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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/texts/XML/echo/la/Bernoulli_1738_AZ870BWE.xml Fri Dec 07 17:05:22 2012 +0100 @@ -0,0 +1,12942 @@ +<?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> + <metadata> + <dcterms:identifier>ECHO:AZ870BWE.xml</dcterms:identifier> + <dcterms:creator identifier="GND:118656503">Bernoulli, Daniel</dcterms:creator> + <dcterms:title xml:lang="la">Hydrodynamica, sive De viribus et motibus fluidorum commentarii</dcterms:title> + <dcterms:date xsi:type="dcterms:W3CDTF">1738</dcterms:date> + <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> + <dcterms:rights>CC-BY-SA</dcterms:rights> + <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> + <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> + <parameters>despecs = 1.1.2</parameters> + <log>pb are correct, no forbidden chars, + subscript/ superscript confusion by data entry + changed opening root tag to root symbol + </log> + </metadata> + <text xml:lang="la" type="free"> +<div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/> +<pb file="0002" n="2"/> +<pb file="0003" n="3"/> +<pb file="0004" n="4"/> +<handwritten/> +<pb file="0005" n="5"/> +<handwritten/> +<pb file="0006" n="6"/> +<pb file="0007" n="7"/> +</div> +<div xml:id="echoid-div2" type="section" level="1" n="2"> +<head xml:id="echoid-head1" xml:space="preserve"><emph style="red">DANIELIS BERNOULLI <emph style="sc">Joh</emph>. <emph style="sc">Fil</emph>.</emph> <lb/><emph style="sc">Med</emph>. <emph style="sc">Prof</emph>. <emph style="sc">Basil</emph>. <lb/>ACAD. SCIENT. IMPER. PETROPOLITANÆ, PRIUS MATHESEOS <lb/>SUBLIMIORIS PROF. ORD. NUNC MEMBRI ET PROF. HONOR. <lb/><emph style="red">HYDRODYNAMICA,</emph> <lb/>SIVE <lb/>DE VIRIBUS ET MOTIBUS FLUIDORUM <lb/>COMMENTARII. <lb/><emph style="red">OPUS ACADEMICUM</emph> <lb/>AB AUCTORE, DUM PETROPOLI AGERET, <lb/>CONGESTUM.</head> + <figure> + <image file="0007-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0007-01"/> + </figure> +</div> +<div xml:id="echoid-div3" type="section" level="1" n="3"> +<head xml:id="echoid-head2" xml:space="preserve">ARGENTORATI, <lb/><emph style="red">Sumptibus JOHANNIS REINHOLDI DULSECKERI,</emph> <lb/>Anno M D CC XXXVIII.</head> +<head xml:id="echoid-head3" xml:space="preserve">Typis <emph style="sc">Joh</emph>. <emph style="sc">Henr</emph>. <emph style="sc">Deckeri</emph>, Typographi Baſilienſis.</head> +<pb file="0008" n="8"/> +<handwritten/> +<pb file="0009" n="9"/> +</div> +<div xml:id="echoid-div4" type="section" level="1" n="4"> +<head xml:id="echoid-head4" xml:space="preserve">CELSISSIMO <lb/>ATQUE <lb/>SERENISSIMO <lb/>PRINCIPI <emph style="sc">ET</emph> DOMINO <lb/>DOMINO <lb/>ERNESTO <lb/>JOHANNI <lb/>DEI GRATIA IN LIVONIA <lb/>CURLANDIÆ <lb/>ET <lb/>SEM - GALLIÆ <lb/>DUCI.</head> +<pb file="0010" n="10"/> +</div> +<div xml:id="echoid-div5" type="section" level="1" n="5"> +<head xml:id="echoid-head5" xml:space="preserve">CELSISSIME <emph style="sc">ATQUE</emph> SERENISSIME <lb/>PRINCEPS, <lb/>DOMINE GRATIOSISSIME.</head> +<p> + <s xml:id="echoid-s1" xml:space="preserve">NOn auſus fuiſſem Sereniſſimo Nomini <lb/>Tuo Hydrodynamicam hanc in-<lb/>ſcribere, niſi illa Academiæ Scien-<lb/>tiarum, ſub umbone Tuo Petropoli <lb/>florentis, conſilio & </s> + <s xml:id="echoid-s2" xml:space="preserve">ſubſidiis a me <lb/>conſcripta fuiſſet. </s> + <s xml:id="echoid-s3" xml:space="preserve">Novimus quan-<lb/>tum Tibi, Sereniſſime Princeps, <lb/>Magnanime Academiæ Protector, poſt Auguſtam illam <lb/>orbis borealis Palladem, debeamus, idque cum toto orbe <lb/>literato, qui præclara ſibi porro ab Academia, amœnis <lb/>benevolentiæ Tuæ radiis colluſtrata, pollicetur, pia & </s> + <s xml:id="echoid-s4" xml:space="preserve"><lb/>immortali recolemus me<unsure/>moria. </s> + <s xml:id="echoid-s5" xml:space="preserve">Florebit in æternitatis <lb/>ſacrario apud Ruſſicam gentem Tuorum in illam merito- +<pb file="0011" n="11"/> +rum magnitudo, apud Curlandos felicium, quæ divina <lb/>illis providentia ſub Sceptro Tuo deſtinavit, fatorum me-<lb/>moria: </s> + <s xml:id="echoid-s6" xml:space="preserve">apud univerſas denique gentes glorioſiſſimæ Tuæ <lb/>vitæ perpetua admiratio. </s> + <s xml:id="echoid-s7" xml:space="preserve">Quam cara ſit ſuperis Ruſſici <lb/>Sceptri Majeſtas populique Tui felicitas, illuſtria tempo-<lb/>rum præſentium fata nos docent. </s> + <s xml:id="echoid-s8" xml:space="preserve">Hi proſperos magno-<lb/>rum conſiliorum eventus; </s> + <s xml:id="echoid-s9" xml:space="preserve">hi vitæ Tibi & </s> + <s xml:id="echoid-s10" xml:space="preserve">Principatus <lb/>diuturnitatem; </s> + <s xml:id="echoid-s11" xml:space="preserve">hi ſucceſſores ex ſanguine Tuo, virtu-<lb/>tum Tuarum æmulos, longa ſerie ad omnem temporum <lb/>profunditatem, orbe univerſo plaudente, largiantur. <lb/></s> + <s xml:id="echoid-s12" xml:space="preserve">lta vovet</s> +</p> +</div> +<div xml:id="echoid-div6" type="section" level="1" n="6"> +<head xml:id="echoid-head6" xml:space="preserve">SERENISSIME & CELSISSIME PRINCEPS <lb/>DOMINE GRATIOSISSIME</head> +<p style="it"> + <s xml:id="echoid-s13" xml:space="preserve">Celſitudinis Tu@</s> +</p> +<p style="it"> + <s xml:id="echoid-s14" xml:space="preserve">Scrib. </s> + <s xml:id="echoid-s15" xml:space="preserve">Baſileæ</s> +</p> +<p> + <s xml:id="echoid-s16" xml:space="preserve">10. </s> + <s xml:id="echoid-s17" xml:space="preserve">Mart. </s> + <s xml:id="echoid-s18" xml:space="preserve">1738.</s> + <s xml:id="echoid-s19" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s20" xml:space="preserve">Humillimus & </s> + <s xml:id="echoid-s21" xml:space="preserve">Obſequioſiſſimus <lb/>Servus <lb/>DANIEL BERNOULLI.</s> + <s xml:id="echoid-s22" xml:space="preserve"/> +</p> +<pb file="0012" n="12"/> +</div> +<div xml:id="echoid-div7" type="section" level="1" n="7"> +<head xml:id="echoid-head7" xml:space="preserve">PRÆFATIO.</head> +<p style="it"> + <s xml:id="echoid-s23" xml:space="preserve">PRodit tandem in publicum Hydrodynamica noſtra, <lb/>ſuperatis omnibus, quæ impreßionem ejus ab octo fere <lb/>annis morata ſunt, obſtaculis; </s> + <s xml:id="echoid-s24" xml:space="preserve">lucem fortaßis haud <lb/>aſpectura, ſiad me ſolum omnis iſte labor pertinuiſſet. </s> + <s xml:id="echoid-s25" xml:space="preserve">Præci-<lb/>puas enim huius operis partes auſpiciis, conſiliis, ſubſidiisque <lb/>Academiæ Scientiarum Petropolitanæ deberi lubens profiteor. <lb/></s> + <s xml:id="echoid-s26" xml:space="preserve">Anſam libro dedit ipſum ejus inſtitutum, quo primi, qui ad <lb/>@am formandam convenerunt, Profeſſores, de argumento <lb/>quodam utili &</s> + <s xml:id="echoid-s27" xml:space="preserve">, quantum fieri poſſet, novo Diatribam con-<lb/>ſcribere tenebantur, certe admonebantur. </s> + <s xml:id="echoid-s28" xml:space="preserve">Theoriam de vi-<lb/>ribus & </s> + <s xml:id="echoid-s29" xml:space="preserve">motibus fluidorum, niſi invita Minerva fuerit <lb/>ſuſcepta, argumentum eſſe nec inutile nec tritum, quisque facile <lb/>largietur. </s> + <s xml:id="echoid-s30" xml:space="preserve">Vt autem Lectoris tædium diſcuterem, rerum va-<lb/>rietati inprimis operam dedi, præſertim in quinque poſteriorbus <lb/>ſectionibus, atque ſpecimina inſerui analytica, phyſica, me-<lb/>chanica, cum theoretica tum practica, nonnulla geome-<lb/>trica, nautica, aſtronomica & </s> + <s xml:id="echoid-s31" xml:space="preserve">alia, quorum tamen ex-<lb/>poſitionem operis ſuſcepti ratio non tam ferre quam poſtu-<lb/>lare viſa fuit. </s> + <s xml:id="echoid-s32" xml:space="preserve">Quæ feſtinanti exciderunt ſphalmata, æquus <lb/>harumque rerum intelligens Lector facile corriget. </s> + <s xml:id="echoid-s33" xml:space="preserve">Vnicus <lb/>hujus ſcripti finis eſt, ut Academiæ inſervirem, cujus <lb/>omnes labores eo collimant, ut bonarum literarum incre-<lb/>menta & </s> + <s xml:id="echoid-s34" xml:space="preserve">publica commoda promoveat.</s> + <s xml:id="echoid-s35" xml:space="preserve"/> +</p> +<pb file="0013" n="13"/> + <figure> + <image file="0013-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0013-01"/> + </figure> +<note position="right" xml:space="preserve"> <lb/># INDEX SECTIONUM. <lb/># SECTIO PRIMA. <lb/>Introitus eſt variaque continet prænotanda. # pag. 1. <lb/># SECTIO SECUNDA. <lb/>Agit de fluidis ſtagnantibus eorundemque æquilibrio tum inter ſe tum <lb/># ad alias potentias relato. # 17. <lb/># SECTIO TERTIA. <lb/>De Velocitatibus fluidorum ex vaſe utcunque formato per foramen <lb/># qualecunque effluentium. # 30. <lb/># SECTIO QUARTA. <lb/>De variis temporibus, quæ in effluxu aquarum deſiderari poſſunt. # 61. <lb/># SECTIO QUINTA. <lb/>De motu aquarum ex vaſis conſtanter plenis. # 90. <lb/># SECTIO SEXTA. <lb/>De fluidorum motu non effluentium ſeu intra latera vaſorum moto-<lb/># rum, ubi præſertim de oſcillationibus fluidorum. # 111. <lb/># SECTIO SEPTIMA. <lb/>De motu aquarum per vaſa ſubmerſa, ubi præſertim exemplis oſten-<lb/># ditur, quam inſigniter utile ſit principium conſervationis virium <lb/># vivarum, vel iis in caſibus, quibus continué aliquid de illis perdi <lb/># cenſendum eſt. # 124. <lb/></note> +<pb file="0014" n="14"/> +<note position="right" xml:space="preserve"> <lb/># SECTIO OCTAVA. <lb/>De motu fluidorum, cum homogeneor<unsure/>um, tum heterogeneorum, per <lb/># vaſa itregularis & præruptæ ſtructuræ, ubi ex theoria virium <lb/># vivarum, quarum pars continué abſor<unsure/>beatur, explicantur præci-<lb/># pue phænomena ſingularia fluidorum per plurima foramina tra-<lb/># jectorum, præmiſſis regulis generalibus pro motibus fluidorum <lb/># ubique definiendis. # 143. <lb/># SECTIO NONA. <lb/>De motu fluidorum, quæ non proprio pondere, ſed potentia aliena <lb/># ejiciuntur, ubi potiſſimum de machinis hydraulicis earundemque <lb/># ultimo, qui dari poteſt, perfectionis gradu. # 163. <lb/># SECTIO DECIMA. <lb/>De affectionibus atque motibus fluidorum claſſicorum, præcipue <lb/># aëris. # 200. <lb/># SECTIO UNDECIMA. <lb/>De fluidis in vorticem actis, tum etiam de iis, quæ in vaſis motis con-<lb/># tinentur. # 244. <lb/># SECTIO DUODECIMA. <lb/>Novam ſtaticam fluidorum motorum, quæ hydraulico - ſtatica vocari <lb/># poteſt, exhibet. # 256. <lb/># SECTIO DECIMA TERTIA. <lb/>De reactione fluidorum ex vaſis effluentium, de menſura effectus, qui <lb/># inde obtineri poteſt ad navigationem, ubi ſimul theoria nova <lb/># pro fluidorum, poſtquam effluxerunt, impetu in plana quibus <lb/># occurrunt definiendo exhibetur. # 278. <lb/></note> +<pb file="0015" n="15"/> + <figure> + <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0015-01"/> + </figure> +</div> +<div xml:id="echoid-div8" type="section" level="1" n="8"> +<head xml:id="echoid-head8" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph></head> +<head xml:id="echoid-head9" xml:space="preserve"><emph style="bf">SECTIO PRIMA.</emph></head> +<head xml:id="echoid-head10" style="it" xml:space="preserve"><emph style="bf">Quæ introitus eſt, variaque continet prænotanda.</emph></head> +<head xml:id="echoid-head11" xml:space="preserve"><emph style="bf">§. 1.</emph></head> +<p> + <s xml:id="echoid-s36" xml:space="preserve">DUplex cum ſit Theoria Fluidorum, quarum altera Hydroſtati-<lb/>ca, liquorum ſtagnantium preſſiones & </s> + <s xml:id="echoid-s37" xml:space="preserve">æquilibria varia, altera <lb/>Hydraulica, fluidorum motum ſpectans, ſeorſum pertractari a <lb/>ſcriptoribus conſueverunt, utramque vero tam arcto nexu in-<lb/>ter ſe cohærere perciperem, ut altera alterius ope plurimum <lb/>egeat, haud dubitavi eas confundere, quantum id ordo rerum <lb/>poſtulare videbatur, ambaſque nomine communi & </s> + <s xml:id="echoid-s38" xml:space="preserve">generaliori Hydrodynami@ <lb/>cæ complecti. </s> + <s xml:id="echoid-s39" xml:space="preserve">Quamvis autem ab antiquiſſimis temporibus fuerit continuo <lb/>exculta Theoria fluidorum, incrementa tamen non admodum notabilia ce-<lb/>pit; </s> + <s xml:id="echoid-s40" xml:space="preserve">veterum quidem Mathematicorum cognitio eo terminabatur, quod æ- +<pb o="2" file="0016" n="16" rhead="HYDRODYNAMICÆ"/> +quilibrium commune fluidorum ſtagnantium, aut etiam corporum cum flui-<lb/>dis, quibus inſident, de quibus Archimedes ſcripſit, intelligebant; </s> + <s xml:id="echoid-s41" xml:space="preserve">& </s> + <s xml:id="echoid-s42" xml:space="preserve">cum <lb/>præterea per ſe pateat, ubi æquilibrium non eſt, motum verſus partem mi-<lb/>noris preſſionis fieri, varios luſus, machinaſque hydraulicas hinc excogitare <lb/>potuerunt, partim oblectationi, partim publicis commodis egregie inſervi-<lb/>entes, qua quidem in re peringenioſos ſe monſtrarunt; </s> + <s xml:id="echoid-s43" xml:space="preserve">videbant etiam, ſed <lb/>quaſi per tranſennam motus illos, qui preſſioni aëris debentur: </s> + <s xml:id="echoid-s44" xml:space="preserve">Veras autem <lb/>rationes accuratasque menſuras in Hydraulicis rebus plane ignorabant, atque <lb/>ſic fere in limine ſubſiſtebant.</s> + <s xml:id="echoid-s45" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s46" xml:space="preserve">§. </s> + <s xml:id="echoid-s47" xml:space="preserve">2. </s> + <s xml:id="echoid-s48" xml:space="preserve">Motui fluidorum determinando inſervit præcipue effluxus aquæ <lb/>ex vaſe per foramen valde parvum: </s> + <s xml:id="echoid-s49" xml:space="preserve">tametſi vero non omnino fugeret Fron-<lb/>tinum alioſque, uti aliqui credunt, velocitatem aquarum ex vaſe vel caſtello <lb/>effluentium creſcere ab aucta altitudine aquæ ſupra effluxus locum, negari <lb/>tamen non poteſt, quin idem Frontinus in computandis aquarum modulis, <lb/>ſeu erogandis aquis turpes & </s> + <s xml:id="echoid-s50" xml:space="preserve">injuſtos commiſerit errores. </s> + <s xml:id="echoid-s51" xml:space="preserve">Benedictus Caſtel-<lb/>lius primus de nexu velocitates inter & </s> + <s xml:id="echoid-s52" xml:space="preserve">altitudines cogitare, falſam autem <lb/>legem ſuſpicatus eſt, putans, ambas eandem rationem ſequi. </s> + <s xml:id="echoid-s53" xml:space="preserve">Poſt hunc de-<lb/>@um Torricellius obſervavit, velocitates creſcere in ſubduplicatâ ratione alti-<lb/>tudinum, quem ſecuti ſunt omnes; </s> + <s xml:id="echoid-s54" xml:space="preserve">nec dum vero conveniebant de abſoluta <lb/>velocitatis menſura, experimenta tamen inſtituerunt, qua iſtam menſuram <lb/>definiri exiſtimarunt, inter quæ potiſſimum allegari ſolet illud, quod a Gulielmino <lb/>ſumtum, octieſque repetitum fuit, quamvis id ab aliis experimentis ex illo tempore <lb/>factis admodum recedat: </s> + <s xml:id="echoid-s55" xml:space="preserve">ſolent autem omnia inter ſe differre, quæ ſub diver-<lb/>ſis fiunt circumſtantiis, nec ſemper tutum eſt, uti ſuo loco dicemus plu-<lb/>ribus, ex quantitate aquæ, definito tempore per definitum lumen effluentis, ju-<lb/>dicium ferre de ejuſdem velocitate. </s> + <s xml:id="echoid-s56" xml:space="preserve">Sic cum ad calculum revocamus expe-<lb/>rimentum Gulielminianum, cujus modo mentionem fecimus, concludendum <lb/>eſſet ex quantitate aquæ, quæ per lumen datum tempore dato effluxit, ve-<lb/>locitatem ejus non majorem fuiſſe illa, quæ debetur quartæ parti altitudinis <lb/>ſuperficiei aqueæ ſupra foramen. </s> + <s xml:id="echoid-s57" xml:space="preserve">Et alia ſunt eodem Auctore experimenta, <lb/>quæ recenſentur Lib. </s> + <s xml:id="echoid-s58" xml:space="preserve">2. </s> + <s xml:id="echoid-s59" xml:space="preserve">prop. </s> + <s xml:id="echoid-s60" xml:space="preserve">1. </s> + <s xml:id="echoid-s61" xml:space="preserve">menſ. </s> + <s xml:id="echoid-s62" xml:space="preserve">aquarum fluent: </s> + <s xml:id="echoid-s63" xml:space="preserve">vi quorum aqua ef-<lb/>fluens velocitate ſua aſcendere poſſit ad duas tertias iſtius altitudinis; </s> + <s xml:id="echoid-s64" xml:space="preserve">Apud <lb/>Mariottum alioſque extant, quæ pro dimidia altitudine faciunt; </s> + <s xml:id="echoid-s65" xml:space="preserve">qua non ob-<lb/>ſtante velocitatum ita æſtimatarum diverſitate, mihi perſuadeo, vix a ſe in- +<pb o="3" file="0017" n="17" rhead="SECTIO PRIMA."/> +vicem veras velocitates diſcrepaſſe, ratione habita ad altitudines aquæ & </s> + <s xml:id="echoid-s66" xml:space="preserve">ubi-<lb/>que tales proxime fuiſſe, quæ integræ altitudini debeantur: </s> + <s xml:id="echoid-s67" xml:space="preserve">illa autem, quæ <lb/>loco ultimo fuere citata, quæque pro dimidia altitudine prima fronte viden-<lb/>tur ſtare, numero apud Authores plurima, movebant procul dubio Newto-<lb/>num, Virum meritis ſuis immortalem, ut paulo confidentius loqueretur de <lb/>Theoria, qua aquam per lumen minimum ex vaſe verticaliter ſurſum exili-<lb/>entem ad dimidiam altitudinem aquæ in vaſe ſtagnantis aſcendere poſſe inve-<lb/>nerat, etſi aſſertum iſtud omnibus experimentis, quæ de his altitudinibus im-<lb/>mediate ſumta fuere, contradicat: </s> + <s xml:id="echoid-s68" xml:space="preserve">Theoriam expoſuit in edit. </s> + <s xml:id="echoid-s69" xml:space="preserve">prima princ. <lb/></s> + <s xml:id="echoid-s70" xml:space="preserve">Math. </s> + <s xml:id="echoid-s71" xml:space="preserve">phil. </s> + <s xml:id="echoid-s72" xml:space="preserve">nat.</s> + <s xml:id="echoid-s73" xml:space="preserve">, eamque petiit ex preſſione, qua aqua præ foramine poſita <lb/>moxque egreſſura ad motum cietur. </s> + <s xml:id="echoid-s74" xml:space="preserve">Quoniam vero natura rei haud ſemper <lb/>permittere videtur, ut a priori definiatur vis aquam ad effluxum animans, a@-<lb/>que potius de ea vix aliter, quam ex phænomenis motus, id eſt, a poſterio@i, <lb/>quod ſæpe expertus ſum, judicare licet, ſuſpectum eſſe debet ratiocinium@ſti <lb/>principio innixum. </s> + <s xml:id="echoid-s75" xml:space="preserve">Hinc etiam Vir modo laudatus ſententiam ſuam muta-<lb/>vit in ſecunda Operis ſui editione, rurſuſque aliquantum in tertia, affirmans <lb/>aquam ad totam quidem altitudinem aſcendere, venam autem, quam efformat, <lb/>præ foramine contrahi ſeu gracileſcere, atque ſic utrique phænomeno velo-<lb/>citatis quantitatisque dato tempore effluentis, quæ ſibi contradicere videban-<lb/>tur, ſatisſaciens. </s> + <s xml:id="echoid-s76" xml:space="preserve">Quamvis autem contractionem iſtam fili aquei veram eſſe <lb/>cauſam, ob quam velocitas a@uæ effluentis non poſſit æſtimari ex quanticate, <lb/>negandum non ſit, puto tamen, Theoriam ipſi non eſſe ſuperinſtruendam, <lb/>quia accidentalis eſt, nec ſibimet ubique conſtans, dum velocitas non variat <lb/>niſi a cauſis alienis veluti attritu, tenacitate aquæ, aliisque ſimilibus. </s> + <s xml:id="echoid-s77" xml:space="preserve">Sic cum <lb/>aqua non per ſimplex foramen, ſed per tubulum cylindricum effluit, vena <lb/>notabiliter non contrahitur ſalva velocitate, excepto eo, quod propter attri-<lb/>tum ei demitur: </s> + <s xml:id="echoid-s78" xml:space="preserve">ſi quis autem hoc non obſtante putet, ex preſſione poſſe re-<lb/>cte & </s> + <s xml:id="echoid-s79" xml:space="preserve">tuto aquarum fluxum deduci, hunc rogarim, ut ad caſus magis com-<lb/>poſitos animum advertat, v. </s> + <s xml:id="echoid-s80" xml:space="preserve">gr. </s> + <s xml:id="echoid-s81" xml:space="preserve">ad fluxum aquæ, quem mira@lem vocat Ma-<lb/>riottus, ex vaſe, quod diaphragma aliquod foramin@ perforatum in duas ca-<lb/>vitates aqua implendas diſpeſcit, ſic ut aqu@ per duo foramina transfluere co-<lb/>gatur: </s> + <s xml:id="echoid-s82" xml:space="preserve">de hoc motu loquitur Mari@@@us in tractatu ſuo egregio de motu aqua-<lb/>rum part. </s> + <s xml:id="echoid-s83" xml:space="preserve">IV. </s> + <s xml:id="echoid-s84" xml:space="preserve">pag. </s> + <s xml:id="echoid-s85" xml:space="preserve">m. </s> + <s xml:id="echoid-s86" xml:space="preserve">442.</s> + <s xml:id="echoid-s87" xml:space="preserve"/> +</p> +<pb o="4" file="0018" n="18" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s88" xml:space="preserve">§. </s> + <s xml:id="echoid-s89" xml:space="preserve">3. </s> + <s xml:id="echoid-s90" xml:space="preserve">Hæc cum ita ſint, facile quiſque ſecum judicabit, quam parum <lb/>ſpei ſuperſit, aliquando Leges motuum pro fluidis ad regulas Geometriæ pu-<lb/>ræ reductum iri, ſine ulla hypotheſi phyſica, cum vel in ipſo limine effuge-<lb/>rint perſpicaciam Viri ingenio præpotentis & </s> + <s xml:id="echoid-s91" xml:space="preserve">incomparabilis: </s> + <s xml:id="echoid-s92" xml:space="preserve">neque ego cre-<lb/>do poſſe ea, quæ in hoc opere expoſiturus ſum, omnem rigorem mathema-<lb/>ticum ſubire: </s> + <s xml:id="echoid-s93" xml:space="preserve">Principia Theoriæ phyſica ſunt & </s> + <s xml:id="echoid-s94" xml:space="preserve">non ſine largitione acceptan-<lb/>da ut proxime vera; </s> + <s xml:id="echoid-s95" xml:space="preserve">admiſſis autem principiis, omnia erunt Geometrica, & </s> + <s xml:id="echoid-s96" xml:space="preserve"><lb/>nullis obnoxia reſtrictionibus, neceſſario nexu inter ſe cohærebunt. </s> + <s xml:id="echoid-s97" xml:space="preserve">Non <lb/>poſſum tamen, quam bene ſentire de phyſicis iſtis poſitionibus, in quas forte <lb/>incidi, quandoquidem me manuduxerunt ad plurimas novas proprietates, <lb/>cum de æquilibrio tum de motu fluidorum detegendas, quæ, niſi me amor <lb/>ſ<unsure/>uſcepti laboris fallit, aliquando Hydrodynamicam inſigniter promovebunt, <lb/>@@ magis excolantur, quam mihi licuit; </s> + <s xml:id="echoid-s98" xml:space="preserve">ubi monuiſſe conveniet, quando mul-<lb/>tis, quicquid novum eſt, ſuſpectum eſſe ſolet, totam me Theoriam animo <lb/>concepiſſe, tractatum conſcripſiſſe, pleraque cum amicis privatim commu-<lb/>nicaſſe, quædam etiam coram Societate noſtra prælegiſſe, priuſquam ullum <lb/>experimentum inſtituerim, ne ex præconceptis menſuris opinione falſa, pro-<lb/>xime tamen illis ſatisfaciente, me falli paterer, quandoque etiam Viros per-<lb/>ſpicaciſſimos intellectis theorematis aperte faſſos eſſe, ſe ſibi talia perſuadere <lb/>non poſſe, nec experimentis confirmatum iri exiſtimare; </s> + <s xml:id="echoid-s99" xml:space="preserve">hisque omnibus ge-<lb/>ſtis, facta demum fuiſſe experimenta coram Amicis, hæcque ita conveniſſe <lb/>cum Theoria, quantum ipſe vix ſperare poteram. </s> + <s xml:id="echoid-s100" xml:space="preserve">Nunc vero redeamus il-<lb/>luc, unde divertimus.</s> + <s xml:id="echoid-s101" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s102" xml:space="preserve">§. </s> + <s xml:id="echoid-s103" xml:space="preserve">4. </s> + <s xml:id="echoid-s104" xml:space="preserve">Poſtquam certi fuerunt Authores de diverſitate velocitatum a mu-<lb/>tatis altitudinibus, vaſa conſiderare cœperunt magis compoſita, fiſtuiis nempe <lb/>varie inclinatis atque inæqualiter amplis inſtructa. </s> + <s xml:id="echoid-s105" xml:space="preserve">Harum autem indolem <lb/>jam ſuo tempore quodammodo cognovit Frontinus, non ignarus, modulum <lb/>augeri a declivitate vel humilitate calicis, id eſt, fiſtulæ ſignatæ, quæ caſtel-<lb/>lo, aut aliquando etiam rivo induebatur: </s> + <s xml:id="echoid-s106" xml:space="preserve">unde etiam calices ad lineam, uti <lb/>loquitur, ordinari<unsure/> & </s> + <s xml:id="echoid-s107" xml:space="preserve">in eadem altitudine poni juſſit. </s> + <s xml:id="echoid-s108" xml:space="preserve">Et hoc quidem reſpe-<lb/>ctu injuſte poſtulatur Fronti@@@s a quibusdam, velocitatis nullam habuiſſe ra-<lb/>tionem; </s> + <s xml:id="echoid-s109" xml:space="preserve">ubi vero calculum ponit om@@is<unsure/> aquæ acceptæ, illamque comparat <lb/>@um eroganda, non video, quomodo excu@@@ poſſit. </s> + <s xml:id="echoid-s110" xml:space="preserve">Experientia quoque <lb/>@@@ctus fuerat, quod notari meretur, plus debito aq@@@ erogari per calicem +<pb o="5" file="0019" n="19" rhead="SECTIO PRIMA."/> +legitimæ tum menſuræ, tum poſitionis, cui ſtatim ſiſtulæ amplioris moduli <lb/>ſubjectæ ſint, quod ita eſſe, recteque a Fabretto indicatum fuiſſe, ſuo loco <lb/>monſtrabo, quamvis Viri alias acutiſſimi, id non ſatis ſibi liquere vel potius <lb/>de eo ſe dubitare, innuerint.</s> + <s xml:id="echoid-s111" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s112" xml:space="preserve">§. </s> + <s xml:id="echoid-s113" xml:space="preserve">5. </s> + <s xml:id="echoid-s114" xml:space="preserve">Quod autem veteres obſcure & </s> + <s xml:id="echoid-s115" xml:space="preserve">ſine veris menſuris viderunt, id <lb/>demum Cl. </s> + <s xml:id="echoid-s116" xml:space="preserve">Gulielminus in Tract. </s> + <s xml:id="echoid-s117" xml:space="preserve">de aquarum fluentium menſura propoſitione ac-<lb/>curatiori & </s> + <s xml:id="echoid-s118" xml:space="preserve">generaliori complexus eſt tali, eandem velocitatem, inquiens, eſſe <lb/>aquæ fluentis per canalem inclinatum, ac ſi fluxerit e vaſe per lumen ſimile, & </s> + <s xml:id="echoid-s119" xml:space="preserve">æquale ſe-<lb/>ctioni, tantundem a ſuperficie aquæ remotum, quantum ſectio ab Horizontali per initium <lb/>alvei, quam propoſitionem impugnavit Dionyſius Papinus, ipſe multum a ve-<lb/>ritate aberrans. </s> + <s xml:id="echoid-s120" xml:space="preserve">Quoniam autem in eo ſumus, ut commenta, tum Hydroſtatica, <lb/>tum Hydraulica præcipua recenſeamus, hoc loco etiam numerandum eſt il-<lb/>lud, de preſſione fluidorum ex impetu cognoſcenda, nempe vim fluidi, in pla-<lb/>num ad angulum rectum irruentis data velocitate, æqualem eſſe ponderi cylindrici fluidi <lb/>ſuper illo plano extructi, cujus altitudo talis ſit, ex qua mobile libere cadendo a quiete <lb/>fluidi velocitatem acquirat. </s> + <s xml:id="echoid-s121" xml:space="preserve">Problematis hujus utiliſſimi ope æſtimare licet vim <lb/>fluidorum machinas agitantium, aut, quale eſt ventus, naves propellentium, <lb/>motus corporum in mediis reſiſtentibus plurimaque alia. </s> + <s xml:id="echoid-s122" xml:space="preserve">De Hydroſtatica au-<lb/>tem, quæ tubulis tenuiſſimis ſeu capillaribus particularis eſt, nihil dico, quia <lb/>hactenus ad Leges generales omnibus fluidis communes reduci non potuit: <lb/></s> + <s xml:id="echoid-s123" xml:space="preserve">Incertus præterea eſt Author, qui primus horum tubulorum indolem obſer-<lb/>vaverit; </s> + <s xml:id="echoid-s124" xml:space="preserve">conſtat tamen recentem eſſe obſervationem, quia de illa in libris an-<lb/>te hos 70. </s> + <s xml:id="echoid-s125" xml:space="preserve">vel 80. </s> + <s xml:id="echoid-s126" xml:space="preserve">annos editis nihil videre eſt.</s> + <s xml:id="echoid-s127" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s128" xml:space="preserve">§. </s> + <s xml:id="echoid-s129" xml:space="preserve">6. </s> + <s xml:id="echoid-s130" xml:space="preserve">Authores præter citatos a Galilæi temporibus, in rebus aquariis <lb/>celebriores ſunt Torricellius, Borellus, Vivianus, Paſcalius, Boilius, recen-<lb/>tioris ætatis ſunt Varignonius, Newtonus, Polen@@@, Hermannus, Jacobus & </s> + <s xml:id="echoid-s131" xml:space="preserve"><lb/>Johannes Bernoulli, quorum inventa extant in Comment. </s> + <s xml:id="echoid-s132" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s133" xml:space="preserve">Reg. </s> + <s xml:id="echoid-s134" xml:space="preserve">Sc. <lb/></s> + <s xml:id="echoid-s135" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s136" xml:space="preserve">Princ. </s> + <s xml:id="echoid-s137" xml:space="preserve">Math. </s> + <s xml:id="echoid-s138" xml:space="preserve">phil. </s> + <s xml:id="echoid-s139" xml:space="preserve">@@@. </s> + <s xml:id="echoid-s140" xml:space="preserve">@ractatu de Caſtellis notiſque ad Frontinum, Pho-<lb/>ronomia, Actis Lipſ.</s> + <s xml:id="echoid-s141" xml:space="preserve">, aliisque operibus variis. </s> + <s xml:id="echoid-s142" xml:space="preserve">Quæ vero circa curvatura@ <lb/>ex preſſione fluidi genitas aliaque hujusmodi inventa a Geometris exhibita fue-<lb/>runt, quia facile ad Geometriam puram reducuntur, utut de reliquo omni<unsure/> <lb/>laude digna ſilentio prætereo.</s> + <s xml:id="echoid-s143" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s144" xml:space="preserve">Expoſitis his, quæ ad alios pertinent, æquum eſſe ſentio, ut meorum quo-<lb/>que ratione ſubducta, dicam ſincere, an aliqua & </s> + <s xml:id="echoid-s145" xml:space="preserve">quanta Hydrodynamicæ in- +<pb o="6" file="0020" n="20" rhead="HYDRODYNAMICÆ"/> +crementa ab illis ſperari poſſint aut debeant. </s> + <s xml:id="echoid-s146" xml:space="preserve">Breviter igitur, quantum po-<lb/>tero, momenta operis ſuſcepti indicabo.</s> + <s xml:id="echoid-s147" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s148" xml:space="preserve">§. </s> + <s xml:id="echoid-s149" xml:space="preserve">7. </s> + <s xml:id="echoid-s150" xml:space="preserve">Exhibentur primo loco Theoremata præcipua, quæ ad æquili-<lb/>brium fluidorum ſtagnantium pertinent: </s> + <s xml:id="echoid-s151" xml:space="preserve">viſa mihi fuit inſtituti ratio id poſtu-<lb/>lare, quamvis libenter fatear, nullas a me novas adjectas fuiſſe propoſitiones: <lb/></s> + <s xml:id="echoid-s152" xml:space="preserve">Demonſtrandi quidem modus, quantum ſcio, mihi proprius eſt, ſed cum facile <lb/>ſit, innumeras ſibi fingere demonſtrationes, parum eſt, hac quoque in parte, <lb/>quod mihi arrogo. </s> + <s xml:id="echoid-s153" xml:space="preserve">Phænomena præterea aliqua tubulorum capillarium obi-<lb/>ter recenſentur, & </s> + <s xml:id="echoid-s154" xml:space="preserve">denique occaſione preſſionis, quam fluida in latera vaſis <lb/>exercent, Theoremata varia & </s> + <s xml:id="echoid-s155" xml:space="preserve">nonnulla nova adduntur, circa figuram veſica-<lb/>rum liquore impletarum, circa earundem potentias ad onera elevanda, circa <lb/>conſtructionem & </s> + <s xml:id="echoid-s156" xml:space="preserve">firmitatem aquæductuum, aliaque affinia.</s> + <s xml:id="echoid-s157" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s158" xml:space="preserve">§. </s> + <s xml:id="echoid-s159" xml:space="preserve">8. </s> + <s xml:id="echoid-s160" xml:space="preserve">Agitur poſtea de motu fluidorum ex vaſe effluentium, & </s> + <s xml:id="echoid-s161" xml:space="preserve">cum om-<lb/>nes, qui hactenus de hacre egerunt, caſum unicum maxime obvium, quo <lb/>foramen ratione amplitudinis vaſis internæ infinite parvum cenſetur, in Theo-<lb/>ri<unsure/>a ſua conſideraverint, noſtra non parum commendatur ſua latitudine; </s> + <s xml:id="echoid-s162" xml:space="preserve">ex-<lb/>tendit enim ſe ad poſitionem foraminis cujuſcunque magnitudinis, imo & </s> + <s xml:id="echoid-s163" xml:space="preserve"><lb/>vaſis cujuscunque figuræ. </s> + <s xml:id="echoid-s164" xml:space="preserve">Quamvis enim figuræ vaſis internæ conſideratio mi-<lb/>nime requiritur, cum foramen ut infinite parvum conſiderari poteſt, attamen <lb/>ſine illa motus aquæ definiri nequit, cum eſt notabilis magnitudinis. </s> + <s xml:id="echoid-s165" xml:space="preserve">Ex Theo-<lb/>ria generali corollaria deducuntur, quæ motum aquarum variabilem ejusdem-<lb/>que affectiones egregie illuſtrant, confirmantque, quicquid aut experientia do-<lb/>cuit, aut rei attributiones per ſe manifeſte indicant. </s> + <s xml:id="echoid-s166" xml:space="preserve">Docet quidem Theoria, <lb/>quando amplitudines internæ vel mediocriter ſuperant amplitudinem luminis, <lb/>errorem eſſe inſen@@@@em, qui ex conſideratione foraminis ut infinite parvi <lb/>naſcitur, atque ſic noſtræ additiones nonnullis fortaſſe videbuntur ſatis inutiles. <lb/></s> + <s xml:id="echoid-s167" xml:space="preserve">Hos vero, ſi modo qui futuri ſint, @@@@m cogitare velim, præter quod non ſo-<lb/>lum aquariis ſcribo, ſed & </s> + <s xml:id="echoid-s168" xml:space="preserve">Geometris, qui veritatibus<unsure/> nudis etiam delectan-<lb/>tur, uſum noſtrarum meditationum aliis in rebus maximum eſſe, quod magis <lb/>intelligent, cum perpenderint, motum incipere a quiete, & </s> + <s xml:id="echoid-s169" xml:space="preserve">per infinitos tran-<lb/>ſire gradus, priuſquam certam celeritatem obtineat, maximas mutationes ſæ-<lb/>pe quidem tam brevi fieri temporis momento, ut ſenſibus nullo plane modo <lb/>percipi poſſint, determinandas tamen eſſe ad ſingula puncta, tum ut motus <lb/>animo recte percipiatur, tum quia exinde varia deduci poſſunt Theoremata.</s> + <s xml:id="echoid-s170" xml:space="preserve"> +<pb o="7" file="0021" n="21" rhead="SECTIO PRIMA."/> +Ita animadverti, (quod exemplum ob rei momentum ſit inſtar omnium,) fie-<lb/>ri non poſſe, ut preſſio aquæ, per canalem data velocitate fluentis, in ejusdem <lb/>latera definiatur, niſi mutationes iſtæ, quas momentaneas dicam, utcunque ſen-<lb/>ſibus inperceptibiles recte animo intelligantur. </s> + <s xml:id="echoid-s171" xml:space="preserve">De his ego, ut primus cogi-<lb/>tavi, ita optatiſſimo cum ſucceſſu novam Theoriæ aquarum partem addidi, quæ, <lb/>quia fluidorum tum motum tum preſſionem ſimul reſpicit, hydraulico - ſtatica <lb/>aptiſſime vocari viſa fuit. </s> + <s xml:id="echoid-s172" xml:space="preserve">Poſt hæc Theoriæ generalis ſpecimina, de vaſis cy-<lb/>lindricis tam ſimplicibus, quam iis, quæ tubis inſtructa ſunt, exhibentur, & </s> + <s xml:id="echoid-s173" xml:space="preserve"><lb/>in his poſterioribus præſertim determinantur mutationes, quæ ab initio fluxus <lb/>oriuntur, dum datus velocitatis gradus attingitur, & </s> + <s xml:id="echoid-s174" xml:space="preserve">id quidem in hypotheſi <lb/>vaſorum ampliſſimorum; </s> + <s xml:id="echoid-s175" xml:space="preserve">notandum autem eſt, has mutationes ſenſibiles ad-<lb/>modum eſſe, etiamſi vaſa ſunt infinitæ amplitudinis, poſſeque illas experimen-<lb/>tis demonſtrari, dum aquæ ex vaſe ampliſſimo per foramen ſimplex effluentes <lb/>primo ſtatim temporis puncto totam, quantam poſſunt, velocitatem habent. <lb/></s> + <s xml:id="echoid-s176" xml:space="preserve">Pendent prædictæ mutationes tum a longitudine tum a figura tubi. </s> + <s xml:id="echoid-s177" xml:space="preserve">Denique <lb/>etiam calculi analytici pro varii generis temporibus inveniendis una cum an-<lb/>notationibus phyſicis eo pertinentibus adjiciuntur. </s> + <s xml:id="echoid-s178" xml:space="preserve">Indicante denique Theo-<lb/>ria, fieri non poſſe, ut aquæ multum ultra ſupremam ſcaturiginis ſuperficiem <lb/>aſcendant, monſtratur ſub fine ſectionis, non pertinere ad hypotheſes noſtras <lb/>phænomenon ſingulare, quod ipſe ſæpius obſervavi, & </s> + <s xml:id="echoid-s179" xml:space="preserve">pro lubitu imitari poſ-<lb/>ſ<unsure/>um, cujusque mentio injicitur in Hiſt. </s> + <s xml:id="echoid-s180" xml:space="preserve">Reg. </s> + <s xml:id="echoid-s181" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s182" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s183" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s184" xml:space="preserve">ad ann. </s> + <s xml:id="echoid-s185" xml:space="preserve">1702. </s> + <s xml:id="echoid-s186" xml:space="preserve">ubi <lb/>dicitur, accidere quandoque, ut aquæ in fontibus ſalientibus aſſurgant ad al-<lb/>titudinem triplam, aut quadruplam ejus, quæ reſpondet aquæ ſuperficiei ſupre-<lb/>mæ, mox tamen enormem aquæ jactum ad confuetam altitudinem deprimi, <lb/>poſteaque genuina iſtius phænomeni ratio cum veris menſuris ex Theoria no-<lb/>ſtra petitis affertur, modusque indicatur ſaltum inſolitum producendi, <lb/>imo & </s> + <s xml:id="echoid-s187" xml:space="preserve">ad lubitum augendi.</s> + <s xml:id="echoid-s188" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s189" xml:space="preserve">§. </s> + <s xml:id="echoid-s190" xml:space="preserve">9. </s> + <s xml:id="echoid-s191" xml:space="preserve">Porro Theoria extenditur ad examen motuum ex vaſis conſtanter <lb/>plenis, quibus nempe tantum aquæ continue affunditur, quantum ex illis ef-<lb/>fluit: </s> + <s xml:id="echoid-s192" xml:space="preserve">horum indoles in eo potiſſimum conſiſtit, ut fluida emanantia magis <lb/>magisque accedant ad illum velocitatis gradum, qui toti altitudini ſuperficiei <lb/>fluidi ſupra foramen debetur, eum vero nunquam omnino attingant, niſi poſt <lb/>tempus infinitum: </s> + <s xml:id="echoid-s193" xml:space="preserve">vergere tamen demonſtrantur aquæ tam cito ad velocitatem <lb/>iſtam, ut poſt tempusculum inſenſibile tantum non totam acquirant, niſi +<pb o="8" file="0022" n="22" rhead="HYDRODYNAMICÆ"/> +cum per longiſſimos rivos aut aquæductus feruntur, magnoque lumine eji-<lb/>ciuntur; </s> + <s xml:id="echoid-s194" xml:space="preserve">tunc enim accelerationes tam celeres non ſunt, quin percipi poſſin@ <lb/>quod ſingulari exemplo ex Cl. </s> + <s xml:id="echoid-s195" xml:space="preserve">Mariotti libro de motu aquarum deſumto con-<lb/>f<unsure/>irmatur. </s> + <s xml:id="echoid-s196" xml:space="preserve">Quoniam vero motus a quiete incipit & </s> + <s xml:id="echoid-s197" xml:space="preserve">perpetuo creſcit, formu-<lb/>læ dantur, quarum ope vel ex fluxus tempore vel ex quantitate aquarum e-<lb/>jectarum velocitas ſingulis temporis punctis definiri poſſit & </s> + <s xml:id="echoid-s198" xml:space="preserve">viciſſim.</s> + <s xml:id="echoid-s199" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s200" xml:space="preserve">§. </s> + <s xml:id="echoid-s201" xml:space="preserve">10. </s> + <s xml:id="echoid-s202" xml:space="preserve">In ſequentibus fluida conſiderantur, quæ intra vaſa moventur, <lb/>@bi præſertim motus fluidorum reciproci ſeu oſcillatorii ad menſuras revocan-<lb/>tur, earumque affectiones indicantur. </s> + <s xml:id="echoid-s203" xml:space="preserve">Dedit autem Newtonus Theorema ſimile <lb/>pro oſcillationibus fluidi, in tubo uniformis amplitudinis (cujus crura duo ex-<lb/>trema verticalia, intermedia pars horizontalis) oſcillantis, quod Theorema <lb/>Pater meus in Comm. </s> + <s xml:id="echoid-s204" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s205" xml:space="preserve">Imp. </s> + <s xml:id="echoid-s206" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s207" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s208" xml:space="preserve">tom. </s> + <s xml:id="echoid-s209" xml:space="preserve">2. </s> + <s xml:id="echoid-s210" xml:space="preserve">p. </s> + <s xml:id="echoid-s211" xml:space="preserve">201. </s> + <s xml:id="echoid-s212" xml:space="preserve">generalius reddidit <lb/>poſita inclinatione qualicunque crurum extremorum verſus horizontem. </s> + <s xml:id="echoid-s213" xml:space="preserve">No-<lb/>ftra Theoria totam rem ſine ulla reſtrictione complectitur, tubos conſiderans <lb/>in ſingulis locis directione ſeu poſitione atque amplitudine pro lubitu variabi-<lb/>les: </s> + <s xml:id="echoid-s214" xml:space="preserve">oſtenditur dein, quibus in caſibus fiat, ut oſcillationes diverſæ excurſionis <lb/>ſ<unsure/>int Iſochronæ, quibus ſtantibus longitudo penduli ſimplicis Iſochroni gene-<lb/>raliſſime determinatur. </s> + <s xml:id="echoid-s215" xml:space="preserve">Sed & </s> + <s xml:id="echoid-s216" xml:space="preserve">præter hoc oſcillationum genus in ſubſequente <lb/>ſectione quædam aliæ examini ſubjiciuntur, veluti illæ, quæ fiunt in tubis a-<lb/>quæ infinitæ vel etiam terminatæ immerſis, in quibus ſingulari circumſpectio-<lb/>ne opus eſt, qua adhibita omnia phænomena calculo ad amuſſim reſpondent, <lb/>eadem vero neglecta tantus fit inter ea diſſenſus, quantus eſt inter leges mo-<lb/>tus, quæ pro corporibus perfecte elaſticis, iisque quæ pro mollibus valent.</s> + <s xml:id="echoid-s217" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s218" xml:space="preserve">§. </s> + <s xml:id="echoid-s219" xml:space="preserve">11. </s> + <s xml:id="echoid-s220" xml:space="preserve">Poſt hæc ad alia magis compoſita progredior, motum nempe flui-<lb/>dorum conſiderans ſive homogeneorum ſive heterogeneorum, quæ per unum <lb/>aut plura foramina transfluere coguntur, priuſquam ejiciantur in aërem, ubi <lb/>regula illa communiter recepta de ſaltu aquæ ad ſupremam aquæ libellam ve-<lb/>liementer fallit, ceſſantibus etiam legibus preſſionis ordinariis. </s> + <s xml:id="echoid-s221" xml:space="preserve">Horum autem <lb/>omnium apud Authores ne veſtigium quidem reperitur, niſi quod Mariottus <lb/>habet, loco ſupra citato part. </s> + <s xml:id="echoid-s222" xml:space="preserve">IV. </s> + <s xml:id="echoid-s223" xml:space="preserve">p. </s> + <s xml:id="echoid-s224" xml:space="preserve">m. </s> + <s xml:id="echoid-s225" xml:space="preserve">442. </s> + <s xml:id="echoid-s226" xml:space="preserve">de motu aquar. </s> + <s xml:id="echoid-s227" xml:space="preserve">ubi quidem fluxum <lb/>aquarum retardari, fuiſſe ſe experientia edoctum, teſtatur, ſimul autem ma-<lb/>nifeſtat, quam procul abfuerit a vera horum motuum Theoria, & </s> + <s xml:id="echoid-s228" xml:space="preserve">videtur ſane <lb/>hæc Theoria omnium fere principiorum, adhuc in rebus ſimilibus adhiberi ſo-<lb/>litorum, vim eludere, ita ut nihil ſit, quod noſtrorum præſtantiam magis con- +<pb o="9" file="0023" n="23" rhead="SECTIO PRIMA."/> +firmet: </s> + <s xml:id="echoid-s229" xml:space="preserve">de eorum veritate enim experimenta inſtituta me amplius dubitare <lb/>non ſinunt. </s> + <s xml:id="echoid-s230" xml:space="preserve">Non deeſt autem hiſce meditatis ſua utilitas, quandoquidem <lb/>magni momenti eſſe poſſint in perficiendis machinis hydraulicis.</s> + <s xml:id="echoid-s231" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s232" xml:space="preserve">§. </s> + <s xml:id="echoid-s233" xml:space="preserve">12. </s> + <s xml:id="echoid-s234" xml:space="preserve">Sequuntur commentationes de machinis hydraulicis, quibus potis-<lb/>ſimum monſtratur, certum quendam perfectionis terminum eſſe, ultra quem <lb/>ire non liceat; </s> + <s xml:id="echoid-s235" xml:space="preserve">defectus autem ab ultimo hoc perfectionis gradu in multis ma-<lb/>chinis maxime receptis calculo numerico ſubjiciuntur, additis regulis ſeu <lb/>præceptis, ad quæ in conſtruendis novis machinamentis animus ſit adverten-<lb/>dus: </s> + <s xml:id="echoid-s236" xml:space="preserve">exempli loco affertur notiſſima per totum orbem machina Marlyenſis, <lb/>de qua monſtratur, ſi modo deſcriptionibus fidendum ſit, quod non ultra <lb/>quinquageſimam ſextam prope partem ſuppeditet ejus aquæ quantitatis, quam <lb/>cæteris paribus machina perfectiſſima theoretice ſubminiſtrare queat. </s> + <s xml:id="echoid-s237" xml:space="preserve">Specia-<lb/>le etiam examen inſtituitur de machina ab antiquiſſimis temporibus ad no-<lb/>ſtram uſque ætatem uſitatiſſima, Cochlea nimirum Archimedis, attentione <lb/>Geometrarum non indigna, tam ratione eorum, quæ ad Geometriam puram, <lb/>quam quæ ad Hydraulicam pertinent.</s> + <s xml:id="echoid-s238" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s239" xml:space="preserve">§. </s> + <s xml:id="echoid-s240" xml:space="preserve">13. </s> + <s xml:id="echoid-s241" xml:space="preserve">Succedunt ſpecimina quædam de motu fluidorum elaſticorum, <lb/>veluti aëris & </s> + <s xml:id="echoid-s242" xml:space="preserve">pulveris pyrii accenſi, præmisſis iis, quæ ad naturam horum <lb/>fluidorum pertinent; </s> + <s xml:id="echoid-s243" xml:space="preserve">quæ vero ipſe non aliter, quam ut hypotheſes phyſicas <lb/>conſidero, de quibus nihil confidenter affirmabo. </s> + <s xml:id="echoid-s244" xml:space="preserve">Propoſitiones & </s> + <s xml:id="echoid-s245" xml:space="preserve">Problema-<lb/>ta hujus ſectionis nova ſunt, & </s> + <s xml:id="echoid-s246" xml:space="preserve">eo ſelecta animo, ut multis quæſtionibus phy-<lb/>ſicis illuſtrandis, aut etiam ſolvendis occaſionem præbere posſint. </s> + <s xml:id="echoid-s247" xml:space="preserve">Adjiciun-<lb/>tur quædam de æſtimatione virium vivarum fluidis elaſticis inſitarum, quæ <lb/>aliquando fortaſſe in praxi mechanica nonnullius uſus erunt: </s> + <s xml:id="echoid-s248" xml:space="preserve">monſtratur enim, <lb/>unius v. </s> + <s xml:id="echoid-s249" xml:space="preserve">gr. </s> + <s xml:id="echoid-s250" xml:space="preserve">libræ pulveris pyrii accenſi effectum in elevandis ponderibus ma-<lb/>jorem eſſe poſſe, quam vel centum homines robuſtisſimi labore continuo in-<lb/>tra unius diei ſpatium efficere posſint.</s> + <s xml:id="echoid-s251" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s252" xml:space="preserve">§. </s> + <s xml:id="echoid-s253" xml:space="preserve">14. </s> + <s xml:id="echoid-s254" xml:space="preserve">Agitur porro de fluidorum motu circulari, ut & </s> + <s xml:id="echoid-s255" xml:space="preserve">de fluidis, quæ <lb/>in vaſis motis ſtagnant; </s> + <s xml:id="echoid-s256" xml:space="preserve">variaque alia intermiſcentur. </s> + <s xml:id="echoid-s257" xml:space="preserve">Quæ autem de motu <lb/>circulari proferuntur, inſervire quodammodo poſſunt ad phænomena gravi-<lb/>tatis per vortices explicanda; </s> + <s xml:id="echoid-s258" xml:space="preserve">cætera valeant, quantum poterunt.</s> + <s xml:id="echoid-s259" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s260" xml:space="preserve">§. </s> + <s xml:id="echoid-s261" xml:space="preserve">15. </s> + <s xml:id="echoid-s262" xml:space="preserve">Præmiſſa Theoria motuum, rurſus ad æquilibria fluidorum deſcen-<lb/>ditur, ſed fluidorum motorum, quorum leges exhibitæ nondum fuerunt. <lb/></s> + <s xml:id="echoid-s263" xml:space="preserve">Mirum eſt, cum alias motus ex presſione definiatur, hic inverſa methodo preſ- +<pb o="10" file="0024" n="24" rhead="HYDRODYNAMICÆ"/> +ſionem ex motu peti, prius ex circumſtantiis definiendo; </s> + <s xml:id="echoid-s264" xml:space="preserve">nec crediderim ali-<lb/>am viam tuto iniri poſſe præter eam, quam ego ſecutus ſum: </s> + <s xml:id="echoid-s265" xml:space="preserve">conſideravi au-<lb/>tem canalem, per quem aquæ fluunt eo in loco eoque temporis puncto, quæ <lb/>quæſtioni conveniunt, amputatum; </s> + <s xml:id="echoid-s266" xml:space="preserve">poſteaque per regulas noſtras præmiſſas <lb/>accelerationem indagavi particulæ aquæ imminentis, proximeque effluxuræ. </s> + <s xml:id="echoid-s267" xml:space="preserve">Ex <lb/>iſta acceleratione colligere licebat compresſionem illius particulæ aqueæ, quæ <lb/>compresſio per naturam fluidorum æqualis eſt presſioni in latera canalis. </s> + <s xml:id="echoid-s268" xml:space="preserve">Co-<lb/>gnita hac presſione apparet, quid fieri debeat, ſi canalis eodem in loco per-<lb/>foratus fuerit, tubulusque foramini reſpondeat; </s> + <s xml:id="echoid-s269" xml:space="preserve">fore nempe, ut aqua in eo <lb/>aſcendat ad certum uſque gradum ſtagnans in tubulo, & </s> + <s xml:id="echoid-s270" xml:space="preserve">ab aqua inferius per <lb/>canalem præterfluente ſuſtenta, ſic, ut hic æquilibrium adſit inter aquas flu-<lb/>entes & </s> + <s xml:id="echoid-s271" xml:space="preserve">ſtagnantes: </s> + <s xml:id="echoid-s272" xml:space="preserve">hoc vero nomine Theoriam iſtam hydraulico-ſtaticam com-<lb/>mode vocari poſſe exiſtimavi. </s> + <s xml:id="echoid-s273" xml:space="preserve">Notari porro meretur, ipſam hanc Theoriam <lb/>fundamentum rurſus eſſe & </s> + <s xml:id="echoid-s274" xml:space="preserve">fontem aliorum motuum antehac incognitorum. <lb/></s> + <s xml:id="echoid-s275" xml:space="preserve">Theoremata, quæ exponuntur, non nova ſolum, ſed & </s> + <s xml:id="echoid-s276" xml:space="preserve">pleraque inexpectata <lb/>ſunt, quorum omnium veritatem nec ipſe plane mihi perſuadere potui, pri-<lb/>uſquam experimenta inſtituiſſem, quæ mihi omnem ſcrupulum demebant. </s> + <s xml:id="echoid-s277" xml:space="preserve"><lb/>Habent autem inſignem uſum, quandoquidem iis innititur vera presſionis <lb/>aquarum, per aquæductus ſeu rivos fluentium, æſtimatio, hincque deducendæ <lb/>tuborum firmitates requiſitæ. </s> + <s xml:id="echoid-s278" xml:space="preserve">Inde quoque pendent accuratæ menſuræ aqua-<lb/>rum per modulos, rivo lateraliter inſertos, erogatarum: </s> + <s xml:id="echoid-s279" xml:space="preserve">in Phyſiologia re-<lb/>ctius jam intelligentur, quæ pertinent ad motum humorum in corpore animali, <lb/>& </s> + <s xml:id="echoid-s280" xml:space="preserve">quæ ſunt alia.</s> + <s xml:id="echoid-s281" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s282" xml:space="preserve">§. </s> + <s xml:id="echoid-s283" xml:space="preserve">16. </s> + <s xml:id="echoid-s284" xml:space="preserve">Denique progredior ad alios quosdam modos, quibus aqua ni-<lb/>ſum facere poteſt, explicandos: </s> + <s xml:id="echoid-s285" xml:space="preserve">ita nempe aqua, dum per foramen effluit, in <lb/>contrarium premit vas non aliter, atque globus retropellit tormentum, ex <lb/>quo exploditur: </s> + <s xml:id="echoid-s286" xml:space="preserve">iſtius retropulſionis plures proprietates deteguntur novæ, <lb/>quæ presſionum naturam egregie illuſtrant, earumque leges, quas affectant, ge-<lb/>nerales in mechanicis rem iſtam ſerio meditantibus indicabunt. </s> + <s xml:id="echoid-s287" xml:space="preserve">Has diſqui-<lb/>ſitiones feci, quia mihi viſum eſt, poſſe ea novæ aliquando navigationi ſine <lb/>remorum, aut venti adminiculo excogitandæ occaſionem præbere; </s> + <s xml:id="echoid-s288" xml:space="preserve">qua de <lb/>re ſuo loco pauca quædam afferam, etſi non ignoro, omnium hujusmodi re-<lb/>rum primordia per ſe pleriſque videri ridicula. </s> + <s xml:id="echoid-s289" xml:space="preserve">Tandem etiam de vi aqua-<lb/>rum ex impulſu hincque nato renixu, quam corpora in fluidis mota offen-<lb/>dunt, Theoremata quædam adjiciuntur.</s> + <s xml:id="echoid-s290" xml:space="preserve"/> +</p> +<pb o="11" file="0025" n="25" rhead="SECTIO PRIMA."/> +<p> + <s xml:id="echoid-s291" xml:space="preserve">§. </s> + <s xml:id="echoid-s292" xml:space="preserve">17. </s> + <s xml:id="echoid-s293" xml:space="preserve">Et hæc quidem ſunt, quæ mihi ex admisſis principiis geometri-<lb/>cam deductionem pati viſa ſunt. </s> + <s xml:id="echoid-s294" xml:space="preserve">Quoniam vero nihil eſt in Theoria tam <lb/>rigoroſe demonſtratum, quod non in applicatione ad corpora reſtrictionem <lb/>aliquam poſtulet, ideo facile apparet, nec ullam Theoriam de fluidis expectan-<lb/>dam eſſe, quæ omnibus menſuris experientia cognitis plenisſime ſatisfaciat; <lb/></s> + <s xml:id="echoid-s295" xml:space="preserve">cujus rei memores eſſe velim, qui Theoremata noſtra experimentis confirmare <lb/>voluerint. </s> + <s xml:id="echoid-s296" xml:space="preserve">Ubique invenient quidem aliquem conſenſum, ſed non perfectum, <lb/>eumque modo ſtrictiorem, modo laxiorem, pro rerum circumſtantiis. </s> + <s xml:id="echoid-s297" xml:space="preserve">Quo-<lb/>ties autem ipſe aliquod experimentum effeci, ante omnia mecum perpendi, <lb/>quousque principia Theoriæ cum caſu propoſito congruerent; </s> + <s xml:id="echoid-s298" xml:space="preserve">atque ſic me <lb/>nunquam aut rarisſime eventus fefellit. </s> + <s xml:id="echoid-s299" xml:space="preserve">Non ſolum enim prævidere ſolebam, <lb/>in quam partem futura eſſet differentia, ſi quæ notabilis eſſe debebat, ſed <lb/>& </s> + <s xml:id="echoid-s300" xml:space="preserve">quanta; </s> + <s xml:id="echoid-s301" xml:space="preserve">quod ipſum, ſi recte judico, ſatis manifeſtat, fluida affectare quidem <lb/>leges, quas ipſis præſcriptas eſſe ponimus, obſtacula autem ubique offendere <lb/>nunc majora, nunc minora. </s> + <s xml:id="echoid-s302" xml:space="preserve">Cæterum experimenta inſtitui non pauca, quo-<lb/>rum ſingula in fine ſectionis, ad quam pertinent, locavi: </s> + <s xml:id="echoid-s303" xml:space="preserve">præſertim vero ſol-<lb/>licitus fui, in propoſitionibus antea incognitis & </s> + <s xml:id="echoid-s304" xml:space="preserve">plerisque ſat paradoxis con-<lb/>firmandis. </s> + <s xml:id="echoid-s305" xml:space="preserve">De experimentorum fide non eſt, quod quis dubitet, cum præ-<lb/>cipua coram Amicis eaque poſt publicatam Theoriam fecerim; </s> + <s xml:id="echoid-s306" xml:space="preserve">magnam ta-<lb/>men experimentorum, quæ animo concepi, partem, quando per ſingula ire <lb/>non licet, aliis relinquens inſtituendam. </s> + <s xml:id="echoid-s307" xml:space="preserve">Perlectis noſtris propoſitionibus quisque <lb/>ſibi finget innumera, neque proin opus eſſe judicavi, omnia, qualia ſunt a <lb/>me deſiderata, exponere; </s> + <s xml:id="echoid-s308" xml:space="preserve">expoſui tamen aliqua.</s> + <s xml:id="echoid-s309" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s310" xml:space="preserve">§. </s> + <s xml:id="echoid-s311" xml:space="preserve">18. </s> + <s xml:id="echoid-s312" xml:space="preserve">Jam vero tandem principiorum, quorum toties mentionem fe-<lb/>cimus, ratio reddenda eſt. </s> + <s xml:id="echoid-s313" xml:space="preserve">Præcipuum eſt conſervatio virium vivarum, ſeu, ut <lb/>ego loquor, æqualitas inter deſcenſum actualem aſcenſumque potentialem: </s> + <s xml:id="echoid-s314" xml:space="preserve">Utar hac <lb/>poſteriore voce, quia idem quod altera ſignificat, ſortem autem apud non-<lb/>nullos Philoſophos, qui vel ad ſolum vis vivæ nomen moventur, magis be-<lb/>nignam fortaſſe experietur. </s> + <s xml:id="echoid-s315" xml:space="preserve">Puto, hic e re noſtra fore, hac de re paulo co-<lb/>pioſius dicere.</s> + <s xml:id="echoid-s316" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s317" xml:space="preserve">§. </s> + <s xml:id="echoid-s318" xml:space="preserve">19. </s> + <s xml:id="echoid-s319" xml:space="preserve">Poſtquam Galilæus docuiſſet, corpus, ſive verticaliter, ſive ſuper <lb/>plano utcunque incurvato, deſcendens eandem velocitatem acquirere, modo <lb/>altitudo lapſus ſit eadem, quod ex natura preſſionum demonſtrari poteſt, <lb/>Hugenius eadem hac propoſitione, ſed generaliori pro hypotheſi feliciter uſus +<pb o="12" file="0026" n="26" rhead="HYDRODYNAMICÆ"/> +eſt in eruendis legibus motuum corporum elaſticorum ex percuſſione, nec <lb/>non in ſtabiliendo centro oſcillationis penduli compoſiti; </s> + <s xml:id="echoid-s320" xml:space="preserve">protulit autem axio-<lb/>ma hoc ſuum talibus verbis: </s> + <s xml:id="echoid-s321" xml:space="preserve">Si pondera quotlibet vi gravitatis ſuæ moveri incipiant <lb/>utcunque, ſmgulaque rurſ{us} ad quietem ſponte reducantur, centrum gravitatis ex ipſis <lb/>c<unsure/>ompoſitæ ad priſtinam altitudinem rediturum eſſe, ubi per vocem utcunque intelligit, <lb/>ſive ſe percutiant inter deſcenſum, ſive premant, aliove modo in ſe invicem agant corpora. <lb/></s> + <s xml:id="echoid-s322" xml:space="preserve">Ex iſto axiomate ſtatim ſequitur principium conſervationis virium vivarum, <lb/>quod ipfe etiam Hugenius demonſtravit, & </s> + <s xml:id="echoid-s323" xml:space="preserve">quo aſſumitur: </s> + <s xml:id="echoid-s324" xml:space="preserve">Si pondera quotli-<lb/>bet vi gravitatis ſuæ moveri incipiant utcunque, ſingulorum velocitates ubique tales fore, <lb/>ut producta, ex earum quadratis in ſu{as} maſſ{as} collecta, ſint proportionalia altitudini <lb/>verticali, per quam centrum gravitatis ex corporib{us} compoſitæ deſcendit, multiplicatæ per <lb/>maſſas omnium. </s> + <s xml:id="echoid-s325" xml:space="preserve">Mirum eſt, quantam habeat hæc hypotheſis in Philoſophia me-<lb/>chanica utilitatem, quod, ſi quis alius, ſane Pater meus recte animadvertit, <lb/>qui id ſparſim, imprimis autem in Diſſertatione Pariſiis edita de legib{us} motuum & </s> + <s xml:id="echoid-s326" xml:space="preserve"><lb/>in Tom. </s> + <s xml:id="echoid-s327" xml:space="preserve">2. </s> + <s xml:id="echoid-s328" xml:space="preserve">Comment. </s> + <s xml:id="echoid-s329" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s330" xml:space="preserve">Imp. </s> + <s xml:id="echoid-s331" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s332" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s333" xml:space="preserve">oſtendit , idemque eſt, quod pro in-<lb/>veſtigandis Legibus motuum, ex propria gravitate ortorum, in fluidis adhibui; </s> + <s xml:id="echoid-s334" xml:space="preserve"><lb/>poſui enim velocitates particularum conſtanter tales eſſe, ut, ſingulis vertica-<lb/>liter ſurſum motis ad ſtatum quietis usque, centrum earum gravitatis com-<lb/>mune ad priſtinam altitudinem aſcendat: </s> + <s xml:id="echoid-s335" xml:space="preserve">malui autem ob rationem ſupra di-<lb/>ctam hanc hypotheſin verbis Hugenianis quam Paternis accommodare, eam-<lb/>que nomine æqualitatis inter deſcenſum actualem aſcenſumque potentialem inſignire, <lb/>quam altero conſervationis virium vivarum, quod etiamnum aliqui, præſertim in <lb/>Anglia, neſcio quo fato, faſtidiunt. </s> + <s xml:id="echoid-s336" xml:space="preserve">Mihi quidem in tota doctrina Leibni-<lb/>tiana de viribus vivis nihil eſſe videtur, de quo non omnes, ſuo tamen qui-<lb/>vis loquendi modo, conveniant, quod, ni fallor, clare oſtendi in Comm. </s> + <s xml:id="echoid-s337" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s338" xml:space="preserve"><lb/>Sc. </s> + <s xml:id="echoid-s339" xml:space="preserve">Imp. </s> + <s xml:id="echoid-s340" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s341" xml:space="preserve">Tom. </s> + <s xml:id="echoid-s342" xml:space="preserve">I. </s> + <s xml:id="echoid-s343" xml:space="preserve">p. </s> + <s xml:id="echoid-s344" xml:space="preserve">131. </s> + <s xml:id="echoid-s345" xml:space="preserve">& </s> + <s xml:id="echoid-s346" xml:space="preserve">ſeq. </s> + <s xml:id="echoid-s347" xml:space="preserve">quem locum hic allegare volui, ne qui@ <lb/>Lectorum ſe verbis offendi patiatur, ſciatque nihil a me accipi, quod in Me-<lb/>chanica receptum non ſit ab omnibus, & </s> + <s xml:id="echoid-s348" xml:space="preserve">quod non neceſſario nexu cohæreat <lb/>cum eo, quod jam Galilæus poſuit, cum ſtatueret, incrementa velocitatum <lb/>proportionem ſequi compoſitam ex preſſionibus & </s> + <s xml:id="echoid-s349" xml:space="preserve">momentis temporum.</s> + <s xml:id="echoid-s350" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s351" xml:space="preserve">§@ 20. </s> + <s xml:id="echoid-s352" xml:space="preserve">De cætero quamvis principium prædictum univerſale ſit, non <lb/>tamen eſt ſine circumſpectione adhibendum, quia ſæpe contingit, ut motus <lb/>tranſeat in materiam alienam. </s> + <s xml:id="echoid-s353" xml:space="preserve">Ita verbi gratia poſitio illius valet pro regu-<lb/>lis motuum ex percuſſione eruendis, ſi modo corpora ſint perfecte elaſtica;</s> + <s xml:id="echoid-s354" xml:space="preserve"> +<pb o="13" file="0027" n="27" rhead="SECTIO PRIMA."/> +ſed cum talia non ſunt, facile eſt videre, partem virium vivarum ſive aſcenſ{us} <lb/>potentialis in compreſſionem corporum impenſam corporibus non reſtitui, ſed <lb/>materiæ cuidam ſubtili, ad quam tranſiit, impreſſam hærere: </s> + <s xml:id="echoid-s355" xml:space="preserve">ſi tamen res <lb/>recte conſideretur, quum ratio cognoſcitur, quæ eſt inter partem corporibus <lb/>reſiduam, eamque quæ ad materiam ſubtilem tranſiit; </s> + <s xml:id="echoid-s356" xml:space="preserve">apparebit, facile occur-<lb/>ri poſſe iſti incommodo, ſicque recte definiri leges motuum pro corporibus <lb/>mollibus. </s> + <s xml:id="echoid-s357" xml:space="preserve">Simile quid ſuccedit in motu aquarum computando, ubi quan-<lb/>doque manifeſtum eſt, partem aſcenſ{us} potentialis continue perdi; </s> + <s xml:id="echoid-s358" xml:space="preserve">cujus utique <lb/>rei in ſubducendo calculo ratio habenda eſt: </s> + <s xml:id="echoid-s359" xml:space="preserve">ad quod probe attento multa <lb/>de aquarum fluxu Theoremata nova mihi contigit detegere, quæ videre eſt in <lb/>Sect. </s> + <s xml:id="echoid-s360" xml:space="preserve">Sext. </s> + <s xml:id="echoid-s361" xml:space="preserve">& </s> + <s xml:id="echoid-s362" xml:space="preserve">Sept. </s> + <s xml:id="echoid-s363" xml:space="preserve">& </s> + <s xml:id="echoid-s364" xml:space="preserve">de quibus nondum video, an ulla alia methodo demon-<lb/>ſtrari nedum excogitari poſſint.</s> + <s xml:id="echoid-s365" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s366" xml:space="preserve">§. </s> + <s xml:id="echoid-s367" xml:space="preserve">21. </s> + <s xml:id="echoid-s368" xml:space="preserve">Sic igitur non incautus principio noſtro uſus ſum, hocque mo-<lb/>do non ſolum de motu aquarum, ſed & </s> + <s xml:id="echoid-s369" xml:space="preserve">de earum preſſione, quod mirum <lb/>videri poteſt, multa antea incognita ſe offerunt, quæ nondum inſtituta Ana-<lb/>lyſi nemo facile præviderit nec expectarit. </s> + <s xml:id="echoid-s370" xml:space="preserve">Quum vero fit, ut aſcenſ{us} poten-<lb/>tialis nec omnis conſervari poſſit ex rei natura, nec prævideri, quanta pars <lb/>abſorbeatur, non ſatis accurate motus fluidorum determinari poteſt, nec pu-<lb/>to, ulla alia methodo poſſe. </s> + <s xml:id="echoid-s371" xml:space="preserve">Igitur Lectorem cautum eſſe velim in corolla-<lb/>riis ex Theoria noſtra deducendis, quæ ſæpe propter mutatas circumſtantias <lb/>non accurate cum experimentis convenire poterunt.</s> + <s xml:id="echoid-s372" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s373" xml:space="preserve">§ 22. </s> + <s xml:id="echoid-s374" xml:space="preserve">Ex præmemoratis jam ſatis liquet, ex noſtra methodo requiri, <lb/>ut ſingularum particularum fluidi definiatur velocitas ex aſſumta velocitate, <lb/>quæ eſt aliquo in loco, veluti in loco effluxus. </s> + <s xml:id="echoid-s375" xml:space="preserve">Neceſſe proin fuit, aliam <lb/>ſuperaddere hypotheſin, quæ hæc eſt: </s> + <s xml:id="echoid-s376" xml:space="preserve">poſtquam ſcilicet mente concepimus <lb/>diviſum fluidum in ſtrata, ad directionem motus perpendicularia, ponemus <lb/>fluidi particulas ejusdem ſtrati eadem velocitate moveri, ita, ut ubique velo-<lb/>citas fluidi reciproce proportionalis ſit amplitudini vaſis reſpondenti. </s> + <s xml:id="echoid-s377" xml:space="preserve">Uſita-<lb/>ta eſt hæc hypotheſis, quamvis cæterum notum ſit, fluidum ad latera vaſis <lb/>paullo tardius, in medio autem velocius moveri, quod ab attritu fit, alias-<lb/>que etiam exceptiones ſubinde eſſe faciendas; </s> + <s xml:id="echoid-s378" xml:space="preserve">error tamen notabilis ab hujus-<lb/>modi defectibus rariſſime poteſt oriri.</s> + <s xml:id="echoid-s379" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s380" xml:space="preserve">§. </s> + <s xml:id="echoid-s381" xml:space="preserve">23. </s> + <s xml:id="echoid-s382" xml:space="preserve">Finiam hæcce de hypotheſibus noſtris præmonita recenſione <lb/>phænomenorum, quæ conſervationem virium vivarum in motu fluidorum ali- +<pb o="14" file="0028" n="28" rhead="HYDRODYNAMICÆ"/> +quantum & </s> + <s xml:id="echoid-s383" xml:space="preserve">illuſtrare & </s> + <s xml:id="echoid-s384" xml:space="preserve">confirmare poterunt: </s> + <s xml:id="echoid-s385" xml:space="preserve">eorum quidem in ipſo opere <lb/>plurima occurrent, quæ autem ob calculum, quem poſtulant, non allegabo. <lb/></s> + <s xml:id="echoid-s386" xml:space="preserve">Triviale autem & </s> + <s xml:id="echoid-s387" xml:space="preserve">obvium eſt, quod de gutta, in aquam ſtagnantem delapſa, <lb/>obſervatur: </s> + <s xml:id="echoid-s388" xml:space="preserve">orbes nempe excitat in ſuperficie aquæ ſtagnantis, horumque eo <lb/>plures, quo vel major fuerit gutta, vel altius delapſa, nec dubium eſt, quin <lb/>iſti orbes ſine fine ſe propagaturi eſſent, niſi tenacitas fluidi, aliaque ſimilia <lb/>obſtaculo eſſent. </s> + <s xml:id="echoid-s389" xml:space="preserve">Quandoque etiam alium effectum ab hujusmodi ſtillis ob-<lb/>ſervare licet, dum plures guttulæ minores a ſuperficie aquæ inferioris in al-<lb/>tum projiciuntur, tuncque conſtanter apparet, quod præſertim huc pertinet, <lb/>eo altius aſſurgere guttulas, quo pauciores numero atque minores volumine <lb/>fuerint, & </s> + <s xml:id="echoid-s390" xml:space="preserve">cum altitudo lapſus eſſet duorum pedum, ſæpius guttulæ minores <lb/>ultra altitudinem lapſus aſcendebant, ſtillante præſertim aqua per foramen <lb/>magnum. </s> + <s xml:id="echoid-s391" xml:space="preserve">Hic notatu quoque dignum eſt illud, quod de particula aquæ per <lb/>canalem tenuem, eumque v. </s> + <s xml:id="echoid-s392" xml:space="preserve">gr. </s> + <s xml:id="echoid-s393" xml:space="preserve">horizontalem, atque in ea extremitate, ver-<lb/>ſus quam aqua fluit, operculo perforato opertum obſervatur. </s> + <s xml:id="echoid-s394" xml:space="preserve">Scilicet eo <lb/>temporis puncto, quo aqua ad operculum usque pervenit, magno impetu <lb/>paucæ guttulæ exiliunt, moxque omnis aquæ motus ſiſtitur; </s> + <s xml:id="echoid-s395" xml:space="preserve">facile autem quis <lb/>ſuſpicari poſſet, aquam foramini imminentem ſua velocitate moveri pergere, <lb/>reliquam ſiſti, id vero conſervationi virium vivarum minime reſponderet; </s> + <s xml:id="echoid-s396" xml:space="preserve">re-<lb/>ſpondet autem egregie vehemens iſte aquæ effluxus momentaneus, vel quaſi <lb/>exploſio: </s> + <s xml:id="echoid-s397" xml:space="preserve">de his alibi plura.</s> + <s xml:id="echoid-s398" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s399" xml:space="preserve">§. </s> + <s xml:id="echoid-s400" xml:space="preserve">24. </s> + <s xml:id="echoid-s401" xml:space="preserve">Hæc ſunt, quæ de hypotheſibus noſtris, earumque tum præſtan-<lb/>tia tum defectu volui in anteceſſum monere. </s> + <s xml:id="echoid-s402" xml:space="preserve">Supereſt ut quædam dicam de <lb/>indole fluidorum, quippe circa quæ lucubrationes noſtræ verſabuntur, non <lb/>quod eam me aliis magis perſpectam habere putem, ſed quod nefas exiſti-<lb/>mem, a more hoc ſcriptoribus omnibus ſolenni recedere. </s> + <s xml:id="echoid-s403" xml:space="preserve">Et primo quidem <lb/>hoc omnes convenire ſolent, motum fluidis quibusvis ineſſe inteſtinum, ſine <lb/>quo nemo profecto tantam fluiditatem, efferveſcentias diverſorum fluidorum, <lb/>diſſolutiones ſolidorum fluidis ſubmerſorum, evaporationes, aliaque phæno-<lb/>mena infinita recte aſſequetur; </s> + <s xml:id="echoid-s404" xml:space="preserve">hinc pleræque res ſolidiſſimæ a ſufficiente ca-<lb/>lore, qui omnia in motum rapit, liqueſcunt: </s> + <s xml:id="echoid-s405" xml:space="preserve">facit autem motus iſte inteſti-<lb/>nus, ut particulæ ſibi non ſint contiguæ, ſed quaſi volitent, quo fit, ut ſine <lb/>frictione a minimo impulſu loco cedant, quod minime ſuccederet, poſitis <lb/>iisdem particulis inter ſe, ſicut in acervo arenæ, contiguis. </s> + <s xml:id="echoid-s406" xml:space="preserve">Ita facile intel- +<pb o="15" file="0029" n="29" rhead="SECTIO PRIMA."/> +lectu eſt, polli<unsure/>nem ex putaminibus ovorum in patella igni ſuperimpoſitum <lb/>lac bulliens, quod dicitur, mentiri. </s> + <s xml:id="echoid-s407" xml:space="preserve">Quo intenſior autem eſt calor, eo ve-<lb/>hementior utique eſt motus particularum, hæque majori intervallo a ſe in-<lb/>vicem diſperſæ; </s> + <s xml:id="echoid-s408" xml:space="preserve">quod convenit cum dilatatione omnium fluidorum ab aucto <lb/>calore, eorundemque contractione ex frigore, cui legi ipſa etiam aqua non-<lb/>dum congelata ſubjicitur: </s> + <s xml:id="echoid-s409" xml:space="preserve">quod autem, dum congelatur, contrariæ ſit in-<lb/>dolis, id ex alia cauſa, fortuito ſuperveniente, deducendum videtur, nempe <lb/>ex eo, quod aqua in interſtitiis ſuis particulas foveat aëreas, quæ ſic volumen <lb/>aquæ non augent, prouti ſaccharum in aqua ſolutum non auget ejusdem vo-<lb/>lumen; </s> + <s xml:id="echoid-s410" xml:space="preserve">quod tempore inſtantis congelationis particularum aquearum motus <lb/>minuatur; </s> + <s xml:id="echoid-s411" xml:space="preserve">quod ſic eædem particulæ magis ad ſe invicem accedant, adeo-<lb/>que ex interſtitiis ſuis particulas aëreas pellant, quæ alibi minus commode <lb/>locatæ volumen augere poſſunt, prouti ſaccharum nondum ſolutum volu-<lb/>men auget aquæ, cui permiſtum eſt. </s> + <s xml:id="echoid-s412" xml:space="preserve">Hinc commode ratio deducitur, cur <lb/>glacies aquæ ab aëre ante congelationem bene purgatæ non ſpecifice levior, <lb/>quin potius gravior fiat. </s> + <s xml:id="echoid-s413" xml:space="preserve">Egregia autem experimenta circa ſolutionem ve-<lb/>ram aëris in aqua ad punctum ſaturationis usque inſtituit Mariottus, eaque <lb/>in Tractatu ſuo de motu aquarum recenſuit. </s> + <s xml:id="echoid-s414" xml:space="preserve">Suſpicioni igitur locus eſt, flui-<lb/>da (ut dixi) congelari, cum ceſſat vel valde diminuitur motus inteſtinus, <lb/>tum enim particulæ in ſe invicem collabuntur, fiuntque contiguæ, ſimulque <lb/>ex interſtitiis particulas heterogeneas, ſi quæ ibi commorentur, expellunt; <lb/></s> + <s xml:id="echoid-s415" xml:space="preserve">nec tamen clarius hinc intelligitur durities corporum congelatorum, quin-<lb/>imo videtur, ceſſante iſto motu corpus mediæ naturæ inter fluidum & </s> + <s xml:id="echoid-s416" xml:space="preserve">ſoli-<lb/>dum,<unsure/> niſi aliud quid accedat, fieri, & </s> + <s xml:id="echoid-s417" xml:space="preserve">comparandum cum acervo arenæ: </s> + <s xml:id="echoid-s418" xml:space="preserve"><lb/>quid autem id rei ſit, ne conjectura quidem aſſequor, licebit interim finge-<lb/>re quaslibet particulas ad ſe gravitare, vel, ut voce Anglis uſitata utar, ſe <lb/>invicem attrahere, attractionemque inſigniter creſcere, accedentibus ad ſe <lb/>invicem particulis; </s> + <s xml:id="echoid-s419" xml:space="preserve">diverſæ eſſe virtutis in diverſis corporibus, minoris v. </s> + <s xml:id="echoid-s420" xml:space="preserve">g. </s> + <s xml:id="echoid-s421" xml:space="preserve"><lb/>in oleis quam in aquis, quarum glacies durior eſt; </s> + <s xml:id="echoid-s422" xml:space="preserve">fluida citius & </s> + <s xml:id="echoid-s423" xml:space="preserve">facilius <lb/>congelari, quorum particulæ vel fortius ſe attrahunt vel motu lentiori agi-<lb/>tantur. </s> + <s xml:id="echoid-s424" xml:space="preserve">Exinde conjicere liceret, aquam ſaccharo vel ſale imprægnatam tar-<lb/>dius congelari, quod particulæ ſacchari vel ſalis, particulis aqueis interpoſitæ, <lb/>harum attractionem diminuant, neque hæ conjungi poſſint, ſiccidumque con-<lb/>gelari, quin particulæ heterogeneæ loco pellantur: </s> + <s xml:id="echoid-s425" xml:space="preserve">& </s> + <s xml:id="echoid-s426" xml:space="preserve">certe omnibus in fluidis, +<pb o="16" file="0030" n="30" rhead="HYDRODYNAMICÆ"/> +quæ particulis heterogeneis ſunt imprægnata, tempore congelationis fit quæ-<lb/>dam partium ex poris expulſio, ſeu ſecretio atque præcipitatio. </s> + <s xml:id="echoid-s427" xml:space="preserve">Infinita ſunt <lb/>alia corporum tum ſolidorum tum fluidorum phænomena, quæ mire admo-<lb/>dum cum principio mutuæ gravitationis conveniunt, ita, ut dolendum ſit, <lb/>principium ipſum tam alte ſupra mentem humanam poſitum eſſe, ut nemi-<lb/>nem eſſe putem, qui id ullo modo intelligere poſſit.</s> + <s xml:id="echoid-s428" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s429" xml:space="preserve">§. </s> + <s xml:id="echoid-s430" xml:space="preserve">25. </s> + <s xml:id="echoid-s431" xml:space="preserve">Denique hic monuiſſe conveniet, tractatum hunc ut Phyſicum <lb/>potius quam Mathematicum mihi conſiderari, nec proin conſultum me duxif-<lb/>ſe, methodum Geometricam in hypotheſibus, definitionibus cæterisque ap-<lb/>paratibus præmittendis nimium affectare, & </s> + <s xml:id="echoid-s432" xml:space="preserve">ubique ordinem ſermonemque <lb/>Geometrarum ſequi, qui ſolent ab ovo ordiri, propoſitionibus complecti, & </s> + <s xml:id="echoid-s433" xml:space="preserve"><lb/>eo ordine omnia pertractare, ut ex primis præmiſſis ſingula rite deducantur, <lb/>nihilque indemonſtratum poſt ſe relinquant, quamvis id a tot aliis jam de-<lb/>monſtratum fuerit. </s> + <s xml:id="echoid-s434" xml:space="preserve">Non mihi hæc cura fuit ratione eorum, quæ ab aliis <lb/>tradita ſunt, ſive definitiones fuerint & </s> + <s xml:id="echoid-s435" xml:space="preserve">axiomata, ſive etiam theoremata, non <lb/>omiſi tamen demonſtrationes eorum, quæ nova ſunt, imo etiam in prima <lb/>ſectione apponuntur demonſtrationes Theorematum, ab aliis paſſim demon-<lb/>ſtratorum; </s> + <s xml:id="echoid-s436" xml:space="preserve">& </s> + <s xml:id="echoid-s437" xml:space="preserve">cum quidam occurrant termini, ab aliis non explicati nec uſi-<lb/>tati, horum definitiones in ipſo textu exhibebo. </s> + <s xml:id="echoid-s438" xml:space="preserve">Cætera modo ſub forma <lb/>Propoſitionum, Theorematum, Problematum, Corollariorum, Scholiorum-<lb/>que pro more Geometrarum proponam, modo etiam ſermone continuo ex-<lb/>plicata dabo.</s> + <s xml:id="echoid-s439" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s440" xml:space="preserve">Unum ſupereſt, de quo Lectorem præmonitum potiſſimum volo: </s> + <s xml:id="echoid-s441" xml:space="preserve">non <lb/>potuiſſe me huic operi eam adhibere ſive diligentiam ſive attentionem, quam <lb/>debuiſſem, & </s> + <s xml:id="echoid-s442" xml:space="preserve">quam ipſe deſideravi. </s> + <s xml:id="echoid-s443" xml:space="preserve">Nullus adeoque dubito, quin nonnul-<lb/>li irrepſerint errores, dum calculos ponerem, quos, ſpero, nemo ſiniſtre <lb/>explicabit: </s> + <s xml:id="echoid-s444" xml:space="preserve">aliquos, qui in oculos incurrerunt, dum tractatum leviter relege-<lb/>rem, ipſe correxi; </s> + <s xml:id="echoid-s445" xml:space="preserve">alios tamen etiamnum ſupereſſe mihi perſuadeo.</s> + <s xml:id="echoid-s446" xml:space="preserve"/> +</p> +<pb o="17" file="0031" n="31" rhead="(o)"/> +</div> +<div xml:id="echoid-div9" type="section" level="1" n="9"> +<head xml:id="echoid-head12" xml:space="preserve">HYDRODYNAMICÆ <lb/>SECTIO SECUNDA,</head> +<head xml:id="echoid-head13" style="it" xml:space="preserve">Quæ agit de fluidis ſtagnantibus eorundemque <lb/>æquilibrio tum inter ſe, tum ad alias po-<lb/>tentias relato.</head> +<head xml:id="echoid-head14" xml:space="preserve">Theorema 1.</head> +<head xml:id="echoid-head15" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s447" xml:space="preserve">SUperficies fluidi ſtagnantis horizonti eſt parallela.</s> + <s xml:id="echoid-s448" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div10" type="section" level="1" n="10"> +<head xml:id="echoid-head16" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s449" xml:space="preserve">Contineat vas A B C D (Fig. </s> + <s xml:id="echoid-s450" xml:space="preserve">1.) </s> + <s xml:id="echoid-s451" xml:space="preserve">fluidum E B C F, cu-<lb/> +<anchor type="note" xlink:label="note-0031-01a" xlink:href="note-0031-01"/> +jus ſuperficies E G F, ſi fieri poſſit, horizonti non ſit parallela: </s> + <s xml:id="echoid-s452" xml:space="preserve">conſi-<lb/>deretur guttula in loco eminentiori a, quæ gravitate ſua verticaliter <lb/>deorſum ſollicitatur vi repræſentata per a c, reſolvatur hæc vis in duas <lb/>collaterales a d & </s> + <s xml:id="echoid-s453" xml:space="preserve">a b alteram perpendicularem ad ſuperficiem, alte-<lb/>ram quæ tangat illam: </s> + <s xml:id="echoid-s454" xml:space="preserve">Cum autem nihil adſit, quod huic vi poſteriori <lb/>reſiſtat, hæc non poteſt non effectum ſuum exerere, ipſamque adeo <lb/>guttulam verſus E trahere, quod eſſet contra hypotheſin ſtagnationis, ſeu <lb/>ſtatus permanentis: </s> + <s xml:id="echoid-s455" xml:space="preserve">Igitur neceſſe eſt, ut vis tangentialis a b ubique nulla <lb/>ſit, quod non aliter contingit, quam cum ſuperficies tota horizonti eſt <lb/>parallela. </s> + <s xml:id="echoid-s456" xml:space="preserve">Q. </s> + <s xml:id="echoid-s457" xml:space="preserve">E. </s> + <s xml:id="echoid-s458" xml:space="preserve">D.</s> + <s xml:id="echoid-s459" xml:space="preserve"/> +</p> +<div xml:id="echoid-div10" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">Fig. 1.</note> +</div> +</div> +<div xml:id="echoid-div12" type="section" level="1" n="11"> +<head xml:id="echoid-head17" xml:space="preserve">Corollarium.</head> +<p> + <s xml:id="echoid-s460" xml:space="preserve">§. </s> + <s xml:id="echoid-s461" xml:space="preserve">2. </s> + <s xml:id="echoid-s462" xml:space="preserve">Hinc intelligitur veritas propoſitionis generalis, quod nempe <lb/>ſuperficies fluidi, cujus partes viribus quibuscunque ſollicitantur, ſe <lb/>ita ſemper componat, ut quælibet guttula, in ſuperficie poſita, trahatur <lb/>ſub directione, ad ſuperficiem perpendiculari.</s> + <s xml:id="echoid-s463" xml:space="preserve"/> +</p> +<pb o="18" file="0032" n="32" rhead="HYDRODYNAMICÆ."/> +</div> +<div xml:id="echoid-div13" type="section" level="1" n="12"> +<head xml:id="echoid-head18" xml:space="preserve">Theorema 2.</head> +<p> + <s xml:id="echoid-s464" xml:space="preserve">§. </s> + <s xml:id="echoid-s465" xml:space="preserve">3. </s> + <s xml:id="echoid-s466" xml:space="preserve">Fluidum homogeneum, duobus tubis @ communicantibus <lb/>utcunque formatis incluſum, ad æquilibrium eſt compoſitum, quando <lb/>ambæ ſuperficies ad libellam poſitæ ſunt, id eſt, æqualem à puncto vaſis <lb/>infimo diſtantiam verticalem ſervant.</s> + <s xml:id="echoid-s467" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div14" type="section" level="1" n="13"> +<head xml:id="echoid-head19" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s468" xml:space="preserve">Sit fluidum vaſi ABC, (Fig. </s> + <s xml:id="echoid-s469" xml:space="preserve">2.) </s> + <s xml:id="echoid-s470" xml:space="preserve">ex duobus cruribus ſeu tubis <lb/> +<anchor type="note" xlink:label="note-0032-01a" xlink:href="note-0032-01"/> +communicantibus compoſito incluſum, ponaturque in utroque crure <lb/>ad eandem altitudinem poſitum: </s> + <s xml:id="echoid-s471" xml:space="preserve">dico non poſſe ſitum hunc mutari, quin <lb/>corpus aliquod grave ex ſitu inferiori in altiorem ſe recipiat, quod eſſet <lb/>contra naturam gravium; </s> + <s xml:id="echoid-s472" xml:space="preserve">Nam ſi ſuperficies E deſcendat in e, & </s> + <s xml:id="echoid-s473" xml:space="preserve">ab al-<lb/>tera parte D ex D elevetur in d, quoniam pars vaſis reliqua eodem fluido <lb/>ante & </s> + <s xml:id="echoid-s474" xml:space="preserve">poſt ſitum mutatum plenum eſt, omnis mutationis effectus in hoc <lb/>ſitus eſt, quod particula E e aſcenderit in D d.</s> + <s xml:id="echoid-s475" xml:space="preserve"/> +</p> +<div xml:id="echoid-div14" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0032-01" xlink:href="note-0032-01a" xml:space="preserve">Fig. 2.</note> +</div> +<p> + <s xml:id="echoid-s476" xml:space="preserve">Cæterum idem quoque liquet ex Theoremate primo, quandoquidem <lb/>in aqua ſtagnante tubus utcunque formatus fingi poteſt, in quo utique aqua <lb/>ſitum ſervabit, quem antea habuit, cum perinde ſit, ſive aqua tubo incluſa, <lb/>coërceatur lateribus tubi, ſive circumſtagnante aqua.</s> + <s xml:id="echoid-s477" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div16" type="section" level="1" n="14"> +<head xml:id="echoid-head20" xml:space="preserve">Scholium 1.</head> +<p> + <s xml:id="echoid-s478" xml:space="preserve">§. </s> + <s xml:id="echoid-s479" xml:space="preserve">4. </s> + <s xml:id="echoid-s480" xml:space="preserve">Si in demonſtratione prima præcedentis paragraphi tota <lb/>maſſa D B E ſitum ſuum commutaſſe concipiatur cum ſitu d B e, facile de-<lb/>monſtratur centrum gravitatis totius maſſæ in ſitum altiorem aſcendiſſe, <lb/>quod non minus abſurdum eſt: </s> + <s xml:id="echoid-s481" xml:space="preserve">Quoniam autem in noſtra demonſtratio-<lb/>ne nulla eſt particula in E e, quæ non aſcenderit poſt mutatum ſitum, exi-<lb/>ſtimavi ſtrictiorem & </s> + <s xml:id="echoid-s482" xml:space="preserve">clariorem fore demonſtrationem, ſi centri gravitatis <lb/>nulla conſideratio habeatur.</s> + <s xml:id="echoid-s483" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div17" type="section" level="1" n="15"> +<head xml:id="echoid-head21" xml:space="preserve">Scholium 2.</head> +<p> + <s xml:id="echoid-s484" xml:space="preserve">§. </s> + <s xml:id="echoid-s485" xml:space="preserve">5. </s> + <s xml:id="echoid-s486" xml:space="preserve">De tubis capillaribus phænomena habemus ſingularia;</s> + <s xml:id="echoid-s487" xml:space="preserve"> +<pb o="19" file="0033" n="33" rhead="SECTIO SECUNDA."/> +aqua enim aſcendit ſupra libellam in tubo ſtrictiori, cujus altera extre-<lb/>mitas aquæ ſubmergitur; </s> + <s xml:id="echoid-s488" xml:space="preserve">Mercurius vero libellam non attingit. </s> + <s xml:id="echoid-s489" xml:space="preserve">Hæc vero <lb/>cum aliquando attente perpenderem, in eandem præter propter incidi cau-<lb/>ſam, quam olim Patruus meus Jacobus Bernoulli, beate defunctus <lb/>dederat in tractatu ſuo de gravitate ætheris, nempe aquam in tubo ſtrictiori <lb/>ideo ultra libellam aſcendere, quod numerus particularum aëreo-ætherea-<lb/>rum in baſi columnæ, quæ aquæ in tubo ſupereminet, minor ſit nume-<lb/>ro particularum in ſimili baſi extra tubum; </s> + <s xml:id="echoid-s490" xml:space="preserve">hoc vero intelligitur ex eo, <lb/>quod poſitis juxta ſe globulis in tabula horizontali, ſi circino cirulus fiat, <lb/>globulorum aliquot neceſſario excludantur, quia dividi nequeunt: </s> + <s xml:id="echoid-s491" xml:space="preserve">Sunt ve-<lb/>ro preſſiones columnarum aëreo- ætherearum (quarum baſis altera eſt in <lb/>tubo, altera extra tubum) ut baſes, id eſt, ut numeri globulorum in baſi-<lb/>bus: </s> + <s xml:id="echoid-s492" xml:space="preserve">unde ſi numerus globulorum in prima baſi ſit = a, in altera = a + b, <lb/>preſſio columnæ prioris = g, erit preſſio alterius columnæ = {a + b/a}g, hinc dif-<lb/>ferentia preſſionum = {b/a}g, cui æquari debet altitudo aquæ ſupra libellam. <lb/></s> + <s xml:id="echoid-s493" xml:space="preserve">Hæc ut rectius intelligantur, conſiderandum erit eſſe g proportionalem qua-<lb/>drato diametri, quæ reſpondet ſuperficiei fluidi tubo incluſi, & </s> + <s xml:id="echoid-s494" xml:space="preserve">eidem <lb/>quadrato ob extremam globulorum parvitatem proportionalem quoque eſſe a, <lb/>ſic ut ratio g ad a cenſenda ſit conſtans, atque proin altitudo aquæ ſupra li-<lb/>bellam proportionem ſequi debeat ipſius b; </s> + <s xml:id="echoid-s495" xml:space="preserve">eſt vero, quod per ſe patet, <lb/>b ut peripheria ſuperficiei fluidi tubo incluſi, erit igitur altitudo ſupra libel-<lb/>lam, ut eadem illa peripheria, id quod experientia jam diu confirmavit. </s> + <s xml:id="echoid-s496" xml:space="preserve">Si <lb/>porro nunc diverſa conſideremus fluida, videbimus eo tortuoſiorem atque <lb/>proin majorem eſſe præmemoratam peripheriam, quo majores ſunt fluidi <lb/>particulæ, & </s> + <s xml:id="echoid-s497" xml:space="preserve">cum à magnitudine hujus peripheriæ pendeat altitudo fluidi <lb/>ſupra libellam, percipimus, cur hæc altitudo in eodem tubo non ſequatur <lb/>rationem gravitatis ſpecificæ inverſam: </s> + <s xml:id="echoid-s498" xml:space="preserve">ita ſi idem tubulus immergatur ſpi-<lb/>ritui vini & </s> + <s xml:id="echoid-s499" xml:space="preserve">aquæ, ille minus aſcendit, quam hæc, cum tamen ob mino-<lb/>rem ſuam gravitatem ſpiritus aſcendere deberet magis; </s> + <s xml:id="echoid-s500" xml:space="preserve">hoc vero indicat, ſi <lb/>recte rem aſſecutus ſum, minores eſſe particulas ſpiritus vini, quam aquæ: </s> + <s xml:id="echoid-s501" xml:space="preserve"><lb/>Nunquam tamen meo judicio aſcenſus ſupra libellam in ullo fluido mutari <lb/>poteſt in deſcenſum, & </s> + <s xml:id="echoid-s502" xml:space="preserve">omnia fluida ejusdem eſſe hac in re indolis, credi-<lb/>derim, niſi alia quædam cauſa, nondum hactenus conſiderata, ſuperve- +<pb o="20" file="0034" n="34" rhead="HYDRODYNAMICÆ."/> +niat, & </s> + <s xml:id="echoid-s503" xml:space="preserve">ſi ex noſtra hypotheſi argumentamur, dicendum erit, Mercurium <lb/>quoque ſupra libellam fuiſſe aſcenſurum, ſi modo particulæ ejus non majo-<lb/>ri vi ſe invicem attraherent, quam particulæ aquæ; </s> + <s xml:id="echoid-s504" xml:space="preserve">huic enim attractioni <lb/>omnia tribuo, quæ Mercurium in diverſa ire faciunt. </s> + <s xml:id="echoid-s505" xml:space="preserve">Experimenta, quæ <lb/>ad hanc ſententiam me manuduxerunt, apponam in fine hujus ſectionis.</s> + <s xml:id="echoid-s506" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div18" type="section" level="1" n="16"> +<head xml:id="echoid-head22" xml:space="preserve">Lemma.</head> +<p> + <s xml:id="echoid-s507" xml:space="preserve">§. </s> + <s xml:id="echoid-s508" xml:space="preserve">6. </s> + <s xml:id="echoid-s509" xml:space="preserve">Sit tubus cylindricus A B D C (Fig. </s> + <s xml:id="echoid-s510" xml:space="preserve">3.) </s> + <s xml:id="echoid-s511" xml:space="preserve">utcunque verſus <lb/> +<anchor type="note" xlink:label="note-0034-01a" xlink:href="note-0034-01"/> +horizontem inclinatus, cujus fundum CD ad latera tubi ſit perpendiculare, <lb/>plenusque intelligatur aquâ usque in AB; </s> + <s xml:id="echoid-s512" xml:space="preserve">dico preſſionem omnis aquæ in <lb/>fundum CD eſſe æqualem ponderi cylindri aquei, cujus baſis eſt CD, & </s> + <s xml:id="echoid-s513" xml:space="preserve"><lb/>cujus altitudo eſt verticalis DE, terminata ab horizontali BE.</s> + <s xml:id="echoid-s514" xml:space="preserve"/> +</p> +<div xml:id="echoid-div18" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0034-01" xlink:href="note-0034-01a" xml:space="preserve">Fig. 3.</note> +</div> +</div> +<div xml:id="echoid-div20" type="section" level="1" n="17"> +<head xml:id="echoid-head23" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s515" xml:space="preserve">Cum forma tubi ſit cylindrica, & </s> + <s xml:id="echoid-s516" xml:space="preserve">fundum inſuper ad late-<lb/>ra tubi perpendiculare, quilibet videt, quod actio fluidi in fundum ea-<lb/>dem ſit, quam haberet cylindrus ſolidus ejusdem ponderis ſuper plano in-<lb/>clinato, conſtat autem ex mechanicis, preſſionem cylindri ſolidi in fundum <lb/>eam eſſe, quæ in propoſitione definitur, ergo & </s> + <s xml:id="echoid-s517" xml:space="preserve">talis erit actio fluidi, ſi <lb/>modo non reſpiciatur adhæſio fluidi in lateribus tubi, ejusdemque indoles <lb/>ratione tubulorum capillarium, à quibus animum abſtrahimus. </s> + <s xml:id="echoid-s518" xml:space="preserve">Q. </s> + <s xml:id="echoid-s519" xml:space="preserve">E. </s> + <s xml:id="echoid-s520" xml:space="preserve">D.</s> + <s xml:id="echoid-s521" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div21" type="section" level="1" n="18"> +<head xml:id="echoid-head24" xml:space="preserve">Theorema 3.</head> +<p> + <s xml:id="echoid-s522" xml:space="preserve">§. </s> + <s xml:id="echoid-s523" xml:space="preserve">7. </s> + <s xml:id="echoid-s524" xml:space="preserve">Sit jam generaliter vas utcunque formatum A H M B (Fig. </s> + <s xml:id="echoid-s525" xml:space="preserve">4.) <lb/></s> + <s xml:id="echoid-s526" xml:space="preserve"> +<anchor type="note" xlink:label="note-0034-02a" xlink:href="note-0034-02"/> +& </s> + <s xml:id="echoid-s527" xml:space="preserve">aqua repletum usque in D E, erit preſſio aquæ in ſingulas vaſis <lb/>particulas, veluti in G aut H, ſemper æqualis ponderi cylindri aquei, cu-<lb/>jus baſis eſt ſuperficies illius particulæ, & </s> + <s xml:id="echoid-s528" xml:space="preserve">cujus altitudo æqualis eſt diſtan-<lb/>tiæ verticali ejusdem particulæ à ſuperficie aquea.</s> + <s xml:id="echoid-s529" xml:space="preserve"/> +</p> +<div xml:id="echoid-div21" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0034-02" xlink:href="note-0034-02a" xml:space="preserve">Fig. 4.</note> +</div> +</div> +<div xml:id="echoid-div23" type="section" level="1" n="19"> +<head xml:id="echoid-head25" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s530" xml:space="preserve">Primo concipiatur in G tubulus cylindricus CG perpendicula-<lb/>riter vaſi inſiſtens, productaque ED, intelligatur hic tubus ſimili liquore +<pb o="21" file="0035" n="35" rhead="SECTIO SECUNDA."/> +plenus usque in C. </s> + <s xml:id="echoid-s531" xml:space="preserve">Si nunc fingatur vas perforatum in G; </s> + <s xml:id="echoid-s532" xml:space="preserve">erit utrumque flui-<lb/>dum in æquilibrio (per §. </s> + <s xml:id="echoid-s533" xml:space="preserve">3.) </s> + <s xml:id="echoid-s534" xml:space="preserve">tantum ergo premit fluidum tubuli C G ver-<lb/>ſus interiora, quantum premit fluidum vaſis verſus exteriora. </s> + <s xml:id="echoid-s535" xml:space="preserve">Sed prior <lb/>preſſio convenit propoſitioni (per §. </s> + <s xml:id="echoid-s536" xml:space="preserve">6.) </s> + <s xml:id="echoid-s537" xml:space="preserve">ergo & </s> + <s xml:id="echoid-s538" xml:space="preserve">altera.</s> + <s xml:id="echoid-s539" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s540" xml:space="preserve">II. </s> + <s xml:id="echoid-s541" xml:space="preserve">Si vero loco puncti G ſumatur aliud H tale, ut linea, quæ eo in <lb/>loco vaſi perpendiculariter inſiſtit, cadat intra vas; </s> + <s xml:id="echoid-s542" xml:space="preserve">tunc poteſt vas integrum <lb/>concipi R H S O N, priori unitum in H, & </s> + <s xml:id="echoid-s543" xml:space="preserve">aqua repletum usque in P O. </s> + <s xml:id="echoid-s544" xml:space="preserve">Sic <lb/>enim apparet, ſi particula H, quæ utrique vaſi communis eſt, perforetur, <lb/>fluida ſic fore in æquilibrio (§. </s> + <s xml:id="echoid-s545" xml:space="preserve">3.) </s> + <s xml:id="echoid-s546" xml:space="preserve">adeoque utriusque preſſionem in H eſſe <lb/>æqualem. </s> + <s xml:id="echoid-s547" xml:space="preserve">Preſſio autem fluidi in R S N ea eſt, quæ indicatur in propoſi-<lb/>tione (per partem primam hujus demonſtrationis) ergo & </s> + <s xml:id="echoid-s548" xml:space="preserve">preſſio fluidi, <lb/>quod eſt in vaſe A M B. </s> + <s xml:id="echoid-s549" xml:space="preserve">Q. </s> + <s xml:id="echoid-s550" xml:space="preserve">E. </s> + <s xml:id="echoid-s551" xml:space="preserve">D.</s> + <s xml:id="echoid-s552" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div24" type="section" level="1" n="20"> +<head xml:id="echoid-head26" xml:space="preserve">Scholion.</head> +<p> + <s xml:id="echoid-s553" xml:space="preserve">§. </s> + <s xml:id="echoid-s554" xml:space="preserve">8. </s> + <s xml:id="echoid-s555" xml:space="preserve">Ex his propoſitionibus facile deducuntur æquilibria flui-<lb/>dorum ſtagnantium in caſibus magis compoſitis. </s> + <s xml:id="echoid-s556" xml:space="preserve">Nolo autem omnia <lb/>proſequi, neque enim inſtituti noſtri ratio id poſtulat, contentus demon-<lb/>ſtrationibus, quas modo dedi, propoſitionum fundamentalium in hydroſtatica. <lb/></s> + <s xml:id="echoid-s557" xml:space="preserve">Quod vero attinet ad preſſiones fluidorum non ſtagnantium, funt certe hæ <lb/>altioris indaginis. </s> + <s xml:id="echoid-s558" xml:space="preserve">Nec dum à quoquam recte determinata fuit preſſio fluido-<lb/>rum, par canales ſeu tubos dato velocitatis gradu fluentium, quamvis id ar-<lb/>gumenti genus tam in rebus aquariis, quam multis aliis ſit utiliſſimum. </s> + <s xml:id="echoid-s559" xml:space="preserve"><lb/>De his vero prius agere non licet, quam de motu fluidorum commentati <lb/>ſimus.</s> + <s xml:id="echoid-s560" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s561" xml:space="preserve">§. </s> + <s xml:id="echoid-s562" xml:space="preserve">9. </s> + <s xml:id="echoid-s563" xml:space="preserve">Patet ex præcedentibus ratio potentiarum veſicularium, qui-<lb/>bus ingentia pondera ſuperari poſſunt: </s> + <s xml:id="echoid-s564" xml:space="preserve">Inde etiam noſcitur vis, quam ſu-<lb/>ſtinent latera tubi, in quo aquæ ſtagnant; </s> + <s xml:id="echoid-s565" xml:space="preserve">quod argumentum, quoniam <lb/>pertractari ſolet ab hydroſtaticæ ſcriptoribus, nunc percurremus, præſer-<lb/>tim cum multa alia eo innitantur, de quibus nobis dicendum erit.</s> + <s xml:id="echoid-s566" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s567" xml:space="preserve">Sit Primo veſicaonmp, (Fig. </s> + <s xml:id="echoid-s568" xml:space="preserve">5.) </s> + <s xml:id="echoid-s569" xml:space="preserve">pavimento & </s> + <s xml:id="echoid-s570" xml:space="preserve">ponderi B interpoſita, in <lb/> +<anchor type="note" xlink:label="note-0035-01a" xlink:href="note-0035-01"/> +quam aqua infundatur per tubum FRo, cujus crus verticale FR brevitatis <lb/>gratia incomparabiliter longius ponemus, quam diametrum veſicæ: </s> + <s xml:id="echoid-s571" xml:space="preserve">Non ele-<lb/>vabitur ſtatim pondus B; </s> + <s xml:id="echoid-s572" xml:space="preserve">Atſi aqua porro infundatur usque v. </s> + <s xml:id="echoid-s573" xml:space="preserve">gr. </s> + <s xml:id="echoid-s574" xml:space="preserve">in F, +<pb o="22" file="0036" n="36" rhead="HYDRODYNAMICÆ"/> +demum attolletur pondus; </s> + <s xml:id="echoid-s575" xml:space="preserve">erit autem æquilibrium, cum locus contactus <lb/>c d ſe habet ad orificium o, ut pondus B ad pondus cylindri aquei altitudi-<lb/>nis FR ſuper baſi o inſiſtentis. </s> + <s xml:id="echoid-s576" xml:space="preserve">Pendet itaque abſoluta elevationis determi-<lb/>natio à ſtructura veſicæ, quæ ſi exempli gratia compoſita fuerit ex filamen-<lb/>tis perfecte flexibilibus, extenſionemque nullam admittentibus, ſimulque <lb/>figuram naturalem habuerit Sphæricam, facile apparet, fore ſpatia conta-<lb/>ctus cnd & </s> + <s xml:id="echoid-s577" xml:space="preserve">gpe æqualia & </s> + <s xml:id="echoid-s578" xml:space="preserve">corrugata, partemque reliquam expanſam, <lb/>habituram eſſe formam Zonæ ſphæricæ; </s> + <s xml:id="echoid-s579" xml:space="preserve">Atque hinc per Geometriam dedu-<lb/>citur quantitas elevationis np, quæ nulla erit, quamdiu circulus maximus <lb/>veſicæ minorem habuerit rationem ad orificium o illa, quæ eſt inter pon-<lb/>dus B & </s> + <s xml:id="echoid-s580" xml:space="preserve">pondus præfati cylindri aquei, nec prius tota explicabitur veſi-<lb/>ca quam altitudo fuerit infinita, id eſt, nunquam. </s> + <s xml:id="echoid-s581" xml:space="preserve">Si vero fibræ alius ſunt <lb/>indolis, aliter ſe res habet, quod multi non ſatis conſiderarunt, quibus de <lb/>figura veſicæ inflatæ ſermo fuit, eamque cavernulis muſcularibus in œcono-<lb/>mia animali applicare voluerunt, quâ de re nunc paullo fuſius agam.</s> + <s xml:id="echoid-s582" xml:space="preserve"/> +</p> +<div xml:id="echoid-div24" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0035-01" xlink:href="note-0035-01a" xml:space="preserve">Fig. 5.</note> +</div> +<p> + <s xml:id="echoid-s583" xml:space="preserve">§. </s> + <s xml:id="echoid-s584" xml:space="preserve">10. </s> + <s xml:id="echoid-s585" xml:space="preserve">Fuerit vèſica DC (Fig. </s> + <s xml:id="echoid-s586" xml:space="preserve">6.) </s> + <s xml:id="echoid-s587" xml:space="preserve">eidemque appenſum pondus P, ſi-<lb/> +<anchor type="note" xlink:label="note-0036-01a" xlink:href="note-0036-01"/> +mulque alligata tubulo DA, cujus rurſus longitudinem compendii ergo in <lb/>comparabiliter majorem longitudine DC fingemus. </s> + <s xml:id="echoid-s588" xml:space="preserve">His poſitis facile qui-<lb/>dem quivis perſpicit, repletis veſica & </s> + <s xml:id="echoid-s589" xml:space="preserve">tubulo fore, ut illa intumeſcat, <lb/>pondusque appenſum P elevet: </s> + <s xml:id="echoid-s590" xml:space="preserve">nemo autem intelliget ſtatum æquilibrii, <lb/>figuramque ventricoſam, niſi plane intelligatur ſtructura veſicæ ejusdemque <lb/>fibrarum, quæ cum ita ſint, caſus aliquot ſingulares examinabimus, qui <lb/>frequentius occurrere poſſunt.</s> + <s xml:id="echoid-s591" xml:space="preserve"/> +</p> +<div xml:id="echoid-div25" type="float" level="2" n="2"> +<note position="left" xlink:label="note-0036-01" xlink:href="note-0036-01a" xml:space="preserve">Fig. 6.</note> +</div> +</div> +<div xml:id="echoid-div27" type="section" level="1" n="21"> +<head xml:id="echoid-head27" xml:space="preserve">Caſus I.</head> +<p> + <s xml:id="echoid-s592" xml:space="preserve">§. </s> + <s xml:id="echoid-s593" xml:space="preserve">11. </s> + <s xml:id="echoid-s594" xml:space="preserve">Si veſica compoſita fuerit ex fibris longitudinalibus DpC, <lb/>DmC &</s> + <s xml:id="echoid-s595" xml:space="preserve">c. </s> + <s xml:id="echoid-s596" xml:space="preserve">inſtar meridianorum in punctis D & </s> + <s xml:id="echoid-s597" xml:space="preserve">C, ceu Polis concurren-<lb/>tibus æqualibus, perfecte flexibilibus & </s> + <s xml:id="echoid-s598" xml:space="preserve">uniformibus, quarum ſingu-<lb/>læ inter ſe proximæ minimis connectantur fibrillis transverſalibus, hisque ita <lb/>laxis, ut minima vel quaſi nulla vi ſufficientem extenſionem admittant. </s> + <s xml:id="echoid-s599" xml:space="preserve">Sic <lb/>quælibet fibra DpC incurvabitur in figuram elaſticæ, totaque veſica formam <lb/>aſſumet ſolidi, quod generatur ex revolutione iſtius curvæ circa axem DC. <lb/></s> + <s xml:id="echoid-s600" xml:space="preserve">Si porro altitudo AD eſt infinita, fit elaſtica DpC rectangula & </s> + <s xml:id="echoid-s601" xml:space="preserve">tunc eſt <lb/>graſſities maxima veſicæ ad longitudinem axis DC ut 25 ad 11 præter pro- +<pb o="23" file="0037" n="37" rhead="SECTIO SECUNDA."/> +pter atque longitudo arcus DpC eſt ad eundem axem proxime ut 5 ad 2, <lb/>ita ut maxima elevatione ponderis veſica tribus quintis partibus decurtetur.</s> + <s xml:id="echoid-s602" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div28" type="section" level="1" n="22"> +<head xml:id="echoid-head28" xml:space="preserve">Caſus II.</head> +<p> + <s xml:id="echoid-s603" xml:space="preserve">§. </s> + <s xml:id="echoid-s604" xml:space="preserve">12. </s> + <s xml:id="echoid-s605" xml:space="preserve">Si poſitis cæteris, ut antea, minima filamenta trans-<lb/>verſalia n o, m p, &</s> + <s xml:id="echoid-s606" xml:space="preserve">c. </s> + <s xml:id="echoid-s607" xml:space="preserve">quæ ſunt perpendiculares ad fibras longitudinales, ex-<lb/>tenſioni reſiſtant, apparet non poſſe figuram fibræ DopC determinari, quin <lb/>duo potentiarum genera unicuique puncto applicata conſiderentur, quo-<lb/>rum alterum curvæ perpendiculariter inſiſtit, & </s> + <s xml:id="echoid-s608" xml:space="preserve">filum extrorſum premit, <lb/>alterum ad axem curvæ DC, eſt perpendiculare & </s> + <s xml:id="echoid-s609" xml:space="preserve">introrſum trahit: </s> + <s xml:id="echoid-s610" xml:space="preserve">faci-<lb/>le etiam intelligitur infinitas poſſe harum preſſionum excogitari leges, ut <lb/>ad curvam quamvis datam fibra DopC ſe componat, atque adeo etiam v. <lb/></s> + <s xml:id="echoid-s611" xml:space="preserve">gr. </s> + <s xml:id="echoid-s612" xml:space="preserve">ad circularem, quæ figura à plerisque Phyſiologis tribuitur fibrillis, quæ <lb/>pertinent ad machinulas muſculares: </s> + <s xml:id="echoid-s613" xml:space="preserve">Sed eſt alius etiam modus, quo fibra <lb/>longitudinalis DopC acquirere poteſt figuram arcus circularis, nempe cum <lb/>omnino abſunt fibrillæ transverſales np, mp, &</s> + <s xml:id="echoid-s614" xml:space="preserve">c. </s> + <s xml:id="echoid-s615" xml:space="preserve">Sic enim dum inflatur ve-<lb/>ſica, hiatus fit inter duas fibras longitudinales proximas DopC & </s> + <s xml:id="echoid-s616" xml:space="preserve">DnmC, <lb/>per quem fluidum erumpit, ſimul autem, cum non ſatis cito effluere poſ-<lb/>ſit, fibras extendit, easque ad figuram circularem componit: </s> + <s xml:id="echoid-s617" xml:space="preserve">atque hoc in <lb/>caſu maxima veſicæ decurtatio, quæ in priori caſu fuit {3/5} totius longitudi-<lb/>nis veſicæ non inflatæ, nunc tantum eſt proxime {4/11}.</s> + <s xml:id="echoid-s618" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s619" xml:space="preserve">§. </s> + <s xml:id="echoid-s620" xml:space="preserve">13. </s> + <s xml:id="echoid-s621" xml:space="preserve">Sequitur ex hiſce, difficile eſſe, ut figura veſicæ inflatæ, cui pon-<lb/>dus appenſum eſt, recte determinetur, quandoquidem nemo ſit, qui indo-<lb/>lem minimarum fibrillarum perfecte cognoſcere poſſit: </s> + <s xml:id="echoid-s622" xml:space="preserve">tranſcribam tamen <lb/>huc exempla quædam, quæ maxime videntur probabilia, ex ſchedis meis <lb/>ſine demonſtratione, quam ſi quis deſideret, reperiet in tom. </s> + <s xml:id="echoid-s623" xml:space="preserve">3. </s> + <s xml:id="echoid-s624" xml:space="preserve">Comm. <lb/></s> + <s xml:id="echoid-s625" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s626" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s627" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s628" xml:space="preserve">Ante omnia autem æquationem dabo ad curvam, quæ ex <lb/>duobus potentiarum generibus, ut dixi in præcedente paragrapho, iisque <lb/>quamcunque legem obſervantibus formatur.</s> + <s xml:id="echoid-s629" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s630" xml:space="preserve">§. </s> + <s xml:id="echoid-s631" xml:space="preserve">14. </s> + <s xml:id="echoid-s632" xml:space="preserve">Sit igitur filum AEG (Fig. </s> + <s xml:id="echoid-s633" xml:space="preserve">7.) </s> + <s xml:id="echoid-s634" xml:space="preserve">duobus punctis A & </s> + <s xml:id="echoid-s635" xml:space="preserve">G affixum; <lb/></s> + <s xml:id="echoid-s636" xml:space="preserve"> +<anchor type="note" xlink:label="note-0037-01a" xlink:href="note-0037-01"/> +ducatur recta AG: </s> + <s xml:id="echoid-s637" xml:space="preserve">ſintque duo puncta in filo infinite propinqua D & </s> + <s xml:id="echoid-s638" xml:space="preserve">E, ex <lb/>quibus agantur ad AG perpendiculares D B & </s> + <s xml:id="echoid-s639" xml:space="preserve">E C; </s> + <s xml:id="echoid-s640" xml:space="preserve">lineola autem D F ſit li-<lb/>neæ AG parallela. </s> + <s xml:id="echoid-s641" xml:space="preserve">Intelligatur ſingulis punctis D vel E applicatas eſſe duas +<pb o="24" file="0038" n="38" rhead="HYDRODYNAMICÆ."/> +potentias utcunque variabiles, quarum altera ſit ubique ad curvam, altera <lb/>ad A G perpendicularis: </s> + <s xml:id="echoid-s642" xml:space="preserve">priorem ponemus in puncto D æqualem A, in <lb/>puncto E æqualem A + dA, alteram in puncto D = C, in puncto E = C + dC: <lb/></s> + <s xml:id="echoid-s643" xml:space="preserve">Sit porro AB = x, BD = y, AD = s, BC = dx, FE = dy, DE = ds, quod <lb/>elementum curvæ conſtantis magnitudinis ponatur; </s> + <s xml:id="echoid-s644" xml:space="preserve">Radius Oſculi in puncto <lb/>D = R, in puncto E = R + dR. </s> + <s xml:id="echoid-s645" xml:space="preserve">Dico æquationem ad curvam fore hanc - AdR <lb/>- R d A = (RdCdx + 2Cdyds + CdxdR) ds, vel poſito CRddx pro Cdyds <lb/>(eſt enim R = {dyds/ddx}) habebitur - AdR - RdA = (RdCdx + CRdds + Cdyds <lb/>+ Cdx dR): </s> + <s xml:id="echoid-s646" xml:space="preserve">ds, ſive {-ARds - RCdx/dx} = ſCdy.</s> + <s xml:id="echoid-s647" xml:space="preserve"/> +</p> +<div xml:id="echoid-div28" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0037-01" xlink:href="note-0037-01a" xml:space="preserve">Fig. 7.</note> +</div> +<p> + <s xml:id="echoid-s648" xml:space="preserve">§. </s> + <s xml:id="echoid-s649" xml:space="preserve">15. </s> + <s xml:id="echoid-s650" xml:space="preserve">Intelligitur ex præcedente æquatione, quod cum potentiæ, <lb/>quæ ſunt ad curvam perpendiculares, ſolæ agunt, fiat AR = conſtanti quan-<lb/>titati, quia nempe ſic fit C = o: </s> + <s xml:id="echoid-s651" xml:space="preserve">tunc igitur radius oſculi ubique ſequitur ra-<lb/>tionem inverſam potentiæ reſpondentis. </s> + <s xml:id="echoid-s652" xml:space="preserve">At ſi potentiæ ad axem perpendi-<lb/>culares ſolæ adſunt, tunc evaneſcente littera A fit - {RCdx/ds} = ſCdy. </s> + <s xml:id="echoid-s653" xml:space="preserve">Po-<lb/>teſt autem hæc æquatio integrari & </s> + <s xml:id="echoid-s654" xml:space="preserve">ad hanc reduci formam RCdx<emph style="super">2</emph> = con-<lb/>ſtanti quantitati; </s> + <s xml:id="echoid-s655" xml:space="preserve">ex qua apparet potentiam ductam in radium oſculi ubique <lb/>eſſe in ratione reciproca quadrati ſinus, quem applicata facit cum curva. <lb/></s> + <s xml:id="echoid-s656" xml:space="preserve">Similiter æquatio canonica integrationem admittit, cum potentiæ, quæ ad <lb/>axem perpendiculares ſunt, omnes inter ſe ſunt æquales ſeu proportionales <lb/>elemento curvæ d s. </s> + <s xml:id="echoid-s657" xml:space="preserve">Ita enim poſito d C = o, obtinetur - AdR - RDA = <lb/>2ndyds + ndxdR, intelligendo per n conſtantem quantitatem, qua æqua-<lb/>tione recte tractata fit nydy + mmdy - nsds = dsſAdx, ubi m conſtans eſt <lb/>ab integratione proveniens.</s> + <s xml:id="echoid-s658" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s659" xml:space="preserve">Si præterea potentiæ ad curvam normales ponantur applicatis y pro-<lb/>portionales, poterit ulterius reduci poſtrema æquatio ad hanc <lb/>- dx = (2ff - {gyy/h}) dy: </s> + <s xml:id="echoid-s660" xml:space="preserve">√(2ny + 2mm)<emph style="super">2</emph> - (2ff - {gyy/h})<emph style="super">2</emph>, <lb/>cujus conſtantes f & </s> + <s xml:id="echoid-s661" xml:space="preserve">m caſibus particularibus erunt applicandæ, dum n & </s> + <s xml:id="echoid-s662" xml:space="preserve">g pen-<lb/>dent à relatione potentiarum in puncto aliquo: </s> + <s xml:id="echoid-s663" xml:space="preserve">unde ſi g = o, oritur catenaria, & </s> + <s xml:id="echoid-s664" xml:space="preserve"><lb/>ſi n = o prodit elaſtica: </s> + <s xml:id="echoid-s665" xml:space="preserve">generaliter vero inſervit æquatio ad curvaturam <lb/>lintei uniformiter gravis, cui fluidum ſuperincumbit, determinandam: </s> + <s xml:id="echoid-s666" xml:space="preserve">Ca- +<pb o="25" file="0039" n="39" rhead="SECTIO SECUNDA."/> +ſus ſimpliciſſimus hujus rei eſt, cum ſupponitur f = m = o, tunc enim fit <lb/>- dx = {-gydy/√(4nnhh - ggyy)} ſeu facta integratione cum additione debitæ conſtan-<lb/>tis, x = - √({4nnhh/gg} - yy) + {2nh/g}, quæ eſt æquatio ad ſemicirculum, ad <lb/>quem nempe ſe linteum accommodabit in ſequenti hypotheſi: </s> + <s xml:id="echoid-s667" xml:space="preserve">Sit filum lin-<lb/>tei gravis AEG (Fig. </s> + <s xml:id="echoid-s668" xml:space="preserve">8.) </s> + <s xml:id="echoid-s669" xml:space="preserve">in ſemicirculum incurvatum. </s> + <s xml:id="echoid-s670" xml:space="preserve">cujus diameter AG <lb/> +<anchor type="note" xlink:label="note-0039-01a" xlink:href="note-0039-01"/> +ad libellam poſita ſit; </s> + <s xml:id="echoid-s671" xml:space="preserve">ſuperincumbat ſilo fluidum usque ad AG, dico ſi <lb/>fluidi pondus ſit æquale ponderi fili, fore ut filum perfecte flexile & </s> + <s xml:id="echoid-s672" xml:space="preserve">uni-<lb/>formis craſſitiei figuram ſemicircularem conſervet. </s> + <s xml:id="echoid-s673" xml:space="preserve">Quomodo autem pon-<lb/>dera fili & </s> + <s xml:id="echoid-s674" xml:space="preserve">fluidi, ut æqualia fiant, efficiendum ſit ex elementis Geometriæ <lb/>conſtat. </s> + <s xml:id="echoid-s675" xml:space="preserve">Denique ſi ſtatuatur tam potentias A quam C eſſe ubique applica-<lb/>tæ reſpondenti y proportionales (quæ hypotheſis ſane maxime convenire vi-<lb/>detur cum vera figura veſicæ in figura ſexta) poterit rurſus æquatio canoni-<lb/>ca, quæ continet differentialia tertii Ordinis, reduci ad æquationem ſimpli-<lb/>citer differentialem eamque per quadraturas facile conſtruendam. </s> + <s xml:id="echoid-s676" xml:space="preserve">Sit nem-<lb/>pe A = my & </s> + <s xml:id="echoid-s677" xml:space="preserve">C = ny, dico naturam curvæ A D G in fig. </s> + <s xml:id="echoid-s678" xml:space="preserve">7. </s> + <s xml:id="echoid-s679" xml:space="preserve">exprimi hâc æquatione <lb/>dx = (g<emph style="super">3</emph> + {1/2} myy) dy: </s> + <s xml:id="echoid-s680" xml:space="preserve">√[(f<emph style="super">3</emph> + {1/2} nyy)<emph style="super">2</emph> - (g<emph style="super">3</emph> + {1/2} myy)<emph style="super">2</emph>] <lb/>in qua literæ conſtantis magnitudinis f & </s> + <s xml:id="echoid-s681" xml:space="preserve">g rurſus ab integrationibus pro-<lb/>dierunt: </s> + <s xml:id="echoid-s682" xml:space="preserve">fit autem valor literæ n negativus, cum æquatio ad veſicæ inflatæ <lb/>figuram determinandam adhibetur.</s> + <s xml:id="echoid-s683" xml:space="preserve"/> +</p> +<div xml:id="echoid-div29" type="float" level="2" n="2"> +<note position="right" xlink:label="note-0039-01" xlink:href="note-0039-01a" xml:space="preserve">Fig. 8.</note> +</div> +<p> + <s xml:id="echoid-s684" xml:space="preserve">§. </s> + <s xml:id="echoid-s685" xml:space="preserve">16. </s> + <s xml:id="echoid-s686" xml:space="preserve">Nolui his nimis inſiſtere, quod non proxime pertinent ad Hy-<lb/>drodynamicam: </s> + <s xml:id="echoid-s687" xml:space="preserve">Nihil etiam addo de fluidis elaſticis, quia horum theoriam <lb/>ſeorſim tradere conſtitui; </s> + <s xml:id="echoid-s688" xml:space="preserve">attamen quod ad preſſiones fluidorum elaſtico-<lb/>rum attinet, poterunt illæ ex natura fluidorum ſimpliciter gravium ſupra ex-<lb/>poſita facile deduci & </s> + <s xml:id="echoid-s689" xml:space="preserve">demonſtrari, fingendo fluidum elaſticitate eſſe deſti-<lb/>tutum, cylindrumque fluidi ſimilis altitudinis infinitæ vel quaſi infinitæ ſu-<lb/>perimcumbere; </s> + <s xml:id="echoid-s690" xml:space="preserve">hæc autem quomodo intelligenda ſint ſuo loco dicemus: <lb/></s> + <s xml:id="echoid-s691" xml:space="preserve">Nunc quidem pergo ad id, quod in rebus aquariis potiſſimum quæri ſolet, <lb/>quanta nempe debeat eſſe firmitas canalium, ut preſſioni aquæ reſiſtere poſ-<lb/>ſint, ubi præſertim conſiderantur canales, qui aquas ad fontes vehunt, de <lb/>quibus ego quoque pauca monebo.</s> + <s xml:id="echoid-s692" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s693" xml:space="preserve">§. </s> + <s xml:id="echoid-s694" xml:space="preserve">17. </s> + <s xml:id="echoid-s695" xml:space="preserve">Probe diſtinguendæ ſunt preſſiones aquarum in canalibus ſtag- +<pb o="26" file="0040" n="40" rhead="HYDRODYNAMICÆ."/> +nantium à preſſionibus fluentium, quamvis id nemo adhuc animadverterit, <lb/>quod ſciam; </s> + <s xml:id="echoid-s696" xml:space="preserve">hinc eſt, quod regulæ à variis exhibitæ valeant tantum pro <lb/>aquis ſtagnantibus, tametſi verbis utantur, quæ perinde eas pertinere ad <lb/>aquas fluentes perſuadere poſſint. </s> + <s xml:id="echoid-s697" xml:space="preserve">Ut vero diſcrimerr utriusque Theoriæ ap-<lb/>pareat in ipſo limine, exemplum quoddam afferam, cujus demonſtratio ex <lb/>inferioribus patebit. </s> + <s xml:id="echoid-s698" xml:space="preserve">Sit loco caſtelli vas ampliſſimum A B C D (Fig. </s> + <s xml:id="echoid-s699" xml:space="preserve">9.) </s> + <s xml:id="echoid-s700" xml:space="preserve">aqua <lb/> +<anchor type="note" xlink:label="note-0040-01a" xlink:href="note-0040-01"/> +repletum usque in EF, & </s> + <s xml:id="echoid-s701" xml:space="preserve">in parte inferiori tubulo cylindrico horizontali <lb/>M O m o inſtructum, per quem aquæ ſine impedimento transfluere poſſe intel-<lb/>ligantur; </s> + <s xml:id="echoid-s702" xml:space="preserve">ducatur verticalis N G terminata ab horizontali E H. </s> + <s xml:id="echoid-s703" xml:space="preserve">His ita præ-<lb/>paratis, dico ſi orificium O o totum digito obſtruatur, punctum N premi <lb/>extrorſum ſecundum totam altitudinem N G; </s> + <s xml:id="echoid-s704" xml:space="preserve">ſi dimidium orificium obtu-<lb/>retur, hanc preſſionem quarta ſui parte diminui, & </s> + <s xml:id="echoid-s705" xml:space="preserve">ſi denique remoto <lb/>digito aquæ liberrime effluant, omnem preſſionem evaneſcere, ſic ut to-<lb/>tum cum parte aut etiam cum nihilo confundi ab Authoribus ſoleat. </s> + <s xml:id="echoid-s706" xml:space="preserve">Sed <lb/>demonſtrabo poſſe preſſionem vel negativam fieri, atque ita in ſuctionem <lb/>mutari. </s> + <s xml:id="echoid-s707" xml:space="preserve">Quoniam vero id agere non poſſum priusquam integram theoriam <lb/>de aquis fluentibus præmiſerim, nunc aquas conſiderabo ſaltem ſtagnantes, <lb/>veluti ſi orificium O o totum fuerit obſtructum.</s> + <s xml:id="echoid-s708" xml:space="preserve"/> +</p> +<div xml:id="echoid-div30" type="float" level="2" n="3"> +<note position="left" xlink:label="note-0040-01" xlink:href="note-0040-01a" xml:space="preserve">Fig. 9.</note> +</div> +<p> + <s xml:id="echoid-s709" xml:space="preserve">§. </s> + <s xml:id="echoid-s710" xml:space="preserve">18. </s> + <s xml:id="echoid-s711" xml:space="preserve">Conſtat autem ex Mechanicis latera tubi M O m o (cujus diame-<lb/>trum incomparabiliter cenſebimus minorem altitudine N G) non aliter ten-<lb/>di, quam ſi explicata eſſent in figuram rectangularem M O m o (Fig. </s> + <s xml:id="echoid-s712" xml:space="preserve">10.) <lb/></s> + <s xml:id="echoid-s713" xml:space="preserve"> +<anchor type="note" xlink:label="note-0040-02a" xlink:href="note-0040-02"/> +appenſumque haberent pondus P, quod ſit æquale ponderi prismatis aquei, <lb/>cujus tria latera ſint 1<emph style="super">0</emph>. </s> + <s xml:id="echoid-s714" xml:space="preserve">radius tubuli, 2<emph style="super">0</emph>. </s> + <s xml:id="echoid-s715" xml:space="preserve">longitudo ejusdem & </s> + <s xml:id="echoid-s716" xml:space="preserve">3°. </s> + <s xml:id="echoid-s717" xml:space="preserve">altitu-<lb/>do aquæ ſupra tubum; </s> + <s xml:id="echoid-s718" xml:space="preserve">Ex hac propoſitione intelligitur nonſolum ratio <lb/>tenſionum, ſi diverſæ fuerint altitudines aquæ aut diametri tuborum, ſed <lb/>& </s> + <s xml:id="echoid-s719" xml:space="preserve">ipſa tenſionum menſura: </s> + <s xml:id="echoid-s720" xml:space="preserve">Quod ſi proin firmitas tuborum major ſit iſta <lb/>tenſione, nullum erit rupturæ periculum; </s> + <s xml:id="echoid-s721" xml:space="preserve">ſi ſecus certo rumpetur tubus. </s> + <s xml:id="echoid-s722" xml:space="preserve">Cæ-<lb/>terum de firmitate tuborum experimenta inſtituta fuerunt à variis; </s> + <s xml:id="echoid-s723" xml:space="preserve">ſunt au-<lb/>tem ejusmodi experimenta difficilia & </s> + <s xml:id="echoid-s724" xml:space="preserve">ſumtuoſa; </s> + <s xml:id="echoid-s725" xml:space="preserve">poterit igitur facilius fir-<lb/>mitas tuborum ſive plumbeorum ſive ferreorum cognoſci, ſi experimento <lb/>innoteſcat, quantum pondus filum plumbeum aut ferreum datæ craſſitiei <lb/>ſuſtinere poſſit ſine rupturæ periculo. </s> + <s xml:id="echoid-s726" xml:space="preserve">Experimentum ſimile à me inſtitu-<lb/>tum apponam in fine ſectionis oſtenſurus quomodo inde firmitas tubi datæ <lb/>craſſitiei & </s> + <s xml:id="echoid-s727" xml:space="preserve">diametri deduci poſſit.</s> + <s xml:id="echoid-s728" xml:space="preserve"/> +</p> +<div xml:id="echoid-div31" type="float" level="2" n="4"> +<note position="left" xlink:label="note-0040-02" xlink:href="note-0040-02a" xml:space="preserve">Fig. 10</note> +</div> +<pb o="27" file="0041" n="41" rhead="SECTIO SECUNDA."/> +</div> +<div xml:id="echoid-div33" type="section" level="1" n="23"> +<head xml:id="echoid-head29" style="it" xml:space="preserve">Sequuntur Experimenta quæ ad Sectionem <lb/>pertinent Secundam.</head> +<head xml:id="echoid-head30" xml:space="preserve">Ad §. 5.</head> +<p> + <s xml:id="echoid-s729" xml:space="preserve">DE tubulis capillaribus: </s> + <s xml:id="echoid-s730" xml:space="preserve">Experimenta innumera de horum tubulorum <lb/>indole à variis ſumta fuerunt, quos inter eminet Georgius Bernhar-<lb/>dus Bulffingerus, qui non ſolum præcipua collegit, ſed & </s> + <s xml:id="echoid-s731" xml:space="preserve">pluri-<lb/>ma de ſuis addidit, vid. </s> + <s xml:id="echoid-s732" xml:space="preserve">Comm. </s> + <s xml:id="echoid-s733" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s734" xml:space="preserve">ſc. </s> + <s xml:id="echoid-s735" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s736" xml:space="preserve">tom. </s> + <s xml:id="echoid-s737" xml:space="preserve">2. </s> + <s xml:id="echoid-s738" xml:space="preserve">pag. </s> + <s xml:id="echoid-s739" xml:space="preserve">233. </s> + <s xml:id="echoid-s740" xml:space="preserve">& </s> + <s xml:id="echoid-s741" xml:space="preserve">ſeqq.</s> + <s xml:id="echoid-s742" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s743" xml:space="preserve">I. </s> + <s xml:id="echoid-s744" xml:space="preserve">Ut oculis recte appareret, quam contrariæ ſint Indolis hâ in par-<lb/>te mercurius & </s> + <s xml:id="echoid-s745" xml:space="preserve">reliqua fluida, confici curavi vas vitreum A B C (Fig. </s> + <s xml:id="echoid-s746" xml:space="preserve">11.) <lb/></s> + <s xml:id="echoid-s747" xml:space="preserve"> +<anchor type="note" xlink:label="note-0041-01a" xlink:href="note-0041-01"/> +ex duobus cruribus verticalibus compoſitum, quorum alterum A B diame-<lb/>trum habebat trium linearum vel quatuor, alterum B C vix tertiæ partis <lb/>lineæ. </s> + <s xml:id="echoid-s748" xml:space="preserve">Cum vas liquore quocunque implebatur, ſuperficies altius erat in <lb/>crure ſtrictiore quam ampliore, veluti in D & </s> + <s xml:id="echoid-s749" xml:space="preserve">G, mercurius autem ſolus <lb/>depreſſior eſt in ſtrictiore quam ampliore, veluti in F & </s> + <s xml:id="echoid-s750" xml:space="preserve">G.</s> + <s xml:id="echoid-s751" xml:space="preserve"/> +</p> +<div xml:id="echoid-div33" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0041-01" xlink:href="note-0041-01a" xml:space="preserve">Fig. 11.</note> +</div> +<p> + <s xml:id="echoid-s752" xml:space="preserve">II. </s> + <s xml:id="echoid-s753" xml:space="preserve">Oſtenſurus mercurium non aliam ob rationem à natura aliorum <lb/>fluidorum recedere, quam ob fortiorem particularum ſuarum mutuam at-<lb/>tractionem cogitavi de his experimentis: </s> + <s xml:id="echoid-s754" xml:space="preserve">tubulum nempe gracilem mercu-<lb/>rio ſuctione implevi eumque horizontaliter poſitum ſenſim erexi; </s> + <s xml:id="echoid-s755" xml:space="preserve">Sic <lb/>effluxit mercurius, nunquam tamen omnis & </s> + <s xml:id="echoid-s756" xml:space="preserve">altitudo verticalis mercurii in <lb/>tubulo reſidui in omni ſitu ſibi conſtabat. </s> + <s xml:id="echoid-s757" xml:space="preserve">Quod ſi autem, cum mercu-<lb/>rius in tubulo ſic ſuſpenditur, extremitas tubi mercurio in vaſe ſtagnanti ad-<lb/>movetur, protinus omnis effluit. </s> + <s xml:id="echoid-s758" xml:space="preserve">Priora Phœnomena, ni fallor, indicant <lb/>mercurio & </s> + <s xml:id="echoid-s759" xml:space="preserve">aliis fluidis idem contingere, cum vi attractrici nullus eſt lo-<lb/>cus; </s> + <s xml:id="echoid-s760" xml:space="preserve">mercurium autem fortiſſime ſe attrahere docet phœnomenon ultimum.</s> + <s xml:id="echoid-s761" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s762" xml:space="preserve">III. </s> + <s xml:id="echoid-s763" xml:space="preserve">Sumatur tubus cylindricus vitreus diametri 3 aut 4 linearum, <lb/>fundo inſtructus ex Charta ſubtili, aut tenuiſſima lamina ferrea parato & </s> + <s xml:id="echoid-s764" xml:space="preserve"><lb/>in medio minimo foraminulo perforato, ut oſtendit (Figura 12.) </s> + <s xml:id="echoid-s765" xml:space="preserve">Inclinetur <lb/> +<anchor type="note" xlink:label="note-0041-02a" xlink:href="note-0041-02"/> +tubus A C B D & </s> + <s xml:id="echoid-s766" xml:space="preserve">impleatur totus mercurio, dein ſenſim erigatur; </s> + <s xml:id="echoid-s767" xml:space="preserve">fiet quod <lb/>antea, & </s> + <s xml:id="echoid-s768" xml:space="preserve">quamvis tubus ſit ampliſſimus, non tamen effluet omnis mercu-<lb/>rius, ſed ſuſpenſa hærebit ejus pars, veluti M C D N, hæcque eo major <lb/>erit quo minus eſt ejus foraminulum o. </s> + <s xml:id="echoid-s769" xml:space="preserve">Dein cum fundum ſubmergitur +<pb o="28" file="0042" n="42" rhead="HYDRODYNAMICÆ."/> +mercurio, in vaſe alio ſervato, tantillum, ſic ut pars ſubmerſa tubi ſit C α, <lb/>non ſolum non aſcendit mercurius in tubo uſque in β (ſumta ſcilicet C α = <lb/>M β) ſed & </s> + <s xml:id="echoid-s770" xml:space="preserve">omnis fere effluit, donec ſuperficies M N pervenit in α. </s> + <s xml:id="echoid-s771" xml:space="preserve">Por-<lb/>ro tubum A C D B vacuum ſat profunde mercurio, qui erat in vaſe alio, <lb/>ſubmerſi, nec tamen prius quicquam influere cœpit ex vaſe in tubum, <lb/>quam ad altitudinem C M eſſet ſubmerſus; </s> + <s xml:id="echoid-s772" xml:space="preserve">& </s> + <s xml:id="echoid-s773" xml:space="preserve">tunc ſtatim eo usque influit <lb/>donec ab utraque parte ad libellam ſit conſtitutus, nempe usque in M N, ſi <lb/>ad eum locum usque erat ſubmerſus. </s> + <s xml:id="echoid-s774" xml:space="preserve">Omnia hæc ex mutua particularum <lb/>mercurialium attractione facile deducuntur. </s> + <s xml:id="echoid-s775" xml:space="preserve">Cæterum dedi operam ut in-<lb/>veſtigarem relationem, quæ eſt inter altitudinem M C & </s> + <s xml:id="echoid-s776" xml:space="preserve">amplitudinem <lb/>foraminuli o; </s> + <s xml:id="echoid-s777" xml:space="preserve">veriſimile utique eſt altitudinem illam eſſe in ratione recipro-<lb/>ca diametri ad foraminulum pertinentis; </s> + <s xml:id="echoid-s778" xml:space="preserve">nec tamen experimento conjectu-<lb/>ram ſatis confirmare potui, tum ob impuritatem mercurii quo utebar, quæ <lb/>faciebat, ut non variato foramine in iteratis experimentis altitudo ſuſpenſi <lb/>mercurii ſibimet ipſi non omnino conſtaret, tum etiam, quod difficile eſt <lb/>foraminula minima accurate metiri; </s> + <s xml:id="echoid-s779" xml:space="preserve">debent enim foraminula eſſe minima, <lb/>quandoquidem altitudo mercurii ſuſpenſi vix eſt ſex octove linearum, cum <lb/>diameter foraminis ſextam partem lineæ æquat, dicam tamen qua metho-<lb/>do uſus fuerim. </s> + <s xml:id="echoid-s780" xml:space="preserve">Filis nempe æneis, quibus in inſtrumentis muſicis utun-<lb/>tur, diverſæ craſſitiei, quorum diametros minimas ex longitudine & </s> + <s xml:id="echoid-s781" xml:space="preserve">pon-<lb/>dere eorum rectiſſime cognovi, chartulam C D perforavi; </s> + <s xml:id="echoid-s782" xml:space="preserve">ſed ſic ſolent <lb/>oriri circa latera foraminis fimbriæ quæ effluxum impediunt, & </s> + <s xml:id="echoid-s783" xml:space="preserve">facile ſucce-<lb/>dit ut foramen majus ſit quam eſt craſſities fili.</s> + <s xml:id="echoid-s784" xml:space="preserve"/> +</p> +<div xml:id="echoid-div34" type="float" level="2" n="2"> +<note position="right" xlink:label="note-0041-02" xlink:href="note-0041-02a" xml:space="preserve">Fig. 12.</note> +</div> +<p> + <s xml:id="echoid-s785" xml:space="preserve">Ad §. </s> + <s xml:id="echoid-s786" xml:space="preserve">18. </s> + <s xml:id="echoid-s787" xml:space="preserve">De firmitate tuborum. </s> + <s xml:id="echoid-s788" xml:space="preserve">Filum æneum rotundum, cujus dia-<lb/>meter erat {2/11} lin. </s> + <s xml:id="echoid-s789" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s790" xml:space="preserve">cui ſucceſſive pondera continue majora appende-<lb/>bantur, prius non diſruptum fuit, quam ad 18. </s> + <s xml:id="echoid-s791" xml:space="preserve">lib. </s> + <s xml:id="echoid-s792" xml:space="preserve">Norimb. </s> + <s xml:id="echoid-s793" xml:space="preserve">pondus ex-<lb/>creviſſet. </s> + <s xml:id="echoid-s794" xml:space="preserve">Dein tenuiſſimam lamellam plumbeam, cui rectangularis figura <lb/>erat, {5/4} lin. </s> + <s xml:id="echoid-s795" xml:space="preserve">latam, {1/131} lin. </s> + <s xml:id="echoid-s796" xml:space="preserve">craſſam rumpi obſervavi cum eidem appenſum <lb/>eſſet pondus trium unciarum cum dimidia. </s> + <s xml:id="echoid-s797" xml:space="preserve">Ex hiſce obſervationibus dua-<lb/>bus ſequitur cæteris paribus filum ex ære plus quam 28. </s> + <s xml:id="echoid-s798" xml:space="preserve">vicibus fortius eſſe, <lb/>quam filum ex plumbo. </s> + <s xml:id="echoid-s799" xml:space="preserve">Ex priori experimento quoque deducitur, ſi tubus <lb/>æreus diametrum habuerit unius pedis, & </s> + <s xml:id="echoid-s800" xml:space="preserve">craſſities laterum fuerit {2/11} lin. </s> + <s xml:id="echoid-s801" xml:space="preserve">poſſe <lb/>eum aquam ſuſtinere ad altitudinem 518. </s> + <s xml:id="echoid-s802" xml:space="preserve">pedum priusquam rumpatur. </s> + <s xml:id="echoid-s803" xml:space="preserve">In <lb/>hoc calculo dedi pedi cubico aquæ pondus 70. </s> + <s xml:id="echoid-s804" xml:space="preserve">librarum. </s> + <s xml:id="echoid-s805" xml:space="preserve">Si vero idem +<pb o="29" file="0043" n="43" rhead="SECTIO SECUNDA."/> +tubus fuerit plumbeus, ſuſtinebit aquam ad altitudinem 18. </s> + <s xml:id="echoid-s806" xml:space="preserve">ped. </s> + <s xml:id="echoid-s807" xml:space="preserve">vi alte-<lb/>rius obſervationis, poteritque altitudinem aquæ ferre 99. </s> + <s xml:id="echoid-s808" xml:space="preserve">ped. </s> + <s xml:id="echoid-s809" xml:space="preserve">ſi latera tubi <lb/>habeant in craſſitie lineam integram. </s> + <s xml:id="echoid-s810" xml:space="preserve">Convenit hoc cum eo quod Mariot-<lb/>tus in tract. </s> + <s xml:id="echoid-s811" xml:space="preserve">de motu aquarum p. </s> + <s xml:id="echoid-s812" xml:space="preserve">472. </s> + <s xml:id="echoid-s813" xml:space="preserve">habet, ubi nempe dicit tubum plum-<lb/>beum, cujus diameter unius erat pedis, & </s> + <s xml:id="echoid-s814" xml:space="preserve">laterum craſſities duarum li-<lb/>nearum cum dimidia ſine ruptura aquam tuliſſe ad altitudinem centum pe-<lb/>dum, quod cum obſervaret abraſit ſenſim latera, donec tandem ad lineæ <lb/>craſſitiem eſſent diminuta, & </s> + <s xml:id="echoid-s815" xml:space="preserve">tum denique vim aquæ tubum disrupiſſe.</s> + <s xml:id="echoid-s816" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s817" xml:space="preserve">Ex obſervata fili ænei firmitate colligitur etiam firmitas tormentorum <lb/>bellicorum: </s> + <s xml:id="echoid-s818" xml:space="preserve">fuerit v. </s> + <s xml:id="echoid-s819" xml:space="preserve">gr. </s> + <s xml:id="echoid-s820" xml:space="preserve">tormentum bellicum cujus animæ diameter habeat <lb/>tres poll. </s> + <s xml:id="echoid-s821" xml:space="preserve">ſolent autem haud procul à lumine accenſorio, ubi maxima eſt <lb/>vis pulveris, craſſities laterum eſſe præterpropter æquales diametro ani-<lb/>mæ, ita ut diameter tota ſit tripla diametri animæ. </s> + <s xml:id="echoid-s822" xml:space="preserve">Quia igitur craſſities <lb/>hæc non eſt negligenda præ diametro animæ, cenſebimus materiam omnem <lb/>concentratam in medio atque ſic ab axe animæ diſtantem tribus pollicibus. <lb/></s> + <s xml:id="echoid-s823" xml:space="preserve">Hoc poſito erit altitudo maxima aquæ quam tormentum haud procul à lu-<lb/>mine accenſorio ferre poteſt = {11/2} x 12 x 3 x 2 x 518 = 205128, quæ vis <lb/>fere ſepties millies ſuperat elaſticitatem aëris naturalis. </s> + <s xml:id="echoid-s824" xml:space="preserve">Oſtendam autem <lb/>in ſequentibus, pulverem pyrium accenſum vim exercere poſſ<unsure/>@ ad rum-<lb/>pendum tormentum aliquantum quidem majorem, quam quæ dicta fuit, <lb/>ſed non multum tamen excedentem. </s> + <s xml:id="echoid-s825" xml:space="preserve">Reliquum autem firmitatis, quod re-<lb/>quirunt tormenta, habent à cingulis ſeu faſciis, quæ dicuntur plattes ban-<lb/>des & </s> + <s xml:id="echoid-s826" xml:space="preserve">moulures, præter id quod in primo ortu tormenti (à l’endroit de <lb/>la culaſſe) craſſities major ſit quam quæ à nobis aſſumta fuit. </s> + <s xml:id="echoid-s827" xml:space="preserve">Interim non <lb/>pauca tormenta diſrumpi, ſic non mirabimur.</s> + <s xml:id="echoid-s828" xml:space="preserve"/> +</p> +<pb o="30" file="0044" n="44" rhead="(o)"/> +</div> +<div xml:id="echoid-div36" type="section" level="1" n="24"> +<head xml:id="echoid-head31" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph> <lb/>SECTIO TERTIA.</head> +<head xml:id="echoid-head32" style="it" xml:space="preserve">De velocitatibus fluidorum ex vaſe utcumque for-<lb/>mato per lumen qualecunque effluentium.</head> +<head xml:id="echoid-head33" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s829" xml:space="preserve">PRiusquam motum aquarum à gravitate propria ortum definire <lb/>tentemus, ruminabimur quod in Sectione prima §. </s> + <s xml:id="echoid-s830" xml:space="preserve">§. </s> + <s xml:id="echoid-s831" xml:space="preserve">18. </s> + <s xml:id="echoid-s832" xml:space="preserve">19. <lb/></s> + <s xml:id="echoid-s833" xml:space="preserve">20. </s> + <s xml:id="echoid-s834" xml:space="preserve">21. </s> + <s xml:id="echoid-s835" xml:space="preserve">& </s> + <s xml:id="echoid-s836" xml:space="preserve">22. </s> + <s xml:id="echoid-s837" xml:space="preserve">à nobis allatum fuit de principiis ad hoc adhi-<lb/>bendis.