a intrare po&longs;&longs;et. Fal&longs;us ergo e&longs;t hic modus. Alium nos trademus infrà Cla&longs;se I. cap. 2. Machina 4. & 11.

Error apud Portam.

§. X.

Alia ratio aquam elevandi in altum propter Corporum impenetrabilitatem.

Modui ele­vandi aquam etiam bru tis notus.

ESt & alia ratio aquam quie&longs;centem atque infu&longs;am va&longs;is ele­vandi, in altum, atque in fontes efformandi, quam vel ip&longs;a bruta do cent animalia. De Cervis quippe Lybicis narratur, quòd dum &longs;itientes foveam offendunt aquam plenam, culus tamen &longs;u­prema &longs;uperficies humilior atque profundior e&longs;t, quàm vt capite ac collo proten&longs;o eam attingere po&longs;&longs;int, calculos ore & vnguibus a&longs;portantes foveæ injiciant, vt de&longs;identibus in fundum lapidi­bus humor &longs;ub&longs;identium corporum occupatione elevetur, &longs;u­perioremque elatus repleat vacuitatem; ita &longs;olerti machinatio­ne reficiuntur. Va&longs;is itaque puteisque &longs;i &longs;olida injicias corpora, quæ fundum petant vel &longs;ponte, vel vi depre&longs;&longs;a; elevabitur aqua, & per tubos &longs;uperiùs applicatos effluens in fontem formabitur. Hoc artificio con&longs;tructus erat olim fons ad S. Georg. Venetijs,

te&longs;te Io&longs;epho Ceredo Di&longs;cur&longs;u primo de Modo elevandi aquas è locishumilibus, pag. 11. Binæ enim fabricatæ erant ci&longs;ternæ, &longs;eu pi&longs;cinæ, humilior ac profundior vna, altera elatior ad prioris latus. Humilior latera habebat angu&longs;ta, & putei in&longs;tar erecta; è cuius fundo v&longs;que ad collateralis &longs;uprema labra deduceban­tur tubi plumbei. Habebat præterea operculum ligneum atque gravi&longs;&longs;imum, quod ci&longs;ternæ latera vacua implebat, & aquæ in­cumbens &longs;uo paulatim pondere de&longs;cendebat, atque aquam con­primens eam intra tubos compellebat; per quos a&longs;cendens, &longs;ub­&longs;equentibus alijs aquis pul&longs;a, exonerabat &longs;e&longs;e intra &longs;uperiorem collateralem ci&longs;ternam, indeque pondere &longs;uo naturaliter dela­bens per fi&longs;tulas efforma batur in fontem. Multas alias Machi­nas hydraulicas eodem con&longs;tructas artificio enumerat ibidem citatus Ceredus, & vnam hac ratione adornatam Machinulam Scaliger Exercitat. 42. contra Cardanum; ex qua tamen fal&longs;um deducit modum o&longs;tendendi fontium originem ex mari. Hac porrò ratione in altum elevatur aqua per antlias, quarum ingen­tem copiam affert Augu&longs;tinus Ramellus in&longs;uo de Machi­nis hydraulicis opere, & nos aliquas adduce­mus Cla&longs;se I. cap. 2. Machina 6.

Fons ad S. Georgium Venetijs.

Protheoria III.

PROTHEORIA TERTIA.

Devi Rarefactiva; &longs;eu de tertio Hydro-pnev­maticarum Machinarum Principio.

§. I.

Rarefactionis vis, & effectus mirabiles.

RArefactionem ingentes &longs;uppeditare vires vt alijs corporibus, ita aquæ & aëri, ad ea in omnem &longs;i­tus differentiam impellendam, multòque magis ad hydro-pnevmatica omnis generis techna&longs;mata mo­lienda; is &longs;olus ignorare pote&longs;t, qui &longs;tupendos ip&longs;ius effectus igno­rat. videmus quotidie, miramurque, ac plangimus &longs;tragem, quam polioxh|tai\ nitrato & &longs;ulphureo pulvere intra bombarda­rum militarium angu&longs;tias rarefacto edunt, in &longs;ternendis non ho­minibus tantùm, fed ædibus, turribus, propugnaculis, atque adeo vrbibus integris. Non Iovis tantummodo fulmina imita­tur rarefactio in bombardis, &longs;ed Plutonis etiam iras æmulatur in &longs;ubterraneis cuniculis; tantò pernicio&longs;iore &longs;trage, quantò mi­nùs patente. Excavant Martiales Architecti à loco ob&longs;idionis ad v&longs;que propugnaculum, cui ruinam minantur, viam anfra­ctuo&longs;am, angu&longs;tamque, magneticæ pixidis ductu, uti inter alios docet P. Athana&longs;ius Kircherus lib. 2. de Arte Magnet. par. 2. cap. 1. Probl 7. & nos in Pantometro Kircherianolib. 5. Ichnographi­co Probl. 8. Sub loco deinde ruinæ de&longs;tinato Cameram con­&longs;truunt laxiorem, eamque implent pulveris tormentarij va&longs;is, & o&longs;tium claudunt, relicto exiguo foramine, perque viam totam ignis pabulum &longs;ternunt, & egre&longs;&longs;i ignem applicant; qui &longs;erpens per viam accendit pulveris ma&longs;&longs;am omnem Camerâ conclu&longs;am, itàque rarefacit, dilatatque, ut angu&longs;tiarum impatiens, & laxio­rem quærens locum, impo&longs;itam &longs;ibi molem di&longs;iiciat horribili fremitu, nec minori terrore, quàm damno. Nihil unquam magis &longs;imile terræ motui, qui & ip&longs;e rarefactionis e&longs;t effectus lu- culenti&longs;&longs;imus, excogitatum fuit. Eodem enim pror&longs;us modo in terræ concu&longs;&longs;ione flamma ex aliqua &longs;ubterranea fornace, vel interno calore, per venulam aliquam nitri aut &longs;ulphuris &longs;erpit, donec perveniat ad locum, in quo maiorem huiu&longs;ce materiæ copiam inveniat; quâ &longs;ubitò accen&longs;a & rarefactâ, dum maio­rem quærit locum, &longs;ibi impo&longs;itas concutit, & di&longs;jicit non rarò, moles. Sæpe etiam iam antea accen&longs;us in terræ vi&longs;ceribus ignis, dum nova augetur aliunde attracta materia, montes &longs;ibi impo&longs;i­ tos concutit, finditque, vt non &longs;emel audivi, vidique in Ætna Si­ciliæ, duorum annorum accola olim, & nunc hoc ip&longs;o anno 1654. quo hæc &longs;cribo, hîc Romæ, quam (raro exe no) concu&longs;&longs;it mo­tusille ingens, qui tantam Soræ &longs;tragem lit, & Romam hinc inde Neapolim, omnes que interiacent regiones tremefecit.

Rarefactio­nis vis in bombardis.

Rarefactio­nis vis in cuniculis &longs;ubterra­neis.

Rarefactio­nis vis in terræ mo­tibus.

Terræmo­tus Romæ anno 1654.

§. II.

Alia Rarefaction Exempla.

Rare&longs;actio­nis exempla d.

NArrat P. Nicolaus Cabæus lib. 4. Meteor. Ari&longs;t. text. 12. que 1. vidi&longs;&longs;e &longs;e marmoreum vas maximum atque duri&longs;&longs;imum ab aqua conglaciata di&longs;ci&longs;&longs;um, eò quòd congelatione rarefactæ fue­rint aliquæ internæ partes aquæ: factà enim illâ &longs;ubitâ congela­tione, ob&longs;titerunt attenuationi & rarefactioni partium latera va­&longs;is, ce&longs;&longs;itque durities vehementiæ rarefactionis, & fractum e&longs;t vas, quod non di&longs;rupi&longs;&longs;ent centum juga boum, &longs;i in diver&longs;as par­tes traxi&longs;&longs;ent. Idem Cabæus ibidem a&longs;&longs;erit, &longs;e vidi&longs;&longs;e columnam marmoream, quam tres homines dilatatis brachijs &longs;imul non poterant complecti, quamque mille juga boum, &longs;i in diver&longs;as partes, toto juncto impetu traxi&longs;&longs;ent, frangere non potui&longs;&longs;ent, fractam tamen pror&longs;us mediam atque truncatam, eò quòd ligna quædam columnæ illi proxima fuerint combu&longs;ta, ideoque ex vi­cino calore &longs;piritulus aliquis in marmore inclu&longs;us intus in meati­bus fuerit rarefactus, & ex rarefactione maiorem &longs;ibi locum quæ­&longs;iverit.

Notum jam pœnè e&longs;t omnibus, quod refert Petrus Ca&longs;tel­lus in Libello de Incendio Ve&longs;uvij Italicè con&longs;cripto, & exip&longs;o Kircherus lib. 3. Artis Magnet, part. 2. cap. 3. circa finem, de pul- vere pyrio ex auro confecto, quod aurum fulminans appellat Ca­

&longs;tellus, & aurum volatile Crollius. E&longs;t id nihil aliud quàm au­rum calcinatum aquâ forti &longs;eu regiâ, &longs;ale armoniaco, & oleo tar­tari præcipitati. Hæc calx quàm primùm &longs;entit etiam calorem ignis, concipit &longs;ponte &longs;ua flammam, & in auras abit, cum ingenti fragore & &longs;trepitu: & &longs;i vel vnum aut alterum huius pulveris gra­num cultro impo&longs;ueris, & candelam infrà applicueris, vtinca. le&longs;cat; tantum edit fragorem, ac &longs;i magnam explo&longs;i&longs;&longs;es bombar­dam. Facit autem effectum contrarium ordinario pulveri pyrio, quia non &longs;ur&longs;um, &longs;ed deor&longs;um tendit; vnde &longs;erupulus vnicus hu­ius volatilis auri po&longs;itus &longs;upra laminam ferream non modicæ cra&longs;­&longs;itiei, & carbone de&longs;uper applicato accen&longs;us, penetrat & per­foratip&longs;am laminam.

Aurum ful­minans.

Nolo hîc ommittere aliud rarefactionis exemplum, quo ego ip&longs;e multis in locis, coram viris etiam Principibus, exiguâ machinulâ non exiguum excita vi fragorem, bombardulæ mi­noris fragori pror&longs;us æqualem ac &longs;imilem. Vitreas Sphærulas nucis avellanæ magnitudine, aut etiam minores, fieri curavi ad lampadem Romæ à quodam Germano artifice, vitrum in omnes colores omnes que figuras florum, fructuum, avium, animalium omnis generis, jcuncularum quarumlibet affabrè elaboratarum efformante, idemque in &longs;ubtili&longs;&longs;ima fila ad rotam circumactam diducente. In eas vitreas Sphærulas acetum, aut nitratum &longs;a­lem aquâ di&longs;&longs;olutum ac liquefactum infundi iu&longs;fi, & ad medieta­tem repleri, ac deinde hermeticè claudi ad eandem lampadem. Harum &longs;phærularum vnam &longs;i carbonibus non nimiùm accen&longs;is, aut calidis cineribus (vt paulatim calefiat) impo&longs;ueris; rarefit &longs;pirito&longs;a materia intus clau&longs;a, & majorem quærens locum, po&longs;t &longs;patium circiter temporis quo P&longs;almus 50. Mi&longs;erere mei Deus, re­citari pote&longs;t, tanto cum impetu ac fragore viam &longs;ibi aperit, vt non tantùm vitrum in mille minuti&longs;&longs;imas particulas diffringat, &longs;ed cineres etiam & carbones toto conclavi di&longs;&longs;ipet, nullius damno, ni&longs;i &longs;i quis vultum propè applicet. Cùm experientiam Romæ coram Sereni&longs;&longs;imi Hol&longs;atiæ Ducis Filio primogenito fa- cerem, foco a&longs;&longs;identi more ibidem &longs;olito, famuli ante cubicu­lum præ&longs;tolantes accurrebant attoniti, bombardulam in Princi­pem &longs;uum explo&longs;am credentes. Alius magnus quidam Prin­ceps cùm nonnullas à me accepi&longs;&longs;et, ac domi die quodam cum multis alijs nobilibus foco a&longs;&longs;ideret, vnam o&longs;tendit, ac videnti­bus omnibus igni iniecit, a&longs;&longs;erens, vbi calefieret, grati&longs;&longs;imum &longs;par&longs;uram odorem. Cùm omnes arrectis naribus, & ore hi­ante odorem ex&longs;pectarent, derepente fit crepitus, & carbonum disiectio, diffugientibus omnibus in timorem primò, deinde in ri&longs;um di&longs;&longs;olutis. Alio tempore coram Cardinale quodam & magno Nobilium comitatu, admonito priùs clanculum &longs;olo Cardinale, vnam explo&longs;i; & ita omnes alij præter dictum Cardinalem territi fuêre, vt auctorem (ignorabant me e&longs;&longs;e) ad pœnam quærerent.

Vitrea &longs;phæ rula mino­ris bombar­dulæ crepi­tum amu­lantes.

Innumeros alios effectus rarefactionis, &longs;icut & conden&longs;a­tionis, adducere po&longs;&longs;em; &longs;ed hi &longs;ufficiunt ad ad&longs;truendam effica­ciam in commovendis corporibus. Quæ efficacia &longs;i tanta e&longs;t, quantam vidimus; quis neget, eandem in aëre & aquis elucere po&longs;&longs;e, dilatando & con&longs;tringendo elementa ip&longs;a, & in fontes a­liaque hydraulica, pnevmaticaque artificia animando? &longs;ed hæc evidenti&longs;&longs;imè o&longs;tendemus in multis ex &longs;equentibus Machinis Partis 2. pertim Cla&longs;. Im. cap. 3. Mach. 1.2.5.6.8. & alijsmultis.

PROTHEORIA

QVARTA.

De fluxu Aquæ naturali;

&longs;eu De quarto Hydro-pnevmaticarum Machina­rum principio.

PLurima, jucundi&longs;&longs;ima, vtili&longs;&longs;imaque circa A­quæ fluxum naturalem, &longs;eu libera ea &longs;it, &longs;eu aquædu­ctibus, canalibus, tubis, &longs;iphonibus, fi&longs;tulis, & qui­bus cunque alijs meatibus con&longs;tricta, dicenda occur­runt noc ioco, quæ Hydro-pnevmaticarum Machinarum Arti-

Protheoria IV.ficibus non minùs nece&longs;&longs;aria, quàm vtilia &longs;unt. Nos ex omni­bus ea tantùm, quæ fini no&longs;tro conducere videbuntur, &longs;elegimus; cætera ad Mechanicam no&longs;tram Magiam re&longs;ervabimus.

CAPVT I.

De proprietatibus Aquæ fluentis liberè.

PROOEMIVM.

LIberè fluit aqua, dum nullis con&longs;tricta e&longs;t meatibus ceu vinculis. Huiu&longs;ce proprie­tates, fini no&longs;tro conducentes, vt a&longs;&longs;ignem, Suppono duo, quæ licet in dubium revocentur ab aliquibus, conceduntur tamen à menotæ Philo&longs;ophis & Mathematicis.

Aqua libe­rè fluere quæ dica­tur.

Primum e&longs;t, Ordinem Elementorum, naturæ eorum con gruentem hunc e&longs;&longs;e, vt Terra &longs;it in medio omnium, Aqua circa Terram, circa Aquam Aër, & circa Aërem putatus lgnis. Ac principio quidem rerum, quando Creavit DEVS cœlum & terram, Genes. C. I. prædictum &longs;ervarunt ordinem, v&longs;que ad diem Mundi tertium, quo die Deus benedictus Terram, anteà &longs;phæricè rotundam, in cavitates varias deformavit, in easque Aquam omnem &longs;ubtercœle&longs;tem, maribus, lacubus, fluminibus, fontibus di&longs;cretam, derivavit, vnumque ex Terra & Aqua globum con&longs;tituit, vnicâ contentum &longs;phæricâ &longs;uperficie, terrenâ hinc, inde aqueâ, prout Aquæ diver&longs;imodè Terram di&longs;ci&longs;&longs;am interfluunt, aut circumfluunt. Qui quidem globus ex Terra & Aqua coagmentatus (quem ob id rectè Terraqueum appellare po&longs;&longs;umus) in prima &longs;ui coagmentatione, quando die Mundi tertio dixit DEUS: congre gentur aquæ, quæ &longs;ub cœlo &longs;unt, in locum vnum, & appareat Arida; & factum e&longs;t ita; in medio Vniver&longs;i, hoc est, in centro &longs;upremi, & omnia ambien­tis Cæli collocatus fui&longs;&longs;e videtur; in medio, inquam, non &longs;olùm quoad &longs;en&longs;um (quod nullus, aut vix vllus, &longs;altem alicuius no­minis, negat) &longs;ed revera ac mathematicè, ita vt centrum gra­vitatis Terraquæ congruerit tunc perfectè centro Vniver&longs;i, exi- gente id naturali ip&longs;ius gravitate. Nunc verò licet ob conti­nuam gravium in Terra Marique mutationem, centrique gra­vitatis variationem prædicta Terraqua non &longs;it forta&longs;&longs;e &longs;emper præcisè & mathematicè in Mundi medio in &longs;en&longs;u dicto (non enim certò constat, ad centrum gravitatis mutatum continuò, &longs;equi continuam, licet exiguam, trepidationem ip&longs;ius, nitentis ad mundi centrum) tam parum tamen ab illo centro distat, quàm multùm di&longs;tat à tota mole Terraquæ quodcunque pon­dus defacto additum aut ablatum de novo. Ex quibus dedu­citur, &longs;i centrum gravitatis Terraquæ differt à centro magni­tudinis eiusdem (vt probabile e&longs;t differre, cùm Terra pondero­&longs;ior &longs;it quàm Aqua, nec omnes Terræ partes, æqualis &longs;int ponde­ris) differre &longs;olùm in&longs;en&longs;ibiliter; ac proinde idem cen&longs;eri po&longs;&longs;e; &longs;altem phy&longs;icè & ad &longs;en&longs;um, vtrumque centrum. Sed de his &longs;usè tractatum vide à Kirchero in Mundo &longs;ubterranneo, libro 1. de Centro&longs;ophia, & à nobis in Mechanica Vniver&longs;ali lib. 1. aut in Magia Mechanica.

o­do naturalis

Terraque­us globus in medio muadi col locat' fuit die tertio creationis rerum.

Terraqua nnuc non e&longs;t &longs;emper in mundi centro ma­thematicè.

Centrum commune omnium gravium ac levium

Alterum quod &longs;uppono est, centrum Vniver&longs;i, adeoque Centrum Terraquei globi, quem in Mundi medio &longs;tatuimus, cen&longs;eri po&longs;&longs;e centrum commune omnium gravium ac levium, quò nimirum omnia gravia per &longs;e tendunt appetitu innato, & à quo levia recedunt, vtraque per rectas ac brevi&longs;&longs;imas lineas, ni&longs;i impediantur, aut cogantur per obliquam ac longiorem viam ver&longs;us illud de&longs;cendere, vel ab illo recedere. Cùm igitur aqua omnis &longs;it gravis, vt experientiâ patet; tendet appetitu naturali ad centrum Terraquæ, id e&longs;t, ad centrum Vniver&longs;i. Idem di­cendum e&longs;t de alijs liquoribus. His præmi&longs;&longs;is accedimus ad propo&longs;itum.

Proprietas I.

Aqua tendit ad loca decliviora.

AQuam, & omnem alium humorem, fluentem iberè, currere ad loca decliviora, & centro Mundi ac Terraquæ propinquiora, &longs;i patetip&longs;is liber aditus, &longs;ive us ac perpendicularis, &longs;ive obliquus; patet expe­rientia quotidiaua, nec indiget probatione. Ex quo deduci­tur, aquæ etiam con&longs;i&longs;tentis partes omnes inclinare innato ap­petitu ad eadem loca decliviora: ideo enim de facto, &longs;ublatis

impedimentis, eò fluit aqua, quia naturaliter eò inclinat. Con­&longs;i&longs;tentem porrò aquam appello, quæ &longs;itum obtinet naturalem. Situs autem &longs;eu po&longs;itio naturalis aquæ e&longs;t, quem dum naturali &longs;uo fluxui & con&longs;i&longs;tentiæ permittitur, &longs;ua &longs;ponte a&longs;&longs;umit, tam in &longs;uperficie &longs;uperiore, quàm in inferiore, & lateralibus.

Aqua con­&longs;i&longs;tens qua dicatur.

Aqua &longs;itus naturalis quis &longs;it.

Proprietas II.

Aquæ &longs;uperficies &longs;uperior, ce&longs;&longs;ante fluxu, &longs;phærica e&longs;t; reliquæ conformantur va&longs;is & rece­ptaculis,

AQua liberè fluens tam diu fluit, donec partium continuata­ rum extima &longs;eu &longs;uperior &longs;uperficies &longs;phærica &longs;it, habens idem centrum cum centro Terraquæ & Vniver&longs;i, hoc e&longs;t, donec omnes extimæ &longs;uperficiei partes æqualiter di&longs;tenta Centro Mundi. De- mon&longs;trat hoc &longs;ubtili&longs;&longs;imè Archimedes lib 1. de in&longs;identibus hu­mido Propo&longs;it. 2. & Ari&longs;tot. lib. 2. de Cœlo text. 31. & &longs;equitur ex præcedente Proprietate. Si enim ce&longs;&longs;ante fluxu, & con&longs;i­&longs;tente iam aquâ, pars vna &longs;uperficiei extimæ altior e&longs;&longs;et, & altera humilior, hoc e&longs;t, &longs;i non omnes æquè di&longs;tarent à centro Mundi (quod e&longs;t &longs;phæricam habere &longs;uperficiem, habentem idem Cen­trum commune Mundi;) non omnes aquæ partes, &longs;ublatis im­pedimentis, fluerent ad loca decliviora, nec aquæ con&longs;i&longs;tentis partes omnes eò naturali appetitu inclinarent; aut certè violen­ter in tali &longs;tatu, & nullo præ&longs;ente impedimento, detinerentur; quod incongruum e&longs;t, & naturis rerum repugnans.

Aqua con&longs;i­&longs;tentis &longs;uper ficies &longs;uperi­or &longs;pharica e&longs;t.

Aquæ va&longs;is contentæ &longs;u perficies con formantur va&longs;orum in ternis figu­ris.

Inferior porrò aquæ &longs;uperficies, & laterales, conforman­tur &longs;uperficiebus internis va&longs;orum & receptaculorum, quibus aqua continetur: Vnde &longs;i vna pars fundi va&longs;orum ac recepta­culorum e&longs;t altior alterâ (prout in mari, lacubus, fluminibus, & va&longs;is ordinariè fit) etiam talis erit aquæ illis contentæ infe­rior &longs;uperficies. Idem intellige de lateralibus &longs;uperficiebus.

Pori&longs;ma I.

Aquarum omnium Su perficies &longs;u­perior e&longs;t &longs;phærica.

COlligitur hinc, Oceani, Marium, lacuum, & aquarum qua­rumcunque continuatarum, & in quibuscunque receptacu­lis contentarum, ac con&longs;i&longs;tentium, &longs;uperficies &longs;uperiores atque externas e&longs;&longs;e &longs;phæricas, habentes idem cum Terraquæ &longs;uperficie

convexa centrum. In va&longs;is tamen & receptaculis exiguis adeo exi­gua e&longs;t & in&longs;en&longs;ibilis&longs;phæricitas i&longs;tius &longs;uperficiei, vt meritò &longs;up­poni po&longs;&longs;it e&longs;&longs;e planam, &longs;eu horizonti parallelam: vnde & nos in &longs;equentibus id nobis concedi po&longs;tulabimus, & ita &longs;uppone­mus.

In va&longs;is ta­men exiguis cen&longs;eri po­te&longs;t plana.

Pori&longs;ma II.

COlligitur iterum, idem vas ad turris aut montis radicem po­ &longs;itum, & aquâ omnino plenum, plùs aquæ continere, mathe­maticè loquendo, quàm po&longs;itum in turris aut montis vertice, & aquâ itidem omnino plenum. Ratio e&longs;t, quia major e&longs;t &longs;phæri­citas aquæ in primo, quàm &longs;ecundo ca&longs;u.

Aquæ plus continet vas plenù infe­riori, quàm &longs;uperiori loco.

Pori&longs;ma III.

COlligitur tertiò, &longs;i aqua ad libellam con&longs;tituta e&longs;&longs;et alicubi, non po&longs;&longs;e con&longs;i&longs;tere, &longs;ed nece&longs;&longs;ariò moveri, quia tunc non omnes partes extimæ & &longs;uperioris &longs;uperficiei e&longs;&longs;ent æquè vicinæ centro Terraquæ. Athoc meliùs explicabitur in Pantometro no&longs;tro Kircheriano lib. 9. Hydragogico, vbi delibellatione a­quarum agemus, iterumque in Mechanica no&longs;tra Vniver&longs;ali, libro quem in&longs;cribemus Mechanicam Hydragogicam.

Aqu con­&longs;i&longs;tere non pote&longs;i ad li­bellam con­&longs;tituta.

Proprietas III.

Aqua minùs pre&longs;&longs;a expellitur à magis pre&longs;&longs;a.

AQuæ, & humidi cuiuscunque, ea e&longs;t natura, inquit Archi­medes lib. 1. de In&longs;identibus humido, Hypothe&longs;i I, vt par­tibus ip&longs;ius æqualiter iacentibus & continuatis inter &longs;e&longs;e; minùs pre&longs;&longs;a à magis pre&longs;&longs;a expellatur; & quidem tantò fortiùs, quantò aut minùs illa, aut magis hæc pre&longs;&longs;a fuerit. Hoc quomodo &longs;it in­telligendum, patebit ex &longs;equentibus Proprietatibus huius capitis, & etiam ex dicendis capite &longs;equenti, Propo&longs;itione 2.

Proprietas IV.

Aquæ, & humidi cuiu&longs;cunque, pars vnaquæ que premi­tur humido &longs;upra ip&longs;am exi&longs;tente ad perpendiculum, &longs;i hu­midum &longs;it de&longs;cendens in aliquo, aut ab alio ali­quo pre&longs;&longs;um.

AS&longs;erit hoc Archimedes loco proximè citato. Itaque quando humidum non e&longs;t de&longs;cendens in aliquo, aut non e&longs;t ab ali­quo alio pre&longs;&longs;um, &longs;ed con&longs;i&longs;tit in &longs;uo &longs;itu naturali, vna pars non premit alteram, vt mox dicemus. Si autem humidum e&longs;t de&longs;cendens in aliquo, vt &longs;i effluit per foramen va&longs;is, aut in­fluit in vas coniunctum: tunc partes &longs;upra foramen premuntur à partibus ad perpendiculum ip&longs;is incumbentibus, non verò ab alijs collateralibus. Similiter &longs;i humidum premitur ab aliquo, partes pre&longs;&longs;æ premunt &longs;olùm quæ ip&longs;is ad perpendiculum &longs;ub­&longs;tant.

Aquæ par­tes inferio­res premun­tur à &longs;upe­rioribus ad perpendicu­lum incum bentibus.

Proprietas V.

Aquâ in &longs;itu naturali con&longs;i&longs;tente, partes &longs;uperiores non premunt inferiores.

PAtet hoc ex multis &longs;ignis. Primò enim alioquin partes aquæ inferiores e&longs;&longs;ent den&longs;iores &longs;uperioribus, quia compre&longs;&longs;æ e&longs;&longs;ent. Secundò, vrinatores infra aquam exi&longs;tentes non &longs;entiunt pon­dus aquarum incumbentium &longs;ibi. Tertiò, herbæ levi&longs;&longs;imæ intra aquam cre&longs;centes, & in altum a&longs;&longs;urgentes, non &longs;ternuntur ab aqua &longs;uperincumbente. Quartò, corpus grave fune alligatum & infra aquam demer&longs;um, &longs;u&longs;tinetur à manu, & extrahitur, tan­ta facilitate, quanta &longs;u&longs;tineretur & extraheretur, &longs;i nulla aqua illi incumberet; imò maiorietiam facilitate, quoniam gravia mi­nùs ponderantintra, quàm extra aquam. Quintò, &longs;itula aquâ plena levior e&longs;t intra aquam, quàm extra: Curhoc, ni&longs;i quia aqua intra aquam non ponderat?

Pori&longs;ma.

Aquæ con &longs;i&longs;tentis par­tes &longs;uperio­res non pre­munt actu inferiores.

COlligitur hinc, quando Archimedes, alijque dicunt, omnes aquæ partes, dum ea in &longs;itu con&longs;i&longs;tit naturali, a qualiter pre­mi, non e&longs;&longs;e &longs;ermonem de actuali pre&longs;&longs;ione, &longs;ed &longs;olùm de aptitu­dinali. Premi vnum ab alio aptitudinaliter, e&longs;t, habere &longs;upra &longs;e grave non &longs;u&longs;tentatum; at premi actualiter e&longs;t ita habere &longs;u­

pra &longs;e grave non &longs;u&longs;tentatum, vt ablato impedimento cedere co­gatur corpori incumbenti; & ni&longs;i cedat, comprimatur. Vel di­cendum e&longs;t, premi quidem partes humidi inferiores à &longs;uperio­ribus &longs;ibi &longs;uperin cumbentibus, at non comprimi. Solùm ergo partes &longs;uperiores aquæ actu premunt inferiores, quando vel ab alio premuntur &longs;uperiores, vel quando &longs;unt altiores alijs partibus contiguis in eodem va&longs;e, aut in duobus va&longs;is inter &longs;e communi­cantibus, vt videbimus, cùm de Siphonibus agetur; vel denique quando infra &longs;e habent inferiores immediatè corpus levius &longs;e in &longs;pecie, v.g. aërem, aut tenui&longs;&longs;imum fundum; in omnibus enim hi&longs;ce ca&longs;ibus partes inferiores cedunt quidem &longs;uperioribus, at minimè ab illis comprimuntur. Hinc e&longs;t, quòd aqua effluatè foramine in fundo aut latere va&longs;is. Hinc etiam e&longs;t, quòd fun- dum tenue va&longs;is incurvetur, ac tandem frangatur. Hinc deni­que e&longs;t, quòd dum partes aquæ &longs;uperiores premuntur, & partes inferiores non habent effugium aut infrà, aut è latere va&longs;is, rum­patur potiùs vas, quàm partes inferiores comprimantur. Vide quæ dicimus infrà parte 2. Cla&longs;se 1. cap. 5. Machina 6.

Premi actu, & premi aptitudina­liter, quid &longs;it.

Proprietas VI.

Aquæ in &longs;itu naturali po&longs;itæ vna pars non expellit alteram.

PAtet experientia. Et ratio e&longs;t, quia vna non premit alteram, vt dictum in præcedenti Proprietate. Quòd &longs;i in va&longs;is con­jugatis altior aqua pellit minùs altam, cau&longs;a e&longs;t, quòd non &longs;it po&longs;ita in &longs;itu naturali, quia &longs;cilicet non habet omnes partes æ­què di&longs;tantes à centro Mundi.

Proprietas VII.

Non omnes aquæ æquales magnitudine, &longs;unt æquales pondere.

SUnt enim &longs;alsæ graviores dulcibus, & frigidæ calidis eiu&longs;dem &longs;peciei, &longs;eu individui; & tam &longs;alsæ, quàm dulces, aliæ alijs gra­viores &longs;unt, vt patet experientia, & fatentur etiam nautæ, vi­dentes naves è fluminibus impuris ad pura, aut è mari ad flumina transeuntes &longs;ub&longs;idere profundiùs, & &longs;ubinde cum &longs;ubmer&longs;ionis periculo.

Aqua &longs;alsa frigida, im­pura, gra­vior e&longs;t quam dulcis &c.

Proprietas VIII.

Aqua naturaliter non a&longs;cendit ad locum altio­rem &longs;uâ origine.

IMò nec fluit naturaliter ab vno ad alterum locum, ni&longs;i terminus à quo altior &longs;it, quàm terminùs ad quem. Vtrumque patet experientiä, & &longs;equitur ex dictis Proprietate I. & II. Hinc Hydragogi a&longs;&longs;e­runt, non po&longs;&longs;e deduciaquam de loco in locum, &longs;ive per alveum, &longs;ive per aquæductus, canales, ac tubos deducenda &longs;it, &longs;i &longs;patium &longs;it æquilibratum, &longs;eu ad libellam con&longs;titutum; nec &longs;ufficere Ma­thematicam, aut quam cunque phy&longs;icam declivitatem, &longs;ed requiri in &longs;ingulis milliaribus cadentiam &longs;eu declivitatem quatuor ad minimùm vnciarum vnius pedis, (e&longs;t vncia pars duodecima pe­dis) ita vt terminus ad quem in fine milliaris &longs;it vicinior centro Terræ quatuorvncijs, quàm terminus à quo; &p o&longs;t alterum mil­liare alijs quatuor vncijs &longs;it vicinior, & &longs;ic deinceps &longs;emper, e&longs;t­que hæc regula à plerisque recepta tanquam Axioma hydra­gogicum. Sed de hac re fusè agemus in Pantometro no&longs;tro Kircheriano lib. 9. vbi agemus ex profe&longs;&longs;o de Hydragogia & li. bellatione aquarum; & in Mechanica Vniver&longs;ali.