</s> + <s xml:id="echoid-s838" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s839" xml:space="preserve">Recordabimur nempe aſcenſum potentialem Syſtematis, cujus ſingulæ <lb/>partes velocitate qualicunque moventur, ſignificare altitudinem verticalem, <lb/>ad quam centrum gravitatis illius Syſtematis pervenit, ſi ſingulæ particulæ <lb/>motu ſurſu@ converſo ſua velocitate, quantum poſſunt, aſcendere intelli-<lb/>gantur, & </s> + <s xml:id="echoid-s840" xml:space="preserve">deſcenſum actualem denotare altitudinem verticalem, per quam <lb/>centrum gravitatis deſcendit, poſtquam ſingulæ particulæ in quiete fuerant. <lb/></s> + <s xml:id="echoid-s841" xml:space="preserve">Tum etiam memores erimus neceſſario aſcenſum potentialem æqualem eſſe <lb/>deſcenſui actuali, quando omnis motus in materia ſubſtrata hæret, nihilque <lb/>de eo in materiam inſenſibilem aut aliam ad ſyſtema non pertinentem tran-<lb/>ſit, & </s> + <s xml:id="echoid-s842" xml:space="preserve">denique motum fluidorum talem proxime eſſe, ut ubique veloci-<lb/>tas reciproce ſit proportionalis amplitudini vaſis reſpondenti, quâ de re ſuo <lb/>loco alia quædam interjiciemus. </s> + <s xml:id="echoid-s843" xml:space="preserve">Nunc convenit examinare ſequentem pro-<lb/>poſitionem.</s> + <s xml:id="echoid-s844" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div37" type="section" level="1" n="25"> +<head xml:id="echoid-head34" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s845" xml:space="preserve">§. </s> + <s xml:id="echoid-s846" xml:space="preserve">2. </s> + <s xml:id="echoid-s847" xml:space="preserve">Si aqua per canalem utcunque formatum fluat, ejusque ve-<lb/>locitas cognita ſit aliquo in loco, invenire aſcenſum potentialem omnis aquæ <lb/>in canali contentæ.</s> + <s xml:id="echoid-s848" xml:space="preserve"/> +</p> +<pb o="31" file="0045" n="45" rhead="SECTIO TERTIA."/> +</div> +<div xml:id="echoid-div38" type="section" level="1" n="26"> +<head xml:id="echoid-head35" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s849" xml:space="preserve">Sit canalis utcunque formatus S T (Fig. </s> + <s xml:id="echoid-s850" xml:space="preserve">13. </s> + <s xml:id="echoid-s851" xml:space="preserve">& </s> + <s xml:id="echoid-s852" xml:space="preserve">14.) </s> + <s xml:id="echoid-s853" xml:space="preserve">per quem aqua fluit <lb/>b c f g; </s> + <s xml:id="echoid-s854" xml:space="preserve">aſſumitur, ſi in axe a e accipiatur punctum quodcunque n, per <lb/> +<anchor type="note" xlink:label="note-0045-01a" xlink:href="note-0045-01"/> +quod planum ad axem perpendiculare p m tranſeat, fore, ut omnes parti-<lb/>culæ aqueæ in illo plano exiſtentes æquali velocitate fluant, & </s> + <s xml:id="echoid-s855" xml:space="preserve">quidem ta-<lb/>li, quæ ſit ubique reciproce proportionalis magnitudini ſectionis p m. </s> + <s xml:id="echoid-s856" xml:space="preserve">Sit <lb/>autem velocitas aquæ in g f talis, quæ debetur altitudini verticali q s, id eſt, <lb/>ſit aſcenſus potentialis ſtrati aquei in g f æqualis lineæ q s, & </s> + <s xml:id="echoid-s857" xml:space="preserve">quoniam hujus-<lb/>modi altitudines ſunt in ratione quadrata velocitatum, ſequitur eſſe aſcen-<lb/>ſum potentialem aquæ in p m æqualem quartæ proportionali ad quadratum <lb/>amplitudinis p m, quadratum amplitudinis g f & </s> + <s xml:id="echoid-s858" xml:space="preserve">altitudinem q s, nempe <lb/>= {gf<emph style="super">2</emph>/pm<emph style="super">2</emph>} X qs. </s> + <s xml:id="echoid-s859" xml:space="preserve">His ita præmonitis ponemus in figura decima quarta eſſe <lb/>curvam B P G, ſcalam amplitudinum canalis, ita ut poſita A N = a n, denotet <lb/>N P amplitudinem in p m: </s> + <s xml:id="echoid-s860" xml:space="preserve">dein curvam H I K eſſe ſcalam aſcenſuum poten-<lb/>tialium, ita ut ſit N I = {EG<emph style="super">2</emph>/NP<emph style="super">2</emph>} X qs. </s> + <s xml:id="echoid-s861" xml:space="preserve">fingatur nunc elementa ſingula curvæ <lb/>H I K habere pondus æquale ponderi ſtrati aquei reſpondentis, & </s> + <s xml:id="echoid-s862" xml:space="preserve">cadere <lb/>centrum gravitatis iſtius curvæ in punctum L, & </s> + <s xml:id="echoid-s863" xml:space="preserve">ducatur L O perpendicu-<lb/>laris ad axem A E; </s> + <s xml:id="echoid-s864" xml:space="preserve">ſic erit L O aſcenſus potentialis totius aquæ quæſitus. </s> + <s xml:id="echoid-s865" xml:space="preserve">Ex <lb/>mechanicis autem conſtat, fi fiat tertia curva U X Z, cujus applicata N X <lb/>ſit ubique æqualis {EG<emph style="super">2</emph>/NP}, fore L O æqualem quartæ proportionali ad ſpa-<lb/>tium A E G B & </s> + <s xml:id="echoid-s866" xml:space="preserve">A E Z U atque lineam q s vel E K. </s> + <s xml:id="echoid-s867" xml:space="preserve">Patet igitur quæſitum. <lb/></s> + <s xml:id="echoid-s868" xml:space="preserve">Q. </s> + <s xml:id="echoid-s869" xml:space="preserve">E. </s> + <s xml:id="echoid-s870" xml:space="preserve">I.</s> + <s xml:id="echoid-s871" xml:space="preserve"/> +</p> +<div xml:id="echoid-div38" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0045-01" xlink:href="note-0045-01a" xml:space="preserve">Fig. 13. <lb/>& 14.</note> +</div> +<p> + <s xml:id="echoid-s872" xml:space="preserve">§. </s> + <s xml:id="echoid-s873" xml:space="preserve">3. </s> + <s xml:id="echoid-s874" xml:space="preserve">Fuerit v. </s> + <s xml:id="echoid-s875" xml:space="preserve">gr. </s> + <s xml:id="echoid-s876" xml:space="preserve">canalis conicus, in quo ſuperficies anterior g f <lb/>& </s> + <s xml:id="echoid-s877" xml:space="preserve">poſterior b c diametros habeant ut m ad n, erit aſcenſus potentialis aquæ <lb/>= {3m3/n(mm + mn + nn)} X qs.</s> + <s xml:id="echoid-s878" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div40" type="section" level="1" n="27"> +<head xml:id="echoid-head36" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s879" xml:space="preserve">§. </s> + <s xml:id="echoid-s880" xml:space="preserve">4. </s> + <s xml:id="echoid-s881" xml:space="preserve">Datis variationibus infinite parvis tam ratione ſitus quam ve-<lb/>locitatis, quæ ſuperficiei aquæ anteriori reſpondent, invenire variationes <lb/>ad aſcenſus potentiales totius aquæ pertinentes.</s> + <s xml:id="echoid-s882" xml:space="preserve"/> +</p> +<pb o="32" file="0046" n="46" rhead="HYDRODYNAMICÆ."/> +</div> +<div xml:id="echoid-div41" type="section" level="1" n="28"> +<head xml:id="echoid-head37" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s883" xml:space="preserve">Sit ſpatium A E G B = M, ſpatium A E Z U = N, qs = v, erit <lb/>aſcenſus potent. </s> + <s xml:id="echoid-s884" xml:space="preserve">= {Nv/M}: </s> + <s xml:id="echoid-s885" xml:space="preserve">quia vero quantitas aquæ in canali conſtanter eadem <lb/>ponitur, erit ſpatium A E G B invariabile, adeoque d M = o ita ut diffe-<lb/>rentiale aſcenſus potent. </s> + <s xml:id="echoid-s886" xml:space="preserve">ſit ſimpliciter = {Ndv + vdN/M}, habetur autem d N <lb/>ex variatione ſitus aquæ. </s> + <s xml:id="echoid-s887" xml:space="preserve">Patet igitur propoſitum. </s> + <s xml:id="echoid-s888" xml:space="preserve">Q. </s> + <s xml:id="echoid-s889" xml:space="preserve">E. </s> + <s xml:id="echoid-s890" xml:space="preserve">I.</s> + <s xml:id="echoid-s891" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div42" type="section" level="1" n="29"> +<head xml:id="echoid-head38" xml:space="preserve">Scholion.</head> +<p> + <s xml:id="echoid-s892" xml:space="preserve">§. </s> + <s xml:id="echoid-s893" xml:space="preserve">5. </s> + <s xml:id="echoid-s894" xml:space="preserve">Poterunt hæ propoſitiones inſervire pro motu |fluidi intra vaſa <lb/>moti, id eſt, non effluentis definiendo, uti ſuo loco oſtendam: </s> + <s xml:id="echoid-s895" xml:space="preserve">at ve-<lb/>ro cum fluidum per foramen effluit, aptius inſtituetur aliter calculus, <lb/>nempe ut ſequitur.</s> + <s xml:id="echoid-s896" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div43" type="section" level="1" n="30"> +<head xml:id="echoid-head39" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s897" xml:space="preserve">§. </s> + <s xml:id="echoid-s898" xml:space="preserve">6. </s> + <s xml:id="echoid-s899" xml:space="preserve">Invenire differentiam aſcenſus potentialis poſtquam guttula <lb/>per foramen effluxit.</s> + <s xml:id="echoid-s900" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div44" type="section" level="1" n="31"> +<head xml:id="echoid-head40" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s901" xml:space="preserve">Fingamus aquam effluere ex vaſe aimb (Fig. </s> + <s xml:id="echoid-s902" xml:space="preserve">15.) </s> + <s xml:id="echoid-s903" xml:space="preserve">utcunque for-<lb/>mato, fundum ſit im perforatum foramine pl: </s> + <s xml:id="echoid-s904" xml:space="preserve">quantitas aquæ, poſtquam <lb/> +<anchor type="note" xlink:label="note-0046-01a" xlink:href="note-0046-01"/> +jam data ejus quantitas effluxit, reſidua in vaſe ſit cimd; </s> + <s xml:id="echoid-s905" xml:space="preserve">effluat autem <lb/>tempusculo infinitè parvo guttula pnol, ſuperficie cd deſcendente in ſitum <lb/>ef: </s> + <s xml:id="echoid-s906" xml:space="preserve">concipiatur in medio aquæ ſectio gh parallela ſuperficiebus cd vel ef <lb/>ipſique fundo im; </s> + <s xml:id="echoid-s907" xml:space="preserve">ſitque velocitas unius cujusvis particulæ in gh talis, <lb/>ut poſſit aſcendere ad altitudinem qs ſeu v, cum nondum effluxit guttula <lb/>& </s> + <s xml:id="echoid-s908" xml:space="preserve">ad altitudinem qz ſive <gap/> + dv, poſtquam ea ipſa guttula effluxit. <lb/></s> + <s xml:id="echoid-s909" xml:space="preserve">Omnibus his ita poſitis, quæritur incrementum aſcenſus potentialis aquæ poſt-<lb/>quam ſitum cimd commutavit cum ſitu eipnolmf, id eſt, poſtquam gut-<lb/>tula emanavit.</s> + <s xml:id="echoid-s910" xml:space="preserve"/> +</p> +<div xml:id="echoid-div44" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0046-01" xlink:href="note-0046-01a" xml:space="preserve">Fig. 15.</note> +</div> +<p> + <s xml:id="echoid-s911" xml:space="preserve">Fiat, ut antea, curva C G I (Fig. </s> + <s xml:id="echoid-s912" xml:space="preserve">16.) </s> + <s xml:id="echoid-s913" xml:space="preserve">ceu ſcala amplitudinum, ubi <lb/> +<anchor type="note" xlink:label="note-0046-02a" xlink:href="note-0046-02"/> +adeoque C D vel E F repræſentabunt magnitudinem ſuperficiei aqueæ ante +<pb o="33" file="0047" n="47" rhead="SECTIO TERTIA."/> +vel poſt effluxum guttulæ, G H amplitudinem illam aſſumtam, I L mag-<lb/>nitudinem fundi, P L magnitudinem foraminis, dum adhærens parallelo-<lb/>grammum minimum P N O L reſpondet guttulæ cylindricæ pnol: </s> + <s xml:id="echoid-s914" xml:space="preserve">dein con-<lb/>ſtruatur alia curva T R U, cujus applicatæ ſint rurſus æquales quadrato lineæ <lb/>G H, diviſo per applicatam reſpondentem curvæ C G I, cui curvæ eadem <lb/>conditione annexum eſt parallelogrammulum L O Y X, cujus nempe latus <lb/>L X eſt æquale quadrato lineæ G H diviſo per lineam PL.</s> + <s xml:id="echoid-s915" xml:space="preserve"/> +</p> +<div xml:id="echoid-div45" type="float" level="2" n="2"> +<note position="left" xlink:label="note-0046-02" xlink:href="note-0046-02a" xml:space="preserve">Fig. 16.</note> +</div> +<p> + <s xml:id="echoid-s916" xml:space="preserve">Jam igitur apparet aſcenſum potent. </s> + <s xml:id="echoid-s917" xml:space="preserve">aquæ ante effluxum guttulæ eſſe = <lb/>quartæ proportionali ad ſpatium D C I P L, ſpatium D T U L & </s> + <s xml:id="echoid-s918" xml:space="preserve">altitudi-<lb/>nem qs, eundemque poſt effluxum guttulæ eſſe = quartæ proportionali <lb/>ad ſpat. </s> + <s xml:id="echoid-s919" xml:space="preserve">FEIPNOL, ſpat. </s> + <s xml:id="echoid-s920" xml:space="preserve">FWUXYOL & </s> + <s xml:id="echoid-s921" xml:space="preserve">altit. </s> + <s xml:id="echoid-s922" xml:space="preserve">qz: </s> + <s xml:id="echoid-s923" xml:space="preserve">ſunt autem in utra-<lb/>que analogia termini primi (nempe ſpat. </s> + <s xml:id="echoid-s924" xml:space="preserve">DCIPL & </s> + <s xml:id="echoid-s925" xml:space="preserve">ſpat. </s> + <s xml:id="echoid-s926" xml:space="preserve">FEIPNOL) in-<lb/>ter ſe æquales, igitur ſi quodvis horum ſpatiorum indicetur per M, ſpa-<lb/>tium D T U L per N, ſpat FWUXYOL per N + dN, altitudo qs per <lb/>v&</s> + <s xml:id="echoid-s927" xml:space="preserve">qz per v + dv, erit incrementum aſcenſus potentialis durante guttulæ efflu-<lb/>xu = {Ndv + vdN/M}. </s> + <s xml:id="echoid-s928" xml:space="preserve">Quod ſi nunc ponatur L D = x, F D = - dx, D C <lb/>= y, H G = m, P L = n, erit D T = {mm/y}, L X = {mm/n}, L O = {-ydx/n} <lb/>(quia ſpatium D F E C = ſpatio L O N P), hincque dN = L O Y X -<lb/>D F W T = - {mmydx/nn} + {mmdx/y}, unde nunc incrementum quæſitum <lb/>aſcenſus petentialis eſt = (Ndv - {mmvydx/nn} + {mmvdx/y}): </s> + <s xml:id="echoid-s929" xml:space="preserve">M. </s> + <s xml:id="echoid-s930" xml:space="preserve">Q.</s> + <s xml:id="echoid-s931" xml:space="preserve">E.</s> + <s xml:id="echoid-s932" xml:space="preserve">I.</s> + <s xml:id="echoid-s933" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div47" type="section" level="1" n="32"> +<head xml:id="echoid-head41" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s934" xml:space="preserve">§. </s> + <s xml:id="echoid-s935" xml:space="preserve">7. </s> + <s xml:id="echoid-s936" xml:space="preserve">Retentis iisdem poſitionibus inven@re deſcenſum actualem infi-<lb/>nitè parvum aquæ, dum guttula effluit.</s> + <s xml:id="echoid-s937" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div48" type="section" level="1" n="33"> +<head xml:id="echoid-head42" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s938" xml:space="preserve">Cum in Figura decima quinta aqua ſitum cdmi mutat cum ſitu efml <lb/>onpi, patet in utroque ſitu centrum gravitatis partis aquæ efmi in eodem <lb/>loco eſſe, poſſeque proin concipi ſolam particulam cdfe, (quæ eſt = - ydx <lb/>dum tota aquæ maſſa eſt = M) deſcendiſſe in lonp. </s> + <s xml:id="echoid-s939" xml:space="preserve">Sit jam altitudo par- +<pb o="34" file="0048" n="48" rhead="HYDRODYNAMICÆ."/> +ticulæ aqueæ cdfe ſupra guttulam lonp = x, altitudo centri gravitatis aquæ <lb/>efmi a fundo = b, erit altitudo centri gravitatis omnis aquæ in ſitu cdmi <lb/>ſupra fundum = b - {ydx/M} X (x - b) & </s> + <s xml:id="echoid-s940" xml:space="preserve">in ſitu efmlonpi erit eadem <lb/>altitudo = ({M + ydx/M}) X b; </s> + <s xml:id="echoid-s941" xml:space="preserve">unde differentia altitudinum ſeu deſcenſus actualis <lb/>quæſitus = - {ydx/M} X x, quæ æquatio indicat, guttulam quæ effluxerit <lb/>multiplicandam eſſe per altitudinem aquæ ſupra foramen, productumque <lb/>dividendum per quantitatem aquæ, ut habeatur deſcenſus actualis, qui fit <lb/>dum guttula effluit, Q. </s> + <s xml:id="echoid-s942" xml:space="preserve">E. </s> + <s xml:id="echoid-s943" xml:space="preserve">I.</s> + <s xml:id="echoid-s944" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div49" type="section" level="1" n="34"> +<head xml:id="echoid-head43" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s945" xml:space="preserve">§. </s> + <s xml:id="echoid-s946" xml:space="preserve">8. </s> + <s xml:id="echoid-s947" xml:space="preserve">Determinare motum fluidi homogenei ex vaſe dato per fo-<lb/>ramen datum effluentis.</s> + <s xml:id="echoid-s948" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div50" type="section" level="1" n="35"> +<head xml:id="echoid-head44" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s949" xml:space="preserve">Quoniam per hypotheſin noſtram aſcenſus potentialis ſingulis mo-<lb/>mentis æqualis eſt Deſcenſui actuali, erit incrementum prioris dum guttula <lb/>effluit æquale incremento poſterioris, quod ſimili tempuſculo oritur. </s> + <s xml:id="echoid-s950" xml:space="preserve">Igi-<lb/>tur ſi rurfus ſuperficies aquæ, poſtquam data ejus quantitas effluxit, pona-<lb/>tur = y, amplitudo vaſis quocunque in loco ad libitum aſſumta = m, am-<lb/>plitudo foraminis = n, altitudo aquæ ſupra foramen = x; </s> + <s xml:id="echoid-s951" xml:space="preserve">ſi præterea <lb/>quantitas N ea lege conſtruatur, quæ §. </s> + <s xml:id="echoid-s952" xml:space="preserve">6. </s> + <s xml:id="echoid-s953" xml:space="preserve">indicata fuit, atque per v in-<lb/>telligatur altitudo debita velocitati aquæ in loco aſſumto, ubi nempe am-<lb/>plitudo vaſis eſt = m, erit per §. </s> + <s xml:id="echoid-s954" xml:space="preserve">6. </s> + <s xml:id="echoid-s955" xml:space="preserve">incrementum aſcenſus potentialis = <lb/>(Ndv - {mmvydx/nn} + {mmvdx/y}): </s> + <s xml:id="echoid-s956" xml:space="preserve">M, minimusque deſcenſus actualis = {- yxdx/M} <lb/>(per præced.</s> + <s xml:id="echoid-s957" xml:space="preserve">§.)</s> + <s xml:id="echoid-s958" xml:space="preserve">; </s> + <s xml:id="echoid-s959" xml:space="preserve">unde habetur (Ndv - {mmvydx/nn} + {mmvdx/y}): </s> + <s xml:id="echoid-s960" xml:space="preserve">M = <lb/>- yxdx: </s> + <s xml:id="echoid-s961" xml:space="preserve">MſeuNdv - {mmvydx/nn} + {mmvdx/y} = - yxdx, quæ æquatio ge-<lb/>neraliter integrari poteſt, quandoquidem litteræ N & </s> + <s xml:id="echoid-s962" xml:space="preserve">y ſunt functiones datæ <lb/>ipſius x & </s> + <s xml:id="echoid-s963" xml:space="preserve">litera v unius tantum dimenſionis eſt.</s> + <s xml:id="echoid-s964" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div51" type="section" level="1" n="36"> +<head xml:id="echoid-head45" xml:space="preserve">Corollarium 1.</head> +<p> + <s xml:id="echoid-s965" xml:space="preserve">§. </s> + <s xml:id="echoid-s966" xml:space="preserve">9. </s> + <s xml:id="echoid-s967" xml:space="preserve">Quum velocitates ſint in ratione reciproca amplitudinum, +<pb o="35" file="0049" n="49" rhead="SECTIO TERTIA."/> +patet fore altitudinem, quæ velocitati aquæ effluentis reſpondet = {mm/nn} v, <lb/>quæ proin, ſi vocetur z, erit nnNdz - mmzydx + {mmnnzdx/y} = mmyxdx.</s> + <s xml:id="echoid-s968" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div52" type="section" level="1" n="37"> +<head xml:id="echoid-head46" xml:space="preserve">Corollarium 2.</head> +<p> + <s xml:id="echoid-s969" xml:space="preserve">§. </s> + <s xml:id="echoid-s970" xml:space="preserve">10. </s> + <s xml:id="echoid-s971" xml:space="preserve">Si foramen ſit valde parvum, ratione amplitudinum vaſis, <lb/>fit n = o, totaque æquatio abit in hanc - mmzydx = - mmyxdx vel <lb/>z = x; </s> + <s xml:id="echoid-s972" xml:space="preserve">tunc igitur aqua ea conſtanter effluit velocitate, qua ad altitudinem <lb/>ſupremæ ſuperficiei usque aſcendere poſſit, quem ſolum caſum Geometræ <lb/>hactenus fuerunt recte aſſecuti: </s> + <s xml:id="echoid-s973" xml:space="preserve">valetque hæc propoſitio pro omnibus vaſis <lb/>utcunque formatis: </s> + <s xml:id="echoid-s974" xml:space="preserve">at cum foramen non ut infinite parvum conſideratur, <lb/>nequaquam negligenda eſt vaſis figura. </s> + <s xml:id="echoid-s975" xml:space="preserve">Notari tamen poteſt, quod niſi fo-<lb/>ramen ſit ampliſſimum, ſine notabili admodum errore idem ut infinitè par-<lb/>vum conſiderari poſſit.</s> + <s xml:id="echoid-s976" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div53" type="section" level="1" n="38"> +<head xml:id="echoid-head47" xml:space="preserve">Corollarium 3.</head> +<p> + <s xml:id="echoid-s977" xml:space="preserve">§. </s> + <s xml:id="echoid-s978" xml:space="preserve">11. </s> + <s xml:id="echoid-s979" xml:space="preserve">Cum fluidum non eſt ubique idem, ſimili modo inſtituen-<lb/>dus eſt calculus, inquirendo nimirum tum in incrementum aſcenſus poten-<lb/>tialis fluidi compoſiti, tum in Deſcenſum actualem, eaque inter ſe æquando. <lb/></s> + <s xml:id="echoid-s980" xml:space="preserve">Quod ſi autem foramen ſit valde parvum, per ſe patet, quod etiam calcu-<lb/>lus oſtendit, fore ut fluidum velocitate exiliat altitudini debita tali, ut ſi vas <lb/>ad eandem altitudinem liquore eodem, qui exilit, repletum ſit, eandem <lb/>preſſionem latera foraminis ſuſtineant.</s> + <s xml:id="echoid-s981" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div54" type="section" level="1" n="39"> +<head xml:id="echoid-head48" xml:space="preserve">Scholium Generale.</head> +<p> + <s xml:id="echoid-s982" xml:space="preserve">§. </s> + <s xml:id="echoid-s983" xml:space="preserve">12. </s> + <s xml:id="echoid-s984" xml:space="preserve">Priusquam Corollaria ſpecialiora ex theoria noſtra dedu-<lb/>camus circa motum fluidorum ex vaſis cylindricis, conveniet hic examina-<lb/>re, quousque hypotheſes aſſumtæ cum rei natura conſpirent & </s> + <s xml:id="echoid-s985" xml:space="preserve">quænam aliæ <lb/>intervenire poſſint cauſæ, quarum in computo nullam rationem habuimus, <lb/>motum fluidum diminuentes.</s> + <s xml:id="echoid-s986" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s987" xml:space="preserve">Quod primo attinet ad Principium conſervationis virium vivarum ſeu <lb/>perpetuæ æqualitatis inter aſcenſum potentialem deſcenſumque actualem nihil hîc vi-<lb/>deo, quod ei notabili impedimento eſſe poſſit, ſi modo à frictionibus, te-<lb/>nacitate, aëris reſiſtentia hujuscemodique aliis obſtaculis mentem abſtra- +<pb o="36" file="0050" n="50" rhead="HYDRODYNAMICÆ."/> +hamus. </s> + <s xml:id="echoid-s988" xml:space="preserve">Sæpe quidem fit, ut principium iſtud non ſine limitatione adhibe-<lb/>ri poſſit, quod in ſequentibus oſtendemus, nempe cum particulæ aquæ <lb/>motu ſingulæ diverſo ſeruntur, quo fit ut ſingulis momentis aliquid de mo-<lb/>tu, vel ſi mavis de aſcenſu potentiali, perdatur. </s> + <s xml:id="echoid-s989" xml:space="preserve">Sed in præſenti caſu nihil <lb/>ſimile accidit, quandoquidem omnes particulæ ſimiliter fere moventur & </s> + <s xml:id="echoid-s990" xml:space="preserve"><lb/>præſertim, quando foramen eſt valde parvum, motus particularum inter-<lb/>narum fere nullus eſt, nihilque adeo inde detrimenti venire poteſt. </s> + <s xml:id="echoid-s991" xml:space="preserve">Alterum <lb/>autem principium, quo aſſumitur velocitatem cujuslibet particulæ eam eſſe, <lb/>quæ reſpondet inverſæ rationi amplitudinum, duplici quidem laborat in-<lb/>comniodo, primo nempe, quod motus circa latera vaſis tardior paulo ſit <lb/>quam in medio nec proin omnes particulæ eidem amplitudini vaſis reſpon-<lb/>dentes, æquali velocitate ferantur, & </s> + <s xml:id="echoid-s992" xml:space="preserve">ſecundo, quod aqua à fundo non ad-<lb/>modum remota motum, quem principium hoc poſtulat, habere non poſ-<lb/>ſit: </s> + <s xml:id="echoid-s993" xml:space="preserve">Utrumque autem nullum ſenſibilem errorem poſt ſe trahit, quando in <lb/>hoc problemate ſimplici figura vaſis interna nihil fere ad motum aquæ efflu-<lb/>entis attineat; </s> + <s xml:id="echoid-s994" xml:space="preserve">Ex eadem ratione intelligitur non multum diverſum eſſe poſ-<lb/>ſe motum aquæ ſub alia quacunque directione effluentis, quia ſcilicet mo-<lb/>tus aquæ internus in ima vaſis parte tantum diverſus fit, hæcque diverſitas <lb/>nullius momenti fere eſſe poteſt. </s> + <s xml:id="echoid-s995" xml:space="preserve">Apparet ergo hypotheſes, quibus calcu-<lb/>lus noſtri hujus Problematis innititur, ita convenire cum natura quæſtionis, <lb/>ut error inde nullus ſenſibus perceptibilis oriri poſſit. </s> + <s xml:id="echoid-s996" xml:space="preserve">At vero impedimen-<lb/>ta ſupra memorata, attritus, tenacitas fluidi aliaque ſimilia majoris efficaciæ <lb/>ſunt, præſertim cum foramen, per quod fluida exiliunt, per quam exi-<lb/>guum, aut altitudo aquæ ſupra foramen admodum magna, aut denique <lb/>tubus valde gracilis eſt, qua de re experimenta plurima extant apud Mariot-<lb/>tum in tract. </s> + <s xml:id="echoid-s997" xml:space="preserve">de mot. </s> + <s xml:id="echoid-s998" xml:space="preserve">aquarum. </s> + <s xml:id="echoid-s999" xml:space="preserve">Jam vero progredior ad examinandum mo-<lb/>tum aquarum ex vaſis Cylindricis per foramina cujuscunque magnitudinis <lb/>effluentium. </s> + <s xml:id="echoid-s1000" xml:space="preserve">Vaſa autem compendii & </s> + <s xml:id="echoid-s1001" xml:space="preserve">elegantioris ſolutionis cauſa conſi-<lb/>derabimus verticaliter poſita.</s> + <s xml:id="echoid-s1002" xml:space="preserve"/> +</p> +<pb o="37" file="0051" n="51" rhead="SECTIO TERTIA."/> +</div> +<div xml:id="echoid-div55" type="section" level="1" n="40"> +<head xml:id="echoid-head49" style="it" xml:space="preserve">De his quæ pertinent ad effluxum aquarum ex Cy-<lb/>lindris verticaliter poſitis, per Lumen quod-<lb/>cunque, quod eſt in fundo horizontali.</head> +<head xml:id="echoid-head50" xml:space="preserve">§. 13.</head> +<p> + <s xml:id="echoid-s1003" xml:space="preserve">GEometræ, quibus de aquis ex vaſe erumpentibus ſermo fuit, con-<lb/>ſiderare potiſſimum ſolent cylindros verticaliter poſitos: </s> + <s xml:id="echoid-s1004" xml:space="preserve">Igitur haud <lb/>abs re erit ex theoria noſtra generali conſectaria illa, quæ huc per-<lb/>tinent, deducere. </s> + <s xml:id="echoid-s1005" xml:space="preserve">Sit amplitudo cylindri ad amplitudinem foraminis ut m <lb/>ad n; </s> + <s xml:id="echoid-s1006" xml:space="preserve">altitudo aquæ ſupra foramen, cum fluxus incipit = a; </s> + <s xml:id="echoid-s1007" xml:space="preserve">altitudo aquæ <lb/>reſiduæ = x, altitudo velocitati aquæ internæ debita = v; </s> + <s xml:id="echoid-s1008" xml:space="preserve">erit in æquatio-<lb/>ne canonica paragraphi octavi y = m, N = mx (per §. </s> + <s xml:id="echoid-s1009" xml:space="preserve">6.) </s> + <s xml:id="echoid-s1010" xml:space="preserve">quæ adeoque <lb/>abit in hanc æquationem. <lb/></s> + <s xml:id="echoid-s1011" xml:space="preserve">mxdv - {m<emph style="super">3</emph>/nn}vdx + mvdx = - mxdx, vel <lb/>(1 - {mm/nn})vdx + xdv = - xdx <lb/>multiplicetur hæc poſterior æquatio per x<emph style="super">{- mm/nn}</emph>, ut habeatur <lb/>(1 - {mm/nn})x<emph style="super">- {mm/nn}</emph> vdx + x<emph style="super">1 - {mm/nn}</emph>dv = - x<emph style="super">1 - {mm/nn}</emph>dx.</s> + <s xml:id="echoid-s1012" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1013" xml:space="preserve">Poteſt jam hæc æquatio integrari: </s> + <s xml:id="echoid-s1014" xml:space="preserve">obſervanda autem eſt in Integratio-<lb/>ne conſtantis additio, talis nempe, ut a fluxus initio, id eſt, cum x = a, <lb/>ſit velocitas fluidi nulla, ipſaque proin v pariter = o: </s> + <s xml:id="echoid-s1015" xml:space="preserve">ita vero oritur: <lb/></s> + <s xml:id="echoid-s1016" xml:space="preserve">x<emph style="super">1 - {mm/nn}</emph> v = {nn/2nn - mm}(a<emph style="super">2 - {mm/nn}</emph> - x<emph style="super">2 - {mm/nn}</emph>) vel <lb/>v = {nna/2nn - mm}(({a/x})<emph style="super">1 - {mm/nn}</emph> - {x/a})</s> +</p> +<p> + <s xml:id="echoid-s1017" xml:space="preserve">§. </s> + <s xml:id="echoid-s1018" xml:space="preserve">14. </s> + <s xml:id="echoid-s1019" xml:space="preserve">Ex hâc igitur æquatione cognoſcitur altitudo generans velocita-<lb/>tem aquæ internæ; </s> + <s xml:id="echoid-s1020" xml:space="preserve">ubi notari meretur, ſi vas ſit ampliſſimum, mox poſſe <lb/>cenſeri v = {nn/mm}x, poſtquam ſcilicet vel tantillum deſcendit aqua, id eſt, +<pb o="38" file="0052" n="52" rhead="HYDRODYNAMICÆ."/> +ſtatim ac x paulo minor eſt quam a. </s> + <s xml:id="echoid-s1021" xml:space="preserve">Regula hæc fallit notabiliter tantum cir-<lb/>ca primum motus initium & </s> + <s xml:id="echoid-s1022" xml:space="preserve">ſi primum iſtud motus elementum conſidera-<lb/>tur (quo nempe altitudo a - x ut infinite parva cenſeri poteſt) indicat æ-<lb/>quatio, eſſe tunc v = a - x. </s> + <s xml:id="echoid-s1023" xml:space="preserve">Unde ſequitur, in omni cylindro, quodcun-<lb/>que fuerit foramen, aquam internam inſtar corporum libere cadentium ac-<lb/>celerari ab initio motus. </s> + <s xml:id="echoid-s1024" xml:space="preserve">Si vero motus aliquantulum continuet, eo minus <lb/>fallet hæc Regula, quo majus fuerit foramen, & </s> + <s xml:id="echoid-s1025" xml:space="preserve">quo altior eſt aqua in tubo; </s> + <s xml:id="echoid-s1026" xml:space="preserve">ſi <lb/>porro deſideretur altitudo ea, quæ velocitati aquæ effluentis reſpondeat, <lb/>quam §. </s> + <s xml:id="echoid-s1027" xml:space="preserve">9. </s> + <s xml:id="echoid-s1028" xml:space="preserve">poſuimus = z, erit z = {mm/nn}v, ſeu <lb/>z = {mma/2nn - mm} (({a/x})<emph style="super">1 - {mm/nn}</emph> - {x/a})</s> +</p> +<p> + <s xml:id="echoid-s1029" xml:space="preserve">§. </s> + <s xml:id="echoid-s1030" xml:space="preserve">15. </s> + <s xml:id="echoid-s1031" xml:space="preserve">Cum n eſt = m, id eſt, cum nullum eſt fundum, apparet <lb/>ex ipſa rei natura, aquam inſtar corporum gravium libere cadere atque ac-<lb/>celerari, id ipſum autem indicat etiam æquatio; </s> + <s xml:id="echoid-s1032" xml:space="preserve">fit enim in hâc poſitione <lb/>z = a - x. </s> + <s xml:id="echoid-s1033" xml:space="preserve">Si vero foramen eſt veluti infinite parvum ratione amplitudinis <lb/>vaſis, quem caſum jam ſupra conſideravimus, ponendum eſt n = o, & </s> + <s xml:id="echoid-s1034" xml:space="preserve">tunc <lb/>fit z = x, quod indicat, aquam ea conſtantur effluere velocitate, qua ad <lb/>totam aquæ altitudinem aſcendere poſſit. </s> + <s xml:id="echoid-s1035" xml:space="preserve">Denique cum mm = 2nn, pro-<lb/>dit z = {mm/o} (x - x), ex quo valore cum nihil cognoſci poſſit, deſcenden-<lb/>dum eſt ad æquationem differentialem §. </s> + <s xml:id="echoid-s1036" xml:space="preserve">13. </s> + <s xml:id="echoid-s1037" xml:space="preserve">quæ nunc hæc eſt: <lb/></s> + <s xml:id="echoid-s1038" xml:space="preserve">- vdx + xdv = - xdx, vel {xdv - vdx/xx} = {- dx/x}, <lb/>quæ integrata cum debitæ conſtantis additione dat {v/x} = log. </s> + <s xml:id="echoid-s1039" xml:space="preserve">{a/x}, vel v = <lb/>xlog.</s> + <s xml:id="echoid-s1040" xml:space="preserve">{a/x}, aut z = 2v = 2xlog.</s> + <s xml:id="echoid-s1041" xml:space="preserve">{a/x}.</s> + <s xml:id="echoid-s1042" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1043" xml:space="preserve">§. </s> + <s xml:id="echoid-s1044" xml:space="preserve">16. </s> + <s xml:id="echoid-s1045" xml:space="preserve">Velocitas aquæ effluentis ab initio creſcit poſteaque decreſcit, <lb/>eſtque alicubi maxima, nempe eo in loco, quo aqua deſcendit ad altitudinem <lb/>a:</s> + <s xml:id="echoid-s1046" xml:space="preserve">({mm - nn/nn})<emph style="super">nn: (mm - 2nn)</emph>; </s> + <s xml:id="echoid-s1047" xml:space="preserve">id quoque experientia edoctus indicavit Ma-<lb/>riottus in tract. </s> + <s xml:id="echoid-s1048" xml:space="preserve">de motu aquarum part. </s> + <s xml:id="echoid-s1049" xml:space="preserve">3. </s> + <s xml:id="echoid-s1050" xml:space="preserve">diſc. </s> + <s xml:id="echoid-s1051" xml:space="preserve">3. </s> + <s xml:id="echoid-s1052" xml:space="preserve">exp. </s> + <s xml:id="echoid-s1053" xml:space="preserve">5, ipſaque velocitas ma-<lb/>xima talis eſt, quæ debetur altitudini +<pb o="39" file="0053" n="53" rhead="SECTIO TERTIA."/> +{mma/mm - 2nn} X [({nn/mm - nn})<emph style="super">nn: (mm - 2nn)</emph> - ({nn/mm - nn})<emph style="super">(mm - nn): (mm - 2nn)</emph>] <lb/>quæ quantitas reducta fit = <lb/>{mma/mm - nn}({nn/mm - nn})<emph style="super">nn: (mm - 2nn)</emph></s> +</p> +<p> + <s xml:id="echoid-s1054" xml:space="preserve">Intelligitur ex iſtis formulis tempus, quo velocitas à nihilo in maxi-<lb/>mam vertitur, plane imperceptibile eſſe, quando foramen vel mediocriter <lb/>parvum tubusque non admodum longus eſt: </s> + <s xml:id="echoid-s1055" xml:space="preserve">notabile autem fieri, cum res <lb/>ſecus ſe habet, quod videmus in fontibus ſalientibus, ad quos aquæ per <lb/>longos tractus vehuntur; </s> + <s xml:id="echoid-s1056" xml:space="preserve">hæc vero quæ ad tempora pertinent, magis in <lb/>ſequenti ſectione explicabuntur, atque ſimul oſtendetur, quam parum aquæ <lb/>ex vaſis ampliſſimis ejiciatur, priusquam maxima velocitate effluant.</s> + <s xml:id="echoid-s1057" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1058" xml:space="preserve">Natura velocitatum melius intelligitur ex appoſita Figura decima ſepti-<lb/> +<anchor type="note" xlink:label="note-0053-01a" xlink:href="note-0053-01"/> +ma, in quâ ſi A B repræſentet totam altitudinem fluidi ſupra foramen ab initio <lb/>fluxus, expriment curvæ A 1 C B, A 2 C B, A 3 C B, A 4 C B, ſcalas altitudi-<lb/>num reſpondentium, ad quas fluidum effluens ſua velocitate aſcendere poſſit in <lb/>diverſis foraminum magnitudinibus: </s> + <s xml:id="echoid-s1059" xml:space="preserve">nempe ſcala accedet ad figuram A 1 C B, ſi <lb/>foramen habeat exiguam rationem ad vaſis amplitudinem & </s> + <s xml:id="echoid-s1060" xml:space="preserve">ad figuram A 2 C B, <lb/>cum aſſumitur fundum majori lumine perforatum; </s> + <s xml:id="echoid-s1061" xml:space="preserve">& </s> + <s xml:id="echoid-s1062" xml:space="preserve">ſi jam ratio foraminis <lb/>ſit ad amplitudinem vaſis ut 1 ad √ 2, erit ſcala illa ut A 3 C B (quo in caſu <lb/>minor fit maxima velocitas quam in quocunque alio, eſtque nominatim ea <lb/>quæ debetur altitudini {2a/c}, intelligendo per c numerum cujus Logarithmus <lb/>eſt unitas, id eſt, altitudini paulo minori quam {3/4}a) ac denique erit ſcala ut <lb/>A 4 C B cum fere nihil fundi ſupereſt.</s> + <s xml:id="echoid-s1063" xml:space="preserve"/> +</p> +<div xml:id="echoid-div55" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0053-01" xlink:href="note-0053-01a" xml:space="preserve">Fig. 17.</note> +</div> +<p> + <s xml:id="echoid-s1064" xml:space="preserve">§. </s> + <s xml:id="echoid-s1065" xml:space="preserve">17. </s> + <s xml:id="echoid-s1066" xml:space="preserve">Jam vero exemplo quodam illuſtrabimus, quod ſupra §. </s> + <s xml:id="echoid-s1067" xml:space="preserve">10. <lb/></s> + <s xml:id="echoid-s1068" xml:space="preserve">indicatum fuit, nempe niſi foramen ſit ampliſſimum, poſſe id ſine valde <lb/>ſenſibili errore in calculo conſiderari ut infinitè parvum, atque adeo aſſumi <lb/>z = x, ut §. </s> + <s xml:id="echoid-s1069" xml:space="preserve">§. </s> + <s xml:id="echoid-s1070" xml:space="preserve">10. </s> + <s xml:id="echoid-s1071" xml:space="preserve">& </s> + <s xml:id="echoid-s1072" xml:space="preserve">15. </s> + <s xml:id="echoid-s1073" xml:space="preserve">dictum fuit. </s> + <s xml:id="echoid-s1074" xml:space="preserve">Videtur id tantum apud nonnullos <lb/>Auctores valuiſſe, ut cenſuerint, nullam magnitudinis in foramine rationem <lb/>unquam eſſe habendam, quantumvis magnum ponatur foramen, quæ res <lb/>certe ridicula eſt: </s> + <s xml:id="echoid-s1075" xml:space="preserve">faltem nemo hactenus quod ſciam magnitudinem forami-<lb/>nis pro hoc negotio recte conſideravit. </s> + <s xml:id="echoid-s1076" xml:space="preserve">Ponamus igitur cylindrum, cujus <lb/>diameter quadrupla tantum ſit diametri foraminis, cujusmodi magna fora- +<pb o="40" file="0054" n="54" rhead="HYDRODYNAMICÆ."/> +mina in inſtrumentis hydraulicis raro occurrere ſolent, & </s> + <s xml:id="echoid-s1077" xml:space="preserve">fingamus ſuperfi-<lb/>ciem aquæ per centeſimam partem deſcendiſſe tantum totius altitudinis ini-<lb/>tialis (deſcendiſſe autem aliquantulum aſſumo, quia à primo initio motus <lb/>aquæ nullus ineſſe poteſt, nedum tantus, ut aqua effluens ad totam alti-<lb/>tudinem aſcendere motu ſuo poſſit) hæ poſitiones faciunt m = 16n & </s> + <s xml:id="echoid-s1078" xml:space="preserve">mm = <lb/>256nn, atque x = {99/100}a, unde prodit <lb/>z = {128/127}({99/100} - ({99/100})<emph style="super">255</emph>)a = {92/100}a, <lb/>quæ quidem aliquantulum differt à quantitate x, ſeu {99/100}a, ſed tamen non <lb/>multum admodum, fitque differentia multo minor, cum minus eſt foramen, <lb/>& </s> + <s xml:id="echoid-s1079" xml:space="preserve">paullo magis deſcendit ſuperficies aquæ. </s> + <s xml:id="echoid-s1080" xml:space="preserve">Igitur differt hæc Theoria à vul-<lb/>gari potiſſimum circa fluxus initium, quo minor eſt motus, quam ſtatutum <lb/>fuit: </s> + <s xml:id="echoid-s1081" xml:space="preserve">è contrario circa fluxus finem majori velocitate aqua ejicitur, quam ſe-<lb/>cundum principia ſolita deberet.</s> + <s xml:id="echoid-s1082" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1083" xml:space="preserve">§. </s> + <s xml:id="echoid-s1084" xml:space="preserve">18. </s> + <s xml:id="echoid-s1085" xml:space="preserve">Hactenus conſideravimus motum aquæ à propria ſua gravitate <lb/>ortum; </s> + <s xml:id="echoid-s1086" xml:space="preserve">ponamus nunc vi aliena aquam ejectam fuiſſe præter vim gravitatis, <lb/>talemque aquæ effluenti communicatam fuiſſe velocitatem, qua ad altitudi-<lb/>nem multo majorem aſcendere poſſit, quam ſi ſola aquæ gravitas motum <lb/>produxiſſet; </s> + <s xml:id="echoid-s1087" xml:space="preserve">dein ſubito vim illam alienam evaneſcere, & </s> + <s xml:id="echoid-s1088" xml:space="preserve">aquam ſibi relin-<lb/>qui; </s> + <s xml:id="echoid-s1089" xml:space="preserve">Id autem ſi fiat, experientia docet citiſſime aquæ velocitatem decreſce-<lb/>re & </s> + <s xml:id="echoid-s1090" xml:space="preserve">mox talem eſſe, ut notabililer non ſuperet velocitatem eam, quæ ex <lb/>ſola aquæ gravitate oritura fuiſſet. </s> + <s xml:id="echoid-s1091" xml:space="preserve">Ita videmus fieri aliquando in fontibus <lb/>ſalientibus (de cujus rei cauſa vera atque menſura alibi dicam) ut aquæ ad <lb/>triplam vel quadruplam majoremve altitudinem aſſiliat, quam eſt ordinaria; <lb/></s> + <s xml:id="echoid-s1092" xml:space="preserve">quod cum ita contingit, ſaltus iſte protinus ceſſat ſolitamque altitudinem, <lb/>quantum id ſenſibus percipi poteſt, non excedit: </s> + <s xml:id="echoid-s1093" xml:space="preserve">loquor autem de tubis <lb/>foraminibus non valde magnis perforatis; </s> + <s xml:id="echoid-s1094" xml:space="preserve">nam cum foramen eſt ali-<lb/>quanto majus, non ita cito decreſcit aquæ ſaltus. </s> + <s xml:id="echoid-s1095" xml:space="preserve">Jam itaque examinabi-<lb/>mus, quousque theoria cum iſtis phænomenis conveniat, accuratasque <lb/>menſuras eorum, quales inde ſequuntur, ſubjungemus. </s> + <s xml:id="echoid-s1096" xml:space="preserve">Ut vero rem ge-<lb/>neraliter proſequamur, ponemus rurſus amplitudinem cylindri ad amplitu-<lb/>dinem foraminis ut m ad n: </s> + <s xml:id="echoid-s1097" xml:space="preserve">aquam ea explodi velocitate qua aſſurgere poſſit <lb/>ad altitudinem a, eoque ipſo temporis puncto altitudinem aquæ ſupra foramen +<pb o="41" file="0055" n="55" rhead="SECTIO TERTIA."/> +eſſe = a, cujus ſola gravitas nunc aquam expellat; </s> + <s xml:id="echoid-s1098" xml:space="preserve">deinde deſcendere ſuper-<lb/>ficiem aquæ in Cylindro per altitudinem verticalem a - x, ita ut altitudo <lb/>reſidua ſit = x & </s> + <s xml:id="echoid-s1099" xml:space="preserve">tunc velocitatem aquæ ejectæ talem eſſe, quæ debeatur al-<lb/>titudini z. </s> + <s xml:id="echoid-s1100" xml:space="preserve">His ita poſitis utemur æquatione generali differentiali §. </s> + <s xml:id="echoid-s1101" xml:space="preserve">9. </s> + <s xml:id="echoid-s1102" xml:space="preserve">quæ <lb/>hæc eſt nn N dz - mmzydx + {mmnnzdx/y} = -mmyxdx (ubi rurſus, ut <lb/>§. </s> + <s xml:id="echoid-s1103" xml:space="preserve">13. </s> + <s xml:id="echoid-s1104" xml:space="preserve">indicatum fuit, eſt y = m & </s> + <s xml:id="echoid-s1105" xml:space="preserve">N = mx) quæque in caſu noſtro particu-<lb/>lari talis fit <lb/>(1 - {mm/nn}) zdx + xdz = - {mm/nn}xdx, <lb/>quæ multiplicata x - {mm/nn} poſteaque ſic integrata, ut poſita x = a, fiat z = α <lb/>dabit æquationem deſideratam finalem <lb/>z = ({mm/2nn - mm + {α/a}) a<emph style="super">{2nn - mm/nn}</emph> X x<emph style="super">{mm - nn/nn}</emph> - {mm/2nn - mm}x <lb/>vel z = {mma/2nn - mm}(({a/x})<emph style="super">1 - {mm/nn}</emph> - {x/a}) + ({x/a})<emph style="super">{mm - nn/nn}</emph>α <lb/>quæ altitudo ſi comparetur cum illa, quæ paragrapho 14. </s> + <s xml:id="echoid-s1106" xml:space="preserve">indicata fuit, in-<lb/>venitur exceſſus unius ſuper alteram = ({x/a})<emph style="super">{mm - nn/nn}</emph>α unde jam omnia ea <lb/>confirmantur Phænomena, quæ modo indicata fuerunt; </s> + <s xml:id="echoid-s1107" xml:space="preserve">exceſſus enim iſte, <lb/>cum m numerus eſt multo major quam n, inſenſibilis ſtatim fit, poſtquam <lb/>aqua vel tantillum deſcendit, id eſt, poſt breviſſimum temporis ſpatium, <lb/>nunquam tamen omnis evaneſcit, quam diu durat fluxus, & </s> + <s xml:id="echoid-s1108" xml:space="preserve">denique eo <lb/>notabilior continue eſt, quo magis ratio numeri m ad n ad æqualitatem ac-<lb/>cedit. </s> + <s xml:id="echoid-s1109" xml:space="preserve">Fuerit v. </s> + <s xml:id="echoid-s1110" xml:space="preserve">gr. </s> + <s xml:id="echoid-s1111" xml:space="preserve">diameter tubi decies major diametro foraminis, expel-<lb/>laturque aqua vi tali, ut velocitate ſua aſſilire poſſit ad altitudinem quæ ſit <lb/>quadrupla altitudinis a ſeu aquæ ſupra foramen, quæritur ad quam altitudinem <lb/>ſua velocitate aqua effluens aſcendere poterit, poſtquam per milleſimam <lb/>partem ipſius a ſuperficies aquea deſcenderit in tubo, ſi interea aqua ſola <lb/>propria gravitate ad effluxum ſolicitetur, dein quænam ſimilis altitudo futu-<lb/>ra fuiſſet, ſi aqua nullum motum ab initio habuiſſet: </s> + <s xml:id="echoid-s1112" xml:space="preserve">eſt autem m = 100n, <lb/>mm = 10000nn, x = {999/1000}a, α = 4a, unde in priori caſu fit +<pb o="42" file="0056" n="56" rhead="HYDRODYNAMICÆ."/> +z = [{10000/9998} ({999/1000} - ({999/1000})<emph style="super">9999</emph>) + 4({999/1000})<emph style="super">9999</emph>] a, <lb/>ſive z = {99915/100000}a + {18/100000}a, in poſteriori caſu autem fit z = {99915/100000}a, <lb/>ex quo exemplo patet, quam exiguus & </s> + <s xml:id="echoid-s1113" xml:space="preserve">plane inſenſibilis ſit exceſſus prio-<lb/>ris altitudinis ſupra alteram, & </s> + <s xml:id="echoid-s1114" xml:space="preserve">quam cito diminuatur jactus ille aqueus, <lb/>quandoquidem tota mutatio fiat, dum ſuperficies aquæ per milleſimem par-<lb/>tem altitudinis a deſcendit, quod tempus in machinis hydraulicis ſolitis non <lb/>poteſt non eſſe admodum breve. </s> + <s xml:id="echoid-s1115" xml:space="preserve">Tum etiam confirmatur, quod ſupra Pa-<lb/>ragrapho 17. </s> + <s xml:id="echoid-s1116" xml:space="preserve">dictum fuit, eſſe ſcilicet proxime z = x, quando foramen eſt <lb/>vel mediocriter parvum, cum in præſenti caſu, ubi motus à quiete incipit, <lb/>differentia inter z & </s> + <s xml:id="echoid-s1117" xml:space="preserve">x ſit tantum quindecim centies milleſimarum partium <lb/>ipſius altitudinis a; </s> + <s xml:id="echoid-s1118" xml:space="preserve">quoniam interim paululum major eſt altitudo z quamx, <lb/>patet ad majorem altitudinem aſcendere poſſe aquam effluentem, poſtquam <lb/>aliquantiſper effluxit aqua, quam eſt altitudo aquæ ſupra foramen.</s> + <s xml:id="echoid-s1119" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1120" xml:space="preserve">§. </s> + <s xml:id="echoid-s1121" xml:space="preserve">19. </s> + <s xml:id="echoid-s1122" xml:space="preserve">Poſtquam ſic ex Theoria noſtra generali deduximus, quæ mo-<lb/>tum fluidorum ex cylindris verticaliter poſitis ſpectant, jam etiam conſi-<lb/>derabimus tubos oblique poſitos, qui prælongi eſſe ſolent in fontibus ſali-<lb/>entibus. </s> + <s xml:id="echoid-s1123" xml:space="preserve">In his enim id ſingulare eſt, quod acceleratio motus non ita repen-<lb/>te fiat, veluti cum Cylindri ſunt verticales atque ſic liceat ſenſibus percipe-<lb/>re conſenſum Theoriæ, cum motu aquarum reali.</s> + <s xml:id="echoid-s1124" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1125" xml:space="preserve">§. </s> + <s xml:id="echoid-s1126" xml:space="preserve">20. </s> + <s xml:id="echoid-s1127" xml:space="preserve">Fingamus canalem utcunque incurvum, ſed tamen Cylindri-<lb/>cum, cujus amplitudo habeatrurſus ad amplitudinem foraminis rationem m ad n-<lb/>Incipiat motus à quiete, ſitque altitudo verticalis aquæ ſupra foramen ab initio <lb/>motus = a; </s> + <s xml:id="echoid-s1128" xml:space="preserve">Effluxerit certa aquæ quantitas, ponaturque altitudo verticalis aquæ <lb/>reſiduæ ſupra foramen = x, longitudo canalis, quæ eo ipſo momento plena eſt <lb/>= ξ, habeatque tunc aqua interna (cujus ſingulas particulas motu axi canalis pa-<lb/>rallelo feri hîc aſſumo) velocitatem, quæ reſpondeat altitudini v; </s> + <s xml:id="echoid-s1129" xml:space="preserve">His ita poſitis, <lb/>ſi ſimili ratiocinio utamur quo ſupra, quærendo nimirum incrementum aſcenſus <lb/>potentialis dum guttula effluit, uti paragrapho 6. </s> + <s xml:id="echoid-s1130" xml:space="preserve">fecimus, idemque ponen-<lb/>do = deſcenſui actuali, obtinetur nunc talis æquatio <lb/>ξdv - {mm/nn} vdξ + vdξ = - xdξ, ſive +<pb o="43" file="0057" n="57" rhead="SECTIO TERTIA."/> +(1 - {mm/nn})vdξ + ξdv = - xdξ <lb/>cujus integralis, quod patet multiplicatis terminis per ξ - {mm/nn} hæc eſt <lb/>v = ξ<emph style="super">{mm/nn} - 1</emph> ſ - xξ<emph style="super">- {mm/nn}</emph> dξ. <lb/></s> + <s xml:id="echoid-s1131" xml:space="preserve">Fuerit v. </s> + <s xml:id="echoid-s1132" xml:space="preserve">gr. </s> + <s xml:id="echoid-s1133" xml:space="preserve">canalis rectus & </s> + <s xml:id="echoid-s1134" xml:space="preserve">ita inclinatus verſus horizontem, ut ſinus anguli <lb/>intercepti inter utrumque ſit ad ſinum totum ut 1 ad g, erit ξ = gx; </s> + <s xml:id="echoid-s1135" xml:space="preserve">unde <lb/>v = {nna/2nn - mm} (({a/x})<emph style="super">{nn - mm/nn}</emph> - {x/a}) <lb/>quæ æquatio cum non differat ab æquatione §. </s> + <s xml:id="echoid-s1136" xml:space="preserve">13. </s> + <s xml:id="echoid-s1137" xml:space="preserve">pro Cylindris verticalibus <lb/>data, ſequitur in utroque caſu velocitates aquæ easdem eſſe, poſtquam deſ-<lb/>cenſus verticales ſuperficiei aquæ iidem ſunt: </s> + <s xml:id="echoid-s1138" xml:space="preserve">Igitur accelerationes in locis <lb/>homologis utrobique ſimiles ſunt ratione altitudinum verticalium, & </s> + <s xml:id="echoid-s1139" xml:space="preserve">hoc tan-<lb/>tum diſcriminis intercedit, quod in canali inclinato lentius fiant, idque in <lb/>ratione ut 1 ad g: </s> + <s xml:id="echoid-s1140" xml:space="preserve">facile igitur ſenſibus percipi poterunt hæ accelerationes in <lb/>canalibus valde inclinatis, quæ in verticalibus ob nimiam mutationum celeri-<lb/>tatem non poſſunt. </s> + <s xml:id="echoid-s1141" xml:space="preserve">Cœterum patet per ſe ex eo, quod frictiones à longitu-<lb/>dine tubi augeantur, non poſſe non velocitates inde diminui, ad quod ani-<lb/>mum advertent ii, quibus experimenta hâc de re inſtituere animus erit.</s> + <s xml:id="echoid-s1142" xml:space="preserve"/> +</p> +<pb o="44" file="0058" n="58" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div57" type="section" level="1" n="41"> +<head xml:id="echoid-head51" style="it" xml:space="preserve">De Effluxu Aquarum ex Cylindris verticaliter po-<lb/>ſitis, qui in alios tubos ſtrictiores pariter <lb/>verticales deſinunt.</head> +<head xml:id="echoid-head52" xml:space="preserve">§. 21.</head> +<p> + <s xml:id="echoid-s1143" xml:space="preserve">COnſtat experientia, inter duos Cylindros omnino æquales ſimiliterque <lb/>poſitos, quorum alterius foramini tubus ſtrictior reſpondeat, hunc <lb/>citius depleri, qui tubum appenſum habet, & </s> + <s xml:id="echoid-s1144" xml:space="preserve">quidem eo citius, quo <lb/>magis tubus à loco inſertionis verſus extremitatem amplitudine creſcit, quæ <lb/>pluribus expoſuit D. </s> + <s xml:id="echoid-s1145" xml:space="preserve">s’Graveſande in Phyſ. </s> + <s xml:id="echoid-s1146" xml:space="preserve">Elem. </s> + <s xml:id="echoid-s1147" xml:space="preserve">Math. </s> + <s xml:id="echoid-s1148" xml:space="preserve">lib. </s> + <s xml:id="echoid-s1149" xml:space="preserve">2. </s> + <s xml:id="echoid-s1150" xml:space="preserve">cap. </s> + <s xml:id="echoid-s1151" xml:space="preserve">8. </s> + <s xml:id="echoid-s1152" xml:space="preserve">Totam rem <lb/>ſequenti Problemate comprehendemus.</s> + <s xml:id="echoid-s1153" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div58" type="section" level="1" n="42"> +<head xml:id="echoid-head53" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s1154" xml:space="preserve">§. </s> + <s xml:id="echoid-s1155" xml:space="preserve">22. </s> + <s xml:id="echoid-s1156" xml:space="preserve">Fuerit vas cylindricum A E H B (Fig. </s> + <s xml:id="echoid-s1157" xml:space="preserve">18.) </s> + <s xml:id="echoid-s1158" xml:space="preserve">verticaliter poſi-<lb/> +<anchor type="note" xlink:label="note-0058-01a" xlink:href="note-0058-01"/> +tum perforatum in F G, quo lumine communicet cum tubo conico F M N G, <lb/>per cujus demum orificium M N aquæ effiuant. </s> + <s xml:id="echoid-s1159" xml:space="preserve">Quæritur velocitas ſuperfi-<lb/>ciei aqueæ C D, poſtquam à quiete deſcendit per A C vel B D.</s> + <s xml:id="echoid-s1160" xml:space="preserve"/> +</p> +<div xml:id="echoid-div58" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0058-01" xlink:href="note-0058-01a" xml:space="preserve">Fig 18.</note> +</div> +</div> +<div xml:id="echoid-div60" type="section" level="1" n="43"> +<head xml:id="echoid-head54" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s1161" xml:space="preserve">Sit altitudo aquæ ſupra M N initialis, nempe N G + H B = a, altitu-<lb/>do ſuperficiei aqueæ in ſitu C D ſupra M N, id eſt, N G + H D = x; </s> + <s xml:id="echoid-s1162" xml:space="preserve">lon-<lb/>gitudo tubi annexi ſeu N G = b; </s> + <s xml:id="echoid-s1163" xml:space="preserve">amplitudo orificii M N = n; </s> + <s xml:id="echoid-s1164" xml:space="preserve">amplitudo <lb/>orificii F G = g, amplitudo Cylindri ſuperioris = m; </s> + <s xml:id="echoid-s1165" xml:space="preserve">ſit velocitas ſuperficiei <lb/>aqueæ in C D talis quæ debeatur altitudini v, erit in æquatione generali §. </s> + <s xml:id="echoid-s1166" xml:space="preserve">8. <lb/></s> + <s xml:id="echoid-s1167" xml:space="preserve">y = m & </s> + <s xml:id="echoid-s1168" xml:space="preserve">N = m (x - b) + {bmm/√gn}, quæ ſubſtitutiones inſtituto calculo con-<lb/>formes eſſe patebunt cum §. </s> + <s xml:id="echoid-s1169" xml:space="preserve">6. </s> + <s xml:id="echoid-s1170" xml:space="preserve">reliquæ autem poſitiones eædem ſunt quæ an-<lb/>te. </s> + <s xml:id="echoid-s1171" xml:space="preserve">Abit igitur æquatio paragraphi 8 in hanc <lb/>m(x - b)dv + {bmm/√gn}dv - {m<emph style="super">3</emph>vdx/nn} + mvdx = - mxdx <lb/>quæ porro diviſa per m factoque x - b + {mb/√gn} = z, dat +<pb o="45" file="0059" n="59" rhead="SECTIO TERTIA."/> +(1 - {mm/nn})vdz + zdv = - zdz - bdz + {mbdz/√gn} <lb/>quæ multiplicata per z<emph style="super">{-mm/nn}</emph> facit <lb/>(1 - {mm/nn})z<emph style="super">- {mm/nn}</emph> vdz + z<emph style="super">1 - {mm/nn}</emph> dv = - z<emph style="super">1 - {mm/nn}</emph> dz - bz<emph style="super">- {mm/nn}</emph> dz + <lb/>{mbz<emph style="super">- {mm/nn}</emph> dz/√gn} <lb/>poſt cujus integrationem addita conſtante Coritur <lb/>z<emph style="super">{nn - mm/nn}</emph> v = C - {nn/2nn - mm} z<emph style="super">{2nn - mm/nn}</emph> - {nnb/nn - mm} z<emph style="super">{nn - mm/nn}</emph> <lb/>+ {mnnb/(nn - mm)√gn} z<emph style="super">{nn - mm/nn}</emph> <lb/>in quo valor quantitatis conſtantis C ex eo definitur quod ab initio fluxus <lb/>(cum nempe x = a ſive z = a - b + {mb/√gn}) ſit v = o quia non poteſt motus <lb/>oriri in inſtanti temporis puncto; </s> + <s xml:id="echoid-s1172" xml:space="preserve">hinc igitur fit C = <lb/>[(a - b + {mb/√gn}) X {nn/2nn - mm} + {nnb√gn - mnnb/(nn - mm)√gn}] X (a - b + {mb/√gn})<emph style="super">{nn - mm/nn}</emph> <lb/>Ex his quidem æquationibus definiuntur omnia; </s> + <s xml:id="echoid-s1173" xml:space="preserve">quia verò calculus fit paullo <lb/>prolixior, niſi amplitudo vaſis ſuperioris indicata per m tanta ſit, ut poſſit ra-<lb/>tione amplitudinum g & </s> + <s xml:id="echoid-s1174" xml:space="preserve">n infinita cenſeri, hunc ſolum conſiderabimus caſum, <lb/>idque eo magis quod error notabilis inde non oriatur, etſi mediocris ſit ma-<lb/>gnitudinis numerus {m/n} aut {m/g}</s> +</p> +<p> + <s xml:id="echoid-s1175" xml:space="preserve">§. </s> + <s xml:id="echoid-s1176" xml:space="preserve">23. </s> + <s xml:id="echoid-s1177" xml:space="preserve">Quod ſi proinde ponamus m = ∞, ſimulque utamur pri-<lb/>mâ æquatione differentiali proximi paragraphi, atque in hâc ponatur <lb/>v = {nn/mm}s, ut ſic inveniatur ex valore litteræ s altitudo ad quam aqua per ori-<lb/>ficium M N effluens ſuâ velocitate aſcendere poſſit, erit primo <lb/>{nn/m} (x - b)ds + {bnn/√gn}ds - msdx + {nn/m}sdx = - mxdx <lb/>& </s> + <s xml:id="echoid-s1178" xml:space="preserve">quia m = ∞ atque facile prævidetur rationem ſore finitam inter s & </s> + <s xml:id="echoid-s1179" xml:space="preserve">x, at-<lb/>que inter ds & </s> + <s xml:id="echoid-s1180" xml:space="preserve">dx, hæc eadem æquatio mutabitur rejectis terminis rejiciendis <lb/>rurſus in hanc - msdx = - mxdx vel s = x, quod pariter paragr. </s> + <s xml:id="echoid-s1181" xml:space="preserve">10.</s> + <s xml:id="echoid-s1182" xml:space="preserve"> +<pb o="46" file="0060" n="60" rhead="HYDRODYNAMICÆ."/> +jam fuit demonſtratam. </s> + <s xml:id="echoid-s1183" xml:space="preserve">E re vero duxi id de novo hic demonſtrare, quia ca-<lb/>ſus præſens diverſus videri poterat ab illo, de quo in præfato paragrapho dici-<lb/>tur. </s> + <s xml:id="echoid-s1184" xml:space="preserve">His intellectis non opus eſt pluribus explicare Phænomena circa hanc <lb/>rem §. </s> + <s xml:id="echoid-s1185" xml:space="preserve">21. </s> + <s xml:id="echoid-s1186" xml:space="preserve">Auctore s’Graveſande indicata; </s> + <s xml:id="echoid-s1187" xml:space="preserve">patet enim, aquam non aliter efflue-<lb/>re per vas compoſitum A E F M N G H B, quam per vas ſimplex A O M N P B, <lb/>cum nempe orificium M N eſt valde parvum, atque hinc majorem eſſe veloci-<lb/>tatem ſuperficiei aqueæ C D, quam ſi per vas A E F G H B aquæ effluerent, po-<lb/>ſito orificio M N = F G, multoque magis ſi M N fuerit majus quam F G, <lb/>quod fit cum tubus verſus inferiora amplitudine creſcit: </s> + <s xml:id="echoid-s1188" xml:space="preserve">attamen obſervari de-<lb/>bet, ab initio motus aquam tardius deſcendere, quam ſic definitum fuit, nec <lb/>regulam iſtam prius locum habere quam ſuperficies C D per ſpatiolum ali-<lb/>quod deſcenderit, quod tamen brevi fit tempore: </s> + <s xml:id="echoid-s1189" xml:space="preserve">mutationes, quæ ab initio <lb/>motus fiunt, in hoc caſu, examinabimus in ſectione ſequente.</s> + <s xml:id="echoid-s1190" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1191" xml:space="preserve">§, 24. </s> + <s xml:id="echoid-s1192" xml:space="preserve">Eodem modo computus eſſet inſtituendus, ſi vaſi, quod ſem-<lb/>per nunc amplitudinis infinitæ ponimus, implantatus eſſet tubulus non verti-<lb/>calis ſed horizontalis, veluti in fig. </s> + <s xml:id="echoid-s1193" xml:space="preserve">19. </s> + <s xml:id="echoid-s1194" xml:space="preserve">aut ſub alia directione qualicunque, ſem-<lb/> +<anchor type="note" xlink:label="note-0060-01a" xlink:href="note-0060-01"/> +per autem reperietur aquas per orificium M N mox, poſtquam ſuperficies aquæ <lb/>in vaſe principali aliquantulum deſcendit, ea proxime effiuere velocitate, quæ re-<lb/>ſpondeat altitudini iſtius ſuperficiei ſupra orificium; </s> + <s xml:id="echoid-s1195" xml:space="preserve">Inde liquet quod manen-<lb/>tibus tam altitudine aquæ ſupra tubulum G N, quam ipſo orificio F G, au-<lb/>geatur quantitas aquæ dato tempore effiuens ab aucta amplitudine orificii M N: <lb/></s> + <s xml:id="echoid-s1196" xml:space="preserve">Sic igitur demonſtratum hic dedimus, quod dictum fuit in fine §. </s> + <s xml:id="echoid-s1197" xml:space="preserve">5, Sect. </s> + <s xml:id="echoid-s1198" xml:space="preserve">1. </s> + <s xml:id="echoid-s1199" xml:space="preserve">Fron-<lb/>tinum experientia fuiſſe edoctum, nempe, plus debito aquæ erogari per cali-<lb/>cem legitimæ tum menſuræ tum poſitionis, cui ſtatim fiſtulæ amplioris moduli <lb/>ſubjectæ ſint. </s> + <s xml:id="echoid-s1200" xml:space="preserve">Et quidem quantitates aquæ cæteris paribus erogari deberent ip-<lb/>ſis orificiis M N proxime proportionales, niſi multa eſſent impedimenta; </s> + <s xml:id="echoid-s1201" xml:space="preserve">quæ <lb/>hanc quantitatem valde diminuant, de quibus proxime dicam: </s> + <s xml:id="echoid-s1202" xml:space="preserve">facere poſſunt <lb/>hæc impedimenta; </s> + <s xml:id="echoid-s1203" xml:space="preserve">ut admodum parum fluxus aquarum promoveatur ab au-<lb/>cto orificio extremo; </s> + <s xml:id="echoid-s1204" xml:space="preserve">ſemper tamen promovebitur aliquantum.</s> + <s xml:id="echoid-s1205" xml:space="preserve"/> +</p> +<div xml:id="echoid-div60" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0060-01" xlink:href="note-0060-01a" xml:space="preserve">Fig. 19.</note> +</div> +<p> + <s xml:id="echoid-s1206" xml:space="preserve">§. </s> + <s xml:id="echoid-s1207" xml:space="preserve">25. </s> + <s xml:id="echoid-s1208" xml:space="preserve">Ex præmiſſis liquet velocitatem, qua ſuperficies aquæ C D in <lb/>utroque, de quo diximus, caſu deſcendit cæteris paribus pendere ab am-<lb/>plitudine orificiorum M N; </s> + <s xml:id="echoid-s1209" xml:space="preserve">Hæc autem ea innituntur hypotheſi, quod aqua <lb/>lateribus tubulorum G N ubique adhæreat & </s> + <s xml:id="echoid-s1210" xml:space="preserve">pleno orificio M N effluat, quæ +<pb o="47" file="0061" n="61" rhead="SECTIO TERTIA."/> +hypotheſis locum amplius habere non poſſet, ſinimium orificium iſtud auge-<lb/>retur. </s> + <s xml:id="echoid-s1211" xml:space="preserve">Dein patet quoque, cum aquæ per tubum verticalem in fig. </s> + <s xml:id="echoid-s1212" xml:space="preserve">18. </s> + <s xml:id="echoid-s1213" xml:space="preserve">effluunt, <lb/>earum fiuxum accelerai à longitudine hujus tubi auctâ: </s> + <s xml:id="echoid-s1214" xml:space="preserve">poſſet tamen hæc <lb/>quoque ita augeri, ut tandem aquæ deſinant eſſe continuæ in tubo, quin po-<lb/>tius in columnas dividantur, quod fiet, ſi tubus longitudinem habeat plus <lb/>quam triginta duorum pedum aut minorem etiam, ſi ſimul amplitudine creſcat <lb/>verſus M N;</s> + <s xml:id="echoid-s1215" xml:space="preserve">ita ſi orificium M N duplum ſit orificii alterius F G, non poterit lon-<lb/>gitudo majoreſſe quam octo pedum, ſine periculo ſubſecuturæ aquarum ſepara-<lb/>tionis in ſuprema tubi parte, quam rem alibi demonſtrabo: </s> + <s xml:id="echoid-s1216" xml:space="preserve">ſed eſt alia inſu-<lb/>per cauſa præter nimiam tubi longitudinem, quæ aquæ ſeparationem produ-<lb/>cere poteſt, nempe quod altitudo aquæ C E H D minor ſit, quam ut ſat <lb/>cito in tubum irrumpere poſſit, quo fit, ut aër una cum aqua ſimul ſuperne <lb/>influat, dum ſuperficies aquæ formam cataractæ ſeu infundibuli cavi aſſumit, <lb/>ſic ut non totum orificium F G aqua obtegatur; </s> + <s xml:id="echoid-s1217" xml:space="preserve">Hæc quidem res facit, ut <lb/>aqua minori copia effluat, non autem ut minori velocitate, quod poſterius <lb/>putavit Auctor quidam Italus, nomine Carolus Fontana, qui hác de re Lin-<lb/>guâſua vernacula ita ſcripſit: </s> + <s xml:id="echoid-s1218" xml:space="preserve">mâ ſe non vifoſſe, inquit, tant’ acqua, che ba-<lb/>ſtaſſe à mantenere piena detta canna, l’acqua attraherà l’aria dentro di ſe in <lb/>tanta quantità, quanto gli mancherá l’acqua intermettendoſi fra l’acqua dà <lb/>ogni banda; </s> + <s xml:id="echoid-s1219" xml:space="preserve">mà la velocità dell’ acqua mancherá tanto, quanto ſará l’altezza <lb/>di tutta l’aria raccolta inſieme che ſarà in eſſa canna. </s> + <s xml:id="echoid-s1220" xml:space="preserve">Rationem ejus, quod <lb/>dixi, non inde velocitatem aquæ diminui poſſe, quilibet perſpicit ex eo, quod <lb/>alias non poſſet aſcenſus poteniialis eſſe æqualis deſ@enſui actuali poteritque res <lb/>facili experimento confirmari, incurvata tubi extremitate M N, ut aquæ ho-<lb/>rizontaliter effluant, & </s> + <s xml:id="echoid-s1221" xml:space="preserve">ex amplitudine jactus velocitas aquæ dignoſci poſſit. <lb/></s> + <s xml:id="echoid-s1222" xml:space="preserve">Quomodo autem pro lubitu fieri poſſit, ut nullis mutatis aliis circumſtantiis <lb/>aër aquis circa ſummitatem tubi miſceatur, ſic habe: </s> + <s xml:id="echoid-s1223" xml:space="preserve">fiat nempe parvulum fo-<lb/>ramen in tubo haud procul ab orificio F G (Fig. </s> + <s xml:id="echoid-s1224" xml:space="preserve">18. </s> + <s xml:id="echoid-s1225" xml:space="preserve">& </s> + <s xml:id="echoid-s1226" xml:space="preserve">19.) </s> + <s xml:id="echoid-s1227" xml:space="preserve">quod ſi autem du-<lb/>rante aquæ fluxu digito obturaveris iſtud foraminulum, aquæ transfluent pu-<lb/>ræ, & </s> + <s xml:id="echoid-s1228" xml:space="preserve">ſi removeris digitum, mox aër per foraminulum idem irrumpet ſeque <lb/>cum aqua præterfluente miſcebit. </s> + <s xml:id="echoid-s1229" xml:space="preserve">His intellectis facile erit rationem reddere <lb/>Phænomenorum, quæ in caminis ſeu fumi-ductibus obſervantur, fumus <lb/>enim altum petit, quia aëre levior eſt, quod conſtat experimentis de fumo <lb/>in vacuo, ubi deſcendiſſe viſus fuit, ſumtis: </s> + <s xml:id="echoid-s1230" xml:space="preserve">idem igitur eſt de fumo aſcen- +<pb o="48" file="0062" n="62" rhead="HYDRODYNAMICÆ."/> +dente, quod de aqua deſcendente: </s> + <s xml:id="echoid-s1231" xml:space="preserve">hæc autem in fig. </s> + <s xml:id="echoid-s1232" xml:space="preserve">18. </s> + <s xml:id="echoid-s1233" xml:space="preserve">eo celerius effluit per <lb/>orificium M N, quo amplius eſt, & </s> + <s xml:id="echoid-s1234" xml:space="preserve">quo humilius poſitum: </s> + <s xml:id="echoid-s1235" xml:space="preserve">ergo etiam fumus <lb/>eo celerius caminum tranſibit, eoque magis ignis in foco accendetur, quo <lb/>altius ducetur caminus, & </s> + <s xml:id="echoid-s1236" xml:space="preserve">quo magis ſuperiora verſus divergit, ſi modo non <lb/>nimis divergat; </s> + <s xml:id="echoid-s1237" xml:space="preserve">quod utrumque experientia confirmat; </s> + <s xml:id="echoid-s1238" xml:space="preserve">Ipſe deinde inſuper <lb/>expertusſum, ſi caminus alicubi perforetur, tantum abeſſe, ut fumus per fora-<lb/>men iſtud exitum tentet, quin potius aër magno impetu irruat, ſeque fumo <lb/>miſcens per caminum aſcendat, non ſecus atque aërem per foraminulum e in <lb/>tubum F G N M (Fig. </s> + <s xml:id="echoid-s1239" xml:space="preserve">18. </s> + <s xml:id="echoid-s1240" xml:space="preserve">& </s> + <s xml:id="echoid-s1241" xml:space="preserve">19.) </s> + <s xml:id="echoid-s1242" xml:space="preserve">irrumpere indicavimus. </s> + <s xml:id="echoid-s1243" xml:space="preserve">Ita vero fumus mino-<lb/>ri certe copia, aut ſaltem difficilius aſcendet ignisque remittet.</s> + <s xml:id="echoid-s1244" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1245" xml:space="preserve">Cæterum duæ ſunt potiſſimum cauſæ, altera aliena altera naturæ rei <lb/>propria, quæ motum aquæ valde retardare poſſunt in fig. </s> + <s xml:id="echoid-s1246" xml:space="preserve">18. </s> + <s xml:id="echoid-s1247" xml:space="preserve">& </s> + <s xml:id="echoid-s1248" xml:space="preserve">19. </s> + <s xml:id="echoid-s1249" xml:space="preserve">Prior eſt <lb/>adhæſio aquæ ad latera tubi, & </s> + <s xml:id="echoid-s1250" xml:space="preserve">altera, quod cum tubus amplitudine creſcit <lb/>velocitas aquæ, nullibi ſibi conſtans in quovis tubi loco mutetur, quæ mutatio <lb/>ſi oriri cenſeatur ab impulſibus infinite parvis aquæ velocius motæ in aquam <lb/>minus velociter motam, apparet ſingulis momentis ab impulſibus his corpo-<lb/>rum mollium aliquid de aſcenſu potentiali perdi, unde neceſſario aquarum ef-<lb/>fluxus notabiliter diminuitur.</s> + <s xml:id="echoid-s1251" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1252" xml:space="preserve">§. </s> + <s xml:id="echoid-s1253" xml:space="preserve">26 Loco ultimo nunc dicam quædam de vaſis recurvis, ex quibus <lb/>aquæ non omnes effluunt: </s> + <s xml:id="echoid-s1254" xml:space="preserve">brevitatis autem gratiâ canalem conſiderabimus <lb/>cylindricum, & </s> + <s xml:id="echoid-s1255" xml:space="preserve">cujus quidem pars, quam ſuperficies aquea non tranſgreditur, <lb/>ſit recta.</s> + <s xml:id="echoid-s1256" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div62" type="section" level="1" n="44"> +<head xml:id="echoid-head55" xml:space="preserve"><emph style="bf">Problema.</emph></head> +<p> + <s xml:id="echoid-s1257" xml:space="preserve">Sit nempe canalis cylindricus C E D B (Fig. </s> + <s xml:id="echoid-s1258" xml:space="preserve">20.) </s> + <s xml:id="echoid-s1259" xml:space="preserve">cujus pars C E quan-<lb/>ta ſufficit eſt recta, reliqua E D B utcunque incurvata; </s> + <s xml:id="echoid-s1260" xml:space="preserve">fuerit canalis totus aqua <lb/> +<anchor type="note" xlink:label="note-0062-01a" xlink:href="note-0062-01"/> +plenus effluxura per foramen B, perveneritque ſuperficies aquæ ex C in F, <lb/>quæritur altitudo reſpondens velocitati aquæ in F.</s> + <s xml:id="echoid-s1261" xml:space="preserve"/> +</p> +<div xml:id="echoid-div62" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0062-01" xlink:href="note-0062-01a" xml:space="preserve">Fig. 20.</note> +</div> +</div> +<div xml:id="echoid-div64" type="section" level="1" n="45"> +<head xml:id="echoid-head56" xml:space="preserve"><emph style="bf">Solutio.</emph></head> +<p> + <s xml:id="echoid-s1262" xml:space="preserve">Ducantur verticalis B H & </s> + <s xml:id="echoid-s1263" xml:space="preserve">horizontales C H, F G, A B, ſitque ſinus <lb/>anguli H C E ad ſinum totum ut 1 ad g: </s> + <s xml:id="echoid-s1264" xml:space="preserve">Jam vero ſi rem recte perpendamus, <lb/>videbimus contineri problema præſens in altero generaliori, quod ſuprà pa-<lb/>ragrapho 20. </s> + <s xml:id="echoid-s1265" xml:space="preserve">tractavimus, ubi habuimus hanc æquationem: <lb/></s> + <s xml:id="echoid-s1266" xml:space="preserve">v = ξ<emph style="super">{mm/nn - 1}</emph> ſ - xξ<emph style="super">{- mm/nn}</emph> dξ +<pb o="49" file="0063" n="63" rhead="SECTIO TERTIA."/> +ubi pro noſtro caſu præſente intelligitur per v altitudo quæſita reſpondens ve-<lb/>locitati ſuperficiei aqueæ in ſitu F, per ξ longitudo B D E F & </s> + <s xml:id="echoid-s1267" xml:space="preserve">per x altitudo <lb/>B G, atque per {m/n} index rationis inter amplitudines tubi & </s> + <s xml:id="echoid-s1268" xml:space="preserve">foraminis B: </s> + <s xml:id="echoid-s1269" xml:space="preserve">Quod <lb/>ſi vero dicatur longitudo B D A = αerit x = {ξ - α/g}, unde nunc habetur <lb/>v = ξ<emph style="super">{mm/nn} - 1}</emph> ſ - ({ξ - α/g}) ξ<emph style="super">{- mm/nn}</emph> dξ</s> +</p> +<p> + <s xml:id="echoid-s1270" xml:space="preserve">Indicetur longitudo totius canalis B D E C per β, & </s> + <s xml:id="echoid-s1271" xml:space="preserve">erit <lb/>ſ - ({ξ - α/g} ξ<emph style="super">{- mm/nn}</emph> dξ = {nnα/g(nn - mm)} (ξ<emph style="super">{nn - mm/nn}</emph> - β<emph style="super">{nn - mm/nn}</emph>}) <lb/>{- nn/g(2nn - mm)} (ξ<emph style="super">{2nn - mm/nn}</emph> - β<emph style="super">{2nn - mm/nn}</emph>) <lb/>atque proinde <lb/>v = {nnα/g(nn - mm)}(1 - ({β/ξ})<emph style="super">{nn - mm/nn}</emph>) <lb/>- {nnξ/g(2nn - mm)}(1 - ({β/ξ})<emph style="super">{2nn - mm/nn}</emph>). </s> + <s xml:id="echoid-s1272" xml:space="preserve">Q. </s> + <s xml:id="echoid-s1273" xml:space="preserve">E. </s> + <s xml:id="echoid-s1274" xml:space="preserve">I.</s> + <s xml:id="echoid-s1275" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div65" type="section" level="1" n="46"> +<head xml:id="echoid-head57" xml:space="preserve"><emph style="bf">Scholium.</emph></head> +<p> + <s xml:id="echoid-s1276" xml:space="preserve">§. </s> + <s xml:id="echoid-s1277" xml:space="preserve">27. </s> + <s xml:id="echoid-s1278" xml:space="preserve">Quoniam hæ æquationes ſunt paullo prolixiores @non immora-<lb/>bimur generali earundem contemplationi, conſideraturi potius caſus iſtos <lb/>particulares, qui calculum abbreviant, nec ultima iſta æquatione definiri <lb/>poſſunt.</s> + <s xml:id="echoid-s1279" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1280" xml:space="preserve">Si operculum in B omne abeſſe ponamus, fit m = n & </s> + <s xml:id="echoid-s1281" xml:space="preserve">(quod ſeorſim <lb/>pro hoc pariter atque @altero caſu mox dicendo erui debet) <lb/>v = {b - ξ + αlog.</s> + <s xml:id="echoid-s1282" xml:space="preserve">ξ - αlog.</s> + <s xml:id="echoid-s1283" xml:space="preserve">β/g} <lb/>tuncque velocitas maxima eſt in A, nominatimquæ talis, quæ reſpondet al-<lb/>titudini {β - α + αlog.</s> + <s xml:id="echoid-s1284" xml:space="preserve">α - αlog.</s> + <s xml:id="echoid-s1285" xml:space="preserve">β.</s> + <s xml:id="echoid-s1286" xml:space="preserve">/g}</s> +</p> +<pb o="50" file="0064" n="64" rhead="HYDRODYNAMICÆ."/> +<p> + <s xml:id="echoid-s1287" xml:space="preserve">Denique punctum E maximo reſpondens deſcenſui obtinetur ope hu-<lb/>jus æquationis, <lb/>ξ - αlog.</s> + <s xml:id="echoid-s1288" xml:space="preserve">ξ = β - αlog.</s> + <s xml:id="echoid-s1289" xml:space="preserve">β</s> +</p> +<p> + <s xml:id="echoid-s1290" xml:space="preserve">Alter caſus ſeorſim ſubducendus calculo eſt, cum mm = 2nn, ubi <lb/>oritur <lb/>v = {αξ - αβ - ξβlog.</s> + <s xml:id="echoid-s1291" xml:space="preserve">ξ + ξβlog.</s> + <s xml:id="echoid-s1292" xml:space="preserve">β/gβ} <lb/>atque ſi capiatur, poſito c pro numero, cujus logarithmus eſt unitas, <lb/>ξ = c<emph style="super">{α - β/β}</emph>β determinabitur ſic locus maximæ velocitatis, cujus altitudo <lb/>generatrix eſt = c<emph style="super">{α - β/β}</emph>β - α, dum maximus deſcenſus, qui proportiona-<lb/>lis eſt toti aquæ effluenti, definitur faciendo <lb/>αξ - αβ - ξβlog.</s> + <s xml:id="echoid-s1293" xml:space="preserve">ξ + ξβlog.</s> + <s xml:id="echoid-s1294" xml:space="preserve">β = o</s> +</p> +<p> + <s xml:id="echoid-s1295" xml:space="preserve">Non dubito, quin hæc ad amuſſim experientiæ eſſent reſponſura, ſi <lb/>modo adhæſio aquæ ad latera tubi motum non retardaret; </s> + <s xml:id="echoid-s1296" xml:space="preserve">puto tamen, even-<lb/>tum experimentorum talem eſſe poſſe, ut intelligenti, qui horum impedi-<lb/>mentorum rationem habeat, ſatis @ſtendant propoſitionum veritatem.</s> + <s xml:id="echoid-s1297" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1298" xml:space="preserve">§. </s> + <s xml:id="echoid-s1299" xml:space="preserve">28. </s> + <s xml:id="echoid-s1300" xml:space="preserve">Ultimo loco communicabo veram ſolutionem phænomeni ali-<lb/>cujus, quod primo aſpectu valde videtur paradoxon. </s> + <s xml:id="echoid-s1301" xml:space="preserve">Poſtquam enim ex <lb/>omnibus hactenus dictis luculenter apparet fieri non poſſe, ut aquæ multo <lb/>majori velocitate effluant quam qualis altitudini aquæ ſupra foramen debetur, <lb/>(poſſunt tamen aliquanto majori, præſertim ſi foramina ſunt magna, con-<lb/>fer ea quæ dixi de velocitatibus maximis §. </s> + <s xml:id="echoid-s1302" xml:space="preserve">16.) </s> + <s xml:id="echoid-s1303" xml:space="preserve">multis mirum fortaſſe videbitur, <lb/>contingere aliquando in fontibus ſalientibus, ut aqua ad temporis momen-<lb/>ium jactum faciat longe altiorem, quam ſecundum regulas noſtras fieri poſſe <lb/>videtur. </s> + <s xml:id="echoid-s1304" xml:space="preserve">Verum tantum abeſt, ut hæ inde aliquid roboris perdant, quin <lb/>potius egregie confirmentur. </s> + <s xml:id="echoid-s1305" xml:space="preserve">Solutio autem paradoxi in eo conſiſtit, quod <lb/>nos hactenus aquas conſideraverimus continuas, & </s> + <s xml:id="echoid-s1306" xml:space="preserve">nullo vacuo aëreo ſepara-<lb/>tas: </s> + <s xml:id="echoid-s1307" xml:space="preserve">Recteque obſervavit D<emph style="super">us</emph>. </s> + <s xml:id="echoid-s1308" xml:space="preserve">De la Hire non fieri hujusmodi ſaltus irrregu-<lb/>lares, niſi aër una cum aqua tubum prope ſcaturiginem fuerit ingreſſus, <lb/>quod ſæpe fieri indicavi §. </s> + <s xml:id="echoid-s1309" xml:space="preserve">25. </s> + <s xml:id="echoid-s1310" xml:space="preserve">Iſte vero aër ſimul cum aqua fertur usque ad <lb/>orificium effluxus, per quod mox erumpit: </s> + <s xml:id="echoid-s1311" xml:space="preserve">id dum fit, maſſa aquea impe- +<pb o="51" file="0065" n="65" rhead="SECTIO TERTIA."/> +tum acquirit, qui in expellendas aquas ſolus impenditur, hocque pacto <lb/>enormem jactum producit. </s> + <s xml:id="echoid-s1312" xml:space="preserve">Hanc phænomeni cauſam mox clarius una cum <lb/>debitis menſuris explicabo, poſtquam præmiſero verba, quæ hâc de re ex-<lb/>tant, in hiſtor. </s> + <s xml:id="echoid-s1313" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s1314" xml:space="preserve">Reg. </s> + <s xml:id="echoid-s1315" xml:space="preserve">ſc. </s> + <s xml:id="echoid-s1316" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s1317" xml:space="preserve">ad An. </s> + <s xml:id="echoid-s1318" xml:space="preserve">1702. </s> + <s xml:id="echoid-s1319" xml:space="preserve">On voit quelques fois, dici-<lb/>tur in loco citato, l’eau qui ſort par un ajutage ſaillir trois ou quatre fois <lb/>plus haut que ne lui permét la hauteur du réſervoir, ausſi ſe rémet - elle bien <lb/>vite à la hauteur, que lui preſcrivent les loix de l’hydroſtatique. </s> + <s xml:id="echoid-s1320" xml:space="preserve">Mais com-<lb/>ment a-t-elle pu en ſortir en un inſtant. </s> + <s xml:id="echoid-s1321" xml:space="preserve">Mſr. </s> + <s xml:id="echoid-s1322" xml:space="preserve">De la Hire l’attribue a de <lb/>l’air enfermè dans la conduite, qui aγant été preſſé & </s> + <s xml:id="echoid-s1323" xml:space="preserve">mis en reſſort par <lb/>l’eau, qui deſcendoit toujours, s’eſt debandé contre celle qui montoit & </s> + <s xml:id="echoid-s1324" xml:space="preserve">lui <lb/>a donné cette viteſſe momentanée.</s> + <s xml:id="echoid-s1325" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1326" xml:space="preserve">Recte itaque animadvertit Dn. </s> + <s xml:id="echoid-s1327" xml:space="preserve">De la Hire aëri ſaltum deberi, dubium-<lb/>que nullum eſt quin veram rationem, quâ aër id producere poſſit, fuiſſet <lb/>eruturus, ſi phænomenon, quod obiter attigit, attentius conſideraſſet, fa-<lb/>cile utique perſperſpecturus, aërem inter medias aquas nullam ſuſtinere preſſio-<lb/>nem, niſi ſuper incumbentis aquæ (imo ne hanc quidem in aquis fluentibus, <lb/>uti inferius in ſect. </s> + <s xml:id="echoid-s1328" xml:space="preserve">XII. </s> + <s xml:id="echoid-s1329" xml:space="preserve">demonſtrabo) nec adeoque aërem compreſſum for-<lb/>tius expellere poſſe aquam ſibi præcedentem, quam ſi ſui loco aqua eſſet. </s> + <s xml:id="echoid-s1330" xml:space="preserve">Ego <lb/>quidem prævidi (quod facillimo experimento ſæpe poſtea ſum expertus) non <lb/>eſſe aquam ante aërem poſitam ſolito altius aſſurgentem, ſed illam, quæ aërem <lb/>ſequitur, quod nunc clarius faciam.</s> + <s xml:id="echoid-s1331" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1332" xml:space="preserve">Sit igitur in Figura vigeſima aquæ ductus C A D B cylindricus, ut eſſe <lb/>ſolet, isque totus aquâ plenus, præter particulam m n B aëre plenam. </s> + <s xml:id="echoid-s1333" xml:space="preserve">Du-<lb/>cantur lineæ horrizontalis & </s> + <s xml:id="echoid-s1334" xml:space="preserve">verticalis C H & </s> + <s xml:id="echoid-s1335" xml:space="preserve">H B: </s> + <s xml:id="echoid-s1336" xml:space="preserve">ponamus brevitatis ergo <lb/>aëris gravitatem præ gravitate aquæ nullam cenſeri poſſe, ita ut tranſitus aëris <lb/>per orificium B nihil reſiſtat fluxui aquæ, quamvis de cætero facile foret in-<lb/>ertiæ aëris rationem habere, niſi calculi prolixitatem evitare vellemus in re, <lb/>ubi nullam quærimus præciſionem. </s> + <s xml:id="echoid-s1337" xml:space="preserve">Sit longitudo canalis C A D f vel C A D m <lb/>(ponimus enim differentiolam mf aëre repletam valde parvam) = β mf vel <lb/>ng = δ: </s> + <s xml:id="echoid-s1338" xml:space="preserve">H B = a; </s> + <s xml:id="echoid-s1339" xml:space="preserve">amplitudo tubi = m, amplitudo orificii B = n; </s> + <s xml:id="echoid-s1340" xml:space="preserve">Denique <lb/>demus aquæ, cum ſuperficies eſt in mn, nullum eſſe motum, quæſituri al- +<pb o="52" file="0066" n="66" rhead="HYDRODYNAMICÆ."/> +titudinem velocitati debitam, quam ſuperficies mn habet, cum pervenit in <lb/>ſitum fg; </s> + <s xml:id="echoid-s1341" xml:space="preserve">ſit iſta altitudo = v, erit aſcenſus potent. </s> + <s xml:id="echoid-s1342" xml:space="preserve">omnis aquæ eo ipſo mo-<lb/>mento pariter = v: </s> + <s xml:id="echoid-s1343" xml:space="preserve">Deſcenſus actualis autem eſt per §. </s> + <s xml:id="echoid-s1344" xml:space="preserve">7. </s> + <s xml:id="echoid-s1345" xml:space="preserve">= tertiæ propor-<lb/>tionali ad totam maſſam aquæ, particulam aquæ mngf & </s> + <s xml:id="echoid-s1346" xml:space="preserve">altitudinem verti-<lb/>calem HB, id eſt, = {δ/β}a; </s> + <s xml:id="echoid-s1347" xml:space="preserve">eſt igitur v = {δ/β}a. </s> + <s xml:id="echoid-s1348" xml:space="preserve">Hæc quidem altitudo dicto <lb/>citius minuitur ſtatim atque aqua per orificium B fluere cogitur, quod de-<lb/>monſtravi §. </s> + <s xml:id="echoid-s1349" xml:space="preserve">18. </s> + <s xml:id="echoid-s1350" xml:space="preserve">ſed primo tamen temporis puncto aqua ſervabit motum <lb/>quem acquiſivit, & </s> + <s xml:id="echoid-s1351" xml:space="preserve">ſic guttula orificio proxima ejicietur velocitate, quæ de-<lb/>beatur altitudini{mmδ/nnß} a. </s> + <s xml:id="echoid-s1352" xml:space="preserve">Poteſt autem hæc altitudo non ſolum eſſe tripla <lb/>aut quadrupla ipſius a, ſed & </s> + <s xml:id="echoid-s1353" xml:space="preserve">quantumcunque magna: </s> + <s xml:id="echoid-s1354" xml:space="preserve">ego certe cum tubis <lb/>vitreis pro lubitu jactus feci decies aut vigeſies altiores ipſius a; </s> + <s xml:id="echoid-s1355" xml:space="preserve">fuerit v. </s> + <s xml:id="echoid-s1356" xml:space="preserve">gr. <lb/></s> + <s xml:id="echoid-s1357" xml:space="preserve">δ = 100 pedum, β = uni pollici, diameter autem tubi decupla diametri, <lb/>quam orificium habet; </s> + <s xml:id="echoid-s1358" xml:space="preserve">& </s> + <s xml:id="echoid-s1359" xml:space="preserve">erit {mmδ/nnß} = {10000/1200} a, ita ut in his circumſtan-<lb/>tiis prima guttula aſſilire demta aëris reſiſtentia debeat ad altitudinem plus-<lb/>quam octies majorem altitudine ſolita a. </s> + <s xml:id="echoid-s1360" xml:space="preserve">Sunt cœterum multa impedimenta <lb/>eaque maximi momenti, quæ jactus enormes cohibeant; </s> + <s xml:id="echoid-s1361" xml:space="preserve">perditur nempe ali-<lb/>quid de motu ab impulſu ſuperficiei aqueæ mn in latera fg, dein etiam ab <lb/>ingenti attritu quem aqua per foraminulum, quod parvulum eſſe debet, <lb/>tam celeriter lata patitur: </s> + <s xml:id="echoid-s1362" xml:space="preserve">multum etiam abeſt, quominus aqua C A D m <lb/>omni ſua celeritate moveatur ob adhæſionem aquæ ad latera tubi, quæ ad-<lb/>hæſio in tam longo tractu valde notabilis eſt.</s> + <s xml:id="echoid-s1363" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1364" xml:space="preserve">Interim veram hanc eſſe ſolutionem phænomeni nullum poteſt eſſe <lb/>dubium, iſtique ſolutioni experimenta quæ feci in omni extenſione ſatisfa-<lb/>ciunt. </s> + <s xml:id="echoid-s1365" xml:space="preserve">Dein hâc theoria etiam recte ſolvitur alterum phænomeni momen-<lb/>tum, quod nempe jactus iſte ſit quaſi momentaneus, poſtque breviſſimum <lb/>tempusculum ad ſenſus non major ſolito: </s> + <s xml:id="echoid-s1366" xml:space="preserve">ita in præſenti, quem modo <lb/>finximus, caſu ſi per regulam §. </s> + <s xml:id="echoid-s1367" xml:space="preserve">18, paullo mutatam (ibi enim de vaſis <lb/>verticaliter poſitis tantum dicitur) exploremus, quantum aquæ effluere de-<lb/>beat ut jactus non amplius milleſimâ parte (quæ utique obſervari in hujus-<lb/>modi experimentis minimè poteſt) ſuperet jactum ſolitum, cum ab initio <lb/>fuerit eodem octies major, invenimus tam parvam eſſe illam quantitatem, ut <lb/>tempus, quo tota ejicitur, nullo modo percipi poſſit.</s> + <s xml:id="echoid-s1368" xml:space="preserve"/> +</p> +<pb o="53" file="0067" n="67" rhead="SECTIO TERTIA."/> +</div> +<div xml:id="echoid-div66" type="section" level="1" n="47"> +<head xml:id="echoid-head58" style="it" xml:space="preserve">Experimenta quæ ad Sect. 3. pertinent.</head> +<head xml:id="echoid-head59" xml:space="preserve">Prænotanda.</head> +<p> + <s xml:id="echoid-s1369" xml:space="preserve">PLurima quidem ſunt in hâc Sectione eaque fere præcipua, quæ vix <lb/>ad experimenta revocari immediate poſſunt; </s> + <s xml:id="echoid-s1370" xml:space="preserve">Etenim cum Auctores ha-<lb/>ctenus motum in fluidis effluentibus alium non conſideraverint, <lb/>quam qui fiunt per foramina valde parva, cumque proin nova ſit theoria <lb/>quam dedimus pro amplitudinibus foraminum qualibuscunque, hæc ipſa <lb/>eſt, cujus confirmatio maxime juvaret. </s> + <s xml:id="echoid-s1371" xml:space="preserve">At non video, quomodo in Cy-<lb/>lindris verticalibus, de quibus potiſſimum egimus, velocitas aquæ effluen-<lb/>tis obſervari poſſit, præſertim cum foramen eſt valde amplum (ſecus enim <lb/>ex tempore depletionis aliquod de velocitatibus judicium ferri poteſt.) </s> + <s xml:id="echoid-s1372" xml:space="preserve">Hæc <lb/>ita perpendens cogitavi demum ſcopo noſtro inſervire poſſe paragraphos 16. <lb/></s> + <s xml:id="echoid-s1373" xml:space="preserve">& </s> + <s xml:id="echoid-s1374" xml:space="preserve">20. </s> + <s xml:id="echoid-s1375" xml:space="preserve">in quorum priore determinata fuit velocitas maxima aquæ effluentis <lb/>ex cylindris verticaliter poſitis, in altero autem demonſtratum eſt, eundem <lb/>eſſe motum ex cylindris oblique poſitis & </s> + <s xml:id="echoid-s1376" xml:space="preserve">verticalibus, ſi utrobique altitudi-<lb/>nes verticales ſimiles aſſumantur: </s> + <s xml:id="echoid-s1377" xml:space="preserve">Commode igitur utemur cylindris oblique <lb/>poſitis, ut ex maxima amplitudine jactus aquei poſſit velocitas maxima aquæ <lb/>ſeu altitudo eidem debita experimento haberi: </s> + <s xml:id="echoid-s1378" xml:space="preserve">& </s> + <s xml:id="echoid-s1379" xml:space="preserve">hâc quidem ratione accu-<lb/>rate velocitas illa maxima, qualis revera eſt, explorari poteſt, etiamſi ſo-<lb/>ramina ſint quantumlibet magna, quæ proin ſi convenire obſervetur cum re-<lb/>gulis noſtris, de integra theoria dubium ſupereſſe nullum poterit.</s> + <s xml:id="echoid-s1380" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1381" xml:space="preserve">Priusquam vero rem ipſam aggrediar, præmittendum erit theore<unsure/>ma <lb/>mechanicum, quod ſequitur.</s> + <s xml:id="echoid-s1382" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div67" type="section" level="1" n="48"> +<head xml:id="echoid-head60" xml:space="preserve">Lemma.</head> +<p> + <s xml:id="echoid-s1383" xml:space="preserve">Sit A B (Fig. </s> + <s xml:id="echoid-s1384" xml:space="preserve">21.) </s> + <s xml:id="echoid-s1385" xml:space="preserve">linea verticalis, B D horizontalis; </s> + <s xml:id="echoid-s1386" xml:space="preserve">linea autem A D <lb/> +<anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/> +directionem habeat qualemcunque, ſub cujus directione corpus in A proje-<lb/>ctum intelligatur, arcum deſcribens parabolium A C, cujus nempe tangens <lb/>in A eſt recta A D, erit altitudo debita velocitati, qua corpus in A proje-<lb/>ctum fuit, = {BC<emph style="super">2</emph> X AD<emph style="super">2</emph>/4AB.</s> + <s xml:id="echoid-s1387" xml:space="preserve">BD.</s> + <s xml:id="echoid-s1388" xml:space="preserve">CD} atque ſi AD fuerit horizontalis ſive angulus B A D <lb/>rectus, erit eadem illa altitudo = {BC<emph style="super">2</emph>/4AB}.</s> + <s xml:id="echoid-s1389" xml:space="preserve"/> +</p> +<div xml:id="echoid-div67" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">Fig. 21.</note> +</div> +<p> + <s xml:id="echoid-s1390" xml:space="preserve">Jam vero quæ mihi obſervata fuerint exponam.</s> + <s xml:id="echoid-s1391" xml:space="preserve"/> +</p> +<pb o="54" file="0068" n="68" rhead="HYDRODYNAMICÆ."/> +</div> +<div xml:id="echoid-div69" type="section" level="1" n="49"> +<head xml:id="echoid-head61" style="it" xml:space="preserve">De Velocitatibus maximis fluidorum per foramina <lb/>valde ampla effluentium.</head> +<head xml:id="echoid-head62" xml:space="preserve">Ad §. 16. & 20.</head> +<head xml:id="echoid-head63" xml:space="preserve">Experimentum Primum.</head> +<p> + <s xml:id="echoid-s1392" xml:space="preserve">TUbum Cylindricum F A (Fig. </s> + <s xml:id="echoid-s1393" xml:space="preserve">22.) </s> + <s xml:id="echoid-s1394" xml:space="preserve">longitudinis quatuor pollicum <lb/> +<anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/> +oblique ad horizontem poſui, in eoque ſitu firmavi, erat autem <lb/>amplitudo tubi ad amplitudinem luminis in A ut 2 ad 1. </s> + <s xml:id="echoid-s1395" xml:space="preserve">& </s> + <s xml:id="echoid-s1396" xml:space="preserve">quidem dia-<lb/>meter tubi præter propter ſeptem lineas exæquabat; </s> + <s xml:id="echoid-s1397" xml:space="preserve">Dein menſuris acceptis <lb/>in particulis æqualibus linearum F E, A B & </s> + <s xml:id="echoid-s1398" xml:space="preserve">B D (quarum lex ex ipſa figura per <lb/>ſe patet) illas inveni 81. </s> + <s xml:id="echoid-s1399" xml:space="preserve">619. </s> + <s xml:id="echoid-s1400" xml:space="preserve">& </s> + <s xml:id="echoid-s1401" xml:space="preserve">740.</s> + <s xml:id="echoid-s1402" xml:space="preserve"/> +</p> +<div xml:id="echoid-div69" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">Fig. 22.</note> +</div> +<p> + <s xml:id="echoid-s1403" xml:space="preserve">His ita præparatis, tubum aquâ replevi, digito interim obturato ori-<lb/>ficio A, eoque confeſtim remoto aquæ breviſſimo tempuſculo effluxere om-<lb/>nes: </s> + <s xml:id="echoid-s1404" xml:space="preserve">obſervare tamen potui, primas & </s> + <s xml:id="echoid-s1405" xml:space="preserve">ultimas propius ad verticalem AB, <lb/>quam medias cecidiſſe; </s> + <s xml:id="echoid-s1406" xml:space="preserve">guttas autem longiſſime projectas incidiſſe in locum <lb/>C invenique poſt ſæpius repetitum experimentum BC particularum, qui-<lb/>bus antea uſus fueram, 235.</s> + <s xml:id="echoid-s1407" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1408" xml:space="preserve">Jam vero ſi per præmiſſum lemma quæratur altitudo E G, ad quam guttæ <lb/>maxima velocitate ejectæ aſcendere poſſint, reperitur E G = 56. </s> + <s xml:id="echoid-s1409" xml:space="preserve">part. </s> + <s xml:id="echoid-s1410" xml:space="preserve">de-<lb/>beret autem vi §. </s> + <s xml:id="echoid-s1411" xml:space="preserve">§. </s> + <s xml:id="echoid-s1412" xml:space="preserve">16. </s> + <s xml:id="echoid-s1413" xml:space="preserve">& </s> + <s xml:id="echoid-s1414" xml:space="preserve">20. </s> + <s xml:id="echoid-s1415" xml:space="preserve">eſſe.</s> + <s xml:id="echoid-s1416" xml:space="preserve"><unsure/> = 62. </s> + <s xml:id="echoid-s1417" xml:space="preserve">niſi attritus aquæ ejusquæ adhæ-<lb/>ſio ad latera tubi impedimentum motui afferret: </s> + <s xml:id="echoid-s1418" xml:space="preserve">majorem conſenſum non <lb/>expectavi.</s> + <s xml:id="echoid-s1419" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1420" xml:space="preserve">II. </s> + <s xml:id="echoid-s1421" xml:space="preserve">Poſitis quæ prius, diminuto tantum ad dimidium foramine A, <lb/>ita ut amplitudo tubi eſſet quadrupla amplitudinis ad lumen pertinentis, ob-<lb/>ſervavi B C = 252; </s> + <s xml:id="echoid-s1422" xml:space="preserve">Hinc deducitur E G = 68 per experimentum; </s> + <s xml:id="echoid-s1423" xml:space="preserve">per <lb/>theoriam autem debuiſſet eſſe = 70; </s> + <s xml:id="echoid-s1424" xml:space="preserve">numeri hi minus differunt quam <lb/>præcedentes, quia hîc multo minus fuit attritus impedimentum ob diminu-<lb/>tam velocitatem internæ aquæ. </s> + <s xml:id="echoid-s1425" xml:space="preserve">Utrumque autem experimentum egregie <lb/>profecto theoriam confirmat.</s> + <s xml:id="echoid-s1426" xml:space="preserve"/> +</p> +<pb o="55" file="0069" n="69" rhead="SECTIO TERTIA,"/> +</div> +<div xml:id="echoid-div71" type="section" level="1" n="50"> +<head xml:id="echoid-head64" style="it" xml:space="preserve">De velocitate aquæ ex vaſe ampliſſimo <lb/>erumpentis.</head> +<head xml:id="echoid-head65" xml:space="preserve">Ad §. 17.</head> +<p> + <s xml:id="echoid-s1427" xml:space="preserve">IN iſto paragrapho dicimus, ſi vas ſit ampliſſimum, aquam mox, poſt-<lb/>quam ſuperficies interna aliquantulum deſcendit, erumpere velocitate, <lb/>quæ conſtanter reſpondeat altitudini aquæ ſupra foramen. </s> + <s xml:id="echoid-s1428" xml:space="preserve">Sinas autem <lb/>ſub quâcunque directione (neque enim in vaſis ampliſſimis directio venæ <lb/>quicquam velocitatem mutare poteſt) aquam effluere, & </s> + <s xml:id="echoid-s1429" xml:space="preserve">obſerves quocun-<lb/>que temporis puncto, in quanta diſtantia ab verticali vena in horizontem <lb/>impingat, & </s> + <s xml:id="echoid-s1430" xml:space="preserve">exinde per præmiſſam regulam quære altitudinem velocitati <lb/>aquæ effluentis eo temporis puncto reſpondentem, ſic ſemper iſtam altitu-<lb/>dinem invenies æqualem altitudini aquæ ſupra centrum foraminis, ſi modo <lb/>excipias primas guttulas, quæ vi §. </s> + <s xml:id="echoid-s1431" xml:space="preserve">16. </s> + <s xml:id="echoid-s1432" xml:space="preserve">minori velocitate effluere debent & </s> + <s xml:id="echoid-s1433" xml:space="preserve"><lb/>actu effluunt: </s> + <s xml:id="echoid-s1434" xml:space="preserve">Neque impedimenta, quorum ſæpius mentionem injecimus, <lb/>ullam notabilem moram fluxui injicient, ſi modo diameter foraminis duas <lb/>aut tres lineas minimum exæquet, & </s> + <s xml:id="echoid-s1435" xml:space="preserve">diameter ipſius vaſis non ſit infra ali-<lb/>quot pollices, & </s> + <s xml:id="echoid-s1436" xml:space="preserve">denique altitudo aquæ nimia non ſit, veluti plurium pedum.</s> + <s xml:id="echoid-s1437" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1438" xml:space="preserve">Hæc omnia ſæpe expertus ſum, experimenti autem genus nimis eſt tri-<lb/>viale, quam ut prolixe deſcribi mereatur.</s> + <s xml:id="echoid-s1439" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div72" type="section" level="1" n="51"> +<head xml:id="echoid-head66" style="it" xml:space="preserve">De vaſis quæ ſunt Tubis verticalibus inſtructa.</head> +<head xml:id="echoid-head67" xml:space="preserve">Ad §. 22. & 23.</head> +<p> + <s xml:id="echoid-s1440" xml:space="preserve">DE his experimenta ſumſit Cel. </s> + <s xml:id="echoid-s1441" xml:space="preserve">s’Graveſande in Phyſ. </s> + <s xml:id="echoid-s1442" xml:space="preserve">Elem. </s> + <s xml:id="echoid-s1443" xml:space="preserve">Math. </s> + <s xml:id="echoid-s1444" xml:space="preserve">quæ <lb/>repetii; </s> + <s xml:id="echoid-s1445" xml:space="preserve">ea vero quæ ad rem præſentem faciunt huc potiſſimum re-<lb/>deunt.</s> + <s xml:id="echoid-s1446" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1447" xml:space="preserve">In Figuris nempe 23. </s> + <s xml:id="echoid-s1448" xml:space="preserve">24. </s> + <s xml:id="echoid-s1449" xml:space="preserve">25. </s> + <s xml:id="echoid-s1450" xml:space="preserve">26. </s> + <s xml:id="echoid-s1451" xml:space="preserve">ſunt ſingulæ aperturæ littera A notatæ, <lb/> +<anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/> +inter ſe æquales, ſolâ B exiſtente paullo majore in ratione ut 16 ad 25, am-<lb/>plitudines quoque, ut & </s> + <s xml:id="echoid-s1452" xml:space="preserve">altitudines cylindrorum ſunt æquales excepto ul-<lb/>timo, cujus longitudo quadrupla eſt: </s> + <s xml:id="echoid-s1453" xml:space="preserve">tubi autem duobus cylindris interme-<lb/>diis annexi, triplam habent longitudinem cylindrorum. </s> + <s xml:id="echoid-s1454" xml:space="preserve">His igitur vaſis <lb/>aqua repletis de ejus effluxu obſervatum fuit.</s> + <s xml:id="echoid-s1455" xml:space="preserve"/> +</p> +<div xml:id="echoid-div72" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">Fig. 23. <lb/>24. 25. <lb/>& 26.</note> +</div> +<p> + <s xml:id="echoid-s1456" xml:space="preserve">I. </s> + <s xml:id="echoid-s1457" xml:space="preserve">Superficiem aquæ à principio non citius deſcendere in Fig. </s> + <s xml:id="echoid-s1458" xml:space="preserve">23. </s> + <s xml:id="echoid-s1459" xml:space="preserve">quam <lb/>Fig. </s> + <s xml:id="echoid-s1460" xml:space="preserve">24. </s> + <s xml:id="echoid-s1461" xml:space="preserve">poſtquam vero utrobique aliquid aquæ effluxit, multo celeriorem</s> +</p> +<pb o="56" file="0070" n="70" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s1462" xml:space="preserve">fieri motum in vaſe compoſito quam in ſimplici; </s> + <s xml:id="echoid-s1463" xml:space="preserve">utrumque præmonui in fine <lb/>§. </s> + <s xml:id="echoid-s1464" xml:space="preserve">23. </s> + <s xml:id="echoid-s1465" xml:space="preserve">Res autem melius & </s> + <s xml:id="echoid-s1466" xml:space="preserve">accuratius intelligitur ex æquationibus differen-<lb/>tialibus, quas §. </s> + <s xml:id="echoid-s1467" xml:space="preserve">§. </s> + <s xml:id="echoid-s1468" xml:space="preserve">22. </s> + <s xml:id="echoid-s1469" xml:space="preserve">& </s> + <s xml:id="echoid-s1470" xml:space="preserve">23. </s> + <s xml:id="echoid-s1471" xml:space="preserve">dedimus, quibus ſi utamur ad prima motuum in-<lb/>crementa invenienda, tam in cylindro ſimplici Fig. </s> + <s xml:id="echoid-s1472" xml:space="preserve">23. </s> + <s xml:id="echoid-s1473" xml:space="preserve">quam in compoſito <lb/>Fig. </s> + <s xml:id="echoid-s1474" xml:space="preserve">24. </s> + <s xml:id="echoid-s1475" xml:space="preserve">atque hunc in finem ponamus amplitudines cylindri & </s> + <s xml:id="echoid-s1476" xml:space="preserve">tubi eſſe ut <lb/>m ad n, erit incrementum, quod vocavimus d v in vaſe ſimplici ad incre-<lb/>mentum in vaſe compoſito, ut 1 + {3m/n} ad 4, adeoque longe majus in iſto <lb/>caſu quam in hoc. </s> + <s xml:id="echoid-s1477" xml:space="preserve">Si proin primum motum recte percipere liceret, cele-<lb/>riorem ſtatim illum obſervaturi eſſemus, qui fit in Cylindro ſimplici; </s> + <s xml:id="echoid-s1478" xml:space="preserve">Cum <lb/>vero in §. </s> + <s xml:id="echoid-s1479" xml:space="preserve">15. </s> + <s xml:id="echoid-s1480" xml:space="preserve">& </s> + <s xml:id="echoid-s1481" xml:space="preserve">23. </s> + <s xml:id="echoid-s1482" xml:space="preserve">porro demonſtratum fuerit, ſuperficiem aquæ, poſt-<lb/>quam paululum deſcendit in utroque vaſe proxime tales eſſe, quæ reſpon-<lb/>deant altitudinibus {nn/mm} x, intelligendo per x altitudines aquæ ſupra orificia, <lb/>per quæ effluit, ſequitur mox multo majori velocitate aquam deſcendere in <lb/>Fig. </s> + <s xml:id="echoid-s1483" xml:space="preserve">24. </s> + <s xml:id="echoid-s1484" xml:space="preserve">quam Fig. </s> + <s xml:id="echoid-s1485" xml:space="preserve">26. </s> + <s xml:id="echoid-s1486" xml:space="preserve">Sic igitur Theoria plane convenit cum obſervatis.</s> + <s xml:id="echoid-s1487" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1488" xml:space="preserve">II. </s> + <s xml:id="echoid-s1489" xml:space="preserve">Superficiem aqueam non parum velocius deſcendere in Figura 26. <lb/></s> + <s xml:id="echoid-s1490" xml:space="preserve">quam 24. </s> + <s xml:id="echoid-s1491" xml:space="preserve">ita ut velocitas in caſu Fig. </s> + <s xml:id="echoid-s1492" xml:space="preserve">24. </s> + <s xml:id="echoid-s1493" xml:space="preserve">ſit quaſi media inter caſus Figuræ <lb/>23. </s> + <s xml:id="echoid-s1494" xml:space="preserve">& </s> + <s xml:id="echoid-s1495" xml:space="preserve">26. </s> + <s xml:id="echoid-s1496" xml:space="preserve">Hic vero rurſus patet, primas quidem accelerationes multo tar-<lb/>dius fieri in cylindro Fig. </s> + <s xml:id="echoid-s1497" xml:space="preserve">24. </s> + <s xml:id="echoid-s1498" xml:space="preserve">quam 26. </s> + <s xml:id="echoid-s1499" xml:space="preserve">Hoc igitur reſpectu ipſa theoria in-<lb/>dicat, quod obſervatum fuit; </s> + <s xml:id="echoid-s1500" xml:space="preserve">at certe differentia multum abeſt, ut tanta <lb/>inde oriri poſſit, quantam expertus fui, neque amplius ſenſibilis eſſe debe-<lb/>ret, poſtquam utrobique ſuperficies paullulum deſcendit, per §. </s> + <s xml:id="echoid-s1501" xml:space="preserve">23. </s> + <s xml:id="echoid-s1502" xml:space="preserve"><lb/>debet autem reliquum impedimento tribui, quod ab attritu aquæ in Fig. </s> + <s xml:id="echoid-s1503" xml:space="preserve">24. </s> + <s xml:id="echoid-s1504" xml:space="preserve"><lb/>oritur: </s> + <s xml:id="echoid-s1505" xml:space="preserve">aqua enim per tubum A A magna velocitate fertur, ſicque tam ob <lb/>velocitatem auctam, quam ob amplitudinem vaſis diminutam impedimentum <lb/>motui aquæ validiſſimum offertur.</s> + <s xml:id="echoid-s1506" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1507" xml:space="preserve">III. </s> + <s xml:id="echoid-s1508" xml:space="preserve">Denique velociſſime, ſi prima temporis puncta excipias, aqueam <lb/>ſuperficiem deſcendere in Cylindro Fig. </s> + <s xml:id="echoid-s1509" xml:space="preserve">25. </s> + <s xml:id="echoid-s1510" xml:space="preserve">& </s> + <s xml:id="echoid-s1511" xml:space="preserve">notanter velocius quam in Fig. </s> + <s xml:id="echoid-s1512" xml:space="preserve">26.</s> + <s xml:id="echoid-s1513" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1514" xml:space="preserve">Id vero conforme eſt cum his quæ §. </s> + <s xml:id="echoid-s1515" xml:space="preserve">23. </s> + <s xml:id="echoid-s1516" xml:space="preserve">demonſtrata ſunt; </s> + <s xml:id="echoid-s1517" xml:space="preserve">deberent <lb/>autem mox poſt commune motus initium, poſitis nempe altitudinibus aquæ <lb/>ſupra orificia effluxus fere æqualibus, velocitates in Figuris 25. </s> + <s xml:id="echoid-s1518" xml:space="preserve">& </s> + <s xml:id="echoid-s1519" xml:space="preserve">26. </s> + <s xml:id="echoid-s1520" xml:space="preserve">pro-<lb/>xime eſſe ut amplitudines orificiorum B & </s> + <s xml:id="echoid-s1521" xml:space="preserve">A, id eſt, ut 25. </s> + <s xml:id="echoid-s1522" xml:space="preserve">ad 16. </s> + <s xml:id="echoid-s1523" xml:space="preserve">& </s> + <s xml:id="echoid-s1524" xml:space="preserve">quod <lb/>minor obſervetur velocitatum differentia, rurſus impedimento frictionis eſt <lb/>tribuendum præter aliam cauſam in fine §. </s> + <s xml:id="echoid-s1525" xml:space="preserve">25. </s> + <s xml:id="echoid-s1526" xml:space="preserve">indicatam.</s> + <s xml:id="echoid-s1527" xml:space="preserve"/> +</p> +<pb o="57" file="0071" n="71" rhead="SECTIO TERTIA."/> +</div> +<div xml:id="echoid-div74" type="section" level="1" n="52"> +<head xml:id="echoid-head68" style="it" xml:space="preserve">De iisdem vaſis, quibus tubi horizontales <lb/>inſeruntur.</head> +<head xml:id="echoid-head69" xml:space="preserve">Ad §. 24.</head> +<p> + <s xml:id="echoid-s1528" xml:space="preserve">CUm aquæ ex vaſe valde amplo veluti C D G (Fig. </s> + <s xml:id="echoid-s1529" xml:space="preserve">19.) </s> + <s xml:id="echoid-s1530" xml:space="preserve">per tubum ho-<lb/>rizontalem G M ampliorem in extremitate N M quam ortu G F fluunt, <lb/>majori velocitate illas ferri per orificium G F (ſi rurſus excipias pri-<lb/>mas guttas) quam ſi vel tubus abeſt, vel Cylindricus eſſet. </s> + <s xml:id="echoid-s1531" xml:space="preserve">Id etiam Frontinus <lb/>experientiâ procul dubio edoctus affirmavit, alii vero moderni negarunt.</s> + <s xml:id="echoid-s1532" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1533" xml:space="preserve">Igitur operæ pretium duxi rem experimento explorare. </s> + <s xml:id="echoid-s1534" xml:space="preserve">Erat autem <lb/>altitudo vaſis, quo uſus ſum, ſupra axem tubi = 5 {1/3} poll. </s> + <s xml:id="echoid-s1535" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s1536" xml:space="preserve">longitudo tu-<lb/>bi G N = 2 poll. </s> + <s xml:id="echoid-s1537" xml:space="preserve">5 lin. </s> + <s xml:id="echoid-s1538" xml:space="preserve">diameter orificii G F erat = 3, 36. </s> + <s xml:id="echoid-s1539" xml:space="preserve">lin. </s> + <s xml:id="echoid-s1540" xml:space="preserve">diameter <lb/>aperturæ M N = 5, 48. </s> + <s xml:id="echoid-s1541" xml:space="preserve">lin. </s> + <s xml:id="echoid-s1542" xml:space="preserve">erant proin amplitudines orificiorum ut 3. </s> + <s xml:id="echoid-s1543" xml:space="preserve">ad <lb/>8 proxime, amplitudo vaſis ſat magna erat, ut infinita cenſeri poſſet præ <lb/>amplitudine tubi. </s> + <s xml:id="echoid-s1544" xml:space="preserve">Volui omnes menſuras allegare, ut quivis experimentum <lb/>repetere poſſit. </s> + <s xml:id="echoid-s1545" xml:space="preserve">Hoc autem vaſe aqua repleto obſervavi amplitudinem jactus, <lb/>& </s> + <s xml:id="echoid-s1546" xml:space="preserve">ex hâc poſtquam omnes menſuras cognoviſſem requiſitas calculum poſui <lb/>de altitudine, quæ velocitati aquæ transfluentis tum in G F, tum in N M <lb/>deberetur: </s> + <s xml:id="echoid-s1547" xml:space="preserve">hanc inveni proxime undecim linearum, atque proin alteram <lb/>= poll. </s> + <s xml:id="echoid-s1548" xml:space="preserve">6. </s> + <s xml:id="echoid-s1549" xml:space="preserve">lin. </s> + <s xml:id="echoid-s1550" xml:space="preserve">cum duabus nonis lineæ partibus, quas easdem altitudines alio <lb/>etiam experimenti genere inveni. </s> + <s xml:id="echoid-s1551" xml:space="preserve">Quoniam autem major eſt altitudo 6. </s> + <s xml:id="echoid-s1552" xml:space="preserve">poll. <lb/></s> + <s xml:id="echoid-s1553" xml:space="preserve">cum 6 {2/9} lin. </s> + <s xml:id="echoid-s1554" xml:space="preserve">quam 5 {1/3} poll. </s> + <s xml:id="echoid-s1555" xml:space="preserve">confirmatur theoria noſtra de acceleratione aquæ <lb/>internæ ab amplificatione tubi verſus extrema, quamvis multum abſit, ut <lb/>duabus potiſſimum rationibus §. </s> + <s xml:id="echoid-s1556" xml:space="preserve">25. </s> + <s xml:id="echoid-s1557" xml:space="preserve">allegatis inductus præmonui, quin tan-<lb/>tum revera acceleretur quantum vi §. </s> + <s xml:id="echoid-s1558" xml:space="preserve">24. </s> + <s xml:id="echoid-s1559" xml:space="preserve">remotis obſtaculis, quorum in cal-<lb/>culo nulla ratio habita fuit, deberet.</s> + <s xml:id="echoid-s1560" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1561" xml:space="preserve">Ad §. </s> + <s xml:id="echoid-s1562" xml:space="preserve">25. </s> + <s xml:id="echoid-s1563" xml:space="preserve">Hoc paragrapho in tranſitu monui, multis modis fieri poſ-<lb/>ſe, ut aër aquæ per tubos fluenti miſceatur. </s> + <s xml:id="echoid-s1564" xml:space="preserve">Inde autem futurum, ut aquæ <lb/>minori copia effluant quidem, ſed non minori velocitate, quod utrumque <lb/>ut experirer primo in tubis A A & </s> + <s xml:id="echoid-s1565" xml:space="preserve">A B (Fig. </s> + <s xml:id="echoid-s1566" xml:space="preserve">24. </s> + <s xml:id="echoid-s1567" xml:space="preserve">& </s> + <s xml:id="echoid-s1568" xml:space="preserve">25.) </s> + <s xml:id="echoid-s1569" xml:space="preserve">non procul ab eorun-<lb/>dem Origine parvulum utrobique feci foramen; </s> + <s xml:id="echoid-s1570" xml:space="preserve">factum eſt, ut aquæ per <lb/>tubos, cum aliquo ſtrepitu ferrentut & </s> + <s xml:id="echoid-s1571" xml:space="preserve">turbidæ effluerent, ſuperficies au-<lb/>tem ſolito multo lentius deſcenderet; </s> + <s xml:id="echoid-s1572" xml:space="preserve">Deinde tubum Figuræ 19. </s> + <s xml:id="echoid-s1573" xml:space="preserve">pariter ali- +<pb o="58" file="0072" n="72" rhead="HYDRODYNAMICÆ."/> +quantulum perforavi, haud procul à G, rurſusque obſervavi, paullo lentius <lb/>deſcendere ſuperficiem internam, cujus rei me certum fecit quod numera-<lb/>bam oſcillationes alicujus penduli, quibus ſuperficies per datum ſpatium <lb/>deſcendit: </s> + <s xml:id="echoid-s1574" xml:space="preserve">ratione autem aquarum effluentium vidi aliquando aquas pleno <lb/>orificio effluere & </s> + <s xml:id="echoid-s1575" xml:space="preserve">tunc aquas ſolito minus eſſe pellucidas, jactum autem or-<lb/>dinarium vel ordinario paullo majorem facere; </s> + <s xml:id="echoid-s1576" xml:space="preserve">ſæpiſſime autem aquam & </s> + <s xml:id="echoid-s1577" xml:space="preserve"><lb/>aërem juxta ſe ferri, illam in parte tubi inferiore juxta latus F M, hunc in <lb/>ſuperiori juxta G N & </s> + <s xml:id="echoid-s1578" xml:space="preserve">tunc aquas eſſe limpidas atque velocitate ejici ſolito <lb/>non ſolum haud minori, ſed & </s> + <s xml:id="echoid-s1579" xml:space="preserve">multo majori, quod fieri poſſe haud obſcu-<lb/>re prævideram. </s> + <s xml:id="echoid-s1580" xml:space="preserve">De hâc re in ſequenti Sectione aliud experimentum majori <lb/>præciſione inſtitutum apponam.</s> + <s xml:id="echoid-s1581" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1582" xml:space="preserve">Dabitur autem fortaſſe alibi locus demonſtrandi aquas ſufficienti aëris <lb/>quantitate permiſtas, ea proxime effluere copia, qua effluerent reſciſſo tu-<lb/>bo eo in loco ubi eſt perforatus, cui rei ipſam quoque experentiam reſpon-<lb/>dere animadverti.</s> + <s xml:id="echoid-s1583" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div75" type="section" level="1" n="53"> +<head xml:id="echoid-head70" style="it" xml:space="preserve">De canalibus recurvis.</head> +<head xml:id="echoid-head71" xml:space="preserve">Ad §. 27.</head> +<p> + <s xml:id="echoid-s1584" xml:space="preserve">DUcta in pariete horizontali M N (Fig. </s> + <s xml:id="echoid-s1585" xml:space="preserve">27.) </s> + <s xml:id="echoid-s1586" xml:space="preserve">tubum cylindricum C D B <lb/> +<anchor type="note" xlink:label="note-0072-01a" xlink:href="note-0072-01"/> +totum aquâ plenum, cruraque ambo inter ſe parallela habentem, ita <lb/>poſui, ut extremitas altera B horizontalem M N raderet, ſimulque <lb/>crura eſſent verticalia, dum interea orificium C digito obturabam aquæ ef-<lb/>fluxum ſic compeſcens.</s> + <s xml:id="echoid-s1587" xml:space="preserve"/> +</p> +<div xml:id="echoid-div75" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0072-01" xlink:href="note-0072-01a" xml:space="preserve">Fig. 27.</note> +</div> +<p> + <s xml:id="echoid-s1588" xml:space="preserve">Dein remoto digito obſervavi, altitudinem maximam B P, ad quam <lb/>aquæ effluentes aſcendebant, aliisque vicibus attendi ad locum E, ad quem <lb/>deſcendit aquæ ſuperficies; </s> + <s xml:id="echoid-s1589" xml:space="preserve">feci autem fub duabus diverſis circumſtantiis ex-<lb/>perimentum; </s> + <s xml:id="echoid-s1590" xml:space="preserve">primo enim loco nullum in B poſueram operculum; </s> + <s xml:id="echoid-s1591" xml:space="preserve">dein-<lb/>de operculum adhibui tali lumine perforatum, quod amplitudinem haberet <lb/>ratione amplitudinis tubi ut 1. </s> + <s xml:id="echoid-s1592" xml:space="preserve">ad √ 2. </s> + <s xml:id="echoid-s1593" xml:space="preserve">Interim menſuræ tales fuere: </s> + <s xml:id="echoid-s1594" xml:space="preserve">C A <lb/>= 345; </s> + <s xml:id="echoid-s1595" xml:space="preserve">A D B = 530; </s> + <s xml:id="echoid-s1596" xml:space="preserve">B P = 33; </s> + <s xml:id="echoid-s1597" xml:space="preserve">& </s> + <s xml:id="echoid-s1598" xml:space="preserve">A E = 88. </s> + <s xml:id="echoid-s1599" xml:space="preserve">particulis, quarum 375 <lb/>longitudinem Pedis Lond. </s> + <s xml:id="echoid-s1600" xml:space="preserve">exæquabant. </s> + <s xml:id="echoid-s1601" xml:space="preserve">Hæc ita fuere in caſu priori, in al-<lb/>tero autem manentibus reliquis vidi B P = 64 & </s> + <s xml:id="echoid-s1602" xml:space="preserve">A E = 54. </s> + <s xml:id="echoid-s1603" xml:space="preserve">Notabo hîcin tran-<lb/>ſitu, quod alio explorare cupiens modo maximum deſcenſum A E, poſt fini-<lb/>tum experimentum inclinaverim tubum, donec aqua jam jam effluxui per B pro- +<pb o="59" file="0073" n="73" rhead="SECTIO TERTIA."/> +xima videbatur, quo temporis puncto diſtantiam menſuravi ſuperficiei à loco <lb/>A antea notato; </s> + <s xml:id="echoid-s1604" xml:space="preserve">diſtantia illa, quam eandem cum maximo deſcenſu A E fore pu-<lb/>tabam, opinione longe minor fuit; </s> + <s xml:id="echoid-s1605" xml:space="preserve">unde edoctus fui partem aquæ, quæ <lb/>in experimento jam per B effluxerat, tubum rurſus ingreſſam fuiſſe.</s> + <s xml:id="echoid-s1606" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1607" xml:space="preserve">His ita obſervatis, magnitudines B P & </s> + <s xml:id="echoid-s1608" xml:space="preserve">A E calculo quæſivi ad normam <lb/>§. </s> + <s xml:id="echoid-s1609" xml:space="preserve">27. </s> + <s xml:id="echoid-s1610" xml:space="preserve">ponendo m primo = n deinde mm = 2nn, inveni autem in caſu prio-<lb/>re B P = 79. </s> + <s xml:id="echoid-s1611" xml:space="preserve">quæ in experimento non ſuperavit 33. </s> + <s xml:id="echoid-s1612" xml:space="preserve">maximumque deſcen-<lb/>ſum AE proxime reperi = 250. </s> + <s xml:id="echoid-s1613" xml:space="preserve">quem experimentum dedit 88. </s> + <s xml:id="echoid-s1614" xml:space="preserve">dein pro <lb/>caſu mm = 2 nn oritur B P præter propter dupla illius, quæ obſervata <lb/>fuit & </s> + <s xml:id="echoid-s1615" xml:space="preserve">A E = 186. </s> + <s xml:id="echoid-s1616" xml:space="preserve">quæ 54. </s> + <s xml:id="echoid-s1617" xml:space="preserve">particularum obſervata fuit.</s> + <s xml:id="echoid-s1618" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1619" xml:space="preserve">Enormes has differentias maxima ex parte adhæſioni aquæ ad latera tu-<lb/>bi tribuo, quæ certe adhæſio in hujusmodi caſibus incredibilem exercere po-<lb/>teſt effectum, uſus enim ſum tubo vix ultra duas lineas in diametro haben-<lb/>tem, majorem utique conſenſum experturus cum tubo ampliore. </s> + <s xml:id="echoid-s1620" xml:space="preserve">Interim <lb/>veriſimile eſt, curvaturam tubi in parte inferiore, aliquid etiam motui derogare.</s> + <s xml:id="echoid-s1621" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1622" xml:space="preserve">Ad §. </s> + <s xml:id="echoid-s1623" xml:space="preserve">28. </s> + <s xml:id="echoid-s1624" xml:space="preserve">Eodem tubo recurvo, quem modo deſcripſi, uſus ſum: </s> + <s xml:id="echoid-s1625" xml:space="preserve">oper-<lb/>culum autem poſui in B minimo foramine pertuſum: </s> + <s xml:id="echoid-s1626" xml:space="preserve">feci ut totus aquâ <lb/>eſſet plenus præter particulam F G B, in quo ſitu aquam detinui ope digiti <lb/>orificio C appoſiti. </s> + <s xml:id="echoid-s1627" xml:space="preserve">Remoto digito deſcendit aqua, & </s> + <s xml:id="echoid-s1628" xml:space="preserve">cum perveniſſet in <lb/>ſitum H D B, guttulæ aliquot tanto impetu per foraminulum in B fuerunt <lb/>veluti exploſæ, ut ad altitudinem plusquam decem pedum aſcenderint, quam-<lb/>vis altitudo H A altitudinem dimidii pedis vix ſuperaret. </s> + <s xml:id="echoid-s1629" xml:space="preserve">Interim ob exigui-<lb/>tatem foraminuli tantam reſiſtentiam offendit aqua dum tranſiret orificium, ut <lb/>fracto impetu aqua non ſolum non ad altitudinem A H aſcenderit (ſupra quam <lb/>tamen remotis omnibus impedimentis aſſilire paullulum continue debuiſſet) <lb/>ſed vix guttula una aut altera notabili temporis mora fuerit expreſſa, ita ut mihi <lb/>perſuadeam, ſi absque impetu ſola aquæ preſſione naturali tantus jactus pro-<lb/>ducendus fuiſſet, id fieri non potuiſſe niſi altitudine minimum centum pedum.</s> + <s xml:id="echoid-s1630" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1631" xml:space="preserve">Dein etiam obſervavi jactum aquæ diminui eo magis quo minus ante <lb/>experimentum relinquitur ſpatium G B; </s> + <s xml:id="echoid-s1632" xml:space="preserve">quæ omnia theoriæ ſunt conformia. <lb/></s> + <s xml:id="echoid-s1633" xml:space="preserve">Menſuras ſuperfluum fuiſſet ſumere, quia ob nimia impedimenta tantus eſſe <lb/>utique nequit jactus aquæ, quantus illis remotis futurus fuiſſet. </s> + <s xml:id="echoid-s1634" xml:space="preserve">Attamen ut <lb/>& </s> + <s xml:id="echoid-s1635" xml:space="preserve">has convenire cum formulis experimento confirmarem, tubum C D B <lb/>ſumſi ampliorem, ut impedimenta adhæſionis maxima parte auferrem, pars +<pb o="60" file="0074" n="74" rhead="HYDRODYNAMICÆ."/> +D F B parvula erat, minorque etiam pars G B, quam in experimento ab aqua <lb/>relinquebam vacuam: </s> + <s xml:id="echoid-s1636" xml:space="preserve">ac denique operculum foramine non admodum parvo <lb/>erat pertuſum. </s> + <s xml:id="echoid-s1637" xml:space="preserve">Et tunc vidi ſaltum non multum admodum defeciſſe ab altitu-<lb/>dine {mmδ/nnß} a, quam §. </s> + <s xml:id="echoid-s1638" xml:space="preserve">28. </s> + <s xml:id="echoid-s1639" xml:space="preserve">pro hoc negotio dedi, imo memini me præſenti <lb/>Amico altitudinem ſaltus recte prædixiſſe, poſtquam perpendiſſem, quantum <lb/>in calculo præter propter impedimentis eſſet dandum.</s> + <s xml:id="echoid-s1640" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1641" xml:space="preserve">Similem aquæ exploſionem momentaneam eamque à ſimili cauſa oriun-<lb/>dam facillime obtinebis cum fontibus, qui aquas per fiſtulam pleno orificio eji-<lb/>ciunt. </s> + <s xml:id="echoid-s1642" xml:space="preserve">Si enim digitum orificio fiſtulæ ſubito ita apponas, ut pars orificii aper-<lb/>ta maneat, protinus aquas magno impetu expelli videbis, moxque tenue aquæ <lb/>filum intra priſtinos velocitatis limites reduci. </s> + <s xml:id="echoid-s1643" xml:space="preserve">Obſervabis etiam aquas eò ma-<lb/>jori impetu atque longius projici quo minus digito relinquas foramen, atque <lb/>pro eodem relicto foramine, jactum inſolitum eò magis protrahi (utut ſemper <lb/>breviſſimum) fierique oculis ſenſibiliorem, quo longior eſt fiſtula, ita ut in <lb/>fontibus ſalientibus, ad quos aquæ ex caſtello per longiſſimos canales ferun-<lb/>tur, ſi canales non eſſent admodum ampli & </s> + <s xml:id="echoid-s1644" xml:space="preserve">aquæ pleno effluerent orificio, <lb/>non dubito quin ſic per notabile temporis ſpatium vehemens aquæ jactus pro-<lb/>trahi poſſet, gradatim ad ſolitam velocitatem rediturus: </s> + <s xml:id="echoid-s1645" xml:space="preserve">Hæc omnia confor-<lb/>mia ſunt cum iis, quæ §. </s> + <s xml:id="echoid-s1646" xml:space="preserve">§. </s> + <s xml:id="echoid-s1647" xml:space="preserve">28. </s> + <s xml:id="echoid-s1648" xml:space="preserve">& </s> + <s xml:id="echoid-s1649" xml:space="preserve">18. </s> + <s xml:id="echoid-s1650" xml:space="preserve">monita fuerunt.</s> + <s xml:id="echoid-s1651" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1652" xml:space="preserve">Experimentum hoc me aliquando & </s> + <s xml:id="echoid-s1653" xml:space="preserve">prima quidem vice feciſſe memi-<lb/>ni coram V. </s> + <s xml:id="echoid-s1654" xml:space="preserve">V. </s> + <s xml:id="echoid-s1655" xml:space="preserve">Cel. </s> + <s xml:id="echoid-s1656" xml:space="preserve">D. </s> + <s xml:id="echoid-s1657" xml:space="preserve">D. </s> + <s xml:id="echoid-s1658" xml:space="preserve">De Maupertuis & </s> + <s xml:id="echoid-s1659" xml:space="preserve">Clairaut, cum quibus antea in ſer-<lb/>monem de rebus iſtis aquariis forte delapſus eram. </s> + <s xml:id="echoid-s1660" xml:space="preserve">Quamvis autem hic nullus <lb/>ſit aër, qui accuſari poſſit, revera tamen phænomenon iſtud ab eo, quod D. <lb/></s> + <s xml:id="echoid-s1661" xml:space="preserve">de la Hire obſervatum fuit, non differt, & </s> + <s xml:id="echoid-s1662" xml:space="preserve">utrumque ab eo provenit, quod <lb/>motus aquæ in canali contentæ, vel ſaltem motus iſtius pars perire non poſſit, <lb/>fine ullo inde proveniente effectu, quem ipſe enormis aquarum jactus con-<lb/>ſtituit.</s> + <s xml:id="echoid-s1663" xml:space="preserve"/> +</p> +<pb file="0075" n="75" rhead="(61)"/> +</div> +<div xml:id="echoid-div77" type="section" level="1" n="54"> +<head xml:id="echoid-head72" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ <lb/>SECTIO QUARTA.</emph></head> +<head xml:id="echoid-head73" style="it" xml:space="preserve">De variis temporibus, quæ in effluxu aquarum <lb/>deſiderari poſſunt.</head> +<head xml:id="echoid-head74" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s1664" xml:space="preserve">REs videbitur multis omnino Geometrica, quæ ſcilicet nulla conſide-<lb/>ratione phyſica opus habeat, ut, cum aquæ ex dato vaſe per lumen co-<lb/>gnitum velocitatibus in omniſitu determinatis effluunt, tempus de-<lb/>finiatur, quo data effluat aquæ quantitas. </s> + <s xml:id="echoid-s1665" xml:space="preserve">Attamen experientia contra-<lb/>rium docet; </s> + <s xml:id="echoid-s1666" xml:space="preserve">nam multo minori quantitate aquæ effluunt perforamina, quæ ſunt <lb/>in lamina tenui, quam ex ſimplici velocitatum conſideratione ſequeretur, idque <lb/>plerumque (nec enim res ſibi conſtat in diverſis circumſtantiis) in ratione ut <lb/>1 ad √ 2; </s> + <s xml:id="echoid-s1667" xml:space="preserve">movit hoc Newtonum, ut affirmaret in prima Princ. </s> + <s xml:id="echoid-s1668" xml:space="preserve">math. </s> + <s xml:id="echoid-s1669" xml:space="preserve">editio-<lb/>ne aquam ex vaſe ea effluere velocitate, quæ generetur altitudine dimidia aquæ <lb/>ſupra foramen, cui opinioni omnia experimenta de velocitatibus immediate <lb/>ſumta, contradicunt. </s> + <s xml:id="echoid-s1670" xml:space="preserve">Explorans poſtmodum ipſe magnus Vir hujus contra-<lb/>dictionis originem, eam poſitam eſſe obſervavit in contractione venæ aqueæ, <lb/>quæ contractio mox præ foramine fieri ſolet. </s> + <s xml:id="echoid-s1671" xml:space="preserve">Alia quoque mihi obſervata fuit <lb/>venæ mutatio priori nunc ſimilis nunc contraria. </s> + <s xml:id="echoid-s1672" xml:space="preserve">Nempe cum aquæ non per <lb/>fimplex foramen, verum per tubulum effluunt, rurſus contrahitur vena, ſi <lb/>tubus exteriora verſus convergit, ſed dilatatur ſi idem divergit. </s> + <s xml:id="echoid-s1673" xml:space="preserve">De contra-<lb/>ctione venæ aqueæ per tubos convergentes effluentis accuratiſſima ſumſit ex-<lb/>perimenta Joh. </s> + <s xml:id="echoid-s1674" xml:space="preserve">Polenus in libro de caſtellis p. </s> + <s xml:id="echoid-s1675" xml:space="preserve">15. </s> + <s xml:id="echoid-s1676" xml:space="preserve">& </s> + <s xml:id="echoid-s1677" xml:space="preserve">ſeqq. </s> + <s xml:id="echoid-s1678" xml:space="preserve">contractio venæ eo <lb/>major à Viro Celeberrimo obſervata fuit, quo amplius erat orificium tubi co-<lb/>nici internum manentibus orificio externo atque longitudine tubi, quæ ratio eſt, <lb/>quod ſimilis aquæ quantitas ceteris paribus eò tardius effluxerit, quò amplius <lb/>fuerit orificium internum, quamvis impedimenta ab adhæſione aquæ ad late- +<pb o="62" file="0076" n="76" rhead="HYDRODYNAMICÆ."/> +ra tubi minorem continue habuerit effectum: </s> + <s xml:id="echoid-s1679" xml:space="preserve">fecerunt autem iſtæ impedimen-<lb/>torum diminutiones, ut aquæ majori velocitate in loco, quo vena maxime <lb/>erat contracta, fluerent, & </s> + <s xml:id="echoid-s1680" xml:space="preserve">nihilominus parcius erogarentur: </s> + <s xml:id="echoid-s1681" xml:space="preserve">verum id eſſe <lb/>colligitur ex obſervatis effluxus temporibus & </s> + <s xml:id="echoid-s1682" xml:space="preserve">venarum, ubi maxime contra-<lb/>huntur, amplitudinibus. </s> + <s xml:id="echoid-s1683" xml:space="preserve">Igitur cum in hiſce venæ mutationibus cardo reiver-<lb/>tatur, è re erit phænomena uberius examinare & </s> + <s xml:id="echoid-s1684" xml:space="preserve">explicare.</s> + <s xml:id="echoid-s1685" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1686" xml:space="preserve">§. </s> + <s xml:id="echoid-s1687" xml:space="preserve">2. </s> + <s xml:id="echoid-s1688" xml:space="preserve">Aſſumamus v. </s> + <s xml:id="echoid-s1689" xml:space="preserve">gr. </s> + <s xml:id="echoid-s1690" xml:space="preserve">cylindrum verticalem, qui in medio fundi <lb/>horizontaliter poſiti, habeat foramen, aqua autem interna diviſa concipiatur <lb/>in ſtrata horizontalia: </s> + <s xml:id="echoid-s1691" xml:space="preserve">His ita poſitis, cenſuimus motum cujusvis ſtrati eun-<lb/>dem eſſe & </s> + <s xml:id="echoid-s1692" xml:space="preserve">talem quidem, ut ſitus horizontalis in illis conſervetur, ubi tamen <lb/>monui, non poſſe hanc hypotheſin extendi ad ſtrata foramini proxima, quo-<lb/>niam vero inde nullus error ſenſibilis oriri poſſit ratione velocitatis aquarum <lb/>effluentium, operæ pretium non eſſe, ut ejus rei ratio habeatur. </s> + <s xml:id="echoid-s1693" xml:space="preserve">Nunc vero, <lb/>quando alia phænomena à motu aquæ internæ obliquo, qualis præſertim in <lb/>prædictis ſtratis foramini proximis eſt, pendent, hunc paucis luſtrabimus.</s> + <s xml:id="echoid-s1694" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1695" xml:space="preserve">§ 3. </s> + <s xml:id="echoid-s1696" xml:space="preserve">Mihi autem videtur motum@ aquæ internæ talem eſſe conci-<lb/>piendum, qualis foret ſi aqua ferretur per tubulos infinitos juxta ſe poſitos, <lb/>quorum intermedii proxime rectà à ſuperficie verſus foramen deſcendunt, re-<lb/>liquis ſenſim ſe incurvantibus prope foramen, uti Fig. </s> + <s xml:id="echoid-s1697" xml:space="preserve">28. </s> + <s xml:id="echoid-s1698" xml:space="preserve">a oſtendit, ex quâ <lb/> +<anchor type="note" xlink:label="note-0076-01a" xlink:href="note-0076-01"/> +apparet, ſingulas particulas hoc modo deſcendere motu tantum non verticali, <lb/>donec fundum prope attingant, eaſque tunc curſum ſuum ſenſim verſus fo-<lb/>ramen inflectere, ita ut particulæ fundo proximæ motu fere horizontali, alte-<lb/>ræ magis verticaliter ad foramen effluant. </s> + <s xml:id="echoid-s1699" xml:space="preserve">Hujuſmodi motus ſæpe oculis ob-<lb/>ſervare potui, cum particulæ ceræ, quam vocant Hiſpanicæ, innatabant aquæ. <lb/></s> + <s xml:id="echoid-s1700" xml:space="preserve">Exinde autem intelligitur non poſſe ſingulas particulas foramini adſtantes dire-<lb/>ctionem ſuam integram ſervare, neque tamen ita eam inflectere, ut motum axi <lb/>plane parallelum aſſumant, ſed fore potius, ut vena aquæ effluentis contra-<lb/>hatur uſque in d e, ubi ſic notabiliter gracilior erit, quam in ortu circa fora-<lb/>men a c. </s> + <s xml:id="echoid-s1701" xml:space="preserve">Hæc autem contractio venæ verticaliter fluentis non confundenda eſt <lb/>cum alia contractione, quæ fit ab acceleratione aquæ. </s> + <s xml:id="echoid-s1702" xml:space="preserve">Dein patet quoque, <lb/>quod cum ſingularum particularum foramini adſtantium diverſa ſit directio, <lb/>neceſſario ab impetu, quem in ſe mutuo faciunt e<unsure/>ædem particulæ, vena com- +<pb o="63" file="0077" n="77" rhead="SECTIO QUARTA."/> +primatur, atque ſic gracileſcat. </s> + <s xml:id="echoid-s1703" xml:space="preserve">Et ab iſta compreſſione fit, quod alias con-<lb/>tradictionem involveret, ut aqua jam jam egreſſa, etiamnum præ foramine ac-<lb/>celeretur, & </s> + <s xml:id="echoid-s1704" xml:space="preserve">ſic aſcenſ{us} potentialis creſcat, etiamſi ad alteram accelerationem <lb/>omnibus corporibus cadentibus communem non attendamus, ceu huc non <lb/>pertinentem, & </s> + <s xml:id="echoid-s1705" xml:space="preserve">cujus deinceps mentionem non faciemus. </s> + <s xml:id="echoid-s1706" xml:space="preserve">Hæc autem niſi me <lb/>fallat opinio, res erit porro hunc in modum tractanda.</s> + <s xml:id="echoid-s1707" xml:space="preserve"/> +</p> +<div xml:id="echoid-div77" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0076-01" xlink:href="note-0076-01a" xml:space="preserve">Fig. 28. a.</note> +</div> +<p> + <s xml:id="echoid-s1708" xml:space="preserve">(I.) </s> + <s xml:id="echoid-s1709" xml:space="preserve">Eouſque vena aquæ conſideranda eſt, donec particularum velo-<lb/>citates amplius non mutentur, quod quamvis nunquam fiat omni rigore, at-<lb/>tamen non procul à foramine fieri cenſendum eſt, veluti in d e. </s> + <s xml:id="echoid-s1710" xml:space="preserve">Hoc <lb/>autem ſi ita fuerit & </s> + <s xml:id="echoid-s1711" xml:space="preserve">aquæ ex vaſe A B C D per foramen a c effluere <lb/>ponantur, erit loco vaſis ſimplicis A B C D concipiendum aliud com-<lb/>poſitum A B a d e c C D.</s> + <s xml:id="echoid-s1712" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1713" xml:space="preserve">Quicquid igitur in præcedente ſectione præmiſſum fuit, pro determi-<lb/>nandis ubique velocitatibus, id omnino locum habebit, ſi loco vaſis ſubjecti <lb/>concipiatur vas, quod dixi tubulo contracto inſtructum. </s> + <s xml:id="echoid-s1714" xml:space="preserve">Nec tamen hæc cor-<lb/>rectio, ratione præmiſſæ noſtræ methodi velocitatum aquæ effluentis determi-<lb/>nandarum, ſenſibilem mutationem producere poteſt ob brevitatem tubuli a d c e, <lb/>poteſt autem valde notabilem ratione quantitatis, quia aquæ non tam per ori-<lb/>ficium a c, quam per d e effluere cenſendæ ſunt.</s> + <s xml:id="echoid-s1715" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1716" xml:space="preserve">(II.) </s> + <s xml:id="echoid-s1717" xml:space="preserve">Sic erunt velocitates in diverſis locis ipſius venæ reciproce ut <lb/>amplitudines ſectionum reſpondentium & </s> + <s xml:id="echoid-s1718" xml:space="preserve">cum in vaſis ampliſſimis velocitas <lb/>in d e talis ſit quæ toti altitudini aquæ conveniat, ſimulque experimentis con-<lb/>ſtet, amplitudines a c & </s> + <s xml:id="echoid-s1719" xml:space="preserve">d e proxime eſſe ut √ 2 ad 1, putavit Newtonus ſic <lb/>confirmari poſſe theoriam ſuam, qua ſtatuit aquam ex foramine vero veloci-<lb/>tate effluere quæ debeatur dimidiæ altitudini aquæ ſupra foramen, quamvis in <lb/>progreſſu velocitas aquæ creſcat: </s> + <s xml:id="echoid-s1720" xml:space="preserve">quâ in re mihi videtur nimium adhæſiſſe præ-<lb/>conceptæ opinioni: </s> + <s xml:id="echoid-s1721" xml:space="preserve">neque enim ratio orificii a c ad d e ſemper eadem eſt, ne-<lb/>que ſic explicari poteſt motus aquarum ex vaſe, cui tubulus adhæret: </s> + <s xml:id="echoid-s1722" xml:space="preserve">verbo! <lb/>attenuatio venæ prorſus accidentalis eſt, poteſt enim tota impediri, apponen-<lb/>do foramini parvulum tubulum cylindricum vel augendo tantum craſſitiem la-<lb/>minæ, cui foramen ineſt, & </s> + <s xml:id="echoid-s1723" xml:space="preserve">tunc ſine ulla correctione locum habent tam ra- +<pb o="64" file="0078" n="78" rhead="HYDRODYNAMICÆ."/> +tione velocitatum quam quantitatum theoremata, quæ in præcedente ſectione <lb/>exhibita fuerunt.</s> + <s xml:id="echoid-s1724" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1725" xml:space="preserve">(III.) </s> + <s xml:id="echoid-s1726" xml:space="preserve">Patet autem ex ipſa explicatione ſupra data de contractione venæ, <lb/>non poſſe non illam à diverſis circumſtantiis mutari; </s> + <s xml:id="echoid-s1727" xml:space="preserve">ita experimenta docent, <lb/>diminui eandem ab auctâ laterum fo<unsure/>raminis craſſitie: </s> + <s xml:id="echoid-s1728" xml:space="preserve">an altitudo aquæ ſupra <lb/>foramen aliquid conferat non ſatis ſcio: </s> + <s xml:id="echoid-s1729" xml:space="preserve">crediderim fere creſcere aliquantulum <lb/>contractionem ab aucta altitudine aquæ internæ, quamvis facile parum id fore <lb/>prævideam: </s> + <s xml:id="echoid-s1730" xml:space="preserve">veriſimile quoque eſt, eo minorem cæteris paribus fore contra-<lb/>ctionem venæ, præſertim verticalis, quo majorem rationem habuerit amplitudo <lb/>foraminis ad amplitudinem cylindri, quia motus aquæ internæ fundo proximæ <lb/>eo minus fit obliquus, ita ut ſi foramen totam amplitudinem cylindri occupet, <lb/>nulla utique attenuatio venæ aqueæ oriri poſſit. </s> + <s xml:id="echoid-s1731" xml:space="preserve">Ad hoc animum advertant ve-<lb/>lim, qui hujus contractionis in ipſa velocitatum determinatione rationem ha-<lb/>bendam eſſe fortaſſe cogitabunt. </s> + <s xml:id="echoid-s1732" xml:space="preserve">Cum enim foramen non multo minus eſt <lb/>amplitudine vaſis, nulla oriri poteſt contractio notabilis & </s> + <s xml:id="echoid-s1733" xml:space="preserve">cum foramen eſt <lb/>parvum, nulla rurſus oritur fere differentia circa velocitates ſive foramen ali-<lb/>quantum augeatur ſive diminuatur.</s> + <s xml:id="echoid-s1734" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1735" xml:space="preserve">§. </s> + <s xml:id="echoid-s1736" xml:space="preserve">4. </s> + <s xml:id="echoid-s1737" xml:space="preserve">Eadem propemodum ratio eſt aquarum horizontaliter, ut de <lb/>aliis directionibus taceam, effluentium: </s> + <s xml:id="echoid-s1738" xml:space="preserve">nam ſimili modo ab omni parte af-<lb/>fluet aqua ad foramen; </s> + <s xml:id="echoid-s1739" xml:space="preserve">imo etiam ex inferiori parte aſcendet uſque ad foramen <lb/>ut effluere poſſit, quod ipſe ſæpe fieri obſervavi. </s> + <s xml:id="echoid-s1740" xml:space="preserve">Simili igitur cauſa ſimilis <lb/>fiet in vena effluente attenuatio, quam eo facilius eſt oculis perſpicere, quod <lb/>hîc locum non habeat altera atteuuatio ab acceleratione aquæ jam egreſſæ <lb/>oriunda. </s> + <s xml:id="echoid-s1741" xml:space="preserve">Et ob hanc rationem, ſi quis obſervationes circa contractionem <lb/>venæ facere inſtituat, is meo judicio melius faciet, utendo venis hori<unsure/>zontali-<lb/>ter, quam ſub aliâ directione effluentibus.</s> + <s xml:id="echoid-s1742" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1743" xml:space="preserve">§. </s> + <s xml:id="echoid-s1744" xml:space="preserve">5. </s> + <s xml:id="echoid-s1745" xml:space="preserve">Quanta autem ſit contractio, id eſt, quænam ratio intercedat <lb/>inter amplitudinem orificii ſectionemque venæ horizontaliter effluentis mini-<lb/>mam experiri licet vel ſumendo actu menſuras diametrorum iſtis amplitudini-<lb/>bus reſpondentium, vel etiam mediante quantitate aquæ dato tempore, da-<lb/>tisque velocitatibus effluentis, ubi tamen velocitates non tam ex altitudine <lb/>aquæ ſupra foramen, quam ex amplitudine jactus deducenda erunt, quando- +<pb o="65" file="0079" n="79" rhead="SECTIO QUARTA."/> +quidem impedimenta nunc majora nunc minora nunquam omnem aquæ ve-<lb/>locitatem permittant, quam vi theoriæ, qua horum impedimentorum ra-<lb/>tio nulla habetur, acqui<unsure/>ere deberet.</s> + <s xml:id="echoid-s1746" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1747" xml:space="preserve">§. </s> + <s xml:id="echoid-s1748" xml:space="preserve">6. </s> + <s xml:id="echoid-s1749" xml:space="preserve">Ex præmiſſis nunc ſatis patere puto perfectum conſenſum fore <lb/>inter quantitatem aquæ effluentis ejuſque velocitatem, ſi modo foramini, quod <lb/>eſt in vaſe, ſubſtituatur aliud foramen eo uſque diminutum, donec ſectionem <lb/>venæ maxime contractæ non ſuperet: </s> + <s xml:id="echoid-s1750" xml:space="preserve">atque perinde erit, in quonam venæ <lb/>loco, aut in quânam profunditate à ſuperficie aquæ foramen hoc eſſe conſti-<lb/>tuatur, ſive in a c ſive in d e, quandoquidem velocitates ſemper proxime re-<lb/>ſpondebunt toti altitudini aquæ ſupra eum locum, quo foramen fingitur: </s> + <s xml:id="echoid-s1751" xml:space="preserve">am-<lb/>plitudinem hujus foraminis mente concipiendi vocabo deinceps Sectionem ve-<lb/>næ aqueæ contractæ.</s> + <s xml:id="echoid-s1752" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1753" xml:space="preserve">§. </s> + <s xml:id="echoid-s1754" xml:space="preserve">7. </s> + <s xml:id="echoid-s1755" xml:space="preserve">Quod ſi jam Sectio iſta, de quâ modo diximus, conſtantem <lb/>haberet rationem ad orificium, in eadem ratione diminuendum cogitatione <lb/>foret foramen effluxus, poſtmodumque calculus de quantitate aquæ dato tem-<lb/>pore effluentis inſtituendus. </s> + <s xml:id="echoid-s1756" xml:space="preserve">Ita nempe poſita iſta ratione = {1/α} nominatâque <lb/>amplitudine orificii n, cenſenda eſſet Sectio venæ ſolidæ = {n/α}.</s> + <s xml:id="echoid-s1757" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1758" xml:space="preserve">At variabilis cum ſit ſub diverſis circumſtantiis, regulas in hanc'<unsure/>rem à <lb/>priori dare non licet: </s> + <s xml:id="echoid-s1759" xml:space="preserve">mutatur autem maxime à craſſitie laminæ, in quâ<unsure/> fora-<lb/>men eſt, aucta vel diminuta: </s> + <s xml:id="echoid-s1760" xml:space="preserve">aliquid etiam, quamvis id parum, conferre po-<lb/>teſt magnitudo foraminis, amplitudines vaſis, hæque tam abſolutæ, quam rela-<lb/>tivæ, ut & </s> + <s xml:id="echoid-s1761" xml:space="preserve">fortaſſe altitudo aquæ ſupra foramen. </s> + <s xml:id="echoid-s1762" xml:space="preserve">Interim aſſumtis lamina te-<lb/>nui, vaſe ampliſſimo, foramine ad 4. </s> + <s xml:id="echoid-s1763" xml:space="preserve">vel 6. </s> + <s xml:id="echoid-s1764" xml:space="preserve">lineas in diametro aſſurgente; </s> + <s xml:id="echoid-s1765" xml:space="preserve">ſolet <lb/>ratio inter foramen & </s> + <s xml:id="echoid-s1766" xml:space="preserve">Sectionem venæ contractæ non multum recedere ab illâ, <lb/>quam Newtonus ſtatuit, nempe ut √ 2 ad 1. </s> + <s xml:id="echoid-s1767" xml:space="preserve">Sæpe autem ab aliis major ob-<lb/>ſervata fuit, atque ab aliis etiam minor.</s> + <s xml:id="echoid-s1768" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1769" xml:space="preserve">§. </s> + <s xml:id="echoid-s1770" xml:space="preserve">8. </s> + <s xml:id="echoid-s1771" xml:space="preserve">Quæcunque vero ſit, in quolibet caſu illam indicabimus, ut an-<lb/>te, per {α/1.</s> + <s xml:id="echoid-s1772" xml:space="preserve">} Huicque poſitioni nunc calculum pro temporibus ſuperinſtruemus; <lb/></s> + <s xml:id="echoid-s1773" xml:space="preserve">brevitatis autem gratia conſiderabimus tantum vaſa cylindrica, atque in his <lb/>duo potiſſimum examinabimus temporum genera; </s> + <s xml:id="echoid-s1774" xml:space="preserve">primum quod punctum <lb/>maximæ velocitatis definit, alterum, quod depletioni reſpondet. </s> + <s xml:id="echoid-s1775" xml:space="preserve">In utroque <lb/>vero caſu motum à quiete incipere ponemus.</s> + <s xml:id="echoid-s1776" xml:space="preserve"/> +</p> +<pb o="66" file="0080" n="80" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s1777" xml:space="preserve">§. </s> + <s xml:id="echoid-s1778" xml:space="preserve">9. </s> + <s xml:id="echoid-s1779" xml:space="preserve">Fuerit igitur vas cylindricum verticaliter poſitum aqua ple-<lb/>num, ſitque altitudo aquæ ab initio fluxus = a, amplitudo cylindri = m, am-<lb/>plitudo foraminis = n, Sectio venæ ſolidæ = {n/α} effluxerit jam aqua per tempus <lb/>t; </s> + <s xml:id="echoid-s1780" xml:space="preserve">ſitque tunc altitudo aquæ reſidua ſupra foramen = x, eodemque temporis <lb/>puncto habeat ſuperficies aquæ internæ velocitatem, quæ reſpondeat altitudini <lb/>v: </s> + <s xml:id="echoid-s1781" xml:space="preserve">erit velocitas ipſa = √ v, eſt autem elementum temporis d t proportio-<lb/>nale elemento ſpatii - d x diviſo per velocitatem √v, unde dt = {- dx/√v}.</s> + <s xml:id="echoid-s1782" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1783" xml:space="preserve">Determinatus @equidem fuit valor ipſius v in ſect. </s> + <s xml:id="echoid-s1784" xml:space="preserve">3. </s> + <s xml:id="echoid-s1785" xml:space="preserve">ubi iisdem denomi-<lb/>nationibus uſi ſumus, quibus nunc utimur. </s> + <s xml:id="echoid-s1786" xml:space="preserve">At quoniam pro recta aquarum <lb/>erogatarum menſura requiritur, ut foramini n ſubſtituatur ſectio venæ con-<lb/>tractæ {n/α}, ſequitur, ut in valore ipſius v eadem fiat ſubſtitutio atque ſic ſta-<lb/>tuatur v = {nna/2nn - mmαα}(({a/x})<emph style="super">{1 - mmαα/nn}</emph> - {x/a})</s> +</p> +<p> + <s xml:id="echoid-s1787" xml:space="preserve">Hic vero valor ſi ſubſtituatur in æquatione <lb/>dt = {- dx/√v}, oritur <lb/>dt = - dx: </s> + <s xml:id="echoid-s1788" xml:space="preserve">√[{nna/2nn - mmαα} (({a/x})<emph style="super">{1 - mmαα/nn}</emph> - {x/a})] <lb/>ope cujus æquationis omnia tempora deſiderata definiri poſſunt per approxi-<lb/>mationes, ſeu ſeries, ſi modo in ſingulis punctis valor ipſius α innoteſcat:</s> + <s xml:id="echoid-s1789" xml:space="preserve"><unsure/> <lb/>Aſſumemus autem eſſe illum conſtantis valoris, quandoquidem in præſenti <lb/>caſu nihil ſit, à quo mutari poſſit præter diverſas altitudines & </s> + <s xml:id="echoid-s1790" xml:space="preserve">velocitates <lb/>fluidi, quæ parum vel nihil quantum ſenſibus percipi poteſt ad id negotii <lb/>conferunt.</s> + <s xml:id="echoid-s1791" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1792" xml:space="preserve">§. </s> + <s xml:id="echoid-s1793" xml:space="preserve">10. </s> + <s xml:id="echoid-s1794" xml:space="preserve">Jam ut æquatio deſiderata per ſeries exhiberi poſſit, conſiderabi-<lb/>mus quantitatem.</s> + <s xml:id="echoid-s1795" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1796" xml:space="preserve">1:</s> + <s xml:id="echoid-s1797" xml:space="preserve">√[{nna/2nn - mmαα} (({a/x})<emph style="super">{1 - mmαα/nn}</emph> - {x/a})] fub hâc forma <lb/>({nnx/mmαα - 2nn})<emph style="super">- {1/2}</emph> X (1 - ({x/a})<emph style="super">{mmαα/nn} - 2</emph>) - <emph style="super">{1/2}</emph> factoremque +<pb o="67" file="0081" n="81" rhead="SECTIO QUARTA."/> +poſteriorem per regulas ſolitas reſolvemus in hanc ſeriem <lb/>1 + {1/2} ({x/a})<emph style="super">{mmαα/nn} - 2</emph> + {1. </s> + <s xml:id="echoid-s1798" xml:space="preserve">3/1. </s> + <s xml:id="echoid-s1799" xml:space="preserve">2. </s> + <s xml:id="echoid-s1800" xml:space="preserve">4} - ({x/a})<emph style="super">{2mmαα/nn}</emph> - 4 + {1.</s> + <s xml:id="echoid-s1801" xml:space="preserve">3.</s> + <s xml:id="echoid-s1802" xml:space="preserve">5/1.</s> + <s xml:id="echoid-s1803" xml:space="preserve">2.</s> + <s xml:id="echoid-s1804" xml:space="preserve">3.</s> + <s xml:id="echoid-s1805" xml:space="preserve">8}({x/a})<emph style="super">{3mmαα/nn} - 6</emph> <lb/>+ &</s> + <s xml:id="echoid-s1806" xml:space="preserve">c. </s> + <s xml:id="echoid-s1807" xml:space="preserve">unde nunc habetur mutata paullulum æquationis forma: <lb/></s> + <s xml:id="echoid-s1808" xml:space="preserve">dt = - {dx√mmαα - 2nn}/n√a} X [({x/a})<emph style="super">- {1/2}</emph> + {1/2} ({x/a})<emph style="super">{mmαα/nn} - {@/z}</emph> <lb/>+ {1.</s> + <s xml:id="echoid-s1809" xml:space="preserve">3/1.</s> + <s xml:id="echoid-s1810" xml:space="preserve">2.</s> + <s xml:id="echoid-s1811" xml:space="preserve">4} ({x/a})<emph style="super">{2mmαα/nn} -{9/2}</emph> + {1.</s> + <s xml:id="echoid-s1812" xml:space="preserve">3, 5/1.</s> + <s xml:id="echoid-s1813" xml:space="preserve">2.</s> + <s xml:id="echoid-s1814" xml:space="preserve">3.</s> + <s xml:id="echoid-s1815" xml:space="preserve">8} ({x/a})<emph style="super">{3mmαα/nn} - {13/2}</emph> + &</s> + <s xml:id="echoid-s1816" xml:space="preserve">c. </s> + <s xml:id="echoid-s1817" xml:space="preserve">] <lb/>Hæc æquatio ita eſt integranda, ut poſita x = a fiat t = 0; </s> + <s xml:id="echoid-s1818" xml:space="preserve">ſic autem oritur <lb/>t = [2 + {nn/2mmαα - 3nn} + {3nn/16mmαα - 28nn} + &</s> + <s xml:id="echoid-s1819" xml:space="preserve">c.</s> + <s xml:id="echoid-s1820" xml:space="preserve">] X {√(mmαα - 2nn).</s> + <s xml:id="echoid-s1821" xml:space="preserve">a/n} <lb/>- [2{(x/a)}<emph style="super">{1/2}</emph> + {nn/2mmαα - 3nn} ({x/a})<emph style="super">{mmαα/nn}-{3/2}</emph> <lb/>+ {3nn/16mmαα - 28nn} ({x/a}) <emph style="super">{2mmαα/nn} - {7/2}</emph> + &</s> + <s xml:id="echoid-s1822" xml:space="preserve">c. </s> + <s xml:id="echoid-s1823" xml:space="preserve">] X <lb/>X {√(mmαα - 2nn).</s> + <s xml:id="echoid-s1824" xml:space="preserve">a/n}, <lb/>ubi 2 √ a exprimit tempus quod corpus impendit dum libere delabitur per <lb/>altitudinem a. </s> + <s xml:id="echoid-s1825" xml:space="preserve">Si vero in iſta æquatione ponatur <lb/>x = a:</s> + <s xml:id="echoid-s1826" xml:space="preserve">({mmαα - nn/nn})<emph style="super">nn: ({mmαα - 2nn})</emph> <lb/>quæ eſt altitudo aquæ cum velocitas maxima eſt (per §. 16. </s> + <s xml:id="echoid-s1827" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s1828" xml:space="preserve">3. </s> + <s xml:id="echoid-s1829" xml:space="preserve">& </s> + <s xml:id="echoid-s1830" xml:space="preserve">§. </s> + <s xml:id="echoid-s1831" xml:space="preserve">8. </s> + <s xml:id="echoid-s1832" xml:space="preserve">ſect <lb/>4.)</s> + <s xml:id="echoid-s1833" xml:space="preserve">, tum obtinetur tempus quod à fluxus principio ad punctum maximæ ve-<lb/>locitatis usque præterit; </s> + <s xml:id="echoid-s1834" xml:space="preserve">& </s> + <s xml:id="echoid-s1835" xml:space="preserve">cum ponitur x = o, oritur tempus, quo vas to-<lb/>tum depletur, ac denique ſi ponatur x = cuicunque quantitati c, exprimet t <lb/>tempus quod ſuperficies inſumit in deſcenſum per altitudinem a - c; </s> + <s xml:id="echoid-s1836" xml:space="preserve">Videbi-<lb/>mus autem pro his caſibus, quid fieri debeat, cum vas eſt valde amplum, <lb/>numerusque m alterum n ſic pluries continet.</s> + <s xml:id="echoid-s1837" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1838" xml:space="preserve">§. </s> + <s xml:id="echoid-s1839" xml:space="preserve">11. </s> + <s xml:id="echoid-s1840" xml:space="preserve">Fuerit primo {m/n} numerus infinitus<unsure/>, erit altitudo aquæ puncto <lb/>maximæ velocitatis reſpondens ſeu</s> +</p> +<pb o="68" file="0082" n="82" rhead="HYDRODYNAMICÆ."/> +<p> + <s xml:id="echoid-s1841" xml:space="preserve">a: </s> + <s xml:id="echoid-s1842" xml:space="preserve">({mmαα - nn/nn})<emph style="super">{nn: (mmαα - 2nn)}</emph> = a: </s> + <s xml:id="echoid-s1843" xml:space="preserve">({mmαα/nn})<emph style="super">nn: mmαα</emph> <lb/>quoniam autem {mmαα/nn} eſt numerus infinitus, poterit cenſeri: <lb/></s> + <s xml:id="echoid-s1844" xml:space="preserve">({mmαα/nn})<emph style="super">nn: mmαα</emph> = 1 + (log.</s> + <s xml:id="echoid-s1845" xml:space="preserve">{mmαα/nn}): </s> + <s xml:id="echoid-s1846" xml:space="preserve">{mmαα/nn}; </s> + <s xml:id="echoid-s1847" xml:space="preserve"><lb/>cujus rei demonſtratio talis eſt: </s> + <s xml:id="echoid-s1848" xml:space="preserve">propoſita ſit quantitas infinita A habeaturq; </s> + <s xml:id="echoid-s1849" xml:space="preserve">ut in <lb/>noſtro exemplo A<emph style="super">1: A</emph>, facile quisque videt eſſe hanc quantitatem paullo majo-<lb/>rem, quam eſt unitas, & </s> + <s xml:id="echoid-s1850" xml:space="preserve">quidem exceſſu infinite parvo, quem vocabimus <lb/>z; </s> + <s xml:id="echoid-s1851" xml:space="preserve">habetur itaque A<emph style="super">1 : A</emph> = 1 + z, ſumantur utrobique logarithmi & </s> + <s xml:id="echoid-s1852" xml:space="preserve">erit <lb/>{log. </s> + <s xml:id="echoid-s1853" xml:space="preserve">A/A} = log. </s> + <s xml:id="echoid-s1854" xml:space="preserve">(1 + z) = (ob infinitè parvum valorem ipſius z) z; </s> + <s xml:id="echoid-s1855" xml:space="preserve">Igitur <lb/>eſt A<emph style="super">1: A</emph> = 1 + {log. </s> + <s xml:id="echoid-s1856" xml:space="preserve">A/A}: </s> + <s xml:id="echoid-s1857" xml:space="preserve">proindeque ſimiliter eſt, ut diximus, <lb/>({mmαα/nn})<emph style="super">nn: mmαα</emph> = 1 + (log.</s> + <s xml:id="echoid-s1858" xml:space="preserve">{mmαα/nn}):</s> + <s xml:id="echoid-s1859" xml:space="preserve">{mmαα/nn}</s> +</p> +<p> + <s xml:id="echoid-s1860" xml:space="preserve">Porro quia quantitas hæc unitati addita eſt infinitè parva, erit <lb/>a:</s> + <s xml:id="echoid-s1861" xml:space="preserve">({mmαα/nn})<emph style="super">nn: mmαα</emph> ſeu <lb/>a:</s> + <s xml:id="echoid-s1862" xml:space="preserve">[1 + (log.</s> + <s xml:id="echoid-s1863" xml:space="preserve">{mmαα/nn}):</s> + <s xml:id="echoid-s1864" xml:space="preserve">{mmαα/nn}) = a - a (log. </s> + <s xml:id="echoid-s1865" xml:space="preserve">{mmαα/nn}):</s> + <s xml:id="echoid-s1866" xml:space="preserve">{mmαα/nn}: <lb/></s> + <s xml:id="echoid-s1867" xml:space="preserve">eſt igitur ſpatium per quod ſuperficies aquæ deſcendit, dum à quiete maxi-<lb/>ma oritur velocitas = a (log. </s> + <s xml:id="echoid-s1868" xml:space="preserve">{mmαα/nn}): </s> + <s xml:id="echoid-s1869" xml:space="preserve">{mmαα/nn}, ſeu = {2nna/mmαα} log. </s> + <s xml:id="echoid-s1870" xml:space="preserve">{mα/n}.</s> + <s xml:id="echoid-s1871" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1872" xml:space="preserve">Indicat hæc æquatio deſcenſum aquæ in vaſe infinite amplo infinite par-<lb/>vum eſſe, cum aqua jam maximum velocitatis gradum attigerit: </s> + <s xml:id="echoid-s1873" xml:space="preserve">Potuiſſet au-<lb/>tem hoc non obſtante dubitari, an non interea quantitas aquæ finita effluat, <lb/>quandoquidem cylindrus ſuper baſi infinita erectus, utut altitudinis infinite <lb/>parvæ magnitudinem poſſit habere infinitam: </s> + <s xml:id="echoid-s1874" xml:space="preserve">at ſequitur ex noſtra æquatio-<lb/>ne, hanc quoque quantitatem infinite parvam eſſe, & </s> + <s xml:id="echoid-s1875" xml:space="preserve">nominatim æqualem <lb/>{@nna/mαα}log.</s> + <s xml:id="echoid-s1876" xml:space="preserve">{mα/n}.</s> + <s xml:id="echoid-s1877" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1878" xml:space="preserve">Atque convenit hoc egregie profecto cum phænomenis, quæ in ef-<lb/>fluxu aquarum ex caſtellis per ſimplex foramen toto die experimur. </s> + <s xml:id="echoid-s1879" xml:space="preserve">Cum +<pb o="69" file="0083" n="83" rhead="SECTIO QUARTA."/> +enim foramen digito obturamus, moxque remoto digito aquas horizontali-<lb/>ter effluere ſinimus, nullam guttulam in terram delapſam obſervamus me-<lb/>diam inter jactum longiſſimum & </s> + <s xml:id="echoid-s1880" xml:space="preserve">locum, qui foramini ad perpendiculum <lb/>reſpondeat.</s> + <s xml:id="echoid-s1881" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1882" xml:space="preserve">§. </s> + <s xml:id="echoid-s1883" xml:space="preserve">12. </s> + <s xml:id="echoid-s1884" xml:space="preserve">Prouti in proximo paragrapho determinavimus quantitates ut-<lb/>ut infinite parvas, deſcenſus aquæ internæ uti & </s> + <s xml:id="echoid-s1885" xml:space="preserve">effluentis aquæ dum maxi-<lb/>ximum velocitatis gradum aqua attingit, ita nunc idem præſtabimus ratione <lb/>tempusculi. </s> + <s xml:id="echoid-s1886" xml:space="preserve">Dico eutem ſufficere in æquatione §. </s> + <s xml:id="echoid-s1887" xml:space="preserve">10. </s> + <s xml:id="echoid-s1888" xml:space="preserve">tempus exprimente, <lb/>ut in utraque ſerie unicus accipiatur terminus primus, quod apparebit cum <lb/>quis calculum ad duos extenderit terminos: </s> + <s xml:id="echoid-s1889" xml:space="preserve">eſt igitur tempuſculum quæſi-<lb/>tum ſive <lb/>t = (2 - 2√{x/a}) X {√(mmαα - 2nn).</s> + <s xml:id="echoid-s1890" xml:space="preserve">a/n} <lb/>hinc poſito pro x valore huc pertinente, qui in præcedente paragrapho fuit <lb/>definitus, fit <lb/>t = [2 - 2√1 - (log.</s> + <s xml:id="echoid-s1891" xml:space="preserve">{mmαα/nn}): </s> + <s xml:id="echoid-s1892" xml:space="preserve">{mmαα/nn}] X √({mmαα - 2 nn/nn})·a <lb/>vel poſito 1 - (log. </s> + <s xml:id="echoid-s1893" xml:space="preserve">{mmαα/nn}): </s> + <s xml:id="echoid-s1894" xml:space="preserve">{2mmαα/nn} pro reſpondente quantitate ſigno ra-<lb/>dicali involuta prodit <lb/>t = [(log.</s> + <s xml:id="echoid-s1895" xml:space="preserve">{mmαα/nn}): </s> + <s xml:id="echoid-s1896" xml:space="preserve">{mmαα/nn}] X √({mmαα - 2nn/nn})·a} <lb/>aut denique rejecta quantitate 2 nn in ſigno radicali, oritur t = {2n√a/mα}.</s> + <s xml:id="echoid-s1897" xml:space="preserve">log.</s> + <s xml:id="echoid-s1898" xml:space="preserve">{mα/n}.</s> + <s xml:id="echoid-s1899" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1900" xml:space="preserve">Eſt autem hoc tempusculum infinite parvum, quia, ut notum eſt, lo-<lb/>garithmus quantitatis infinitæ infinities minor eſt ipsâ quantitate. </s> + <s xml:id="echoid-s1901" xml:space="preserve">At vero <lb/>cum ſic ſtatim ab initio fluxus, aqua maxima ſua velocitate expellitur, mi-<lb/>rum prima fronte videbitur fortaſſe aliquibus, motum in inſtanti generari <lb/>finitum: </s> + <s xml:id="echoid-s1902" xml:space="preserve">nemo tamen abſurdum putabit, maſſam infinitam, cujusmodi <lb/>eſt quantitas aquæ in vaſe infinito contentæ, poſſe tempuſculo infinitè parvo <lb/>motum producere finitum, idque ſolâ gravitatis actione.</s> + <s xml:id="echoid-s1903" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1904" xml:space="preserve">§. </s> + <s xml:id="echoid-s1905" xml:space="preserve">13. </s> + <s xml:id="echoid-s1906" xml:space="preserve">Si præterea in iſta vaſis infinite ampli poſitione tempus deple-<lb/>tionis, quod utique infinitum erit, exprimere velimus, erit, ut ſupra indi- +<pb o="70" file="0084" n="84" rhead="HYDRODYNAMICÆ."/> +catum fuit, in æquatione paragraphi decimi ponendum x = o, ſint<unsure/>ulque <lb/>ſolus primus ſeriei terminus adhibendus rurſusque ponendum m α pro <lb/>√(mmαα - 2nn); </s> + <s xml:id="echoid-s1907" xml:space="preserve">atque ſic fit <lb/>t = {2mα/n}√a.</s> + <s xml:id="echoid-s1908" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1909" xml:space="preserve">Tum denique tempus, quod impenditur in deſcenſum ſuperficiei per <lb/>altitudinem a - c exprimetur in ſimili hypotheſi hac æquatione <lb/>t = {2ma/n} (√a - √c).</s> + <s xml:id="echoid-s1910" xml:space="preserve">}</s> +</p> +<p> + <s xml:id="echoid-s1911" xml:space="preserve">§. </s> + <s xml:id="echoid-s1912" xml:space="preserve">14. </s> + <s xml:id="echoid-s1913" xml:space="preserve">Præmiſſæ æquationes non accurate quidem, proxime tamen <lb/>ſatisfacient, cum vas non infinitæ, permagnæ tamen amplitudinis eſt: </s> + <s xml:id="echoid-s1914" xml:space="preserve">imo <lb/>non multum admodum defie<unsure/>ient, cum numerus m vel mediocriter ſuperat <lb/>numerum n. </s> + <s xml:id="echoid-s1915" xml:space="preserve">Liceat quædam hic verba adjicere circa experimentum quod in <lb/>fine paragraphi undecimi indicavi, deturque hæc venia inſtituto noſtro, <lb/>quod in phænomenis motuum experientia cognitis potiſſimum verſatur il-<lb/>luſtrandis examinandisque. </s> + <s xml:id="echoid-s1916" xml:space="preserve">Dixi autem in citato paragrapho cum aqua ho-<lb/>rizontaliter effluit, primam guttulam totam ſtatim obtinere amplitudinem <lb/>jactus; </s> + <s xml:id="echoid-s1917" xml:space="preserve">atque idem hoc quidem indicat theoria pro vaſis ampliſſimis; </s> + <s xml:id="echoid-s1918" xml:space="preserve">at ve-<lb/>ro in vaſis mediocriter amplis, quædam guttulæ minori impetu effluere de-<lb/>berent, priusquam punctum maximæ velocitatis adſit, hæque guttulæ in-<lb/>cidere deberent in locum aliquem medium inter maximum jactum & </s> + <s xml:id="echoid-s1919" xml:space="preserve">pun-<lb/>ctum, quod foramini verticaliter reſpondet; </s> + <s xml:id="echoid-s1920" xml:space="preserve">atque hoc etiam ita fieri ob-<lb/>ſervavi, ex vaſis amplitudinis veluti decies foramine majoris. </s> + <s xml:id="echoid-s1921" xml:space="preserve">Verum cum <lb/>experimentum aliquando ſumerem de vaſe pedem dimidium alto, quod am-<lb/>plitudinem præter propter centuplam haberet foraminis, ne minima quidem <lb/>particula aquæ, quantum videre potui, notabiliter à jactu aquæ pleno de-<lb/>fecit. </s> + <s xml:id="echoid-s1922" xml:space="preserve">Videamus itaque quænam aquæ quantitas in hoc caſu effluere deberet <lb/>ante punctum maximæ velocitatis; </s> + <s xml:id="echoid-s1923" xml:space="preserve">erit autem tanta, quantam continet cy-<lb/>lindrus ejusdem amplitudinis in altitudine <lb/>a - a: </s> + <s xml:id="echoid-s1924" xml:space="preserve">({mmαα - nn/nn})<emph style="super">nn: (mmαα - 2nn)</emph> <lb/>(vid. </s> + <s xml:id="echoid-s1925" xml:space="preserve">§. </s> + <s xml:id="echoid-s1926" xml:space="preserve">10. </s> + <s xml:id="echoid-s1927" xml:space="preserve">ſub. </s> + <s xml:id="echoid-s1928" xml:space="preserve">fin.)</s> + <s xml:id="echoid-s1929" xml:space="preserve">; </s> + <s xml:id="echoid-s1930" xml:space="preserve">nec differt fere hæc minima altitudo ab hac multo com-<lb/>pendioſiori, nempe {2nna/mmαα} log. </s> + <s xml:id="echoid-s1931" xml:space="preserve">{mα/n} (vid. </s> + <s xml:id="echoid-s1932" xml:space="preserve">§. </s> + <s xml:id="echoid-s1933" xml:space="preserve">11.) </s> + <s xml:id="echoid-s1934" xml:space="preserve">ubi nunc per {n/m} intelligi- +<pb o="71" file="0085" n="85" rhead="SECTIO QUARTA."/> +tur {1/100} & </s> + <s xml:id="echoid-s1935" xml:space="preserve">per a pes dimidius, dum pro a ſubſtitui poteſt √2. </s> + <s xml:id="echoid-s1936" xml:space="preserve">(non deſide-<lb/>ramus enim hic ſummam accurationem) & </s> + <s xml:id="echoid-s1937" xml:space="preserve">per log. </s> + <s xml:id="echoid-s1938" xml:space="preserve">indicatur logarithmus <lb/>hyperbolicus, ita vero fit, <lb/>{2nna/mmαα}log.</s> + <s xml:id="echoid-s1939" xml:space="preserve">{mα/n} = {1/20000} (log. </s> + <s xml:id="echoid-s1940" xml:space="preserve">100. </s> + <s xml:id="echoid-s1941" xml:space="preserve">+ {1/2} log. </s> + <s xml:id="echoid-s1942" xml:space="preserve">2.) </s> + <s xml:id="echoid-s1943" xml:space="preserve">= <lb/>0, 0002475 ped. </s> + <s xml:id="echoid-s1944" xml:space="preserve">ſeu, 0, 000297 poll. </s> + <s xml:id="echoid-s1945" xml:space="preserve">& </s> + <s xml:id="echoid-s1946" xml:space="preserve">quoniam amplitudinem vaſis <lb/>æqualem inveneram 6 {1/5} poll. </s> + <s xml:id="echoid-s1947" xml:space="preserve">quadratis, intellexi quantitatem aquæ quæſitam, <lb/>quæ nempe effluere debuiſſet priusquam jactus maximus oriretur, exæquare <lb/>circiter partem quinquageſimam ſecundam unius pollicis cubici, ſeu, poſito <lb/>guttam mediocrem ſex lineas cubicas efficere, plusquam quinque guttas. </s> + <s xml:id="echoid-s1948" xml:space="preserve">In ex-<lb/>perimento autem nullam obſervavi, cujus rei rationem eſſe ſuſpicor, quod primæ <lb/>guttulæ, quam vis jam ejectæ ab aqua ſubſequente tamen etiamnum propellantur; <lb/></s> + <s xml:id="echoid-s1949" xml:space="preserve">nimis enim celeriter alteræ ſubſequuntur, quam ut primæ ab illis interea divelli <lb/>poſſint. </s> + <s xml:id="echoid-s1950" xml:space="preserve">Huc autem facit, quod tempusculum à fluxus initio ad maximam ex-<lb/>Pulſionem usque (quod nempe per §. </s> + <s xml:id="echoid-s1951" xml:space="preserve">13. </s> + <s xml:id="echoid-s1952" xml:space="preserve">eſt proxime = {2n@√a/mα} log. </s> + <s xml:id="echoid-s1953" xml:space="preserve">{mα/n}, ubi <lb/>per 2√a hic intelligitur tempus, quo corpus per altitudinem dimidii pedis <lb/>labitur, id eſt, circiter {2/11} unius minuti ſecundi) quod inquam tempuſculum <lb/>illud non ultra partem centeſimam quinquageſimam octavam unius minuti <lb/>ſecundi excurrat.</s> + <s xml:id="echoid-s1954" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1955" xml:space="preserve">Fortaſſe aliquid contribuit, quod non poſſit digitus ſat celeriter à fo-<lb/>ramine removeri. </s> + <s xml:id="echoid-s1956" xml:space="preserve">Præſertim vero huc pertinet, quod maxima pars illius <lb/>aquæ, quæ ante præſentem maximam velocitatem erumpit, ita ad maximam <lb/>jactum accedat, ut nulla differentia obſervari poſſit & </s> + <s xml:id="echoid-s1957" xml:space="preserve">ſic vix unica guttula <lb/>notabili diſcrimine ab illo defectura fuiſſet, ſi ſe libere ab aqua ſubſequente <lb/>ſeparare potuiſſet.</s> + <s xml:id="echoid-s1958" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1959" xml:space="preserve">§. </s> + <s xml:id="echoid-s1960" xml:space="preserve">15. </s> + <s xml:id="echoid-s1961" xml:space="preserve">Hactenus de aquis per foramina effluentibus: </s> + <s xml:id="echoid-s1962" xml:space="preserve">progrediamur <lb/>nunc ad effluxum aquarum ex vaſis per conos ſeu convergentes ſeu diver-<lb/>gentes. </s> + <s xml:id="echoid-s1963" xml:space="preserve">Quod ſi autem aquæ effluant per tubum convergentem, dictat ea-<lb/>dem ratio à motu particularum convergente petita §. </s> + <s xml:id="echoid-s1964" xml:space="preserve">3. </s> + <s xml:id="echoid-s1965" xml:space="preserve">pro foraminibus ſim-<lb/>plicibus expoſita, fore ut aquæ vena præ foramine contrahatur etiam-<lb/>num ejusque particulæ accelerentur & </s> + <s xml:id="echoid-s1966" xml:space="preserve">ſic quantitas aquæ dato tem-<lb/>pore effluentis minor ſit quam menſuræ orificii effluxus & </s> + <s xml:id="echoid-s1967" xml:space="preserve">velocitatum, <lb/>nulla habita ratione ad contractionem venæ, indicant. </s> + <s xml:id="echoid-s1968" xml:space="preserve">Parva autem ſolet +<pb o="72" file="0086" n="86" rhead="HYDRODYNAMICÆ."/> +eſſe iſta contractio in tubis longioribus. </s> + <s xml:id="echoid-s1969" xml:space="preserve">In tubis divergentibus omnia fiunt <lb/>modo contrario: </s> + <s xml:id="echoid-s1970" xml:space="preserve">dilatatur enim vena præ foramine; </s> + <s xml:id="echoid-s1971" xml:space="preserve">aquæ motus retarda-<lb/>tur & </s> + <s xml:id="echoid-s1972" xml:space="preserve">major aquæ quantitas dato tempore effluit, quam ſine iſta dilatatione <lb/>ſequeretur ex obſervatis amplitudine orificii & </s> + <s xml:id="echoid-s1973" xml:space="preserve">velocitatibus aquæ per illud <lb/>effluentis. </s> + <s xml:id="echoid-s1974" xml:space="preserve">Ex tubis denique cylindricis effluens vena aquea nec contrahitur <lb/>nec dilatatur.</s> + <s xml:id="echoid-s1975" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1976" xml:space="preserve">Probe eſt itaque attendendum ad has ſive contractiones ſive dilatatio-<lb/>nes in æſtimandis quantitatibus aquæ dato tempore effluentis, quam quæ-<lb/>ſtionem obiter tractabimus in fine ſectionis.</s> + <s xml:id="echoid-s1977" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1978" xml:space="preserve">Nunc autem libet examini ſubjicere mutationes quæ in effluxus aquarum <lb/>ſuccedunt ab initio motus. </s> + <s xml:id="echoid-s1979" xml:space="preserve">In his vero compendii cauſa non attendemus ad <lb/>mutationes venæ; </s> + <s xml:id="echoid-s1980" xml:space="preserve">neque enim res ita eſt comparata ut poſſit experimentis ſatis <lb/>accurate confirmari neque magni momenti hic ſunt præfatæ mutationes; </s> + <s xml:id="echoid-s1981" xml:space="preserve">res <lb/>autem ipſa digna eſt, quæ ſollicite perquiratur ut ejus natura animo recte in-<lb/>telligi poſſit.</s> + <s xml:id="echoid-s1982" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1983" xml:space="preserve">De vaſis, quæ tubos hahent annexos, jamjam egimus in ſuperiori ſe-<lb/>ctiones §. </s> + <s xml:id="echoid-s1984" xml:space="preserve">31. </s> + <s xml:id="echoid-s1985" xml:space="preserve">§. </s> + <s xml:id="echoid-s1986" xml:space="preserve">32. </s> + <s xml:id="echoid-s1987" xml:space="preserve">§. </s> + <s xml:id="echoid-s1988" xml:space="preserve">33. </s> + <s xml:id="echoid-s1989" xml:space="preserve">& </s> + <s xml:id="echoid-s1990" xml:space="preserve">quidem paragrapho 31. </s> + <s xml:id="echoid-s1991" xml:space="preserve">æquationes dedimus ge-<lb/>neraliores, quæcunque fuerit ratio inter amplitudines vaſis & </s> + <s xml:id="echoid-s1992" xml:space="preserve">tubi: </s> + <s xml:id="echoid-s1993" xml:space="preserve">ſed ni-<lb/>mis ſunt perplexæ calculumque poſtulant admodum operoſum: </s> + <s xml:id="echoid-s1994" xml:space="preserve">In paragra-<lb/>pho, qui hunc ſequitur, hypotheſin pertractavi, quæ vas ubique amplitudi-<lb/>nis infinitæ ratione tubi facit, in qua hypotheſi dixi, aquam effluere velocita-<lb/>te, qua ad integram altitudinem aquæ ſupra orificium effluxus aſcendere poſ-<lb/>ſit; </s> + <s xml:id="echoid-s1995" xml:space="preserve">ſed tamen in fine paragraphi expreſſe monui, ab initio motus aquam <lb/>tardius deſcendere, quam ſic definitum fuit, nec regulam iſtam prius locum <lb/>habere, quam ſuperficies per ſpatiolum aliquod deſcenderit, quæ res per ſe ſa-<lb/>tis patet, quandoquidem non poſſit in inſtanti velocitas maxima produci à <lb/>ſtatu quietis in tubo, quamvis fiat in vaſe foramine ſimplici perforato.</s> + <s xml:id="echoid-s1996" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s1997" xml:space="preserve">Hæc ita perpendens animo concepi mutationes initiales explorare, eas-<lb/>que ad certas menſuras reducere. </s> + <s xml:id="echoid-s1998" xml:space="preserve">Ad hoc autem minime ſufficit præmemorata <lb/>regula, quâ iſtarum mutationum initialium nulla ratio habetur, quamvis cæ-<lb/>terum exacte vera in vaſe infinite amplo; </s> + <s xml:id="echoid-s1999" xml:space="preserve">omnes enim mutationes quæ ſta-<lb/>tum maximæ velocitatis præcedunt, fiunt dum ſuperficies per ſpatiolum infi-<lb/>nite parvum deſcendunt; </s> + <s xml:id="echoid-s2000" xml:space="preserve">attamen deſcenſus iſte, ſi modo vas fuerit ſenſu +<pb o="73" file="0087" n="87" rhead="SECTIO QUARTA."/> +Geometrico infinitum, non ſolum non fit tempore infinite parvo, prouti in <lb/>caſu foraminis ſimplicis, ſed tempore infinitè magno, intereaque etiam quan-<lb/>titas aquæ infinita effluit, cum per foramen quantitas cæteris paribus infinite <lb/>parva effluat. </s> + <s xml:id="echoid-s2001" xml:space="preserve">Hæc autem ut eruerem, opus habui aliam elicere æquationem <lb/>ex æquatione generali §. </s> + <s xml:id="echoid-s2002" xml:space="preserve">23. </s> + <s xml:id="echoid-s2003" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s2004" xml:space="preserve">3. </s> + <s xml:id="echoid-s2005" xml:space="preserve">quam ſimpliciſſimam hanc s = x, poſita <lb/>s pro altitudine, quæ velocitati aquæ effluentis reſpondeat & </s> + <s xml:id="echoid-s2006" xml:space="preserve">x pro altitudi-<lb/>ne aquæ ſupra orificium effluxus; </s> + <s xml:id="echoid-s2007" xml:space="preserve">intelliget autem quisque rem pro inſtitu-<lb/>to noſtro ita eſſe efficiendam, ut habeatur ratio incrementorum velocitatis, <lb/>quod antea non requirebatur.</s> + <s xml:id="echoid-s2008" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2009" xml:space="preserve">§. </s> + <s xml:id="echoid-s2010" xml:space="preserve">16. </s> + <s xml:id="echoid-s2011" xml:space="preserve">Fuerit igitur ut in paragrapho 22. </s> + <s xml:id="echoid-s2012" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s2013" xml:space="preserve">3. </s> + <s xml:id="echoid-s2014" xml:space="preserve">cylindrus A E H B <lb/>(Fig. </s> + <s xml:id="echoid-s2015" xml:space="preserve">18.) </s> + <s xml:id="echoid-s2016" xml:space="preserve">is que cenſeatur infinite amplus & </s> + <s xml:id="echoid-s2017" xml:space="preserve">aqua plenus, habeatque tubum <lb/>annexum F M N G finitæ amplitudinis formæ coni truncati, ſive creſcentis <lb/>amplitudine ſive decreſcentis verſus orificium M N, per quod aquæ effluunt: <lb/></s> + <s xml:id="echoid-s2018" xml:space="preserve">ſit ut ibi altitudo initialis aquæ ſupra foramen M N, nempe N G + H B = a; </s> + <s xml:id="echoid-s2019" xml:space="preserve"><lb/>altitudo ſuperficiei aqueæ in ſitu C D ſupra M N, id eſt, N G + H D = x; </s> + <s xml:id="echoid-s2020" xml:space="preserve"><lb/>longitudo tubi annexi ſeu N G = b, amplitudo orificii M N = n, amplitudo <lb/>orificii F G = g, amplitudo cylindri, quæ eſt infinita, = m; </s> + <s xml:id="echoid-s2021" xml:space="preserve">ſitque tandem <lb/>velocitas ſuperficiei aquæ in ſitu C D talis quæ conveniat altitudini v, quæ <lb/>altitudo utique infinite parva erit. </s> + <s xml:id="echoid-s2022" xml:space="preserve">His poſitis vidimus loco citato obtinere <lb/>generaliter hanc æquationem: </s> + <s xml:id="echoid-s2023" xml:space="preserve"><lb/>m(x - b)dv + {bmm/√gn}dv - {m<emph style="super">3</emph>/nn}vdx + mvdx = - mxdx <lb/>in quâ patet, poſſe nunc negligi terminum primum m(x - b)dv præ ſe-<lb/>cundo {bmm/√gn}dv, ut & </s> + <s xml:id="echoid-s2024" xml:space="preserve">quartum mvdx præ tertio - {m<emph style="super">3</emph>/nn}vdx, atque ſic aſſumi <lb/>{bmm/√gn}dv - {m<emph style="super">3</emph>v/nn}dx = - mxdx. </s> + <s xml:id="echoid-s2025" xml:space="preserve"><lb/>in qua æquatione ſi rurſus negligatur primus terminus, quod fieri poteſt, <lb/>niſi mutationes etiam deſiderentur, quæ durante primo deſcenſu, etſi infi-<lb/>nite parvo fiunt, orietur regula vulgaris aſcenſus potentialis aquæ effluentis ad <lb/>altitudinem integram aquæ: </s> + <s xml:id="echoid-s2026" xml:space="preserve">nunc vero pro noſtro negotio, quo mutatio-<lb/>nes illas primas deſideramus, terminus iſte retinendus erit, atque ſic æqua-<lb/>tio ultima in tota ſua extenſione pertractanda.</s> + <s xml:id="echoid-s2027" xml:space="preserve"/> +</p> +<pb o="74" file="0088" n="88" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s2028" xml:space="preserve">Ponatur autem pro ſeparandis ab invicem indeterminatis {mm/nn}v - x = s, ſive <lb/>v = {nn/mm}(s + x), atque dv = {nn/mm} (ds + dx) ſicque fiet <lb/>dx = {- nnbds/nnb - ms√gn}, <lb/>quæ ita eſt integranda, ut facta x = a, prodeat v = o, hincque s = - a, <lb/>ita vero fit <lb/>x - a = {nnb/m√gn}log.</s> + <s xml:id="echoid-s2029" xml:space="preserve">{nnb - ms√gn/nnb + ma√gn} <lb/>& </s> + <s xml:id="echoid-s2030" xml:space="preserve">poſito pro s valore ejus aſſumto {mm/nn}v - x, prodit <lb/>x - a = {nnb/m√gn}log.</s> + <s xml:id="echoid-s2031" xml:space="preserve">{n<emph style="super">4</emph>b - m<emph style="super">3</emph>v√gn + mnnx√gn/n<emph style="super">4</emph>b + mnna√gn}</s> +</p> +<p> + <s xml:id="echoid-s2032" xml:space="preserve">Hic rurſus in quantitate ſigno logarithmicali involuta poteſt ex nume-<lb/>ratore eliminari terminus n<emph style="super">4</emph>b, infinities nempe minor termino mnnx√gn <lb/>nec non ex denominatore terminus n<emph style="super">4</emph>b infinities pariter minor altero <lb/>mnna√gn. </s> + <s xml:id="echoid-s2033" xml:space="preserve">Et ſic fit <lb/>x - a = {nnb/m√gn}log.</s> + <s xml:id="echoid-s2034" xml:space="preserve">{nnx - mma/nna}</s> +</p> +<p> + <s xml:id="echoid-s2035" xml:space="preserve">Inde habetur, poſito c pro numero cujus logarithmus eſt unitas: <lb/></s> + <s xml:id="echoid-s2036" xml:space="preserve">v = {nnx/mm} - {nna/mm} X c {m.</s> + <s xml:id="echoid-s2037" xml:space="preserve">(x - a)√gn/nnb} <lb/>aut poſita a - x = z, ſic ut z denotet ſpatium, per quod ſuperficies aquæ <lb/>jam deſcendit, poterit æquationi hæc conciliari forma: </s> + <s xml:id="echoid-s2038" xml:space="preserve"><lb/>v = {nn.</s> + <s xml:id="echoid-s2039" xml:space="preserve">(a - z)/mm} - {nna/mm}:</s> + <s xml:id="echoid-s2040" xml:space="preserve">c<emph style="super">{mz/nb}</emph>√{g/n} <lb/>de qua iterum liquet quod cum z vel minimam habuerit rationem ad b, fiat <lb/>denominator alterius termini infinitus & </s> + <s xml:id="echoid-s2041" xml:space="preserve">v = {nn.</s> + <s xml:id="echoid-s2042" xml:space="preserve">(a - z)/mm} = {nnx/mm}: </s> + <s xml:id="echoid-s2043" xml:space="preserve">at vero ali-<lb/>ter ſe res habet, quamdiu deſcenſus z infinite parvus eſt, quem caſum nunc <lb/>conſideramus.</s> + <s xml:id="echoid-s2044" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2045" xml:space="preserve">§. </s> + <s xml:id="echoid-s2046" xml:space="preserve">17. </s> + <s xml:id="echoid-s2047" xml:space="preserve">Hiſce præmiſſis facile nunc eſt definire per quantulum ſpatium <lb/>deſcendat fluidum, dum maximam velocitatem acquirit, faciendo nempe +<pb o="75" file="0089" n="89" rhead="SECTIO QUARTA."/> +dv = o, ſive - {nndz/mm} + {na/mb}√{g/n}:</s> + <s xml:id="echoid-s2048" xml:space="preserve">c<emph style="super">{mz/nb}√{g/n} = o</emph>, id eſt, <lb/>z = {nb/m}√{n/g}, X log.</s> + <s xml:id="echoid-s2049" xml:space="preserve">({ma/nb}√{g/n})</s> +</p> +<p> + <s xml:id="echoid-s2050" xml:space="preserve">Hæc autem altitudo multiplicata per altitudinem cylindri m dat quan-<lb/>titatem aquæ interea effluentis, nempe nb√{n/g} X log.</s> + <s xml:id="echoid-s2051" xml:space="preserve">({ma/nb}√{g/n},) quæ quan-<lb/>titas, ut ſupra §. </s> + <s xml:id="echoid-s2052" xml:space="preserve">15. </s> + <s xml:id="echoid-s2053" xml:space="preserve">præmonui, eſt infinita, quamvis tantum logarithmica-<lb/>liter, cujusmodi infinitum minus eſt, quam radix cujuscunque dimenſionis <lb/>datæ ex eodem infinito; </s> + <s xml:id="echoid-s2054" xml:space="preserve">eſt ſcilicet log. </s> + <s xml:id="echoid-s2055" xml:space="preserve">∞ minor quam ∞ {1/n}, quantuscunque <lb/>fuerit numerus n aſſignabilis. </s> + <s xml:id="echoid-s2056" xml:space="preserve">Atque hoc ideo moneo, ut ſic intelliga-<lb/>tur, qui fiat, ut, ſi à vero infinito ratiocinamur ad quantitates valde ma-<lb/>gnas, quantitas iſta aquæ ſat parva evadat. </s> + <s xml:id="echoid-s2057" xml:space="preserve">Cæterum corollaria formulæ <lb/>hæc ſunt.</s> + <s xml:id="echoid-s2058" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2059" xml:space="preserve">(I) Si tubus annexus eſt cylindricus, fit z = {nb/m}log.</s> + <s xml:id="echoid-s2060" xml:space="preserve">{ma/nb}: <lb/></s> + <s xml:id="echoid-s2061" xml:space="preserve">Igitur cæteris paribus hæc quantitas ſe habet, ut longitudo tubi annexi, quod <lb/>generaliter etiam verum eſt: </s> + <s xml:id="echoid-s2062" xml:space="preserve">nam à mutato valore ipſius b cenſenda eſt non <lb/>mutari quantitas log.</s> + <s xml:id="echoid-s2063" xml:space="preserve">{ma/nb}√{g/n} ob valorem infinitum numeri {m/n}.</s> + <s xml:id="echoid-s2064" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2065" xml:space="preserve">(II) Pro eodem orificio g cæterisque etiam paribus, ſequitur quantitas z <lb/>ſesquiplicatam rationem orificii extremi: </s> + <s xml:id="echoid-s2066" xml:space="preserve">atque ſi idem tubus modo orifi-<lb/>cio ſtrictiori modo ampliori vaſi applicetur, erit quantitas aquæ in caſu prio-<lb/>ri ad ſimilem quantitatem in poſteriori, ut quadratum orificii amplioris, ad <lb/>quadratum orificii minoris.</s> + <s xml:id="echoid-s2067" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2068" xml:space="preserve">(III) Denique obſervandum eſt valere totum ratiocinium pro omnibus <lb/>directionibus tubi, quod quivis perſpiciet qui §. </s> + <s xml:id="echoid-s2069" xml:space="preserve">22. </s> + <s xml:id="echoid-s2070" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s2071" xml:space="preserve">3. </s> + <s xml:id="echoid-s2072" xml:space="preserve">recte examinabit. <lb/></s> + <s xml:id="echoid-s2073" xml:space="preserve">Poterit igitur tubus adhiberi etiam horizontalis aut ſub quâcunque alia di-<lb/>rectione & </s> + <s xml:id="echoid-s2074" xml:space="preserve">utcunque incurvus, ad quod præſertim in inſtituendis experimen-<lb/>tis animus erit advertendus. </s> + <s xml:id="echoid-s2075" xml:space="preserve">Semper autem intelligetur per b longitudo tu-<lb/>bi, per a vero altitudo aquæ verticalis ſupra orificium extremum.</s> + <s xml:id="echoid-s2076" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2077" xml:space="preserve">§. </s> + <s xml:id="echoid-s2078" xml:space="preserve">18. </s> + <s xml:id="echoid-s2079" xml:space="preserve">Venio nunc ad tempus, quo iſtæ mutationes à quiete ad ma- +<pb o="76" file="0090" n="90" rhead="HYDRODYNAMICÆ"/> +ximam velocitatem fiunt: </s> + <s xml:id="echoid-s2080" xml:space="preserve">Dico autem poſſe in calculo hujusmodi tempo-<lb/>rum ſimpliciter poni v = {nn/mm}a; </s> + <s xml:id="echoid-s2081" xml:space="preserve">Reliquæ enim quantitates in æquatione ul-<lb/>tima §. </s> + <s xml:id="echoid-s2082" xml:space="preserve">16. </s> + <s xml:id="echoid-s2083" xml:space="preserve">evaneſcunt, quantumlibet parva ſumatur altitudo z, modo ha-<lb/>beat rationem vel minimam aſſignabilem ad altitudinem illam infinite par-<lb/>vam, quæ reſpondet maximæ velocitati, nempe ad {nb/m}√{n/g} X log.</s> + <s xml:id="echoid-s2084" xml:space="preserve">({ma/nb}√{g/n}). <lb/></s> + <s xml:id="echoid-s2085" xml:space="preserve">Sequitur exinde eſſe prædictum tempus, quod vocabo <lb/>t = {b√n/√ga} X log.</s> + <s xml:id="echoid-s2086" xml:space="preserve">({ma/nb}√{g/n}) <lb/>& </s> + <s xml:id="echoid-s2087" xml:space="preserve">proinde infinitum, quamvis idem tempus admodum exiguum ſit, quum <lb/>amplitudo vaſis non eſt infinita, ſed utcunque magna, quod rurſus ex na-<lb/>tura infiniti logarithmicalis eſt deducendum.</s> + <s xml:id="echoid-s2088" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2089" xml:space="preserve">§. </s> + <s xml:id="echoid-s2090" xml:space="preserve">19. </s> + <s xml:id="echoid-s2091" xml:space="preserve">Quia altitudo velocitatis, ut vidimus in proximo paragrapho, poteſt <lb/>ſtatim cenſeri = {nn/mm}a, id eſt, æqualis maximæ, cum ſuperficies per minimam <lb/>partem aſſignabilem deſcenſus infinite parvi, poſt quem velocitas maxima <lb/>plena adeſt, deſcendit, ſequitur mutationes plerasque à quiete usque ad ſta-<lb/>tum maximæ velocitatis eſſe inſenſibiles, id eſt, infinite parvas, imo non <lb/>ſolum plerasque, ſed & </s> + <s xml:id="echoid-s2092" xml:space="preserve">omnes præter particulam infinite parvam: </s> + <s xml:id="echoid-s2093" xml:space="preserve">res ſci-<lb/>licet ſic ſe habet: </s> + <s xml:id="echoid-s2094" xml:space="preserve">velocitas à primo initio plane nulla eſt, & </s> + <s xml:id="echoid-s2095" xml:space="preserve">poſtquam aqua <lb/>per ſpatiolum infinite parvum deſcendit, jam eſt tantum non maxima; </s> + <s xml:id="echoid-s2096" xml:space="preserve">dein <lb/>dum per aliud ſpatiolum rurſus quidem infinite parvum priori tamen infinite <lb/>majus, deſcendit, pergit velocitate ſua moveri, incrementa ſumens infinitè <lb/>parva, & </s> + <s xml:id="echoid-s2097" xml:space="preserve">tunc demum vere maximam velocitatem attingit: </s> + <s xml:id="echoid-s2098" xml:space="preserve">Cum vero po-<lb/>ſteriores illæ mutationes ceu infinite parvæ non poſſint ſenſibus percipi, aliter <lb/>pertractabimus ea quæ à §. </s> + <s xml:id="echoid-s2099" xml:space="preserve">17. </s> + <s xml:id="echoid-s2100" xml:space="preserve">dedimus theoremata, conſiderando loco mu-<lb/>tationum à quiete usque ad punctum maximæ velocitatis, easdem mutatio-<lb/>nes usque ad datum gradum velocitatis.</s> + <s xml:id="echoid-s2101" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2102" xml:space="preserve">§. </s> + <s xml:id="echoid-s2103" xml:space="preserve">20. </s> + <s xml:id="echoid-s2104" xml:space="preserve">Indagabimus itaque, per quantum ſpatiolum z ſuperficies aquæ <lb/>à ſtatu quietis deſcendere, quantaque aqua effluere, ac denique quantum <lb/>tempus præterire debeat, ut aqua interna velocitate moveatur, quæ gene-<lb/>retur lapſu libero per datam altitudinem, quam vocabimus {nn/mm}e, ita ut ip-<lb/>fa e denotet ſimilem altitudinem pro velocitate aquæ effluentis. </s> + <s xml:id="echoid-s2105" xml:space="preserve">Ad hoc re- +<pb o="77" file="0091" n="91" rhead="SECTIO QUARTA."/> +quiritur, ut in æquatione ultimâ paragraphi decimi ſexti ponatur {nne/mm} pro <lb/>v, ſic autem erit <lb/>{nne/mm} = {nn(a - z)/mm} - {nna/mm}:</s> + <s xml:id="echoid-s2106" xml:space="preserve">c{mz/nb}√{g/n} <lb/>hincque deducitur {mz/nb}√{g/n} = log.</s> + <s xml:id="echoid-s2107" xml:space="preserve">{a/a - e - z}; </s> + <s xml:id="echoid-s2108" xml:space="preserve">hic vero cum e ponatur defice-<lb/>re notabiliter ab a poteſt rejici littera z ſigno logarithmicali involuta, unde <lb/>obtinetur <lb/>z = {nb/m}√{n/g} X log.</s> + <s xml:id="echoid-s2109" xml:space="preserve">{a/a - e}</s> +</p> +<p> + <s xml:id="echoid-s2110" xml:space="preserve">Hæc vero æquatio jam indicat ſpatiolum, quod eſt infinite parvum, <lb/>& </s> + <s xml:id="echoid-s2111" xml:space="preserve">per quod deſcendit ſuperficies aquæ, dum à quiete velocitas aquæ effluen-<lb/>tis tanta ſit, quæ debeatur altitudini e; </s> + <s xml:id="echoid-s2112" xml:space="preserve">ſeque habet hoc ſpatiolum ad illud pa-<lb/>ragrapho decimo ſeptimo indicatum, quo nempe velocitas maxima oritur, ut <lb/>log. </s> + <s xml:id="echoid-s2113" xml:space="preserve">{a/a - e} ad log. </s> + <s xml:id="echoid-s2114" xml:space="preserve">({ma/nb}√{g/n}) ita ut primum ſit infinities minus altero, etſi <lb/>pariter infinite parvo.</s> + <s xml:id="echoid-s2115" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2116" xml:space="preserve">Si porro definita quantitas z multiplicetur per m, obtinetur quantitas <lb/>aquæ effluentis dum illa velocitas altitudini e debita producitur, quæ proin <lb/>quantitas eſt æqualis <lb/>nb√{n/g} X log.</s> + <s xml:id="echoid-s2117" xml:space="preserve">{a@/a - e} <lb/>atque ſic finitæ magnitudinis, & </s> + <s xml:id="echoid-s2118" xml:space="preserve">quidem eo majoris, quo longior ſumitur tu-<lb/>bus, & </s> + <s xml:id="echoid-s2119" xml:space="preserve">quo major jactus expectatur.</s> + <s xml:id="echoid-s2120" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2121" xml:space="preserve">Denique tempus, quo idem fit, ſi recte ſeligantur termini rejiciendi, <lb/>reperitur æquale <lb/>2√({nbb/ag} log.</s> + <s xml:id="echoid-s2122" xml:space="preserve">{a/a - e}) <lb/>atque ſic finitum ſed admodum parvum & </s> + <s xml:id="echoid-s2123" xml:space="preserve">in nullo exemplo ultra minutum ſe-<lb/>cundum facile extendendum.</s> + <s xml:id="echoid-s2124" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2125" xml:space="preserve">§. </s> + <s xml:id="echoid-s2126" xml:space="preserve">21. </s> + <s xml:id="echoid-s2127" xml:space="preserve">Hæc omnia accurate examinare ac proſequi volui, tum quod <lb/>multorum phænomenorum, quæ in effluxu aquarum obſervari ſolent, ſolu-<lb/>tio inde pendeat, tum etiam ut illas mutationes, quæ ſenſibus plane ſunt im-<lb/>perceptibiles, animo recte aſſequeremur. </s> + <s xml:id="echoid-s2128" xml:space="preserve">Multi fuerunt, qui tranſitus ab infini- +<pb o="78" file="0092" n="92" rhead="HYDRODYNAMICÆ"/> +to ad finitum aut viciſſim à finito ad infinitum in aquis fluentibus non recte aſ-<lb/>ſecuti à plurimis difficultatibus ſe extricare non potuerunt, quæ aliâs facilè ad-<lb/>mittunt ſolutionem, ſi autem loco vaſis fere infiniti, cujuſmodi nulla ſunt, ſu-<lb/>matur vas valde amplum, aut etiam quod in multis caſibus ſufficit, medio-<lb/>criter amplum, erunt formulæ proxime veræ, & </s> + <s xml:id="echoid-s2129" xml:space="preserve">modo magis modo minus <lb/>ad verum accedent pro indole quæſtionis: </s> + <s xml:id="echoid-s2130" xml:space="preserve">de his quædam monebo in ſequen-<lb/>tibus experimentis. </s> + <s xml:id="echoid-s2131" xml:space="preserve">Interim ſic ſatis jam apparet ex theoria, quod potiſſimum <lb/>explicare conſtitueram, cur aqua ex vaſe ampliſſimo ſimplici omni ſtatim ve-<lb/>locitate effluat, & </s> + <s xml:id="echoid-s2132" xml:space="preserve">cur ſecus ſit de aquis ex vaſe per tubum ejectis: </s> + <s xml:id="echoid-s2133" xml:space="preserve">Menſuræ <lb/>vero præcilæ de his quæſtionibus ex æquationibus ipſis erunt deducendæ.</s> + <s xml:id="echoid-s2134" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2135" xml:space="preserve">§. </s> + <s xml:id="echoid-s2136" xml:space="preserve">22. </s> + <s xml:id="echoid-s2137" xml:space="preserve">Tandem quod pertinet ad tempus depletionis, patet cum am-<lb/>plitudo vaſis vel mediocriter ſuperat amplitudinem tubi annexi, poſſe ſine <lb/>ſenſibili errore cenſeri illud = {mα/n} θ intelligendo per θ tempus, quo corpus à <lb/>quiete libere cadendo abſolvit altitudinem, quam aqua ab initio fluxus habuit <lb/>ſupra orificium tubi extremum, atque ſumendo pro {mα/n} rationem quæ eſt <lb/>inter amplitudinem vaſis & </s> + <s xml:id="echoid-s2138" xml:space="preserve">ſectionem venæ, ſive contractam ſive dilatatam. </s> + <s xml:id="echoid-s2139" xml:space="preserve">Impe-<lb/>dimenta vero, quæ in his caſibus fortuito ſuperveniunt, tempus iſtud admo-<lb/>dum augent. </s> + <s xml:id="echoid-s2140" xml:space="preserve">Si vero tempus deſideretur, quo ſuperficies aquæ per datam de-<lb/>ſcendat altitudinem erit illud ſumendum = {mα/n} (θ - Τ) ſumto pro Τ tem-<lb/>pore quod corpus inſumit libere cadendo per altitudinem, quam aqua in fine <lb/>fluxus ſupra foramen habet.</s> + <s xml:id="echoid-s2141" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div79" type="section" level="1" n="55"> +<head xml:id="echoid-head75" style="it" xml:space="preserve">Experimenta quœ ad Sect. IV. pertinent.</head> +<p> + <s xml:id="echoid-s2142" xml:space="preserve">QUum magna pars hujus ſectionis poſita ſit in contractione venæ aqueæ <lb/>per foramen in lamina tenui factum fluentis, animo concepi de iſta <lb/>contractione experimenta inſtituere accurata, non quidem menſuras <lb/>accipiendo diametrorum, quam methodum non ſuſſicienti accuratione fieri <lb/>poſſe expertus ſum, ſed obſervando velocitates actuales ex amplitudine jactus, <lb/>& </s> + <s xml:id="echoid-s2143" xml:space="preserve">quantitates datis temporibus effluentes; </s> + <s xml:id="echoid-s2144" xml:space="preserve">In experimentis automato uſus <lb/>ſum, quod tempore unius minuti primi 144. </s> + <s xml:id="echoid-s2145" xml:space="preserve">vicibus pulſabat, atque ſic ſe-<lb/>quentia ſumſi.</s> + <s xml:id="echoid-s2146" xml:space="preserve"/> +</p> +<pb o="79" file="0093" n="93" rhead="SECTIO QUARTA."/> +</div> +<div xml:id="echoid-div80" type="section" level="1" n="56"> +<head xml:id="echoid-head76" style="it" xml:space="preserve">Ad Theoriam Contractionis Venarum aquearum</head> +<head xml:id="echoid-head77" xml:space="preserve">Experimentum 1.</head> +<p> + <s xml:id="echoid-s2147" xml:space="preserve">Tubum cylindricum adhibui, cujus diameter erat 4. </s> + <s xml:id="echoid-s2148" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2149" xml:space="preserve">3. </s> + <s xml:id="echoid-s2150" xml:space="preserve">lin. </s> + <s xml:id="echoid-s2151" xml:space="preserve">menſ. <lb/></s> + <s xml:id="echoid-s2152" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s2153" xml:space="preserve">è lamina tenui factum quique foramen habebat in latere, id eſt, in ſuper-<lb/>ficie cylindrica: </s> + <s xml:id="echoid-s2154" xml:space="preserve">erat diameter foraminis = 4 {52/125} lin. </s> + <s xml:id="echoid-s2155" xml:space="preserve">aquæ effluebant hori-<lb/>zontaliter ex cylindro verticaliter poſito, & </s> + <s xml:id="echoid-s2156" xml:space="preserve">fuit ab initio fluxus altitudo aquæ <lb/>ſupra centrum foraminis = 4. </s> + <s xml:id="echoid-s2157" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2158" xml:space="preserve">8. </s> + <s xml:id="echoid-s2159" xml:space="preserve">lin. </s> + <s xml:id="echoid-s2160" xml:space="preserve">ſimiliſque altitudo in fine fluxus = <lb/>3. </s> + <s xml:id="echoid-s2161" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2162" xml:space="preserve">duravit autem omnis fluxus intervallo undecim automati pulſuum, quæ <lb/>proxime efficiunt tempus 4. </s> + <s xml:id="echoid-s2163" xml:space="preserve">minutorum ſecundorum cum dimidio.</s> + <s xml:id="echoid-s2164" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2165" xml:space="preserve">Porro repetito ſæpius experimento obſervatiſque tum altitudine fora-<lb/>minis ſupra tabulam horizontaliter poſitam, tum amplitudine jactus, hacque <lb/>tam in principio quam in fine fluxus, vidi ex Lemm. </s> + <s xml:id="echoid-s2166" xml:space="preserve">in principio Experimento-<lb/>rum præcedentis Sect. </s> + <s xml:id="echoid-s2167" xml:space="preserve">indicato velocitatem aquæ effluentis in loco venæ maxime <lb/>contractæ conſtanter talem fuiſſe, quantum quidem ſenſibus dijudicari potuit, <lb/>quæ deberetur altitudini aquæ ſupra eundem locum, qui in eadem altitudine <lb/>eſt quâ foramen.</s> + <s xml:id="echoid-s2168" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2169" xml:space="preserve">Igitur ſi contractionem venæ aqueæ ubique eandem fuiſſe ponamus & </s> + <s xml:id="echoid-s2170" xml:space="preserve"><lb/>huic caſui applicemus æquationem ultimam paragraphi decimi tertii, nempe <lb/>t = {2mα/n}(√a - √c) erit ponendum t = 4 {1/2} min. </s> + <s xml:id="echoid-s2171" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s2172" xml:space="preserve">{m/n} = 133; </s> + <s xml:id="echoid-s2173" xml:space="preserve">2√a(= tem-<lb/>pori quod corpus inſumit libere cadendo per altitudinem aquæ initialem) = <lb/>o, 1483 & </s> + <s xml:id="echoid-s2174" xml:space="preserve">2 √c (= tempori ſimili pro altitudine aquæ ultima) = o, 1246: </s> + <s xml:id="echoid-s2175" xml:space="preserve">fit <lb/>4 {1/2} = 3, 15 α unde α = 1, 43. </s> + <s xml:id="echoid-s2176" xml:space="preserve">Exinde conſequens eſt, amplitudinem fora-<lb/>minis fuiſſe ad ſectionem venæ contractæ ut 143. </s> + <s xml:id="echoid-s2177" xml:space="preserve">ad 100; </s> + <s xml:id="echoid-s2178" xml:space="preserve">hæc ratio tantillo <lb/>major eſt quam quæ intercedit inter √ 2 & </s> + <s xml:id="echoid-s2179" xml:space="preserve">1 nempe inter 141 & </s> + <s xml:id="echoid-s2180" xml:space="preserve">100; </s> + <s xml:id="echoid-s2181" xml:space="preserve">ſed ſi <lb/>accuratiſſime velocitates obſervari potuiſſent, dubium non eſt, quin illæ paul-<lb/>lo minores futuræ fuiſſent, quam quæ toti altitudini aquæ debeantur; </s> + <s xml:id="echoid-s2182" xml:space="preserve">& </s> + <s xml:id="echoid-s2183" xml:space="preserve">cum <lb/>hujus rei ratio habetur, deprehenditur valorem ipſius α ſic pauxillum dimi-<lb/>nuendum eſſe; </s> + <s xml:id="echoid-s2184" xml:space="preserve">poteſt igitur ex toto experimento colligl<unsure/> tutiſſime rationem <lb/>præmemoratam fuiſſe ut √ 2 ad 1.</s> + <s xml:id="echoid-s2185" xml:space="preserve"/> +</p> +<pb o="80" file="0094" n="94" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div81" type="section" level="1" n="57"> +<head xml:id="echoid-head78" xml:space="preserve">Experimentum 2.</head> +<p> + <s xml:id="echoid-s2186" xml:space="preserve">Deinde experimento explorare volui, an in omnibus jactibus ſub qua-<lb/>cunque directione contractio eadem ſit, & </s> + <s xml:id="echoid-s2187" xml:space="preserve">hunc in finem exiſtimavi rem ſic <lb/>eſſe aggrediendam, ut præter directionis iſtius mutationem circumſtantiæ cæ-<lb/>ræ omnes eſſent prorſus ſimiles. </s> + <s xml:id="echoid-s2188" xml:space="preserve">Id vero ſic obtinui.</s> + <s xml:id="echoid-s2189" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2190" xml:space="preserve">Eodem ſcilicet, quo antea cylindro uſus ſum, eum autem arcæ pris-<lb/>maticæ verticaliter poſitæ implantavi, ita, ut axis cylindri eſſet horizontalis, <lb/>ſicque implantatum circumverti, ut centrum foraminis, aqua@um effluxui de-<lb/>ſtinati, modo locum ſummum, modo medium, modo imum occuparet: </s> + <s xml:id="echoid-s2191" xml:space="preserve">in <lb/>primo caſu aquæ verticaliter ſurſum effluebant, in ſecundo horizontaliter, in <lb/>tertio verticaliter deorſum ejiciebantur; </s> + <s xml:id="echoid-s2192" xml:space="preserve">in ſingulis vero feci ut altitudines aquæ <lb/>in arca ſupra centrum foraminis eſſent perfecte æquales: </s> + <s xml:id="echoid-s2193" xml:space="preserve">ſucceſſus hic fuit.</s> + <s xml:id="echoid-s2194" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2195" xml:space="preserve">Obſervavi æqualibus temporibus ſuperficiem aquæ in ſingulis caſibus per <lb/>ſpatia æqualia in arca deſcendere. </s> + <s xml:id="echoid-s2196" xml:space="preserve">Igitur in venis ſurſum projectis aqua uperior <lb/>non reſiſtit ſenſibiliter aquæ inferiori ſubſequenti, quod idem alio intellexi modo, <lb/>quod ſcilicet, ſi ad parvam à foramine diſtantiam veluti 3. </s> + <s xml:id="echoid-s2197" xml:space="preserve">linearum nummo <lb/>aliquo venam aqueam cujuſcunque directionis excipiebam, ita ut vena in num-<lb/>mum perpendiculariter incideret, effluxus aquarum non fuerit retardatus. <lb/></s> + <s xml:id="echoid-s2198" xml:space="preserve">Porro nec aqua in venis verticaliter deſcendentibus anterior poſteriorem poſt ſe <lb/>trahit; </s> + <s xml:id="echoid-s2199" xml:space="preserve">ipſaque venæ contractio ſimilis ubique eſt, non conſiderata retarda-<lb/>tione accelerationeque aquarum ſurſum vel deorſum ejectarum, quæ faciunt <lb/>ut vena in aliqua à foramine diſtantia vel intumeſcat, vel gracileſcat. </s> + <s xml:id="echoid-s2200" xml:space="preserve">Hic <lb/>enim ſermo eſt de illa modo contractione, quæ oritur à motu particularum <lb/>obliquo in regione foraminis.</s> + <s xml:id="echoid-s2201" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div82" type="section" level="1" n="58"> +<head xml:id="echoid-head79" xml:space="preserve">Experimentum 3.</head> +<p> + <s xml:id="echoid-s2202" xml:space="preserve">Eadem machina prædicto modo præparata uſus ſum ad explorandum, <lb/>num contractio venæ cæteris paribus mutaretur ab aucta altitudine aquæ ſupra <lb/>foramen. </s> + <s xml:id="echoid-s2203" xml:space="preserve">Hunc in finem duas acus infixi lateribus internis arcæ ad perpen-<lb/>diculum ſibi reſpondentes, prior eminebat ſupra centrum foraminis 13 poll. <lb/></s> + <s xml:id="echoid-s2204" xml:space="preserve">cum 10. </s> + <s xml:id="echoid-s2205" xml:space="preserve">lineis, altera 12. </s> + <s xml:id="echoid-s2206" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2207" xml:space="preserve">1 {3/5} lin. </s> + <s xml:id="echoid-s2208" xml:space="preserve">menſ. </s> + <s xml:id="echoid-s2209" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s2210" xml:space="preserve">amplitudo arcæ erat ad am-<lb/>plitudinem foraminis ut 404. </s> + <s xml:id="echoid-s2211" xml:space="preserve">ad 1. </s> + <s xml:id="echoid-s2212" xml:space="preserve">vidi autem ſuperficiem aquæ à ſuperiore +<pb o="81" file="0095" n="95" rhead="SECTIO QUARTA."/> +acu ad inferiorem deſcendiſſe poſt intervalla 24. </s> + <s xml:id="echoid-s2213" xml:space="preserve">automati pulſuum, quæ faciunt <lb/>tempus 10. </s> + <s xml:id="echoid-s2214" xml:space="preserve">minutorum ſecundorum.</s> + <s xml:id="echoid-s2215" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2216" xml:space="preserve">Quod ſi vero tempus idem quæratur ad Hypotheſin, venam ſe nihil <lb/>contraxiſſe, ſimulque aquas omni velocitate, quam vi theoriæ nullo præſen-<lb/>te impedimento alieno habere debuiſſent, effluxiſſe, reperitur illud = 6 {7/8} <lb/>min. </s> + <s xml:id="echoid-s2217" xml:space="preserve">ſec.</s> + <s xml:id="echoid-s2218" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2219" xml:space="preserve">Sic igitur concludi poteſt, fuiſſe amplitudinem foraminis ad ſectionem <lb/>venæ contractæ ut 10. </s> + <s xml:id="echoid-s2220" xml:space="preserve">ad 6 {7/8}, id eſt, α = 1, 45, cum in primo experimento fue-<lb/>rit pro eodem foramine perpenſis omnibus circumſtantiis α = 1, 41.</s> + <s xml:id="echoid-s2221" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2222" xml:space="preserve">Poſtquam hæc ita expertus fuiſſem, reſiduum erat explorare, an aquæ <lb/>omni velocitate ad ſenſus effluxerint, qua de re eo magis dubitavi, quod <lb/>creſcentibus velocitatibus aquæ, creſcant ſimul impedimenta, hæcque proin <lb/>notabilia eſſe poſſint in majoribus aquæ altitudinibus, qualia in minoribus <lb/>non ſunt.</s> + <s xml:id="echoid-s2223" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2224" xml:space="preserve">Feci itaque omni adhibita cura (quod potiſſimum ad præciſionem ex-<lb/>perimenti requiritur) ut aquæ ſub directione perfecte horizontali effluerent, <lb/>& </s> + <s xml:id="echoid-s2225" xml:space="preserve">acceptis menſuris tum amplitudinis jactus, tum altitudinis foraminis ſupra <lb/>tabulam horizontalem, vidi ſubducto calculo, quod cum altitudo aquæ erat <lb/>= 13. </s> + <s xml:id="echoid-s2226" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2227" xml:space="preserve">cum 10. </s> + <s xml:id="echoid-s2228" xml:space="preserve">lin. </s> + <s xml:id="echoid-s2229" xml:space="preserve">ſeu 166. </s> + <s xml:id="echoid-s2230" xml:space="preserve">lin. </s> + <s xml:id="echoid-s2231" xml:space="preserve">aquæ effluxerint, ſeu potius per ſectionem <lb/>venæ contractam transfluxerint, velocitate, quæ convenit altitudini 158 {1/2} lin. <lb/></s> + <s xml:id="echoid-s2232" xml:space="preserve">igitur velocitas in calculo diminuenda eſt in ratione ſubduplicata harum altitu-<lb/>dinum atque in eadem ratione proxime decreſcit valor inventus litteræ α, qui <lb/>ita fit paullo minor quam 1, 42 ſeu rurſus 1, 41 & </s> + <s xml:id="echoid-s2233" xml:space="preserve">ſic colligere licet, ſolam al-<lb/>titudinem aquæ mutatam ad ſenſus non mutare contractionem venæ.</s> + <s xml:id="echoid-s2234" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div83" type="section" level="1" n="59"> +<head xml:id="echoid-head80" xml:space="preserve">Experimentum 4.</head> +<p> + <s xml:id="echoid-s2235" xml:space="preserve">Tubum habui cylindricum altitudinis 4 poll. </s> + <s xml:id="echoid-s2236" xml:space="preserve">cujus ſectio per axem re-<lb/>preſentatur per (Fig. </s> + <s xml:id="echoid-s2237" xml:space="preserve">28. </s> + <s xml:id="echoid-s2238" xml:space="preserve">b.) </s> + <s xml:id="echoid-s2239" xml:space="preserve">C A B D, amplitudo cylindri erat ad amplitudinem <lb/>foraminis a c ut 110 ad 1. </s> + <s xml:id="echoid-s2240" xml:space="preserve">Cylindrus iſte aqua plenus omnis evacuatus fuit tem-<lb/> +<anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/> +pore 21. </s> + <s xml:id="echoid-s2241" xml:space="preserve">minutorum ſecundorum cum dimidio. </s> + <s xml:id="echoid-s2242" xml:space="preserve">Notari autem debet, non <lb/>prius aquis effluxum concedendum eſſe, quam nullus in illis motus turbina-<lb/>torius obſervetur; </s> + <s xml:id="echoid-s2243" xml:space="preserve">ſecus enim aqua mox in turbinem vertitur, durante effluxu +<pb o="82" file="0096" n="96" rhead="HYDRODYNAMICÆ"/> +ſat celerem, effluxuſque valde retardatur, eoque magis, quo celerius aqua <lb/>interna in Gyrum agitur: </s> + <s xml:id="echoid-s2244" xml:space="preserve">quia porro nunquam omnis aqua effluit, effluxus <lb/>tempus conſideravi, uſquedum ſtillatim effluere inciperet.</s> + <s xml:id="echoid-s2245" xml:space="preserve"/> +</p> +<div xml:id="echoid-div83" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">Fig. 28. b</note> +</div> +<p> + <s xml:id="echoid-s2246" xml:space="preserve">Indicat hoc experimentum minorem hic aquæ fuiſſe contractionem <lb/>quam pro ratione √2 ad 1; </s> + <s xml:id="echoid-s2247" xml:space="preserve">Expectaveram tempus evacuationis fore admodum <lb/>23. </s> + <s xml:id="echoid-s2248" xml:space="preserve">min. </s> + <s xml:id="echoid-s2249" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s2250" xml:space="preserve">ſed eventus paullo alius fuit ut dixi, cujus rei rationem eſſe poſt-<lb/>modum animadverti, quod labia foraminis elongata tubulum fere quamvis <lb/>breviſſimum formarent, ut Figura oſtendit, qui venæ aqueæ contractionem <lb/>impediebat: </s> + <s xml:id="echoid-s2251" xml:space="preserve">interim latitudo iſtorum labiorum duas tertias lineæ non attin-<lb/>gebat.</s> + <s xml:id="echoid-s2252" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div85" type="section" level="1" n="60"> +<head xml:id="echoid-head81" xml:space="preserve">Experimentum 5.</head> +<p> + <s xml:id="echoid-s2253" xml:space="preserve">Feci ut aquæ ex vaſe ampliſſimo per tubulum effluerent horizontali-<lb/>ter: </s> + <s xml:id="echoid-s2254" xml:space="preserve">erat autem tubus breviſſimus, longitudinem nempe 3. </s> + <s xml:id="echoid-s2255" xml:space="preserve">lin. </s> + <s xml:id="echoid-s2256" xml:space="preserve">non excedens, <lb/>habebatque in diametro fere 5. </s> + <s xml:id="echoid-s2257" xml:space="preserve">lin.</s> + <s xml:id="echoid-s2258" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2259" xml:space="preserve">Effluxit data aquæ quantitas tempore 11 {1/4} min. </s> + <s xml:id="echoid-s2260" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s2261" xml:space="preserve">quæ effluere debuiſ-<lb/>ſet tempore 10 {2/3} min. </s> + <s xml:id="echoid-s2262" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s2263" xml:space="preserve">ſi neque contractam fuiſſe venam, neque ulla adfuiſſe <lb/>impedimenta ſtatuatur.</s> + <s xml:id="echoid-s2264" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2265" xml:space="preserve">Velocitates reales aquæ non cenſui opus eſſe ut experirer, nullus du-<lb/>bitans tales fuiſſe, quales eſſe debeant, ut obſervato tempore per obſervatum <lb/>orificium data quantitas aquæ, nulla facta ad contractionem venæ attentione, <lb/>efflueret.</s> + <s xml:id="echoid-s2266" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2267" xml:space="preserve">Alios inſuper alîus diametri longitudinisque adhibui tubulos & </s> + <s xml:id="echoid-s2268" xml:space="preserve">vidi <lb/>quantitates aquæ dato tempore datisque velocitatibus effluentis recte reſponde-<lb/>re orficiis effluxus: </s> + <s xml:id="echoid-s2269" xml:space="preserve">velocitates autem eo magis defeciſſe à velocitate integræ <lb/>altitudini aquæ debita, quo ſtrictior & </s> + <s xml:id="echoid-s2270" xml:space="preserve">quo longior erat tubus, ut & </s> + <s xml:id="echoid-s2271" xml:space="preserve">quo altior <lb/>erat aqua.</s> + <s xml:id="echoid-s2272" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div86" type="section" level="1" n="61"> +<head xml:id="echoid-head82" style="it" xml:space="preserve">Ad Theoriam aquarum per tubos effluentium.</head> +<head xml:id="echoid-head83" xml:space="preserve">Experimentum 6.</head> +<p> + <s xml:id="echoid-s2273" xml:space="preserve">Vaſa, quorum ſectiones per axem repreſentant Fig. </s> + <s xml:id="echoid-s2274" xml:space="preserve">24. </s> + <s xml:id="echoid-s2275" xml:space="preserve">& </s> + <s xml:id="echoid-s2276" xml:space="preserve">25. </s> + <s xml:id="echoid-s2277" xml:space="preserve">cylin-<lb/>drica, altitudinem habebant 4. </s> + <s xml:id="echoid-s2278" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2279" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s2280" xml:space="preserve">tubosque annexos longitudinis unius <lb/>pedis, amplitudines cylindrorum erant ad amplitudines orificiorum A, ut 110 +<pb o="83" file="0097" n="97" rhead="SECTIO QUARTA."/> +ad 1; </s> + <s xml:id="echoid-s2281" xml:space="preserve">Orificium autem B eratad orificium A proxime ut 25 ad 16; </s> + <s xml:id="echoid-s2282" xml:space="preserve">tempus eva-<lb/>cuationis repletis antea cylindris fuit in Fig. </s> + <s xml:id="echoid-s2283" xml:space="preserve">24, ſex min. </s> + <s xml:id="echoid-s2284" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s2285" xml:space="preserve">cum dimidio, in <lb/>altera præterpropter 4 hujuſmodi minutorum cum triente.</s> + <s xml:id="echoid-s2286" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2287" xml:space="preserve">In his caſibus vaſa ſatis ampla ſuere ratione tuborum annexorum, ut <lb/>veluti infinita cenſeri poſſent; </s> + <s xml:id="echoid-s2288" xml:space="preserve">debuiſſetque proin per Regulas paſſim à no-<lb/>bis indicatas aqua effluere per orificia extrema velocitatibus reſpondentibus <lb/>toti altltudini aquæ, ſi modo excipias prima fluxus momenta, quæ ipſa tam <lb/>brevia hic ſunt, ut obſervari non poſſint. </s> + <s xml:id="echoid-s2289" xml:space="preserve">Et cum præterea, ut paſſim mo-<lb/>nui, quantitas aquæ dato tempore per tubos effluentis ſimpliciter æſtiman-<lb/>da ſit ex celeritatibus & </s> + <s xml:id="echoid-s2290" xml:space="preserve">magnitudine orificiorum inveni per regnlam §. </s> + <s xml:id="echoid-s2291" xml:space="preserve">22. <lb/></s> + <s xml:id="echoid-s2292" xml:space="preserve">exhibitam, tempus evacuationis in primo caſu 4 {1/3} minſec. </s> + <s xml:id="echoid-s2293" xml:space="preserve">in poſteriori = <lb/>fere 3. </s> + <s xml:id="echoid-s2294" xml:space="preserve">m. </s> + <s xml:id="echoid-s2295" xml:space="preserve">ſec.</s> + <s xml:id="echoid-s2296" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2297" xml:space="preserve">Quod in experimento majora paullo fuerint obſervata in Fig. </s> + <s xml:id="echoid-s2298" xml:space="preserve">24. </s> + <s xml:id="echoid-s2299" xml:space="preserve">ma-<lb/>ximam partem adhæſioni aquæ ad latera tubi, in Fig. </s> + <s xml:id="echoid-s2300" xml:space="preserve">autem 25. </s> + <s xml:id="echoid-s2301" xml:space="preserve">alii in ſuper <lb/>rationi in paragrapho 34. </s> + <s xml:id="echoid-s2302" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s2303" xml:space="preserve">3. </s> + <s xml:id="echoid-s2304" xml:space="preserve">indicatæ eſt tribuendum.</s> + <s xml:id="echoid-s2305" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2306" xml:space="preserve">Phænomena alia in his vaſis ſunt notanda: </s> + <s xml:id="echoid-s2307" xml:space="preserve">nempe cum vaſa ſunt tan-<lb/>tum non evacuata, percipitur ſonus quidem ab aëre, qui tunc aquæ in ori-<lb/>ficio ſuperiori ſe miſcet. </s> + <s xml:id="echoid-s2308" xml:space="preserve">hunc vero ſonum pro ultimo fluxus momento acce-<lb/>pi: </s> + <s xml:id="echoid-s2309" xml:space="preserve">facile fit porro, ut aquæ effluxus concedatur priusquam ad perfectam <lb/>quietem fuerit reducta (nam ab impletione agitantur & </s> + <s xml:id="echoid-s2310" xml:space="preserve">in turbinem mo-<lb/>ventur aquæ); </s> + <s xml:id="echoid-s2311" xml:space="preserve">tunc autem effluxus admodum retardatur & </s> + <s xml:id="echoid-s2312" xml:space="preserve">cataractæ ſpecies <lb/>interne formatur, continueque aër aquæ effluenti ſe permiſcet. </s> + <s xml:id="echoid-s2313" xml:space="preserve">Ita poteſt <lb/>pro lubitu retardari effluxus, ſi in vorticem aquæ agantur antequam effluant.</s> + <s xml:id="echoid-s2314" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div87" type="section" level="1" n="62"> +<head xml:id="echoid-head84" xml:space="preserve">Experimentum 7.</head> +<p> + <s xml:id="echoid-s2315" xml:space="preserve">Vaſe uſus ſum Prismatico, cui tubulus infixus erat horizontaliter ut in <lb/>Fig. </s> + <s xml:id="echoid-s2316" xml:space="preserve">19. </s> + <s xml:id="echoid-s2317" xml:space="preserve">Habebat orificium G F in Diametro præciſe quinque lineas; </s> + <s xml:id="echoid-s2318" xml:space="preserve">alterum <lb/>N M 6 {1/2} lin. </s> + <s xml:id="echoid-s2319" xml:space="preserve">Erant proin ipſæ amplitudines orificiorum G F & </s> + <s xml:id="echoid-s2320" xml:space="preserve">N M ut 100. <lb/></s> + <s xml:id="echoid-s2321" xml:space="preserve">ad 169. </s> + <s xml:id="echoid-s2322" xml:space="preserve">amplitudo vero vaſis continebat amplitudinem orificii N M ducentis <lb/>& </s> + <s xml:id="echoid-s2323" xml:space="preserve">una vicibus. </s> + <s xml:id="echoid-s2324" xml:space="preserve">Longitudo tubuli G N erat 4. </s> + <s xml:id="echoid-s2325" xml:space="preserve">poll.</s> + <s xml:id="echoid-s2326" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2327" xml:space="preserve">Denide vas aquâ implevi usque in C D, cujus altitudo ſupra axem tu-<lb/>bi erat 13. </s> + <s xml:id="echoid-s2328" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2329" xml:space="preserve">10, lin. </s> + <s xml:id="echoid-s2330" xml:space="preserve">Aperto orificio N M effluxerunt aquæ deſcenditque +<pb o="84" file="0098" n="98" rhead="HYDRODYNAMICÆ"/> +ſuperficies usque in E H tempore 8 {1/3} min. </s> + <s xml:id="echoid-s2331" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s2332" xml:space="preserve">erat vero altitudinum differentia <lb/>C E vel D H duorum pollicum cum octo lineis.</s> + <s xml:id="echoid-s2333" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2334" xml:space="preserve">Subducto calculo ad normam paragraphi 22. </s> + <s xml:id="echoid-s2335" xml:space="preserve">ubi neque ad impedimen-<lb/>ta, neque ad mutationem venæ attenditur, videmus prædictum tempus de-<lb/>ſcenſus eſſe debuiſſe proxime = 5 min. </s> + <s xml:id="echoid-s2336" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s2337" xml:space="preserve">cum fere dimidio. </s> + <s xml:id="echoid-s2338" xml:space="preserve">Igitur ſtatu-<lb/>endum eſt hoc modo, velocitatem mediam totalem ſe habuiſſe ad velocita-<lb/>tem integram, quam theoria indicat, ut 5 {1/2} ad 8 {1/3} ſeu proxime ut 2 ad 3; <lb/></s> + <s xml:id="echoid-s2339" xml:space="preserve">hincque concludi poteſt, aquam per orificium M N effluxiſſe velocitate, quæ <lb/>conveniat ({2/3})<emph style="super">2</emph>, ſeu quatuor nonis partibus altitudinis aquæ ſupra foramen <lb/>M N, per alterum vero orificium G F transfluxiſſe velocitate quinque præter <lb/>propter quartis ejusdem altitudinis partibus debita.</s> + <s xml:id="echoid-s2340" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2341" xml:space="preserve">Apparet itaque rurſus effluxum aquarum promoveri ab auctâ amplitu-<lb/>dine orificii tubi verſus exteriora, quamvis nec orificium quo tubus in vas <lb/>eſt implantatus, nec ſitus tubi ſit mutatus.</s> + <s xml:id="echoid-s2342" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2343" xml:space="preserve">Porro in tabula horizontaliter poſita P Q obſervavi amplitudinem jactus <lb/>P Q pro altitudine o P, quæ erat 4. </s> + <s xml:id="echoid-s2344" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2345" xml:space="preserve">8. </s> + <s xml:id="echoid-s2346" xml:space="preserve">lin. </s> + <s xml:id="echoid-s2347" xml:space="preserve">Inveni autem P Q = 9. </s> + <s xml:id="echoid-s2348" xml:space="preserve">poll. <lb/></s> + <s xml:id="echoid-s2349" xml:space="preserve">6. </s> + <s xml:id="echoid-s2350" xml:space="preserve">lin.</s> + <s xml:id="echoid-s2351" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2352" xml:space="preserve">Sequitur ex iſta obſervatione, quod ſi dilatationis venæ conſideratione <lb/>ſepoſita aquæ in N M velocitatem debuerint habere, qualis debetur altitudi-<lb/>ni 4. </s> + <s xml:id="echoid-s2353" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2354" xml:space="preserve">10. </s> + <s xml:id="echoid-s2355" xml:space="preserve">lin. </s> + <s xml:id="echoid-s2356" xml:space="preserve">cum tamen vi præmiſſi experimenti certe habuerit velocita-<lb/>tem debitam altitudini fere 6. </s> + <s xml:id="echoid-s2357" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2358" xml:space="preserve">2. </s> + <s xml:id="echoid-s2359" xml:space="preserve">lin. </s> + <s xml:id="echoid-s2360" xml:space="preserve">Confirmat hæc obſervatio id quod <lb/>§. </s> + <s xml:id="echoid-s2361" xml:space="preserve">15. </s> + <s xml:id="echoid-s2362" xml:space="preserve">dixi, nempe in tubis divergentibus venam aqueam dilatari veluti in <lb/>m, ipſiusque motum retardari. </s> + <s xml:id="echoid-s2363" xml:space="preserve">In præſenti vero caſu, ut ambæ obſervatio-<lb/>nes concilientur, dicendum erit venam ita dilatatam fuiſſe, ut amplitudi-<lb/>nem haberet ratione orificii N M reciproce ut prædictæ velocitates ſeu reci-<lb/>proce ut radices altitudinum iſtis velocitatibus debitarum, nempe ut √ 74. <lb/></s> + <s xml:id="echoid-s2364" xml:space="preserve">ad √ 58. </s> + <s xml:id="echoid-s2365" xml:space="preserve">proindeque diametros venæ dilatatæ & </s> + <s xml:id="echoid-s2366" xml:space="preserve">orificii fuiſſe ut ∜74 ad ∜ 58. </s> + <s xml:id="echoid-s2367" xml:space="preserve"><lb/>ſeu ut 100 ad 941.</s> + <s xml:id="echoid-s2368" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div88" type="section" level="1" n="63"> +<head xml:id="echoid-head85" xml:space="preserve">Experimentum 8.</head> +<p> + <s xml:id="echoid-s2369" xml:space="preserve">Aliud feci experimentum quod, quamvis huc nondum pertineat, ni-<lb/>hilominus recenſebo: </s> + <s xml:id="echoid-s2370" xml:space="preserve">nempe in ortu prope orificium G F tubum perforavi <lb/>foramine e duarum fere linearum, rurſusque deſcenſum ſuperficiei ex C D in +<pb o="85" file="0099" n="99" rhead="SECTIO QUARTA."/> +E H obſervavi effluente aqua per N M, ſimulque amplitudinem jactus exa-<lb/>minavi.</s> + <s xml:id="echoid-s2371" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2372" xml:space="preserve">Duo hæc vidi, quæ prima fronte ſibi contradicere fere videntur; </s> + <s xml:id="echoid-s2373" xml:space="preserve">de-<lb/>ſcenſus ex C D in E H tardior factus eſt quam in præcedenti experimento <lb/>fuerat, & </s> + <s xml:id="echoid-s2374" xml:space="preserve">nunc duravit 10. </s> + <s xml:id="echoid-s2375" xml:space="preserve">min. </s> + <s xml:id="echoid-s2376" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s2377" xml:space="preserve">& </s> + <s xml:id="echoid-s2378" xml:space="preserve">tamen amplior fuit jactus P Q pro ea-<lb/>dem altitudine o P; </s> + <s xml:id="echoid-s2379" xml:space="preserve">jam enim erat P Q = 10. </s> + <s xml:id="echoid-s2380" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2381" xml:space="preserve">10. </s> + <s xml:id="echoid-s2382" xml:space="preserve">lin.</s> + <s xml:id="echoid-s2383" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2384" xml:space="preserve">Ambo Phænomena ita explico: </s> + <s xml:id="echoid-s2385" xml:space="preserve">ob foramen e, quod fuit factum prope <lb/>G F quodque aëri liberum tranſitum concedit, ſolvitur nexus, quem alias <lb/>inter ſe habent aquæ in tubo, nec proin aliter transfluunt aquæ ubi eſt fo-<lb/>raminulum e, quam ſi eo ipſo in loco eſſet reſciſſus tubus; </s> + <s xml:id="echoid-s2386" xml:space="preserve">fluerent autem <lb/>tardius, quod paſſim demonſtravi, ſi tubus G N M F ceu divergens brevior <lb/>fieret. </s> + <s xml:id="echoid-s2387" xml:space="preserve">Quod porro aquæ quamvis minori quantitate, tamen majori impetu per <lb/>orificium N M non mutatum fluere poſſint ſine implicita contradictione, ra-<lb/>tio eſt permixtio aëris cum aqua; </s> + <s xml:id="echoid-s2388" xml:space="preserve">nam aër perpetuo irruit in tubum per fo-<lb/>raminulum e & </s> + <s xml:id="echoid-s2389" xml:space="preserve">una cum aqua effluit per N M. </s> + <s xml:id="echoid-s2390" xml:space="preserve">Denique phænomenon illud, <lb/>quod aquæ actu celerius fluant per M N aperto, quam clauſo foramine e, <lb/>aliter explicari non poſſe mihi videtur, quam quod impedimenta extrinſeca <lb/>minus agant in aquam aëre rarefactam quam naturalem.</s> + <s xml:id="echoid-s2391" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div89" type="section" level="1" n="64"> +<head xml:id="echoid-head86" style="it" xml:space="preserve">Ad theoriam aquarum, quæ ex vaſis ampliſsi-<lb/>mis à puncto quietis usque ad datum veloci-<lb/>tatis gradum effluunt.</head> +<head xml:id="echoid-head87" xml:space="preserve">Experimentum 9.</head> +<p> + <s xml:id="echoid-s2392" xml:space="preserve">Quum aquæ per foramen in lamina tenui factum ex vaſe ampliſſimo <lb/>effluunt, prima ſtatim guttula omni velocitate, quæ altitudini aquæ ſupra fo-<lb/>ramen debetur, erumpit.</s> + <s xml:id="echoid-s2393" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2394" xml:space="preserve">Conforme hoc eſt cum theoria §. </s> + <s xml:id="echoid-s2395" xml:space="preserve">11. </s> + <s xml:id="echoid-s2396" xml:space="preserve">indicata, ſi vas ſit revera <lb/>infinitum, & </s> + <s xml:id="echoid-s2397" xml:space="preserve">quamvis etiam non fuerit ſenſu Geometrico infinitum, <lb/>dummodo ſit valde amplum, nulla pariter guttula ab initio fluxus ob-<lb/>ſervari poteſt, quæ non maxima velocitate effluxerit: </s> + <s xml:id="echoid-s2398" xml:space="preserve">Phænomenon hoc <lb/>explicui §. </s> + <s xml:id="echoid-s2399" xml:space="preserve">14. </s> + <s xml:id="echoid-s2400" xml:space="preserve">cum nempe vi theoriæ in caſu particulari aliquo ibidem re-<lb/>cenſito vix una aut duæ guttulæ ſenſibiliter à jactu maximo deficere debuif- +<pb o="86" file="0100" n="100" rhead="HYDRODYNAMICÆ"/> +ſent, dixi non poſſe tantillam aquæ quantitatem ſe ab aqua ſubſequente ſepa-<lb/>rare ob mutuam aquearum particularum attractionem ſeu adhæſionem.</s> + <s xml:id="echoid-s2401" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div90" type="section" level="1" n="65"> +<head xml:id="echoid-head88" xml:space="preserve">Experimentum 10.</head> +<p> + <s xml:id="echoid-s2402" xml:space="preserve">Quum vero aquæ ex vaſe ampliſſimo per tubum vaſi horizontaliter in-<lb/>ſertum effluebant, obſervavi priusquam vena effluens jactum formaret, ma-<lb/>ximum o m Q (vid. </s> + <s xml:id="echoid-s2403" xml:space="preserve">Fig. </s> + <s xml:id="echoid-s2404" xml:space="preserve">19,) ſat notabilem aquæ quantitatem in tabulam ho-<lb/>rizontalem ſubjectam delabi mediam inter P. </s> + <s xml:id="echoid-s2405" xml:space="preserve">& </s> + <s xml:id="echoid-s2406" xml:space="preserve">Q. </s> + <s xml:id="echoid-s2407" xml:space="preserve">eo majorem eſſe hanc quan-<lb/>titatem quo longior eſt tubus G N & </s> + <s xml:id="echoid-s2408" xml:space="preserve">quo magis verſus N divergit, ac deni-<lb/>que inæqualiter aquam illam diſtribui, multo copioſius ſcilicet decidere in <lb/>locum, qui eſt remotior à puncto P, quam qui eidem eſt propior; </s> + <s xml:id="echoid-s2409" xml:space="preserve">Ratio-<lb/>ne autem temporis, quo omnes iſtæ mutationes fiunt, vidi illud breviſſimum <lb/>eſſe, & </s> + <s xml:id="echoid-s2410" xml:space="preserve">tale ut ejus menſura percipi non poſſet.</s> + <s xml:id="echoid-s2411" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2412" xml:space="preserve">Omnia iſta phænomena ex aſſe ſatisfaciunt propoſitionibus, quas dedi-<lb/>mus à paragrapho undecimo usque ad finem ſectionis. </s> + <s xml:id="echoid-s2413" xml:space="preserve">Menſuræ autem ibi-<lb/>dem exhibitæ experimentis recte confirmari non poſſunt, præſertim illæ, <lb/>quæ §, §. </s> + <s xml:id="echoid-s2414" xml:space="preserve">15. </s> + <s xml:id="echoid-s2415" xml:space="preserve">16. </s> + <s xml:id="echoid-s2416" xml:space="preserve">& </s> + <s xml:id="echoid-s2417" xml:space="preserve">17. </s> + <s xml:id="echoid-s2418" xml:space="preserve">indicatæ ſunt, ubi ſcilicet formulæ communicantur, <lb/>quæ exprimant quantitatem aquæ effluentis, dum à quiete maximus fit ja-<lb/>ctus: </s> + <s xml:id="echoid-s2419" xml:space="preserve">ratio eſt primò, quod primæ guttulæ quæ prope punctum P in tabu-<lb/>lam decidere deberent ab aqua ſubſequente non libere ſe ſeparent; </s> + <s xml:id="echoid-s2420" xml:space="preserve">ſecundo, <lb/>quod aquæ quantitas venæ O Q proxima (quæ quidem maximam vi ipſius <lb/>theoriæ partem conſtituit) intercipi non queat, & </s> + <s xml:id="echoid-s2421" xml:space="preserve">denique, quod motus <lb/>aquarum per tubos admodum tetardari ſolet, ab impedimentis extrinſecis, <lb/>imprimis ſi tubi divergant, atque ſic motus realis ſit admodum diverſus à <lb/>motu quem aquæ habituræ eſſent, remotis omnibus impedimentis. </s> + <s xml:id="echoid-s2422" xml:space="preserve">Reli-<lb/>quæ menſuræ à nobis indicatæ paucioribus iisque minoris momenti difficnl-<lb/>tatibus ſunt ſubjectæ; </s> + <s xml:id="echoid-s2423" xml:space="preserve">continentur autem §. </s> + <s xml:id="echoid-s2424" xml:space="preserve">20. </s> + <s xml:id="echoid-s2425" xml:space="preserve">& </s> + <s xml:id="echoid-s2426" xml:space="preserve">exprimunt potiſſimum aquæ <lb/>quantitatem, quæ à primo motus puncto effluit, dum aqua datum velocita-<lb/>tis gradum attingit.</s> + <s xml:id="echoid-s2427" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2428" xml:space="preserve">Quamvis ob rationes modo dictas, præſertim in caſu tuborum diver-<lb/>gentium perfectus conſenſus theoriæ cum experimentis minime expectari <lb/>poſſit, talem tamen expertus fui ſucceſſum, ut facile intellexerim integrum <lb/>futurum fuiſſe conſenſum ſi impedimenta omnia una cum aquearum parti- +<pb o="87" file="0101" n="101" rhead="SECTIO QUARTA."/> +cularum mutua adhæſione præveniri potuiſſent. </s> + <s xml:id="echoid-s2429" xml:space="preserve">Experimenta autem ſunmſi <lb/>tum de tubo divergente, tum de cylindrico: </s> + <s xml:id="echoid-s2430" xml:space="preserve">ſingula nunc exponam:</s> + <s xml:id="echoid-s2431" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div91" type="section" level="1" n="66"> +<head xml:id="echoid-head89" xml:space="preserve">Experimentum 11.</head> +<p> + <s xml:id="echoid-s2432" xml:space="preserve">In Figuræ 19. </s> + <s xml:id="echoid-s2433" xml:space="preserve">tubus formâ coni truncati horizontaliter vaſi erat in ſer-<lb/>tus, vas ipſum aqua implevi usque in C D, ita, ut altitudo ejus ſupra axem <lb/>tubi eſſet æqualis 433. </s> + <s xml:id="echoid-s2434" xml:space="preserve">particulis æqualibus, quibus in toto experimento <lb/>uſus ſum. </s> + <s xml:id="echoid-s2435" xml:space="preserve">Pro illa altitudine experimento inquiſivi in punctum Q maximo <lb/>jactui reſpondens, & </s> + <s xml:id="echoid-s2436" xml:space="preserve">fuit P Q = 287. </s> + <s xml:id="echoid-s2437" xml:space="preserve">part. </s> + <s xml:id="echoid-s2438" xml:space="preserve">dum altitudo o P erat = 146. <lb/></s> + <s xml:id="echoid-s2439" xml:space="preserve">part. </s> + <s xml:id="echoid-s2440" xml:space="preserve">Sic vidi motum aquæ tum propter aquæ adhæſionem, tum propter Fi-<lb/>guram tubi fuiſſe valde retardatum, quod in his caſibus fieri debere aliquo-<lb/>ties monui. </s> + <s xml:id="echoid-s2441" xml:space="preserve">Debuiſſet autem, ſi nihil obſtitiſſet motui, eſſe P Q = 503. </s> + <s xml:id="echoid-s2442" xml:space="preserve">part.</s> + <s xml:id="echoid-s2443" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2444" xml:space="preserve">Deinde Patinam poſui in tabulam horizontalem, cujus ora erant in S <lb/>& </s> + <s xml:id="echoid-s2445" xml:space="preserve">R: </s> + <s xml:id="echoid-s2446" xml:space="preserve">Patinam autem prius madefeci, omnemque aquam ex illa depluere rur-<lb/>ſus ſivi: </s> + <s xml:id="echoid-s2447" xml:space="preserve">ſumtaque menſura P R, illam inveni 206 part.</s> + <s xml:id="echoid-s2448" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2449" xml:space="preserve">Denique diameter G F erat = 13. </s> + <s xml:id="echoid-s2450" xml:space="preserve">part & </s> + <s xml:id="echoid-s2451" xml:space="preserve">M N = 17 part. </s> + <s xml:id="echoid-s2452" xml:space="preserve">longitudo <lb/>tubi autem erat = 125 part.</s> + <s xml:id="echoid-s2453" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2454" xml:space="preserve">His omnibus ita præparatis, dum orificium M N dignito obturarem, <lb/>remoto confeſtim digito aquæ ejiciebantur, earumque pars aliqua in patinam <lb/>decidebat: </s> + <s xml:id="echoid-s2455" xml:space="preserve">hanc ſollicite in tubum vitreum collegi cylindricum, cujus diame-<lb/>ter erat = 8 {1/2} part. </s> + <s xml:id="echoid-s2456" xml:space="preserve">tubus iſte impletus fuit ad altitudinem 210 part. </s> + <s xml:id="echoid-s2457" xml:space="preserve">fuit igitur <lb/>quantitas aquæ in patinam delapſæ = 11922 particulis cubicis.</s> + <s xml:id="echoid-s2458" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2459" xml:space="preserve">Jam vero deberet iſta quantitas per §. </s> + <s xml:id="echoid-s2460" xml:space="preserve">20. </s> + <s xml:id="echoid-s2461" xml:space="preserve">eſſe = nb√{n/g} X log. </s> + <s xml:id="echoid-s2462" xml:space="preserve">{a/a - e}, <lb/>ubi per n intelligitur amplitudo orificii N M ſeu 227. </s> + <s xml:id="echoid-s2463" xml:space="preserve">part. </s> + <s xml:id="echoid-s2464" xml:space="preserve">quadratæ per g am-<lb/>plitudo orificii G F = 133. </s> + <s xml:id="echoid-s2465" xml:space="preserve">part. </s> + <s xml:id="echoid-s2466" xml:space="preserve">quadr. </s> + <s xml:id="echoid-s2467" xml:space="preserve">denotat porro b longitudinem tubi, <lb/>quæ fuit = 125 part. </s> + <s xml:id="echoid-s2468" xml:space="preserve">per a proprie intelligitur altitudo ſuperficiei C D, ſupra <lb/>axem tubi, hic vero intelligenda potius eſt altitudo conveniens velocitati <lb/>aquæ in punctum Q incidentis, ſeu 141. </s> + <s xml:id="echoid-s2469" xml:space="preserve">part. </s> + <s xml:id="echoid-s2470" xml:space="preserve">ſimiliterque pro e ſumenda eſt <lb/>altitudo conveniens velocitati particulæ in punctum R incidentis, nempe 73 part. <lb/></s> + <s xml:id="echoid-s2471" xml:space="preserve">Denique vox abbreviata log. </s> + <s xml:id="echoid-s2472" xml:space="preserve">ſignificat logarithmum Hyperbolicum. </s> + <s xml:id="echoid-s2473" xml:space="preserve">Factis <lb/>iſtis ſubſtitutionibus numericis, fit <lb/>nb√{n/g} X log. </s> + <s xml:id="echoid-s2474" xml:space="preserve">{a/a - e} = 227 X 125 X {17/13} X log. </s> + <s xml:id="echoid-s2475" xml:space="preserve">{141/68} = 26830.</s> + <s xml:id="echoid-s2476" xml:space="preserve"/> +</p> +<pb o="88" file="0102" n="102" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s2477" xml:space="preserve">Fuit igitur quantitas aquæ experimento inventa ad quantitatem, quam <lb/>theoria ſepoſita impedimentorum conſideratione indicat, ut 11922 ad 26830; <lb/></s> + <s xml:id="echoid-s2478" xml:space="preserve">qui numeri, quamvis non parum differant, tamen egregie theoriam confir-<lb/>mant, quod ipſum nunc clare ob oculos ponam.</s> + <s xml:id="echoid-s2479" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2480" xml:space="preserve">In formula nb√{n/g} X log. </s> + <s xml:id="echoid-s2481" xml:space="preserve">{a/a - e}, poſuimus pro a altitudinem velo-<lb/>citati maximæ aquæ effluentis debitam, qualis revera fuit in experimento, <lb/>non qualis remotis obſtaculis futura fuiſſet; </s> + <s xml:id="echoid-s2482" xml:space="preserve">fecimus nempe a = 141: </s> + <s xml:id="echoid-s2483" xml:space="preserve">in theo-<lb/>ria vero eſt a = 433. </s> + <s xml:id="echoid-s2484" xml:space="preserve">Quod ſi autem valor iſte poſterior aſſumatur, retinen-<lb/>do valorem altitudinis e = 73, fit nb√{n/g} log. </s> + <s xml:id="echoid-s2485" xml:space="preserve">{a/a - e} proxime = 6700, qui <lb/>numerus nunc multo minor eſt numero per experimentum eruto, cum antea <lb/>fuerit admodum major. </s> + <s xml:id="echoid-s2486" xml:space="preserve">Talis autem fit cum altitudo e ſervare valorem ponitur: <lb/></s> + <s xml:id="echoid-s2487" xml:space="preserve">Verum prouti altitudo a aucta fuit ab 141 uſque ad 433, ita certe etiam altitu-<lb/>do e eſt augenda, foretque utraque altitudo in eadem ratione augenda, ſi im-<lb/>pedimenta primis guttulis æqualiter reſiſterent & </s> + <s xml:id="echoid-s2488" xml:space="preserve">ſequentibus: </s> + <s xml:id="echoid-s2489" xml:space="preserve">ſed minorem <lb/>reſiſtentiam offendunt cæteris paribus particulæ, quo tardius moventur, atque <lb/>proin etiam guttulæ quæ cadunt cis terminum R minus retardantur, quam quæ <lb/>terminum iſtum tranſgrediuntur: </s> + <s xml:id="echoid-s2490" xml:space="preserve">Facile eſt exinde colligere in minori ratione <lb/>augendam eſſe altitudinem e quam alteram a, ipſam vero rationem dicere non <lb/>poſſumus, niſi à poſteriori, faciendo ſcilicet, ut theoria conveniat cum ex-<lb/>perimento; </s> + <s xml:id="echoid-s2491" xml:space="preserve">ita reperitur ponendum eſſe e = 120, qui numerus animo ad om-<lb/>nes circumſtantias bene attento plane ſatisfacit.</s> + <s xml:id="echoid-s2492" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2493" xml:space="preserve">Sic igitur manifeſtum mihi videtur, experimenti ſucceſſum talem <lb/>fuiſſe, ut plane cum theoria conveniat. </s> + <s xml:id="echoid-s2494" xml:space="preserve">Hujusmodi autem exempla omni-<lb/>no demonſtrant, veras motuum leges in fluidis nos tradidiſſe, eaque inter <lb/>infinita alia ſelegi, quod nullam habent nexum neque affinitatem cum regu-<lb/>la communi, quæ fluida ubique velocitate effluere ſtatuit, toti altitudini <lb/>aquæ ſupra foramen debita, neque poſſint principiis conſuetis ſolvi. </s> + <s xml:id="echoid-s2495" xml:space="preserve">Cæte-<lb/>rum quoniam in hoc experimento motus aquæ retardatus fuit, aliud inſtitue-<lb/>re volui, quo omnia impedimenta admodum diminuerentur, ut ſic appare-<lb/>ret eo magis ad ſe invicem accedere numeros experimenti & </s> + <s xml:id="echoid-s2496" xml:space="preserve">regulæ, quo <lb/>minora eſſent impedimenta.</s> + <s xml:id="echoid-s2497" xml:space="preserve"/> +</p> +<pb o="89" file="0103" n="103" rhead="SECTIO QUARTA."/> +</div> +<div xml:id="echoid-div92" type="section" level="1" n="67"> +<head xml:id="echoid-head90" xml:space="preserve">Experimentum 12.</head> +<p> + <s xml:id="echoid-s2498" xml:space="preserve">Jam itaque uſus fui tubo cylindrico per quem facilior fit transfluxus eo-<lb/>que ob eandem rationem ampliore: </s> + <s xml:id="echoid-s2499" xml:space="preserve">erat præterea arca cui tubus inſertus fuit <lb/>multo amplior, & </s> + <s xml:id="echoid-s2500" xml:space="preserve">denique altitudo aquæ in arca contentæ ſupra axem tubi <lb/>multo minor fuit, ut minori velocitate aquæ transfluerent, ſicque obſtacu-<lb/>la minoris momenti offenderent: </s> + <s xml:id="echoid-s2501" xml:space="preserve">Cætera fuerunt, ut ante.</s> + <s xml:id="echoid-s2502" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2503" xml:space="preserve">Fuit igitur altitudo aquæ ſupra axem tubi = 130. </s> + <s xml:id="echoid-s2504" xml:space="preserve">part. </s> + <s xml:id="echoid-s2505" xml:space="preserve">o P = 553. </s> + <s xml:id="echoid-s2506" xml:space="preserve">part. <lb/></s> + <s xml:id="echoid-s2507" xml:space="preserve">P Q = 453. </s> + <s xml:id="echoid-s2508" xml:space="preserve">part. </s> + <s xml:id="echoid-s2509" xml:space="preserve">P R = 297. </s> + <s xml:id="echoid-s2510" xml:space="preserve">diameter G F vel M N = 19. </s> + <s xml:id="echoid-s2511" xml:space="preserve">part. </s> + <s xml:id="echoid-s2512" xml:space="preserve">tubique longi-<lb/>tudo 130. </s> + <s xml:id="echoid-s2513" xml:space="preserve">part.</s> + <s xml:id="echoid-s2514" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2515" xml:space="preserve">Vidi aquam in patinam delapſam cylindrum expleviſſe, qui 8 {1/2} part. <lb/></s> + <s xml:id="echoid-s2516" xml:space="preserve">in diametro continebat ad altitudinem 281. </s> + <s xml:id="echoid-s2517" xml:space="preserve">part. </s> + <s xml:id="echoid-s2518" xml:space="preserve">& </s> + <s xml:id="echoid-s2519" xml:space="preserve">cujus proinde capacitas <lb/>erat 15950. </s> + <s xml:id="echoid-s2520" xml:space="preserve">part. </s> + <s xml:id="echoid-s2521" xml:space="preserve">cub. </s> + <s xml:id="echoid-s2522" xml:space="preserve">In hoc caſu ponendum eſt a = {453. </s> + <s xml:id="echoid-s2523" xml:space="preserve">453/4. </s> + <s xml:id="echoid-s2524" xml:space="preserve">553} = 93. </s> + <s xml:id="echoid-s2525" xml:space="preserve">part. </s> + <s xml:id="echoid-s2526" xml:space="preserve"><lb/>e = 40. </s> + <s xml:id="echoid-s2527" xml:space="preserve">part. </s> + <s xml:id="echoid-s2528" xml:space="preserve">n = g = 284. </s> + <s xml:id="echoid-s2529" xml:space="preserve">particulis quadratis & </s> + <s xml:id="echoid-s2530" xml:space="preserve">b = 130. </s> + <s xml:id="echoid-s2531" xml:space="preserve">His vero factis <lb/>ſubſtitutionibus fit <lb/>nb√{n/g} X log. </s> + <s xml:id="echoid-s2532" xml:space="preserve">{a/a - e} = 284. </s> + <s xml:id="echoid-s2533" xml:space="preserve">130. </s> + <s xml:id="echoid-s2534" xml:space="preserve">log. </s> + <s xml:id="echoid-s2535" xml:space="preserve">{93/53} = 20760, <lb/>cui numerus in experimento reſpondet, ut vidimus, 15950. </s> + <s xml:id="echoid-s2536" xml:space="preserve">Hic vero nu-<lb/>merus fere quatuor quintas alterius explet, ſicque eidem proxime accedit, <lb/>cum in præcedenti exemplo ob rationes allatas ſimilis numerus à ſimili plus <lb/>quam dimidio defecerit,</s> +</p> +<p> + <s xml:id="echoid-s2537" xml:space="preserve">Jam igitur abunde patet, ſolis obſtaculis extrinſecis attribuendum eſſe, <lb/>quod experimenta non ad amuſſim reſpondeant formulis; </s> + <s xml:id="echoid-s2538" xml:space="preserve">interim tamen ta-<lb/>lia eſſe, ut non poſſint melius harum formularum robur demonſtrare.</s> + <s xml:id="echoid-s2539" xml:space="preserve"/> +</p> +<pb file="0104" n="104" rhead="(90)"/> +</div> +<div xml:id="echoid-div93" type="section" level="1" n="68"> +<head xml:id="echoid-head91" xml:space="preserve">HYDRODYNAMICÆ <lb/>SECTIO QUINTA.</head> +<head xml:id="echoid-head92" style="it" xml:space="preserve">De motu aquarum ex vaſis conſtanter plenis.</head> +<head xml:id="echoid-head93" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s2540" xml:space="preserve">VAſa plena ſervantur, cum continue totidem affunduntur aquæ, <lb/>quot effluunt; </s> + <s xml:id="echoid-s2541" xml:space="preserve">affuſio autem eſſe poteſt vel in eadem cum mo-<lb/>tus ſuperficiei aqueæ directione eademque ſingulis momentis <lb/>velocitate, quaſi ſcilicet nova continue crearetur ſuperficies, <lb/>cui velocitas aquæ proximæ jam inſit, vel lateralis & </s> + <s xml:id="echoid-s2542" xml:space="preserve">ſine im-<lb/>petu, veluti ſi ſuperficies, quæ continue nova creari fingitur, nullo motu <lb/>prædita ſit & </s> + <s xml:id="echoid-s2543" xml:space="preserve">demum ab aqua inferiore ad motum cienda. </s> + <s xml:id="echoid-s2544" xml:space="preserve">Reliquos affun-<lb/>dendi novas aquas, qui infiniti ſunt, modos præteribo.</s> + <s xml:id="echoid-s2545" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2546" xml:space="preserve">Regula interim circa hunc motum, præſertim poſteriorem, recepta eſt, <lb/>aquam effluere velocitate conveniente altitudini ſuperficiei ſupra lumen: </s> + <s xml:id="echoid-s2547" xml:space="preserve">fa-<lb/>cile tamen eſt prævidere illam valere non poſſe, niſi pro vaſe ubique infini-<lb/>te amplo, in reliquis autem fore, ut motus à quiete incipiens ſenſim ſenſim-<lb/>que per aliqua temporis intervalla augeatur, & </s> + <s xml:id="echoid-s2548" xml:space="preserve">poſt infinitum demum tem-<lb/>pus omnem velocitatem acquirat. </s> + <s xml:id="echoid-s2549" xml:space="preserve">Attamen, ſi dicendum, quod res eſt, <lb/>fiunt iſtæ accelerationes plerunque tam celeriter, ut minimo tempusculo tan-<lb/>tum non tota velocitas adſit: </s> + <s xml:id="echoid-s2550" xml:space="preserve">Verum res ſecus ſe habet in prælongis aquæ <lb/>ductibus, in quibus velocitatum augmenta oculos non effugiunt & </s> + <s xml:id="echoid-s2551" xml:space="preserve">cum <lb/>diſtinctis menſuris obſervari poſſunt.</s> + <s xml:id="echoid-s2552" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2553" xml:space="preserve">Quicquid autem ejus rei ſit, cum nullibi diſplicere poſſit accuratio <lb/>mathematica, conſtitui motum aquarum à principio ad quemvis datum ter-<lb/>minum conſiderare & </s> + <s xml:id="echoid-s2554" xml:space="preserve">proſequi.</s> + <s xml:id="echoid-s2555" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2556" xml:space="preserve">§. </s> + <s xml:id="echoid-s2557" xml:space="preserve">2. </s> + <s xml:id="echoid-s2558" xml:space="preserve">Omnes hujus motus proprietates ad tres præcipue æquationes ſe <lb/>reduci patiuntur 1⁰. </s> + <s xml:id="echoid-s2559" xml:space="preserve">inter quantitatem aquæ ejectæ reſpondentisque velocitatis; <lb/></s> + <s xml:id="echoid-s2560" xml:space="preserve">2⁰. </s> + <s xml:id="echoid-s2561" xml:space="preserve">inter tempus & </s> + <s xml:id="echoid-s2562" xml:space="preserve">velocitatem & </s> + <s xml:id="echoid-s2563" xml:space="preserve">3⁰. </s> + <s xml:id="echoid-s2564" xml:space="preserve">inter quantitatem aquæ & </s> + <s xml:id="echoid-s2565" xml:space="preserve">tempus. </s> + <s xml:id="echoid-s2566" xml:space="preserve"><lb/>Harum æquationum ſi una habeatur reliquæ inde ſua ſponte fluunt.</s> + <s xml:id="echoid-s2567" xml:space="preserve"/> +</p> +<pb o="91" file="0105" n="105" rhead="SECTIO QUINTA."/> +<p> + <s xml:id="echoid-s2568" xml:space="preserve">Primam igitur ſolam accuratius ſcrutabimur: </s> + <s xml:id="echoid-s2569" xml:space="preserve">Hic vero memores ſimus <lb/>eorum, quæ in præcedente ſectione monita fuerunt circa contractionem venæ <lb/>per ſimplicia orificia, aut tubos convergentes effluentis, & </s> + <s xml:id="echoid-s2570" xml:space="preserve">dilatationem ejuſ-<lb/>dem, cum per tubos divergentes ejicitur. </s> + <s xml:id="echoid-s2571" xml:space="preserve">Indicavimus autem §. </s> + <s xml:id="echoid-s2572" xml:space="preserve">3. </s> + <s xml:id="echoid-s2573" xml:space="preserve">Art. </s> + <s xml:id="echoid-s2574" xml:space="preserve">1. </s> + <s xml:id="echoid-s2575" xml:space="preserve">Sect. <lb/></s> + <s xml:id="echoid-s2576" xml:space="preserve">IV. </s> + <s xml:id="echoid-s2577" xml:space="preserve">eò uſque venam conſiderandam eſſe, donec particularum velocitates (ab-<lb/>ſtrahendo animum à mutationibus quas gravitas in particulis extra vas producit) <lb/>amplius non mutentur, & </s> + <s xml:id="echoid-s2578" xml:space="preserve">omnem illam venæ partem ceü intra vas motam <lb/>æſtimandam eſſe, quaſi ſcilicet ſuperficies venæ eouſque indureſcat. </s> + <s xml:id="echoid-s2579" xml:space="preserve">Igitur dein-<lb/>ceps cum de vaſe per quod aquæ effluunt ſermo erit, ſubintelligendum erit <lb/>vas illud ideale, cujus orificium effluxus ſit ſectio venæ nulli deinceps muta-<lb/>tioni ſubjectæ, niſi quæ deſcenſui vel aſcenſui venæ debetur.</s> + <s xml:id="echoid-s2580" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div94" type="section" level="1" n="69"> +<head xml:id="echoid-head94" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s2581" xml:space="preserve">§. </s> + <s xml:id="echoid-s2582" xml:space="preserve">3. </s> + <s xml:id="echoid-s2583" xml:space="preserve">Invenire velocitatem aquæ effluentis ex vaſe conſtanter pleno, poſtquam <lb/>jam data aquæ quantitas effluxit.</s> + <s xml:id="echoid-s2584" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div95" type="section" level="1" n="70"> +<head xml:id="echoid-head95" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s2585" xml:space="preserve">Duo ſunt modi affundendæ aquæ præcipue conſideratu digni, quorum <lb/>quivis aliam poſtulat problematis ſolutionem: </s> + <s xml:id="echoid-s2586" xml:space="preserve">vel enim aqua verticaliter in <lb/>vas depluere ponitur & </s> + <s xml:id="echoid-s2587" xml:space="preserve">ita quidem, ut eâdem præciſe affluat velocitate, quam <lb/>habet aquæ ſuperficies, vel lateraliter affluit aqua, ſicque caret impetu, quo <lb/>ſua ſponte aquæ ſuperficiem inſequi poſſit & </s> + <s xml:id="echoid-s2588" xml:space="preserve">in motum demum eſt cienda.</s> + <s xml:id="echoid-s2589" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div96" type="section" level="1" n="71"> +<head xml:id="echoid-head96" xml:space="preserve">Caſus 1.</head> +<p> + <s xml:id="echoid-s2590" xml:space="preserve">Ut pro primo caſu æquationem inveniamus inter quantitatem aquæ <lb/>ejectæ, velocitatemque reſpondentem, iiſdem unica mutata circumſtantia ve-<lb/>ſtigiis inſiſtendum erit, quæ in primis paragraphis ſectionis tertiæ ſecuti ſumus.</s> + <s xml:id="echoid-s2591" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2592" xml:space="preserve">Sit igitur ut in §. </s> + <s xml:id="echoid-s2593" xml:space="preserve">6. </s> + <s xml:id="echoid-s2594" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s2595" xml:space="preserve">3. </s> + <s xml:id="echoid-s2596" xml:space="preserve">vas propoſitum aimb (Fig. </s> + <s xml:id="echoid-s2597" xml:space="preserve">15. </s> + <s xml:id="echoid-s2598" xml:space="preserve">& </s> + <s xml:id="echoid-s2599" xml:space="preserve">16.) <lb/></s> + <s xml:id="echoid-s2600" xml:space="preserve">quod affuſione aquarum conſtanter plenum ſervatur uſque in c d; </s> + <s xml:id="echoid-s2601" xml:space="preserve">effluant au-<lb/>tem aquæ per foramen pl; </s> + <s xml:id="echoid-s2602" xml:space="preserve">ponaturque eam aquæ quantitatem jam effluxiſſe, <lb/>quæ contineri poſſit in cylindro ſuper foramine p l erecto altitudinis x, ulti-<lb/>mam autem guttulam effluxiſſe velocitate, qua aſcendere poſſit ad altitudinem <lb/>q s ſeu v; </s> + <s xml:id="echoid-s2603" xml:space="preserve">ſic jam exhibenda erit æquatio inter x & </s> + <s xml:id="echoid-s2604" xml:space="preserve">v.</s> + <s xml:id="echoid-s2605" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2606" xml:space="preserve">Sit curva C G I ſcala amplitudinum, talis nempe, ut, denotante H L +<pb o="92" file="0106" n="106" rhead="HYDRODYNAMICÆ"/> +altitudinem ſupra foramen, exprimat H G amplitudinem vaſis in illo loco. <lb/></s> + <s xml:id="echoid-s2607" xml:space="preserve">Deinde fiat tertia curva t r u, cujus applicata H r ſit ubique æqualis tertiæ con-<lb/>tinue proportionali ad G H & </s> + <s xml:id="echoid-s2608" xml:space="preserve">P L ſeu cujus applicata H rſit = P L<emph style="super">2</emph>: </s> + <s xml:id="echoid-s2609" xml:space="preserve">G H.</s> + <s xml:id="echoid-s2610" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2611" xml:space="preserve">Dicatur ſpatium D C I L = M, ſpatium D t u L = N, & </s> + <s xml:id="echoid-s2612" xml:space="preserve">erit aſcen-<lb/>ſus potentialis aquæ in vaſe contentæ, poſtquam prædicta quantitas jam efflu-<lb/>xit (per §. </s> + <s xml:id="echoid-s2613" xml:space="preserve">2. </s> + <s xml:id="echoid-s2614" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s2615" xml:space="preserve">3.) </s> + <s xml:id="echoid-s2616" xml:space="preserve">= {N/M}v. </s> + <s xml:id="echoid-s2617" xml:space="preserve">Effluere porro intelligatur particula p l o n, ſu-<lb/>perficiesque c d deſcendere in e f, erit jam velocitatis altitudo pro particula p l o n <lb/>= v + d v; </s> + <s xml:id="echoid-s2618" xml:space="preserve">atque ſi nunc conſtruatur parallelogrammum L x y O, cujus latus <lb/>L O ſit = l o & </s> + <s xml:id="echoid-s2619" xml:space="preserve">alterum L x = P L, erit aſcenſus potentialis ejusdem aquæ <lb/>in ſitu e f m l o n p i e æqualis tertiæ proportionali ad ſpatium E F L O N P I E, <lb/>(quod rurſus eſt = M, quia P L O N exprimit magnitudinem guttulæ p l o n, <lb/>dum C D F E exprimit quantitatem minimam c d f e iſti guttulæ æqualem) <lb/>ſpatium w u x y O L F (quod eſt = ſpatio N - D t w F + L x yO, unde ſi <lb/>P L ſeu L x ponatur = n, C D = m, L O = lo = dx, erit D t = {nn/m}, <lb/>D F = {n/m} dx, hinc ſpatiolum D tw F = {n<emph style="super">3</emph>/mm} dx & </s> + <s xml:id="echoid-s2620" xml:space="preserve">ſpatium L xy O = <lb/>ndx & </s> + <s xml:id="echoid-s2621" xml:space="preserve">denique ſpatium w uxy O L F = N - {n<emph style="super">3</emph>/mm} dx + ndx) & </s> + <s xml:id="echoid-s2622" xml:space="preserve">altitudi-<lb/>nem v + dv. </s> + <s xml:id="echoid-s2623" xml:space="preserve">Eſt igitur aſcenſus potentialis modo dictus = (N - {n<emph style="super">3</emph>/mm} dx + ndx) X <lb/>(v + dv): </s> + <s xml:id="echoid-s2624" xml:space="preserve">M = rejectis differentialibus ſecundi ordinis {N/M} v + {N/M} dv <lb/>- {n<emph style="super">3</emph>/mmM} vdx + {n/M}vdx, ſic ut incrementum aſcenſus potentialis, quod aquæ <lb/>acceſſit dum guttula plon effluxit, ſit = {N/M}dv - {n<emph style="super">3</emph>/mmM}vdx + {n/M}vdx, ubi <lb/>ſpatia N & </s> + <s xml:id="echoid-s2625" xml:space="preserve">M ſunt conſtantis magnitudinis ob aquæ continuam affuſionem. </s> + <s xml:id="echoid-s2626" xml:space="preserve">Non <lb/>conſideramus in hoc caſu primo aſcenſum potentialem guttulæ cdfe, quæ af-<lb/>funditur dum altera æqualis plon effluit, quia iſte aſcenſus non generatur vi <lb/>interna, neque enim aqua inferior poſt ſe trahere ponitur particulam cdfe, <lb/>quin potius hanc vi quadam extrinſeca continue affundi conſideramus, idque <lb/>nec ma<unsure/>jori nec minore velocitate quam quæ eſt ſuperficiei ef. </s> + <s xml:id="echoid-s2627" xml:space="preserve">Ergo omne <lb/>incrementum hic conſiderandum, eſt ut diximus <lb/>{N/M}dv - {n<emph style="super">3</emph>/mmM}vdx + {n/M} vdx.</s> + <s xml:id="echoid-s2628" xml:space="preserve"/> +</p> +<pb o="93" file="0107" n="107" rhead="SECTIO QUINTA."/> +<p> + <s xml:id="echoid-s2629" xml:space="preserve">Debet vero iſtud incrementum æquari deſcenſui actuali centri gravitatis; <lb/></s> + <s xml:id="echoid-s2630" xml:space="preserve">Atqui iſte deſcenſus, poſita D L = a, eſt per paragraphum ſeptimum ſect. </s> + <s xml:id="echoid-s2631" xml:space="preserve">3. </s> + <s xml:id="echoid-s2632" xml:space="preserve"><lb/>= {nadx/M}; </s> + <s xml:id="echoid-s2633" xml:space="preserve">habetur igitur talis æquatio <lb/>{N/M}dv - {n<emph style="super">3</emph>/mmM}vdx + {n/M}vdx = {nadx/M}, ſeu <lb/>dx = Ndv: </s> + <s xml:id="echoid-s2634" xml:space="preserve">(na - nv + {n<emph style="super">3</emph>/mm} v);</s> + <s xml:id="echoid-s2635" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2636" xml:space="preserve">Hæc vero ſi ita integretur, ut v & </s> + <s xml:id="echoid-s2637" xml:space="preserve">x ſimul evaneſcant, dat <lb/>x = {mmN/n<emph style="super">3</emph> - nmm} log. </s> + <s xml:id="echoid-s2638" xml:space="preserve">{mma - mmv + nnv/mma} <lb/>quæ æquatio, poſito c pro numero cujus logarithmus eſt unitas, æquivalet <lb/>huic @alteri <lb/>v = {mma/mm - nn} X (1 - c{n<emph style="super">3</emph> - nmm/mmN} x)</s> +</p> +<p> + <s xml:id="echoid-s2639" xml:space="preserve">Hæc vero ſolutio quadrat pro caſu primo, ubi aqua ſuperne motu af-<lb/>ſunditur communi cum deſcenſu ſuperficiei proximæ.</s> + <s xml:id="echoid-s2640" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div97" type="section" level="1" n="72"> +<head xml:id="echoid-head97" xml:space="preserve">Caſus II.</head> +<p> + <s xml:id="echoid-s2641" xml:space="preserve">Quod ſi jam particula c d f e lateraliter continue affundi ponatur, tunc <lb/>propter inertiam ſuam motui aquæ inferioris reſiſtit atque proinde aſcenſus <lb/>potentialis ipſius aliter in computum venit. </s> + <s xml:id="echoid-s2642" xml:space="preserve">Tunc autem prius conſideran-<lb/>dus eſt aſcenſus potentialis maſſæ aqueæ c d m l p i c auctæ guttula mox affunden-<lb/>da; </s> + <s xml:id="echoid-s2643" xml:space="preserve">deinde indagandus aſcenſus potent. </s> + <s xml:id="echoid-s2644" xml:space="preserve">ejusdem aquæ in ſitu c d m l o n p i c, <lb/>poſtquam nempe guttula jam effluxit, eorumque differentia eſt æquanda cum <lb/>deſcenſu actuali. </s> + <s xml:id="echoid-s2645" xml:space="preserve">{nadx/M}. </s> + <s xml:id="echoid-s2646" xml:space="preserve">Verum aſcenſus potentialis omnis prædictæ aquæ ante <lb/>affuſionem particulæ ejusdemque poſt affuſionem ita invenitur: </s> + <s xml:id="echoid-s2647" xml:space="preserve">nempe aſcen-<lb/>ſus potentialis aquæ c d m l p i c eſt = {Nv/M}, & </s> + <s xml:id="echoid-s2648" xml:space="preserve">aſcenſus potent. </s> + <s xml:id="echoid-s2649" xml:space="preserve">particulæ affundi <lb/>paratæ nullus eſt, quia lateraliter affuſa motum communem nondum habet <lb/>cum maſſa inferiore; </s> + <s xml:id="echoid-s2650" xml:space="preserve">Igitur aſcenſus potentialis utriusque aquæ (qui ſcilicet <lb/>habetur multiplicando maſſam reſpective per ſuum aſcenſum potentialem, di-<lb/>videndoque productorum aggregatum per aggregatum maſſarum) eſt = +<pb o="94" file="0108" n="108" rhead="HYDRODYNAMICÆ"/> +(M X {Nv/M} + ndx X o): </s> + <s xml:id="echoid-s2651" xml:space="preserve">(M + ndx) = {Nv/M + ndx}. </s> + <s xml:id="echoid-s2652" xml:space="preserve">Poſtquam vero particula <lb/>n d x ſuperne jam affuſa eſt, communem acquiſivit motum cum aqua proxi-<lb/>me inferiori, ſicque fit aſcenſus potentialis ejusdem aquæ in ſitu c d m l o n p i c <lb/>æqualis tertiæ proportionali ad ſpatium C D L O N P I C (M + ndx), ſpa-<lb/>tium D t u x y O L D (N + ndx) & </s> + <s xml:id="echoid-s2653" xml:space="preserve">altitudinem v + dv, id eſt, = <lb/>{(N + ndx) x (v + dv)/M + ndx}, cujus exceſſus ſupra priorem aſcenſum potentialem eſt = <lb/>{Ndv + nvdx + ndxdv/M + dx} =, rejectis differentialibus ſecundi ordinis, {Ndv + nvdx/M}. <lb/></s> + <s xml:id="echoid-s2654" xml:space="preserve">Habetur igitur talis æquatio {Ndv + nvdx/M} = {nadx/M}, quæ ut prior per tra-<lb/>ctata & </s> + <s xml:id="echoid-s2655" xml:space="preserve">ad finem deducta dat <lb/>x = {N/n} log. </s> + <s xml:id="echoid-s2656" xml:space="preserve">{a/a - v}, vel <lb/>v = a X (1 - c {-nx/N}) <lb/>quæ ſolutio valet pro affuſione laterali.</s> + <s xml:id="echoid-s2657" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div98" type="section" level="1" n="73"> +<head xml:id="echoid-head98" xml:space="preserve">Scholion 1.</head> +<p> + <s xml:id="echoid-s2658" xml:space="preserve">§. </s> + <s xml:id="echoid-s2659" xml:space="preserve">4. </s> + <s xml:id="echoid-s2660" xml:space="preserve">Sunt hæ æquationes inter ſe admodum diverſæ; </s> + <s xml:id="echoid-s2661" xml:space="preserve">diverſitas au-<lb/>tem eo major quo minoris eſt amplitudinis vas; </s> + <s xml:id="echoid-s2662" xml:space="preserve">& </s> + <s xml:id="echoid-s2663" xml:space="preserve">ſi quidem amplitudo va-<lb/>ſis ſuprema in cd quaſi infinita ſit præ amplitudine foraminis, evaneſcit n <lb/>præ m fitque in priori caſu ſicut in poſteriori. <lb/></s> + <s xml:id="echoid-s2664" xml:space="preserve">v = a X (1 - c<emph style="super">{-n/N}x</emph>) <lb/>Eſt igitur hâc in hypotheſi motus utrobique idem quod haud difficulter <lb/>quisque prævidere potuerit. </s> + <s xml:id="echoid-s2665" xml:space="preserve">Celerior autem ſemper eſt cæteris paribus mo-<lb/>tus in priori affuſione, quam in altera.</s> + <s xml:id="echoid-s2666" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2667" xml:space="preserve">Conveniet hic rem etiam phyſice explicare, ut eam diſtinctius in omni-<lb/>bus phænomenis percipere poſſimus.</s> + <s xml:id="echoid-s2668" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2669" xml:space="preserve">Sit loco vaſis cujuſcunque & </s> + <s xml:id="echoid-s2670" xml:space="preserve">quamcunque directionem habentis bre-<lb/>vioris delineationis gratia cylindrus verticalis cum foramine in fundo, nempe <lb/>G H N D (Fig. </s> + <s xml:id="echoid-s2671" xml:space="preserve">29.) </s> + <s xml:id="echoid-s2672" xml:space="preserve">ſitque dein vas E F P Q perforatum in R S; </s> + <s xml:id="echoid-s2673" xml:space="preserve">fingantur orifi-<lb/> +<anchor type="note" xlink:label="note-0108-01a" xlink:href="note-0108-01"/> +cia RS & </s> + <s xml:id="echoid-s2674" xml:space="preserve">GD perfecte æqualia, & </s> + <s xml:id="echoid-s2675" xml:space="preserve">ad minimam diſtantiam ſibi perfecte re- +<pb o="95" file="0109" n="109" rhead="SECTIO QUINTA."/> +ſpondentia, ita ut aquæ ex ſuperiori vaſe effluentes omnes in cylindrum ſubje-<lb/>ctum influant.</s> + <s xml:id="echoid-s2676" xml:space="preserve"/> +</p> +<div xml:id="echoid-div98" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0108-01" xlink:href="note-0108-01a" xml:space="preserve">Fig. 29,</note> +</div> +<p> + <s xml:id="echoid-s2677" xml:space="preserve">Incipiant aquæ ex utroque vaſe effluere, ex ſuperiori autem conſtanter ea <lb/>effluere velocitate ponantur, quam habet ſuperficies aquæ in cylindro ſuppoſito.</s> + <s xml:id="echoid-s2678" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2679" xml:space="preserve">Ita patet ſatisfieri primæ affuſionis conditioni. </s> + <s xml:id="echoid-s2680" xml:space="preserve">Jam vero hujus motus <lb/>phænomena inveſtigabimus, viſuri num cum præcedentibus conveniant.</s> + <s xml:id="echoid-s2681" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2682" xml:space="preserve">Conſideremus igitur vas ſuperius eſſe veluti infinitum, ut aquæ per R S <lb/>effluentes ſingulis momentis habeant velocitatem quæ conveniat altitudini P B <lb/>ſeu F A: </s> + <s xml:id="echoid-s2683" xml:space="preserve">ſic fingendum erit eſſe hanc altitudinem P B ab initio infinite parvam, <lb/>quia tunc aquæ velocitate infinite parva effluere debent, deinde vero ſenſim <lb/>creſcere, idque continue magis magisque, donec poſt tempus infinitum mo-<lb/>tus uniformis maneat, quæritur autem an altitudo aquæ P B tandem infinita <lb/>futura ſit an vero certum terminum non tranſgreſſura. </s> + <s xml:id="echoid-s2684" xml:space="preserve">Id ſic cognoſcetur.</s> + <s xml:id="echoid-s2685" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2686" xml:space="preserve">Sit altitudo G H vel R H (neque enim illas inter ſe differre cenſendum <lb/>eſt) = a, A F = x, amplitudo orificii L M = n, amplitudo orificii R S = m; <lb/></s> + <s xml:id="echoid-s2687" xml:space="preserve">quia vero, ut manifeſtum eſt, utrumque vas cohærere & </s> + <s xml:id="echoid-s2688" xml:space="preserve">unum efficere puta-<lb/>ri poteſt, erit poſt tempus infinitum (per §. </s> + <s xml:id="echoid-s2689" xml:space="preserve">23. </s> + <s xml:id="echoid-s2690" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s2691" xml:space="preserve">III.) </s> + <s xml:id="echoid-s2692" xml:space="preserve">velocitas <lb/>aquæ in L M = √a + x, & </s> + <s xml:id="echoid-s2693" xml:space="preserve">in R S = √ x, (quod poſterius patet, ſi nunc iterum <lb/>ſeparata vaſa cenſentur, nam utrumque ſine errore fingi poteſt) debent autem <lb/>velocitates eſſe in inverſa ratione amplitudinum orificiorum: </s> + <s xml:id="echoid-s2694" xml:space="preserve">eſt itaque <lb/>√a + x.</s> + <s xml:id="echoid-s2695" xml:space="preserve">√x:</s> + <s xml:id="echoid-s2696" xml:space="preserve">:m. </s> + <s xml:id="echoid-s2697" xml:space="preserve">n, unde a + x. </s> + <s xml:id="echoid-s2698" xml:space="preserve">x: </s> + <s xml:id="echoid-s2699" xml:space="preserve">mm. </s> + <s xml:id="echoid-s2700" xml:space="preserve">nn, vel a.</s> + <s xml:id="echoid-s2701" xml:space="preserve">x:</s> + <s xml:id="echoid-s2702" xml:space="preserve">: mm - nn. </s> + <s xml:id="echoid-s2703" xml:space="preserve">nn, ergo <lb/>x = {nna/mm - nn} & </s> + <s xml:id="echoid-s2704" xml:space="preserve">a + x = {mma/mm - nn}, videmus igitur altitudinem, velocitati <lb/>aquæ in LM debitam, eſſe hoc modo = {mma/mm - nn}, poſtquam ſcilicet infi-<lb/>nita aquæ quantitas jam effluxit: </s> + <s xml:id="echoid-s2705" xml:space="preserve">ſuperius autem habuimus eandem altitudinem, <lb/>ſeu v = {mma/mm - nn} X (1 - c{n<emph style="super">3</emph> - nmm/mmN}x), ubi ſi ponitur x = ∞ (infinito <lb/>enim tempore infinita quantitas transfluit) evaneſcit terminus exponentialis, ſi <lb/>modo m major ſit quam n & </s> + <s xml:id="echoid-s2706" xml:space="preserve">ſic fit pariter v = {mma/mm - nn}. </s> + <s xml:id="echoid-s2707" xml:space="preserve">Mirabilis eſt iſte con-<lb/>ſenſus, quia valde diverſæ ſunt viæ, quas ſecuti ſumus. </s> + <s xml:id="echoid-s2708" xml:space="preserve">Cæterum ſi m non ſit ma-<lb/>jor quam n motus nunquam fit permanens nequidem poſt tempus infinitum, <lb/>creſcit enim tunc velocitas in infinitum cum ſecus altitudo velocitatis nunquam +<pb o="96" file="0110" n="110" rhead="HYDRODYNAMICÆ"/> +tranſgrediatur altitudinem {mma/mm - nn}. </s> + <s xml:id="echoid-s2709" xml:space="preserve">De his igitur caſibus nihil eſt quod di-<lb/>camus.</s> + <s xml:id="echoid-s2710" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div100" type="section" level="1" n="74"> +<head xml:id="echoid-head99" xml:space="preserve">Scholion 2.</head> +<p> + <s xml:id="echoid-s2711" xml:space="preserve">§. </s> + <s xml:id="echoid-s2712" xml:space="preserve">5. </s> + <s xml:id="echoid-s2713" xml:space="preserve">Quæſtio hic nunc alia occurrit notatu digna; </s> + <s xml:id="echoid-s2714" xml:space="preserve">nempe quis eſſe <lb/>poſſit modus affuſionis mechanicus, ut vas ſuperius ad debitam durante toto <lb/>fluxu altitudinem plenum ſervetur. </s> + <s xml:id="echoid-s2715" xml:space="preserve">Difficile foret iſtud Problema ob incon-<lb/>ſtantiam altitudinis quæſitæ, niſi peculiare hic artificium occurreret, quod <lb/>nunc tradam.</s> + <s xml:id="echoid-s2716" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2717" xml:space="preserve">Nititur autem ſuper eo, quod aqua in ſpatio minimo RSDG nullam <lb/>patiatur compreſſionem neque affirmativam neque negativam, quia ex hypo-<lb/>theſi communi velocitate movetur cum aqua proxime ſubſtrata, atque ſic nul-<lb/>la particula nullam nec propellere nec retinere tentet.</s> + <s xml:id="echoid-s2718" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2719" xml:space="preserve">Fiat igitur vas quod dixi utrumque, ſitque tubus cum vaſe ſuperiore <lb/>firmatus (neque enim aliter quam demonſtrationis gratia illa poſuimus antea <lb/>ſeparata) habeat autem tubus in ſummitate a (Fig. </s> + <s xml:id="echoid-s2720" xml:space="preserve">30.) </s> + <s xml:id="echoid-s2721" xml:space="preserve">foraminulum, cui re-<lb/> +<anchor type="note" xlink:label="note-0110-01a" xlink:href="note-0110-01"/> +ſpondeat tubulus a m, in hunc tubulum immittatur tubus vitreus recurvus <lb/>a b c d g, obtectis cera oris m n: </s> + <s xml:id="echoid-s2722" xml:space="preserve">ducatur horizontalis a e noteturque punctum e. <lb/></s> + <s xml:id="echoid-s2723" xml:space="preserve">His ſic præparatis, ſic erit faciendum, ut durante toto experimento ſummitas <lb/>aquæ conſtanter permaneat in puncto e; </s> + <s xml:id="echoid-s2724" xml:space="preserve">& </s> + <s xml:id="echoid-s2725" xml:space="preserve">ad hoc requiri videbis, ut ab initio <lb/>ſuperficies aquæ ſit fundo F P proxima, deinde, ut continue elevetur, & </s> + <s xml:id="echoid-s2726" xml:space="preserve">de-<lb/>nique ut poſt tempus etſi infinitum nunquam tamen tranſcendat altitudinem <lb/>{nna/mm - nn}, facile autem erit aquarum affuſionem ita moderari, ut ſuperficies <lb/>à puncto e non admodum divagetur, @ſi modo circumſtantiæ non ſint ita com-<lb/>paratæ, ut aquæ ab initio nimis celeriter ſint affundendæ.</s> + <s xml:id="echoid-s2727" xml:space="preserve"/> +</p> +<div xml:id="echoid-div100" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0110-01" xlink:href="note-0110-01a" xml:space="preserve">Fig. 30.</note> +</div> +<p> + <s xml:id="echoid-s2728" xml:space="preserve">Quod ſi autem ſuperficiem in tubulo ſupra e elevatam animadvertis, in-<lb/>hibe paullo affuſionem, quod faciendum eſſe alibi demonſtrabo, ſi ſecus fuerit, <lb/>largius aquas affunde.</s> + <s xml:id="echoid-s2729" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2730" xml:space="preserve">Nihil habet difficultatis iſtud experimenti genus cujuſmodi ſæpe feci, <lb/>ſed ne error in experimentum irrepat, examinandus eſt tubi vitrei effectus ca-<lb/>pillaris; </s> + <s xml:id="echoid-s2731" xml:space="preserve">hunc effectum invenies, ſi obturato orificio L M, priusque madefa-<lb/>cto tubo, cylindrus aqua impleatur uſque ad ſummitatem, atque ſic invenies <lb/>ſuperficiem aquæ in tubo pertingere uſque in f, locum nempe altiorem quam e, +<pb o="97" file="0111" n="111" rhead="SECTIO QUINTA."/> +hoc autem punctum f illi, de quo modo diximus, abſtrahendo animum à <lb/>natura tubulorum capillarium, ſubſtitues.</s> + <s xml:id="echoid-s2732" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2733" xml:space="preserve">Hoc igitur modo recte efficietur affuſio ad normam hypotheſeos no-<lb/>ſtræ & </s> + <s xml:id="echoid-s2734" xml:space="preserve">ſic deinceps de hoc motu experimenta ſumi poterunt. </s> + <s xml:id="echoid-s2735" xml:space="preserve">Poſt quam <lb/>vero ſic prolixe ſatis rem explicuimus, non opus puto monere vas ſuperius <lb/>non aliter pertinere ad vas cylindricum inferius, quod ſolum conſideramus, <lb/>quam ut cylindrus eo, quo fieri debet, modo plenus ſervetur atque ſic per m <lb/>non intelligendam eſſe amplitudinem vaſis ſuperioris ſed amplitudinem orificii <lb/>R S, quæ proprie nobis eſt ſuperficies aquæ, cum aquæ ſupra R S tantum de-<lb/>bitæ affuſioni in cylindrum inferiorem inſerviant.</s> + <s xml:id="echoid-s2736" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div102" type="section" level="1" n="75"> +<head xml:id="echoid-head100" xml:space="preserve">Scholion 3.