Aqua non pote&longs;t decur rereper &longs;pa­tium æqui­libratum.

CAPVT II. DE Proprietatibus Aquæ fluentis per &longs;iphones.

QUamvis in Machinis no&longs;tris Hydro-pnevmati­cis, Parte 2. indifferenter & pro eodem fumamus no­men &longs;iphonis, & tubi, pro fi&longs;tula &longs;cilicet, per quam aqua fluit, &longs;ive recta illa &longs;it, &longs;ive curva; hîc tamen, maioris di&longs;tinctionis gratia, &longs;iphonem à tubo di&longs;tinguimus. Egitur Siphon nihil aliud quàm tubus inflexus. Huius variæ &longs;unt &longs;pecies, vt &longs;equitur.

Siphon quid

Propo&longs;itio I.

Siphonis varias divi&longs;iones a&longs;&longs;ignare.

SIphon igitur, &longs;eu tubus in flexus, dividitur variè, & habet diver­ &longs;as figuras, vt apparet in appo&longs;itis &longs;chemati&longs;mis. Alius enim dicitur &longs;iphon erectus, alius in ver&longs;us, aliusmixtus, alius æquali­um, alius in æqualium crurium.

Siphonis &longs;pecies va­ria.

Siphon ere­ctus.

Siphon erectus e&longs;t, cuius omnia crura &longs;ur&longs;um &longs;unt. Huius­modi &longs;unt &longs;eptem priores, & vltimus ex hîc appo&longs;itis.

Siphon in­ver&longs;us.

Siphon inver&longs;us e&longs;t, cuius omnia crura deor&longs;um &longs;unt. Hu­iusmodi &longs;untijdem, &longs;i inver &longs;i intelligantur, hoc e&longs;t, &longs;i B &longs;ur&longs;um, C & A deor&longs;um &longs;pectent.

Siphon mi­xtus.

Siphon mixtus e&longs;t, cuius crura partim &longs;ur&longs;um, partim de­or&longs;um &longs;unt. Huiu&longs;modi &longs;unt tres penultimi ex appo&longs;itis, &longs;eu 8. 9. & 10.

Siphones tam erecti, quàm inver&longs;i, vel habent vtrumque erus æquale, vel inæquale. Si æqualia &longs;unt crura, vellongitu­dine & capacitate &longs;untæqualia, vt apparet in prima, tertia, &

&longs;eptima figura: vel &longs;ola longitudine æqualia, capacitate verò inæqualia, vt apparet in &longs;ecunda, quarta, quinta, &longs;exta, & vlti­ma figura.

Siphones a­qualium, & inaqua­lium cruri­um.

Siphones mixti, vel habent omnia crura æqualia longitu­dine, vt in octava figura apparet; vel aliqua tantùm, utin nona figura; vel nulla, ut in decima figura patet. Hæc eadem crura po&longs;&longs;unt e&longs;&longs;e diver&longs;æ capacitatis inter &longs;e.

Siphon con­iugatus.

Nota. Siphonem quartum aliqui vocant coniugatum, re­liquos verò inflexos appellant.

Propo&longs;itio II.

Siphonis erecti proprietates a&longs;&longs;ignare.

Siphonis e­recti proprie tates.

PRima. Aqua vni &longs;iphonis erecti cruri infu&longs;a de&longs;cendit pri­mùm, deinde a&longs;cendit per alterum crus, donec &longs;upremæ &longs;u­perficies in vtroque crure &longs;int in eadem linea horizontali. Sint &longs;iphones erecti ABCDEF, &longs;ive æ­qualium, &longs;ive inæ­qualium crurium, tam quoad longitu­dinem, quàm quoad capacitatem; vnque crutium, &longs;ive longi­ori, &longs;ive breviori, &longs;ive æquali, & &longs;ive largiori, &longs;ive &longs;tricti­ori, v.g. cruri AB, infundatur aqua: de&longs;cendet ea primum v&longs;que ad B, indeque per C & D a&longs;cen­det, donec &longs;upremæ aqu&ecedil; &longs;uperficies &longs;int in vtroque crure in eadem linea Hori- zontali, v.g. in linea HIK; hoc e&longs;t, donec omnes dictarum &longs;u­perficierum partes di&longs;tent æqualiter à centro terræ, juxta dicta cap. 1. Proprietat. 2. Ratio de&longs;umitur ex dictis ibidem. Ex his formo &longs;equens.

Axioma Hydraulicum I.

QVantùm aqua de&longs;cendit per vnum &longs;iphonis erecti crus, tantundem

a&longs;cendit per alterum. Idem contingit in omnibus canalibus, alveis, & meatibus quibuscunque incurvatis.

Axioma hy draulicum.

Secunda. Si vni crurium, &longs;ive longiori, &longs;ive breviori, &longs;ive æquali, & &longs;ive capaciori, &longs;ive minùs capaci, addas aliquid aquæ, v.g. cruri AB; attollitur etiam alterius cruris aqua, donec rur­&longs;us &longs;uperficies di&longs;tent æqualiter a centrro terræ, &longs;eu &longs;int in eadem linea horizontali, v.g. in linea GFE. Ratio e&longs;t eadem.

Tertia. Si cruri longiori, &longs;ive id capacius &longs;it altero iam pleno, &longs;ive non, addas plùs aquæ, v.g. cruri AB; de&longs;cendet ea p B, elevabitque aquam cruris CDEF, & expellet per os EF, donec ce&longs;&longs;ante infu&longs;ione &longs;it iterum vtraque &longs;uperficies in li­nea GFE. Patet experientiâ, & ratio e&longs;t eadem, ne &longs;cilicet &longs;u­perficies vnius cruris di&longs;tet plùs aut minùs, à centro terræ, quàm alterius.

Quarta. Si cruri breviori, licet capaciori, nempe cruri EFCD, jam pleno addas plùs aquæ |; ea non attollet aquam cruris longioris vltra punctum G, &longs;ed effluet ex ore EF, licet aqua cruris EFCD &longs;it longè maior & pondero&longs;ior, quàm aqua cruris AB. Patet itidem experientiâ, & ratio e&longs;t eadem.

Corollarium I.

COlligitur hinc primò, aquam maioris perpendiculi pellere aquam minoris perpendiculi, non ob&longs;tante maiore copia, & maiore pondere huius: ideo enim aqua cruris AB primæ & tertiæ figuræ expellit aquam cruris CDEF, licet longè maiorem & pondero&longs;iorem, quia perpendiculum illius e&longs;t maius &longs;eu longius, quàm perpendiculum huius. Aquæ perpendiculum hic, & in &longs;equentibus, appello altitudinem aquæ &longs;upra horizontem, &longs;eu &longs;upra centrum Terræ, ita vt illa dicatur habere maius perpendi- culum, quàm altera, cuius altitudo ab infima ad &longs;upremam &longs;u­perficiem maior e&longs;t, in eodem horizonte.

Aqua ma­ioru perpen­diculi pellit aquam mi­noris perpen diculi in Si­phonibus e­rectis.

Perpendicu lum aquæ quid.

Corollarium II.

Aqua ma­gis aut, mi nùs pre&longs;&longs;a in &longs;iphoni­bus, quanam dicatur.

COlligitur &longs;ecundò, Quænam aqua in &longs;iphone erecto dicatur magis aut minùs pre&longs;&longs;a: Nam cùm con&longs;tet, Aquam minùs pre&longs;&longs;am expelli à magis pre&longs;&longs;a, per dicta Capite primo, Propriet. 3. con&longs;tet præterea, Aquam minoris perpendiculi in &longs;iphone e­recto premi atque expelli ab aqua maioris perpendiculi; clarum e&longs;t, Aquam minoris perpendiculi dici minùs, aquam verò maioris perpendiculi dici magis pre&longs;sam. Ex his formo &longs;equens

Axioma Hydraulicum II.

IN hydraulicis non aquarum copiâ, &longs;ed perpendiculis pugnatur, & vin­

citur. Illa enim vincit pellendo, vel trahendo alteram, cuius perpendiculum e&longs;t maius.

Axiomo hy draulicum.

Corollarium III.

Siphonis e­recti ope a­quam ex v­no monte in alterum de rivare.

COlligitur tertiò, Quomodo &longs;iphonis erectiope (hoc e&longs;t,­nale, tubo autin modum &longs;iphonis erecti con&longs;tructo) ex monte per vallem ad alterum montem, priori paulò humiliotem, aut ad locum in latere montis, derivari po&longs;&longs;it aqua; &longs;i nimirum fiat &longs;iphon erectus, cuius os vnum immergatur aquæ montis altioris, & dein &longs;iphon deducatur per vallem, & latera montium, usque adlocum de&longs;tinatum, humiliorem tamen origine aquæ.

Corollarium IV.

Dolium quo­que &longs;it ple num, Si­phone ex­plorare.

COlligltur quar­ tò, Quomodo &longs;cire po&longs;&longs;is præcisè quou&longs;que dolium aliquod exinanitum &longs;it, manente obtu­ramento &longs;uperiore clauso, nulloque alio foramine facto præterid, cui epi­&longs;tomium immitti &longs;olet. Sit enim dolium DC, exinanitum v&longs;que ad lineam FG; quod tu ignores: &longs;cies id, &longs;i tubum oblongum AB incurves in­feriùs iuxta B, & doleo iuxta C in&longs;eras epi&longs;tomij loco; liquor enim dolij effluens a&longs;cendet per tubum, donec perveniat v&longs;que ad E, & æquetur &longs;upremæ &longs;uperficiei liquoris in dolio contenti.

Corollarium V.

Dolium im plere per e­pi&longs;tomium ope &longs;iphonis.

COlligitur quintò, Quomodo repleri po&longs;&longs;it prædictum doli­um per foramen C epi&longs;tomij. Si enim tubum AB incur­vatum in&longs;eras dolio iuxta C, & &longs;uperiùs iuxta A infundas li­quorem; a&longs;cendet is in dolio, donec æquetur &longs;ummitati tubi, aut donec dolium &longs;it omnino repletum, &longs;i tubus altior e&longs;t dolio. Bene tamen Schvventerus monet in fuis Recreationibus Mathe­maticis parte 13. quæ&longs;t. 3. debere in &longs;ummitate dolij foramen a­periri, per quod aër elabatur, impediturus alioquin liquoris in­fu&longs;ionem, &longs;i non po&longs;&longs;it erumpere per foramen C & tubum BA.

Annotatio.

In Siphone erecto plurium quàm duorum crurium eadem con­tingunt, quæ in &longs;iphone duorum crurium, vt con&longs;ideranti patet.

Propo&longs;itio III.

Siphonis inver&longs;i proprietates a&longs;&longs;ignare.

Siphonis in­ver&longs;i pro­prietates.

PLurimas proprietates, easque plerumque miras habet Siphon ever&longs;us, tam æqualium quoad longitudinem & capacitatem crurium, quàm inæqualium. Nos præcipuas referemus.

Prima Proprietas.

SI crura &longs;iphonis inver&longs;i inæqualia &longs;unt quoad longitudinem, & totus &longs;iphon repleatur aquâ, & invertatur; effluit ea tota per cruslongius, licet &longs;it minus capacitate. E&longs;to Siphon inver&longs;us ABC inæqualium crurium, aquâ plenus, & crus AB longius &longs;it quàm crus CB, hoc e&longs;t, extremæ aquarum &longs;uperficies A & C inæqualiter di&longs;tent à Centro Mundi, &longs;cilicet AB minùs, CB plùs; dico, totam aquam effluere per crus AB, licet capacitate &longs;it minus quàm CB. Ratio e&longs;t, quia cùm aquæ AB perpendi­culum, &longs;it longius, quàm aquæ CB perpendiculum, ma­gis premitur aqua apud A, quàm apud C, per corollarlum I. Propo&longs;itione. II. præcedentis; ac pro­indeper Propr. III. cap. 1. aqua AB ex­pellet aquam CB, non quidem per pre&longs;­&longs;ionem, &longs;ed per at­tractionem; hoc e&longs;t, effluetaqua AB, &longs;e­cumque trahetaquam CB, & quidem to­tam, tum propter rationem dictam, tum nè detur vacu­um in &longs;iphone.

Annotatio I.

MOnuit hîc P. Athana&longs;ius Kircherus, ea quæ diximus hoc loco, &longs;o­lùm e&longs;&longs;e intelligenda in eo ca&longs;u, in quo tubi ABC laxior portio BC inferiùs e&longs;t clau&longs;a, &longs;olo foramine I. relicto. Si enim totum orificium a­pertum fuerit; aqua nulla ratione &longs;i&longs;tipoterit quin effluat &longs;ecumque tra­hat aquam cruris BA. Rationem ad&longs;ignat, quia quando &longs;olum fora­men I e&longs;t apertum, &longs;ola columna aquea BI in fig. I. habet rationem per­pendiculi, eò quòd &longs;olum illa premat &longs;upra I. At quando totum orifi­cium e&longs;t apertum, tota aqua BHICG habet rationem perpendiculi, quia tota premit &longs;upra orificium apertum, ac proinde æquivalet pluribus perpendiculis æqualibus perpendiculo BI; quæ &longs;imul &longs;umpta &longs;uperant perpendiculum longius BA. Addidit, hæc &longs;e longa experientia didici&longs;&longs;e.

Opportuna monitio, nec menti meæcontraria. Puto tamen, a­quam ex orificio CI penitus aperto effluere non tam obrationem ad&longs;ig­natam, quàm quòd aëri facillimus pateat ingre&longs;&longs;us ob laxitatem orificij; quo ingrediente di&longs;continuatur aqua, & per vtrumque orificium apertum effluit.

Pori&longs;ma.

COnfirmatur hinc Axioma hydraulicum præcedens, quòd in &longs;iphonibus (& etiam tubis, & quibuscunque va&longs;is) non pu­gnatur ponderibus, &longs;ed altitudinibus, &longs;eu perpendiculis. In­tellige, in ca&longs;u & &longs;en&longs;u explicato.

Annotatio II.

DIcere quis po&longs;&longs;et, quando &longs;olùm patet foramen I, tunc attracto cy­lindrulo aqueo BI, faciliùs &longs;uccedit aër per foramen attractus, quàm aqua lateralis HI, vel GI; immi&longs;&longs;oque aëre, aqua illa lateralis facillimè effluet per foramen I. Quare in nullo ca&longs;u videtur po&longs;&longs;e ve­rificari, quòd aqua AB attrahat totam aquam BC Nego tamen id recte dici, quia aër ingrediens per foramen I adhærebit &longs;olùm vltimæ &longs;uperficiei aquæ fugientis.

Secunda Proprietas.

SI crura &longs;iphonis inver&longs;i æqualia &longs;int quoad longitudinem, li­cetinæqualia capacitate (cum limitatione tamen po&longs;ita) & totus repleatur aquâ, invertaturque; nihil effluit. Decurtetur in præcedente &longs;chemate crus AB in D, & repleatur totus&longs;iphon aquâ, ita vt extremæ aquæ &longs;uperficies C & D æqualiter di&longs;tent à centro mundi; dico, nihil effluere, licet cruris CB maior aqua & pondero&longs;ior &longs;it, quàm aqua cruris AB. Idem fiet, &longs;i absque cru­ris AB decurtatione, extrema aquæ &longs;uperficies in illo æquè di­&longs;tetà centro Mundi, atque in crure CB, vt &longs;i vtro e&longs;&longs;et in linea horizontali CD, aut in linea GHE. Ratio e&longs;t, quia cùm in vtroque crure æquale &longs;it perpendiculum &longs;eu altitudo aquæ, pre­metur vtrobique æqualiter; vel ergo vtriu&longs;que cruris aqua &longs;imul de&longs;cendere deberet, quod fieri non pote&longs;t absque vacuo; vel neutrius. Exhac, & præcedente Proprietate, formari pote&longs;t aliud.

Axioma hydraulicum III.

IN hydraulicis æqualia perpendicula æqualiter premunt, inæqualia

inæqualiter, &longs;cilicet longius plùs, brevius minùs.

Axioma by draulicum.

Tertia Proprietas.

Aqua per longius crus &longs;iphonis ce­leriùs cur­rit, quàm per brevius.

AQua per longius crus &longs;iphonis inver&longs;i eò currit celeriùs, quò longius fuerit crus. Ratio e&longs;t, quia quò longius e&longs;t crus, eò longius e&longs;t aquæ de&longs;cendentis perpendiculum, ac proinde eò ma­ius pondus incumbit aquæ prope os, eòque magis premitur a­qua in egre&longs;lu, celeriúsque propterea effluit, & con&longs;equenter ce­leriùs de&longs;cendit. Hinc formo &longs;equens

Axioma Hydraulicum IV.

Axioma hy draulicum.

In hydraulicis per &longs;iphones & tubos longiores celeriùs fluit aqua, quàm per breviores, po&longs;ito eodem, aut æquali orificio per quod effluit.

Quarta Proprietas.

Aqua per &longs;iphonem fluit difformi celeritate.

AQua per longius crus &longs;iphonis inver&longs;i non effluit celeritate v­niformi, &longs;ed difformi; augetur enim continuò celeritas de­&longs;cen&longs;us, donec exhau&longs;tum &longs;it crus oppo&longs;itum: nunquam tamen de&longs;cendit tam celeriter aqua, quàm &longs;i liberè, hoc e&longs;t, non per &longs;iphonem, &longs;ed per tubum, aut extra tubum vel canalem de&longs;cen­deret. Ratio e&longs;t, quia aqua effluens per crus longius AB, in an­tea po&longs;ito, & in quocunque alio &longs;iphone, debet &longs;ecum trahere contranitentem aquam cruris brevioris CB, nè detur vacuum intra &longs;iphonem; quæ aqua cruris brevioris quoniam continuò minuitur effluente aqua per os A, ideo faciliùs &longs;emper ac faci­liùs attrahitur ab aqua cruris longioris AB; cuius proinde de&longs;cen­dentis celeritas continuò intenditur, donec exhau&longs;tum &longs;it to­tum crus CB. Non æquat tamen celeritatem eam, quâ liberè de&longs;cenderet, non trahendo &longs;ecum aquam alterius cruris; quia quàm diu re&longs;tat aliquid aquæ in crure breviori CB, attrahere illud debet, vt po&longs;&longs;it effluere, ac proin de &longs;emper aliquid retinet impedimenti; quo caret, dum liberè cadit. Adde quòd quò diutiùs aqua effluit per A, eò aquæ cruris CB &longs;uperficies ex­trema C magis a&longs;cendit ver&longs;us B, adeoque & perpendiculum aquæ cruris CB continuò minuitur, aquæ interim, quæ ex crure AB fluit, perpendiculo manente eodem. Augetur ergo a&longs;&longs;iduè proportio perpendiculi BA ad perpendiculum BC, adeoque & proportio pre&longs;&longs;ionis aquæ BA ad pre&longs;&longs;ionem aquæ BC au­getur; ergo, &c. Dixi, donec exhau&longs;tum &longs;it totum crus CB oppo&longs;itum: nam illo exhau&longs;to, minuitur &longs;emper magis ac magis celeritas effluxus per os A, quoniam minuitur perpendiculum aquæ, ac proinde &longs;emper minùs ac minùs premitur aqua apud A.

Annotatio.

In &longs;iphone inver&longs;o pluribus tubis quàm duobus compo&longs;ito eadem. & ob eandem cau&longs;am contingunt, vt pat&ecedil;bit con&longs;ideranti.

Propo&longs;itio IV.

Siphonis mixti inæqualium crurium proprie­tates a&longs;&longs;ignare.

SIphonum mixtorum crura vel &longs;unt omnia æqualia, vel omnia

inæqualia, vel aliqua æqualia, aliqua inæqualia. De proprie­tatibus &longs;iphonis, cuius omnia crura &longs;unt æqualia, agemus Pro­po&longs;it. &longs;equenti; de reliquorum proprietatibus hîc agemus. Sit igitur &longs;iphon AKDI. Hic &longs;i integer maneat, habebit omnia cru­ra inæqualia: &longs;i truncetur in E, aut in B, habebit aliqua crura æqualia, alia inæqualia: &longs;i truncetur in F, aut N, habebit iterum omnia crura inæqualia.

Siphonis mi­xti variæ &longs;pecies.

Prima Proprietas.

SI&longs;iphon AKDI maneatinteger, & repleatur totus aquâ, efflu­et ea totaper os I. Ratio e&longs;t, quia perpendiculum DI lon­gius e&longs;t quàm perpendiculum DK, ut &longs;upponitur; ergo plùs premitur aqua apud I, quàm apud K, per dicta Propo&longs;it. III. præcedente, Propriet. I. & Capite I. Propriet. III. ac proinde aqua DI trahit aquam DK; quâ tractâ, &longs;equitur etiam aquKA, tum proprio pondere, tum ad vacuum impediendum.

Secunda Proprietas.

SI crus DI truncetur in B, ita vtos B &longs;it eiusdem altitudinis &longs;upra horizontom cum ore A, & totus &longs;iphon AKDB replea­tur aquá; e&longs;t qui putat non effluere ex B, &longs;ed effluere ex A, du­rareque fluxum donec &longs;uperficies aquæ B pervenerit ad C, hoc e&longs;t, donec effluxerit tantum aquæ, quantum e&longs;tintra BDC, tunc enim, ait, ce&longs;&longs;abit fluxus ex A, &longs;tabitque aqua in æquilibrio intra AKC. Quòd non effluat aqua ex B, clarum e&longs;t, quia perpendiculum BD brevius e&longs;t quàm perpendiculum DK, ergò aqua DB non pote&longs;t trahere a­quam DK, & multò minùs aquam DKA. Quòd autem effluat ex A, rationem a&longs;&longs;ignat, quia perdendiculum DK longius e&longs;t quàm perpendiculum AK; ergo aqua DK fortiùs premit deor­&longs;um ver&longs;us K, quàm aqua AK, ac proinde aqua AK debet cedere, & effluere ex A, ac &longs;equi tota reliqua aqua, nè vacuum detur. Quòd de­nique ce&longs;&longs;et fluxus aquæ ex A, quan­do &longs;uperficies B pervenerit ad C, rationem dat, quia tunc perpendicu­la CK, AK, &longs;unt æqualia.

Experentiâ tamen con&longs;tat, in dicto ca&longs;u nullam effluere aquam ex Siphone AKDB, neque per os B, neque per os A. Rationem ad&longs;igno, quia aqua quæ e&longs;t in AKC, ita e&longs;t librata, vt à poten­tia quantumvis parva moveri, & ex vna parte attolli po&longs;&longs;it; qua­re humido CD &longs;upra &longs;e po&longs;ito non re&longs;i&longs;tit. I am verò quoniam perpendicula crurium DC, DB æqualia &longs;unt, aqua in C æquè premitur ac in B; quare cùm nec infra B, nec infra C vllum &longs;it impedimentum extrin&longs;ecum, præter aërem (iam enim o&longs;ten­&longs;um e&longs;t, aquam CKA non re&longs;i&longs;tere humido CD de&longs;cendenti) pariter ex A & B aqua deberet effluere; quod fieri non pote&longs;t absque vacuo. Patet ergo cur aqua con&longs;i&longs;tat. Dices, in hoc di&longs;cur&longs;u committitur manife&longs;tus paralogi&longs;mus; dividitur enim mente crus DK in puncto C, & di&longs;curritur ac &longs;i realiter ibidem e&longs;&longs;et facta divi&longs;io; Natura non dividit aquam cruris DK in pun­cto C, nec propter no&longs;tras intellectuales divi&longs;iones mutat ope- randi vim, cùm non ob&longs;tante illa divi&longs;ione revera crus DK lon­gius &longs;it, quàm crus AK. Re&longs;pondeo, quamvis DK longius &longs;it quàm crus AK, eius tamen activitati &longs;eu pre&longs;&longs;ioni deor&longs;um non &longs;olùm re&longs;i&longs;tit crus AK, &longs;ed etiam crus DB; illud, nè aqua KA pellatur &longs;ur&longs;um; hoc, nèaqua AD trahatur &longs;ur&longs;um: vnde per­pendiculum DK habet duplicem re&longs;i&longs;tentiam, &longs;cilicet DB, & AK; quæ duæ re&longs;i&longs;tentiæ æquant eius activitatem, & con&longs;equenter impediunt motum aquæ DK, eiusque fluxum, &longs;ive per os A, &longs;i­ve per os B. Divi&longs;io autem intellectualis cruris DK in puncto C fit &longs;olùm ad o&longs;ten dendum quomodo duplex prædicta re&longs;i&longs;ten­tia æquivaleat activitati totius cruris DK.

Tertia proprietas.

SI crus DI truncetur in F, vt os F &longs;it inferius quàm os A; idem qui &longs;uprà putat non effluere aquam ex F, &longs;ed ex A, du­rareque ut antea fluxum, donec &longs;uperficies F pervenerit v&longs;que ad C. Rationem eandem dat cum illa, quam ad&longs;ignavit paulò antè.

Eâdem tamen experientiä con&longs;tat, aquam in hoc ca&longs;u ef­fluere ex F, donec aqua cruris AK de&longs;cenderit in H ad pa­rem cum F altitudinem. Ratio e&longs;t, quia humidum in F ma­gis premitur, quàm in C, id e&longs;t, quàm in A, ob longius per­pendiculum DF, quàm DC; quare humidum in F de&longs;cen­det, trahetque humidum reliquum, nè detur vacuum. Nec re­fert, quòd dividatur intellectu crus DK in puncto C, & compa­retur crus DF cum parte DC &longs;olùm, ac &longs;i & Natura ibidem divi&longs;i&longs;&longs;et crus DK; hoc enim &longs;olùm fit ob rationem paulò antè allatam.

Quarta Proprietas.

SI crus DI truncetur in E, vt os E habeat eandem altitudinem cum puncto K; ille idem qui &longs;uprà exi&longs;timat nihil effluere, neque ex E, neque ex A. Ratio eius e&longs;t, quia perpendicula DE, DK &longs;unt æqualia; ergo aqua apud E non plùs premitur quàm apud K, nec apud K plùs quàm apud E; ergo neque ex E, neque ex A effluere pote&longs;t aqua, nè æqualis potentia dicatur &longs;upera&longs;&longs;e æ­qualem.

Dicendum tamen, eâdem experientiá duce, effluxuram aquam ex E, duraturumque fluxum, donec &longs;uperficies A per­venlat ad K, & non vltrà. Rationem ad&longs;igno, quia perpen­diculum DE longius e&longs;t, quàm DC, dum plenus e&longs;t tubus. Dum autem aqua de&longs;cenderit ad K, perpendicula DK, DE &longs;unt æ­qualia.

Quinta Proprietas.

Si crus DI truncetur in N, effluet aqua ex A, donec &longs;uperficies N perveniat in C, vbi con&longs;i&longs;tet. Ratio e&longs;t, quia altitudo aquæ DK, vel potiùs DC, maior e&longs;t, quàm altitudo DN.

Sexta Proprietas.

Effluxus aquæ in prædictis ca&longs;ibus, &longs;ive ex ore A, &longs;ive ex ore F, E, I, non e&longs;t vniformis, &longs;ed difformis, eo pror&longs;us modo, quo diximus in Propo&longs;it. III. præced. Proprietate IV.

Propo&longs;itio V.

Siphonis mixti æqualium crurium proprieta­tes a&longs;&longs;ignare.

EX &longs;iphone mixto æqualium crurium, qua­lis e&longs;t ABCD, in appo&longs;ito &longs;chemate, &longs;i a­quâ totus repleatur; effluit aqua ex D. Ra­tio e&longs;t, quia licet tam perpendicula AB, CB, quàm perpendicula CD, CB, &longs;int æqualia; quia tamen aqua CB æquilibrata e&longs;t ab aqua AB, poterit à qualibet potentia facillimè moveri, ideoque ab aqua CD deor&longs;um ni­tente poterittrahi, & effluere ex D; & ita de facto contingit, vt experientia docet.

Propo&longs;itio VI.

Inver&longs;i Siphonis, va&longs;i vno crure impo&longs;i­to, proprietates a&longs;&longs;ignare.

Crus inter­num, & crus exter­num &longs;ipho­nis

OBvia &longs;unt quæ dicam, ideò breviter expediam. E&longs;to igitur vas aquâ plenum ABCD, cui inver&longs;i &longs;iphonis crus vnum FE immergatur, alterum FN foris maneat. Vocetur autem crus FE internum, crus verò FN externum. Huius fiphonis variæ &longs;unt proprietates, provt variè difponi poterit crus eius externum.

Prima Proprietas.

Si enim crus exter­num FN maneat integrum, & ex ore N ex&longs;ugatur aër; &longs;e­quetur a va&longs;is, ob vacui metum, & tota effluet, aut pœ­nè tota (donec nimirum aquæ &longs;uperficies &longs;uperior CD de&longs;cen­derit v&longs;que ad o&longs;culum E) &longs;i internum crus fundum pœnè tangat. Ratio e&longs;t, quia per­pendiculum externi cruris e&longs;t longius quàm interni. Si to­tus &longs;iphon aquâ impleatur, & vtrumque orificium claudatur, ac deinde minus crus EF aquæ imponatur, maiori foras prominente, & &longs;imul vtrumque orifi­cium aperiatur; idem fiet.

Effluet tamen inæqualiter, cum perpetuo nimirum celerita­tis decremento, quia perpendiculum aquæ cruris interni, altitu­do nempe aquæ à puncto P u&longs;que ad &longs;uperficiem &longs;uperiorem aquæ va&longs;is (hæc enim &longs;ola premit actu, reliquæ verò inferiores actu non premunt, per dicta cap. 1. Proprietate V.) &longs;emper ma­gis ac magis cre&longs;cit, ac proinde &longs;emper magis magi&longs;que re&longs;i&longs;tit perpendiculo aquæ cruris externi.

Quò verò crus externum longius fuerit infra &longs;uperficiem

&longs;uperiorem aquæ va&longs;is, eò fluet aqua celeriùs, & æqualibus tem­poribus copio&longs;iùs: quia tunc eò magis perpendiculum externi &longs;uperabit perpendiculum interni cruris in longitudine, & vi pre­mendi deor&longs;um. Non tamen &longs;ufficit, ut crus externum &longs;it qua- cunque ratione longius crure interno, &longs;ed nece&longs;&longs;arium omnino e&longs;t ut perpendiculum externi cruris longius &longs;it perpendiculo in­terni, juxta dicta &longs;uprà Protheoria I. §. VI.

Aqua inquali cele­ritate efflu­it ex crure externo &longs;i­phonis,

Secunda Proprietas.

Si crus externum FN decurtetur in K, & internum tan­gat ferè fundum in E, & ex ore K extrahatur aer; &longs;equetur aqua u&longs;que ad os K, propter vacui metum, & tota ferè effluet, donec nimirum perveniat &longs;uprema aquæ &longs;uperficies u&longs;que ad o&longs;cu­lum E. Ratio e&longs;t, quia perpendiculum cruris externi longius e&longs;t quàm interni. At po&longs;tquam aqua pervenerit u&longs;que ad o&longs;cu­lum E, nihil amplius effluit, quia perpendicula FE, FK &longs;unt æqualia. Hærebit ergo aqua in æquilibrio in utroque crure.

Tertia Proprietas.

Si crus externum FN decurtetur in H, & internum infra lineam LM, aut pertingat &longs;olummodò u&longs;que ad dlineam; vas tamen &longs;it plenum u&longs;que ad lineam CD, autaqua &longs;it &longs;upra lineam LM; effluet aqua per os H, donec &longs;upre­ma aquæ va&longs;is &longs;uperficies perveniat u&longs;que ad lineam LM, & non ampliùs. Ratio e&longs;t, quia antequam dicta aquæ &longs;uperficies per­veniat ad lineam LM, perpendiculum cruris externi e&longs;t &longs;em­per longius perpendiculo cruris interni; quando verò de&longs;cendit jam ad dictam lineam, am bo perpendicula &longs;untæqualia. Fluxus porrò difformitas quoad celeritatem aut tarditatem eadem e&longs;t, quæ antea.

Quarta Proprietas.

Si crus externum decurtetur in G, ut &longs;it eju&longs;dem altitudi­nis &longs;upra Horizontem cum &longs;uperficie humidi CD, & aqua at­trahatur in G; nihil effluet, quia perpendicula &longs;unt æqualia: quare aqua hærebit in æquilibrio in utroque crure.

Corollarium I.

Ex his colligitur primò, Si vas &longs;it plenum &longs;olummodò u&longs;que ad lineam LM, & &longs;iphonis crus externum decurtetur in H, vel G, vel O, internum verò pertingat u&longs;que ad fundum; nihil po&longs;&longs;e effluere ex crure externo.

Perpendicu li exce&longs;&longs;um tuetur Na­tura in a­quæ fluxu.

Corollarium II.

Colligitur &longs;ecundò, quàm pertinax &longs;it Natura, vel con&longs;tans potiùs, in retinendo atque tuendo perpendiculi exce&longs;&longs;u, nè mi­ nor aut æqualis potentia dicatur vici&longs;&longs;e majorem.

Perpendicu­lo, non copia aquæ Natu­ra pugnat, ac vincit, in hydrauli­cis.

Corollarium III.

Colligitur tertiò, quomodo Natura in Hydraulicis pugnet, ac &longs;uperet, non aquarum copiâ, &longs;ed altitudine perpendiculari.