</head> +<p> + <s xml:id="echoid-s2737" xml:space="preserve">§. </s> + <s xml:id="echoid-s2738" xml:space="preserve">6. </s> + <s xml:id="echoid-s2739" xml:space="preserve">Non debeo hic præterire, quod ſic caſus habeatur qui pertinet <lb/>ad hydraulico-ſtaticam, de qua ſcientia quædam monui in Sect. </s> + <s xml:id="echoid-s2740" xml:space="preserve">I. </s> + <s xml:id="echoid-s2741" xml:space="preserve">§. </s> + <s xml:id="echoid-s2742" xml:space="preserve">8. </s> + <s xml:id="echoid-s2743" xml:space="preserve">cognoſci-<lb/>mus nempe nunc quanta velocitate aqua in a præterfluere debeat ut preſſio <lb/>ejus in latera tubi præciſe nulla ſit. </s> + <s xml:id="echoid-s2744" xml:space="preserve">Hæc vero dum ſcriberem, jam detexeram <lb/>leges hydraulico-ſtaticæ generales, & </s> + <s xml:id="echoid-s2745" xml:space="preserve">non ſine voluptate vidi, quod iſte caſus <lb/>ceu corollarium ex theoria plane alia deductus ſimilem acquirat ſolutionem ex <lb/>theoria generali. </s> + <s xml:id="echoid-s2746" xml:space="preserve">Sic omnia ubique mutuo cohærent nexu, legitimamque <lb/>principiorum applicationem demonſtrant.</s> + <s xml:id="echoid-s2747" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div103" type="section" level="1" n="76"> +<head xml:id="echoid-head101" xml:space="preserve">Scholion 4.</head> +<p> + <s xml:id="echoid-s2748" xml:space="preserve">§. </s> + <s xml:id="echoid-s2749" xml:space="preserve">7. </s> + <s xml:id="echoid-s2750" xml:space="preserve">Sequuntur nunc quædam de alio aquæ affundendo modo. </s> + <s xml:id="echoid-s2751" xml:space="preserve">Ponatur <lb/>cylindrus R H N G pro vaſe quocunque, ſitque is conſtanter plenus conſervan-<lb/>dus affuſione laterali: </s> + <s xml:id="echoid-s2752" xml:space="preserve">poterit id fieri injiciendo ſufficientem aquæ quantitatem <lb/>per tubulum m a; </s> + <s xml:id="echoid-s2753" xml:space="preserve">quamvis autem id non fiat ſine motu, attamen, quia hic <lb/>horizontalis eſt, moxl<unsure/> omnis tollitur, & </s> + <s xml:id="echoid-s2754" xml:space="preserve">per ſe neque promovet fluxum per <lb/>cylindrum neque eundem retardat; </s> + <s xml:id="echoid-s2755" xml:space="preserve">ſed eſt alius inſuper modus, quem ſub-<lb/>ducto recte calculo eodem recidere intelligimus: </s> + <s xml:id="echoid-s2756" xml:space="preserve">nempe ſi vas E F P Q infini-<lb/>te amplum cenſemus, & </s> + <s xml:id="echoid-s2757" xml:space="preserve">ejus fundum aqua continue obtectum intelligimus, <lb/>ſed ita, ut aquæ altitudo in vaſe ſuperiori ſit pro infinite parva habenda; </s> + <s xml:id="echoid-s2758" xml:space="preserve">ſubmi-<lb/>niſtrabit vas ſuperius aquam tubo ſibi annexo, neque alius inde motus orietur, <lb/>quam ab affuſione laterali, ſi modo orificium R S ſemper obtectum maneat; <lb/></s> + <s xml:id="echoid-s2759" xml:space="preserve">facile autem fit ut ibi cataracta quædam formetur, ſi orificium L M amplum, <lb/>tubusque R S N H longus ſit. </s> + <s xml:id="echoid-s2760" xml:space="preserve">Quod hic alter modus eundem cum priori effe- +<pb o="98" file="0112" n="112" rhead="HYDRODYNAMICÆ"/> +ctum in motum aquarum exerere debeat, quiſque videt ex eo, quod in utro-<lb/>que modo omnis aquæ tubum ingredientis inertia ſit ab aqua inferiore ſupe-<lb/>randa. </s> + <s xml:id="echoid-s2761" xml:space="preserve">Sed idem etiam à priori demonſtrari poterit inquirendo in motum, qui <lb/>inde oriri debeat, ſecundum æquationem paragraphi octavi Sect. </s> + <s xml:id="echoid-s2762" xml:space="preserve">III. </s> + <s xml:id="echoid-s2763" xml:space="preserve">quæ hæc eſt: <lb/></s> + <s xml:id="echoid-s2764" xml:space="preserve">Ndv - {mmvydx/nn} + {mmvdx/y} = - yxdx; </s> + <s xml:id="echoid-s2765" xml:space="preserve"><lb/>accommodabitur autem ad præſentem caſum, ſi pro m, x & </s> + <s xml:id="echoid-s2766" xml:space="preserve">- d x ſubſtituas re-<lb/>ſpective n, a, & </s> + <s xml:id="echoid-s2767" xml:space="preserve">{ndx/y}, (cujus rei ratio patebit, ſi hæc cum illis contuleris) ſi-<lb/>mulque y infinitum ponas; </s> + <s xml:id="echoid-s2768" xml:space="preserve">tunc enim evaneſcit tertius æquationis terminus, <lb/>fitque omnino, ut pro præſenti negotio ſupra invenimus, <lb/>Ndv + nvdx = nadx.</s> + <s xml:id="echoid-s2769" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2770" xml:space="preserve">Poſtquam in his ſcholiis motus utriuſque indolem, quantum ſimplexrei <lb/>conſideratio phyſica permittit, eorumque differentiam oſtendimus, ſimulque <lb/>modum illos producendi ad legem hypotheſeos mechanicum tradidimus, ſu-<lb/>pereſt, ut reliqua phænomena notabiliora etiam indicentur, quod nunc faciam.</s> + <s xml:id="echoid-s2771" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div104" type="section" level="1" n="77"> +<head xml:id="echoid-head102" xml:space="preserve">Corollarium 1.</head> +<p> + <s xml:id="echoid-s2772" xml:space="preserve">§. </s> + <s xml:id="echoid-s2773" xml:space="preserve">8. </s> + <s xml:id="echoid-s2774" xml:space="preserve">Si in vaſe R S N H omnè fundum abſit, erit orificium L M = <lb/>orificio R S; </s> + <s xml:id="echoid-s2775" xml:space="preserve">poteſt etiam hoc ab illo ſuperari, ſi nempe vaſis divergant late-<lb/>ra. </s> + <s xml:id="echoid-s2776" xml:space="preserve">In his autem caſibus nullum habet terminum altitudo v in æquatione <lb/>v = {mma/mm - nn} X (1 - c{n<emph style="super">3</emph> - nmm/mmN} x) <lb/>& </s> + <s xml:id="echoid-s2777" xml:space="preserve">fit infinita, ſi quantitas aquæ ejectæ indicata per n x eſt infinita.</s> + <s xml:id="echoid-s2778" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2779" xml:space="preserve">Id quidem per ſe patet ex æquatione, cum n eſt major quam m; </s> + <s xml:id="echoid-s2780" xml:space="preserve">at <lb/>cum amplitudines orificiorum ſunt æquales, recurrendum eſt ad æquationem <lb/>differentialem paragraphi tertii, ex qua iſta æquatio proxima deducta fuit, nempe <lb/>{N/M}dv - {n<emph style="super">3</emph>/mmM}vdx + {n/M}vdx = {n/M}adx, <lb/>quæ poſito n = m dat N d v = n a d x, id eſt, v = {nax/N}, ubi v fit manifeſte in-<lb/>finita ſi x eſt infinita.</s> + <s xml:id="echoid-s2781" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2782" xml:space="preserve">§. </s> + <s xml:id="echoid-s2783" xml:space="preserve">9. </s> + <s xml:id="echoid-s2784" xml:space="preserve">Sin autem vaſi propoſito fundum ſit, atque in eo foramen, cujus +<pb o="99" file="0113" n="113" rhead="SECTIO QUINTA."/> +amplitudo indicata per n minor ſit amplitudine orificii R S expreſſa per m, <lb/>habet v valorem quem nunquam attingit quidem, ſed tamen proxime aſſe-<lb/>quitur, & </s> + <s xml:id="echoid-s2785" xml:space="preserve">ad quem tam cito convergit, niſi data opera vaſa huic rei contra-<lb/>ria excogitata adhibeantur, ut poſt minimum fluxus tempuſculum, quod <lb/>ſenſibus percipi poſſit, notabiliter ab eo non deficiat. </s> + <s xml:id="echoid-s2786" xml:space="preserve">Eſt autem terminus il-<lb/>le talis, v = {mma/mm - nn}: </s> + <s xml:id="echoid-s2787" xml:space="preserve">igitur in caſu Scholii ſecundi §. </s> + <s xml:id="echoid-s2788" xml:space="preserve">5. </s> + <s xml:id="echoid-s2789" xml:space="preserve">ultimus ter-<lb/>minus P B eſt = v - a = {nna/mm - nn}. </s> + <s xml:id="echoid-s2790" xml:space="preserve">Exemplo citiſſimam velocitatis ad ultimum <lb/>ſuum terminum acceſſionem illuſtrabo, poſtquam æquationem inter v & </s> + <s xml:id="echoid-s2791" xml:space="preserve"><lb/>tempus altitudini v reſpondens appoſuero.</s> + <s xml:id="echoid-s2792" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div105" type="section" level="1" n="78"> +<head xml:id="echoid-head103" xml:space="preserve">Corollarium 3.</head> +<p> + <s xml:id="echoid-s2793" xml:space="preserve">§. </s> + <s xml:id="echoid-s2794" xml:space="preserve">10. </s> + <s xml:id="echoid-s2795" xml:space="preserve">In caſu affuſionis, quam vocamus, lateralis, fit ultima altitu-<lb/>do v = a, quæcunque inter utrumque vaſis orificium ratio interceſſerit.</s> + <s xml:id="echoid-s2796" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div106" type="section" level="1" n="79"> +<head xml:id="echoid-head104" xml:space="preserve">Corollarium 4.</head> +<p> + <s xml:id="echoid-s2797" xml:space="preserve">§. </s> + <s xml:id="echoid-s2798" xml:space="preserve">11. </s> + <s xml:id="echoid-s2799" xml:space="preserve">Si vas eſt cylindricum ejusque longitudo ponatur = b, fit (vid. <lb/></s> + <s xml:id="echoid-s2800" xml:space="preserve">§. </s> + <s xml:id="echoid-s2801" xml:space="preserve">3.) </s> + <s xml:id="echoid-s2802" xml:space="preserve">N = {nnb/m}: </s> + <s xml:id="echoid-s2803" xml:space="preserve">notetur autem non confundendos eſſe valores litterarum a <lb/>& </s> + <s xml:id="echoid-s2804" xml:space="preserve">b, primus enim exprimit altitudinem ſupremi orificii ſupra inferius, alter <lb/>longitudinem canalis; </s> + <s xml:id="echoid-s2805" xml:space="preserve">Sic itaque conveniunt inter ſe valores in hoc ſaltem <lb/>caſu, cum axis vaſis linea eſt recta & </s> + <s xml:id="echoid-s2806" xml:space="preserve">verticalis; </s> + <s xml:id="echoid-s2807" xml:space="preserve">at ſi axis tortuoſus eſt, vel <lb/>ſaltem non verticalis, differunt à ſe invicem: </s> + <s xml:id="echoid-s2808" xml:space="preserve">Hæc ideo expreſſe monere <lb/>volui, ne quis ſibi a figuris vaſorum, quorum axes ubique rectos & </s> + <s xml:id="echoid-s2809" xml:space="preserve">verti-<lb/>cales feci, imponi patiatur.</s> + <s xml:id="echoid-s2810" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2811" xml:space="preserve">Quod ſi igitur pro vaſis cylindricis ponatur N = {nn/m}b fit pro affuſio-<lb/>ne verticali <lb/>v = {mma/mm - nn} X (1 - c<emph style="super">{nn - mm/mnb} x</emph>) <lb/>& </s> + <s xml:id="echoid-s2812" xml:space="preserve">pro altera laterali fit v = a (1 - c<emph style="super">{- mx/nb}</emph>).</s> + <s xml:id="echoid-s2813" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div107" type="section" level="1" n="80"> +<head xml:id="echoid-head105" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s2814" xml:space="preserve">§. </s> + <s xml:id="echoid-s2815" xml:space="preserve">12. </s> + <s xml:id="echoid-s2816" xml:space="preserve">Invenire velocitatem aquæ, ex vaſe conſtanter pleno effluentis, <lb/>poſtquam fluxus per datum tempus duravit.</s> + <s xml:id="echoid-s2817" xml:space="preserve"/> +</p> +<pb o="100" file="0114" n="114" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div108" type="section" level="1" n="81"> +<head xml:id="echoid-head106" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s2818" xml:space="preserve">Retentis hypotheſibus & </s> + <s xml:id="echoid-s2819" xml:space="preserve">denominationibus omnibus, quas in §. </s> + <s xml:id="echoid-s2820" xml:space="preserve">3. </s> + <s xml:id="echoid-s2821" xml:space="preserve">adhi-<lb/>buimus, poſitoque inſuper tempore à fluxus initio præterito = t, mutan-<lb/>das habebimus æquationes in dicto paragrapho datas in alias, quæ relatio-<lb/>nem exprimant inter t & </s> + <s xml:id="echoid-s2822" xml:space="preserve">v, eliminatis quantitatibus x & </s> + <s xml:id="echoid-s2823" xml:space="preserve">d x. </s> + <s xml:id="echoid-s2824" xml:space="preserve">Eſt vero elemen-<lb/>tum tempuſculi d t proportionale minimo ſpatiolo d x, quod percurritur, di-<lb/>viſo per velocitatem √v: </s> + <s xml:id="echoid-s2825" xml:space="preserve">ponemus igitur d t = {γdx/√v}, & </s> + <s xml:id="echoid-s2826" xml:space="preserve">ſic mutabitur æquatio <lb/>dx = Ndv: </s> + <s xml:id="echoid-s2827" xml:space="preserve">(na - nv + {n<emph style="super">3</emph>/mm} v) <lb/>quæ data fuit pro affuſione verticali debita velocitate inſtituenda in hanc <lb/>(I) dt = N γdv:</s> + <s xml:id="echoid-s2828" xml:space="preserve">(na√v - nv√v + {n<emph style="super">3</emph>/mm} v√v) <lb/>altera vero affuſioni inſerviens laterali, nempe dx = Ndv: </s> + <s xml:id="echoid-s2829" xml:space="preserve">(na - nv) <lb/>abit in hanc poſt eandem ſubſtitutionem <lb/>(II) dt = N γdv:</s> + <s xml:id="echoid-s2830" xml:space="preserve">(na√v - nv√v) <lb/>Hæ vero æquationes debito modo integratæ dant pro prima <lb/>(α) t = {mNγ/n√(mma - nna)} X log. </s> + <s xml:id="echoid-s2831" xml:space="preserve">{m√a + √(mmv - nnv)/m√a - √(mmv - nnv)} <lb/>& </s> + <s xml:id="echoid-s2832" xml:space="preserve">pro altera, quæ ex priori deducitur, poſito m = ∞ <lb/>(β) t = {Nγ/n√a} X log. </s> + <s xml:id="echoid-s2833" xml:space="preserve">{√a + √v/√a - √v}. </s> + <s xml:id="echoid-s2834" xml:space="preserve">Q. </s> + <s xml:id="echoid-s2835" xml:space="preserve">E. </s> + <s xml:id="echoid-s2836" xml:space="preserve">I.</s> + <s xml:id="echoid-s2837" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div109" type="section" level="1" n="82"> +<head xml:id="echoid-head107" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s2838" xml:space="preserve">§. </s> + <s xml:id="echoid-s2839" xml:space="preserve">13. </s> + <s xml:id="echoid-s2840" xml:space="preserve">Si vas de quo ſermo eſt ſit cylindricum utcunque intortum & </s> + <s xml:id="echoid-s2841" xml:space="preserve"><lb/>inclinatum, cujus longitudo ponatur = b, manente altitudine ſuperficiei <lb/>aqueæ ſupra foramen = a, erit rurſus, ut §. </s> + <s xml:id="echoid-s2842" xml:space="preserve">11. </s> + <s xml:id="echoid-s2843" xml:space="preserve">N = {nn/m}b.</s> + <s xml:id="echoid-s2844" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2845" xml:space="preserve">Quoniam autem, ut conſtat, 2γ√A exprimit tempus, quod corpus <lb/>inſumit cadendo libere & </s> + <s xml:id="echoid-s2846" xml:space="preserve">à quiete per altitudinem A, patet quantitatem <lb/>{2mNγ/nn√a} (= 2γ√{bb/a}) exprimere tempus quo corpus moveri incipiens à <lb/>quiete liberè deſcendit per altitudinem {bb/a}: </s> + <s xml:id="echoid-s2847" xml:space="preserve">accipiemus iſtud tempus pro +<pb o="101" file="0115" n="115" rhead="SECTIO QUINTA."/> +communi menſura idemque ponemus = θ, & </s> + <s xml:id="echoid-s2848" xml:space="preserve">mutabitur pro vaſis ſeu ca-<lb/>nalibus cylindricis æquatio (α) in hanc <lb/>t = {nθ/2√(mm - nn)} X log. </s> + <s xml:id="echoid-s2849" xml:space="preserve">{m√a + √(mmv - nnv)/m√a - √(mmv - nnv)} <lb/>altera vera ſignata (β) talis fit <lb/>t = {nθ/2m} X log. </s> + <s xml:id="echoid-s2850" xml:space="preserve">{√a + √v/√a - √v}, <lb/>ex quarum utraque apparet, non poſſe non breviſſimo tempore aquas om-<lb/>nem fere velocitatem acquirere, idque eo citius quo amplior eſt tubus, <lb/>quo brevior, & </s> + <s xml:id="echoid-s2851" xml:space="preserve">quo magis verticalis: </s> + <s xml:id="echoid-s2852" xml:space="preserve">Neque accelerationes ullo modo eſſe <lb/>perceptibiles, niſi prælongi ſtatuantur aquæ ductus & </s> + <s xml:id="echoid-s2853" xml:space="preserve">tunc quoque brevi <lb/>tempore omnes fere accelerationum gradus percurri, quod utrumque nunc <lb/>exemplo illuſtrabo.</s> + <s xml:id="echoid-s2854" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2855" xml:space="preserve">(I) Quæritur tempus quo fluidum ex cylindro conſtanter pleno verticali, <lb/>ſedecim pedes anglicos longo & </s> + <s xml:id="echoid-s2856" xml:space="preserve">cujus diameter quintupla ſit diametri fo-<lb/>raminis, velocitatem acquirit quæ debeatur altitudini {99/100}a, idque in hypo-<lb/>theſi, ad quam æquatio ſecunda pertinet; </s> + <s xml:id="echoid-s2857" xml:space="preserve">ſic eſt {n/m} = {1/25}, v = {99/100}a, <lb/>b = a, unde tempus quod corpus inſumit cadendo libere per ſpatium {bb/a}, <lb/>ſeu θ = uni minuto ſecundo; </s> + <s xml:id="echoid-s2858" xml:space="preserve">hinc fit t = {1/50} log. </s> + <s xml:id="echoid-s2859" xml:space="preserve">399. </s> + <s xml:id="echoid-s2860" xml:space="preserve">id eſt, proxime no-<lb/>næ parti unius minuti ſecundi, quod tempusculum utique imperceptibile <lb/>eſt; </s> + <s xml:id="echoid-s2861" xml:space="preserve">Cum vero tempus notabile aſſumitur, fiunt mutationes altitudinum v, <lb/>inſenſibiles. </s> + <s xml:id="echoid-s2862" xml:space="preserve">Si tempus ſimile (quo nempe velocitas pariter nonaginta no-<lb/>vem centeſimis partibus altitudinis, quanta poſt tempus infinitum fit, debi-<lb/>ta generetur) in prima hypotheſi quæratur, nempe tempus quo obtinetur <lb/>v = {99/100} X ({mma/mm - nn}) reperitur illud præcedente paullulum majus, <lb/>ſed exceſſu inſenſibili: </s> + <s xml:id="echoid-s2863" xml:space="preserve">unde patet in hujusmodi vaſis non poſſe fere aquas <lb/>ſat celeriter affundi in vas ſuperius, ut hypotheſi ſatisfiat, nec adeoque ratio-<lb/>ne ejusdem hypotheſeos experimenta alia ſumi poſſe, quam ut exploretur, <lb/>num revera tanta ſit altitudo B P in figura trigeſima, quanta vi paragraphi <lb/>quinti eſſe debet, ut punctum e aut f, durante fluxu ſitum ſervet, quem <lb/>ante fluxum obturato orificio L M, nullaque exiſtente aqua in vaſe ſuperiore <lb/>habuit.</s> + <s xml:id="echoid-s2864" xml:space="preserve"/> +</p> +<pb o="102" file="0116" n="116" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s2865" xml:space="preserve">(II) Quæritur nunc idem tempus pro ſecunda rurſus hypotheſi, ſi <lb/>tubus ejusdem fuerit amplitudinis eodemque foramine inſtructus, ſed oblique <lb/>ſitus longitudinemque b habuerit 184 perticarum ſeu 1104 pedum Pariſ. </s> + <s xml:id="echoid-s2866" xml:space="preserve">dum al-<lb/>titudo ſuperficiei aqueæ ſupra orificium effluxus ſit 16. </s> + <s xml:id="echoid-s2867" xml:space="preserve">ped. </s> + <s xml:id="echoid-s2868" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s2869" xml:space="preserve">Ita fiet b = <lb/>1104, & </s> + <s xml:id="echoid-s2870" xml:space="preserve">{bb/a} = 76176. </s> + <s xml:id="echoid-s2871" xml:space="preserve">atque præterpropter θ = 72 ſec. </s> + <s xml:id="echoid-s2872" xml:space="preserve">min. </s> + <s xml:id="echoid-s2873" xml:space="preserve">unde tempus <lb/>quæſitum medium eſt inter octo novemque minuta ſecunda, quod certe ſatis<unsure/> <lb/>notabile eſt. </s> + <s xml:id="echoid-s2874" xml:space="preserve">Si vero tempus deſideretur, quo altitudo v exæquet tantum quar-<lb/>tam partem altitudinis a, reperietur illud æquale {72/50} log. </s> + <s xml:id="echoid-s2875" xml:space="preserve">3 = proxime uni mi-<lb/>nuto ſecundo cum dimidio.</s> + <s xml:id="echoid-s2876" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2877" xml:space="preserve">Neſcio an hæc conveniant cum iis, quæ Mariottus à ſe obſervata refert <lb/>in tract. </s> + <s xml:id="echoid-s2878" xml:space="preserve">de mot. </s> + <s xml:id="echoid-s2879" xml:space="preserve">aquar. </s> + <s xml:id="echoid-s2880" xml:space="preserve">part. </s> + <s xml:id="echoid-s2881" xml:space="preserve">5. </s> + <s xml:id="echoid-s2882" xml:space="preserve">diſc. </s> + <s xml:id="echoid-s2883" xml:space="preserve">1. </s> + <s xml:id="echoid-s2884" xml:space="preserve">ubi mentionem facit alicujus fontis ſa-<lb/>lientis, qui eſt à Chantilly, ad quem aquæ devehuntur per canalem 184. </s> + <s xml:id="echoid-s2885" xml:space="preserve">pertic{as} <lb/>longum, ſi modo recte ex antecedentibus conjeci, eratque ſumma ſuperficiei <lb/>aqueæ altitudo ſupra orificium effluxus indicata per a ſedecim pedum: </s> + <s xml:id="echoid-s2886" xml:space="preserve">diame-<lb/>ter aquæductus erat 5. </s> + <s xml:id="echoid-s2887" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2888" xml:space="preserve">orificium autem habebat diametrum unius pollicis. <lb/></s> + <s xml:id="echoid-s2889" xml:space="preserve">Videtur mihi Mariottus ita loqui ac ſi accelerationes multo fuiſſent tardiores, <lb/>quam ab formula noſtra indicantur, quod neſcio an tribuendum ſit huic quod <lb/>fortaſſe alium, præter orificium de quo hic ſermo eſt, exitum habuerint aquæ, <lb/>an, quod aquæ ductus dum fluxus inciperet non fuerit aqua plenus, quod <lb/>poſterius multa faciunt, ut credam; </s> + <s xml:id="echoid-s2890" xml:space="preserve">ſi neutrum fuerit, confido phænomena <lb/>qualia à Mariotto obſervata fuerunt & </s> + <s xml:id="echoid-s2891" xml:space="preserve">quotidie de novo obſervari poterunt pla-<lb/>ne conveniſſe cum calculo noſtro. </s> + <s xml:id="echoid-s2892" xml:space="preserve">Cæterum verba Mariotti hæc ſunt: </s> + <s xml:id="echoid-s2893" xml:space="preserve"><lb/>Illud inſuper, ait, ſingulari eidem jactui accidit, quod obturato manu orifici@ <lb/>per decem aut duodecim ſcrupulorum ſecundorum temporis ſpatium eodem{q́ue}; </s> + <s xml:id="echoid-s2894" xml:space="preserve">po-<lb/>ſtea reſerato, aqua non protin{us} erumpat, ſed paullatim aſſurgens jact{us} aſcen-<lb/>dat ad 3. </s> + <s xml:id="echoid-s2895" xml:space="preserve">poll. </s> + <s xml:id="echoid-s2896" xml:space="preserve">poſtea ad pedis altitudinem & </s> + <s xml:id="echoid-s2897" xml:space="preserve">deni ad du@s pedes ſucceſsive no-<lb/>tabilibus intervallis.</s> + <s xml:id="echoid-s2898" xml:space="preserve">..</s> + <s xml:id="echoid-s2899" xml:space="preserve">..</s> + <s xml:id="echoid-s2900" xml:space="preserve">.. </s> + <s xml:id="echoid-s2901" xml:space="preserve">Sedtandem tamen toto impetu ſuo aquæ exiliebant.</s> + <s xml:id="echoid-s2902" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div110" type="section" level="1" n="83"> +<head xml:id="echoid-head108" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s2903" xml:space="preserve">§. </s> + <s xml:id="echoid-s2904" xml:space="preserve">14. </s> + <s xml:id="echoid-s2905" xml:space="preserve">Invenire quantitatem aquæ per datum vas, conſtanter plenum <lb/>conſervandum, dato tempore transfluentem.</s> + <s xml:id="echoid-s2906" xml:space="preserve"/> +</p> +<pb o="103" file="0117" n="117" rhead="SECTIO QUINTA."/> +</div> +<div xml:id="echoid-div111" type="section" level="1" n="84"> +<head xml:id="echoid-head109" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s2907" xml:space="preserve">Adhibitis rurſus poſitionibus & </s> + <s xml:id="echoid-s2908" xml:space="preserve">denominationibus paragraphi tertii & </s> + <s xml:id="echoid-s2909" xml:space="preserve"><lb/>duodecimi, invenienda nunc erit æquatio inter x & </s> + <s xml:id="echoid-s2910" xml:space="preserve">t: </s> + <s xml:id="echoid-s2911" xml:space="preserve">quia vero, ut vidi-<lb/>mus §. </s> + <s xml:id="echoid-s2912" xml:space="preserve">12. </s> + <s xml:id="echoid-s2913" xml:space="preserve">eſt d t = {γdx/√v}, erit √ v = {γdx/dt}, hicque valor ſubſtituendus <lb/>erit in æquationibus, quas dedimus §. </s> + <s xml:id="echoid-s2914" xml:space="preserve">3. </s> + <s xml:id="echoid-s2915" xml:space="preserve">integratis; </s> + <s xml:id="echoid-s2916" xml:space="preserve">prior harum æquationum <lb/>hæc fuit: </s> + <s xml:id="echoid-s2917" xml:space="preserve">v = {mma/mm - nn} X (1 - c{n<emph style="super">3</emph> - nmm/mmN} x) <lb/>quæ pro præſecuti inſtituto mutatur in hanc <lb/>(I) {γγdx<emph style="super">2</emph>/dt<emph style="super">2</emph>} = {mma/mm - nn} X (1 - c{n<emph style="super">3</emph> - nmm/mmN} x) <lb/>altera ex §. </s> + <s xml:id="echoid-s2918" xml:space="preserve">3. </s> + <s xml:id="echoid-s2919" xml:space="preserve">allegatarum æquationum talis fuit <lb/>v = a X (1 - c<emph style="super">{- n/N} x</emph>) <lb/>quæ adeoque ſubminiſtrat in præſenti caſu ſequentem <lb/>(II) {γγdx<emph style="super">2</emph>/dt<emph style="super">2</emph>} = a X (1 - c<emph style="super">{- n/N} x</emph>)</s> +</p> +<p> + <s xml:id="echoid-s2920" xml:space="preserve">Erunt nunc æquationes (I) & </s> + <s xml:id="echoid-s2921" xml:space="preserve">(II) integrandæ, quod quidem facile <lb/>eſt & </s> + <s xml:id="echoid-s2922" xml:space="preserve">quia prior alteram continet (utraque enim eadem eſt ſi m = ∞) <lb/>hanc ſolam pertractabimus, eamque nunc ſub hâc forma conſiderabimus. <lb/></s> + <s xml:id="echoid-s2923" xml:space="preserve">dt = {γ√(mm - nn)/m√a}dx:</s> + <s xml:id="echoid-s2924" xml:space="preserve">√(1 - c{n<emph style="super">3</emph> - nmm/mmN}x)</s> +</p> +<p> + <s xml:id="echoid-s2925" xml:space="preserve">Ponatur autem ut integrationis modus eo magis pateſcat <lb/>c{n<emph style="super">3</emph> - nmm/mmN}x = z, atque proin dx = {mmNdz/(n<emph style="super">3</emph> - nmm)z}, <lb/>dein brevitatis ergo indice<unsure/>tur quantitas conſtans <lb/>{γ√(mm - nn)/m√a} X {mmN/n<emph style="super">3</emph> - nmm}, ſeu {- γmN/n√(mm - nn) a} per α, <lb/>& </s> + <s xml:id="echoid-s2926" xml:space="preserve">habebitur dt = {αdz/z√(1 - z)}, +<pb o="104" file="0118" n="118" rhead="HYDRODYNAMICÆ."/> +in quâ ſi præterea fiat 1 - z = qq, ſeu z = 1 - qq, dz = - 2qdq, <lb/>oritur <lb/>dt = {- 2αdq/1 - qq} = {- αdq/1 + q} {- αdq/1 - q} <lb/>cujus integralis eſt <lb/>t = - α log. </s> + <s xml:id="echoid-s2927" xml:space="preserve">(1 + q) + α log. </s> + <s xml:id="echoid-s2928" xml:space="preserve">(1 - q) = α log. </s> + <s xml:id="echoid-s2929" xml:space="preserve">{1 - q/1 + q}.</s> + <s xml:id="echoid-s2930" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2931" xml:space="preserve">Nec opus eſt conſtante, quandoquidem ex natura rei t & </s> + <s xml:id="echoid-s2932" xml:space="preserve">x, ſimul <lb/>evaneſcere debent, poſito autem x = o, fit z = 1, & </s> + <s xml:id="echoid-s2933" xml:space="preserve">q = o, igitur pa-<lb/>riter t & </s> + <s xml:id="echoid-s2934" xml:space="preserve">q ſimul à nihilo incipere debent, cui conditioni ſatisfacit æquatio <lb/>inventa t = α log. </s> + <s xml:id="echoid-s2935" xml:space="preserve">{1 - q/1 + q}: </s> + <s xml:id="echoid-s2936" xml:space="preserve">Supereſt ut retrogrado ordine valores priſtinos <lb/>reaſſumamus, ita vero fit <lb/>t = α log. </s> + <s xml:id="echoid-s2937" xml:space="preserve">{1 - √(1 - z)/1 + √(1 - z)} vel <lb/>t = {γmN/n√(mm - nn)a} X log. </s> + <s xml:id="echoid-s2938" xml:space="preserve">{1 + √(1 - z)/1 - √(1 - z)} vel denique <lb/>(I) t = {γmN/n√(mm - nn) a} X [log. </s> + <s xml:id="echoid-s2939" xml:space="preserve">[1 + √(1 - c{n<emph style="super">3</emph> - nmm/mmN} x)] <lb/>- log. </s> + <s xml:id="echoid-s2940" xml:space="preserve">[1 - √(1 - c{n<emph style="super">3</emph> - nmm/mmN} x)]] <lb/>Iſtaque æquatio poſito m = ∞ dat alteram æquationem quæſitam <lb/>(II) t = {γN/n√a} X [log. </s> + <s xml:id="echoid-s2941" xml:space="preserve">[1 + √(1 - c<emph style="super">{- n/N} x</emph>)] <lb/>- log. </s> + <s xml:id="echoid-s2942" xml:space="preserve">[1 - √(1 - c<emph style="super">{-n/N} x</emph>)]] Q. </s> + <s xml:id="echoid-s2943" xml:space="preserve">E. </s> + <s xml:id="echoid-s2944" xml:space="preserve">I.</s> + <s xml:id="echoid-s2945" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div112" type="section" level="1" n="85"> +<head xml:id="echoid-head110" xml:space="preserve">Corollarium 1.</head> +<p> + <s xml:id="echoid-s2946" xml:space="preserve">§. </s> + <s xml:id="echoid-s2947" xml:space="preserve">15. </s> + <s xml:id="echoid-s2948" xml:space="preserve">Si ponatur x = ∞, ut appareat natura rei, cum infinita jam <lb/>transfluxit aquæ quantitas aſſumaturque m major quam n, prouti plerumque <lb/>eſſe ſolet, evaneſcere cenſenda eſt, in utroque logarithmo affirmative ſum-<lb/>to, quantitas exponentialis & </s> + <s xml:id="echoid-s2949" xml:space="preserve">habebitur utrobique log. </s> + <s xml:id="echoid-s2950" xml:space="preserve">2. </s> + <s xml:id="echoid-s2951" xml:space="preserve">At vero in logarith-<lb/>mo negative ſumto ſtatuenda eſt <lb/>√(1 - c{n<emph style="super">3</emph> - nmm/mmN} x) = 1 - {1/2} c{n<emph style="super">3</emph> - nmm/mmN} x & </s> + <s xml:id="echoid-s2952" xml:space="preserve">proinde, +<pb o="105" file="0119" n="119" rhead="SECTIO QUINTA."/> +log. </s> + <s xml:id="echoid-s2953" xml:space="preserve">[1 - √(1 - c{n<emph style="super">3</emph> - nmm/mmN} x)] = log.</s> + <s xml:id="echoid-s2954" xml:space="preserve">{1/2}c{n<emph style="super">3</emph> - nmm/mmN} x = {n<emph style="super">3</emph> - nmm/mmN} x - log. </s> + <s xml:id="echoid-s2955" xml:space="preserve">2@</s> +</p> +<p> + <s xml:id="echoid-s2956" xml:space="preserve">Hæ ſubſtitutiones ſi recte fiant, erit pro primo quem finximus affuſio-<lb/>nis modo <lb/>(I) t = {γmN/n√(mm - nn) a} X (2 log. </s> + <s xml:id="echoid-s2957" xml:space="preserve">2 + {mmn - n<emph style="super">3</emph>/mmN} x) <lb/>quæ poſito rurſus m = ∞ dat pro altero caſu <lb/>(II) t = {γN/n√a} X (2. </s> + <s xml:id="echoid-s2958" xml:space="preserve">log. </s> + <s xml:id="echoid-s2959" xml:space="preserve">2 + {n/N} x).</s> + <s xml:id="echoid-s2960" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2961" xml:space="preserve">Sequitur ex iſtis formulis, minori quidem quantitate transfluere aquas, <lb/>ac ſi ſtatim ab initio omni velocitate, quam in utroque caſu poſt tempus <lb/>infinitum acquirunt, effluerent: </s> + <s xml:id="echoid-s2962" xml:space="preserve">differentiam tamen nunquam certum trans-<lb/>gredi terminum & </s> + <s xml:id="echoid-s2963" xml:space="preserve">poſt tempus infinitum finitis comprehendi terminis.</s> + <s xml:id="echoid-s2964" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div113" type="section" level="1" n="86"> +<head xml:id="echoid-head111" xml:space="preserve">Corollarium 2.</head> +<p> + <s xml:id="echoid-s2965" xml:space="preserve">§. </s> + <s xml:id="echoid-s2966" xml:space="preserve">16. </s> + <s xml:id="echoid-s2967" xml:space="preserve">Quum convertimus æquationes inventas, obtinemus <lb/>(I) x = {2mmN/mmn - n<emph style="super">3</emph>} - [log. </s> + <s xml:id="echoid-s2968" xml:space="preserve">(1 + c<emph style="super">{-t/α</emph>}) - log. </s> + <s xml:id="echoid-s2969" xml:space="preserve">2 + {t/2α}], & </s> + <s xml:id="echoid-s2970" xml:space="preserve"><lb/>(II) x = {2N/n} X [log. </s> + <s xml:id="echoid-s2971" xml:space="preserve">(1 + c<emph style="super">{-t/β}</emph>) - log. </s> + <s xml:id="echoid-s2972" xml:space="preserve">2 + {t/2β}] <lb/>ubi α, ut ſupra, = {-γmN/n√(mm - nn)a} & </s> + <s xml:id="echoid-s2973" xml:space="preserve">β = {-γN/n√a}.</s> + <s xml:id="echoid-s2974" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2975" xml:space="preserve">Si præterea, ut in proximo Corollario, ponatur t = ∞, evaneſcit <lb/>unitas præ quantitatibus, exponentialibus, quæ ſupra omnem ordinem infinitæ <lb/>ſunt, & </s> + <s xml:id="echoid-s2976" xml:space="preserve">fit log. </s> + <s xml:id="echoid-s2977" xml:space="preserve">(1 + c<emph style="super">{-t/α}</emph>) = -{t/α} atque log. </s> + <s xml:id="echoid-s2978" xml:space="preserve">(1 + c<emph style="super">{-t/β}</emph>) = -{t/β}: <lb/></s> + <s xml:id="echoid-s2979" xml:space="preserve">unde tunc erit reſumtis valoribus litterarum α & </s> + <s xml:id="echoid-s2980" xml:space="preserve">β. </s> + <s xml:id="echoid-s2981" xml:space="preserve"><lb/>(I) x = {mt√a/γ√(mm - nn)} - {2mmN/mmn - n<emph style="super">3</emph>} log. </s> + <s xml:id="echoid-s2982" xml:space="preserve">2. </s> + <s xml:id="echoid-s2983" xml:space="preserve">& </s> + <s xml:id="echoid-s2984" xml:space="preserve"><lb/>(II) x = {t√a/γ} - {2N/n} log. </s> + <s xml:id="echoid-s2985" xml:space="preserve">2.</s> + <s xml:id="echoid-s2986" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s2987" xml:space="preserve">Igitur ſi ſtatim à fluxus initio utrobique aquæ omni, quam acquirere +<pb o="106" file="0120" n="120" rhead="HYDRODYNAMICÆ"/> +poſſunt, velocitate conſtanter effluerent, non excederet earum quantitas poſt <lb/>tempus infinitum quantitatem pro eodem tempore theoriæ reſpondentem, niſi <lb/>parvula quantitate, quæ in prino caſu exprimitur per {2mmN/mm - nn} log. </s> + <s xml:id="echoid-s2988" xml:space="preserve">2. </s> + <s xml:id="echoid-s2989" xml:space="preserve">& </s> + <s xml:id="echoid-s2990" xml:space="preserve">in <lb/>ſecundo per {aN/n} log. </s> + <s xml:id="echoid-s2991" xml:space="preserve">2. </s> + <s xml:id="echoid-s2992" xml:space="preserve">Atque ſi loco temporis infiniti ſumas tempus tantum <lb/>aliquot ſcrupulorum ſecundorum, idem theorema proxime locum habebit; </s> + <s xml:id="echoid-s2993" xml:space="preserve">ita <lb/>ut ſi v. </s> + <s xml:id="echoid-s2994" xml:space="preserve">gr. </s> + <s xml:id="echoid-s2995" xml:space="preserve">poſt decem prima minuta ſecunda effluxerit quantitas Q, effluxura <lb/>fere ſit totidem minutis ſecundis proxime ſequentibus Q + {2mmN/mmn - n<emph style="super">3</emph>} log. </s> + <s xml:id="echoid-s2996" xml:space="preserve">2. </s> + <s xml:id="echoid-s2997" xml:space="preserve">vel <lb/>in altero caſu Q + {2N/n} log. </s> + <s xml:id="echoid-s2998" xml:space="preserve">2.</s> + <s xml:id="echoid-s2999" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div114" type="section" level="1" n="87"> +<head xml:id="echoid-head112" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s3000" xml:space="preserve">§. </s> + <s xml:id="echoid-s3001" xml:space="preserve">17. </s> + <s xml:id="echoid-s3002" xml:space="preserve">Ad theoriam hacte<unsure/>nus expoſitam pertine<unsure/>t etiam motus aquarum per <lb/>ſiphones. </s> + <s xml:id="echoid-s3003" xml:space="preserve">Indicat autem theoria, poſſe ſiphonis axem utcunque inflecti, ne-<lb/>que inde motum aquarum deturbatum iri, modo altitudo ſuperficiei aqueæ ſu-<lb/>pra orificium effluxus eadem maneat; </s> + <s xml:id="echoid-s3004" xml:space="preserve">cum præterea aquæductus, ſiphones <lb/>aut diabetæ hujuſcemodique vaſa alia ſoleant eſſe cylindrica erit ut monui §. <lb/></s> + <s xml:id="echoid-s3005" xml:space="preserve">13. </s> + <s xml:id="echoid-s3006" xml:space="preserve">quoties id contingit, ponendum N = {nn/m} b, intelligendo per b longitu-<lb/>dinem canalis aut ſiphonis: </s> + <s xml:id="echoid-s3007" xml:space="preserve">in formulis quoque paragraphorum 14, 15, & </s> + <s xml:id="echoid-s3008" xml:space="preserve">16, <lb/>erunt quantitates ſic interpretandæ, ubi de temporibus quæſtio eſt, ut 2 γ √ A <lb/>repræſentet tempus quod corpus impendit in deſcenſum per altitudinem ver-<lb/>ticalem A à quiete cœptum.</s> + <s xml:id="echoid-s3009" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3010" xml:space="preserve">Cæterum, ut dixi paſſim, nihil indicat ſingulare theoria hujus ſectionis, <lb/>quod ſub ſenſus cadat, niſi in aquæ ductibus admodum longis, ad horizonta-<lb/>lem valde obliquis & </s> + <s xml:id="echoid-s3011" xml:space="preserve">orificium non admodum ſtrictum habentibus; </s> + <s xml:id="echoid-s3012" xml:space="preserve">hæctria <lb/>enim concurrunt ad retardandas ſicque notabiles efficiendas accelerationes, <lb/>quarum menſuræ potiſſimum theoriam commendant.</s> + <s xml:id="echoid-s3013" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3014" xml:space="preserve">Eſt tamen & </s> + <s xml:id="echoid-s3015" xml:space="preserve">in his circumſtantiis medium aliquod obſervandum, ne <lb/>impedimenta ab adhæſione aquæ oriunda nimia ſint</s> +</p> +<p> + <s xml:id="echoid-s3016" xml:space="preserve">Quod attinet ad affuſionem aquarum, mihi viſus ſum animadvertere, ſi +<pb o="107" file="0121" n="121" rhead="SECTIO QUINTA."/> +verticaliter fiat & </s> + <s xml:id="echoid-s3017" xml:space="preserve">cum impetu, tantum abeſſe, ut inde motus acceleretur, quin <lb/>potius retardetur, niſi aquarum affuſio fiat in totam ſuperficiem æquabiliter eo, <lb/>quem §. </s> + <s xml:id="echoid-s3018" xml:space="preserve">4. </s> + <s xml:id="echoid-s3019" xml:space="preserve">expoſui, modo, ſi enim aliter affundantur, motus aquarum in va-<lb/>ſe perturbatur, iſque motus confuſus effluxum retardat.</s> + <s xml:id="echoid-s3020" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3021" xml:space="preserve">§. </s> + <s xml:id="echoid-s3022" xml:space="preserve">18. </s> + <s xml:id="echoid-s3023" xml:space="preserve">Denique huc quodammodo pertinent experimenta ab Clar. <lb/></s> + <s xml:id="echoid-s3024" xml:space="preserve">Joanne Poleno inſtituta, ut refert in libro primo de motu aquæ mixto, p. </s> + <s xml:id="echoid-s3025" xml:space="preserve">21. </s> + <s xml:id="echoid-s3026" xml:space="preserve">& </s> + <s xml:id="echoid-s3027" xml:space="preserve"><lb/>ſeqq. </s> + <s xml:id="echoid-s3028" xml:space="preserve">quæ ideo hic alleganda eſſe cenſui, quod egregie demonſtrant, ubique <lb/>celeritatem ultimam in vaſis conſtanter plenis eam eſſe, quæ integræ aquæ al-<lb/>titudini conveniat, ſi vaſa non ſint ſubmerſa, aut differentiæ altitudinum aquæ <lb/>internæ & </s> + <s xml:id="echoid-s3029" xml:space="preserve">externæ in vaſis ſubmerſis, quamvis de cætero nihil in illis ſit, quod <lb/>nunc novum adhuc ſit, quia nullæ illic conſiderantur accelerationes.</s> + <s xml:id="echoid-s3030" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3031" xml:space="preserve">Finge cylindrum, cujus axis habeat ſitum verticalem, amplitudinis ve-<lb/>luti infinitæ; </s> + <s xml:id="echoid-s3032" xml:space="preserve">fundum integrum ſit: </s> + <s xml:id="echoid-s3033" xml:space="preserve">in pariete autem fiſſura ſit axi parallela, fo-<lb/>ramen habens parallelogrammi rectanguli, quæ à fundo ad cylindri uſque ſum-<lb/>mitatem extendatur. </s> + <s xml:id="echoid-s3034" xml:space="preserve">Puta porro aquam in cylindrum affundi æquabiliter, ita, <lb/>ut æqualibus temporibus quantitates injiciantur æquales, effluent aquæ ex cy-<lb/>lindro per fiſſuram: </s> + <s xml:id="echoid-s3035" xml:space="preserve">nec tamen ab initio eadem effluent quantitate, qua ſuper-<lb/>ne affunduntur, ſed minori: </s> + <s xml:id="echoid-s3036" xml:space="preserve">igitur aſſurget ſuperficies aquæ in cylindro ad <lb/>certam uſque altitudinem aſſymptoton; </s> + <s xml:id="echoid-s3037" xml:space="preserve">ſi vero is jam intelligatur adeſſe ter-<lb/>minus, immutata manebit altitudo aquæ & </s> + <s xml:id="echoid-s3038" xml:space="preserve">eadem quantitate effluent conſtan-<lb/>ter aquæ, qua affunduntur: </s> + <s xml:id="echoid-s3039" xml:space="preserve">Apparet quoque, altitudinem aquæ in cylindro <lb/>eo majorem fore, quo largius affundantur: </s> + <s xml:id="echoid-s3040" xml:space="preserve">Quæritur itaque auctis quantitati-<lb/>bus aquarum dato tempore affundendis, in quanam ratione creſcere debeant <lb/>altitudines, ad quas aquæ in cylindro aſſurgent.</s> + <s xml:id="echoid-s3041" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3042" xml:space="preserve">Solutio hæc eſt. </s> + <s xml:id="echoid-s3043" xml:space="preserve">Sit altitudo aquæ, cum eſt in ſtatu permanente = α: <lb/></s> + <s xml:id="echoid-s3044" xml:space="preserve">& </s> + <s xml:id="echoid-s3045" xml:space="preserve">abſcindatur à ſuperficie pars quæ ſit = x, una cum differentiali d x: </s> + <s xml:id="echoid-s3046" xml:space="preserve">ſit lati-<lb/>tudo rimæ = n, habebimus veluti foramen amplitudinis = n d x, per quod <lb/>aquæ effluunt velocitate √ x: </s> + <s xml:id="echoid-s3047" xml:space="preserve">igitur quantitas aquæ dato tempore ibi effluen-<lb/>tis eſt ut n d x √ x, cujus integralis eſt {2/3} n x √ x; </s> + <s xml:id="echoid-s3048" xml:space="preserve">quæ exprimit quantitatem <lb/>aquæ dato tempore per rimæ longitudinem abſciſſam x effluentem: </s> + <s xml:id="echoid-s3049" xml:space="preserve">& </s> + <s xml:id="echoid-s3050" xml:space="preserve">ſic quan-<lb/>titas aquæ eodem tempore per rimam integram effluens exprimetur per {2/3} n α <lb/>√ α: </s> + <s xml:id="echoid-s3051" xml:space="preserve">tantum autem effluit, quantum affunditur; </s> + <s xml:id="echoid-s3052" xml:space="preserve">hinc ſi quantitas aquæ dato <lb/>illo tempore affuſæ dicatur q, erit {2/3} n α √ α = q. </s> + <s xml:id="echoid-s3053" xml:space="preserve">Id indicat quantitates aqua- +<pb o="108" file="0122" n="122" rhead="HYDRODYNAMICÆ"/> +rum dato tempore affundendarum ſequi rationem ſeſquiplicatam altitudinum, <lb/>ad quas aquæ à fundo cylindri aſcendunt: </s> + <s xml:id="echoid-s3054" xml:space="preserve">aut viciſſim altitudines ſequi ra-<lb/>tionem ſubtriplicatam quadratorum quantitatum, quibus aquæ dato tempore <lb/>affunduntur.</s> + <s xml:id="echoid-s3055" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3056" xml:space="preserve">§. </s> + <s xml:id="echoid-s3057" xml:space="preserve">19. </s> + <s xml:id="echoid-s3058" xml:space="preserve">Soluto hoc problemate venio ad alterum Cl. </s> + <s xml:id="echoid-s3059" xml:space="preserve">Poleno conſide-<lb/>ratum.</s> + <s xml:id="echoid-s3060" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3061" xml:space="preserve">Sit idem cylindrus, ſed aquis in foſſa veluti vaſe infinito ſtagnantibus, <lb/>ſubmerſus; </s> + <s xml:id="echoid-s3062" xml:space="preserve">dicaturque altitudo ſubmerſionis = a, quæritur nunc iiſdem po-<lb/>ſitis, ut antea, rurſus æquatio inter altitudinem α ſuperficiei aqueæ internæ ſu-<lb/>pra externam, & </s> + <s xml:id="echoid-s3063" xml:space="preserve">quantitatem q dato tempore affundendam.</s> + <s xml:id="echoid-s3064" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3065" xml:space="preserve">Quod ad illam rimæ partem α, quæ aquas ejicit & </s> + <s xml:id="echoid-s3066" xml:space="preserve">ſupra aquam exter-<lb/>nam eminet, illam jam vidimus dato tempore erogare quantitatem {2/3} n α √ α: <lb/></s> + <s xml:id="echoid-s3067" xml:space="preserve">reſidua autem rimæ pars ſubmerſa aquas ubique communi velocitate tranſimit-<lb/>tit, ut ex infra dicendis patebit, & </s> + <s xml:id="echoid-s3068" xml:space="preserve">quidem velocitate √ α, ita, ut multiplica-<lb/>ta hâc velocitate per magnitudinem rimæ ſubmerſæ n a, habeatur quantitas, <lb/>quam dato tempore ejicit = n a √ α. </s> + <s xml:id="echoid-s3069" xml:space="preserve">Si utraque quantitas in ſummam conji-<lb/>ciatur, habebitur ({2/3} α + a)n√α = q.</s> + <s xml:id="echoid-s3070" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3071" xml:space="preserve">Ope hujus æquationis cognoſcitur q ex datis altitudinibus a & </s> + <s xml:id="echoid-s3072" xml:space="preserve">α: </s> + <s xml:id="echoid-s3073" xml:space="preserve">aut <lb/>viciſſim altitudo α ex cognitis quantitatibus a & </s> + <s xml:id="echoid-s3074" xml:space="preserve">q.</s> + <s xml:id="echoid-s3075" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3076" xml:space="preserve">Convenire autem hanc æquationem admodum accurate cum experi-<lb/>mentis, ipſe oſtendit celeberrimus eorum auctor, cujus ſolutio ab hâc no-<lb/>ſtra non differt. </s> + <s xml:id="echoid-s3077" xml:space="preserve">Sequitur ex iſta æquatione, elevationes α eo majores eſſe pro <lb/>iiſdem aquarum affuſionibus, quo minor eſt altitudo ſubmerſionis a.</s> + <s xml:id="echoid-s3078" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div115" type="section" level="1" n="88"> +<head xml:id="echoid-head113" style="it" xml:space="preserve">Experimenta quæ ad Sectionem V. pertinent.</head> +<head xml:id="echoid-head114" xml:space="preserve">Ad §. 5.</head> +<p> + <s xml:id="echoid-s3079" xml:space="preserve">VAſe uſus ſum §. </s> + <s xml:id="echoid-s3080" xml:space="preserve">5. </s> + <s xml:id="echoid-s3081" xml:space="preserve">deſcripto cum tubulo vitreo (Fig. </s> + <s xml:id="echoid-s3082" xml:space="preserve">30.) </s> + <s xml:id="echoid-s3083" xml:space="preserve">Primo autem <lb/>obturavi orificium L M, tubumque R N aqua implevi, donec ſuper-<lb/>ficies ejus raderet foraminulum in a: </s> + <s xml:id="echoid-s3084" xml:space="preserve">aquam tunc tubo ingreſſam ob-<lb/>fervavi extremitate attigiſſe punctum f: </s> + <s xml:id="echoid-s3085" xml:space="preserve">poſtea reſerato orificio L M, & </s> + <s xml:id="echoid-s3086" xml:space="preserve">aquis ef-<lb/>fluentibus novas affundebam in vas ſuperius E F P Q adhibita diligentia, ut <lb/>extremitas aquæ in f interea nec aſcenderet nec deſcenderet. </s> + <s xml:id="echoid-s3087" xml:space="preserve">Hæc dum fie- +<pb o="109" file="0123" n="123" rhead="SECTIO QUINTA."/> +rent elevabatur ſuperficies A B, nunquam autem certum terminum tranſgredie-<lb/>batur; </s> + <s xml:id="echoid-s3088" xml:space="preserve">fuit nempe quantum videre potui, maxima altitudo P B ſeu F A = <lb/>{nn/mm - nn}a, denotante {n/m} rationem inter orificium inferius L M & </s> + <s xml:id="echoid-s3089" xml:space="preserve">ſuperius <lb/>R S, & </s> + <s xml:id="echoid-s3090" xml:space="preserve">a altitudinem verticalem orificii poſterioris ſupra alterum.</s> + <s xml:id="echoid-s3091" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3092" xml:space="preserve">Id vero ſolum eſt, quod ipſemet inſtitui experimentum, quamvis mul-<lb/>tæ ſint propoſitiones in hâc ſectione contentæ, quæ mereantur attentionem <lb/>eæque ſatis inexpectatæ, non potui tamen de illis experimenta ſumere; </s> + <s xml:id="echoid-s3093" xml:space="preserve">ſunt <lb/>enim ita comparatæ in vaſis brevioribus, ut quod ſingulare habent, id ſenſus <lb/>effugiat, rem autem experiri in longis aquæductibus commode non potui: <lb/></s> + <s xml:id="echoid-s3094" xml:space="preserve">cum aliis hæc dabitur occaſio, theoriam hanc examinaturis, animum adver-<lb/>tent ad ſequentia:</s> + <s xml:id="echoid-s3095" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3096" xml:space="preserve">I<emph style="super">0</emph>. </s> + <s xml:id="echoid-s3097" xml:space="preserve">In fontibus ſalientibus obſervetur altitudo jactus integra; </s> + <s xml:id="echoid-s3098" xml:space="preserve">poſtmo-<lb/>dum obturato prius orificio eodemque mox reſerato videatur aquæ quanti-<lb/>tas, quæ effluat, dum aqua ad dimidiam altitudinem jactus integri, aut <lb/>aliam partem quamcunque perveniat, quod quidem breviſſimo eveniet tem-<lb/>pore, illius quantitatis menſura ſit longitudo cylindri ſuper foramine, per <lb/>quod aquæ exiliunt, exſtructi, quam longitudinem vocavimus x, alti-<lb/>tudinem vero jactus integram nominavimus a, altitudinemque jactus qui <lb/>nondum totam attigerit altitudinem, obſervatam deſignavimus per v. </s> + <s xml:id="echoid-s3099" xml:space="preserve">Tum <lb/>denique inſtituto calculo exploreter, num hæ quantitates recte reſpondeant <lb/>æquationibus pro utroque affundendi modo exhibitis in paragrapho tertio.</s> + <s xml:id="echoid-s3100" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3101" xml:space="preserve">II<emph style="super">0</emph>. </s> + <s xml:id="echoid-s3102" xml:space="preserve">Fiant omnia, ut ante, hoc ſaltem diſcrimine, quod loco quanti-<lb/>tatis effluentis tempus effluxus notetur, ut ſic examinari poſſint formulæ <lb/>paragraphi decimi tertii, & </s> + <s xml:id="echoid-s3103" xml:space="preserve">denique comparetur quantitas cum tempore <lb/>fluxus, ut appareat num recte reſpondeat formulæ §. </s> + <s xml:id="echoid-s3104" xml:space="preserve">14.</s> + <s xml:id="echoid-s3105" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3106" xml:space="preserve">III<emph style="super">0</emph>. </s> + <s xml:id="echoid-s3107" xml:space="preserve">Tum præcipue fiat id experimenti genus, quod indicavi para-<lb/>grapho decimo ſexto, obſervando ſcilicet, quantitates aquarum dimidiis <lb/>temporibus reſpondentes; </s> + <s xml:id="echoid-s3108" xml:space="preserve">dixi autem, quantumvis magnum ſumatur tem-<lb/>pus, differentiam harum quantitatum nunquam exæquare {2mmN/mmn - n<emph style="super">3</emph>} log. </s> + <s xml:id="echoid-s3109" xml:space="preserve">2. <lb/></s> + <s xml:id="echoid-s3110" xml:space="preserve">in priori, quem finximus, affundendi modo; </s> + <s xml:id="echoid-s3111" xml:space="preserve">aut {2N/n} log. </s> + <s xml:id="echoid-s3112" xml:space="preserve">2. </s> + <s xml:id="echoid-s3113" xml:space="preserve">in poſteriori.</s> + <s xml:id="echoid-s3114" xml:space="preserve"> +<pb o="110" file="0124" n="124" rhead="HYDRODYNAMICÆ"/> +Iſtas autem differentias, utut nunquam perfecte orituras, minimo tamen <lb/>tempore proxime adfuturas eſſe.</s> + <s xml:id="echoid-s3115" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3116" xml:space="preserve">Quæ reliqua funt in hâc ſectione Corollaria & </s> + <s xml:id="echoid-s3117" xml:space="preserve">Scholia quisque facile <lb/>videbit, quo modo ad experimenta vocari poſſint: </s> + <s xml:id="echoid-s3118" xml:space="preserve">Velim autem, prius-<lb/>quam judicium ferat, attentus ſit ad omnes circumſtantias ratione impedi-<lb/>mentorum, contractionis venæ, aliorumque, quas nolo ubique repetere. <lb/></s> + <s xml:id="echoid-s3119" xml:space="preserve">Ad §.</s> + <s xml:id="echoid-s3120" xml:space="preserve">§. </s> + <s xml:id="echoid-s3121" xml:space="preserve">17. </s> + <s xml:id="echoid-s3122" xml:space="preserve">& </s> + <s xml:id="echoid-s3123" xml:space="preserve">18. </s> + <s xml:id="echoid-s3124" xml:space="preserve">Experimenta pro confirmatione problematis §. </s> + <s xml:id="echoid-s3125" xml:space="preserve">17. </s> + <s xml:id="echoid-s3126" xml:space="preserve">ad <lb/>vaſa non ſubmerſa pertinentis, vide p. </s> + <s xml:id="echoid-s3127" xml:space="preserve">26. </s> + <s xml:id="echoid-s3128" xml:space="preserve">lib. </s> + <s xml:id="echoid-s3129" xml:space="preserve">cit. </s> + <s xml:id="echoid-s3130" xml:space="preserve">Jll. </s> + <s xml:id="echoid-s3131" xml:space="preserve">Poleni.</s> + <s xml:id="echoid-s3132" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3133" xml:space="preserve">Cum vero in vaſe ſubmerſo eſſet altitudo a = 55. </s> + <s xml:id="echoid-s3134" xml:space="preserve">lin. </s> + <s xml:id="echoid-s3135" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s3136" xml:space="preserve">(quæ altitu-<lb/>do ei dicitur mortua) quinque inſtituit experimenta, in quibus altitudo, <lb/>quam dicit, viva ſeu α erat ſucceſſive linearum 8 {3/4}; </s> + <s xml:id="echoid-s3137" xml:space="preserve">25; </s> + <s xml:id="echoid-s3138" xml:space="preserve">42; </s> + <s xml:id="echoid-s3139" xml:space="preserve">58 & </s> + <s xml:id="echoid-s3140" xml:space="preserve">73 {1/2}: <lb/></s> + <s xml:id="echoid-s3141" xml:space="preserve">his ſubſtitutis valoribus in æquatione §. </s> + <s xml:id="echoid-s3142" xml:space="preserve">18. </s> + <s xml:id="echoid-s3143" xml:space="preserve">exhibita ſequitur, quantitates <lb/>aquarum dato tempore affuſarum fuiſſe ut 100; </s> + <s xml:id="echoid-s3144" xml:space="preserve">199; </s> + <s xml:id="echoid-s3145" xml:space="preserve">299; </s> + <s xml:id="echoid-s3146" xml:space="preserve">396 & </s> + <s xml:id="echoid-s3147" xml:space="preserve">495: </s> + <s xml:id="echoid-s3148" xml:space="preserve"><lb/>actu affuſæ fuerunt in ratione ut 100, 200, 300, 400, & </s> + <s xml:id="echoid-s3149" xml:space="preserve">500: </s> + <s xml:id="echoid-s3150" xml:space="preserve">differen-<lb/>tia tantilla eſt, ut dubitari poſſit, an non perfectus conſenſus futurus <lb/>fuiſſet, ſi omnes menſuræ rectiſſime haberi potuiſſent.</s> + <s xml:id="echoid-s3151" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3152" xml:space="preserve">Reliqua etiam experimenta à viro Cl. </s> + <s xml:id="echoid-s3153" xml:space="preserve">inſtituta cum theoria perfecte <lb/>conſentiunt: </s> + <s xml:id="echoid-s3154" xml:space="preserve">calculum eorum videre eſt apud ipſum Auctorem. </s> + <s xml:id="echoid-s3155" xml:space="preserve">E re au-<lb/>tem duxi eadem hic apponere, quia ad argumentum hujusce ſectionis per-<lb/>tinent, quamvis cæterum libenter fatear, me magis deſiderare illa experi-<lb/>menta, quæ à calculo mutationem momentanearum, nemini quod ſciam ad-<lb/>huc conſideratarum, pendent, quam quæ ſtatum permanentem ſupponunt.</s> + <s xml:id="echoid-s3156" xml:space="preserve"/> +</p> +<pb o="111" file="0125" n="125"/> +</div> +<div xml:id="echoid-div116" type="section" level="1" n="89"> +<head xml:id="echoid-head115" xml:space="preserve">HYDRODYNAMICÆ <lb/>SECTIO SEXTA.</head> +<head xml:id="echoid-head116" style="it" xml:space="preserve">De fluidis non effluentibus ſeu intra latera <lb/>vaſorum motis.</head> +<head xml:id="echoid-head117" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s3157" xml:space="preserve">HActenus conſideravimus aquas effluentes; </s> + <s xml:id="echoid-s3158" xml:space="preserve">nunc vero contem-<lb/>plabimur motus aquarum, quæ vaſorum limites non præterfluunt. <lb/></s> + <s xml:id="echoid-s3159" xml:space="preserve">Omnes hos motus ad duo reducam genera, ambo ſeorſim per-<lb/>tractanda:</s> + <s xml:id="echoid-s3160" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3161" xml:space="preserve">1<emph style="super">0</emph>. </s> + <s xml:id="echoid-s3162" xml:space="preserve">Cum fluidum in tubo infinite longo continue movetur verſus <lb/>eandem plagam.</s> + <s xml:id="echoid-s3163" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3164" xml:space="preserve">2<emph style="super">0</emph>. </s> + <s xml:id="echoid-s3165" xml:space="preserve">Cum motibus reciprocis ſeu oſcillatoriis agitatur.</s> + <s xml:id="echoid-s3166" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div117" type="section" level="1" n="90"> +<head xml:id="echoid-head118" style="it" xml:space="preserve">De motu aquarum per canales <lb/>indefinite longos.</head> +<head xml:id="echoid-head119" xml:space="preserve">Caſus 1.</head> +<p> + <s xml:id="echoid-s3167" xml:space="preserve">§. </s> + <s xml:id="echoid-s3168" xml:space="preserve">2. </s> + <s xml:id="echoid-s3169" xml:space="preserve">Sit primo canalis horizontaliter poſitus, ſed amplitudinibus data <lb/>quacunque varians lege: </s> + <s xml:id="echoid-s3170" xml:space="preserve">ponatur fluidum in illo ita poſitum, quod fieri <lb/>ſolet in tubis ſtrictioribus, ut ambæ ſuperficies extremæ ſitum obtineant <lb/>ad axem canalis perpendicularem & </s> + <s xml:id="echoid-s3171" xml:space="preserve">ſic datâ quadam velocitate moveri in-<lb/>cipere. </s> + <s xml:id="echoid-s3172" xml:space="preserve">Hæc ſi ita ſint, nullaque plane motus impedimenta adeſſe fingan-<lb/>tur, perſpicuum eſt, motui aquarum nullum finem fore, quemadmodum <lb/>globus ſuper tabula horizontali liberrime progrediens motum ſine fine con-<lb/>continuat. </s> + <s xml:id="echoid-s3173" xml:space="preserve">Attamen inſignis inter utrumque motum intercedit differentia: <lb/></s> + <s xml:id="echoid-s3174" xml:space="preserve">globi nempe partes omnes uniformi continue progrediuntur velocitate, in <lb/>aqua perpetuo motum mutant: </s> + <s xml:id="echoid-s3175" xml:space="preserve">Neque difficile erit motum iſtum definire, <lb/>cum conſiderabimus, motum talem eſſe debere, ut aſcenſus potentialis totius <lb/>aquæ idem conſervetur, qui ab initio motus fuit: </s> + <s xml:id="echoid-s3176" xml:space="preserve">Determinavimus autem <lb/>aſcenſum potent, aquæ certâ velocitate in canali quocunque motæ in ſectionis ter- +<pb o="112" file="0126" n="126" rhead="HYDRODYNAMICÆ"/> +tiæ paragrapho ſecundo: </s> + <s xml:id="echoid-s3177" xml:space="preserve">Igitur nihil ad ſolutionem quæſtionis amplius re-<lb/>ſiduum eſt: </s> + <s xml:id="echoid-s3178" xml:space="preserve">Neque tamen abs re erit unum alterumve ejus rei exemplum <lb/>attuliſſe.</s> + <s xml:id="echoid-s3179" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div118" type="section" level="1" n="91"> +<head xml:id="echoid-head120" xml:space="preserve"><emph style="bf">Exemplum 1.</emph></head> +<p> + <s xml:id="echoid-s3180" xml:space="preserve">Si v. </s> + <s xml:id="echoid-s3181" xml:space="preserve">gr. </s> + <s xml:id="echoid-s3182" xml:space="preserve">canalis B g f C (Fig. </s> + <s xml:id="echoid-s3183" xml:space="preserve">31.) </s> + <s xml:id="echoid-s3184" xml:space="preserve">qui figuram habeat coni-truncati; </s> + <s xml:id="echoid-s3185" xml:space="preserve">in <lb/>telligatur pars ejus B G F C fluido plena moto verſus g f; </s> + <s xml:id="echoid-s3186" xml:space="preserve">habeantque parti-<lb/>culæ fluidi in G F velocitatem debitam altitudini v; </s> + <s xml:id="echoid-s3187" xml:space="preserve">ac denique pervenerit <lb/>fluidum in ſitum b g f c: </s> + <s xml:id="echoid-s3188" xml:space="preserve">His poſitis quæritur velocitas fluidi in g f. </s> + <s xml:id="echoid-s3189" xml:space="preserve">Voca-<lb/>bo autem altitudinem velocitati aquæ in g f debitam = V; </s> + <s xml:id="echoid-s3190" xml:space="preserve">Sit vertex coni <lb/>in H; </s> + <s xml:id="echoid-s3191" xml:space="preserve">diameter in B C = n; </s> + <s xml:id="echoid-s3192" xml:space="preserve">diameter in G F = m: </s> + <s xml:id="echoid-s3193" xml:space="preserve">longitudo B G = a; <lb/></s> + <s xml:id="echoid-s3194" xml:space="preserve">Gg = b, erit diameter g f = {m a - m b + n b/a}. </s> + <s xml:id="echoid-s3195" xml:space="preserve">Deinde quia ſolidum B G F C <lb/>eſt æquale ſolido b g f c erit B C<emph style="super">2</emph> X B H - G F<emph style="super">2</emph> X G H = b c<emph style="super">2</emph> X b H <lb/>- g f<emph style="super">2</emph> X g H: </s> + <s xml:id="echoid-s3196" xml:space="preserve">unde b c<emph style="super">2</emph> X b H = B C<emph style="super">2</emph> X B H - G F<emph style="super">2</emph> X G H <lb/>+ g f<emph style="super">2</emph> X g H: </s> + <s xml:id="echoid-s3197" xml:space="preserve">eſt vero b H = {BH/BC} X b c: </s> + <s xml:id="echoid-s3198" xml:space="preserve">igitur b c<emph style="super">3</emph> = B C<emph style="super">3</emph>-. </s> + <s xml:id="echoid-s3199" xml:space="preserve"><lb/>{GF<emph style="super">2</emph> X GH X BC/BH} + {gf<emph style="super">2</emph> X gH X BC/BH} = B C<emph style="super">3</emph> - G F<emph style="super">3</emph> + g f<emph style="super">3</emph>, ſeu <lb/>b c = √Cub.</s> + <s xml:id="echoid-s3200" xml:space="preserve">n<emph style="super">3</emph> - m<emph style="super">3</emph> + ({m a - m b + n b/a})<emph style="super">3</emph>},</s> +</p> +<p> + <s xml:id="echoid-s3201" xml:space="preserve">Eſt vero per §. </s> + <s xml:id="echoid-s3202" xml:space="preserve">3. </s> + <s xml:id="echoid-s3203" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s3204" xml:space="preserve">3. </s> + <s xml:id="echoid-s3205" xml:space="preserve">aſcenſus potent. </s> + <s xml:id="echoid-s3206" xml:space="preserve">aquæ in ſitu B G F C <lb/>= {3 m<emph style="super">3</emph> v/n(mm + mn + nn)}; </s> + <s xml:id="echoid-s3207" xml:space="preserve">pariterque aſcenſus potent. </s> + <s xml:id="echoid-s3208" xml:space="preserve">ejusdem aquæ in ſitu b g f c <lb/>reperitur = {3 α<emph style="super">3</emph> v;</s> + <s xml:id="echoid-s3209" xml:space="preserve">/β(αα + αβ + ββ)}, poſito brevitatis ergo α & </s> + <s xml:id="echoid-s3210" xml:space="preserve">β pro inventis valo-<lb/>ribus diametrorum g f & </s> + <s xml:id="echoid-s3211" xml:space="preserve">b c. </s> + <s xml:id="echoid-s3212" xml:space="preserve">Erit igitur <lb/>V = {m<emph style="super">3</emph> X (αα + αβ + ββ) X β X v/α<emph style="super">3</emph> X (mm + mn + nn) n}.</s> + <s xml:id="echoid-s3213" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3214" xml:space="preserve">Ex hâc formula facile colligitur, majori continue velocitate moveri <lb/>particulas anteriores, minori poſteriores, & </s> + <s xml:id="echoid-s3215" xml:space="preserve">ſic, ut ſi foraminulum g f cen-<lb/>ſeatur infinite parvum, fiat velocitas aquæ in g f infinita & </s> + <s xml:id="echoid-s3216" xml:space="preserve">in b c infinite parva.</s> + <s xml:id="echoid-s3217" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div119" type="section" level="1" n="92"> +<head xml:id="echoid-head121" xml:space="preserve"><emph style="bf">Exemplum 2.</emph></head> +<p> + <s xml:id="echoid-s3218" xml:space="preserve">Fuerit canalis compoſitus ex duobus tubis cylindricis B N & </s> + <s xml:id="echoid-s3219" xml:space="preserve">O P +<pb o="113" file="0127" n="127" rhead="SECTIO SEXTA."/> +(Fig. </s> + <s xml:id="echoid-s3220" xml:space="preserve">32.) </s> + <s xml:id="echoid-s3221" xml:space="preserve">inæqualis amplitudinis; </s> + <s xml:id="echoid-s3222" xml:space="preserve">in ramo ampliore moveri ponatur flui-<lb/> +<anchor type="note" xlink:label="note-0127-01a" xlink:href="note-0127-01"/> +dum B G F C verſus P velocitate quæ reſpondeat altitudini v. </s> + <s xml:id="echoid-s3223" xml:space="preserve">Ita perſpi-<lb/>cuum eſt nullam motus mutationem adfore, priusquam ſuperficies G F <lb/>pervenerit in M N; </s> + <s xml:id="echoid-s3224" xml:space="preserve">ab hoc autem temporis puncto motum continue variari <lb/>donec fluidum omne ſubingreſſum fuerit tubum ſtrictiorem. </s> + <s xml:id="echoid-s3225" xml:space="preserve">Quæritur ita-<lb/>que cum fluidum fitum tenet b g f c, quænam futura ſit velocitas ſuperficiei <lb/>f g; </s> + <s xml:id="echoid-s3226" xml:space="preserve">altitudinem autem hujus velocitatis deſignabimus per V.</s> + <s xml:id="echoid-s3227" xml:space="preserve"/> +</p> +<div xml:id="echoid-div119" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">Fig. 32.</note> +</div> +<p> + <s xml:id="echoid-s3228" xml:space="preserve">Sint diametri G F & </s> + <s xml:id="echoid-s3229" xml:space="preserve">g f ut n & </s> + <s xml:id="echoid-s3230" xml:space="preserve">m: </s> + <s xml:id="echoid-s3231" xml:space="preserve">longitudo B G vocetur = a; <lb/></s> + <s xml:id="echoid-s3232" xml:space="preserve">b M = b, erit O g = {nn/mm} X (a - b); </s> + <s xml:id="echoid-s3233" xml:space="preserve">aſcenſus potent. </s> + <s xml:id="echoid-s3234" xml:space="preserve">aquæ B G F C = v; </s> + <s xml:id="echoid-s3235" xml:space="preserve"><lb/>aſcenſus potent. </s> + <s xml:id="echoid-s3236" xml:space="preserve">aquæ b g f c = {n<emph style="super">4</emph> a - n<emph style="super">4</emph>b + m<emph style="super">4</emph>b/n<emph style="super">4</emph>a} X V; </s> + <s xml:id="echoid-s3237" xml:space="preserve">ergo <lb/>V = {n<emph style="super">4</emph>a/n<emph style="super">4</emph>a - n<emph style="super">4</emph>b + m<emph style="super">4</emph>b} v.</s> + <s xml:id="echoid-s3238" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3239" xml:space="preserve">Ex his intelligitur velocitatem primæ guttulæ in tubum ſtrictiorem ir-<lb/>rumpentis reſpondere altitudini {n<emph style="super">4</emph>/m<emph style="super">4</emph>} v, hanc vero velocitatem citiſſime decre-<lb/>ſcere, ita ut poſtquam parvula fluidi pars transfluxit, jam poſſit cenſeri V = {a/a - b} v, <lb/>& </s> + <s xml:id="echoid-s3240" xml:space="preserve">cum omne fluidum transfluxerit, priſtinam aſſumat velocitatem. </s> + <s xml:id="echoid-s3241" xml:space="preserve">Fuerit <lb/>v. </s> + <s xml:id="echoid-s3242" xml:space="preserve">gr. </s> + <s xml:id="echoid-s3243" xml:space="preserve">diameter tubi amplioris decupla@ alterius, & </s> + <s xml:id="echoid-s3244" xml:space="preserve">effluet prima guttula ex <lb/>tubo ampliore in ſtrictiorem velocitate debita altitudini 10000 v: </s> + <s xml:id="echoid-s3245" xml:space="preserve">ſi vero de-<lb/>cimam fluidi partem jam transfluxiſſe ponas, invenies altitudinem, quæ con-<lb/>veniat velocitati fluidi in tubo ſtrictiori progredientis, proxime æqualem {10/9} v.</s> + <s xml:id="echoid-s3246" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3247" xml:space="preserve">Sitempus quæras, quo fiat transfluxus fluidi O f, invenies illud æquale <lb/>{2(n<emph style="super">4</emph>a - n<emph style="super">4</emph>b + m<emph style="super">4</emph>b){3/2} - 2m<emph style="super">6</emph>a√a/3mm(n<emph style="super">4</emph> - m<emph style="super">4</emph>)√av}. </s> + <s xml:id="echoid-s3248" xml:space="preserve">Igitur omne fluidum transfluit tempore <lb/>{2n<emph style="super">6</emph>a√a = 2m<emph style="super">6</emph>a√a/3mm(n<emph style="super">4</emph> - m<emph style="super">4</emph>)√av} = {2(n<emph style="super">4</emph> + mmnn + m<emph style="super">4</emph>)a/3mm(nn + mm)√v}, ubi per {a/√v} intelligitur tem-<lb/>pus, quo fluidum in tubo ampliori libere motum abſolvit ſpatium a. </s> + <s xml:id="echoid-s3249" xml:space="preserve">Hæc <lb/>vero, ut dixi, ſe ita habebunt ſi nulla ſint motus impedimenta, ſimulque <lb/>in toto tractu canalis compoſiti velocitates amplitudinibus reciproce propor-<lb/>tionales ponantur. </s> + <s xml:id="echoid-s3250" xml:space="preserve">Interim jam alibi monui non poſſe aquas lateri M N pro- +<pb o="114" file="0128" n="128" rhead="HYDRODYNAMICÆ"/> +ximas hanc legem ſervare. </s> + <s xml:id="echoid-s3251" xml:space="preserve">Cum itaque talis caſus occurrit, eo magis con-<lb/>veniet motus realis cum theoria, quo longior fuerit pars b m & </s> + <s xml:id="echoid-s3252" xml:space="preserve">quo paucio-<lb/>ra adfuerint obſtacula.</s> + <s xml:id="echoid-s3253" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3254" xml:space="preserve">§. </s> + <s xml:id="echoid-s3255" xml:space="preserve">3. </s> + <s xml:id="echoid-s3256" xml:space="preserve">Quod ſi nunc canalis fuerit non horizontaliter ſed oblique ad <lb/>horizontem poſitus, apparet omnia ſimiliter ſe habere, niſi quod aſcenſus potent. <lb/></s> + <s xml:id="echoid-s3257" xml:space="preserve">aquæ in omni ſitu æquandus ſit aſcenſui potent. </s> + <s xml:id="echoid-s3258" xml:space="preserve">initiali aucto deſcenſu actuali, id <lb/>eſt, deſcenſui verticali centri gravitatis. </s> + <s xml:id="echoid-s3259" xml:space="preserve">Atque ſi nullo impulſu aqua ſua ſponte <lb/>ſe movere incipiat, erit ſimpliciter deſcenſus actualis æqualis aſcenſui potent.</s> + <s xml:id="echoid-s3260" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3261" xml:space="preserve">Igitur aqua continue progredi perget, quamdiu centrum gravitatis lo-<lb/>co humiliori poſitum eſt, ac fuit ab initio motus. </s> + <s xml:id="echoid-s3262" xml:space="preserve">At vero cum tubus ita <lb/>fuerit formatus & </s> + <s xml:id="echoid-s3263" xml:space="preserve">inflexus eaque fluidi quantitate repletus, ut centrum gravi-<lb/>tatis priſtinam altitudinem reaſſumere poſſit, tunc fluidum motum obtinebit <lb/>retrogradum & </s> + <s xml:id="echoid-s3264" xml:space="preserve">ſine fine oſcillabitur. </s> + <s xml:id="echoid-s3265" xml:space="preserve">De iſto motu præcipuam huj<unsure/>us ſectionis <lb/>partem faciente mox dicemus. </s> + <s xml:id="echoid-s3266" xml:space="preserve">Interea obſervare licet, fieri poſſe, ut aqua <lb/>omnis ex loco humiliore per altiorem ſua ſponte ſine prævia ſuctione præter-<lb/>fluat, ſi modo omnia debito modo ſe habeant.</s> + <s xml:id="echoid-s3267" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div121" type="section" level="1" n="93"> +<head xml:id="echoid-head122" style="it" xml:space="preserve">De oſcillationibus fluidorum in tubisrecurvis.</head> +<head xml:id="echoid-head123" xml:space="preserve">Caſus II.</head> +<p> + <s xml:id="echoid-s3268" xml:space="preserve">§. </s> + <s xml:id="echoid-s3269" xml:space="preserve">4. </s> + <s xml:id="echoid-s3270" xml:space="preserve">Dedit Pater meus in Comm. </s> + <s xml:id="echoid-s3271" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s3272" xml:space="preserve">Scient. </s> + <s xml:id="echoid-s3273" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s3274" xml:space="preserve">tom. </s> + <s xml:id="echoid-s3275" xml:space="preserve">2. </s> + <s xml:id="echoid-s3276" xml:space="preserve">theore-<lb/>mata quædam, quæ inſignem manifeſtant uſum quem theoria virium vivarum <lb/>habet in rebus mechanicis. </s> + <s xml:id="echoid-s3277" xml:space="preserve">Illud vero quod tertio loco poſitum eſt ita ſe <lb/>habet.</s> + <s xml:id="echoid-s3278" xml:space="preserve"/> +</p> +<p style="it"> + <s xml:id="echoid-s3279" xml:space="preserve">Sit tubus cylindricus A B C H (Fig. </s> + <s xml:id="echoid-s3280" xml:space="preserve">33.) </s> + <s xml:id="echoid-s3281" xml:space="preserve">utrobi{q́ue} apertus at{q́ue} infle-<lb/> +<anchor type="note" xlink:label="note-0128-01a" xlink:href="note-0128-01"/> +xus in duo crura B A & </s> + <s xml:id="echoid-s3282" xml:space="preserve">C H ad partem horizontalem B C; </s> + <s xml:id="echoid-s3283" xml:space="preserve">ſit ſinus anguli <lb/>A B C = p, & </s> + <s xml:id="echoid-s3284" xml:space="preserve">ſinus anguli H C B = q; </s> + <s xml:id="echoid-s3285" xml:space="preserve">exiſtente nimirum ſinu toto = 1; <lb/></s> + <s xml:id="echoid-s3286" xml:space="preserve">ſit porro ille tubus aqua plenus uſ{q́ue} ad horizontalem M N; </s> + <s xml:id="echoid-s3287" xml:space="preserve">vocetur{q́ue} L longi-<lb/>tudo partis tubi M B C N aqua plenæ: </s> + <s xml:id="echoid-s3288" xml:space="preserve">Erunt agitati liquoris in hoc tubo oſ-<lb/>cillationes tam majores, quam minores omnes tautochronæ at{q́ue} ejuſdem duratio-<lb/>nis cum oſcillationibus minimis penduli alicujus ſimplicis, cujus longitudo <lb/>= {L/p + q}.</s> + <s xml:id="echoid-s3289" xml:space="preserve"/> +</p> +<div xml:id="echoid-div121" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0128-01" xlink:href="note-0128-01a" xml:space="preserve">Fig. 33.</note> +</div> +<p> + <s xml:id="echoid-s3290" xml:space="preserve">Huic theoremati eodem auctore ſubnectitur tale corollarium.</s> + <s xml:id="echoid-s3291" xml:space="preserve"/> +</p> +<pb o="115" file="0129" n="129" rhead="SECTIO SEXTA."/> +<p style="it"> + <s xml:id="echoid-s3292" xml:space="preserve">Si anguli A B C & </s> + <s xml:id="echoid-s3293" xml:space="preserve">H C B ſunt recti, qui unicus caſus est; </s> + <s xml:id="echoid-s3294" xml:space="preserve">à Newta-<lb/>no ſolutus, erit longitudo penduli ſimplicis, quod oſcillanti aquæ iſochronum eſt, <lb/>= {1/2} L, ut invenit Newtonus.</s> + <s xml:id="echoid-s3295" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3296" xml:space="preserve">§. </s> + <s xml:id="echoid-s3297" xml:space="preserve">5. </s> + <s xml:id="echoid-s3298" xml:space="preserve">Hæc ſunt quæ adhuc cum publico communicata fuerunt circa <lb/>oſcillationes fluidorum, & </s> + <s xml:id="echoid-s3299" xml:space="preserve">quidem primo à Newtono, ut undarum naturam, <lb/>à Patre meo, ut fertilitatem principii virium vivarum oſtenderet. </s> + <s xml:id="echoid-s3300" xml:space="preserve">Quia vero <lb/>noſtrum inſtitutum eſt pleniorem dare de motibus aquarum theoriam, è re <lb/>erit iſtud argumenti genus in tota ſua extenſione proſequi: </s> + <s xml:id="echoid-s3301" xml:space="preserve">Igitur diſquiram, <lb/>quibus modis oſcillationes fluidi inæquales fiant iſochronæ, & </s> + <s xml:id="echoid-s3302" xml:space="preserve">quibus non <lb/>item? </s> + <s xml:id="echoid-s3303" xml:space="preserve">Dein pro prioribus dabo longitudinem penduli ſimplicis tautochroni, <lb/>pro alteris tempus durationis indicabo: </s> + <s xml:id="echoid-s3304" xml:space="preserve">tubos autem utcunque inflexos & </s> + <s xml:id="echoid-s3305" xml:space="preserve">inæ <lb/>qualiter amplos conſiderabo.</s> + <s xml:id="echoid-s3306" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div123" type="section" level="1" n="94"> +<head xml:id="echoid-head124" xml:space="preserve">Lemma.</head> +<p> + <s xml:id="echoid-s3307" xml:space="preserve">§. </s> + <s xml:id="echoid-s3308" xml:space="preserve">6. </s> + <s xml:id="echoid-s3309" xml:space="preserve">Sit c A d (Fig. </s> + <s xml:id="echoid-s3310" xml:space="preserve">34.) </s> + <s xml:id="echoid-s3311" xml:space="preserve">uter ſeu canalis aqua plenus formæ cujuſcun-<lb/> +<anchor type="note" xlink:label="note-0129-01a" xlink:href="note-0129-01"/> +que datæ deſinens utrobique in duos canales cylindricos a c & </s> + <s xml:id="echoid-s3312" xml:space="preserve">f d, utcunque ad <lb/>horizontem inclinatos & </s> + <s xml:id="echoid-s3313" xml:space="preserve">cujuſcunque amplitudinis, quorum alterum plenum <lb/>aqua ponam uſque in a, alterum uſque in f; </s> + <s xml:id="echoid-s3314" xml:space="preserve">oporteat determinare altitudinem <lb/>centri gravitatis omnis aquæ, ex data altitudine centri gravitatis aquæ in u-<lb/>tre c A d contentæ, cæteriſque quantum ſufficit præcognitis.</s> + <s xml:id="echoid-s3315" xml:space="preserve"/> +</p> +<div xml:id="echoid-div123" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0129-01" xlink:href="note-0129-01a" xml:space="preserve">Fig. 34.</note> +</div> +</div> +<div xml:id="echoid-div125" type="section" level="1" n="95"> +<head xml:id="echoid-head125" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s3316" xml:space="preserve">Fuerit centrum gravitatis aquæ in vaſe c A d contentæ in C, ductaque in-<lb/>telligatur per iſtud punctum C verticalis A B, deinde ducantur horizontales <lb/>a m, c g, f n, & </s> + <s xml:id="echoid-s3317" xml:space="preserve">d h una cum verticalibus c b & </s> + <s xml:id="echoid-s3318" xml:space="preserve">d e. </s> + <s xml:id="echoid-s3319" xml:space="preserve">Ponatur a c = a: </s> + <s xml:id="echoid-s3320" xml:space="preserve">f d = α: <lb/></s> + <s xml:id="echoid-s3321" xml:space="preserve">b c = b; </s> + <s xml:id="echoid-s3322" xml:space="preserve">e d = β: </s> + <s xml:id="echoid-s3323" xml:space="preserve">amplitudo tubi a c = g; </s> + <s xml:id="echoid-s3324" xml:space="preserve">amplitudo tubi f d = γ: </s> + <s xml:id="echoid-s3325" xml:space="preserve">ſit porro <lb/>maſſa aquea ſeu capacitas canalis c A d = M, linea A g = f; </s> + <s xml:id="echoid-s3326" xml:space="preserve">A h = φ: </s> + <s xml:id="echoid-s3327" xml:space="preserve">A C =m: </s> + <s xml:id="echoid-s3328" xml:space="preserve"><lb/>Dividantur lineæ m g & </s> + <s xml:id="echoid-s3329" xml:space="preserve">n h bifariam punctis D & </s> + <s xml:id="echoid-s3330" xml:space="preserve">E & </s> + <s xml:id="echoid-s3331" xml:space="preserve">ſic erunt centra gravitatis <lb/>aquarum in tubis cylindricis contentarum in altitudinibus punctorum D & </s> + <s xml:id="echoid-s3332" xml:space="preserve">E.</s> + <s xml:id="echoid-s3333" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3334" xml:space="preserve">His poſitis fit A D = f + {1/2}@b; </s> + <s xml:id="echoid-s3335" xml:space="preserve">A E = φ + {1/2}β: </s> + <s xml:id="echoid-s3336" xml:space="preserve">maſſa aquæ in a c = <lb/>g a: </s> + <s xml:id="echoid-s3337" xml:space="preserve">in f d = γ α: </s> + <s xml:id="echoid-s3338" xml:space="preserve">Igitur ſi centrum gravitatis quæſitum pro omni aqua a c A d f <lb/>intelligatur in altitudine F poſitum, habebitur, ut conſtat in mechanicis, A F <lb/>multiplicando maſſam aquæ in a c per D A, maſſam aquæ f d per E A & </s> + <s xml:id="echoid-s3339" xml:space="preserve">maſſam +<pb o="116" file="0130" n="130" rhead="HYDRODYNAMICÆ"/> +aquæ in c A d per C A, aggregatumque horum productorum dividendo per <lb/>ſummam harum maſſarum. </s> + <s xml:id="echoid-s3340" xml:space="preserve">Unde invenitur.</s> + <s xml:id="echoid-s3341" xml:space="preserve"> +A F = {ga X (f + {1/2}<emph style="super">b</emph>) + γα X (φ + {1/2}<emph style="super">β</emph>) + Mm/ga + γα + M}</s> +</p> +</div> +<div xml:id="echoid-div126" type="section" level="1" n="96"> +<head xml:id="echoid-head126" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s3342" xml:space="preserve">§. </s> + <s xml:id="echoid-s3343" xml:space="preserve">7. </s> + <s xml:id="echoid-s3344" xml:space="preserve">Determinare ubique velocitates aquæ oſcillantis, poſito oſcilla-<lb/>tiones ultra terminos tuborum cylindricorum non divagari.</s> + <s xml:id="echoid-s3345" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div127" type="section" level="1" n="97"> +<head xml:id="echoid-head127" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s3346" xml:space="preserve">Sit aqua oſcillationem inchoans in ſitu a c A d f perveneritque poſtmo-<lb/>dum in ſitum o c A d p, retentiſque denominationibus |in præcedente paragra-<lb/>pho factis, ponatur a o = x; </s> + <s xml:id="echoid-s3347" xml:space="preserve">erit f p = {gx/γ}: </s> + <s xml:id="echoid-s3348" xml:space="preserve">unde (ſi nempe centrum gravita-<lb/>tis omnis aquæ deſcendiſſe putetur ex F in O) erit vi præcedentis paragraphi <lb/>A O = {g X (a - x) X (f + {1/2}<emph style="super">b</emph> - {bx/2a}) + γ X (a + {gx/γ}) X (φ + {1/2}<emph style="super">β</emph> + {βgx/2αγ}) + Mm/ga + γα + M}</s> +</p> +<p> + <s xml:id="echoid-s3349" xml:space="preserve">Inde deducitur deſcenſus centri gravitatis ſeu deſcenſus actualis <lb/>F O = {(b - β + f - φ)gx - ({bg/2a} + {bgg/2αγ}) xx/ga + γα + M}</s> +</p> +<p> + <s xml:id="echoid-s3350" xml:space="preserve">Sit nunc velocitas aquæ in tubo a c (cum nempe ſuperficies eſt in o) ta-<lb/>lis quæ reſpondeat altitudini v, & </s> + <s xml:id="echoid-s3351" xml:space="preserve">erit tunc aſcenſus potent. </s> + <s xml:id="echoid-s3352" xml:space="preserve">aquæ in altero tubo <lb/>= {gg/γγ} v: </s> + <s xml:id="echoid-s3353" xml:space="preserve">pariterque aſcenſus potent. </s> + <s xml:id="echoid-s3354" xml:space="preserve">aquæ c A d, erit proportionalis altitudini v, <lb/>eamque proinde ponemus = N v (ubi N pendet à figura utris c A d & </s> + <s xml:id="echoid-s3355" xml:space="preserve">deter-<lb/>minari poteſt per §. </s> + <s xml:id="echoid-s3356" xml:space="preserve">2. </s> + <s xml:id="echoid-s3357" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s3358" xml:space="preserve">3.) </s> + <s xml:id="echoid-s3359" xml:space="preserve">Jam vero ſi multiplicatis ubique aſcenſibus po-<lb/>tentialibus per ſuas maſſas producta dividantur per ſummam maſſarum, habebi-<lb/>tur aſcenſ{us} potent. </s> + <s xml:id="echoid-s3360" xml:space="preserve">omnis aquæ o c A d p = <lb/>{(ga - gx + {αgg/γ} + {g<emph style="super">3</emph>x/γγ} + MN)v/ga + γα + M}</s> +</p> +<p> + <s xml:id="echoid-s3361" xml:space="preserve">Et quia hic aſcenſus potentialis eſt æqualis deſcenſui actuali F O paullo ante <lb/>invento, erit +<pb o="117" file="0131" n="131" rhead="SECTIO SEXTA."/> +v = {(b - β + f - φ) gx - ({bg/2a} + {bgg/2αγ)} xx/ga - gx + {αgg/γ} + {g<emph style="super">3</emph>/γ} {x/γ} + MN} Q.</s> + <s xml:id="echoid-s3362" xml:space="preserve">E.</s> + <s xml:id="echoid-s3363" xml:space="preserve">I.</s> + <s xml:id="echoid-s3364" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div128" type="section" level="1" n="98"> +<head xml:id="echoid-head128" xml:space="preserve">Corollarium 1.</head> +<p> + <s xml:id="echoid-s3365" xml:space="preserve">§. </s> + <s xml:id="echoid-s3366" xml:space="preserve">8. </s> + <s xml:id="echoid-s3367" xml:space="preserve">Quia linea mn = mg - nh + gh = h - β + f - m, ponemus <lb/>mn = c, ſimulque multiplicabimus denominatorem & </s> + <s xml:id="echoid-s3368" xml:space="preserve">numeratorem per <lb/>2γγαα: </s> + <s xml:id="echoid-s3369" xml:space="preserve">Ita vero habebimus <lb/>v = {2gγγaαcx - (gγγαb + ggγaß)xx/2gγγaaα - 2gγγaαx + 2ggγaαα + 2g<emph style="super">3</emph>aαx + 2γγaαMN}</s> +</p> +</div> +<div xml:id="echoid-div129" type="section" level="1" n="99"> +<head xml:id="echoid-head129" xml:space="preserve">Corollarium 2.</head> +<p> + <s xml:id="echoid-s3370" xml:space="preserve">§. </s> + <s xml:id="echoid-s3371" xml:space="preserve">9. </s> + <s xml:id="echoid-s3372" xml:space="preserve">Si fiat v =o, patet tunc valorem x denotare totam fluidi ſuper-<lb/>ficiei excurſionem in tubo ac, quæ ſic invenitur æqualis {2γaαc/γαb + gαβ}, in altero <lb/>vero tubo fit = {2gaαc/γαb + gαβ}.</s> + <s xml:id="echoid-s3373" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3374" xml:space="preserve">Igitur poterit aqua in tubo ſtrictiori ad quamcunque elevari altitudi-<lb/>nem, ſi modo ratio amplitudinum g & </s> + <s xml:id="echoid-s3375" xml:space="preserve">γ ſat magna ſumatur.</s> + <s xml:id="echoid-s3376" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div130" type="section" level="1" n="100"> +<head xml:id="echoid-head130" xml:space="preserve">Corollarium 3.</head> +<p> + <s xml:id="echoid-s3377" xml:space="preserve">§. </s> + <s xml:id="echoid-s3378" xml:space="preserve">10. </s> + <s xml:id="echoid-s3379" xml:space="preserve">Pars illa vaſis c A d, quam neutra ſuperficierum unquam attin-<lb/>gi ponimus, nihil pertinet ad iſtas fluidi excurſiones ſive augendas ſive dimi-<lb/>nuendas: </s> + <s xml:id="echoid-s3380" xml:space="preserve">facere tamen poteſt, ut inferius oſtendetur, ad accelerandas retar-<lb/>dandaſque oſcillationes.</s> + <s xml:id="echoid-s3381" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div131" type="section" level="1" n="101"> +<head xml:id="echoid-head131" xml:space="preserve">Corollarium 4.</head> +<p> + <s xml:id="echoid-s3382" xml:space="preserve">§. </s> + <s xml:id="echoid-s3383" xml:space="preserve">11. </s> + <s xml:id="echoid-s3384" xml:space="preserve">Ponatur uterque tubus communis amplitudinis, erit, poſito <lb/>nempe g = γ, <lb/>v = 2gaαcx - (gαb + gaβ)xx/2gaaα + 2gaαα + 2aαMN}.</s> + <s xml:id="echoid-s3385" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3386" xml:space="preserve">In hoc caſu maxima ſuperficiei utriuſque velocitas eſt, cum in medio <lb/>totius excurſionis poſitæ ſunt, ſecus ac fit, cum tubi ſunt inæqualis amplitu-<lb/>dinis.</s> + <s xml:id="echoid-s3387" xml:space="preserve"/> +</p> +<pb o="118" file="0132" n="132" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s3388" xml:space="preserve">Notandum quoque eſt, ſimiles eſſe inter ſe retardationes & </s> + <s xml:id="echoid-s3389" xml:space="preserve">accelera-<lb/>tiones in diſtantiis ſimilibus ſuperficierum à punctis mediarum excurſionum, <lb/>id eſt, à locis maximarum velocitatum.</s> + <s xml:id="echoid-s3390" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div132" type="section" level="1" n="102"> +<head xml:id="echoid-head132" xml:space="preserve">Theorema.</head> +<p> + <s xml:id="echoid-s3391" xml:space="preserve">§. </s> + <s xml:id="echoid-s3392" xml:space="preserve">12. </s> + <s xml:id="echoid-s3393" xml:space="preserve">Cum amplitudines tuborum cylindricorum prædicto modo <lb/>ſunt æquales, erunt oſcillationes tam majores quam minores inter ſe Iſochro-<lb/>næ, modo ſuperficies nunquam deſcendant infra orificia eorundem tuborum.</s> + <s xml:id="echoid-s3394" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div133" type="section" level="1" n="103"> +<head xml:id="echoid-head133" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s3395" xml:space="preserve">Ex mechanicis conſtat, quod ſi mobile oſcillans ſpatium perfecerit <lb/>= x, habeatque in ſingulis locis elementum temporis dt = {mdx/√nx - xx}, intel-<lb/>ligendo per m & </s> + <s xml:id="echoid-s3396" xml:space="preserve">n quantitates conſtantes, id faciat oſcillationes ſuas tam majo-<lb/>res quam minores eodem tempore.</s> + <s xml:id="echoid-s3397" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3398" xml:space="preserve">Quia vero in noſtro caſu eſt <lb/>v = {2gaαcx - (gαb + gaβ)xx/2gaaα + 2gaαα + 2aαMN}, <lb/>& </s> + <s xml:id="echoid-s3399" xml:space="preserve">quia velocitas ipſa eſt æqualis √ v, erit <lb/>dt = dx√({2gaaα + 2gaαα + 2aαMN/gαb + gaβ}):</s> + <s xml:id="echoid-s3400" xml:space="preserve">√({2aαcx/gαb + gaβ} - xx), <lb/>ubi pariter omnes litteræ conſtantem habent valorem præter x, quæ ſpatium <lb/>percurſum denotat; </s> + <s xml:id="echoid-s3401" xml:space="preserve">patet has quoque fluidi oſcillationes iſochronas fore <lb/>Q. </s> + <s xml:id="echoid-s3402" xml:space="preserve">E. </s> + <s xml:id="echoid-s3403" xml:space="preserve">D.</s> + <s xml:id="echoid-s3404" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div134" type="section" level="1" n="104"> +<head xml:id="echoid-head134" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s3405" xml:space="preserve">§. </s> + <s xml:id="echoid-s3406" xml:space="preserve">13. </s> + <s xml:id="echoid-s3407" xml:space="preserve">Invenire longitudinem penduli ſimplicis, quod ſit tautochro-<lb/>num cum oſcillationibus fluidi præfatis.</s> + <s xml:id="echoid-s3408" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div135" type="section" level="1" n="105"> +<head xml:id="echoid-head135" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s3409" xml:space="preserve">In mechanicis demonſtratur, quod, cum dt = {mdx/√nx - xx}, ſit longitu-<lb/>do penduli ſimplicis tautochroni = {1/2} mm: </s> + <s xml:id="echoid-s3410" xml:space="preserve">Erit igitur in noſtro caſu de quo <lb/>ſermo eſt longitudo penduli quæſita = {gaaα + gaαα + aαMN/gαb + gaβ}. </s> + <s xml:id="echoid-s3411" xml:space="preserve"># Q.</s> + <s xml:id="echoid-s3412" xml:space="preserve">E.</s> + <s xml:id="echoid-s3413" xml:space="preserve">I.</s> + <s xml:id="echoid-s3414" xml:space="preserve"/> +</p> +<pb o="119" file="0133" n="133" rhead="SECTIO SEXTA."/> +</div> +<div xml:id="echoid-div136" type="section" level="1" n="106"> +<head xml:id="echoid-head136" xml:space="preserve">Corollarium. 1.</head> +<p> + <s xml:id="echoid-s3415" xml:space="preserve">§. </s> + <s xml:id="echoid-s3416" xml:space="preserve">14. </s> + <s xml:id="echoid-s3417" xml:space="preserve">Si ponatur canalis c A d ejuſdem amplitudinis cum tubis con-<lb/>junctis, ejuſque longitudo vocetur l, erit maſſa aquæ in eo contentæ, quam <lb/>vocavimus M = gl; </s> + <s xml:id="echoid-s3418" xml:space="preserve">& </s> + <s xml:id="echoid-s3419" xml:space="preserve">aſcenſuspotent. </s> + <s xml:id="echoid-s3420" xml:space="preserve">aquæ in illo contentæ, quem poſuimus = <lb/>N v, erit = v, ita ut habeatur N = 1. </s> + <s xml:id="echoid-s3421" xml:space="preserve">Subſtitutis autem, iſtis valoribus pro <lb/>litteris M & </s> + <s xml:id="echoid-s3422" xml:space="preserve">N, prodit longitudo penduli tautochroni pro iſto caſu particulari = <lb/>{aaα + aαα + aαl/αb + aβ} = {aα/αb + aβ} X (a + α + l) = {a + α + l/{b/a} + {β/α}</s> +</p> +<p> + <s xml:id="echoid-s3423" xml:space="preserve">Quia vero a + α + l eſt longitudo totius tractus aqua pleni & </s> + <s xml:id="echoid-s3424" xml:space="preserve">{b/a} ſigni-<lb/>ficat rationem ſinus anguli bac ad ſinum totum pariter atque {β/α} denotat ra-<lb/>tionem ſinus anguli efd ad ſinum totum, videmus non differre noſtram ſo-<lb/>lutionem ab illa, quam Pater meus pro iſto caſu dedit, quamque ſupra <lb/>recenſui §. </s> + <s xml:id="echoid-s3425" xml:space="preserve">4.</s> + <s xml:id="echoid-s3426" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div137" type="section" level="1" n="107"> +<head xml:id="echoid-head137" xml:space="preserve">Corollarium 2.</head> +<p> + <s xml:id="echoid-s3427" xml:space="preserve">§. </s> + <s xml:id="echoid-s3428" xml:space="preserve">15. </s> + <s xml:id="echoid-s3429" xml:space="preserve">Si ponatur canalis c A d infinitæ ubique amplitudinis, erit <lb/>MN = o (per §. </s> + <s xml:id="echoid-s3430" xml:space="preserve">2. </s> + <s xml:id="echoid-s3431" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s3432" xml:space="preserve">3.) </s> + <s xml:id="echoid-s3433" xml:space="preserve">& </s> + <s xml:id="echoid-s3434" xml:space="preserve">longitudo penduli tantochroni = {a + α/{b/a} + {β/α}}, qua-<lb/>ſi nempe totus canalis intermedius c A d abeſſet, tubique cylindrici inter ſe <lb/>immediate eſſent conjuncti.</s> + <s xml:id="echoid-s3435" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3436" xml:space="preserve">Eſt tamen hîc ſpeciale aliquid conſiderandum, quod infra monebo.</s> + <s xml:id="echoid-s3437" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div138" type="section" level="1" n="108"> +<head xml:id="echoid-head138" xml:space="preserve">Scholion.</head> +<p> + <s xml:id="echoid-s3438" xml:space="preserve">§. </s> + <s xml:id="echoid-s3439" xml:space="preserve">16. </s> + <s xml:id="echoid-s3440" xml:space="preserve">Complectitur hoc theorema omnes caſus, qui oſcillationes tan-<lb/>tochronas faciunt, ubi tubi a c & </s> + <s xml:id="echoid-s3441" xml:space="preserve">p d ſunt recti: </s> + <s xml:id="echoid-s3442" xml:space="preserve">cum vero hi tubi, in qui-<lb/>bus fluidi ſuperficies excurrunt, incurvati ſunt, dantur alii inſuper tanto-<lb/>chronismi caſus, quos facile foret determinare, ſi hiſce diutius immorari <lb/>vellemus. </s> + <s xml:id="echoid-s3443" xml:space="preserve">Cæterum cum tubi hi inæqualis amplitudinis ſunt, fiunt quoque <lb/>tempora oſcillationbus diverſarum magnitudinum reſpondentia inæqualia, <lb/>& </s> + <s xml:id="echoid-s3444" xml:space="preserve">quomodo tempus tale definiri debeat unicuique apparet ex §. </s> + <s xml:id="echoid-s3445" xml:space="preserve">8. </s> + <s xml:id="echoid-s3446" xml:space="preserve">ubi velo-<lb/>citatem fiuidi in quolibet puncto dedimus.</s> + <s xml:id="echoid-s3447" xml:space="preserve"/> +</p> +<pb o="120" file="0134" n="134" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s3448" xml:space="preserve">Hæc autem de oſcillationibus finitis. </s> + <s xml:id="echoid-s3449" xml:space="preserve">Si nunc oſcillationes minimas <lb/>eſſe cenſeamus, videbimus illas fieri omnes inter ſe tantochronas, manen-<lb/>te eadem fluidi quantitate, eodemque canali, quæcunque interea ſint cana-<lb/>lis figura & </s> + <s xml:id="echoid-s3450" xml:space="preserve">amplitudines. </s> + <s xml:id="echoid-s3451" xml:space="preserve">Id exponam in ſequenti paragrapho.</s> + <s xml:id="echoid-s3452" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div139" type="section" level="1" n="109"> +<head xml:id="echoid-head139" xml:space="preserve">Theorema.</head> +<p> + <s xml:id="echoid-s3453" xml:space="preserve">§. </s> + <s xml:id="echoid-s3454" xml:space="preserve">17. </s> + <s xml:id="echoid-s3455" xml:space="preserve">Oſcillationes minimæ fluidi in quocunque canali oſcillantis, <lb/>quamvis inæquales inter ſe, ſunt omnes Iſochronæ.</s> + <s xml:id="echoid-s3456" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div140" type="section" level="1" n="110"> +<head xml:id="echoid-head140" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s3457" xml:space="preserve">Cum oſcillationes ſunt minimæ, poſſunt illæ canalis particulæ, in qui-<lb/>bus ſuperficies fluidi agitantur, pro cylindricis haberi, igitur manentibus <lb/>denominationibus iisdem, manebit valor, quem aſſignavimus litteræ v in <lb/>§. </s> + <s xml:id="echoid-s3458" xml:space="preserve">8. </s> + <s xml:id="echoid-s3459" xml:space="preserve">& </s> + <s xml:id="echoid-s3460" xml:space="preserve">ex eadem ratione ſequitur, litteras a, b, α, β & </s> + <s xml:id="echoid-s3461" xml:space="preserve">x ceu infinite parvi <lb/>valoris negligi poſſe præ {M/g}, ſic ut in præſenti caſu cenſeri debeat <lb/>v = {2gγaαcx - (gγαb + ggab)xx/2γaαMN}</s> +</p> +<p> + <s xml:id="echoid-s3462" xml:space="preserve">Sunt igitur vi paragraphi duodecimi oſcillationes omnes, quoad mi-<lb/>nimæ ſunt, inter ſe Iſochronæ. </s> + <s xml:id="echoid-s3463" xml:space="preserve">Q.</s> + <s xml:id="echoid-s3464" xml:space="preserve">E.</s> + <s xml:id="echoid-s3465" xml:space="preserve">D.</s> + <s xml:id="echoid-s3466" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div141" type="section" level="1" n="111"> +<head xml:id="echoid-head141" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s3467" xml:space="preserve">§. </s> + <s xml:id="echoid-s3468" xml:space="preserve">18. </s> + <s xml:id="echoid-s3469" xml:space="preserve">Determinare longitudinem penduli ſimplicis tautochroni cum <lb/>oſcillationibus minimuis fluidi in canali quocunque agitati.</s> + <s xml:id="echoid-s3470" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div142" type="section" level="1" n="112"> +<head xml:id="echoid-head142" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s3471" xml:space="preserve">Quia in omni motu eſt elementum temporis dt = {dx/√v}, erit nunc <lb/>dt = dx√(2γaαMN/gγαb + ggab}):</s> + <s xml:id="echoid-s3472" xml:space="preserve">√({2γaαcx/γαb + gaβ} - xx) <lb/>Igitur vi Paragraphi decimi tertii erit longitudo quæſita penduli cum præ-<lb/>dictis oſcillationibus tautochroni = {γaαMN/gγαb + ggaβ}. </s> + <s xml:id="echoid-s3473" xml:space="preserve">Q. </s> + <s xml:id="echoid-s3474" xml:space="preserve">E. </s> + <s xml:id="echoid-s3475" xml:space="preserve">I.</s> + <s xml:id="echoid-s3476" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div143" type="section" level="1" n="113"> +<head xml:id="echoid-head143" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s3477" xml:space="preserve">§. </s> + <s xml:id="echoid-s3478" xml:space="preserve">19. </s> + <s xml:id="echoid-s3479" xml:space="preserve">Quamvis jam paſſim monuerim, quid intelligendum ſit per +<pb o="121" file="0135" n="135" rhead="SECTIO SEXTA."/> +quantitates M & </s> + <s xml:id="echoid-s3480" xml:space="preserve">N, tamen hic apponam totam conſtructionem, ut natura <lb/>rei eo magis unicuique pateat.</s> + <s xml:id="echoid-s3481" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3482" xml:space="preserve">Fuerit canalis qualiscunque A B C D E, (Fig. </s> + <s xml:id="echoid-s3483" xml:space="preserve">35. </s> + <s xml:id="echoid-s3484" xml:space="preserve">a & </s> + <s xml:id="echoid-s3485" xml:space="preserve">b) aqua plenus us-<lb/> +<anchor type="note" xlink:label="note-0135-01a" xlink:href="note-0135-01"/> +que in B & </s> + <s xml:id="echoid-s3486" xml:space="preserve">D; </s> + <s xml:id="echoid-s3487" xml:space="preserve">ponatur ſinus totus = 1, ſinus anguli D B C = {b/a} = m, <lb/>ſinus anguli B D C = {β/α} = n, erit longitudo penduli tautochroni = {γMN/mgγ + ngg}, <lb/>ubi g denotat amplitudinem canalis in B & </s> + <s xml:id="echoid-s3488" xml:space="preserve">γ amplitudinem ejus in D.</s> + <s xml:id="echoid-s3489" xml:space="preserve"/> +</p> +<div xml:id="echoid-div143" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">Fig. 35. <lb/>a & b.</note> +</div> +<p> + <s xml:id="echoid-s3490" xml:space="preserve">Concipiatur nunc longitudo canalis B C D fluido plena in rectam ex-<lb/>tenſa bcd, ſuper qua ceu axe fiat curva F G H, quæ ſit ſcala amplitudinum <lb/>in locis homologis, ita, ut poſita bc = B C ſit c G ad b F, ut amplitudo in <lb/>C ad amplitudinem in B. </s> + <s xml:id="echoid-s3491" xml:space="preserve">Igitur ſi b F repræſentet amplitudinem in B, tunc <lb/>ſpatium bd H F repræſentabit magnitudinem M. </s> + <s xml:id="echoid-s3492" xml:space="preserve">Deinde ſuper eodem axe bd <lb/>conſtruatur alia curva L M N, cujus applicata c M ſit ubique {bF<emph style="super">2</emph>/cG} & </s> + <s xml:id="echoid-s3493" xml:space="preserve">erit <lb/>(per §. </s> + <s xml:id="echoid-s3494" xml:space="preserve">2. </s> + <s xml:id="echoid-s3495" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s3496" xml:space="preserve">3.) </s> + <s xml:id="echoid-s3497" xml:space="preserve">N = ſpatio b d N L diviſo per ſpatium bd H F, ita ut ſit <lb/>M X N = ſpatio b d N L, quod multiplicatum per {γ/mgγ + ngg} dabit longitu-<lb/>dinem penduli tautochroni.</s> + <s xml:id="echoid-s3498" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div145" type="section" level="1" n="114"> +<head xml:id="echoid-head144" xml:space="preserve">Corollarium 1.</head> +<p> + <s xml:id="echoid-s3499" xml:space="preserve">§. </s> + <s xml:id="echoid-s3500" xml:space="preserve">20. </s> + <s xml:id="echoid-s3501" xml:space="preserve">Si tubus B C D ſit ubique ejusdem amplitudinis, ejusque lon-<lb/>gitudo dicatur l, erit F H linea recta ipſi bd parallela, pariter atque L N: <lb/></s> + <s xml:id="echoid-s3502" xml:space="preserve">hinc ſpatium bd N L = gl & </s> + <s xml:id="echoid-s3503" xml:space="preserve">longitudo penduli tautochroni = {l/m + n}.</s> + <s xml:id="echoid-s3504" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div146" type="section" level="1" n="115"> +<head xml:id="echoid-head145" xml:space="preserve">Corollarium 2.</head> +<p> + <s xml:id="echoid-s3505" xml:space="preserve">§. </s> + <s xml:id="echoid-s3506" xml:space="preserve">21. </s> + <s xml:id="echoid-s3507" xml:space="preserve">Sit B C D canalis conicus longitudinis l; </s> + <s xml:id="echoid-s3508" xml:space="preserve">erit c G (poſita bc = x) <lb/>= ({x/l}[√γ - √g] + √g)<emph style="super">2</emph>; </s> + <s xml:id="echoid-s3509" xml:space="preserve">unde cM = gg:</s> + <s xml:id="echoid-s3510" xml:space="preserve">({x/l}[√γ - √g] + √g)<emph style="super">2</emph>; <lb/></s> + <s xml:id="echoid-s3511" xml:space="preserve">ergo ſpatium bcML = {ggl/√gγ - g} - {ggl/√γ - γg}:</s> + <s xml:id="echoid-s3512" xml:space="preserve">({x/l}[√γ - √g] + √g) & </s> + <s xml:id="echoid-s3513" xml:space="preserve"><lb/>proinde totum ſpatium bdN L = {ggl/√gγ - g} + {ggl/√gγ - γ} = {ggl/√gγ}: </s> + <s xml:id="echoid-s3514" xml:space="preserve">Eſt <lb/>igitur longitudo penduli tautochroni cum oſcillante aqua = {l√gγ/mγ + ng}.</s> + <s xml:id="echoid-s3515" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3516" xml:space="preserve">Hinc intelligitur cæteris paribus oſcillari aquam tardiſſime cum ampli- +<pb o="122" file="0136" n="136" rhead="HYDRODYNAMICÆ"/> +tudines in B & </s> + <s xml:id="echoid-s3517" xml:space="preserve">D ſunt in ratione reciproca ſinuum angulorum reſpondentium <lb/>D B C & </s> + <s xml:id="echoid-s3518" xml:space="preserve">B D C: </s> + <s xml:id="echoid-s3519" xml:space="preserve">dein quo longior ſit pars aqua plena & </s> + <s xml:id="echoid-s3520" xml:space="preserve">quo minores angu-<lb/>li modo dicti, eò pariter tardiores fieri oſcillationes.</s> + <s xml:id="echoid-s3521" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3522" xml:space="preserve">Porro comparatis inter ſe tubis cylindricis & </s> + <s xml:id="echoid-s3523" xml:space="preserve">conicis, poſitisque an-<lb/>gulis B D C & </s> + <s xml:id="echoid-s3524" xml:space="preserve">D B C æqualibus, perſpicuum eſt, citius oſcillari aquam cæ-<lb/>teris paribus in conicis quam cylindricis, quia nempe {l√gγ/γ + g} ſemper mi-<lb/>nor eſt quam {1/2}l, quæcunque ratio inæqualis intercedatinter g & </s> + <s xml:id="echoid-s3525" xml:space="preserve">γ. </s> + <s xml:id="echoid-s3526" xml:space="preserve">Si porro <lb/>prædicti anguli inæquales ponantur, fieri poteſt tam ut tardius quam ut ci-<lb/>tius oſcilletur aqua in uno tuborum genere reſpectu alterius, quod ut exem-<lb/>plo confirmem, ponam angulum D B C rectum, id eſt, m = 1, & </s> + <s xml:id="echoid-s3527" xml:space="preserve">ſinum <lb/>alterius anguli B D C ſeu n = {1/4}, ita erit longitudo penduli pro tubis cylin-<lb/>dricis = {4/5}l: </s> + <s xml:id="echoid-s3528" xml:space="preserve">Si vero ſub iisdem reliquis circumſtantiis tubo cylindrico ſub-<lb/>ſtituas conicum, qui amplitudinem in B habeat quadruplo majorem, quam eſt <lb/>amplitudo in D, habebis, poſito γ = {1/4}g, longitudinem penduli = l: </s> + <s xml:id="echoid-s3529" xml:space="preserve">longius eſt <lb/>itaque cæteris paribus pendulum tautochronum pro tubo conico quam pro cy-<lb/>lindrico, & </s> + <s xml:id="echoid-s3530" xml:space="preserve">tardius fiunt oſcillationes in illo, quam in hoc: </s> + <s xml:id="echoid-s3531" xml:space="preserve">ſed ſi nunc, <lb/>manentibus rurſus reliquis, tubum conicum ſtrictiorem ponamus in B quam <lb/>in D, contrarium erit: </s> + <s xml:id="echoid-s3532" xml:space="preserve">fuerit v. </s> + <s xml:id="echoid-s3533" xml:space="preserve">gr. </s> + <s xml:id="echoid-s3534" xml:space="preserve">γ = 4g, erit longitudo penduli = {8/17}l, <lb/>& </s> + <s xml:id="echoid-s3535" xml:space="preserve">proinde minor, quam ſi tubus cylindricus foret; </s> + <s xml:id="echoid-s3536" xml:space="preserve">rurſusque minor erit, <lb/>ſi amplitudinem in B admodum majorem ponas, quam eſt in D: </s> + <s xml:id="echoid-s3537" xml:space="preserve">ita ſi fuerit <lb/>γ = {1/64}g, erit longitudo penduli = {8/17}l, ut ante. </s> + <s xml:id="echoid-s3538" xml:space="preserve">Notabile eſt, ut in præ-<lb/>cedente etiam vidimus exemplo, quod, manentibus amplitudine in B, ſitu <lb/>canalis B C D ejusdemque longitudine, duæ ſemper diverſæ definiri poſſint <lb/>amplitudines in D pro eadem penduli tautochroni longitudine, niſi cum an-<lb/>guli D B C & </s> + <s xml:id="echoid-s3539" xml:space="preserve">B D C ſunt æquales. </s> + <s xml:id="echoid-s3540" xml:space="preserve">Hujus rei exemplum eſt particulare, quod, <lb/>ſive amplitudo in D æqualis ſit amplitudini in B, ſive rationem ad eandem ha-<lb/>@eat quadratam ſinus ang. </s> + <s xml:id="echoid-s3541" xml:space="preserve">B D C & </s> + <s xml:id="echoid-s3542" xml:space="preserve">ſin. </s> + <s xml:id="echoid-s3543" xml:space="preserve">ang. </s> + <s xml:id="echoid-s3544" xml:space="preserve">D B C, eodem tempore oſcilla-<lb/>tiones fluidi abſolvantur in tubo utroque.</s> + <s xml:id="echoid-s3545" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div147" type="section" level="1" n="116"> +<head xml:id="echoid-head146" xml:space="preserve">Scholion Generale.</head> +<p> + <s xml:id="echoid-s3546" xml:space="preserve">§. </s> + <s xml:id="echoid-s3547" xml:space="preserve">22. </s> + <s xml:id="echoid-s3548" xml:space="preserve">Experimenta de oſcillantibus fluidis ita ſumpſi, ut crebra tenta-<lb/>tione longitudinem penduli ſimplicis Iſochroni invenirem, hancque longitu-<lb/>dinem in diverſis caſibus talem præter propter eſſe obſervare potui, quam +<pb o="123" file="0137" n="137" rhead="SECTIO SEXTA."/> +theoria in hâc ſectione indicat; </s> + <s xml:id="echoid-s3549" xml:space="preserve">aliquando tamen longitudinem illam debitâ <lb/>paullo majorem inveni; </s> + <s xml:id="echoid-s3550" xml:space="preserve">cujus rei rationem haud @ difficulter hanc eſſe vidi, <lb/>quod frictiones fluidi excurſiones non ſolum diminuant, ſed & </s> + <s xml:id="echoid-s3551" xml:space="preserve">retardent@, ut <lb/>&</s> + <s xml:id="echoid-s3552" xml:space="preserve">, quod tubi eo in loco, quo inflectuntur, ſtrictiores eſſe ſoleant: </s> + <s xml:id="echoid-s3553" xml:space="preserve">Id poſte-<lb/>rius, ſi omni cura evitetur, ſique ipſæ @inflexiones non uno angulo ſed lente <lb/>fiant, & </s> + <s xml:id="echoid-s3554" xml:space="preserve">ſi denique pro liquore oſcillante mercurius puriſſimus adhibeatur, <lb/>dubium mihi nullum ſupereſt, fore ut experimenta præmiſſam theoriam ad <lb/>amuſſim confirment, ita, ut operæ pretium non duxerim anxie de illis in-<lb/>quirere.</s> + <s xml:id="echoid-s3555" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3556" xml:space="preserve">Id tamen ratione experimentorum à me inſtitutorum ſuperaddam, <lb/>quod amplitudines tuborum ante experimentum in diverſis eorum locis accu-<lb/>rate exploraverim ope columellæ mercurii, quæ dum gradatim totam longi-<lb/>tudinem tubi percurreret, longitudinibus ſuis diverſis, quarum menſuras aſ-<lb/>ſiduè accipiebam, amplitudinum variationes ubique manifeſtabat: </s> + <s xml:id="echoid-s3557" xml:space="preserve">Et hæ <lb/>quidem amplitudines ita in tubo erunt explorandæ, poſtquam jam fuerit in-<lb/>curvatus, nam ab incurvatione amplitudines admodum decreſcunt. </s> + <s xml:id="echoid-s3558" xml:space="preserve">Hæc <lb/>ratio fuit, quod in primo hanc in rem à me ſumto experimento, ſucceſſus <lb/>expectationem meam fefellerit: </s> + <s xml:id="echoid-s3559" xml:space="preserve">Tubum nempe vitreum, cujusmodi pro <lb/>barometris conficiendis adhibere ſolent, ſatis amplum eundemque fere per-<lb/>fecte cylindricum, incurvare feci, ut oſtendit propemodum Figura vigeſi-<lb/>ma ſeptima, eoque deinde mercurio maximam partem repleto, oſcillatio-<lb/>nes ejus longe tardius fieri vidi, quam expectaveram, quia non attendi, <lb/>tubum ab incurvatione in D inſigniter fuiſſe conſtrictum, præſertim ubi an-<lb/>guli formantur. </s> + <s xml:id="echoid-s3560" xml:space="preserve">Hujus igitur rei, ut rationem haberem, tubis deinceps <lb/>lente incurvatis uſus fui, quales oſtendit Fig. </s> + <s xml:id="echoid-s3561" xml:space="preserve">35. </s> + <s xml:id="echoid-s3562" xml:space="preserve">a. </s> + <s xml:id="echoid-s3563" xml:space="preserve">in iisque amplitudines <lb/>poſt incurvationem diligenter exploravi.</s> + <s xml:id="echoid-s3564" xml:space="preserve"/> +</p> +<pb file="0138" n="138" rhead="(124)"/> +</div> +<div xml:id="echoid-div148" type="section" level="1" n="117"> +<head xml:id="echoid-head147" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph></head> +<head xml:id="echoid-head148" xml:space="preserve">SECTIO SEPTIMA.</head> +<head xml:id="echoid-head149" style="it" xml:space="preserve">De motu aquarum per vaſa ſubmerſa, ubi exem-<lb/>plis oſtenditur, quam inſigniter utile ſit princi-<lb/>pium conſervationis virium vivarum, veliis in caſibus, quibus continue <lb/>aliquid de illis perdi cenſendum eſt.</head> +<head xml:id="echoid-head150" xml:space="preserve">PARS PRIMA.</head> +<head xml:id="echoid-head151" style="it" xml:space="preserve">De deſcenſu aquarum.</head> +<head xml:id="echoid-head152" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s3565" xml:space="preserve">FInge cylindrum aquâ plenum, cujus fundum perforatum ſit, illudque <lb/>ad certam altitudinem aquæ ſtagnanti veluti infinitæ ſubmerſum, & </s> + <s xml:id="echoid-s3566" xml:space="preserve"><lb/>facile intelliges ſuperficiem aquæ in cylindro contentæ deſcenſuram, <lb/>& </s> + <s xml:id="echoid-s3567" xml:space="preserve">quidem infra ſuperficiem aquæ exterioris, dein rurſus aſcenſuram <lb/>& </s> + <s xml:id="echoid-s3568" xml:space="preserve">ſic porro. </s> + <s xml:id="echoid-s3569" xml:space="preserve">Hæ vero oſcillationes admodum differunt ab oſcillationibus in præ-<lb/>cedente ſectione conſideratis, in quibus nempe motus reciproci ſemper ſunt <lb/>inverſo ordine iidem cum motibus, qui præceſſerunt. </s> + <s xml:id="echoid-s3570" xml:space="preserve">Quis autem hic præſumat <lb/>refluxum aquarum ſeu aſcenſum eundem fore, qui fuerat deſcenſus. </s> + <s xml:id="echoid-s3571" xml:space="preserve">Talia <lb/>ſi quis ſtatueret, is certe vehementer falleretur, etiamſi vel nihil motus di-<lb/>minuatur ab adhæſione aquarum ad latera vaſis hujuscemodique aliis impe-<lb/>dimentis, non ſecus atque regulæ motuum à percuſſione pro corporibus <lb/>elaſticis valde diverſæ ſunt ab iis, quæ pro corporibus mollibus valent, utut <lb/>in utroque caſu corpora liberrime moveri cenſeantur. </s> + <s xml:id="echoid-s3572" xml:space="preserve">Utor hoc ſimili, quod <lb/>argumentum noſtrum egregie illuſtrat: </s> + <s xml:id="echoid-s3573" xml:space="preserve">Prouti enim regulæ motuum in cor-<lb/>poribus mollibus recte determinatur, ſi poſt colliſionem ea vis vivæ <lb/>pars deperdita cenſeatur quæ in compreſſionem corporum impenſa fuit <lb/>(neque enim hæc ut in corporibus elaſticis reſtituitur motui progreſſivo) ita <lb/>aſcenſus fluidi non minus recte definietur, ſi accurate examinetur, quantum vis <lb/>vivæ ſingulis momentis motui particularum aquearum inteſtino communi-<lb/>cetur, nunquam rediturum ad motum progreſſivum, de quo ſermo eſt.</s> + <s xml:id="echoid-s3574" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3575" xml:space="preserve">§. </s> + <s xml:id="echoid-s3576" xml:space="preserve">2. </s> + <s xml:id="echoid-s3577" xml:space="preserve">Cum itaque res eo deducta ſit, ut exploretur, quantum vis vivæ <lb/>in motibus iſtis reciprocis continue perdatur, diſquiſitionem ab hoc incipie-<lb/>mus.</s> + <s xml:id="echoid-s3578" xml:space="preserve"/> +</p> +<pb o="125" file="0139" n="139" rhead="SECTIO SEPTIMA."/> +<p> + <s xml:id="echoid-s3579" xml:space="preserve">Primò autem patet omnem vim vivam quæ particulis effluentibus ineſt <lb/>tranſire ad aquam externam nec ullo modo promovere ſubſequentem aſcenſum <lb/>ſeu influxum aquæ externæ in tubum: </s> + <s xml:id="echoid-s3580" xml:space="preserve">Nimis hæc eſt clara hypotheſis, quam <lb/>ut majori explicatione opus habeat: </s> + <s xml:id="echoid-s3581" xml:space="preserve">reſpicit autem aquarum effluxum & </s> + <s xml:id="echoid-s3582" xml:space="preserve">in hoc <lb/>unica eſt conſideranda. </s> + <s xml:id="echoid-s3583" xml:space="preserve">Venit jam altera, quæ pertinet ad aquarum influxum.</s> + <s xml:id="echoid-s3584" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3585" xml:space="preserve">Secundò igitur non minus perſpicuum mihi quidem eſt, quod ir-<lb/>ruente aqua per foramen majori velocitate, quam quæ aquæ internæ aſcen-<lb/>denti ineſt, exceſſus ille rurſus motum quendam inteſtinum in eadem aqua <lb/>interna cieat, parum aut nihil ad aſcenſum conferentem. </s> + <s xml:id="echoid-s3586" xml:space="preserve">Hoc ſi ita ſit, pona-<lb/>turque amplitudo foraminis = 1, amplitudo cylindri = n, aſcenſus potent. <lb/></s> + <s xml:id="echoid-s3587" xml:space="preserve">guttulæ irrumpentis = n n v, ejusque velocitas = n√v, retinebit hæc par-<lb/>ticula motu ſuo, quem cum reliqua aqua interna communem habet, velocitatem <lb/>√v, conſervabitque proinde aſcenſum potent. </s> + <s xml:id="echoid-s3588" xml:space="preserve">v; </s> + <s xml:id="echoid-s3589" xml:space="preserve">reliquum autem aſcenſus potent. </s> + <s xml:id="echoid-s3590" xml:space="preserve"><lb/>nempe n n v - v ad motum particularum inteſtinum transiiſſe cenſendum eſt. </s> + <s xml:id="echoid-s3591" xml:space="preserve"><lb/>Hypotheſis iſta, quamvis Phyſica ſit & </s> + <s xml:id="echoid-s3592" xml:space="preserve">proxime tantum vera, tamen mag-<lb/>nam habet utilitatem ad motus fluidorum ſine notabili errore determinandos, <lb/>quoties in vaſe uniformis continuitas, quæ hactenus aſſumta fuit, prærum-<lb/>pitur, veluti cum aqua per plura foramina tranſire cogitur; </s> + <s xml:id="echoid-s3593" xml:space="preserve">Imo credide-<lb/>rim unicam eſſe, cujus ope hujusmodi motus mira phænomena recte expli-<lb/>cari poſſint. </s> + <s xml:id="echoid-s3594" xml:space="preserve">Quapropter velim, ut recte animo perpendatur, antequam ad <lb/>alia divertatur lector.</s> + <s xml:id="echoid-s3595" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3596" xml:space="preserve">§. </s> + <s xml:id="echoid-s3597" xml:space="preserve">3. </s> + <s xml:id="echoid-s3598" xml:space="preserve">Jam igitur quæſtionem ipſam examinabimus, incipiendo ab a-<lb/>quarum deſcenſu. </s> + <s xml:id="echoid-s3599" xml:space="preserve">Concipiatur cylindrus A I M B, (Fig. </s> + <s xml:id="echoid-s3600" xml:space="preserve">36.) </s> + <s xml:id="echoid-s3601" xml:space="preserve">aqua plenus <lb/> +<anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/> +usque in X Y & </s> + <s xml:id="echoid-s3602" xml:space="preserve">aquæ infinitæ R T V S ſubmerſus, ita ut longitudo ejus ſit <lb/>in ſitu verticali habeat ejus fundum lumen P L, per quod aqua ex vaſe in <lb/>aquam circumfluam effluere poſſit. </s> + <s xml:id="echoid-s3603" xml:space="preserve">Quæritur velocitas aquæ internæ, poſt-<lb/>quam ſuperficies ejus per datum ſpatium X C vel Y D deſcendit, poſita <lb/>M Y vel I X = a, M V = b, M D = x, amplitudine foraminis = 1, & </s> + <s xml:id="echoid-s3604" xml:space="preserve"><lb/>denique amplitudine cylindri = n.</s> + <s xml:id="echoid-s3605" xml:space="preserve"/> +</p> +<div xml:id="echoid-div148" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">Fig. 36.</note> +</div> +<p> + <s xml:id="echoid-s3606" xml:space="preserve">Solutio eadem erit, quam pro ſimili quæſtione, ſed ea admodum <lb/>generali, dedimus in ſectione tertia: </s> + <s xml:id="echoid-s3607" xml:space="preserve">obſervetur tantum, quod ſumta par-<lb/>ticula aquæ infinitè parva C D F E æquali guttulæ P L O N eo ipſo tempore <lb/>ejectæ, deſcenſus actualis ſit nunc æſtimandus ex altitudine D V vel C T, <lb/>cum in altero caſu definiendus erat ex tota altitudine D M.</s> + <s xml:id="echoid-s3608" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3609" xml:space="preserve">Sit nempe velocitas ſuperficiei aqueæ C D ea, quæ debetur +<pb o="126" file="0140" n="140" rhead="HYDRODYNAMICÆ"/> +altitudini v, & </s> + <s xml:id="echoid-s3610" xml:space="preserve">in ſitu infinite propinquo E F reſpondebit eadem velocitas <lb/>altitudini v - d v; </s> + <s xml:id="echoid-s3611" xml:space="preserve">Et cum aſcenſus potentialis aquæ C D M L P I C ſit v, obti-<lb/>nebitur aſcenſus potent. </s> + <s xml:id="echoid-s3612" xml:space="preserve">ejusdem aquæ in ſitu proximo E F M L O N P I E, ſi <lb/>multiplicetur maſſa E F M L P I E (n x - n d x) per ſuum aſcenſum potent. <lb/></s> + <s xml:id="echoid-s3613" xml:space="preserve">(v - d v) ut etiam guttula L O N P (n d x) per ſuum itidem aſcenſum poten-<lb/>tialem n n v, aggregatumque productorum dividatur per ſummam maſſarum <lb/>(n x): </s> + <s xml:id="echoid-s3614" xml:space="preserve">habetur itaque iste aſcenſus potentialis = {(n x - n d x) x (v - d v) + n d x x n n v/nx} <lb/>ſeu {xv - vdx - xdv + nnvdx/x}.</s> + <s xml:id="echoid-s3615" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3616" xml:space="preserve">Eſt proinde incrementum aſcenſus potent. </s> + <s xml:id="echoid-s3617" xml:space="preserve">= {- vdx - xdv + nnvdx/x}. <lb/></s> + <s xml:id="echoid-s3618" xml:space="preserve">(conf. </s> + <s xml:id="echoid-s3619" xml:space="preserve">§. </s> + <s xml:id="echoid-s3620" xml:space="preserve">6. </s> + <s xml:id="echoid-s3621" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s3622" xml:space="preserve">3.) </s> + <s xml:id="echoid-s3623" xml:space="preserve">Iſtud vero incrementum æquale cenſendum eſt cum de-<lb/>ſcenſu actuali infinitè parvo, qui (per §. </s> + <s xml:id="echoid-s3624" xml:space="preserve">7. </s> + <s xml:id="echoid-s3625" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s3626" xml:space="preserve">3. </s> + <s xml:id="echoid-s3627" xml:space="preserve">& </s> + <s xml:id="echoid-s3628" xml:space="preserve">per annotationem modo <lb/>datam) eſt = {(x - b)dx/x}. </s> + <s xml:id="echoid-s3629" xml:space="preserve">Habetur itaque talis æquatio <lb/>- vdx - xdv + nnvdx = (x - b)dx, <lb/>quæ debito modo integrata mutatur in hanc <lb/>v = {1/nn - 2} X (x - {x<emph style="super">nn - 1</emph>/a<emph style="super">nn - 2</emph>}) - {b/nn - 1} X (1 - {x<emph style="super">nn - 1</emph>/a<emph style="super">nn - 1</emph>}).</s> + <s xml:id="echoid-s3630" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3631" xml:space="preserve">Ex iſta vero æquatione talia ſequuntur corollaria.</s> + <s xml:id="echoid-s3632" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3633" xml:space="preserve">§. </s> + <s xml:id="echoid-s3634" xml:space="preserve">4. </s> + <s xml:id="echoid-s3635" xml:space="preserve">Fuerit amplitudo cylindri veluti infinita ratione foraminis, & </s> + <s xml:id="echoid-s3636" xml:space="preserve"><lb/>erit cènſendum v = {x - b/nn}; </s> + <s xml:id="echoid-s3637" xml:space="preserve">ipſaque altitudo pro velocitate aquæ, dum <lb/>effluit, eſt = x - b. </s> + <s xml:id="echoid-s3638" xml:space="preserve">Unde conſequens eſt, aquam effluere velocitate, <lb/>quam grave acquirit cadendo ex altitudine ſuperficiei internæ ſupra externam, <lb/>& </s> + <s xml:id="echoid-s3639" xml:space="preserve">eo usque effluet, donec ambæ ſuperficies ſint ad libellam poſitæ, tunc-<lb/>que omnis motus ceſſabit: </s> + <s xml:id="echoid-s3640" xml:space="preserve">adeoque eadem lege aquæ effluunt, quaſi fun-<lb/>dum ſitum I M mutaret cum T V.</s> + <s xml:id="echoid-s3641" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3642" xml:space="preserve">Cum vero foramen non poteſt ceu infinite parvum conſiderari, deſcen-<lb/>dit ſuperficies aquæ internæ infra externam; </s> + <s xml:id="echoid-s3643" xml:space="preserve">atque ut innoteſcatad quamnam <lb/>profunditatem x y ſit deſcenſura ſuperficies C D, facienda eſt v = o, ſeu <lb/>(nn - 1)(a<emph style="super">nn - 1</emph>x - x<emph style="super">nn - 1</emph>a) = (nn - 2) X (a<emph style="super">nn - 1</emph>b - x<emph style="super">nn - 1</emph>b), <lb/>nunquam autem ſuperficies interna tantum deſcendet infra ſuperficiem exter- +<pb o="127" file="0141" n="141" rhead="SECTIO SEPTIMA."/> +nam, quantum ſuper eandem elevata fuerat, provenit iſte defectus ab aſcenſu <lb/>pot. </s> + <s xml:id="echoid-s3644" xml:space="preserve">aquæ durante deſcenſu ejectæ, cui debet eſſe proportionalis.</s> + <s xml:id="echoid-s3645" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3646" xml:space="preserve">§. </s> + <s xml:id="echoid-s3647" xml:space="preserve">5. </s> + <s xml:id="echoid-s3648" xml:space="preserve">Notabile eſt, quod cum eo profundius deſcendat aqua in cylin-<lb/>dro, quo magis ab initio deſcenſus fuerit elevata & </s> + <s xml:id="echoid-s3649" xml:space="preserve">quo majori lumine perfo-<lb/>ratum eſtfundum, nunquam tamen omnis aqua ex cylindro effluere poſſit <lb/>quantumvis fuerit ante deſcenſum elevata & </s> + <s xml:id="echoid-s3650" xml:space="preserve">pars cylindri ſubmerſa utlibet <lb/>parva, ipſumque ſimul foramen vel totum fundum exhaurire ponatur.</s> + <s xml:id="echoid-s3651" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3652" xml:space="preserve">§. </s> + <s xml:id="echoid-s3653" xml:space="preserve">6. </s> + <s xml:id="echoid-s3654" xml:space="preserve">Velocitas ſuperficiei aquæ internæ maxima eſt, cum ſumitur <lb/>x = ({a<emph style="super">nn - 1</emph>/nna - nnb - a + 2b})<emph style="super">1: (nn - 2)</emph></s> +</p> +<p> + <s xml:id="echoid-s3655" xml:space="preserve">Si proinde n = 1, exiſtente ſcilicet orificio cylindri toto aperto, fit <lb/>x = b, & </s> + <s xml:id="echoid-s3656" xml:space="preserve">maxima eſt velocitas, cum ambæ ſuperficies ſunt in eadem altitu-<lb/>dine poſitæ.</s> + <s xml:id="echoid-s3657" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3658" xml:space="preserve">Quia vero multa ſunt, quæ ex hiſce æquationibus dignoſci nequeunt <lb/>in duobus caſibus, nempe nn = 1 & </s> + <s xml:id="echoid-s3659" xml:space="preserve">nn = 2, hique multa habent particula-<lb/>ria, eoſdem ſeorſim jam attingam.</s> + <s xml:id="echoid-s3660" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3661" xml:space="preserve">§. </s> + <s xml:id="echoid-s3662" xml:space="preserve">7. </s> + <s xml:id="echoid-s3663" xml:space="preserve">Sit primo nn = 1, & </s> + <s xml:id="echoid-s3664" xml:space="preserve">erit - xdv = (x - b) dx (per §. </s> + <s xml:id="echoid-s3665" xml:space="preserve">3.) </s> + <s xml:id="echoid-s3666" xml:space="preserve">vel <lb/>- dv = dx - {bdx/x}, quæ ſic integrata, ut ſit ſimul v = o & </s> + <s xml:id="echoid-s3667" xml:space="preserve">x = a, dat - v = <lb/>x - a + b log. </s> + <s xml:id="echoid-s3668" xml:space="preserve">{a/x}, ſeu v = a - x - b log. </s> + <s xml:id="echoid-s3669" xml:space="preserve">{a/x}: </s> + <s xml:id="echoid-s3670" xml:space="preserve">Exinde talia deduci poſſunt.</s> + <s xml:id="echoid-s3671" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3672" xml:space="preserve">I<emph style="super">0</emph>. </s> + <s xml:id="echoid-s3673" xml:space="preserve">Ut obtineatur maximus deſcenſus, faciendum eſt a - x - b log. </s> + <s xml:id="echoid-s3674" xml:space="preserve">{a/x} <lb/>= o; </s> + <s xml:id="echoid-s3675" xml:space="preserve">patet autem ex iſta æquatione, nunquam negativum valorem obtinere <lb/>litteram x, imo nequidem totam evaneſcere ſine contradictione, niſi pona-<lb/>tur {a/b} = ∞, quod indicat fieri non poſſe, ut omnis effluat aqua durante de-<lb/>ſcenſu in iſto caſu & </s> + <s xml:id="echoid-s3676" xml:space="preserve">multo minus in reliquis, quod confirmat paragraphum <lb/>quintum.</s> + <s xml:id="echoid-s3677" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3678" xml:space="preserve">II<emph style="super">0</emph>. </s> + <s xml:id="echoid-s3679" xml:space="preserve">Velocitas maxima talis eſt, quæ debetur altitudini a - b - b log. </s> + <s xml:id="echoid-s3680" xml:space="preserve">{a/b}, <lb/>atque ſi differentia inter a & </s> + <s xml:id="echoid-s3681" xml:space="preserve">b, quam ponam = c, ſit valde parva, exiſten-<lb/>tibus nimirum excurſionibus fluidi perexiguis ratione longitudinis, ad quam +<pb o="128" file="0142" n="142" rhead="HYDRODYNAMICÆ"/> +cylindrus eſt ſubmerſus, poterit log. </s> + <s xml:id="echoid-s3682" xml:space="preserve">{a/b} cenſeri = {c/b} - {cc/2bb} ipſaque proinde <lb/>altitudo maximæ debita velocitati ſeu a-b-blog. </s> + <s xml:id="echoid-s3683" xml:space="preserve">{a/b} = {cc/2b}, quod motum ad-<lb/>modum lentum fore arguit.</s> + <s xml:id="echoid-s3684" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3685" xml:space="preserve">Demonſtrabo autem in ſequentibus, totum motum cæteris paribus <lb/>eundem manere, cum cylindri cenſentur infinite ſubmerſi, quocunque fora-<lb/>mine fundum fuerit perforatum, ita ut motus aquæ internæ à diminuto fora-<lb/>mine non retardetur; </s> + <s xml:id="echoid-s3686" xml:space="preserve">quod quamvis prima fronte admodum paradoxum vi-<lb/>deatur, non poterit tamen vera ejus ratio phyſica effugere animum l<unsure/>ad hæc <lb/>attentiorem. </s> + <s xml:id="echoid-s3687" xml:space="preserve">In eo ſcilicet verſatur, quod vis viva, quæ in tubo generatur, <lb/>veluti infinita ſit præ vi viva aquæ per foramen tranſeuntis nec adeoque hujus <lb/>foraminis conſideratio computum diverſum faciat.</s> + <s xml:id="echoid-s3688" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3689" xml:space="preserve">Demonſtrabimus etiam ſimiles eſſe motus reciprocos & </s> + <s xml:id="echoid-s3690" xml:space="preserve">oſcillationes <lb/>tam majores quam minores inter ſe eſſe Iſochronas, atque pro hiſce longitu-<lb/>dinem penduli ſimplicis tautochroni determinabimus.</s> + <s xml:id="echoid-s3691" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3692" xml:space="preserve">§. </s> + <s xml:id="echoid-s3693" xml:space="preserve">8. </s> + <s xml:id="echoid-s3694" xml:space="preserve">Fuerit nunc nn = 2; </s> + <s xml:id="echoid-s3695" xml:space="preserve">Ita vero habetur vi §. </s> + <s xml:id="echoid-s3696" xml:space="preserve">3. </s> + <s xml:id="echoid-s3697" xml:space="preserve">v d x - x d v = <lb/>(x - b) dx, vel {xdv - vdx/xx} = {(b - x)dx/x x}, quæ recte integrata abit in hanc v = <lb/>{bx/a} - b + x log. </s> + <s xml:id="echoid-s3698" xml:space="preserve">{a/x}. </s> + <s xml:id="echoid-s3699" xml:space="preserve">Si fiat {bx/a} - b + x log. </s> + <s xml:id="echoid-s3700" xml:space="preserve">{a/x} = o, dabit x locum maximi de-<lb/>ſcenſus; </s> + <s xml:id="echoid-s3701" xml:space="preserve">locus autem maximæ velocitatis habebitur, faciendo x = c<emph style="super">{b - a/a}</emph>a, <lb/>ubi per c intelligitur numerus, cujus logarithmus eſt unitas.</s> + <s xml:id="echoid-s3702" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3703" xml:space="preserve">Poſtquam ſic varios perſtrinximus caſus pro diverſis foraminum ma-<lb/>gnitudinibus, ſupereſt ut etiam conſideremus, quid in diverſis altitudinum <lb/>a & </s> + <s xml:id="echoid-s3704" xml:space="preserve">b caſibus ſuccedere poſſit.</s> + <s xml:id="echoid-s3705" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3706" xml:space="preserve">§. </s> + <s xml:id="echoid-s3707" xml:space="preserve">9. </s> + <s xml:id="echoid-s3708" xml:space="preserve">Et primo quidem ſi b nulla ſtatuatur præ a, quod fit cum cylin-<lb/>dri fundum tantum radit ſuperficiem aquæ exterioris, tunc prodit <lb/>v = {1/nn - 2}(x - {x<emph style="super">nn - 1</emph>/a<emph style="super">nn - 2</emph>}) <lb/>quæ quidem æquatio non niſi forma differt ab illa, quæ §. </s> + <s xml:id="echoid-s3709" xml:space="preserve">14. </s> + <s xml:id="echoid-s3710" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s3711" xml:space="preserve">3. </s> + <s xml:id="echoid-s3712" xml:space="preserve">data fuit <lb/>pro eo caſu, quo aquæ ex cylindro in aërem ejici ponuntur. </s> + <s xml:id="echoid-s3713" xml:space="preserve">Et ſæpe etiam +<pb o="129" file="0143" n="143" rhead="SECTIO SEPTIMA."/> +expertus ſum cylindrum eodem tempore evacuari, ſive aquæ in aërem eji-<lb/>ciantur, ſive fundum aquæ ſtagnanti tantillum ſubmergatur. </s> + <s xml:id="echoid-s3714" xml:space="preserve">Docet hæc ex-<lb/>perientia parum aut nihil obſtare aërem externum effluxui, cum reſiſtentia <lb/>plus quam octingenties major notabiliorem effectum non exerat. </s> + <s xml:id="echoid-s3715" xml:space="preserve">Quia adeo-<lb/>que iſte caſus nihil particulare habet, quod non loco citato monitum fuerit, <lb/>huic non ulterius immorabimur: </s> + <s xml:id="echoid-s3716" xml:space="preserve">Inquiremus potius, quid fieri debeat, cum <lb/>elevatio aquæ internæ ſuper externam, quanta ab initio deſcenſus eſt, ſumi-<lb/>tur valde parva & </s> + <s xml:id="echoid-s3717" xml:space="preserve">negligenda præ immerſione cylindri; </s> + <s xml:id="echoid-s3718" xml:space="preserve">cui hypotheſi ſatisfit, <lb/>cum exceſſus altitudinis a ſuper altitudinem b (quem exceſſum rurſus vocabi-<lb/>mus (ut §. </s> + <s xml:id="echoid-s3719" xml:space="preserve">7.) </s> + <s xml:id="echoid-s3720" xml:space="preserve">c) eſt admodum parvus.</s> + <s xml:id="echoid-s3721" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3722" xml:space="preserve">§. </s> + <s xml:id="echoid-s3723" xml:space="preserve">10. </s> + <s xml:id="echoid-s3724" xml:space="preserve">Cum itaque ponitur a - b = c, ponendum etiam erit a - x = z, <lb/>tumque utraque quantitas, nempe c & </s> + <s xml:id="echoid-s3725" xml:space="preserve">z, erunt negligendæ præ quantitatibus <lb/>a & </s> + <s xml:id="echoid-s3726" xml:space="preserve">b, ſed ſi a - x = z, erit x = a - z & </s> + <s xml:id="echoid-s3727" xml:space="preserve">x<emph style="super">nn - 1</emph> = (a - z)<emph style="super">nn - 1</emph> = <lb/>a<emph style="super">nn - 1</emph> - (nn - 1)a<emph style="super">nn - 2</emph>z + ({<emph style="ol">nn - 1. nn -2</emph>/2})a<emph style="super">nn - 3</emph>zz <lb/>- ({<emph style="ol">nn - 1. nn - 2. nn - 3</emph>/2. </s> + <s xml:id="echoid-s3728" xml:space="preserve">3.</s> + <s xml:id="echoid-s3729" xml:space="preserve">})a<emph style="super">nn - 4</emph> z<emph style="super">3</emph> + &</s> + <s xml:id="echoid-s3730" xml:space="preserve">c.</s> + <s xml:id="echoid-s3731" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3732" xml:space="preserve">Hæc ſeries quantum ad inſtitutum noſtrum ſufficit eſt continuanda; <lb/></s> + <s xml:id="echoid-s3733" xml:space="preserve">ſufficiet autem ad tres usque terminos. </s> + <s xml:id="echoid-s3734" xml:space="preserve">Igitur in æquatione integrata quam <lb/>dedimus §. </s> + <s xml:id="echoid-s3735" xml:space="preserve">3. </s> + <s xml:id="echoid-s3736" xml:space="preserve">ponemus, x = a - z & </s> + <s xml:id="echoid-s3737" xml:space="preserve"><lb/>x<emph style="super">nn - 1</emph> = a<emph style="super">nn - 1</emph> - (nn - 1)a<emph style="super">nn - 2</emph> z + ({<emph style="ol">nn - 1. nn - 2</emph>/2})a<emph style="super">nn - 3</emph>zz & </s> + <s xml:id="echoid-s3738" xml:space="preserve"><lb/>ſic erit <lb/>v = {1/nn -2} [a - z - a + (nn - 1) z - ({<emph style="ol">nn - 1. nn -2</emph>/2}){zz/a}] <lb/>- {b/nn - 1}[1 - 1 + (nn - 1){z/a} - ({<emph style="ol">nn - 1. nn - 2</emph>/2}){zz/aa}]</s> +</p> +<p> + <s xml:id="echoid-s3739" xml:space="preserve">In qua æquatione ſi termini ſe deſtruentes deleantur, atque ponatur a - c <lb/>pro b, rejiciaturque terminus qui affectatur quantitate {czz/aa}, prodit ſimpliciter <lb/>v = {2cz - zz/2a}. <lb/></s> + <s xml:id="echoid-s3740" xml:space="preserve">ex quâ formula, cum littera n evanuerit, indicium habemus, nihil magni- +<pb o="130" file="0144" n="144" rhead="HYDRODYNAMICÆ"/> +tudinem orificii pertinere ad motum aquæ internæ, cujus rei originem jam <lb/>ſupra (§. </s> + <s xml:id="echoid-s3741" xml:space="preserve">7.) </s> + <s xml:id="echoid-s3742" xml:space="preserve">indicavi.</s> + <s xml:id="echoid-s3743" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3744" xml:space="preserve">In ſequentibus autem demonſtrabimus, non differre hunc motum à <lb/>ſubſequente motu refluo, hincque oſcillationes fieri tautochronas. </s> + <s xml:id="echoid-s3745" xml:space="preserve">Prius-<lb/>quam vero ad alia pergam monendum duxi, in iſto calculo quantitates <lb/>{c/a} & </s> + <s xml:id="echoid-s3746" xml:space="preserve">{z/a} non ſolum præ unitate, ſed & </s> + <s xml:id="echoid-s3747" xml:space="preserve">præ {1/nn} ceu infinite parvas poſitas fuiſ-<lb/>ſe, ad quod animus probe eſt advertendus in inſtituendis experimentis; <lb/></s> + <s xml:id="echoid-s3748" xml:space="preserve">licet utique theoriam infinite parvorum ad experimenta, ſine notabili erro-<lb/>re revocare diminuendo admodum quantitates, quæ in theoria ceu infinite <lb/>parvæ conſideratæ fuerunt, ſed faciendum eſt, ut in experimento omnia <lb/>huic legi ſint ſubjecta. </s> + <s xml:id="echoid-s3749" xml:space="preserve">Ita v. </s> + <s xml:id="echoid-s3750" xml:space="preserve">gr. </s> + <s xml:id="echoid-s3751" xml:space="preserve">ſi in cylindro omne fundum abſit, poſito <lb/>n = 1, idque ſubmerſum ponatur ad altitudinem triginta quinque pollicum, <lb/>ſatis accurate ſumetur experimentum, cum aqua ante oſcillationes elevata <lb/>tantum fuerit ad altitudinem unius pollicis ſupra ſuperficiem aquæ circum-<lb/>fluæ nec dum error notabilis erit, ſi vel orificiium inferius ad dimidium <lb/>obſtruatur exiſtente tunc {c/a} ad {1/nn} ut 1. </s> + <s xml:id="echoid-s3752" xml:space="preserve">9, quæ ratio in noſtro experimento <lb/>tuto adhuc negligi poteſt: </s> + <s xml:id="echoid-s3753" xml:space="preserve">at ſi jam diametrum tubi duplam ponas diame-<lb/>tri orificii, occluſis tribus quartis aperturæ integræ partibus, jam fiet n = 4 <lb/>& </s> + <s xml:id="echoid-s3754" xml:space="preserve">{c/a} ad {1/nn} ut 4 ad 9, quæ ratio non ſatis parva amplius erit, ut experimentum <lb/>conditionibus theoriæ cum ſufficienti præciſione ſatisfacere affirmari poſſit.