Corollarium IV.

Colligitur quartò, &longs;iphonis operationes hydraulicas ad li­bræ operationes commodè revocari po&longs;le, uti mox dicemus Propo&longs;it: VIII. &longs;eque

Pori&longs;ma.

Fonticulus duplex ope &longs;iphonis in­ver&longs;i.

Hinc etiam patet, qua ratione &longs;iphonis inver&longs;i præ&longs;idio fonti­culus duplex exhiberi po&longs;&longs;it, &longs;i nimirum crus brevius AB indatur vitreo protuberanti ventri BC, habeatque intra ventrem &longs;tricti&longs;&longs;i­mum o&longs;culum B aper­tum; deinde ex eodem ventre derivetur crus longius DE in curva­tum inferiùs, & apud E de&longs;inens in o&longs;culum an­gu&longs;tum. Si enim hu­ju&longs;modi &longs;iphon impo­natur va&longs;i pleno, & ex o&longs;culo E &longs;ugatur aër; a&longs;cendet aqua per crus AB, & in ventre exiliet radiosâ virgulâ vix a&longs;pectabili, indeque &longs;e demittet in crus alterum longius DE, cogeturque &longs;ua propendente demi&longs;&longs;ione ad ejectionem eò altio­rem, quò profundiùs de&longs;cendet.

Propo&longs;itio VII.

Cau&longs;am ad&longs;ignare, cur quando o&longs;culum externi cruris &longs;iphonis inver&longs;i, e&longs;t altius, aut æquè altum ac aquæ &longs;uperficies &longs;uprema in va&longs;e, aqua non fluat; quando verò e&longs;t demi&longs;&longs;ius, fluat.

VIdimus, experientiam docere, quando perpendiculum cru­ris externi &longs;iphonis inver&longs;i, e&longs;t minus, aut æquale perpendi­culo cruris interni, u&longs;que ad &longs;upremam aquæ &longs;uperficiem aquam non effluere po&longs;&longs;e per os externi cruris, ce&longs;&longs;ante &longs;uctu, &longs;eu attra­ctione; quando verò perpendiculum externi e&longs;t majus quàm in­terni cruris in &longs;en&longs;u expo&longs;ito, fluere donec æqualia fiant perpen­dicula, aut donec tota effluxerit aqua ex va&longs;e. Quæritur nunc hujus phænomeni cau&longs;a.

Siphonis crus longius &longs;uperat bre­vius, non capacius mi capax.

Hero Alexandrinus in libello de Spiritalibus cap 1. ait, quo&longs;dam putare, cau&longs;am e&longs;&longs;e, quòd aqua cruris externi debeat trahere aquam cruris interni, ac proinde illa debeat e&longs;&longs;e major ac copio&longs;ior, quàm hæc. Sed hoc minimè verum e&longs;&longs;e, patet ex eo (quod etiam Hero advertit) quòd licet crus externum &longs;it lon­gè amplius atque capacius interno, multoque plùs aquæ conti­neat; &longs;i tamen longius non e&longs;t, nequaquam effluit ex ip&longs;o aqua, &longs;ed ubi ce&longs;&longs;at violenta &longs;uctio, &longs;eu tractio, tota refluit per crus in­ternum longius intra vas. Non ergo externum crus &longs;uperat in­ternum, quòd aquæ plùs contineat. Itaque videamus natura­lem cau&longs;am & genuinam, ex Heronis primò, deinde ex no&longs;tra &longs;ententia.

Diximus cap. 1. Propriet. II. aquæ continuæ & quie&longs;centis &longs;uperficiem &longs;uperiorem e&longs;&longs;e &longs;phæricam (etiam in parvis va&longs;is, & receptaculis, licet in&longs;enfibiliter) centrum habentem idem quod Terra, &longs;eu globus Terraqueus; non quie&longs;centem verò aquam tam diu fluere, donec partes continuæ in una &longs;uperficie &longs;phærica con&longs;tituantur, hoc e&longs;t, donec omnes extimæ &longs;uperficiei partes continuæ æqualiter di&longs;tent à centro Terræ. Hoc po&longs;ito putat Hero, cau&longs;am ob quam, quando externi cruris orificium e&longs;t æquè altum ac &longs;uperficies &longs;uperior aquæ va&longs;is, hoc e&longs;t, quando

externum crus e&longs;t æquale interno quoad longitudinem, u&longs;que ad aquæ &longs;uperficiem &longs;upremam va&longs;is, aqua non effluat ex crure externo, etiam&longs;i &longs;iphon &longs;it plenus aquâ, e&longs;&longs;e, quia tunc aqua &longs;i­phonis continuatur cum aqua va&longs;is, & utriu&longs;que &longs;uperficies ex­trema æqualiter di&longs;tat à centro Mundi: Cau&longs;am verò cur, quan­do externi cruris orificium e&longs;t demi&longs;&longs;ius &longs;uperficie extrema aquæ va&longs;is, effluat, e&longs;&longs;e, quia tunc &longs;uperficies aquæ va&longs;is e&longs;t altior, quàm &longs;uperficies aquæ cruris externi, cum cujus aqua continuatur, ac proinde tam diu fluere debet aqua va&longs;is, po&longs;tquam &longs;emel fluere cœpit, donec &longs;uperficies fiant æqualiter di&longs;tantes à centro Ter­ræ. Ob contrariam ergo cau&longs;am, &longs;ecundùm Heronem non effluit aqua ex crure externo breviore, &longs;ed refluit intra vas quia nimirum &longs;uprema &longs;uperficies aquæ va&longs;is e&longs;t declivior, quàm &longs;u­perficies aquæ cruris externi.

Heronis opi­nio circa a­quæ fluxum è crure ex­terno &longs;ipho­nis.

Cœterùm cur aqua in &longs;ecundo ca&longs;u, ex va&longs;e a&longs;cendat ad &longs;iphonis &longs;ummitatem, contra ejus naturam, indeque de&longs;cendat ad orificium u&longs;que externi cruris, non e&longs;t ob cau&longs;am jam ad&longs;igna­tam, &longs;ed ut impleat, aut impediat vacuum, &longs;eu quia trahitur vio­lenter ad impediendum, aut implendum vacuum. Quoniam enim ex crure externo extrahitur aër, trahit is &longs;ecum, nè vacuus maneat &longs;iphon, & di&longs;continuentur corpora, nempe aër & aqua, trahit inquam &longs;ecum aquam u&longs;que ad orificium cruris externi; quò ubi pervenerit, nullam ampliùs violentiam patitur ab aëre, nec ab alio extrin&longs;eco agente, &longs;ed naturæ &longs;uæ relinquitur, ut pars illa incipiat, aut pro&longs;equatur fluxum, cujus &longs;uperficies extre­ma minùs di&longs;tat à centro Mundi, duretque fluxus, donec am­bæ &longs;uperficies æquales fiant.

Experimentum.

ES&longs;e porrò hanc, quam ad&longs;ignavit Hero, genuinam dicti phænomeni cau&longs;am, confirmat ip&longs;e tali Experimento. Si &longs;umentes, inquit, duo va&longs;a (cuju&longs;modi &longs;unt A & B in appo­&longs;itis figuris) injiciemus humidum in utrumque ip&longs;orum, & implentes &longs;iphonem, digiti&longs;que ip&longs;ius o&longs;cula comprehendentes, demittemus alte­rum crus in unum dictorum va&longs;orum adeo, ut aquâ demergatur; alte­rum verò in alterum vas, (ut aqua etiam demergatur, prout in appo&longs;itis figuris apparet;) fiet tota aqua continua; utrumque enim humidorum, quæ &longs;unt in væ&longs;ibus, (ita interpres) conjungitur cum eo, quod e&longs;t in &longs;iphone, ita ut totum continuum fiat. Si igitur &longs;uperficies humidorum, quæ priùs erant in va­&longs;ibus, in una &longs;uperficie con&longs;i&longs;tant,ut in prima figura; quie&longs;cet & &longs;ic utraque ip&longs;arum demer&longs;o &longs;iphone: &longs;in minùs, ut in &longs;ecunda figura; quoniam aqua continua facta e&longs;t, nece&longs;&longs;ariò ob continuitatem in hu­miliorem locum fluet, quou&longs;&queacute; vel omnis aqua, quæ in va&longs;ibus, in una &longs;uperficie &longs;it, vel alterum va&longs;orum exinaniatr, illud videlicet, quod altiori fuerit loco. Fiant igitur in una &longs;uperficie humida, quæ &longs;unt in va&longs;ibus; quie&longs;cent utique, ita ut etiam humidum, quod e&longs;t in &longs;i­phone, &longs;imul quie&longs;cat. Quòd &longs;i quis mente concipiat &longs;iphonem &longs;­&longs;ectum juxta &longs;uperficies humidorum, quæ &longs;unt in va&longs;ibus; & ita qui­e&longs;cet humidum, quod e&longs;t in &longs;iphone: ergo & &longs;iphone ip&longs;o &longs;u&longs;pen&longs;o, & in neutram partem inclinato, rur&longs;us quie&longs;cet humidum, &longs;ive æqualem omnino habens latitudinem, &longs;ive alterum crus altero multo majus &longs;it; non enim in hoc cau&longs;a e&longs;t, cur humidum quie&longs;cat, &longs;ed in eo, quòd ip­&longs;ius o&longs;cula æqualiter jaceant. Hæc Hero. lege etiam quæ &longs;equun­tur apud ip&longs;um in eodem cap. 1.

Experimen tum circa aquæ fluxum ex &longs;iphone.

Siphon in­ver&longs;us æ­qualium crurium longitu di­ne, inæqua lium capa­citate, reti­net aquam &longs;u&longs;pen&longs;am &longs;ecundum Heronem.

Sed hæc cau&longs;a ex Herone ad&longs;ignata non videtur e&longs;&longs;e ve­ ra, aut &longs;altem non e&longs;t adæquata. Primò enim, &longs;i ideo fluit aqua ex crure longiore, AD, &longs;eu potius CD, &longs;ecundæ figuræ, quia &longs;uperfies extrema aquæ ip&longs;ius non e&longs;t æqualis &longs;uperficiei extremæ aquæ alterius cruris, BF, &longs;eu potius EF, &longs;ed demi&longs;&longs;ior; ergo quando &longs;iphon &longs;ive æqualium, &longs;ive inæqualium crurium, aqu&adot; plenus, & utrimque compre&longs;&longs;is digitis obturatus, imponi­tur duobus prædictis va&longs;is aqu&adot; plenis, quorum unius &longs;uperficies &longs;uprema aquæ altior e&longs;t, quàm alterius, effluitque aqua intra vas demi&longs;&longs;ius, eò quòd alterius &longs;uperficies &longs;uprema altior e&longs;t, & ad decliviorem locum tendit, nititurque &longs;uperficiem &longs;uam adæ­quare &longs;uperficiei alterius va&longs;is humilioris; ergo inquam, quando hoc contingit, aqua ex va&longs;e altiore a&longs;cendit naturaliter & inna­to appetitu ad &longs;iphonis &longs;uperiorem partem, indeque de&longs;cendit intra vas humilius; naturaliter enim aquæ partes altiores fluunt ad loca decliviora. Deinde &longs;i illa e&longs;t cau&longs;a adæquata, cur quan­do &longs;iphon ina qualium crurium repletur aquâ, & &longs;ecundùm longius crus immergitur aquæ, revius verò extra in aëre relinquitur, curquam, nouit aqua per externum, ubi nullum e&longs;t impedimentum prteerem? pote&longs;t enim aqua per exte num fluere d loca decliviora, & fundare unam conti­nuam &longs;uperficiem di&longs;tinctam à &longs;uperficie, quæ in va&longs;e e&longs;t; & præterea inclinationem habet ad de&longs;cendendum, non verò ad a&longs;cendendum. Præterea, cur quando &longs;iphon inæqualium cru­rium repletur aqua, & tenetur inver&longs;us in aëre, non effluit aqua ex crure breviori, cum ejus aqua plùs di&longs;tet à centro Terræ, quàm aqua cruris longioris? Alia ergo cau&longs;a ad&longs;ignari debet, cur aqua cruris longioris, &longs;eu externi, &longs;eu interni, vincataquam cru­ris brevioris; quam paulò po&longs;t dabimus.

Heronis opi­nio exami­natur.

Qui tamen Heronis &longs;ententiam tueri volet, ad has ratio­nes in contrarium allatas re&longs;pondere poterit, ut &longs;equitur. Nem­pe ad primam, aquam cruris brevioris non a&longs;cendere naturali­ter, &longs;ed tractam ab aqua cruris longioris AD, &longs;eu CD, ad va­cuum in &longs;iphone impediendum, & ad tuendam continuitatem partium aquæ aqua enim cruris longioris AD, &longs;eu CD, de&longs;cen­dit naturaliter (utpote tendens ad locum decliviorem) &longs;ecum­que trahit reliquam aquam, quia &longs;uprema &longs;uperficies va&longs;is A, cum qua continuatur aqua &longs;iphonis, & va&longs;is B, e&longs;t declivior, &longs;uprema verò &longs;uperficies va&longs;is B e&longs;t altior, & debet fieri æqua­litas inter ip&longs;as: quæ non fieret, &longs;i aqua &longs;iphonis BF, &longs;eu EF, de­&longs;cenderet.

Ad &longs;ecundam, non effluere aquam per brevius, &longs;ed per lon­gius crus, quia aqua amat continuitatem &longs;uarum partium, & ap­petit habere &longs;uperficiem &longs;upremam omnium partium æqualiter di&longs;tantem à centro Terræ; quorum utrumque obtinet, &longs;i ef­fluit aqua per longius crus; neutrum, &longs;i per brevius.

Ad Tertiam, ideo non effluere aquam è crure breviore, &longs;ed è longiore, quia aqua cruris brevioris e&longs;t altior, & aqua cruris longioris declivior: curruntautem aquæ continuatæ partes alti­ ores ad decliviores naturaliter, & non hæ ad illas.

Authoris o­pinio circa aquæ fluxum è crure ex­terno longio re &longs;iphonis.

Sed quidquid &longs;it de hac Heronis &longs;ententia, mihi magis pla­cet &longs;equens ratio allati phænomeni, quæ univer&longs;alis e&longs;t & in omnibus &longs;imilibus ca&longs;ibus locum habet, ut vidimus in præce­dentibus, & iterum videbimus infrà hoc eodem capite Propo­&longs;it. 13. & 14. Pro qua recolendum e&longs;t, quod diximus &longs;uprà Ca­pite 1. Proprietate IV. ex Archimede, aquæ & humidi cuju&longs;cun­que partem unamquamque premi humido &longs;upra ip&longs;am exi&longs;tente ad perpendiculum, &longs;i humidum &longs;it de&longs;cendens in aliquo, aut cer­tè nullum habeat impedimentum, ut de&longs;cendat.

Dico itaque, cau&longs;am hujus rei e&longs;&longs;e eandem cum ea, quæ ad&longs;ignata fuit in præcedentibus, quia nimirum aqua cruris lon­gioris potentior e&longs;t ad de&longs;cendendum, &longs;ecumque trahendum aquam cruris brevioris, quàm aqua cruris alterius; & hoc ideo, quia aqua premit ac premitur, dum premit ac premitur, ad per­pendiculum, hoc e&longs;t, &longs;ecundùm lineam quæ tran&longs;it per centrum Mundi. Cujus ergo aquæ perpendiculum longius e&longs;t, illa ma­gis premit ac vincit, hoc e&longs;t, de&longs;cendit, &longs;i nihil ob&longs;tet de&longs;cen­&longs;um impediens, ut in no&longs;tro ca&longs;u contingit. Hîc ergo verum etiam e&longs;t, quòd in Hydraulicis pugnatur & vincitur non pondere, aut copi aquarum, &longs;ed perpendiculis, juxta Axioma hydraulicum, 2. quod &longs;uprà Propo&longs;itione 2, formatum fuit.

Proprietas VIII.

Siphonem ad Libram &longs;eu vectem revocare, filtrum ad &longs;iphonem.

ESto recta linea AB, divi&longs;a bifariam in puncto C. Si huic puncto C &longs;upponatur hypomochlion G, aut alligetur fi­ lum FG; &longs;tabit, hærebitque li­nea AB in æquilibrio horizon­ti parallela, & neutra pars præ­valebit. Si prolongetur pars CB u&longs;que ad D, prævalebit pars CD parti CA, & de&longs;cendet. Si prolongetur eadem pars CB, u&longs;­que ad E, adhuc magis prævale­bit pars C parti CA, cele­riu&longs;que de&longs;cendet; & quò ma­gis prolongabitur pars longior, eò magis prævalebit breviori, magi&longs;que celeriter de&longs;cendet. Linea porrò AB, aut AD, aut AE, &longs;i hypomochlio G impo­natur, repræ&longs;entatvectem; &longs;i è filo FC &longs;u&longs;pendatur, repræ&longs;entat libram, aut &longs;tateram.

E&longs;to jam &longs;iphon ACB, &longs;ive erectus, &longs;ive inver&longs;us, divi&longs;us mente in duas æquales partes AC, BC, & repleatur totus aquâ; ex neutro crure effluet, ut vidimus atque probavimus, &longs;ed aqua &longs;tabit, hærebitve in duobus cruribus AC, BC, in æquilibrio. Si prolongetur crus CB, u&longs;que ad D, & repleatur totus &longs;i­phon aquâ; prævalebit crus CD, & aqua ejus in inver&longs;o &longs;ipho­ne effluens &longs;ecum trahet reliquam; in erecto verò &longs;iphone de­&longs;cendet & expellet reliquam per os A. Si prolongetur idem crus CB u&longs;que ad E, adhuc magis prævalebit crus CE, ma­gi&longs;que celeriter de&longs;cendet ejus aqua, & vel &longs;ecum trahet reli­quam, vel illam &longs;ub&longs;idendo expellet.

Siphon tam erectus. quinver&longs;us, re vecatur ad libram.

Patet ergo, quomodo &longs;iphon tam erectus, quàm inver&longs;us, re­vocetur ad libram, aut vectem

Filtrum appellamus panni laciniam ACB, pendentem ex va&longs;e A, ita ut pars AC &longs;it in­tra vas, & aquæ immer&longs;a, &longs;al­tem ex parte; pars verò altera CB extra vas pendeat. Hæc panni lacinia &longs;i priùs madefiat tota, deinde modo dicto ex va­&longs;e aquâ pleno &longs;u&longs;pendatur; tra­hit aquam non &longs;ecus ac &longs;iphon, &longs;i pars exterior fuerit longior quàm interior; &longs;i æqualis, aut brevior, non trahit. Con&longs;tat ergo, filtrum ad &longs;iphonem re­duci.

Filtrum re­vocatur ad &longs;iphonem.

Propo&longs;itio IX.

Diabetem &longs;piritalem de&longs;cribere, eju&longs;que proprietates ad&longs;ignare.

EO&longs;dem ferè effectus, quos præ&longs;tat &longs;iphon inflexus, cujus unum crus va&longs;i imponitur, præ&longs;tat etiam tubus certo mo­do va&longs;i implantatus, quem Hero Alexandrinus vocat medium &longs;piritalem diabetem, qua&longs;i dicas, pera&longs;cen&longs;orem; alij verò vo­cant tubum &longs;uffocabilem. Sic autem con&longs;truitur.

Sit vas ABCD. Per ejus fundum CD trajiciatur tubus EF utrimque apertus, ip&longs;i fundo coarctatus, & in partem inferiorem excedens apud F; at o&longs;culum ejus &longs;uperi­us E non adæquet altitudinem va&longs;is, &longs;ed &longs;it paulò inferius. Huic tubo EF circumponatur alius tu­bus GHI, &longs;uperius clau&longs;us, qui tamen o&longs;culum E non attingat, &longs;ed tantum ab ip&longs;o di&longs;tet, quantum &longs;ufficit ad aquæ ingre&longs;&longs;um per o&longs;culum E: inferiùs verò vel non attingat undequaque fun­dum va&longs;is; vel attingit, relinquatur à latere foramen I, per quod aqua intrare po&longs;&longs;it.

Diabetes &longs;piritalis quomodo con&longs;trua­tur.

Si jam vas impleas aliquò u&longs;que aquâ, v. g. u&longs;que ad KL, in­ &longs;inuabit &longs;e&longs;e illa per foramen I, & a&longs;cendet u&longs;que ad H, in­tra tubum IGH. Si igitur per o&longs;culum F attrahas aërem, qui e&longs;t in tubo FE; &longs;equetur &longs;imul aër, qui e&longs;t in tubi &longs;uperpo­&longs;iti parte GH; & po&longs;t aërem &longs;equetur aqua, quæ e&longs;t in eju&longs;dem tubi parte IH; nec ce&longs;&longs;abit fluxus ex F, donec tota aqua va­&longs;is effluxerit, alioquin daretur vacuum intra tubos, ut patet. Quòd &longs;i totum vas impleas aquâ, a&longs;cendet illa per foramen I, intra tubum IHG, u&longs;que dum &longs;upergrediatur o&longs;culum E; quo &longs;upergre&longs;&longs;o, in&longs;inuabit &longs;e&longs;e, naturali pondere, aqua per dictum o&longs;culum, & effluet per F, nec ce&longs;&longs;abit, donec tota effluxerit aqua. Hac arte docebimus infrà Cla&longs;&longs;e 1. cap. 1. Machina IV. con&longs;truere vas, quod effundat aquam per fundum, dum ple­num e&longs;t; dum verò plenum non e&longs;t, nihil effundat.

Diabetis &longs;piritalis proprieta­tes.

Annotatio I.

FLuxus aquæ etiam hîc inæqualis est, cum continuo celeritatis de- cremento: & quò tubus EF lon­giùs exce&longs;&longs;erit extra fundum, eò celeriùs, adeoque & copio&longs;iùs hu­midum effluet. Harum autem proprietatum ratio e&longs;t eadem cum illa quæ &longs;uprà Propo&longs;it. 7. pro &longs;i­phone va&longs;i impo&longs;ito e&longs;t allata; tu­bus enim IG æquivalet cruri in­terno, & FE, cruri externo.

Aquæ flu­xus per dia­betem &longs;piri­talem inæ­qualis e&longs;t.

Annotatio II.

IDem effectus habebitur, &longs;i in­ tra vas fi &longs;iphon inver&longs;usABC, cujus dor&longs;um B &longs;it paulò infra &longs;umma labra va&longs;is. Si enim aqua infundatur va&longs;i, a&longs;cendet ea per crus AB; & cùm &longs;upergre&longs;&longs;a fuerit dor&longs;um B, effluet per crus BC; nec ce&longs;&longs;abit, donec dota effluxerit. Vide quæ dicemus infrà Parte 2. loc. cit. Machina IV. Si verò non totum vas implea­tur aquâ, & aër &longs;ugatur ex o&longs;culo C; &longs;equetur &longs;imiliter aqua va&longs;is, nec ce&longs;&longs;abit fluxus, donec tota fuerit elap&longs;a, quoniam perpendiculum aquæ cadentis (&longs;icut & in Diabete) majus e&longs;t quàm perpendiculum aquæ a&longs;cendentis.

Diabetes &longs;piritalis a­lia ratione con&longs;tructus.

Propo&longs;itio X.

Siphonis inver&longs;i fluxum æqualem reddere.

SUprà Propo&longs;it. VI. Propriet. I. o&longs;tendimus, aquam per exter­num &longs;iphonis inver&longs;i crus fluere inæqualiter, cum perpetuo nimirum celeritatis decremento. Oporteat igitur efficere ut fluxus &longs;emper æqualis &longs;it. Docet id Hero Alexandrinus cap. 3. de &longs;piritalibus in hunc &longs;en&longs;um.

Sit vas ABCD, aqua plenum v&longs;que ad EF, gra­tia exempli (pote&longs;t enim plùs & minùs e&longs;&longs;e repletum) habens &longs;uas an&longs;ulas LL per­foratas. Fiat lebetarium G, &longs;ive apertum, &longs;ive clau­&longs;um, quod po&longs;&longs;it innatare &longs;uperficiei aquæ EF; per eius operculum ac fundum trajice &longs;iphonis HIK crus internum ac brevius HI, & &longs;tanno, aliavè materia ad foramina benè coarcta, nè aqua &longs;ubintrare in le­betarium po&longs;&longs;it. Alterum crus externum ac longius IK trajice per foramina an&longs;ularum LL, &longs;ic ut fa­cilè per eas moveri a&longs;cen­dendo ac de&longs;cendendo po&longs;­&longs;it. His factis, &longs;i per o&longs;cu­lum K attrahas aërem, qui e&longs;t in &longs;iphone, &longs;equetur, ob vacui fugam, aqua va&longs;is, eâque &longs;ub­&longs;idente &longs;ub&longs;idebit &longs;imul & lebetarium, & &longs;iphon infixus lebeta­rio, fluetque aqua per K, donec o&longs;culum H tangat fundum CD. Erit autem ille fluxus &longs;emper æqualis, eò quòd exce&longs;&longs;us perpendiculi cruris externi & longioris &longs;upra perpendicu­lum cruris interni & brevioris, &longs;it &longs;emper æqualis, hoc e&longs;t, idem, nempe in &longs;chemate po&longs;ito exce&longs;&longs;us KM u&longs;que ad &longs;upremam aquæ va&longs;is &longs;uperficiem EF, propterea quòd efflu­ente aqua ex K, & de&longs;cendente &longs;uperficie EF ver&longs;us fundum va&longs;is, de&longs;cendat & lebes cum &longs;iphone &longs;ibicoarctato, ut dixi; quo fit, ut crus IK eandem &longs;emper aquæ quantitatem ex va&longs;e per crus HI attrahat, nempe illam quæ e&longs;t in GI, à G &longs;uprema aquæ &longs;uperficie, v&longs;que ad 1; ac proinde eâdem &longs;emper fa­cilitate difficultatevè efflu­it aqua ex K ergo & eâ­dem celeritate.

Aquæ flu­xum per crus exter­num &longs;ipho­nis æqualem reddere.

Annotatio I.

Pote&longs;t loco lebetarij G accipi &longs;uberis, alteriusvè levi&longs;­&longs;imi ligni fru&longs;tum, & per id trajici &longs;iphonis crus internum.

Annotatio II.

Aquaeffluit ex crure ex­terno &longs;ipho­nis eò cele­rius, quo maior e&longs;t i­p&longs;i' exce&longs;&longs;us &longs;upra inter­num crus.

Quantò maior erit ex­ce&longs;&longs;us KM cruris externi, tantò celeriùs effluet aqua exK, &longs;emper tamen æqualiter. Ratio e&longs;t, quia tunc magis pre­mitur aqua apud o&longs;culum K, & aqua cruris externi maio­rem exce&longs;&longs;um habet &longs;upra aquam cruris interni, ac proinde mi­norem difficultatem ad illam trahendam. Neque hoc e&longs;t contra id, quod diximus Pro­po&longs;it. VII. præcedente: aliud enim e&longs;t, cau&longs;am dare, cur fluat aqua ex crure longiore, aliud cur celeriùs fluat.

Annotatio III.

Quomodo fieri po&longs;&longs;it fluxus aquæ ex &longs;iphone partim æqualis, par­tim inæqualis, hoc e&longs;t, vno tempore æqualis, altero inæqualis, pront no­bis placuerit, docet Hero &longs;upra citatus cap. 4.

Propo&longs;itio XI.

Siphone inver&longs;o aquam ex vno montis latere in alterum, per verticem deducere.

SItmons A, in cuius latere vno &longs;it fons, alia vè aqua, &longs;eu flu­ens, &longs;eu &longs;tagnans B, aut C, in pede nimirum, aut in latere montis; &longs;itque in latere altero locus E paulò humilior, quàm

aqua B, aut C; & &longs;it in hunc locum E deducenda prædicta aqua per montis verticem. Fiat &longs;iphon inver&longs;us inæqualium crurium BCDE, ex plumbo, aut metallo, cuius orificium B, aut C cruris brevioris &longs;it immer&longs;um fonti B, aut C, alterum verò orificium E cruris longioris de&longs;inat intra ci&longs;ternam, ali­udvè receptorium E in loco de&longs;tinato. Hoc facto, obture­tur vtrumque &longs;iphonis os, & per foramen D in vertice montis factum in &longs;iphone, repleatur totus &longs;iphon aqua, & foramen di­ligenti&longs;&longs;imè obturetur, nè aërem recipere po&longs;&longs;it. Si iam aperia­tur eodem tempore &longs;imul vtrumque &longs;iphonis orificium, effluet aqua per E cruris longioris intra receptaculum, &longs;ecumque tra­het, nè vacuum intra &longs;iphonem fiat, aquam fontis B, aut C; nec ce&longs;&longs;abit fluxus, quàm diu orificium B, aut C, fuerit aquæ im­mer&longs;um. Ratio patet ex dictis Propo&longs;itione 6. & alijs pa&longs;&longs;im ex præcedentibus.

Aquam per montis ver­ticem à ba­&longs;e ad basè deducere &longs;phone.

Annotatio I.

OMninò nece&longs;&longs;arium e&longs;t, ut locus E &longs;it humilior, &longs;altem aliquan­tulùm, quàm locus fontis aut aquæ B, &longs;eu , ut &longs;iphonis crus DE longius fieri po&longs;&longs;it quàm BD, aut DC. Quod &longs;i fieri non po&longs;&longs;it, &longs;pes nulla e&longs;t ad effectum de&longs;ideratum con&longs;equendum. Quare diligenter priùs libellandum e&longs;t &longs;patium inter B & E, aut C & E. Qu` autem humilior fuerit locus E quàm aqua B aut C, eòmcliorem res &longs;ucce&longs;&longs;um habebit, & eò velociùs qua effluet ex orificio E, vti ex di­ctis patet.

Annotatio II.

PO&longs;&longs;e etiam alia ratione trahi aquam exfonte aut C per &longs;iphonem, v&longs;que ad locum , absque eo quòd in vertice &longs;t foramen D, & &longs;i­phon repleatur aqunulli p; &longs;i nimirum ex orificio E tubi extrahatur aër follibus, aut Æolopila, quam de&longs;cribimus infrà parte 2. cla&longs;&longs;e 1. cap. 3. Mach. VI. Quam tamen rationem nullo modo &longs;ub&longs;i&longs;tere po&longs;&longs;e, hinc o&longs;tendo, quòd fieri non po&longs;&longs;it, ut follis per canalem vacuum tantum aquarum pondus, quantum canalem ex attactu replere debet, attrahers po&longs;&longs;it: montis enim in&longs;tar follem e&longs;&longs;e oporteret, qui id præ&longs;taret, cùm omnis attractio proportionem &longs;uam babeat ad pondus attrahendum; quæ &longs;i im­par&longs;it, rumpentur folles potiùs quàm pond9 attrahatur. Vt vel hinc appæ­reat, quàm multa in &longs;peculatione vera appareant, quæ tamen nullam effe­ctum in praxi exhibeant. Marinus Mer&longs;ennus in Pbænomenis Hydrau­licis Propo&longs;it. 34. addit alium modum, &longs;i nimirum in loco de&longs;tinato fiat pi&longs;cinæ E vndique clau&longs;a, eaque repleatur aquâ, & orificium E &longs;iphonis adferruminetur operculo perforato pi&longs;cinæ, ita ut aquam non attingat: &longs;i enim re&longs;eratur epi&longs;tomium pi&longs;cinæ, effluet aqua, inquit, & in aquæ lo­cum &longs;uccedet aër &longs;iphonis, in aëris locum aqua B, aut C, nè vacuum admittatur in &longs;iphone: dummodò, inquit, pi&longs;cina vacua totum &longs;iphonis aërem recipere po&longs;&longs;it. Sed mirum, quantum hallucinatur Mer&longs;ennus, cùm, ut diximus &longs;uprà Protheoria I. & videbimus infrà &longs;uo loco, & ip&longs;emet Mer&longs;ennus in&longs;inuat, aqua fontis B, aut C trahi, atque a&longs;cendere per &longs;iphonis crus BD, aut CD, nullatenus po&longs;&longs;it, ni&longs;i infra pi&longs;cinam E de&longs;cendat tubus, per quem aqua fluat, paulò maioris per­pendiculi quàm &longs;it perpendiculum BD, aut CD. quod in hac & &longs;imi­libus Machinis e&longs;t &longs;ummè nece&longs;&longs;arium, vt diximus, cùm de vi Attracti­va egimus, & in &longs;ecunda Parte &longs;æpius repetemus, præ&longs;ertim cla&longs;&longs;e 1. cap. 1. Machina 2. Pote&longs;t tamen longitudo tubi, qui infra pi&longs;cinam E de­&longs;cendere deberet, compen&longs;ari multis brevioribus tubis, ut diximus eo­dem loco de vi Attractiva, & loco citato Machina 7. iterum dicemus. Accedit & illud, quòd aqua in ci&longs;terna non &longs;ufficiat ad effectum inten­tum præ&longs;tandum, ni&longs;i ci&longs;terna tantum aquæ contineat, quantæ ip&longs;e ca­nalis capax e&longs;t; at quis tantam &longs;abricari volet ci&longs;ternam?

Annotatio III.