</s> + <s xml:id="echoid-s3755" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3756" xml:space="preserve">Hic itaque jam porro inquirere conveniet, quid de his caſibus ſtatuen-<lb/>dum ſit, quibus {c/a} & </s> + <s xml:id="echoid-s3757" xml:space="preserve">{1/nn} notabilem quidem inter ſe habent rationem, utra-<lb/>que vero quantitas fit admodum exigua, quod nimirum fit, cum cylindrus <lb/>profundiſſime ſubmergitur, ſimul autem fundum parvulo eſt pertuſum fo-<lb/>ramine.</s> + <s xml:id="echoid-s3758" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3759" xml:space="preserve">§. </s> + <s xml:id="echoid-s3760" xml:space="preserve">11. </s> + <s xml:id="echoid-s3761" xml:space="preserve">Sed iſte, quem modo finximus, caſus melius ex æquatione <lb/>differentiali paragraphi tertii, quam ex integrali, ut antea factum, deduci-<lb/>tur: </s> + <s xml:id="echoid-s3762" xml:space="preserve">poteſt autem pro his circumſtantiis rejici terminus - v d x præ n n v d x, <lb/>atque ſic aſſumi - x d v + n n v d x = (x - b) d x, in quâ ſi rurſus ponitur <lb/>a - b = c & </s> + <s xml:id="echoid-s3763" xml:space="preserve">a - x = z, prodit <lb/>adv + zdv + nnvdz = (c - z) dz +<pb o="131" file="0145" n="145" rhead="SECTIO SEPTIMA."/> +cujus ſecundus terminus z d v rurſus præ primo negligi poteſt, ita vero <lb/>habetur <lb/>adv + nnvdz = (c - z)dz.</s> + <s xml:id="echoid-s3764" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3765" xml:space="preserve">Ponatur hic (ſumto α pro numero, cujus logarithmus hyperbolicus eſt <lb/>unitas) v = {1/nn}α<emph style="super">{-nnz/a}</emph>q; </s> + <s xml:id="echoid-s3766" xml:space="preserve">hoc modo mutabitur poſtrema æquatio in hanc <lb/>α{-nnz/a}adq = nn (c - z)dz, vel <lb/>adq = nnα<emph style="super">{nnz/a}</emph> X (c - z)dz:</s> + <s xml:id="echoid-s3767" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3768" xml:space="preserve">Hæc vero ita eſt integranda, ut z & </s> + <s xml:id="echoid-s3769" xml:space="preserve">v vel etiam z & </s> + <s xml:id="echoid-s3770" xml:space="preserve">q ſimul evane-<lb/>ſcant; </s> + <s xml:id="echoid-s3771" xml:space="preserve">habebitur igitur <lb/>q = (c + {a/nn} - z)α<emph style="super">{nnz/a}</emph> - c - {a/nn}, vel denique <lb/>v = {1/nn} (c + {a/nn} - z) - {1/nn} (c + {a/nn})α<emph style="super">{-nnz/a}</emph>;</s> + <s xml:id="echoid-s3772" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3773" xml:space="preserve">Ex iſta vero æquatione deducitur:</s> + <s xml:id="echoid-s3774" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3775" xml:space="preserve">I. </s> + <s xml:id="echoid-s3776" xml:space="preserve">Oriri rurſus, ut paragrapho decimo alia mathodo inventum fuit, <lb/>v = {2cz - zz/2a}, ſi nempe rurſus ponatur {nnz/a} numerus valde parvus, Id ve-<lb/>ro ut pateat, reſolvenda eſt quantitas exponentialis α<emph style="super">{-nnz/a}</emph> in ſeriem, quæ <lb/>eſt ipſi æqualis, 1 - {nnz/a} + {n<emph style="super">4</emph>zz/2aa} - {n<emph style="super">6</emph>z<emph style="super">3</emph>/2. </s> + <s xml:id="echoid-s3777" xml:space="preserve">3a<emph style="super">3</emph>} + &</s> + <s xml:id="echoid-s3778" xml:space="preserve">c. </s> + <s xml:id="echoid-s3779" xml:space="preserve">ex quâ pro noſtro <lb/>ſcopo tres priores termini ſufficiunt; </s> + <s xml:id="echoid-s3780" xml:space="preserve">eo autem ſubſtituto valore rejectoque <lb/>termino rejiciendo, reperitur ut dixi <lb/>v = {2cz - zz/2a}</s> +</p> +<p> + <s xml:id="echoid-s3781" xml:space="preserve">II. </s> + <s xml:id="echoid-s3782" xml:space="preserve">At ſi viciſſim {nn/1} infinites major ponatur quam {a/z} aut {a/c}, quia tunc <lb/>α{-nnz/a} = o, ut & </s> + <s xml:id="echoid-s3783" xml:space="preserve">{a/nn} = o, fieri intelligitur v = c - z, ſive v = x - b, <lb/>ut §. </s> + <s xml:id="echoid-s3784" xml:space="preserve">4.</s> + <s xml:id="echoid-s3785" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3786" xml:space="preserve">III. </s> + <s xml:id="echoid-s3787" xml:space="preserve">Neutram vero præmiſſarum formularum ſine notabili errore lo-<lb/>cum habere patet, cum {nnc/a}, numerus eſt mediocris, nempe nec infinitus, <lb/>nec infinite parvus, & </s> + <s xml:id="echoid-s3788" xml:space="preserve">tamen utraque quantitas {nn/1} & </s> + <s xml:id="echoid-s3789" xml:space="preserve">{a/c} infinita.</s> + <s xml:id="echoid-s3790" xml:space="preserve"/> +</p> +<pb o="132" file="0146" n="146" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s3791" xml:space="preserve">Fuerit v. </s> + <s xml:id="echoid-s3792" xml:space="preserve">gr. </s> + <s xml:id="echoid-s3793" xml:space="preserve">elevatio indicata per c unius pollicis, immerſio cylindri b <lb/>80. </s> + <s xml:id="echoid-s3794" xml:space="preserve">poll. </s> + <s xml:id="echoid-s3795" xml:space="preserve">ipſaque a 81. </s> + <s xml:id="echoid-s3796" xml:space="preserve">poll. </s> + <s xml:id="echoid-s3797" xml:space="preserve">dein ponatur diameter tubi tripla diametri forami-<lb/>nis, id eſt, nn = 81, erit v = {2 - z - 2α<emph style="super">- z</emph>/nn}, atque ſi porro ponatur <lb/>z = c = 1, ut habeatur altitudo velocitatis, cum utraque ſuperficies eſt ad <lb/>libellam poſita, erit v = {α - 2/nnα}, id eſt, proxime v = {1/307} poll. </s> + <s xml:id="echoid-s3798" xml:space="preserve">cum ſecundum <lb/>paragraphum decimum debuiſſet oriri v = {1/162} poll. </s> + <s xml:id="echoid-s3799" xml:space="preserve">& </s> + <s xml:id="echoid-s3800" xml:space="preserve">ſecundum paragraphum <lb/>quartum v = o. </s> + <s xml:id="echoid-s3801" xml:space="preserve">In eodem exemplo fit ſpatium integrum, quod ſuperficies <lb/>percurrit non omnino octo quintarum partium unius pollicis, locusque <lb/>maximæ velocitatis eſt præterpropter ſexaginta novem centeſimarum partium <lb/>ejusdem menſuræ infra altitudinem initialem.</s> + <s xml:id="echoid-s3802" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3803" xml:space="preserve">§. </s> + <s xml:id="echoid-s3804" xml:space="preserve">12. </s> + <s xml:id="echoid-s3805" xml:space="preserve">Non difficilius eſſet ad omnes vaſorum figuras extendere, quæ <lb/>hactenus dicta ſunt, imo etiam ad ſpatia finita, quibus aqua externa deter-<lb/>minetur: </s> + <s xml:id="echoid-s3806" xml:space="preserve">fiunt autem formulæ plerumque adeo prolixæ, ut conſultius du-<lb/>xerim easdem ſilentio præterire, & </s> + <s xml:id="echoid-s3807" xml:space="preserve">ſpecimine ſaltem aliquo particularem oſten-<lb/>dere modum, quo theoria ad quoslibet caſus alios eruendos applicanda ſit.</s> + <s xml:id="echoid-s3808" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3809" xml:space="preserve">Attentionem particulariorem merentur, quæ de motu aquarum in tu-<lb/>bis inferius largiter apertis, & </s> + <s xml:id="echoid-s3810" xml:space="preserve">profundiſſime ſubmerſis indicavi, quia in his <lb/>motus oſcillatorius, ut in pendulis, conſtantis durationis eſt, & </s> + <s xml:id="echoid-s3811" xml:space="preserve">undarum <lb/>in mari fluxus illuſtratur ab illis. </s> + <s xml:id="echoid-s3812" xml:space="preserve">Exiſtimavi autem prius de refluxu aquarum <lb/>in cylindris ſubmerſis generaliter tractandum eſſe, atque oſtendendum in iſta <lb/>hypotheſi refluxum non differre à præcedente fluxu, quam motus totus <lb/>oſcillatorius examinetur. </s> + <s xml:id="echoid-s3813" xml:space="preserve">Jam igitur de iſto refluxu commentabimur, dein-<lb/>ceps utrumque motum in diverſis caſibus combinaturi, ne aliquid in argu-<lb/>mento deſiderari poſſit.</s> + <s xml:id="echoid-s3814" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div150" type="section" level="1" n="118"> +<head xml:id="echoid-head153" xml:space="preserve">PARS SECUNDA.</head> +<head xml:id="echoid-head154" style="it" xml:space="preserve">De aſcenſu aquarum.</head> +<p> + <s xml:id="echoid-s3815" xml:space="preserve">§. </s> + <s xml:id="echoid-s3816" xml:space="preserve">13. </s> + <s xml:id="echoid-s3817" xml:space="preserve">Poſtquam aquæ deſcenderunt in vaſe ſubmerſo, quantum id <lb/>ipſis natura rei permittit, duo potiſſimum conſideranda ſe offerunt; </s> + <s xml:id="echoid-s3818" xml:space="preserve">primo <lb/>exceſſus altitudinis ſuperficiei externæ ſupra internam & </s> + <s xml:id="echoid-s3819" xml:space="preserve">ſecundo vis viva ſeu <lb/>productum ex aſcenſu potentiali in maſſam illius aquæ, quæ ex cylindro in aquam +<pb o="133" file="0147" n="147" rhead="SECTIO SEPTIMA."/> +circumſtagnantem durante deſcenſu ejecta fuit: </s> + <s xml:id="echoid-s3820" xml:space="preserve">hæc enim vis viva, quæ redi-<lb/>re non poteſt ad aquam in cylindro, facit potiſſimum ut aquæ multum abſint, <lb/>quo minus priſtinam, ex quâ ceciderant, in refluxu attingant altitudinem: <lb/></s> + <s xml:id="echoid-s3821" xml:space="preserve">nec tamen unica eſt hæc ratio, etiamſi vel nihil obſtent impedimenta tenaci-<lb/>tatis, adhæſionis, hujuſmodique alia: </s> + <s xml:id="echoid-s3822" xml:space="preserve">Altera ratio indicata fuit §. </s> + <s xml:id="echoid-s3823" xml:space="preserve">2. </s> + <s xml:id="echoid-s3824" xml:space="preserve">Iſtius <lb/>vero rationis menſura ex ipſo aſcenſu eſt deducenda, cum prior ad deſcenſum <lb/>pertineat & </s> + <s xml:id="echoid-s3825" xml:space="preserve">ſola, abſtrahendo animum ab impedimentis extrinſecis, in cauſa <lb/>eſt, cur non aqua in aſcenſu tantum ſupra ſuperficiem externam elevetur, <lb/>quantum infra eandem depreſſa fuerat. </s> + <s xml:id="echoid-s3826" xml:space="preserve">Notandum enim eſt, futurum fuiſſe, <lb/>aquis vel per minimum foramen influentibus, ut eadem velocitate aſcende-<lb/>rent, tanquam ſi omne fundum deeſſet, plenoque orificio irrumperent, ſimo-<lb/>do poſt influxum impetum, quem in aquas internas faciunt, totum exererent <lb/>ad earum aſcenſum promovendum: </s> + <s xml:id="echoid-s3827" xml:space="preserve">Verum quicunque hanc rem recte perpen-<lb/>dit facile videt, plerumque impetum iſtum totum fere impendi in motum ali-<lb/>quem inteſtinum, qui nihil aſcenſum promoveat; </s> + <s xml:id="echoid-s3828" xml:space="preserve">dico autem notanter ple-<lb/>rumque (quod bene notetur velim) quia cum foramen magnum admodum <lb/>eſt, non difficulter prævidetur, impetum aquarum influentium ita apte fieri, <lb/>ut motus internus haud parum inde promoveatur; </s> + <s xml:id="echoid-s3829" xml:space="preserve">at cum foramen minus eſt, <lb/>liquet, rem ſecus ſe habere. </s> + <s xml:id="echoid-s3830" xml:space="preserve">Recte igitur adhibetur hypotheſis noſtra, cum vel <lb/>fundum omne abeſt, aut fere totum eſt perforatum (ſic enim exceſſus velocita-<lb/>tis aquæ influentis ſupra velocitatem aquæ internæ nullus, aut valde exiguus eſt, <lb/>& </s> + <s xml:id="echoid-s3831" xml:space="preserve">nullum illa in hanc impetum facit) vel etiam cum foramen minimum eſt, quia <lb/>ſic omnis impetus infringitur. </s> + <s xml:id="echoid-s3832" xml:space="preserve">Sed ſi foramen rationem habuerit ad amplitudi-<lb/>nem tubi, veluti ut √ 2. </s> + <s xml:id="echoid-s3833" xml:space="preserve">ad 1, vel ut 2. </s> + <s xml:id="echoid-s3834" xml:space="preserve">ad 1, aut circiter, major paululum <lb/>erit motus quam qui ex iſta hypotheſi ſequitur, quia tunc notabilem impetum <lb/>faciunt aquæ irruentes, nec is omnis per rei naturam perditur.</s> + <s xml:id="echoid-s3835" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3836" xml:space="preserve">Facile igitur eſt ſine inſtituto calculo prævidere ſequentes in aquarum, <lb/>poſtquam ex certa altitudine delapſæ fuerunt, refluxu affectiones.</s> + <s xml:id="echoid-s3837" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3838" xml:space="preserve">I. </s> + <s xml:id="echoid-s3839" xml:space="preserve">Nullum nempe fore refluxum ſenſibilem, ſi foramen ſit valde par-<lb/>vum.</s> + <s xml:id="echoid-s3840" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3841" xml:space="preserve">II. </s> + <s xml:id="echoid-s3842" xml:space="preserve">Cum pars cylindri ſubmerſa non mutata maneat, nunquam aquas in <lb/>refluxu certum terminum prætergreſſuras, ſi vel in infinitum elevatæ fuerint <lb/>aquæ in prævio deſcenſu: </s> + <s xml:id="echoid-s3843" xml:space="preserve">nunquam enim, ex quâcunque altitudine incipiat <lb/>deſcenſus, omnes aquæ ex cylindro effluunt, ut vidimus, §. </s> + <s xml:id="echoid-s3844" xml:space="preserve">§. </s> + <s xml:id="echoid-s3845" xml:space="preserve">5. </s> + <s xml:id="echoid-s3846" xml:space="preserve">& </s> + <s xml:id="echoid-s3847" xml:space="preserve">7.</s> + <s xml:id="echoid-s3848" xml:space="preserve"/> +</p> +<pb o="134" file="0148" n="148" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s3849" xml:space="preserve">III. </s> + <s xml:id="echoid-s3850" xml:space="preserve">Cum deſcenſus incipere intelligatur ab altitudine X Y, ſubſe-<lb/>quenſque aſcenſus fieri uſque in CD, fore productum deſcenſus actualis maſſæ aquæ <lb/>X Y D C uſque ad T V in maſſam, menſuram rationis utriuſque combinatæ, <lb/>quæ, ut §. </s> + <s xml:id="echoid-s3851" xml:space="preserve">2. </s> + <s xml:id="echoid-s3852" xml:space="preserve">dictum, aſcenſum à præcedente deſcenſu differre faciunt, & </s> + <s xml:id="echoid-s3853" xml:space="preserve">cum <lb/>ratio ſecundo loco recenſita evaneſcat, ſi omne auferatur fundum IM, fore <lb/>tunc iſtud productum æquale vi vivæ omnis aquæ, durante deſcenſu ejectæ, ita <lb/>ut ſine alio calculo, præter hactenus jam poſitos, aſcenſus aquarum in cylin-<lb/>dro toto aperto definiri poſſit.</s> + <s xml:id="echoid-s3854" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3855" xml:space="preserve">IV. </s> + <s xml:id="echoid-s3856" xml:space="preserve">Aſcenſum fore æqualem deſcenſui, cum cylindrus infinite ſub-<lb/>merſus intelligitur evaneſcentibus tunc præfatis diminutionis cauſis.</s> + <s xml:id="echoid-s3857" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3858" xml:space="preserve">V. </s> + <s xml:id="echoid-s3859" xml:space="preserve">Hinc igitur oſcillationes ſine fine fore, quia poſtremæ oſcillatio-<lb/>nes ſemper ſint veluti infinite parvæ ratione ſubmerſionis altitudinum: </s> + <s xml:id="echoid-s3860" xml:space="preserve">faciunt <lb/>autem impedimenta aliena, quorum nullam hucuſque rationem habuimus, ut <lb/>omnis motus cito admodum ceſſet.</s> + <s xml:id="echoid-s3861" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3862" xml:space="preserve">§. </s> + <s xml:id="echoid-s3863" xml:space="preserve">14. </s> + <s xml:id="echoid-s3864" xml:space="preserve">His generatim præmonitis, problema accuratiori calculo ſub-<lb/>jiciemus: </s> + <s xml:id="echoid-s3865" xml:space="preserve">duplicem autem dabo ſolutionem, alteram ad principia modo ex-<lb/>poſita accommodatam, alteram ſpecie quodammodo diverſam.</s> + <s xml:id="echoid-s3866" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3867" xml:space="preserve">Igitur retentis tum figura, tum denominationibus §. </s> + <s xml:id="echoid-s3868" xml:space="preserve">3. </s> + <s xml:id="echoid-s3869" xml:space="preserve">conſiderabi-<lb/>mus aquam ex altitudine X Y deſcendiſſe uſque in x y, & </s> + <s xml:id="echoid-s3870" xml:space="preserve">ab hoc termino aſ-<lb/>cenſum ſuum inchoare; </s> + <s xml:id="echoid-s3871" xml:space="preserve">dicatur M y vel I x = α & </s> + <s xml:id="echoid-s3872" xml:space="preserve">poſtquam jam aſcendit uſ-<lb/>que ad c d vel e f, ponatur M d = ξ, df = dξ: </s> + <s xml:id="echoid-s3873" xml:space="preserve">His ita ad calculum præpa-<lb/>ratis, deſignataque rurſus per v altitudine debita velocitati aquæ in c d & </s> + <s xml:id="echoid-s3874" xml:space="preserve">per <lb/>v + d v ſimili altitudine in ſitu proximo e f, inquiremus in incrementum aſcen. <lb/></s> + <s xml:id="echoid-s3875" xml:space="preserve">ſus potentialis aquæ accedens, dum cylindrum ſubit guttula L O N P, ſuperfi-<lb/>cieſque ex c d aſcendit in e f; </s> + <s xml:id="echoid-s3876" xml:space="preserve">Perſpicuum autem eſt, cum ubique aſcenſus po-<lb/>tent. </s> + <s xml:id="echoid-s3877" xml:space="preserve">aquæ internæ multiplicatus per ſuam maſſam exprimatur per n ξ v (nec <lb/>enim ulla attentio adhibenda eſt ad motum inteſtinum) fore ejusdem produ-<lb/>cti incrementum n ξ d v + n v d ξ: </s> + <s xml:id="echoid-s3878" xml:space="preserve">Si vero præterea conſideretur aſcenſus po-<lb/>tent. </s> + <s xml:id="echoid-s3879" xml:space="preserve">n n v - v, (vid. </s> + <s xml:id="echoid-s3880" xml:space="preserve">§. </s> + <s xml:id="echoid-s3881" xml:space="preserve">2.) </s> + <s xml:id="echoid-s3882" xml:space="preserve">quem guttula influens n d ξ perdit, quique pariter <lb/>debetur deſcenſui actuali particulæ aqueæ n d ξ per altitudinem b - x, patet eſſe <lb/>ponendum <lb/>nξdv + nvdξ + (nnv - v) ndξ = (b - ξ) ndξ, vel <lb/>ξdv + nnvdξ = (b - ξ) dξ.</s> + <s xml:id="echoid-s3883" xml:space="preserve"/> +</p> +<pb o="135" file="0149" n="149" rhead="SECTIO SEPTIMA."/> +<p> + <s xml:id="echoid-s3884" xml:space="preserve">Idem vero aliter ſic invenitur.</s> + <s xml:id="echoid-s3885" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3886" xml:space="preserve">Conſideretur ſcilicet guttulæ L O N P quaſi nullam velocitatem fuiſſe, <lb/>priuſquam influere inciperet, eandem vero ſtatim atque influere incipiat, ac-<lb/>quirere aſcenſum potentialem, qui ſit = n n v, quamvis mox poſt ſui influxum <lb/>(per annot. </s> + <s xml:id="echoid-s3887" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s3888" xml:space="preserve">§. </s> + <s xml:id="echoid-s3889" xml:space="preserve">2.) </s> + <s xml:id="echoid-s3890" xml:space="preserve">cenſenda ſit motum continuare velocitate communi <lb/>√ v. </s> + <s xml:id="echoid-s3891" xml:space="preserve">Quo facto ſic erit ratiocinandum. </s> + <s xml:id="echoid-s3892" xml:space="preserve">Ante influxum guttulæ, eſt aſcenſus <lb/>potent. </s> + <s xml:id="echoid-s3893" xml:space="preserve">aquæ c d M L P I c (cujus maſſa = n ξ) = v. </s> + <s xml:id="echoid-s3894" xml:space="preserve">& </s> + <s xml:id="echoid-s3895" xml:space="preserve">aſcenſ. </s> + <s xml:id="echoid-s3896" xml:space="preserve">potent. </s> + <s xml:id="echoid-s3897" xml:space="preserve">guttulæ <lb/>L O N P (cujus maſſa = n d ξ) = o; </s> + <s xml:id="echoid-s3898" xml:space="preserve">ergo aſcenſus potentialis omnis aquæ <lb/>c d M L O N P I c = {nξv/nξ = ndξ} = {ξv/ξ + dξ}.</s> + <s xml:id="echoid-s3899" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3900" xml:space="preserve">At vero poſtquam guttula L O N P influxit ſitumque aſſumſit L on P, <lb/>eſt ejus aſcenſ. </s> + <s xml:id="echoid-s3901" xml:space="preserve">potent. </s> + <s xml:id="echoid-s3902" xml:space="preserve">= n n v, reliquæ autem aquæ e f M L o n P I e (cujus <lb/>quidem maſſa rurſus = n ξ) aſcenſus potent. </s> + <s xml:id="echoid-s3903" xml:space="preserve">eſt = v + d v; </s> + <s xml:id="echoid-s3904" xml:space="preserve">igitur aſcenſus <lb/>potent. </s> + <s xml:id="echoid-s3905" xml:space="preserve">omnis aquæ hic conſideratæ poſt influxum guttulæ eſt <lb/>= {ndξ x nnv + nξx(v + dv)/nξ + ndξ} = {ξv + ξdv + nnvdξ/ξ + dξ}, cum ante eundem influ-<lb/>xum fuerit {ξv/ξ + dξ}: </s> + <s xml:id="echoid-s3906" xml:space="preserve">cepit igitur incrementum {ξdv + nnvdξ/ξ + dξ}, vel ſimplicius <lb/>{ξdv + nnvdξ/ξ}. </s> + <s xml:id="echoid-s3907" xml:space="preserve">Iſtud vero incrementum æquandum eſt cum deſcenſu actuali <lb/>quem aqua facit mutando ſitum c d M L O N P I c ſitu e f M L O N P I e, qui <lb/>deſcenſus æqualis eſt quartæ proportionali ad maſſam aquæ internæ n ξ, ad <lb/>guttulam n d ξ & </s> + <s xml:id="echoid-s3908" xml:space="preserve">altitudinem V f vel b - ξ, ſic ut præfatus deſcenſus ſit = <lb/>{(b - ξ)dξ/ξ}: </s> + <s xml:id="echoid-s3909" xml:space="preserve">unde iterum habetur talis æquatio <lb/>ξdv + nnvdξ = (b - ξ)dξ;</s> + <s xml:id="echoid-s3910" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3911" xml:space="preserve">Hujus vero integralis poſt debitæ conſtantis additionem talis fit <lb/>v = {b/nn} (1 - ({α/ξ})<emph style="super">nn</emph>) - {1/nn + 1} (ξ - ({α/ξ})<emph style="super">nn</emph> α), <lb/>quam nunc pro diverſis ejus circumſtantiis perpendemus.</s> + <s xml:id="echoid-s3912" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3913" xml:space="preserve">§. </s> + <s xml:id="echoid-s3914" xml:space="preserve">15. </s> + <s xml:id="echoid-s3915" xml:space="preserve">Et quidem cum fuerit amplitudo tubi infinities major, quam <lb/>amplitudo foraminis; </s> + <s xml:id="echoid-s3916" xml:space="preserve">patet fieri v = {b - ξ/nn}, & </s> + <s xml:id="echoid-s3917" xml:space="preserve">irruere proinde aquam velo-<lb/>citate quæ debeatur altitudini ſuperficiei externæ fuper internam, neque <lb/>tunc ultra ſuperficiem aquæ externæ fiet aſcenſus.</s> + <s xml:id="echoid-s3918" xml:space="preserve"/> +</p> +<pb o="136" file="0150" n="150" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s3919" xml:space="preserve">Cum vero amplitudo foraminis rationem habet finitam ad amplitudi-<lb/>nem tubi, aſcenſus fit ultra ſuperficiem R S veluti usque in s t: </s> + <s xml:id="echoid-s3920" xml:space="preserve">minor au-<lb/>tem ſemper erit Vt quam Vy, niſi cum omne fundum abeſt, tunc enim <lb/>erit V t = V y. </s> + <s xml:id="echoid-s3921" xml:space="preserve">Prouti monuimus §. </s> + <s xml:id="echoid-s3922" xml:space="preserve">5. </s> + <s xml:id="echoid-s3923" xml:space="preserve">in deſcenſu differentiam inter V Y & </s> + <s xml:id="echoid-s3924" xml:space="preserve"><lb/>V y, proportionalem eſſe & </s> + <s xml:id="echoid-s3925" xml:space="preserve">originem debere aſcenſui potentiali aquæ durante <lb/>deſcenſu ejectæ, ita nunc obſervari poteſt in aſcenſu differentiam inter V y <lb/>& </s> + <s xml:id="echoid-s3926" xml:space="preserve">V t originem habere ab illiſione guttularum L o n P in maſſam aquæ ſu-<lb/>perjacentis, quæ quidem illiſio non promovet aſcenſum, ſed in inutilem mo-<lb/>tum inteſtinum impenditur, prouti indicatum fuit §. </s> + <s xml:id="echoid-s3927" xml:space="preserve">2. </s> + <s xml:id="echoid-s3928" xml:space="preserve">Ergo cum omne <lb/>fundum I M abeſt, aqua tubum eadem velocitate ingreditur, qua jam gau-<lb/>det aqua tubum antea ingreſſa & </s> + <s xml:id="echoid-s3929" xml:space="preserve">nulla fit colliſio, quæ cauſa eſt cur in iſto <lb/>caſu tantum aſcendat aqua ultra ſuperficiem R S, quantum fuerat infra il-<lb/>lam depreſſa, quod æquatio, uti mox videbimus, indicat.</s> + <s xml:id="echoid-s3930" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3931" xml:space="preserve">§. </s> + <s xml:id="echoid-s3932" xml:space="preserve">16. </s> + <s xml:id="echoid-s3933" xml:space="preserve">Determinabitur maximus aſcenſus s t, faciendo v = o. </s> + <s xml:id="echoid-s3934" xml:space="preserve">Igitur <lb/>ut motus omnis recte definiatur, alternatim adhibendæ erunt formulæ §. </s> + <s xml:id="echoid-s3935" xml:space="preserve">§. </s> + <s xml:id="echoid-s3936" xml:space="preserve">3. <lb/></s> + <s xml:id="echoid-s3937" xml:space="preserve">& </s> + <s xml:id="echoid-s3938" xml:space="preserve">14. </s> + <s xml:id="echoid-s3939" xml:space="preserve">erutæ, quod nunc hoc unico illuſtrabo exemplo, quo nn = 1.</s> + <s xml:id="echoid-s3940" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3941" xml:space="preserve">Si proinde nn = 1, fit v = b (1 - {α/ξ} - {1/2} (ξ - {αα/ξ}): </s> + <s xml:id="echoid-s3942" xml:space="preserve">eritque <lb/>v = o, cum ſumitur ξ = 2b - α, id eſt, cum ſumitur V t = V y. </s> + <s xml:id="echoid-s3943" xml:space="preserve">Igi-<lb/>tur ſi verbi gratia tubus A B M I aqua plenus, omnique fundo deſtitutus fue-<lb/>rit ad medietatem usque immerſus aquæ exteriori, atque tota ipſius longi-<lb/>tudo dicatur a, aqua ſic agitabitur ut primo infra T V deſcendat, ſpatio <lb/>o, 297a, deinde ſimili ſpatio ſuper eandem T V elevetur, rurſusque infra eam <lb/>deprimatur ſpatio o, 240a, eodemque lineam illam iterum tranſcendat, & </s> + <s xml:id="echoid-s3944" xml:space="preserve"><lb/>ſic porro.</s> + <s xml:id="echoid-s3945" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3946" xml:space="preserve">§. </s> + <s xml:id="echoid-s3947" xml:space="preserve">17. </s> + <s xml:id="echoid-s3948" xml:space="preserve">Patet etiam cum α eſt = o, tubo ſcilicet ab omni aqua va-<lb/>cuo, fore generaliter v = {b/nn} - {ξ/nn + 1}: </s> + <s xml:id="echoid-s3949" xml:space="preserve">aſcenſumquè integrum conſequen-<lb/>ter fore {nn + 1/nn}b vel aſcenſum ſupra ſuperficiem exteriorem aquæ = {b/nn}.</s> + <s xml:id="echoid-s3950" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3951" xml:space="preserve">§. </s> + <s xml:id="echoid-s3952" xml:space="preserve">18. </s> + <s xml:id="echoid-s3953" xml:space="preserve">Venio nunc ad tubos infinite ſubmerſos, in quibus deſcenſum <lb/>cum ſuis affectionibus determinavimus §. </s> + <s xml:id="echoid-s3954" xml:space="preserve">10. </s> + <s xml:id="echoid-s3955" xml:space="preserve">Utemur autem eadem plane <lb/>methodo ad hunc caſum definiendum quâ ibi uſi ſumus: </s> + <s xml:id="echoid-s3956" xml:space="preserve">erit nobis igitur <lb/>depreſſio initialis V y(= b - α) = c, aſcenſus inde factus y d (= ξ - α) = z.</s> + <s xml:id="echoid-s3957" xml:space="preserve"> +<pb o="137" file="0151" n="151" rhead="SECTIO SEPTIMA."/> +Sic eſt ξ = α + z & </s> + <s xml:id="echoid-s3958" xml:space="preserve">b = α + c, ubi quantitates z & </s> + <s xml:id="echoid-s3959" xml:space="preserve">c ſunt ceu infinite par-<lb/>væ conſiderandæ ratione quantitatis α. </s> + <s xml:id="echoid-s3960" xml:space="preserve">Habetur hinc <lb/>({α/ξ})<emph style="super">nn</emph> = ({α/α + z})<emph style="super">nn</emph> = (1 + {z/α})<emph style="super">-nn</emph> = adhibendo ſeriem notam <lb/>& </s> + <s xml:id="echoid-s3961" xml:space="preserve">ex illa ſumendo tres primos terminos 1 - {nnz/α} + {nn.</s> + <s xml:id="echoid-s3962" xml:space="preserve"><emph style="ol">nn + 1</emph>zz/2αα}. </s> + <s xml:id="echoid-s3963" xml:space="preserve">Subſtitu-<lb/>tis iſtis valoribus pro b, ξ & </s> + <s xml:id="echoid-s3964" xml:space="preserve">({α/ξ})<emph style="super">nn</emph> mutatur æquatio ultima paragraphi de-<lb/>cimi quarti in hanc, v = {α + c/nn} X ({nnz/α} - {nn x <emph style="ol">nn + 1</emph>zz/2αα}) -<lb/>{1/nn + 1} X (α + z - α + nnz - {nn.</s> + <s xml:id="echoid-s3965" xml:space="preserve"><emph style="ol">nn + 1</emph>zz/2α}) = <lb/>(α + c) X ({z/α} - {<emph style="ol">nn + 1</emph>zz/2αα}) - (z - {nnzz/2α}) = <lb/>{cz/α} - {zz/2α} - {<emph style="ol">nn + 1</emph>czz/2αα}: </s> + <s xml:id="echoid-s3966" xml:space="preserve">Poteſt autem negligi iſte ultimus terminus & </s> + <s xml:id="echoid-s3967" xml:space="preserve">ſic <lb/>fit ſimpliciter <lb/>v = {2cz - zz/2α}, <lb/>quam æquationem n non amplius ingreditur: </s> + <s xml:id="echoid-s3968" xml:space="preserve">Neque illa differt ab æquatio-<lb/>ne pro deſcenſu §. </s> + <s xml:id="echoid-s3969" xml:space="preserve">10. </s> + <s xml:id="echoid-s3970" xml:space="preserve">data, nempe v = {2cz - zz/2a}, quandoquidem quan-<lb/>titas a & </s> + <s xml:id="echoid-s3971" xml:space="preserve">α non differunt niſi quantitate minima 2 c.</s> + <s xml:id="echoid-s3972" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3973" xml:space="preserve">Cæterum hic omnia etiam ſunt ſubintelligenda, quæ eodem §. </s> + <s xml:id="echoid-s3974" xml:space="preserve">10. </s> + <s xml:id="echoid-s3975" xml:space="preserve">de <lb/>tubo non nimis obſtruendo dicta ſunt.</s> + <s xml:id="echoid-s3976" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3977" xml:space="preserve">§. </s> + <s xml:id="echoid-s3978" xml:space="preserve">19. </s> + <s xml:id="echoid-s3979" xml:space="preserve">Sunt igitur deſcenſus & </s> + <s xml:id="echoid-s3980" xml:space="preserve">aſcenſus ſibi æquales; </s> + <s xml:id="echoid-s3981" xml:space="preserve">nam ex æquatio-<lb/>nibus noſtris patet, liquorem æqualiter librari ultra ſuperficiem aquæ externæ. <lb/></s> + <s xml:id="echoid-s3982" xml:space="preserve">Deinde vero potiſſimum ſequitur ex iſtis formulis, eſſe vel oſcillationes inæqua-<lb/>les inter ſe iſochronas, modo omnes poſſint infinite parvæ cenſeri ratione ſub-<lb/>merſionis: </s> + <s xml:id="echoid-s3983" xml:space="preserve">Pendulum autem ſimplex tautochronum eſſe ejuſdem longitudinis <lb/>cum parte tubi ſubmerſa.</s> + <s xml:id="echoid-s3984" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3985" xml:space="preserve">Differt iſtud theorema ab illo, quod §. </s> + <s xml:id="echoid-s3986" xml:space="preserve">4. </s> + <s xml:id="echoid-s3987" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s3988" xml:space="preserve">6. </s> + <s xml:id="echoid-s3989" xml:space="preserve">de oſcillationibus in <lb/>tubo cylindrico ex duobus cruribus verticalibus compoſito citatum fuit, in eo, <lb/>quod ibi oſcillationes omnes non excluſis oſcillationibus finitæ magnitudinis <lb/>ſint tautochronæ, cum@in præſenti caſu oſcillationes finitæ ſint inæqualis dura- +<pb o="138" file="0152" n="152" rhead="HYDRODYNAMICÆ"/> +tionis; </s> + <s xml:id="echoid-s3990" xml:space="preserve">deinde quod ibi longitudo penduli ſit æqualis dimidiæ longitudini tubi, <lb/>cum hîc ſit æqualis integræ, quamvis ſi recte res perpendatur, hic potius ſit con-<lb/>ſenſus quam diſſenſus dicendus ob tubi, quæ in priori caſu eſt, duplicationem.</s> + <s xml:id="echoid-s3991" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s3992" xml:space="preserve">§. </s> + <s xml:id="echoid-s3993" xml:space="preserve">20. </s> + <s xml:id="echoid-s3994" xml:space="preserve">Utroque oſcillationum genere illuſtratur natura undarum ven-<lb/>to agitatarum: </s> + <s xml:id="echoid-s3995" xml:space="preserve">neque enim aliter moventur, quam quod aquæ in illis conti-<lb/>nue aſcendant rurſuſque deſcendant. </s> + <s xml:id="echoid-s3996" xml:space="preserve">Ita patet quod dicit Newtonus, tem-<lb/>pora undulationum eſſe in ratione dimidiata latitudinum undarum, quia ponit <lb/>undarum formam ſibi conſtanter eſſe ſimilem & </s> + <s xml:id="echoid-s3997" xml:space="preserve">proinde earum latitudinem <lb/>proportionalem profunditati, ad quam aquæ agitantur. </s> + <s xml:id="echoid-s3998" xml:space="preserve">Veriſimile autem eſt <lb/>profunditatem eam eſſe, quæ pendulo ſimplici cum undis tautochrono, nempe <lb/>v.</s> + <s xml:id="echoid-s3999" xml:space="preserve">gr. </s> + <s xml:id="echoid-s4000" xml:space="preserve">60 {1/3} ped. </s> + <s xml:id="echoid-s4001" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s4002" xml:space="preserve">ſi ſingulis binis ſecundis fiat undarum aſcenſus deſcenſuſve.</s> + <s xml:id="echoid-s4003" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4004" xml:space="preserve">§. </s> + <s xml:id="echoid-s4005" xml:space="preserve">21. </s> + <s xml:id="echoid-s4006" xml:space="preserve">Quamvis noluerim ad prolixitatem calculi evitandam, hoc ar-<lb/>gumentum in omni ſua extenſione proſequi, propterque ea de cylindricis va-<lb/>ſis tantum egerim, attamen quia in caſu ſubmerſionis infinitæ, enunciationes <lb/>& </s> + <s xml:id="echoid-s4007" xml:space="preserve">theoremata parum de ſua concinnitate perdunt, ſuperaddam theorema ge-<lb/>nerale pro oſcillationibus aquæ in tubo utcunque inæquali, omiſſa tamen de-<lb/>monſtratione, quæ ex alibi dictis unicuique obvia erit, præſertim vero ex iis <lb/>quæ in Sect. </s> + <s xml:id="echoid-s4008" xml:space="preserve">6. </s> + <s xml:id="echoid-s4009" xml:space="preserve">§. </s> + <s xml:id="echoid-s4010" xml:space="preserve">§. </s> + <s xml:id="echoid-s4011" xml:space="preserve">6. </s> + <s xml:id="echoid-s4012" xml:space="preserve">7. </s> + <s xml:id="echoid-s4013" xml:space="preserve">& </s> + <s xml:id="echoid-s4014" xml:space="preserve">ſeqq. </s> + <s xml:id="echoid-s4015" xml:space="preserve">uſque ad 20. </s> + <s xml:id="echoid-s4016" xml:space="preserve">expoſita fuerunt. </s> + <s xml:id="echoid-s4017" xml:space="preserve">Faciendum au-<lb/>tem eſt, ut cylindricæ ſit ſtructuræ pars illa vaſis ſuperior, in quâ excurſiones <lb/>fiunt.</s> + <s xml:id="echoid-s4018" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4019" xml:space="preserve">§. </s> + <s xml:id="echoid-s4020" xml:space="preserve">22. </s> + <s xml:id="echoid-s4021" xml:space="preserve">Fuerit igitur bd longitudo vaſis ſubmerſi (Fig. </s> + <s xml:id="echoid-s4022" xml:space="preserve">35. </s> + <s xml:id="echoid-s4023" xml:space="preserve">b) Repræſentet <lb/>b F ejus amplitudinem in loco ſuperficiei, ponaturque vas ita formatum, ut ſit <lb/>curva FGH ſcala amplitudinum: </s> + <s xml:id="echoid-s4024" xml:space="preserve">ſumatur linea b c fiatque curva L M N, <lb/>cujus applicata c M ſit ubique = {bF<emph style="super">2</emph>/cG}, & </s> + <s xml:id="echoid-s4025" xml:space="preserve">erit longitudo penduli iſochro-<lb/>ni cum oſcillationibus aqueæ ſuperficiei = ſpatio bd NL diviſo per b L.</s> + <s xml:id="echoid-s4026" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div151" type="section" level="1" n="119"> +<head xml:id="echoid-head155" xml:space="preserve">Corollarium.</head> +<p> + <s xml:id="echoid-s4027" xml:space="preserve">§. </s> + <s xml:id="echoid-s4028" xml:space="preserve">23. </s> + <s xml:id="echoid-s4029" xml:space="preserve">Ex præcedente paragrapho ſequitur, ſi tubus ſubmerſus coni-<lb/>cus fuerit, habeatque amplitudinem in regione aquæ ſuperficiei, quæ ſit ad <lb/>orificium ſubmerſum ut m ad n, fore longitudinem penduli Iſochroni cum <lb/>vibrante aqua ad longitudinem ſubmerſi tubi, ut √m ad √n, id eſt, ut ra-<lb/>dices prædictarum amplitudinum, atque ſi tubus idem ſitu, modo recto mo- +<pb o="139" file="0153" n="153" rhead="SECTIO SEPTIMA."/> +do inverſo, ſubmergatur tantum non totus, fore longitudines pendulorum <lb/>iſochronorum in ratione contraria orificiorum ſubmerſorum.</s> + <s xml:id="echoid-s4030" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div152" type="section" level="1" n="120"> +<head xml:id="echoid-head156" xml:space="preserve">Scholium Generale.</head> +<p> + <s xml:id="echoid-s4031" xml:space="preserve">§. </s> + <s xml:id="echoid-s4032" xml:space="preserve">24. </s> + <s xml:id="echoid-s4033" xml:space="preserve">Quæ in hac ſectione continentur, quia novis hypotheſibus inni-<lb/>tuntur pleraque, eo magis operæ pretium erit experimentis tentare. </s> + <s xml:id="echoid-s4034" xml:space="preserve">Ego <lb/>quidem diverſa inſtitui, non vacavit autem ſingula quæ mente conceperam <lb/>exequi: </s> + <s xml:id="echoid-s4035" xml:space="preserve">quæ feci inferius recenſebo; </s> + <s xml:id="echoid-s4036" xml:space="preserve">Interim ut tutius judicium ferri poſſit <lb/>de conſenſu experimentorum cum theoria, diſpiciendum prius erit pro re-<lb/>rum circumſtantiis, an & </s> + <s xml:id="echoid-s4037" xml:space="preserve">quantum fere contractio venæ effluentis (cujus <lb/>naturam expoſui in ſect. </s> + <s xml:id="echoid-s4038" xml:space="preserve">4.) </s> + <s xml:id="echoid-s4039" xml:space="preserve">calculum turbare poſſit: </s> + <s xml:id="echoid-s4040" xml:space="preserve">quod incommodum <lb/>maxima parte tolli poterit, ſi fiat ut orificii inferioris latera parvulum ali-<lb/>quem cylindrum efforment, vix dimidiæ lineæ altitudinis, qua de re animo <lb/>revolvatur experimentum quartum ad ſectionem quartam pertinens. </s> + <s xml:id="echoid-s4041" xml:space="preserve">Deinde <lb/>etiam animus advertendus ad reſiſtentias ab adhæſione aquæ oriundas, quæ <lb/>quidem parum retardant motus, ſitempora oſcillationum reſpicias, multum <lb/>autem excurſionibus detrahunt, præſertim ſi tubi ſtrictiores & </s> + <s xml:id="echoid-s4042" xml:space="preserve">longiores ſu-<lb/>mantur. </s> + <s xml:id="echoid-s4043" xml:space="preserve">Igitur magis fidendum erit experimentis, quæ circa oſcillationum <lb/>tempora facta fuerint, quia hæc tempora à diminutione excurſionum non <lb/>multum admodum alterantur. </s> + <s xml:id="echoid-s4044" xml:space="preserve">Ratione primi experimentorum generis, quo <lb/>excurſiones fluidorum in tubis, tam deſcenſus quam aſcenſus inquirendi ob-<lb/>ſervandique veniunt, hâc uſus fui circumſpectione, ut filum tubo circumvol-<lb/>verem eo in loco, ad quem aquas deſcenſuras vel aſcenſuras eſſe expectabam, <lb/>idemque filum poſt ſæpe repetitum experimentum ita tandem locavi, ut ſu-<lb/>perficies fluidi oſcillantis nec ultra nec citra excurreret. </s> + <s xml:id="echoid-s4045" xml:space="preserve">Reliqua etiam loca, <lb/>quæ in tubo obſervanda erant, pariter filo circumvoluto notavi. </s> + <s xml:id="echoid-s4046" xml:space="preserve">Quod deinde <lb/>ad tempora oſcillationum pertinet, quia hæ citiſſime decreſcunt fiuntque im-<lb/>perceptibiles & </s> + <s xml:id="echoid-s4047" xml:space="preserve">plane nullæ, non potui illa aliter inquirere, quam exploran-<lb/>do poſt ſæpiſſime iteratum experimentum longitudinem penduli ſimplicis iſo-<lb/>chroni, quod dum oſcillabat digitum orificio tubi ſuperimpoſui eumque eo <lb/>præciſe temporis puncto removi, ut & </s> + <s xml:id="echoid-s4048" xml:space="preserve">pendulum & </s> + <s xml:id="echoid-s4049" xml:space="preserve">fluidum oſcillationem ſi-<lb/>mul inciperent.</s> + <s xml:id="echoid-s4050" xml:space="preserve"/> +</p> +<pb o="140" file="0154" n="154" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div153" type="section" level="1" n="121"> +<head xml:id="echoid-head157" xml:space="preserve">EXPERIMENTA</head> +<head xml:id="echoid-head158" xml:space="preserve">Ad ſect. ſept. referenda.</head> +<head xml:id="echoid-head159" xml:space="preserve">Experimentum 1.</head> +<p> + <s xml:id="echoid-s4051" xml:space="preserve">TUbum adhibui vitreum cylindricum diametri fere quatuor linearum, <lb/>inferius totum apertum. </s> + <s xml:id="echoid-s4052" xml:space="preserve">Eum aquæ, in vaſe pellucido ampliſſimo <lb/>ſtagnanti, ſubmerſi ad altitudinem 44. </s> + <s xml:id="echoid-s4053" xml:space="preserve">lin. </s> + <s xml:id="echoid-s4054" xml:space="preserve">digitumque orificio admo-<lb/>vi ſuperno, ne extrahendo tubi partem deſcenderet in illo aqua: </s> + <s xml:id="echoid-s4055" xml:space="preserve">extraxi <lb/>deinceps tubum ad alt. </s> + <s xml:id="echoid-s4056" xml:space="preserve">22. </s> + <s xml:id="echoid-s4057" xml:space="preserve">lin. </s> + <s xml:id="echoid-s4058" xml:space="preserve">ita ut tam pars tubi ſubmerſa, quam altitudo <lb/>aquæ internæ@ſupra externam eſſet 22. </s> + <s xml:id="echoid-s4059" xml:space="preserve">lin. </s> + <s xml:id="echoid-s4060" xml:space="preserve">moxque remoto digito obſervavi <lb/>deſcenſum ſuperficiei in tubo infra ſuperficiem aquæ ſtagnantis eumque vidi <lb/>fuiſſe 9 {1/2} lin,</s> +</p> +<p> + <s xml:id="echoid-s4061" xml:space="preserve">Debuiſſet autem vi §. </s> + <s xml:id="echoid-s4062" xml:space="preserve">§. </s> + <s xml:id="echoid-s4063" xml:space="preserve">7. </s> + <s xml:id="echoid-s4064" xml:space="preserve">& </s> + <s xml:id="echoid-s4065" xml:space="preserve">17. </s> + <s xml:id="echoid-s4066" xml:space="preserve">deſcendere tredecim lineis; </s> + <s xml:id="echoid-s4067" xml:space="preserve">Defectus <lb/>trium linearum cum dimidia unice fere adhæſioni aquæ ad latera tubi tribuen-<lb/>dus videtur.</s> + <s xml:id="echoid-s4068" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4069" xml:space="preserve">Obſervato deſcenſu totum experimentum repetii, ut aſcenſum quoque <lb/>proximum experirer: </s> + <s xml:id="echoid-s4070" xml:space="preserve">Viſus autem mihi fuit 8. </s> + <s xml:id="echoid-s4071" xml:space="preserve">lin. </s> + <s xml:id="echoid-s4072" xml:space="preserve">qui vi paragraphi deci-<lb/>mi ſexti, habito reſpectu ad prævium deſcenſum, eſſe debuerat 9 {1/2} lin. <lb/></s> + <s xml:id="echoid-s4073" xml:space="preserve">nempe tantus, quantus fuit præcedens deſcenſus. </s> + <s xml:id="echoid-s4074" xml:space="preserve">Hic vero experimentum <lb/>unica tantum linea cum dimidia defecit, cum in prima experimenti parte ad <lb/>tres uſque lineas cum dimidia defectus adfuit, quia nimirum major ibi facta <lb/>fuit excurſio eaque velocitate majori, ita ut impedimenta, quæ una cum velo-<lb/>citatibus creſcunt, admodum majora offenderit.</s> + <s xml:id="echoid-s4075" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div154" type="section" level="1" n="122"> +<head xml:id="echoid-head160" xml:space="preserve">Experimentum 2.</head> +<p> + <s xml:id="echoid-s4076" xml:space="preserve">Eodem tubo uſus ſum, ſed eo lamina munito, quæ foramine erat per-<lb/>tuſa amplitudine √ {1/2} ratione amplitudinis tubi, cum ſuperficies tubi eſſet <lb/>octodecim lineis elevata ſupra aquam ſtagnantem, totidemque lineis fundum <lb/>ſubmerſum, vidi ſuperficiem tubi in deſcenſu quinque fere lineis infra aquam <lb/>ſtagnantem deſcendiſſe.</s> + <s xml:id="echoid-s4077" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4078" xml:space="preserve">Paragraphus octavus autem deſcenſum arguit 7 {1/2} lin. </s> + <s xml:id="echoid-s4079" xml:space="preserve">defectum, qui <lb/>plusquam 2 {1/2} lin. </s> + <s xml:id="echoid-s4080" xml:space="preserve">fuit, rurſus adhæſioni aquæ ad latera tubi adſcribo.</s> + <s xml:id="echoid-s4081" xml:space="preserve"/> +</p> +<pb o="141" file="0155" n="155" rhead="SECTIO SEPTIMA."/> +<p> + <s xml:id="echoid-s4082" xml:space="preserve">Deinde tubum hunc eadem lamina inſtructum admoto ſuperius digi-<lb/>to aquæ immiſi@ad profunditatem 18. </s> + <s xml:id="echoid-s4083" xml:space="preserve">lin. </s> + <s xml:id="echoid-s4084" xml:space="preserve">totum ab aquâ vacuum: </s> + <s xml:id="echoid-s4085" xml:space="preserve">remoto <lb/>digito emerſit ſuperficies tubi ſupra aquam ſtagnantem integris octo lineis, <lb/>cum §. </s> + <s xml:id="echoid-s4086" xml:space="preserve">17. </s> + <s xml:id="echoid-s4087" xml:space="preserve">earum novem indicat pro iſto caſu.</s> + <s xml:id="echoid-s4088" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4089" xml:space="preserve">Quod hic defectus minor admodum fuerit, quam in deſcenſu, ratio-<lb/>ni adſcripſi, quam prolixe paragrapho decimo tertio indicavi, cum dicerem <lb/>motum paullo majorem oriturum, cum foramen amplitudinem reſpectu tu-<lb/>bi notabilem veluti in ratione √ {1/2} ad 1, aut circiter habuerit, quam qui ex <lb/>hypotheſi ſequitur: </s> + <s xml:id="echoid-s4090" xml:space="preserve">atque ut ea de re certus plane fierem, tubum adhibui <lb/>breviorem & </s> + <s xml:id="echoid-s4091" xml:space="preserve">ampliorem, ut omnis fere impedimentis alienis effectus præri-<lb/>peretur, & </s> + <s xml:id="echoid-s4092" xml:space="preserve">experimentum cepi, quod ſequitur.</s> + <s xml:id="echoid-s4093" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div155" type="section" level="1" n="123"> +<head xml:id="echoid-head161" xml:space="preserve">Experimentum 3.</head> +<p> + <s xml:id="echoid-s4094" xml:space="preserve">Tubum adhibui cujus diameter erat plus quam ſeptem linearum, quem <lb/>ex ferro confieri curavi, quia vitreus bene cylindricus non fuit ad manus: <lb/></s> + <s xml:id="echoid-s4095" xml:space="preserve">longitudo ejus fuit quatuor pollicum cum ſex lineis & </s> + <s xml:id="echoid-s4096" xml:space="preserve">ſemiſſe: </s> + <s xml:id="echoid-s4097" xml:space="preserve">amplitudo <lb/>ejus ratione foraminis indicata per n fuit = 1, 860 & </s> + <s xml:id="echoid-s4098" xml:space="preserve">nn = 3, 4 5 8.</s> + <s xml:id="echoid-s4099" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div156" type="section" level="1" n="124"> +<head xml:id="echoid-head162" xml:space="preserve">De iſto tubo experimentum ita ſumſi:</head> +<p> + <s xml:id="echoid-s4100" xml:space="preserve">Obturato ſcilicet orificio ſuperiori identidem tentavi, ad quam pro-<lb/>funditatem ſubmergendus eſſet aquæ in arca ampliſſima ſtagnanti, ut re-<lb/>moto protinus digito, qui orificium obtegebat, aqua ad limbum ejus-<lb/>dem orificii præciſe aſcenderet, nihilque præterflueret. </s> + <s xml:id="echoid-s4101" xml:space="preserve">Iſtam vero pro-<lb/>funditatem expertus ſum 3. </s> + <s xml:id="echoid-s4102" xml:space="preserve">poll. </s> + <s xml:id="echoid-s4103" xml:space="preserve">cum tribus lineis; </s> + <s xml:id="echoid-s4104" xml:space="preserve">fuit igitur aſcenſus ſupra <lb/>aquam externam unius pollicis & </s> + <s xml:id="echoid-s4105" xml:space="preserve">trium linearum cum dimidia, cum vel <lb/>omnibus remotis impedimentis parum ultra undecim lineas aſcenſus fieri <lb/>debuerit vi paragraphi 17. </s> + <s xml:id="echoid-s4106" xml:space="preserve">Recte igitur præmonitum fuit §. </s> + <s xml:id="echoid-s4107" xml:space="preserve">13. </s> + <s xml:id="echoid-s4108" xml:space="preserve">non poſſe <lb/>non aſcenſus fieri paullo majores in iſtiusmodi caſibus, quam hypotheſis <lb/>poſtulat. </s> + <s xml:id="echoid-s4109" xml:space="preserve">Mox eidem tubo aliud applicui fundum; </s> + <s xml:id="echoid-s4110" xml:space="preserve">erat jam n = 3, 68, <lb/>& </s> + <s xml:id="echoid-s4111" xml:space="preserve">nn = 13, 54: </s> + <s xml:id="echoid-s4112" xml:space="preserve">difficile fuit experimenti ſucceſſum recte dignoſcere, quia <lb/>ſuperficies in tubo aſcendens ſemper fuit bullata: </s> + <s xml:id="echoid-s4113" xml:space="preserve">viſum tamen fuit, tubum <lb/>nunc immergendum fuiſſe ad altitudinem 4. </s> + <s xml:id="echoid-s4114" xml:space="preserve">poll. </s> + <s xml:id="echoid-s4115" xml:space="preserve">cum duabus tribuſue lineis, <lb/>manentibus ſic extra aquam præterpropter quatuor lineis, prorſus ut theo-<lb/>ria indicat.</s> + <s xml:id="echoid-s4116" xml:space="preserve"/> +</p> +<pb o="142" file="0156" n="156" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div157" type="section" level="1" n="125"> +<head xml:id="echoid-head163" xml:space="preserve">Experimentum 4.</head> +<p> + <s xml:id="echoid-s4117" xml:space="preserve">Tubum cylindricum vitreum, qui tres præterpropter lineas habebat <lb/>in diametro immerſi ad altitudinem 20. </s> + <s xml:id="echoid-s4118" xml:space="preserve">poll. </s> + <s xml:id="echoid-s4119" xml:space="preserve">fecique, ut aqua in illo libraretur, <lb/>elevata prius aquâ ad altitudinem unius fere pollicis. </s> + <s xml:id="echoid-s4120" xml:space="preserve">Ultra quatuor vel <lb/>quinque itus reditusque bene notabiles non fecit, nec adeoque omni rigore <lb/>longitudinem penduli ſimplicis iſochroni examinare potui; </s> + <s xml:id="echoid-s4121" xml:space="preserve">mihi tamen illa <lb/>viſa fuit 22. </s> + <s xml:id="echoid-s4122" xml:space="preserve">aut 23. </s> + <s xml:id="echoid-s4123" xml:space="preserve">pollicum; </s> + <s xml:id="echoid-s4124" xml:space="preserve">ex quo intuli adhæſionem aquæ ad latera tu-<lb/>bi non ſolum diminuere excurſiones, ſed & </s> + <s xml:id="echoid-s4125" xml:space="preserve">morari pauliſper tempora <lb/>oſcillationum: </s> + <s xml:id="echoid-s4126" xml:space="preserve">debuiſſet enim ſecundum § 19. </s> + <s xml:id="echoid-s4127" xml:space="preserve">eſſe præfata longitudo vi-<lb/>ginti tantummodo pollicum. </s> + <s xml:id="echoid-s4128" xml:space="preserve">Idem expertus ſum in oſcillationibus, quas <lb/>in ſuperiori ſectione pertractavimus.</s> + <s xml:id="echoid-s4129" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4130" xml:space="preserve">Cæterum obturato vel ad dimidium fere orificio inferiori, obſervare <lb/>non potui, excurſiones inde fuiſſe diminutas aut oſcillationes retardatas, <lb/>quod conforme eſt cum iis, quæ §. </s> + <s xml:id="echoid-s4131" xml:space="preserve">§. </s> + <s xml:id="echoid-s4132" xml:space="preserve">7. </s> + <s xml:id="echoid-s4133" xml:space="preserve">& </s> + <s xml:id="echoid-s4134" xml:space="preserve">18. </s> + <s xml:id="echoid-s4135" xml:space="preserve">habentur.</s> + <s xml:id="echoid-s4136" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div158" type="section" level="1" n="126"> +<head xml:id="echoid-head164" xml:space="preserve">Experimentum 5.</head> +<p> + <s xml:id="echoid-s4137" xml:space="preserve">Tubum conicum longitudine 21. </s> + <s xml:id="echoid-s4138" xml:space="preserve">poll. </s> + <s xml:id="echoid-s4139" xml:space="preserve">immerſi aquæ orificio ampliore, <lb/>ita ut unicus pollex extra aquam emineret: </s> + <s xml:id="echoid-s4140" xml:space="preserve">fuit autem alterum orificium al-<lb/>terius paululum plusquam duplum. </s> + <s xml:id="echoid-s4141" xml:space="preserve">Longitudinem penduli iſochroni cum <lb/>vibrationibus aquæ in tubo libratæ inveni quindecim poll. </s> + <s xml:id="echoid-s4142" xml:space="preserve">debuiſſet autem ſe-<lb/>cundum §. </s> + <s xml:id="echoid-s4143" xml:space="preserve">23. </s> + <s xml:id="echoid-s4144" xml:space="preserve">eſſe eadem longitudo paullo minor quatuordecim pollicibus. <lb/></s> + <s xml:id="echoid-s4145" xml:space="preserve">Denique ſimiliter eodem tubo uſus, ſed ſitu inverſo, deprehendi longitudi-<lb/>nem penduli iſochroni tantillo plusquam duplam ejus, quæ antea fuerat, <lb/>prouti citato paragrapho indicatur.</s> + <s xml:id="echoid-s4146" xml:space="preserve"/> +</p> +<pb file="0157" n="157" rhead="(143)"/> +</div> +<div xml:id="echoid-div159" type="section" level="1" n="127"> +<head xml:id="echoid-head165" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph></head> +<head xml:id="echoid-head166" xml:space="preserve">SECTIO OCTAVA.</head> +<head xml:id="echoid-head167" style="it" xml:space="preserve">De motu fluidorum cum homogeneorum tum hetero-<lb/>geneorum per vaſa irregularis & præruptæ ſtru-<lb/>cturæ, ubi ex theoria virium vivarum, quarum pars <lb/>continue abſorbeatur, explicantur præcipue Phæno-<lb/>mena ſingularia fluidorum, per plurima foramina trajecto-<lb/>rum, præmiſsis regulis generalibus pro motibus fluido-<lb/>rum ubique definiendis.</head> +<head xml:id="echoid-head168" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s4147" xml:space="preserve">ALiis adhuc principiis præter quam in ſectione proxime præceden-<lb/>te uſi non ſumus, quam hiſce duobus quod velocitates fluidorum <lb/>ſint ubique reciproce proportionales amplitudinibus vaſorum, cujus <lb/>ope invenitur aſcenſus potentialis totius aquæ ex dato aſcenſu po-<lb/>tentiali cujusvis particulæ; </s> + <s xml:id="echoid-s4148" xml:space="preserve">tum quod aſcenſus pot. </s> + <s xml:id="echoid-s4149" xml:space="preserve">totius aquæ perpetuo æqua-<lb/>lis maneat deſcenſui actuali. </s> + <s xml:id="echoid-s4150" xml:space="preserve">Quoties ambo hæc principia locum habent, mi-<lb/>nime dubitandum eſt, quin methodo à nobis adhibita motus fluidorum <lb/>recte definiatur. </s> + <s xml:id="echoid-s4151" xml:space="preserve">Non diffitebor tamen, hujusmodi fieri poſſe ſtructuræ va-<lb/>ſa, in quibus fluida moventur, ut neutrum iſtorum principiorum recte pro-<lb/>cedat. </s> + <s xml:id="echoid-s4152" xml:space="preserve">Prius equidem raro aut nunquam notabiliter à vero abducit, quia <lb/>ubicunque locum non habet, ibi nullum fere aquæ habere ſolent motum, <lb/>poſſuntque ſine ſenſibili errore ceu ſtagnantes conſiderari: </s> + <s xml:id="echoid-s4153" xml:space="preserve">Longe vero ali-<lb/>ter comparatum eſt alterum principium, quod apparebit exinferioribus ex-<lb/>emplis, & </s> + <s xml:id="echoid-s4154" xml:space="preserve">cujus rei luculentum eſſe poſſunt teſtimonium ea, quæ in ſupe-<lb/>riori ſectione protulimus circa refluxum aquarum; </s> + <s xml:id="echoid-s4155" xml:space="preserve">tantum enim abeſt, ut <lb/>aquæ in vaſe ſubmerſo ex data altitudine delapſæ, ad hanc altitudinem re-<lb/>gredi poſſint, prouti vi iſtius principii deberent, ſublatis impedimentis ex-<lb/>trinſecis, quin potius plerunque vix ſenſibilis ſit earum aſcenſus præ deſcen-<lb/>ſu, quem antea fecerunt: </s> + <s xml:id="echoid-s4156" xml:space="preserve">imo nequidem aſcendere ſuperficies aquæ poteſt +<pb o="144" file="0158" n="158" rhead="HYDRODYNAMICÆ"/> +tantum ſupra aquam, cui tubus immergitur, quantum infra eandem de-<lb/>preſſa fuerat, niſi cum tubus totus eſt apertus: </s> + <s xml:id="echoid-s4157" xml:space="preserve">iſta vero ſuperficies multo <lb/>minus deprimitur quam antea fuerat elevata. </s> + <s xml:id="echoid-s4158" xml:space="preserve">Horum rationem dedimus in <lb/>ſuperiori ſectione: </s> + <s xml:id="echoid-s4159" xml:space="preserve">Hæc quia ita ſunt, regulas nunc dabo duas pro motu <lb/>aquarum ubique definiendo, easque porro exemplis illuſtrabo talibus, quæ <lb/>nulla adhuc theoria explicari potuerunt, cum noſtra autem egregie admo-<lb/>dum conveniunt.</s> + <s xml:id="echoid-s4160" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div160" type="section" level="1" n="128"> +<head xml:id="echoid-head169" xml:space="preserve">Regula 1.</head> +<p> + <s xml:id="echoid-s4161" xml:space="preserve">§. </s> + <s xml:id="echoid-s4162" xml:space="preserve">2. </s> + <s xml:id="echoid-s4163" xml:space="preserve">Diſpiciendum eſt, aſſumta alicubi in vaſe propoſito velocitate <lb/>fluidi ceu cognita, quænam reliquis fluidi partibus futura ſit velocitas. </s> + <s xml:id="echoid-s4164" xml:space="preserve">Ita <lb/>enim cognoſcetur aſcenſus potentialis totius fluidi ejusque incrementum. </s> + <s xml:id="echoid-s4165" xml:space="preserve">Ha-<lb/>ctenus conſideravimus fluida in infinita ſtrata parallela vel potius ad latera <lb/>vaſis ubique perpendicularia diviſa, ſtatuimusque velocitates hiſce ſtratis re-<lb/>ciproce proportionales: </s> + <s xml:id="echoid-s4166" xml:space="preserve">Facile quidem eſt vaſa effingere, ubi aliter moven-<lb/>tur fluida; </s> + <s xml:id="echoid-s4167" xml:space="preserve">crediderim autem his in locis motum notabilem nunquam ha-<lb/>bere fluida ita, ut error ex iſta hypotheſi ſenſibilis naſci fere non poſſit: <lb/></s> + <s xml:id="echoid-s4168" xml:space="preserve">poterit tamen majoris accurationis ergo præfata regula adhiberi. </s> + <s xml:id="echoid-s4169" xml:space="preserve">Præſertim <lb/>vero huc pertinet contractio venarum, cum fluida per foramina in tenuibus <lb/>admodum laminis facta transire coguntur, qua in re magna eſt adhibenda <lb/>circumſpectio: </s> + <s xml:id="echoid-s4170" xml:space="preserve">Effectus hujusmodi contractionum haud male, puto, prævi-<lb/>debuntur, cum recte perpenſa fuerint, quæ in ſectione quarta de illis monui.</s> + <s xml:id="echoid-s4171" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div161" type="section" level="1" n="129"> +<head xml:id="echoid-head170" xml:space="preserve">Regula 2.</head> +<p> + <s xml:id="echoid-s4172" xml:space="preserve">§. </s> + <s xml:id="echoid-s4173" xml:space="preserve">3. </s> + <s xml:id="echoid-s4174" xml:space="preserve">Singulis momentis diſpiciendum eſt, quantum vis vivæ, ſeu <lb/>quodnam productum ex aſcenſu potentiali in maſſam oriatur ad fluxum præ-<lb/>cipuum, cujus natura quæritur, nihil conferens. </s> + <s xml:id="echoid-s4175" xml:space="preserve">Id vero rurſus uniuscu-<lb/>jusque circumſpectæ æſtimationi relinquendum eſt. </s> + <s xml:id="echoid-s4176" xml:space="preserve">Quod ſic oritur, ad-<lb/>dendum eſt facto ex aſcenſu potentiali, quem motus præcipuus involvit, in <lb/>maſſam, aggregatumque productorum demum æquale cenſendum eſt facto <lb/>ex maſsâ omnis aquæ in ejusdem deſcenſum actualem.</s> + <s xml:id="echoid-s4177" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4178" xml:space="preserve">Magni profecto eſt momenti hæc regula, & </s> + <s xml:id="echoid-s4179" xml:space="preserve">ut puto, fere unica ad mo-<lb/>tuum menſuras obtinendas, quiin vaſis irregularibus, pluribusque cavitatibus <lb/>inter ſe communicantibus diviſis fiunt, quod nunc pluribus illuſtrabo exemplis.</s> + <s xml:id="echoid-s4180" xml:space="preserve"/> +</p> +<pb o="145" file="0159" n="159" rhead="SECTIO OCTAVA."/> +</div> +<div xml:id="echoid-div162" type="section" level="1" n="130"> +<head xml:id="echoid-head171" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s4181" xml:space="preserve">§. </s> + <s xml:id="echoid-s4182" xml:space="preserve">4. </s> + <s xml:id="echoid-s4183" xml:space="preserve">Propoſitum fuerit vas A C R B (Fig. </s> + <s xml:id="echoid-s4184" xml:space="preserve">37.) </s> + <s xml:id="echoid-s4185" xml:space="preserve">infinitæ quaſi ratione <lb/> +<anchor type="note" xlink:label="note-0159-01a" xlink:href="note-0159-01"/> +foraminum mox dicendorum ubique amplitudinis & </s> + <s xml:id="echoid-s4186" xml:space="preserve">diaphragmate aliquo E F <lb/>in duas diſtinctum cavitates inter ſe communicantes, mediante foramine G: <lb/></s> + <s xml:id="echoid-s4187" xml:space="preserve">habeat præterea vas iſtud in infima ſui parte aliud foramen D: </s> + <s xml:id="echoid-s4188" xml:space="preserve">deinde pona-<lb/>tur vas aquâ plenum uſque in P Q, ſic ut cavitas inferior C E F R tota ſit hu-<lb/>mido repleta, atque inſuper diaphragmati ſuperjaceat pars altera P Q F E. </s> + <s xml:id="echoid-s4189" xml:space="preserve">His <lb/>poſitis, fluidoque jam moveri incipiente, quæritur velocitas aquæ per foramen <lb/>D in aërem effluentis velaltitudo genitri<unsure/>x hujus velocitatis.</s> + <s xml:id="echoid-s4190" xml:space="preserve"/> +</p> +<div xml:id="echoid-div162" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0159-01" xlink:href="note-0159-01a" xml:space="preserve">Fig. 37.</note> +</div> +</div> +<div xml:id="echoid-div164" type="section" level="1" n="131"> +<head xml:id="echoid-head172" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s4191" xml:space="preserve">Fuerit altitudo ſuperficiei P Q ſupra foramen D = x, amplitudo fo-<lb/>raminis D = n, alteriusque G = m. </s> + <s xml:id="echoid-s4192" xml:space="preserve">Perſpicuum autem eſt aſcenſum potentia-<lb/>lem cujuſvis guttæ per G transfluentis nihil promovere effluxum per D, totum-<lb/>que impendi in motum aliquem excitandum inteſtinum, qui mox abſorbetur <lb/>ſine alio effectu: </s> + <s xml:id="echoid-s4193" xml:space="preserve">neceſſe igitur eſt ut ſingulis momentis motus generetur no-<lb/>vus in particulis foramen G tranſeuntibus, non minus atque in particulis per <lb/>D effiuentibus. </s> + <s xml:id="echoid-s4194" xml:space="preserve">Sed ſi aſcenſus potentialis guttulæ per D effluentis dicatur v, id <lb/>eſt, ſi aqua exilire ponatur per D velocitate, cujus altitudo genitrix ſit v, erit <lb/>ſimilis altitudo ratione guttulæ mole ſua priori æqualis, per G eodem tempo-<lb/>re transfluentis {nnv/mm}. </s> + <s xml:id="echoid-s4195" xml:space="preserve">Multiplicatis iſtis aſcenſibus potentialibus per maſſam, quam <lb/>æqualem habent, quamque vocabo M, erit aggregatum productorum = <lb/>Mv + {Mnnv/mm}. </s> + <s xml:id="echoid-s4196" xml:space="preserve">Et cum ob infinitam amplitudinem vaſis alius motus non <lb/>generetur, erit præfatum aggregatum (per reg. </s> + <s xml:id="echoid-s4197" xml:space="preserve">2.) </s> + <s xml:id="echoid-s4198" xml:space="preserve">cenſendum æquale facto ex <lb/>maſſa omnis aquæ in ejusdem deſcenſum actualem. </s> + <s xml:id="echoid-s4199" xml:space="preserve">At vero ſi maſſa omnis aquæ <lb/>dicatur μ, erit (per § 7. </s> + <s xml:id="echoid-s4200" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s4201" xml:space="preserve">3.) </s> + <s xml:id="echoid-s4202" xml:space="preserve">deſcenſus actualis, qui fit dum guttula M ef-<lb/>fluit = {Mx/μ}, ita ut productum commune ſit = M x. </s> + <s xml:id="echoid-s4203" xml:space="preserve">Igitur habetur <lb/>Mv + {Mnnv/mm} = Mx, ſive v = {mmx/nn + mm}. </s> + <s xml:id="echoid-s4204" xml:space="preserve">Q. </s> + <s xml:id="echoid-s4205" xml:space="preserve">E. </s> + <s xml:id="echoid-s4206" xml:space="preserve">F.</s> + <s xml:id="echoid-s4207" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div165" type="section" level="1" n="132"> +<head xml:id="echoid-head173" xml:space="preserve">Scholium 1.</head> +<p> + <s xml:id="echoid-s4208" xml:space="preserve">§. </s> + <s xml:id="echoid-s4209" xml:space="preserve">5. </s> + <s xml:id="echoid-s4210" xml:space="preserve">Apparet ex iſto exemplo, motum ſine calculo differentiali de- +<pb o="146" file="0160" n="160" rhead="HYDRODYNAMICÆ"/> +terminari poſſe, cum figura vaſis ubique ampliſſimi motum hunc mutare non <lb/>poteſt. </s> + <s xml:id="echoid-s4211" xml:space="preserve">Interim difficile futurum non fuiſſet, conſideratione quoque habita ad <lb/>amplitudines vaſis, fluxum definire, & </s> + <s xml:id="echoid-s4212" xml:space="preserve">ſolo brevitatis ſtudio id vitavimus pa-<lb/>riterque omittemus in ſequentibus, niſi fortaſſe motus notabiliter à figura vaſis <lb/>varia mutetur, quod fieri poteſt in tubis ſatis amplis, ſed iis longiſſimis, in <lb/>quibus fluidum movetur, præſertim ſi motus determinandi ſint oſcillatorii. <lb/></s> + <s xml:id="echoid-s4213" xml:space="preserve">lmo vidimus in præcedente Sectione, ſi oſcillationes ſint valde parvæ in tubis <lb/>profundiſſime ſubmerſis, tunc tantum abeſſe, ut ad ſolum foramen fundi ſit <lb/>attendendum, neglectis amplitudinibus etiamſi ſatis magnis, quin potius ad <lb/>has ſolas fere ſit reſpiciendum.</s> + <s xml:id="echoid-s4214" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div166" type="section" level="1" n="133"> +<head xml:id="echoid-head174" xml:space="preserve">Scholium 2.</head> +<p> + <s xml:id="echoid-s4215" xml:space="preserve">§. </s> + <s xml:id="echoid-s4216" xml:space="preserve">6. </s> + <s xml:id="echoid-s4217" xml:space="preserve">Quia in calculo, quem poſuimus, vis viva cujuſvis guttulæ per <lb/>G transfluentis ab aqua cavitatis inferioris abſorberi debet, perſpicuum eſt, <lb/>propoſitionem non eſſe extendendam ad illos caſus, qui hypotheſi repugnent, <lb/>veluti cum diaphragma E F fundo C R proximum eſt ſimulque foramina ſibi <lb/>directe reſpondent: </s> + <s xml:id="echoid-s4218" xml:space="preserve">ita enim non arduum eſt providere, motum longe diver-<lb/>ſum fore ab eo, quem præſens theoria indicat. </s> + <s xml:id="echoid-s4219" xml:space="preserve">At vero, ſi diſtantia D G ma-<lb/>gna ſit, ſique ſimul foraminum ſitus ſit obliquus & </s> + <s xml:id="echoid-s4220" xml:space="preserve">latera foraminum venis <lb/>aqueis negent contractionem; </s> + <s xml:id="echoid-s4221" xml:space="preserve">dubium nullum eſt, quin theoria accurate om-<lb/>@@bus phænomenis reſpondeat.</s> + <s xml:id="echoid-s4222" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div167" type="section" level="1" n="134"> +<head xml:id="echoid-head175" xml:space="preserve">Corollarium.</head> +<p> + <s xml:id="echoid-s4223" xml:space="preserve">§. </s> + <s xml:id="echoid-s4224" xml:space="preserve">7. </s> + <s xml:id="echoid-s4225" xml:space="preserve">Si foramen G eſt admodum amplum præ altero, fit fere v = x, <lb/>ſed hæc altitudo v, cui nimirum reſpondet velocitas aquæ per D effluentis, <lb/>non parum decreſcit, creſcente foramine D, ita ut ſi fuerit v. </s> + <s xml:id="echoid-s4226" xml:space="preserve">gr. </s> + <s xml:id="echoid-s4227" xml:space="preserve">duplum fo-<lb/>raminis G, ſit v = {1/5}x & </s> + <s xml:id="echoid-s4228" xml:space="preserve">tantum non tota evaneſcat, cum foramen G eſt <lb/>valde exiguum reſpectu foraminis D.</s> + <s xml:id="echoid-s4229" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4230" xml:space="preserve">His ita inventis, jam quivis veram perſpiciet rationem motuum illo-<lb/>rum, quos Mariottus primus obſervavit, & </s> + <s xml:id="echoid-s4231" xml:space="preserve">quibus ceu valde admirabilibus te-<lb/>ſtatur ſe ſupra modum fuiſſe delectatum, ſimulque intelliget, quam longe <lb/>Auctor iſte in reliquis perſpicaciſſimus à viâ aberraverit in hiſce diſquiſitioni-<lb/>bus. </s> + <s xml:id="echoid-s4232" xml:space="preserve">Non abs re fore puto obſervata Mariotti hic apponere.</s> + <s xml:id="echoid-s4233" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4234" xml:space="preserve">§. </s> + <s xml:id="echoid-s4235" xml:space="preserve">8. </s> + <s xml:id="echoid-s4236" xml:space="preserve">Vas adhibuit, quale repræſentat Figura trigeſima octava, quæ <lb/> +<anchor type="note" xlink:label="note-0160-01a" xlink:href="note-0160-01"/> +<pb o="147" file="0161" n="161" rhead="SECTIO OCTAVA."/> +non differt â priori niſi in eo, quod in ima parte cylindro A B C tubus hori-<lb/>zontalis M D inſertus ſit perforatus lumine D, per quod aquæ verticaliter exi-<lb/>liunt: </s> + <s xml:id="echoid-s4237" xml:space="preserve">Diaphragma vero E F in medio perforatum eſt lumine G ut antea: </s> + <s xml:id="echoid-s4238" xml:space="preserve">in-<lb/>fra illud parvulum erat foramen K, ut facilius cavitas inferior aquis impleri poſ-<lb/>ſet, quo facto idem obturabatur, reliquumque vaſis replebatur.</s> + <s xml:id="echoid-s4239" xml:space="preserve"/> +</p> +<div xml:id="echoid-div167" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0160-01" xlink:href="note-0160-01a" xml:space="preserve">Fig. 38.</note> +</div> +<p> + <s xml:id="echoid-s4240" xml:space="preserve">His ita præparatis, effluentibusque aquis per D, obſervavit Mariottus, <lb/>mox illas aſcendiſſe uſque in I, deinde ſenſim imminuta velocitate uſque in <lb/>N & </s> + <s xml:id="echoid-s4241" xml:space="preserve">tandem, imminente depletione tota cavitatis ſuperioris, A B F E uſque <lb/>in O, tuncque aſſumtis confeſtim novis viribus aſſiliviſſe fere uſque in F. </s> + <s xml:id="echoid-s4242" xml:space="preserve">Animad-<lb/>vertit etiam, ſi bene memini, altitudinem jactus initialis eo minorem eſſe, quo mi-<lb/>nus ſit foramen G, ratione alterius D. </s> + <s xml:id="echoid-s4243" xml:space="preserve">Videatur ejus tract. </s> + <s xml:id="echoid-s4244" xml:space="preserve">de motu aquarum part. </s> + <s xml:id="echoid-s4245" xml:space="preserve">IV. <lb/></s> + <s xml:id="echoid-s4246" xml:space="preserve">diſc. </s> + <s xml:id="echoid-s4247" xml:space="preserve">1. </s> + <s xml:id="echoid-s4248" xml:space="preserve">Putat autem horum motuum mutationes explicari poſſe fingendo vaſi <lb/>A B F E ampliſſimo tubum ſtrictiorem adhærere G L M D, per quem aquæ fluant. </s> + <s xml:id="echoid-s4249" xml:space="preserve"><lb/>At vero demonſtravimus & </s> + <s xml:id="echoid-s4250" xml:space="preserve">experientia quotidie docet, motum aquarum ex <lb/>vaſe A B G L M D admodum diverſum eſſe ab eo, qui modo indicatus fuit. </s> + <s xml:id="echoid-s4251" xml:space="preserve"><lb/>Non minus falleretur ſi quis putaret aquam eadem velocitate exilire per fora-<lb/>men D, quaſi illud in diaphragmate E F poſitum eſſet, nam fieri poteſt, ut <lb/>altitudo jactus initialis ſit major & </s> + <s xml:id="echoid-s4252" xml:space="preserve">minor altitudine F B. </s> + <s xml:id="echoid-s4253" xml:space="preserve">Nec denique ea effluent <lb/>aquæ quantitate, uti facile quis ſuſpicari poſſet, qua eodem tempore effluerent <lb/>ex vaſe ſuperiori ſimplici reſciſſa parte E F D C quamvis ita proxime ſe res ha-<lb/>beat, cum foramen G admodum minus eſt foramine D.</s> + <s xml:id="echoid-s4254" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4255" xml:space="preserve">§. </s> + <s xml:id="echoid-s4256" xml:space="preserve">9. </s> + <s xml:id="echoid-s4257" xml:space="preserve">Noſtra vero æquatio, nempe v = {mmx/nn + mm}, recte omnino <lb/>reſpondet phænomenis: </s> + <s xml:id="echoid-s4258" xml:space="preserve">indicat enim aquam mox ab initio fluxus ad certam aſ-<lb/>cendere altitudinem, eamque tanto minorem, quanto minus eſt foramen <lb/>diaphragmatis p@æ foramine altero; </s> + <s xml:id="echoid-s4259" xml:space="preserve">dein iſtum aſcenſum ſenſim diminui, do-<lb/>nec aqua omnis ex cavitate ſuperiori effluxerit, quo ipſo momento protinus <lb/>augmentum capit, totamque aquæ ſuperincumbentis altitudinem tantum non <lb/>attingit, quia tunc ex vaſe ſimplici eoque infinite amplo effluere cenſendæ ſunt <lb/>aquæ: </s> + <s xml:id="echoid-s4260" xml:space="preserve">pauliſper tamen etiamnum retardantur aquæ à tranſitu aëris per fora-<lb/>men G, & </s> + <s xml:id="echoid-s4261" xml:space="preserve">ſane notabiliter retardantur, cum foramen ſuperius valde parvum <lb/>eſt, de quo argumento mox quædam dicemus, cum de fluidis heterogeneis <lb/>ſermo erit. </s> + <s xml:id="echoid-s4262" xml:space="preserve">Si figura Mariotti debita proportione reſpondeat argumento in-<lb/>ſtituto, oportet, ut foramen G alterius fecerit paullo pluſqnam dimidium.</s> + <s xml:id="echoid-s4263" xml:space="preserve"/> +</p> +<pb o="148" file="0162" n="162" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s4264" xml:space="preserve">§. </s> + <s xml:id="echoid-s4265" xml:space="preserve">10. </s> + <s xml:id="echoid-s4266" xml:space="preserve">Indicat porro formula noſtra, quod multis fortaſſe nondum <lb/>perſpectâ hâc theoriâ ſatis paradoxum videri potuiſſet, ſitum diaphragmatis <lb/>E F ſive altiorem ſive humiliorem nullo modo mutare impetum ſive veloci-<lb/>tatem aquæ effluentis; </s> + <s xml:id="echoid-s4267" xml:space="preserve">ratio autem iſtius phænomeni omnibus nunc, puto, <lb/>manifeſta eſt.</s> + <s xml:id="echoid-s4268" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4269" xml:space="preserve">§. </s> + <s xml:id="echoid-s4270" xml:space="preserve">11. </s> + <s xml:id="echoid-s4271" xml:space="preserve">Jam vero examinabimus inſuper motum aquarum, cum plura <lb/>ſunt diaphragmata foraminibus pertuſa, per quæ aquæ tranſire cogantur, ut <lb/>effluxus per foramen D fieri poſſit. </s> + <s xml:id="echoid-s4272" xml:space="preserve">Poterit id eodem abſolvi modo, quo uſi <lb/>ſumus in problemate §. </s> + <s xml:id="echoid-s4273" xml:space="preserve">4. </s> + <s xml:id="echoid-s4274" xml:space="preserve">Ita autem inſtituto recte calculo retentisque deno-<lb/>minationibus ibidem adhibitis apparebit eſſe <lb/>v = x: </s> + <s xml:id="echoid-s4275" xml:space="preserve">(1 + {nn/αα} + {nn/ββ} + {nn/γγ} + &</s> + <s xml:id="echoid-s4276" xml:space="preserve">c.) <lb/></s> + <s xml:id="echoid-s4277" xml:space="preserve">ubi per α, β, γ &</s> + <s xml:id="echoid-s4278" xml:space="preserve">c. </s> + <s xml:id="echoid-s4279" xml:space="preserve">intelliguntur amplitudines foraminum, quæ ſunt in dia-<lb/>phragmatibus, dum n exprimit ut antea amplitudinem foraminis D, per quod <lb/>aquæ effluunt.</s> + <s xml:id="echoid-s4280" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4281" xml:space="preserve">§. </s> + <s xml:id="echoid-s4282" xml:space="preserve">12. </s> + <s xml:id="echoid-s4283" xml:space="preserve">Si proinde loco unius diaphragmatis ſint in ſimili vaſe, quale <lb/>(Fig. </s> + <s xml:id="echoid-s4284" xml:space="preserve">39.) </s> + <s xml:id="echoid-s4285" xml:space="preserve">repræſentat, plura diaphragmata veluti in B, C, R &</s> + <s xml:id="echoid-s4286" xml:space="preserve">c. </s> + <s xml:id="echoid-s4287" xml:space="preserve">per quæ <lb/> +<anchor type="note" xlink:label="note-0162-01a" xlink:href="note-0162-01"/> +aqua transfluat, dum per infimum foramen D effluit, mutabitur & </s> + <s xml:id="echoid-s4288" xml:space="preserve">augebi-<lb/>tur confeſtim velocitas aquæ effluentis, quoties aliqua cavitas depletur: </s> + <s xml:id="echoid-s4289" xml:space="preserve">talis <lb/>autem eſſe poteſt proportio inter altitudines A B, B C, C R, R E &</s> + <s xml:id="echoid-s4290" xml:space="preserve">c. <lb/></s> + <s xml:id="echoid-s4291" xml:space="preserve">atque amplitudines foraminum D, G, F, H &</s> + <s xml:id="echoid-s4292" xml:space="preserve">c. </s> + <s xml:id="echoid-s4293" xml:space="preserve">ut ſemper, quoties <lb/>nova depleri incipit concameratio, vena effluens ad eandem altitudinem <lb/>O aſſurgat, ſeu eadem velocitate effluat. </s> + <s xml:id="echoid-s4294" xml:space="preserve">Id vero obtinetur (deſignatis am-<lb/>plitudinibus foraminum D, G, F, H &</s> + <s xml:id="echoid-s4295" xml:space="preserve">c. </s> + <s xml:id="echoid-s4296" xml:space="preserve">per n, α, β, γ, &</s> + <s xml:id="echoid-s4297" xml:space="preserve">c.) </s> + <s xml:id="echoid-s4298" xml:space="preserve">faciendo <lb/>B C = {nn/αα} A B; </s> + <s xml:id="echoid-s4299" xml:space="preserve">C R = {nn/ββ} A B; </s> + <s xml:id="echoid-s4300" xml:space="preserve">R E = {nn/γγ} A B &</s> + <s xml:id="echoid-s4301" xml:space="preserve">c. </s> + <s xml:id="echoid-s4302" xml:space="preserve"><lb/>ita ut poſitis ſoraminibus inter ſe æqualibus ſint pariter lineæ A B, B C, C R, <lb/>R E &</s> + <s xml:id="echoid-s4303" xml:space="preserve">c. </s> + <s xml:id="echoid-s4304" xml:space="preserve">inter ſe æquales faciendæ. </s> + <s xml:id="echoid-s4305" xml:space="preserve">Facile quoque erit in vaſe cylindrico <lb/>eam conciliare foraminibus magnitudinem, ut ſuperficies fluidi eodem tem-<lb/>pore ab uno diaphragmate ad ſubſequens quodcunque deſcendat, & </s> + <s xml:id="echoid-s4306" xml:space="preserve">cum <lb/>hæc diaphragmata æqualiter à ſe invicem & </s> + <s xml:id="echoid-s4307" xml:space="preserve">à fundo diſtant, uniformis clep-<lb/>ſydrarum ſtructura excogitari poteſt.</s> + <s xml:id="echoid-s4308" xml:space="preserve"/> +</p> +<div xml:id="echoid-div168" type="float" level="2" n="2"> +<note position="left" xlink:label="note-0162-01" xlink:href="note-0162-01a" xml:space="preserve">Fig. 39.</note> +</div> +<p> + <s xml:id="echoid-s4309" xml:space="preserve">§. </s> + <s xml:id="echoid-s4310" xml:space="preserve">13. </s> + <s xml:id="echoid-s4311" xml:space="preserve">Si vero omnia diaphragmata altiſſime poſita ſint, jucundus erit +<pb o="149" file="0163" n="163" rhead="SECTIO OCTAVA."/> +luſus hydraulicus, venam proſilientem D O videre, quæ æqualibus incre-<lb/>mentis æqualibusque temporum intervallis, quod utrumque fieri poteſt, <lb/>ſubſultim creſcat.</s> + <s xml:id="echoid-s4312" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4313" xml:space="preserve">§. </s> + <s xml:id="echoid-s4314" xml:space="preserve">14. </s> + <s xml:id="echoid-s4315" xml:space="preserve">Propoſitum nunc ſit motum fluidi exilientis indagare, cum per <lb/>ſingula foramina alia atque alia fluida transfluunt. </s> + <s xml:id="echoid-s4316" xml:space="preserve">Fluida autem leviora con-<lb/>tinue ponenda eſſe apparet, quo ſunt altius poſita, ne motus turbetur, <lb/>quod fit cum eodem tempore fluidum inferius aſcendit, ſuperiore deſcen-<lb/>dente, per commune foramen. </s> + <s xml:id="echoid-s4317" xml:space="preserve">Innoteſcet hoc modo quisnam ſit motus <lb/>in aquis ex vaſe effluentibus undique clauſo præter foraminulum aliquod ſu-<lb/>perne exiſtens, quod aëri tranſitum concedit. </s> + <s xml:id="echoid-s4318" xml:space="preserve">Hypotheſin vero infinitæ va-<lb/>ſis cylindrici amplitudinis ratione foraminum retinebimus, atque porro <lb/>gravitatem ſpecificam fluidi per D exilientis deſignabimus per A, illiusque <lb/>quod per G transfluit notabimus littera B, ſimiliterque gravitates ſpecificas <lb/>fluidorum per foramina, F, H, &</s> + <s xml:id="echoid-s4319" xml:space="preserve">c. </s> + <s xml:id="echoid-s4320" xml:space="preserve">fluentium indicabimus reſpective litte-<lb/>ris C, D, &</s> + <s xml:id="echoid-s4321" xml:space="preserve">c. </s> + <s xml:id="echoid-s4322" xml:space="preserve">Denique cum etiam conſiderandæ hic ſint altitudines diver-<lb/>ſorum fluidorum, quorum quidem, ob figuram vaſis cylindricam ſolum <lb/>infimum effluens altitudinem mutat, vocabimus x altitudinem fluidi infimi <lb/>ſupra foramen D, fluidorum reliquorum, eo quo ſibi ſuperincumbunt or-<lb/>dine, altitudines deſignabimus reſpective per b, c, d &</s> + <s xml:id="echoid-s4323" xml:space="preserve">c. </s> + <s xml:id="echoid-s4324" xml:space="preserve">reliquas denomi-<lb/>nationes paragraphi undecimi retinebimus; </s> + <s xml:id="echoid-s4325" xml:space="preserve">quibus ita præparatis compu-<lb/>tus inſtituetur ut §. </s> + <s xml:id="echoid-s4326" xml:space="preserve">4. </s> + <s xml:id="echoid-s4327" xml:space="preserve">factum eſt, neque enim quicquam aliud inſuper ob-<lb/>ſervandum eſt, quam ut maſſæ guttularum iisdem tempusculis per diverſa <lb/>foramina transeuntium non ſimpliciter ex mole, ſed etiam ex gravitate ſpe-<lb/>cificia æſtimentur: </s> + <s xml:id="echoid-s4328" xml:space="preserve">deſcenſus autem actualis pro ſingulis fluidis erit ſeorſim ſu-<lb/>mendus: </s> + <s xml:id="echoid-s4329" xml:space="preserve">Hiſce veſtigiis inſiſtendo reperitur talis primo æquatio <lb/>A v + {nn/αα} B v + {nn/ββ} C v + {nn/γγ} D v + &</s> + <s xml:id="echoid-s4330" xml:space="preserve">c. </s> + <s xml:id="echoid-s4331" xml:space="preserve">= A x + B b + C c + D d, + &</s> + <s xml:id="echoid-s4332" xml:space="preserve">c. <lb/></s> + <s xml:id="echoid-s4333" xml:space="preserve">quæ reducta dat <lb/>v = (A x + B b + C c + D d + &</s> + <s xml:id="echoid-s4334" xml:space="preserve">c.)</s> + <s xml:id="echoid-s4335" xml:space="preserve">: </s> + <s xml:id="echoid-s4336" xml:space="preserve">(A + {nn/αα} B + {nn/ββ} C + {nn/γγ} D + &</s> + <s xml:id="echoid-s4337" xml:space="preserve">c.)</s> + <s xml:id="echoid-s4338" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4339" xml:space="preserve">§. </s> + <s xml:id="echoid-s4340" xml:space="preserve">15. </s> + <s xml:id="echoid-s4341" xml:space="preserve">Si duo ſint liquores, erunt duo termini tam in numeratore <lb/>quam in denominatore ſumendi & </s> + <s xml:id="echoid-s4342" xml:space="preserve">tres termini cum tres fuerint liquores, atque <lb/>ſic porro: </s> + <s xml:id="echoid-s4343" xml:space="preserve">Si proinde liquor effluens ſit, v. </s> + <s xml:id="echoid-s4344" xml:space="preserve">gr. </s> + <s xml:id="echoid-s4345" xml:space="preserve">mercurius, ipſique ſuperin-<lb/>cumbat aqua ſtatuanturque gravitates ſpecificæ horum liquorum ut 14. </s> + <s xml:id="echoid-s4346" xml:space="preserve">ad 1. </s> + <s xml:id="echoid-s4347" xml:space="preserve">fiet +<pb o="150" file="0164" n="164" rhead="HYDRODYNAMICÆ"/> +v = {14x + b/14 + {nn/αα}} <lb/>atque ſi ratio foraminum D & </s> + <s xml:id="echoid-s4348" xml:space="preserve">G ſuerit ex gr. </s> + <s xml:id="echoid-s4349" xml:space="preserve">ut 3 ad 1, fiet <lb/>v = {14x + b/23}</s> +</p> +<p> + <s xml:id="echoid-s4350" xml:space="preserve">§. </s> + <s xml:id="echoid-s4351" xml:space="preserve">16. </s> + <s xml:id="echoid-s4352" xml:space="preserve">Patet quoque ratiocinium iſtud non excludere eos caſus, qui-<lb/>bus fluida ſuperiora ſunt inferioribus ſpecifice graviora, modo fluida infe-<lb/>riora non aſcendant per eadem foramina, per quæ ſuperiora deſcendunt: <lb/></s> + <s xml:id="echoid-s4353" xml:space="preserve">neque vero id futurum eſſe præſumo (nec tamen affirmo) cum loco ſimpli-<lb/>cis foraminis tubulus ſit quamvis exiguæ altitudinis, per quem liquor ſupe-<lb/>rior deſcendat in inferiorem cavitatem, velutiin fig. </s> + <s xml:id="echoid-s4354" xml:space="preserve">40. </s> + <s xml:id="echoid-s4355" xml:space="preserve">ubi quidem duo tantum <lb/>liquores conſiderantur.</s> + <s xml:id="echoid-s4356" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4357" xml:space="preserve">Hic autem altitudo C R variabilis eſt, & </s> + <s xml:id="echoid-s4358" xml:space="preserve">altitudo A C conſtans; </s> + <s xml:id="echoid-s4359" xml:space="preserve">in-<lb/>terim tamen uniformitatis litterarum gratia vocabimus alt@tudinem A C = x, <lb/>alteram C R = b; </s> + <s xml:id="echoid-s4360" xml:space="preserve">gravitatem ſpecificam fluidi per D erumpentis faciemus rur-<lb/>ſus = A, alteriusque fluidi per G transeuntis = B, & </s> + <s xml:id="echoid-s4361" xml:space="preserve">erit altitudo D O <lb/>ſeu <lb/>v = {Ax + Bb/A + {nn/αα}B} <lb/>Igitur ſi per foramina D & </s> + <s xml:id="echoid-s4362" xml:space="preserve">G reſpective fluant aqua & </s> + <s xml:id="echoid-s4363" xml:space="preserve">me<unsure/>rcuri<unsure/>us erit nunc <lb/>v = {x + 14b/1 + {14nn/αα}}</s> +</p> +<p> + <s xml:id="echoid-s4364" xml:space="preserve">§. </s> + <s xml:id="echoid-s4365" xml:space="preserve">17. </s> + <s xml:id="echoid-s4366" xml:space="preserve">Ut porro innoteſcat motus fluidi ſimplicis ex vaſe ſuperne <lb/>parvulo foramine aërem admittente, obſervandum eſt, nullam hic altitudi-<lb/>nem eſſe b; </s> + <s xml:id="echoid-s4367" xml:space="preserve">quia aër utrique orificio incumbere ad eandem altitudinem cen-<lb/>ſeri poteſt, erit proinde <lb/>v = {Ax/A + {nn/αα}B <lb/>atque ſi fuerit {A/B} = 850, quæ præterpropter ſolet eſſe proportio inter <lb/>gravitates ſpecificas aquæ & </s> + <s xml:id="echoid-s4368" xml:space="preserve">aëris, erit <lb/>v = {850x/850 + {nn/αα}};</s> + <s xml:id="echoid-s4369" xml:space="preserve"/> +</p> +<pb o="151" file="0165" n="165" rhead="SECTIO OCTAVA."/> +<p> + <s xml:id="echoid-s4370" xml:space="preserve">§, 18. </s> + <s xml:id="echoid-s4371" xml:space="preserve">Omnia hæc principia, quæ hactenus adhibuimus, facile ut <lb/>jam dixi extenduntur ad vaſa, quæ finitam ratione foraminum habent am-<lb/>plitudinem; </s> + <s xml:id="echoid-s4372" xml:space="preserve">Poteſt autem eorum veritas alio etiam modo admodum diverſo <lb/>evinci, uti oſtendam, cum ad hydraulico-ſtaticam pervenero, quia altero il-<lb/>lo demonſtrandi modo preſſiones fluidorum in ſingulis vaſis partibus magis <lb/>fiunt perſpicuæ; </s> + <s xml:id="echoid-s4373" xml:space="preserve">differunt autem horum fluidorum regulæ ſtaticæ vehemen-<lb/>ter à legibus, quæ fluidis ſtagnantibus debentur.</s> + <s xml:id="echoid-s4374" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4375" xml:space="preserve">Cæterum habent hæc ſuam utilitatem ad machinas hydraulicas recte <lb/>perſpiciendas; </s> + <s xml:id="echoid-s4376" xml:space="preserve">neque enim ſatis ad hæc attenti fuiſſe videntur artifices: </s> + <s xml:id="echoid-s4377" xml:space="preserve">da-<lb/>bitur autem occaſio de iis uberius diſſerendi in ſequenti ſectione, ubi cal-<lb/>culum ponemus, quantum vis in propellendis aquis adhibitæ perdatur à <lb/>tranſitu aquæ per plura foramina, oſtenſuri ſimul remedia adhibenda, ut illud <lb/>virium detrimentum, quantum fieri poteſt, diminuatur. </s> + <s xml:id="echoid-s4378" xml:space="preserve">Prius vero alia quæ-<lb/>dam vaſa compoſita in hâc Sectione conſiderabimus, quam ad hæc deſcen-<lb/>damus.</s> + <s xml:id="echoid-s4379" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4380" xml:space="preserve">§. </s> + <s xml:id="echoid-s4381" xml:space="preserve">19. </s> + <s xml:id="echoid-s4382" xml:space="preserve">Fit aliquando, ut vaſa juxta ſe poſita aquas unum ex altero re-<lb/>cipiant effluxuras demum ex ultimo. </s> + <s xml:id="echoid-s4383" xml:space="preserve">Hoſce vero motus jam exemplo illuſtra-<lb/>bimus.</s> + <s xml:id="echoid-s4384" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4385" xml:space="preserve">Propoſitum fuerit vas cujuſcunque formæ A G M B (Fig. </s> + <s xml:id="echoid-s4386" xml:space="preserve">41.) </s> + <s xml:id="echoid-s4387" xml:space="preserve">quod <lb/> +<anchor type="note" xlink:label="note-0165-01a" xlink:href="note-0165-01"/> +nova aquarum affuſione conſtanter plenum conſervatur uſque in A B. </s> + <s xml:id="echoid-s4388" xml:space="preserve">Ex eo-<lb/>dem interim vaſe fluidum tranſire intelligatur per foramen M in aliud vas con-<lb/>tiguum B M N C & </s> + <s xml:id="echoid-s4389" xml:space="preserve">ex hoc rurſus in aliud C N R D per foramen N & </s> + <s xml:id="echoid-s4390" xml:space="preserve">ſic porro, <lb/>donec tandem aquæ in aërem ejiciantur, quæranturque loca ſuperficierum H L, <lb/>P Q, &</s> + <s xml:id="echoid-s4391" xml:space="preserve">c. </s> + <s xml:id="echoid-s4392" xml:space="preserve">poſtquam fuerunt ad ſtatum permanentiæ reducta. </s> + <s xml:id="echoid-s4393" xml:space="preserve">Quæſtio autem <lb/>ſic ſolvetur.</s> + <s xml:id="echoid-s4394" xml:space="preserve"/> +</p> +<div xml:id="echoid-div169" type="float" level="2" n="3"> +<note position="right" xlink:label="note-0165-01" xlink:href="note-0165-01a" xml:space="preserve">Fig. 41.</note> +</div> +<p> + <s xml:id="echoid-s4395" xml:space="preserve">Perſpicuum nempe eſt ex eo, quod ſuperficies A B, H L, P Q, &</s> + <s xml:id="echoid-s4396" xml:space="preserve">c. </s> + <s xml:id="echoid-s4397" xml:space="preserve">in eo-<lb/>dem loco permanent, aquas iis tranſire per foramina M, N, R velocitatibus, quæ <lb/>debeantur altitudinibus B H, L P, Q R, ſi modo tranſitus aquarum per unum fo-<lb/>ramen non acceleret earundem fluxum per foramen proximum, quod certe non <lb/>fiet, niſi expreſſe opera detur, ut id aliquantum fiat. </s> + <s xml:id="echoid-s4398" xml:space="preserve">Præterea vero conſiderandum <lb/>eſt, velocitates aquarum per foramina transfluentium reciproce eſſe forami-<lb/>nibus proportionales, quia in ſtatu permanentiæ eodem tempore eædem aqua-<lb/>rum quantitates per fingula foramina trajiciuntur. </s> + <s xml:id="echoid-s4399" xml:space="preserve">Ex iſtis intelligitur, deſig- +<pb o="152" file="0166" n="166" rhead="HYDRODYNAMICÆ"/> +natis amplitudinibus foraminum M, N, R, per m, n, p, fore L P <lb/>= {mm/nn} X B H; </s> + <s xml:id="echoid-s4400" xml:space="preserve">Q R = {mm/pp} X B H: </s> + <s xml:id="echoid-s4401" xml:space="preserve">Eſt vero B H + L P + Q R æqualis al-<lb/>titudini ſuperficiei A B ſupra foramen ultimum R ſeu D R; </s> + <s xml:id="echoid-s4402" xml:space="preserve">erit igitur <lb/>B H + {mm/nn} X B H + {mm/pp} X B H = D R, <lb/>& </s> + <s xml:id="echoid-s4403" xml:space="preserve">proinde B H = D R: </s> + <s xml:id="echoid-s4404" xml:space="preserve">(1 + {mm/nn} + {mm/pp}); </s> + <s xml:id="echoid-s4405" xml:space="preserve">pariterque <lb/>L P = {mm/nn} X D R: </s> + <s xml:id="echoid-s4406" xml:space="preserve">(1 + {mm/nn} + {mm/pp}) atque <lb/>Q R = {mm/pp} X D R: </s> + <s xml:id="echoid-s4407" xml:space="preserve">(1 + {mm/nn} + {mm/pp}), ſeu <lb/>B H = D R: </s> + <s xml:id="echoid-s4408" xml:space="preserve">(1 + {mm/nn} + {mm/pp}) <lb/>L P = D R: </s> + <s xml:id="echoid-s4409" xml:space="preserve">(1 + {nn/mm} + {nn/pp}) <lb/>Q R = D R: </s> + <s xml:id="echoid-s4410" xml:space="preserve">(1 + {pp/nn} + {pp/mm}) <lb/>atque ſic determinantur ſitus invariabiles ſuperficierum H L, P Q, &</s> + <s xml:id="echoid-s4411" xml:space="preserve">c. </s> + <s xml:id="echoid-s4412" xml:space="preserve">At <lb/>quanto tempore id fiat, ſi aliter ſuperficies illæ ſint poſitæ & </s> + <s xml:id="echoid-s4413" xml:space="preserve">quænam inte-<lb/>rea aquæ quantitas per ſingula foramina fluat, inferius examinabimus unà cum <lb/>aliis quæſtionibus eo pertinentibus: </s> + <s xml:id="echoid-s4414" xml:space="preserve">Jam vero ex allatis valoribus altitudi-<lb/>num B H, L P, Q R &</s> + <s xml:id="echoid-s4415" xml:space="preserve">c. </s> + <s xml:id="echoid-s4416" xml:space="preserve">præcipuas affectiones deducemus.</s> + <s xml:id="echoid-s4417" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4418" xml:space="preserve">§. </s> + <s xml:id="echoid-s4419" xml:space="preserve">20. </s> + <s xml:id="echoid-s4420" xml:space="preserve">I. </s> + <s xml:id="echoid-s4421" xml:space="preserve">Cum ſingula foramina ſunt inter ſe æque ampla, erit B H = <lb/>L P = Q R &</s> + <s xml:id="echoid-s4422" xml:space="preserve">c. </s> + <s xml:id="echoid-s4423" xml:space="preserve">& </s> + <s xml:id="echoid-s4424" xml:space="preserve">quævis iſtarum altitudinum toties continebitur in altitu-<lb/>dine D R, quoties vaſa replicantur.</s> + <s xml:id="echoid-s4425" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4426" xml:space="preserve">II. </s> + <s xml:id="echoid-s4427" xml:space="preserve">Si vero aliquod foraminum ſit infinite parvum ratione reliquorum, <lb/>erunt omnes ſuperficies, quæ ſunt cis foramen poſitæ, in eadem altitudine <lb/>cum prima ſuperficie A B: </s> + <s xml:id="echoid-s4428" xml:space="preserve">reliquæ autem fundo G R erunt proximæ.</s> + <s xml:id="echoid-s4429" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4430" xml:space="preserve">III. </s> + <s xml:id="echoid-s4431" xml:space="preserve">Si canalis fingatur continuus per ſingula foramina M, N, R &</s> + <s xml:id="echoid-s4432" xml:space="preserve">c. <lb/></s> + <s xml:id="echoid-s4433" xml:space="preserve">tranſiens, intelligitur, aquam per orificium canalis effluere debere velocitate, <lb/>quæ debeatur toti altitudini D R. </s> + <s xml:id="echoid-s4434" xml:space="preserve">In noſtro vero caſu ea velocitas reſpondet <lb/>tantum altitudini Q R, cujus rei ratio & </s> + <s xml:id="echoid-s4435" xml:space="preserve">origo eſt, quod aſcenſus pot ntialis ſin-<lb/>gularum guttularum per foramina, excepto ſolo foramine effluxus, transfluen- +<pb o="153" file="0167" n="167" rhead="SECTIO OCTAVA."/> +tium abſorbeatur. </s> + <s xml:id="echoid-s4436" xml:space="preserve">Igitur vis viva quæ ſingulis momentis perditur eſt ad vim <lb/>vivam quæ ſingulis momentis generatur, ut D Q ad D R. </s> + <s xml:id="echoid-s4437" xml:space="preserve">Altitudines vero <lb/>B H, L P, &</s> + <s xml:id="echoid-s4438" xml:space="preserve">c. </s> + <s xml:id="echoid-s4439" xml:space="preserve">repræſentant reſpective vim vivam, quæ continue guttulis per <lb/>foramina M, N transfluentibus ſeparatim demitur. </s> + <s xml:id="echoid-s4440" xml:space="preserve">Puto tamen ſi foramina <lb/>fuerint fere æqualia, eorumque centra in rectam lineam poſita, ac denique pa-<lb/>rietes B M, C N, D R non admodum à ſe invicem remoti ſint, fieri poſſe, ut <lb/>aliquanto majori velocitate aquæ erumpant, quam theoria iſta indicat: </s> + <s xml:id="echoid-s4441" xml:space="preserve">In re-<lb/>liquis caſibus non dubito de ejusdem accuratione, abſtrahendo animum ab im-<lb/>pedimentis ſæpe indicatis.</s> + <s xml:id="echoid-s4442" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4443" xml:space="preserve">IV. </s> + <s xml:id="echoid-s4444" xml:space="preserve">Denique perſpicuum eſt, quoties ſuperficies aquæ H L, P Q &</s> + <s xml:id="echoid-s4445" xml:space="preserve">c. <lb/></s> + <s xml:id="echoid-s4446" xml:space="preserve">ſitum ſuum mutant ſive plures, ſive una ſola, mox omnes ſuperficies loca <lb/>mutaturas eſfe, donec eo quo dictum fuit modo fuerint ad æquilibrium repo-<lb/>ſitæ. </s> + <s xml:id="echoid-s4447" xml:space="preserve">Mutationes autem iſtas generaliter definire nodoſi æque ac prolixi eſt <lb/>calculi, niſi vaſa ponantur priſmatica & </s> + <s xml:id="echoid-s4448" xml:space="preserve">infinitæ quaſi amplitudinis ratione <lb/>foraminum, ut nempe incrementa aſcenſuum potentialium aquarum M L, N Q &</s> + <s xml:id="echoid-s4449" xml:space="preserve">c. </s> + <s xml:id="echoid-s4450" xml:space="preserve"><lb/>quæ locum mutant, negligi poſſint ratione aſcenſuum potentialium, qui in <lb/>guttulis per M, N, R transfluentibus perpetuo generantur. </s> + <s xml:id="echoid-s4451" xml:space="preserve">Neque profecto <lb/>reſtrictio hæc afficere nos debet, cum paſſim jam viderimus in vaſis vel me-<lb/>diocriter admodum amplis poſſe ſine ſenſibili errore incrementa motus maſſa-<lb/>rum internarum rejici in calculo. </s> + <s xml:id="echoid-s4452" xml:space="preserve">Omittam igitur ſolutionem generalem, <lb/>quæ mihi eſt, ob nimiam ejus prolixitatem, atque ut in hâc ſectione adhuc <lb/>feci, vaſa ceu infinite ampla & </s> + <s xml:id="echoid-s4453" xml:space="preserve">quidem ad majorem concinnitatem priſmatica <lb/>ponam. </s> + <s xml:id="echoid-s4454" xml:space="preserve">Incipiam autem à vaſe bifido.</s> + <s xml:id="echoid-s4455" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4456" xml:space="preserve">§. </s> + <s xml:id="echoid-s4457" xml:space="preserve">21. </s> + <s xml:id="echoid-s4458" xml:space="preserve">Repræſentatur hujusmodi vas bifidum (Fig. </s> + <s xml:id="echoid-s4459" xml:space="preserve">42.) </s> + <s xml:id="echoid-s4460" xml:space="preserve">cujus pars A M <lb/> +<anchor type="note" xlink:label="note-0167-01a" xlink:href="note-0167-01"/> +aquis plena, altera B N ſaltem usque ad H L repleta ponitur, cum jam flu-<lb/>xus per utrumque orificium M & </s> + <s xml:id="echoid-s4461" xml:space="preserve">N incipit: </s> + <s xml:id="echoid-s4462" xml:space="preserve">affundanturque aquæ in A H, <lb/>ut vas conſtanter plenum ſervetur, ſic autem fiet, ut aquæ in B N aſſurgant <lb/>(aut etiam deſcendant pro rerum circumſtantiis) quod cum ita ſit, quære-<lb/>mus velocitatem ſuperficiei aqueæ, cum perveniet in ſitum h l.</s> + <s xml:id="echoid-s4463" xml:space="preserve"/> +</p> +<div xml:id="echoid-div170" type="float" level="2" n="4"> +<note position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">Fig. 42.</note> +</div> +<p> + <s xml:id="echoid-s4464" xml:space="preserve">Hunc in finem exprimemus amplitudinem orificii M per m, orificii N <lb/>per n & </s> + <s xml:id="echoid-s4465" xml:space="preserve">amplitudinem h l (quæ quidem ubique eadem ponitur) per g. </s> + <s xml:id="echoid-s4466" xml:space="preserve">Dein-<lb/>de ponemus B M = a, H M = b, B h = x, atque proinde h M = a - x. <lb/></s> + <s xml:id="echoid-s4467" xml:space="preserve">Sic vero patet ex poſitione infinitæ veluti vaſorum A M & </s> + <s xml:id="echoid-s4468" xml:space="preserve">B N amplitudinis, +<pb o="154" file="0168" n="168" rhead="HYDRODYNAMICÆ"/> +cum ſuperficies aquæ variabilis eſt in h l, fore altitudinem debitam velocitati <lb/>aquæ per M transfluentis = B b = x, velocitatemque ipſam = √x, ſi-<lb/>milemque altitudinem ratione orificii N = h M = a - x, atque velocita-<lb/>tem aquæ per N transfluentis = √a - x; </s> + <s xml:id="echoid-s4469" xml:space="preserve">eſt igitur quantitas dato tempu-<lb/>ſculo per M in vas B N influentis ad quantitatem eodem tempuſculo ex vaſe <lb/>effluentis ut m√x ad n√a - x, harumque quantitatum differentia diviſa <lb/>per amplitudinem g dat velocitatem ſuperficiei h l, quæ proinde velocitas, <lb/>quam vocabimus v, exprimetur hâc æquatione, <lb/>v = {m√x - n√a - x/g}</s> +</p> +<p> + <s xml:id="echoid-s4470" xml:space="preserve">§. </s> + <s xml:id="echoid-s4471" xml:space="preserve">22. </s> + <s xml:id="echoid-s4472" xml:space="preserve">Ut jam innoteſcat tempus, quo ſuperficies fluidi ex H L venit in <lb/>h l, vocabimus illud tempus t: </s> + <s xml:id="echoid-s4473" xml:space="preserve">quia autem eſt dt = {-dx/v}, erit, poſito <lb/>pro v valore modo invento, <lb/>dt = {-gdx/m√x - n√a - x} <lb/>Poteſt quidem hæc formula immediate rationalis fieri ponendo x = {4aqq/(1 + qq)<emph style="super">2</emph>}, <lb/>atque deinde debito modo conſtrui: </s> + <s xml:id="echoid-s4474" xml:space="preserve">Iſta vero methodus paullo prolixior eſt <lb/>hâc altera, qua quantitas reducenda dividitur in duo membra ſeorſim inte-<lb/>granda, nempe præmiſſa æquatio non differt ab hâc: <lb/></s> + <s xml:id="echoid-s4475" xml:space="preserve">dt = {mgdx√x/nna - (mm + nn) x} + {ngdx√a - x/nna - (mm + nn) x}: </s> + <s xml:id="echoid-s4476" xml:space="preserve"><lb/>Et autem ſ{mgdx√x/nna - (mm + nn) x} = - {2mg/mm + nn}√x + {mng√a/(mm + nn)√(mm + nn)} X <lb/>log.</s> + <s xml:id="echoid-s4477" xml:space="preserve">{n√a + √mm + nn√x/n√a - √mm + nn√x}; </s> + <s xml:id="echoid-s4478" xml:space="preserve">alteriusque membri integrale <lb/>nempe ſ{ngdx√a - x/nna - (mm + nn) x} fit = {-2ng/mm + nn}√(a - x) + <lb/>{mng√a/(mm + nn) X √(mm + nn)} log. </s> + <s xml:id="echoid-s4479" xml:space="preserve">{m√a + √mm + nn X √a - x/m√a - √mm + nn X √a - x}; </s> + <s xml:id="echoid-s4480" xml:space="preserve"><lb/>Patet exinde addita debita conſtante fore <lb/>t = {2mg√a - b - 2mg√x + 2ng√b - 2ng√a - x/mm + nn} + <lb/>{mng√a/(mm + nn) X √(mm + nn)} X +<pb o="155" file="0169" n="169" rhead="SECTIO OCTAVA."/> +log. </s> + <s xml:id="echoid-s4481" xml:space="preserve">{mna + (mm + nn) X √(ax - xx) + m√(mm + nn)√ax + n√(mm + nn)√(aa - ax)/mna + (mm + nn) X √(ax - xx) - m√(mm + nn)√ax - n√(mm + nn)√(aa - ax)} <lb/>- {mng√a/(mm + nn) X √(mm + nn)} X <lb/>log. </s> + <s xml:id="echoid-s4482" xml:space="preserve">{mna + (mm + nn) X √(ab - bb) + m√(mm + nn)√(aa - ab) + n√(mm + nn)√ab/mna + (mm + nn) X √(ab - bb) - m√(mm + nn)√(aa - ab) - n√(mm + nn)√ab}:</s> + <s xml:id="echoid-s4483" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4484" xml:space="preserve">§. </s> + <s xml:id="echoid-s4485" xml:space="preserve">23. </s> + <s xml:id="echoid-s4486" xml:space="preserve">Ex paragrapho 19. </s> + <s xml:id="echoid-s4487" xml:space="preserve">liquet ſuperficiem h l in ſitu ſuo permanere <lb/>cum eſt B h (= x) = {nna/mm + nn}. </s> + <s xml:id="echoid-s4488" xml:space="preserve">At vero ſi in æquatione integrata præce-<lb/>dentis paragraphi ponitur x = {nna/mm + nn}, fit denominator in quantitate lo-<lb/>garithmicali = o, ipſaque proinde quantitas infinita: </s> + <s xml:id="echoid-s4489" xml:space="preserve">tempus igitur totius <lb/>motus infinities majus eſt, quam cujuscunque partis.</s> + <s xml:id="echoid-s4490" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4491" xml:space="preserve">Sed ut alium inſuper caſum determinemus, videbimus quanto tempo-<lb/>re ſuperficies aquæ ex infimo ſitu M N (poſito nempe b = o) aſcendat quan-<lb/>titate {1/2} a, poſito m:</s> + <s xml:id="echoid-s4492" xml:space="preserve">n = 4:</s> + <s xml:id="echoid-s4493" xml:space="preserve">3. </s> + <s xml:id="echoid-s4494" xml:space="preserve">fit autem <lb/>t = {8g√a - 14g√{1/2}a/25} + {12g√a/125} log. </s> + <s xml:id="echoid-s4495" xml:space="preserve">({49 + 35√2/49 - 35√2}) - {12g√a/125} log. </s> + <s xml:id="echoid-s4496" xml:space="preserve">- 4, ſeu <lb/>t = {8g√a - 7g√2a/25} + {12g√a/125} log. </s> + <s xml:id="echoid-s4497" xml:space="preserve">({49 + 35√2/140√2 - 196}), <lb/>id eſt, proxime t = {15g/100} X 2√a, quod indicat, eſſe tempus iſtud ad tem-<lb/>pus quo grave libere cadit per altitudinem B M proxime ut 15g ad 100: <lb/></s> + <s xml:id="echoid-s4498" xml:space="preserve">Pariter tempus deſcenſus invenitur, ſi ab initio ſuperficies h l fuerit ultra ſitum <lb/>æquilibrii poſita. </s> + <s xml:id="echoid-s4499" xml:space="preserve">Fuerit v. </s> + <s xml:id="echoid-s4500" xml:space="preserve">gr. </s> + <s xml:id="echoid-s4501" xml:space="preserve">utrumque vas aquis totum repletum, orificia <lb/>autem M & </s> + <s xml:id="echoid-s4502" xml:space="preserve">N rationem nunc habeant quæ eſt inter 3 & </s> + <s xml:id="echoid-s4503" xml:space="preserve">4, ſitque tempus <lb/>determinandum, quo ſuperficies ex B deſcendat per dimidiam B M: </s> + <s xml:id="echoid-s4504" xml:space="preserve">hypothe-<lb/>ſes hæ faciunt m = 3; </s> + <s xml:id="echoid-s4505" xml:space="preserve">n = 4; </s> + <s xml:id="echoid-s4506" xml:space="preserve">b = a, atque x = {1/2}a, ita vero fit <lb/>t = {8g√a - 7g√2a/25} + {12g√a/125} log. </s> + <s xml:id="echoid-s4507" xml:space="preserve">({49 + 35√2/49 - 35√2}) - {12g√a/125} log. </s> + <s xml:id="echoid-s4508" xml:space="preserve">- 4. </s> + <s xml:id="echoid-s4509" xml:space="preserve">Ex <lb/>quo apparet in utroque exemplo idem eſſe tempus.</s> + <s xml:id="echoid-s4510" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4511" xml:space="preserve">§. </s> + <s xml:id="echoid-s4512" xml:space="preserve">24. </s> + <s xml:id="echoid-s4513" xml:space="preserve">Priusquam deſcendamus ad vaſa multifida indagaſſe conveniet, <lb/>quænam aquæ quantitas per utrumque orificium M & </s> + <s xml:id="echoid-s4514" xml:space="preserve">N fluat, dum ſuperfi-<lb/>cies aquæ ex ſitu H L venit in h l. </s> + <s xml:id="echoid-s4515" xml:space="preserve">Et primo quidem, quod ad orificium M +<pb o="156" file="0170" n="170" rhead="HYDRODYNAMICÆ"/> +pertinet, perſpicuum eſt quantitatem aquæ dato tempuſculo (dt) per illud <lb/>transfluentem proportionalem eſſe velocitati (√x) ductæ in magnitudinem <lb/>orificii (m) ipſumque tempuſculum d t, ita ut hæc quantitas ſit <lb/>(ob dt = {gdx/m√x - n√a - x} per §. </s> + <s xml:id="echoid-s4516" xml:space="preserve">22.) </s> + <s xml:id="echoid-s4517" xml:space="preserve">= {-mgdx√x/m√x - n√a - x}, <lb/>atque proinde omnis quantitas quæ ab initio effluxerit <lb/>= - ſ{mgdx√x/m√x - n√a - x}. </s> + <s xml:id="echoid-s4518" xml:space="preserve">Eſt autem - ſ{mgdx√x/m√x - n√a - x} = <lb/>{mnga/(m + n)<emph style="super">2</emph>} log. </s> + <s xml:id="echoid-s4519" xml:space="preserve">({ma - mb - nb/mx + nx - na}) + {mg/m + n} X (a - b - x).</s> + <s xml:id="echoid-s4520" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4521" xml:space="preserve">Eodem modo eruitur quantitas aquæ interea per orificium N effluen-<lb/>tis (quæ ſcilicet eſt = - ſ{ngdx√a - x/m√x - n√a - x}) = <lb/>{mnga/(m + n)<emph style="super">2</emph>} log. </s> + <s xml:id="echoid-s4522" xml:space="preserve">({ma - mb - nb/mx + nx - na}) - {ng/m + n} X (a - b - x).</s> + <s xml:id="echoid-s4523" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4524" xml:space="preserve">Atque inde etiam innoteſcit quantitas aquæ, quæ in A B affunditur, ne-<lb/>que enim differt ab illa, quæ per M transfluit: </s> + <s xml:id="echoid-s4525" xml:space="preserve">aqua denique in vaſe B N col-<lb/>lecta exprimitur per g (a - b - x,) & </s> + <s xml:id="echoid-s4526" xml:space="preserve">cum differentia ſumitur aquarum per <lb/>M & </s> + <s xml:id="echoid-s4527" xml:space="preserve">N transfluentium, oritur eadem iſta quantitas g (a - b - x).</s> + <s xml:id="echoid-s4528" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4529" xml:space="preserve">§. </s> + <s xml:id="echoid-s4530" xml:space="preserve">25. </s> + <s xml:id="echoid-s4531" xml:space="preserve">Prouti §. </s> + <s xml:id="echoid-s4532" xml:space="preserve">21. </s> + <s xml:id="echoid-s4533" xml:space="preserve">velocitatem ſuperficiei locum continue mutantis <lb/>determinavimus pro vaſe bifido, ita nunc in vaſis multifidis velocitatès ſingu-<lb/>larum ſuperficierum definiemus. </s> + <s xml:id="echoid-s4534" xml:space="preserve">Fuerit nempe altitudo ſuperficiei ſupremæ ſu-<lb/>pra proximam = x, altitudo hujus ſupra ſequentem = y, deinde = z, rur-<lb/>ſuſque altitudo proxima = s, & </s> + <s xml:id="echoid-s4535" xml:space="preserve">ſic porro. </s> + <s xml:id="echoid-s4536" xml:space="preserve">Amplitudines vero orificiorum <lb/>deſignentur per m, n, p, q. </s> + <s xml:id="echoid-s4537" xml:space="preserve">&</s> + <s xml:id="echoid-s4538" xml:space="preserve">c. </s> + <s xml:id="echoid-s4539" xml:space="preserve">amplitudines vaſis ſecundi, tertii, quarti &</s> + <s xml:id="echoid-s4540" xml:space="preserve">c. <lb/></s> + <s xml:id="echoid-s4541" xml:space="preserve">ſint M, N, P. </s> + <s xml:id="echoid-s4542" xml:space="preserve">&</s> + <s xml:id="echoid-s4543" xml:space="preserve">c. </s> + <s xml:id="echoid-s4544" xml:space="preserve">Sic patet fore velocitatem ſuperficiei ſecundæ = {m√x - n√y/M}; </s> + <s xml:id="echoid-s4545" xml:space="preserve"><lb/>veloc. </s> + <s xml:id="echoid-s4546" xml:space="preserve">ſuperf. </s> + <s xml:id="echoid-s4547" xml:space="preserve">tert. </s> + <s xml:id="echoid-s4548" xml:space="preserve">= {n√y - p√z/N}; </s> + <s xml:id="echoid-s4549" xml:space="preserve">velocit. </s> + <s xml:id="echoid-s4550" xml:space="preserve">ſuperfic. </s> + <s xml:id="echoid-s4551" xml:space="preserve">quartæ = {p√z - q√s/P} &</s> + <s xml:id="echoid-s4552" xml:space="preserve">c.</s> + <s xml:id="echoid-s4553" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4554" xml:space="preserve">Porro cum ſpatiola iiſdem tempuſculis à ſuperficiebus percurſa ſint ut <lb/>velocitates, apparet fic ſingulis momentis determinari ſitus iſtarum ſuperfi-<lb/>cierum, quamvis æquationes ſint intractabiles fere. </s> + <s xml:id="echoid-s4555" xml:space="preserve">Id ex ſe patet, ſi vel uni- +<pb o="157" file="0171" n="171" rhead="SECTIO OCTAVA."/> +ca ſuperficies extra ſitum æquilibrii, ſupra §. </s> + <s xml:id="echoid-s4556" xml:space="preserve">19. </s> + <s xml:id="echoid-s4557" xml:space="preserve">definiti poſita fuerit, fore ut <lb/>omnes reliquæ motibus reciprocis agitentur, donec poſt tempus infinitum in <lb/>priſtinum ſitum redierint ſimul.</s> + <s xml:id="echoid-s4558" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4559" xml:space="preserve">§. </s> + <s xml:id="echoid-s4560" xml:space="preserve">26. </s> + <s xml:id="echoid-s4561" xml:space="preserve">Sit porro vas ita formatum, ut oſtendit Fig. </s> + <s xml:id="echoid-s4562" xml:space="preserve">43. </s> + <s xml:id="echoid-s4563" xml:space="preserve">diviſum ſcilicet <lb/> +<anchor type="note" xlink:label="note-0171-01a" xlink:href="note-0171-01"/> +in duas partes A B E G & </s> + <s xml:id="echoid-s4564" xml:space="preserve">L Q N E inter ſe, mediante foramine M communi-<lb/>cantes; </s> + <s xml:id="echoid-s4565" xml:space="preserve">ſintque præterea foramina H & </s> + <s xml:id="echoid-s4566" xml:space="preserve">N per quæ aquæ exiliant, dum in A B <lb/>totidem affunduntur. </s> + <s xml:id="echoid-s4567" xml:space="preserve">Sint autem amplitudines in utroque vaſe veluti infinite <lb/>amplæ ratione foraminum M, H & </s> + <s xml:id="echoid-s4568" xml:space="preserve">N; </s> + <s xml:id="echoid-s4569" xml:space="preserve">Hiſque poſitis propoſitum ſit veloci-<lb/>tates invenire, quibus aquæ tam per H, quam per N ejiciantur ſeu altitudines <lb/>iſtis velocitatibus debitas. </s> + <s xml:id="echoid-s4570" xml:space="preserve">Erunt autem velocitates invariabiles, quia vas aquis <lb/>plenum conſervatur, ſimulque vaſis amplitudines reſpectu foraminum infini-<lb/>tæ cenſentur.</s> + <s xml:id="echoid-s4571" xml:space="preserve"/> +</p> +<div xml:id="echoid-div171" type="float" level="2" n="5"> +<note position="right" xlink:label="note-0171-01" xlink:href="note-0171-01a" xml:space="preserve">Fig. 43.</note> +</div> +<p> + <s xml:id="echoid-s4572" xml:space="preserve">Solutio iſtius problematis ex præcedentibus facile colligetur, ſi modo <lb/>concipiatur foramen M in duas diviſum partes o & </s> + <s xml:id="echoid-s4573" xml:space="preserve">p, quarum altera o aquas <lb/>foramini H, altera p foramini N mittat: </s> + <s xml:id="echoid-s4574" xml:space="preserve">partes autem o & </s> + <s xml:id="echoid-s4575" xml:space="preserve">p (quia per utram-<lb/>que eadem fluunt velocitate aquæ) eam habebunt rationem, quam inter ſe ha-<lb/>bent quantitates aquarum eodem tempore per H & </s> + <s xml:id="echoid-s4576" xml:space="preserve">N effluentium, id eſt, ra-<lb/>tionem compoſitam ex ratione amplitudinis H ad amplitudinem N & </s> + <s xml:id="echoid-s4577" xml:space="preserve">veloci-<lb/>tatis in H ad velocitatem in N. </s> + <s xml:id="echoid-s4578" xml:space="preserve">Quibus præmonitis perſpicuum eſt, fi amplitu-<lb/>dines foraminum M, H & </s> + <s xml:id="echoid-s4579" xml:space="preserve">N indicentur per α, β, γ, altitudines autem velo-<lb/>citatibus in H & </s> + <s xml:id="echoid-s4580" xml:space="preserve">N debitæ deſignentur per x & </s> + <s xml:id="echoid-s4581" xml:space="preserve">y, ipſæque proinde velocitates <lb/>per √x & </s> + <s xml:id="echoid-s4582" xml:space="preserve">√y fore amplitudinem o = {β√x/β√x + γ√y} α & </s> + <s xml:id="echoid-s4583" xml:space="preserve">amplitudinem <lb/>p = {γ√y/β√x + γ√y} α.</s> + <s xml:id="echoid-s4584" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4585" xml:space="preserve">Ponatur nunc altitudo ſuperficiei A B ſupra orificium H = a, & </s> + <s xml:id="echoid-s4586" xml:space="preserve">habebi-<lb/>tur@, ut demonſtratum fuit §. </s> + <s xml:id="echoid-s4587" xml:space="preserve">4. </s> + <s xml:id="echoid-s4588" xml:space="preserve">ſi quadratum foraminis o dividatur per ſum-<lb/>mam quadratorum foraminum o & </s> + <s xml:id="echoid-s4589" xml:space="preserve">H & </s> + <s xml:id="echoid-s4590" xml:space="preserve">quod oritur multiplicetur per a; </s> + <s xml:id="echoid-s4591" xml:space="preserve">ſic <lb/>igitur fit x = {ααax/ααx + (β√x + γ√y)<emph style="super">2</emph>}, ex quo oritur hæc æquatio <lb/>(A) ααx + (β√x + γ√y)<emph style="super">2</emph> = ααa.</s> + <s xml:id="echoid-s4592" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4593" xml:space="preserve">Eodem modo ratione foraminum p & </s> + <s xml:id="echoid-s4594" xml:space="preserve">N, poſita altitudine A B ſupra <lb/>N = a + b, obtinetur hæc altera æquatio:</s> + <s xml:id="echoid-s4595" xml:space="preserve"> +<pb o="158" file="0172" n="172" rhead="HYDRODYNAMICÆ"/> +(B) ααy + (β√x + γ√y)<emph style="super">2</emph> = αα X (a + b).</s> + <s xml:id="echoid-s4596" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4597" xml:space="preserve">Subtractâ æquatione (B) ab æquatione (A) prodity = x + b, ex quo <lb/>ſequitur, ſi venæ ambæ verticaliter ſurſum dirigantur, utramque ad eundem lo-<lb/>cum aſſilire. </s> + <s xml:id="echoid-s4598" xml:space="preserve">Deinde ſi in æquatione (A) ſubſtituatur pro y valor ejus x + b, <lb/>erit <lb/>(C) ααx + (β√x + γ√x + b)<emph style="super">2</emph> = ααa, <lb/>unde deducitur valor ipſius x æquatione quadrata.</s> + <s xml:id="echoid-s4599" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4600" xml:space="preserve">§. </s> + <s xml:id="echoid-s4601" xml:space="preserve">27. </s> + <s xml:id="echoid-s4602" xml:space="preserve">Ex præcedentis paragraphi æquationibus ſequentes fluunt affe-<lb/>ctiones.</s> + <s xml:id="echoid-s4603" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4604" xml:space="preserve">I. </s> + <s xml:id="echoid-s4605" xml:space="preserve">Quia velocitas aquæ per M transfluentis eſt = {β√x + γ√y/α}, eritalti-<lb/>tudo generans hanc velocitatem = ({β√x + γ√y/α})<emph style="super">2</emph>; </s> + <s xml:id="echoid-s4606" xml:space="preserve">ſed ſi addantur æqua-<lb/>tiones (A) & </s> + <s xml:id="echoid-s4607" xml:space="preserve">(B) fit: <lb/></s> + <s xml:id="echoid-s4608" xml:space="preserve">({β√x + γ√y/α})<emph style="super">2</emph> = {2a + b - x - y/2} = ob(y = x + b)a - x.</s> + <s xml:id="echoid-s4609" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4610" xml:space="preserve">II. </s> + <s xml:id="echoid-s4611" xml:space="preserve">Si foramen H ſit valde exiguum ratione foraminum M & </s> + <s xml:id="echoid-s4612" xml:space="preserve">N, id eſt, ſi <lb/>β poſſit cenſeri nulla ratione α & </s> + <s xml:id="echoid-s4613" xml:space="preserve">γ, abit æquatio (C) in hanc <lb/>ααx + γγx + γγb = ααa, ſeu <lb/>x = {ααa - γγb/αα + γγ};</s> + <s xml:id="echoid-s4614" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4615" xml:space="preserve">Id vero egregie convenit cum paragrapho decimo nono, cum manife-<lb/>ſtum ſit aquam per foramen valde exiguum ad eandem altitudinem aſſilire, <lb/>quam haberet aqua, ſi hæc laminam L Q tantum deorſum premat, quantum <lb/>ab aqua interna ſurſum premitur; </s> + <s xml:id="echoid-s4616" xml:space="preserve">Iſta vero præfata altitudo vi paragraphi 19. <lb/></s> + <s xml:id="echoid-s4617" xml:space="preserve">eſt {ααa - γγb/αα + γγ}; </s> + <s xml:id="echoid-s4618" xml:space="preserve">Eſt porro in iſta hypotheſi altitudo velocitatis aquarum in N <lb/>ſeu x + b = {ααa + ααb/αα + γγ} <lb/>& </s> + <s xml:id="echoid-s4619" xml:space="preserve">denique altitudo velocitatis aquarum in M, ſeu <lb/>a - x = {γγa + γγb/αα + γγ}; </s> + <s xml:id="echoid-s4620" xml:space="preserve"><lb/>quæ poſteriores æquationes in iſto caſu particulari pariter ex §. </s> + <s xml:id="echoid-s4621" xml:space="preserve">19. </s> + <s xml:id="echoid-s4622" xml:space="preserve">immediate <lb/>colligi aut prævideri potuiſſent.</s> + <s xml:id="echoid-s4623" xml:space="preserve"/> +</p> +<pb o="159" file="0173" n="173" rhead="SECTIO OCTAVA."/> +<p> + <s xml:id="echoid-s4624" xml:space="preserve">III. </s> + <s xml:id="echoid-s4625" xml:space="preserve">Si vero nunc alterum foramen N admodum exiguum præ ambo-<lb/>bus reliquis ponatur, erit facto γ = o <lb/>x = {ααa/αα + ββ}; </s> + <s xml:id="echoid-s4626" xml:space="preserve">deinde <lb/>x + b = {ααa + ααb + ββb/αα + ββ}, & </s> + <s xml:id="echoid-s4627" xml:space="preserve"><lb/>a - x = {ββa/αα + ββ}.</s> + <s xml:id="echoid-s4628" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4629" xml:space="preserve">IV. </s> + <s xml:id="echoid-s4630" xml:space="preserve">Si γγb = ααa, fit x = o. </s> + <s xml:id="echoid-s4631" xml:space="preserve">Nullam igitur in hoc caſu preſſionem <lb/>ſuſtinent partes laminæ L Q: </s> + <s xml:id="echoid-s4632" xml:space="preserve">imo inferiora verſus premitur, ſi γ ſit majus <lb/>quam {ααa/b}, & </s> + <s xml:id="echoid-s4633" xml:space="preserve">lamina nullibi ſit perforata.</s> + <s xml:id="echoid-s4634" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4635" xml:space="preserve">Iſta vero omnia ſimiliter ex §. </s> + <s xml:id="echoid-s4636" xml:space="preserve">19. </s> + <s xml:id="echoid-s4637" xml:space="preserve">facile colliguntur.</s> + <s xml:id="echoid-s4638" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4639" xml:space="preserve">V. </s> + <s xml:id="echoid-s4640" xml:space="preserve">Ita quoque ope ejusdem paragraphi ſine calculo novo prævideri po-<lb/>tuiſſet, quid fieri debeat, cum poſitis foraminibus H & </s> + <s xml:id="echoid-s4641" xml:space="preserve">N in eadem altitudi-<lb/>ne ſumma foraminum eorum, ceu unicum amplitudinis β + γ conſiderari <lb/>poteſt: </s> + <s xml:id="echoid-s4642" xml:space="preserve">Indicant nempe tam §. </s> + <s xml:id="echoid-s4643" xml:space="preserve">19. </s> + <s xml:id="echoid-s4644" xml:space="preserve">quam §. </s> + <s xml:id="echoid-s4645" xml:space="preserve">26. </s> + <s xml:id="echoid-s4646" xml:space="preserve">eſſe <lb/>x = {ααa/αα + (β + γ)<emph style="super">2</emph>},</s> +</p> +<p> + <s xml:id="echoid-s4647" xml:space="preserve">VI. </s> + <s xml:id="echoid-s4648" xml:space="preserve">Notari etiam poteſt, cum valor ipſius x fit imaginarius, id pro-<lb/>venire ex eo, quod aquæ non ſolum non effluant, in aliquibus caſibus per <lb/>H, ſed quod ſuperficies L Q etiam deſcendat; </s> + <s xml:id="echoid-s4649" xml:space="preserve">unde fieri poteſt, ut infra <lb/>orificium M deſcendat, quo ipſo ceſſat aqua@um contiguitas contra hypothe-<lb/>ſin propoſitionis. </s> + <s xml:id="echoid-s4650" xml:space="preserve">Si autem valor x eſt realis, tum dupliciter exprimitur, <lb/>ſed alter valor inutilis eſt reputandus; </s> + <s xml:id="echoid-s4651" xml:space="preserve">ſic igitur cavendum ne præpoſtera <lb/>radix ceu utilis aſſumatur.</s> + <s xml:id="echoid-s4652" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4653" xml:space="preserve">VII. </s> + <s xml:id="echoid-s4654" xml:space="preserve">Denique ut caſum ſpecialiſſimum attingamus, ponemus om-<lb/>nia foramina inter ſe æqualia, & </s> + <s xml:id="echoid-s4655" xml:space="preserve">prodibit 5xx + (2b - 6a) x = - aa + <lb/>2ab - bb, ſeu x = {3a - b - 2√ (aa + ab - bb)/5}; </s> + <s xml:id="echoid-s4656" xml:space="preserve">atque ſi fuerit præterea <lb/>a = 3b, erit x = (proxime) {4/15} b, deinde altitudo velocitatis in forami-<lb/>ne N ſeu x + b = {19/15}b atque altitudo velocitati in M debita ſeu a - x = {41/15}b. <lb/></s> + <s xml:id="echoid-s4657" xml:space="preserve">Sunt itaque velocitates ſeu etiam, quia foramina æqualia ſunt, quantitates +<pb o="160" file="0174" n="174" rhead="HYDRODYNAMICÆ"/> +aquarum iisdem temporibus per foramina M, H & </s> + <s xml:id="echoid-s4658" xml:space="preserve">N transfluentium proxi-<lb/>me ut √ 41. </s> + <s xml:id="echoid-s4659" xml:space="preserve">2. </s> + <s xml:id="echoid-s4660" xml:space="preserve">& </s> + <s xml:id="echoid-s4661" xml:space="preserve">√ 19.</s> + <s xml:id="echoid-s4662" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4663" xml:space="preserve">§. </s> + <s xml:id="echoid-s4664" xml:space="preserve">28. </s> + <s xml:id="echoid-s4665" xml:space="preserve">Ex his omnibus patet methodus determinandi motum in fluidis <lb/>tum etiam, cum quantitas virium vivarum non conſervatur; </s> + <s xml:id="echoid-s4666" xml:space="preserve">& </s> + <s xml:id="echoid-s4667" xml:space="preserve">ſimili modo <lb/>ſemper abſolvetur computus, quoties ex natura ſubjectæ quæſtionis præſu-<lb/>mi poteſt (uti in quæſtionibus hujus ſectionis accurate potuit) quantum vis <lb/>vivæ ſingulis momentis inutilis ad motum determinandum evaneſcat. </s> + <s xml:id="echoid-s4668" xml:space="preserve">Neque <lb/>enim ſoli ſunt caſus, quos adhuc examinavimus: </s> + <s xml:id="echoid-s4669" xml:space="preserve">lubet itaque alium addere, <lb/>qui oſcillationes fluidorum ſpectat, ut innoteſcat quantum inde decremen-<lb/>tum excurſiones fluidi capiant.</s> + <s xml:id="echoid-s4670" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4671" xml:space="preserve">Sint duo tubi amplitudine æquales & </s> + <s xml:id="echoid-s4672" xml:space="preserve">cylindrici A L & </s> + <s xml:id="echoid-s4673" xml:space="preserve">B H (Fig. </s> + <s xml:id="echoid-s4674" xml:space="preserve">44.) <lb/></s> + <s xml:id="echoid-s4675" xml:space="preserve"> +<anchor type="note" xlink:label="note-0174-01a" xlink:href="note-0174-01"/> +verticaliter inſerti vaſi ampliſſimo horizontali A B O P. </s> + <s xml:id="echoid-s4676" xml:space="preserve">Sit vas iſtud totum <lb/>aqua repletum: </s> + <s xml:id="echoid-s4677" xml:space="preserve">tubi autem aquam habeant usque in C & </s> + <s xml:id="echoid-s4678" xml:space="preserve">F; </s> + <s xml:id="echoid-s4679" xml:space="preserve">deinde ſubla-<lb/>to æquilibrio hæreat altera ſuperficies in G altera in E; </s> + <s xml:id="echoid-s4680" xml:space="preserve">moxque aqua ſibi re-<lb/>licta moveri incipiat. </s> + <s xml:id="echoid-s4681" xml:space="preserve">His poſitis tantum deberet ſuperficies G deſcendere <lb/>infra locum C, alteraque ſuperficies E aſcendere ſupra F, quanta eſt altitudo <lb/>G C ſeu E F ſi omnis vis viva conſervaretur (ab impedimentis frictionum aliis-<lb/>que ſimilibus nunc animum abſtrahimus): </s> + <s xml:id="echoid-s4682" xml:space="preserve">Verum patet, vim vivam omnis <lb/>aquæ per A in vas horizontale fluentis abſumi ſine alio effectu ab aqua ibi-<lb/>dem ſtagnante, indeque ſequitur deſcenſum ſuperficiei G alteriusque aſcen-<lb/>ſum minorem fore, quam modo dictum fuit: </s> + <s xml:id="echoid-s4683" xml:space="preserve">id igitur decrementum nunc <lb/>explorabimus.</s> + <s xml:id="echoid-s4684" xml:space="preserve"/> +</p> +<div xml:id="echoid-div172" type="float" level="2" n="6"> +<note position="left" xlink:label="note-0174-01" xlink:href="note-0174-01a" xml:space="preserve">Fig. 44.</note> +</div> +<p> + <s xml:id="echoid-s4685" xml:space="preserve">Ponatur ad hunc finem ſuperficiem ex G perveniſſe in M, ponaturque <lb/>G M = x, G C = b, C A = a: </s> + <s xml:id="echoid-s4686" xml:space="preserve">erit B E = a - b, E N = x; </s> + <s xml:id="echoid-s4687" xml:space="preserve">M C = F N = <lb/>b - x; </s> + <s xml:id="echoid-s4688" xml:space="preserve">Deinde fiat altitudo debita velocitati ſuperficiei in M = v, in ſitu <lb/>proximo m = v + dv; </s> + <s xml:id="echoid-s4689" xml:space="preserve">eritque incrementum vis vivæ aquæ (dum ſuperficies <lb/>percurrunt elementa M m, N n, ſeu dx) = 2 adv, cui addenda eſt vis viva <lb/>guttulæ, quæ ab aqua vaſis horizontalis abſumitur, nempe v d x, & </s> + <s xml:id="echoid-s4690" xml:space="preserve">eritſum@ <lb/>ma 2adv + vdx æqualis deſcenſui actuali aquæ multiplicato per maſſam aquæ, <lb/>quod productum eſt æquale deſcenſui actuali guttulæ dx, multiplicato per <lb/>2b - 2x. </s> + <s xml:id="echoid-s4691" xml:space="preserve">Eſt igitur <lb/>2adv + vdx = 2bdx - 2xdx.</s> + <s xml:id="echoid-s4692" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4693" xml:space="preserve">Hæc vero æquatio recte integrata abit in hanc +<pb o="161" file="0175" n="175" rhead="SECTIO OCTAVA."/> +v = 4a + 2b - 2x - c<emph style="super">{- x/2a}</emph> X (2b + 4a)</s> +</p> +<p> + <s xml:id="echoid-s4694" xml:space="preserve">unde ſi ponatur 4a + 2b - 2x - c<emph style="super">{- x/2a}</emph> X (2b + 4a) = o, <lb/>dabit valor ipſius x totam excurſionem, à qua ſi auferatur b, reſiduum indi-<lb/>cabit deſcenſum infra punctum æquilibrii C.</s> + <s xml:id="echoid-s4695" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4696" xml:space="preserve">§. </s> + <s xml:id="echoid-s4697" xml:space="preserve">29. </s> + <s xml:id="echoid-s4698" xml:space="preserve">Ut vero exemplo quodam appareat, quantum hâc ratione oſ-<lb/>cillationes diminuantur, ponemus a = b, facta ſcilicet C A = G C & </s> + <s xml:id="echoid-s4699" xml:space="preserve">B E = o.</s> + <s xml:id="echoid-s4700" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4701" xml:space="preserve">Ita oritur <lb/>3a - x = c<emph style="super">{- x/2a}</emph> X (3a) ſive <lb/>c<emph style="super">{x/2a}</emph> = {3a/3a - x} vel x = 2a log. </s> + <s xml:id="echoid-s4702" xml:space="preserve">3a/3a - x}, <lb/>cui æquationi prope admodum ſatisfacit valor x = {7/4} a. </s> + <s xml:id="echoid-s4703" xml:space="preserve">Eſt igitur decremen-<lb/>tum excurſionis ſeu a - b = quartæ parti elevationis fluidi ſupra punctum me-<lb/>dium: </s> + <s xml:id="echoid-s4704" xml:space="preserve">ſi majus obſervetur experimento, reliquum adhæſioni aquæ ad latera <lb/>tuborum tribuendum erit.</s> + <s xml:id="echoid-s4705" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4706" xml:space="preserve">§. </s> + <s xml:id="echoid-s4707" xml:space="preserve">30. </s> + <s xml:id="echoid-s4708" xml:space="preserve">Neque iſta diminutarum excurſionum ratio plane, ut ſuſpicor, <lb/>auferetur, ſi vel æqualis fiat amplitudinis tubus horizontalis cum verticalibus, <lb/>ob mutatam fluidi directionem in punctis A & </s> + <s xml:id="echoid-s4709" xml:space="preserve">B.</s> + <s xml:id="echoid-s4710" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4711" xml:space="preserve">Cæterum infiniti alii fingi poſſent caſus iiſdem principiis ſolvendi, velu-<lb/>ti ſi natura oſcillationum indaganda ſit in vaſe Fig. </s> + <s xml:id="echoid-s4712" xml:space="preserve">44. </s> + <s xml:id="echoid-s4713" xml:space="preserve">cum id in parte horizon-<lb/>tali diaphragmate in duas diſpeſcitur partes ſolo lumine, quod diaphragma ha-<lb/>beat, inter ſe communicantes & </s> + <s xml:id="echoid-s4714" xml:space="preserve">hujuſmodi alii. </s> + <s xml:id="echoid-s4715" xml:space="preserve">Puto autem hæc jam ſufficere, <lb/>ut quiſque ſibi facile regulas generales pro iſtiuſmodi quæſtionibus ſolvendis <lb/>formare poſſit.</s> + <s xml:id="echoid-s4716" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div174" type="section" level="1" n="135"> +<head xml:id="echoid-head176" xml:space="preserve">EXPERIMENTA</head> +<head xml:id="echoid-head177" style="it" xml:space="preserve">Ad ſectionem octavam pertinentia.</head> +<head xml:id="echoid-head178" xml:space="preserve">Experimentum 1.</head> +<p> + <s xml:id="echoid-s4717" xml:space="preserve">PAragraphum quartum, quo dicitur altitudinem velocitati aquæ per <lb/>orificium D effluentis (Fig. </s> + <s xml:id="echoid-s4718" xml:space="preserve">37.) </s> + <s xml:id="echoid-s4719" xml:space="preserve">eſſe {mmx/nn + mm} eo confirmavi modo, ut +<pb o="162" file="0176" n="176" rhead="HYDRODYNAMICÆ"/> +utrumque orificium G & </s> + <s xml:id="echoid-s4720" xml:space="preserve">D limbum haberet inſtar Zonulæ paullulum <lb/>elevatum, ne contractioni venarum locus eſſet, tutumque fieri poſſet judi-<lb/>cium à quantitate aquæ dato tempore effluentis ad velocitates. </s> + <s xml:id="echoid-s4721" xml:space="preserve">Deinde ſum-<lb/>tis accuratè menſuris, obſervatoque tempore quo ſuperficies per datum ſpa-<lb/>tium A P deſcenderet, vidi tempus iſtud recte reſpondere velocitatibus dicto <lb/>paragrapho definitis: </s> + <s xml:id="echoid-s4722" xml:space="preserve">obſervavi etiam nihilo mutari motum ab elevatione <lb/>aut depreſſione diaphragmatis. </s> + <s xml:id="echoid-s4723" xml:space="preserve">Reliqua ad experimentum pertinentia me-<lb/>moria exciderunt, neque ea in chartam conjeci: </s> + <s xml:id="echoid-s4724" xml:space="preserve">ſuperfluum autem duxi ex-<lb/>perimentum repetere, quod unicuique facile erit imitari: </s> + <s xml:id="echoid-s4725" xml:space="preserve">fundamentum au-<lb/>tem id eſt reliquis, quæ adeoque ulteriori diſquiſitione experimentali vix <lb/>opus habent: </s> + <s xml:id="echoid-s4726" xml:space="preserve">volui tamen ſequentia præterea tentare.</s> + <s xml:id="echoid-s4727" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div175" type="section" level="1" n="136"> +<head xml:id="echoid-head179" xml:space="preserve">Experimentum 2.</head> +<p> + <s xml:id="echoid-s4728" xml:space="preserve">Vaſe uſus|ſum, quale fere adhibuit Mariottus (vid. </s> + <s xml:id="echoid-s4729" xml:space="preserve">fig. </s> + <s xml:id="echoid-s4730" xml:space="preserve">38.) </s> + <s xml:id="echoid-s4731" xml:space="preserve">rurſusque <lb/>confirmavi æquationem noſtram hunc in modum: </s> + <s xml:id="echoid-s4732" xml:space="preserve">feci ut aquæ per orificium <lb/>D horizontaliter effluerent, tuncque menſuras cepi altitudinis orificii D ſu-<lb/>pra pavimentum & </s> + <s xml:id="echoid-s4733" xml:space="preserve">diſtantiam loci, ubi vena in pavimentum incidebat à <lb/>puncto in eodem pavimento, cui orificium D verticaliter imminebat; </s> + <s xml:id="echoid-s4734" xml:space="preserve">Inde <lb/>cognovi altitudinem velocitati aquæ in D effluentis debitam: </s> + <s xml:id="echoid-s4735" xml:space="preserve">eandem autem <lb/>hanc altitudinem experimento proxime inveneram, quam theoria hujus ſe-<lb/>ctionis indicat §. </s> + <s xml:id="echoid-s4736" xml:space="preserve">IV. </s> + <s xml:id="echoid-s4737" xml:space="preserve">Similia experimenta apponam in fine experimentorum <lb/>ad ſectionem duodecimam pertinentium, quæ ſimul theoriam noſtram hy-<lb/>draulico - ſtaticam confirmabunt.</s> + <s xml:id="echoid-s4738" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4739" xml:space="preserve">Denique cum multa ſint in §. </s> + <s xml:id="echoid-s4740" xml:space="preserve">§. </s> + <s xml:id="echoid-s4741" xml:space="preserve">26. </s> + <s xml:id="echoid-s4742" xml:space="preserve">& </s> + <s xml:id="echoid-s4743" xml:space="preserve">27. </s> + <s xml:id="echoid-s4744" xml:space="preserve">quæ ſingulari calculo eruta <lb/>fuerunt, operæ pretium erit de illis quoque experimenta ſumere, præſertim <lb/>cum alia ſimul eadem opera ſumi poterunt experimenta, quæ in ſect. </s> + <s xml:id="echoid-s4745" xml:space="preserve">XII. <lb/></s> + <s xml:id="echoid-s4746" xml:space="preserve">recenſebuntur, ſi vas, quale Fig. </s> + <s xml:id="echoid-s4747" xml:space="preserve">43. </s> + <s xml:id="echoid-s4748" xml:space="preserve">ſiſtit, ad hunc finem fieri curetur.</s> + <s xml:id="echoid-s4749" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4750" xml:space="preserve">Cæterum hæc theoria etiam confirmatur experimentis in Sectione Septima <lb/>recenſitis, quæ de oſcillationibus fluidorum in tubos per foramina influen-<lb/>tium ſumſi.</s> + <s xml:id="echoid-s4751" xml:space="preserve"/> +</p> +<pb file="0177" n="177" rhead="(163)"/> +</div> +<div xml:id="echoid-div176" type="section" level="1" n="137"> +<head xml:id="echoid-head180" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph></head> +<head xml:id="echoid-head181" xml:space="preserve"><emph style="bf">SECTIO NONA.</emph></head> +<head xml:id="echoid-head182" style="it" xml:space="preserve">De motu fluidorum, quæ non proprio pondere, <lb/>ſed potentia aliena ejiciuntur, ubi præſertim de <lb/>Machinis Hydraulicis earundemque ultimo qui da-<lb/>ri poteſt perfectionis gradu, & quomodo mecha-<lb/>nica tam ſolidorum quam fluidorum ulterius perſici poſsit.