MVlta alia circa fluxum aquæ ex &longs;iphone dici po&longs;&longs;ent, præ&longs;ertim quòd de&longs;cen&longs;us aquæ fluentis per crus imitetur leges gravium de­&longs;cendentium motu naturali; &longs;ed hæc, & &longs;imilia alia, intelligi facilè po&longs;&longs;unt ex dicendis Capite &longs;equenti, de fluxu aquæ per tubos Propo&longs;it. 9.

Propo&longs;itio XII.

Siphone inver&longs;o aquam ex montis radice ad ejus verticem elevare.

MIrabilis e&longs;t &longs;equens modus, & antiquis minimè cognit imò ab omnibus pro impo&longs;&longs;ibili habitus, inquit Portomnes enim putârunt, &longs;i in &longs;iphonis inver&longs;i collo &longs;eu curvara &longs;uprema foramen fieret, & vel minimum aëris ingredetur, fore ut &longs;iphonis aqua di&longs;continuaretur, & aqua divi&longs;a putrumque &longs;iphonis crus hinc atque inde delaberetur, totu&longs;q&longs;iphon evacuaretur. Contrarium mon&longs;trat &longs;equens modquem præ&longs;cribit Joannes Bapti&longs;ta Porta lib. 2. Spirital. cap. & 14. in hunc &longs;en&longs;um.

Vide præ cedentem figuram pag 99. vel 100.

Aquam è montis ra­dice ad ver­ticem ele­vare.

Sit ut antea mons A, in eju&longs;que radice fons B, cujaqua &longs;it elevanda u&longs;que ad verticem D. Fiat &longs;iphon inverinæqualium, ut antea, crurium BDE, cujus orificium B immer&longs;um fonti, orificium verò E vel liberum omninò vel de&longs;inat in pi&longs;cinam &longs;ubjectam. Habebit autem res tanmeliorem &longs;ucce&longs;ium, quantò longius fuerit crus DE, quàalterum DB. Fiat deinde infra collum &longs;iphonis, ubi D, ceptaculum aquarum H, undique clau&longs;um quàm diligent &longs;imè, &longs;uoque epi&longs;tomio I in&longs;tructum, ex quo depromi aqpo&longs;&longs;it. In hoc receptaculum derivetur ex collo &longs;eu curvatu&longs;iphonis tubulus G, habens cla viculam volubilem, &longs;itque tbulus arcti&longs;&longs;imè adferruminatus & &longs;iphonis collo, & receptculi operculo priùs perforatis. His factis, repleatur per infudibulum D &longs;iphon BDE, ob&longs;tructis priùs orificijs B & eoque repleto obturetur foramen D, & aperiatur &longs;imul utruque orificium B & E; dumque per tubum fluit aqua, & exerumpit; aperiatur modici&longs;&longs;imè, & momentaneâ qua&longs;i mola clavicula tubuli G; &longs;tatimque per dictum tubulum de&longs;cedet modica aqua intra receptaculum H, a&longs;cendetque mocus aër intra &longs;iphonem, ibique aquæ mi&longs;tus delabetur unà cuip&longs;a per crus DE, & per os E erumpet. Hoc &longs;i &longs;æpius fec ris, tandem replebitur receptaculum H, ut aqua per ejus epi &longs;tomium I depromi po&longs;&longs;it magna copia. Hæc Porta.

Annotatio.

VVlt igitur Porta, tubulum G aperiendum e&longs;&longs;e modici&longs;&longs;imè, ut ni­mirum ingrediatur modicus aër intra &longs;iphonem, & ita mi&longs;cea­tur aquæ, ut ea non di&longs;continuetur. Si enim ingrederetur aër copio­&longs;us, po&longs;&longs;et ita interrumpi aqua &longs;iphonis aëre interpo&longs;ito, ut perpendi­culum BD e&longs;&longs;et majus, quàm perpendiculum reliquæ aquæ intra crusDE; quod ubi contingeret, &longs;tatim di&longs;continuaretur aquæ fluxus, & aqua utriu&longs;que cruris de&longs;cenderet, unaquæque ad &longs;uum orificium, & efflueret. Hoc igitur incommodum ut evitetur, intromittendus e&longs;t, inquit Porta, repetitis vicibus modici&longs;&longs;imus aër, ut &longs;ine interruptio­nis periculo po&longs;&longs;it mi&longs;ceri aquæ. Nam quantumvis modicus intro­mittatur aër, &longs;i tamen is aquam cruris DE di&longs;continuat, infallibili­ter interrumpetur aquæ fluxus, ni&longs;i pars FE &longs;it paulò longior quàm crus BD. Ponamus cnim aërem interceptum e&longs;&longs;e KF. Hic aër dum

erit in L, aut in M, non interrumpet quidem aquæ cur&longs;um, quia perpendiculum LE, aut EM, adhuc longius e&longs;t, ut &longs;uppono, quàm perpendiculum DB. At quamprimùm aër KF fuerit infra M, jam interrumpetur aquæ fluxus, eò quòd perpendiculum aquæ ab aëre u&longs;que ad E brevius e&longs;t, quàm perpendiculum à vertice D u&longs;que adB. Ne igitur aquæ interruptio, &longs;i fortè contingeret, ejus influxum per &longs;iphonem impediat, crus DE adeo prolongandum e&longs;&longs;et, ut facta interruptione etiam notabili aquæ &longs;iphonis per interpo&longs;itionem aëris, perpendiculum tamen aquæ ab aëre u&longs;que ad E maneret &longs;em­per longius perpendiculo cruris DB. Meliorem igitur &longs;ucce&longs;&longs;um res habebit, &longs;i fons &longs;it in latere montis, v. g. in C, & non in radice apud B. Praxim hanc ut dixi, docet Ioannes Bapti&longs;ta Porta lib. 2. Spiritalium cap. 13. & 14. pendetque ex praxi, quam ex codem re­tulimus &longs;uprà Protheoria I. §. X. Refert eandem praxim Mer&longs;ennus in hydraulicis Phænomenis Propo&longs;it. 34. ubi tamen nonnulla de &longs;uo adjungit non u&longs;quequaque veritati congrua. Veruntamen fal&longs;um omnino e&longs;t, quod habetur apud Portam loco cit. cap. 15. editionis Ita­licæ, de modo elevandi aquam ex flumine præterfluente per &longs;iphonem u&longs;que ad turris vicinæ altitudinem; quem quidem modum &longs;u&longs;picer non e&longs;&longs;e ip&longs;ius Portæ, &longs;ed e&longs;&longs;e interpretis &longs;eu traductoris ex Latino in Italicum idioma Ioannis E&longs;crivani Hi&longs;pani, qui in Dedicatoria fatetur &longs;e adjunxi&longs;&longs;e nonnulla, quæ oretenus ab ip&longs;o Porta didicerat, ut &longs;u­pra in Præloquio dicebam.

Aër in tulo di&longs;continu­ans aquam, interrum­pit fluxum æ­quæ.

Portæ error in elevan­da aqua ad turris al­titudinem.

Propo&longs;itio XIII.

Cau&longs;am a&longs;&longs;ignare, cur in &longs;iphone inver&longs;o interrupto perpendiculum aquæ de&longs;cendentis debeat &longs;upera­re perpendiculum aquæ a&longs;cendentis.

Siphon in­terruptus. & cau&longs;a ef­fectus ip&longs;ius

I. Figura.

INfrà Parte 2. Cla&longs;­&longs;e 1. cap. 1. Ma­china 2. damus Ma­chinam hîc po&longs;itam, eamque appellam9Siphonem inver&longs;um interruptum. Ean­dem Machinam ex­hibuimus etiam &longs;u­prà Protheoria 1. §. V. ubi diximus, tu­bum EF, per quem aqua de&longs;cendit, de­bere e&longs;&longs;e paulò lon­giorem tubo BK, per quem aqua a­&longs;cendit. Quod ta­met&longs;i veri&longs;&longs;imum &longs;it, & multis experi­mentis à Porta pri­mùm, deinde à Mer &longs;enno, Kirchero, at­que à me etiam con­probatum; &longs;emper tamen vi&longs;um mihi fuit mirabile, præ- &longs;ertim, &longs;i de primo aquæ a&longs;cen&longs;u per tubum BK &longs;it &longs;ermo: nec ullum vidi, aut audivi unquam, qui hujus rei cau&longs;am ad&longs;ignet, cui acquie&longs;cere po&longs;&longs;im. Aliam etiam Machinam hîc po­

II. Fig. &longs;itam, & priori non multùm ab&longs;imilem, adduxi eâdem Pro­theoria I. §. VI. quam etiam habet Salomon à Caus in &longs;uis hydraulicis lib. 1. in qua ut aqua a&longs;cendat per crus AB, debet tubus rectus DE e&longs;&longs;e paulò longior quàm dictum crus AB, ut quilibet facilè experi­ ri pote&longs;t. Quæritur igitur, quæ­nam &longs;it hujus rei cau&longs;a in utraque Machina, & in alijs hi&longs;ce &longs;imili­bus? Neque enim hîc valetratio, quam ad&longs;ignavimus &longs;uprà Propo­&longs;it. VII. hujus capitis: nam hîc non agimus de &longs;iphone continuato, &longs;eu noninterrupto, in quo partes omnes aqu&ecedil; &longs;unt continuatæ, tum inter &longs;e, tum etiam cum aqua va&longs;is, ex quo extrahitur aqua per crus externum ac longius de&longs;cendens; &longs;ed agimus de &longs;iphone interrupto, in quo &longs;ci­licet aër intercedit inter aquam & aquam. Adde quod hîc, licet tu­bi EF, & DE breviores e&longs;&longs;ent, quàm tubus BK & crus AB; tamen extrema &longs;uperficies aquæ illorum tuborum &longs;emper e&longs;t vicinior centro Mundi, ac proinde per ip&longs;os effluere deberet aqua, &longs;ecumque trahere & aërem & aquam oppo&longs;itam va&longs;orum.

Dubitatio. & rationes dubitandi circa &longs;ipho­nem inter­ruptum.

Dices, aqua quæ trahit alteram, gravior debet e&longs;&longs;e aquâ quæ trahitur, ac proinde longiores debent e&longs;&longs;e prædicti tubi EF, & DE, ut plùs aquæ contineant. Sed contra e&longs;t mani­fe&longs;ta experientia allata Propo&longs;it. III. hujus capitis Propriet. I. & alia etiam allata Protheoria I. §. VIII.

Solutiones in&longs;ufficien­tes propo&longs;itæ dubitatio­nis.

Dices iterum, in hydraulicis pugnari ac vinci non aqua­rum copiâ, &longs;ed aquarum perpendiculis, ut dictum Propo&longs;it. 2. hujus capitis in Axiomate 2. & alibi &longs;æpe; ideoque longiores debent e&longs;&longs;e prædicti tubi EF, & DE, ut perpendicula aqua­rum de&longs;cendentium &longs;int longiora, quàm perpendicula aqua­rum a&longs;cendentium. Re&longs;oondeo, veri&longs;&longs;imum e&longs;&longs;e citatum Axioma; & cau&longs;am hujus rei, loquendo de &longs;iphone inver&longs;o non interrupto, ad&longs;ignavimus Propo&longs;it 7. quæ tamen cau&longs;a in ca&longs;u præ&longs;enti de &longs;iphone interrupto non militat; ideoque ulte­riùs quæritur, quænam &longs;it hujus rei cau&longs;a in &longs;iphone interru­pto, atque adeo in no&longs;tro ca&longs;u. Difficultatem agnovit Mer­&longs;ennus in Phænom. Hydraul. Propo&longs;it 38. &longs;ed non &longs;olvit.

Solutio Au­ctoris ratio num dubi­tandi circæ &longs;iphonem in terruptum.

Dico igitur, cau&longs;am e&longs;&longs;e, quòd aquæ pars unaquæque premitur aquâ &longs;upra ip&longs;am exi&longs;tente ad perpendiculum, &longs;i aqua illa &longs;it de&longs;cendens in aliquo, (hoc e&longs;t, &longs;i nullum impedimen­tum extrin&longs;ecum habeat, quò minùs de&longs;cendere po&longs;&longs;it) pro­ut a&longs;&longs;eruimus &longs;uprà cap. 1. hujus Protheoriæ 4. Propriet. 4. ex Archimede. Illa igitur aqua, cujus perpendiculum e&longs;t majus, &longs;eu longius, magis premitur, majoremque habet vim ad de­&longs;cendendum, &longs;i patet aditus ad de&longs;cendendum. Quoniam igitur aperto epi&longs;tomio E tubi EF, in I. Fig. & aperto ore tubi recti DE, in II. Fig. patet aditus aquæ ad defcendendum; de&longs;cendet illa nece&longs;&longs;ariò, innato pondere, & intra tubum, propter partes continuatas, fundabit &longs;uum perpendiculum; & nè vacuum detur, trahet &longs;ecum aërem, & coget aquam oppo­fitorum va&longs;orum a&longs;cendere, &longs;uumque &longs;imiliter perpendiculum fundare. Nunquam tamen a&longs;cendere pote&longs;t aqua ad majus &longs;patium in tubo BK, & crure AB, quàm de&longs;cenderit per tu­bum EF, & DE; quia &longs;i a&longs;cenderet ad majus &longs;patium, jam perpendiculum aquæ a&longs;cendentis majus e&longs;&longs;et, quàm perpen­diculum aquæ de&longs;cendentis, ac proinde illa vinceret hanc, & retraheret. Itaque &longs;i vas C haberet in fundo foramen angu­itum &longs;ine tubo adnexo, nihil aquæ efflueret, quia nullum fun­dare po&longs;&longs;et perpendiculum; ideoque nihil a&longs;cenderet per tu- bum BK. Si idem vas C haberet foramen laxum, efflueret quidem aqua, &longs;ed &longs;imul &longs;ubintraret aër, &longs;icque ce&longs;&longs;ante metu vacui, aut di&longs;continuitatis corporum, nihil a&longs;cenderet per tu­bum KB. Si tubus EF e&longs;&longs;et brevictubo BK, de&longs;cenderet aqua u&longs;que ad orificium F, & ad æquale &longs;patium a&longs;cenderet per tu­bum BK; &longs;ed facta hac æqualitate, &longs;i&longs;teretur utrimque aqua, propter perpendiculorum æqualitatem. Si tubus EF e&longs;&longs;et præcisè æqualis tubo BK; de&longs;cenderet &longs;imiliter aqua u&longs;que ad o&longs;culum F, & ex parte altera a&longs;cenderet u&longs;que ad K, &longs;ed neutra ulteriùs pergeret, propter cau&longs;am jam dictam de per­pendiculorum æqualitate. Si denique tubus EF, e&longs;t vel pau­lulum longior tubo BK, de&longs;cendit aqua u&longs;que ad F, & cùm nullum inveniat impedimentum quò minùs egrediatur, nec detineatur ab altera parte à perpendiculo æquali; effluit inna­to pondere ex F, & ut vitetur vacuum, corporumque di&longs;con­tinuitas, &longs;ecum rapit & aěrem va&longs;is KA, & aquam tubi BK, va&longs;i&longs;que B.

Notandum tamen e&longs;t, quod jam &longs;æpius inculcavi, alti­tudinem perpendiculi aquæ tubi BK &longs;umendam e&longs;&longs;e à &longs;upre­ma aquæ va&longs;is B, &longs;uperficie, u&longs;que ad &longs;upremam &longs;uperficiem aquæ eju&longs;dem tubi.

Quod diximus de perpendiculis tuborum BK & EF, in­telligi etiam debet de perpendiculis AB, & DE alterius Machinæ.

Propo&longs;itio XIV.

Cau&longs;am ad&longs;ignare cur in Fonte Heronis perpendicu­lum aquæ de&longs;cendentis debeat e&longs;&longs;e longius perpendi­culo aquæ a&longs;cendentis.

INfrà Parte 2. Cla&longs;&longs;e 1. cap. 2. Machina 1. Fontem damus,

eumque Fontem Heronis in va&longs;is immediatis appellamus; & ita con&longs;truitur. Vas fit oblongum NOPQ, diaphra­gmate &longs;eu &longs;epto IM di­&longs;criminatum in duo rece­ptacula &longs;eu va&longs;a, NIMQ &longs;uperius, & IMOP inferius. Superius vas labrum habet aliquantulùm concavum BD, in eoque foramen K. In hac Machina di&longs;ponun­tur tubi &longs;eu canales ABC, DEL, & GF, uti figura mon&longs;trat, & fu&longs;ius dicitur loco citato; impletoque receptaculo NIMQ aquâ, aliove liquore per K fora­men, & diligenter clau&longs;o foramine K, infunditur aqua labro BD, quæ per canalem DEL de&longs;cendens in vas vacuum IOPM, premit ibidem aërem, atque expellit per tubum FG in vas plenum NIMque aër verò ibidem pre­mens aquam, expellit ip&longs;am per tubum CBA &c. prout fu­fiùs dicitur loco citato. Similes alias Machinas damus ibidem Machina 2. 3. & 4. Advertimus autem in Notis ad dictam Machinam 1. tubum DEL, per quem aqua de&longs;cendit, longio­rem e&longs;&longs;e debere tubo CBA, per quem afcendit aqua. Præ­dicto Heronis fonti omnino &longs;imilis e&longs;t Lucerna, quam attuli­mus &longs;uprà Protheoria 2. §. 5. in qua tamen errorem aliquem Heronis notavimus circa claviculam R, diximu&longs;que prædi­ctam claviculam debere de&longs;inere in tubulum paulò longiorem, quàm e&longs;t tubulus OX. Quærimus nunc hujus rei cau&longs;am.

Fons Hero­nis in va&longs;is immediatis

Dicam quod &longs;entio. Vt autem meliùs mentem meam in­telligas, Lector, recole quæ diximus &longs;uprà Propo&longs;it. 2. hujus ca- pitis de Proprietatibus &longs;iphonis erecti; ubi a&longs;&longs;eruimus, aquam cruris longioris de&longs;cendentem elevare, atque expellere aquam cruris brevioris, cogereque in altum exilire, &longs;i o&longs;culum &longs;it an­gu&longs;tum. Cau&longs;am ejus rei diximus e&longs;&longs;e exce&longs;&longs;um perpendi­culi aquæ de&longs;cendentis atque pellentis, &longs;upra perpendiculum aquæ a&longs;cendentis & pul&longs;æ. His &longs;uppo&longs;itis, ita di&longs;curro. Fons Heronis, & Lucerna ad ip&longs;ius &longs;imilitudinem con&longs;tructa, omne&longs;­ que Machinæ &longs;imili artificio adornatæ, nihil aliud &longs;unt, quàm &longs;iphon erectus interruptus. Nam tubus DEL habet rationem cruris erecti, per quod aqua de&longs;cendit; tubus verò CBA ha­bet rationem alterius cruris erecti, per quod aqua a&longs;cendit; Aqua de&longs;cendens per DEL, cogit a&longs;cendere aquam per CBA, mediante aëre aquam premente. Sicut igitur in &longs;iphone erecto non interrupto, ut aqua de&longs;cendens per unum crus, expellat aquam a&longs;cendentem per alterum, perpendiculum aquæ de­&longs;cendentis majus debet e&longs;&longs;e perpendiculo a&longs;cendentis, ideo­que crus, per quod aqua de&longs;cendit, longius debet e&longs;&longs;e, quàm crus per quod a&longs;cendit, alioquin non effluet aqua a&longs;cendens; ita in ca&longs;u noftro tubus DEL debet e&longs;&longs;e longior tubo ABC, ut perpendiculum aquæ de&longs;cendentis majus fit, quàm perpendi­culum aquæ a&longs;cendentis. Tantò autem, cæteris paribus, al­tiùs a&longs;cendet aqua per tubum ABC, quantò is magis &longs;upera­ bitur à tubo DEL. Itaque hîc etiam, uti alibi ubique in hydraulicis, non aquarum copiâ, fed perpendiculo pugnatur ac vincitur.

Perpendicu lum aquæ eadentis in fonte Hero­nis cur exce dere debeat perpendicu lum aquæ a&longs;cendentis

Perpendicu lo, non aqua­rum copia, pugnatur, ac vincitur in hydrau­licis.

Hæc mea e&longs;t opinio: cui meliùs quid occurrerit, profe ratin medium; nullus enim adhuc rei huius cau&longs;am attigit, quod &longs;ciam. Non nego tamen, & aëris prementis copiam ab aquæ de&longs;cendentis copia maiore vehementiùs pre&longs;&longs;am ac pul&longs;am, & o&longs;culi A, vel potiùs totius tubuli CBA angu&longs;tiam, multùm conferre ad aquam altiùs ex ofculo A eiaculandam, tum ob urgentis impetus incrementum, tum ob decrementum gravi­tatis aquæ deor&longs;um in tubo, po&longs;t egre&longs;&longs;um è tubo, nitentis. Ne­go tamen, erupturam aquam è tubo CBA, &longs;i longior is fuerit tubo DEL; idque ob rationem a&longs;&longs;ignatam.

CAPVT III.

De Proprietatibus Aquæ fluentis per tubos.

PROOEMIVM.

Tubus quid &longs;it.

TVbos hîc appello Columnas concavas, &longs;ive cylindri­cæ illæ &longs;int, &longs;iv&ecedil; pri&longs;maticæ. Hos in Machinis Par-

te 2. pa&longs;&longs;im voco in differenter tubos, &longs;iphones, ca­nales, fi&longs;tulas, & alijs etiam nominibus. Foramen per quod effluit aqua tubo contenta, appellat Hero Alexandrinus in &longs;uis Pnevmaticis, os, o&longs;culum, & orificium; Marinus verò Mer&longs;ennus in hydraulicis &longs;uis Phænomenis vocat lumen. Hoc foramen fieri pote&longs;t vel in ba&longs;e tubi, vel in latere. Fieri etiam po&longs;&longs;unt tubi conici ad in&longs;tar conorum truncatorum, quo­rum os &longs;eu lumen &longs;it vel in vertice, vel in ba&longs;i coni, prout ap­paret in figuris Propo&longs;itionis primæ &longs;equentis.

Tubi fora­men, os, o&longs;culum, lumen, idem &longs;unt.

Tuborum variæ for­mæ.

Tub' &longs;em­per plenus.

Tubi per quos fluit aqua, aut &longs;unt &longs;emper pleni, aut non &longs;unt &longs;emper pleni. Tubum &longs;emper plenum vocamus, quem fons aut vas aliquod &longs;uperimpo&longs;itum ita implet per orificium &longs;uperius, ut dum aqua effluit per inferius, &longs;emper tamen plenus maneat v&longs;que ad &longs;uperius orificium; quod fit, dum tantum,

aut plùs influit, quàm effluit. Tubum non &longs;emper plenum appellamus, qui, dum effluit aqua per foramen, paulatim exin­anitur, nullâ interim aliâ influente aquâ.

Tubus non &longs;emper ple nus.

Tubi quicunque, per quos aqua decurrit, aut &longs;unt erecti perpendiculariter, aut inclinati, aut pro&longs;trati horizontaliter. Priores vocabimus verticales; medios, inclinatos; ultimos horizontales.

Tubi verti ales.

His ita expo&longs;itis, nunc afferemus varia Phænomena, id e&longs;t, apparentias &longs;eu ob&longs;ervationes circa fluxum aquæ per tu­bos verticales, cuiuscunque formæ & conditionis; ex quibus deinde deducemus varia Problemata atque Theoremata; quo­rum multa applicari etiam poterunt fluxui aquæ per &longs;iphones, de quibus cap. præcedente. Agimus autem hoc Capite &longs;olùm de fluxu aquæ per tubos ex foramine in ba&longs;i facto; de fluxu enim ex foramine facto in latere, loquemur capite &longs;equenti.

PROPOSITIO I.

PHÆNOMENON I.

Aqua decurrit per tubi verticalis foramen ba&longs;is in&longs;tar Columnæ aqueæ, cuius ba&longs;is æqualis foramini, altitudo perpen­diculares à ba&longs;i erectæ; &longs;ive tubi &longs;int &longs;emper pleni, &longs;ive non.

SInt tubi verticales, AB, cuiuscunque figuræ, haben­ tes foramen B in ba&longs;i. Dico aquam decurrere per foramen B in&longs;tar columnæ aqueæ AB, cuius ba&longs;is e&longs;t foramen B, altitudo verò perpendiculares BA, a ba&longs;i erectæ, &longs;ive tubi &longs;int &longs;emper pleni, &longs;ive non. Patet expe­rientia, & &longs;equitur ex dictis Cap. 1. Propriet. IV. & ex doctrina Archimedis lib. 1. de In&longs;identibus humido, Suppo&longs;itione I.

Aqua per foramen ba &longs;is tuborum verticalium fluit in&longs;tar­colu mnæ a­queæ.

Ac primò quidem, &longs;i tubus e&longs;t columnaris, & tota ba&longs;is e&longs;t aperta, prout e&longs;t tubus primæ, ex appo&longs;itis, Figuræ ad &longs;ini&longs;tram; manife&longs;ta e&longs;t experientia: videmus enim totam aquam &longs;imul paulatim de&longs;cendere, &longs;i tubus non e&longs;t &longs;emper plenus. Idem au­tem fit, &longs;i tubus e&longs;t &longs;emper plenus; de&longs;cendit enim &longs;emper nova & nova columna aquea, & effluit ex ba&longs;i B.

Si autem tubus e&longs;t quidem columnaris, &longs;ed aqua non ef­fluit per totam ba&longs;im apertam, &longs;ed per foramen in ba&longs;i factum, prout apparet in &longs;ecunda Figura; item &longs;i e&longs;t conoidalis in&longs;tar co­ni truncati inver&longs;i, prout in tertia Figura apparet, idem contin­git: Nam videmus, effluente aquâ per B, &longs;ub&longs;idere paulatim aquam A, incumbentem ip&longs;i B ad perpendiculum, fierique; fo&longs;&longs;am in &longs;uperiori &longs;uperficie vbi A, reliquas verò partes collaterales e A, f A, confluere intra fo&longs;&longs;am; ad eum modum, quo in arenarijs horologijs, dum è &longs;uperiori phiala defluit arena per foramen diaphragmatis in inferiorem phialam, efficitur fo&longs;&longs;a in &longs;uperiori arenæ &longs;uperficie, in eamque incidunt circumia centes arenæ par­tes. Quod &longs;ignum e&longs;t, aquam de&longs;cendere & effluere per B in modum columnæ AB. Et hoc quidem contingit apertè in tu­bis non &longs;emper plenis, ut experientia docet. Idem autem con­tingere nece&longs;&longs;e e&longs;t in tubis &longs;emper plenis; ni&longs;i quòd in his de&longs;cen­dat &longs;emper nova ac nova columna aquea.

Quod diximus de tubis columnaris & conoidalis figuræ; intelligendum etiam e&longs;t de columnis aliarum figurarum, pro­pter paritatem rationis, dummodo &longs;int verticaliter erecti.

Si foramen e&longs;t quidem in ba&longs;i tubi (cuiuscunque figuræ & conditionis &longs;it tubus,) at &longs;uperior aquæ fuperficies non &longs;u­pereminet foramini ad perpendiculum, ut in appo&longs;ita Figura ap­paret; adhuc effluit aqua columnaliter, &longs;eu in&longs;tar columnæ AB, cuius ba&longs;is e&longs;t foramen B. altitudo perpendiculares à ba&longs;i u&longs;que ad &longs;ummitatem i&longs;tius aquæ; quæ ba&longs;i perpendiculariter imminet. Ratio de&longs;umitur ex dictis cap 1. Proprietate IV.

Corollarium.

EX his colligitur, &longs;olam columnam aqueam AB premere &longs;upra foramen B, quoniam illa &longs;ola ad perpendiculum im­minet ba&longs;is foramini.

Columna aquea &longs;ola premit &longs;u­pra foramen per quod ef­fluit.

Propo&longs;itio II. Phænomenon II.

Per tubos tam &longs;emper, quàm non &longs;emper plenos æqua­lis altitudinis, & æqualium foraminum, effluit æqualis aquæ copia, eodem vel æquali tempore, cuiuscunque capaci­tatis & formæ &longs;int tubi.

ALtitudo tuborum, tum hac, tum &longs;equentibus Propo&longs;itioni bus, &longs;umitur à lumine &longs;eu foramine, u&longs;que ad &longs;uperius orifi­cium, includendo etiam ip&longs;ius foraminis altitudinem &longs;eu cra&longs;­&longs;itiem, ab interiori ad exteriorem &longs;uperficiem ba&longs;is, &longs;i in ba&longs;i &longs;eu fundo &longs;it foramen.

Tuborum altitudo vde &longs;umatur

Sint igitur, ut antea, tubi, AB, æquè ali, & æqualium fo­raminum B, &longs;ive &longs;emper pleni, &longs;ive non &longs;emper pleni, cuius­cunque capacitatis, & figuræ. Dico, per foramina illa effluere eodem, vel æquali tempore, æqualem aquæ copiam. Quoni­am enim, per præcedentem Propo&longs;itionem, &longs;upra foramem B in omnibus æqualis aquæ copia æquali vi premit, nempe colu­mna aquea AB, cuius ba&longs;is e&longs;t foramen B in omnibus æquale, ut &longs;upponitur; & altitudo e&longs;t perpendicularis AB, in omnibus itidem æqualis; nece&longs;&longs;e e&longs;t, æqualem aquæ copiam eodem tem­pore decurrere ex omnibus: ubi enim omnia &longs;unt æqualia, ef­fectus &longs;unt æquales.

Corollarium I.

Aquæ flux' è foramine va&longs;is non e&longs;t celerior pro­pter va&longs;is capacitem.

COlligitur hinc, ad aquæ effluxum maiorem, aut celeriorem è foramine eodem, aut æquali, nihil facere capacitatem va&longs;is aut tubi; adeo ut &longs;i totus Oceanus e&longs;&longs;et inclu&longs;us in uno tubo, aut va&longs;e, & in altero modica aqua, vterque tamen tubus e&longs;&longs;et æquè altus, & haberet æqualia foramina; æqualis aquæ copia ex vtroque efflueret eodem, vel æquali tempore.

Corollarium II.

COlligitur præterea, per foramina æqualia in eadem ba&longs;i eius­dem tubi, æqualem aquam effluere eodem tempore. Sed de hoc agemus infrà cap. 5. Propo&longs;. 1.

Propo&longs;itio III. Phænomenon III.

Per tubos tam &longs;emper, quàm non &longs;emper plenos æqua­lium luminum, &longs;ed inæqualium altitudinum, effluit eodem, vel æquali tempore, inæqualis aquæ copia.

NEmpe per tubum magis altum maior, & per tubum minùs altum, minor. Ratio e&longs;t, quia &longs;upra lumen altioris tubi maior aquæ copia, & maiori vi ac celeritate; & &longs;upra lumen mi­noris minor, & minori vi ac celeritate premit, nempe aquea co­lumna magis aut minùs alta.

Propo&longs;itio IV. Phænomenon IV.

Per tubos &longs;emper, & non &longs;emper plenos inæqualium luminum, &longs;ed æqualium altitudinum, effluit eodem, vel æ­quali tempore, inæqualis aquæ copia.

NEmpe per maius lumen maior, & per minus minor. Ratio e&longs;t eadem, quia &longs;cilicet &longs;upra maius lumen premit maior a­quæ copia, & maiori vi; & &longs;upra minus minor, & minori vi, &longs;ci­licet columna aquea æquè alta, &longs;ed non æquè cra&longs;&longs;a.

Pori&longs;ma.

PEr tubos vtro&longs;que, hoc e&longs;t, tam &longs;emper plenos, quàm non &longs;emper plenos, inæqualium luminum, & inæqualium altitu- dinum, effluere pote&longs;t eodem, vel æquali tempore, æqualis, & inæqualis aquæ copia, hoc e&longs;t, æqualis per aliquos, inæqualis per alios. Ratio e&longs;t, quia defectus luminis in vno pote&longs;t &longs;up­pleri per altitudinem; & defectus altitudinis in altero pote&longs;t &longs;up­pleri per lumen, vt experientia etiam docet.

Propo&longs;itio V. Phænomenon. V.

Per tubos æquè altos, & æqualium luminum, non &longs;em­per plenos, fluit eodem tempore æqualis aquæ copia; &longs;ed tantò fluit vnus diutiùs altero, quantò plùs aquæ continet vnus quàm alter.

PAtet ex hactenus dictis, & experientiâ, nec indiget alia pro­batione. Ille autem plùs aquæ continet, qui amplior e&longs;t: loquimur enim hîc de ijs tubis, quorum orificia &longs;unt æqualia, etiam&longs;i ip&longs;i tubi &longs;int inæqualis amplitudinis.

Corollarium.

ERgo & per æqualia foramina ba&longs;is eiusdem tubi, æqualis eo­dem tempore effluit aqua. Vide cap. 5. Propo&longs;. 1.

Propo&longs;itio VI. Phænomenon VI.

Per tubos non &longs;emper plenos, & non æquè altos, æqua­lium tamen luminum, eodem vel æquali tempore non fluit æqualis aquæ copia.

EAdem e&longs;t ratio de quibu&longs;cunque va&longs;is. Patet experientiâ,

quæ mon&longs;trat, ex altiori tubo &longs;eu va&longs;e fluere maiorem aquæ, copiam tempore eodem, vel æquali, quàm ex tubo vel va&longs;e mi­nùs alto habente lumen æquale. Et ratio e&longs;t, quia in altiori magis preinitur aqua &longs;upra lumen, quàm in minùs alto. Vide Propo&longs;it. III.

Vas magis plenum plus aquæ effun­dit per fora­men, quàm minus ple­num.

Pori&longs;ma I.

SEquitur hinc, ex eodem tubo, aut va&longs;e, non &longs;emper pleno, æqualibus temporibus, non æqualem effluere aquam ex eo­dem foramine, &longs;ed in &longs;ecundo tempore minorem, quàm in pri­mo, & in tertio minorem, quàm in &longs;ecundo, & &longs;ic deinceps. Vide Propo&longs;it. 24.

Pori&longs;ma II.

SEquitur præterea, &longs;patia quæ æqualibus temporibus evacu­antur in dicto ca&longs;u, non e&longs;&longs;e æqualia, &longs;ed &longs;ecundum &longs;patium corre&longs;pondens &longs;ecundo tempori, e&longs;&longs;e minus primo; & tertium &longs;patium corre&longs;pondens tertio tempori, minus &longs;ecundo; & &longs;ic de cæteris. Vide ibidem.

Propo&longs;itio VII. Phænomenon VII.

Tubus altitudine quadrupedalis, cui pro ba&longs;is diametro pedis Pari&longs;ien&longs;is vncia, aquâ &longs;emper plenus, effundit per lumen lineare in ba&longs;i &longs;itum aquæ libram &longs;patio tredecim mi­nutorum &longs;ecundorum temporis.

OB&longs;ervavit hoc Marinus Mer&longs;ennus, ut a&longs;&longs;erit ip&longs;e in Hydrau­licis &longs;uis Phænomenis Propo&longs;it. 1.

Annotatio I.

VTitur autem Mer&longs;ennus tam pede, quàm libra (heminam vocat ip&longs;e) Pari&longs;ien&longs;i. Et pedem quidem, uti in aliarum Nationum pedibus fie­ri&longs;olet, dividit in duodecim æquales partes, quas uncias appellat; & quamlíbet vnciam &longs;ubdividit in alias duodecim partes æquales, quas ip&longs;e lineas, alij &longs;crupulos vocant. Vncia, &longs;eu duodecima pedis pars ap­pellatur etiam pollex; ab aliquibus verò cum Mer&longs;enno digitus: rectiùs tamen digitus e&longs;t decima &longs;exta pars pedis, vt con&longs;tat ex Vitruvio, Iulio Frontino, Columella, & alijs apud Villalpandum tom. 3. Apparatus Part. 2. lib. 3. cap. 16.

Pes dividitur in vnci­ 12. & di­gitos 16.

Linea e&longs;t duodecima pars vncia.

Minuti &longs;e­undi dura tio, quanta &longs;it.

Minutum &longs;ecundum e&longs;t &longs;exage&longs;ima pars unius minuti primi, &longs;eu3600. pars vnius horæ, & proximè re&longs;pondet lento arteriæ &longs;eu cordis pul&longs;ui, ut Mer&longs;ennus a&longs;&longs;erit.

Annotatio II.

NOn e&longs;t nece&longs;&longs;e, ut ba&longs;is tubi quadrupedalis, quo fit ob&longs;ervatio, &longs;it

pedalis, aut uncialis, aut alterius determinatæ amplitudinis atque capacitatis, &longs;ed &longs;ufficit quantalibet latitudo tubi, dummodò foramen &longs;it lineare. Ratio e&longs;t, quia; ut vidimus Propo&longs;itione 2. huius capitis, ea­dem aquæ quantitas eodem, vel æquali tempore, fluit ex tubo &longs;emper pleno quantum vis arcto, vel lato, dummodò foramen &longs;it lineare, & al- titudo &longs;it quadrupedalis; adeo ut &longs;i totus Oceanus, ut &longs;uprà innuimus in Corollario I. Propo&longs;it. 2. e&longs;&longs;et inclu&longs;us va&longs;i quatuor pedum æltitudinis & luminis linearis, non effunderet ni&longs;i vnicam heminam &longs;eu libram Pari­&longs;ien&longs;em &longs;patio tredecim minutorum &longs;ecundorum; Si tamen addas, inquit Mer&longs;ennus, pro horis &longs;ingulis vnam quadrage&longs;imam quintam libræ partem, quâ&longs;olet aqua marina &longs;uperare aquam fontium ac fluviorum in pondere, & con&longs;equenter vi premendi, & vehementiùs è foramine e­rumpendi.

Vas ampli­us non effu­dit plùs a quæ per fora men, quàm minù, am-

plum, &longs;i al­titudo & fo­ramè æqua lia &longs;unt.

Annotatio III.

QVod dicit Mer&longs;ennus de tubo quadrupedali, & de libra aquæ, in­telligi debet &longs;olùm loquendo de pede & libra Pari&longs;ien&longs;i, non ve­rò aliarum Nationum, &longs;ed unu&longs;qui&longs;que in &longs;ua natione experientiam facere debet, ut &longs;ciat quantum aquæ, quanto tempore, ex quantæ al­ titudinis, quantique foraminis tubo, ejiciatur. Cæterùm cùm diffe­rentia pedum, atque librarum apud diver&longs;as Nationes tanta &longs;it, quan­ta ip&longs;arum Nationum; cum&queacute; difficulter exacta men&longs;ura unius certi ac determinati pedis, v. g. Romani, aut Pari&longs;ien&longs;is, ad alias Nationes tran&longs;mitti libris impre&longs;&longs;is po&longs;&longs;it, quòd chartæ priùs madefactæ, ac dein­de &longs;iccatæ, non reddant fideliter longitudinem linearum impre&longs;&longs;arum; nolo diutiùs huic rei inhærere, comparando pedem Pari&longs;ien&longs;em cum pedibus aliarum Nationum. Faciat, qui volet, aut cujus intere&longs;t, experientiam, ut videat quanto tempore tubus quatuor pedum regio­nis &longs;uæ ejiciat per lumen lineare aquæ libram itidem regionis &longs;uæ. De exacta pedis Romani men&longs;ura, & de modo tran&longs;mittendi illum ad alias nationes, di&longs;&longs;eram in Pantometro Kircheriano Lib. 1. Technico Parte 2. Cap. 4.

Podis men­&longs;ura diver­&longs;a apud di­ver&longs;as na­tiones.

Propo&longs;itio VIII. Phænomenon VIII.

Aquæ fluentes ex tubis tam &longs;emper, quàm non &longs;em­per plenis æqualium foraminum, &longs;ed inæqualium altitudinum, habent rationem &longs;ubduplicatam altitudinum tuborum; habentque dicti tubi duplicatam rationem aquarum, quas fundunt.

RAtio &longs;eu proportio duplicata quæcunque, e&longs;t ratio quæ­ cunque &longs;implex &longs;emel repetita, &longs;eu bis continuè &longs;umpta: quemadmodum ratio quæcunque triplicata, quadruplicata, &c. e&longs;t ratio quæcunque &longs;implex bis, ter &c. repetita, &longs;eu ter, quater continuè &longs;umpta. Exemplum. Inter 2 & 1 reperitur ratio dupla; hæc ratio &longs;i &longs;emel repetatur, &longs;eu adhuc &longs;emel accipiatur, hoc e&longs;t, &longs;i bis continuè &longs;umatur hoc modo, 4, 2, 1; erit inter 4 & 1 ratio &longs;eu proportio duplicata illius pro­portionis, quæ e&longs;t inter 2 & 1, quandoquidem inter 4 & 1 reperitur ratio dupla &longs;emel repetita, &longs;eu bis continuè &longs;umpta, hoc e&longs;t, duplicata, &longs;cilicet &longs;emel inter 4 & 2, & ite­rum inter 2 & 1. Similiter inter 8 & 1 e&longs;t ratio triplicata il­lius, quæ e&longs;t inter 2 & 1, quia inter 8 & 1, intercedit ter ratio dupla, nempe 8 ad 4, 4 ad 2, 2 ad 1. Sic 16 ad 1 ha­bet rationem quadruplicatam, & 32 ad 1 rationem quintu­plicatam rationis illius, quam habet 2 ad 1. Aliud exemplum. Inter 6 ad 4 reperitur ratio &longs;e&longs;quialtera &longs;implex; hæc ratio duplicatur, &longs;i adhuc &longs;emel repetatur, &longs;eu &longs;i bis continuè &longs;uma­tur, ut apparet in his numeris 9, 6, 4: nam quia ut 6 ad 4, ita 9 ad 6; ideo inter 9 & 4 bis reperitur ratio &longs;e&longs;quialtera. Si verò eadem ratio &longs;e&longs;quialtera bis repetatur, &longs;eu ter conti­nuè ponatur; erit inter extremos terminos ratio &longs;e&longs;quialtera triplicata, ut apparet in his numeris, 13, 9, 6, 4; quam pro­portionem ab&longs;que fractione habebis, &longs;i duplicaveris ho&longs;ce numeros &longs;ic, 27, 18, 12, 8: nam ut 12 continet 8 &longs;emel cum dimidio, ita 18 continet 12 &longs;emel cum dimidio, & 27 etiam continet 18 &longs;emel cum dimidio.

Duplicatproportio quæ.

Subduplica ta proportio quæ.

Ex ratione duplicata, triplicata, quadruplicata, &c. facilè intelligitur ratio &longs;ubduplicata, &longs;ubtriplicata, &longs;ubquadruplica­ta, &c. Nam per rationem &longs;ubduplicatam intelligimus dimidium rationis duplicatæ. Verbi gratia, 4 ad 1 habet rationem dupli­catam rationis duplæ; 2 ad 1, aut 4 ad 2, con&longs;tituunt dimidium rationis 4 ad 1; ideo 2 ad 1, & 4 ad 2, habent rationem &longs;ub­duplicatam. Similiter 9 ad 4 habet rationem duplicatam rationis &longs;e&longs;quialteræ; dimidium talis rationis e&longs;t 9 ad 6, vel 6 ad 4; ideo 9 ad 6, & 6 ad 4 habent rationem &longs;ubduplica­tam prædictæ rationis &longs;e&longs;quialteræ.

His explicatis, e&longs;to tubus AB unius pedis, &

tubus CD quatuor pedum, æqualium foraminum, & uterque &longs;eu &longs;emper, &longs;eu non &longs;emper plenus; qui quidem eodem, vel æquali tempore inæqualem effundunt aquæ copiam, nempe major majorem, & minor minorem, ut con&longs;tat ex Propo&longs;itione III. præcedenti. Dico, aquam tubi CD, ad aquam tubi AB eodem aut æquali tempore effu&longs;am, ha­bere rationem &longs;ubduplicatam tuborum, hoc e&longs;t, aquam effluentem è tubo CD e&longs;&longs;e duplam aquæ effluentis è tubo BA. Etidem dicendum e&longs;t de qua­cunque alia ratione &longs;eu proportione; ut &longs;i unus tu­bus &longs;it 9 pedum, alter unius pedis, erit aqua ma­joris ad aquam minoris, ut 3 ad 1. Con&longs;tat ex ob&longs;ervatione, ut a&longs;&longs;erit Mer&longs;ennus in &longs;uis Hydraulicis, Propo­&longs;it. 2 po&longs;t medium. Ratio phænomeni dependet ex velocita­te aquæ de&longs;cendentis & effluentis ex tubo CD, &longs;upra veloci­tatem æquæ de&longs;cendentis & effluentis ex tubo AB; de qua vide Propo&longs;it. IX. & X. &longs;eque ubi dicemus, illam ad hanc e&longs;&longs;e du­plam, hoc e&longs;t, &longs;ubduplicatam altitudinum tuborum haben­tium æqualia foramina; quo demon&longs;trato, demon&longs;trabimus deinde Propo&longs;itione XI. hanc præ&longs;entem Propo&longs;itionem.

Aquæ dupli catam ra­tionem ha­bent tubo­rum æqua­lium lumi­num, at in­æqualium altitudinum.

Pori&longs;ma I.

COlligitur ex his, tuborum æqualium foraminum altitudi­ nes debere e&longs;&longs;e in duplicata ratione aquarum inæqualium quas debent eodem tempore fundere. Verbi gratia, tubus pe­dalis determinato tempore dat unam aquæ libram ex &longs;uo fo­ramine; ut alius tubus ex æquali foramine æquali tempore det duas libras, debet habere duplicatam rationem ad illum, nem­pe debet e&longs;&longs;e altus quatuor pedibus. Sic etiam quia tubus qua­tuor pedum per lineare lumen &longs;patio 13 minutorum &longs;ecundo­rum fundit unam libram aquæ, ut diximus Propo&longs;it. VII. ut alius tubus eodem tempore per lumen lineare fundat centum libras, debet habere altitudinem duplicatam 1 ad 100; nem­pe 40000. pedum. Atque hæc e&longs;t conver&longs;a præcedentis Pro­po&longs;itionis.

Tubi æqua­lium lumi­num, at in­æqualium altitudinum, habent du­plicatam ra­tionem aqua­rum.

Pori&longs;ma II.

COlligitur præterea, datis tubis æqualium foraminum, at inæqualium altitudinum, in numeris aut lineis, mediam proportionalem dare aquas; & datis aquis in numeris aut men­&longs;uris, tertiam proportionalem dare tuborum altitudinem æqua­lium foraminum.

Annotatio.

Mediæ, ac Tertiæ pro­portionalis quantitatis in ventio.

DEinventione mediæ ac tertiæ proportionalis magnitudinis in nume­ris ac lineis, agemus infrà in Parergo hujus capitis, & in Arith­metica ac Geometria practica, & in Pantometro Kircheriano lib. 8. cap. 1. Propo&longs;it. 1. & 3.

Propo&longs;itio IX. Theorema I.

Aqua naturali motu de&longs;cendens & effluens per tubos, imitatur leges aliorum gravium naturali motu de&longs;cendentium.

NOn uni mirabile vi&longs;um, inquit Mer&longs;ennus in Phænome­nis Hydraulicis Propo&longs;it. III. quòd non &longs;it eadem ratio quantitatis aquæ fluentis ex tubo pedali, ad aquæ quantitatem ex quadrupedali tubo &longs;alientis, po&longs;ito æquali foramine in utro­que, quæ e&longs;t 1 ad 4; cùm aqua quadrupedalis tubi quadruplo magis, quàm pedalis premere videatur fundum, & aquam ex lumine &longs;alientem. Verùm mirari de&longs;ines, ubi noveris, aquam eo &longs;olùm modo premere, vel ea duntaxat velocitate tubum egredi, quâ moveretur, &longs;i ex eadem tubi altitudine cecidi&longs;&longs;et; adeo ut &longs;it eadem i&longs;tius phænomeni ratio, quæ de&longs;cen&longs;us aquæ per tubum; & eadem hujus de&longs;cen&longs;us, quæ de&longs;cen&longs;us aliorum m; de quo proinde hîc agendum paulò accuratiùs, quan­dem aquæ motus per tubum, & ex tubo eruptio, &longs;equi­ges gravium motu naturali de&longs;cendentium.

Leges gravium naturali motu de&longs;cendentium.

LEges porrò, quas gravia in de&longs;cen&longs;u naturali motu facto &longs;ervant, incredibili cura ac diligentia indagavit & ob&longs;er­vavit Bononiæ P. Joannes Bapti&longs;ta Ricciolus, unà cum alijs è no&longs;tra Societate Patribus, ea&longs;que clari&longs;&longs;imè ac di&longs;tincti&longs;&longs;imè proponit in Tomo 1. Almage&longs;ti &longs;ui novi; fusè quidem lib. 9. &longs;ect. 4. cap. 16. num. 24. breviter verò lib. 2. cap. 21. . Quibus Riccioli ob&longs;ervationibus tantam ego fidem adhibeo, ut licet hîc Romæ ea&longs;dem repetere opportuni&longs;&longs;imum mihi foret, de­mi&longs;&longs;is gravibus per cochleatas Divi Petri in Vaticano &longs;calas alti&longs;&longs;imas, & ab omni aëris commotione liberas, fene&longs;tris in &longs;uper plurimis ab ima ba&longs;i ad &longs;ummitatem u&longs;que refertas; n­luerim tamen me fru&longs;tra fatigare, cùm nec accuratiores ad­hibere me po&longs;&longs;e putem, nec &longs;i adhibeam, alias præter ip&longs;ius leges reperturum me certò &longs;ciam. Ex multis igitur de&longs;cen­&longs;us gravium legibus à Ricciolo locis citatis adductis, duas &longs;e­quentes affero, quoniam eæ propo&longs;ito no&longs;tro &longs;atisfacient.

Gravium naturali motu de­&longs;cendentium leges.

Riccioli di­ligentia ex­acta in ob­&longs;ervandis le gibus gra­viu

I. Gravia naturali motu per lineam perpendicularem in aëre de­ &longs;cendentia, moventur difformi velocitate, eâque majore & majore ver&longs;us finem motus. Lex hæc nota fuit jam inde à viginti & am­pliùs &longs;æculis in Academijs Phy&longs;icorum, præ&longs;ertim Peripateti­corum cum Ari&longs;totele lib. 1. de cœlo cap. 88. inquit Ricciolus, patetque oculis, auribus, & tactui; videmus enim gravia de­&longs;cendentia accelerare magis ac magis motum; audimus vehe­mentiorem &longs;onum, percipimus validiorem percu&longs;&longs;ionem, ex lap&longs;u gravis ac duri è loco altiori, quàm minùs alto.

Lex I. gra­vium de­&longs;cendentium.

II. Motus gravium naturaliter de&longs;cendentium per aërem, cre­ &longs;cit eo incremento velocitatis, quod e&longs;t inter numeros impares ab uni­tate numeratos; &longs;eu ita ut &longs;patia æqualibus temporibus tran&longs;mi&longs;&longs;a &longs;int inter &longs;e, ut quadrata temporum; &longs;eu ita, ut &longs;patia tran&longs;mi&longs;&longs;a certis temporibus habeant inter &longs;e duplicatam proportionem illius, quam ha­bent tempora quibus &longs;patia illa men&longs;urata fuerint. ita Ricciolus lo­cis citatis, & ante ip&longs;um Galilæus Dialogo 2. de Sy&longs;temate Mundi, & Balianus lib 1. de Motu naturali gravium, Propo&longs;it. 6. (licet deinde lib. 4. à pag. 110. ad 113. aliter &longs;tatuat) & po&longs;tip&longs;os Ga&longs;&longs;endus in Epi&longs;tolis de motu impre&longs;&longs;o à motore, & Kirche­rus in Mu&longs;urgia lib. 6. parte 1. Lemmate 3. & con&longs;tat experien­ tiâ. Numeri pariter impares ab unitate numerati, &longs;unt hi: 1, 3, 5, 7, 9, 11, 13, 15 &c. Itaque &longs;i grave aliquod de&longs;cendens con­ficit in primo minuto temporis unum &longs;tadium &longs;patij; in &longs;ecundo minuto conficiet tria ftadia, in tertio quinque, in quarto &longs;e­ptem, &c. Sic enim &longs;patia æqualibus temporibus confecta, erunt inter &longs;e, ut quadrata temporum: nam quadratum mi­nuti primi e&longs;t 1; & quadratum minuti &longs;ecundi, 4; & quadra­tum minuti tertij, 9; & quadratum minuti quarti, 16 &c. ut apparet inappo&longs;ita tabella. Cùm igitur grave de&longs;cen­dens in primo minuto con­ficiat unum &longs;patium, & in &longs;ecundo minuto tria &longs;patia, quæ cum uno conficiunt quatuor; & in tertio minu­to quinque &longs;patia, quæ cum quatuor efficiunt novem; & in quarto minuto &longs;eptem &longs;patia, quæ cum novem efficiunt &longs;exdecim; & in quinto minuto no­vem &longs;patia, quæ cum &longs;exdecim efficiunt viginti quinque, &c. apparet manife&longs;tè, &longs;patia tran&longs;mi&longs;&longs;a à gravi de&longs;cendente, ha­bere &longs;e, ut quadrata temporum æqualium, quibus percurrun­tur illa &longs;patia. In idem autem recidit, &longs;i dicas, &longs;patia tran&longs;­mi&longs;&longs;a certis temporibus habere inter &longs;e duplicatam rationem &longs;eu proportionem illius proportionis, quam habent tempora, quibus &longs;patia illa men&longs;urata fuerint: nam tempus &longs;eu minutum &longs;ecundum ad primum, ut apparet ex tabella præcedente, ha­bet rationem &longs;eu proportionem duplam, &longs;patia verò &longs;ecundo minuto confecta, nempe 1 ad 3 (quæ &longs;imul juncta efficiunt 4) ad primum minutum habent rationem duplicatam rationis plæ. Eadem e&longs;t ratio de cæteris. Lege etiam Mundum &longs;u- terraneum P. Athana&longs;ij Kircheri, ubi in Libro Centro&longs;ophico fu&longs;i&longs;&longs;imè pertractat hanc materiam. Lege præterea Joannem Caramuelem in Cruce Sublimium ingeniorum.

Lex II. gra­vium de­&longs;cendentium.

Numeri pa­riter impa­res ab unita te numera­ti.

TemporaQuadrataSpatia1112433954167525963611

Leges aquæ naturali motu de&longs;cendentis & effluentis per tubos.

SIt jam tubus ABCD, &longs;ive &longs;emper, &longs;ive non &longs;emper plenus, cujus foramen B; tempu&longs;que in quo aqua ab A de&longs;cendit u&longs;que ad B, dividatur in quinque æquales partes, in quarum prima confecerit &longs;patium A1; Dico, eandem aquam in &longs;e­cunda parte temporis conficere &longs;patium A4, in tertia parte A9, in quarta &longs;patium A16, in quinta &longs;patium A25; atque adeo &longs;patia æqualibus quinque temporibus confecta, habere &longs;e ut quadrata quinque illorum temporum, ut ex appo&longs;itis utrimque numeris ap­paret; &longs;eu velocitatem de&longs;cen&longs;us aquæ cre&longs;cere &longs;e­cundùm progre&longs;&longs;ionem numerorum imparium, prout indicant numeri intra tubum notati; &longs;eu &longs;patia confecta habere duplicatam rationem illius, quam habent tempora, ut apparet &longs;i conferas nu­meros externos lateris AB, cum numeris lateris CD. Ratio e&longs;t, quia aqua gravis e&longs;t, & intra tu­bum naturali motu de&longs;cendit, nullumque ade&longs;t impedimentum motus; ergo imitari debet leges aliorum gra­vium naturali motu de&longs;cendentium.

Leges aqua naturali motu de­&longs;cendentis per tubos.

Corollarium I.

COlligitur hinc primò, Aquam è foramine B, tubi &longs;emper pleni, effluere tantâ velocitate, quantâ de&longs;cendit ab A ad B. Ratio e&longs;t, quia ut &longs;upponitur, tantum effluit per B, quan­tum influit per A, & è contrario tantum per A influit, quan­tum per B effluit, & in de&longs;cen&longs;u ab A ad B nulla fit inter­ruptio; ergo tam velociter effluit per B, quàm velociter de­&longs;cendit ab A ad B.

Leges aquæ effluentis per foramen tubi &longs;emper pleni.

Corollarium II.

COlligitur &longs;ecundò, Aquam apud B, in utroque tubo, hoc e&longs;t, &longs;emper & non &longs;emper pleno, tantâ &longs;olùm vi premi prope foramen B, quanta e&longs;t vis &longs;eu pondus aquæ à B u&longs;que ad &longs;u­premam aquæ &longs;uperficiem, id e&longs;t, columnæ aqueæ habentis pro ba&longs;i latitudinem foraminis B, pro altitudine perpendicu­larem à B ad dictam &longs;uperficiem. Ratio e&longs;t, quia &longs;olùm illa columna premit &longs;upra B, per dicta Propo&longs;it. 1. Intellige, ni&longs;i illa aliunde impetum recipiat, ut &longs;i ip&longs;i affunderetur aqua ex tubo inclinato, &longs;ive conjuncto, &longs;ive di&longs;iuncto.

Corollarium. III.

COlligitur tertiò, Leges prædictas de&longs;cen&longs;us gravium &longs;ervari ab aqua in tubo &longs;emper pleno, de&longs;cendendo ab A ad B, &longs;o­lùm in primo de&longs;cen&longs;u, non verò in de&longs;cen&longs;u continuato; item &longs;olùm à prima parte aquæ de&longs;cendentis, non à partibusip&longs;i conti­guis & &longs;ub&longs;equentibus. Explico De&longs;cendat aqua ab A ad B, effi­ciatque de&longs;cen&longs;us columnam aqueam AB, quæ dividatur in vi­ginti quinque æquales partes &longs;ecundùm longitudinem &longs;eu altitu­dinem ab A ad B. Dico, &longs;olùmvige&longs;imam quintam partem, quæ e&longs;t apud B, &longs;ervare prædictam legem de&longs;cen&longs;us gravium cre­&longs;centis in velocitate &longs;ecundùm numeros impares ab unitate numeratos; & quidem illam &longs;olam vige&longs;imam quintam par­tem, quæ e&longs;t in prima columna aquea à de&longs;cen&longs;u aquæ for­mata, non verò quæ e&longs;t in reliquis columnis &longs;ub&longs;equentibus. Ratio e&longs;t, quia cùm nulla pars ex viginti quinque egrediatur è foramine B, quin ingrediatur alia per orificium A priori æqualis; & nulla tran&longs;eat vige&longs;imum quintum & ultimum &longs;pa­tiolum lateris AB, quin eodem tempore tran&longs;eat alia æqualis primum &longs;patiolum, & alia &longs;ecundum, & alia tertium, ac reli­qua omma; nece&longs;sè e&longs;t ut omnes reliquæ partes eâdem velo­citate conficiant &longs;ingula &longs;patiola, qua velocitate ultima con­ficit ultimum; & eâdem velocitate unaquæque pertran&longs;eat &longs;patiolum &longs;ibi corre&longs;pondens, qua velocitate ultima pars per­tran&longs;it ultimum &longs;patiolum. Patet ergo propo&longs;itum.

Lex de&longs;cen &longs;us aquæ è tubo expli­catur.

Corollarium IV.

COlligitur quartò, Aquam è foramine B efflue­ re eò majori velocitate, quò altior fuerit tu­bus. Ratio e&longs;t, quia quò altior e&longs;t tubus, eò ma­jorem velocitatem adquirit prima pars aquæ de­&longs;cendentis in &longs;uo de&longs;cen&longs;u, & con&longs;equenter reli­quæ &longs;ub&longs;equentes; item quò altior e&longs;t tubus, eò altior e&longs;t columna aquea premens apud B. ideo­que eò vehementiùs premit apud idem B. Vide quæ diximus Propo&longs;it. VI.

Aqua è tu­bo eò velo­ciùs effluit, quò altior e&longs;t tubus.

Corollarium V.

COlligitur quintò, Aquam ex æqualibus tubis, quoad fora­ mina & altitudines, effluere æquali velocitate.

Aqua ex æ­qualibus tu­bis quoad o­mnia, æqua­li velocita­te effluit,

Propo&longs;itio X. Theorema II.

Velocitates motus aquæ de&longs;cendentis & effluentis per tubos æqualium foraminum, &longs;ed inæqualium altitu­dinum, habent &longs;ubduplicatam rationem altitudinum.

ESto tubus AB altus uno pede, & alius CD al­tus quatuor pedibus, &longs;ed uterque æqualium foraminum, &longs;ive &longs;emper, &longs;ive non &longs;emper pleni, è quibus effluat aqua inæquali velocitate, juxta di­cta Propo&longs;itione præcedente, Corollario IV. Dico, velocitates motus & effluxus aquarum e&longs;&longs;e in &longs;ub­duplicata ratione tuborum, hoc e&longs;t, aquam per tubum CD de&longs;cendere, & effluere duplo velo­ciùs, quàm per tubum AB. Quoniam enim aquæ motus per tubos æmulatur leges gravium de&longs;cen­dentium motu naturali, per dicta Propo&longs;it. IX. præcedente, hoc e&longs;t, cre&longs;cit in velocitate &longs;ecun- dùm progre&longs;&longs;ionem numerorumim parium ab untate numeratorum; &longs;i aqua in primo tempore dterminato, v.g. in primo minuto de&longs;cendit à C iE, hoc e&longs;t, ab A in B; in &longs;ecundo minuto d&longs;cendet ab E ad D, ac proinde in fine quarti pami mota erit duplo velociùs, quàm in fine prinpalmi; Ergo velocitas motus aquæ per tubum CIad velocitatem aquæ per tubum AB, habet &longs;ulduplicatam rationem altitudinum. Eadem eratio in alijs inæqualibus altitudinibus, quamcuque habeant inter &longs;e proportionem.

Proportio velocitatis motus aquæ de&longs;cenden­tis per tubos: inæqualium foraminum.

Pori&longs;ma.

TUbi ergo duplicatam habent rationem illius, quam habenvelocitates motus de&longs;cendentis, & effluentis per ip&longs;os aquæ

Propo&longs;itio XI. Theorema III.

Cau&longs;am a&longs;&longs;ignare, cur aquæ fluentes per tubos æqua lium luminum, &longs;ed inæqualium altitudinum, habeantra­tionem &longs;ubduplicatam altitudinum tuborum.

Proportio­nis prædicta cau&longs;a.

QUod a&longs;&longs;eruimus Propo&longs;it. VIII. huius capitis, demon&longs;trandum hîc e&longs;t, utfidemibi obligatam &longs;olvamus.

Ratio igitur ad&longs;ignati hîc & ibi Phænomeni e&longs;t, quòd fluxus aquæ per foramen, &longs;eu quantitas aquæ effluentis, pendetvelocitate aquæ eiusdem de&longs;cendentis per tubum, per dictPropo&longs;it. IX. Corollario. I. Velocitas autem illa e&longs;t &longs;ubduplicataaltitudinum, per dicta Propo&longs;it. X; ergo & quantitas aquæ efflu­entis ex tubis inæqualium altitudinum &longs;ubduplicata erit tuborum.

Propo&longs;itio XII. Theorema IV.

Tempora quibus æqualis aquæ quantitas è tubis æqua­lium luminum, &longs;ed inæqualium altitudinum effluit, ha­bent &longs;ubduplicatam rationem tuborum.

Proportio temporum quibus a­qua effluit è tubis.

SItut antea tubus AB vnius, & tubus CD quatuor pedum in altitudine, &longs;ed æqualium luminum, &longs;ive &longs;emper, &longs;ive non &longs;em- per pleni; fluatque ex tubo AB vna libra aquæ &longs;patio duorum minutorum. Dico, eandem aquam effluere ex tubo CD &longs;pa­tio vnius minuti, e&longs;&longs;eque propterea tempus fluxus aquæ ex tubo CD, ad tempus fluxus aquæ ex tubo AB, ut 1 ad 2, quæ e&longs;t ratio &longs;ubduplicata tuborum. Ratio ex dictis patet. Quoniam enim velocitates effluxûs aquæ ex æquali lumine habent &longs;ubduplica­tam rationem tuborum, ut demon&longs;travimus Propo&longs;it. X, fluet in dimidio temporis tantum ex tubo CD, quantum in toto tem­pore ex tubo AB, po&longs;ita æqualitate luminum; ac proinde in toto tempore duplum effluet ex tubo CD. Eadem e&longs;t ratio in alijs tuborum æqualium luminum proportionibus.

Pori&longs;ma.

TUbi ergo quilibet æqualium luminum, &longs;ed inæqualium alti­

tudinum, habent rationem duplicatam temporum, quibus effluit æqualis aquæ quantitas.

Proportio tuborum tempora flxus aquæ.

Propo&longs;itio XIII. Theorema V.

Si tubi, &longs;ive &longs;emper pleni, &longs;ive non &longs;emper pleni, &longs;int eiu&longs;dem altitudinis, &longs;ed inæqualium foraminum, e&longs;t eadem ra­tio aquæ ad aquam, quæ foraminis ad foramen, phy&longs;icè &longs;eu ad &longs;en&longs;um.

ESt Mer&longs;enni in hydraulicis Phænomenis Propo&longs;it. IV. quem &longs;equitur P. Andreas Tacquet in &longs;uis hydraulicis Manu&longs;cri­ptis cap. 4. Phænomen. 7. Ratio e&longs;t, quòd licet ex maiori lu­mine &longs;eu foramine fluat eodem tempore maior aquea columna quàm ex minori, tantòque maior exillo quàm ex hoc, quantò fuerit maius lumen &longs;eu foramen illud quàm hoc, quoad aream, per dicta Propo&longs;it. 4. huius eapitis; tamen vtraque columna fluit ex æquali altitudine tuborum æquâ velocitate phy&longs;icè & ad &longs;en&longs;um.

Proportio quæ ad quam eadem. quæ forami­num in tu­bu æquè ­tis.