</head> +<head xml:id="echoid-head183" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s4752" xml:space="preserve">IN hâc ſectione, qua Machinas examinare hydraulicas, uſumque <lb/>earum, quantum fieri poteſt, perficere potiſſimum conſtitui, <lb/>animum abſtrahemus à variationibus motus, quæ originem du-<lb/>cunt à potentia vel inertia fluidi interni, quia ut vidimus mo-<lb/>tus aquæ internæ tantum non æquabilis eſt à primo fere fluxus initio, ſi ori-<lb/>ficium exile ſit, uti eſt in Machinis hydraulicis plerisque ratione amplitudi-<lb/>num internarum. </s> + <s xml:id="echoid-s4753" xml:space="preserve">Res enim foret ridicula in rebus practicis ſollicitos eſſe <lb/>de mutationibus, quæ primis fluxus momentis fiunt, quasque jam determi-<lb/>navimus in ſectione quarta, quod ibi operæ pretium eſſe poterat ut omnis <lb/>theoriæ vis inde eluceſceret. </s> + <s xml:id="echoid-s4754" xml:space="preserve">Igitur durante toto motu, brevitatis gratiâ, po-<lb/>nemus aquam conſtanter velocitate expelli, quæ ſe habeat ut radix potentiæ <lb/>internæ prementis, poſtquam hæc potentia ad pondus cylindri aquei foramini <lb/>ſuperincumbentis reducta fuerit: </s> + <s xml:id="echoid-s4755" xml:space="preserve">nam quæcunque fuerit iſta potentia, con-<lb/>ſiderandum erit pondus cylindri verticalis aquei ſuperficiei aqueæ internæ ſu-<lb/>perincumbentis, atque altitudo iſtius cylindri dabit altitudinem velocitati <lb/>aquæ exilientis debitam, ſi modo nulla adſint obſtacula extrinſeca, & </s> + <s xml:id="echoid-s4756" xml:space="preserve">aqua <lb/>ex@vaſe ampliſſimo ejiciatur. </s> + <s xml:id="echoid-s4757" xml:space="preserve">Hoc ita intelligendum eſt, ut ſi operculum <lb/>A B pondere P oneratum (Fig. </s> + <s xml:id="echoid-s4758" xml:space="preserve">45.) </s> + <s xml:id="echoid-s4759" xml:space="preserve">aquam per orificium F expellat, pon-<lb/> +<anchor type="note" xlink:label="note-0177-01a" xlink:href="note-0177-01"/> +dus autem P æquale ſit ponderi cylindri aquei H A B I, tunc vena aquea F G <lb/>altitudinem H I attingere debeat.</s> + <s xml:id="echoid-s4760" xml:space="preserve"/> +</p> +<div xml:id="echoid-div176" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0177-01" xlink:href="note-0177-01a" xml:space="preserve">Fig. 45.</note> +</div> +<pb o="164" file="0178" n="178" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div178" type="section" level="1" n="138"> +<head xml:id="echoid-head184" xml:space="preserve">Definitiones.</head> +<p> + <s xml:id="echoid-s4761" xml:space="preserve">§. </s> + <s xml:id="echoid-s4762" xml:space="preserve">2. </s> + <s xml:id="echoid-s4763" xml:space="preserve">Per potentiam moventem deinceps intelligam principium illud agens, <lb/>quod conſiſtit in pondere, preſſione animata aliisve hujuscemodi viribus, <lb/>uti dicuntur, mortuis.</s> + <s xml:id="echoid-s4764" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4765" xml:space="preserve">Productum autem quod oritur à multiplicatione potentiæ iſtius moventis <lb/>per ejusdem velocitatem æque ac tempus durante quo preſſionem ſuam exe-<lb/>rit, deſignabo per potentiam abſolutam. </s> + <s xml:id="echoid-s4766" xml:space="preserve">Vel quia productum ex velocitate & </s> + <s xml:id="echoid-s4767" xml:space="preserve"><lb/>tempore proportionale eſt ſimpliciter ſpatio percurſo, licebit etiam potentiam <lb/>abſolutam colligere ex potentia mouente multiplicata per ſpatium, quod eadem <lb/>percurrit. </s> + <s xml:id="echoid-s4768" xml:space="preserve">Id vero productum ideo voco potentiam abſolutam, quia ex illo de-<lb/>mum æſtimandi ſunt labores hominum operariorum in elevandis aquis exant-<lb/>lati, quod mox demonſtratum dabo in regulis, quæ mihi in hanc rem ob-<lb/>ſervatæ fuerunt. </s> + <s xml:id="echoid-s4769" xml:space="preserve">Interim viſæ mihi fuerunt machinæ hydraulicæ commode ſe <lb/>reduci pati ad duo genera, quorum alterum aquas cum impetu ejicit, alte-<lb/>rum de loco in locum placide veluti transportat. </s> + <s xml:id="echoid-s4770" xml:space="preserve">Utrumque ordine ſuo <lb/>pertractabo genus & </s> + <s xml:id="echoid-s4771" xml:space="preserve">denique ſub finem quædam addam de diverſis poten-<lb/>tiis moventibus.</s> + <s xml:id="echoid-s4772" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div179" type="section" level="1" n="139"> +<head xml:id="echoid-head185" style="it" xml:space="preserve">(A) De machinis aquas cum impetu in altum projicientibus.</head> +<head xml:id="echoid-head186" xml:space="preserve">Regula 1.</head> +<p> + <s xml:id="echoid-s4773" xml:space="preserve">§. </s> + <s xml:id="echoid-s4774" xml:space="preserve">3. </s> + <s xml:id="echoid-s4775" xml:space="preserve">Labores hominum operariorum, qui machinis hydraulicis pro <lb/>aquis elevandis apponuntur, æſtimandi ſunt ex potentia abſoluta, id eſt, ex <lb/>potentia movente ſeu preſſione quam exerunt, ex tempore & </s> + <s xml:id="echoid-s4776" xml:space="preserve">ex velocitate <lb/>puncti, cui potentia movens applicatur.</s> + <s xml:id="echoid-s4777" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div180" type="section" level="1" n="140"> +<head xml:id="echoid-head187" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s4778" xml:space="preserve">(α) De potentia movente res eſt perſpicua: </s> + <s xml:id="echoid-s4779" xml:space="preserve">labores enim cæteris omni-<lb/>bus paribus ſunt utique proportionales numero operariorum ſeu potentiæ mo-<lb/>venti. </s> + <s xml:id="echoid-s4780" xml:space="preserve">(β) Ratione temporis res eſt non minus manifeſta ex omnium cir-<lb/>cumſtantiarum replicatione, quæ ex duplicatione temporis oritur. </s> + <s xml:id="echoid-s4781" xml:space="preserve">(γ) De-<lb/>nique quod ad velocitatem attinet res ex eo eſt deducenda, quod ſive poten-<lb/>tiam moventem duplices, ſive ejus velocitatem non diverſus oriatur effectus, +<pb o="165" file="0179" n="179" rhead="SECTIO NONA."/> +nempe duplus ab utraque parte. </s> + <s xml:id="echoid-s4782" xml:space="preserve">Finge pondus P deſcenſu ſuo aquam per ori-<lb/>ficium F ejicere ad altitudinem F G: </s> + <s xml:id="echoid-s4783" xml:space="preserve">deinde manentibus reliquis duplica-<lb/>tum puta orificium F, & </s> + <s xml:id="echoid-s4784" xml:space="preserve">vides ad eandem altitudinem F G eodemque tem-<lb/>pore duplam aquæ quantitatem ejectum iri ab eadem potentia movente P, ſed <lb/>ea duplo celerius deſcendente. </s> + <s xml:id="echoid-s4785" xml:space="preserve">Pariter quantitas aquæ manentibus reliquis <lb/>duplicabitur, ſi & </s> + <s xml:id="echoid-s4786" xml:space="preserve">orificium F & </s> + <s xml:id="echoid-s4787" xml:space="preserve">amplitudinem A B & </s> + <s xml:id="echoid-s4788" xml:space="preserve">pondus ſeu potent. </s> + <s xml:id="echoid-s4789" xml:space="preserve">mo-<lb/>vent. </s> + <s xml:id="echoid-s4790" xml:space="preserve">P duplices, tunc vero velocitas hujus potentiæ duplicatæ invariata ma-<lb/>net. </s> + <s xml:id="echoid-s4791" xml:space="preserve">Igitur utroque modo effectus geminatur. </s> + <s xml:id="echoid-s4792" xml:space="preserve">Q. </s> + <s xml:id="echoid-s4793" xml:space="preserve">E. </s> + <s xml:id="echoid-s4794" xml:space="preserve">D.</s> + <s xml:id="echoid-s4795" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div181" type="section" level="1" n="141"> +<head xml:id="echoid-head188" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s4796" xml:space="preserve">§. </s> + <s xml:id="echoid-s4797" xml:space="preserve">4. </s> + <s xml:id="echoid-s4798" xml:space="preserve">Propoſitio præcedens non ſenſu phyſiologico ſed morali eſt in-<lb/>terpretanda: </s> + <s xml:id="echoid-s4799" xml:space="preserve">moraliter neque plus neque minus æſtimo laborem hominis, qui <lb/>eadem celeritate conatum duplum exercet, quam ejus qui eodem conatu ce-<lb/>leritatem duplicat, quia nempe uterque eundem edit effectum, fieri tamen po-<lb/>teſt, ut alterius labor, quamvis altero non minus robuſti, ſenſu phyſiologi-<lb/>co ſit admodum major. </s> + <s xml:id="echoid-s4800" xml:space="preserve">Si quis conatu 20. </s> + <s xml:id="echoid-s4801" xml:space="preserve">librarum ſingulis minutis primis ſpa-<lb/>tium 200. </s> + <s xml:id="echoid-s4802" xml:space="preserve">ped. </s> + <s xml:id="echoid-s4803" xml:space="preserve">faciat, is facile conatum geminabit, difficillime vero velocita-<lb/>tem. </s> + <s xml:id="echoid-s4804" xml:space="preserve">Ex hoc conſequens eſt in omni machinarum genere diſpiciendum præ-<lb/>ſertim eſſe, quomodo debeant eſſe conſtitutæ, ut pro eodem tempore minima <lb/>hominum defatigatione productum ex conatu eorum in velocitatem omnium <lb/>maximum ſit: </s> + <s xml:id="echoid-s4805" xml:space="preserve">atque exinde patebit, quænam in ergatis longitudo vectibus <lb/>ſit tribuenda, quantus in rotis ſeu tympanis calcatoriis radius ſit faciendus, <lb/>quanta remis longitudo ſit concilianda, & </s> + <s xml:id="echoid-s4806" xml:space="preserve">ſic de aliis machinis.</s> + <s xml:id="echoid-s4807" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4808" xml:space="preserve">Ratione uſus autem tympanorum calcatoriorum, quæ frequentiſſime <lb/>adhibentur ut momentum noſtræ animadverſionis eo magis fiat perſpicuum, <lb/>hoc experimentum intelligatur:</s> + <s xml:id="echoid-s4809" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4810" xml:space="preserve">Ponamus in Fig. </s> + <s xml:id="echoid-s4811" xml:space="preserve">46. </s> + <s xml:id="echoid-s4812" xml:space="preserve">altitudinem verticalem multorum milliarum, ad <lb/> +<anchor type="note" xlink:label="note-0179-01a" xlink:href="note-0179-01"/> +quam homo dato tempore aſcendere debeat: </s> + <s xml:id="echoid-s4813" xml:space="preserve">tempus autem ſumemus decem <lb/>horarum, quia talis laboribus diurnis terminus eſſe ſolet, dein fingamus plu-<lb/>res vias, A C, A D &</s> + <s xml:id="echoid-s4814" xml:space="preserve">c. </s> + <s xml:id="echoid-s4815" xml:space="preserve">diverſe ad horizontalem B D inclinatas: </s> + <s xml:id="echoid-s4816" xml:space="preserve">His poſitis <lb/>intelligimus eò celerius viatori progrediendum eſſe, quo viam ſelegerit mi-<lb/>nus inclinatam, ut eodem tempore culmen montis A attingat, & </s> + <s xml:id="echoid-s4817" xml:space="preserve">patet viam <lb/>aliquam fore veluti A C, ſuper quâ minima defatigatione iter abſolvet, quan- +<pb o="166" file="0180" n="180" rhead="HYDRODYNAMICÆ"/> +doquidem nemo nec ſuper plano verticali incedere nec dato tempore viam in-<lb/>finitam abſolvere poteſt; </s> + <s xml:id="echoid-s4818" xml:space="preserve">Statuamus viam hanc minimæ defatigationis cum <lb/>horizontali angulum facere A C B 30. </s> + <s xml:id="echoid-s4819" xml:space="preserve">graduum.</s> + <s xml:id="echoid-s4820" xml:space="preserve"/> +</p> +<div xml:id="echoid-div181" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0179-01" xlink:href="note-0179-01a" xml:space="preserve">Fig. 46.</note> +</div> +<p> + <s xml:id="echoid-s4821" xml:space="preserve">Quod ſi ita ſit, erit tympanum calcatorium ita fabricandum, ut pon-<lb/>dus deſiderata velocitate ſuperetur, cum calcator perpetuo triginta gradib{us} à <lb/>puncto tympani infimo diſtat.</s> + <s xml:id="echoid-s4822" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4823" xml:space="preserve">Ex eodem principio etiam inter machinas diverſi generis ſelectus eſt <lb/>faciendus: </s> + <s xml:id="echoid-s4824" xml:space="preserve">ita v. </s> + <s xml:id="echoid-s4825" xml:space="preserve">gr. </s> + <s xml:id="echoid-s4826" xml:space="preserve">ſi in ergatis vectiarius potentiam exerat, ſeu preſſionem <lb/>horizontalem, quæ efficiat quartam ſui proprii ponderis partem, hocque niſu <lb/>ſingulis minutis primis ſpatium 200. </s> + <s xml:id="echoid-s4827" xml:space="preserve">ped. </s> + <s xml:id="echoid-s4828" xml:space="preserve">abſolvat, is fere ut puto eodem de-<lb/>fatigabitur modo, ac ſi eadem velocitate tympanum rotatorium ad angulum <lb/>30. </s> + <s xml:id="echoid-s4829" xml:space="preserve">grad. </s> + <s xml:id="echoid-s4830" xml:space="preserve">calcet; </s> + <s xml:id="echoid-s4831" xml:space="preserve">interim tamen pondus duplum eodem tempore ad eandem al-<lb/>titudinem hoc modo feret calcator, quia cæteris paribus preſſionem duplam <lb/>exerit.</s> + <s xml:id="echoid-s4832" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div183" type="section" level="1" n="142"> +<head xml:id="echoid-head189" xml:space="preserve">Regula 2.</head> +<p> + <s xml:id="echoid-s4833" xml:space="preserve">§. </s> + <s xml:id="echoid-s4834" xml:space="preserve">5. </s> + <s xml:id="echoid-s4835" xml:space="preserve">Exiſtente eadem potentia abſoluta dico omnes machinas, quæ <lb/>nullas patiuntur frictiones & </s> + <s xml:id="echoid-s4836" xml:space="preserve">quæ nullos motus ad propoſitum finem inutiles <lb/>generant, eundem effectum præſtare neque adeo unam alteri præferendam <lb/>eſſe.</s> + <s xml:id="echoid-s4837" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div184" type="section" level="1" n="143"> +<head xml:id="echoid-head190" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s4838" xml:space="preserve">Ex mechanicis conſtat machinam utcunque compoſitam reduci poſſe <lb/>ad vectem ſimplicem: </s> + <s xml:id="echoid-s4839" xml:space="preserve">igitur omnem machinationem hydraulicam repræſen-<lb/>tare licebit ſimplici antlia vecte inſtructa Fig. </s> + <s xml:id="echoid-s4840" xml:space="preserve">47. </s> + <s xml:id="echoid-s4841" xml:space="preserve">ubi nempe embolus ope ve-<lb/> +<anchor type="note" xlink:label="note-0180-01a" xlink:href="note-0180-01"/> +ctis M N mobilis circa punctum M detruditur, atque ſic aqua per orificium F <lb/>expellitur. </s> + <s xml:id="echoid-s4842" xml:space="preserve">At vero ſi potentia movens P vecti applicata intelligatur in N, vi-<lb/>demus ex præcedente propoſitione nihil lucri accedere potentiæ abſolutæ ab aucta <lb/>vel diminuta longitudine vectis M N: </s> + <s xml:id="echoid-s4843" xml:space="preserve">& </s> + <s xml:id="echoid-s4844" xml:space="preserve">certe quæcunque ſit iſta longitudo <lb/>fieri poteſt, ut potentia movens eadem atque invariata velocitate mota eandem <lb/>aquæ quantitatem eodem impetu expellat, ſi modo amplitudo antliæ A B ra-<lb/>tionem habeat conſtantem ad longitudinem vectis M N. </s> + <s xml:id="echoid-s4845" xml:space="preserve">Ex quibus perſpi-<lb/>cuum eſt, omnes machinas eadem potentia abſoluta eundem effectum præſtare, <lb/>ſi modo à frictionibus motibuſque ad deſtinatum finem inutilibus animus ab-<lb/>ſtrahatur.</s> + <s xml:id="echoid-s4846" xml:space="preserve"/> +</p> +<div xml:id="echoid-div184" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0180-01" xlink:href="note-0180-01a" xml:space="preserve">Fig. 47.</note> +</div> +<pb o="167" file="0181" n="181" rhead="SECTIO NONA."/> +</div> +<div xml:id="echoid-div186" type="section" level="1" n="144"> +<head xml:id="echoid-head191" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s4847" xml:space="preserve">§. </s> + <s xml:id="echoid-s4848" xml:space="preserve">6. </s> + <s xml:id="echoid-s4849" xml:space="preserve">Non deſunt qui putent machinam excogitari poſſe, cujus ope <lb/>minimo labore maxima aquæ quantitas ad quamcunque altitudinem elevari <lb/>poſſit, animumque excrucient, in anquirendis rotis, vectibus, ponderibus <lb/>appendendis: </s> + <s xml:id="echoid-s4850" xml:space="preserve">ſed operam perdunt, neque audiendi ſunt hujuſmodi promiſſo-<lb/>res, cum magni quid ſibi videntur inveniſſe: </s> + <s xml:id="echoid-s4851" xml:space="preserve">Optima machina eſt, ſi ſolum <lb/>ejus effectum reſpiciamus, quæ minimas patitur frictiones, nullosque gene-<lb/>rat motus inutiles, de quo utroque evitando præcepta trademus infrà,</s> +</p> +</div> +<div xml:id="echoid-div187" type="section" level="1" n="145"> +<head xml:id="echoid-head192" xml:space="preserve">Regula 3.</head> +<p> + <s xml:id="echoid-s4852" xml:space="preserve">§. </s> + <s xml:id="echoid-s4853" xml:space="preserve">7. </s> + <s xml:id="echoid-s4854" xml:space="preserve">In antliis, quales Figuris 45. </s> + <s xml:id="echoid-s4855" xml:space="preserve">& </s> + <s xml:id="echoid-s4856" xml:space="preserve">47. </s> + <s xml:id="echoid-s4857" xml:space="preserve">repræſentantur, in quibus ſu-<lb/>perficies aquæ interna A B in eadem propemodum altitudine eſt cum foramine <lb/>F, ſunt potentiæ abſolutæ pro iiſdem temporibus in triplicata ratione velocita-<lb/>tum aquarum exilientium.</s> + <s xml:id="echoid-s4858" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div188" type="section" level="1" n="146"> +<head xml:id="echoid-head193" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s4859" xml:space="preserve">Sunt enim potentiæ moventes in duplicata ratione velocitatum, quibus <lb/>aquæ per foramen F erumpunt & </s> + <s xml:id="echoid-s4860" xml:space="preserve">velocitates potentiarum moventium ſequuntur <lb/>ipſam rationem velocitatum aquarum exilientium: </s> + <s xml:id="echoid-s4861" xml:space="preserve">Sed pro iiſdem temporibus <lb/>ſunt potentiæ abſolutæ ut potentiæ moventes multiplicatæ per ſuas velocitates, <lb/>ergo patet propoſitio.</s> + <s xml:id="echoid-s4862" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div189" type="section" level="1" n="147"> +<head xml:id="echoid-head194" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s4863" xml:space="preserve">§. </s> + <s xml:id="echoid-s4864" xml:space="preserve">8. </s> + <s xml:id="echoid-s4865" xml:space="preserve">Sequitur ex iſta regula, ſi animus ſit aquam per foramen F ad <lb/>altitudinem F G elevare, magnam potentiæ abſolutæ partem ſine fructu perdi, <lb/>cum aquæ majori impetu erumpunt, quam quæ altitudini F G reſpondeat; <lb/></s> + <s xml:id="echoid-s4866" xml:space="preserve">fac enim aquas dupla velocitate expelli, requiretur potentia abſoluta octupla, <lb/>neque tamen ratione finis propoſiti effectus plus quam duplus eſt cenſendus, <lb/>quia nempe eodem tempore dupla aquarum quantitas elevatur: </s> + <s xml:id="echoid-s4867" xml:space="preserve">potuiſſetque <lb/>iſte effectus obtineri potentia abſoluta ſubquadrupla exprimendo aquas ſimplici <lb/>velocitate per foramen duplum; </s> + <s xml:id="echoid-s4868" xml:space="preserve">Hoc igitur nomine tres quartæ partes iſtius <lb/>potentiæ inutiliter impenſæ dicendæ ſunt. </s> + <s xml:id="echoid-s4869" xml:space="preserve">Originem hujus detrimenti indicavi <lb/>§. </s> + <s xml:id="echoid-s4870" xml:space="preserve">5. </s> + <s xml:id="echoid-s4871" xml:space="preserve">eaque conſiſtit in motu qui generatur ad propoſitum finem inutili: </s> + <s xml:id="echoid-s4872" xml:space="preserve">nem- +<pb o="168" file="0182" n="182" rhead="HYDRODYNAMICÆ"/> +pe omnis motus qui aquis reſiduus eſt poſtquam altitudinem G attigerunt in <lb/>noſtro caſu ſuperfluus eſt dicendus.</s> + <s xml:id="echoid-s4873" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div190" type="section" level="1" n="148"> +<head xml:id="echoid-head195" xml:space="preserve">Regula 4.</head> +<p> + <s xml:id="echoid-s4874" xml:space="preserve">§. </s> + <s xml:id="echoid-s4875" xml:space="preserve">9. </s> + <s xml:id="echoid-s4876" xml:space="preserve">Cum aquæ expelluntur per canalem D F (Fig. </s> + <s xml:id="echoid-s4877" xml:space="preserve">48.) </s> + <s xml:id="echoid-s4878" xml:space="preserve">habentque <lb/> +<anchor type="note" xlink:label="note-0182-01a" xlink:href="note-0182-01"/> +in orificio F velocitatem quæ debeatur altitudini verticali G F, eſt potentia abſo-<lb/>luta eodem tempore impenſa proportionalis velocitati aquæ in F ductæ in alti-<lb/>tudinem G ſupra A B.</s> + <s xml:id="echoid-s4879" xml:space="preserve"/> +</p> +<div xml:id="echoid-div190" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0182-01" xlink:href="note-0182-01a" xml:space="preserve">Fig. 48.</note> +</div> +</div> +<div xml:id="echoid-div192" type="section" level="1" n="149"> +<head xml:id="echoid-head196" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s4880" xml:space="preserve">Eſt enim potentia movens P proportionalis præfatæ altitudini & </s> + <s xml:id="echoid-s4881" xml:space="preserve">velo-<lb/>citas iſtius potentiæ eſt ut velocitas aquæ in F.</s> + <s xml:id="echoid-s4882" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div193" type="section" level="1" n="150"> +<head xml:id="echoid-head197" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s4883" xml:space="preserve">§. </s> + <s xml:id="echoid-s4884" xml:space="preserve">10. </s> + <s xml:id="echoid-s4885" xml:space="preserve">Pòtentiæ abſolutæ majori ratione creſcunt quam velocitates <lb/>aquarum effluentium, id eſt, quam quantitates eodem tempore ejectæ: </s> + <s xml:id="echoid-s4886" xml:space="preserve">atta-<lb/>men differentia rationum fere inſenſibilis eſt, cum altitudo F G parva admo-<lb/>dum eſt ratione altitudinis canalis F D: </s> + <s xml:id="echoid-s4887" xml:space="preserve">Sit ex. </s> + <s xml:id="echoid-s4888" xml:space="preserve">gr. </s> + <s xml:id="echoid-s4889" xml:space="preserve">F G æqualis {1/4} F D (negli-<lb/>gendo altitudinem B D) mox vero ejiciantur aquæ velocitate dupla, ita, ut <lb/>nunc ſit F D = F G; </s> + <s xml:id="echoid-s4890" xml:space="preserve">ſic erunt potentiæ abſolutæ ut 1 X {@/4} ad 2 X 2 ſeu ut 5 ad 16 <lb/>ſic ut ad ejiciendam duplam aquæ quantitatem potentia abſoluta requiratur pluſ-<lb/>quam tripla: </s> + <s xml:id="echoid-s4891" xml:space="preserve">Si vero F G ſtatuatur prius = {1/100} F D, & </s> + <s xml:id="echoid-s4892" xml:space="preserve">deinde aquæ rurſus <lb/>dupla velocitate exprimi ponantur, erunt nunc potentiæ abſolutæ ut 1 X 101 <lb/>ad 2 X 204 ſeu ut 101 ad 208, quæ ratio à ſubdupla parum deficit. </s> + <s xml:id="echoid-s4893" xml:space="preserve">Sequitur <lb/>inde, quo minori velocitate aquæ hauriantur, eo majori cum fructu potentiam <lb/>abſolutam impendi, & </s> + <s xml:id="echoid-s4894" xml:space="preserve">tunc demum eam propemodum omnem utiliter impen-<lb/>di, cum fere inſenſibili velocitate aquæ per orificium F effluunt: </s> + <s xml:id="echoid-s4895" xml:space="preserve">poterit au-<lb/>tem magnitudo orificii compenſare velocitatis exiguitatem, ut dato tempore <lb/>notabilis aquarum quantitas hauriri poſſit. </s> + <s xml:id="echoid-s4896" xml:space="preserve">Diſpendium potentiæ abſolutæ ſic de-<lb/>finietur.</s> + <s xml:id="echoid-s4897" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div194" type="section" level="1" n="151"> +<head xml:id="echoid-head198" xml:space="preserve">Regula 5.</head> +<p> + <s xml:id="echoid-s4898" xml:space="preserve">§. </s> + <s xml:id="echoid-s4899" xml:space="preserve">11. </s> + <s xml:id="echoid-s4900" xml:space="preserve">Conſtitutum fuerit ope antliæ A B D F, valvula in fundo in-<lb/>ſtructæ & </s> + <s xml:id="echoid-s4901" xml:space="preserve">aquæ impoſitæ, aquas ex loco humiliori A D in altiorem F trans-<lb/>fundere, fueritque velocitas media aquæ in F effluentis debita altitudini F G, +<pb o="169" file="0183" n="183" rhead="SECTIO NONA."/> +erit diſpendium potentiæ abſolutæ ad integram hanc potentiam ut F G ad alti-<lb/>tudinem G ſupra A B.</s> + <s xml:id="echoid-s4902" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div195" type="section" level="1" n="152"> +<head xml:id="echoid-head199" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s4903" xml:space="preserve">Fingamus augeri admodum orificium F diminuta in eadem ratione ve-<lb/>locitate aquarum effluentium in F; </s> + <s xml:id="echoid-s4904" xml:space="preserve">ſic non mutabitur quantitas aquæ dato <lb/>tempore effluentis, ſi velocitas potentiæ moventis eadem ſit, atque proinde idem <lb/>erit effectus. </s> + <s xml:id="echoid-s4905" xml:space="preserve">Sed ſi velocitas ita diminuatur, ut altitudo ipſi debita ſit inſenſi-<lb/>bilis, exprimetur potentia movens per altitudinem F ſupra A B, cum antea po-<lb/>tentia movens erat æqualis altitudini G ſupra A B; </s> + <s xml:id="echoid-s4906" xml:space="preserve">& </s> + <s xml:id="echoid-s4907" xml:space="preserve">cum in utroque caſu ea-<lb/>dem ſit velocitas potentiæ moventis, erunt potentiæ abſolutæ pro iiſdem tempori-<lb/>bus ut altitudo G ad altitudinem F ſupra communem A B. </s> + <s xml:id="echoid-s4908" xml:space="preserve">Igitur differentia <lb/>altitudinum G & </s> + <s xml:id="echoid-s4909" xml:space="preserve">F exprimet diſpendium, cum integra altitudo G ſupra A B <lb/>repræſentat totam potentiam abſolutam.</s> + <s xml:id="echoid-s4910" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4911" xml:space="preserve">§. </s> + <s xml:id="echoid-s4912" xml:space="preserve">12. </s> + <s xml:id="echoid-s4913" xml:space="preserve">Idem ratiocinium valet pro omni machinationum genere: </s> + <s xml:id="echoid-s4914" xml:space="preserve">Quo-<lb/>ties nempe aquæ in locum, ad quem elevandæ ſunt, evectæ notabilem habent <lb/>velocitatem, magnum fit potentiæ abſolutæ diſpendium: </s> + <s xml:id="echoid-s4915" xml:space="preserve">poſita enim altitudine <lb/>elevationis = A; </s> + <s xml:id="echoid-s4916" xml:space="preserve">altitudine debita velocitati aquarum in loco quo effundun-<lb/>tur = B, integra potentia abſoluta = P, perdetur {B/A + B} X P.</s> + <s xml:id="echoid-s4917" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4918" xml:space="preserve">Notari etiam poteſt, cum aquæ trans altitudinem aliquam, cujus cul-<lb/>men in F poſitum ſit, fundi debent ope antliæ tubo inſtructæ, continuandum <lb/>eſſe tubum D F inferiora verſus quantum id liceat, nec abrumpendum in F, <lb/>prouti id apparet ex Fig. </s> + <s xml:id="echoid-s4919" xml:space="preserve">49. </s> + <s xml:id="echoid-s4920" xml:space="preserve">Nam ſi v. </s> + <s xml:id="echoid-s4921" xml:space="preserve">gr. </s> + <s xml:id="echoid-s4922" xml:space="preserve">punctum F duplo altius poſitum ſit, <lb/> +<anchor type="note" xlink:label="note-0183-01a" xlink:href="note-0183-01"/> +quam extremitas tubi G, duplo major potentia abſoluta requiritur pro transfun-<lb/>dendis aquis per canalem abruptum in F, quam per continuatum uſque in G; <lb/></s> + <s xml:id="echoid-s4923" xml:space="preserve">ſi parvula utrobique velocitate effluant, cujus nempe altitudo genitrix parva <lb/>ſit ratione altitudinum F D vel G D.</s> + <s xml:id="echoid-s4924" xml:space="preserve"/> +</p> +<div xml:id="echoid-div195" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0183-01" xlink:href="note-0183-01a" xml:space="preserve">Fig. 49</note> +</div> +</div> +<div xml:id="echoid-div197" type="section" level="1" n="153"> +<head xml:id="echoid-head200" xml:space="preserve">Regula 6.</head> +<p> + <s xml:id="echoid-s4925" xml:space="preserve">§. </s> + <s xml:id="echoid-s4926" xml:space="preserve">13. </s> + <s xml:id="echoid-s4927" xml:space="preserve">Cum in antliis quas hucusque conſideravimus opercula A B <lb/>ſeu potius emboli non bene lateribus machinarum reſpondent, hiatus relin-<lb/>quitur, & </s> + <s xml:id="echoid-s4928" xml:space="preserve">ab hoc aliud diſpendii genus in potentiis abſolutis oritur, quod <lb/>in antliis, in quibus altitudo orificii ſuprà embolum negligi poteſt, +<pb o="170" file="0184" n="184" rhead="HYDRODYNAMICÆ"/> +ſic determinatur. </s> + <s xml:id="echoid-s4929" xml:space="preserve">Ut aggregatum ex foramine effluxus & </s> + <s xml:id="echoid-s4930" xml:space="preserve">prædicto hiatu, <lb/>ad eundem hiatum, ita potentia abſoluta, quæ impenditur, ad partem illius <lb/>quæ inutilis eſt, ſeu ad ejusdem diſpendium.</s> + <s xml:id="echoid-s4931" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div198" type="section" level="1" n="154"> +<head xml:id="echoid-head201" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s4932" xml:space="preserve">Nam aquæ per foramen & </s> + <s xml:id="echoid-s4933" xml:space="preserve">hiatum æqualiter premuntur, & </s> + <s xml:id="echoid-s4934" xml:space="preserve">æqualive-<lb/>locitate fluunt; </s> + <s xml:id="echoid-s4935" xml:space="preserve">perditur autem omnis potentia abſoluta, quæaquas per hiatum <lb/>cogit, & </s> + <s xml:id="echoid-s4936" xml:space="preserve">hæc ſe habet ad integram potentiam abſolutam, ut hiatus ad ſum-<lb/>mam foraminis & </s> + <s xml:id="echoid-s4937" xml:space="preserve">hiatus.</s> + <s xml:id="echoid-s4938" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div199" type="section" level="1" n="155"> +<head xml:id="echoid-head202" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s4939" xml:space="preserve">§. </s> + <s xml:id="echoid-s4940" xml:space="preserve">14. </s> + <s xml:id="echoid-s4941" xml:space="preserve">Convenit utique embolis uti bene formatis & </s> + <s xml:id="echoid-s4942" xml:space="preserve">politis; </s> + <s xml:id="echoid-s4943" xml:space="preserve">neceſſe <lb/>quoque eſt ut cavitas antliæ ſit plane cylindrica, ejusdemque latera pariter <lb/>perpolita. </s> + <s xml:id="echoid-s4944" xml:space="preserve">Vix autem crediderim, niſi id fiat alio fine, è re eſſe, ut embo-<lb/>li cavitates@ultima accuratione expleant, quia fortaſſe ſic majus oritur virium <lb/>diſpendium à frictionibus, quam ſi circumcirca parvulus relictus fuiſſet hia-<lb/>tus: </s> + <s xml:id="echoid-s4945" xml:space="preserve">Si enim hiatus ille centeſimam v. </s> + <s xml:id="echoid-s4946" xml:space="preserve">gr. </s> + <s xml:id="echoid-s4947" xml:space="preserve">partem foraminis effluxus efficiat, <lb/>vix amplius locus erit frictionibus & </s> + <s xml:id="echoid-s4948" xml:space="preserve">non niſi centeſima præterpropter poten-<lb/>tiæ abſolutæ pars inde perditur, & </s> + <s xml:id="echoid-s4949" xml:space="preserve">fortaſſe à frictione emboli cavitatem antliæ <lb/>exacte occupantis majus diſpendium oritur. </s> + <s xml:id="echoid-s4950" xml:space="preserve">Igitur hoc reſpectu non eſt quod <lb/>nimis ſollicite evitemus tranſitum aquæ per hiatum ab embolo relictum. </s> + <s xml:id="echoid-s4951" xml:space="preserve">Non <lb/>reſpicit autem hæc animadverſio illas machinas, in quibus emboli retractio-<lb/>ne aquæ in antliam attrahendæ ſunt. </s> + <s xml:id="echoid-s4952" xml:space="preserve">Hic enim juſta & </s> + <s xml:id="echoid-s4953" xml:space="preserve">plena emboli ma-<lb/>gnitudo omnino eſt neceſſaria.</s> + <s xml:id="echoid-s4954" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div200" type="section" level="1" n="156"> +<head xml:id="echoid-head203" xml:space="preserve">Regula 7.</head> +<p> + <s xml:id="echoid-s4955" xml:space="preserve">§. </s> + <s xml:id="echoid-s4956" xml:space="preserve">15. </s> + <s xml:id="echoid-s4957" xml:space="preserve">In machinis quæ plura habent foramina aquas transmittentia ex una <lb/>cavitate in alteram, aliquid de potentia abſoluta perditur, cujus rei rationem <lb/>in præcedente ſectione eſſe diximus, quod ſingularum guttularum ex una <lb/>cavitate in alteram per foramen commune fluentium aſcenſus potentialis perit.</s> + <s xml:id="echoid-s4958" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4959" xml:space="preserve">Quo plura ſunt & </s> + <s xml:id="echoid-s4960" xml:space="preserve">quo minora hujusmodi foramina, eo majus oritur <lb/>potentiæ abſolutæ diſpendium, quod magni momenti eſſe ſolet, idque fortaſſe <lb/>præter communem opinionem, in machinis, quas Vitruvius ab inventore +<pb o="171" file="0185" n="185" rhead="SECTIO NONA."/> +vocat, Cteſibianis. </s> + <s xml:id="echoid-s4961" xml:space="preserve">Loquor autem de foraminibus ita diſpoſitis, ut omnis <lb/>aqua effluxura per illa tranſire debeat. </s> + <s xml:id="echoid-s4962" xml:space="preserve">Iſtud jam detrimenti genus tali de-<lb/>finietur calculo.</s> + <s xml:id="echoid-s4963" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s4964" xml:space="preserve">Sit amplitudo foraminis ultimi aquas in aërem emittentis = n, ampli-<lb/>tudines autem reliquorum foraminum, per quæ aquæ trajiciuntur intra ma-<lb/>chinam, deſignentur litteris α, β, γ, &</s> + <s xml:id="echoid-s4965" xml:space="preserve">c. </s> + <s xml:id="echoid-s4966" xml:space="preserve">& </s> + <s xml:id="echoid-s4967" xml:space="preserve">erit, poſita utrobique eadem <lb/>potentia movente, altitudo debita velocitati aquæ effluentis ad ſimilem altitu-<lb/>dinem nullis obſtantibus foraminibus internis, ut <lb/>1 ad 1 + {nn/αα} + {nn/ββ} + {nn/γγ} + &</s> + <s xml:id="echoid-s4968" xml:space="preserve">c. </s> + <s xml:id="echoid-s4969" xml:space="preserve">(per §. </s> + <s xml:id="echoid-s4970" xml:space="preserve">11. </s> + <s xml:id="echoid-s4971" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s4972" xml:space="preserve">8.) </s> + <s xml:id="echoid-s4973" xml:space="preserve">ſequitur inde factis iſtis <lb/>altitudinibus inter ſe æqualibus, fore potentias moventes ut <lb/>1 + {nn/αα} + {nn/ββ} + {nn/γγ} + &</s> + <s xml:id="echoid-s4974" xml:space="preserve">c.</s> + <s xml:id="echoid-s4975" xml:space="preserve">ad 1, & </s> + <s xml:id="echoid-s4976" xml:space="preserve">quia utrobique velocitates potentiarum <lb/>moventium eædem ſunt, ſimilem quoque pro iisdem temporibus rationem <lb/>habebunt potentiæ abſolutæ. </s> + <s xml:id="echoid-s4977" xml:space="preserve">Superflua igitur eſt pars ejus {nn/αα} + {nn/ββ} + {nn/γγ} + &</s> + <s xml:id="echoid-s4978" xml:space="preserve">c. <lb/></s> + <s xml:id="echoid-s4979" xml:space="preserve">unde diſpendium potentiæ abſolutæ erit ad totam hanc potentiam ut <lb/>{nn/αα} + {nn/ββ} + {nn/γγ} + &</s> + <s xml:id="echoid-s4980" xml:space="preserve">c. </s> + <s xml:id="echoid-s4981" xml:space="preserve">ad 1 + {nn/αα} + {nn/ββ} + {nn/γγ} + &</s> + <s xml:id="echoid-s4982" xml:space="preserve">c.</s> + <s xml:id="echoid-s4983" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div201" type="section" level="1" n="157"> +<head xml:id="echoid-head204" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s4984" xml:space="preserve">§. </s> + <s xml:id="echoid-s4985" xml:space="preserve">16. </s> + <s xml:id="echoid-s4986" xml:space="preserve">Quoties idea machinæ foramina poſtulat, per quæ aquæ ex <lb/>uno modiolo in alium transfluant (quod fit in omni antliarum genere; </s> + <s xml:id="echoid-s4987" xml:space="preserve">velu-<lb/>ti aſpirantium, aſpirantes gallice aut prementium, foulantes &</s> + <s xml:id="echoid-s4988" xml:space="preserve">c.) </s> + <s xml:id="echoid-s4989" xml:space="preserve">ſunt illa <lb/>foramina quantum id reliquæ circumſtantiæ permittunt, ampliſſima facienda, <lb/>ita ut amplitudo orificii effluxus parva admodum ſit reſpectu illorum forami-<lb/>num internorum: </s> + <s xml:id="echoid-s4990" xml:space="preserve">Ut vero uſus regulæ clarius pateat, exempla conſidera-<lb/>bimus machinarum aliarum non minus uſitatarum.</s> + <s xml:id="echoid-s4991" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div202" type="section" level="1" n="158"> +<head xml:id="echoid-head205" xml:space="preserve">Exemplum 1.</head> +<p> + <s xml:id="echoid-s4992" xml:space="preserve">Propoſita ſit machina (quam repræſentat Figura 50.) </s> + <s xml:id="echoid-s4993" xml:space="preserve">in qua emboli C <lb/> +<anchor type="note" xlink:label="note-0185-01a" xlink:href="note-0185-01"/> +& </s> + <s xml:id="echoid-s4994" xml:space="preserve">F alternatim deprimuntur, atque per diabetas A B, D E aquæ in modio-<lb/>lum B E H intruduntur, ut ſic jactus fiat continuus per orificium H. </s> + <s xml:id="echoid-s4995" xml:space="preserve">Cum <lb/>hic emboli alternatim agant, alterutrum conſiderabimus quaſi ſolum ſed con-<lb/>tinue agentem; </s> + <s xml:id="echoid-s4996" xml:space="preserve">ita vero conſiderandum eſt foramen effluxus H, amplitudi- +<pb o="172" file="0186" n="186" rhead="HYDRODYNAMICÆ"/> +nis n, & </s> + <s xml:id="echoid-s4997" xml:space="preserve">alterutrum foraminum o, p, quibus ſingulis ſit amplitudo α; </s> + <s xml:id="echoid-s4998" xml:space="preserve">ita erit <lb/>diſpendium potentiæ abſolutæ = {nn/αα}, poſita potentiâ integra = 1 + {nn/aα}, <lb/>quæ quantitates ſunt ut n n ad n n + α α. </s> + <s xml:id="echoid-s4999" xml:space="preserve">Conſiderabile certe eſt hoc diſpen-<lb/>dium, ſi iconibus harum machinarum fidere licet, in quibus fæpe orificia <lb/>o & </s> + <s xml:id="echoid-s5000" xml:space="preserve">p minora ſunt orificio effluxus H, quod ſi foret plus quam dimidium <lb/>perderetur potentiæ abſolutæ. </s> + <s xml:id="echoid-s5001" xml:space="preserve">Erunt autem canales A B & </s> + <s xml:id="echoid-s5002" xml:space="preserve">D E per totum tra-<lb/>ctum, quantum id licet, amplificandi, ut machina parum de ſuâ præſtantia <lb/>perdat.</s> + <s xml:id="echoid-s5003" xml:space="preserve"/> +</p> +<div xml:id="echoid-div202" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0185-01" xlink:href="note-0185-01a" xml:space="preserve">Fig. 50.</note> +</div> +<p> + <s xml:id="echoid-s5004" xml:space="preserve">Ceterum fuit hæc machina excogitata, ut jactus fieret continuus per H. <lb/></s> + <s xml:id="echoid-s5005" xml:space="preserve">Quia tamen fieri non poteſt, quin aliquod temporis intervallum intercedat <lb/>inter ultimum emboli elevationis punctum, inſtantisque ejusdem depreſſio-<lb/>nis initium, non poterit jactus omnino eſſe continuus & </s> + <s xml:id="echoid-s5006" xml:space="preserve">æquabilis. </s> + <s xml:id="echoid-s5007" xml:space="preserve">Huic <lb/>vero incommodo optimum remedium attulit auctor machinæ illius, cujus <lb/>mentionem facit D. </s> + <s xml:id="echoid-s5008" xml:space="preserve">Perrault in Comment. </s> + <s xml:id="echoid-s5009" xml:space="preserve">ad Vitruvium pag. </s> + <s xml:id="echoid-s5010" xml:space="preserve">318. </s> + <s xml:id="echoid-s5011" xml:space="preserve">edit. </s> + <s xml:id="echoid-s5012" xml:space="preserve">2. </s> + <s xml:id="echoid-s5013" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s5014" xml:space="preserve"><lb/>quamque in Bibliotheca Regia Paris. </s> + <s xml:id="echoid-s5015" xml:space="preserve">aſſervari dicit; </s> + <s xml:id="echoid-s5016" xml:space="preserve">inſerviet nobis hæc ma-<lb/>china alterius exempli loco: </s> + <s xml:id="echoid-s5017" xml:space="preserve">figuram autem deſumam una cùm ejusdem de-<lb/>ſcriptione ex ipſo Perraultio.</s> + <s xml:id="echoid-s5018" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div204" type="section" level="1" n="159"> +<head xml:id="echoid-head206" xml:space="preserve">Exemplum 2.</head> +<p> + <s xml:id="echoid-s5019" xml:space="preserve">„ Machina eſt referente præfato Perraultio, in quâ aqua expellitur ex <lb/>modiolo A (Fig. </s> + <s xml:id="echoid-s5020" xml:space="preserve">51.) </s> + <s xml:id="echoid-s5021" xml:space="preserve">mediante embolo B in catinum F G, ex quo aër, <lb/> +<anchor type="note" xlink:label="note-0186-01a" xlink:href="note-0186-01"/> +ſi modo aliquid aquæ jam adſit, egredi non valet; </s> + <s xml:id="echoid-s5022" xml:space="preserve">quia tubus E F us-<lb/>que ad fundum fere deſcendit: </s> + <s xml:id="echoid-s5023" xml:space="preserve">ſic enim fit, ut aqua propulſa ex modio-<lb/>lo A per diabeten D, imumque catini occupans claudat orificium tubæ in <lb/>F, aërique tranſitum neget. </s> + <s xml:id="echoid-s5024" xml:space="preserve">Igitur cum embolus novas intrudit aquas <lb/>in mediolum, partim aëre partim aqua, repletum, hæ aquæ de novo af-<lb/>fuſæ vim exerunt in utrumque fluidum, & </s> + <s xml:id="echoid-s5025" xml:space="preserve">cum aqua non poſſit exilire <lb/>per tubum F E eadem velocitate qua irruit ex antlia per diabeten D, quia <lb/>ſcilicet (ſunt verba Perraultii) tubus F E in extremitate ſua E orificio per-<lb/>forata eſt multo minori, quam eſt orificium tubi D, aqua in catino ac-<lb/>cumulata aërem comprimit, ab eodemque reciproce preſſa, etiam dum <lb/>embolus elevatur, per tubam F E exilit.”</s> + <lb/></p><p> + <s xml:id="echoid-s5026" xml:space="preserve">Perditur in hâc machina magna potentiæ abſolutæ pars à tranſitu aquæ per dia-<lb/>beten D, hocque diſpendium eo majus erit, quo anguſtior eſt iſte tubulus:</s> + <s xml:id="echoid-s5027" xml:space="preserve"> +<pb o="173" file="0187" n="187" rhead="SECTIO NONA."/> +fiat igitur amplus aut etiam plures tubi conſtruantur aquas transmittentes: </s> + <s xml:id="echoid-s5028" xml:space="preserve">ma-<lb/>joris eſt momenti hæc annotatio in præſenti caſu, quod multo majus diſpen-<lb/>dium ab anguſtia diabetes D oritur, quam in aliis machinis; </s> + <s xml:id="echoid-s5029" xml:space="preserve">fac enim am-<lb/>plitudinem hujus diabetes eandem, quæ eſt orificio E, & </s> + <s xml:id="echoid-s5030" xml:space="preserve">pone inſuper æqua-<lb/>libus temporis intervallis embolum deprimi, retrahique non perdetur jam ſo-<lb/>lum dimidia potentiæ abſolutæ pars, ut aliàs, ſed plane quatuor quintæ partes <lb/>inutiles fient. </s> + <s xml:id="echoid-s5031" xml:space="preserve">Quia vero multa ſunt in hâc machina, quæ ſingularem po-<lb/>ſtulant calculum, placet illam ſeorſim perluſtrare.</s> + <s xml:id="echoid-s5032" xml:space="preserve"/> +</p> +<div xml:id="echoid-div204" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0186-01" xlink:href="note-0186-01a" xml:space="preserve">Fig. 51.</note> +</div> +</div> +<div xml:id="echoid-div206" type="section" level="1" n="160"> +<head xml:id="echoid-head207" style="it" xml:space="preserve">Digreſſus continens aliquas commentationes in Ma-<lb/>chinam Hydraulicam quam repræſent at figura 51.</head> +<p> + <s xml:id="echoid-s5033" xml:space="preserve">(α) Non poteſt jactus aqueus per E eſſe omnino æquabill<unsure/>s, durante <lb/>tota emboli agitatione: </s> + <s xml:id="echoid-s5034" xml:space="preserve">Dum enim embolus elevatur, novæ aquæ non acce-<lb/>dunt, atque ſic diminuitur quantitas aquæ in catino G E contentæ, aërque <lb/>eidem ſuperincumbens dilatatur ac denique elater ipſius diminuitur: </s> + <s xml:id="echoid-s5035" xml:space="preserve">hinc <lb/>quoque velocitate continue minori aqua erumpit donec rurſus ab embolo <lb/>intruſo acceleretur.</s> + <s xml:id="echoid-s5036" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5037" xml:space="preserve">Verum ſi ponatur ſpatium, quod aër in catino occupat longe ma-<lb/>jus ſpatio illo ab aqua, quæ durante una emboli elevatione ejicitur, occu-<lb/>pato, ceſſat fere tota hæc inæqualitas, poſito embolum uniformiter agitari <lb/>& </s> + <s xml:id="echoid-s5038" xml:space="preserve">diu ante fuiſſe agitatum, quæ poſterior hypothſis ideo neceſſaria eſt, quod <lb/>primæ agitatione valde differant à ſequentibus. </s> + <s xml:id="echoid-s5039" xml:space="preserve">Igitur brevitatis ergo om-<lb/>nibus hiſce hypotheſibus ſatitfaciemus, ideſt, ubique ſtatum, qui dicitur, <lb/>permanentiæ ponemus.</s> + <s xml:id="echoid-s5040" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5041" xml:space="preserve">(β) Cum igitur primis emboli agitationibus ſenſim augeatur velocitas <lb/>aquæ per E effluentis, mox fit ut jactus aqueus velocitatem tantum non in-<lb/>tegram attingat; </s> + <s xml:id="echoid-s5042" xml:space="preserve">quo rei ſtatu poſito, patet tantum aquæ depreſſ<unsure/>ione em-<lb/>boli impelli in catinum, quantum ex eodem tota emboli agitatione ejicitur.</s> + <s xml:id="echoid-s5043" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5044" xml:space="preserve">Primis autem agitationibus plus intruditur, quam ejicitur, idque non <lb/>ideo, ut putavit Dn. </s> + <s xml:id="echoid-s5045" xml:space="preserve">Perrault, quod orificium in E altero in G minus ſit <lb/>(idemque enim ſuccederet ſi vel majus eſſet) ſed quod cauſa efficiens non p of-<lb/>ſit ſtatim omnem ſuum exerere effectum in ejiciendis aquis.</s> + <s xml:id="echoid-s5046" xml:space="preserve"/> +</p> +<pb o="174" file="0188" n="188" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s5047" xml:space="preserve">(γ) Videbitur forta@@e rem non ſatis perluſtrantibus fore, ut omni-<lb/>bus in ſtatu permanente jam poſitis, nullisque præſentibus obſtaculis alienis, <lb/>aqua per foramen E velocitate exiliat, qua aſcendere poſſit ad altitudinem co-<lb/>lumnæ aqueæ in æquilibrio poſitam cum preſſione emboli: </s> + <s xml:id="echoid-s5048" xml:space="preserve">atque ita ſane fo-<lb/>ret, ſi preſſio emboli ſine interr uptione adeſſet, nullusque in aqua aſcenſus po-<lb/>tentialis perderetur: </s> + <s xml:id="echoid-s5049" xml:space="preserve">quia vero in utroque res aliter ſe habet, non poteſt non <lb/>alia oriri in jactu aqueo velocitatis æſtimatio: </s> + <s xml:id="echoid-s5050" xml:space="preserve">Hinc quiſque non obſcure videt <lb/>animum advertendum eſſe ad temporum rationem, quibus embolus deprimi-<lb/>tur, retrahiturque, tum etiam ad rationem amplitudinum in canaliculo D & </s> + <s xml:id="echoid-s5051" xml:space="preserve"><lb/>orificio E.</s> + <s xml:id="echoid-s5052" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5053" xml:space="preserve">(δ) Ponamus igitur tempus quo embolus deprimitur = θ tempus <lb/>unius integræ agitationis = t, amplitudinem orificii E = μ, & </s> + <s xml:id="echoid-s5054" xml:space="preserve">diabetes D = m: <lb/></s> + <s xml:id="echoid-s5055" xml:space="preserve">deinde comparata potentia embolum detrudente cum ſuperincumbente colum-<lb/>na aquea, faciamus hujus columnæ altitudinem = a, altitudinem vero aquæ <lb/>exilientis velocitati debitam = x. </s> + <s xml:id="echoid-s5056" xml:space="preserve">His ita ad calculum præparatis licebit duo-<lb/>bus indagare modis rationem quæ futura ſit inter velocitates aquarum in <lb/>orificio E & </s> + <s xml:id="echoid-s5057" xml:space="preserve">diabete D, atque hinc valorem incognitæ x; </s> + <s xml:id="echoid-s5058" xml:space="preserve">elicere. </s> + <s xml:id="echoid-s5059" xml:space="preserve">Primò enim <lb/>patet tempore θ (quo ſcilicet embolus detruditur) tantum aquæ fluere per <lb/>diabeten D, quantum tempore t (quo embolus deprimitur retrahiturque) ef-<lb/>fluit per E. </s> + <s xml:id="echoid-s5060" xml:space="preserve">Eſt igitur velocitas in D ad velocitatem in E ut {1/mθ} ad {1/μt}: </s> + <s xml:id="echoid-s5061" xml:space="preserve">& </s> + <s xml:id="echoid-s5062" xml:space="preserve"><lb/>quum poſterior hæc velocitas ſit = √ x, erit altera = {μt/mθ} √ x. </s> + <s xml:id="echoid-s5063" xml:space="preserve">Secundò quia <lb/>velocitas aquæ effluentis debetur preſſioni aëris in catino, ſequitur hanc preſ-<lb/>ſionem æquivalere ponderi columnæ aqueæ altitudinis x; </s> + <s xml:id="echoid-s5064" xml:space="preserve">ſed ſi à preſſione <lb/>emboli auferas preſſionem aëris, habebis preſſionem, quæ velocitatem aquæ <lb/>in D generet; </s> + <s xml:id="echoid-s5065" xml:space="preserve">hinc quia differentia preſſionum exprimitur per a - x, repræ-<lb/>ſentabitur velocitas aquæ in D per √ (a - x); </s> + <s xml:id="echoid-s5066" xml:space="preserve">Igitur nunc eſt velocitas aquæ <lb/>in D ad velocitatem aquæ in orificio E ut √ (a - x) ad √ x. </s> + <s xml:id="echoid-s5067" xml:space="preserve">Combinatis ratio-<lb/>nibus utroque modo inventis, fit <lb/>√ (a - x):</s> + <s xml:id="echoid-s5068" xml:space="preserve">√x = {1/mθ}: </s> + <s xml:id="echoid-s5069" xml:space="preserve">{1/μt}, ſive <lb/>x = {mmθθ/mmθθ + μμtt} X a.</s> + <s xml:id="echoid-s5070" xml:space="preserve"/> +</p> +<pb o="175" file="0189" n="189" rhead="SECTIO NONA."/> +<p> + <s xml:id="echoid-s5071" xml:space="preserve">Patet ex iſta æquatione altitudinem jactus duplici titulo deficere ab alti-<lb/>tudine columnæ prementis a, magis nempe deficit, cum celerius deprimitur, <lb/>tardiuſve elevatur embolus tum etiam cum orificium E ratione canaliculi D <lb/>amplitudine creſcit. </s> + <s xml:id="echoid-s5072" xml:space="preserve">Fuerit v. </s> + <s xml:id="echoid-s5073" xml:space="preserve">gr. </s> + <s xml:id="echoid-s5074" xml:space="preserve">amplitudo iſtius orificii æqualis amplitudini <lb/>tubuli D atque pari celeritate embolus deprimatur eleveturque & </s> + <s xml:id="echoid-s5075" xml:space="preserve">prodibit <lb/>x = {1/5} a, ſic ut ad quintam partem tantum aſſurgat vena effluens altitudinis a.</s> + <s xml:id="echoid-s5076" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5077" xml:space="preserve">(ε) Diſpendium potentiæ abſolutæ jam hoc modo eruetur, poſito prius <lb/>nullum laborem in elevandum embolum impendi. </s> + <s xml:id="echoid-s5078" xml:space="preserve">Sit velocitas quâ embolus <lb/>deprimitur = v, & </s> + <s xml:id="echoid-s5079" xml:space="preserve">erit potentia abſoluta tempore unius agitationis integræ im-<lb/>penſa = a v θ (per paragraphum tertium) quia vero effectus in eo conſi-<lb/>ſtit, ut effluxus fiat per E durante tempore t ipſaque aqua ad altitudinem <lb/>{mmθθ/mmθθ + μμ tt} X a elevetur, potuiſſet id antlia ſimplex figuræ quadrageſimæ <lb/>quintæ efficere, ſi pro potentia premente in illa ſumtus fuiſſet cylindrus aqueus <lb/>altitudinis {mmθθ/mmθθ + μμtt} X a, atque hæc potentia durante tempore t velocitate <lb/>{θ/t} v egiſſet; </s> + <s xml:id="echoid-s5080" xml:space="preserve">unde potentia abſoluta in hâc machina ſimplici, qua nihil de illa <lb/>perditur, requiſita futura fuiſſet = <lb/>{mmθθ/mmθθ + μμtt} X a X {θ/t} v X t = {mmθθ/mmθθ + μμtt} X a v θ. <lb/></s> + <s xml:id="echoid-s5081" xml:space="preserve">Eſt igitur tota potentia abſoluta ad partem ejus inutiliter perditam ut a v θ ad <lb/>a v θ - {mmθθ/mmθθ + μμtt} X a v θ ſeu ut mm θθ + μμtt ad μμtt. </s> + <s xml:id="echoid-s5082" xml:space="preserve">Igitur ſi in-<lb/>tegra potentia abſoluta deſignetur per P, erit ejus diſpendium = {μμtt/mmθθ + μμtt} X P.</s> + <s xml:id="echoid-s5083" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5084" xml:space="preserve">Neceſſe igitur eſt in hâc præ aliis antliis, ut diabetes amplitudine ad-<lb/>modum ſuperet orificium E, vel ut multiplex adſit. </s> + <s xml:id="echoid-s5085" xml:space="preserve">Si enim unicus adeſſet, <lb/>isque amplitudine orificio E æqualis, ſimulque uniformi velocitate ſurſum de-<lb/>orſumque agitari ponatur embolus, diſpendium oriretur quatuor quintarum <lb/>totius partium: </s> + <s xml:id="echoid-s5086" xml:space="preserve">atque ſi vel duplo amplior fiat, etiamnum perdetur dimi-<lb/>dium potentiæ abſolutæ.</s> + <s xml:id="echoid-s5087" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5088" xml:space="preserve">(ς) Denique perſpicuum eſt minorem preſſionem ſuſtinere latera catini <lb/>G E, quam modioli A A, quippe preſſiones iſtæ ſint ut x ad a, id eſt, ut +<pb o="176" file="0190" n="190" rhead="HYDRODYNAMICÆ"/> +mmθθ + μμtt ad m m θ θ, ex qua ratione artifices judicabunt de firmitate <lb/>laterum, quæ pro utroque requiritur.</s> + <s xml:id="echoid-s5089" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div207" type="section" level="1" n="161"> +<head xml:id="echoid-head208" xml:space="preserve">Regula 8.</head> +<p> + <s xml:id="echoid-s5090" xml:space="preserve">§. </s> + <s xml:id="echoid-s5091" xml:space="preserve">17. </s> + <s xml:id="echoid-s5092" xml:space="preserve">Quando embolus in antliis retrahitur & </s> + <s xml:id="echoid-s5093" xml:space="preserve">aqua in modiolum in-<lb/>fluit, non ſolum proprio pondere ſolicitata ſed maximam partem ab embo-<lb/>lo attracta, tunc omnis potentia abſoluta in hanc attractionem impenſa caſu <lb/>ſupervenit, quia antlia, ſub aquis, ut fit, poſita, ſua ſponte impleretur ſi ſuf-<lb/>ficiens huic impletioni tempus concederetur; </s> + <s xml:id="echoid-s5094" xml:space="preserve">nec adeoque attractio illa ita <lb/>pertinet ad ejiciendas aquas certa cum velocitate, quin tota vitari poſſit, hoc-<lb/>que nomine labor in illam impenſus mihi inutilis dicitur.</s> + <s xml:id="echoid-s5095" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5096" xml:space="preserve">Quia vero influxus aquarum partim proprio pondere fit, partim <lb/>etiam elevatione emboli, non poteſt diſpendium potentiæ abſolutæ ab effectu <lb/>æſtimari: </s> + <s xml:id="echoid-s5097" xml:space="preserve">Quin potius calculus ita eſt ponendus, ut poſitis potentia embo-<lb/>lum in certo ſitu elevante = π, velocitate emboli = v, tempuſculoque <lb/>quantitatibus π & </s> + <s xml:id="echoid-s5098" xml:space="preserve">v reſpondente d t, dicatur omnis potentia abſoluta in eleva-<lb/>tionem emboli impenſa = ſ π v d t vel = ſ π d x, ſi per d x intelligatur ele-<lb/>mentum ſpatioli tempuſculo d t percurſi. </s> + <s xml:id="echoid-s5099" xml:space="preserve">Sequitur inde, ſi conſtantis mag-<lb/>nitudinis ſit, uti fere eſt conatus, quo embolus elevatur, fore potentiam abſo-<lb/>lutam æqualem potentiæ moventi ductæ in ſpatium percurſum: </s> + <s xml:id="echoid-s5100" xml:space="preserve">ſimile autem ra-<lb/>tiocinium cum valeat etiam pro depreſſione emboli ſimulque tantum eleve-<lb/>tur embolus quantum deprimitur, apparet potenti{as} abſolut{as}, quæ in attrahen-<lb/>das expellendaſque alternatim aquas impenduntur, proxime eſſe ut potentiæ <lb/>utrobique moventes; </s> + <s xml:id="echoid-s5101" xml:space="preserve">unde diſpendium oritur quod eſt = {π/π + p} X P, factis ſci-<lb/>licet potentia elevante = π, potentia deprimente = p & </s> + <s xml:id="echoid-s5102" xml:space="preserve">potentia abſoluta in <lb/>elevationem depreſſionemque emboli impenſa = P.</s> + <s xml:id="echoid-s5103" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5104" xml:space="preserve">Poteſt aliter diſpendium potentiœ abſolutæ proxime æſtimari ex eo, quod <lb/>omnis aſcenſ{us} potentialis aquæ in antliam influentis inutiliter generatus cenſeri <lb/>debeat. </s> + <s xml:id="echoid-s5105" xml:space="preserve">Sed ſi iiſdem temporibus, ſive eadem velocitate embolus ſurſum de-<lb/>orſumque movetur, erit velocitas quâ aquæ admittuntur ad velocitatem quâ <lb/>ejiciuntur reciproce ut foramina reſpondentia, ipſique aſcenſus potentiales utro-<lb/>bique erunt in ratione quadrata inverſa foraminum reſpondentium. </s> + <s xml:id="echoid-s5106" xml:space="preserve">Si deinde +<pb o="177" file="0191" n="191" rhead="SECTIO NONA."/> +diverſis temporibus fiant emboli elevatio & </s> + <s xml:id="echoid-s5107" xml:space="preserve">depreſſio, ſunt velocitates recipro-<lb/>ce ut tempora & </s> + <s xml:id="echoid-s5108" xml:space="preserve">aſcenſus potentiales reciproce ut quadrata temporum. </s> + <s xml:id="echoid-s5109" xml:space="preserve">Eſt igi-<lb/>tur aſcenſus potentialis aquæ influxu generatus ad aſcenſum potent. </s> + <s xml:id="echoid-s5110" xml:space="preserve">qui ab effluxu <lb/>oritur ſolusque intenditur, in ratione reciproca quadrata compoſita ex ratio-<lb/>ne foraminis influxus ad foramen effluxus & </s> + <s xml:id="echoid-s5111" xml:space="preserve">temporis, quo hauriuntur aquæ ad <lb/>tempus quo expelluntur.</s> + <s xml:id="echoid-s5112" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div208" type="section" level="1" n="162"> +<head xml:id="echoid-head209" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s5113" xml:space="preserve">§. </s> + <s xml:id="echoid-s5114" xml:space="preserve">18. </s> + <s xml:id="echoid-s5115" xml:space="preserve">Ex utraque æſ<unsure/>timandi ratione ſequitur lente embolum eſſe ele-<lb/>vandum: </s> + <s xml:id="echoid-s5116" xml:space="preserve">ita enim parva fit potentia movens ratione primæ methodi aut magnum <lb/>fit tempus elevationis ratione ſecundæ, atque ſic operarii ſingulis elevationis <lb/>emboli intervallis à conatu præcedentis depreſſionis exantlato reficientur. </s> + <s xml:id="echoid-s5117" xml:space="preserve">Po-<lb/>ſterior porro methodus indicat foramina, per quæ aquæ attrahuntur amplian-<lb/>da & </s> + <s xml:id="echoid-s5118" xml:space="preserve">multiplicanda eſſe; </s> + <s xml:id="echoid-s5119" xml:space="preserve">id vero etiam priori conforme eſt methodo, quia <lb/>ſic ſufficiens fere aquæ quantitas ſua ſponte influit, minorique adeo potentia mo-<lb/>vente opus eſt.</s> + <s xml:id="echoid-s5120" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div209" type="section" level="1" n="163"> +<head xml:id="echoid-head210" xml:space="preserve">Regula 9.</head> +<p> + <s xml:id="echoid-s5121" xml:space="preserve">§. </s> + <s xml:id="echoid-s5122" xml:space="preserve">19. </s> + <s xml:id="echoid-s5123" xml:space="preserve">Denique jactum aqueum verticaliter aſſurgentem nunquam <lb/>eam attingere altitudinem obſervandum eſt, quæ debeatur aquæ velocitati ini-<lb/>tiali, id eſt, ſi vena fluidi verticaliter aſſurgere incipiat ab ſua origine veloci-<lb/>tate tali, quam grave libere cadendo ex altitudine a acquirat, non poterit flui-<lb/>dum aſcendere ad totam altitudinem a, etiamſi aëris reſiſtentiam removeas, aut <lb/>quicquid excogitare velis, quod caſu motum retardare queat. </s> + <s xml:id="echoid-s5124" xml:space="preserve">Ipſa enim rei na-<lb/>tura defectum aliquem exigit neceſſario, cujus rei ratio phyſica hæc eſt: </s> + <s xml:id="echoid-s5125" xml:space="preserve">Nem-<lb/>pe quælibet guttula etiamſi aſcenſum incipiens verticalem, non poteſt tamen, <lb/>quin ſenſim ad latera deflectatur & </s> + <s xml:id="echoid-s5126" xml:space="preserve">tandem, cum ad ſummum pervenit, motu <lb/>feratur horizontali, qui notabilis eſſe debet, quia per ſupremum limbum vel <lb/>ſectionem venæ aqueæ omnis aqua tranſit, quæ per foramen effluxit: </s> + <s xml:id="echoid-s5127" xml:space="preserve">fac igi-<lb/>tur unicuique guttulæ eo temporis puncto quo horizontaliter movetur veloci-<lb/>tatem ineſſe, quam grave lapſu libero per altitudinem b acquirit: </s> + <s xml:id="echoid-s5128" xml:space="preserve">ita vides non <lb/>poſſe venam ultra altitudinem a - b aſſurgere: </s> + <s xml:id="echoid-s5129" xml:space="preserve">Atque hoc titulo diſpendium <lb/>oritur ratione potentiæ abſolutæ totius ut b ad a.</s> + <s xml:id="echoid-s5130" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div210" type="section" level="1" n="164"> +<head xml:id="echoid-head211" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s5131" xml:space="preserve">§. </s> + <s xml:id="echoid-s5132" xml:space="preserve">20. </s> + <s xml:id="echoid-s5133" xml:space="preserve">Obſervatum fuit inter aquas communi velocitate ex tubulis di- +<pb o="178" file="0192" n="192" rhead="HYDRODYNAMICÆ"/> +verſimode formatis ejectas alias aliis altius aſſurgere: </s> + <s xml:id="echoid-s5134" xml:space="preserve">Ergo hic attendendum eſt <lb/>ad ultimorum tubulorum aquas emittentium (des ajutages) conformationem <lb/>aptiſſimam.</s> + <s xml:id="echoid-s5135" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5136" xml:space="preserve">Hâc de re experimenta inſtituit D. </s> + <s xml:id="echoid-s5137" xml:space="preserve">Mariotte in tract. </s> + <s xml:id="echoid-s5138" xml:space="preserve">de mot. </s> + <s xml:id="echoid-s5139" xml:space="preserve">aquar.</s> + <s xml:id="echoid-s5140" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div211" type="section" level="1" n="165"> +<head xml:id="echoid-head212" xml:space="preserve">Scholium Generale.</head> +<p> + <s xml:id="echoid-s5141" xml:space="preserve">§. </s> + <s xml:id="echoid-s5142" xml:space="preserve">21. </s> + <s xml:id="echoid-s5143" xml:space="preserve">Examinavimus adhuc impedimenta, quæ caſu ſuperveniunt in <lb/>machinis hydraulicis aquas cum impetu ejicientibus: </s> + <s xml:id="echoid-s5144" xml:space="preserve">Præcipua illa eſſe puto, <lb/>quæ expoſui; </s> + <s xml:id="echoid-s5145" xml:space="preserve">poterunt tamen alia inſuper excogitari, ſed, ut credo, mino-<lb/>ris admodum momenti. </s> + <s xml:id="echoid-s5146" xml:space="preserve">Ubique fere menſuras dedimus omnino geometricas <lb/>ſimulque modum indicavimus, quo iiſdem impedimentis maximâ ex parte <lb/>obviam iri poſſit. </s> + <s xml:id="echoid-s5147" xml:space="preserve">Qui majoribus intendit, putans poſſe minimo labore ſeu <lb/>(quod eodem recidere demonſtravi § 3.) </s> + <s xml:id="echoid-s5148" xml:space="preserve">minima potentia abſoluta quemvis ef-<lb/>fectum in elevandis aquis deſideratum præſtari, opinione fallitur, atque oleum <lb/>& </s> + <s xml:id="echoid-s5149" xml:space="preserve">operam perdet. </s> + <s xml:id="echoid-s5150" xml:space="preserve">Si enim ab impedimentis iſtis expoſitis aliiſve ſimilibus for-<lb/>taſſe excogitandis animum abſtrahas, machina in rerum natura perfectiſſima <lb/>erit ſimplex antlia figuræ quadrageſimæ quintæ, atque ſi aquæ ejus ope in al-<lb/>tum projectæ colligantur in G, dico fieri non potuiſſe ut minori labore eadem <lb/>aquarum quantitas ad eandem altitudinem F G elevarentur.</s> + <s xml:id="echoid-s5151" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5152" xml:space="preserve">Eſt deinde aliud machinarum genus, quod à machinationibus adhuc <lb/>pertractatis differt in eo, quod hæ aquas cum impetu ejiciant, illæ placide ſi-<lb/>ne motu notabili transferant. </s> + <s xml:id="echoid-s5153" xml:space="preserve">Sed & </s> + <s xml:id="echoid-s5154" xml:space="preserve">in his ultimus perfectionis qui dari poteſt <lb/>gradus eodem recidit. </s> + <s xml:id="echoid-s5155" xml:space="preserve">Sunt autem pleræque multis obſtaculis iiſque maximi <lb/>momenti obnoxiæ. </s> + <s xml:id="echoid-s5156" xml:space="preserve">De his igitur nunc directe nobis erit agendum.</s> + <s xml:id="echoid-s5157" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div212" type="section" level="1" n="166"> +<head xml:id="echoid-head213" style="it" xml:space="preserve">(B) De machinis hydraulicis aquas ſine not abili impetu ex loco humiliori in <lb/>altiorem tranſportantibus.</head> +<head xml:id="echoid-head214" xml:space="preserve">Regula 10.</head> +<p> + <s xml:id="echoid-s5158" xml:space="preserve">§. </s> + <s xml:id="echoid-s5159" xml:space="preserve">22. </s> + <s xml:id="echoid-s5160" xml:space="preserve">Si pondus aliquod per datam altitudinem verticalem (a) potentia <lb/>movente utcunque variabili ſed directe applicata elevetur nulluſque motus in <lb/>fummitate altitudinis propoſitæ corpori ſuperſit, conſtanter erit eadem potentia <lb/>abſoluta in elevationem ponderis impenſa, nempe æqualis producto ex ponde-<lb/>re corporis elevati & </s> + <s xml:id="echoid-s5161" xml:space="preserve">altitudine elevationis a.</s> + <s xml:id="echoid-s5162" xml:space="preserve"/> +</p> +<pb o="179" file="0193" n="193" rhead="SECTIO NONA."/> +</div> +<div xml:id="echoid-div213" type="section" level="1" n="167"> +<head xml:id="echoid-head215" xml:space="preserve">Demonſtratio.</head> +<p> + <s xml:id="echoid-s5163" xml:space="preserve">Nam ſi pondus, quod vocabo A, aſcenderit per altitudinem y, eoque <lb/>in loco animari ponatur potentia movente variabili P directe applicata, move-<lb/>rique velocitate v, erit tempuſculum, quo pondus per elementum d y eleva-<lb/>tur = {dy/v}, quod ductum in potentiam moventem P, ejuſdemque velocitatem <lb/>v, dat elementum potentiæ abſolutæ (per defin. </s> + <s xml:id="echoid-s5164" xml:space="preserve">§. </s> + <s xml:id="echoid-s5165" xml:space="preserve">2.) </s> + <s xml:id="echoid-s5166" xml:space="preserve">= P d y, ergo ſ P dy dabit <lb/>totam potentiam abſolutam, ſi poſt integrationem fiat y = a; </s> + <s xml:id="echoid-s5167" xml:space="preserve">in omni vero motu <lb/>incrementum velocitatis d v eſt æquale potentiæ animanti ſeu moventi, quæ <lb/>hîc eſt {P - A/A} ductæ in tempuſculum quod nunc eſt {dy/v}; </s> + <s xml:id="echoid-s5168" xml:space="preserve">habemus igitur d v = <lb/>({P - A/A}) X {dy/v} vel A v d v = P d y - A dy, id eſt, {1/2} A v v = ſ P d y - A y, ſive <lb/>ſ P d y = {1/2} A v v + Ay, ubi faciendum eſt y = a & </s> + <s xml:id="echoid-s5169" xml:space="preserve">v = o (per hypoth.) </s> + <s xml:id="echoid-s5170" xml:space="preserve">ita ut <lb/>ſit ſ P d y = A a.</s> + <s xml:id="echoid-s5171" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5172" xml:space="preserve">Quia autem, ut vidimus, ſ P d y exprimit integram potentiam abſolu-<lb/>tam in elevandum pondus impenſam @ erit eadem hæc potentia conſtanter <lb/>eadem, nominatimque æqualis producto ex pondere A & </s> + <s xml:id="echoid-s5173" xml:space="preserve">altitudine a, ut <lb/>habet propoſito. </s> + <s xml:id="echoid-s5174" xml:space="preserve">Q. </s> + <s xml:id="echoid-s5175" xml:space="preserve">E. </s> + <s xml:id="echoid-s5176" xml:space="preserve">D.</s> + <s xml:id="echoid-s5177" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div214" type="section" level="1" n="168"> +<head xml:id="echoid-head216" xml:space="preserve">Corollarium.</head> +<p> + <s xml:id="echoid-s5178" xml:space="preserve">§. </s> + <s xml:id="echoid-s5179" xml:space="preserve">23. </s> + <s xml:id="echoid-s5180" xml:space="preserve">Ex demonſtratione noſtra apparet, eſſe quoque potentiam abſo-<lb/>lutam eandem, quoties velocitas in ſummitate eſt eadem, id eſt, quoties <lb/>altitudo ad quam corpus velocitate ſua reſidua aſcendere poteſt, nempe {1/2} vv <lb/>eſt conſtans: </s> + <s xml:id="echoid-s5181" xml:space="preserve">atque ſi altitudo iſta dicatur b, erit potentia abſoluta = A (a + b). <lb/></s> + <s xml:id="echoid-s5182" xml:space="preserve">Igitur patet nunc, quanta pars potentiæ abſolutæ perdatur, cum animus ſit <lb/>pondus A ad altitudinem a elevare, idemque in ſummitate velocitatem reſi-<lb/>duam habeat debitam altitudini b; </s> + <s xml:id="echoid-s5183" xml:space="preserve">erit nempe diſpendium potentiæ ad in-<lb/>tegram potentiam ut b ad b + a.</s> + <s xml:id="echoid-s5184" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div215" type="section" level="1" n="169"> +<head xml:id="echoid-head217" xml:space="preserve">Scholium 1.</head> +<p> + <s xml:id="echoid-s5185" xml:space="preserve">§. </s> + <s xml:id="echoid-s5186" xml:space="preserve">24. </s> + <s xml:id="echoid-s5187" xml:space="preserve">Cavendum itaque eſt, ne machinæ ita ſint conſtructæ, ut ve-<lb/>hementi motu aquæ ad locum deſtinatum transportentur. </s> + <s xml:id="echoid-s5188" xml:space="preserve">Parvum autem <lb/>eſſe ſolet iſtud diſpendii genus in plerisque machinis.</s> + <s xml:id="echoid-s5189" xml:space="preserve"/> +</p> +<pb o="180" file="0194" n="194" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div216" type="section" level="1" n="170"> +<head xml:id="echoid-head218" xml:space="preserve">Scholium 2.</head> +<p> + <s xml:id="echoid-s5190" xml:space="preserve">§. </s> + <s xml:id="echoid-s5191" xml:space="preserve">25. </s> + <s xml:id="echoid-s5192" xml:space="preserve">Omnia ſimiliter ſe habent ſi corpus non verticaliter, ſed ſu-<lb/>per plano utcunque inclinato, aut etiam curva qualicunque elevetur, ſem-<lb/>per enim tota potentia abſoluta erit æqualis A (a + b), id eſt, producto ex <lb/>pondere in altitudinem elevationis auctam altitudine velocitati corporis in <lb/>ſummitate reſiduæ debita, cujus rei demonſtratione ſuperſedeo, quod pa-<lb/>rum differt à præcedente demonſtratione.</s> + <s xml:id="echoid-s5193" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div217" type="section" level="1" n="171"> +<head xml:id="echoid-head219" xml:space="preserve">Scholium Generale.</head> +<p> + <s xml:id="echoid-s5194" xml:space="preserve">§. </s> + <s xml:id="echoid-s5195" xml:space="preserve">26. </s> + <s xml:id="echoid-s5196" xml:space="preserve">Quia omnium machinarum utcunque compoſitarum effectus <lb/>reduci poſſunt ad naturam plani inclinati, perſpicuum eſt omnes machi-<lb/>nas, ſi à frictionibus iisque potentiarum abſolutarum diſpendiis, quæ hactenus <lb/>recenſuimus, animum removeamus eodem recidere, quia potentia abſoluta ſim-<lb/>pliciter pendet ab altitudine ad quam corpus eſt elevandum ejusdemque pon-<lb/>dere. </s> + <s xml:id="echoid-s5197" xml:space="preserve">Habet hoc commune potentia abſoluta cum vi viva ſeu cum aſcenſu de-<lb/>ſcenſuve actuali. </s> + <s xml:id="echoid-s5198" xml:space="preserve">Isque ultimus eſt perfectionis machinarum gradus, quem <lb/>transgredi non licet, imo nec attingere quidem, ſemper enim remotis om-<lb/>nibus frictionibus diſpendiisque, potuiſſet eadem potentia abſoluta majus pon-<lb/>dus ad eandem altitudinem elevari. </s> + <s xml:id="echoid-s5199" xml:space="preserve">Ut jam comparatio inſtitui poſſit quæ-<lb/>dam circa machinarum defectum, tam illarum quæ aquas ad deſideratam <lb/>altitudinem veluti projiciunt, quam quæ easdem transportant, nunc ha-<lb/>rum poſteriorum defectus maxime notabiles quoque indicabimus.</s> + <s xml:id="echoid-s5200" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5201" xml:space="preserve">(I.) </s> + <s xml:id="echoid-s5202" xml:space="preserve">Frictiones tanto obſtaculo ſunt in plerisque hujusmodi machinis, <lb/>ut ſolæ maximam potentiæ partem abſorbeant, præſertim autem cum aſſerculi <lb/>quadrati aut globi ovales, catena in circulum redeunte connexi, per cana-<lb/>lem, cui ſunt accommodati, transeuntis aquas elevant.</s> + <s xml:id="echoid-s5203" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5204" xml:space="preserve">(II.) </s> + <s xml:id="echoid-s5205" xml:space="preserve">Pleræque machinæ, præſertim vero rurſus quas modo indicavi-<lb/>mus, roſariorum nomine deſignari ſolitæ ita ſunt comparatæ, ut aqua dum <lb/>elevatur continue pars ejus deſtillet, ſive plane decidat in locum ex quo <lb/>hauſta fuit ſive ſaltem ex loco ſuperiori in inferiorem, uti in roſariis; </s> + <s xml:id="echoid-s5206" xml:space="preserve">ſi in <lb/>his globuli aut aſſerculi canali ſunt bene adaptati frictio fit fere inſuperabi-<lb/>lis, ſin minus maxima aquæ quantitas per hiatus relictos deſtillat, ex ſupe- +<pb o="181" file="0195" n="195" rhead="SECTIO NONA."/> +rioribus diviſionibus in inferiores, ita ut minima aquæ pars in illis ſuperſit, <lb/>cum culmen attigerunt, ejus quantitatis quam in toto itinere receperunt. <lb/></s> + <s xml:id="echoid-s5207" xml:space="preserve">Videntur itaque vel ſolo hoc nomine iſtæ machinæ admodum improbandæ, <lb/>præſertim vero ſi aquæ limpidæ ſint elevandæ, quæ antliis hauriri poſſint.</s> + <s xml:id="echoid-s5208" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5209" xml:space="preserve">(III.) </s> + <s xml:id="echoid-s5210" xml:space="preserve">Solent quoque machinæ ejus eſſe indolis, ut aquam ultra altitu-<lb/>dinem propoſitam attollant: </s> + <s xml:id="echoid-s5211" xml:space="preserve">Perditur autem potentia quæ exceſſui reſpon-<lb/>det, atque ſi aquæ trans molem ſunt evehendæ, difficulter id obtinetur, <lb/>quod indicavi §. </s> + <s xml:id="echoid-s5212" xml:space="preserve">12.</s> + <s xml:id="echoid-s5213" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5214" xml:space="preserve">(IV.) </s> + <s xml:id="echoid-s5215" xml:space="preserve">Sunt & </s> + <s xml:id="echoid-s5216" xml:space="preserve">machinæ, quæ directam potentiæ moventis applicatio-<lb/>nem non admittunt, ex quâ obliquitate rurſus diſpendium aliquod oritur.</s> + <s xml:id="echoid-s5217" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5218" xml:space="preserve">§. </s> + <s xml:id="echoid-s5219" xml:space="preserve">27. </s> + <s xml:id="echoid-s5220" xml:space="preserve">Iſtaque fere ſunt, quæ notabilis momenti mihi viſa fuerunt, ob-<lb/>ſtacula; </s> + <s xml:id="echoid-s5221" xml:space="preserve">neſcio autem an illis in tantum obviam iri poſſit, quantum de pri-<lb/>mo machinarum genere demonſtravimus: </s> + <s xml:id="echoid-s5222" xml:space="preserve">frictionum diminuendarum artifi-<lb/>cia quædam norunt mechanici: </s> + <s xml:id="echoid-s5223" xml:space="preserve">machinas quæ ſitulis aquas hauriunt atque <lb/>elevant prætulerim roſariis: </s> + <s xml:id="echoid-s5224" xml:space="preserve">ſitulæ autem ita ſint fabricatæ, ſi modo id fieri <lb/>poſſit, ut in ſitu infimo ſtatim impleantur nihilque emittant priusquam <lb/>ad ſitum ſupremum pervenerint. </s> + <s xml:id="echoid-s5225" xml:space="preserve">Cum aqua transfundenda eſt trans locum <lb/>altiorem in alium minus altum, opera danda eſt, ut impetus aquæ labentis <lb/>promoveat motum tympani ſeu rotæ in gyrum agendæ, quamvis multum <lb/>abſit ut ſic omnis potentia abſoluta utiliter impendatur, prouti fieri antlia figu-<lb/>ræ 49. </s> + <s xml:id="echoid-s5226" xml:space="preserve">indicavimus (§. </s> + <s xml:id="echoid-s5227" xml:space="preserve">12.) </s> + <s xml:id="echoid-s5228" xml:space="preserve">Principium actionis conſiſtet, ſi recte judicio, <lb/>aptiſſime in calcatura: </s> + <s xml:id="echoid-s5229" xml:space="preserve">homines enim iſti labori maxime ſunt aſſueti; </s> + <s xml:id="echoid-s5230" xml:space="preserve">perti-<lb/>net huc, quod monui §. </s> + <s xml:id="echoid-s5231" xml:space="preserve">4. </s> + <s xml:id="echoid-s5232" xml:space="preserve">occaſione regulæ primæ de angulo acclivitatis, <lb/>ſub quo viator dato tempore minima defatigatione certam attingere poſſit <lb/>altitudinem verticalem. </s> + <s xml:id="echoid-s5233" xml:space="preserve">Crediderim hominem mediocris ſtaturæ, ſanum <lb/>& </s> + <s xml:id="echoid-s5234" xml:space="preserve">robuſtum ſuper via ad 30. </s> + <s xml:id="echoid-s5235" xml:space="preserve">gradus acclivi incedentem non dificulter ſin-<lb/>gulis horis 3600. </s> + <s xml:id="echoid-s5236" xml:space="preserve">pedes confecturum, atque proinde ad altitudinem vertica-<lb/>lem 1800. </s> + <s xml:id="echoid-s5237" xml:space="preserve">pedum pondus corporis ſui, quod ponam 144 librarum ſeu <lb/>duorum pedum cubicorum aquæ, elevaturum. </s> + <s xml:id="echoid-s5238" xml:space="preserve">Talis igitur homo poterit ope <lb/>machinæ calcatura circumagendæ & </s> + <s xml:id="echoid-s5239" xml:space="preserve">perfectiſſimæ (in qua ſcilicet nihil de <lb/>potentia abſoluta perdatur) ſingulis horis duos pedes cubicos aquæ ad altitu-<lb/>dinem verticalem 1800. </s> + <s xml:id="echoid-s5240" xml:space="preserve">pedum elevare, ſeu quod idem eſt, ſingulis minutis <lb/>ſecundis unum ped. </s> + <s xml:id="echoid-s5241" xml:space="preserve">cub. </s> + <s xml:id="echoid-s5242" xml:space="preserve">ad alt. </s> + <s xml:id="echoid-s5243" xml:space="preserve">unius pedis: </s> + <s xml:id="echoid-s5244" xml:space="preserve">machinas quæ multo minoris +<pb o="182" file="0196" n="196" rhead="HYDRODYNAMICÆ"/> +ſunt effectus, officium facientibus operariis, parum puto commendabiles: <lb/></s> + <s xml:id="echoid-s5245" xml:space="preserve">Interim inſtituto experimento in ædibus Ill. </s> + <s xml:id="echoid-s5246" xml:space="preserve">D. </s> + <s xml:id="echoid-s5247" xml:space="preserve">General de Coulon cum antlia, <lb/>quod in fine ſectionis apponam, effectum haud parum minorem expertus <lb/>ſum, quo confirmatus ſum in ſententia mea operarios calcatura plurimum <lb/>præſtare: </s> + <s xml:id="echoid-s5248" xml:space="preserve">facile autem prævideo in machinis admodum compoſitis longe <lb/>minorem effectum prodire, quia in his maxima potentiæ abſolutæ pars inutilis <lb/>impenditur. </s> + <s xml:id="echoid-s5249" xml:space="preserve">Notabile iſtius rei nunc afferam exemplum à notiſſima machi-<lb/>na Marlyenſi, oſtenſurus quam incredibile fere potentiæ abſolutæ diſpendium <lb/>ab omnibus impedimentis collectis oriatur.</s> + <s xml:id="echoid-s5250" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5251" xml:space="preserve">Tractatum edidit Weidlerus de machinis hydraulicis in quo plenam de-<lb/>ſcriptionem facit machinæ Marlyenſis, atque refert omnes aquas elevari à <lb/>motu 14 rotarum, quarum alæ ab impetu ſequanæ propellantur: </s> + <s xml:id="echoid-s5252" xml:space="preserve">hunc <lb/>impetum facit pro omnibus rotis æqualem ponderi 1000594 librarum, is-<lb/>que eſt quem nos deſignavimus nomine potentiæ moventis. </s> + <s xml:id="echoid-s5253" xml:space="preserve">Præterea alas mo-<lb/>tu ferri ex aliquibus circumſtantiis colligere potui, quo conficiant 3 {3/4} pe-<lb/>des ſingulis minutis ſecundis, atque hæc velocitas habenda eſt pro velocita-<lb/>te potentiæ moventis; </s> + <s xml:id="echoid-s5254" xml:space="preserve">deinde addit ſingulis diebus elevari vi illius machinæ <lb/>11700000 libras aquæ ad altit. </s> + <s xml:id="echoid-s5255" xml:space="preserve">500 ped. </s> + <s xml:id="echoid-s5256" xml:space="preserve">His ita poſitis videamus nunc in <lb/>machina ſimpliciſſima fig. </s> + <s xml:id="echoid-s5257" xml:space="preserve">45, qua nihil de potentia abſoluta perdi intelligatur, <lb/>quanta ad iſtam effectum potentia P pariter velocitate ut 3 {3/4} mota requira-<lb/>tur. </s> + <s xml:id="echoid-s5258" xml:space="preserve">Erit autem altitudo F G = 500 ped. </s> + <s xml:id="echoid-s5259" xml:space="preserve">& </s> + <s xml:id="echoid-s5260" xml:space="preserve">quoniam tempore 24 horarum <lb/>ejici debeant per lumen F 11700000 libræ, id eſt, 162500 ped. </s> + <s xml:id="echoid-s5261" xml:space="preserve">cub. </s> + <s xml:id="echoid-s5262" xml:space="preserve">ma-<lb/>gnitudo iſtius luminis ponenda erit = 0, 0108 partium pedis unius qua-<lb/>drati: </s> + <s xml:id="echoid-s5263" xml:space="preserve">Velocitas aquæ in F tanta eſt, ut abſolvat ſingulis minutis ſecundis <lb/>173 ped. </s> + <s xml:id="echoid-s5264" xml:space="preserve">Igitur continet velocitatem 3 {3/4}, quam pondus P habere ponitur, <lb/>46 vicibus & </s> + <s xml:id="echoid-s5265" xml:space="preserve">toties ſuperare debet amplitudo antliæ A B amplitudinem lu-<lb/>minis F: </s> + <s xml:id="echoid-s5266" xml:space="preserve">Erit proinde amplitudo A B fingenda 0, 4968, part. </s> + <s xml:id="echoid-s5267" xml:space="preserve">ped, quadrat. <lb/></s> + <s xml:id="echoid-s5268" xml:space="preserve">ex quo conſequens eſt, pondus P æquale futurum ponderi cylindri aquei <lb/>ſuper baſi A B ad altitudinem 500 ped. </s> + <s xml:id="echoid-s5269" xml:space="preserve">conſtructi ſeu ponderi 248, 4 pe-<lb/>dum cub. </s> + <s xml:id="echoid-s5270" xml:space="preserve">aquæ, id eſt, ponderi 17885 librarum, quæ tantum quinqua-<lb/>geſimam ſextam partem efficit potentiæ moventis quam eadem velocitate mo-<lb/>tam applicari oſtendit Weidlerus. </s> + <s xml:id="echoid-s5271" xml:space="preserve">Sic igitur in tota machina diſpendium <lb/>fit quod {55/56} integræ potentiæ abſolutæ. </s> + <s xml:id="echoid-s5272" xml:space="preserve">exæquat.</s> + <s xml:id="echoid-s5273" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5274" xml:space="preserve">Poſtquam ita naturam machinarum hydraulicarum, quantum illud in +<pb o="183" file="0197" n="197" rhead="SECTIO NONA."/> +generalibus fieri poteſt, examinavimns, haud abs re erit exemplum aliquod <lb/>ſpeciale accuratius pertractare, & </s> + <s xml:id="echoid-s5275" xml:space="preserve">quia cochlea Archimedis multis gaudet <lb/>egregiis proprietatibus, quas nemo ſatis, quantum ſcio, aperuit, ab ha<unsure/>c <lb/>exemplum deſumam idque eo libentius, quod multi ſint, qui contra no-<lb/>ſtras regulas putant ſingularem huic cochleæ virtutem ineſſe pro elevanda <lb/>magna aquæ quantitate brevi tempore parvaque vi: </s> + <s xml:id="echoid-s5276" xml:space="preserve">falluntur autem qui ita <lb/>cogitant: </s> + <s xml:id="echoid-s5277" xml:space="preserve">nam ſi obſtaculorum accidentalium nulla habeatur ratio, idem <lb/>præſtat eadem potentia abſoluta, quod reliquæ machinæ omnes.</s> + <s xml:id="echoid-s5278" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div218" type="section" level="1" n="172"> +<head xml:id="echoid-head220" style="it" xml:space="preserve">Commentationes ſpeciales de Cochlea Archimedis.</head> +<p> + <s xml:id="echoid-s5279" xml:space="preserve">(I.) </s> + <s xml:id="echoid-s5280" xml:space="preserve">Varii ſunt auctores, qui modum docuerunt conſtruendi hanc co-<lb/>chleam: </s> + <s xml:id="echoid-s5281" xml:space="preserve">ſumma huc redit, ut canalis quidam aut plures ſuperficiei cylindricæ <lb/>circumflectantur, & </s> + <s xml:id="echoid-s5282" xml:space="preserve">ita quidem ut canalis ubique eandem habeat inclinationem <lb/>ratione axis cylindri, quam Vitruvius præter neceſſitatem in omnibus cochleis <lb/>fieri jubet ad angulum ſemirectum. </s> + <s xml:id="echoid-s5283" xml:space="preserve">Requiritur ergo ante omnia, ut in ſuperfi-<lb/>cie cylindri linea ſpiralis ducatur ad cujus normam canalis ſit ponendus, id quod <lb/>facillime meo judicio in ſuperficie admodum polita fieri poterit (præſertim cum <lb/>helices non parum à ſe diſtare debent) circumvolvendo eidem aliquoties funi-<lb/>culum: </s> + <s xml:id="echoid-s5284" xml:space="preserve">hic enim tenſus ſua ſponte deſideratam lineam faciet, neque enim ſpi-<lb/>ralis ſibi ſimilis ubique eſſe pote ſt, aut conſtantem habere ad axem cylindri in-<lb/>clinationem, quin arcus inter duo puncta interceptus ſit omnium arcuum eoſ-<lb/>dem terminos habentium minimus, quam indolem funiculo extenſo compe-<lb/>tere palam eſt: </s> + <s xml:id="echoid-s5285" xml:space="preserve">ſi vero frictiones impedimento ſint, filum ad minora interval-<lb/>la extendi poterit. </s> + <s xml:id="echoid-s5286" xml:space="preserve">Sed non eſt, cur in re per ſe pluribus modis facillima ſcru-<lb/>puloſi ſimus.</s> + <s xml:id="echoid-s5287" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5288" xml:space="preserve">Lex ſpiralis primaria eſt, ut ubique æqualiter ad axem cylindri incli-<lb/>net, cui legi ſequens innititur conſtructio, quam in gratiam infra dicendorum <lb/>apponam:</s> + <s xml:id="echoid-s5289" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5290" xml:space="preserve">Finge cylindrum rectum M a f N (Fig. </s> + <s xml:id="echoid-s5291" xml:space="preserve">52. </s> + <s xml:id="echoid-s5292" xml:space="preserve">(1)) cujus ſuperficiei ſit in-<lb/> +<anchor type="note" xlink:label="note-0197-01a" xlink:href="note-0197-01"/> +ſcribenda deſiderata ſpiralis a 1 b 2 c 3 d &</s> + <s xml:id="echoid-s5293" xml:space="preserve">c. </s> + <s xml:id="echoid-s5294" xml:space="preserve">eandemque ſuperficiem puta ex-<lb/>plicatam in planam figura præditam parallelogrammi rectanguli A a f F <lb/>(Fig. </s> + <s xml:id="echoid-s5295" xml:space="preserve">52. </s> + <s xml:id="echoid-s5296" xml:space="preserve">(2)), ſumantur hic ab una parte A B, B C, C D, D E, & </s> + <s xml:id="echoid-s5297" xml:space="preserve">E F, ab al-<lb/> +<anchor type="note" xlink:label="note-0197-02a" xlink:href="note-0197-02"/> +<pb o="184" file="0198" n="198" rhead="HYDRODYNAMICÆ"/> +tera ab, bc, cd, de, & </s> + <s xml:id="echoid-s5298" xml:space="preserve">ef, ſingulæ ſingulis æquales; </s> + <s xml:id="echoid-s5299" xml:space="preserve">jungantur lineis rectis <lb/>puncta B, C, D, E & </s> + <s xml:id="echoid-s5300" xml:space="preserve">F cum punctis a, b, c, d, & </s> + <s xml:id="echoid-s5301" xml:space="preserve">e: </s> + <s xml:id="echoid-s5302" xml:space="preserve">his ita factis, ſi ſuperfi-<lb/>cies plana rurſus in cylindricam convolvatur, junctis lineis A F & </s> + <s xml:id="echoid-s5303" xml:space="preserve">a f, coinci-<lb/>dentibuſque punctis A & </s> + <s xml:id="echoid-s5304" xml:space="preserve">a; </s> + <s xml:id="echoid-s5305" xml:space="preserve">B & </s> + <s xml:id="echoid-s5306" xml:space="preserve">b &</s> + <s xml:id="echoid-s5307" xml:space="preserve">c. </s> + <s xml:id="echoid-s5308" xml:space="preserve">fiet ut lineæ a B, b C, c D &</s> + <s xml:id="echoid-s5309" xml:space="preserve">c. </s> + <s xml:id="echoid-s5310" xml:space="preserve">in ſuper-<lb/>ficie cylindrica lineam continuam forment, quæ ipſa erit ſpiralis deſiderata. </s> + <s xml:id="echoid-s5311" xml:space="preserve">Ad <lb/>faciliorem intellectum in utraque figura puncta homologa communibus litteris <lb/>diſtinxi.</s> + <s xml:id="echoid-s5312" xml:space="preserve"/> +</p> +<div xml:id="echoid-div218" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0197-01" xlink:href="note-0197-01a" xml:space="preserve">Fig. 52. <lb/>(1.)</note> +<note position="right" xlink:label="note-0197-02" xlink:href="note-0197-02a" xml:space="preserve">Fig. 52. <lb/>(2.)</note> +</div> +<p> + <s xml:id="echoid-s5313" xml:space="preserve">(II.) </s> + <s xml:id="echoid-s5314" xml:space="preserve">Propoſitus jam fuerit cylindrus M a f N (Fig. </s> + <s xml:id="echoid-s5315" xml:space="preserve">52. </s> + <s xml:id="echoid-s5316" xml:space="preserve">(1)), habens ad <lb/>ductum ſpiralis modo deſcriptæ circumflexum canalem, cujus diametrum ve-<lb/>luti infinite parvum cenſebimus ratione diametri ad cylindrum pertinentis: </s> + <s xml:id="echoid-s5317" xml:space="preserve">at-<lb/>que ſic habebitur cochlea Archimedis, quâ ſi uti velimus ad elevandas aquas ex <lb/>M in N, cylindrus erit horizontem verſus inclinandus, & </s> + <s xml:id="echoid-s5318" xml:space="preserve">ita quidem ut an-<lb/>gulus a M H (interceptus inter diametrum baſeos M a, quæ eſt in plano verti-<lb/>cali, & </s> + <s xml:id="echoid-s5319" xml:space="preserve">horizontalem M H) ſit major quam angulus s a o, quem faciunt tan-<lb/>gentes circuli & </s> + <s xml:id="echoid-s5320" xml:space="preserve">ſpiralis in communi puncto a. </s> + <s xml:id="echoid-s5321" xml:space="preserve">Deinde converſo cylindro cir-<lb/>ca axem ſuum in directione a g h M s aquæ influent per inferius canalis circum-<lb/>ducti orificium effluentque per ſuperius.</s> + <s xml:id="echoid-s5322" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5323" xml:space="preserve">(III) Ut naturam hujus elevationis recte intelligamus, tria ſe nobis of-<lb/>ferunt puncta in qualibet ſpiralis helice examinanda, nempe puncta o, p & </s> + <s xml:id="echoid-s5324" xml:space="preserve">q, <lb/>quorum primum o maxime diſtat ab horizonte, alterum p eidem proximum eſt, <lb/>& </s> + <s xml:id="echoid-s5325" xml:space="preserve">q in eadem altitudine poſitum eſt cum puncto o in helice proxime inferio-<lb/>ri ſumto: </s> + <s xml:id="echoid-s5326" xml:space="preserve">per ſingula puncta o ducta eſt recta g n; </s> + <s xml:id="echoid-s5327" xml:space="preserve">per puncta p recta h m & </s> + <s xml:id="echoid-s5328" xml:space="preserve">per <lb/>puncta q recta s t. </s> + <s xml:id="echoid-s5329" xml:space="preserve">Situs vero harum linearum determinabuntur in ſequentibus.</s> + <s xml:id="echoid-s5330" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5331" xml:space="preserve">(IV) Sit radius, qui pertinet ad baſin cylindri, = 1 ſumatur-<lb/>que pro ſinu toto; </s> + <s xml:id="echoid-s5332" xml:space="preserve">ſinus anguli sao = m, ejuſdemque coſinus = M, ſinus an-<lb/>guli a M H = n, ejuſdemque coſinus = N; </s> + <s xml:id="echoid-s5333" xml:space="preserve">arcus a g = X; </s> + <s xml:id="echoid-s5334" xml:space="preserve">coſinus illius arcus <lb/>= x, erit perpendiculum ex o in horizontem demiſſum, nempe o r = {mNX/M} <lb/>+ n (1 + x). </s> + <s xml:id="echoid-s5335" xml:space="preserve">Quia vero or maxima eſt, fit {mNdX/M} + ndx = o, & </s> + <s xml:id="echoid-s5336" xml:space="preserve">cum ex <lb/>natura circuli ſit dX = {-dx/√1 - xx}, erit {- mNdx/M√(1 - xx)} + ndx = o, ergo <lb/>√1 - xx = {mN/Mn}. </s> + <s xml:id="echoid-s5337" xml:space="preserve">Eſt igitur ſinus arcus quæſiti a g = {mN/Mn} aut coſinus +<pb o="185" file="0199" n="199" rhead="SECTIO NONA."/> +x = ± {√(nn - mm)/Mn}: </s> + <s xml:id="echoid-s5338" xml:space="preserve">ſignum ſuperius dat arcum a g, inferius arcum a b de-<lb/>terminantem puncta infima p.</s> + <s xml:id="echoid-s5339" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5340" xml:space="preserve">Atque ſic determinavimus tum puncta ſuprema o, tum ima p, patetque <lb/>arcus M b & </s> + <s xml:id="echoid-s5341" xml:space="preserve">a g eſſe inter ſe æquales, ſimul autem ex quantitate irrationali <lb/>√ (nn - mm) valorem litteræ x afficiente colligitur fieri non poſſe, ut m ſit <lb/>major quam n: </s> + <s xml:id="echoid-s5342" xml:space="preserve">neque enim in hoc caſu punctum datur infimum, quod tota <lb/>ſpiralis ubique aſcendit continue: </s> + <s xml:id="echoid-s5343" xml:space="preserve">Neque etiam inſerviet ſic cochlea ad ele-<lb/>vandas aquas; </s> + <s xml:id="echoid-s5344" xml:space="preserve">unde jam patet ratio ejus, quod monui in articulo hujus di-<lb/>greſſionis ſecundo, de requiſito exceſſu anguli a M H ſupra angulum sao.</s> + <s xml:id="echoid-s5345" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5346" xml:space="preserve">(V) Ponamus nunc globum alicubi eſſe in cavitate canalis, cochleam-<lb/>que in ſitu ſuo firmari: </s> + <s xml:id="echoid-s5347" xml:space="preserve">ſic minime quieſcet globus, quin exiſtat in puncto <lb/>aliquo p. </s> + <s xml:id="echoid-s5348" xml:space="preserve">Quod ſi vero cochlea non retineri ponatur, globus deſcendet, <lb/>deſcenſuque cochleam circumaget, atque ſi præterea fingatur, nullius eſſe <lb/>ponderis cochleam motumque globi liberrime fieri nihil obſtantibus ſrictio-<lb/>nibus, deſcendet globus ſuper recta m b non alia lege, quam globus libere <lb/>ſuper plano inclinato deſcendens. </s> + <s xml:id="echoid-s5349" xml:space="preserve">Apparet itaque potentiam requiri ad im-<lb/>pediendum globi deſcenſum, firmandamque cochleam. </s> + <s xml:id="echoid-s5350" xml:space="preserve">Iſtam potentiam <lb/>applicatam ponemus in puncto f in plano circuli & </s> + <s xml:id="echoid-s5351" xml:space="preserve">perpendiculariter ad ra-<lb/>dium inquiſituri in rationem, quam habeat ad pondus globi in puncto ali-<lb/>quo p quieſcentis.</s> + <s xml:id="echoid-s5352" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5353" xml:space="preserve">Sit pondus globi = p: </s> + <s xml:id="echoid-s5354" xml:space="preserve">quia vero actio globi eſt verticalis, reſolven-<lb/>da erit in duas alias ad perpendiculum ſibi inſiſtentes, quarum una commu-<lb/>nem habeat cum axe cochleæ directionem, altera eidem perpendicularis ſit, <lb/>prior cum nihil ad circumagendam cochleam conferat rejicienda, poſterior-<lb/>que ſola conſideranda erit; </s> + <s xml:id="echoid-s5355" xml:space="preserve">eſt vero actio illa reſidua = n p & </s> + <s xml:id="echoid-s5356" xml:space="preserve">agit in ve-<lb/>ctem, qui eſt = ſinui arcus M b ſeu arcus a g, hicque ſinus (per. </s> + <s xml:id="echoid-s5357" xml:space="preserve">art. </s> + <s xml:id="echoid-s5358" xml:space="preserve">IV.) </s> + <s xml:id="echoid-s5359" xml:space="preserve">eſt <lb/>={mN/Mn}. </s> + <s xml:id="echoid-s5360" xml:space="preserve">Eſt igitur momentum actionis = {mN/Mn} X np = {mNp/M}; </s> + <s xml:id="echoid-s5361" xml:space="preserve">hoc ſi di-<lb/>vidas per radium baſeos, qui eſt vectis pertinens ad potentiam applicatam <lb/>in f in æquilibrio pofitam cum actione globi, habebis iſtam potentiam quæ-<lb/>ſitam = {mNp/M}. </s> + <s xml:id="echoid-s5362" xml:space="preserve">Sic igitur directe ex natura vectis deducere licet, quod +<pb o="186" file="0200" n="200" rhead="HYDRODYNAMICÆ"/> +alii ex principio alieno petere ſolent. </s> + <s xml:id="echoid-s5363" xml:space="preserve">Præmiſſis iſtis præmittendis uſum ma-<lb/>chinæ conſiderare nunc incipiemus, quem habet pro elevandis aquis.</s> + <s xml:id="echoid-s5364" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div220" type="section" level="1" n="173"> +<head xml:id="echoid-head221" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s5365" xml:space="preserve">(VI) Quæritur quænam maxima ſit aquæ quantitas quam cochlea qua-<lb/>vis revolutione ejicere poteſt.</s> + <s xml:id="echoid-s5366" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div221" type="section" level="1" n="174"> +<head xml:id="echoid-head222" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s5367" xml:space="preserve">Conſideremus helicem integram a 1 b, ſitque quantitas aquæ quam <lb/>plena continet = q: </s> + <s xml:id="echoid-s5368" xml:space="preserve">Notandum autem eſt non poſſe helicem eſſe totam aqua re-<lb/>pletam, ſi enim totus canalis plenus eſſet, effluerent aquæ per orificium <lb/>inferius, igitur quivis ramus, qualis eſt a 1 b, partim aëre partim aqua oc-<lb/>cupatur, erit autem altera aquæ extremitas in o ceu puncto ſupremo, alte-<lb/>ra in q, ceu puncto ad libellam cum priori compoſito: </s> + <s xml:id="echoid-s5369" xml:space="preserve">pars igitur aqua ple-<lb/>na eſt o p q, atque ſi hæc pars ponatur ad longitudinem totius helicis a 1 b <lb/>ut g ad h, erit maxima aquæ quantitas una revolutione ejicienda = {g q/h}. </s> + <s xml:id="echoid-s5370" xml:space="preserve">Q.</s> + <s xml:id="echoid-s5371" xml:space="preserve">E.</s> + <s xml:id="echoid-s5372" xml:space="preserve">I.</s> + <s xml:id="echoid-s5373" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div222" type="section" level="1" n="175"> +<head xml:id="echoid-head223" xml:space="preserve">Scholium 1.</head> +<p> + <s xml:id="echoid-s5374" xml:space="preserve">(VII) Quoniam, ut diximus, fieri non poteſt ut aq<unsure/>a per totum ca-<lb/>nalis tractum ſit contigua, cavendum eſt, ne ſeparatio aquæ impediatur, <lb/>quod facile fieri poteſt cum totum cylindri fundum aquæ immergitur, quia <lb/>ſic aëri prohibetur ingreſſus per orificium inferius canalis: </s> + <s xml:id="echoid-s5375" xml:space="preserve">Neque faciendum <lb/>eſt, ut nimia fundi pars extra aquam promineat, quia ſic cochlea non om-<lb/>nem, quam una revolutione alias poſſet, aquam haurit; </s> + <s xml:id="echoid-s5376" xml:space="preserve">imo nihil hauriet, <lb/>ſi immerſio punctum h non attingat: </s> + <s xml:id="echoid-s5377" xml:space="preserve">Debita autem fiet immerſio usque ad <lb/>punctum g, quia ſic arcus helicis o p q, qui aquam retinere valet, maximus <lb/>fit. </s> + <s xml:id="echoid-s5378" xml:space="preserve">Etſi enim nunquam rei periculum fecerim, & </s> + <s xml:id="echoid-s5379" xml:space="preserve">plerique auctores aliter <lb/>de illa loqui videantur, malim tamen rationi, quam auctoritati illorum, qui <lb/>ad immerſionem hanc animum non adverterunt, credere.</s> + <s xml:id="echoid-s5380" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5381" xml:space="preserve">Regula igitur ratione immerſionis hæc obſervabitur, fundum nempe ſub-<lb/>mergetur, donec chorda arcus extra aquam eminentis ſit = {2mN/Mn}, ubi lit-<lb/>teræ m, N, M, & </s> + <s xml:id="echoid-s5382" xml:space="preserve">n idem ſignificant, quod in articulo quarto.</s> + <s xml:id="echoid-s5383" xml:space="preserve"/> +</p> +<pb o="187" file="0201" n="201" rhead="SECTIO NONA."/> +</div> +<div xml:id="echoid-div223" type="section" level="1" n="176"> +<head xml:id="echoid-head224" xml:space="preserve">Scholium 2.</head> +<p> + <s xml:id="echoid-s5384" xml:space="preserve">(VIII) Apparet quidem poſt levem rei contemplationem eò majorem <lb/>eſſe rationem inter arcum helicis o p q & </s> + <s xml:id="echoid-s5385" xml:space="preserve">integram helicem a 1 b, id eſt, inter <lb/>g & </s> + <s xml:id="echoid-s5386" xml:space="preserve">h, atque proinde eo majorem ceteris paribus aquæ quantitatem ſingulis <lb/>revolutionibus ejici, quo minor eſt angulus s a o & </s> + <s xml:id="echoid-s5387" xml:space="preserve">quo major angulus a M H, <lb/>ſeu quo minor eſt diſtantia inter duas proximas helices & </s> + <s xml:id="echoid-s5388" xml:space="preserve">quo magis cochlea <lb/>verſus horizontem inclinat: </s> + <s xml:id="echoid-s5389" xml:space="preserve">Veram autem illam rationem algebraice expri-<lb/>mere non licet: </s> + <s xml:id="echoid-s5390" xml:space="preserve">In omni tamen caſu particulari id facili appropinquatione <lb/>obtinetur.</s> + <s xml:id="echoid-s5391" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5392" xml:space="preserve">Exemplum præcedentis regulæ deſumam à cochlea, qualem Vitruvius ad-<lb/>hibere & </s> + <s xml:id="echoid-s5393" xml:space="preserve">conſtruere docet. </s> + <s xml:id="echoid-s5394" xml:space="preserve">Facit autem angulum s a o ſemirectum & </s> + <s xml:id="echoid-s5395" xml:space="preserve">ſic <lb/>m = M = √{1/2} = o, 70710: </s> + <s xml:id="echoid-s5396" xml:space="preserve">deinde inter N G & </s> + <s xml:id="echoid-s5397" xml:space="preserve">M G rationem ſtatuit, <lb/>quæ eſt ut 3 ad 4; </s> + <s xml:id="echoid-s5398" xml:space="preserve">inde deducitur angulus G N M vel a M H = 53<emph style="super">0</emph>, 8<emph style="super">1</emph>, ejus-<lb/>que ſinus n = o, 80000 atque conſinus N = o, 60000: </s> + <s xml:id="echoid-s5399" xml:space="preserve">ergo (per art. </s> + <s xml:id="echoid-s5400" xml:space="preserve">III.) <lb/></s> + <s xml:id="echoid-s5401" xml:space="preserve">eſt ſinus arcus a g altiſſimum punctum o definientis = {m N/M n} = {3/4}, ipſeque <lb/>arcus a g = 48<emph style="super">0</emph>, 35<emph style="super">1</emph>. </s> + <s xml:id="echoid-s5402" xml:space="preserve">Debet adeoque vi regulæ art. </s> + <s xml:id="echoid-s5403" xml:space="preserve">VII. </s> + <s xml:id="echoid-s5404" xml:space="preserve">arcus extra aquam <lb/>eminens in fundo eſſe 97<emph style="super">0</emph>, 10<emph style="super">1</emph>; </s> + <s xml:id="echoid-s5405" xml:space="preserve">immergeturque arcus 262<emph style="super">0</emph>, 50<emph style="super">1</emph>.</s> + <s xml:id="echoid-s5406" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5407" xml:space="preserve">Ut jam præterea definiamus rationem inter arcum helicis o p q & </s> + <s xml:id="echoid-s5408" xml:space="preserve">helicem <lb/>integram a 1 b, notandum eſt, eandem eſſe illam rationem, quæ intercedit in-<lb/>ter arcum circularem g h M s & </s> + <s xml:id="echoid-s5409" xml:space="preserve">circumferentiam circuli, quod ex figura ſocia <lb/>manifeſtum eſt. </s> + <s xml:id="echoid-s5410" xml:space="preserve">Determinatur autem arcus g h M s hunc in modum. </s> + <s xml:id="echoid-s5411" xml:space="preserve">Eſt nem-<lb/>pe arc. </s> + <s xml:id="echoid-s5412" xml:space="preserve">g h M s = arc. </s> + <s xml:id="echoid-s5413" xml:space="preserve">a g h M s - arc. </s> + <s xml:id="echoid-s5414" xml:space="preserve">a g. </s> + <s xml:id="echoid-s5415" xml:space="preserve">Sed vidimus in articulo tertio, ſi ex <lb/>quocunque puncto ſpiralis, veluti o & </s> + <s xml:id="echoid-s5416" xml:space="preserve">q perpendicula ad horizontem punctum <lb/>M radentem demittantur, qualia ſunt o r & </s> + <s xml:id="echoid-s5417" xml:space="preserve">q x, fore iſtud perpendiculum <lb/>= {mNX/M} + n (1 + x) ſeu in noſtro caſu = o, 60000 X + o, 80000(1 + x), <lb/>denotante X arcum circularem, puncto in ſpirali aſſumto reſponden-<lb/>tem, nempe arcum a g aut arc. </s> + <s xml:id="echoid-s5418" xml:space="preserve">a g h M s & </s> + <s xml:id="echoid-s5419" xml:space="preserve">x ſignificante ejusdem arcus co-<lb/>ſinum. </s> + <s xml:id="echoid-s5420" xml:space="preserve">Eſt vero arc. </s> + <s xml:id="echoid-s5421" xml:space="preserve">a g = 48<emph style="super">0</emph>, 35<emph style="super">1</emph> = (quia radius exprimitur unitate) <lb/>o, 84797, ejuſque coſinus = o, 66153: </s> + <s xml:id="echoid-s5422" xml:space="preserve">Igitur in noſtro caſu fit or = <lb/>o, 50878 + 1, 32922 = 1, 83800. </s> + <s xml:id="echoid-s5423" xml:space="preserve">Quia porro puncta o & </s> + <s xml:id="echoid-s5424" xml:space="preserve">q ſunt in eadem <lb/>altitudine poſita, atque lineæ o r & </s> + <s xml:id="echoid-s5425" xml:space="preserve">q x inter ſe æquales, apparet quæſtionem <lb/>nunc eo eſſe reductam, ut alius arcus a g h M s inveniatur puncto q reſpondens, +<pb o="188" file="0202" n="202" rhead="HYDRODYNAMICÆ"/> +qui ſi vocetur X, ejuſque coſinus x, ſit o, 60000X + o, 80000 (1 + x) <lb/>= or = 1, 83800: </s> + <s xml:id="echoid-s5426" xml:space="preserve">pro iſta conditione invenitur arcus a g h M s proxime <lb/>175 {1/2} grad. </s> + <s xml:id="echoid-s5427" xml:space="preserve">incidente puncto s in plagam a g M: </s> + <s xml:id="echoid-s5428" xml:space="preserve">Et cum arcus a g fuerit 48<emph style="super">0</emph>, <lb/>35<emph style="super">1</emph>, erit tandem arcus g h M s 126<emph style="super">0</emph>, 55<emph style="super">1</emph>, qui proinde erit ad circumfe-<lb/>rentiam circuli præterpropter ut 10 ad 29: </s> + <s xml:id="echoid-s5429" xml:space="preserve">ſimiliſque ratio intercedit inter ar-<lb/>cum helicis o p q integramque helicem.</s> + <s xml:id="echoid-s5430" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5431" xml:space="preserve">Conſequens inde eſt, ſingulis revolutionibus cochlea à Vitruvio de-<lb/>ſcripta proxime ejici {10/29} illius quantitatis, quam helix integra & </s> + <s xml:id="echoid-s5432" xml:space="preserve">plena con-<lb/>tinet, ſeu paullulum ultra trientem.</s> + <s xml:id="echoid-s5433" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div224" type="section" level="1" n="177"> +<head xml:id="echoid-head225" xml:space="preserve">Scholium 3.</head> +<p> + <s xml:id="echoid-s5434" xml:space="preserve">(IX) Notandum tamen eſt, quæcunque ſit aquæ quantitas, quæ qua-<lb/>libet cochleæ revolutione canalem inferius ingreditur, ſuperiuſque ex eodem <lb/>eſſluit, nullum nec detrimentum nec lucrum propterea cadere in potentiam ab-<lb/>ſolutam ſi nulla habeatur frictionum ration, quia potentia movens cæteris paribus <lb/>illi quantitati proportionalis eſt. </s> + <s xml:id="echoid-s5435" xml:space="preserve">At vero quia frictiones ſemper obſtant, eædem-<lb/>que fere ſunt ob pondus machinæ proprium, ſive major ſive minor quantitas <lb/>aquæ hauriatur, opera utique danda eſt, ut iſta quantitas cæteris paribus fiat <lb/>maxima: </s> + <s xml:id="echoid-s5436" xml:space="preserve">Hâc de re nunc agam paullo diſertius.</s> + <s xml:id="echoid-s5437" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div225" type="section" level="1" n="178"> +<head xml:id="echoid-head226" xml:space="preserve">Scholium 4.</head> +<p> + <s xml:id="echoid-s5438" xml:space="preserve">(X) Jam innui ſuprà, creſcere rationem arcus g h M s ad circumferen-<lb/>tiam circuli decreſcentibus angulis s a o & </s> + <s xml:id="echoid-s5439" xml:space="preserve">N M G: </s> + <s xml:id="echoid-s5440" xml:space="preserve">uterque igitur minimus eſſet <lb/>conſtruendus, niſi alia obſtarent incommoda, præſertim ratione anguli N M G. <lb/></s> + <s xml:id="echoid-s5441" xml:space="preserve">Quod ad angulum s a o attinet, poteſt is fere ad lubitum diminui, neque aliud <lb/>inde incommodum reſultat, niſi quod latera canalis circumflectendi nimis ad <lb/>ſe invicem accedere poſſunt: </s> + <s xml:id="echoid-s5442" xml:space="preserve">E contrario à diminutione iſtius anguli aliud ob-<lb/>tinetur compendium, nempe quod tunc eo verticalius poſſit erigi machina ip-<lb/>ſaque aqua eo altius elevari, etenim angulus a M H ſemper major eſſe debet <lb/>angulo s a o: </s> + <s xml:id="echoid-s5443" xml:space="preserve">à verticaliori autem cochleæ poſitione ſimul obtinetur, ut mino-<lb/>ri incommodo ſit machinæ proprium pondus eaque facilius ſuſtineatur.</s> + <s xml:id="echoid-s5444" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5445" xml:space="preserve">Hæc ita perpendens crediderim fere ſufficere poſſe angulum 5 graduum, <lb/>quem faciat canalis cum baſe nuclei. </s> + <s xml:id="echoid-s5446" xml:space="preserve">Cardanus quoque minorem iſtum fecit +<pb o="189" file="0203" n="203" rhead="SECTIO NONA."/> +angulum quam Vitruvius, & </s> + <s xml:id="echoid-s5447" xml:space="preserve">cum eo pauciores ſuper eodem nucleo circum-<lb/>flecti poſſint canales, quo obliquius ſunt inſerti, Vitruvius octo, Cardanus <lb/>tres tantum ponendos ſtatuit: </s> + <s xml:id="echoid-s5448" xml:space="preserve">ſunt autem canales longiores in cochlea Car-<lb/>dani, ita ut longitudinibus accedat, quod numero canalium decedit. </s> + <s xml:id="echoid-s5449" xml:space="preserve">Ra-<lb/>tione alterius anguli N M G obſervari meretur, aquam altius elevari poſſe, <lb/>quo major iſte fiat angulus, ſed e contrario minorem aquæ quantitatem <lb/>ſingulis ejici revolutionibus. </s> + <s xml:id="echoid-s5450" xml:space="preserve">Juſtum fortaſſe tenebunt medium, qui angu-<lb/>lum iſtum 60. </s> + <s xml:id="echoid-s5451" xml:space="preserve">facient gradum.</s> + <s xml:id="echoid-s5452" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5453" xml:space="preserve">(XI.) </s> + <s xml:id="echoid-s5454" xml:space="preserve">Subducemus nunc hujus noſtræ quoque ad normam præceden-<lb/>tis articuli conſtructæ cochleæ calculum, prouti fecimus de cochlea ad Vi-<lb/>truvii præceptum conſtructa, art. </s> + <s xml:id="echoid-s5455" xml:space="preserve">VIII. </s> + <s xml:id="echoid-s5456" xml:space="preserve">Quia vero per hypotheſin angulus <lb/>s a o eſt 5<emph style="super">0</emph> & </s> + <s xml:id="echoid-s5457" xml:space="preserve">angulus N M G = 60<emph style="super">0</emph>; </s> + <s xml:id="echoid-s5458" xml:space="preserve">reperietur per art. </s> + <s xml:id="echoid-s5459" xml:space="preserve">IV. </s> + <s xml:id="echoid-s5460" xml:space="preserve">arcus a g 8<emph style="super">0</emph>, 43<emph style="super">1</emph>, <lb/>& </s> + <s xml:id="echoid-s5461" xml:space="preserve">linea verticalis o r = 1, 00574, cui æqualis erit altera verticalis q x, ſi <lb/>dentur arcui a g h M s 284<emph style="super">0</emph>, 57<emph style="super">1</emph>, a quo ſi ſubtrahatur arcus a g, remanet ar-<lb/>cus g h M s 276<emph style="super">0</emph>, 14<emph style="super">1</emph>: </s> + <s xml:id="echoid-s5462" xml:space="preserve">qui reſpondet arcui helicis aquam retinere valenti: </s> + <s xml:id="echoid-s5463" xml:space="preserve">eſt <lb/>igitur hæc pars ad totam helicem ut 16574 ad 21600 vel ut 8287 ad 10800, <lb/>ſic ut ſingulis revolutioniqus ejici poſſint plus quam quatuor quintæ partes <lb/>integræ helicis capacitatis, duplumque cum triente præterpropter hac ma-<lb/>china efficiatur, quam obtinetur ſimili machinatione ad mentem Vitruvii fa-<lb/>bricata: </s> + <s xml:id="echoid-s5464" xml:space="preserve">altius quoque eodem nucleo elevantur aquæ in ratione ut √3 ad √2. <lb/></s> + <s xml:id="echoid-s5465" xml:space="preserve">Venio jam ad potentiam tum moventem tum abſolutam, quæ in elevandis aquis <lb/>impenditur.</s> + <s xml:id="echoid-s5466" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div226" type="section" level="1" n="179"> +<head xml:id="echoid-head227" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s5467" xml:space="preserve">(XII.) </s> + <s xml:id="echoid-s5468" xml:space="preserve">Dato pondere aquæ in helice quieſcentis, invenire potentiam <lb/>tangentialem in f in æquilibrio cum illo pondere poſitam.</s> + <s xml:id="echoid-s5469" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div227" type="section" level="1" n="180"> +<head xml:id="echoid-head228" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s5470" xml:space="preserve">Vidimus quomodo problema hoc geometrice ſolvendum ſit ratione <lb/>globi in puncto infimo p quieſcentis. </s> + <s xml:id="echoid-s5471" xml:space="preserve">In præſenti vero caſu paullo aliter ſe <lb/>res habet, quod pondus aquæ per magnum helicis arcum eſt diſtributum, <lb/>neque in puncto aliquo dato concentratum. </s> + <s xml:id="echoid-s5472" xml:space="preserve">Facile quidem eſt in anteceſ-<lb/>ſum prævidere, in utroque caſu easdem fore potentias ex regulis mechani- +<pb o="190" file="0204" n="204" rhead="HYDRODYNAMICÆ"/> +cæ indirectis; </s> + <s xml:id="echoid-s5473" xml:space="preserve">placet tamen hujus rei demonſtrationem dare ex natura vectis <lb/>petitam, quia mechanici eo omnia reducere amant.</s> + <s xml:id="echoid-s5474" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5475" xml:space="preserve">Helicem conſiderabimus a 1 b ex figura quinquageſima ſecunda ſeor-<lb/>ſim deſumtam, ad evitandam linearum confuſionem, conſervatis denomi-<lb/>nationibus art. </s> + <s xml:id="echoid-s5476" xml:space="preserve">IV. </s> + <s xml:id="echoid-s5477" xml:space="preserve">adhibitis. </s> + <s xml:id="echoid-s5478" xml:space="preserve">Sic igitur in Figura 53. </s> + <s xml:id="echoid-s5479" xml:space="preserve">erit rurſus angulus <lb/> +<anchor type="note" xlink:label="note-0204-01a" xlink:href="note-0204-01"/> +N M G angulus quem facit nucleus cum horizonte, cujus ſinus = N, ſi-<lb/>nusque anguli a M H = n; </s> + <s xml:id="echoid-s5480" xml:space="preserve">a 1 b eſt una ſpiralis circumvolutio: </s> + <s xml:id="echoid-s5481" xml:space="preserve">baſis nuclei <lb/>eſt circulus a c M p a; </s> + <s xml:id="echoid-s5482" xml:space="preserve">ſinus anguli p a l eſt ut ante = m, ejusque coſinus M; <lb/></s> + <s xml:id="echoid-s5483" xml:space="preserve">puncta vero l & </s> + <s xml:id="echoid-s5484" xml:space="preserve">o ſunt extremitates aquæ in ſpirali quieſcentis & </s> + <s xml:id="echoid-s5485" xml:space="preserve">in ea-<lb/>dem altitudine ab horizonte poſita, ex iſtis punctis ductæ ſunt ad periphe-<lb/>riam baſis rectæ l c & </s> + <s xml:id="echoid-s5486" xml:space="preserve">o p ad baſin perpendiculares. </s> + <s xml:id="echoid-s5487" xml:space="preserve">In parte helicis quam <lb/>aqua occupat ſumta ſunt duo puncta infinite propinqua m & </s> + <s xml:id="echoid-s5488" xml:space="preserve">n & </s> + <s xml:id="echoid-s5489" xml:space="preserve">per hæc du-<lb/>ctæ ſunt rectæ n f & </s> + <s xml:id="echoid-s5490" xml:space="preserve">m g rurſus ad baſin perpendiculares. </s> + <s xml:id="echoid-s5491" xml:space="preserve">Denique ex pun-<lb/>ctis c, f, g, p ductæ ſunt ad diametrum a M perpendiculares c d, f h, g i & </s> + <s xml:id="echoid-s5492" xml:space="preserve"><lb/>p q; </s> + <s xml:id="echoid-s5493" xml:space="preserve">atque centrum baſis ponitur in e, radiusque e a = 1. </s> + <s xml:id="echoid-s5494" xml:space="preserve">Sit jam arcus <lb/>ſpiralis l 1 o aqua plenus = c & </s> + <s xml:id="echoid-s5495" xml:space="preserve">conſequenter arcus circularis eidem reſpon-<lb/>dens c M p = M c; </s> + <s xml:id="echoid-s5496" xml:space="preserve">a l = e; </s> + <s xml:id="echoid-s5497" xml:space="preserve">a c = M e; </s> + <s xml:id="echoid-s5498" xml:space="preserve">a d (ſeu ſinus verſus arcus ac) = f; </s> + <s xml:id="echoid-s5499" xml:space="preserve"><lb/>a q = g; </s> + <s xml:id="echoid-s5500" xml:space="preserve">pondus aquæ in l s o = p: </s> + <s xml:id="echoid-s5501" xml:space="preserve">arcus a l n = x; </s> + <s xml:id="echoid-s5502" xml:space="preserve">n m = d x; </s> + <s xml:id="echoid-s5503" xml:space="preserve">a c f = M x; </s> + <s xml:id="echoid-s5504" xml:space="preserve"><lb/>f g = M d x; </s> + <s xml:id="echoid-s5505" xml:space="preserve">a b = y; </s> + <s xml:id="echoid-s5506" xml:space="preserve">h i = d y; </s> + <s xml:id="echoid-s5507" xml:space="preserve">h f = √2y - yy, erit pondus guttulæ in <lb/>nm = {p d x/c}; </s> + <s xml:id="echoid-s5508" xml:space="preserve">ſi vero linea h f multiplicetur per ſinum anguli a M H, divida-<lb/>turque per ſinum totum, habetur vectis quo particula n m cochleam circum-<lb/>agere tentat: </s> + <s xml:id="echoid-s5509" xml:space="preserve">eſtigitur vectis iſte = n √ (2y - yy) qui multiplicatus per præ-<lb/>fatum guttulæ pondus {p d x/c} dat ejusdem momentum {n p d x/c} √ (2y - y y)}. </s> + <s xml:id="echoid-s5510" xml:space="preserve"><lb/>Sed ex natura circuli eſt M d x = {dy√ (2y - yy): </s> + <s xml:id="echoid-s5511" xml:space="preserve">hoc igitur valore ſubſtituto <lb/>pro d x, fit idem guttulæ n m momentum = {n p d y/M c}, cujus integralis, ſub-<lb/>tracta debita conſtante, eſt {n p (y - f)/Mc}, denotatque momentum aquæ in ar-<lb/>cu l n; </s> + <s xml:id="echoid-s5512" xml:space="preserve">hinc igitur momentum omnis aquæ in l 1 o eſt = {n p (g - f)/Mc}: </s> + <s xml:id="echoid-s5513" xml:space="preserve">quod <lb/>diviſum per vectem potentiæ in f applicatæ ſeu per 1 relinquit potentiam <lb/>iſtam quæſitam pariter = {n p (g - f)/Mc}. </s> + <s xml:id="echoid-s5514" xml:space="preserve">Q. </s> + <s xml:id="echoid-s5515" xml:space="preserve">E. </s> + <s xml:id="echoid-s5516" xml:space="preserve">I.</s> + <s xml:id="echoid-s5517" xml:space="preserve"/> +</p> +<div xml:id="echoid-div227" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0204-01" xlink:href="note-0204-01a" xml:space="preserve">Fig. 53.</note> +</div> +<pb o="191" file="0205" n="205" rhead="SECTIO NONA."/> +</div> +<div xml:id="echoid-div229" type="section" level="1" n="181"> +<head xml:id="echoid-head229" xml:space="preserve">Scholium 1.</head> +<p> + <s xml:id="echoid-s5518" xml:space="preserve">(XIII.) </s> + <s xml:id="echoid-s5519" xml:space="preserve">Ut appareat, non differre valorem iſtius potentiæ ab illa, quam <lb/>pro globo ejusdem ponderis p invenimus articulo V. </s> + <s xml:id="echoid-s5520" xml:space="preserve">nempe {m N p/M}, demon-<lb/>ſtranda eſt æqualitas inter {n p (g - f)/Mc} & </s> + <s xml:id="echoid-s5521" xml:space="preserve">{m N p/M} ſeu inter n (g - f) & </s> + <s xml:id="echoid-s5522" xml:space="preserve">m N c: </s> + <s xml:id="echoid-s5523" xml:space="preserve">iſta <lb/>vero æqualitas deducenda eſt ex eo, quod extremitates aquæ l & </s> + <s xml:id="echoid-s5524" xml:space="preserve">o in eadem <lb/>ab horizonte altitudine poſitæ ſint; </s> + <s xml:id="echoid-s5525" xml:space="preserve">inde enim ſequitur, ut demonſtravi-<lb/>mus art. </s> + <s xml:id="echoid-s5526" xml:space="preserve">IV. </s> + <s xml:id="echoid-s5527" xml:space="preserve">eſſe aggregatum ex arcu a c multiplicato per {m N/M} & </s> + <s xml:id="echoid-s5528" xml:space="preserve">ex linea M d <lb/>multiplicata per n = aggregato ex arcu a c M p pariter multiplicato per {m N/M} <lb/>& </s> + <s xml:id="echoid-s5529" xml:space="preserve">ex linea M q multiplicata per n. </s> + <s xml:id="echoid-s5530" xml:space="preserve">Adhibitis itaque denominationibus præ-<lb/>cedentis articuli, fit M e X {m N/M} + (2 - f) X n = (M e + M c) X {m N/M} + <lb/>(2 - g) X n, vel n (g - f) = m N c; </s> + <s xml:id="echoid-s5531" xml:space="preserve">quæ æqualitas demonſtranda erat ad <lb/>demonſtrandam æqualitatem potentiarum tum pro globo tum pro aqua in <lb/>f applicandarum.</s> + <s xml:id="echoid-s5532" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div230" type="section" level="1" n="182"> +<head xml:id="echoid-head230" xml:space="preserve">Scholium 2.</head> +<p> + <s xml:id="echoid-s5533" xml:space="preserve">(XIV) Quia potentia {n p (g - f)/M c} non differt ab {m N p/M} & </s> + <s xml:id="echoid-s5534" xml:space="preserve">quantitas {m N/M} <lb/>eadem manet, quæcunque aquæ quantitas una revolutione hauriatur aut eji-<lb/>ciatur, erit potentia iſta proportionalis eidem quantitati aquæ ſingulis revolu-<lb/>tionibus ejectæ ſeu ponderi p. </s> + <s xml:id="echoid-s5535" xml:space="preserve">Facile quoque demonſtratu eſt, ſi eadem aqua-<lb/>rum quantitas, eadem potentia movente eademque velocitate ad parem altitudi-<lb/>nem verticalem elevetur ſuper ſimplici plano, quod ad hunc finem debite ver-<lb/>ſus horizontem inclinatum ſit, fore ut tempus elevationis quoque idem ſit.</s> + <s xml:id="echoid-s5536" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5537" xml:space="preserve">Igitur eadem potentia abſoluta requiritur in cochlea Archimedis, quam <lb/>ſuper plano inclinato, ad quod omnes machinæ reduci poſſunt, nec ullam <lb/>habet iſta cochlea prærogativam præ reliquis machinis in theoria ſpectatis. <lb/></s> + <s xml:id="echoid-s5538" xml:space="preserve">Fortaſſe in praxi minus eſt obnoxia incommodis §. </s> + <s xml:id="echoid-s5539" xml:space="preserve">26. </s> + <s xml:id="echoid-s5540" xml:space="preserve">indicatis: </s> + <s xml:id="echoid-s5541" xml:space="preserve">nequaquam <lb/>improbo ejus uſum, ſed nec eam præfero præ antliis Cteſibianis.</s> + <s xml:id="echoid-s5542" xml:space="preserve"/> +</p> +<pb o="192" file="0206" n="206" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s5543" xml:space="preserve">§. </s> + <s xml:id="echoid-s5544" xml:space="preserve">28. </s> + <s xml:id="echoid-s5545" xml:space="preserve">Intelligitur ex hactenus dictis, quibus titulis una machina alte-<lb/>ri præferenda ſit, quemnam machinæ perfectionis gradum admittant; </s> + <s xml:id="echoid-s5546" xml:space="preserve">ad quid <lb/>potiſſimum attendendum ſit in illarum conſtructione & </s> + <s xml:id="echoid-s5547" xml:space="preserve">uſu; </s> + <s xml:id="echoid-s5548" xml:space="preserve">quanta potentiæ ab-<lb/>ſolutæ pars perdatur, aliaque ſimilia: </s> + <s xml:id="echoid-s5549" xml:space="preserve">Equidem machinas tantum conſidera-<lb/>vimus potentiis ut dicuntur animatis motas: </s> + <s xml:id="echoid-s5550" xml:space="preserve">facile autem apparet iiſdem legibus <lb/>ſubjectas eſſe machinas, quæ ab impetu aquarum, venti, aut ab aquarum gra-<lb/>vitatione hujusmodique aliis principiis ſunt movendæ; </s> + <s xml:id="echoid-s5551" xml:space="preserve">ſemper enim potentia <lb/>movens ducta in tempus & </s> + <s xml:id="echoid-s5552" xml:space="preserve">velocitatem puncti cui potentia eſt applicata, dabit <lb/>productum ex quantitate aquæ & </s> + <s xml:id="echoid-s5553" xml:space="preserve">altitudine ad quam iſta quantitas aſſumto <lb/>tempore elevari poſſit ope machinæ propoſitæ, ſepoſitis impedimentis alienis. <lb/></s> + <s xml:id="echoid-s5554" xml:space="preserve">Loquor autem de machinis, quibus nihil de potentia abſoluta perditur; </s> + <s xml:id="echoid-s5555" xml:space="preserve">fieri enim <lb/>poteſt, ut maxima pars pereat, quod ſatis oſtendimus in ſuperioribus.</s> + <s xml:id="echoid-s5556" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5557" xml:space="preserve">§. </s> + <s xml:id="echoid-s5558" xml:space="preserve">29. </s> + <s xml:id="echoid-s5559" xml:space="preserve">Apparet exinde aquam ad certam altitudinem elevatam poſſe <lb/>rurſus ſuo deſcenſu eundem præſtare effectum: </s> + <s xml:id="echoid-s5560" xml:space="preserve">effectus autem erit æſtimandus <lb/>ex quantitate aquarum elevandarum & </s> + <s xml:id="echoid-s5561" xml:space="preserve">ex altitudine elevationis, ſic ut v. </s> + <s xml:id="echoid-s5562" xml:space="preserve">gr. <lb/></s> + <s xml:id="echoid-s5563" xml:space="preserve">deſcenſu 8. </s> + <s xml:id="echoid-s5564" xml:space="preserve">pedum cubicorum ex altitudine unius pedis poſſint totidem rur-<lb/>ſus elevari pedes cubici ad ſimilem altitudinem aut 4. </s> + <s xml:id="echoid-s5565" xml:space="preserve">pedes cubici ad altitudi-<lb/>nem duorum pedum, aut unus pes cubicus ad altitudinem 8. </s> + <s xml:id="echoid-s5566" xml:space="preserve">pedum & </s> + <s xml:id="echoid-s5567" xml:space="preserve">ſic ut-<lb/>cunque libuerit. </s> + <s xml:id="echoid-s5568" xml:space="preserve">Specimen machinæ, quæ poſſit aquam ad quamcunque al-<lb/>titudinem elevare minimo aquarum deſcenſu, videre eſt apud D. </s> + <s xml:id="echoid-s5569" xml:space="preserve">Perrault in <lb/>Comment. </s> + <s xml:id="echoid-s5570" xml:space="preserve">ad Vitruvium lib. </s> + <s xml:id="echoid-s5571" xml:space="preserve">10. </s> + <s xml:id="echoid-s5572" xml:space="preserve">cap. </s> + <s xml:id="echoid-s5573" xml:space="preserve">12. </s> + <s xml:id="echoid-s5574" xml:space="preserve">quam machinam ut incredibile fere para-<lb/>doxon affert ejusque inventorem facit D. </s> + <s xml:id="echoid-s5575" xml:space="preserve">Franchini Italum, cujus induſtria & </s> + <s xml:id="echoid-s5576" xml:space="preserve"><lb/>conſiliis in horto Bibliothecæ Regiæ cum ſucceſſu conſtructa fuit. </s> + <s xml:id="echoid-s5577" xml:space="preserve">Fundamen-<lb/>tum machinæ in eo conſiſtit, ut ſitulæ concatenatæ, & </s> + <s xml:id="echoid-s5578" xml:space="preserve">in circulum redeuntes <lb/>aquam excipiant eamque in locum tranſportent infimum, ibique effundant, <lb/>dum alia ſitularum ſeries aquas hauriunt & </s> + <s xml:id="echoid-s5579" xml:space="preserve">ad locum longe altiorem, minori <lb/>tamen copia ferunt atque effundunt: </s> + <s xml:id="echoid-s5580" xml:space="preserve">perſpicuum autem eſt, ſeriem priorem <lb/>ſi omnes ſitulæ deſcendentes graviores ſint omnibus ſitulis aſcendentibus, alte-<lb/>ram perpetuo in gyrum acturam eſſe; </s> + <s xml:id="echoid-s5581" xml:space="preserve">Machinæ etiam ſunt, quæ idem præ-<lb/>ſtant per ſimplices tubos ope epiſtomiorum ſtatis temporibus convertendorum, <lb/>in quam quidem converſionem nulla potentia impenditur. </s> + <s xml:id="echoid-s5582" xml:space="preserve">Hujuſmodi ma-<lb/>chinationes deſcribit Carolus Fontana.</s> + <s xml:id="echoid-s5583" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5584" xml:space="preserve">At ſi quis credat poſſe ex impetu aquarum ex certa altitudine delapſa-<lb/>rum & </s> + <s xml:id="echoid-s5585" xml:space="preserve">in machinæ alas impingentium idem obtineri, is longe aberrabit. </s> + <s xml:id="echoid-s5586" xml:space="preserve">Ta- +<pb o="193" file="0207" n="207" rhead="SECTIO NONA."/> +lis machinatio pertineret ad illarum claſſem, quibus maxima potentiæ obſolutæ <lb/>pars evaneſcit ſine fructu.</s> + <s xml:id="echoid-s5587" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5588" xml:space="preserve">Non abs re erit iſtud argumentum accuratius proſequi, & </s> + <s xml:id="echoid-s5589" xml:space="preserve">oſtendere <lb/>quantus effectus ab impetu aquarum aut venti obtineri poſſit & </s> + <s xml:id="echoid-s5590" xml:space="preserve">ſub quibus cir-<lb/>cumſtantiis effectus iſte ſit omnium maximus dicendus.</s> + <s xml:id="echoid-s5591" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div231" type="section" level="1" n="183"> +<head xml:id="echoid-head231" style="it" xml:space="preserve">(C) De Machinis, quæ ab impetu fluidi, veluti vi venti moventur.</head> +<p> + <s xml:id="echoid-s5592" xml:space="preserve">§. </s> + <s xml:id="echoid-s5593" xml:space="preserve">30. </s> + <s xml:id="echoid-s5594" xml:space="preserve">Poſtquam aquæ ad certam altitudinem elevatæ ex eâdem rur-<lb/>ſus decidunt, continueque in alas rotæ circumagendæ impingunt, fieri aliter <lb/>non poteſt, quin potentia abſoluta ad rotam ſic circumagendam requiſita multo <lb/>minor ſit illa, quæ in elevationem aquarum impenſa fuit, cujus rei præci-<lb/>pua ratio eſt, quod aquæ poſt impulſum ad latera deſilientes velocitatem <lb/>etiamnum conſervent, quæ ad rotæ rotationem nihil confert. </s> + <s xml:id="echoid-s5595" xml:space="preserve">Igitur magna <lb/>potentiæ abſolutæ pars inutilis fieret, ſi elevatione aquarum efficiendum eſſet, <lb/>ut ab impetu earundem machina circumagatur & </s> + <s xml:id="echoid-s5596" xml:space="preserve">ab hac denique aquæ rurſus <lb/>aliæ ad certam altitudinem eleventur; </s> + <s xml:id="echoid-s5597" xml:space="preserve">& </s> + <s xml:id="echoid-s5598" xml:space="preserve">quidem major minorve pars perit <lb/>pro diverſis circumſtantiis, nunquam vero, ut monſtrabo, minus quam {23/27} <lb/>totius perdetur, ſi ad normam vulgaris impulſus aquarum æſtimationis com-<lb/>putus fiat.</s> + <s xml:id="echoid-s5599" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5600" xml:space="preserve">§. </s> + <s xml:id="echoid-s5601" xml:space="preserve">31. </s> + <s xml:id="echoid-s5602" xml:space="preserve">Statuitur autem communiter ſi aquæ ex cylindro valde amplo <lb/>per ſimplex foramen tota ſua velocitate, id eſt, quæ toti altitudini aquæ ſu-<lb/>pra foramen debeatur, fluant, atque vena ſtatim præ foramine directe impin-<lb/>gat in planum, fore ut impetus fluidi contra planum in æquilibrio ſit cum pon-<lb/>dere cylindri aquei, ſuper foramine ad altitudinem aquæ erecti. </s> + <s xml:id="echoid-s5603" xml:space="preserve">Experimento <lb/>quidem fallaci auctores ſeducti hanc ſtabiliverunt theoriam omnino falſam. <lb/></s> + <s xml:id="echoid-s5604" xml:space="preserve">Nolui tamen hîc ab illa recedere, quia veram theoriam nondum expoſui at-<lb/>que deinceps facile erit expoſita noſtra theoria calculum corrigere. </s> + <s xml:id="echoid-s5605" xml:space="preserve">Liceat igi-<lb/>tur, donec ſuo loco rem rectius perpenderimus, vulgari ſententiæ, quamvis <lb/>erroneæ, adhærere. </s> + <s xml:id="echoid-s5606" xml:space="preserve">Quo major eſt impetus fluidi, eo majori ratione erit po-<lb/>tentia abſoluta, quam dabimus, augenda.</s> + <s xml:id="echoid-s5607" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5608" xml:space="preserve">§. </s> + <s xml:id="echoid-s5609" xml:space="preserve">32. </s> + <s xml:id="echoid-s5610" xml:space="preserve">Finge nunc (Fig. </s> + <s xml:id="echoid-s5611" xml:space="preserve">54.) </s> + <s xml:id="echoid-s5612" xml:space="preserve">vas A B C ceu antliam quæ aquas per <lb/> +<anchor type="note" xlink:label="note-0207-01a" xlink:href="note-0207-01"/> +foramen C in directione tantum non verticali expellat: </s> + <s xml:id="echoid-s5613" xml:space="preserve">aquas autem, cum ad <lb/>ſummum pervenerint, ab alio vaſe E D F excipi. </s> + <s xml:id="echoid-s5614" xml:space="preserve">In alterius hujus vaſis fundo +<pb o="194" file="0208" n="208" rhead="HYDRODYNAMICÆ"/> +concipe foramen D, priori C æquale, & </s> + <s xml:id="echoid-s5615" xml:space="preserve">in eadem altitudine poſitum, ita ut <lb/>tanta aquarum copia effluat per D, quanta fuperius injicitur, ipſumque vas <lb/>E D F conſtanter plenum ſervetur. </s> + <s xml:id="echoid-s5616" xml:space="preserve">Porro puta aquas per D effluentes perpe-<lb/>tuo impingere in alas alicujus rotæ, quæ hoc modo circumacta aquas alias ele-<lb/>vet: </s> + <s xml:id="echoid-s5617" xml:space="preserve">Loco iſtius machinæ deſcribitur in figura ſimplex vectis volubilis circa H, <lb/>ponendo talem vectem continue alium atque alium adeſſe præ foramine D, <lb/>qui aquas excipiat, atque altera ſua extremitate aquas hauriat, eaſdemque ad <lb/>datam altitudinem elevet.</s> + <s xml:id="echoid-s5618" xml:space="preserve"/> +</p> +<div xml:id="echoid-div231" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0207-01" xlink:href="note-0207-01a" xml:space="preserve">Fig. 54.</note> +</div> +<p> + <s xml:id="echoid-s5619" xml:space="preserve">His ita poſitis inquiram primo in potentiam abſolutam, quæ aquas per fo-<lb/>ramen C ad altitudinem C E elevat; </s> + <s xml:id="echoid-s5620" xml:space="preserve">deinde quoque in potentiam abſolutam, quæ <lb/>requiritur in G ad vectem eadem velocitate movendum, quâ movetur ab im-<lb/>pulſu aquarum D G.</s> + <s xml:id="echoid-s5621" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5622" xml:space="preserve">§. </s> + <s xml:id="echoid-s5623" xml:space="preserve">33. </s> + <s xml:id="echoid-s5624" xml:space="preserve">Sit amplitudo foraminis C vel D = n, amplitudo A B = m, ve-<lb/>locitas aquarum in C vel D = v, pondus cylindri ſuper foramine C aut D ad <lb/>altitudinem C E extructi = p: </s> + <s xml:id="echoid-s5625" xml:space="preserve">tempus fluxus = t; </s> + <s xml:id="echoid-s5626" xml:space="preserve">erit pondus P = {m/n} p: </s> + <s xml:id="echoid-s5627" xml:space="preserve">ve-<lb/>locitas, qua pondus dum aquæ expelluntur deſcendit = {n/m} v: </s> + <s xml:id="echoid-s5628" xml:space="preserve">eſt igitur (per <lb/>§. </s> + <s xml:id="echoid-s5629" xml:space="preserve">3.) </s> + <s xml:id="echoid-s5630" xml:space="preserve">potentia abſoluta in aquas per C ejectas impenſa = {m/n} p X {n/m} v X t = p v t.</s> + <s xml:id="echoid-s5631" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5632" xml:space="preserve">§. </s> + <s xml:id="echoid-s5633" xml:space="preserve">34. </s> + <s xml:id="echoid-s5634" xml:space="preserve">Ut jam potentia abſoluta in gyrationem vectis G L circa punctum <lb/>Himpenfa determinetur, notandum eſt illam minime ſibimet conſtare; </s> + <s xml:id="echoid-s5635" xml:space="preserve">mutari <lb/>enim à mutata velocitate, quacum vectis circumagitur. </s> + <s xml:id="echoid-s5636" xml:space="preserve">Igitur faciemus ve-<lb/>locitatem qua extremitas ejus in G movetur = V. </s> + <s xml:id="echoid-s5637" xml:space="preserve">Hoc autem modo aquæ <lb/>impingere cenſendæ ſunt in G velocitate v - V, atque ſic preſſionem exerce-<lb/>re, quæ fit = ({v - V/v})<emph style="super">2</emph> p: </s> + <s xml:id="echoid-s5638" xml:space="preserve">(ſunt enim preſſiones in ratione quadrata velo-<lb/>citatum fluidi impingentis atque pro velocitate v ponitur preſſio = p). </s> + <s xml:id="echoid-s5639" xml:space="preserve">Iſta <lb/>vero preſſio eſt loco potentiæ moventis; </s> + <s xml:id="echoid-s5640" xml:space="preserve">poſſumus nempe loco preſſionis fluidi <lb/>ponere pondus vecti ſuperincumbens in G, quod ſit = ({v - V/v})<emph style="super">2</emph> p. </s> + <s xml:id="echoid-s5641" xml:space="preserve">Iſtud <lb/>vero pondus eadem velocitate movebitur quâ punctum G, nempe velocitate V, <lb/>agitque durante tempore t: </s> + <s xml:id="echoid-s5642" xml:space="preserve">Eſt igitur potentia abſoluta ad rotationem vectis du-<lb/>rante tempore t & </s> + <s xml:id="echoid-s5643" xml:space="preserve">velocitate V requiſita = ({v - V/v})<emph style="super">2</emph> p X V X t.</s> + <s xml:id="echoid-s5644" xml:space="preserve"/> +</p> +<pb o="195" file="0209" n="209" rhead="SECTIO NONA."/> +<p> + <s xml:id="echoid-s5645" xml:space="preserve">§. </s> + <s xml:id="echoid-s5646" xml:space="preserve">35. </s> + <s xml:id="echoid-s5647" xml:space="preserve">Quod ſi igitur vectis L G non immediate circumagitur, ſed <lb/>fluidum ad altitudinem C E elevatur, eo animo, ut vena fluidi ſuo impulſu <lb/>in G vectem circumagendo ab altera parte aquam elevet, erit potentia abſoluta <lb/>integra ad potentiam abſolutam utilem, ut p v t ad ({v - V/v})<emph style="super">2</emph> p V t, ſeu ut v<emph style="super">3</emph><unsure/> <lb/>ad (v - V)<emph style="super">2</emph> V: </s> + <s xml:id="echoid-s5648" xml:space="preserve">eademque ſe habebit ad partem ſui inutilem ut v<emph style="super">3</emph> ad v<emph style="super">3</emph> -<lb/>vv V + 2 v V V - V<emph style="super">3</emph>.