Dixi phy&longs;icè & ad &longs;en&longs;um, quia licet gravia eiusdem &longs;pe­ciei, &longs;ed inæqualis molis ac ponderis, ab eodem ad eundem terminum non de&longs;cendant æquè velociter, revera & mathema­ticè, quemadmodum &longs;entiunt Galilæus Dialogo 2. de Sy&longs;tem. Mundi, Ioannes Bapti&longs;ta Balianus lib. 1. de motu naturali gravi­um &longs;olidorum in Præfat. Nicolaus Cabæus lib. 1. Meteoror. textu 17. que 5. & 6. Arriaga di&longs;put. 4. de Generat. &longs;ect. 5. &longs;ub&longs;ect. 3. Ma&longs;trius, Bellutus, & alij (quod ego fal&longs;um exi&longs;timo, mathe. maticè loquendo, cum Patre Ioanne Bapti&longs;ta Ricciolo, qui tom. 1. Almage&longs;ti Novi lib. 2. cap. 21. Propo&longs;it. 2. & lib. 9. &longs;ect. 4. num. 24. a&longs;&longs;erit, duorum gravium eiusdem &longs;peciei & figuræ &longs;ea inæ­qualis molis ac ponderis, ex eadem altitudine momento eodem dimi&longs;­&longs;orum, illud naturali motu citiùs de&longs;cendere ad eundem terminum, quod e&longs;t gravius; ubi etiam Experimenta multa diver&longs;is annis coram multis viris doctis incredibili diligentiâ peracta Bononiæ refert num. 13.) tamen in parvis altitudinibus, quales &longs;unt tuborum in omni ferè ca&longs;u, tam exigua e&longs;t differentia velocitatum, ut pro eâdem &longs;eu æquali cen&longs;eri meritò po&longs;&longs;it.

Pori&longs;ma I.

SEquitur hinc, tubos non &longs;emper plenos, æquales quoad alti­tudines, & ba&longs;es, inæquales tamen quoad foramina, evacua­ri inæqualibus temporibus, hoc e&longs;t, citiùsillum, qui maius ha­bet lumen; e&longs;&longs;eque tempora, quibus evacuantur, inter &longs;e ut lu­mina, hac tamen conditione, vt per foramen maius citiùs efflu­at tota aqua, quàm per foramen minus, tantoque; citiùs per majus, quàm per minus, quantò foramen maius &longs;uperat minus. Atque hoc e&longs;t quod dicemus Propo&longs;it. XVI. &longs;equente, tempora &longs;cili­cetin dicto ca&longs;u e&longs;&longs;e reciprocè vt lumina.

Pori&longs;ma II.

SEquitur præterea, ex tubis non &longs;emper plenis, quorum æ­quales &longs;unt altitudines, at inæquales ba&longs;es, &longs;ed totæ apertæ, effluere totam aquam æquali tempore; quandoquidem vtrobi­que columna aquea, licet inæqualis ponderis ac molis, æquè velociter de&longs;cendit quoad &longs;en&longs;um, per idem &longs;patium.

Propo&longs;itio XIV. Theorema VI.

Tubinon &longs;emper pleni æquè alti, & æqualium. forami­num, &longs;ed inæqualium ba&longs;ium, evacuantur inæqualibus temporibus; e&longs;tque eadem ratio temporum, quæ ba&longs;ium.

PRimum patet per &longs;e, quia cæteris omnibus paribus major a­ quæ quantitas maius requirit tempus ad effluendum, quàm minor. Alterum demon&longs;tratur, vel potiùs explicatur &longs;ic.

Proportio temporum eadem quæ ba&longs;ium tu­borum, quo­ad aquæ fluxum.

E&longs;to tubus AB minoris ba&longs;is, & alius ACBD majoris, uterque cylindricus, uterque æquè altus, & æqualis foraminis B; &longs;itque diameter ba&longs;is BD tripla diametri ba&longs;is B: eritigitur area ba&longs;is BD noncupla areæ ba&longs;is B, quoniam circuli inter &longs;e &longs;unt, ut qua­drata diametrorum, per Propo&longs;it. 2. lib. 12. Element. Euclidis; quadratum autem diametri BD triplo maio­ris diametro B, e&longs;t nonies maius, quàm quadratum diametri B, ut ex Geometria practica patet. Cùm igitur cylindri æquè alti &longs;int inter &longs;e, ut illorum ba&longs;es, ut patet ex eadem Geomet. practica; &longs;equitur, aquam tubi ACBD e&longs;&longs;e noncuplam aquæ tubi AB; ac proinde tempus quo exhauritur per foramen B tubus ACDB, noncuplum erit temporis, quo exhauritur per idem foramen B, tubus AB, quandoquidem cylindrus aqueus ACBD, non magis premit &longs;upra foramen B, quàm cylindrulus aqueus AB, per dicta Propo&longs;it. II. in Corollario, & Propo&longs;it. VII. Annot. 2.

Annotatio I.

MEr&longs;ennus in Hydraulicis Phænomenis Propo&longs;it. 8. ait, con&longs;tare ex ob&longs;ervatione, tubum quadrupedalem, cuius ba&longs;is digitalis, uno mi­nuto temporis totum per lineare lumen exhauriri; tubum verò quadru­pedalem cuius ba&longs;is pedalis, &longs;patio 144. minutorum, &longs;eu duabus horis, &24. minutis. Servatur ergò inter temporaratio ba&longs;ium, vt demon&longs;tra­vimus: Nam cùm pedis longitudo contineat, ex Mer&longs;enni mente, ut vidimus Propo&longs;it. VII. huius Capitis Annot. I. digiti latitudinem duodecies; continebit quadratum pedis quadrata digiti 144. ac proinde ba&longs;is pedalis ba&longs;im digitalem centies quadragies quater continebit, diximus in Propo&longs;itione.

Mer&longs;enni ob&longs;ervatio in tuborum evacuati­one.

Annotatio II.

P. Pauli Ca &longs;ati dubita­tio contra Mer&longs;enni ob&longs;ervatio­nem.

ACuti&longs;&longs;imè notavit hîc P.Paulus Ca&longs;atus in cen&longs;ura huius Operis, po&longs;­&longs;e dubitari de veritate huius Propo&longs;itionis 14. & po&longs;&longs;e aliquem &longs;u&longs;pi­cari, an experimentum allatum à Mer&longs;enno &longs;it in gratiam Propo&longs;itionis confictum. Ratio dubitandi e&longs;t, inquit, quia quo tempore par­vulus tubus AB exhauritur, ex maiori tubo ABCD effluit maior aquæ quantitas eodem tempore, quia &longs;cilicet effluit in progre&longs;&longs;u maiori velocitate ex maiori, quàm ex minori: nam in majore tubo e&longs;t maius aquæ perpendiculum in progre&longs;&longs;u. Fac enim in minori de&longs;cendi&longs;&longs;e &longs;emi&longs;&longs;em; aquæ altitudo e&longs;t &longs;olùm dimidia totius altitudinis: fac ex majori eodem tempore de&longs;cendi&longs;&longs;e æ­qualem aquam, hoc e&longs;t (quia maior ad minorem e&longs;t ut 9. ad 1.) (1/18) totius aquæ; ergo remanet altitudo aquæ in majori (17/18) totius altitudinis; ergo plus aquæ effluet vbi maior e&longs;t velocitas ratione maioris perpendiculi. Debet igitur hæc ratio conciliari cum experimento; & opus e&longs;&longs;et o&longs;tendere, quomodo, licet id veri­ficetur in priotibus cylindrulis aquæ effluentibus, tamen maior illa velocitas priorum compen&longs;entur maiori tarditate po&longs;terio­rum qui habent minus perpendiculum.

Hocidem dicitur de Propo&longs;itione 16, in qua con&longs;ideratur columna vt duodecupla minoris, nulla habita ratione quòd in progre&longs;&longs;u de&longs;cen&longs;us fiunt altitudines valde inæquales: neque vi­detur valere illa argumentatio, facta præci&longs;ione per intellectum; nam &longs;ermo e&longs;t de re prout à parte rei. Hæc P. Ca&longs;atus. Nolui ego in dubium revocare experimentum Mer&longs;enni, quoniam id nefas exi&longs;ti­mavi; nec experimenti cau&longs;am indagare, ac multò minùs demon&longs;tra­tionem afferre, quoniam id ad propo&longs;itum meum nihil conducit. Ratio tamen huius rei &longs;ine dubio refundenda e&longs;t in illam, quam ip&longs;emet Ca&longs;atus in&longs;inuavit.

Propo&longs;itio XV. Phænomenon. IX.

Tempora quibus deplentur tubi non &longs;emper pleni æque lati, &longs;ed non æquè alti, per æqualia foramina, &longs;untin al­titudinum ratione &longs;ubduplicata.

COn&longs;tat enim ex ob&longs;ervatione, inquit Mer&longs;ennus in Hydrau­

licis Phænomenis Propo&longs;it. IX. tubum pedalem 30 &longs;ecundis, quadrupedalem verò latitudinis eiusdem 60 &longs;ecundis, per fora­men lineare, totum exhauriri.

Proportio temporum aquæ fluxus ex tubis, ad eorum alti­tudinem.

Corollarium I.

HInc colligitur primò, tuborum altitudines e&longs;&longs;e in ratione duplicata temporum, quibus deplentur æqualia lumina, vt diximus etiam in Pori&longs;mate Propo&longs;itionis XII. præcedentis.

Corollarium II

COlligitur &longs;ecundò, tam tubos non &longs;emper plenos, quàm tu­bos &longs;emper plenos, juxta eandem rationem tribuere aquas ex æqualibus foraminibus, juxta dicta hac, & octava Propo­&longs;itione.

Propo&longs;itio XVI. Theorema VII.

Tempora quibus evacuantur tubi non &longs;emper pleni &longs;i­miles, & æquales quoad altitudines & ba&longs;es, per lumina &longs;imilia inæqualia, &longs;unt reciprocè vt lumina.

PRopo&longs;itio intelligenda e&longs;t in eo &longs;en&longs;u, quem explicavimus in Porismate 1. Propo&longs;it. 13. præcedentis.

Sint itaque tubi cylindrici ABFQ, &longs;imiles, & æquales in altitudinibus, humido pleni, quorum lumina &longs;int EC, KN, cir­cularia, at inæqualia (eadem e&longs;t ratio de tubis pri&longs;maticis, & foraminibus quadratis, alteriusvè Figuræ;) &longs;itque lumen &longs;eu foramen KN duodecies maius quoad aream, quàm lumen EC. Dico, tempus quo exhauritur tubus AB, per lumen EC, e&longs;&longs;e ad tempus, quo exhauritur tubus FQ, per lumen KN, ut e&longs;t reciprocèlumen KN ad lumen EC; id e&longs;t, tantò plùs tempo­ris requiri ut exhauriatur tubus AB, per foramen EC, quàm tubus FQ, per foramen KN, quantò maius e&longs;t lumen KN, quàm lumen EC, nempe duodecies plùs.

Proportio temporum effluxus a­quæ ad fora mina tubo­rum.

Ad hoc o&longs;tendendum, intelligan­tur &longs;uper luminibus EC, KN, tanquam &longs;uper ba&longs;ibus, cylindri DC, LN, eju&longs;dem altitudinis cum cylindricis tubis AB, Fque Patet ex dictis Propo&longs;it. XIII. præce­dente, Porismate 2, hos duos tubos DC, LN, per lumina EC, KN, eodem &longs;eu æquali tempore exhauriri. Iam &longs;ic. Per Propo&longs;itionem XIII, huius capi­tis, aqua quæ effluit ex tubo FQ, per lumen KN, e&longs;t ad aquam, quæ eodem &longs;eu æquali tempore effluit ex tubo AB, per foramen EC, ut foramen KN ad foramen EC; hoc e&longs;t, eodem &longs;eu æquali tempore, quo ex lumine EC effluit una columna aquea DC, effluunt ex lumine KN duodecim columnæ aqueæ DC: Ergo dum ex lumine KN effluxit tota aqua tubi FQ, effluxit ex lumine EC &longs;olùm duodecima pars aquæ tubi AB; ac proinde tantò plùs temporis requiritur, ut evacuetur tubus AB per lumen EC, quàm ut evacuetur tubus FQ per lumen KN, quantò maius e&longs;t lumen KN quàm lumen EC. Ergo tempora &longs;unt reciprocè ut lumina.

Pori&longs;ma.

SEquitur hinc, etiam conver&longs;am e&longs;&longs;e veram, nempe lumina, per quæ evacuantur tubi prædicti, e&longs;&longs;einter &longs;e ut reciprocè tempora, quibus evacuantur: Vnde data ratione temporum, da­bitur ratio luminum; &longs;icut è contrario, data ratione luminum, datur ratio temporum &longs;eu durationum, quibus evacuantur prædicti tubi.

Propo&longs;itio XVII. Problema I.

Datis altitudine & foramine tubi &longs;emper pleni, invenire quantitatem aquæ quam dato tempore effundat; vel, datis ij&longs;dem, invenire magnitudinem ci&longs;ternæ quæ dato tempore repleatur.

SUppono hîc id, quod notavi &longs;uprà Pro­po&longs;it. XIII. tam exiguam e&longs;&longs;e differen­tiam inter velocitates aquarum ex tubis eiusdem altitudinis, & diver&longs;orum fora­minum, effluentium, ut cen&longs;eri meritò po&longs;­&longs;it, aquam ad aquam e&longs;&longs;e, utforamen ad foramen. Quo po&longs;ito.

Data alti­tudine & foramine tu bi, invenire aquæ quan­titatem da­to tempore effluentem.

Sit, exempli gratia, altitudo tubi ADB pedum 64, luminis &longs;eu foraminis diameter DB digitalis, tempus hora una. Oporteat igitur reperire, quantum aquæ tribuat, &longs;eu quantam ci&longs;ternam impleat, prædictus tubus, intra unam horam. So­lutio pendet ex dictis Propo&longs;it. VII. XIII. & VIII. & præterea ex Propo&longs;it. 2. lib. 12. Eu­clidis. Igitur tubum 4 pedes altum, & luminis linearis, repræ&longs;entet CPK. Ex tubo ADB. &longs;ume EDB portionem etiam 4 pedes altam. Et quoniam diameter DB digitalis, e&longs;t ad linearem PK, ut 12 ad 1; eritper Propo&longs;it. &longs;ecundam lib. 12. Eucli­dis, lumen DB ad lumen KP, utquadra­tum diametri DB, ad quadratum diame­tri PK, hoc e&longs;t, ut 144 ad 1. Quare cùm per Propo&longs;it. XIII. præcedentem, in tubis æquè altis & &longs;emper plenis, &longs;ed inæquali­um luminum, aqua ab uno effu&longs;a, fit ad a­quam ab altero effu&longs;am, tempore eodem, utlumen ad lumen; dabit EDB. centies quadragies quater plus aquæ, quàm det eodem tempore tubus CPK: Sed tubus CPK, per Propo&longs;it. VII. præcedentem, &longs;patio tredecim &longs;ecundorum temporis dat unam libram aquæ: Ergo tubus EDB &longs;patio tredecim &longs;ecundorum tomporis da­bit 144 libras. Quoniam autem in una hora, &longs;eu in 60. minutis primis, hoc e&longs;t, in 3600 minutis &longs;ecundis, tredecim &longs;ecunda con­tinentur ducenties &longs;eptuagies &longs;exies, rema­nentque , &longs;ive (facilioris calculi gratia) ducenties &longs;eptuagies &longs;epties; &longs;i 144 libras, quas tredecim &longs;ecundis dat tubus EDB, multiplices per 277, fient ferè 39888. li­bræ, quas horæ &longs;patio fundit tubus EDB. Inveniatur iam inter 64, altitudinem nem­pe tubi ADB, & inter 4, altitudinem ni­mirum portionis EDB, media proporti­onalis 16. Quoniam igitur per Propo&longs;it. VIII. præcedentem, aquæ quas tubi ADB, & EDB &longs;emper pleni, eodem tempore fundunt, &longs;unt in &longs;ubduplicata ratione eius quam habent altitudines ADB, & EDB; erit aqua quam horæ &longs;patio dat tubus EDB, ad aquam quam eodem tempore dabit tubus ADB, ut 4 ad 16, &longs;eu ut 1 ad 4. Itaque &longs;i fiat, ut 1 ad 4, ita 39888 (tot enim aquæ libras iam o&longs;tendimus horæ &longs;patio dare tubum EDP) ad alium numerum, nempe ad 159552, quot hic continet vni­tates, tot aquæ libras horæ &longs;patio dabit tu­bus ABD &longs;emper plenus, lumen habens digitale, altitudinem pedum 64; atque adeo tubus prædictus implebit horæ &longs;pa­tio ci&longs;ternam capacem librarum aquæ 159552.

Annotatio.

Pari&longs;ien&longs;es libra 72. ef­ficiunt pe­dem cubisum aquæ.

MEr&longs;ennus ait, heminas &longs;eu libras Pari&longs;ien&longs;es 72. efficere pedem cubicum aquæ. Itaque &longs;i placet prædictas libras ad pedes cubi­cos reducere, divide numerum 159552 per 72, & quotiens 2216 da­bit aquæ pedes cubicos quos horæ &longs;patio fundet tubus ADB.

Propo&longs;itio XVIII. Problema II.

Datis altitudine & lumine tubi &longs;emper pleni, invenire tempus quo datam aquæ quantitatem effundat, &longs;i­ve quo datam ci&longs;ternam impleat.

SIt tubus &longs;emper plenus ADB, v.g. 16 pedes altus, lumen

habens digitale DB; &longs;it verò quantitas aquæ à tubo ADB effundendæ, aut ci&longs;terna ab eodem replenda, 3000 pedum cu­bicorum. Oporteat invenire tempus quo dictam aquam ef­fundat, aut dictam ci&longs;ternam impleat.

Data alti­tudine & foramine tubi, inve­nire tempus quo data a­qua effluit.

Revocentur primò 3000 pedes cubici aquæ ad heminas &longs;eu libras (quod fiet, &longs;i 3000 multiplica­bis per 72. fiatq, numerus librarum 216000, (quandoquidem unus pes cubicus conficit 72 libras, ut diximus Propo&longs;it: præceden­te in Annotat:) Deinde &longs;umatur ex ADB, tubus quadrupedalis EDB: Demum inter ADB altitudinem, id e&longs;t, 16, & EDB alti­tudinem, id e&longs;t, 4, inveniatur medius nu­merus proportionalis 8. Igitur, per Pro­po&longs;it. VIII. præcedentem, aqua fluens per tubum EDB, e&longs;t ad aquam per tubum ADB eodem tempore fluentem, ut 4 ad 8, in ratione videlicet &longs;ubduplicata altitudi­num EDB 4 pedum, & ADB 16 pedum. Quo ergo tempore dat tubus ADB 3000 pedes cubicos aquæ, id e&longs;t, 216000 hemi­nas &longs;eu libras; eodem tempore EDB dabit duplò minus, &longs;eu heminas &longs;olùm 108000. A&longs;&longs;umatur jam, ut antea, tubus quadrupedalis CPK lu­minis linearis. Tubi EDB, CPK, &longs;unt æ­què alti; & lumina DB, PK, dantur, per Propo&longs;it. lib. 12. Euclid. nempe 1, & 144 lineæ quadratæ; aqua etiam quam CPK fundit&longs;patio 13 &longs;ecundorum, datur, per VII. præcedentem, una nimirum libra: Qua­re cùm quantitates aquæ à tubis æquè al­tis eodem tempore effu&longs;æ &longs;int ut lumina, per Propo&longs;it. XIII. præcedentem, &longs;i fiat ut lumen PK, ad lumen DB, id e&longs;t, ut 1 ad 144, ita aqua unius ad aquam alterius; pro­ducetur numerus heminarum &longs;eu libra­rum, quas effundit tubus EDB &longs;patio tre­decim &longs;ecundorum, nempe 144. Quo­niam igitur tubus EDB 144 libras dat &longs;patio tredecim &longs;ecundo­rum; libras 108000 quanto tempore dabit? Fiat, ut 144 libræ ad hoc e&longs;t, ad tredecim &longs;ecunda, ita 108000 ad 9750; dabit hic numerus &longs;ecunda, quibus tubus EDB dat libras 108000. Sed o&longs;ten&longs;um e&longs;t &longs;uprà, quo tempote tubus EDB dat 108000 libras, tubum ADB dare libras 216000, id e&longs;t, 3000 pedes cu­bicos aquæ; Inventum e&longs;t igitur tempus, quo tubus ADB &longs;em­per plenus fundit datam aquam 3000 pedum cubicorum, nem­pe 9750 &longs;ecunda, &longs;eu 162 prima; quæ faciunt horas 2, & 42 minuta prima.

Propo&longs;itio XIX. Problema III.

Datis tempore, quantitate aquæ, &longs;eu Ci&longs;ternâ, & lu­mine tubi, invenire altitudinem tubi, qui &longs;emper plenus ci&longs;ternam dato tempore repleat.

Dato tem­pore, & quantitate aquæ, inve­nire tubum ex quo ef­fluat.

SIt datum tempus unius horæ & 21 minutorum primorum, aqua data 3000 pedum cubicorum, &longs;eu 216000 librarum capax ci&longs;terna, & lumen &longs;eu foramen tubi digitale, &longs;eu duo- decim linearum. Oporteat invenire altitudinem tubi qui unâ horâ & 21 minutis primis ex foramine digitali ejiciat 216000 libras aquæ, &longs;eu repleat ci&longs;teruam 3000 pedum cubicorum.

Quoniam tubus quadrupedalis, cujus lumen digitale, eji­cit tredecim &longs;ecundorum &longs;patio libras 144, ut vidimus Pro­po&longs;it. XVII. præcedente, & idem tubus &longs;patio duarum hora­rum, ac 42 minutorum primorum ejicit libras 108000, ut vi­dimus Propo&longs;it. XVIII. præcedente; ergo duplum hujus aquæ, hoc e&longs;t, libræ 216000, ejicientur à tubo eju&longs;dem luminis alto pedes 16. eò quòd tuborum æqualium foraminum altitudines debeant e&longs;&longs;e in duplicata ratione aquarum, ut duplo plùs aquæ eodem tempore fundant, prout diximus Propo&longs;it. VIII. Pori­&longs;mate 1. Ut verò eadem aqua effluat ex tubo eju&longs;dem fora­minis duplo velociùs, hoc e&longs;t, intra horam unam, minuta 21; debet tubus e&longs;&longs;e altus pedes 64, ut patet ex Pori&longs;mate Propo­&longs;it. X. & XII.

Pori&longs;ma.

SImili ratione datis tempore, ci&longs;terna, & altitudine tubi, in­venitur foramen tubi, qui ci&longs;ternam datam repleat in illo tempore.

Propo&longs;itio XX. Problema IV.

Dato Va&longs;e, & foramine per quod effluit aqua, invenire tempus quo evacuatur.

SIt datum vas pri&longs;maticum ABCD, cujus tam longitudo,

quàm latitudo &longs;it decem pedum, altitudo vero 16 pedum, ac proinde capacitas totius va&longs;is &longs;it 1600 pedum cubicorum, ba&longs;is verò &longs;it 100 pedum quadratorum; lumen denique &longs;eu foramen E in fundo va&longs;is &longs;it digitale. Oporteat invenire tem­pus quo dictum vas, &longs;i aquâ fuerit plenum, per lumen E eva­cuetur.

Dato va&longs;e, & forami­ne, inveni­re tempus quo evacua tur.

Con&longs;tat ex dictis Pro­po&longs;it. XIV. in Annotat. tu­bum quadrupedalem, cu­jus ba&longs;is pedalis, & lumen lineare, &longs;patio 144. minu­torum, &longs;eu duabus horis & 24 minutis, totum ex­hauriri, &longs;i non &longs;it &longs;emper plenus. Con&longs;tat præterea ex eadem Propo&longs;itione, tubos non &longs;emper plenos, æquè altos, & æqualium foraminum, &longs;ed inæqua­lium ba&longs;ium, evacuari in­æqualibus temporibus; & tempora quibus evacuantur, e&longs;&longs;e inter &longs;e, vt ba&longs;es. A&longs;&longs;uma­tur igitur tubus pri&longs;maticus quadrupedalis KL, ba&longs;im habens pedalem, & lumen lineare. A&longs;&longs;umatur præterea ex va&longs;e dato ABCD, pars CFDH alta pedes quatuor, ba&longs;&longs;im habens ean­dem cumtoto va&longs;e centum pedum, lumen verò 1 lineare. Quo­niam igitur, per Propo&longs;itionem XIV. dictam, tempora quibus vas CFDH, & tubus KL exhauriuntur per lumen lineare, &longs;unt inter &longs;e ut ba&longs;es ip&longs;orum; ba&longs;es autem ex hypothe&longs;i habent pro­portionem ut 100. ad 1; nece&longs;&longs;e e&longs;t, tempus quo evacuatur vas CFDH per lineare lumen 1, ad tempus quo evacuatur tubus KL per lumen etiam lineare, e&longs;&longs;e ut 100 ad 1. Quare cùm tu­bus KL per lineare lumen evacuetur &longs;patio 144 minutorum, ut vidimus; evacuabitur vas CFDH, per lineare lumen 1, &longs;pa­tio 14400 minutorum, (tot enim producuntur, &longs;i 144 multi­plicentur per 100) &longs;eu horarum 240, vel dierum decem. Iam verò quoniam per Propo&longs;it. XVI. tempus quo effluitaqua ex va­&longs;e CFDH per lumen digitale E, ad tempus quo effluit eadem aqua ex eodem va&longs;e per lumen lineare 1, e&longs;t reciprocè, &longs;icut e&longs;t foramen 1 ad foramen E, nempe ut 1 ad 144; &longs;i dividatur tempus quo effluit aqua per foramen 1, nempe 14400 minuta, per 144; hoc e&longs;t, &longs;i accipiatur cente&longs;ima quadrage&longs;ima quarta pars minutorum 14400, nempe minuta 100; habebitur tem­pus quo exhauritur vas CFDH, per digitale lumen E. Tan­dem quoniam per Propo&longs;it. XV. præcedentem, tempora qui­bus deplentur va&longs;a ABCD, CFDH. æqualium ba&longs;ium, &longs;ed non æqualium altitudinum, per idem foramen E, &longs;unt in &longs;ubdu­plicata ratione altitudinum, nempe pedum 4, & pedum 16; &longs;i inter 4 & 16 inveniatur medius numerus proportionalis, nem­pe 8; erit tempus quo evacuatur vas CFDH, per lumen digita­le E, nimirum 100 minutorum, ad tempus quo evacuatur vas ABCD per idem lumen digitale E, ut 4 ad 8, &longs;eu ut 1 ad 2, nempe ut minuta 100, ad minuta 200, quæ efficiunt horas 3. min. 20.

Annotatio.

QVod diximus de va&longs;e pri&longs;matico, debet etiam intelligi de cylindrico, & cuiuscunque alterius figuræ; &longs;ed tunc a&longs;&longs;umi debet tubus quadru­pedalis va&longs;i &longs;imilis.

Propo&longs;itio XXI. Problema V.

Dato va&longs;e, & tempore, invenire foramen per quod evacuetur tempore dato.

SIt data ci&longs;terna A, 1600. pe­ dum cubicorum; & tempus B, quo evacuandum e&longs;t totum, &longs;it horarum 40 & minutorum 16; &longs;itque inveniendum foramen, per quod evacuetur ci&longs;ter­na A data, in tempore B dato. A&longs;&longs;umatur quodvis lumen &longs;eu foramen notæ magnitudinis, v.g. lineare; & per Propo&longs;itionem XX. præcedentem inveniatur tempus C, quo ci&longs;terna A hu­mido plena, evacuetur tota per lumen lineare. Quoniam igi­tur, per Propo&longs;it. XVI. præcedentem, tempora quibus evacuan­tur va&longs;a non &longs;emper plena, &longs;imilia, & æqualia quoad altitudinem & ba&longs;es, per lumina &longs;imilia inæqualia, &longs;unt reciprocè ut lumina; & è contrario, lumina prædictorum va&longs;orum &longs;unt reciprocè ut tempora, quibus evacuantur, ut diximus in Pori&longs;mate dictæ Propo&longs;itionis XVI; &longs;i fiat ut tempus B, ad tempus C, quo per lineare lumen effluit ci&longs;terna A, ita lumen lineare ad aliud; hoc ip&longs;um erit lumen quod quærebatur.

Dato va&longs;e & tempore, invenire foramen.

ci&longs;t.temp.temp. --------------- ABC

Propo&longs;itio XXII. Problema VI.

Altitudinem &longs;caturiginis dati fontis per tubos fluentis invenire.

Altitudinem Scaturigi­nis fontis in venire per tubos fluen­tis.

FIat notum lumen, per quod fontis a­qua fluat; aut fonti lumen notæ ma­gnitudinis applica, v.g. lineare. Ob­&longs;erva deinde quot aquæ libras fons per lineare lumen effundat &longs;patio unius minuti primi, &longs;eu 60 minuto­rum &longs;ecundorum; &longs;itque numerus ille librarum B. Quoniam igitur per Propo&longs;it. VII. huius capitis, tubus qua drupedalis &longs;emper plenus per lumen lineare effundit &longs;patio tiedecim &longs;ecundorum vnam libram, & con&longs;equenter &longs;patio 60 &longs;ecundorum, &longs;eu unius minuti primi, libras 4: & præterea, quoniam per Pori&longs;ma I. Propo&longs;it. VIII. huius capitis, altitudines tuborum, habentium idem &longs;eu æquale lumen, &longs;unt in duplicata ratione eius quam ha­bent aquæ quantitates per tubos eodem tempore effusæ: &longs;i fiat, ut 4 lib. ad numerum librarum B, ita altitudo 4 pedum, ad aliud, nempe ad altitudinem numeri M; & iterum, ut 4 ad M, ita M ad N; dabit numerus N altitudinem &longs;caturiginis in pedibus, eò quòd ratio 4 ad N &longs;it duplicata rationis 4 ad M, &longs;eu ratic­nis 4 ad B, nimirum aquæ ad aquam.

Propo&longs;itio XXIII. Problema VII.

Data alicuius tubi, aut va&longs;is erogatorij altitudine, ac tempore, quo determinatam aquæ quantitatem è &longs;uo lumine effundit, invenire altitudinem eiusdem autalterius tubi, qui æquali tempore, per æquale lumen, aliam determi­natam aquæ quantitatem effundat.

SIt tubi vel alterius va&longs;is erogatorij aquâ &longs;emper pleni altitudo 9. pedum, ex cuius lumine &longs;patio unius minuti &longs;aliat una aquæ libra, &longs;it autem producenda altitudo eò u&longs;que, ut æquali &longs;patio minuti, per idem aut æquale lumen effundat 16. libras aquæ. Du­plicetur ratio 16 ad 1, & proveniet ratio 256 ad 1; nam 16 ducta in 16 efficiunt 256: cumque 9 referat unitatem, multiplica 256 per 9, & provenient 2304, pro tubi aut alterius va&longs;is quæ&longs;iti alti­tudine. Ratio e&longs;t, quia tubi habent duplicatam rationem a­quarum, per Propo&longs;it. VIII. huius capitis. Si itaque fiat, ut 1 ad 256, ita 9 ad aliud; provenient 2304.

Data tubi altitudine, ac tempore effiuentis a­quæ deter­minatæ, in­venire alti­tudinem pro alia a­quæ quanti­tate.

Propo&longs;itio XXIV. Problema VIII.

In tubo &longs;eu va&longs;e non &longs;emper pleno determinare &longs;patia, quæ temporibus æqualibus &longs;ibi &longs;ucced entibus evacuantur; vti & men&longs;uram &longs;eu pondus aquæ quæ effluit.

PArte 2. Cla&longs;&longs;e 1. cap. 4. inter alias Machinas afferemus varia liy­drologia, &longs;eu horologia aquatica, quibus per fluxum aquæ è foramine alicui9 tubi, aut va&longs;is, metimur horas æquales &longs;eu &ecedil;qua­les temporis partes, &longs;ignando in va&longs;is latere lineas determinantes fluxum æqualibus temporibus corre&longs;pondentem. At quoniam ex dictis &longs;uprà Propo&longs;itione VI. con&longs;tat, &longs;patia quæ æqualibus temporibus evacuantur, non e&longs;&longs;e æqualia, &longs;ed &longs;emper minora atque minora evadere, eò quòd æqualibus temporibus non ef­fluat æqualis aquæ copia, &longs;ed &longs;emper minor ac minor; ideo de­terminandum híc e&longs;t, quomodo geometricè inveniendum &longs;it in quolibet va&longs;e dictum &longs;patiorum decrementum, &longs;eu quomodo dividendum &longs;it latus va&longs;is, ut &longs;patia ad&longs;ignata æqualibus tem­poribus evacuentur. Iterum quoniam per dicta eâdem Pro­po&longs;itione VI, aqua quæ æqualibus temporibus effluit è dictis va­&longs;is, non e&longs;t æqualis, &longs;ed in æqualis; determinandum e&longs;t, quan­tum quovis æquali tempore effluat.

In tubo de­terminare &longs;patia que temporibus æqualibus evacuan­tur.

Dico itaque, aquam æqualibus temporibus effluere è tu­bis non &longs;emper plenis ea ratione, ut &longs;ingulis temporibus decre- &longs;eat & aquæ effu&longs;æ quantitas, & va&longs;is &longs;eu tubi evacuati &longs;patium, & aquæ de&longs;cendentis ac effluentis velocitas eo decremento, quod e&longs;t inter numeros impares, ver&longs;us unitatem. Explico. Sit vas quodcunque aquâ plenum, eius fluxus dividere debeat di­em, aut quam cunque diei partem, in quatuor æquales partes; &longs;intque in va&longs;e 16 v.g. men&longs;uræ &longs;eu libræ aquæ, quæ dato tempore effluant. Dico, primo tempore effluere &longs;eptem men­&longs;uras, &longs;ecundo quinque, tertiò tres, quarto & ultimo unam.

Dico iterum, &longs;i tota va&longs;is altitudo ab infima ad &longs;upremam aquæ &longs;uperficiem dividatur in 16 æquales partes, aquam in pri­mo tempore de&longs;cendere ad 7 &longs;patia, in &longs;ecundo ad quinque in ter­tio ad tria, in quarto ad unum. Dico tandem, aquam in pri­ma hora de&longs;cendere atque effluere ut &longs;eptem, in &longs;ecunda ut quin­que, in tertia ut tria, in quarta ut unum. Sequitur ex dictis &longs;uprà Propo&longs;itione 8, 9, & 10, quas vide. Eadem ratio in omnibus alijs e&longs;t.

Corollarium.

Vas parare, cuius flux9dividat da­tum tempus in partes æquales.

EX his patet, quomodo parandum &longs;it vas, cuius fluxus dato tempore durans dividat illud tempus in partes æquales. Sit enim vas, cuius fluxus duret per &longs;patium duodecim horarum, & hoc &longs;patium temporis &longs;it dividendum, beneficio fluxus aquæ, in 12 æquales partes &longs;eu horas. Duc 12. in &longs;eip&longs;a, & produces 144. Deinde divide va&longs;is altitudinem à &longs;uprema ad infimam aquæ &longs;u­perficiem, in 144 partes &longs;eu &longs;patia æqualia, & inferiùs prope fundum va&longs;is incipiendo, deputa ultimæ horæ ex duodecim vnum &longs;patiolum, penultimæ tria, ante penultimæ quinque, & &longs;ic ul­teriùs progredere &longs;ecun dùm &longs;eriem &longs;eu progre&longs;&longs;ionem numero­rum impatium, prout apparet in numeris lateri dextro CD &longs;equen­tis figuræ appo&longs;itis. His pr&ecedil;&longs;titis, divi&longs;um erit vas in &longs;patia in&ecedil;qua­lia, qu&ecedil; &longs;ingulis æqualibus temporibus, nempe horis, evacuantur, videbi&longs;que aquam in prima hora de&longs;cendere à &longs;patio ultimo &longs;eu 144, u&longs;que ad &longs;patium 121; in &longs;ecunda hora u&longs;que ad &longs;patium 100; in tertia u&longs;que ad 81; & &longs;ic porrò u&longs;que ad finem; prout apparet in numeris lateri &longs;ini&longs;tro AB eiu&longs;dem figuræ appo&longs;itis. Lege Torricellum lib. 2. de motu projectorum.

Eadem ratione reperies, quot partes, aut men&longs;uræ certæ, &longs;eu pondera aquæ effluant ex va­&longs;e in &longs;ingulis horis. Item qua velocitate effluat atque de&longs;cendatin ijsdem, &longs;ingulis horis.

Annotatio I.

NVmeri dextri lateris CD indicant etiam partes &longs;eu &longs;patiola va&longs;is, quæ percurrit aqua &longs;ingulis horis &longs;uo motu de&longs;censûs; &longs;imiliter & partes aquæ, quæ &longs;ingu­lis horis effluunt, & velocitatem, quâ effluunt. Nam in prima hora percurrit &longs;patia 23, in &longs;ecunda 21, in tertia 19, in quarta, 17, &c.

Numeri &longs;ini&longs;tri lateris AB &longs;unt duodecim qua­drata temporis in duodecim horas divi&longs;i. Nam 1 e&longs;t quadratum horæ primæ, incipiendo ab ultima; 4 e&longs;t quadratum horæ &longs;ecundæ, 9 e&longs;t quadratum horæ ter­tiæ &c. vt patet, &longs;i &longs;inguli numerimedio figuræ in&longs;eri­pti multiplicentur per &longs;eip&longs;os. Et hoc e&longs;t, quod dixi­mus &longs;uprà Propo&longs;it. IX. in motu gravium naturaliter de&longs;cendentium &longs;patia æqualibus temporibus transmi&longs;­&longs;a e&longs;&longs;e inter &longs;e, ut quadrata temporum.

Annotatio II.

SI vas aliquod dividere velles in partes 24, multi­plica 24 per 24, producentur 576. Divide ergo altitudinem va&longs;is in 576 partes æquales, & procede ut dictum in Corollario, juxta &longs;eriem numerorum im­parium. Eadem ratione quotlibet vas in quotlibet partes partieris.

Annotatio III.

QVoniam quando in va&longs;e exigua e&longs;t aqua, ut vix fundum operiat, fluxus ip&longs;ius guttatim & non continuè &longs;it, ideo&queacute; aqua fluens guttulis &longs;uis errorem in­ducere pote&longs;t; meliùs e&longs;t, &longs;i duæ ultimæ divi&longs;iones &longs;eu &longs;patiola va&longs;is divi&longs;i, notata numeris 1 & 3, non ve­niant in numerum &longs;patiolorum, & potiùs ver&longs;us prin­cipium adjiciantur alia duo &longs;patia; &longs;icque vas, de quo in Corollario, non dividatur in 12, &longs;ed in 14 partes, modo ibi dicto, dividendo totam ip&longs;ius altitudinem in partes æquales 196 (qui numerus re&longs;ultat ex ductu numeri 14 in &longs;eip&longs;um) & addendo &longs;patium 13, &14, quæ demptis duobus infimis efficiant duodecim &longs;patia.

PARERGVM CAPITIS TERTII.

De inventione mediæ, ac tertiæ propor­tionalis quantitatis, in numeris, & lineis.

EXdictis hoc Capite patet, quàm &longs;æpe nece&longs;&longs;a­ria &longs;it inventio mediæ, ac tertiæ, quartæque proportionalis quantitatis, ut ex nota aquæ men­&longs;ura, aut pondere, venire po&longs;&longs;imus in cognitio­nem de&longs;ideratæ altitudinis tubi, aut va&longs;is; autè contrario, ut ex nota tubi altitudine veniamus in cognitionem men&longs;uræ aut ponderis aquæ, & quamcunque tuborum altitudinem, aquarumvè copiam reperiamus facilè per quamcunque ra­tionem &longs;ubduplicatam, aut duplicatam, &longs;ive nu­meris exprimi po&longs;&longs;it, &longs;ive non po&longs;&longs;it exprimi abs­que &longs;urdis. Operæ ergo pretium duxi, Regu­las brevi&longs;&longs;imas a&longs;&longs;ignare, vel &longs;altem in&longs;inuare, prædictas quantitates inveniendi, mediam dico, ac tertiam, quartamque proportionalem, tam arithmeticè, quàm geometricè, libantes brevi&longs;­&longs;imè nonnulla ex ij, quæ fu&longs;iùs dicemus in Arith­metica, & in Geometria practica, &longs;euin Pantometro Kircheriano.

Propo&longs;itio I.

Inter duos numeros medium proportiona­lem invenire.

DUos numeros propo&longs;itos multiplica inter&longs;e, & ex producto

erue radicem quadratam; erit hæc radix medio loco pro­portionalis inter duos numeros datos. Exemplum. Sit inter 4 & 16 inveniendus medius proportionalis numerus: multiplica 16 per 4, fiunt 64; cuius radix quadrata e&longs;t 8, e&longs;tque medio loco proportionalis inter 4 & 16; quia ut e&longs;t 4 ad 8, ita 8 ad 16.

Numerum medium in­ter duos in­venire.

Propo&longs;itio II.

Datis duobus numeris, tertium continuè proportionalem invenire.

TRes numeri continuè proportionales dicuntur, quando e&longs;t, ut primus ad &longs;ecundum, ita &longs;ecundus ad tertium. Huiusmodi &longs;unt 1, 3, 9, & 1, 2, 4: item 2, 4, 8: item 4, 8, 16. Propo&longs;itisigi­tur quibuscunque duobus numeris, in venietur tertius, qui ad &longs;ecundum &longs;it ut ip&longs;e &longs;ecundus ad primum, &longs;eu ad quem &longs;ecun­dus &longs;it ut primus ad &longs;ecundum; &longs;i &longs;ecundum ducas in &longs;eip&longs;um; productus enim erit tertius proportionalis.

Numerum tertium pro portionalem po&longs;t duos in­venire.

Propo&longs;itio III.

Inter duas rectas lineas datas invenire mediam proportionalem.

SInt datæ duæ rectæ AB, BC, inter quas invenienda &longs;it media propor­tionalis. Coniungantur rectæ AB, CB in unam rectam continuam in puncto B, ut fiat recta ABC; eâque divisâ bi­fariam in D, de&longs;cribatur &longs;emicirculus aut circulus AEC, ad intervallum DA, vel DC; tandemque ex B pun­cto erigatur perpendicularis BE ad circumferentiam u&longs;que; eritque BE media proportionalis quæ&longs;ita. Demon&longs;trationem vide apud Euclidem lib. 6. Propo&longs;it. 13.

Lineam me diam pro­portionalem inter duas invenire.

Propo&longs;itio IV.

Datis duabus rectis, invenire tertiam pro­portionalem.

Lineam ter tiam propor tionalem po&longs;t duas in venire.

SInt datæ duæ rectæ AB, & BE, præcedentis figuræ, &longs;itque invenienda tertia, ad quam ita &longs;e habeat &longs;ecunda, &longs;icut pri­ma ad &longs;ecundam. Coniungantur rectæ AB, BE, in puncto B ad angulum rectum, ducaturque recta EA; eáque bifariam di­visâ in F, ducatur recta FD perpendicularis ad AE; & facto centro D, intervallo DA de&longs;cribatur circulus, qui nece&longs;&longs;ariò tran&longs;ibit per punctum E, per quintam Quarti Euclid. Siiam producatur recta AB u&longs;que ad circumferentiam circuli, hoc e&longs;t, u&longs;que ad punctum C; erit BC tertia proportionalis quæ&longs;ita.

Sint iterum datæ duæ rectæ BC, & BE, &longs;itque invenien­da tertia proportionalis. Coniungantur, ut antea, rectæ illæ in B, ut efficiantangulum rectum, & ducatur recta EC; at­que ex puncto medio G demittatur perpendicularis GD, & producta recta CB in continuum, de&longs;cribatur centro D, in­tervallo DC, circulus, qui iterum tran&longs;ibit per punctum E, & &longs;ecabit rectam CB productam in A; eritque hæc recta BA tertia proportionalis quæ&longs;ita.

Annotatio

QVod dictum e&longs;t de lineis hîc po&longs;itis, dicendum e&longs;t de quibuscunque

line is propo&longs;itis. Itaque &longs;ipropo&longs;itis duobus tubis inveniendus &longs;it velmedius, veltertius proportionalis; coniunge lineas rectas tubis da­tis æquales; & operare ut dictum, & invenies quod quæris. Quòd &longs;i tubi propo&longs;iti, ac lineæ ip&longs;is æquales nimis e&longs;&longs;ent longæ, ac proinde minùs commodè circulo includi po&longs;&longs;ent; accipe ip&longs;arum &longs;ubmultiplices, v.g. dimidiam, tertiam, quartam, &c. partem, & cum ip&longs;is procede ut dictum; eritque inventa linea æquè &longs;ubmultiplex lineæ aut tubi quæ&longs;iti.

Tubum me dium t­tium pr tionalem in venire.

Propo&longs;itio V.

Aliter invenire tertiam proportionalem.

QUod fecimus in pr&ecedil;cedenti Propo&longs;itione mediante circulo, fieri pote&longs;t faciliùs absque circulo &longs;ic.

Lineam ter tiam pro­portionalem aliter inve­nire.

Duæ rectæ datæ, AB, & BE, con­&longs;tituantur ad angulum rectum ABE, ut dictum, & ducatur recta EA. Pro­ductâ deinde AB, antecedente &longs;eu primâ duarum datarum, ducatur ex E ad AE perpendicularis EC, occur­rensip&longs;i AB productæ in C; eritque BC tertia proportionalis.

Si e&longs;&longs;ent propo&longs;itæ duæ rectæ CB, BE, deberent ip&longs;æ coniungi ad angu­lum rectum CBE, & ductâ rectâ CE, productaque CB, deberet duci per­pendicularis AE, ad rectam CE, e&longs;&longs;etque AB tertia quæ&longs;ita.

Annotatio.

INventâ alterutro modorum tertiâ continuè proportionali, &longs;i primam omi&longs;eris, & alijs duabus tertiam inveneris; habebis quatuor leneas continuè proportionales. Eodemque modo invenies quintam, &longs;extam, &longs;eptimam, & quamcunque aliam.

Propo&longs;itio VI.

Adhuc aliter invenire tertiam proportionalem.

SI difficile tibi videtur ducere tot perpendiculares ad inveni­ endamtertiam proportionalem, uti poteris modo præ&longs;cripto ab Euclide lib. 6. Propo&longs;it. 11. &longs;ic:

Adhuc ali­ter item pre &longs;tare.

Sint duæ rectæ AB, AE. Di&longs;ponantur eæita, ut efficiant angulum A quemcunque, producaturque AB quam vis e&longs;&longs;e antecedentem &longs;eu primam, & capiatur BC æqualis ip&longs;i AE, quæ con&longs;equens e&longs;&longs;e de- bet, &longs;ive &longs;ecunda, aut media. Deinde ductâ rectâ AB, aga­tur illi ex C parallela CD, occurrensip&longs;i AE productæ, in D; eritque ED tertia proportionalis quæ&longs;ita.

Annotatio.

SIduæ datæ&longs;int nimis longæ, &longs;ervetur id, quod diximus in Annota­tione Propo&longs;itionis quartæ præcedentis.

Propo&longs;itio VII.

Datis tribus numeris, invenire quartum pro­portionalem.

Numeris tribus datis quartum proportiona­lem inveni­re.

UTere Regulâ proportionum apud Arithmeticos noti&longs;&longs;ima, quam Auteam vocant; & habebis intentum.

Propo&longs;itio VIII.

Datis tribus rectia lineis, quartam propor­tionaleminvenirc.

Lineis tri­bus datis, quartam pro­portiona­lem inveni­re.

SInt tres lineæ rectæ, AB, BC, AD, quibus invenienda &longs;it quarta. proportionalis, ad quàm &longs;it tertia AD, ut e&longs;t prima AB, ad &longs;ecundam BC. Di&longs;ponantur primæ duæ, AB, BC, &longs;ecundùm lineam rectam quæ &longs;it AC; tertia verò AD, cum prima AB, faciatangulum A quemcunque: deinde ex B ad D ducatur recta BD, cui per C ducatur parallela CE, oc­currens rectæ AD productæ, in E puncto. Dico, DE, e&longs;&longs;e quartam proportionalem. Demon&longs;trationem vide apud Euclidem lib. 6. Propo&longs;it. 12.

CAPVT IV.

De proprietatibus aquæ &longs;alientis ex tubis. PROOEMIVM.

CApite præceden­ ti egimuo de pro­prietatibus a­quæ fluentis per tubos verticales, hoc e&longs;t, efflu­entis ex foramine facto in ba&longs;ibus tuborum: hoc Ca­pite agemus de proprieta­tibus eju&longs;dem aquæ &longs;ali­entis ex tubis verticali­bus, hoc e&longs;t, effluentis ex foraminibus lateralib9 tu­borum, &longs;eu per tubos inferi­ùs inflexos. Po&longs;&longs;unt autem tubi verticales inferiùs inflecti triplici modo, ho­ rizontaliter, verticali­ter, & medio modo. Ho­rizontaliter inflexus dici­tur, &longs;i pars inferior infle­xa &longs;it horizonti parallela re&longs;pectupartis reliquæ ver ticaliter erectæ; &longs;eu&longs;i pars inferior inflexa faciat cum reliquæ angulum rectum. Talis e&longs;t in appo&longs;ito &longs;che­mate tubus ABC. Ver­ticaliter inflexus dicitur, &longs;i pars inferior inflexa &longs;it horizonti perpendicularis re&longs;pectu reliquæ partis et­iam perpendiculariter e­rectæ. Talis e&longs;t in &longs;che­mate præ&longs;enti tubus ABE. Medio modo inflexus di­citur, &longs;ipars inferior ne­que parallela e&longs;t horizon­ti, neque perpendicula­ris, &longs;ed inter utrumque &longs;itum facit angulum &longs;e­mirectum &longs;eu 45 gradu­um. Talis e&longs;t tubus ABD.

Proprieta­tes aquæ &longs;a­lientis ex tu bis vertica­libus.

Tubi verti­cale, tripli­citer inflecti po&longs;&longs;unt in­feriùs.

Easdem appellatio­nes &longs;ubibit tubusinflexus, &longs;i pars inferior habeat tria epi&longs;tomia, E, D, C, quorum E &longs;it erectum perpendiculariter,C pro&longs;tratum horizontaliter, D inclinatum inter utrum­que, ad dictum angulum. Quod fit per tria epi&longs;tomia dicta, fieri pote&longs;t per unum, &longs;i ita in&longs;eratur parti inferiori tubi inflexi, ut verti atque dirigi po&longs;&longs;it in quemcunque &longs;itum.

Radij aquæ, qui ex inflexo tubo, aut ex epi&longs;tomiis præ- dictis exiliunt, quales &longs;unt in præcedenti diagramnate radij EF, DI, DK, CG, CH, appellari po&longs;&longs;unt aqueæ virgulæ, aquei ecdro­mi, hoc e&longs;t, excur&longs;us &longs;eu &longs;altus aquæ, & &longs;imilibus nominibus. Marinus tamen Mer&longs;ennus in Hydraulicis &longs;uis Phænomenis appellat eos &longs;alientes, aitque eo nomine appellari à Iulio Fron­tino, aliis que veteribus; quare & nos eo vocabulo utemur. Radius igitur EF dicatur &longs;aliens verticalis, Radius CG, & CH, &longs;aliens Horizontalis; radius denique DI, & DK, &longs;aliens Medius, et licet &longs;alientes CG, CH, nonprocedant ho­rizontaliter, &longs;ed ob aquæ gravitatem &longs;tatim deprimantur, at­que curventur; vocantur tamen &longs;alientes horizontales, propter epi&longs;tomium aut tubulum C horizontaliter di&longs;po&longs;itum.

Radij aqu&ecedil; ex inflexo tubo exili­entes.

Salientes aquæ ex tu bis quid &longs;int.

His præm&longs;&longs;is, afferemus nunc nonnullas ex multis ob&longs;er- vationes, &longs;eu Phænomena aquæ &longs;alientis ex tubis, præ&longs;ertim ex Marino Mer&longs;enno, qui eâ in re diligentiam multorum vincit. His deinde addemus unum aut alterum Problema. Atque ex his patebunt proprietates aquæ &longs;alientis ex tubis.

Ph&ecedil;nome­na aquæ &longs;a lientis.

Propo&longs;itio I. Phænomenum. I.

Salientium &longs;ive ecdromorum horizontalium, & medio­rum, &longs;uper eodem horizonte, longitudines &longs;unt in ratione &longs;ub­duplicata tuborum, ex quibus exiliunt.

DIxi, &longs;uper eodem horizonte; nam &longs;uper diver&longs;ishorizon­tibus diver&longs;a e&longs;t eiu&longs;dem &longs;alientis longitudo, etiam re&longs;pectu eiu&longs;dem tubi, ut dicemus Propo&longs;itione &longs;e­quenti. Voco autem longitudinem horizontalium &longs;alientium, di&longs;tantiam in præcedenti &longs;chemate à puncto z v.g. adpunctum H, aut G, &C.

Salientium| proportio ad duos tubos.

Sint igitur in præcedenti &longs;chemate duo tubi, BL pedalis, & BA quadrupedalis; &longs;ithorizon OHG; &longs;itque tubi pedalis BL &longs;aliens horizontalis CH, tubi verò quadrupedalis BA &longs;aliens horizontalis &longs;it CG. Dico, longitudinem &longs;alientis tubi BL pedalis, &longs;cilicet CH, e&longs;&longs;e &longs;ubduplam &longs;alientis tubi BA, &longs;cilicet CG. Ob&longs;erva vit id Mer&longs;ennus, ut a&longs;&longs;erit ip&longs;e in &longs;uis Phænome­nis hydraulicis Propo&longs;it. XVI. Quare, &longs;i longitudo &longs;alientis tu­bi pedalis e&longs;t unius pedis, longitudo &longs;alientis tubi quadrupedalis erit duorum pedum; & longitudo &longs;alientis tubi &longs;exde cupedalis erit quatuor pedum; & longitudo &longs;alientistubi 64 pedum erit octo pedum; & ita de reliquis. Ratio huius Phænomeni vide­tur de&longs;umenda e&longs;&longs;e à velocitate aquæ de&longs;cendentis per tubos, quæ etiam e&longs;t in ratione tuborum &longs;ubduplicata, ut con&longs;tat ex Propo&longs;it. X. Capitis præcedentis.

Pori&longs;ma.

SEquitur hinc, altitudines tuborum habere duplicatam ratio­nem eius, quam habent longitudines &longs;alientium horizonta­lium.

Annotatio I.

QVod dixi in Propo&longs;itione de &longs;alientibus horizontalibus CG, CH, debet etiam intelligi de &longs;alientibus medijs DK, DI.

Annotatio II.

Salientium longitudines ob&longs;ervare.

LOngitudines &longs;ælientium &longs;uper horizontem quemcun&queacute; pote&longs;t quilibet ob&longs;ervare facillimè, & ubivis locorum, &longs;iconficiat &longs;ibi tubulum præcedentis figuræ ABC portatilem, determinatæ altitudinis & lati­tudinis, v.g. altumpede vno, & latum digito. Si enim apud C epi&longs;to­mium in&longs;erat, aut etiam &longs;olum digitum apponat foramini C, & de­inde repleto tubo epi&longs;tomium aperiat, aut digitum amoveat, ut paucu­læ &longs;olùm guttulæ exiliant, iterumque obturet foramen; notare poterit in horizonte &longs;ignum à &longs;aliente impre&longs;&longs;um; cuiusmodi &longs;igna &longs;unt in &longs;uperiori &longs;chemate K, I, H, G, &C.

Propo&longs;itio II. Phænomenon II.

Salientes horizontales, & mediæ, eju&longs;dem tubi, co &longs;unt longiores, quò lumen tubi fuerit altius &longs;upra horizontem.

SIntin præcedenti diagrammate diverfi horizontes, OG, PS,

QT, RV &C. &longs;upra quos elevatus &longs;it tubus ABC; Dico &longs;ali­entem CHSTV (& idem intelligendum e&longs;t de reliquis &longs;alienti­bus horizontalibus, ac medijs) &longs;upra horizontem RV, e&longs;&longs;e longiorem quàm &longs;upra horizontem QT, & &longs;upra hunc longio­rem quàm &longs;upra PS, & longiorem &longs;upra hunc, quàm &longs;upra OG. Cre&longs;cit itaque eò magis &longs;alientium horizontalium ac mediarum longitudo, quo magis cre&longs;cit luminis tubi &longs;uper horizontem ele­vatio. Si verò in ex ce&longs;&longs;iva altitudine orificium poneretur; cer­tum e&longs;t aquæ ecdromen u&longs;que ad terminum con&longs;titutum flu­xum non continuaturam, &longs;ed cum tempore in guttas, & tandem in aërem re&longs;olutum iri, ut experientia docuit in Cuppula S. Petri interiori, ex quo &longs;cyphus plenus vino vel aqua effu&longs;us ita evanuit in intermedio aëre, ut nullum pror&longs;us ve&longs;tigium nec quidem gut­tarum in inferiori pavimento ijs, qui &longs;tudio id attendebant, ap­paruerit, ut P. Kircherus mihi a&longs;&longs;eruit. Ex nubibus tamen de­cidit guttatim in terram aqua, quia in magna copia & continua­tæ decidunt guttæ.

Salientes eo sunt longiores quò altiù foramen tubi.

Annotatio I.

QVanta verò &longs;it &longs;alientium in quavis tubi &longs;upra horizontem eleva­tione longitudo, inve&longs;tigandum e&longs;t ob&longs;ervatione, quam quilibet facilè poterit facere modo dicto Propo&longs;itione præcedente. Mer&longs;ennus in &longs;uis Phænomenis Propo&longs;it. XVI. proponit paucas ex multis in &longs;equen­ti tabella, in quâ Prima Columna continet varias tubi pedalis, velpo­tiùs luminis tubi, &longs;upra horizontem elevationes in pedibus, ita ut prima elevatio &longs;it unius pedis, &longs;ecunda duorum pedum, &c: Secunda verò co­lumna exhibet &longs;alientium longitudines in pedibus & digitis. Addidi ego ex eodem Mer&longs;enno Propo&longs;it. 29. &longs;alientem tubi pedalis elevati puno cum dimidio &longs;upra horizontem.

Salientium tabella ex Mer&longs;enno.

Altitudines tubi &longs;upra Horizontem.Longitudines &longs;alientiumPedes.PedesDigi1--------------------1--------------------1 1/2--------------------2--------------------2--------------------2--------------------3--------------------3--------------------4--------------------3--------------------5--------------------4--------------------6--------------------4--------------------12--------------------5--------------------18--------------------6--------------------26--------------------8--------------------50--------------------10--------------------
Altitudines tubi &longs;upra Horizontem.Longitudines &longs;alientiumPedes.PedesDigi1--------------------1--------------------1 1/2--------------------2--------------------2--------------------2--------------------3--------------------3--------------------4--------------------3--------------------5--------------------4--------------------6--------------------4--------------------12--------------------5--------------------18--------------------6--------------------26--------------------8--------------------50--------------------10--------------------

Annotatio II.

Idem Mer&longs;ennus loco citato Propo&longs;it. XVII. ait tubum quadrupeda lem habentem altitudinem unius pedis &longs;upra horizontem, ejicere &longs;ali entem horizontalem ad pedes 3 2/3 proximè.

Propo&longs;itio III. Phænomenon. III.

Salientes horizontales & mediæ, eandem longitudinis rationem &longs;ervant, quàm altitudines tuborum &longs;upra eun­dem horizontem.

DIver&longs;as tubi eiu&longs;dem altitudines &longs;upra horizontem referant in &longs;equenti diagrammate lineæ OG, PS, QT, RV. Dico, ut altitudo BO ad altitudi­ nem BR, ita e&longs;&longs;e longitu­nem &longs;alientis CH ad lon­gitudinem &longs;alientis CV. Intelligendum autem hoc e&longs;t de illis &longs;alientium lon­gitudinibus, quas ex datis altitudinibus, ob&longs;ervare commodè po&longs;&longs;umus: nam cùm ne&longs;ciamus, utrum, & ubi &longs;alientium incremen­tum &longs;it de&longs;iturum, &longs;i ele­vatio tubi &longs;upra horizon­tem æquaretur terræ &longs;e­midiametro; non pote&longs;t Propo&longs;itio præ&longs;ens e&longs;&longs;e u­niver&longs;alis, ni&longs;i in &longs;en&longs;u ex­explicato.

Salientes e­andem lon­gitudinis proportionem &longs;ervant, quam tubo­rum altitu­dines.

Propo&longs;itio IV.

Phænomenon IV.

Salientium verticali­um in quacunque elevatio­ne tubi &longs;upra horizontem &longs;emper eadem e&longs;t alti­tudo.

ESto tubus ABC præ­ cedentis diagramma­tis, & &longs;aliens EF, &longs;itque horizon RV, & tubus &longs;it modò in Q, modò in P, modò in O, modò in B. Dico, &longs;alientis EF altitudinem e&longs;&longs;e &longs;emper eandem in quacunque tubi elevatione &longs;upra horizontem RV. In hori­zontalibus &longs;alientibus contrarium contingit, ut vidimus Pro­po&longs;itione II. Ratio Phænomeni e&longs;t, quòd altitudo &longs;alientis EF dependeat &longs;olùm à tubilongitudine, & lumine E, po&longs;ito æquali &longs;emper medio &longs;eu aëre; at in horizontalibus longitudo dependet etiam à figura &longs;alientis, de qua Mer&longs;ennus, & Tor­ricellus.

Salientium verticali­um altitu­do quæ.

Propo&longs;itio V. Phænomenon V.

Saliens verticalis nunquam adæquat altitu­dinem originis.

Saliens ver ticalis nun­quam adæ­quat altitu dinem ori­ginis.

NUnquam enim &longs;aliens EF æquare pote&longs;t altitudinem tubi BA. Et eadem e&longs;t ratio de alijs omnibus &longs;alientibus, qua­rum altitudo nunquam æquare pote&longs;t altitudinem originis, &longs;ive origo &longs;tatim tubo includatur, &longs;ive non. Et ratio e&longs;t, quia gra­vitas humidi, & re&longs;i&longs;tentia aëris impediunt prædictam adæqua­tionem.

Propo&longs;itio VI. Phænomenon VI.

Saliens verticalis tubi quadrupedalis proximè æquat 5/6 tubi &longs;ui.

Saliens ver ticalis tubi quadrupe­dalis.

OB&longs;ervavit id Mer&longs;ennus, a&longs;&longs;eritque in &longs;uis Phænomenis Pro­po&longs;it. XVII. ubi ait, hanc &longs;alientem po&longs;&longs;e e&longs;&longs;e reliquarum modulum, &longs;eu men&longs;uram. Vide quæ diximus Propo&longs;itione II. Annot. 2. ubi idem quod hîc a&longs;&longs;eruimus.

Propo&longs;itio VII. Phænomenon VII.

Salientes verticales eò &longs;unt longiores, quò tubi &longs;unt longiores; &longs;ed non eâdem proportione illæ ac hi cre&longs;cunt.

Salientium verticalium longitudo qu&adot; propor tione cre­&longs;cat.

Vnde licet tuborum non adeo longorum &longs;alientes verticales &longs;int proximè 5/6, aut 2/3 &longs;uorum tuborum; tamen in tubis longis &longs;alientes verticales vix excedunt 1/4 tubi. Et hoc prove­nit tum ob aëris re&longs;i&longs;tentiam, tum ob aquæ gravitatem. Ita Mer&longs;ennus in Phænomenis Propo&longs;it. XVII.

Propo&longs;itio VIII. Problema I.

Data tubi altitudine, & &longs;upra horizontem elevatione, invenire longitudinem &longs;alientis horizontalis, & mediæ.

SIt altitudo tubi alta pedes 9, cuius os &longs;it elevatum &longs;upra horizon­

tem pedes quinque, & &longs;itinvenienda longitudo &longs;alientis hori­zontalis, aut mediæ, huius tubi. Fieri hoc pote&longs;t duplici viâ. Primò per ob&longs;ervationem &longs;ic. Applica orificio tubi epi&longs;tomi­um, aut tubulum horizontaliter, aut medio modo, prout opus fuerit, & nota &longs;alientis punctum pavimento impre&longs;&longs;um, iuxta dicta Propo&longs;it. 1. huius Capitis Annotat. II. Secundò per calculum &longs;ic. Quoniam, per Propo&longs;itionem I. hujus Capitis, &longs;alientium horizontalium & mediarum, &longs;uper eodem horizonte longitudines, &longs;untin ratione &longs;ubduplicata tuborum; & per di­cta Propo&longs;it. 11. huius eiu&longs;dem Capitis, tubus pedalis pedes quin­que &longs;upra horizontem elevatus habet &longs;alientem longam pedes quatuor; &longs;i inveniatur media proportionalis inter 1 & 9, nempe 3, erit hæc longitudo quæ&longs;ita.

Salientis ho rizontalis longitudi­nem inve­nire, data tubi altitu­dine.

Propo&longs;itio IX. Problema II.

Data longitudine &longs;alientis horizontalis, aut mediæ, invenire altitudinem tubi, cognitâ eius elevatione &longs;u­pra horizontem.

SIt data longitudo &longs;alientis horizontalis, aut mediæ, pedum octo, &longs;it que tubus ip&longs;ius elevatus &longs;upra horizontem pedes quinque, & in venienda &longs;it altitudo talis tubi. Quoniam, per Pori&longs;ma Propo&longs;it. 1. huius Capitis, altitudines tuborum habent duplicatam rationem eius, quam habent longitudines &longs;alienti­um horizontalium, & mediarum; & quoniam &longs;aliens horizon­talis tubi unius pedis, elevati &longs;upra horizontem quinque pedi­bus, e&longs;t pedum quatuor; &longs;i rationem prædictarum &longs;alientium, nempe 8 ad 4, duplices, &longs;eu bis &longs;umas &longs;ic: 16, 8. 4; erit tertius numerus 16, altitudo tubi quæ&longs;ita, hic enim numerus 16 ad 4, habet duplicatam rationem eius quam habet 8 ad 4.

Salientis ho rizontalis longitudine data inve­nire tubi al titudinem.

Propo&longs;itio X. Problema III.

Ex nota &longs;caturiginis altitudine fontis unius ex tubo exi­lientis horizontaliter, altitudinem &longs;caturiginis cuiuscun­que alterius æqualiter &longs;upra horizontem elevati invenire.

Fontis alti­tudine &longs;ca­turiginis da ta, in venire alias.

A10.B --8.12.10.15.22 1/2.
A10.B --8.12.10.15.22 1/2.