</s> + <s xml:id="echoid-s5649" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5650" xml:space="preserve">§. </s> + <s xml:id="echoid-s5651" xml:space="preserve">36. </s> + <s xml:id="echoid-s5652" xml:space="preserve">In omnibus fere machinis, quarum principium motus conſiſtit <lb/>in impulſu fluidi fieri ſolet, ut velocitas vectis, ubi fluidi impetum ſuſtinet, <lb/>ſeu V ſit admodum parva ratione velocitatis fluidi v; </s> + <s xml:id="echoid-s5653" xml:space="preserve">in his autem maxima <lb/>pars effectus, qui ab eadem fluidi quantitate pari velocitate moti obtineri poſ-<lb/>ſet, perditur.</s> + <s xml:id="echoid-s5654" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5655" xml:space="preserve">§. </s> + <s xml:id="echoid-s5656" xml:space="preserve">37. </s> + <s xml:id="echoid-s5657" xml:space="preserve">Maximus oritur ab impulſu fluidi effectus, ſive, quod idem <lb/>eſt, maxima fit potentia abſoluta §. </s> + <s xml:id="echoid-s5658" xml:space="preserve">34. </s> + <s xml:id="echoid-s5659" xml:space="preserve">definita, ſi ſit V = {1/3} v; </s> + <s xml:id="echoid-s5660" xml:space="preserve">& </s> + <s xml:id="echoid-s5661" xml:space="preserve">tunc eſt iſta <lb/>potentia abſoluta = {4/27} p v t, atque etiamnum viginti tribus vigeſimis ſeptimis <lb/>partibus deficit, à potentia ſimili, quæ in elevandas aquas ex C in E F im-<lb/>penditur.</s> + <s xml:id="echoid-s5662" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5663" xml:space="preserve">Si proinde naturalis habeatur aquarum deſcenſus, atque illo utendum <lb/>ſit ad elevandas aquas aliudve ſimile quid præſtandum, faciendum eſt ut ma-<lb/>china eo in loco, quo fit impulſus, velocitate moveatur ſubtripla velocitatis flui-<lb/>di impingentis. </s> + <s xml:id="echoid-s5664" xml:space="preserve">Huic vero conditioni ſemper ſatisfieri poteſt, quod ex alla-<lb/>to vectis exemplo patet. </s> + <s xml:id="echoid-s5665" xml:space="preserve">Si enim majori velocitate moveatur punctum G, di-<lb/>minue partem H G manentibus reliquis aut eam auge, ſi minori moveatur <lb/>velocitate punctum G. </s> + <s xml:id="echoid-s5666" xml:space="preserve">Vel etiam ſalva longitudine H G fac, ut aquæ in ex-<lb/>tremitate L majori minorive quantitate hauriantur.</s> + <s xml:id="echoid-s5667" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5668" xml:space="preserve">§. </s> + <s xml:id="echoid-s5669" xml:space="preserve">38. </s> + <s xml:id="echoid-s5670" xml:space="preserve">Iſta vero ratione fluidorum ad perpendiculum in alas impin-<lb/>gentium: </s> + <s xml:id="echoid-s5671" xml:space="preserve">alius eſt computus pro fluidis oblique incidentibus in alas moletri-<lb/>narum vi venti agitandarum aliarumque ſimilium machinarum. </s> + <s xml:id="echoid-s5672" xml:space="preserve">De his nunc <lb/>pauca quædam ſuperaddam atque iis ſectioni huic finem imponam.</s> + <s xml:id="echoid-s5673" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5674" xml:space="preserve">Quum fluidum in ſuperficiem totius alæ utcunque poſitæ & </s> + <s xml:id="echoid-s5675" xml:space="preserve">in dire-<lb/>ctione ad motum fluidi perpendiculari rotaturæ impingit, docent auctores, flui-<lb/>dum maximum in alam exercere niſum ad promovendam rotationem, quando <lb/>ala cum directione venti angulum facit, cujus ſinus ſit ad ſinum totum ut √ 2 +<pb o="196" file="0210" n="210" rhead="HYDRODYNAMICÆ"/> +ad √ 3; </s> + <s xml:id="echoid-s5676" xml:space="preserve">Si vero vena fluidi eadem atque tota excipiatur ab ala, modo ſic mo-<lb/>do aliter ad directionem fluidi inclinatâ, maximam preſſionem ſuſtinebit in <lb/>directione rotationis ala, quæ facit angulum ſemirectum cum directione fluidi.</s> + <s xml:id="echoid-s5677" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5678" xml:space="preserve">Prima Regula pertinet ad machinas quæ à vento omnia ambiente cir-<lb/>cumaguntur: </s> + <s xml:id="echoid-s5679" xml:space="preserve">altera ad illas, quæ à vena ſolitaria & </s> + <s xml:id="echoid-s5680" xml:space="preserve">à certa determinataque flui-<lb/>di quantitate moventur. </s> + <s xml:id="echoid-s5681" xml:space="preserve">Utraque vero hypotheſi innititur, quod motus ala-<lb/>rum admodum parvus ſit reſpectu motus fluidi, ſi enim ad motum alarum re-<lb/>ſpicias, ambæ regulæ falſæ ſunt; </s> + <s xml:id="echoid-s5682" xml:space="preserve">neque profecto iſte motus negligendus eſt, in <lb/>moletrinis enim ſæpe obſervavi, extremitates alarum velocitate ferri, ipſam <lb/>fere venti velocitatem exæquante.</s> + <s xml:id="echoid-s5683" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5684" xml:space="preserve">Hæc cum ita ſint, calculum nunc ita ponemus, ut utriuſque motus <lb/>rationem habeamus.</s> + <s xml:id="echoid-s5685" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5686" xml:space="preserve">§. </s> + <s xml:id="echoid-s5687" xml:space="preserve">39. </s> + <s xml:id="echoid-s5688" xml:space="preserve">Sit igitur fluidum D E B A (Fig. </s> + <s xml:id="echoid-s5689" xml:space="preserve">55.) </s> + <s xml:id="echoid-s5690" xml:space="preserve">quod ſub directione E B <lb/> +<anchor type="note" xlink:label="note-0210-01a" xlink:href="note-0210-01"/> +impingit in totum planum A B: </s> + <s xml:id="echoid-s5691" xml:space="preserve">moveri autem ponitur planum motu paral-<lb/>lelo in directione B b ad E B perpendiculari: </s> + <s xml:id="echoid-s5692" xml:space="preserve">Sint porro velocitates ejusmo-<lb/>di, ut dum particula fluidi percurrit lineam E B, punctum plani B abſol-<lb/>vat lineam B b. </s> + <s xml:id="echoid-s5693" xml:space="preserve">His poſitis fingere licet totum ſyſtema, fluidum nempe <lb/>cum plano moveri à b verſus B & </s> + <s xml:id="echoid-s5694" xml:space="preserve">quidem velocitate b B: </s> + <s xml:id="echoid-s5695" xml:space="preserve">Ita vero fiet, ut <lb/>planum A B quieſcat, particula autem fluidi in punctum B incidens cenſen-<lb/>da ſit veniſſe expuncto e, ſumta E e = B b, & </s> + <s xml:id="echoid-s5696" xml:space="preserve">ſic de omnibus guttulis. <lb/></s> + <s xml:id="echoid-s5697" xml:space="preserve">Igitur loco fluidi D E B A in planum motum A B incidentis velocitate E B <lb/>concipiendum erit fluidum d e B A in idem planum A B ſed immotum inci-<lb/>dens velocitate e B: </s> + <s xml:id="echoid-s5698" xml:space="preserve">Producatur jam A B usque in b agaturque D E d e b per-<lb/>pendicularis ad E B, erit motus particulæ fluidi repræſentatus per e B reſol-<lb/>vendus in e g & </s> + <s xml:id="echoid-s5699" xml:space="preserve">g B, ſibi invicem perpendiculariter inſiſtentes, quorum po-<lb/>ſterior nihil in planum A B agit, alter vero e g rurſus ex duobus compoſi-<lb/>tus eſt motibus e f & </s> + <s xml:id="echoid-s5700" xml:space="preserve">f g, quorum poſterior f g planum A B inutiliter in di-<lb/>rectione E B propellere tentat; </s> + <s xml:id="echoid-s5701" xml:space="preserve">dum prior e f ſolus idem planum in dire-<lb/>ctione B b propellit. </s> + <s xml:id="echoid-s5702" xml:space="preserve">Demonſtratum itaque eſt, quamlibet particulam face-<lb/>re impulſum proportionalem lineæ e f: </s> + <s xml:id="echoid-s5703" xml:space="preserve">Dein patet quoque, ſi linea A B <lb/>repræſentet totum planum, fore numerum particularum dato tempore in <lb/>planum impingentium repræſentandum per lineam B N perpendicularem ad <lb/>A d ſeu B e. </s> + <s xml:id="echoid-s5704" xml:space="preserve">Unde tandem niſus aquarum ad movendum planum in dire-<lb/>ctione B b eſt proportionalis lineæ e f ductæ in B N.</s> + <s xml:id="echoid-s5705" xml:space="preserve"/> +</p> +<div xml:id="echoid-div232" type="float" level="2" n="2"> +<note position="left" xlink:label="note-0210-01" xlink:href="note-0210-01a" xml:space="preserve">Fig. 55.</note> +</div> +<pb o="197" file="0211" n="211" rhead="SECTIO NONA."/> +<p> + <s xml:id="echoid-s5706" xml:space="preserve">Ut jam determinetur inclinatio plani ad fluidum ſub his circumſtantiis <lb/>maxime favorabilis ut motus plani in directione B b promoveatur: </s> + <s xml:id="echoid-s5707" xml:space="preserve">ponemus <lb/>A B = 1, D E ſeu A C = x, B C = √1 - xx; </s> + <s xml:id="echoid-s5708" xml:space="preserve">lineam E B, quæ repræ-<lb/>ſentat motum fluidi, = v, & </s> + <s xml:id="echoid-s5709" xml:space="preserve">B b ceu menſuram motus plani = V; </s> + <s xml:id="echoid-s5710" xml:space="preserve">atque <lb/>ſic inſtituto calculo invenitur <lb/>ef = xv √ (1 - xx) - (1 - xx) V, atque BN = [xv - V √ (1 - xx]: <lb/></s> + <s xml:id="echoid-s5711" xml:space="preserve">√ (vv + VV); </s> + <s xml:id="echoid-s5712" xml:space="preserve">unde e f X B N = [xv - V √ (1 - xx)]<emph style="super">2</emph> X {√ (1 - xx)/√ (vv + VV)}, quæ <lb/>quantitas maxima erit, cum fit <lb/>(9v<emph style="super">4</emph> + 18vvVV + 9V<emph style="super">4</emph>)x<emph style="super">6</emph> - (12v<emph style="super">4</emph> + 30vvVV + 18V<emph style="super">4</emph>) x<emph style="super">4</emph> <lb/>+ (4v<emph style="super">4</emph> + 16vvVV + 9V<emph style="super">4</emph>) xx - 4vvVV = o.</s> + <s xml:id="echoid-s5713" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5714" xml:space="preserve">§. </s> + <s xml:id="echoid-s5715" xml:space="preserve">40. </s> + <s xml:id="echoid-s5716" xml:space="preserve">Calculus ratione inclinationis alarum in moletrinis alius eſt, <lb/>quia velocitates in diverſis alarum locis variæ ſunt; </s> + <s xml:id="echoid-s5717" xml:space="preserve">ſunt enim proportiona-<lb/>les diſtantiis à centro, facile autem nunc cuivis erit computum pro mole-<lb/>trinis inſtituere, huic caſui non ulterius inſiſtam, ſufficiat id notaſſe, quod <lb/>non ſatis accurate ſtatuatur ab auctoribus x x = {2/3}, & </s> + <s xml:id="echoid-s5718" xml:space="preserve">quod verus valor ip-<lb/>ſius x ſemper minor ſit quam √ {2/3}. </s> + <s xml:id="echoid-s5719" xml:space="preserve">Si fuerit v. </s> + <s xml:id="echoid-s5720" xml:space="preserve">gr. </s> + <s xml:id="echoid-s5721" xml:space="preserve">V = v, & </s> + <s xml:id="echoid-s5722" xml:space="preserve">omnia alæ <lb/>puncta ſimili velocitate moveri cenſeantur, fiet x = √ {1/2}, quod indicat in-<lb/>clinandam eſſe alam ad directionem venti ſub angulo ſemirecto. </s> + <s xml:id="echoid-s5723" xml:space="preserve">Optima <lb/>alarum conſtructio foret, ſi incurvarentur, ita, ut ſub angulo minori ventus <lb/>in illas impingat ſuperius quam inferius, aut ſi fiat ut alæ ubique ventum <lb/>ſub angulo medio quinquaginta præterpropter graduum excipiant.</s> + <s xml:id="echoid-s5724" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5725" xml:space="preserve">§. </s> + <s xml:id="echoid-s5726" xml:space="preserve">41. </s> + <s xml:id="echoid-s5727" xml:space="preserve">Pergo ad alterum caſum, quo omne fluidum à plano, utcun-<lb/>que id inclinatum ſit, excipi ponitur. </s> + <s xml:id="echoid-s5728" xml:space="preserve">Hic autem patet; </s> + <s xml:id="echoid-s5729" xml:space="preserve">quia numerus <lb/>particularum dato tempore impellentium ſemper idem eſt, nullam eſſe at-<lb/>tentionem faciendam ad linem B N, atque ſic niſum quem aquæ faciunt ad <lb/>movendum planum A B in directione B b ſimpliciter repræſentari per e f ſeu <lb/>xv√1 - xx - (1 - xx) V. </s> + <s xml:id="echoid-s5730" xml:space="preserve">Igitur niſus iſte maximus obtinebitur ſumendo <lb/>xx = {1/2} + {V/2√(vv + VV)}, atque erit ipſe niſus tunc = {1/2}√(vv + VV) <lb/>- {1/2} V, ſi per v intelligatur preſſio directa, quam vena exerit in planum cui <lb/>perpendiculariter occurrit.</s> + <s xml:id="echoid-s5731" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5732" xml:space="preserve">§. </s> + <s xml:id="echoid-s5733" xml:space="preserve">42. </s> + <s xml:id="echoid-s5734" xml:space="preserve">Conſideremus nunc venam D E B A tanquam immediate ex ori- +<pb o="198" file="0212" n="212" rhead="HYDRODYNAMICÆ"/> +ficio D in figura 54. </s> + <s xml:id="echoid-s5735" xml:space="preserve">egreſſam & </s> + <s xml:id="echoid-s5736" xml:space="preserve">vocemus rurſus directam preſſionem venæ <lb/>ita conſideratæ p, ſicut §. </s> + <s xml:id="echoid-s5737" xml:space="preserve">33; </s> + <s xml:id="echoid-s5738" xml:space="preserve">atque erit niſus iſtius aquæ, quo conatur <lb/>planum debito modo, ut niſus maximus fiat, inclinatum propellere in di-<lb/>rectione ad venam perpendiculari = {p/2 v} X (√vv + VV - V): </s> + <s xml:id="echoid-s5739" xml:space="preserve">Et ſi porro <lb/>iſte niſus multiplicatur per velocitatem plani V atque tempus, obtinetur <lb/>potentia abſoluta, qua planum eadem velocitate per idem temporis ſpatium <lb/>moveri queat; </s> + <s xml:id="echoid-s5740" xml:space="preserve">ſic igitur præfata potentia abſoluta erit = {pVt/2v} X (√vv + VV - V).</s> + <s xml:id="echoid-s5741" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5742" xml:space="preserve">§. </s> + <s xml:id="echoid-s5743" xml:space="preserve">43. </s> + <s xml:id="echoid-s5744" xml:space="preserve">Potentia abſoluta, quam modo definivimus, ita eſt comparata, <lb/>ut continue creſcat creſcente V, atque ſi velocitas V infinita ſumatur, fit <lb/>eadem potentia = {1/4} X p v t. </s> + <s xml:id="echoid-s5745" xml:space="preserve">Igitur cum in figura 54 vena D G uti volu-<lb/>mus ad rotandam machinam per impulſum obliquum, nunquam plusquam <lb/>quarta pars obtineri poteſt illius potentiæ abſolutæ, quæ in elevationem aqua-<lb/>rum ex C in E F impenditur. </s> + <s xml:id="echoid-s5746" xml:space="preserve">Impulſu vero directo, nunquam plus quam <lb/>{4/27} obtineri vidimus §. </s> + <s xml:id="echoid-s5747" xml:space="preserve">37. </s> + <s xml:id="echoid-s5748" xml:space="preserve">Ergo effectus fere duplo major impulſu obliquo <lb/>ſeu motu rotæ horizontali quam impulſu directo, ſeu motu rotæ verticali <lb/>obtineri poteſt.</s> + <s xml:id="echoid-s5749" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5750" xml:space="preserve">Si vero impulſus fluidorum aliter æſtimetur quam §. </s> + <s xml:id="echoid-s5751" xml:space="preserve">31. </s> + <s xml:id="echoid-s5752" xml:space="preserve">indicatum <lb/>fuit, erit ubique in eadem ratione mutandus valor litteræ p, qua impulſus <lb/>æſtimatio fuit mutata.</s> + <s xml:id="echoid-s5753" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5754" xml:space="preserve">Experimentum, de quo §. </s> + <s xml:id="echoid-s5755" xml:space="preserve">27. </s> + <s xml:id="echoid-s5756" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s5757" xml:space="preserve">9, mentionem feci, hoc eſt. </s> + <s xml:id="echoid-s5758" xml:space="preserve">Nem-<lb/>pe unus operarius ope antliæ intra ſeptem minuta prima cum dimidio pe-<lb/>des cubicos ſedecim cum dimidio ad altitudinem quatuordecim pedum evexit.</s> + <s xml:id="echoid-s5759" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5760" xml:space="preserve">Iſte vero effectus æqualiter diſtributus æquivalet huic actioni, qua di-<lb/>midius præter propter pes cubicus ſingulis minutis ſecundis elevatur ad alti-<lb/>tudinem unius pedis: </s> + <s xml:id="echoid-s5761" xml:space="preserve">Hic igitur effectus dimidius admodum eſt illius, quem <lb/>hominem ſanum & </s> + <s xml:id="echoid-s5762" xml:space="preserve">robuſtum calcatura dare poſſe ex aliis deduxi principiis <lb/>in paragrapho decimo ſeptimo. </s> + <s xml:id="echoid-s5763" xml:space="preserve">Non crediderim defectum petendum eſſe <lb/>omnem à decrementis, quæ in potentiam abſolutam ex variis cauſis in iſta ſe-<lb/>ctione expoſitis cadere poſſunt, ſed potius ab eo, quod plus defatigentur <lb/>homines ab agitatione emboli in antlia, quam à calcatura in rota calcatoria.</s> + <s xml:id="echoid-s5764" xml:space="preserve"/> +</p> +<pb o="199" file="0213" n="213" rhead="SECTIO NONA."/> +<p> + <s xml:id="echoid-s5765" xml:space="preserve">Experimentum plane ſimile, ſed cum antlia longe perfectiori artifi-<lb/>cioque ſingulari fabricata, ante aliquot demum menſes Genevæ ſumſi præ-<lb/>ſentibus Viris Clariſſimis D. </s> + <s xml:id="echoid-s5766" xml:space="preserve">D. </s> + <s xml:id="echoid-s5767" xml:space="preserve">De la Rive, Calendrin, Cramer & </s> + <s xml:id="echoid-s5768" xml:space="preserve">Jala-<lb/>bert Acad. </s> + <s xml:id="echoid-s5769" xml:space="preserve">Genev. </s> + <s xml:id="echoid-s5770" xml:space="preserve">Profeſſ. </s> + <s xml:id="echoid-s5771" xml:space="preserve">ſucceſſus experimenti talis fuit, ut intellexerim <lb/>operarium unum ſingulis minutis ſecundis quatuor quintas partes unius pe-<lb/>dis cubici ad altitudinem unius pedis elevaſſe vel potius effectum æqualem <lb/>præſtitiſſe. </s> + <s xml:id="echoid-s5772" xml:space="preserve">Notabile eſt experimentum, nec puto ulla alia machina effe-<lb/>ctum obtineri poſſe admodum majorem. </s> + <s xml:id="echoid-s5773" xml:space="preserve">Mirabile quoque id eſt, quod <lb/>ſic omnis generis machinas, quacunque potentia animatas, ſi obſtacula demas <lb/>effectum haud multo diſſimilem præſtare appareat. </s> + <s xml:id="echoid-s5774" xml:space="preserve">Re bene perpenſa ſta-<lb/>tuo, hominem machina perfectiſſima ſingulis minutis ſecundis pedem cu-<lb/>bicum aquæ ad altitudinem unius pedis elevare poſſe aut effectum ſimilem <lb/>producere.</s> + <s xml:id="echoid-s5775" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5776" xml:space="preserve">Huc etiam pertinerent, præſertim ratione paragraphi trigeſimi primi, <lb/>experimenta quæ accuratiſſime inſtitui ad æſtimandum impetum venæ flui-<lb/>dæ in planum impingentis, quibus confirmatus fui in theoria nova, quam <lb/>hac de re ſtabiliveram ſimulque edoctus, errorem è Mariotti temporibus <lb/>communem fuiſſe commiſſum. </s> + <s xml:id="echoid-s5777" xml:space="preserve">Quia vero in fine hujus ſectionis hac de re <lb/>non diſertè ſermo fuit, atque in fectione decima tertia expreſſe eam pertra-<lb/>ctare animus eſt, ideo eo usque diſquiſitiones haſce, ex principiis mecha-<lb/>nicis nondum obſervatis, erutas differemus.</s> + <s xml:id="echoid-s5778" xml:space="preserve"/> +</p> +<pb o="200" file="0214" n="214"/> +</div> +<div xml:id="echoid-div234" type="section" level="1" n="184"> +<head xml:id="echoid-head232" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph></head> +<head xml:id="echoid-head233" xml:space="preserve">SECTIO DECIMA.</head> +<head xml:id="echoid-head234" style="it" xml:space="preserve">De affectionibus atque motibus fluidorum elaſti-<lb/>corum, præcipue autem aëris.</head> +<head xml:id="echoid-head235" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s5779" xml:space="preserve">FLuida nunc elaſtica conſideraturis licebit nobis talem iis affinge-<lb/>re conſtitutionem, quæ cum omnibus adhuc cognitis conveniat <lb/>affectionibus, ut ſic ad reliquas etiam nondum ſatis exploratas <lb/>detur aditus. </s> + <s xml:id="echoid-s5780" xml:space="preserve">Fluidorum autem elaſticorum præcipuæ affectio-<lb/>nes in eo poſitæ ſunt: </s> + <s xml:id="echoid-s5781" xml:space="preserve">1<emph style="super">0</emph>. </s> + <s xml:id="echoid-s5782" xml:space="preserve">ut ſint gravia, 2<emph style="super">0</emph>. </s> + <s xml:id="echoid-s5783" xml:space="preserve">ut ſe in omnes plagas expli-<lb/>cent, niſi contineantur, & </s> + <s xml:id="echoid-s5784" xml:space="preserve">3<emph style="super">0</emph>. </s> + <s xml:id="echoid-s5785" xml:space="preserve">ut ſe continue magis magisque comprimi <lb/>patiantur creſcentibus potentiis compreſſionis: </s> + <s xml:id="echoid-s5786" xml:space="preserve">Ita comparatus eſt aër, ad <lb/>quem potiſſimum præſentes noſtræ pertinent cogitationes.</s> + <s xml:id="echoid-s5787" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5788" xml:space="preserve">§. </s> + <s xml:id="echoid-s5789" xml:space="preserve">2. </s> + <s xml:id="echoid-s5790" xml:space="preserve">Finge itaque vas cylindricum verticaliter poſitum A C D B <lb/>(Fig. </s> + <s xml:id="echoid-s5791" xml:space="preserve">56.) </s> + <s xml:id="echoid-s5792" xml:space="preserve">atque in illo operculum mobile E F, cui pondus P ſuper in-<lb/> +<anchor type="note" xlink:label="note-0214-01a" xlink:href="note-0214-01"/> +cumbat: </s> + <s xml:id="echoid-s5793" xml:space="preserve">contineat cavitas E C D F corpuſcula minima motu rapidiſſimo <lb/>hinc inde agitata: </s> + <s xml:id="echoid-s5794" xml:space="preserve">ſic corpuſcula, dum impingunt in operculum E F idem-<lb/>que ſuis ſuſtinent impetibus continue repetitis fluidum componunt elaſticum <lb/>quod remoto aut diminuto pondere P ſeſe expandit: </s> + <s xml:id="echoid-s5795" xml:space="preserve">quod eodem aucto <lb/>condenſatur & </s> + <s xml:id="echoid-s5796" xml:space="preserve">quod in fundum horizontalem C D haud aliter gravitat, ac ſi <lb/>nulla virtute elaſtica eſſet præditum: </s> + <s xml:id="echoid-s5797" xml:space="preserve">ſive enim quieſcant corpusſcula ſive agi-<lb/>tentur, non mutant gravitatem, ita ut fundum tum pondus tum elaſticita-<lb/>tem fluidi ſuſtineat. </s> + <s xml:id="echoid-s5798" xml:space="preserve">Tale igitur fluidum quod cum primariis convenit flui-<lb/>dorum elaſticorum affectionibus ſubſtituemus aëri, atque ſic alias, quæ jam <lb/>in aëre detectæ fuerunt explicabimus aliasque nondum ſatis perpenſas ulte-<lb/>rius illuſtrabimus proprietates.</s> + <s xml:id="echoid-s5799" xml:space="preserve"/> +</p> +<div xml:id="echoid-div234" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0214-01" xlink:href="note-0214-01a" xml:space="preserve">Fig. 56.</note> +</div> +<p> + <s xml:id="echoid-s5800" xml:space="preserve">§. </s> + <s xml:id="echoid-s5801" xml:space="preserve">3. </s> + <s xml:id="echoid-s5802" xml:space="preserve">Corpuſcula cavitati cylindri incluſa conſiderabimus tanquam nu-<lb/>mero infinita, & </s> + <s xml:id="echoid-s5803" xml:space="preserve">cum ſpatium E C D F occupant, tunc aërem illa dicemus <lb/>formare naturalem, ad cujus menſuras omnia ſunt referenda: </s> + <s xml:id="echoid-s5804" xml:space="preserve">atque ſic pon- +<pb o="201" file="0215" n="215" rhead="SECTIO DECIMA."/> +dus P operculum detinens in ſitu E F non differt à preſſione Atmoſphæræ ſuper-<lb/>incumbentis, quam proinde per P in ſequentibus deſignabimus.</s> + <s xml:id="echoid-s5805" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5806" xml:space="preserve">Notetur autem hanc preſſionem minime æqualem eſſe ponderi abſo-<lb/>luto cylindri verticalis aërei operculo E F in atmoſphæra ſuperincumbentis, <lb/>quod hactenus inconſiderate affirmarunt auctores: </s> + <s xml:id="echoid-s5807" xml:space="preserve">ſed eſt preſſio iſta æqualis <lb/>quartæ proportionali ad ſuperficiem terræ, magnitudinem operculi E F & </s> + <s xml:id="echoid-s5808" xml:space="preserve">pon-<lb/>deri totius atmoſphæræ in ſuperficiem terræ.</s> + <s xml:id="echoid-s5809" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5810" xml:space="preserve">§. </s> + <s xml:id="echoid-s5811" xml:space="preserve">4. </s> + <s xml:id="echoid-s5812" xml:space="preserve">Quæratur jam pondus π, quod aërem E C D F in ſpatium e C <lb/>D f condenſare valeat, poſitis velocitatibus particularum in utroque aëre, <lb/>naturali ſcilicet & </s> + <s xml:id="echoid-s5813" xml:space="preserve">condenſato, iisdem: </s> + <s xml:id="echoid-s5814" xml:space="preserve">ſit autem E C = 1 & </s> + <s xml:id="echoid-s5815" xml:space="preserve">e C = s: </s> + <s xml:id="echoid-s5816" xml:space="preserve">Cum <lb/>vero operculum E F transponitur in e f, majorem à fluido patitur niſum duplici <lb/>modo: </s> + <s xml:id="echoid-s5817" xml:space="preserve">primo quod numerus particularum ratione ſpatii, cui includuntur, <lb/>major nunc eſt, & </s> + <s xml:id="echoid-s5818" xml:space="preserve">ſecundo quod quævis particula ſæpius impulſum repetit: <lb/></s> + <s xml:id="echoid-s5819" xml:space="preserve">ut recte calculum ponamus incrementi, quod à prima pendet cauſa, parti-<lb/>culas conſiderabimus ceu quieſcentes, atque numerum earum, quæ opercu-<lb/>lo in ſitu E F ſunt contiguæ, faciemus = n, & </s> + <s xml:id="echoid-s5820" xml:space="preserve">erit numerus ſimilis pro ſi-<lb/>tu operculi in e f = n: </s> + <s xml:id="echoid-s5821" xml:space="preserve">({eC/EC})<emph style="super">{2/3}</emph>, ſeu = n: </s> + <s xml:id="echoid-s5822" xml:space="preserve">s<emph style="super">{2/3}</emph>:</s> + <s xml:id="echoid-s5823" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5824" xml:space="preserve">Notetur autem fluid<unsure/>um à nobis conſiderari non magis condenſatum <lb/>in parte inferiori, quam in ſuperiori, quale eſt, cum pondus P veluti infi-<lb/>nitè majus eſt pondere proprio fluidi: </s> + <s xml:id="echoid-s5825" xml:space="preserve">Perſpicuum hinc eſt, hoc nomine <lb/>vim fluidi eſſe, ut ſunt numeri n & </s> + <s xml:id="echoid-s5826" xml:space="preserve">n: </s> + <s xml:id="echoid-s5827" xml:space="preserve">s<emph style="super">{2/3}</emph>, id eſt, ut s<emph style="super">{2/3}</emph> ad 1. </s> + <s xml:id="echoid-s5828" xml:space="preserve">Quod vero <lb/>attinet ad alterum incrementum à ſecunda proveniens cauſa, invenitur id re-<lb/>ſpiciendo motum particularum; </s> + <s xml:id="echoid-s5829" xml:space="preserve">atque ſic apparet impulſus eo ſæpius fieri, <lb/>quo propius ad ſe invicem ſitæ ſunt particulæ: </s> + <s xml:id="echoid-s5830" xml:space="preserve">Erunt ſcilicet impulſuum nu-<lb/>meri reciproce ut diſtantiæ mediæ inter ſuperficies particularum: </s> + <s xml:id="echoid-s5831" xml:space="preserve">Iſtæque di-<lb/>ſtantiæ mediæ ita determinabuntur.</s> + <s xml:id="echoid-s5832" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5833" xml:space="preserve">Particulas ponemus eſſe ſphæricas, diſtantiamque mediam inter cen-<lb/>tra globulorum pro ſitu operculi E F vocabimus D; </s> + <s xml:id="echoid-s5834" xml:space="preserve">diametrumque globuli <lb/>deſignabimus per d: </s> + <s xml:id="echoid-s5835" xml:space="preserve">ita erit diſtantia media inter ſuperficies globulorum = <lb/>D - d: </s> + <s xml:id="echoid-s5836" xml:space="preserve">patet vero in ſitu operculi e f fore diſtantiam mediam inter centra <lb/>globulorum = D ∛ s, atque proinde diſtantiam mediam inter ſuperficies <lb/>globulorum = D ∛ s - d. </s> + <s xml:id="echoid-s5837" xml:space="preserve">Igitur reſpectu ſecundæ cauſæ erit vis aëris na- +<pb o="202" file="0216" n="216" rhead="HYDRODYNAMICÆ"/> +turalis E C D F ad vim aëris compreſſi e C D f ut {1/D - d} ad {1/D∛s - d}, ſeu ut <lb/>D∛s - d ad D - d: </s> + <s xml:id="echoid-s5838" xml:space="preserve">Conjunctis vero ambabus cauſis erunt prædictæ vires, <lb/>ut s<emph style="super">{2/3}</emph> X (D ∛s - d) ad D - d.</s> + <s xml:id="echoid-s5839" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5840" xml:space="preserve">Rationi D ad d aliam ſubſtituere poſſumus magis intelligibilem: <lb/></s> + <s xml:id="echoid-s5841" xml:space="preserve">nempe ſi putemus operculum E F pondere infinito depreſſum deſcendere <lb/>usque in ſitum mn, in quo particulæ omnes ſe tangunt, atque lineam mC <lb/>vocemus m, erit D ad d ut 1 ad ∛ m, quâ ratione ſubſtituta, erunt tandem <lb/>viresaëris naturalis E C D F & </s> + <s xml:id="echoid-s5842" xml:space="preserve">compreſſi e C D fut s<emph style="super">{2/3}</emph> X (∛s - ∛m) ad 1 - ∛m, <lb/>ſeu ut s - ∛mss ad 1 - ∛m. </s> + <s xml:id="echoid-s5843" xml:space="preserve">Eſt igitur π = {1 - ∛m/s - ∛mss} X P.</s> + <s xml:id="echoid-s5844" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5845" xml:space="preserve">§ 5. </s> + <s xml:id="echoid-s5846" xml:space="preserve">Ex omnibus phænomenis judicare poſſumus aërem naturalem <lb/>admodum condenſari poſſe, & </s> + <s xml:id="echoid-s5847" xml:space="preserve">fere in ſpatiolum infinite parvum comprimi; <lb/></s> + <s xml:id="echoid-s5848" xml:space="preserve">facta igitur m = o, fit π = {P/s}, ita ut pondera comprimentia ſint fere <lb/>in ratione inverſa ſpatiorum, quæ aër diverſimode compreſſus occupat; </s> + <s xml:id="echoid-s5849" xml:space="preserve"><lb/>quod multiplex experientia confirmavit. </s> + <s xml:id="echoid-s5850" xml:space="preserve">Et poteſt certe hæc regula tuto <lb/>accipi in aëre rariore quam eſt naturalis; </s> + <s xml:id="echoid-s5851" xml:space="preserve">an vero etiam poſſit in aëre ad-<lb/>modum denſiori, non ſatis exploratum habeo: </s> + <s xml:id="echoid-s5852" xml:space="preserve">nec dum enim fuerunt ex-<lb/>perimenta ea accuratione, quæ hic requiritur, inſtituta: </s> + <s xml:id="echoid-s5853" xml:space="preserve">unico opus eſt ad <lb/>definiendum valorem litteræ m, ſed eo accuratiſſime inſtituendo & </s> + <s xml:id="echoid-s5854" xml:space="preserve">quidem <lb/>cum aëre vehementer compreſſo; </s> + <s xml:id="echoid-s5855" xml:space="preserve">gradus autem caloris in aëre, dum com-<lb/>primitur, ſollicitè invariatus conſervetur.</s> + <s xml:id="echoid-s5856" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5857" xml:space="preserve">§. </s> + <s xml:id="echoid-s5858" xml:space="preserve">6. </s> + <s xml:id="echoid-s5859" xml:space="preserve">Elaſticitas interim aëris nonſolum à condenſatione augetur, ſed <lb/>& </s> + <s xml:id="echoid-s5860" xml:space="preserve">ab aucto calore, & </s> + <s xml:id="echoid-s5861" xml:space="preserve">quia conſtat calorem intendi ubique creſcente motu par-<lb/>ticularum inteſtino, ſequitur, elaſticitatem aëris ſpatium non mutantis auctam, <lb/>intenſiorem arguere motum in particulis aëris, quod cum hypotheſi noſtra re-<lb/>cte convenit: </s> + <s xml:id="echoid-s5862" xml:space="preserve">perſpicuum enim eſt, eo majus requiri pondus P ad continen-<lb/>dum aërem in ſitu E C D F, quo majori velocitate particulæ aëreæ agitantur: <lb/></s> + <s xml:id="echoid-s5863" xml:space="preserve">lmo non difficile eſt videre pondus P ſecuturum rationem duplicatam iſtius ve-<lb/>locitatis, ideo quod ab aucta velocitate tum numerus impetuum tum intenſitas <lb/>corundem æqualiter creſcat, utrumq; </s> + <s xml:id="echoid-s5864" xml:space="preserve">veroſeorſim proportionale ſit ponderi P.</s> + <s xml:id="echoid-s5865" xml:space="preserve"/> +</p> +<pb o="203" file="0217" n="217" rhead="SECTIO DECIMA."/> +<p> + <s xml:id="echoid-s5866" xml:space="preserve">Igitur ſi velocitas particularum aërearum dicatur v, erit pondus, quod <lb/>in ſitu operculi E F ſuſtinere valet, = v v P & </s> + <s xml:id="echoid-s5867" xml:space="preserve">in ſitu ef = {1 - ∛m - ∛mss} X vvP, <lb/>vel proxime = {vvP/s}, quia ut vidimus m numerus admodum exiguus eſt ra-<lb/>tione unitatis & </s> + <s xml:id="echoid-s5868" xml:space="preserve">numeri s.</s> + <s xml:id="echoid-s5869" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5870" xml:space="preserve">§. </s> + <s xml:id="echoid-s5871" xml:space="preserve">7. </s> + <s xml:id="echoid-s5872" xml:space="preserve">Iſtud theorema, quod in præcedente paragrapho appoſui, quo <lb/>nempe indicatur, in omni æëre cujuſcun denſitatis ſed eodem caloris gradu<unsure/> <lb/>prædito elaſticitates eſſe ut denſitates, at{q́ue} proinde etiam incrementa elaſticita-<lb/>tum, quæ fiunt à calore æqualiter aucto proportionalia eſſe denſitatibus, <lb/>Iſtud, inquam, theorema experientia edoctus fuit D. </s> + <s xml:id="echoid-s5873" xml:space="preserve">Amontons idemque re-<lb/>cenſuit dans les mémoires de l’Acad. </s> + <s xml:id="echoid-s5874" xml:space="preserve">R. </s> + <s xml:id="echoid-s5875" xml:space="preserve">des Sc. </s> + <s xml:id="echoid-s5876" xml:space="preserve">de Paris pour l’année 1702. </s> + <s xml:id="echoid-s5877" xml:space="preserve">Senſusiſtius <lb/>theorematis eſt, ſi v. </s> + <s xml:id="echoid-s5878" xml:space="preserve">gr. </s> + <s xml:id="echoid-s5879" xml:space="preserve">aër naturalis mediocris caloris pondus 100lb. </s> + <s xml:id="echoid-s5880" xml:space="preserve">datæ <lb/>ſuperficiei impoſitum ſuſtinere valeat, atque deinde calor ipſius augeatur do-<lb/>nec 120 lb. </s> + <s xml:id="echoid-s5881" xml:space="preserve">eadem ſuperficie<unsure/> eodemque volumine ferre poſſit, fore ut idem <lb/>aër in dimidium ſpatium condenſatus, & </s> + <s xml:id="echoid-s5882" xml:space="preserve">iiſdem caloris gradibus præditus re-<lb/>ſpective ferre poſſit 200 lb. </s> + <s xml:id="echoid-s5883" xml:space="preserve">& </s> + <s xml:id="echoid-s5884" xml:space="preserve">240 lb. </s> + <s xml:id="echoid-s5885" xml:space="preserve">ita ut incrementa 20 lb. </s> + <s xml:id="echoid-s5886" xml:space="preserve">& </s> + <s xml:id="echoid-s5887" xml:space="preserve">40 lb, utrobique <lb/>ab aucto calore genita ſint denſitatibus proportionalia. </s> + <s xml:id="echoid-s5888" xml:space="preserve">Affirmat porro aëris, <lb/>quem vocat temperatum, elaterem eſſe ad elaterem aëris ejusdem cum aqua <lb/>bulliente caloris, proxime ut 3. </s> + <s xml:id="echoid-s5889" xml:space="preserve">ad 4 vel accuratius ut 55 ad 73. </s> + <s xml:id="echoid-s5890" xml:space="preserve">At ego inſtitu-<lb/>tis experimentis cognovi aërem calidiſſimum, qualis maxime fervente in hiſce <lb/>terris eſt æſtate, tanti nondum eſſe elateris, quantum D. </s> + <s xml:id="echoid-s5891" xml:space="preserve">Amontons aëri tribuit <lb/>temperato; </s> + <s xml:id="echoid-s5892" xml:space="preserve">imo nec ſub ipſo æquatore aërem unquam ejus eſſe caloris mihi <lb/>perſuadeo. </s> + <s xml:id="echoid-s5893" xml:space="preserve">Meis autem magis fidendum eſſe puto experimentis quam Amon-<lb/>tonianis, ideo quod in his aër non conſervarit ſuum volumen ejuſque variatio-<lb/>nis nulla ab Auctore habita fuerit ratio in calculo. </s> + <s xml:id="echoid-s5894" xml:space="preserve">Aëris qui hic Petropoli frigi-<lb/>diſſimus fuit die 25. </s> + <s xml:id="echoid-s5895" xml:space="preserve">Decembr. </s> + <s xml:id="echoid-s5896" xml:space="preserve">1731. </s> + <s xml:id="echoid-s5897" xml:space="preserve">s<emph style="super">t</emph>. </s> + <s xml:id="echoid-s5898" xml:space="preserve">vet. </s> + <s xml:id="echoid-s5899" xml:space="preserve">elaterem deprehendi eſſe ad elate-<lb/>rem ſimilis aëris, communi cum aqua bulliente calore præditi, ut 523 ad 1000.</s> + <s xml:id="echoid-s5900" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5901" xml:space="preserve">Sed anno 1733. </s> + <s xml:id="echoid-s5902" xml:space="preserve">d. </s> + <s xml:id="echoid-s5903" xml:space="preserve">21. </s> + <s xml:id="echoid-s5904" xml:space="preserve">Jan. </s> + <s xml:id="echoid-s5905" xml:space="preserve">multo intenſius fuit frigus eique reſpondere <lb/>obſervavi aëris elaſticitatem infra dimidiam ejus quam habet ſimilis aër ad <lb/>aquam bullientem calefactus. </s> + <s xml:id="echoid-s5906" xml:space="preserve">Sed cum eſſet maximus aëris calor in loco um-<lb/>broſo ann. </s> + <s xml:id="echoid-s5907" xml:space="preserve">1731. </s> + <s xml:id="echoid-s5908" xml:space="preserve">elaſticitatem habuit proxime {4/3} & </s> + <s xml:id="echoid-s5909" xml:space="preserve">accuratius {100/76}, ejus quam <lb/>habuit aër frigidiſſimus & </s> + <s xml:id="echoid-s5910" xml:space="preserve">{2/3} ejus quam habet aër ejusdem cum aqua bulliente +<pb o="204" file="0218" n="218" rhead="HYDRODYNAMICÆ"/> +caloris: </s> + <s xml:id="echoid-s5911" xml:space="preserve">maximæ igitur caloris variationes in aëre hic locorum continentur <lb/>intra terminos 3 & </s> + <s xml:id="echoid-s5912" xml:space="preserve">4, quos in Anglia non ultra terminos 7 & </s> + <s xml:id="echoid-s5913" xml:space="preserve">8 excurrere legi. <lb/></s> + <s xml:id="echoid-s5914" xml:space="preserve">Calor autem aëris, cujus elaſticitas tres quartas exæquet partes elaſticitatis aë-<lb/>ris inſtar aquæ bullientis calidi, corpori animali fere intolerabilem eſſe puto.</s> + <s xml:id="echoid-s5915" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5916" xml:space="preserve">§. </s> + <s xml:id="echoid-s5917" xml:space="preserve">8. </s> + <s xml:id="echoid-s5918" xml:space="preserve">Ex cognita ratione inter diverſas ejusdem aëris eodemque ſpatio <lb/>incluſi elaſticitates, facile eſt deducere menſuram caloris, qui ad aërem perti-<lb/>neat, ſi modo conveniamus in definiendo calore duplo, triplo &</s> + <s xml:id="echoid-s5919" xml:space="preserve">c. </s> + <s xml:id="echoid-s5920" xml:space="preserve">quæ defi-<lb/>nitio arbitraria eſt, neque in rerum natura poſita; </s> + <s xml:id="echoid-s5921" xml:space="preserve">mihi quidem videtur non <lb/>incongrue aëris calorem ſi communis ſit denſitatis proportionalem ſtatui ejus <lb/>elaſticitati. </s> + <s xml:id="echoid-s5922" xml:space="preserve">Primus autem caloris gradus, à quo reliqui menſuram accipiant, <lb/>ſumetur ab aqua pluviali bulliente, quia huic procul dubio ubique terrarum <lb/>idem proxime caloris gradus eſt.</s> + <s xml:id="echoid-s5923" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5924" xml:space="preserve">His ita acceptis erunt calores aquæ bullientis, aëris tempore æſtivo cali-<lb/>diſſimi & </s> + <s xml:id="echoid-s5925" xml:space="preserve">aëris tempore hyemali frigidiſſimi in hiſce terris proxime ut 6, 4 & </s> + <s xml:id="echoid-s5926" xml:space="preserve">3. <lb/></s> + <s xml:id="echoid-s5927" xml:space="preserve">Dicam nunc quemadmodum hoſce invenerim numeros, ut de accuratione ex-<lb/>perimentorum, quorum ſucceſſus ab Amontonianis diverſus admodum eſt, ju-<lb/>dicium ferri poſſit.</s> + <s xml:id="echoid-s5928" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5929" xml:space="preserve">§. </s> + <s xml:id="echoid-s5930" xml:space="preserve">9. </s> + <s xml:id="echoid-s5931" xml:space="preserve">Barometro nempe uſus ſum ordinario A C B E, (Fig. </s> + <s xml:id="echoid-s5932" xml:space="preserve">57.) </s> + <s xml:id="echoid-s5933" xml:space="preserve">id-<lb/> +<anchor type="note" xlink:label="note-0218-01a" xlink:href="note-0218-01"/> +que hermetice ſigillari curavi in m; </s> + <s xml:id="echoid-s5934" xml:space="preserve">hoc modo inſtrumentum mutavi in ther-<lb/>mometrum aëreum mutationibus barometricis non obnoxium: </s> + <s xml:id="echoid-s5935" xml:space="preserve">Creſcente <lb/>enim calore intenditur elaterium aëris A m F altiorque fit columna mercurii <lb/>B D, quam aër captus ſuſtinet & </s> + <s xml:id="echoid-s5936" xml:space="preserve">ſi ſpatium A m F veluti infinitum cen-<lb/>ſeri poſſet, eſſet calor in ratione altitudinis B D (per §. </s> + <s xml:id="echoid-s5937" xml:space="preserve">§. </s> + <s xml:id="echoid-s5938" xml:space="preserve">7. </s> + <s xml:id="echoid-s5939" xml:space="preserve">& </s> + <s xml:id="echoid-s5940" xml:space="preserve">8.) </s> + <s xml:id="echoid-s5941" xml:space="preserve">atque <lb/>hujus thermometri ope poterit menſura caloris ubique ſpecifice definiri. </s> + <s xml:id="echoid-s5942" xml:space="preserve">Si <lb/>enim immergatur inſtrumentum aquæ bullienti pluviali in ſitu verticali obſer-<lb/>veturque punctum G ad quod ſuperficies mercurii aſcendit; </s> + <s xml:id="echoid-s5943" xml:space="preserve">fueritque dein-<lb/>de alius caloris gradus qualiscunque definiendus, qui mercurium ſuſtinuiſſe <lb/>ad punctum D usque obſervatus fuerit, erit utique calor iſte ad calorem <lb/>aquæ ferventis ut B D ad B G. </s> + <s xml:id="echoid-s5944" xml:space="preserve">Et cum ratio B D ad B G conſtans ſit, quæ-<lb/>c<unsure/>unque fuerit altitudo B G, erit idem caloris gradus, de quo ſermo eſt, <lb/>ubique locorum facile imitabilis. </s> + <s xml:id="echoid-s5945" xml:space="preserve">Poterit autem B G in centum aut mille <lb/>dividi particulas atque hujusmodi particulis altitudo B D definiri.</s> + <s xml:id="echoid-s5946" xml:space="preserve"/> +</p> +<div xml:id="echoid-div235" type="float" level="2" n="2"> +<note position="left" xlink:label="note-0218-01" xlink:href="note-0218-01a" xml:space="preserve">Fig. 57.</note> +</div> +<pb o="205" file="0219" n="219" rhead="SECTIO DECIMA."/> +<p> + <s xml:id="echoid-s5947" xml:space="preserve">Nihil dico de modis hujusmodi thermometra ſenſibiliora reddendi; </s> + <s xml:id="echoid-s5948" xml:space="preserve">eo-<lb/>rum quisque facile excogitabit plures, qui volet. </s> + <s xml:id="echoid-s5949" xml:space="preserve">Curetur autem, ut alti-<lb/>tudo B E non ſit infra 4 pedes, imo ut major ſit, ſi etiam aliorum fluidorum <lb/>bullientium gradus caloris, qui ſæpe major eſt quam in aqua, experiri ani-<lb/>mus ſit. </s> + <s xml:id="echoid-s5950" xml:space="preserve">Si minora hujusmodi thermometra deſiderentur, poterunt ea ita <lb/>fieri, ut tempore ſigillationis in m ampulla vitrea A F igni lampadis appona-<lb/>tur ad rarefaciendum aërem in illa contentum, tuncque protinus ſigillatio <lb/>fiat, & </s> + <s xml:id="echoid-s5951" xml:space="preserve">ne ſigillationi mora injiciatur, poterit prius ampulla vitrea in tubu-<lb/>lum capillarem duci, qui vel leviter flammæ admotus illico colliqueſcat. </s> + <s xml:id="echoid-s5952" xml:space="preserve">Hoc <lb/>modo thermometra obtinui non ultra quatuor aut ſex pollices longa, ſed <lb/>parvæ virtutis. </s> + <s xml:id="echoid-s5953" xml:space="preserve">Cæterum multum refert, ut ſpatium E D ſit ab omni aëre, <lb/>quantum fieri poteſt, vacuum, neque de iſto vacuo ſatis certi erimus cum <lb/>viderimus in ſitu inſtrumenti horizontali mercurium extremitatem Eattinge-<lb/>re, quia fieri poteſt, ut aër, qui antea in ſpatio E D fuit, ſeſe in poros <lb/>mercurii recipiat, rurſusque priſtinum ſpatium occupet deſcendente mercu-<lb/>rio: </s> + <s xml:id="echoid-s5954" xml:space="preserve">tutius erit examen admovendo partem D E flammæ: </s> + <s xml:id="echoid-s5955" xml:space="preserve">ſi enim à calore <lb/>flammæ ſuperficies D locum non mutet, indicium erit certum vacuum eſſe ab <lb/>aëre ſpatium E D.</s> + <s xml:id="echoid-s5956" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5957" xml:space="preserve">§. </s> + <s xml:id="echoid-s5958" xml:space="preserve">10. </s> + <s xml:id="echoid-s5959" xml:space="preserve">In præcedente paragrapho conſideravimus ſpatium A m F ab <lb/>aëre occupatum veluti infinitum ratione ſpatii D G aut D E: </s> + <s xml:id="echoid-s5960" xml:space="preserve">Quod ſi vero <lb/>fuerit tantum octuplo vel decuplo majus, nondum licebit illud ſine notabi-<lb/>li errore tanquam infinitum conſiderare: </s> + <s xml:id="echoid-s5961" xml:space="preserve">atque hinc conjicio ortum eſſe <lb/>errorem aliquem in definiendo elatere aëris mediocriter calidi in experimen-<lb/>tis Amontonianis.</s> + <s xml:id="echoid-s5962" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5963" xml:space="preserve">Ut igitur accuratiſſime fiat experimentum, ita procedendum erit: </s> + <s xml:id="echoid-s5964" xml:space="preserve">Fue-<lb/>rit ſuperficies mercurii inferior in A F ducaturque horizontatis in A L: </s> + <s xml:id="echoid-s5965" xml:space="preserve">dein-<lb/>de pro caloris gradu qualicunque definiendo inclinetur inſtrumentum, donec <lb/>ſuperficies mercurii ſit in puncto g, (quod idem eſt in quo mercurius ſubſi-<lb/>ſtebat à gradu caloris aquæ ferventis in ſitu thermometri verticali) tuncque <lb/>capiatur menſura altitudinis verticalis gh, quæ erit ad altitudinem G B vere <lb/>ut elater aëris, cujus calor definiendus eſt, ad elaterem aëris inſtar aquæ fer-<lb/>ventis calidi. </s> + <s xml:id="echoid-s5966" xml:space="preserve">Sic igitur calores erunt proprie in ratione altitudinem gh. <lb/></s> + <s xml:id="echoid-s5967" xml:space="preserve">Priusquam hoc argumentum abrumpam, notaſſe conveniet (quandoquidem <lb/>aliquibus fortaſſe videbitur primum, qui à nobis poſitus fuit, caloris gradum ab <lb/>aqua bulliente deſumtum non ſemper nec ubique ſibi omnino conſtare) quod +<pb o="206" file="0220" n="220" rhead="HYDRODYNAMICÆ"/> +loco caloris aquæ bullientis thermometrum etiam poſſit certis & </s> + <s xml:id="echoid-s5968" xml:space="preserve">fixis men-<lb/>ſuris fieri, ſi experimento denſitas aëris exploretur ſeu ejus gravitas ſpecifi-<lb/>ca ſimulque altitudo barometri notetur. </s> + <s xml:id="echoid-s5969" xml:space="preserve">Si enim thermometrum inclinetur, <lb/>donec ſuperficies mercurii fuerit in g & </s> + <s xml:id="echoid-s5970" xml:space="preserve">eo tempore altitudo barometri fue-<lb/>rit 28. </s> + <s xml:id="echoid-s5971" xml:space="preserve">poll. </s> + <s xml:id="echoid-s5972" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s5973" xml:space="preserve">atque pes cubicus aëris, in quo thermometrum poſitum eſt, <lb/>pondus habuerit 600. </s> + <s xml:id="echoid-s5974" xml:space="preserve">gran. </s> + <s xml:id="echoid-s5975" xml:space="preserve">Norimb, poterit altitudo verticalis gh ceu pri-<lb/>mus caloris gradus conſiderari. </s> + <s xml:id="echoid-s5976" xml:space="preserve">Si autem alio loco & </s> + <s xml:id="echoid-s5977" xml:space="preserve">tempore altitudo baro-<lb/>metri fuerit 29. </s> + <s xml:id="echoid-s5978" xml:space="preserve">poll. </s> + <s xml:id="echoid-s5979" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s5980" xml:space="preserve">& </s> + <s xml:id="echoid-s5981" xml:space="preserve">pondus pedis cub. </s> + <s xml:id="echoid-s5982" xml:space="preserve">aëris, qui ambit aliud ther-<lb/>mometrum (in quo primum caloris gradum definire animus eſt) ſit 500. </s> + <s xml:id="echoid-s5983" xml:space="preserve">gran. <lb/></s> + <s xml:id="echoid-s5984" xml:space="preserve">Norimb. </s> + <s xml:id="echoid-s5985" xml:space="preserve">ac denique ſuperficies mercurii in thermometro rurſus ſiting, erit <lb/>altitudo verticalis primo caloris gradui conveniens {29.</s> + <s xml:id="echoid-s5986" xml:space="preserve">600/28.</s> + <s xml:id="echoid-s5987" xml:space="preserve">500} X gh. </s> + <s xml:id="echoid-s5988" xml:space="preserve">In uſu <lb/>thermometri inclinetur ſemper inſtrumentum, donec ſuperficies mercurii ſit <lb/>ing: </s> + <s xml:id="echoid-s5989" xml:space="preserve">Volui methodum hanc apponere ut appareret quam facile ſit in theo-<lb/>ria fixam dare caloris menſuram: </s> + <s xml:id="echoid-s5990" xml:space="preserve">In praxi vero alteram multo faciliorem <lb/>ſatisque accuratam huic prætulerim.</s> + <s xml:id="echoid-s5991" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5992" xml:space="preserve">§. </s> + <s xml:id="echoid-s5993" xml:space="preserve">11. </s> + <s xml:id="echoid-s5994" xml:space="preserve">Veniamus nunc ad aëris conſiderandam atmoſphæram, quæ <lb/>non à ſuperincumbente pondere alieno, ſed propria coërcetur mole: </s> + <s xml:id="echoid-s5995" xml:space="preserve">Primè <lb/>autem examinabimus preſſiones columnarum aërearum verticalium atque æqui-<lb/>libria earum tum inter ſe tum cum columna mercuriali in barometris: </s> + <s xml:id="echoid-s5996" xml:space="preserve">Secundò <lb/>elaſticitates aëris in variis atmoſphæræ altitudinibus ſupra mare atque altitudi-<lb/>nes reſpondentes barometricas rimabimur: </s> + <s xml:id="echoid-s5997" xml:space="preserve">Atque his præmiſſis, plurimis ſa-<lb/>tisfaciemus phænomenis aliis ad mutationes atmoſphæræ pertinentibus.</s> + <s xml:id="echoid-s5998" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s5999" xml:space="preserve">§. </s> + <s xml:id="echoid-s6000" xml:space="preserve">12. </s> + <s xml:id="echoid-s6001" xml:space="preserve">Sint duo tubi æqualis amplitudinis verticales A C & </s> + <s xml:id="echoid-s6002" xml:space="preserve">B D <lb/>(Fig. </s> + <s xml:id="echoid-s6003" xml:space="preserve">58.) </s> + <s xml:id="echoid-s6004" xml:space="preserve">uterque indefinitæ altitudinis: </s> + <s xml:id="echoid-s6005" xml:space="preserve">Deinde finge tubulos ſtrictiores ho-<lb/> +<anchor type="note" xlink:label="note-0220-01a" xlink:href="note-0220-01"/> +rizontales ab, cd, ef, gh, lm, & </s> + <s xml:id="echoid-s6006" xml:space="preserve">c. </s> + <s xml:id="echoid-s6007" xml:space="preserve">numero veluti infinitos, utrinque apertos & </s> + <s xml:id="echoid-s6008" xml:space="preserve"><lb/>hiantes in tubos verticales. </s> + <s xml:id="echoid-s6009" xml:space="preserve">Puta præterea ubique aëreas particulas hos tubos <lb/>occupantes eadem velocitate agitari, eundemque adeo caloris gradum habe-<lb/>re: </s> + <s xml:id="echoid-s6010" xml:space="preserve">Ita dubium nullum eſt, quin funda A & </s> + <s xml:id="echoid-s6011" xml:space="preserve">B æqualiter premantur ſimulque <lb/>ipſis æquale pondus (quod ſcilicet ipſum eſt pondus columnæ aëreæ indefini-<lb/>tæ A C vel B D) ſuperincumbat.</s> + <s xml:id="echoid-s6012" xml:space="preserve"/> +</p> +<div xml:id="echoid-div236" type="float" level="2" n="3"> +<note position="left" xlink:label="note-0220-01" xlink:href="note-0220-01a" xml:space="preserve">Fig. 58.</note> +</div> +<p> + <s xml:id="echoid-s6013" xml:space="preserve">Intelligis etiam, ſi in æqualibus altitudinibus veluti in g & </s> + <s xml:id="echoid-s6014" xml:space="preserve">h diaphrag-<lb/>mata fingas atque abeſſe putes aërem inferiorem g A & </s> + <s xml:id="echoid-s6015" xml:space="preserve">h B, etiamnum iſta dia- +<pb o="207" file="0221" n="221" rhead="SECTIO DECIMA."/> +phragmata utrinque æqualiter premi & </s> + <s xml:id="echoid-s6016" xml:space="preserve">æqualia eſſe pondera columnarum aë-<lb/>rearum g C atque h D diaphragmatibus ſuperjacentium. </s> + <s xml:id="echoid-s6017" xml:space="preserve">Siigitur pondus totius <lb/>columnæ aëreæ A C vel B D dicatur A, & </s> + <s xml:id="echoid-s6018" xml:space="preserve">pondus columnæ aëreæ g C vel h D <lb/>ponatur B, erit pondus aëris inter A & </s> + <s xml:id="echoid-s6019" xml:space="preserve">g ſive B & </s> + <s xml:id="echoid-s6020" xml:space="preserve">h intercepti = A - B, pon-<lb/>dus fundo A vel B ſuperjacens = A, & </s> + <s xml:id="echoid-s6021" xml:space="preserve">pondus diaphragmatiing vel h incum-<lb/>bens = B.</s> + <s xml:id="echoid-s6022" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6023" xml:space="preserve">§. </s> + <s xml:id="echoid-s6024" xml:space="preserve">13. </s> + <s xml:id="echoid-s6025" xml:space="preserve">At ſi inæquali velocitate in tubis A C & </s> + <s xml:id="echoid-s6026" xml:space="preserve">B D particulæ agitentur, <lb/>res alia erit: </s> + <s xml:id="echoid-s6027" xml:space="preserve">tamen quæcunque fingatur velocitatum & </s> + <s xml:id="echoid-s6028" xml:space="preserve">calorum in ſingulis lo-<lb/>cis diverſitas, patet nihilominus utrobique æqualiter preſſum iri partes tubi <lb/>in eadem altitudine poſitas, velutiing & </s> + <s xml:id="echoid-s6029" xml:space="preserve">h, atque proinde diaphragmata, ſi <lb/>fingantur utrobique in eadem altitudine poſita, æqualem preſſionem ſuſtentu-<lb/>ra eſſe. </s> + <s xml:id="echoid-s6030" xml:space="preserve">Si enim dicas minorem eſſe preſſioneming quam in h, nihil erit <lb/>quod fluxum aëris ex B D in A C per tubulum tranſverſum hg impediat, ſicque <lb/>iſta poſitio contra ſtatum permanentiæ, quem ſupponimus, pugnabit.</s> + <s xml:id="echoid-s6031" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6032" xml:space="preserve">Cum itaque loca in eadem altitudine poſita æqualiter à ſuperincumben-<lb/>te aëre premantur, erunt (p. </s> + <s xml:id="echoid-s6033" xml:space="preserve">§. </s> + <s xml:id="echoid-s6034" xml:space="preserve">6.) </s> + <s xml:id="echoid-s6035" xml:space="preserve">denſitates in locis homologis quibuſcun-<lb/>que, velutiing & </s> + <s xml:id="echoid-s6036" xml:space="preserve">h, proxime in reciproca ratione quadrata velocitatum, quibus <lb/>in illis locis particulæ agitantur.</s> + <s xml:id="echoid-s6037" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6038" xml:space="preserve">§. </s> + <s xml:id="echoid-s6039" xml:space="preserve">14. </s> + <s xml:id="echoid-s6040" xml:space="preserve">Conſequens eſt ex præcedente paragrapho, ubique locorum <lb/>eandem eſſe aëris preſſionem in æqualibus à ſuperficie maris altitudinibus, ſi <lb/>atmoſphæra in ſtatu permanente æquilibrii poſrta nulliſque agitata ventis pute-<lb/>tur, quæcunque fuerit caloris differentia in diverſis atmoſphæræ partibus: </s> + <s xml:id="echoid-s6041" xml:space="preserve">Igi-<lb/>tur ubique terrarum ſub æquatore & </s> + <s xml:id="echoid-s6042" xml:space="preserve">ſub polo eadem ſit oportet altitudo mer-<lb/>curii in barometris, quæ in ſuperficie maris aut in æqualibus ſuper illam alti-<lb/>tudinibus poſita ſunt, ſi atmoſphæra nullis obnoxia ſit mutationibus. </s> + <s xml:id="echoid-s6043" xml:space="preserve">Pono <lb/>autem aquas à ſuperficie maris terminatas ad commune æquilibrium eſſe poſi-<lb/>tas, non quod id omnino neceſſe ſit, ſed quod nulla adhuc obſervata fuerit <lb/>differentia: </s> + <s xml:id="echoid-s6044" xml:space="preserve">imo curſus (les courans) aquarum in multis oceani locis, qui ad <lb/>eandem perpetuo diriguntur plagam, hanc hypotheſin non omni rigore ac-<lb/>cipiendam eſſe oſtendunt.</s> + <s xml:id="echoid-s6045" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6046" xml:space="preserve">§. </s> + <s xml:id="echoid-s6047" xml:space="preserve">15. </s> + <s xml:id="echoid-s6048" xml:space="preserve">Jam notavi denſitatem aëris in quovis tuborum verticalium loco <lb/>pendere à calore reſpondente: </s> + <s xml:id="echoid-s6049" xml:space="preserve">Et cum diverſi eſſe poſſint caloris gradus ma- +<pb o="208" file="0222" n="222" rhead="HYDRODYNAMICÆ"/> +nente æquilibrio, diverſæ quoque eſſe poterunt denſitates: </s> + <s xml:id="echoid-s6050" xml:space="preserve">ponantur itaque <lb/>denſitates in g = D, in h = δ; </s> + <s xml:id="echoid-s6051" xml:space="preserve">finganturque utrobique duo ſtrata altitudinis <lb/>æqualis & </s> + <s xml:id="echoid-s6052" xml:space="preserve">infinitè parvæ dx, poſita altitudine A g vel B h = x: </s> + <s xml:id="echoid-s6053" xml:space="preserve">Ita erit pon-<lb/>dus columnæ aëreæ A g = ſD dx & </s> + <s xml:id="echoid-s6054" xml:space="preserve">columnæ B h = ſδdx: </s> + <s xml:id="echoid-s6055" xml:space="preserve">atque hoc mo-<lb/>do poterit tum integræ columnæ tum cujusvis partis pondus definiri: </s> + <s xml:id="echoid-s6056" xml:space="preserve">Interim <lb/>apparet, minime requirere rei naturam, ut ſint pondera columnarum A C <lb/>& </s> + <s xml:id="echoid-s6057" xml:space="preserve">B D vel A g & </s> + <s xml:id="echoid-s6058" xml:space="preserve">B h vel denique g C & </s> + <s xml:id="echoid-s6059" xml:space="preserve">h D inter ſe æqualia, quamvis (per <lb/>§. </s> + <s xml:id="echoid-s6060" xml:space="preserve">13.) </s> + <s xml:id="echoid-s6061" xml:space="preserve">preſſiones tam in funda A & </s> + <s xml:id="echoid-s6062" xml:space="preserve">B quam in diaphragmatr g & </s> + <s xml:id="echoid-s6063" xml:space="preserve">h ſint inter <lb/>ſe æquales; </s> + <s xml:id="echoid-s6064" xml:space="preserve">mirum id primo intuitu quibusdam fortaſſe erit, fieri poſſe ut <lb/>fundum A aliam ſuſtineat preſſionem quam eſt pondus columnæ aëreæ inde-<lb/>finitæ A C ei ſuperincumbentis, quandoquidem omnibus in ſtatu ſuo perma-<lb/>nentibus, ut fere videtur, concipi poſſint orificia a, c, e, g, &</s> + <s xml:id="echoid-s6065" xml:space="preserve">c. </s> + <s xml:id="echoid-s6066" xml:space="preserve">ſingula <lb/>obturata, quo ſane in caſii dubium nullum eſt, quin preſſio fundi A ſit ip-<lb/>ſum columnæ aëreæ ſuperjacentis pondus: </s> + <s xml:id="echoid-s6067" xml:space="preserve">hunc vero ſcrupulum ſibi quisque <lb/>eximet hunc in modum: </s> + <s xml:id="echoid-s6068" xml:space="preserve">fingamus utramque columnam terminatæ altitudi-<lb/>nis (quamvis enim ſine fine aſſurgant quamdiu particulæ motum aliquem ſer-<lb/>vant, attamen terminatæ erunt, ſi eædem particulæ in ſuprema columnarum <lb/>parte motu deſtitutæ ſint, ſicque ſimplex fluidum grave omni elaſticitate de-<lb/>ſtitutum efficient) hoc poſito apparet 1<emph style="super">0</emph>. </s> + <s xml:id="echoid-s6069" xml:space="preserve">columnam utramque ad commu-<lb/>nem aſſurgere altitudinem apertis tubulis transverſalibus, qui ubique adſunt. <lb/></s> + <s xml:id="echoid-s6070" xml:space="preserve">2<emph style="super">0</emph>. </s> + <s xml:id="echoid-s6071" xml:space="preserve">ſuprema ſtrata utrobique eſſe æque denſa, quia ſunt ad æquilibrium po-<lb/>ſita & </s> + <s xml:id="echoid-s6072" xml:space="preserve">communem habent altitudinem. </s> + <s xml:id="echoid-s6073" xml:space="preserve">Ex hoc jam obvium eſt, quare non <lb/>liceat tubulos transverſales conſiderare ceu obturatos, quod oſtendere con-<lb/>ſtitui. </s> + <s xml:id="echoid-s6074" xml:space="preserve">Perſpicuum quoque eſt exſe, preſſiones ubique proportionales eſſe <lb/>ponderi ſupremi ſtrati, ex quo conſequens eſt, quod jam §. </s> + <s xml:id="echoid-s6075" xml:space="preserve">13. </s> + <s xml:id="echoid-s6076" xml:space="preserve">indicatum <lb/>fuit, preſſiones ab utraque parte æquales inter ſe eſſe ſub æqualibus altitudi-<lb/>nibus. </s> + <s xml:id="echoid-s6077" xml:space="preserve">Si jam columnæ nusquam terminatæ ſint, licebit mente ultima con-<lb/>cipere ſtrata aut ſub æqualibus altitudinibus diaphragmata fingere utrobique <lb/>æquali pondere onerata, ſic ut nihil vi demonſtrationis inde decedat.</s> + <s xml:id="echoid-s6078" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6079" xml:space="preserve">§. </s> + <s xml:id="echoid-s6080" xml:space="preserve">16. </s> + <s xml:id="echoid-s6081" xml:space="preserve">Igitur quum in barometro ex loco humiliori veluti A in altiorem <lb/>g transportato mercurius deſcendit, non ſequitur pondus columnæ mercu-<lb/>rialis, quæ in barometro deſcendit æquale eſſe ponderi columnæ aëreæ ejus-<lb/>dem diametri & </s> + <s xml:id="echoid-s6082" xml:space="preserve">altitudinis A g, qnod ab aliquibus ita aſſeritur. </s> + <s xml:id="echoid-s6083" xml:space="preserve">Et profe-<lb/>cto cæteris paribus columna mercurii deſcendens eadem erit tam tempore +<pb o="209" file="0223" n="223" rhead="SECTIO DECIMA."/> +hyemali quam æſtivo cum ex ſententia illa deberet tempore calido eſſe mi-<lb/>nor, quam tempore frigido: </s> + <s xml:id="echoid-s6084" xml:space="preserve">Eadem quoque erit in locis meridionalibus & </s> + <s xml:id="echoid-s6085" xml:space="preserve">ſep-<lb/>tentrionalibus.</s> + <s xml:id="echoid-s6086" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6087" xml:space="preserve">Patet exinde quid cenſendum ſit de illa methodo, qua in Anglia ali-<lb/>quando uſos eſſe recenſet D. </s> + <s xml:id="echoid-s6088" xml:space="preserve">Du Hamel in hiſt. </s> + <s xml:id="echoid-s6089" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s6090" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s6091" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s6092" xml:space="preserve">ad indagandam ra-<lb/>tionem inter gravitates ſpecificas aëris & </s> + <s xml:id="echoid-s6093" xml:space="preserve">mercurii: </s> + <s xml:id="echoid-s6094" xml:space="preserve">Obſervata nimirum altitu-<lb/>dine mercurii in loco humiliori, tum etiam in altiori, gravitates ſpecificas in <lb/>aëre & </s> + <s xml:id="echoid-s6095" xml:space="preserve">mercurio ſtatuerunt, ut erat differentia altitudinum mercurii in baro-<lb/>metro ad altitudinem inter locos obſervationum interceptam: </s> + <s xml:id="echoid-s6096" xml:space="preserve">Etiamſi aër <lb/>ejuſdem denſitatis ponatur ab imo obſervationis loco ad alterum uſque, non li-<lb/>cet tamen inde judicare de ejus gravitate ſpecifica ratione mercurii. </s> + <s xml:id="echoid-s6097" xml:space="preserve">Quicquid <lb/>ab experimento colligere licet, hoc ſolum eſt:</s> + <s xml:id="echoid-s6098" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6099" xml:space="preserve">Conſideremus ſcilicet integram cruſtam aëream terram ambientem at-<lb/>que inter ambo obſervationis loca interceptam, & </s> + <s xml:id="echoid-s6100" xml:space="preserve">erit pondus iſtius cruſtæ <lb/>ad ſuperficiem terræ, ut pondus columnæ mercurialis, qualis in barometro <lb/>deſcendit ad baſin ejus; </s> + <s xml:id="echoid-s6101" xml:space="preserve">Manifeſta hæc ſunt ex eo quod ſumma baſium A & </s> + <s xml:id="echoid-s6102" xml:space="preserve">B <lb/>ſuſtinent quidem ſummam ponderum, quæ habent columnæ aëreæ A C & </s> + <s xml:id="echoid-s6103" xml:space="preserve">B D, <lb/>neque tamen quævis baſis premitur ſuæ columnæ pondere ſeorſim, & </s> + <s xml:id="echoid-s6104" xml:space="preserve">quod <lb/>idem reſectis columnis A g & </s> + <s xml:id="echoid-s6105" xml:space="preserve">B h intelligi debet de columnis g C & </s> + <s xml:id="echoid-s6106" xml:space="preserve">h D, dia-<lb/>phragmatis in g & </s> + <s xml:id="echoid-s6107" xml:space="preserve">h poſitis, incumbentibus. </s> + <s xml:id="echoid-s6108" xml:space="preserve">Igitur experimentum non tam <lb/>gravitatem ſpecificam aëris, in quo factum eſt, indicat quam omnis aëris terræ <lb/>proximi gravitatem ſpecificam mediam determinat; </s> + <s xml:id="echoid-s6109" xml:space="preserve">prior admodum variabilis <lb/>eſt, altera procul dubio conſtanter eadem fere permanet.</s> + <s xml:id="echoid-s6110" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6111" xml:space="preserve">Faciamus computum gravitatis ſpecificæ iſtius mediæ aëris omnis, quiter-<lb/>ram ambit: </s> + <s xml:id="echoid-s6112" xml:space="preserve">Multis vero experimentis, quæ in diverſis locis parum ſupra mare <lb/>elevatis ſumta fuerunt, id conſtat, elevationi 66 pedum proxime deſcenſum <lb/>reſpondere unius lineæ in barometro. </s> + <s xml:id="echoid-s6113" xml:space="preserve">Sequitur inde, quod aëris gravitas ſpe-<lb/>cifica media ratione mercurii ſit, ut altitudo unius lineæ ad altitudinem 66. </s> + <s xml:id="echoid-s6114" xml:space="preserve">ped. <lb/></s> + <s xml:id="echoid-s6115" xml:space="preserve">id eſt, ut ut 1 ad 9504; </s> + <s xml:id="echoid-s6116" xml:space="preserve">ergo poſita gravitate ſpecifica mercurii = 1, erit <lb/>gravitas ſpecifica media aëris = 0, 000105. </s> + <s xml:id="echoid-s6117" xml:space="preserve">Notabile eſt profecto tantam <lb/>eſſe hanc gravitatem mediam aëris: </s> + <s xml:id="echoid-s6118" xml:space="preserve">certus enim ſum vel maxime ſæviente hic <lb/>locorum frigore, aëris gravitatem ſpecificam vixdum tantam eſſe, quantam <lb/>nunc exhibuimus pro ſtatu medio omnis aëris terram ambientis: </s> + <s xml:id="echoid-s6119" xml:space="preserve">at ſub æqua- +<pb o="210" file="0224" n="224" rhead="HYDRODYNAMICÆ"/> +tore multo erit minor & </s> + <s xml:id="echoid-s6120" xml:space="preserve">omnibus recte perpenſis non crediderim gravitatem me-<lb/>diam aëris, qui inter utramque latitudinem 60. </s> + <s xml:id="echoid-s6121" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6122" xml:space="preserve">continetur, ultra 0, 000090 <lb/>excurrere; </s> + <s xml:id="echoid-s6123" xml:space="preserve">quo poſito erit gravitas media aëris ab utroque polo ad 30. </s> + <s xml:id="echoid-s6124" xml:space="preserve">gradus, <lb/>terram cingentis, (quod ſpatium paullo pluſquam octavam totius terræ ſuper-<lb/>ficiei efficit partem) = 0, 000210, quæ dupla eſt aëris hic locorum denſiſſi-<lb/>mi: </s> + <s xml:id="echoid-s6125" xml:space="preserve">ſub ipſo autem polo, præſertim antarctico admodum gravior erit aër & </s> + <s xml:id="echoid-s6126" xml:space="preserve"><lb/>fortaſſe aqua vix decies levior, cum eſt frigidiſſimus atque denſiſſimus.</s> + <s xml:id="echoid-s6127" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6128" xml:space="preserve">§. </s> + <s xml:id="echoid-s6129" xml:space="preserve">17. </s> + <s xml:id="echoid-s6130" xml:space="preserve">Veniamus nunc ad mutationes tum atmoſphæræ tum barometri: <lb/></s> + <s xml:id="echoid-s6131" xml:space="preserve">Conſiderabimus ergo duo barometra utrobique in imo aëris loco poſita, alte-<lb/>rum in A, alterum in B, & </s> + <s xml:id="echoid-s6132" xml:space="preserve">in utroque mercurium ad eandem altitudinem ſu-<lb/>ſpenſum ponemus: </s> + <s xml:id="echoid-s6133" xml:space="preserve">Poſtea in A ſubito aërem admodum calefieri fingamus: </s> + <s xml:id="echoid-s6134" xml:space="preserve">Ita <lb/>videmus fore, ut idem aër rarefiat: </s> + <s xml:id="echoid-s6135" xml:space="preserve">neque tamen inde ulla barometri mutatio <lb/>proditura eſſet, ſi nullam aër haberet inertiam ad motum, etiamſi omnis aër <lb/>ex A C in B D tranſpellatur: </s> + <s xml:id="echoid-s6136" xml:space="preserve">poſita autem iſta inertia ſupervenit quædam preſ-<lb/>ſio in omnes plagas eaque maxime ſenſibilis in regione A. </s> + <s xml:id="echoid-s6137" xml:space="preserve">Creſcet igitur ad <lb/>tempus altitudo mercurii in utroque barometro, magiſque creſcet in A quam <lb/>in B. </s> + <s xml:id="echoid-s6138" xml:space="preserve">Contrarium erit, ſi extemplo magna quædam aëris maſſa barometro A <lb/>vel B vicina à frigore condenſetur.</s> + <s xml:id="echoid-s6139" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6140" xml:space="preserve">§. </s> + <s xml:id="echoid-s6141" xml:space="preserve">18. </s> + <s xml:id="echoid-s6142" xml:space="preserve">Hæc unica videtur cauſa, quæ aliquam in barometris in A vel <lb/>B poſitis, efficere poſſit mutationem, quia hâc remotâ funda A & </s> + <s xml:id="echoid-s6143" xml:space="preserve">B ſemper <lb/>æqualiter premuntur, nempe unuſquiſque pondere, quod ſit dimidium co-<lb/>lumnarum aërearum A C & </s> + <s xml:id="echoid-s6144" xml:space="preserve">B D ſimul ſumtarum, quæ quidem ponderum <lb/>ſumma conſtans eſt. </s> + <s xml:id="echoid-s6145" xml:space="preserve">Si hæc ad atmoſphæram applicare velimus, notandum <lb/>eſt funda A & </s> + <s xml:id="echoid-s6146" xml:space="preserve">B repræſentare loca ima atmoſphæræ, quæ quidem in ſuperficie <lb/>terræ poſita forent, ſi aër terræ viſcera penetrare nequiret: </s> + <s xml:id="echoid-s6147" xml:space="preserve">quia vero res ſecus <lb/>ſe habet, erunt loca fundis A & </s> + <s xml:id="echoid-s6148" xml:space="preserve">B analoga intra ſuperficiem terræ cenſenda.</s> + <s xml:id="echoid-s6149" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6150" xml:space="preserve">§. </s> + <s xml:id="echoid-s6151" xml:space="preserve">19. </s> + <s xml:id="echoid-s6152" xml:space="preserve">Putentur nunc barometra in g & </s> + <s xml:id="echoid-s6153" xml:space="preserve">h poſita; </s> + <s xml:id="echoid-s6154" xml:space="preserve">ſitque in ambobus <lb/>mercurius ad eandem altitudinem ſuſpenſus: </s> + <s xml:id="echoid-s6155" xml:space="preserve">his poſitis cauſa fingatur ſuper-<lb/>venire, qua columna A g ſive ſola ſive conjunctim cum ſocia B h calefiat atque <lb/>ſeſe expandat. </s> + <s xml:id="echoid-s6156" xml:space="preserve">His perſpicuum eſt, ſi vel nulla aëris ſit inertia fore, ut preſ-<lb/>fiones aëris in g & </s> + <s xml:id="echoid-s6157" xml:space="preserve">h creſcant, quia his locis major nunc aëris quantitas ſuper-<lb/>eminet quam antea; </s> + <s xml:id="echoid-s6158" xml:space="preserve">acceſſit nimirum pondus omnis aëris, qui ex A g & </s> + <s xml:id="echoid-s6159" xml:space="preserve">B h à <lb/>calore fuit ſurſum propulſus. </s> + <s xml:id="echoid-s6160" xml:space="preserve">Atque ut hæc ſymbolis indicemus, faciemus pon- +<pb o="211" file="0225" n="225" rhead="SECTIO DECIMA."/> +dus columnæ A g, antequam novus caloris gradus ſuperveniret, = A, alte-<lb/>rius B h = α, pondus columnæ g C = B, columnæ h D = β: </s> + <s xml:id="echoid-s6161" xml:space="preserve">pondus co-<lb/>lumnæ A g rarefactæ = C, pondus columnæ B h itidem rarefactæ = γ: </s> + <s xml:id="echoid-s6162" xml:space="preserve">al-<lb/>titudo mercurii in g ante expanſionem aëris A g & </s> + <s xml:id="echoid-s6163" xml:space="preserve">B h = l, altitudo ſimilis <lb/>poſt iſtam expanſionem = x & </s> + <s xml:id="echoid-s6164" xml:space="preserve">habebimus hanc analogiam <lb/>B + β: </s> + <s xml:id="echoid-s6165" xml:space="preserve">l:</s> + <s xml:id="echoid-s6166" xml:space="preserve">: B + A - C + β + α - γ: </s> + <s xml:id="echoid-s6167" xml:space="preserve">x: </s> + <s xml:id="echoid-s6168" xml:space="preserve">unde eſt <lb/>x = {B + A - C + β + α - γ/B + β}l.</s> + <s xml:id="echoid-s6169" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6170" xml:space="preserve">Igitur aſcendet mercurius ab rarefacto aëre inferiore per altitudinem <lb/>x - l = {A - C + α - γ/B + β} l = (poſitis omnibus in utroque tubo paribus) {A - C/B} l.</s> + <s xml:id="echoid-s6171" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6172" xml:space="preserve">Refrigeſcente autem rurſus aëre in A g & </s> + <s xml:id="echoid-s6173" xml:space="preserve">B h iterum deſcendet mercu-<lb/>rius in utroque barometro.</s> + <s xml:id="echoid-s6174" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6175" xml:space="preserve">Notandum hic eſt, poſſe hoc modo à parvula caloris mutatione in A g <lb/>atque B h notabilem oriri in barometro variationem ob inſignem aëris denſi-<lb/>tatem in partibus inferioribus, qua fieri poteſt, ut in parte A g multo plus <lb/>aëris contineatur (imo infinities, ſi aër vi infinita preſſus in infinitè parvum <lb/>ſpatium condenſari ponatur) quam in reliqua g C, etiamſi longitudine infini-<lb/>ta. </s> + <s xml:id="echoid-s6176" xml:space="preserve">Unde ſi pondus A admodum majus ſit pondere B, ſimulque manente <lb/>cauſa aërem rarefaciente, pondus C datam ſervet rationem ad A, quod ita <lb/>fere fit, apparet aſcenſum mercurii à minimo caloris gradu ſuperveniente in <lb/>A g poſſe utcunque magnum eſſe.</s> + <s xml:id="echoid-s6177" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6178" xml:space="preserve">Equidem ſi fingatur, partes A g & </s> + <s xml:id="echoid-s6179" xml:space="preserve">B h ſtrictiores admodum eſſe præ <lb/>amplitudinibus in g C & </s> + <s xml:id="echoid-s6180" xml:space="preserve">h D, intelligitur variationes barometi ab aucto di-<lb/>minutove caloris gradu in A g & </s> + <s xml:id="echoid-s6181" xml:space="preserve">B h ita fieri minus notabiles, quia ponde-<lb/>ra A & </s> + <s xml:id="echoid-s6182" xml:space="preserve">α ipſaque C & </s> + <s xml:id="echoid-s6183" xml:space="preserve">γ prioribus proportionalia hocmodo decreſcunt; </s> + <s xml:id="echoid-s6184" xml:space="preserve">atta-<lb/>men variationes barometricæ, quæ ab hac cauſa proveniant, etiamnum ut-<lb/>cunque magnæ concipi poterunt.</s> + <s xml:id="echoid-s6185" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6186" xml:space="preserve">§. </s> + <s xml:id="echoid-s6187" xml:space="preserve">20. </s> + <s xml:id="echoid-s6188" xml:space="preserve">Hæc dum ita perpenduntur, veriſimile fit variationes barome-<lb/>tricas maxima parte petendas eſſe à celeribus caloris mutationibus in cryp-<lb/>tis ſubterraneis. </s> + <s xml:id="echoid-s6189" xml:space="preserve">Multas eſſe eaſque permagnas hujuſmodi cryptas jam diu <lb/>notum eſt: </s> + <s xml:id="echoid-s6190" xml:space="preserve">in terra etiam ſolida pori facere poſſunt quod cryptæ: </s> + <s xml:id="echoid-s6191" xml:space="preserve">ſi om-<lb/>nes cavitates (tum quæ à cavernis, tum quæ à poris aërem continentibus for- +<pb o="212" file="0226" n="226" rhead="HYDRODYNAMICÆ"/> +mantur) ad altitudinem infra ſuperficiem terræ 20000. </s> + <s xml:id="echoid-s6192" xml:space="preserve">aut 30000. </s> + <s xml:id="echoid-s6193" xml:space="preserve">pedum col-<lb/>ligas earumque capacitatem compares cum ſoliditate cruſtæ terreſtris ejuſ-<lb/>dem altitudinis, hancque vel millies aut centies millies altera majorem ponas, <lb/>erit profecto etiamnum ſufficiens cauſa iſta ad maximas barometri mutatio-<lb/>nes explicandas. </s> + <s xml:id="echoid-s6194" xml:space="preserve">Hæc ut puto ex præcedente paragrapho unicuique perſpi-<lb/>cua erunt.</s> + <s xml:id="echoid-s6195" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6196" xml:space="preserve">Cæterum loca quæ ſunt cryptis propiora, ea magis & </s> + <s xml:id="echoid-s6197" xml:space="preserve">ventis & </s> + <s xml:id="echoid-s6198" xml:space="preserve">baro-<lb/>metri mutationibus erunt obnoxia, ob aëris ad motum inertiam, quæ for-<lb/>taſſe ratio eſt, quod verſus æquatorem, ubi omnia fere pontus, minores <lb/>variationes in barometro obſerventur quam in locis hiſce ſeptentrionalibus.</s> + <s xml:id="echoid-s6199" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6200" xml:space="preserve">§. </s> + <s xml:id="echoid-s6201" xml:space="preserve">21. </s> + <s xml:id="echoid-s6202" xml:space="preserve">Ex eodem fonte deducitur, aliquid etiam ad variationes baro-<lb/>metricas conferre poſſe exhalationes aqueas ex terræ poris: </s> + <s xml:id="echoid-s6203" xml:space="preserve">ſed certe parum <lb/>id erit: </s> + <s xml:id="echoid-s6204" xml:space="preserve">ſi enim tantum aquæ vapores ſuppeditarint, quantum maxima plu-<lb/>ria decidere poteſt, vix inde unica linea mercurius aſcendet in barometro, <lb/>præterquam quod hæc cauſa non ſit ita celeris, quin illius effectus in totam <lb/>atmoſphæram ſimul fere diſtribuatur, atque ſic pro certo quodam loco to-<lb/>tus evaneſcat. </s> + <s xml:id="echoid-s6205" xml:space="preserve">Si enim totam conſideramus Atmoſphæram, quæ terram am-<lb/>bit, animadverti certe non poterit eſſe eam vaporibus nunc minus nunc ma-<lb/>gis oneratam. </s> + <s xml:id="echoid-s6206" xml:space="preserve">Equidem rationem §. </s> + <s xml:id="echoid-s6207" xml:space="preserve">20. </s> + <s xml:id="echoid-s6208" xml:space="preserve">expoſitam omnibus reliquis prætu-<lb/>lerim, magnas enim & </s> + <s xml:id="echoid-s6209" xml:space="preserve">celeres in terræ viſceribus fieri poſſe mutationes indi-<lb/>cant terræ motus, qui ſæpe ad centum usque milliaria eodem tempore ſen-<lb/>tiuntur, & </s> + <s xml:id="echoid-s6210" xml:space="preserve">alia hujuscemodi phænomena.</s> + <s xml:id="echoid-s6211" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6212" xml:space="preserve">Ad mutationes barometricas explicandas imprimis requiritur cauſa quæ-<lb/>dam ſubita; </s> + <s xml:id="echoid-s6213" xml:space="preserve">jam enim monui lentas in integram diſtribui aëris maſſam nul-<lb/>liusque eſſe effectus, idque demonſtravi §. </s> + <s xml:id="echoid-s6214" xml:space="preserve">14. </s> + <s xml:id="echoid-s6215" xml:space="preserve">Atque hanc ob cauſam parvi <lb/>faciendas eſſe mutationes, quæ immediate fiant in atmoſphæra ſupra terræ <lb/>ſuperficiem.</s> + <s xml:id="echoid-s6216" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6217" xml:space="preserve">§. </s> + <s xml:id="echoid-s6218" xml:space="preserve">22. </s> + <s xml:id="echoid-s6219" xml:space="preserve">Et hæc videtur pariter cauſa quod luna, quæ tantæ eſt efficaciæ <lb/>ad oceani aquas agitandas, nullum, qui obſervationibus diligentiſſimis ob-<lb/>ſervari potuerit, effectum exerat in barometrum: </s> + <s xml:id="echoid-s6220" xml:space="preserve">ſique cauſæ etiam reli-<lb/>quæ, quæ mutationem aliquam alicubi in Atmoſphæra producere valent <lb/>paullatim agerent, foret procul dubio in omnibus locis à ſuperficie maris æque <lb/>diſtantibus eadem conſtanter mercurii altitudo ad ſenſus. </s> + <s xml:id="echoid-s6221" xml:space="preserve">Hæc altitudo media +<pb o="213" file="0227" n="227" rhead="SECTIO DECIMA."/> +vocari poteſt & </s> + <s xml:id="echoid-s6222" xml:space="preserve">proxime determinabitur eo modo quo uſus eſt Joh. </s> + <s xml:id="echoid-s6223" xml:space="preserve">Jacobus <lb/>Scheuchzer, obſervando quotidie altitudinem barometricam per longum <lb/>temporis tractum ſumendoque inter omnes mediam.</s> + <s xml:id="echoid-s6224" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6225" xml:space="preserve">Atque hâc circumſpectione uſus celeberrimus Auctor ex multis obſerva-<lb/>tionibus, quæ ad ipſum ex pluribus transmiſſæ fuerunt locis, poſuit altitu-<lb/>dinem mediam.</s> + <s xml:id="echoid-s6226" xml:space="preserve"/> +</p> +<note position="right" xml:space="preserve"> <lb/>Patavii ------ # 27 poll. 11 {1/2} lin. Pariſ. <lb/>Pariſiis ------ # 27 poll. 9 {1/2} l. <lb/>Turini ------ # 27 poll. 1 {1/4} l. <lb/>Baſileæ ------ # 26 poll. 10 {1/8} l. <lb/>Tiguri ------ # 26 poll. 6 {1/2} l. <lb/>In monte Gothardi - # 21 poll. 27 {1/2} l. <lb/></note> +<p> + <s xml:id="echoid-s6227" xml:space="preserve">§. </s> + <s xml:id="echoid-s6228" xml:space="preserve">23. </s> + <s xml:id="echoid-s6229" xml:space="preserve">Diverſitates iſtarum altitudinum mediarum ab inæqualibus loco-<lb/>rum ſupra mare elevationibus provenire notum eſt. </s> + <s xml:id="echoid-s6230" xml:space="preserve">Jam enim Paſcalii tempo-<lb/>re experimenta ſumta fuere de deſcenſu mercurii in barometro ex loco profun-<lb/>diori in altiorem lato. </s> + <s xml:id="echoid-s6231" xml:space="preserve">Inde Philoſophi in mutuam cauſæ & </s> + <s xml:id="echoid-s6232" xml:space="preserve">effectus propor-<lb/>tionem inquirere: </s> + <s xml:id="echoid-s6233" xml:space="preserve">Diverſæ in hanc rem variis auctoribus prodiere regulæ: <lb/></s> + <s xml:id="echoid-s6234" xml:space="preserve">Præcipua, cui etiamnum plurimi adhærent, hæc eſt, quod altitudines loco-<lb/>rum proportionem ſequantur logarithmorum, qui altitudinibus barometri re-<lb/>ſpondent. </s> + <s xml:id="echoid-s6235" xml:space="preserve">Fundata eſt hæc regula præcipue ſuper eo, quod denſitas aëris ubi-<lb/>que proportionalis ſit ponderi aëris ſuperincumbentis: </s> + <s xml:id="echoid-s6236" xml:space="preserve">male autem hic appli-<lb/>catur iſtud principium, quod pro aëre ejuſdem caloris tantum valet, neque <lb/>res certa eſt in omni altitudine aëris, quamvis in eadem columna verticali exi-<lb/>ſtentis; </s> + <s xml:id="echoid-s6237" xml:space="preserve">ſi vero ita ſit, calorem æqualem eſſe, fatendum eſt, ſic ſatis recte regu-<lb/>lam ſe habere.</s> + <s xml:id="echoid-s6238" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6239" xml:space="preserve">At experimenta regulæ plane ſunt contraria; </s> + <s xml:id="echoid-s6240" xml:space="preserve">igitur non eſt ubiquè idem <lb/>caloris gradus per totam columnæ aëreæ verticalis altitudinem, quod ut nunc <lb/>planum faciam, apponam experimenta quædam accurate, ut mihi perſuadeo, <lb/>inſtituta, ſed tamen, quod doleo, diverſis temporibus lociſque: </s> + <s xml:id="echoid-s6241" xml:space="preserve">magis utique <lb/>inſtituto noſtro convenirent experimenta eodem tempore in eodemque mon-<lb/>te, diverſis tantum altitudinibus, ſumta; </s> + <s xml:id="echoid-s6242" xml:space="preserve">talia autem, niſi pro mediocribus <lb/>locorum altitudinibus, nulla adhuc quantum ſcio extant cum omnibus quæ <lb/>ſcire oportet circumſtantiis.</s> + <s xml:id="echoid-s6243" xml:space="preserve"/> +</p> +<pb o="214" file="0228" n="228" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s6244" xml:space="preserve">(I) In altitudine 1070 ped. </s> + <s xml:id="echoid-s6245" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s6246" xml:space="preserve">à ſuperficie maris barometrum deſcen-<lb/>dit 16 {1/3} lin. </s> + <s xml:id="echoid-s6247" xml:space="preserve">cum in ſuperficie maris altitudinem teneret 28 poll. </s> + <s xml:id="echoid-s6248" xml:space="preserve">4 {2/3} lin. </s> + <s xml:id="echoid-s6249" xml:space="preserve">(alii po-<lb/>nunt ſimpliciter 28 poll. </s> + <s xml:id="echoid-s6250" xml:space="preserve">in ſchedis autem quas D. </s> + <s xml:id="echoid-s6251" xml:space="preserve">De Lisle mecum communi-<lb/>cavit habetur 28 poll. </s> + <s xml:id="echoid-s6252" xml:space="preserve">4 {2/3} lin.)</s> + <s xml:id="echoid-s6253" xml:space="preserve">. </s> + <s xml:id="echoid-s6254" xml:space="preserve">Igitur poſita elaſticitate aëris in ſuperficie ma-<lb/>ris, uti deinceps ſemper ponam, = 1; </s> + <s xml:id="echoid-s6255" xml:space="preserve">inventa fuit elaſticitas in loco ſuperiori <lb/>quam deſignabo per E = 0, 9520.</s> + <s xml:id="echoid-s6256" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6257" xml:space="preserve">(II) In altitudine à ſuperficie maris 1542 ped. </s> + <s xml:id="echoid-s6258" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s6259" xml:space="preserve">deſcendit Mercurius in <lb/>barometro 21 {1/2} lin. </s> + <s xml:id="echoid-s6260" xml:space="preserve">qui in mari ad altitudinem 28 poll. </s> + <s xml:id="echoid-s6261" xml:space="preserve">2 lin. </s> + <s xml:id="echoid-s6262" xml:space="preserve">ſuſpenſus hæſit: </s> + <s xml:id="echoid-s6263" xml:space="preserve">hic <lb/>igitur fuit E = 0, 9364.</s> + <s xml:id="echoid-s6264" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6265" xml:space="preserve">(III) In altitudine montis Pici ſuper Inſula Teneriffa 13158 ped. </s> + <s xml:id="echoid-s6266" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s6267" xml:space="preserve">à ſu-<lb/>perficie maris ſtetit mercurius ad altitudinem 17 poll. </s> + <s xml:id="echoid-s6268" xml:space="preserve">5. </s> + <s xml:id="echoid-s6269" xml:space="preserve">lin. </s> + <s xml:id="echoid-s6270" xml:space="preserve">dum in ſuperficie <lb/>maris teneret altit. </s> + <s xml:id="echoid-s6271" xml:space="preserve">27 poll. </s> + <s xml:id="echoid-s6272" xml:space="preserve">10 lin. </s> + <s xml:id="echoid-s6273" xml:space="preserve">unde eo in loco fuit E = 0, 6257.</s> + <s xml:id="echoid-s6274" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6275" xml:space="preserve">(IV) Si in minoribus altitudinibus accurate deſcenſus Mercurii obſer-<lb/>ventur, reperitur deſcenſum unius lineæ reſpondere altitudini 65 aut 66 ped. <lb/></s> + <s xml:id="echoid-s6276" xml:space="preserve">Igitur in altitudine 65 ped. </s> + <s xml:id="echoid-s6277" xml:space="preserve">eſt E = 0, 9970. </s> + <s xml:id="echoid-s6278" xml:space="preserve">Extant paſſim hæ obſervationes: </s> + <s xml:id="echoid-s6279" xml:space="preserve"><lb/>tertiam autem habeo à D<emph style="super">no</emph>. </s> + <s xml:id="echoid-s6280" xml:space="preserve">De Lisle fuitque à R. </s> + <s xml:id="echoid-s6281" xml:space="preserve">P. </s> + <s xml:id="echoid-s6282" xml:space="preserve">Feuillée inſtituta atque co-<lb/>ram Societate Reg. </s> + <s xml:id="echoid-s6283" xml:space="preserve">Scient. </s> + <s xml:id="echoid-s6284" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s6285" xml:space="preserve">prælecta: </s> + <s xml:id="echoid-s6286" xml:space="preserve">eſtque illa ſcopulus, ad quem omnes, <lb/>quæ adhuc lucem aſpexerunt, theoriæ illidunt.</s> + <s xml:id="echoid-s6287" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6288" xml:space="preserve">§. </s> + <s xml:id="echoid-s6289" xml:space="preserve">24. </s> + <s xml:id="echoid-s6290" xml:space="preserve">Ut jam pateat, quouſque hæc cum poſitione logarithmicæ, <lb/>ceu ſcalæ altitudinum elaſticitatibus reſpondentium conveniant, ponemus al-<lb/>titudinem loci à ſuperficie maris certo numero pedum Pariſinorum definien-<lb/>dam = x: </s> + <s xml:id="echoid-s6291" xml:space="preserve">elaterem aëris in ſuperficie maris deſignabimus per 1, & </s> + <s xml:id="echoid-s6292" xml:space="preserve">elaterem <lb/>aëris in altitudine x ponemus = E. </s> + <s xml:id="echoid-s6293" xml:space="preserve">Notetur autem atmoſphæram nunc nobis <lb/>conſiderari invariatam aut ſaltem ſibi conſtanter ſimilem, ita ut elateres aëris <lb/>in ſuperficie maris & </s> + <s xml:id="echoid-s6294" xml:space="preserve">in altitudine quacunque x conſtantem ſervent rationem. <lb/></s> + <s xml:id="echoid-s6295" xml:space="preserve">Si enim admodum inæqualiter in diverſis atmoſphæræ altitudinibus, nulla ſer-<lb/>vata proportione elateres inconſtantia temporis mutentur, ſane nulla excogi-<lb/>tari poterit regula. </s> + <s xml:id="echoid-s6296" xml:space="preserve">His præmiſſis ponamus nunc æquationem α log. </s> + <s xml:id="echoid-s6297" xml:space="preserve">E = x ubi <lb/>coëfficiens α unica determinabitur obſervatione: </s> + <s xml:id="echoid-s6298" xml:space="preserve">utamur obſervatione prima <lb/>& </s> + <s xml:id="echoid-s6299" xml:space="preserve">erit α log. </s> + <s xml:id="echoid-s6300" xml:space="preserve">0, 9520 = 1070, hincque α (ſecundum logarithmos Vlacquia-<lb/>nos) = - 50194. </s> + <s xml:id="echoid-s6301" xml:space="preserve">Igitur pro hoc negotio, ſi logarithmica ſatisfacere de-<lb/>beat, ponendum eſſet - 50194 log. </s> + <s xml:id="echoid-s6302" xml:space="preserve">E = x, ſive log. </s> + <s xml:id="echoid-s6303" xml:space="preserve">{1/E} = {x/50194}: </s> + <s xml:id="echoid-s6304" xml:space="preserve">Ad +<pb o="215" file="0229" n="229" rhead="SECTIO DECIMA."/> +hujus autem æquationis normam, ſi ponatur pro ſecunda obſervatione <lb/>x = 1542, invenitur E = 0, 9317, ipſa autem obſervatio indicat E = 0, <lb/>9364: </s> + <s xml:id="echoid-s6305" xml:space="preserve">differentia inter hypotheſin & </s> + <s xml:id="echoid-s6306" xml:space="preserve">obſervationem eſt plus quam ſeſquilineæ, <lb/>quæ ſane notabilis eſt reſpectu habito ad differentiam parvam altitudinum ver-<lb/>ticalium.</s> + <s xml:id="echoid-s6307" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6308" xml:space="preserve">Si jam porro pro tèrtia obſervatione ponatur x = 13158, fit ex hypo-<lb/>theſi E = 0, 5469, dum experimentum indicavit E = 0, 6257: </s> + <s xml:id="echoid-s6309" xml:space="preserve">quæ diffe-<lb/>rentia nimia eſt, quam ut ullo modo logarithmica ſervari poſſit: </s> + <s xml:id="echoid-s6310" xml:space="preserve">valet enim <lb/>hæc differentia plus quam duos pollices cum duabus lineis.</s> + <s xml:id="echoid-s6311" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6312" xml:space="preserve">§. </s> + <s xml:id="echoid-s6313" xml:space="preserve">25. </s> + <s xml:id="echoid-s6314" xml:space="preserve">Rejecta logarithmica conſequens eſt elaſticitates in diverſis at-<lb/>moſphæræ altitudinibus nequaquam eſſe denſitatibus proportionales, aut quod <lb/>eodem recidit, diverſum eſſe in diverſis altitudinibus medium caloris gradum. <lb/></s> + <s xml:id="echoid-s6315" xml:space="preserve">Aliæ igitur ab aliis, quibus defectus iſte probe fuit notatus, fuerunt excogita-<lb/>tæ regulæ: </s> + <s xml:id="echoid-s6316" xml:space="preserve">earum tamen nulla ad experimentum III. </s> + <s xml:id="echoid-s6317" xml:space="preserve">(§. </s> + <s xml:id="echoid-s6318" xml:space="preserve">23.) </s> + <s xml:id="echoid-s6319" xml:space="preserve">ſatis accommo-<lb/>data dici poteſt. </s> + <s xml:id="echoid-s6320" xml:space="preserve">Veram, quam natura ſequatur, legem invenire, rem eſſe pu-<lb/>to vix ſperandam: </s> + <s xml:id="echoid-s6321" xml:space="preserve">quis enim aliter quam levibus conjecturis aſſequetur@ ra-<lb/>tionem velocitatum mediarum in particulis aëreis: </s> + <s xml:id="echoid-s6322" xml:space="preserve">Incidi tamen forte in ali-<lb/>quam hypotheſin, quæ phænomenis non male reſpondet: </s> + <s xml:id="echoid-s6323" xml:space="preserve">prius autem pro <lb/>quacunque velocitatum lege curvam dabo, quam ad ſpecialem iſtam hypothe-<lb/>ſin deſcendam.</s> + <s xml:id="echoid-s6324" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6325" xml:space="preserve">§. </s> + <s xml:id="echoid-s6326" xml:space="preserve">26. </s> + <s xml:id="echoid-s6327" xml:space="preserve">Sit linea verticalis A D (Fig. </s> + <s xml:id="echoid-s6328" xml:space="preserve">59,); </s> + <s xml:id="echoid-s6329" xml:space="preserve">Q F horizontalis radat ſu-<lb/> +<anchor type="note" xlink:label="note-0229-01a" xlink:href="note-0229-01"/> +perficiem maris: </s> + <s xml:id="echoid-s6330" xml:space="preserve">Denotet B F velocitatem mediam particularum aërearum in <lb/>ſuperficie maris: </s> + <s xml:id="echoid-s6331" xml:space="preserve">B M denſitatem mediam & </s> + <s xml:id="echoid-s6332" xml:space="preserve">B Q elaſticitatem, quæ in omni <lb/>loco æque alto eadem eſt. </s> + <s xml:id="echoid-s6333" xml:space="preserve">Deinde per puncta F, M, Q ductæ concipiantur <lb/>curvæ E F H, L M O, P Q S ceu ſcalæ, quæ in omnibus altitudinibus, veluti <lb/>B C, applicatis C G, C N, C R denotent velocitates medias particularum aë-<lb/>rearum, denſitates medias & </s> + <s xml:id="echoid-s6334" xml:space="preserve">elaſticitates medias. </s> + <s xml:id="echoid-s6335" xml:space="preserve">Datis nunc duabus curvis ter-<lb/>tiam licet determinare ex eo, quod elaſticitates (ceu experientia docuit & </s> + <s xml:id="echoid-s6336" xml:space="preserve"><lb/>§. </s> + <s xml:id="echoid-s6337" xml:space="preserve">§. </s> + <s xml:id="echoid-s6338" xml:space="preserve">3. </s> + <s xml:id="echoid-s6339" xml:space="preserve">4 5. </s> + <s xml:id="echoid-s6340" xml:space="preserve">& </s> + <s xml:id="echoid-s6341" xml:space="preserve">6. </s> + <s xml:id="echoid-s6342" xml:space="preserve">explicatum fuit) ſint proxime in ratione compoſita ex qua-<lb/>drato velocitatum modo dictarum & </s> + <s xml:id="echoid-s6343" xml:space="preserve">ſimplici denſitatum.</s> + <s xml:id="echoid-s6344" xml:space="preserve"/> +</p> +<div xml:id="echoid-div237" type="float" level="2" n="4"> +<note position="right" xlink:label="note-0229-01" xlink:href="note-0229-01a" xml:space="preserve">Fig. 59.</note> +</div> +<p> + <s xml:id="echoid-s6345" xml:space="preserve">Ipſe quidem monui prædicto loco hanc proportionem non poſſe exa-<lb/>cte eſſe veram, quia aër quidem elaterem poteſt habere infinitum ſeu vi in-<lb/>finita comprimi, non poteſt autem in ſpatium plane infinite parvum conden- +<pb o="216" file="0230" n="230" rhead="HYDRODYNAMICÆ"/> +ſari: </s> + <s xml:id="echoid-s6346" xml:space="preserve">quia tamen in aëre qui ſit naturali vel quadruplo denſior, hæc proprie-<lb/>tas, quod nempe elaſticitates ſint in ratione compoſita ex quadrato velocita-<lb/>tum particularum & </s> + <s xml:id="echoid-s6347" xml:space="preserve">ſimplici denſitatum experimentis etiamnum ad ſenſus <lb/>omnino reſpondere viſa fuit, illa ſine ullo ſenſibili errore uti poterimus <lb/>pro aëre naturali atmoſphæræ mari incumbentis, ſiquidem eo accuratius <lb/>vera ſit quo rarior eſt aër.</s> + <s xml:id="echoid-s6348" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6349" xml:space="preserve">His ad calculum præparatis ponemus B F = a, B M = b, B Q = c, <lb/>B C = x, C c = dx; </s> + <s xml:id="echoid-s6350" xml:space="preserve">C G = v, C N = z, C R = y, & </s> + <s xml:id="echoid-s6351" xml:space="preserve">erit y: </s> + <s xml:id="echoid-s6352" xml:space="preserve">c = vvz: <lb/></s> + <s xml:id="echoid-s6353" xml:space="preserve">aab ſeu y = {cvvz/aab}. </s> + <s xml:id="echoid-s6354" xml:space="preserve">Quia porro elaſticitatis menſura eſt pondus ſuperin-<lb/>cumbentis aëris, erit q R (- dy) = ponderi ſtrati aërei intercepti inter C & </s> + <s xml:id="echoid-s6355" xml:space="preserve"><lb/>c, quod proportionale eſt aëris denſitati z & </s> + <s xml:id="echoid-s6356" xml:space="preserve">altitudini ſtrati dx: </s> + <s xml:id="echoid-s6357" xml:space="preserve">eſt igitur <lb/>- dy = {zdx/n} ſeu z = {- ndy/dx}, quo valore ſubſtituto in æquatione <lb/>(y = {cvvz/aab}) habetur y = {cvv/aab} X {- ndy/dx} vel <lb/>- {dy/y} = {aabdx/ncvv}.</s> + <s xml:id="echoid-s6358" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6359" xml:space="preserve">§. </s> + <s xml:id="echoid-s6360" xml:space="preserve">27. </s> + <s xml:id="echoid-s6361" xml:space="preserve">Si ponatur velocitas particularum aërearum in omni altitudine <lb/>eadem, nempe = a, fiet {- dy/y} = {bdx/nc}, vel, facta debita integratione, <lb/>log.</s> + <s xml:id="echoid-s6362" xml:space="preserve">{c/y} = {bx/nc}; </s> + <s xml:id="echoid-s6363" xml:space="preserve">Iſtam vero hypotheſin non ſatis experimentis confirmari vi-<lb/>dimus §. </s> + <s xml:id="echoid-s6364" xml:space="preserve">24. </s> + <s xml:id="echoid-s6365" xml:space="preserve">Igitur alia tentata, poſui v = √(aa + mx) vel vv = aa + mx, <lb/>quæ lex eſt in motibus corporum libere cadentium: </s> + <s xml:id="echoid-s6366" xml:space="preserve">neque id ſine ſucceſſu; <lb/></s> + <s xml:id="echoid-s6367" xml:space="preserve">ita vero fit <lb/>{- dy/y} = {aabdx/naac + mncx} <lb/>vel log. </s> + <s xml:id="echoid-s6368" xml:space="preserve">{c/y} = {aab/mnc} log. </s> + <s xml:id="echoid-s6369" xml:space="preserve">{aa + mx/aa}.</s> + <s xml:id="echoid-s6370" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6371" xml:space="preserve">In hac æquatione paullo generaliori in qua m & </s> + <s xml:id="echoid-s6372" xml:space="preserve">n etiamnum arbitra-<lb/>riæ ſunt, porro periculum feci, num non poſſet poni {aab/mnc} = 1, atque id <lb/>etiam apte fieri vidi: </s> + <s xml:id="echoid-s6373" xml:space="preserve">ſic vero obtinui <lb/>log. </s> + <s xml:id="echoid-s6374" xml:space="preserve">{c/y} = log. </s> + <s xml:id="echoid-s6375" xml:space="preserve">{aa + mx/aa} vel {c/y} = {aa + mx/aa} aut {y/c} = {aa/aa + mx}.</s> + <s xml:id="echoid-s6376" xml:space="preserve"/> +</p> +<pb o="217" file="0231" n="231" rhead="SECTIO DECIMA."/> +<p> + <s xml:id="echoid-s6377" xml:space="preserve">Indicat iſta hypotheſis eſſe elaſticitates aeris ubique in ratione reciproca qua-<lb/>drata velocitatum, quibus particulæ aëreæ agitantur, ſive eſſe C R ad B Q <lb/>ut B F² ad C G², atque cum E F H ex hypotheſi parabola eſt ſuper axe <lb/>A D verticem habens infra punctum B ad diſtantiam {aa/m}, ſequitur eſſe cur-<lb/>vam P Q S hyperbolam; </s> + <s xml:id="echoid-s6378" xml:space="preserve">Dictam vero diſtantiam {aa/m} ſumendam eſſe = 22000 <lb/>pedum animadverti, ut obſervationibus §. </s> + <s xml:id="echoid-s6379" xml:space="preserve">23. </s> + <s xml:id="echoid-s6380" xml:space="preserve">proxime ſatisfiat. </s> + <s xml:id="echoid-s6381" xml:space="preserve">Inde talis <lb/>jam prodit æquatio ſpecifica <lb/>{y/c} = {22000/22000 + x}.</s> + <s xml:id="echoid-s6382" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6383" xml:space="preserve">Pro curva vero LMO invenitur {z/b} = (per §. </s> + <s xml:id="echoid-s6384" xml:space="preserve">26.) </s> + <s xml:id="echoid-s6385" xml:space="preserve">{aay/cvv}, ſeu <lb/>(quia {aa/vv} = {22000/22000 + x} = {y/c}) prodit poſt hanc ſubſtitutionem <lb/>{z/b} = ({22000/22000 + x})<emph style="super">2</emph>.</s> + <s xml:id="echoid-s6386" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6387" xml:space="preserve">§. </s> + <s xml:id="echoid-s6388" xml:space="preserve">28. </s> + <s xml:id="echoid-s6389" xml:space="preserve">Ut appareat, quousque hypotheſis noſtra conveniat cum expe-<lb/>rimentis §. </s> + <s xml:id="echoid-s6390" xml:space="preserve">23. </s> + <s xml:id="echoid-s6391" xml:space="preserve">ponemus in æquatione pro elaſticitatibus ſucceſſive pro x, <lb/>1070; </s> + <s xml:id="echoid-s6392" xml:space="preserve">1542; </s> + <s xml:id="echoid-s6393" xml:space="preserve">13158, & </s> + <s xml:id="echoid-s6394" xml:space="preserve">65; </s> + <s xml:id="echoid-s6395" xml:space="preserve">ita invenitur reſpective {y/c} = o, 9536; <lb/></s> + <s xml:id="echoid-s6396" xml:space="preserve">{y/c} = o, 9345; </s> + <s xml:id="echoid-s6397" xml:space="preserve">{y/c} = o, 6257, atque {y/c} = o, 99705: </s> + <s xml:id="echoid-s6398" xml:space="preserve">obſervatio-<lb/>nes autem indicant {y/c} = o, 9520; </s> + <s xml:id="echoid-s6399" xml:space="preserve">{y/c} = o, 9364; </s> + <s xml:id="echoid-s6400" xml:space="preserve">{y/c} = o, 6257, <lb/>atque {y/c} = o, 9970. </s> + <s xml:id="echoid-s6401" xml:space="preserve">Obſervatio tertia aliis hypotheſibus inimiciſſima <lb/>cum noſtra plane conſpirat, nec reliquæ plusquam o, 0019 particulis diſ-<lb/>ſentiunt, quæ in altitudine barometri tres quintas unius lineæ partes valent. </s> + <s xml:id="echoid-s6402" xml:space="preserve"><lb/>Nemo autem qui expertus fuerit, quam vagæ & </s> + <s xml:id="echoid-s6403" xml:space="preserve">parum inter ſe conſentien-<lb/>tes fuerint obſervationes barometricæ, tantillam differentiam admodum cu-<lb/>rabit. </s> + <s xml:id="echoid-s6404" xml:space="preserve">Ipſe interim hanc rem non aliter quam hypotheſin precariam conſi-<lb/>dero, neque aliam ob cauſam calculum §. </s> + <s xml:id="echoid-s6405" xml:space="preserve">§. </s> + <s xml:id="echoid-s6406" xml:space="preserve">26. </s> + <s xml:id="echoid-s6407" xml:space="preserve">& </s> + <s xml:id="echoid-s6408" xml:space="preserve">27. </s> + <s xml:id="echoid-s6409" xml:space="preserve">præmiſi, quam ut <lb/>rationem darem, quâ fieri poſſit ut altitudines verticales non reſpondeant <lb/>logarithmis altitudinum barometricarum, prouti deberet fieri, ſi per totam <lb/>atmoſphæram uniformis eſſet calor: </s> + <s xml:id="echoid-s6410" xml:space="preserve">inſtituto enim calculo factaque compa-<lb/>ratione ejus cum experimentis mihi videre viſus ſum, non poſſe rem hanc à +<pb o="218" file="0232" n="232" rhead="HYDRODYNAMICÆ"/> +diverſa particularum aërearum gravitatione in diverſis à centro terræ diſtantiis <lb/>ſufficienter explicari, prouti Newtonus tentavit ſtatuendo gravitationes ha-<lb/>rum particularum decreſcere in ratione quadrata diſtantiarum à centro terræ, <lb/>quæ hypotheſis in altitudinibus 13000 pedes Pariſ. </s> + <s xml:id="echoid-s6411" xml:space="preserve">non excurrentibus ſenſibi-<lb/>lem differentiam non efficit ab hypotheſi uniformis gravitationis. </s> + <s xml:id="echoid-s6412" xml:space="preserve">Similiter <lb/>ego aliquando incidi in opinionem auctam vim centrifugam particularum <lb/>aërearum in majoribus altitudinibus aliquid hic contribuere poſſe; </s> + <s xml:id="echoid-s6413" xml:space="preserve">at pariter <lb/>inſtituto calculo opinioni huic non amplius adhæſi. </s> + <s xml:id="echoid-s6414" xml:space="preserve">Interim non puto, ab-<lb/>ſurdum eſſe, ſi dicamus calorem aëris medium eo majorem eſſe, quo ma-<lb/>gis à ſuperficie maris diſtet. </s> + <s xml:id="echoid-s6415" xml:space="preserve">Velim autem ut probe notetur, hic ſermonem <lb/>eſſe de calore medio in libera atmoſphæra: </s> + <s xml:id="echoid-s6416" xml:space="preserve">ſic enim fieri poteſt, ut calor realis <lb/>quidem in montibus non creſcat ex cauſis aliis, nec tamen inde hypo-<lb/>theſis evertatur, quandoquidem §. </s> + <s xml:id="echoid-s6417" xml:space="preserve">§. </s> + <s xml:id="echoid-s6418" xml:space="preserve">15. </s> + <s xml:id="echoid-s6419" xml:space="preserve">& </s> + <s xml:id="echoid-s6420" xml:space="preserve">16. </s> + <s xml:id="echoid-s6421" xml:space="preserve">jam demonſtratum fuerit, <lb/>pondus columnæ mercurii in barometro non præciſe cenſendum eſſe æquale <lb/>ponderi columnæ aëreæ in illa regione fumtæ, ſed ponderi medio omnium <lb/>columnarum terræ inſiſtentium: </s> + <s xml:id="echoid-s6422" xml:space="preserve">De diverſis denſitatibus itaque ſic ſentio.</s> + <s xml:id="echoid-s6423" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6424" xml:space="preserve">§. </s> + <s xml:id="echoid-s6425" xml:space="preserve">29. </s> + <s xml:id="echoid-s6426" xml:space="preserve">Si æqualis eſſet ubique calor, forent utique denſitates elaſticita-<lb/>tibus ad ſenſus proportionales, reſponderentque altitudines verticales loga-<lb/>rithmis altitudinum barometricarum: </s> + <s xml:id="echoid-s6427" xml:space="preserve">At vero id experimentis repugnare po-<lb/>no: </s> + <s xml:id="echoid-s6428" xml:space="preserve">neque tamen crediderim in duobus locis parum à ſe invicem diſſitis no-<lb/>tabilem intercedere poſſe caloris differentiam, quia calor in corpore rariore <lb/>ut eſt aër, mox uniſormiter diſtribuitur, niſi perpetua adſit cauſa, quæ aërem <lb/>vicinum calefaciat.</s> + <s xml:id="echoid-s6429" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6430" xml:space="preserve">Alia autem res eſt in locis remotioribus, nec enim abſurdum puto aë-<lb/>rem vel decies denſiorem ſtatuere ſub polis, quam ſub æquatore, ſi modo <lb/>aër utrobique accipiatur ſuperficiei terræ proximus; </s> + <s xml:id="echoid-s6431" xml:space="preserve">at in magnis altitudini-<lb/>bus minor utique erit differentia inter denſitatem aëris qui polis & </s> + <s xml:id="echoid-s6432" xml:space="preserve">ejus qui <lb/>æquatori reſpondet cæteris paribus, & </s> + <s xml:id="echoid-s6433" xml:space="preserve">propterea inæqualiter admodum <lb/>decreſcent à ſuperficie terræ denſitates aëris & </s> + <s xml:id="echoid-s6434" xml:space="preserve">multo magis decreſcent ſub <lb/>polis quam ſub æquatore: </s> + <s xml:id="echoid-s6435" xml:space="preserve">hoc igitur modo fieri poſſet, ut ſub polis denſitates <lb/>aëris reales in parvis altitudinibus v. </s> + <s xml:id="echoid-s6436" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6437" xml:space="preserve">decreſcant in ratione ut (22000 + x)<emph style="super">4</emph> <lb/>ad 22000<emph style="super">4</emph> ob auctum calorem, & </s> + <s xml:id="echoid-s6438" xml:space="preserve">ſub æquatore vix ſenſibiliter decre-<lb/>ſcant, ob diminutum calorem, quæ caloris diminutio prope æquatorem +<pb o="219" file="0233" n="233" rhead="SECTIO DECIMA."/> +confirmatur ex eo quod culmen montis Pici per decem fere menſium ſpa-<lb/>tium ſit nive obtectum, dum in ipſa Teneriffæ inſula nunquam ut ferunt <lb/>ningit. </s> + <s xml:id="echoid-s6439" xml:space="preserve">Igitur non abſurde denſitates mediæ cenſeri poſſunt diminui in ratio-<lb/>ne ut (22000 + x)<emph style="super">2</emph> ad 22000<emph style="super">2</emph>, ut §. </s> + <s xml:id="echoid-s6440" xml:space="preserve">27. </s> + <s xml:id="echoid-s6441" xml:space="preserve">aſſumtum fuit; </s> + <s xml:id="echoid-s6442" xml:space="preserve">dum elaſtici-<lb/>tates ubique decreſcant in ratione ut 22000 + x ad 22000; </s> + <s xml:id="echoid-s6443" xml:space="preserve">neque enim hæ <lb/>in iisdem à ſuperficie terræ altitudinibus differre poſſunt, niſi à cauſis fortuito <lb/>ſupervenientibus & </s> + <s xml:id="echoid-s6444" xml:space="preserve">parum durantibus.</s> + <s xml:id="echoid-s6445" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6446" xml:space="preserve">§. </s> + <s xml:id="echoid-s6447" xml:space="preserve">30. </s> + <s xml:id="echoid-s6448" xml:space="preserve">In terris, quæ intra quadrageſimum & </s> + <s xml:id="echoid-s6449" xml:space="preserve">ſexageſimum latitudi-<lb/>nis gradum continentur, probabile eſt denſitates in eadem proxime ratione <lb/>decreſcere qua elaſticitates; </s> + <s xml:id="echoid-s6450" xml:space="preserve">hancque ob rationem volui periculum facere, <lb/>quænam inde refractionum theoria oriatur, qua de re nunc quædam adjiciam.</s> + <s xml:id="echoid-s6451" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div239" type="section" level="1" n="185"> +<head xml:id="echoid-head236" style="it" xml:space="preserve">Digreſsio de refractione radiorum per atmoſphæ-<lb/>ram transeuntium.</head> +<p> + <s xml:id="echoid-s6452" xml:space="preserve">(α) Proprietas eſt notiſſima radiorum ex uno medio in aliud inciden-<lb/>tium eaque innumeris experimentis confirmata, quod angulus incidentiæ ad <lb/>angulum refractionis conſtantem ſervat rationem: </s> + <s xml:id="echoid-s6453" xml:space="preserve">præterea etiam patet, <lb/>ſi refractio fiat infinite parva, id eſt, ſi differentia utriusque ſinus rationem <lb/>habeat infinite parvam ad alterutrum ſinum, fore ut ſinus anguli, qui inter-<lb/>cipitur inter radium incidentiæ prolongatum & </s> + <s xml:id="echoid-s6454" xml:space="preserve">radium refractum, eandem <lb/>habeat rationem ad ſinum totum, quam habet differentia ſinuum angulo-<lb/>rum incidentiæ & </s> + <s xml:id="echoid-s6455" xml:space="preserve">refractionis ad coſinum anguli incidentiæ. </s> + <s xml:id="echoid-s6456" xml:space="preserve">Illum vero, <lb/>quem modo allegavi, angulum interceptum inter radium incidentiæ prolon-<lb/>gatum & </s> + <s xml:id="echoid-s6457" xml:space="preserve">radium refractum, deinceps vocabo angulum refractionis differentia-<lb/>lem. </s> + <s xml:id="echoid-s6458" xml:space="preserve">Exinde ſequitur, quod ſit cæteris paribus ſinus anguli refractionis differen-<lb/>tialis proportionalis ſinui anguli incidentiæ diviſo per coſinum ejusdem anguli.</s> + <s xml:id="echoid-s6459" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6460" xml:space="preserve">(β) Experimenta porro docent, ſi radius ex aëre in aërem diverſæ ab <lb/>altero denſitatis incidat, eſſe angulum refractionis differentialem cæteris pari-<lb/>bus differentiæ denſitatum proportionalem.</s> + <s xml:id="echoid-s6461" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6462" xml:space="preserve">Experimenta autem hanc in rem, quantum fieri poteſt, ſumta fuerunt <lb/>à D. </s> + <s xml:id="echoid-s6463" xml:space="preserve">Hauksbée, accuratiſſime, tum de aëre admodum condenſato, tum +<pb o="220" file="0234" n="234" rhead="HYDRODYNAMICÆ"/> +etiam de aëre rariſſimo, qui tandem pro nullo haberi poterat: </s> + <s xml:id="echoid-s6464" xml:space="preserve">modus quo <lb/>inſtituta fuerunt deſcribitur in transactionibus Anglicanis: </s> + <s xml:id="echoid-s6465" xml:space="preserve">Succeſſus autem om-<lb/>nium experimentorum huc redit, ut arguant fuiſſe ſinum anguli refractionis <lb/>differentialis ad ſinum totum ut 5 {1/8} pollices ad 2588. </s> + <s xml:id="echoid-s6466" xml:space="preserve">pedes, cum radius inci-<lb/>deret ex aëre naturali in ſpatium ab aëre vacuum ſub angulo triginta duorum <lb/>graduum, id eſt, ut 1 ad 6060 & </s> + <s xml:id="echoid-s6467" xml:space="preserve">iisdem poſitis, mutato angulo triginta <lb/>duorum graduum in ſemirectum, ut 1 ad 3787 (per §. </s> + <s xml:id="echoid-s6468" xml:space="preserve">a). </s> + <s xml:id="echoid-s6469" xml:space="preserve">Inde deducitur, <lb/>ſi radius ex aëre naturali in vacuum ſub angulo quocunque incidat, eſſe ſi-<lb/>num anguli incidentiæ ad ſinum anguli refractionis ut 3787 ad 3786.</s> + <s xml:id="echoid-s6470" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6471" xml:space="preserve">Neutonus loco hujus rationis aſſumit in tract. </s> + <s xml:id="echoid-s6472" xml:space="preserve">ſuo optico illam, quæ eſt <lb/>inter 3201 & </s> + <s xml:id="echoid-s6473" xml:space="preserve">3200, eamque deducit ex refractionum quantitate ab Aſtrono-<lb/>mis obſervata: </s> + <s xml:id="echoid-s6474" xml:space="preserve">ſtatuit autem quantitatem refractionis eandem eſſe, ſi ſtrata ra-<lb/>dium refringentia ſint parallela, in quacunque cæterum ratione denſitates <lb/>medii decreſcant, ſi modo in primo & </s> + <s xml:id="echoid-s6475" xml:space="preserve">ultimo ſtrato denſitatum differentia <lb/>eadem maneat (vid. </s> + <s xml:id="echoid-s6476" xml:space="preserve">Neut. </s> + <s xml:id="echoid-s6477" xml:space="preserve">tract. </s> + <s xml:id="echoid-s6478" xml:space="preserve">opt. </s> + <s xml:id="echoid-s6479" xml:space="preserve">pag. </s> + <s xml:id="echoid-s6480" xml:space="preserve">321. </s> + <s xml:id="echoid-s6481" xml:space="preserve">edit. </s> + <s xml:id="echoid-s6482" xml:space="preserve">gall.)</s> + <s xml:id="echoid-s6483" xml:space="preserve">. </s> + <s xml:id="echoid-s6484" xml:space="preserve">De reliquo ſub di-<lb/>verſis circumſtantiis non poteſt non admodum eſſe variabilis refractio, quod <lb/>aër, quem vocamus naturalem, multis mutationibus ſit obnoxius, tum à <lb/>calore & </s> + <s xml:id="echoid-s6485" xml:space="preserve">frigore, tum à preſſione atmoſphæræ, quæ ambo concurrunt <lb/>ad denſitatem aëris formandam, cui denſitati refractiones radiorum in va-<lb/>cuum incidentium ſunt proportionales cæteris paribus. </s> + <s xml:id="echoid-s6486" xml:space="preserve">Eadem etiam mo-<lb/>nuit D. </s> + <s xml:id="echoid-s6487" xml:space="preserve">Hauksbée in recenſione experimentorum, quæ modo allegavimus, <lb/>eamque ob rationem ſtatum aëris, qui erat, cum experimenta ſumeret, pro-<lb/>be definivit.</s> + <s xml:id="echoid-s6488" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6489" xml:space="preserve">(γ) Fuerit nunc A C (Fig. </s> + <s xml:id="echoid-s6490" xml:space="preserve">60.) </s> + <s xml:id="echoid-s6491" xml:space="preserve">arcus circuli terreſtris centro B ductus, in <lb/> +<anchor type="note" xlink:label="note-0234-01a" xlink:href="note-0234-01"/> +cujus plano radius luminis A G eſt: </s> + <s xml:id="echoid-s6492" xml:space="preserve">erit autem iſte radius incurvatus AG ejus in-<lb/>dolis, ut convergat ad aſymptoton, huicque aſymptotæ parallela putetur <lb/>AH; </s> + <s xml:id="echoid-s6493" xml:space="preserve">ducatur horizontalis A E, rectaque A F quæ tangat in A curvam AG. <lb/></s> + <s xml:id="echoid-s6494" xml:space="preserve">Ita videmus fore angulum H A E menſuram altitudinis aſtri veræ, & </s> + <s xml:id="echoid-s6495" xml:space="preserve">angu-<lb/>lum F A E menſuram altitudinis apparentis, angulumque F A H fore angu-<lb/>lum refractionis: </s> + <s xml:id="echoid-s6496" xml:space="preserve">eſt autem angulus F A H idem quod ſumma omnium an-<lb/>gulorum refractionis differentialium, ſeu augulorum contactus qualis eſt angu-<lb/>lus c b o.</s> + <s xml:id="echoid-s6497" xml:space="preserve"/> +</p> +<div xml:id="echoid-div239" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0234-01" xlink:href="note-0234-01a" xml:space="preserve">Fig. 60.</note> +</div> +<p> + <s xml:id="echoid-s6498" xml:space="preserve">Conſiderentur duo elementa curvæ ab, bo, & </s> + <s xml:id="echoid-s6499" xml:space="preserve">per puncta a, b, o, <lb/>ducti inteligantur centro communi B arcus αα, ββ, γγ: </s> + <s xml:id="echoid-s6500" xml:space="preserve">ſitque denſitas +<pb o="221" file="0235" n="235" rhead="SECTIO DECIMA."/> +aëris ααββ = D; </s> + <s xml:id="echoid-s6501" xml:space="preserve">denſitas aëris ββγγ = D - d D, erit (per §.</s> + <s xml:id="echoid-s6502" xml:space="preserve">§. </s> + <s xml:id="echoid-s6503" xml:space="preserve">α, β) <lb/>ſinus anguli contactus in b diviſus per ſinum totum, ſeu ipſe angulus conta-<lb/>ctus proportionalis differentiæ denſitatum d D multiplicatæ per rationem ſi-<lb/>nuum angulorum incidentiæ & </s> + <s xml:id="echoid-s6504" xml:space="preserve">refractionis, id eſt, multiplicatæ per {be/eo}. </s> + <s xml:id="echoid-s6505" xml:space="preserve">Si <lb/>vero ducatur B D perpendicularis ad FA productam, perſpicuum eſt, vix <lb/>differre {be/eo} & </s> + <s xml:id="echoid-s6506" xml:space="preserve">{BD/Do}, ideo quod radius fere ſit rectus ſicque poſſit trian-<lb/>gulum B D o pro rectilineo haberi & </s> + <s xml:id="echoid-s6507" xml:space="preserve">ſimili cum triangulo beo. </s> + <s xml:id="echoid-s6508" xml:space="preserve">Igitur erit <lb/>angulus quæſitus F A H proportionalis ſ{BD/Do} X dD.</s> + <s xml:id="echoid-s6509" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6510" xml:space="preserve">(δ) Hiſce veſtigiis inſiſtendo ponendoque eſſe ubiquel<unsure/> denſitatem <lb/>D = {22000/22000 + x}G, ubix exprimit lineam na numero pedum Pariſinorum <lb/>& </s> + <s xml:id="echoid-s6511" xml:space="preserve">G denotat denſitatem aëris in loco obſervationis, inveni quod ſequitur. <lb/></s> + <s xml:id="echoid-s6512" xml:space="preserve">Sit ſinus altitudinis aſtri apparentis = f, coſinus = F, radius terræ = r <lb/>numero pedum Pariſinorum exprimendus: </s> + <s xml:id="echoid-s6513" xml:space="preserve">indicetur numerus 22000 per a: </s> + <s xml:id="echoid-s6514" xml:space="preserve"><lb/>ponatur porro ſinus totus = 1, angulus refractionis differentialis pro radio ex <lb/>aëre naturali in vacuum ſub angulo ſemirecto incidentis = g: </s> + <s xml:id="echoid-s6515" xml:space="preserve">Denique bre-<lb/>vitatis ergo fiat 2r - 2a = α; </s> + <s xml:id="echoid-s6516" xml:space="preserve">- FFrr + 2ar - aa = β: </s> + <s xml:id="echoid-s6517" xml:space="preserve">& </s> + <s xml:id="echoid-s6518" xml:space="preserve">erit β aut nu-<lb/>merus affirmativus aut negativus; </s> + <s xml:id="echoid-s6519" xml:space="preserve">affirmativus erit, ſi altitudo apparens ſide-<lb/>ris parva fuerit & </s> + <s xml:id="echoid-s6520" xml:space="preserve">quidem infra 2<emph style="super">0</emph>, 44<emph style="super">1</emph>: </s> + <s xml:id="echoid-s6521" xml:space="preserve">ſecus erit negativus: </s> + <s xml:id="echoid-s6522" xml:space="preserve">In priori ca-<lb/>ſu obtinebitur angulus quæſitus F A H hunc in modum: </s> + <s xml:id="echoid-s6523" xml:space="preserve">Fiat nempe ſemicir-<lb/>culus M L F (Fig. </s> + <s xml:id="echoid-s6524" xml:space="preserve">61.) </s> + <s xml:id="echoid-s6525" xml:space="preserve">cujus radius A M = 1: </s> + <s xml:id="echoid-s6526" xml:space="preserve">ſumatur A C = {α/2fr}; </s> + <s xml:id="echoid-s6527" xml:space="preserve"><lb/> +<anchor type="note" xlink:label="note-0235-01a" xlink:href="note-0235-01"/> +AB = {2β - αa/2afr}, ducanturque C D, B T ad M C perpendiculares & </s> + <s xml:id="echoid-s6528" xml:space="preserve">erit an-<lb/>gulus F A H = {- fFrr/2β}g + {far/β}g + {farα x DT/2β√β}g. <lb/></s> + <s xml:id="echoid-s6529" xml:space="preserve">In caſu, quo β eſt negativus, erit idem angulus <lb/>F A H = {-far/β}g + {fFrr/β}g + {farα/2β√β}g x log. </s> + <s xml:id="echoid-s6530" xml:space="preserve">{(α - 2√β) x (Fr - a + √β)/(α + 2√β) x (Fr - a - √β)}.</s> + <s xml:id="echoid-s6531" xml:space="preserve"/> +</p> +<div xml:id="echoid-div240" type="float" level="2" n="2"> +<note position="right" xlink:label="note-0235-01" xlink:href="note-0235-01a" xml:space="preserve">Fig. 61.</note> +</div> +<p> + <s xml:id="echoid-s6532" xml:space="preserve">(ε) Secundum iſtas hypotheſes ponendo pro radio terræ 19600000. <lb/></s> + <s xml:id="echoid-s6533" xml:space="preserve">poterit pro omni altitudine ſideris apparentis ejus determinari refractio aſtro-<lb/>nomica, ſi bene experimento inventus fuerit valor anguli g: </s> + <s xml:id="echoid-s6534" xml:space="preserve">quia vero difficile <lb/>admodum eſt hunc valorem cum ſufficiente accuratione definire, conſultius +<pb o="222" file="0236" n="236" rhead="HYDRODYNAMICÆ"/> +erit in caſu aliquo particulari aſtronomice refractionem definire, & </s> + <s xml:id="echoid-s6535" xml:space="preserve">ex hoc re-<lb/>liquos calculo ſubducere. </s> + <s xml:id="echoid-s6536" xml:space="preserve">Aſſumamus v. </s> + <s xml:id="echoid-s6537" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6538" xml:space="preserve">in altitudine decem graduum re-<lb/>fractionem eſſe 5 min. </s> + <s xml:id="echoid-s6539" xml:space="preserve">28 ſec. </s> + <s xml:id="echoid-s6540" xml:space="preserve">cui hypotheſi plerique Aſtronomi Pariſiis adhærent. <lb/></s> + <s xml:id="echoid-s6541" xml:space="preserve">Inveniemus hancce refractionis tabulam.</s> + <s xml:id="echoid-s6542" xml:space="preserve"/> +</p> +<note position="right" xml:space="preserve"> <lb/>altit. ſid. appar. # refract. # altit. ſid. appar. # refract. <lb/>0 grad. # 34 min. # 53 ſec. # 50 grad. # 0 min. # 53 ſec. <lb/>5 # 9 - - - - # 59 - - # 55 # - - - - - # 44 {1/3} <lb/>10 # 5 - - - - # 28 - - # 60 # - - - - - # 36 {1/2} <lb/>15 # 3 - - - - # 44 - - # 65 # - - - - - # 29 {1/2} <lb/>20 # 2 - - - - # 52 - - # 70 # - - - - - # 23 <lb/>25 # 2 - - - - # 12 - - # 75 # - - - - - # 17 <lb/>30 # 1 - - - - # 47 - - # 80 # - - - - - # 11 {1/4} <lb/>35 # 1 - - - - # 29 - - # 85 # - - - - - - # 5 {1/2} <lb/>40 # 1 - - - - # 15 - - # 90 # - - - - - - # 0. <lb/>45 # 1 - - - - - # 3- -<lb/></note> +<p> + <s xml:id="echoid-s6543" xml:space="preserve">Quia vero @refractiones ſequuntur rationem litteræ g. </s> + <s xml:id="echoid-s6544" xml:space="preserve">id eſt, anguli re-<lb/>fractionis differentialis radii ſub angulo ſemirecto ex aëre naturali in vacuum in-<lb/>cidentis & </s> + <s xml:id="echoid-s6545" xml:space="preserve">quia iſte angulus proportionalis eſt denſitati aëris naturalis, ſeu aëris, <lb/>quem obſervator reſpirat, patet ſi vel aër conſtanter ſimiliter vaporibus eſſet <lb/>oneratus (à quibus animum adhuc abſtraximus) non poſſe tamen fieri, quin <lb/>refractiones aſtronomicæ ſint admodum variabiles. </s> + <s xml:id="echoid-s6546" xml:space="preserve">Majores nempe erunt in <lb/>ſuperficie maris quam in montibus, eritque notabilis differentia vel in me-<lb/>diocribus montium altitudinibus: </s> + <s xml:id="echoid-s6547" xml:space="preserve">majores præterea erunt tempore frigido <lb/>quam calido, hæcque ſola cauſa in hiſce terris refractiones minimum quarta <lb/>parte augere poteſt: </s> + <s xml:id="echoid-s6548" xml:space="preserve">denique majores etiam erunt refractiones barometro al-<lb/>to quam humili. </s> + <s xml:id="echoid-s6549" xml:space="preserve">Poterunt autem ſi vapores nullo ſint obſtaculo, refractiones <lb/>omni tempore recte definiri, ſi inſtrumentum, quod §. </s> + <s xml:id="echoid-s6550" xml:space="preserve">9. </s> + <s xml:id="echoid-s6551" xml:space="preserve">deſcriptum fuit <lb/>quodque Fig. </s> + <s xml:id="echoid-s6552" xml:space="preserve">57. </s> + <s xml:id="echoid-s6553" xml:space="preserve">repræſentat ſimul adhibeatur cum barometro; </s> + <s xml:id="echoid-s6554" xml:space="preserve">ſi enim alti-<lb/>tudinem mercurii in barometro dividas per altitudinem mercurii in altero in-<lb/>ſtrumento, habebis denſitatem aëris, cui cæteris paribus refractio proportio-<lb/>nalis eſt facienda. </s> + <s xml:id="echoid-s6555" xml:space="preserve">Neque dubito, quin refractio ſolis minor ſit refractionibus <lb/>reliquorum ſiderum, quod calor ſolis aërem non mediocriter expandit aëriſ-<lb/>que denſitatem diminuit.</s> + <s xml:id="echoid-s6556" xml:space="preserve"/> +</p> +<pb o="223" file="0237" n="237" rhead="SECTIO DECIMA."/> +<p> + <s xml:id="echoid-s6557" xml:space="preserve">§. </s> + <s xml:id="echoid-s6558" xml:space="preserve">31. </s> + <s xml:id="echoid-s6559" xml:space="preserve">Ex iis quæ de agitatione particularum aërearum, à quâ utique <lb/>calor aëris pendet, præſertim vero, quæ §. </s> + <s xml:id="echoid-s6560" xml:space="preserve">10. </s> + <s xml:id="echoid-s6561" xml:space="preserve">monita fuerunt, apparet gra-<lb/>dum eundem caloris aëri ineſſe, quoties eadem ratio intercedit inter ejus ela-<lb/>ſticitatem atque denſitatem; </s> + <s xml:id="echoid-s6562" xml:space="preserve">elaſticitatem indicat barometrum; </s> + <s xml:id="echoid-s6563" xml:space="preserve">denſitatem <lb/>concludimus ex gravitate aëris ſpecifica; </s> + <s xml:id="echoid-s6564" xml:space="preserve">atque inde ut vidimus §. </s> + <s xml:id="echoid-s6565" xml:space="preserve">10, gradus <lb/>obtineri poterit caloris fixus, ſi aquæ bullientis calor incertus videatur, prouti <lb/>D°. </s> + <s xml:id="echoid-s6566" xml:space="preserve">Fahrenheid obſervatus fuit pendere à pondere atmoſphæræ incumbentis. <lb/></s> + <s xml:id="echoid-s6567" xml:space="preserve">Inſtrumenta quæ ſingulis momentis denſitatem aëris indicant facile excogitari <lb/>poſſunt atque à multis deſcripta fuerunt.</s> + <s xml:id="echoid-s6568" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6569" xml:space="preserve">Notandum hic eſt rationem illam modo dictam inter aëris elaſticitatem <lb/>ejuſque denſitatem ſimul exhibere altitudinem aëris homogenei, & </s> + <s xml:id="echoid-s6570" xml:space="preserve">quia nobis <lb/>deinceps ſermo erit de iſta altitudine, convenit illam recte prius definire, quam <lb/>ad alia pergamus.</s> + <s xml:id="echoid-s6571" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6572" xml:space="preserve">§. </s> + <s xml:id="echoid-s6573" xml:space="preserve">32. </s> + <s xml:id="echoid-s6574" xml:space="preserve">Si fingamus columnam aëream verticalem uniformis denſitatis <lb/>& </s> + <s xml:id="echoid-s6575" xml:space="preserve">cum mercurio barometri ad æquilibrium compoſitam, erit altitudo illius <lb/>columnæ altitudo quam voco aëris homogenei pro data denſitate.</s> + <s xml:id="echoid-s6576" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6577" xml:space="preserve">Et quia aëris mediocriter denſi gravitas ſpecifica eſt ad gravitatem ſpe-<lb/>cificam mercurii ut 1 ad 11000 ipſaque altitudo media mercurii in barometro <lb/>pro locis parum à ſuperficie maris elevatis ſit 2 {1/3} ped. </s> + <s xml:id="echoid-s6578" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s6579" xml:space="preserve">erit altitudo aëris <lb/>homogenei mediocriter denſi 25666 pedum.</s> + <s xml:id="echoid-s6580" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6581" xml:space="preserve">Patet ex iſta definitione altitudines illas, de quibus nunc dicimus, eo <lb/>minores eſſe, quo denſior eſt aër, cui altitudo reſpondere debet, & </s> + <s xml:id="echoid-s6582" xml:space="preserve">quo mi-<lb/>nor eſt altitudo mercurii in barometro. </s> + <s xml:id="echoid-s6583" xml:space="preserve">Igitur ſi idem ſit caloris gradus in mon-<lb/>tibus & </s> + <s xml:id="echoid-s6584" xml:space="preserve">in ſuperficie maris, eadem quoque erit utrobique altitudo aëris ho-<lb/>mogenei, quia pro eodem caloris gradu aëris denſitas rationem ſequitur aëris <lb/>elaſticitatis ſeu altitudinis mercurii in barometro. </s> + <s xml:id="echoid-s6585" xml:space="preserve">Apparet porro altitudinem <lb/>aëris homogenei in ſuperficie maris admodum decreſcere ab æquatore verſus <lb/>polos, quia frigus intenditur denſitaſque aëris augetur manente elaſticitate & </s> + <s xml:id="echoid-s6586" xml:space="preserve"><lb/>in iiſdem regionibus minorem eſſe tempore hyemali quam æſtivo.</s> + <s xml:id="echoid-s6587" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6588" xml:space="preserve">§. </s> + <s xml:id="echoid-s6589" xml:space="preserve">33. </s> + <s xml:id="echoid-s6590" xml:space="preserve">Multa ſunt quæ ad motum aëris definiendum pertinent, quo-<lb/>rum ſolutio pendet ab altitudine aëris homogenei: </s> + <s xml:id="echoid-s6591" xml:space="preserve">Inter hæc etiam eſt propa-<lb/>gatio ſoni ejuſque celeritas: </s> + <s xml:id="echoid-s6592" xml:space="preserve">Quamvis enim celeritas ſoni diverſimode defi- +<pb o="224" file="0238" n="238" rhead="HYDRODYNAMICÆ"/> +niatur à diverſis, quos concipere poſſumus de ejus propagatione modis ita, ut <lb/>nunc videatur celeritatem eam eſſe quæ debeatur altitudini aëris homogenei, <lb/>nunc quæ dimidiæ altitudini reſpondeat, aut etiam dimidiæ altitudini multipli-<lb/>catæ per rationem quadrati circulo circumſcripti ad aream circuli, omnes ta-<lb/>men opiniones in eo conveniunt, quod celeritas ſoni proportionalis ſit radici <lb/>altitudinis aëris homogenei cum eo, in quo propagatur. </s> + <s xml:id="echoid-s6593" xml:space="preserve">Si ita ſe res habeat, <lb/>celerius propagatur ſonus in aëre calido quam frigido, barometro alto quam <lb/>humili, (nihil dicam de ventis ſecundis aut contrariis); </s> + <s xml:id="echoid-s6594" xml:space="preserve">multa in hanc rem <lb/>partim in Italia partim in Anglia ſumta fuerunt experimenta, hæcque poſterio-<lb/>ra docuerunt celeritatem ſoni mediam reſpondere 1140 ped. </s> + <s xml:id="echoid-s6595" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s6596" xml:space="preserve">intra minu-<lb/>@um ſecundum perficiendis. </s> + <s xml:id="echoid-s6597" xml:space="preserve">At quia in uno eodemque loco variabilis eſt al-<lb/>titudo atmoſphæræ homogeneæ nominatimque hic locorum excurrit à muta-<lb/>tionibus barometricis junctis cum mutationibus caloris à 3 uſque ad 4, variabi-<lb/>lis erit ubique celeritas ſoni, ſi vel nihil mutent venti, eaque celeritas in hiſce <lb/>terris continebitur intra terminos √ 3 & </s> + <s xml:id="echoid-s6598" xml:space="preserve">√ 4, ſeu 173 & </s> + <s xml:id="echoid-s6599" xml:space="preserve">200.</s> + <s xml:id="echoid-s6600" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6601" xml:space="preserve">§. </s> + <s xml:id="echoid-s6602" xml:space="preserve">34. </s> + <s xml:id="echoid-s6603" xml:space="preserve">Venio jam ad varias quæ fingi poſſunt de motu aëris quæſtio-<lb/>nes ſolvendas ſimiles illis, quas de motu fluidorum non elaſticorum in præce-<lb/>dentibus habuimus.</s> + <s xml:id="echoid-s6604" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div242" type="section" level="1" n="186"> +<head xml:id="echoid-head237" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s6605" xml:space="preserve">Sit motus definiendus aëris ex vaſe per foramen exiguum erumpentis <lb/>in ſpatium infinitum ab aëre vacuum.</s> + <s xml:id="echoid-s6606" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div243" type="section" level="1" n="187"> +<head xml:id="echoid-head238" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s6607" xml:space="preserve">Apparet ex natura quæſtionis inſenſibilem eſſe motum localem aëris <lb/>interni quo ſeſe expandit, dum certa ſui quantitas per foramen erumpit: </s> + <s xml:id="echoid-s6608" xml:space="preserve">Igi-<lb/>tur hic ſolus aſcenſus potentialis, quem particula aërea, dum expellitur, acqui-<lb/>rit conſiderandus eſt, atque comparandus cum deſcenſu actuali vel potius cum <lb/>diminutione elaſticitatis, quam aër internus habet. </s> + <s xml:id="echoid-s6609" xml:space="preserve">Ut vero totam rem ad <lb/>methodum noſtram pro fluidis non elaſticis adhibitam reducamus, conſidera-<lb/>bimus cylindrum verticalem communis cum vaſe propoſito amplitudinis atque <lb/>tantæ altitudinis, quanta eſt altitudo aëris homogenei cum aëre interno, is ve-<lb/>ro cylindrus, ſi ſimili aëre plenus cenſeatur, ſed non elaſtico, eadem veloci- +<pb o="225" file="0239" n="239" rhead="SECTIO DECIMA."/> +tate ſuo pondere aërem infimum expellet per foramen, qua aër in vaſe pro-<lb/>poſito ſua elaſticitate ſe ipſum expellit. </s> + <s xml:id="echoid-s6610" xml:space="preserve">In priori autem caſu ejicitur veloci-<lb/>tate quæ debetur ipſi altitudini cylindri, ergo & </s> + <s xml:id="echoid-s6611" xml:space="preserve">in poſteriori. </s> + <s xml:id="echoid-s6612" xml:space="preserve">Notandum <lb/>autem eſt, altitudinem quam pro cylindro finximus, perpetuo eandem eſſe, <lb/>quia aëris elaſticitas & </s> + <s xml:id="echoid-s6613" xml:space="preserve">denſitas in eadem ratione diminuuntur, calorem autem <lb/>non mutari ponimus. </s> + <s xml:id="echoid-s6614" xml:space="preserve">Igitur ſi altitudo aëris homogenei (quæ à calore aëris <lb/>interni pendet) dicatur A, effluet aër conſtanter velocitate √ A. </s> + <s xml:id="echoid-s6615" xml:space="preserve">Nec tamen, <lb/>quod calculus oſtendit, vas ipſum unquam evacuatur, quia aër effluens fit <lb/>continue rarior, quod ut æquatione comprehendamus, ponemus denſitatem ſeu <lb/>quantitatem aëris à fluxus initio = 1; </s> + <s xml:id="echoid-s6616" xml:space="preserve">denſitatem ſeu quantitatem aëris poſt de-<lb/>finitum tempus reſidui = x, tempusque ipſum = t, erit, quia velocitas <lb/>conſtans eſt, - d x = a x d t, ubi per a intelligitur quantitas conſtans defi-<lb/>nienda ex magnitudine vaſis, amplitudine foraminis & </s> + <s xml:id="echoid-s6617" xml:space="preserve">altitudine A: </s> + <s xml:id="echoid-s6618" xml:space="preserve">hinc <lb/>{- dx/x} = adt & </s> + <s xml:id="echoid-s6619" xml:space="preserve">log. </s> + <s xml:id="echoid-s6620" xml:space="preserve">{1/x} = at. </s> + <s xml:id="echoid-s6621" xml:space="preserve">reperitur autem valor coëfficientis a hoc modo. <lb/></s> + <s xml:id="echoid-s6622" xml:space="preserve">Quia poſitum à nobis fuit - d x = a x d t; </s> + <s xml:id="echoid-s6623" xml:space="preserve">erit ab initio effluxus - dx = a d t. </s> + <s xml:id="echoid-s6624" xml:space="preserve"><lb/>Jam mutetur elementum primum (- d x) in cylindrum foramini ceu baſi ſu-<lb/>perinſtructum; </s> + <s xml:id="echoid-s6625" xml:space="preserve">erit autem altitudo iſtius cylindruli = - L d x, ſi L ſit altitu-<lb/>do cylindri ſuper eodem foramine extructi & </s> + <s xml:id="echoid-s6626" xml:space="preserve">communem cum vaſe propo-<lb/>ſito capacitatem habentis: </s> + <s xml:id="echoid-s6627" xml:space="preserve">hæc porro longitudo - L d x illa eſt, quæ tem-<lb/>puſculo d t percurritur, & </s> + <s xml:id="echoid-s6628" xml:space="preserve">quia poni ſolet tempuſculum æquale ſpatio percur-<lb/>ſo diviſo per velocitatem, erit hic d t = {- L d x/√ A}; </s> + <s xml:id="echoid-s6629" xml:space="preserve">ſubſtituatur iſte valor in <lb/>æquatione - d x = a d t & </s> + <s xml:id="echoid-s6630" xml:space="preserve">habebitur - d x = {- a L d x/√A}, ſive a = {√A/L}. </s> + <s xml:id="echoid-s6631" xml:space="preserve">Eſt <lb/>proinde æquatio finalis hæc: </s> + <s xml:id="echoid-s6632" xml:space="preserve"><lb/>log. </s> + <s xml:id="echoid-s6633" xml:space="preserve">{1/x} = {t√A/L}.</s> + <s xml:id="echoid-s6634" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6635" xml:space="preserve">Si tempus exprimere lubeat per certum minutorum ſecundorum nu-<lb/>merum, quem vocabimus n, & </s> + <s xml:id="echoid-s6636" xml:space="preserve">intelligatur per s ſpatium quod mobile ab-<lb/>ſolvit cadendo libere à quiete intra unum minutum ſecundum, erit ponen-<lb/>dum t = 2n√s, ſicque fiet <lb/>log. </s> + <s xml:id="echoid-s6637" xml:space="preserve">{1/x} = {2n√As/L}.