SIt &longs;caturigo fontis unius A alta pedes 10; O. porteat invenire altitudinem &longs;caturiginis fon­tis B. Idem lumen, &longs;eu eundem tubulum ap­plica horizontaliter ad utriu&longs;que fontis o&longs;cu­lum, & ob&longs;erva &longs;alientium horizontalium longitudinem; &longs;itque fontis A longitudo pedum 8, fontis verò B longitudo &longs;it pedum 12. Fiat igitur, ut 8 ad 12, ita 10 ad aliud, nempe ad 15; & iterum, ut 10 ad 15, ita 15 ad aliud, nempe ad (22 3/10), &longs;ive 1/2; eritque altitudo &longs;caturiginis fontis B pedum 22 1/2.

Demon&longs;tratio. Per Pori&longs;ma Propo&longs;it. 1. hujus Capitis, altitudines tuborum habent duplicatam rationem ejus, quam habent longitudines &longs;alientium horizontalium. Quoniam ergo 10, nempe altitudo &longs;caturiginis A, e&longs;t ad 22 1/2 in duplicata ra­tione eius, quam habet 10 ad 15, hoc e&longs;t, quam habet 8 ad 12, &longs;aliens nempe fontis A, ad &longs;alientem fontis B; erit nece&longs;&longs;ariò 22 1/2 altitudo &longs;caturiginis fontis B quæ&longs;ita.

Atque hæc &longs;ufficiant de tuborum &longs;eu fontium &longs;alientibus; qui plura de&longs;iderat, legat Mer&longs;ennum in Hydraulicis Phæno­menis, & Torricellum lib. 2. de Motu projectorum à fol. 191. ubi incipit agere de Motu aquarum.

CAPVT V.

De fluxu Aquæ per diver&longs;a eiu&longs;dem va&longs;is aut tubi foramina.

Proprieta­tes uxus aqu per di­ver&longs;a eius­dem tubs au: va&longs;is foramma.

HActenus locuti &longs;umus de aquæ fluxu per foramina tu­borum verticalium tum horizontalia, tum lateralia, & tubos cum tubis, aquas cum aquis, foramina cum fo­raminibus diver&longs;orum tuborum comparavimus; nunc eiu&longs;dem tubi & va&longs;is cuiu&longs;cunque foramina diver&longs;a inter &longs;e&longs;e conferemus breviter & methodicè, in&longs;i&longs;tentes ve&longs;tigijs Ioannis Bapti&longs;tæ Baliani lib. 6. de Motu naturali gravium & liquidorum. Quibus tamen multa addi po&longs;&longs;unt ex dictis Capite præcedenti. Sermo autem e&longs;t hîc etiam, uti in præcedentibus, de tu­bis ac va&longs;is verticaliter erectis, &longs;ive cylindrica illa &longs;int, &longs;ive pri&longs;­matica; & &longs;ive foramina &longs;int rectangula, &longs;ive circularia. Præmit­tamus ergo &longs;equentes.

Petitiones.

Petitiones pro fluxu a­quæ ex fora minibus tu borum.

1. Vbi omnia &longs;unt æqualia, effectus &longs;unt æquales.

2. Quantitates eiusdem generis &longs;unt omnes commen­&longs;urabiles, &longs;altem phy&longs;icè, ut dicemus Propo&longs;it. 2. in Annotat.

3. Aqua tran&longs;iens per tubi &longs;eu va&longs;is foramen, decurrit à &longs;ummo va&longs;is ad foramen, tanquam per canalem columnarem, cuius ba&longs;is foramen.

Annotatio I.

TErtia hæc Petitio eadem e&longs;t cum Propo&longs;it. 1. Capitis tertij præceden­

tis. Et licet ibi locuti tantùm fuerimus de foraminibus exi&longs;tenti­bus in ba&longs;i tuborum verticaliter erectorum, hîc tamen idem concedi no­bis petimus etiam de foraminibus in tuborum va&longs;orumque lateribus exi­&longs;tentibus, propter paritatem rationis. Petimus itaque nobis concedi, quòd &longs;i in latere tubi, aut va&longs;is AB fiat foramen B, decurrat aqua à va&longs;is &longs;umitate ad foramen u&longs;que in&longs;tar columnæ, cu­ius ba&longs;is e&longs;t æqualis foramini, altitudo perpendicula­res &longs;uper foraminis diametro erectæ, prout in appo­&longs;ita figura e&longs;t columna ABCD. Ratio pote&longs;t e&longs;&longs;e, quia ex dictis Capite 1 patet aquam non premere ni&longs;i partes &longs;ibi perpendiculariter &longs;ubiectas, dum aqua e&longs;t de&longs;cendens in aliquo, &longs;eu ex aliquo effluens, prout hîc &longs;it.

Aqua in­&longs;tar colum­na effluit ex forami­ne ba&longs;is in va&longs;e.

Propo&longs;itio I. Theorema I.

Per foramina æqualia, æquè à &longs;ummo tubi di&longs;tantia, &longs;ive in ba&longs;i, &longs;ive in latere, æquali tempore æquales fluunt aquarum quantitates.

IN va&longs;e, &longs;eu tubo AB, &longs;int foramina C & D æqualia, & horizontalia (& eadem e&longs;t ra­tio, &longs;i lateralia e&longs;&longs;ent, æquè à &longs;ummitate di­&longs;tantia) per quæ aqua æquali, vel potiùs eo­dem tempore decurrat. Dico, aquas de­ cur&longs;as (liceat ita loqui) e&longs;&longs;e æquales inter &longs;e. Vbi enim omnia &longs;unt æqualia, effectus &longs;unt æquales, per Petitionem primam hujus Capitis: at hîc omnia &longs;unt æqualia, &longs;cilicet foramina, columnæ aqueæ, vis premendi, & &longs;imilia; ergo effectus, qui &longs;unt aquæ decur&longs;æ, æquales &longs;unt. Per foramina ergo æqualia, &c. Quod erat o&longs;ten dendum.

Aquæ flu­xus exæqua libus fora­minibus va forum.

Annotatio.

ET hoc verum e&longs;t, &longs;ive vas &longs;it &longs;emper plenum, &longs;ive non. Eadem e&longs;t ratio, &longs;i va&longs;a &longs;int cylindrica, & foramina rotunda, ut diximus et­iam paulò antè, & &longs;emper in &longs;equentibus dictum volumus.

Pori&longs;ma.

SEquitur hinc, &longs;i ex duobus eiu&longs;dem va&longs;is foraminibus æqua­liter à &longs;ummo va&longs;is di&longs;tantibus aquæ decurrentes eodem tem­pore æquales &longs;unt, foramina e&longs;&longs;e æqualia.

Propo&longs;itio II. Theorema II.

Aquæ è foraminibus æqualiter à &longs;ummo tubi di&longs;tanti­bus decurrentes, &longs;unt inter &longs;e ut foramina.

IN tubo &longs;eu va&longs;e AB, &longs;int duo foramina rectangula inæqualia, C minus, & D maius, &longs;ed ambo horizontalia, atque adeo æ­quèalta, &longs;eu æqualiter à va&longs;is &longs;ummitate remota; & aqua de­cur&longs;a per C &longs;it E, aqua verò decur&longs;a per D &longs;it F. Dico, a­quam E decur&longs;am per C, habere &longs;e ad aquam F decur&longs;am per D, ut&longs;e habet foramen C ad foramen D. Quoniam enim, per Petitionem &longs;e­cundam hujus, quantitates eiusdem gene­ris &longs;unt commen&longs;urabiles, erunt longitu­dines C & D commen&longs;urabiles. Siter­go communis earum men&longs;ura G, & &longs;ecen­tur dictæ longitudines C & D in partes, quæ &longs;int æquales ip&longs;i G; quibus divi&longs;is à transver&longs;alib9 lineis perpendicularib9, fiant totforamina, quot &longs;unt dictæ partes. Tum &longs;ic. Huiu&longs;modi foramina erunt omnia inter &longs;e æqualia, ex con­&longs;tructione, & per 36. Primi Euclid. Ergo ex eis effluent eodem tempore quantitates aquæ æquales, per Propo&longs;it 1. huius. Quot igitur &longs;unt foramina in C & D, toterunt quantitates a­ quarum æquales in E & F. Cùm igitur &longs;int qua­tuor magnitudines, C, D, E, F, quarum prima C e&longs;t ad &longs;ecundam E, ut e&longs;t tertia D, ad quar­tam F; erit etiam vici&longs;&longs;im &longs;eu permutando ut C ad D, ita E ad F, per decimam &longs;extam Quinti Euclidis.

Proportio aquæ efflu­entis ad fo­ramina.

Annotatio

QVamvis contingere po&longs;&longs;it, ut longitudines CD non &longs;int commen­&longs;urabiles, ac proinde G non &longs;it earum communis men&longs;ura; nihil refert, quia hîc non &longs;umus in Mathematicis, &longs;ed in Phy&longs;icis, ubi non ha­betur ratio in&longs;en&longs;ibilium.

Propo&longs;itio III. Theorema III.

Aqua per foramina va&longs;is eo impetu &longs;eu velocitate de­currit, quo per tubos æqualium foraminum & alti­tudinum.

SIt vas CD, in quo foramen D, al­titudo va&longs;is CD; & &longs;it tubus AB perpendicularis, in quo foramen B &longs;it æquale foramini D, & altitudo AB æqualis altitudini CD. Dico, in B & D impetus &longs;eu velocitates aquæ effluentis e&longs;&longs;e æquales. Quo­niam enim aqua fluens è foramine D, decurrit per &longs;patium CD, ac &longs;i de­curreret per tubum AB perpendi­cularem eiusdem longitudinis, per Petitionem tertiam huius; in D & B &longs;ortitur impetus &longs;eu velocitates &ecedil;qua­les, per Corollarium quintum Propo&longs;itio. 9. Capitis III.

Aquæ flu­xus exfora­minibus ve lociter quæ.

Propo&longs;itio IV. Theorema IV.

Velocitates aquæ decurrentis per foramina æqualia eiu&longs;dem va&longs;is, inæqualiter di&longs;tantia à &longs;ummo va&longs;is, &longs;untin &longs;ubduplicata ratione di&longs;tantiæ.

SInt in va&longs;e AC foraminum æqualium B & C, di&longs;tantiæ à &longs;ummo va&longs;is AB, AC; &longs;itque media inter illas AD. Dico, impetum &longs;eu velocitatem in C, ad impe­tum &longs;eu velocitatem in B, e&longs;&longs;e ut AD ad AB. Quoniam enim, per Propo&longs;it. tertiam hu­jus, aqua per foramina B & C decurrit eo impetu, quo per tubos æqualium foraminum & altitudinum; erunt per Propo&longs;it. 10. Capitis tertij, velocitates aquæ in C & B, ut AD ad AB, hoc e&longs;t, velocitas apud C erit media pro­portionalis inter altitudines AC, AB, atque adeo &longs;ubduplica­tam rationem habebit prædictarum altitudinum.

Proportio dictæ velo­citatis ad foraminum di&longs;tantiam.

Propo&longs;itio V. Theorema V.

Aquæ per æqualia foramina inæqualiter à &longs;ummitate va&longs;is di&longs;tantia fluentes, &longs;unt in &longs;ubduplicata ratione di&longs;tantiarum.

IN va&longs;e AC, præcedentis Propo&longs;itionis, altitudines à forami­nibus æqualibus B & C, ad &longs;ummum va&longs;is A, &longs;int AB, AC, quarum media &longs;it AD. Dico, aquam fluentem per C, ad aquam fluentem per B, e&longs;&longs;e ut AD, ad AB, hoc e&longs;t, e&longs;&longs;e mediam proportionalem, ideoque habere &longs;ubduplicatam rationem altitudinum prædictarum. Quoniam enim aqua flu­it per B & C veluti per tubos, per Petitionem tertiam; a­quæ autem per tubos æqualium altitudinum, &longs;unt in &longs;ubdupli­cata ratione altitudinum tuborum, per propo&longs;it. 8. Capitis tertij; con&longs;tat propo&longs;itum.

Propo&longs;itio VI. Problema I.

Secto foramine laterali va&longs;is in partes æquales, à rectis horizontalibus, invenire rationes aquarum ex eis fluentium.

IN va&longs;e ST &longs;ecetur foramen laterale AB in partes æquales AC, CD, DB, à rectis lineis hori­zontalibus, ut fiant totidem fo ramina, quorum altitudines &longs;int notæ; & per AC fluat aqua E, per CD aqua F, per DB aqua G, tempore æquali &longs;eu eodem: oporteatque venari proportio­nem aquarum E, F, G. Fiant HI, KL, MN, altitudines forami­num AC, CD, DB, à &longs;ummo va­&longs;is; & inter ip&longs;as mediæ proportionales OP, QR, per decimam tertiam Sexti Euclidis, aut per propo&longs;it. 3. Parergi Capitis tertij præcedentis. Quoniam igitur aqua E ad aquam F, e&longs;t ut HI ad OP, per Propo&longs;it. 5. huius Capitis, nota e&longs;t ratio aquæ E ad aquam F. Item quoniam aqua F ad aquam G, e&longs;t ut KL ad QR, per eandem Propo&longs;it. 5. huius Capitis, nota e&longs;t pariter ra­tio aquæ F ad aquam G: At ratioaquæ E ad aquam G com­po&longs;ita rationum inter EF & FG notarum, e&longs;t pariter nota; Reperta e&longs;t igitur ratio aquarum E, F, G.

Problemata varia de fluxu aquæ per foramina va&longs;orum.

Propo&longs;itio VII. Problema II.

Secto foramine laterali va&longs;is in partes inæquales, à re­ctis horizontalibus, reperire rationes aquarum efflu entium ex ip&longs;is.

FOramen CD va&longs;is AB &longs;ecetur à recta E in partes inæquales CE, ED; & effluat ex parte &longs;uperiore e E aqua F, ex inferiori verò ED aqua G, eo­dem tempore. Quæritur proportio F ad G. Si ED foramen minus non men&longs;urat CE, reperiatur eorum ma­xima communis men&longs;ura, per tertiam Decimi Euclidis, quæ &longs;it H; & juxta eam &longs;ecetur CE in partes CQ, QK, KE; item ED in partes EI, ID. Quoniam igitur foramen CD &longs;ectum e&longs;t in partes CQ, QK, KE, EI, ID, æquales per con&longs;tructionem; reperietur per Propo&longs;it. 6. hujus Capitis, ratio aquarum per eas fluentium, ac proinde aquarum fluentium per CE, & ED.

Propo&longs;itio VIII. Problema III.

Datis foraminibus inæqualibus &longs;uper eadem horizon­tali, venari rationes aquarum.

SInt foramina AB, & CD, &longs;uper ho­rizontali BD, utrumque laterale; Quærendaque &longs;it proportio aquarum ex eis eodem tempore fluentium. Produ­catur recta CEG, parallela rectæ DB, dividaturque foramen AB in duo fo­ramina AG, GB. Quoniam igitur no­ta e&longs;t ratio aquarum fluentium ex CD, & GB, per Propo&longs;it. 2. hujus Capitis; item ratio aquarum BG, & GA, per Propo&longs;it. 7. præcedentem; nota pariter e&longs;t ratio ex eis compo&longs;ita inter aquas fluentes per CD, & AG. Cùm igitur nota &longs;it ratio aquæ fluentis per CD, ad aquam fluentem per BG, & per GA partes; nota erit ratio eiu&longs;dem ad totam fluentem per AB.

Propo&longs;itio IX. Problema IV.

Datis foraminibus eiu&longs;dem va&longs;is, quorum vnum &longs;u­perius, alterum inferius, inter easdem parallelas perpen­diculares, reperire rationes aquarum.

DEntur foramina AB, CD, inter parallelas easdem perpendiculares AC, & BD, &longs;itque; venanda ratio aquarum ex eis, æquali tem­pore, fluentium. Concipiatur BC, tan­quam foramen inter easdem parallelas. Quo­niam igitur nota e&longs;t ratio aquarum fluentium ex CD, & ex CB, per Propot. 7. hujus Ca­pitis; item ex CB, & BA, per eandem Pro­po&longs;itionem &longs;eptimam, nota erit pariter ratio aquarum fluenti­um per CD, & AB.

Propo&longs;itio X. Problema V.

Datis foraminibus eiu&longs;dem va&longs;is, quorum vnum &longs;u­perius, alterum inferius, non inter ea&longs;dem parallelas, repire rationes aquarum.

DAta &longs;int foramina AD, EH; oporte­atque reperire rationes aquarum per il­la æquo tempore fluentium. Duc hori­zontales HI, EK, & producta DB in L, concipiatur IL tanquam foramen inter ea&longs;dem parallelas cum AD; & quæratur ratio aquarum per AD, IL fluentium, per Propo&longs;it. 9. hujus Capitis, & &longs;it ut M ad N. Item quæratur ratio IL ad EH, per Pro­po&longs;it. 2. hujus Capitis, & &longs;it ut N ad O. Dico, M ad O e&longs;&longs;e rationem aquarum per AD, & HE fluentium. Quoniam enim ut M ad N, ita e&longs;t AD ad IL; & ut N ad O, ita IL ad EH, per con&longs;tructionem: Erit ex æquo, per vige&longs;imam &longs;ecundam Quinti Euclid. ut M ad O, ita aqua per AD adaquam per EH fluentem.

Propo&longs;itio XI. Problema VI.

Dato foramine, & linea horizontali, in aliquo va&longs;e, con&longs;tituere &longs;uper illa foramen, è quo æqualis aqua fluat eodem tempore.

SIt datum foramen AM, & hori­zontalis CD; &longs;itque de&longs;criben­dum foramen &longs;uper CD, è quo ef­fluat eodem &longs;eu æquali tempore, a­qua ut per AM. Erigantur per­pendiculares AE, MC, & produ­catur DC in E, & &longs;uper EC fiat fo­ramen æquale foramini AM, & &longs;it FC; & ducta FG parallela ip&longs;i CD, fiat HI media inter KB, & KE: & tandem fiat, ut HI ad KE, ita DL ad EC. Dico, foramen LG e&longs;&longs;e foramen quæ&longs;itum, per quod æquali tempore fluat aqua ut per AM. Quoniam e­nim, per Propo&longs;it. 2. hujus Capitis, aqua foraminis LG ad a­quam foraminis FC, e&longs;t ut DL ad CE, hoc e&longs;t, HI, ad KE; & per Propo&longs;it. 5. hujus Capitis, aqua foraminis AM ad aquam foraminis CF, e&longs;t ut eadem HI, ad KE; erit ut aqua LG ad CF, ita aqua AM ad CF, per undecimam Quinti Euclidis, ac proinde aqua foraminis AM erit æqualis aquæ foraminis LG, per nonam Quinti Euclid. & per 1. Axio. lib. 1. Euclid.

Annotatio.

POte&longs;t tamen horizontalis data e&longs;&longs;e tam vicina &longs;ummitati va&longs;is, ut quæ&longs;itum foramen ficri non po&longs;&longs;it, nec hoc, nec alio modo: & tunc Problema e&longs;t in&longs;olubile.

Propo&longs;itio XII. Problema VII.

Dato foramine, & latere alterius in eodem va&longs;e, reperi­re foramen, è quo æqualis aqua effluat.

DAtum &longs;it foramen AB, & latus DC, in eodem va&longs;e; oporteatque de&longs;cribere foramen, cujus latus &longs;it DC, & è quo effluat eodem tempore aqua ut ab AB. Ductis CN, & DF ho­rizontalibus, & &ptail;ducta BE, reperiatur ratio aquarum fluentium ex AB, & FE, per Propo&longs;it. 9. hujus Capitis, quæ &longs;it ut G ad H; & fiat, ut H ad G, ita FI ad FK, per decimam tertiam Sex­ti Euclid. & Propo&longs;it. 3. Parergi Capitis 3; & à K erigatur perpendicularis KL, & tandem fiat foramen, cujus latus DC, æquale & &longs;imile ip&longs;i FL foramini, & &longs;it DM. Dico, è foramine DM fluere aquam, ut ex AB foramine. Quoniam enim aqua fluens ex AB, ad aquam fluentem ex FE, e&longs;tut G ad H, per con­&longs;tructionem; item aqua fluens per FL, &longs;eu ei æquale DM, ad fluentem per idem FE, e&longs;t itidem ut G ad H, per Propo&longs;it. 2. hujus Capitis; erunt aquæ fluentes per AB, & per DM, inter &longs;e æquales, per nonam Quinti Euclidis. Igitur DM erit foramen quæ&longs;itum.

Propro&longs;itio XIII. Problema VIII.

Dato foramine, reperire aliud æquale in eodem va&longs;e, è quo fluat aqua in ratione data.

DEtur in va&longs;e AB foramen C, & da­ta &longs;it ratio aquarum D, E, quarum D fluat in dato tempore per foramen C; reperiendum verò &longs;it, ubi fiat æquale foramen, è quo fluat in æquali tempore aqua E. Fiat ad D, E, AC, quarta pro­portionalis AF, per duodecimam Sexti Euclid. aut Propo&longs;it. 8. Parergi Capitis tertij præcedentis; & ad AC, AF, ter­tia proportionalis AG, per vndecimam &longs;exti Euclid. aut per Propo&longs;it. 6. Paret­gi citati; & in G fiat foramen, (quod &longs;i fieri nequit, Problema e&longs;t in&longs;olubile.) Dico, G e&longs;&longs;e locum foraminis quæ&longs;iti. Quoniam enim aquæ fluentes per dicta foramina &longs;unt in &longs;ubduplicata ratione altitudinum AC, AG, per Propo&longs;it. 5. hujus capitis; & aquæ D, & E &longs;unt pa­riter in &longs;ubduplicata ratione datæ altitu­dinis AC, & inveniendæ altitudinis, per eandem Propo&longs;itionem 5. hujus capitis; &longs;equitur aquas fluen­tes per dicta foramina C & G, e&longs;&longs;e ut aquas D & E.

Propo&longs;itio XIV. Problema IX.

Dato foramine, aptare in eodem va&longs;e aliud datum &longs;i­mile, magnitudinis diver&longs;æ, à quo aqua fluens cum fluente à primo, habeat rationem datam.

IN va&longs;e AB, datum &longs;it foramen C, & aliud ad&longs;ignatum D &longs;imile, &longs;ed diver­&longs;æ magnitudinis, &longs;ive majoris, &longs;ive mi­noris; ratio verò aquarum data &longs;it E, F. Aptandum &longs;it foramen D ad&longs;ignatum eâ lege, ut aqua per illud fluens, cum aqua fluente ex C, &longs;it ut E ad F. Super horizontali ducta CG, fiat fora­men G, æquale foramini D; & perqui­ratur ratio aquarum fluentium per C & G, per Propo&longs;it. 8. hujus capitis, & &longs;it ut E ad H. Quæ &longs;i e&longs;t eadem quæ e&longs;t inter E & F, habemus intentum: Sin minùs, fiat aliud foramen infra aut &longs;upra G, ei &longs;imile & æquale, à quo fluat aqua, quæ cum fluente ab ip&longs;o G habeat rationem ut H ad F, & &longs;it I, Quod &longs;i fieri nequit, Problema e&longs;t in- &longs;olubile.) Dico, I e&longs;&longs;e foramen quæ&longs;itum. Quoniam enim aqua C ad aquam G, e&longs;t ut E ad H; & aqua G ad aquam I, e&longs;t ut H ad F, ex con&longs;tructione & operatione facta; erit ex æquo etiam aqua C ad aquam I, ut E ad F, per vige­&longs;imam &longs;ecundam Quinti Euclidis.

CAPUT VI.

De Aëris gravitate, rarefactione, & conden&longs;atione.

QUoniam in Pneumaticis ac Hydropneumaticis Ma­chinis magni refert, aëris naturalis pondus &longs;eu gra­vitatem, rarefactique ac conden&longs;ati quantitatem ae vires cogno&longs;cere; de ijs nonnihil parergi loco ad­dendum hîc cen&longs;ui. Sit itaque

Propo&longs;itio I.

Aëris gravitatem invenire.

TAmet&longs;i aêr levis &longs;it inter reliqua elementa, &longs;uaque levitate &longs;ur&longs;um tendat, ut &longs;it &longs;upra terram & aquam; admixtum tamen habet aliquid gravitatis. Patet ex eo, quòd folles lu­&longs;orij aëre adden&longs;ato bene inflati, ut & ve&longs;icæ inflatæ, plùs pon­derent, quàm flaccidi, ut exqui&longs;ita trutina aut libra deprehen­di pote&longs;t. P. Joannes Bapti&longs;ta Ricciolus Tomo 1. Almge&longs;ti Novi lib. 2. cap. 5. num. 4. ait expendi&longs;&longs;e &longs;e ve&longs;icam bovinam quæ flaccida erat &longs;crupulorum 4, & granorum 4, & deprehen­di&longs;&longs;e inflatam fui&longs;&longs;e &longs;crupulorum 4, & granorum 6. Quare aër additus per inflationem, appendebat grana 2. Marinus Mer&longs;ennus in Phænomenis &longs;uis Propo&longs;it. XXIX. a&longs;&longs;erit, &longs;e Geo­metris præ&longs;entibus & adjuvantibus, pondera&longs;&longs;e bilance Æo­lopilam æneam (qualem de&longs;cribimus infrà Parte 2. Cla&longs;&longs;e 1. cap. 3. Machina 6.) &longs;atis calefactam, & propemodum candentem, omnique humore de&longs;titutam, & quàm minimùm aêris conti­nentem; deprehendi&longs;&longs;eque pondus fui&longs;&longs;e unciarum 4, drach­marum 6, & granorum 15: po&longs;tquam verò naturaliter refrixi&longs;&longs;et æolopila, & aër antea vehementer rarefactus redij&longs;&longs;et ad pri­&longs;tinum ac naturalem &longs;uum &longs;tatum; iterum pondera&longs;&longs;e ip&longs;am, & inveni&longs;&longs;e pondus præcedente pondere majus fui&longs;&longs;e 4 gra­nis. (Apud Mer&longs;ennum habetur, minus; utique typographi­co errore.) Erat autem bilanx, inquit Mer&longs;ennus, quæ dimi­dio grani perdebat æquilibrium.

Aëris gra­vitatem in­venire.

Atque ex his patet, qua ratione aëris gravitas inveniatur. Alij aliter explorant pondus aëris, præ&longs;ertim Illu&longs;tri&longs;&longs;imus Do­minus Carolus Vintimillia Eques Panormitanus in Sicilia, & Ami­cus Uranicus &longs;incei i&longs;&longs;imus, qui legendam mihi dedit demon­&longs;trationem &longs;ubtili&longs;&longs;imam, praximque hydro&longs;taticam, qua olim Adole&longs;cens coram Sereni&longs;&longs;imo Philiberto Siciliæ Prorege, No­bilitateque Panormitana univer&longs;a aërem ponderaverat. Vide etiam quæ habet Daniel Lip&longs;torpius part. 3. Speciminum Phi­lo&longs;ophiæ Carthe&longs;ianæ cap. 2. in fine, ubiadducit modum Galilæi & Joann. Chry&longs;o&longs;t. Magneni. Vide præterea Appendicem in fine Operis.

Propo&longs;itio II.

Quantum conden&longs;ari aër po&longs;&longs;it, invenire.

Vide infrà Iconi&longs;mi VIII. Figu­ram IX. a.

AS&longs;umatur fons pneumaticus, qualem de&longs;cribimus infrà, Parte 2. Cla&longs;&longs;e 1. cap. 2. Machina 8. eju&longs;que capacitas exactè exploretur in men&longs;ura cubica aquæ. Capiat autem dictus fons, gratiâ exempli, unum pedem cubicum, &longs;eu libras 72. In hunc fontem immitte &longs;yringe, per foramen C, aquam quantum fieri, &longs;ine ruptionis periculo, pote&longs;t. Si jam notam feceris in men&longs;ura, quantitatem aquæ immi&longs;&longs;æ, eamque à fon­

tis capacitate detraxeris; re&longs;iduum fontis indicabit locum aëris conden&longs;ati, qui priùs totam fontis capacitatem imple- bat. Ut &longs;i immi&longs;&longs;æ &longs;untduæ tertiæ pedis cubici, occupabit aër unam tertiam partem.

Aër quan­tum conden &longs;ari po&longs;&longs;it, invenire.

Atque hæc e&longs;t expediti&longs;&longs;ima ratio men&longs;urandi conden­&longs;ationem aëris. Fieri quoque idip&longs;um pote&longs;t &longs;clopeto pneu­matico, de quo infrà loco cit. cap. 2. Machina 13. & 14.

Credunt aliqui, inquit Mer&longs;ennus in Hydraulicis Propo­&longs;it. 31. aërem per conden&longs;ationem non po&longs;&longs;e ni&longs;i ad tertiam &longs;pa­tij, quod naturaliter occupat, partem redigi, quòd videant aquam in vas infu&longs;am tres heminas &longs;eu libras continens, non &longs;uperare duas heminas, ob aërem intus manentem. Certum e&longs;t tamen, &longs;ubdit idem, magis conden&longs;ari po&longs;&longs;e; ip&longs;eque ex­perientiâ didicit, po&longs;&longs;e ad &longs;patium quindecuplo minus in &longs;clo­peto pneumatico cogi, idque &longs;olâ vi manus &longs;yringe utentis; quæ vis cùm intendi multò magis po&longs;&longs;it, dubium non e&longs;t, aë­rem adhuc multò magis conden&longs;ati po&longs;&longs;e. Vide Mer&longs;ennum in Hydraulicis loco citato.

Propo&longs;itio III.

Quantum rarefieri aër po&longs;&longs;it, invenire.

MArinus Mer&longs;ennus loco citato Propo&longs;it. 31. ait, con&longs;tare, aërem ita rarefieri in ignitis æolopilis (quas de&longs;cribimus Parte 2. Cla&longs;&longs;e 1. cap. 2. Machina 6.) ut &longs;eptuage&longs;ima pars aë­ris priùs inclu&longs;i, & nondum rarefacti, æolopilas impleat. Hæc autem rarefactionis quantitas inveniri pote&longs;t præ&longs;idio Æolo­pilæ hunc in modum.

Aër quan­tum po&longs;&longs;it rarefieri.

Æolopila notæ capacitatis calefiat eò u&longs;que, donec ma­jorem vim caloris &longs;u&longs;tinere ampliùs non po&longs;&longs;e judicetur citra liquefactionis periculum: &longs;ic enim fiet, ut quàm minimùm aë­ris rarefacti totam ip&longs;ius capacitatem expleat. Quo facto, igni extracta &longs;tatim aquæ frigidæ immergatur. Continget enim ut aër, qui rarefactus totam Æolopilam occupabat, ad naturalem redeat con&longs;titutionem; ac proinde Æolopila, nè detur vacuum, aquam &longs;uget. Hujus aquæ quantitas &longs;i fiat nota, & ab Æolopilæ capacitate detrahatur; notam relinquet aëris quantitatem & nunc adden&longs;ati, & antea rarefacti.

Hæc ex Mer&longs;enni &longs;ententia; quæ tamen valde incerta mi­hi videntur, tum quia non &longs;citur quando Æolopila e&longs;t &longs;ummè calefacta, tum quia aqua ab&longs;umitur à calore Æolopilæ ante & po&longs;t attractionem, tum denique quia con&longs;tare non pote&longs;t, utrum aër re&longs;iduus po&longs;t aquæ attractionem &longs;it ad naturalem &longs;tatum re­dactus, an verò præter naturam adden&longs;atus, aut rarefactus. Vi­de quæ dicimus infrà in Appendice.

Conclu&longs;io

Protheoriæ Quartæ.

DIci hîc nonnulla po&longs;&longs;ent de &longs;alientium ex fontibus pneuma­ticis, quos de&longs;cribimus Parte 2. Cla&longs;&longs;e 1. cap. 2. Machina 8. longitudine, altitudineque, & cum tuborum &longs;alientibus compa­rari, ut facit Mer&longs;ennus in &longs;uis Hydraulicis Propo&longs;it. 28. &longs;ed quoniam valde incerta &longs;unt & quæ Mer&longs;ennus affert, & quæ à me, & à quocunque alio, varijs experientijs inventa, afferri po&longs;­&longs;unt; &longs;atiùs judicavi, omnino ab&longs;tinere. Dependet enim &longs;alien­tium huju&longs;modi longitudo atque altitudo, non à tuborumlongi­tudine, &longs;ed poti&longs;&longs;imùm ab aëris intra fontem compre&longs;&longs;i vehe­mentia, quæ infinitis modis variari pote&longs;t; & cùm omnia exper­tus fueris, dubius adhuc hærebis.

Dici præterea multa po&longs;&longs;ent de fluxu aquæ per tubos incli­natos & horizontales; examinarique, quæ habet Joannes Bapti­&longs;ta Balianus lib 4. & 5. de Motu gravium, &longs;eu lib. 1. & 2. de motu liquidorum: at quoniam hæc non admodum conducunt fini no­&longs;tro in hoc Opere intento, & longiorem &longs;ermonem requirunt; omittenda potiùs omnino quàm paucis per&longs;tringenda pu­tavi, acturus de illis in Mechanica Univer&longs;ali. Ad praxin igitur, mi&longs;sâ theoriâ, procedamus.