</s> + <s xml:id="echoid-s6638" xml:space="preserve"/> +</p> +<pb o="226" file="0240" n="240" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div244" type="section" level="1" n="188"> +<head xml:id="echoid-head239" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s6639" xml:space="preserve">§. </s> + <s xml:id="echoid-s6640" xml:space="preserve">35. </s> + <s xml:id="echoid-s6641" xml:space="preserve">Quæritur motus aëris denſioris in aërem externum rariorem <lb/>infinitum ex vaſe per foramen valde parvum erumpentis, poſito in utroque <lb/>aëre eodem caloris gradu.</s> + <s xml:id="echoid-s6642" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div245" type="section" level="1" n="189"> +<head xml:id="echoid-head240" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s6643" xml:space="preserve">Sit denſitas aëris interni initialis = D; </s> + <s xml:id="echoid-s6644" xml:space="preserve">denfitas aëris externi = δ: <lb/></s> + <s xml:id="echoid-s6645" xml:space="preserve">denſitas aëris interni poſt datum tempus t reſidui = x, altitudo aëris homo-<lb/>genei, (ſive ratione aëris interni ſive externi, nec enim diverſa eſſe poteſt, <lb/>ſi uterque aër eodem calore præditus ſit, ſicque denſitates & </s> + <s xml:id="echoid-s6646" xml:space="preserve">elaſticitates in <lb/>pari ratione decreſcant) = A. </s> + <s xml:id="echoid-s6647" xml:space="preserve">Quæratur ubique altitudo aëris homogenei, <lb/>qui habeat eandem preſſionem ſeu elaterem cum aëre externo & </s> + <s xml:id="echoid-s6648" xml:space="preserve">cujus den-<lb/>ſitas eadem ſit cum aëre interno: </s> + <s xml:id="echoid-s6649" xml:space="preserve">hæc altitudo ab initio erit {δA/D}, & </s> + <s xml:id="echoid-s6650" xml:space="preserve">poſt <lb/>tempus t erit {δA/x}. </s> + <s xml:id="echoid-s6651" xml:space="preserve">Patet autem velocitatem aëris erumpentis talem ubique <lb/>fore, quæ reſpondeat differentiæ definitarum altitudinum A & </s> + <s xml:id="echoid-s6652" xml:space="preserve">{δA/x}; </s> + <s xml:id="echoid-s6653" xml:space="preserve">eſt itaque <lb/>poſt tempus t velocitas aëris erumpentis = √(A - {δA/x}).</s> + <s xml:id="echoid-s6654" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6655" xml:space="preserve">Sunt porro decrementa denſitatum (- d x) proportionalia quantitati-<lb/>bus aëris erumpentis, quæ rationem habent compoſitam ex velocitate <lb/>(√(A - {δA/x})) ex denſitate (x) & </s> + <s xml:id="echoid-s6656" xml:space="preserve">ex tempuſculo (d t): </s> + <s xml:id="echoid-s6657" xml:space="preserve">ſic igitur eſt - d x <lb/>= a (√(A - {δA/x})) x d t, ubi a eſt numerus conſtans qui per metho-<lb/>dum præcedentis paragraphi fit = {1/L}, retenta ſignificatione hujus litteræ <lb/>ibidem adhibita; </s> + <s xml:id="echoid-s6658" xml:space="preserve">hocque valore ſubſtituto oritur <lb/>- d x = {dt/L} X √ (Axx - δAx) ſeu {- dx/√ (xx - δx)} = {dt√A/L}: <lb/></s> + <s xml:id="echoid-s6659" xml:space="preserve">Factaque debita integratione fit: </s> + <s xml:id="echoid-s6660" xml:space="preserve"><lb/>log.</s> + <s xml:id="echoid-s6661" xml:space="preserve">{[√x - √(x - δ)] x [√D + √(D - δ)]/[√x + √(x - δ)] x [√D - √(D - δ)]} = {t√A/L}, aut poſito rurſus, ut in <lb/>præcedente paragragho, t = 2 n √ s, erit +<pb o="227" file="0241" n="241" rhead="SECTIO DECIMA."/> +log.</s> + <s xml:id="echoid-s6662" xml:space="preserve">{[√x - √(x - δ)] x [√D + √(D - δ)]/[√x + √(x - δ)] X [√D - √(D - δ)]} = {2n√As/L}</s> +</p> +</div> +<div xml:id="echoid-div246" type="section" level="1" n="190"> +<head xml:id="echoid-head241" xml:space="preserve">Corollarium 1.</head> +<p> + <s xml:id="echoid-s6663" xml:space="preserve">§. </s> + <s xml:id="echoid-s6664" xml:space="preserve">36. </s> + <s xml:id="echoid-s6665" xml:space="preserve">Omnis effluxus fit tempore finito quâ in re iſta quæſtio ab alte-<lb/>ra præcedente differt: </s> + <s xml:id="echoid-s6666" xml:space="preserve">Ceſſat autem aër effluere, cum eſt x = δ, & </s> + <s xml:id="echoid-s6667" xml:space="preserve">tunc fit <lb/>n = {L/2√As} X log. </s> + <s xml:id="echoid-s6668" xml:space="preserve">{√D + √(D - δ)/√D - √(D - δ)}.</s> + <s xml:id="echoid-s6669" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6670" xml:space="preserve">Sit v. </s> + <s xml:id="echoid-s6671" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6672" xml:space="preserve">A = 26000 ped. </s> + <s xml:id="echoid-s6673" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s6674" xml:space="preserve">contineat vas propoſitum unum pedem <lb/>cubicum, foramen autèm habeat amplitudinem unius lineæ quadratæ, erit <lb/>L = 20736; </s> + <s xml:id="echoid-s6675" xml:space="preserve">ponatur inſuper aërem intèrnum ab initio duplo fuiſſe den-<lb/>ſiorem externo; </s> + <s xml:id="echoid-s6676" xml:space="preserve">eſt autem ut conſtat s = 15 {1/12} ped. </s> + <s xml:id="echoid-s6677" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s6678" xml:space="preserve">Fiet igitur <lb/>n = {20736√3/√(181.</s> + <s xml:id="echoid-s6679" xml:space="preserve">26000)} log. </s> + <s xml:id="echoid-s6680" xml:space="preserve">{√2 + 1/√2 - 1} = 29, 2, <lb/>quod ſignificat aërem utrumque ad æquilibrium compoſitum iri tempore <lb/>paullo majori quam viginti novem minutorum ſecundorum, poſt idque <lb/>omnem effluxum ceſſaturum. </s> + <s xml:id="echoid-s6681" xml:space="preserve">Fieri autèm poteſt à contractione, quam flui-<lb/>da præ foramine patiuntur (vid. </s> + <s xml:id="echoid-s6682" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s6683" xml:space="preserve">IV.) </s> + <s xml:id="echoid-s6684" xml:space="preserve">& </s> + <s xml:id="echoid-s6685" xml:space="preserve">ad quam nullam fecimus in com-<lb/>puto attentionem, ut tempus iſtud augeatur fere in in ratione ut 1 ad √ 2.</s> + <s xml:id="echoid-s6686" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div247" type="section" level="1" n="191"> +<head xml:id="echoid-head242" xml:space="preserve">Corollarium 2.</head> +<p> + <s xml:id="echoid-s6687" xml:space="preserve">§. </s> + <s xml:id="echoid-s6688" xml:space="preserve">37. </s> + <s xml:id="echoid-s6689" xml:space="preserve">Si fingatur aërem non immediate per foramen effluere, ſed <lb/>per longum tubum, non mutabitur propterea velocitas, ſi modo totius tubi <lb/>capacitas ſit veluti infinite parva ratione capacitatis, quæ in vaſe ipſo eſt; <lb/></s> + <s xml:id="echoid-s6690" xml:space="preserve">Videtur autem denſitatem aëris, quamdiu in tubo eſt, eandem eſſe cum denſitate <lb/>aëris vaſi incluſi, nectamen, quod demonſtrabo inferius, elaſticitas aëris in tubo <lb/>major eſt elaſticitate aëris externi, qui tubum circumdat. </s> + <s xml:id="echoid-s6691" xml:space="preserve">Conſequens inde <lb/>eſt, ventum aërem eſſe denſiorem aëre quieſcente, ſed non magis elaſticum: </s> + <s xml:id="echoid-s6692" xml:space="preserve"><lb/>attamen denſitatum differentia parvula quoque erit; </s> + <s xml:id="echoid-s6693" xml:space="preserve">ventus enim, qui vel <lb/>30. </s> + <s xml:id="echoid-s6694" xml:space="preserve">pedes ſingulis minutis ſecundis conficit, aërem vicinum, æque calidum <lb/>& </s> + <s xml:id="echoid-s6695" xml:space="preserve">quietum, vix una milleſima ſeptingentiſſima parte denſitate ſuperabit.</s> + <s xml:id="echoid-s6696" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div248" type="section" level="1" n="192"> +<head xml:id="echoid-head243" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s6697" xml:space="preserve">§. </s> + <s xml:id="echoid-s6698" xml:space="preserve">38. </s> + <s xml:id="echoid-s6699" xml:space="preserve">Definire influxum aëris per foramen valde parvum in vas aëre <lb/>rariore plenum, poſito rurſus utrobique eodem caloris gradu.</s> + <s xml:id="echoid-s6700" xml:space="preserve"/> +</p> +<pb o="228" file="0242" n="242" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div249" type="section" level="1" n="193"> +<head xml:id="echoid-head244" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s6701" xml:space="preserve">Fuerit vas ab initio omnino vacuum, & </s> + <s xml:id="echoid-s6702" xml:space="preserve">poſt tempus t ponatur denſitas <lb/>aëris interni = x; </s> + <s xml:id="echoid-s6703" xml:space="preserve">ſic reperietur iiſdem fere veſtigiis inſiſtendo, quibus in tri-<lb/>geſimo quinto paragrapho uſi ſumus retentisque iisdem denominationibus <lb/>{dx/√(δ - x)} = {dt√AD/L} ſive t = 2n√s = {2L/√A} - {2L√(D - x)/√AD}.</s> + <s xml:id="echoid-s6704" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6705" xml:space="preserve">Numerus igitur minutorum ſecundorum, quo totum vas impletur, <lb/>donec inter utrumque aërem æquilibrium ſit exprimitur per {L/√As}: </s> + <s xml:id="echoid-s6706" xml:space="preserve">& </s> + <s xml:id="echoid-s6707" xml:space="preserve">eſt tem-<lb/>pus repletionis duplum illius quo repleretur ſi velocitate initiali conſtanter in-<lb/>flueretaër. </s> + <s xml:id="echoid-s6708" xml:space="preserve">In caſu quo capacitas vaſis pedem cubicum continet & </s> + <s xml:id="echoid-s6709" xml:space="preserve">foramen lineam <lb/>quadratam æquat, fit repletio tempore propemodum triginta trium minutorum <lb/>ſecundorum, niſi contractione venæ aëreæ influentis repletio retardetur.</s> + <s xml:id="echoid-s6710" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6711" xml:space="preserve">§. </s> + <s xml:id="echoid-s6712" xml:space="preserve">39. </s> + <s xml:id="echoid-s6713" xml:space="preserve">Expoſuimus varias fluidorum elaſticorum ſive motorum ſive <lb/>quieſcentium proprietates: </s> + <s xml:id="echoid-s6714" xml:space="preserve">Unum ſupereſt non omittendum, quo fluida ela-<lb/>ſtica differunt à non - elaſticis, hoc ſcilicet, quod fluido elaſtico vel quieſ-<lb/>centi vis viva inſita ſit, non quod inſtar aliorum corporum motorum ſe ad cer-<lb/>tam altitudinem elevare poſſit, neque enim motum localem in illo hic conſi-<lb/>deramus, ſed quod elatere ſuo talem aſcenſum in aliis corporibus gravibus ge-<lb/>nerare poſſit. </s> + <s xml:id="echoid-s6715" xml:space="preserve">Licebit autem, quod ſpero, in ſequentibus uti vocabulo vis vi-<lb/>væ corpori elaſtico compreſſo inſitæ, quando nihil aliud eo intelligitur quam aſcen-<lb/>ſus potentialis, quem corpus elaſticum aliis corporibus communicare poteſt <lb/>priuſquam totam ſuam vim elaſticam perdiderit.</s> + <s xml:id="echoid-s6716" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6717" xml:space="preserve">Meretur hic in anteceſſum notari, quod ſicut deſcenſus corporis dati <lb/>per datam altitudinem, utcunque fiat, eandem conſtanter vim vivam in cor-<lb/>pore producit, ita quoque elaſtrum ſive fluidum elaſticum poſtquam à dato <lb/>tenſionis ſeu condenſationis gradu ad datum alium gradum fuit reductum ut-<lb/>cunque, id ſemper eandem vim vivam in ſe recipiat rurſuſque contraria muta-<lb/>tione alii corpori communicare poſſit.</s> + <s xml:id="echoid-s6718" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6719" xml:space="preserve">De hujuſmodi viribus vivis fluido elaſtico compreſſo inſitis earundem-<lb/>que menſuris paucis nunc agam: </s> + <s xml:id="echoid-s6720" xml:space="preserve">dignum attentione argumentum eſt, quod <lb/>eo reducantur menſuræ virium prò machinis aëre, aut igne aut aliis hujuſmo- +<pb o="229" file="0243" n="243" rhead="SECTIO DECIMA."/> +di viribus motricibus, quarum fortaſſe plures novæ non ſine inſigni mechani-<lb/>cæ practicæ incremento & </s> + <s xml:id="echoid-s6721" xml:space="preserve">perfectione excogitari poterunt, movendis.</s> + <s xml:id="echoid-s6722" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6723" xml:space="preserve">§. </s> + <s xml:id="echoid-s6724" xml:space="preserve">40. </s> + <s xml:id="echoid-s6725" xml:space="preserve">Ut incipiamus ab aëre in vacuo, conſiderabimus cylindrum ver-<lb/>ticaliter poſitum A B C D (Fig. </s> + <s xml:id="echoid-s6726" xml:space="preserve">62.) </s> + <s xml:id="echoid-s6727" xml:space="preserve">cum ſuſtentaculo E F, quod omni pon-<lb/> +<anchor type="note" xlink:label="note-0243-01a" xlink:href="note-0243-01"/> +dere deſtitutum liberrime ſurſum dèorſumque moveri poſſit. </s> + <s xml:id="echoid-s6728" xml:space="preserve">Sit ſpatio E B C F <lb/>aër incluſus, totus autem cylindrus in vacuo poſitus fingatur: </s> + <s xml:id="echoid-s6729" xml:space="preserve">Sit preſſio aëris <lb/>E B C F tanta qua ſuſtinere poſſit pondus p, quod æquale erit preſſioni colum-<lb/>næ atmoſphæræ, ſi aër iſte ſit naturalis. </s> + <s xml:id="echoid-s6730" xml:space="preserve">Superveniat jam aliud pondus P: </s> + <s xml:id="echoid-s6731" xml:space="preserve">ita <lb/>fiet ut operculum deſcendat in G H motibuſque reciprocis ad puncta H & </s> + <s xml:id="echoid-s6732" xml:space="preserve">F <lb/>agitetur. </s> + <s xml:id="echoid-s6733" xml:space="preserve">Ut motum definiamus, utemur hypotheſi ordinaria, quod preſſio-<lb/>nes aëris cæteris paribus ſint denſitatibus proportionales.</s> + <s xml:id="echoid-s6734" xml:space="preserve"/> +</p> +<div xml:id="echoid-div249" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0243-01" xlink:href="note-0243-01a" xml:space="preserve">Fig. 62.</note> +</div> +<p> + <s xml:id="echoid-s6735" xml:space="preserve">Fuerit itaque F C = a, F H = x; </s> + <s xml:id="echoid-s6736" xml:space="preserve">velocitas ſuſtentaculi in ſitu G H = v, <lb/>erit preſſio, qua ſuſtentaculum G H ad ulteriorem deſcenſum urgetur = P + p <lb/>- {a/a - x} p, huicque preſſioni æqualis cenſenda eſt vis, quæ pondus ſuſtenta-<lb/>culo incumbens animat; </s> + <s xml:id="echoid-s6737" xml:space="preserve">igitur ſi hanc vim dividas per maſſam habebis vim <lb/>accelerantem, quæ multiplicata per tempuſculum ſeu per {dx/v}, dabit incre-<lb/>mentum velocitatis dv, eſt itaque <lb/>dv = (P + p - {ap/a - x}) X {dx/v}: </s> + <s xml:id="echoid-s6738" xml:space="preserve">(P + p), vel <lb/>{1/2} (P + p) vv = (P + p) x - ap log. </s> + <s xml:id="echoid-s6739" xml:space="preserve">{a/a - x}.</s> + <s xml:id="echoid-s6740" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6741" xml:space="preserve">Sed ex deſcenſu ponderis (P + p) per altitudinem x generatur vis viva <lb/>potentialis (P + p) x, & </s> + <s xml:id="echoid-s6742" xml:space="preserve">cum ſuſtentaculum eſt in ſitu G H, ineſt corpori (P + p) <lb/>vis viva actualis {1/2} (P + p) vv, id eſt, (P + p) x - ap log. </s> + <s xml:id="echoid-s6743" xml:space="preserve">{a/a - x}, quæ à prio-<lb/>ri deficit quantitate ap log. </s> + <s xml:id="echoid-s6744" xml:space="preserve">{a/a - x}, hæcque in compreſſionem aëris tranſiit.</s> + <s xml:id="echoid-s6745" xml:space="preserve"/> +</p> +<p style="it"> + <s xml:id="echoid-s6746" xml:space="preserve">Dico itaque non poſſe aërem occupantem ſpatium a condenſari in ſpa-<lb/>tium a - x, quin vis viva impendatur, quæ generatur ex deſcenſu ponderis <lb/>p per altitudinem a log. </s> + <s xml:id="echoid-s6747" xml:space="preserve">{a/a - x} quocunque modo illa compreſsio facta fuerit; </s> + <s xml:id="echoid-s6748" xml:space="preserve">po-<lb/>t<unsure/>eſt autem modis fieri infinitis. </s> + <s xml:id="echoid-s6749" xml:space="preserve">Iſtam vero regulam uno nunc alterove exem-<lb/>plo illuſtrabo.</s> + <s xml:id="echoid-s6750" xml:space="preserve"/> +</p> +<pb o="230" file="0244" n="244" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s6751" xml:space="preserve">Sit baſis cylindri unius pedis quadrati, altitudo initialis F C duorum <lb/>pedum: </s> + <s xml:id="echoid-s6752" xml:space="preserve">contineaturque in ſpatio B F aër qualis in ſuperficie terræ medius eſſe <lb/>ſolet, qui ferre poſſit ſuperficie E F 2240 libras: </s> + <s xml:id="echoid-s6753" xml:space="preserve">ponatur x = 1, ut ſic ha-<lb/>beatur vis viva, qua duo pedes cubici aëris naturalis in ſpatium unius pedis <lb/>cubici coërceri poſſunt in vacuo: </s> + <s xml:id="echoid-s6754" xml:space="preserve">eritque iſta vis viva = 2 X 2240 X log. </s> + <s xml:id="echoid-s6755" xml:space="preserve">2 <lb/>= 3105, id eſt, talis quæ generatur lipſu libero corporis 3105 librarum <lb/>per altitudinem unius pedis. </s> + <s xml:id="echoid-s6756" xml:space="preserve">Ergo & </s> + <s xml:id="echoid-s6757" xml:space="preserve">viciſſim, ſi habeatur pes cubicus aëris <lb/>naturali duplo denſioris, poterit illius ope pondus elevari 3105 librarum ad <lb/>altitudinem unius pedis in vacuo, dum aëris naturalis denſitatèm aſſumit.</s> + <s xml:id="echoid-s6758" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6759" xml:space="preserve">Sit porro ſub iisdem reliquis circumſtantiis idem aër in ſpatium duplum, <lb/>quam antea fuit, expanſum, occupans nunc in cylindro altitudinem quatuor <lb/>pedum, iſque rurſus condenſetur in ſpatium unius pedis cubici, requiretur <lb/>ad hanc compreſſionem vis viva, quæ exprimitur per 4 X 1120 log. </s> + <s xml:id="echoid-s6760" xml:space="preserve">4, quæ <lb/>priore duplo major eſt. </s> + <s xml:id="echoid-s6761" xml:space="preserve">Igitur in vacuo ſi habeatur pes cubicus aëris naturali <lb/>duplo denſioris, poterit illius ope pondus elevari 6210 librarum ad altitud. <lb/></s> + <s xml:id="echoid-s6762" xml:space="preserve">unius pedis, dum aëris naturalis dimidiam denſitatem aſſumit, aut pondus <lb/>9315 lib. </s> + <s xml:id="echoid-s6763" xml:space="preserve">dum aëre naturali fit quadruplo rarior.</s> + <s xml:id="echoid-s6764" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6765" xml:space="preserve">Conſequens inde eſt, ſi aër in ſpatium expandere ſe poſſit infinitum & </s> + <s xml:id="echoid-s6766" xml:space="preserve"><lb/>ubique elaſticitatem ſervet denſitati proportionalem, quantitati aëris finitæ <lb/>vim vivam ineſſe infinitam.</s> + <s xml:id="echoid-s6767" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6768" xml:space="preserve">§. </s> + <s xml:id="echoid-s6769" xml:space="preserve">41. </s> + <s xml:id="echoid-s6770" xml:space="preserve">Hæc autem pertinent ad æſtimationem vis vivæ, quæ aëri in <lb/>vacuo poſito inſita ſit: </s> + <s xml:id="echoid-s6771" xml:space="preserve">paullo alius fit computus pro aëre denſiore, qui in at-<lb/>moſphæra poſitus eſt: </s> + <s xml:id="echoid-s6772" xml:space="preserve">hic enim maximus expanſionis gradus non ultra æquili-<lb/>brium cum aëre atmoſphæræ extendi poteſt: </s> + <s xml:id="echoid-s6773" xml:space="preserve">facile hinc eſt in anteceſſum præ-<lb/>videre, ſi v. </s> + <s xml:id="echoid-s6774" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6775" xml:space="preserve">habeatur pes cubicus aëris naturali duplo denſioris, vim vi-<lb/>vam quæ in atmoſphæra ab hoc aëre compreſſo elici poſſit, minime eſſe in-<lb/>finitam. </s> + <s xml:id="echoid-s6776" xml:space="preserve">Poterunt autem hujuſmodi vires vivæ hunc in modum determinari.</s> + <s xml:id="echoid-s6777" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6778" xml:space="preserve">§. </s> + <s xml:id="echoid-s6779" xml:space="preserve">42. </s> + <s xml:id="echoid-s6780" xml:space="preserve">Sit aër E B C F naturalis & </s> + <s xml:id="echoid-s6781" xml:space="preserve">in æquilibrio cum aëre externo; </s> + <s xml:id="echoid-s6782" xml:space="preserve">in-<lb/>telligatur autem per p preſſio atmoſphæræ, in ſuſtentaculum E F, quæ quidem <lb/>cum preſſione aëris interni nondum condenſati in æquilibrio eſt. </s> + <s xml:id="echoid-s6783" xml:space="preserve">Imponatur <lb/>eidem ſuſtentaculo pondus P; </s> + <s xml:id="echoid-s6784" xml:space="preserve">fuerit jam aër condenſatus in ſpatium G B C H; <lb/></s> + <s xml:id="echoid-s6785" xml:space="preserve">habeatque ſuſtentaculum pondere P oneratum in ſitu G H velocitatem v, erit <lb/>retentis reliquis denominationibus +<pb o="231" file="0245" n="245" rhead="SECTIO DECIMA."/> +dv = (P + p - {ap/a - x}) X {dx/v}: </s> + <s xml:id="echoid-s6786" xml:space="preserve">P, vel <lb/>Pvdv = (P - {xp/a - x}) dx, quæ integrata dat <lb/>{1/2} P vv = Px + px - ap log. </s> + <s xml:id="echoid-s6787" xml:space="preserve">{a/a - x}.</s> + <s xml:id="echoid-s6788" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6789" xml:space="preserve">Jam vero deſcenſu ponderis P per altitudinem x genita fuit vis viva P x, <lb/>de qua eidem ponderi ceu velocitate v moto ineſt pars {1/2} P v v ſeu P x + p x -<lb/>ap log. </s> + <s xml:id="echoid-s6790" xml:space="preserve">{a/a - x}; </s> + <s xml:id="echoid-s6791" xml:space="preserve">pars igitur vis vivæ quæ ad aërem tranſiit, eſt = - p x + <lb/>ap log. </s> + <s xml:id="echoid-s6792" xml:space="preserve">{a/a - x}, quæ minor eſt altera §. </s> + <s xml:id="echoid-s6793" xml:space="preserve">40. </s> + <s xml:id="echoid-s6794" xml:space="preserve">definita.</s> + <s xml:id="echoid-s6795" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6796" xml:space="preserve">Habeatur v. </s> + <s xml:id="echoid-s6797" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6798" xml:space="preserve">pes cubicus aëris naturali duplo denſioris, inveniètur <lb/>vis viva, quam iſte aër amittit, dum aëris naturalis circumfuſi denſitatem aſſu-<lb/>mit, ea quæ lapſu libero corporis 865. </s> + <s xml:id="echoid-s6799" xml:space="preserve">lib. </s> + <s xml:id="echoid-s6800" xml:space="preserve">per altitudinem unius pedis gene-<lb/>ratur.</s> + <s xml:id="echoid-s6801" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6802" xml:space="preserve">Pari ſenſu pes cubicus aëris naturali triplo denſioris vim vivam habere <lb/>intelligitur talem quæ reſpondeat lapſui libero corporis 2898 lib. </s> + <s xml:id="echoid-s6803" xml:space="preserve">per altitud. <lb/></s> + <s xml:id="echoid-s6804" xml:space="preserve">unius pedis, qui numerus nempe prodit cum ponitur p = 2240, ut §. </s> + <s xml:id="echoid-s6805" xml:space="preserve">40; </s> + <s xml:id="echoid-s6806" xml:space="preserve"><lb/>a = 3. </s> + <s xml:id="echoid-s6807" xml:space="preserve">& </s> + <s xml:id="echoid-s6808" xml:space="preserve">x = 2.</s> + <s xml:id="echoid-s6809" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6810" xml:space="preserve">§. </s> + <s xml:id="echoid-s6811" xml:space="preserve">43. </s> + <s xml:id="echoid-s6812" xml:space="preserve">Perſpicuum eſt ex hoc conſenſu inter conſervationem virium vi-<lb/>varum aëri compreſſo & </s> + <s xml:id="echoid-s6813" xml:space="preserve">corpori à data altitudine delapſo inſitarum, nullam <lb/>eſſe ad uſum machinarum perficiendum prærogativam ſperandam ex principio <lb/>aëris comprimendi, & </s> + <s xml:id="echoid-s6814" xml:space="preserve">ubique valere regulas in præcedente ſectione exhibi-<lb/>tas. </s> + <s xml:id="echoid-s6815" xml:space="preserve">Quia vero multis modis fit, ut aër non vi ſed natura ſit compreſſus aut <lb/>elaterem naturali majorem acquirat, ſpes certe eſt, poſſe hujuſmodi rebus na-<lb/>turalibus magna ad machinas movendas compendia excogitari, prouti D. <lb/></s> + <s xml:id="echoid-s6816" xml:space="preserve">Amontons jamjam docuit modum movendarum machinarum vi ignis. </s> + <s xml:id="echoid-s6817" xml:space="preserve">Mihi <lb/>perſuadeo ſi omnis vis viva, quæ in carbonum pede cubico latet, ex eodem-<lb/>que combuſtione elicitur, utiliter ad machinam movendam impendatur, quod <lb/>plus inde profici poſſit, quam labore diurno octo aut decem hominum. </s> + <s xml:id="echoid-s6818" xml:space="preserve">Etenim <lb/>carbones dum comburuntur aëris elafticitatem nonſolum inſigniter augent, ſed <lb/>& </s> + <s xml:id="echoid-s6819" xml:space="preserve">ingentem aëris novi quantitatem generant.</s> + <s xml:id="echoid-s6820" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6821" xml:space="preserve">Ita Haleſius in veget. </s> + <s xml:id="echoid-s6822" xml:space="preserve">ſtatiks deprehendit ex ſemipollice carbonis 180.</s> + <s xml:id="echoid-s6823" xml:space="preserve"> +<pb o="232" file="0246" n="246" rhead="HYDRODYNAMICÆ"/> +pollices aëris ejuſdem cum aëre naturali elaſticitatis fuiſſe generatos; </s> + <s xml:id="echoid-s6824" xml:space="preserve">ergo pes <lb/>cubicus carbonum aërem dabit ad 360. </s> + <s xml:id="echoid-s6825" xml:space="preserve">ped. </s> + <s xml:id="echoid-s6826" xml:space="preserve">cub. </s> + <s xml:id="echoid-s6827" xml:space="preserve">Sed ſi §. </s> + <s xml:id="echoid-s6828" xml:space="preserve">41. </s> + <s xml:id="echoid-s6829" xml:space="preserve">quæratur vis vi-<lb/>va quæ generari poſſit à pede cubico aëris naturali 360. </s> + <s xml:id="echoid-s6830" xml:space="preserve">vicibus denſioris, in-<lb/>venietur illam convenire cum pondere 3938000. </s> + <s xml:id="echoid-s6831" xml:space="preserve">librarum ab altitudine unius <lb/>pedis delapſo: </s> + <s xml:id="echoid-s6832" xml:space="preserve">atque ſi præterea aëris illius elaſticitas à calore carbonum in-<lb/>cenſorum quadruplo fieri major ponatur, conveniet iſta vis viva cum pondere <lb/>15752000. </s> + <s xml:id="echoid-s6833" xml:space="preserve">lib. </s> + <s xml:id="echoid-s6834" xml:space="preserve">ab eadem altitudine delapſo. </s> + <s xml:id="echoid-s6835" xml:space="preserve">Difficile autem eſt machinam ad <lb/>hunc finem aptam excogitare. </s> + <s xml:id="echoid-s6836" xml:space="preserve">Multæ præterea aliæ ſuntres naturales, quæ non-<lb/>ſolum aërem fovent compreſſum, ſed & </s> + <s xml:id="echoid-s6837" xml:space="preserve">aërem circumfuſum calefaciendo eun-<lb/>dem magis elaſticum reddere valent: </s> + <s xml:id="echoid-s6838" xml:space="preserve">tales ſunt calx viva cum aqua dulci miſta, <lb/>omniaque fermentantia, aquæ in vapores vi ignis redactæ incredibilis vis ineſt; <lb/></s> + <s xml:id="echoid-s6839" xml:space="preserve">machina ad hoc eſt Londini ingenioſiſſima quæ hoc principio motus aquas toti <lb/>urbi erogat eamque deſcripſit Cl. </s> + <s xml:id="echoid-s6840" xml:space="preserve">Weidlerus. </s> + <s xml:id="echoid-s6841" xml:space="preserve">Præſertim vero conſiderari mere-<lb/>tur ſtupendus, qui à pulvere pyrio expectari poſſit effectus: </s> + <s xml:id="echoid-s6842" xml:space="preserve">Calculo enim quo-<lb/>rundam ſumtorum experimentorum ſubducto, quem infra adjiciam, edoctus <lb/>fui elaſticitatem pulveris pyrii accenſi plus decies millies ſuperare elaſticita-<lb/>tem aëris naturalis, imo omnibus bene perpenſis probabile fit, elaſticitatem ejus <lb/>eſſe incredibiliter majorem: </s> + <s xml:id="echoid-s6843" xml:space="preserve">ponamus autem auræ pulveris pyrii accenſi ex-<lb/>panſæ elaſticitatem decreſcere in ſimili ratione cum denſitate: </s> + <s xml:id="echoid-s6844" xml:space="preserve">hiſce poſitis in-<lb/>venietur vis viva pedi cubico pulveris pyrii inſita, ſi in §. </s> + <s xml:id="echoid-s6845" xml:space="preserve">42. </s> + <s xml:id="echoid-s6846" xml:space="preserve">ponatur a = 10000; </s> + <s xml:id="echoid-s6847" xml:space="preserve"><lb/>x = 9999, p = 2240 & </s> + <s xml:id="echoid-s6848" xml:space="preserve">ſumatur - p x + a p log. </s> + <s xml:id="echoid-s6849" xml:space="preserve">{a/a - x}, quæ quantitas ſic fit <lb/>æqualis 183913864. </s> + <s xml:id="echoid-s6850" xml:space="preserve">Igitur machina datur in theoria, quæ ope unius pedis cu-<lb/>bici pulveris pyrii poſſit elevare 183913864 libras ad altitudinem unius pedis, <lb/>quem laborem vel centum homines robuſtiſſimi intra unius diei ſpatium perfi-<lb/>cere poſſe non crediderim, quâcunque machina utantur. </s> + <s xml:id="echoid-s6851" xml:space="preserve">Probabile autem eſt, <lb/>ut dixi, effectum pulveris pyrii longe majorem eſſe; </s> + <s xml:id="echoid-s6852" xml:space="preserve">certe autem non mi-<lb/>nor eſt, calculus enim innititur altitudini, ad quam globus ferreus ex tor-<lb/>mento bellico ejectus in vacuo aſcendere poſſit, in quo experimentorum ge-<lb/>nere maxima pulveris pyrii pars perit.</s> + <s xml:id="echoid-s6853" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6854" xml:space="preserve">Iſta vero magis percipientur, ſi notetur eundem calculum (quem an-<lb/>tea fecimus pro effectu, qui ex aëre condenſato ſeſe reſtituente oritur, de-<lb/>monſtrando) procedere etiam pro aëre qui naturali circumfuſo non quidem <lb/>magis denſus ſed tamen ab aucto calore magis elaſticus fit: </s> + <s xml:id="echoid-s6855" xml:space="preserve">ita v. </s> + <s xml:id="echoid-s6856" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6857" xml:space="preserve">quoties <lb/>pes cubicus aëris ordinarii augmento caloris duplum elaterem acquiſivit, +<pb o="233" file="0247" n="247" rhead="SECTIO DECIMA."/> +poteſt ejus ope pondus 865 librarum ad altitudinem unius pedis elevari, ſi <lb/>modo machina adhibeatur perfectiſſima.</s> + <s xml:id="echoid-s6858" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6859" xml:space="preserve">Ab auctis autem aëris tum denſitate tum calore pendent omnium re-<lb/>rum hic expoſitarum effectus.</s> + <s xml:id="echoid-s6860" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6861" xml:space="preserve">§. </s> + <s xml:id="echoid-s6862" xml:space="preserve">44. </s> + <s xml:id="echoid-s6863" xml:space="preserve">Interim non ſolum ab aëre condenſato calefactove vis viva pro <lb/>machinis movendis impendenda obtineri poteſt, ſed & </s> + <s xml:id="echoid-s6864" xml:space="preserve">ab aëre rariore aut <lb/>frigidiore. </s> + <s xml:id="echoid-s6865" xml:space="preserve">Ubicunque enim æquilibrium ſublatum eſt, vis viva adeſt, quæ <lb/>impendi poteſt, ſi debita machina excogitetur, ad onera elevanda machina-<lb/>mentaque circumagenda. </s> + <s xml:id="echoid-s6866" xml:space="preserve">Methodus autem determinans vim vivam, quæ ab <lb/>aëre datæ denſitatis datique caloris ſpatium datum occupante elici poteſt, <lb/>mutatis mutandis eadem eſt cum illa quam §. </s> + <s xml:id="echoid-s6867" xml:space="preserve">42. </s> + <s xml:id="echoid-s6868" xml:space="preserve">adhibuimus.</s> + <s xml:id="echoid-s6869" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6870" xml:space="preserve">45. </s> + <s xml:id="echoid-s6871" xml:space="preserve">Fuerit nempe rurſus cylindrus verticalis A B C D (Fig. </s> + <s xml:id="echoid-s6872" xml:space="preserve">63.) </s> + <s xml:id="echoid-s6873" xml:space="preserve">cum dia-<lb/> +<anchor type="note" xlink:label="note-0247-01a" xlink:href="note-0247-01"/> +phragmate mobili E F: </s> + <s xml:id="echoid-s6874" xml:space="preserve">puta aërem E B C F, ut §. </s> + <s xml:id="echoid-s6875" xml:space="preserve">42. </s> + <s xml:id="echoid-s6876" xml:space="preserve">naturalem & </s> + <s xml:id="echoid-s6877" xml:space="preserve">in æquili-<lb/>brio cum aëre externo: </s> + <s xml:id="echoid-s6878" xml:space="preserve">preſſio autem aëris cujusvis in E F dicatur p<emph style="super">0</emph>: </s> + <s xml:id="echoid-s6879" xml:space="preserve">Finge <lb/>dein pondus P, quod mediante fune trans duas trochleas M & </s> + <s xml:id="echoid-s6880" xml:space="preserve">N ducto cum <lb/>diaphragmate cohæreat, idemque verſus A D trahat, perveneritque ſic dia-<lb/>phragma ex ſitu E F in G H: </s> + <s xml:id="echoid-s6881" xml:space="preserve">Denique ponatur rurſus F C = a, F H = x: <lb/></s> + <s xml:id="echoid-s6882" xml:space="preserve">velocitas diaphragmatis in ſitu G H ſeu ponderis in ſitu P = v; </s> + <s xml:id="echoid-s6883" xml:space="preserve">His poſitis <lb/>ſi conferantur §. </s> + <s xml:id="echoid-s6884" xml:space="preserve">§. </s> + <s xml:id="echoid-s6885" xml:space="preserve">40. </s> + <s xml:id="echoid-s6886" xml:space="preserve">& </s> + <s xml:id="echoid-s6887" xml:space="preserve">42. </s> + <s xml:id="echoid-s6888" xml:space="preserve">patebit fore nunc <lb/>dv = (P + {ap/a + x} - p) X {dx/v}: </s> + <s xml:id="echoid-s6889" xml:space="preserve">P vel <lb/>Pvdv = (P - {px/a + x}) dx, quæ integrata dat <lb/>{1/2}Pvv = Px - px + ap log.</s> + <s xml:id="echoid-s6890" xml:space="preserve">{a + x/a}.</s> + <s xml:id="echoid-s6891" xml:space="preserve"/> +</p> +<div xml:id="echoid-div250" type="float" level="2" n="2"> +<note position="right" xlink:label="note-0247-01" xlink:href="note-0247-01a" xml:space="preserve">Fig. 63.</note> +</div> +<p> + <s xml:id="echoid-s6892" xml:space="preserve">At rurſus deſcenſus ponderis P per altitudinem x producta fuit vis viva <lb/>P x, dum ipſi interim ponderi velocitate v moto ineſt tantum vis viva {1/2} P v v <lb/>ſeu Px - px + ap log.</s> + <s xml:id="echoid-s6893" xml:space="preserve">{a + x/a}, Igitur vis viva, quæ reſidua eſt, nempe p x -<lb/>a p log.</s> + <s xml:id="echoid-s6894" xml:space="preserve">{a + x/a}, ad aërem tranſiit rurſusque reſtitutione æquilibrii inter aërem <lb/>internum & </s> + <s xml:id="echoid-s6895" xml:space="preserve">externum, illa vis viva ad alia corpora pro lubitu transfundi po-<lb/>terit: </s> + <s xml:id="echoid-s6896" xml:space="preserve">Igitur ſi habeas ſpatium G B C H aëre plenum cujus denſitas ſit ad <lb/>denſitatem aëris externi ut C F ad C H, in poteſtate erit vis viva p x -<lb/>a p log.</s> + <s xml:id="echoid-s6897" xml:space="preserve">{a + x/a}.</s> + <s xml:id="echoid-s6898" xml:space="preserve"/> +</p> +<pb o="234" file="0248" n="248" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s6899" xml:space="preserve">An vero iſta vis viva aëri inhæreat proprie externo an interno, logoma@ <lb/>chia eſt; </s> + <s xml:id="echoid-s6900" xml:space="preserve">ſufficit quod à ſublato æquilibrio inter utrumque aërem talis vis viva <lb/>obtineri poteſt, dum reſtitutio permittitur.</s> + <s xml:id="echoid-s6901" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6902" xml:space="preserve">Habeatur v. </s> + <s xml:id="echoid-s6903" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6904" xml:space="preserve">pes cubicus aëris naturali duplo rarioris, cui hypotheſi <lb/>quadrabunt poſitiones p = 2240 lib. </s> + <s xml:id="echoid-s6905" xml:space="preserve">a = {1/2} ped. </s> + <s xml:id="echoid-s6906" xml:space="preserve">& </s> + <s xml:id="echoid-s6907" xml:space="preserve">x = {1/2} ped. </s> + <s xml:id="echoid-s6908" xml:space="preserve">& </s> + <s xml:id="echoid-s6909" xml:space="preserve">erit vis viva, de <lb/>qua ſermo eſt, = 1120 - 1120 log. </s> + <s xml:id="echoid-s6910" xml:space="preserve">2 = 344, id eſt, ea quæ generatur lapſu li-<lb/>bero 344 lib. </s> + <s xml:id="echoid-s6911" xml:space="preserve">ab altitudine unius pedis.</s> + <s xml:id="echoid-s6912" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6913" xml:space="preserve">Si pes cubicus ſit aëre repletus, qui naturali ſit quadruplo rarior, erit <lb/>jam vis viva quæſita (poſito nempe p = 2240, & </s> + <s xml:id="echoid-s6914" xml:space="preserve">a = {1/4}, x = {3/4}) = 1680 -<lb/>560. </s> + <s xml:id="echoid-s6915" xml:space="preserve">log. </s> + <s xml:id="echoid-s6916" xml:space="preserve">4 = 904, ſeu talis quæ oritur lapſu libero ponderis 904 lib. </s> + <s xml:id="echoid-s6917" xml:space="preserve">per altit. <lb/></s> + <s xml:id="echoid-s6918" xml:space="preserve">unius pedis.</s> + <s xml:id="echoid-s6919" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6920" xml:space="preserve">Si denique habeatur pes cubicus ab aëre omnino vacuus, ponendum <lb/>eſt p = 2240; </s> + <s xml:id="echoid-s6921" xml:space="preserve">a = 0, & </s> + <s xml:id="echoid-s6922" xml:space="preserve">x = 1: </s> + <s xml:id="echoid-s6923" xml:space="preserve">atque ſic erit vis viva quæſita = 2240 X <lb/>(1 - 0 log. </s> + <s xml:id="echoid-s6924" xml:space="preserve">{1/0}) conſtat autem eſſe 0 log. </s> + <s xml:id="echoid-s6925" xml:space="preserve">{1/0} infinite parvum præ unitate; </s> + <s xml:id="echoid-s6926" xml:space="preserve">eſt igitur <lb/>numerus iſte = 2240, qui indicat poſſe hac vi viva 2240 libras ad altitudinem <lb/>unius pedis elevari.</s> + <s xml:id="echoid-s6927" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6928" xml:space="preserve">§. </s> + <s xml:id="echoid-s6929" xml:space="preserve">46. </s> + <s xml:id="echoid-s6930" xml:space="preserve">Pertinet ad præſens argumentum ſtupenda vis aëris admodum <lb/>condenſati, ſed præſertim auræ pulveris pyrii accenſi in uſu ſclopetorum pneu-<lb/>maticorum & </s> + <s xml:id="echoid-s6931" xml:space="preserve">tormentorum bellicorum. </s> + <s xml:id="echoid-s6932" xml:space="preserve">De his quæ ſeorſim commentatus ſum <lb/>huic ſectioni adjiciam.</s> + <s xml:id="echoid-s6933" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div252" type="section" level="1" n="194"> +<head xml:id="echoid-head245" style="it" xml:space="preserve">De vi aëris condenſati & auræ pulveris pyrii ac-<lb/>cenſi ad globos projiciendos in uſu ſclopetorum <lb/>pneumaticorum & tormentorum bellicorum.</head> +<p> + <s xml:id="echoid-s6934" xml:space="preserve">(I) Sit A G (Fig. </s> + <s xml:id="echoid-s6935" xml:space="preserve">64.) </s> + <s xml:id="echoid-s6936" xml:space="preserve">longitudo animæ in tormento ſclopetove hori-<lb/> +<anchor type="note" xlink:label="note-0248-01a" xlink:href="note-0248-01"/> +zontaliter poſito, voceturque = a: </s> + <s xml:id="echoid-s6937" xml:space="preserve">denotet A C longitudinem ſpatii, quod <lb/>aër condenſatus ſeu aura pulveris pyrii accenſi occupat ab initio exploſionis, <lb/>ſitque A C = b: </s> + <s xml:id="echoid-s6938" xml:space="preserve">pondus globi ejiciendi E = 1; </s> + <s xml:id="echoid-s6939" xml:space="preserve">ponimus autem, globum ca-<lb/>vitatem animæ exacte replere & </s> + <s xml:id="echoid-s6940" xml:space="preserve">liberrime in illa moveri: </s> + <s xml:id="echoid-s6941" xml:space="preserve">denſitas aëris con-<lb/>denſati in ſpatio A D ſe habeat ad denſitatem aëris naturalis ut n ad 1: </s> + <s xml:id="echoid-s6942" xml:space="preserve">Deni- +<pb o="235" file="0249" n="249" rhead="SECTIO DECIMA."/> +que ponatur pondus columnæ mercurii (cujus baſis eſt C D & </s> + <s xml:id="echoid-s6943" xml:space="preserve">cujus altitudo <lb/>eadem ſit quæ in barometro) = P. </s> + <s xml:id="echoid-s6944" xml:space="preserve">Utemur autem hypotheſi, ſive globus pro-<lb/>pellatur ab aëre condenſato ſive à pulveris pyrii aura, potentiam illius fluidi <lb/>propellentis proportionalem eſſe denſitati.</s> + <s xml:id="echoid-s6945" xml:space="preserve"/> +</p> +<div xml:id="echoid-div252" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0248-01" xlink:href="note-0248-01a" xml:space="preserve">Fig. 64.</note> +</div> +<p> + <s xml:id="echoid-s6946" xml:space="preserve">His ad calculum præparatis, globum conſiderabimus in ſitu e, poneu-<lb/>do A c = x, velocitatemque globi in hoc ſitu = v, ſic erit potentia globum <lb/>in ſitu e propellens = ({nb/x} - 1) X P, quæ diviſa per maſſam 1 ductaque in ele-<lb/>mentum ſpatii d x dat incrementum dimidium quadrati velocitatis; </s> + <s xml:id="echoid-s6947" xml:space="preserve">unde fit v d v <lb/>= ({nb/x} - 1) X P d x, ſive {1/2} v v = (b - x + nb log. </s> + <s xml:id="echoid-s6948" xml:space="preserve">{x/b})P. </s> + <s xml:id="echoid-s6949" xml:space="preserve">Ponatur x = a, <lb/>habetur altitudo debita velocitati, quacum globus exploditur; </s> + <s xml:id="echoid-s6950" xml:space="preserve">vocetur iſta <lb/>altitudo α & </s> + <s xml:id="echoid-s6951" xml:space="preserve">erit <lb/>α = (b - a + nb log. </s> + <s xml:id="echoid-s6952" xml:space="preserve">{a/b}) X P.</s> + <s xml:id="echoid-s6953" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6954" xml:space="preserve">(II) Sit v. </s> + <s xml:id="echoid-s6955" xml:space="preserve">gr. </s> + <s xml:id="echoid-s6956" xml:space="preserve">in ſclopeto pneumatico longitudo animæ ſeu a = 3 ped. <lb/></s> + <s xml:id="echoid-s6957" xml:space="preserve">Paris. </s> + <s xml:id="echoid-s6958" xml:space="preserve">longitudo A C = 4 poll. </s> + <s xml:id="echoid-s6959" xml:space="preserve">fueritque aër captus in A D naturali decies den-<lb/>ſior ſeu n = 10, diameter animæ ſeu globuli ejiciendi trium linearum ejus-<lb/>que gravitas ſpecifica ratione mercurii ut 10 ad 17. </s> + <s xml:id="echoid-s6960" xml:space="preserve">Erit P præterpropter = <lb/>286; </s> + <s xml:id="echoid-s6961" xml:space="preserve">indeque invenitur α = 2788, indicio globum ejectum iri velocitate <lb/>qua in vacuo ad altitudinem 2788 ped. </s> + <s xml:id="echoid-s6962" xml:space="preserve">aſcendere poſſit. </s> + <s xml:id="echoid-s6963" xml:space="preserve">Ex præcedente for-<lb/>mula colligitur jactum globi vehementiſſimum fore pro eadem auræ elaſticæ <lb/>quantitate, ſi longitudo animæ fiat = n b. </s> + <s xml:id="echoid-s6964" xml:space="preserve">Si vero animus ad impedimenta <lb/>alia, quæ globus præter inertiam ſuam & </s> + <s xml:id="echoid-s6965" xml:space="preserve">reſiſtentiam aëris externi in tranſitu <lb/>ſuo per Sclopeti animam patitur, advertatur, apparet longitudinem animæ <lb/>ad jactum vehementiſſimum producendum requiri longe minorem. </s> + <s xml:id="echoid-s6966" xml:space="preserve">Si longi-<lb/>tudo n b admodum major ſit longitudine a, quod ita eſt in jactibus fortiori-<lb/>bus, erit ſine ſenſibili errore α = n b P log. </s> + <s xml:id="echoid-s6967" xml:space="preserve">{a/b}.</s> + <s xml:id="echoid-s6968" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6969" xml:space="preserve">Si tormentum ſit verticaliter erectum, fit aliquantum diverſus calculus <lb/>ſed pro vehementioribus jactibus differentia nequit eſſe ſenſibilis. </s> + <s xml:id="echoid-s6970" xml:space="preserve">Igitur quia <lb/>jactus deinceps conſiderabimus tantum vehementiſſimos, brevitatis ergo po-<lb/>nemus a = nb P X log. </s> + <s xml:id="echoid-s6971" xml:space="preserve">{a/b}.</s> + <s xml:id="echoid-s6972" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6973" xml:space="preserve">(III) Prouti in præcedentibus altitudinem determinavimus debitam ve- +<pb o="236" file="0250" n="250" rhead="HYDRODYNAMICÆ"/> +locitati qua globus exploditur, ex data vi elaſtica auræ globum ejicientis, ita <lb/>viciſſim patet, ex obſervata illa altitudine vim auræ elaſticam deduci poſſe, <lb/>eſt enim <lb/>n = α: </s> + <s xml:id="echoid-s6974" xml:space="preserve">(b P log. </s> + <s xml:id="echoid-s6975" xml:space="preserve">{a/b}).</s> + <s xml:id="echoid-s6976" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6977" xml:space="preserve">Exinde poterit vis elaſtica pulveris pyrii ſi non accurate definiri, ſaltem <lb/>ad terminos reduci, quos certe ſuperabit. </s> + <s xml:id="echoid-s6978" xml:space="preserve">At quæres, qui altitudo a expe-<lb/>rimento determinari poſſit; </s> + <s xml:id="echoid-s6979" xml:space="preserve">ad quod reſpondeo, poſſe eam ſat accurate col-<lb/>ligi ex tempore, quod globus verticaliter ſurſum ejectus ab exploſionis pun@ <lb/>cto inſumit, dum in terram delabitur habita in calculo aëris reſiſtentiæ ratio-<lb/>ne. </s> + <s xml:id="echoid-s6980" xml:space="preserve">Transſcribam huc experimenta in comm. </s> + <s xml:id="echoid-s6981" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s6982" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s6983" xml:space="preserve">tom. </s> + <s xml:id="echoid-s6984" xml:space="preserve">2. </s> + <s xml:id="echoid-s6985" xml:space="preserve">p. </s> + <s xml:id="echoid-s6986" xml:space="preserve">p. </s> + <s xml:id="echoid-s6987" xml:space="preserve">338 & </s> + <s xml:id="echoid-s6988" xml:space="preserve"><lb/>339 recenſita, quorum calculum inſtitui factis, ratione aëris reſiſtentiæ hy-<lb/>potheſibus, gravitates ſpecificas ferri & </s> + <s xml:id="echoid-s6989" xml:space="preserve">aëris eſſe ut 7650 ad 1 & </s> + <s xml:id="echoid-s6990" xml:space="preserve">aërem, in <lb/>quo globus aſcendit, uniformis eſſe denfitatis: </s> + <s xml:id="echoid-s6991" xml:space="preserve">gravitatum ſpecificarum ratio <lb/>paullo major aſſumta fuiſſe videtur quam debebat, ſed compenſabitur in al-<lb/>tiſſimis jactibus error à diminutione aëris denſitatum verſus ſuperiora.</s> + <s xml:id="echoid-s6992" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s6993" xml:space="preserve">„Tormenti ſitus omni accuratione ad perpendiculum erat accommo-<lb/>datus & </s> + <s xml:id="echoid-s6994" xml:space="preserve">ſingulis vicibus in hunc ſitum reponebatur atque firmabatur: </s> + <s xml:id="echoid-s6995" xml:space="preserve">ſin-<lb/>gula experimenta fuerunt repetita: </s> + <s xml:id="echoid-s6996" xml:space="preserve">Erat autem longitudo animæ 7, 7. </s> + <s xml:id="echoid-s6997" xml:space="preserve">ped. <lb/></s> + <s xml:id="echoid-s6998" xml:space="preserve">angl. </s> + <s xml:id="echoid-s6999" xml:space="preserve">diameter globi erat 0, 2375 ped. </s> + <s xml:id="echoid-s7000" xml:space="preserve">diameter animæ menſurata non fuit <lb/>neque magnitudo luminis accenſorii: </s> + <s xml:id="echoid-s7001" xml:space="preserve">qualibet vice ponderabatur quantitas <lb/>pulveris pyrii adhibiti & </s> + <s xml:id="echoid-s7002" xml:space="preserve">pendulo definiebatur tempus à puncto exploſionis <lb/>ad punctum, quo globus in terram cecidit: </s> + <s xml:id="echoid-s7003" xml:space="preserve">tabula ſequens exhibet, tum <lb/>quæ obſervata, tum quæ calculo inde eruta fuerunt.</s> + <s xml:id="echoid-s7004" xml:space="preserve"/> +</p> +<note position="right" xml:space="preserve"> <lb/>quant. pulv. \\ pyr. numero \\ unciar. holl. \\ expreſſ. # tempus aſc. \\ & deſcens. in \\ min. ſec. ob-\\ ſerv. # altit. jactus \\ in aëre reſiſt, \\ per calculum \\ in ped. Angl. # temp. aſc. in \\ aëre reſiſt. \\ per calculum \\ in min. ſec. # temp. deſc. in \\ aëre reſiſt. \\ per calculum \\ in min. ſec. # altit. jactus in \\ vacuo per \\ calculum in \\ ped. Angl. # temp. aſcenſ. \\ & deſc. in va- \\ cuo per calc. \\ in min. ſec. <lb/>I # II # III # IV # V # VI # VII <lb/>{1/2} # 11 # 486 # 5, 42 # 5, 58 # 541 # 11, 6 <lb/>2 # 34 # 4550 # 14, 37 # 19, 63 # 13694 # 58 <lb/>4 # 45 # 7819 # 16, 84 # 28, 16 # 58750 # 121 <lb/></note> +<pb o="237" file="0251" n="251" rhead="SECTIO DECIMA."/> +<p> + <s xml:id="echoid-s7005" xml:space="preserve">„Pro eodem tormento eodemque globo, ſed priori diminuto pede uno <lb/>cum ſeptem decimis partibus, ſic ut longitudo animæ reſidua eſſet præciſe <lb/>6. </s> + <s xml:id="echoid-s7006" xml:space="preserve">ped. </s> + <s xml:id="echoid-s7007" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s7008" xml:space="preserve">inſervit ſequens tabula eadem lege conſtructa.</s> + <s xml:id="echoid-s7009" xml:space="preserve"/> +</p> +<note position="right" xml:space="preserve"> <lb/>I # II # III # IV # V # VI # VII <lb/>{1/2} # 8 # 257 # 3, 95 # 4, 05 # 274 # 8, 2 <lb/>2 # 20, 5 # 1665 # 9, 74 # 10, 76 # 2404 # 24, 5 <lb/>4 # 28 # 3187 # 12, 5 # 15, 5 # 6604 # 40, 5 <lb/>6 # 32, 5 # 4304 # 13, 9 # 18, 6 # 11810 # 54, 3 <lb/>8 # 38 # 5643 # 15, 54 # 22, 46 # 22394 # 74 <lb/></note> +<p> + <s xml:id="echoid-s7010" xml:space="preserve">Multa ſunt, quæ ſucceſſum horum experimentorum ita reddunt dubium, <lb/>ut nullum ſit, quod eandem auræ elaſticitatem arguat. </s> + <s xml:id="echoid-s7011" xml:space="preserve">Maximam ego in-<lb/>æqualitatem ex eo oriri crediderim; </s> + <s xml:id="echoid-s7012" xml:space="preserve">quod minima pars pulveris inflammetur <lb/>ſtatim ab exploſionis initio, quod magna pars tum demum accendatur, cum <lb/>globus orificio tormenti jam proximus eſt, & </s> + <s xml:id="echoid-s7013" xml:space="preserve">quod maxima denique pars non <lb/>inflammata ejiciatur: </s> + <s xml:id="echoid-s7014" xml:space="preserve">facit fortaſſe hæc ſola ratio, ut vis elaſtica auræ globum <lb/>propellentis ſit centies major, quam quæ vi experimenti, nulla habita iſtius <lb/>rei ratione, prodit: </s> + <s xml:id="echoid-s7015" xml:space="preserve">id mihi valde probabile fit, ex eo quod adhibito in tor-<lb/>mento 7, 7 ped. </s> + <s xml:id="echoid-s7016" xml:space="preserve">longo pulvere ad 4 uncias globus in vacuo jactu ſuo aſcen-<lb/>dere potuerit ad altitudinem 58750 ped. </s> + <s xml:id="echoid-s7017" xml:space="preserve">cum eadem pulveris quantitate eo-<lb/>demque tormento ſed 1, 7 pede decurtato jactus reſponderit altitudini in va-<lb/>cuo 6604 pedum, quæ altitudo vix ultra nonam partem prioris excurrit: </s> + <s xml:id="echoid-s7018" xml:space="preserve">Ex <lb/>comparatione utriusque experimenti conjicio, maximam pulveris quantita-<lb/>tem in tormento longiore inflammatam fuiſſe dum globus jamjam eſſet ori-<lb/>ficio proximus neque ab ipſo ultra 1, 7 ped. </s> + <s xml:id="echoid-s7019" xml:space="preserve">amplius diſtaret.</s> + <s xml:id="echoid-s7020" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7021" xml:space="preserve">Diminuitur quoque jactus globi à magnitudine luminis accenſorii, ut <lb/>& </s> + <s xml:id="echoid-s7022" xml:space="preserve">ab hiatu qui inter globum & </s> + <s xml:id="echoid-s7023" xml:space="preserve">internam animæ ſuperficiem relinquitur, per <lb/>quod utrumque notabilis auræ pars inutilis avolat: </s> + <s xml:id="echoid-s7024" xml:space="preserve">tanta autem inde dimi-<lb/>nutio non oritur, quantam illam nondum poſito calculo præſumſeram: </s> + <s xml:id="echoid-s7025" xml:space="preserve">ad-<lb/>jiciam tamen inſequentibus calculum, ut methodus habeatur vi pulveris <lb/>pyrii longiſſimos ſtatuendi limites, quos etiamnum certe transgrediatur.</s> + <s xml:id="echoid-s7026" xml:space="preserve"/> +</p> +<pb o="238" file="0252" n="252" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s7027" xml:space="preserve">(IV) Quod maximam oſtendit auræ elaſticitatem eſt experimentum ter-<lb/>tium cum tormento nondum decurtato ſumtum, quod indicat aſcendere <lb/>potuiſſe globum accepto impetu ad altitudinem α = 58750 ped. </s> + <s xml:id="echoid-s7028" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s7029" xml:space="preserve">Erat <lb/>autem longitudo animæ A G ſeu a = 7, 7: </s> + <s xml:id="echoid-s7030" xml:space="preserve">longitudo A C (quantum ex <lb/>amplitudine animæ & </s> + <s xml:id="echoid-s7031" xml:space="preserve">gravitate pulveris pyrii conjicio) erat = 0, 08. </s> + <s xml:id="echoid-s7032" xml:space="preserve">De-<lb/>nique valor ipſius P (ſeu ponderis columnæ mercurialis, cujus baſis ſit cir-<lb/>culus maximus globi & </s> + <s xml:id="echoid-s7033" xml:space="preserve">cujus altitudo ſit 30. </s> + <s xml:id="echoid-s7034" xml:space="preserve">poll. </s> + <s xml:id="echoid-s7035" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s7036" xml:space="preserve">ratione ponderis glo-<lb/>bi ferri deſignati per unitatem) invenitur poſita gravitate ſpecifica inter mer-<lb/>curium & </s> + <s xml:id="echoid-s7037" xml:space="preserve">ferrum ut 17 ad 10 = 26, 8: </s> + <s xml:id="echoid-s7038" xml:space="preserve">Et cum per §. </s> + <s xml:id="echoid-s7039" xml:space="preserve">III. </s> + <s xml:id="echoid-s7040" xml:space="preserve">ſit proxime n = <lb/>α: </s> + <s xml:id="echoid-s7041" xml:space="preserve">(b P log {a/b}) erit n = 6004. </s> + <s xml:id="echoid-s7042" xml:space="preserve">Unde ſequitur, ſi aura pulveris pyrii inflamma-<lb/>ti elaſticitatem habeat ſuæ denſitati proportionalem, eſſe illius maximam ela-<lb/>ſticitatem minimum ſexies millies majorem elaſticitate aëris ordinarii.</s> + <s xml:id="echoid-s7043" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7044" xml:space="preserve">(V) At vero ſi jam conſideremus partem auræ inutilem, quæ avolat per <lb/>lumen accenſorium & </s> + <s xml:id="echoid-s7045" xml:space="preserve">hiatum à globo relictum, majorem elaſticitatem inve-<lb/>niemus: </s> + <s xml:id="echoid-s7046" xml:space="preserve">Calculus qui ad hanc quæſtionem ſolvendam requiritur, cum non <lb/>parum prolixus atque intricatus ſit, non hæſitavi hypotheſes adhibere paul-<lb/>lo liberiores, quibus admodum facilitatur: </s> + <s xml:id="echoid-s7047" xml:space="preserve">quamvis ipſæ hypotheſes non <lb/>ſint omni rigore veræ, errorem tamen notabilem producere non poſſunt. <lb/></s> + <s xml:id="echoid-s7048" xml:space="preserve">Primo ponam utramque aperturam, per quam aura evolare poſſit, eſſe ve-<lb/>luti infinite parvam ratione animæ amplitudinis; </s> + <s xml:id="echoid-s7049" xml:space="preserve">hoc poſito poterit ſingulis <lb/>momentis velocitas, cum qua aura avolat, æſtimari immediate ex preſſione <lb/>ſola: </s> + <s xml:id="echoid-s7050" xml:space="preserve">hujusmodi autem hypotheſin ſine ullo ſenſibili errore fieri poſſe pro <lb/>omni fluido, tunc etiam cum foramina non ſunt admodum exigua, paſſim <lb/>ut corollarium ex theoria noſtra deduximus, & </s> + <s xml:id="echoid-s7051" xml:space="preserve">multo facilius aſſumi poſſe in <lb/>fluido valde elaſtico facile quisque videbit ex eo, quod incrementum aſcen-<lb/>ſus potentialis ratione motus interni longe minus eſt ratione aſcenſus potentialis <lb/>particulæ per foramen exilientis in fluido, quod à propria elaſticitate ex-<lb/>pellitur, quam quod gravitatis vi ejicitur: </s> + <s xml:id="echoid-s7052" xml:space="preserve">in priori enim minor eſt motus <lb/>localis internus quam in altero. </s> + <s xml:id="echoid-s7053" xml:space="preserve">Secundo auræ pulveris pyrii inflammati vim <lb/>elaſticam tantam eſſe, ut niſus atmoſphæræ contrarius attendi non mereatur: </s> + <s xml:id="echoid-s7054" xml:space="preserve"><lb/>tertio velocitatem globi in tormento utut permagnam, tamen minimam cen-<lb/>ſeri poſſe ratione velocitatis, qua aura per hiatum utrumque avolat, quia <lb/>nempe inertia iſtius auræ non poteſt non admodum eſſe exigua ratione in- +<pb o="239" file="0253" n="253" rhead="SECTIO DECIMA."/> +ertiæ quæ globo ineſt: </s> + <s xml:id="echoid-s7055" xml:space="preserve">vi iſtius hypotheſeos avolabit aura per utramque aper-<lb/>turam eadem velocitate, cum alias poſita velocitate in lumine accenſorio <lb/>= √ A, & </s> + <s xml:id="echoid-s7056" xml:space="preserve">velocitate globi = v, velocitas auræ in hiatu a<unsure/> globo ad ſuperfi-<lb/>ciem animæ relicto dicenda eſſet = √ A - v. </s> + <s xml:id="echoid-s7057" xml:space="preserve">Venio nunc ad ſolutionem.</s> + <s xml:id="echoid-s7058" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7059" xml:space="preserve">(VI) Primo notandum eſt, ſi elaſticitates auræ cenſeantur denſitatibus <lb/>proportionales, fore ut aura<unsure/> conſtanter eadem velocitate per utramque <lb/>aperturam avolet, uti vidimus in problemate §. </s> + <s xml:id="echoid-s7060" xml:space="preserve">34. </s> + <s xml:id="echoid-s7061" xml:space="preserve">iſtaque velocitas no-<lb/>minatim talis erit, quæ generetur ab altitudine auræ homogeneæ, cu-<lb/>jus pondus auram captam coërcere poſſit, ne ſe expandat. </s> + <s xml:id="echoid-s7062" xml:space="preserve">Igitur deter-<lb/>minabitur dicta velocitas hoc modo: </s> + <s xml:id="echoid-s7063" xml:space="preserve">ſit gravitas globi = 1, elaſticitas <lb/>ſeu pondus quod auram pulveris modo inflammati A C D B in illo com-<lb/>preſſionis ſtatu coërcere poſſit = P: </s> + <s xml:id="echoid-s7064" xml:space="preserve">pondus pulveris adhibiti = p; <lb/></s> + <s xml:id="echoid-s7065" xml:space="preserve">erit pondus auræ pulveris modo inflammati etiam = p: </s> + <s xml:id="echoid-s7066" xml:space="preserve">ſique lon-<lb/>gitudo A C ponitur = b, patet altitudinem auræ homogeneæ, quæ pondus <lb/>P habeat, fore = {P/p} b; </s> + <s xml:id="echoid-s7067" xml:space="preserve">Igitur velocitas quacum aura recens nata per lumen <lb/>accenſorium avolat eſt = √({P/p} b), eademque velocitate durante tota ex-<lb/>ploſione ejicietur, idque non ſolum per lumen accenſorium, ſed & </s> + <s xml:id="echoid-s7068" xml:space="preserve">proxime <lb/>per hiatum inter globum & </s> + <s xml:id="echoid-s7069" xml:space="preserve">animam relictum.</s> + <s xml:id="echoid-s7070" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7071" xml:space="preserve">(VII) Sit nunc porro amplitudo animæ = F; </s> + <s xml:id="echoid-s7072" xml:space="preserve">hiatus interceptus inter <lb/>globum & </s> + <s xml:id="echoid-s7073" xml:space="preserve">animam = f: </s> + <s xml:id="echoid-s7074" xml:space="preserve">amplitudo luminis accenſorii = Φ: </s> + <s xml:id="echoid-s7075" xml:space="preserve">longitudo ani-<lb/>mæ = a, quantitas auræ ab initio exploſionis = g. </s> + <s xml:id="echoid-s7076" xml:space="preserve">Intelligatur deinde glo-<lb/>bus perveniſſe ex E in e, dicaturque A C = x: </s> + <s xml:id="echoid-s7077" xml:space="preserve">quantitas auræ eo temporis <lb/>puncto in tormento reſidua = z: </s> + <s xml:id="echoid-s7078" xml:space="preserve">velocitas globi in iſto ſitu = v, reliquæ de-<lb/>nominationes fuerunt jam antea explicatæ.</s> + <s xml:id="echoid-s7079" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7080" xml:space="preserve">Quoniam elaſticitas per hypotheſin eſt directe ut quantitas & </s> + <s xml:id="echoid-s7081" xml:space="preserve">recipro-<lb/>ce ut ſpatium, erit elaſticitas auræ in A c d B reſiduæ = {zb/gx} P: </s> + <s xml:id="echoid-s7082" xml:space="preserve">quæ quidem <lb/>non tota in propellendum globum impenditur, ſed tantum pars ejus, quæ <lb/>ſe habeat ad totam ut F - f ad f. </s> + <s xml:id="echoid-s7083" xml:space="preserve">Eſt itaque poſito d t pro elemento temporis <lb/>dv = {F - f/F} X {zb/gx} P X dt. <lb/></s> + <s xml:id="echoid-s7084" xml:space="preserve">Per methodum autem §. </s> + <s xml:id="echoid-s7085" xml:space="preserve">34. </s> + <s xml:id="echoid-s7086" xml:space="preserve">exhibitam, ubi quantitas aëris dato tempuſculo <lb/>effluens ſpecifice definita fuit, invenitur +<pb o="240" file="0254" n="254" rhead="HYDRODYNAMICÆ"/> +- dz = {f + φ/F} X {z/x} X √ ({P/p} b) xdt; <lb/></s> + <s xml:id="echoid-s7087" xml:space="preserve">Ex comparatione harum duarum æquationum oritur <lb/>- dz = {f + φ/F - f} X {g/b} X {√b/√Pp} X dv, <lb/>quæ cum debitæ conſtantis additione integrata dat <lb/>z = g - {f + φ/F - f} X {g/b} X {√b/√Pp} X v. </s> + <s xml:id="echoid-s7088" xml:space="preserve"><lb/>Si jam in æquatione prima ſubſtituatur valor iſte inventus pro z, ſimulque <lb/>ponatur {dx/v} pro dt, fiet <lb/>vdv = {F - f/F} X {b/x} X P X dx - {f + φ/F} X {√(bP)/x√p} X vdx, ſive <lb/>{Fvdv√p/(F - f) X bP√p - (f + φ) X v√ (bP)} = {dx/x}, <lb/>quæ æquatio poſt debitam ſui integrationem, facta x = a, abit in hanc <lb/>log. </s> + <s xml:id="echoid-s7089" xml:space="preserve">{a/b} = [-F(f + φ) v√ p - F (F - f) p√ (Pb) X log.</s> + <s xml:id="echoid-s7090" xml:space="preserve">(1 - {(f + φ)v/(F - f) √ (bPp)})]: </s> + <s xml:id="echoid-s7091" xml:space="preserve"><lb/>(f + φ)<emph style="super">2</emph> X √Pb.</s> + <s xml:id="echoid-s7092" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7093" xml:space="preserve">(VIII) Si jam per experimentum innotuerit valor ipſius v, poterit in-<lb/>de deduci valor ipſius P, qui denotat elaſticitatem auræ pulveris pyrii non-<lb/>dum expanſæ: </s> + <s xml:id="echoid-s7094" xml:space="preserve">Quod ut exemplo illuſtremus, eodem utemur experimento, <lb/>quod jam articulo IV. </s> + <s xml:id="echoid-s7095" xml:space="preserve">expoſuimus, ut appareat inde, quodnam ab avolatione <lb/>auræ elaſticitatis augmentum arguat. </s> + <s xml:id="echoid-s7096" xml:space="preserve">Sic igitur ponetur calculus.</s> + <s xml:id="echoid-s7097" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7098" xml:space="preserve">Quia pondus globi, quod erat trium librarum, indicavimus per uni-<lb/>tatem, erunt quatuor unicæ pulveris adhibitæ exprimendæ per {1/12}: </s> + <s xml:id="echoid-s7099" xml:space="preserve">igitur <lb/>p = {1/12}. </s> + <s xml:id="echoid-s7100" xml:space="preserve">Menſuras aperturarum, quas conſideramus, non accepi: </s> + <s xml:id="echoid-s7101" xml:space="preserve">ſolet autem <lb/>hiatus à globo r<unsure/>elictus conſtituere in ſimili tormento præterpropter partem <lb/>decimam quintam amplitudinis animæ; </s> + <s xml:id="echoid-s7102" xml:space="preserve">amplitudinem luminis accenſoriihic <lb/>fere negligi poſſe puto; </s> + <s xml:id="echoid-s7103" xml:space="preserve">itaque ſtatuam F = 15; </s> + <s xml:id="echoid-s7104" xml:space="preserve">f = 1; </s> + <s xml:id="echoid-s7105" xml:space="preserve">φ = 0: </s> + <s xml:id="echoid-s7106" xml:space="preserve">Deinde <lb/>habetur rurſus a = 7, 7; </s> + <s xml:id="echoid-s7107" xml:space="preserve">b = 0, 08; </s> + <s xml:id="echoid-s7108" xml:space="preserve">altitudo ad quam globus in vacuo <lb/>aſcendere poſſit ſeu {1/2} vv = 58750, ſeuv = 343: </s> + <s xml:id="echoid-s7109" xml:space="preserve">Igitur æquatio ultima <lb/>ſuperioris articuli hæc erit <lb/>log.</s> + <s xml:id="echoid-s7110" xml:space="preserve">96 = { - 5251/√P} + 17, 5 log. </s> + <s xml:id="echoid-s7111" xml:space="preserve">{√P/√P-300}, <lb/>cui proxime ſatisfit cum ſumitur √ P = 534 & </s> + <s xml:id="echoid-s7112" xml:space="preserve">proinde P = 285156, <lb/>quod efficit pondus columnæ mercurialis ejusdem cum anima tormenti am- +<pb o="241" file="0255" n="255" rhead="SECTIO DECIMA."/> +plitudinis, cujus altitudo ſit plusquam 10000 vicibus major altitudine com-<lb/>muni barometri, invenimus autem ſupra art. </s> + <s xml:id="echoid-s7113" xml:space="preserve">IV. </s> + <s xml:id="echoid-s7114" xml:space="preserve">numerum n ( qui idem ſi-<lb/>gnificabat) = 6004. </s> + <s xml:id="echoid-s7115" xml:space="preserve">Ergo jam tuto affirmabimus ( ubique enim quæ negle-<lb/>ximus majorem vim pulveri arguunt) ineſſe pulveri pyrio vim elaſticam, <lb/>minimum decies millies majorem vi elaſtica aëris ordinarii. </s> + <s xml:id="echoid-s7116" xml:space="preserve">Apparet autem <lb/>ſimul ex comparatione numerorum 10000 & </s> + <s xml:id="echoid-s7117" xml:space="preserve">6004, quantum circiter vi pul-<lb/>veris decedat ab hiatibus ſæpe dictis. </s> + <s xml:id="echoid-s7118" xml:space="preserve">Equidem iſtud decrementum majus pu-<lb/>taſſem: </s> + <s xml:id="echoid-s7119" xml:space="preserve">Confirmatus autem ſum hoc calculo in re de qua aliquando me cer-<lb/>tiorem voluit vir harum rerum gnarus, nullum nempe ſe in tormentis nota-<lb/>bile obſervaſſe decrementum, cum lumen accenſorium diuturno uſu ſupra <lb/>modum amplificatum eſſet in obſidio.</s> + <s xml:id="echoid-s7120" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7121" xml:space="preserve">(IX) Verum ut ex æquatione noſtra quædam corollaria deduci poſ-<lb/>ſint faciliora quam vis proxime tantum vera, mutabimus quantitatem lo-<lb/>garithmicalem in ſeriem. </s> + <s xml:id="echoid-s7122" xml:space="preserve">Eſt autem <lb/>- log. </s> + <s xml:id="echoid-s7123" xml:space="preserve">(1 - {(f + φ)v/(F - f)√(bPp)}) = {(f + φ)v/(F - f)√(b P p)} <lb/>+ {(f + φ)<emph style="super">2</emph> vv/2(F - f)<emph style="super">2</emph> X b P p} + {(f + φ)<emph style="super">3</emph>v<emph style="super">3</emph>/3(F - f)<emph style="super">3</emph> X b P p√(b P p)} + &</s> + <s xml:id="echoid-s7124" xml:space="preserve">c. <lb/></s> + <s xml:id="echoid-s7125" xml:space="preserve">Iſtoque valore ſubſtituto in æquatione ultima art. </s> + <s xml:id="echoid-s7126" xml:space="preserve">(VII) fit <lb/>log. </s> + <s xml:id="echoid-s7127" xml:space="preserve">{a/b} = {Fvv/2(F - f). </s> + <s xml:id="echoid-s7128" xml:space="preserve">b P} + {F.</s> + <s xml:id="echoid-s7129" xml:space="preserve">(f + φ)v<emph style="super">3</emph>/3.</s> + <s xml:id="echoid-s7130" xml:space="preserve">(F - f)<emph style="super">2</emph>bP√(bPp)} + &</s> + <s xml:id="echoid-s7131" xml:space="preserve">c. </s> + <s xml:id="echoid-s7132" xml:space="preserve"><lb/>Notabimus hic iſtam æquationem perfecte convenire cum æquatione ultima <lb/>art. </s> + <s xml:id="echoid-s7133" xml:space="preserve">(II) ſi aperturæ f & </s> + <s xml:id="echoid-s7134" xml:space="preserve">φ ponantur = 0: </s> + <s xml:id="echoid-s7135" xml:space="preserve">quod enim hic indicatur per {1/2} vv <lb/>& </s> + <s xml:id="echoid-s7136" xml:space="preserve">n P ibi eſt α & </s> + <s xml:id="echoid-s7137" xml:space="preserve">P, convenientibus denominationibus reliquis.</s> + <s xml:id="echoid-s7138" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7139" xml:space="preserve">(X) Ut appareat, quantum proxime altitudo jactus ab aperturis dimi-<lb/>nuatur, ſi iſtæ aperturæ ſint minimæ, inſerviet hæc æquatio. </s> + <s xml:id="echoid-s7140" xml:space="preserve">Intelligatur per <lb/>α altitudo ad quam globus pervenire poſſit in vacuo, ſi nulla auræ quantitas <lb/>per aperturas avolare ponatur, & </s> + <s xml:id="echoid-s7141" xml:space="preserve">erit decrementum iſtius altitudinis ab erup-<lb/>tione auræ per easdem aperturas oriundum proxime hoc <lb/>[(2α)<emph style="super">{3/2}</emph> X (f + φ)]: </s> + <s xml:id="echoid-s7142" xml:space="preserve">[3F X √ (bPp].</s> + <s xml:id="echoid-s7143" xml:space="preserve"/> +</p> +<pb o="242" file="0256" n="256" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s7144" xml:space="preserve">Unde in eodem tormento adhibitaque eadem pulveris quantitate & </s> + <s xml:id="echoid-s7145" xml:space="preserve"><lb/>manente globi pondere, erunt decrementa jactuum proportronalia ampli-<lb/>tudinibus aperturarum.</s> + <s xml:id="echoid-s7146" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7147" xml:space="preserve">Decrementa eadem fere ſequuntur rationem ſubduplicatam quantita-<lb/>tum pulveris adhibitarum cæteris paribus; </s> + <s xml:id="echoid-s7148" xml:space="preserve">quia enim logarithmi magno-<lb/>rum numerorum in multo minori creſcunt ratione ac numeri ipſi & </s> + <s xml:id="echoid-s7149" xml:space="preserve">quo-<lb/>niam inſuper eſt α = b P log. </s> + <s xml:id="echoid-s7150" xml:space="preserve">{a/b}, poterit cæteris paribus ſtatui α propor-<lb/>tionale ipſi b, quia P non afficitur à b. </s> + <s xml:id="echoid-s7151" xml:space="preserve">Sed decrementum, de quo ſermo eſt, <lb/>ceteris paribus rationem ſequitur quantitatis (α{3/2}): </s> + <s xml:id="echoid-s7152" xml:space="preserve">(√ bp) ſeu rationem <lb/>quantitatis {b/√p}; </s> + <s xml:id="echoid-s7153" xml:space="preserve">ipſum vero p, quod pondus denotat pulveris adhibiti eſt ut b; <lb/></s> + <s xml:id="echoid-s7154" xml:space="preserve">igitur decrementum prædictum ſequitur proxime rationem √ b, quæ ſub-<lb/>duplicata eſt quantitatis pulveris adhibiti. </s> + <s xml:id="echoid-s7155" xml:space="preserve">Igitur ratione habita jactuum, de-<lb/>crementa multo majora ſunt in jactibus debilibus, quam vehementioribus, <lb/>idque etiam experimenta art. </s> + <s xml:id="echoid-s7156" xml:space="preserve">(III) recenſita confirmare videntur: </s> + <s xml:id="echoid-s7157" xml:space="preserve">non video <lb/>enim aliam rationem, cur in prima tabula experimentorum globi jactus in <lb/>vacuo, ſumtis duabus pulveris unciis, plus quam vigeſies ſexies altior eſſe <lb/>debuerit, quam cum uncia dimidia ſumeretur, & </s> + <s xml:id="echoid-s7158" xml:space="preserve">cur mox duplicata pul-<lb/>veris quantitate ad 4. </s> + <s xml:id="echoid-s7159" xml:space="preserve">uncias jactus tantum quadruplo altior poſt calculum pro-<lb/>deat, quam quantitate duarum unciarum.</s> + <s xml:id="echoid-s7160" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7161" xml:space="preserve">(XI) Quæ reliquæ in utraque tabula comparent experime<unsure/>ntorum in-<lb/>æqualitates, eas ut ſupra dixi, maximam partem derivo ab eo, quod pulvis <lb/>non omnis inflammatur, nec is qui inflammetur omnis ſtatim ab initio ex-<lb/>ploſionis flammam concipiat. </s> + <s xml:id="echoid-s7162" xml:space="preserve">Neque certe id mirabimur, cum perpendimus <lb/>totum exploſionis tempus in exper. </s> + <s xml:id="echoid-s7163" xml:space="preserve">4. </s> + <s xml:id="echoid-s7164" xml:space="preserve">tab. </s> + <s xml:id="echoid-s7165" xml:space="preserve">1. </s> + <s xml:id="echoid-s7166" xml:space="preserve">nequidem centeſimam unius minuti <lb/>ſecundi partem efficere. </s> + <s xml:id="echoid-s7167" xml:space="preserve">Igitur cum certum ſit maximam pulveris partem non <lb/>inflammatam ejici, nec exiguam partem reliqui tardius inflammari, quam in <lb/>calculo poſitum fuit; </s> + <s xml:id="echoid-s7168" xml:space="preserve">cumque præterea notabilis pulveris pars fucata ſit vapo-<lb/>ribus materiaque terreſtri, quæ non accenditur, ſequitur longe majorem in-<lb/>efſe elaſticitatem partibus accenſis, quam quæ experimenti calculo art. </s> + <s xml:id="echoid-s7169" xml:space="preserve">(X.) <lb/></s> + <s xml:id="echoid-s7170" xml:space="preserve">determinata fuit, fortaſſe decies aut centies major eſt.</s> + <s xml:id="echoid-s7171" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7172" xml:space="preserve">At vero ſit tantum talis, quam experimentum oſtendit, elaſticitate +<pb o="243" file="0257" n="257" rhead="SECTIO DECIMA."/> +nempe aëris ordinarii decies millies major; </s> + <s xml:id="echoid-s7173" xml:space="preserve">ſequitur inde auram illam elaſti-<lb/>cam, quæ ex pulvere pyrio accenſo elicitur aut non aërem eſſe communem <lb/>aut elaſticitates in majori ratione creſcere quam denſitates: </s> + <s xml:id="echoid-s7174" xml:space="preserve">non poteſt enim <lb/>denſitas aëris, qui à pulvere modo inflammato oritur, eſſe plus quam millies <lb/>denſitate aëris ordinarii major, ſi pulvis vel totus ex aëre compreſſo compo-<lb/>ſitus ſit, quod ex gravitate pulveris ſpecifica ratione aëris concludo.</s> + <s xml:id="echoid-s7175" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7176" xml:space="preserve">Quæſtio interim jamdudum eſt agitata, an aura elaſtica factitia, quæ ex <lb/>corporibus deducitur, aër ſit ordinarius nec ne, quam ego quæſtionem non <lb/>decidam.</s> + <s xml:id="echoid-s7177" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7178" xml:space="preserve">Si tamen ponatur, pulverem pyrium aërem eſſe naturali millies den-<lb/>ſiorem & </s> + <s xml:id="echoid-s7179" xml:space="preserve">decies millies magis elaſticum, tum ex §. </s> + <s xml:id="echoid-s7180" xml:space="preserve">4. </s> + <s xml:id="echoid-s7181" xml:space="preserve">ſequetur, aërem vi infi-<lb/>nita compreſſum non poſſe pluribus quam 1331. </s> + <s xml:id="echoid-s7182" xml:space="preserve">vicibus condenſari & </s> + <s xml:id="echoid-s7183" xml:space="preserve">ſecun-<lb/>dum eandem regulam foret aëris naturali quadruplo denſioris elaſticitas ad ela-<lb/>ſticitatem aëris naturalis ut 4 {1/4} ad 1.</s> + <s xml:id="echoid-s7184" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7185" xml:space="preserve">An vero experimenta ab aliis inſtituta, quæ harum elaſticitatum ratio-<lb/>nem faciunt accurate ut 4 ad 1 ſufficiente accuratione facta fuerint & </s> + <s xml:id="echoid-s7186" xml:space="preserve">an calor <lb/>aëris dum comprimebatur idem permanſerit? </s> + <s xml:id="echoid-s7187" xml:space="preserve">neſcio. </s> + <s xml:id="echoid-s7188" xml:space="preserve">Veroſimile autem eſt. <lb/></s> + <s xml:id="echoid-s7189" xml:space="preserve">eandem auram quæ in poris pulveris pyrii latet, cauſam eſſe elaſticitatis cor-<lb/>porum elaſticorum aut villorum contractilium: </s> + <s xml:id="echoid-s7190" xml:space="preserve">dum enim in cavernulis ſcatet, <lb/>ſi corpora in figuram inſolitam vi quadam redigantur, comprimitur aura ela-<lb/>ſtica, cavernuliſque dum reddit figuram capaciſſimam corpus reſtituit in pri-<lb/>ſtinam figuram & </s> + <s xml:id="echoid-s7191" xml:space="preserve">longitudinem.</s> + <s xml:id="echoid-s7192" xml:space="preserve"/> +</p> +<pb o="244" file="0258" n="258"/> +</div> +<div xml:id="echoid-div254" type="section" level="1" n="195"> +<head xml:id="echoid-head246" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph></head> +<head xml:id="echoid-head247" xml:space="preserve">SECTIO UNDECIMA.</head> +<head xml:id="echoid-head248" style="it" xml:space="preserve">De fluidis in vorticem actis, tum etiam de iis, quæ <lb/>in vaſis motis continentur.</head> +<head xml:id="echoid-head249" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s7193" xml:space="preserve">EX eo tempore quo Keplerus & </s> + <s xml:id="echoid-s7194" xml:space="preserve">Carteſius vortices adhibuere pro <lb/>variis naturæ phænomenis explicandis, multi operam ſuam haud <lb/>male ſe collocaturos rati ſollicite iſtud argumentum ruminati ſunt: <lb/></s> + <s xml:id="echoid-s7195" xml:space="preserve">primus autem, ni fallor, naturam ejus recte penetravit Huge-<lb/>nius in tract. </s> + <s xml:id="echoid-s7196" xml:space="preserve">ſur la peſanteur; </s> + <s xml:id="echoid-s7197" xml:space="preserve">ſuperaddam quædam, quæ ad in-<lb/>ſtitutum meum pertinent, ab aliis fortaſſe non ſatis examinata.</s> + <s xml:id="echoid-s7198" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7199" xml:space="preserve">Poni autem ſolent vortices ad ſtatum permanentiæ ſeu durationis redu-<lb/>cti, ita ut nulli mutationi ſubjectum lege conſtanter eadem moveatur flui-<lb/>dum.</s> + <s xml:id="echoid-s7200" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7201" xml:space="preserve">§. </s> + <s xml:id="echoid-s7202" xml:space="preserve">2. </s> + <s xml:id="echoid-s7203" xml:space="preserve">Sit cylindrus A B C D (Fig. </s> + <s xml:id="echoid-s7204" xml:space="preserve">65. </s> + <s xml:id="echoid-s7205" xml:space="preserve">& </s> + <s xml:id="echoid-s7206" xml:space="preserve">66.) </s> + <s xml:id="echoid-s7207" xml:space="preserve">verticaliter poſitus, cujus <lb/> +<anchor type="note" xlink:label="note-0258-01a" xlink:href="note-0258-01"/> +axis G H, iſque ad certam altitudinem plenus ſit, concipiatur aqua in vorti-<lb/> +<anchor type="note" xlink:label="note-0258-02a" xlink:href="note-0258-02"/> +cem acta ſintque omnia jam ad ſtatum durationis reducta: </s> + <s xml:id="echoid-s7208" xml:space="preserve">Ita ſuperficies aquæ <lb/>deprimetur verſus axem & </s> + <s xml:id="echoid-s7209" xml:space="preserve">elevabitur verſus latera: </s> + <s xml:id="echoid-s7210" xml:space="preserve">Sectionem per axem termi-<lb/>natam à ſuperficie aquæ repræſentabimus curva E O F, hujuſque curvæ nunc <lb/>indolem dabimus ex data relatione, quam inter ſe habent velocitates ſub certis <lb/>ab axe diſtantiis.</s> + <s xml:id="echoid-s7211" xml:space="preserve"/> +</p> +<div xml:id="echoid-div254" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0258-01" xlink:href="note-0258-01a" xml:space="preserve">Fig. 65.</note> +<note position="left" xlink:label="note-0258-02" xlink:href="note-0258-02a" xml:space="preserve">& 66.</note> +</div> +<p> + <s xml:id="echoid-s7212" xml:space="preserve">Ducantur g a & </s> + <s xml:id="echoid-s7213" xml:space="preserve">f n infinite propinquæ & </s> + <s xml:id="echoid-s7214" xml:space="preserve">horizontales, agaturque a m <lb/>verticalis: </s> + <s xml:id="echoid-s7215" xml:space="preserve">Sit O g = x, gf ſeu am = dx, ga = y, mn = dy: </s> + <s xml:id="echoid-s7216" xml:space="preserve">Patet autem <lb/>quamlibet guttulam in ſuperficie poſitam niſu ſuo, ex vi centriſuga horizontali <lb/>& </s> + <s xml:id="echoid-s7217" xml:space="preserve">vi gravitatis verticali, compoſito perpendiculariter ſuperficiei inſiſtere, quia <lb/>ſi oblique contranitatur nihil ſit, quod guttulam in loco ſuo conſervet.</s> + <s xml:id="echoid-s7218" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7219" xml:space="preserve">Igitur ſi vis centrifuga guttulæ in a poſitæ exprimatur per horizontalem +<pb o="245" file="0259" n="259" rhead="SECTIO UNDECIMA."/> +b a & </s> + <s xml:id="echoid-s7220" xml:space="preserve">vis gravitatis per verticalem c a compleaturque rectangulum a b e c, erit <lb/>diagonalis a e ad curvam perpendicularis; </s> + <s xml:id="echoid-s7221" xml:space="preserve">unde triangulum e c a ſimile eſt tri-<lb/>angulo a m n & </s> + <s xml:id="echoid-s7222" xml:space="preserve">ſic d x: </s> + <s xml:id="echoid-s7223" xml:space="preserve">dy = ec: </s> + <s xml:id="echoid-s7224" xml:space="preserve">ca = ba: </s> + <s xml:id="echoid-s7225" xml:space="preserve">ca, vel ut vis centrifuga in <lb/>puncto a ad vim gravitatis.</s> + <s xml:id="echoid-s7226" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7227" xml:space="preserve">Demonſtravit autem Hugenius vim centrifugam corporis in gyrum acti <lb/>celeritate, quam lapſu libero per altitudinem dimidii radii acquirere poſſit, æqua-<lb/>lem eſſe vi ſuæ gravitatis: </s> + <s xml:id="echoid-s7228" xml:space="preserve">quod ſi proinde altitudo reſpondens guttulæ veloci-<lb/>tati gyratoriæ dicatur V; </s> + <s xml:id="echoid-s7229" xml:space="preserve">vis gravitalis g: </s> + <s xml:id="echoid-s7230" xml:space="preserve">erit vis centrifuga = {2gV/y}, unde <lb/>dx: </s> + <s xml:id="echoid-s7231" xml:space="preserve">dy = {2gV/y}: </s> + <s xml:id="echoid-s7232" xml:space="preserve">g, vel dx = {2Vdy/y}.</s> + <s xml:id="echoid-s7233" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7234" xml:space="preserve">§. </s> + <s xml:id="echoid-s7235" xml:space="preserve">3. </s> + <s xml:id="echoid-s7236" xml:space="preserve">Si ponatur V = {1/2} y, fiet x = y & </s> + <s xml:id="echoid-s7237" xml:space="preserve">proinde linea E O erit <lb/>recta conſtituens cum axe G H angulum ſemirectum habebitque cavitas for-<lb/>mam coni: </s> + <s xml:id="echoid-s7238" xml:space="preserve">Si vero ſervata eadem proportione velocitatum, quæ nempe ſint <lb/>ubique radicibus diſtantiarum ab axe proportionales, aquæ celerius tardiuſve <lb/>circumagantur, fiet angulus E O G eo acutior, quo celerius moventur, ita ut <lb/>ſi infinita fuerit velocitas, tunc aquæ perpendiculariter fundo inſiſtere debeant <lb/>inſtar muri, cavitatemque cylindricam interius formare, ſi modo operculum <lb/>ſit in A D, quod impediat, quominus aquæ omnes ejiciantur.</s> + <s xml:id="echoid-s7239" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7240" xml:space="preserve">§. </s> + <s xml:id="echoid-s7241" xml:space="preserve">4. </s> + <s xml:id="echoid-s7242" xml:space="preserve">Si ponatur paullo generalius 2 V = fy<emph style="super">e</emph>, fiet dx = fy<emph style="super">e - 1</emph> dy <lb/>vel x = {f/e}y<emph style="super">e</emph>: </s> + <s xml:id="echoid-s7243" xml:space="preserve">Hinc ſequitur curvam ſemper fore verſus axem conca-<lb/>vam, ut in figura 65, ſi ſit e major unitate atque convexam, ut in fig. </s> + <s xml:id="echoid-s7244" xml:space="preserve">66. </s> + <s xml:id="echoid-s7245" xml:space="preserve">ſi ſit <lb/>minor. </s> + <s xml:id="echoid-s7246" xml:space="preserve">In priori cafu eſt angulus E O G ſemper rectus, in altero ſemper nul-<lb/>lus: </s> + <s xml:id="echoid-s7247" xml:space="preserve">in ſolo caſu quo e = 1 poteſt angulus iſte eſſe qualiſcunque.</s> + <s xml:id="echoid-s7248" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7249" xml:space="preserve">§. </s> + <s xml:id="echoid-s7250" xml:space="preserve">5. </s> + <s xml:id="echoid-s7251" xml:space="preserve">Inſervire poſſunt hæc ad dignoſcendam quodammodo ſcalam <lb/>velocitatum in vortice artificioſe producto: </s> + <s xml:id="echoid-s7252" xml:space="preserve">ſi enim ſuperficiem videas conca-<lb/>vam, recte judicabis velocitates majori creſcre ratione, quam diſtantiæ ab axe <lb/>creſcant, ſi convexam contrarium deduces. </s> + <s xml:id="echoid-s7253" xml:space="preserve">Si curva non videatur ad parabo-<lb/>licum genus pertinere, indicium erit velocitates non poſſe comparari cum di-<lb/>ſtantiarum determinata aliqua potentia. </s> + <s xml:id="echoid-s7254" xml:space="preserve">Quo major obſervata fuerit linea E M <lb/>terminata ab horizontali O M, eo major putabitur velocitas particularum ab-<lb/>ſoluta ſeu littera f.</s> + <s xml:id="echoid-s7255" xml:space="preserve"/> +</p> +<pb o="246" file="0260" n="260" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s7256" xml:space="preserve">§. </s> + <s xml:id="echoid-s7257" xml:space="preserve">6. </s> + <s xml:id="echoid-s7258" xml:space="preserve">Exiſtimo autem non poſſe vorticem in ſtatu ſuo per tempus aliquod <lb/>notabile permanere, ſi vires centrifugæ partium æqualium in fluido homoge-<lb/>neo creſcant ab axe verſus peripheriam: </s> + <s xml:id="echoid-s7259" xml:space="preserve">hoc enim ſi eſſet, cum nihil ſit, quod <lb/>partium axi viciniorum vim centrifugam ſufficienter coërceat, fieret utique, ut <lb/>partes illæ viciniores perpetuo ab ax@ recederent, remotioresque ad illum <lb/>propellerent, neque unquam in hoc ſtatu æquilibrium aut ſtatus durationis <lb/>obtineri poſſet. </s> + <s xml:id="echoid-s7260" xml:space="preserve">Apparet inde quantitatem hanc {2gV/y} (quæ nempe in fluidis <lb/>homogeneis vim centrifugam partium æqualium exprimit) aut una creſcere <lb/>cum γ aut ſaltem non decreſcere, atque ſic ſi rurſus ad ſpecialem hypotheſin <lb/>antea factam (2V = fy<emph style="super">e</emph>) deſcendamus, non poterit e eſſe minor unitate. </s> + <s xml:id="echoid-s7261" xml:space="preserve">Igi-<lb/>tur in omnibus vorticibus, de quibus hic ſermo eſt, ad ſtatum durationis re-<lb/>ductis, ſuperficies nunquam convexa erit, ut in figura 66, ſed ſemper aut con-<lb/>cava, ut in figura 65. </s> + <s xml:id="echoid-s7262" xml:space="preserve">aut conica: </s> + <s xml:id="echoid-s7263" xml:space="preserve">& </s> + <s xml:id="echoid-s7264" xml:space="preserve">quia e vel major eſt unitate vel eidem æqua-<lb/>lis, aliter fieri non poteſt, quin velocitates aut æquali aut majori ratione cre-<lb/>ſcant cum radicibus diſtantiarum ab axe. </s> + <s xml:id="echoid-s7265" xml:space="preserve">Hæc cum ita mecum perpendo, non <lb/>intelligo quemadmodum Newtonus fingere ſibi potuerit duos vortices fluidi <lb/>ubique homogenei ad ſtatum perpetuæ durationis reductos, in quorum altero <lb/>tempora periodica partium ſint ut earum diſtantiæ ab axe cylindri, in altero au-<lb/>tem ut quadrata diſtantiarum à centro ſphæræ: </s> + <s xml:id="echoid-s7266" xml:space="preserve">Nam in horum vorticum altero <lb/>velocitates ubique eſſent æquales, & </s> + <s xml:id="echoid-s7267" xml:space="preserve">in altero plane decreſcerent ab axe ver-<lb/>ſus peripheriam.</s> + <s xml:id="echoid-s7268" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7269" xml:space="preserve">Magis veroſimile eſt, in pleriſque vorticibus, qui ſtatum perduratio-<lb/>nis jam attigerint, fluidi ſive homogenei ſive heterogenei partium ſingularum <lb/>tempora periodica eadem fore, quaſi totus cylindrus ſolidus fuerit, partes au-<lb/>tem quæ ſint ſpecifice graviores circumferentiæ, viciniores futuras eſſe. </s> + <s xml:id="echoid-s7270" xml:space="preserve">In hoc <lb/>caſu fit v proportionale ipſi y & </s> + <s xml:id="echoid-s7271" xml:space="preserve">V proportionale ejusdem quadrato, curvaque <lb/>E O F erit parabola Apolloniana, cujus vertex in O & </s> + <s xml:id="echoid-s7272" xml:space="preserve">cujus axis ſit O G.</s> + <s xml:id="echoid-s7273" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7274" xml:space="preserve">Præſertim hæc ita proxime fore præſumo, ſi vortex generetur à rota-<lb/>tione vaſis cylindrici circa axem H G, vel etiam ab agitatione uniformi baculi <lb/>juxta latera vaſis, cujuſmodi vorticum phænomena expoſuit D. </s> + <s xml:id="echoid-s7275" xml:space="preserve">Saulmon i@ <lb/>Comm. </s> + <s xml:id="echoid-s7276" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s7277" xml:space="preserve">Reg. </s> + <s xml:id="echoid-s7278" xml:space="preserve">ſc. </s> + <s xml:id="echoid-s7279" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s7280" xml:space="preserve">a. </s> + <s xml:id="echoid-s7281" xml:space="preserve">1716.</s> + <s xml:id="echoid-s7282" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7283" xml:space="preserve">§. </s> + <s xml:id="echoid-s7284" xml:space="preserve">7. </s> + <s xml:id="echoid-s7285" xml:space="preserve">Preſſiones quas diverſæ cylindri A B C D partes à fluido ſuſtinent, +<pb o="247" file="0261" n="261" rhead="SECTIO UNDECIMA."/> +proportionales ſunt altitudinibus columnarum verticalium iisdem partibus re-<lb/>ſpondentium: </s> + <s xml:id="echoid-s7286" xml:space="preserve">neque enim requiritur, ut huic ponderi conatum fluidi à vi <lb/>centrifuga oriundum addamus, quia conatus iſte effectum jam obtinuit in ele-<lb/>vandis aquis: </s> + <s xml:id="echoid-s7287" xml:space="preserve">Atque ſi vas non fuerit cylindricum ſed irregularis utcunque <lb/>ſtructuræ, licebit cylindrum fingere, cujus axis coincidat cum axe rotatio-<lb/>nis, fluido ita plenum, ut punctum Otam in vaſe propoſito quam in cylindro <lb/>fictitio in eodem loco poſitum ſit: </s> + <s xml:id="echoid-s7288" xml:space="preserve">tanta enim in quovis cylindri puncto preſ-<lb/>ſio erit, quanta eſt in eodem puncto, quatenus id ad vas propoſitum pertinet. <lb/></s> + <s xml:id="echoid-s7289" xml:space="preserve">Apparet ex hoc ipſo, poſſe ſuperficies vorticum ex alio principio quam quo an-<lb/>te uſi ſumus definiri: </s> + <s xml:id="echoid-s7290" xml:space="preserve">Ducta nempe linea horizontali O M & </s> + <s xml:id="echoid-s7291" xml:space="preserve">verticali N a cum <lb/>ſua infinite propinque p n ſequitur altitudinem N a ſeu O g proportionalem eſſe <lb/>vi centrifugæ omnium particularum quæ ſunt in O N & </s> + <s xml:id="echoid-s7292" xml:space="preserve">differentiam altitudi-<lb/>num duarum proximarum, nempe a m ſeu gf, proportionalem vi centrifugæ <lb/>particulæ N p: </s> + <s xml:id="echoid-s7293" xml:space="preserve">Unde rurſus derivatur æquatio finalis, quam §. </s> + <s xml:id="echoid-s7294" xml:space="preserve">2. </s> + <s xml:id="echoid-s7295" xml:space="preserve">dedimus, nem-<lb/>pe dx = {2 V dy/y}.</s> + <s xml:id="echoid-s7296" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7297" xml:space="preserve">§. </s> + <s xml:id="echoid-s7298" xml:space="preserve">8. </s> + <s xml:id="echoid-s7299" xml:space="preserve">Videamus nunc quid accidere debeat corporibus vortici inna-<lb/>tantibus; </s> + <s xml:id="echoid-s7300" xml:space="preserve">ut autem quæſtio eo diſtinctior atque ſimplicior fiat, corporis loco <lb/>conſiderabimus globulum parvum ejusdem cum fluido vorticoſo gravitatis ſpe-<lb/>cificæ.</s> + <s xml:id="echoid-s7301" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7302" xml:space="preserve">Globulus talis fluido commiſſus duabus ſtatim potentiis ſollicitatur, <lb/>altera tangentiali ab impetu fluidi ortum trahente, altera centripeta, quæ à <lb/>vi fluidi centrifuga naſcitur. </s> + <s xml:id="echoid-s7303" xml:space="preserve">Iſtæ vires conſtantem ſervant inter ſe rationem, <lb/>quadratam nempe velocitatis fluidi reſpectivæ; </s> + <s xml:id="echoid-s7304" xml:space="preserve">ſive quieſcat corpus ſive motu <lb/>circulari feratur.</s> + <s xml:id="echoid-s7305" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7306" xml:space="preserve">Notari autem meretur ab iis, qui in explicandis gravitatis phænomenis, <lb/>adhærent principiis Carteſianis, vim tangentialem eſſe incomparabiliter majo-<lb/>rem vi centripeta: </s> + <s xml:id="echoid-s7307" xml:space="preserve">eſt enim illa ad hanc, ut diſtantia corporis ab axe vorticis <lb/>ad octo tertias partes diametri globi; </s> + <s xml:id="echoid-s7308" xml:space="preserve">demonſtrationem videre eſt in Comment. <lb/></s> + <s xml:id="echoid-s7309" xml:space="preserve">Acad, Petrop. </s> + <s xml:id="echoid-s7310" xml:space="preserve">tom. </s> + <s xml:id="echoid-s7311" xml:space="preserve">II. </s> + <s xml:id="echoid-s7312" xml:space="preserve">p. </s> + <s xml:id="echoid-s7313" xml:space="preserve">318. </s> + <s xml:id="echoid-s7314" xml:space="preserve">& </s> + <s xml:id="echoid-s7315" xml:space="preserve">319.</s> + <s xml:id="echoid-s7316" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7317" xml:space="preserve">§. </s> + <s xml:id="echoid-s7318" xml:space="preserve">9. </s> + <s xml:id="echoid-s7319" xml:space="preserve">Quamvis ſciam multa à variis allegata fuiſſe, ut oſtenderent, ma-<lb/>t<unsure/>eriam ſubtilem celerrime in vorticem actam corpora quidem verſus axem de-<lb/>trudere poſſe neque tamen inde ſequi, ut ſimul à vortice deferantur iſta corpo- +<pb o="248" file="0262" n="262" rhead="HYDRODYNAMICÆ"/> +ra, non potui tamen hunc mihi ſcrupulum eximere, poſtquam cognovi vim <lb/>tangentialem vi centripeta eſſe pene infinite majorem. </s> + <s xml:id="echoid-s7320" xml:space="preserve">An non melius huic dif-<lb/>ficultati occurritur, ſi duos ſuper eodem axe vortices ſtatuamus contrarios & </s> + <s xml:id="echoid-s7321" xml:space="preserve"><lb/>æqualis virtutis: </s> + <s xml:id="echoid-s7322" xml:space="preserve">Videtur enim, phænomena naturæ plurima conciliari non <lb/>poſſe cum vorticum hypotheſi, niſi ponamus duos plureſve vortices liberri-<lb/>me ſub qualicunque directione ſe invicem trajicere poſſe: </s> + <s xml:id="echoid-s7323" xml:space="preserve">vel ſola gravitatio <lb/>communis omnium corporum cæleſtium verſus ſe invicem, quæ in dubium <lb/>vocari nequit, ſatis oſtendit aut valedicendum eſſe hypotheſi vorticum, aut li-<lb/>berrimam vorticum plurium in omnes plagas decusſationem ſtatuendam eſſe-<lb/>Si igitur duo vortices æqualis virtutis contrarii ſuper eodemque axe fingerentur, <lb/>tunc impetus contrarii deſtruerent vires utriuſque vorticis tangentiales; </s> + <s xml:id="echoid-s7324" xml:space="preserve">ſimul <lb/>autem uterque vortex concurreret ad corpus verſus axem communem depri-<lb/>mendum.</s> + <s xml:id="echoid-s7325" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7326" xml:space="preserve">§. </s> + <s xml:id="echoid-s7327" xml:space="preserve">10. </s> + <s xml:id="echoid-s7328" xml:space="preserve">Altera accedit difficultas, quominus poſſit corporum gravitas <lb/>peti ex effectu duorum vorticum contrariorum ſuper eodem axe motorum. </s> + <s xml:id="echoid-s7329" xml:space="preserve">Ita <lb/>enim corpora non verſus punctum commune aut quaſi punctum ſed verſus <lb/>axem gravitarent, motuque ad eundem perpendiculari laberentur, quod cum <lb/>deſcenſu corporum verticali & </s> + <s xml:id="echoid-s7330" xml:space="preserve">rotunditate vel quaſi rotunditate terræ corpo-<lb/>rumque cœleſtium pugnat.</s> + <s xml:id="echoid-s7331" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7332" xml:space="preserve">Huic alteri quoque difficultati occurretur, ſi fingantur duo axes ad ſe <lb/>invicem perpendiculares aut proxime tales, circa quorum utrumque duo vor-<lb/>tices contrarii æqualis virtutis circumagantur. </s> + <s xml:id="echoid-s7333" xml:space="preserve">Namque vis compoſita omnium <lb/>vorticum ita intelligi poteſt comparata, ut corpus detrudat proxime verſus pun-<lb/>ctum, quo ambo axes ſe invicem interſecant; </s> + <s xml:id="echoid-s7334" xml:space="preserve">ſemper tamen foret terra ali-<lb/>quantum compreſſa verſus planum per ambos axes tranſiens. </s> + <s xml:id="echoid-s7335" xml:space="preserve">Poterit autem vel <lb/>huic incommodo, ſi modo incommodum ſit, obviam iri, multiplicando ad-<lb/>modum vorticum numerum: </s> + <s xml:id="echoid-s7336" xml:space="preserve">nam ſi vel infiniti fere ſtatuantur vortices, pote-<lb/>runt omnes eadem facilitate ſe trajicere, ac radii luminis, qui ſe minime im-<lb/>pediunt.</s> + <s xml:id="echoid-s7337" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7338" xml:space="preserve">Volui iſta hic adjicere in gratiam eorum, qui vorticibus delectantur, <lb/>ut videant, an motus iſte facilius concipi poſſit eo, quem Hugenius finxit: <lb/></s> + <s xml:id="echoid-s7339" xml:space="preserve">utroque enim phænomena naturæ æqualiter explicari poſſunt. </s> + <s xml:id="echoid-s7340" xml:space="preserve">Hanc ſenten-<lb/>tiam paullo accuratius expoſui in diſſertatione, quam Academia Reg. </s> + <s xml:id="echoid-s7341" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s7342" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s7343" xml:space="preserve"><lb/>præmio a. </s> + <s xml:id="echoid-s7344" xml:space="preserve">1734. </s> + <s xml:id="echoid-s7345" xml:space="preserve">affectam imprimi curavit.</s> + <s xml:id="echoid-s7346" xml:space="preserve"/> +</p> +<pb o="249" file="0263" n="263" rhead="SECTIO UNDECIMA."/> +<p> + <s xml:id="echoid-s7347" xml:space="preserve">§. </s> + <s xml:id="echoid-s7348" xml:space="preserve">11. </s> + <s xml:id="echoid-s7349" xml:space="preserve">Quia dubitari nequit, quin omnes planetæ verſus ſolem & </s> + <s xml:id="echoid-s7350" xml:space="preserve">ſatellites ver-<lb/>ſus ſuos planetas ad mentem Newtoni gravitent, hujusque gravitatis cauſa affinis <lb/>ſit cum illa qua corpora terreſtria verſus centrum terræ tendunt, erit vorticum <lb/>hypotheſis ad totum ſyſtema mundi extendenda, ſi pro gravitate corporum <lb/>terreſtrium explicanda adhibeatur. </s> + <s xml:id="echoid-s7351" xml:space="preserve">Ita vero planetæ, materiæ ſubtili innatan-<lb/>tes, moverentur in medio reſiſtente, paulatimque de motu ſuo aliquid per-<lb/>dentes ad centrum ſolis accedere ſub forma ſpiralis deberent: </s> + <s xml:id="echoid-s7352" xml:space="preserve">hoc vero cum ex <lb/>antiquiſſimis obſervationibus non appareat, poſtulat vorticum hypotheſis, ut <lb/>fluidum vorticoſum ponatur ſupra modum rarum atque ſubtile idque veloci-<lb/>tate, quam mens humana vix aſſequi poſſit, motum: </s> + <s xml:id="echoid-s7353" xml:space="preserve">quo enim rarius flui-<lb/>dum, eo celerius motum fingas neceſſe eſt. </s> + <s xml:id="echoid-s7354" xml:space="preserve">Fortaſſe oportunius motuum <lb/>perennitas explicabitur à communicatione quadam motus reciproca, ita ut <lb/>quas modo corpus cœleſte propulſit particulas, ab his alio tempore vi ſimili <lb/>propellatur.</s> + <s xml:id="echoid-s7355" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7356" xml:space="preserve">§. </s> + <s xml:id="echoid-s7357" xml:space="preserve">12. </s> + <s xml:id="echoid-s7358" xml:space="preserve">Venio jam ad reliquas corporum gravitantium proprietates, quæ <lb/>ex hypotheſi vorticum ſequuntur. </s> + <s xml:id="echoid-s7359" xml:space="preserve">Ponamus itaque corpus in fluido vortico-<lb/>ſo quieſcens, quod nullas fluidi particulas per poros ſuos tranſmittat: </s> + <s xml:id="echoid-s7360" xml:space="preserve">ita ten-<lb/>det corpus verſus centrum vorticis, eritque vis ejus centripeta præciſe æqua-<lb/>lis vi centrifugæ fluidi vorticoſi, quod ſub ſimili volumine in eadem à centro <lb/>diſtantia poſitum ſit. </s> + <s xml:id="echoid-s7361" xml:space="preserve">Ergo corpora quæcunque in ſimili vorticis loco conſtitu-<lb/>ta eandem habent vim centripetam ſi idem habeant volumen, etiamſi quanti-<lb/>tates materiæ in uno quoque corpore ſint utcunque inæquales, & </s> + <s xml:id="echoid-s7362" xml:space="preserve">ſi hujusmo-<lb/>di corpora libere verſus centrum vorticis moveri poſſint, ferentur velocitati-<lb/>bus inæqualibus reciproce ſcilicet proportionalibus quantitatum materiæ ra-<lb/>dicibus quadratis, ſi ſpatia emenſa ſint æqualia.</s> + <s xml:id="echoid-s7363" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7364" xml:space="preserve">§. </s> + <s xml:id="echoid-s7365" xml:space="preserve">13. </s> + <s xml:id="echoid-s7366" xml:space="preserve">Quæ in præcedente paragrapho monita ſunt, facile applicantur <lb/>gravitati corporum, ſi modo principium gravitatis ſit vis centrifuga alicujus <lb/>materiæ ſubtilis celerrime in vorticem actæ. </s> + <s xml:id="echoid-s7367" xml:space="preserve">Quia vero experientia docet om-<lb/>nia corpora terreſtria in vacuo ſimili deſcendere velocitate omniaque corpora <lb/>è filo ſuſpenſa æquali vibrationes facere tautochronas, inde concludemus, par-<lb/>ticulas ultimas graves, per quas nempe fluidum gravificum penetrare nequeat, <lb/>in omnibus corporibus terreſtribus eſſe æqualis denſitatis ſpecificæ, id eſt, ſub <lb/>æqualibus voluminibus æquales materiæ ſolidæ quantitates continere, idque non <lb/>minus in particulis gravibus, quæ aurum quam quæ plumas componunt. </s> + <s xml:id="echoid-s7368" xml:space="preserve">Ne +<pb o="250" file="0264" n="264" rhead="HYDRODYNAMICÆ"/> +vero hæc ſecus ac volo explicentur dicendum mihi erit, quid intelligam per <lb/>ultimas particulas graves & </s> + <s xml:id="echoid-s7369" xml:space="preserve">per ma@e<unsure/>riam ſolidam ipſis inſitam.</s> + <s xml:id="echoid-s7370" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7371" xml:space="preserve">§. </s> + <s xml:id="echoid-s7372" xml:space="preserve">14. </s> + <s xml:id="echoid-s7373" xml:space="preserve">Sunt igitur particulæ graves proprie ſic dictæ illæ, quæ impene-<lb/>trabiles ſunt materiæ ſubtili vorticoſæ: </s> + <s xml:id="echoid-s7374" xml:space="preserve">hujuſmodi enim particulæ idem faciunt, <lb/>quod corpora in vortice poſita, de quibus §. </s> + <s xml:id="echoid-s7375" xml:space="preserve">12. </s> + <s xml:id="echoid-s7376" xml:space="preserve">diximus: </s> + <s xml:id="echoid-s7377" xml:space="preserve">quamvis autem <lb/>impenetrabiles ſint materiæ ſubtili modo dictæ, non crediderim tamen illas <lb/>perfecte ſolidas, quales Hugenius præſumſiſſe videtur in tract. </s> + <s xml:id="echoid-s7378" xml:space="preserve">ſuo de gravitate, <lb/>id eſt tales quorum ſpatium totum materia repletum ſit ſine poris aut fluido <lb/>interfluo: </s> + <s xml:id="echoid-s7379" xml:space="preserve">exiſtimo potius has particulas graves ſuos rurſus habere poros, at-<lb/>que in illis fluidum aliud eſſe longum ſubtilius, quod particulas graves ea-<lb/>dem libertate trajicit, qua fluidum gravificum fluit per corpora ſenſibilia: </s> + <s xml:id="echoid-s7380" xml:space="preserve">re-<lb/>ſiduum vero quod in particulis gravibus ſibi cohæret voco materiam ſolidam ad <lb/>particulas easdem pertinentem.</s> + <s xml:id="echoid-s7381" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7382" xml:space="preserve">§. </s> + <s xml:id="echoid-s7383" xml:space="preserve">15. </s> + <s xml:id="echoid-s7384" xml:space="preserve">Perſpicuum ex his eſt, diverſas corporum gravitates ſpecificas <lb/>minime petendas eſſe ex diverſa denſitate particularum gravium, ſed ex eo, <lb/>quod hæ particulæ poſſint eſſe in diverſis corporibus ſub eodem volumine nu-<lb/>mero inæquales, aut etiam magnitudine, ſic ut in corporibus compactiori-<lb/>bus majorisve gravitatis ſpecificæ particulæ graves, vel minoribus interſtitiis <lb/>poſitæ vel volumine majores ſint.</s> + <s xml:id="echoid-s7385" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7386" xml:space="preserve">Etſi vero diverſas denſitates ſpecificas habuiſſent particulæ graves in diver-<lb/>ſis corporibus, non propterea diverſas habitura fuiſſent gravitates ſpecificas <lb/>corpora cæteris poſitis paribus: </s> + <s xml:id="echoid-s7387" xml:space="preserve">talia autem corpora ex alto delapſa diverſa in-<lb/>ter ſe velocitate fuiſſent deſcenſura verſus centrum terræ: </s> + <s xml:id="echoid-s7388" xml:space="preserve">Fieri itaque potuiſ-<lb/>ſet, ut corpora æqualis gravitatis ſpecificæ, vel in vacuo communiter ita di-<lb/>cto inæquali velocitate deſcendiſſent non minus atque corpora videmus diver-<lb/>ſæ gravitatis ſpecificæ æquali velocitate deſcendentia: </s> + <s xml:id="echoid-s7389" xml:space="preserve">In hujuſmodi autem cor-<lb/>poribus leges motuum longe aliæ forent, atque nunc ſunt, ubi mafſæ ex ſo-<lb/>lis ponderibus æſtimantur.</s> + <s xml:id="echoid-s7390" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7391" xml:space="preserve">§. </s> + <s xml:id="echoid-s7392" xml:space="preserve">16. </s> + <s xml:id="echoid-s7393" xml:space="preserve">Cæterum quia omnia, quantum experientia conſtat, corpora <lb/>terreſtria habent ſuas particulas graves æqualis denſitatis ſpecificæ, ut §. </s> + <s xml:id="echoid-s7394" xml:space="preserve">13. </s> + <s xml:id="echoid-s7395" xml:space="preserve">mo-<lb/>nitum fuit, facile quidem inducar, ut credam idem in omnibus planetis fieri <lb/>ſeorſim conſideratis: </s> + <s xml:id="echoid-s7396" xml:space="preserve">Planetas vero inter ſe comparatos particulas ſuas graves di- +<pb o="251" file="0265" n="265" rhead="SECTIO UNDECIMA."/> +verſæ habere denſitatis ſpecificæ mihi admodum eſt probabile, quia nullam vi-<lb/>deo rationem, cur in omnibus planetis ſimiles eſſe debeant iſtæ particulæ. </s> + <s xml:id="echoid-s7397" xml:space="preserve">Sed <lb/>à particularum gravium denſitate in quolibet planeta pendet hujus vis centrifuga <lb/>ſeu conatus recedendi à ſole. </s> + <s xml:id="echoid-s7398" xml:space="preserve">Igitur nondum licet colligere planetarum vires <lb/>centrifagas ſe habere, in ratione quadr ata reciproca eorundem diſtantiarum à <lb/>ſole ex eo, quod tempora periodica rationem ſequantur ſeſquiplicatam diſtan-<lb/>tiarum: </s> + <s xml:id="echoid-s7399" xml:space="preserve">talis enim concluſio ſupponit ſimilem in omnibus planetis particula-<lb/>rum gravium denſitatem.</s> + <s xml:id="echoid-s7400" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7401" xml:space="preserve">§. </s> + <s xml:id="echoid-s7402" xml:space="preserve">17. </s> + <s xml:id="echoid-s7403" xml:space="preserve">Planetarum vires centrifugæ æquales utique ſunt viribus contra-<lb/>riis quibus verſus ſolem trahuntur: </s> + <s xml:id="echoid-s7404" xml:space="preserve">Quia autem, ut dixi in ſuperiori paragra-<lb/>pho, nondum certum eſt, in quanam ratione reſpectu diſtantiarum à ſole vi-<lb/>res planetarum centrifugæ mutentur, ideo neque de eorum viribus gravitatis <lb/>verſus ſolem aliquid certi ſtatuere licet; </s> + <s xml:id="echoid-s7405" xml:space="preserve">Et plurima quidem ſunt in vorticum <lb/>hypotheſi, quæ vires gravitatis in diverſis diſtantiis conſtituunt & </s> + <s xml:id="echoid-s7406" xml:space="preserve">determinant.</s> + <s xml:id="echoid-s7407" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7408" xml:space="preserve">Cum enim vis gravitatis ſit æqualis vi centrifugæ materiæ ſubtilis, quæ <lb/>particulas corporis graves penetrare nequit, ſequitur eo majorem eſſe vim gra-<lb/>vitatis, quo majori materiæ ſubtilis quantitati tranſitus negatur; </s> + <s xml:id="echoid-s7409" xml:space="preserve">quia vero ſci-<lb/>mus corpus ſæpe fluido uni impenetrabile eſſe, quod alii fluido ſubtiliori li-<lb/>berrimum concedit transfluxum, fieri poteſt, ſi modo materiam vorticoſam <lb/>in diverſis à centro vorticis diſtantiis inæqualiter ſubtilem putemus, ut unus <lb/>idemque planeta in inæqualibus à ſole diſtantiis inæqualiter ad ſolem pellatur, <lb/>quod idem facilius contingere poteſt @in diverſis planetis, quia accedit diverſa <lb/>quæ eſſe poteſt particularum gravium, ſtructura.</s> + <s xml:id="echoid-s7410" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7411" xml:space="preserve">Præter hæc ſunt etiam diverſa materiæ vorticoſæ denſitas, velocitas di-<lb/>ftantiaque à centro, quæ concurrunt ad vim gravitatis formandam. </s> + <s xml:id="echoid-s7412" xml:space="preserve">Si vero <lb/>eorum ratio habeatur, apparebit poſſe quidem vires gravitatis decreſcere <lb/>creſcentibus diſtantiis à centro virium, neque tamen propterea vires centrifugas <lb/>æqualium materiæ vorticoſæ voluminum pariter decreſcere, quod poſterius <lb/>ob rationem §. </s> + <s xml:id="echoid-s7413" xml:space="preserve">6. </s> + <s xml:id="echoid-s7414" xml:space="preserve">expoſitam fieri non poſſe exiſtimo.</s> + <s xml:id="echoid-s7415" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7416" xml:space="preserve">l<unsure/>ſta vero quæ generaliter & </s> + <s xml:id="echoid-s7417" xml:space="preserve">obiter diſputavimus de natura vorticum eo-<lb/>rumque ad Phænomena gravitatis applicatione, ſufficiant: </s> + <s xml:id="echoid-s7418" xml:space="preserve">animus non fuit +<pb o="252" file="0266" n="266" rhead="HYDRODYNAMICÆ"/> +vorticum commendare hypotheſin, ſed quasdam tantum inde concluſiones <lb/>facere, ſine quibus ipſam hypotheſin ſubſiſtere non poſſe crediderim.</s> + <s xml:id="echoid-s7419" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7420" xml:space="preserve">Venio jam ad alteram ſectionis partem, qua breviter conſiderabimus <lb/>ſtatum fluidorum, quæ intra vaſa mota continentur: </s> + <s xml:id="echoid-s7421" xml:space="preserve">Argumentum eſt fertiliſ-<lb/>ſimum infinitiſque modis variabile: </s> + <s xml:id="echoid-s7422" xml:space="preserve">Sed pauca attingemus, ceu exempla, ad <lb/>quæ multa alia revocari poterunt.</s> + <s xml:id="echoid-s7423" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7424" xml:space="preserve">§. </s> + <s xml:id="echoid-s7425" xml:space="preserve">18. </s> + <s xml:id="echoid-s7426" xml:space="preserve">Si aqua in vaſe perforato contineatur ipſumque vas libere ca-<lb/>dat, ex ſe patet, nihil aquæ durante vaſis lapſu eſſe effluxurum, quia nem-<lb/>pe particulæ ſuperiores non gravitant in inferiores: </s> + <s xml:id="echoid-s7427" xml:space="preserve">Si vas motu quidem acce-<lb/>lerato deſcendat ſed tardiore quam quo corpora naturaliter in vacuo accele-<lb/>rantur, effluet aqua, ſed minori velocitate ac ſi vas quiescat: </s> + <s xml:id="echoid-s7428" xml:space="preserve">Contrarium erit, <lb/>ſi vas motu accelerato ſurſum trahatur: </s> + <s xml:id="echoid-s7429" xml:space="preserve">Denique ſi vas horizontaliter accele-<lb/>rato motu feratur (jam enim ad reliquas non attendemus directiones) fieri po-<lb/>teſt, ut velocitas aquæ effluentis major ſit vel minor velocitate ordinaria pro <lb/>ratione ſitus foraminis: </s> + <s xml:id="echoid-s7430" xml:space="preserve">Velocitates autem aquæ ſic determinabuntur.</s> + <s xml:id="echoid-s7431" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7432" xml:space="preserve">§. </s> + <s xml:id="echoid-s7433" xml:space="preserve">19. </s> + <s xml:id="echoid-s7434" xml:space="preserve">Sit v. </s> + <s xml:id="echoid-s7435" xml:space="preserve">gr. </s> + <s xml:id="echoid-s7436" xml:space="preserve">cylindus A C D B (Fig. </s> + <s xml:id="echoid-s7437" xml:space="preserve">67.) </s> + <s xml:id="echoid-s7438" xml:space="preserve">aqua plenus usque in A B, <lb/> +<anchor type="note" xlink:label="note-0266-01a" xlink:href="note-0266-01"/> +cujus fundum C D foramen habeat in E valde parvum per quod aquæ effluant, <lb/>dum interea totum vas ſurſum trahaturà pondere P deſcendente mediante funi-<lb/>culo ſuper duabus trochleis H & </s> + <s xml:id="echoid-s7439" xml:space="preserve">G excurrente. </s> + <s xml:id="echoid-s7440" xml:space="preserve">Denique conſtanter tantum <lb/>aquæ ſuperius affundi ponatur, quantum effluit per foramen E: </s> + <s xml:id="echoid-s7441" xml:space="preserve">pondus vero <lb/>cylindri & </s> + <s xml:id="echoid-s7442" xml:space="preserve">aquæ in eo contentæ indicetur per p. </s> + <s xml:id="echoid-s7443" xml:space="preserve">lta apparet quamlibet gut-<lb/>tam aquæ in vaſe veluti ſtagnantis vi animari ad aſcenſum quæ ſe habeat ad vim <lb/>gravitatis naturalem ut {P - p/P + p} ad 1: </s> + <s xml:id="echoid-s7444" xml:space="preserve">Quia vero reactio guttulæ in fundum æqua-<lb/>lis eſt vi, qua ad aſcenſum animatur quævis guttula, præter preſſionem natu-<lb/>ralem aliam exeret in fundum, quæ exprimenda erit per {P - p/P + p}. </s> + <s xml:id="echoid-s7445" xml:space="preserve">Utraquè vero <lb/>preſſio ſimul ſumta erit ad preſſionem ſolam naturalem ut {2P/P + p} ad 1, adeo <lb/>ut fundum haud ſecus ab incumbente aqua prematur, quam ſi cylindrus quieſ-<lb/>ceret eſſetque altitudo aquæ = {2P/P + p} X A C, ex quo ipſo ſequitur altitudinem <lb/>velocitati aquæ uniformiter effluentis debitam eſſe = {2P/P + p} X A C.</s> + <s xml:id="echoid-s7446" xml:space="preserve"/> +</p> +<div xml:id="echoid-div255" type="float" level="2" n="2"> +<note position="left" xlink:label="note-0266-01" xlink:href="note-0266-01a" xml:space="preserve">Fig. 67.</note> +</div> +<pb o="253" file="0267" n="267" rhead="SECTIO UNDECIMA."/> +<p> + <s xml:id="echoid-s7447" xml:space="preserve">Igitur ſi P = o, nulla effluet aqua, cadente vaſe motu naturaliter acce-<lb/>lerato: </s> + <s xml:id="echoid-s7448" xml:space="preserve">ſi P = p, effluet aqua, velocitate ordinaria, quia tunc vas quieſcit; <lb/></s> + <s xml:id="echoid-s7449" xml:space="preserve">atque ſi P = ∞, erit velocitas aquæ effluentis ad velocitatem ordinariam ut <lb/>√ 2 ad 1.</s> + <s xml:id="echoid-s7450" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7451" xml:space="preserve">§. </s> + <s xml:id="echoid-s7452" xml:space="preserve">20. </s> + <s xml:id="echoid-s7453" xml:space="preserve">Quæritur nunc quid accidere debeat fluido, quod in vaſe con-<lb/>tinetur, cui motus horizontalis uniformiter acceleratus imprimitur. </s> + <s xml:id="echoid-s7454" xml:space="preserve">Id vero <lb/>facillimum eſt videre ex hoc ſolo, quod nunc inertia particularum ceu dire-<lb/>ctioni, ſub qua vas movetur, contraria ſit horizontalis, dum gravitatis ea-<lb/>rundem eſt verticalis: </s> + <s xml:id="echoid-s7455" xml:space="preserve">Utraque vero manet conſtanter eadem.</s> + <s xml:id="echoid-s7456" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7457" xml:space="preserve">Igitur poſtquam fluidum ad ſtatum durationis ſeu permanentiæ perve-<lb/>nit, ſuperficies ejus plana erit, ſed inclinata verſus plagam motus. </s> + <s xml:id="echoid-s7458" xml:space="preserve">Angulus <lb/>autem inclinationis determinabitur ut ſequitur.</s> + <s xml:id="echoid-s7459" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7460" xml:space="preserve">Sit vas cylindricum A C D L (Fig. </s> + <s xml:id="echoid-s7461" xml:space="preserve">68.) </s> + <s xml:id="echoid-s7462" xml:space="preserve">verticaliter poſitum, quod ſu-<lb/> +<anchor type="note" xlink:label="note-0267-01a" xlink:href="note-0267-01"/> +per plano horizontali C D H, mediante pondere P ope trochleæ G vaſi anne-<lb/>xo in S movetur motu uniformiter accelerato, ſitque pondus vaſis & </s> + <s xml:id="echoid-s7463" xml:space="preserve">aquæ in <lb/>illo contentæ ad pondus P ut p ad P: </s> + <s xml:id="echoid-s7464" xml:space="preserve">gravitatio naturalis = 1; </s> + <s xml:id="echoid-s7465" xml:space="preserve">eritque niſus <lb/>cujuslibet guttulæ in directione G S ratione ſuæ gravitationis = {P/P + p}: </s> + <s xml:id="echoid-s7466" xml:space="preserve">Igi-<lb/>tur ſi A B ſit in eodem plano cum S G & </s> + <s xml:id="echoid-s7467" xml:space="preserve">cum ſuperficie aquæ, ducaturque A L, <lb/>patet actionem gravitatis naturalis fore ad reactionem à pondere P oriundam, <lb/>ut B L ad A L ſeu ut 1 ad {P/P + p}: </s> + <s xml:id="echoid-s7468" xml:space="preserve">vocatoque ſinu toto 1, fore ſinum anguli <lb/>L A B = {P/√(2PP + 2P p + pp)}.</s> + <s xml:id="echoid-s7469" xml:space="preserve"/> +</p> +<div xml:id="echoid-div256" type="float" level="2" n="3"> +<note position="right" xlink:label="note-0267-01" xlink:href="note-0267-01a" xml:space="preserve">Fig. 68.</note> +</div> +<p> + <s xml:id="echoid-s7470" xml:space="preserve">Hinc etiam intelligitur fundum C D majorem ab incumbente aqua <lb/>preſſionem pati in C quam in D, idque in ratione altitudinum A C & </s> + <s xml:id="echoid-s7471" xml:space="preserve">B D: <lb/></s> + <s xml:id="echoid-s7472" xml:space="preserve">ſique idem fundum perforetur minimo foraminulo, aquam ejectum iri velo-<lb/>citate, quæ reſpondeat altitudini columnæ verticalis ſuperincumbentis. </s> + <s xml:id="echoid-s7473" xml:space="preserve">Ita <lb/>vero erit, poſtquam omnia jam ad ſtatum permanentiæ pervenerint; </s> + <s xml:id="echoid-s7474" xml:space="preserve">ſi pon-<lb/>dus P veriabile ſit, nunquam in eodem ſitu permanebit ſuperficies A B: </s> + <s xml:id="echoid-s7475" xml:space="preserve">à <lb/>pondere autem iſto pendet velocitas, qua vas movetur in ſingulis locis. </s> + <s xml:id="echoid-s7476" xml:space="preserve">Igi-<lb/>tur ſi totum pondus auferatur, poſtquam vas jam motum acquiſiverit, per- +<pb o="254" file="0268" n="268" rhead="HYDRODYNAMICÆ"/> +get vas ſuâ velocitate moveri, ſuperficies autem aquæ declivitatem deponet, <lb/>rurſusque ad ſitum horizontalem componetur, veluti ſi quieſcat vas; </s> + <s xml:id="echoid-s7477" xml:space="preserve">in his <lb/>adeoque caſibus non eſt vaſis motus, qui fluidorum ſtatum permutet, ſed <lb/>motus variatio.</s> + <s xml:id="echoid-s7478" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7479" xml:space="preserve">§. </s> + <s xml:id="echoid-s7480" xml:space="preserve">21. </s> + <s xml:id="echoid-s7481" xml:space="preserve">Quod in præcedente paragrapho monuimus de vaſe cylindrico<unsure/> <lb/>verticaliter poſito facile extenditur ad vas cujuscunque figuræ: </s> + <s xml:id="echoid-s7482" xml:space="preserve">qualis enim <lb/>eſt inclinatio ſuperficiei aqueæ A B ad horizontem in vaſe cylindrivo, talis <lb/>erit in omnibus reliquis vaſis: </s> + <s xml:id="echoid-s7483" xml:space="preserve">preſſio autem aquæ in latera vaſis ubique de-<lb/>finietur, ſi columna concipiatur verticalis ab eo puncto, pro quo preſſio <lb/>aquæ definienda eſt, usque ad ſuperficiem aquæ, quæ cogitatione producen-<lb/>da erit, ſi id opus fuerit. </s> + <s xml:id="echoid-s7484" xml:space="preserve">Si loco vaſis ſumatur v. </s> + <s xml:id="echoid-s7485" xml:space="preserve">gr. </s> + <s xml:id="echoid-s7486" xml:space="preserve">tubus ab utraque par-<lb/>te inflexus, veluti A C D L (Fig. </s> + <s xml:id="echoid-s7487" xml:space="preserve">69.) </s> + <s xml:id="echoid-s7488" xml:space="preserve">isque moveatur in directione C D, <lb/> +<anchor type="note" xlink:label="note-0268-01a" xlink:href="note-0268-01"/> +tum utraque ſuperficies M, N ſitum mutabit in A, B, donec recta A B debi-<lb/>tam obtineat inclinationem antea definitam; </s> + <s xml:id="echoid-s7489" xml:space="preserve">fieri etiam poteſt ut pars aquæ <lb/>efflat per A, priusquam æquilibrium adſit: </s> + <s xml:id="echoid-s7490" xml:space="preserve">ſi crus D L deorſum ſpectet, <lb/>ut in figura 70. </s> + <s xml:id="echoid-s7491" xml:space="preserve">aqua manebit veluti ſuſpenſa: </s> + <s xml:id="echoid-s7492" xml:space="preserve">in utroque enim caſu inclina-<lb/>tio lineæ A B cæteris paribus eadem erit.</s> + <s xml:id="echoid-s7493" xml:space="preserve"/> +</p> +<div xml:id="echoid-div257" type="float" level="2" n="4"> +<note position="left" xlink:label="note-0268-01" xlink:href="note-0268-01a" xml:space="preserve">Fig. 69.</note> +</div> +<p> + <s xml:id="echoid-s7494" xml:space="preserve">In figura autem 69. </s> + <s xml:id="echoid-s7495" xml:space="preserve">erit linea M A eo major, quo longius eſt crus ho-<lb/>rizontale C D: </s> + <s xml:id="echoid-s7496" xml:space="preserve">ſic ut minimæ accelerationes aut etiam retardationes obſer-<lb/>vari poſſint, quod ſæpe aliis rebus inſervire poteſt, veluti dignoſcendis ac-<lb/>celerationibus navium, niſibusque quos exercent ſingulis remorum ſub-<lb/>merſionibus remiges; </s> + <s xml:id="echoid-s7497" xml:space="preserve">in his tamen caſibus, quia non poteſt ſtatus ſuppo-<lb/>ni durationis ſeu permanentiæ, omnis fluidi motus, qui ſingulis vicibus re-<lb/>plicatur, eſſet inquirendus.</s> + <s xml:id="echoid-s7498" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7499" xml:space="preserve">Facit eadem hæc ratio, ut nondum liceat omnino ex præmiſſis deter-<lb/>minare, quid fieri debeat cum vaſa fluidum continentia percutiuntur.</s> + <s xml:id="echoid-s7500" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7501" xml:space="preserve">Poſſunt autem regulæ percuſſionum ex ordinariis legibus preſſionum de-<lb/>duci, quandoquidem percuſſio nihil aliud ſit, niſi ingens preſſio parum durans.</s> + <s xml:id="echoid-s7502" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7503" xml:space="preserve">§. </s> + <s xml:id="echoid-s7504" xml:space="preserve">22. </s> + <s xml:id="echoid-s7505" xml:space="preserve">Sit v. </s> + <s xml:id="echoid-s7506" xml:space="preserve">gr. </s> + <s xml:id="echoid-s7507" xml:space="preserve">tubus cylindricus horizontaliter ſitus A B C D (Fig. </s> + <s xml:id="echoid-s7508" xml:space="preserve">71.) <lb/></s> + <s xml:id="echoid-s7509" xml:space="preserve"> +<anchor type="note" xlink:label="note-0268-02a" xlink:href="note-0268-02"/> +aqua plenus, impingatque globus P in tubi prominentiam A P: </s> + <s xml:id="echoid-s7510" xml:space="preserve">tunc aqua <lb/>ſubito premet vehementer fundum B A verſus P: </s> + <s xml:id="echoid-s7511" xml:space="preserve">ut hanc preſſionem recte +<pb o="255" file="0269" n="269" rhead="SECTIO UNDECIMA."/> +intelligamus, ponemus primo nullum ineſſe pondus tubo: </s> + <s xml:id="echoid-s7512" xml:space="preserve">ita apparet ex <lb/>æqualitate inter actionem & </s> + <s xml:id="echoid-s7513" xml:space="preserve">reactionem fundum durante globi impulſu non <lb/>aliter impelli ab aqua, quam pelleretur in contrariam partem à globo, ſi <lb/>hic immediate in fundum impingat. </s> + <s xml:id="echoid-s7514" xml:space="preserve">Si vero pondera aquæ & </s> + <s xml:id="echoid-s7515" xml:space="preserve">tubi rationem <lb/>habere ponantur ut p ad π, diminuetur impulſus aquæ in fundum, eritque <lb/>impulſus totus ad impulſum reſiduum ut p + π ad p; </s> + <s xml:id="echoid-s7516" xml:space="preserve">diſtribuitur enim im-<lb/>pulſus æqualiter in omnem tum aquæ tum tubi materiam, ſolumque fluidum <lb/>in fundum reagit.</s> + <s xml:id="echoid-s7517" xml:space="preserve"/> +</p> +<div xml:id="echoid-div258" type="float" level="2" n="5"> +<note position="left" xlink:label="note-0268-02" xlink:href="note-0268-02a" xml:space="preserve">Fig. 71.</note> +</div> +<p> + <s xml:id="echoid-s7518" xml:space="preserve">Nunc autem in fundo B A parvulum fingamus foramen m, ſed per id <lb/>tamen aqua liberrime fluere putetur; </s> + <s xml:id="echoid-s7519" xml:space="preserve">ita intelligimus, particulam aquæ per <lb/>foraminulum m ejectum iri durante impulſu; </s> + <s xml:id="echoid-s7520" xml:space="preserve">neque tamen quantitas iſtius <lb/>aquæ determinari poterit; </s> + <s xml:id="echoid-s7521" xml:space="preserve">pendet enim à rigiditate materiæ A P impulſum <lb/>recipientis: </s> + <s xml:id="echoid-s7522" xml:space="preserve">ſi nempe materia iſta rigidiſſima ſit, fortior preſſio ſubſtituenda <lb/>eſt impetui, ſed minus durans; </s> + <s xml:id="echoid-s7523" xml:space="preserve">conſideretur v. </s> + <s xml:id="echoid-s7524" xml:space="preserve">gr. </s> + <s xml:id="echoid-s7525" xml:space="preserve">idem impetus in duobus <lb/>diverſis caſibus: </s> + <s xml:id="echoid-s7526" xml:space="preserve">ſit autem in uno preſſio quadrupla, in altero duratio preſ-<lb/>ſionis quadrupla, quod fieri poteſt cum materia rigidior eſt in caſu priori <lb/>quam poſteriori: </s> + <s xml:id="echoid-s7527" xml:space="preserve">ita effluet in impulſu preſſionis minoris magisque durantis <lb/>dupla circiter quantitas quam in altero. </s> + <s xml:id="echoid-s7528" xml:space="preserve">Poſſunt hoc modo rigiditates ma-<lb/>teriarum explorari: </s> + <s xml:id="echoid-s7529" xml:space="preserve">ſed poſſunt etiam ex ſono.</s> + <s xml:id="echoid-s7530" xml:space="preserve"/> +</p> +<pb o="256" file="0270" n="270"/> +</div> +<div xml:id="echoid-div260" type="section" level="1" n="196"> +<head xml:id="echoid-head250" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph></head> +<head xml:id="echoid-head251" xml:space="preserve"><emph style="bf">SECTIO DUODECIMA.</emph></head> +<head xml:id="echoid-head252" style="it" xml:space="preserve">Quæ ſtaticam fluidorum motorum, quam hy-<lb/>draulico - ſtaticam voco, exhibet.</head> +<head xml:id="echoid-head253" xml:space="preserve">§ 1.</head> +<p> + <s xml:id="echoid-s7531" xml:space="preserve">INter eos, qui preſſionis fluidorum intra vaſa ſubſiſtentium men-<lb/>ſuras dederunt, pauci regulas Hydroſtaticæ vulgares, quas in <lb/>ſectione ſecunda demonſtravimus, transgreſſi ſunt: </s> + <s xml:id="echoid-s7532" xml:space="preserve">multa tamen <lb/>alia ſunt, quæ ad Hydroſtaticam proprie ſic dictam pertinent, <lb/>veluti cum actioni gravitatis vis centrifuga conjuncta eſt, aut vis inertiæ, <lb/>quod utrumque in præcedente ſectione commentati ſumus: </s> + <s xml:id="echoid-s7533" xml:space="preserve">poſſentque hu-<lb/>jusmodi vires mortuæ excogitari & </s> + <s xml:id="echoid-s7534" xml:space="preserve">combinari infinitis aliis modis. </s> + <s xml:id="echoid-s7535" xml:space="preserve">Non <lb/>vero hæc ſunt, quæ maxime deſideranda mihi videntur: </s> + <s xml:id="echoid-s7536" xml:space="preserve">cum difficile non <lb/>ſit regulas ad id negotium dare generales. </s> + <s xml:id="echoid-s7537" xml:space="preserve">Deſidero potius fluidorum ſtati-<lb/>cam, quæ intra vaſa moventur motu progreſſivo, veluti aquarum per cana-<lb/>les ad fontes ſalientes fluentium: </s> + <s xml:id="echoid-s7538" xml:space="preserve">multiplicis enim uſus eſt, nec ab ullo tra-<lb/>ctata aut ſi qui mentionem de illa feciſſe dici poſſunt, ab his minime rectè <lb/>fuit explicata: </s> + <s xml:id="echoid-s7539" xml:space="preserve">qui enim de preſſione aquarum per aquæ ductus fluentium <lb/>horumque requiſita firmitate ad preſſionem illam ſuſtinendam dixerunt, non <lb/>alias, quam pro fluidis nullo motu latis leges tradiderunt.</s> + <s xml:id="echoid-s7540" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7541" xml:space="preserve">§. </s> + <s xml:id="echoid-s7542" xml:space="preserve">2. </s> + <s xml:id="echoid-s7543" xml:space="preserve">Singulare eſt in iſta hydraulico - ſtatica, quod niſus aquarum <lb/>prius definiri non poſſit, quam motus recte fuerit cognitus, quæ ratio eſt, <lb/>quod tam diu latuit hæc doctrina; </s> + <s xml:id="echoid-s7544" xml:space="preserve">parum enim ſolliciti hactenus fuerunt Au-<lb/>ctores in motu aquarum diſquirendo, & </s> + <s xml:id="echoid-s7545" xml:space="preserve">velocitates ubique fere ex ſola aquæ <lb/>altitudine æſtimarunt: </s> + <s xml:id="echoid-s7546" xml:space="preserve">quamvis autem ſæpe motus tam cito ad hanc veloci-<lb/>tatem tendat, ut accelerationes ſenſibus plane diſtingui nequeant, & </s> + <s xml:id="echoid-s7547" xml:space="preserve">in in-<lb/>ſtanti omnis motus generari videatur, intereſt tamen, ut hæ accelerationes <lb/>recte intelligantur, quia aliter preſſiones aquarum fluentium definiri ſæpe <lb/>non poſſunt, proptereaque exiſtimavi, rem eſſe maximi momenti à motus <lb/>principio usque ad datum terminum mutationes illas utcunque momentaneas +<pb o="257" file="0271" n="271" rhead="SECTIO DUODECIMA."/> +omni cura perpendere, experimentisque confirmare, quod paſſim in hoc <lb/>tractatu, præſértim autem in ſectione tertia, feci.</s> + <s xml:id="echoid-s7548" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7549" xml:space="preserve">§. </s> + <s xml:id="echoid-s7550" xml:space="preserve">3. </s> + <s xml:id="echoid-s7551" xml:space="preserve">Si ubique motus definiri poſſet, facile foret ſtaticam in fluidis <lb/>motis generaliſſimam formare: </s> + <s xml:id="echoid-s7552" xml:space="preserve">ſi enim foramen, ſed id infinite parvum fin-<lb/>gas, eo ipſo in loco pro quo preſſio aquarum deſideratur, quæres primo <lb/>quanta velocitate aquæ per illud foraminulum ſint erupturæ & </s> + <s xml:id="echoid-s7553" xml:space="preserve">cui altitudini <lb/>illa velocitas debeatur: </s> + <s xml:id="echoid-s7554" xml:space="preserve">intelligis autem huic ipſi altitudini proportionalem <lb/>eſſe preſſionem, quam quæris.</s> + <s xml:id="echoid-s7555" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7556" xml:space="preserve">Ex hoc principio petenda eſt preſſio quam ſuſtinet lamina horizonta-<lb/>lis L Q in figura quadrageſima tertia, ſi perforata non fuerit: </s> + <s xml:id="echoid-s7557" xml:space="preserve">poſtquam enim <lb/>demonſtratum à nobis fuit in corollario ſecundo paragraphi trigeſimi primi <lb/>Sectionis octavæ, ſi foraminulum H infinite parvum fuerit ratione foraminum <lb/>M & </s> + <s xml:id="echoid-s7558" xml:space="preserve">N: </s> + <s xml:id="echoid-s7559" xml:space="preserve">ratioque horum foraminum M & </s> + <s xml:id="echoid-s7560" xml:space="preserve">N indicetur per α & </s> + <s xml:id="echoid-s7561" xml:space="preserve">γ, fore altitu-<lb/>dinem velocitati aquæ per H erumpentis debitam = {αα X LB - γγ X NQ/αα + γγ}, inde <lb/>judicabimus niſum aquæ in laminam L Q non perforatam huic ipſi altitudini <lb/>proportionalem eſſe: </s> + <s xml:id="echoid-s7562" xml:space="preserve">quod idem alio modo demonſtratum dedimus in para-<lb/>grapho decimo nono citatæ Sectionis: </s> + <s xml:id="echoid-s7563" xml:space="preserve">Hinc ſequitur fieri poſſe, ut lamina L Q <lb/>nullam preſſionem patiatur, quantumvis magna ſupra eam fuerit altitudo aquæ, <lb/>ſcilicet quando γ = α √ (L B: </s> + <s xml:id="echoid-s7564" xml:space="preserve">N Q), imo preſſionem in ſuctionem mutari <lb/>poſſe.</s> + <s xml:id="echoid-s7565" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7566" xml:space="preserve">§. </s> + <s xml:id="echoid-s7567" xml:space="preserve">4. </s> + <s xml:id="echoid-s7568" xml:space="preserve">Similiter obtinetur preſſio aquæ in laminam L Q, ſi vel hæc per-<lb/>forata fuerit foramine finitæ ratione amborum reliquorum magnitudinis. </s> + <s xml:id="echoid-s7569" xml:space="preserve">Si <lb/>enim foraminulo infinite parvo lamina præter illud, quod eſt in H, perforata <lb/>fuerit, non poteſt non velocitate communi aqua per utrumque erumpere: </s> + <s xml:id="echoid-s7570" xml:space="preserve">Et <lb/>cum hæc velocitas cognita ſit (per §. </s> + <s xml:id="echoid-s7571" xml:space="preserve">30. </s> + <s xml:id="echoid-s7572" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s7573" xml:space="preserve">8.) </s> + <s xml:id="echoid-s7574" xml:space="preserve">pro foramine H, habetur <lb/>quoque velocitas, qua aqua per foraminulum, quod nempe concipimus, <lb/>erumpere debeat, atque ſic preſſionem aquæ cognoſcimus. </s> + <s xml:id="echoid-s7575" xml:space="preserve">Fuerint v. </s> + <s xml:id="echoid-s7576" xml:space="preserve">gr. </s> + <s xml:id="echoid-s7577" xml:space="preserve">fora-<lb/>mina M, H & </s> + <s xml:id="echoid-s7578" xml:space="preserve">N inter ſe æqualia, altitudo autem B L habuerit ad altitudi-<lb/>nem L Q rationem ut 10 ad 3, erit preſſio in laminam L Q ſubdecupla illius, <lb/>quæ eſt obturatis foraminibus H & </s> + <s xml:id="echoid-s7579" xml:space="preserve">N.</s> + <s xml:id="echoid-s7580" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7581" xml:space="preserve">Denique ſi in alio loco preſſionem aquæ deſideres, addes ſaltem alti-<lb/>tudinem, qua lamina L Q ſupra illum locum eminet, altitudini jactus per ori- +<pb o="258" file="0272" n="272" rhead="HYDRODYNAMICÆ"/> +ficium H. </s> + <s xml:id="echoid-s7582" xml:space="preserve">Eadem methodus inſervit ad preſſiones aquarum in reliquis vaſis, <lb/>quæ in Sectione octava tractavimus, determinandas. </s> + <s xml:id="echoid-s7583" xml:space="preserve">Differunt autem omnes <lb/>hæ quæſtiones ab iis, quæ ad motum fluidorum per canales pertinent, quod <lb/>aquæ ob infinitam vaſorum à nobis poſitam amplitudinem veluti quieſcant in <lb/>cavitatibus & </s> + <s xml:id="echoid-s7584" xml:space="preserve">nihilominus preſſionem longe aliam exerceant, quam aliter ſo-<lb/>lent. </s> + <s xml:id="echoid-s7585" xml:space="preserve">In canalibus autem aquæ preſſionem ſuam eo magis mutant, quo majo-<lb/>ri velocitate præterfluunt, & </s> + <s xml:id="echoid-s7586" xml:space="preserve">omnem fere conſuetam preſſionem exerunt, ſi <lb/>velocitas iſta ſit valde parva.</s> + <s xml:id="echoid-s7587" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7588" xml:space="preserve">Hæc ita, cum velocitates fluidorum determinari poſſunt per methodos <lb/>jam ſuprà à nobis traditas. </s> + <s xml:id="echoid-s7589" xml:space="preserve">Singulari autem methodo res pertractanda eſt, cum <lb/>aquæ per canales fluunt, hancque doctrinam potiſſimum titulo hydraulico-ſtati-<lb/>cæ intelligo: </s> + <s xml:id="echoid-s7590" xml:space="preserve">Hic non tam preſſio ex velocitate quam reciproce velocitas, ſi <lb/>foraminulum in lateribus canalis fiat, ex preſſione definiri poteſt. </s> + <s xml:id="echoid-s7591" xml:space="preserve">Et de iſta <lb/>hydraulico-statica, cujus uſus ampliſſimus eſt, in præſenti ſectione potiſſimum <lb/>agere conſtitui.</s> + <s xml:id="echoid-s7592" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div261" type="section" level="1" n="197"> +<head xml:id="echoid-head254" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s7593" xml:space="preserve">§. </s> + <s xml:id="echoid-s7594" xml:space="preserve">5. </s> + <s xml:id="echoid-s7595" xml:space="preserve">Fuerit vas ampliſſimum A C E B (Fig. </s> + <s xml:id="echoid-s7596" xml:space="preserve">72.) </s> + <s xml:id="echoid-s7597" xml:space="preserve">aqua conſtanter ple-<lb/> +<anchor type="note" xlink:label="note-0272-01a" xlink:href="note-0272-01"/> +num conſervandum, tubo inſtructum cylindrico & </s> + <s xml:id="echoid-s7598" xml:space="preserve">horizontali E D; </s> + <s xml:id="echoid-s7599" xml:space="preserve">ſitque in <lb/>extremitate tubi foramen o aquas velocitate uniformi emittens; </s> + <s xml:id="echoid-s7600" xml:space="preserve">quæritur preſſio <lb/>aquæ in latera tubi E D.</s> + <s xml:id="echoid-s7601" xml:space="preserve"/> +</p> +<div xml:id="echoid-div261" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0272-01" xlink:href="note-0272-01a" xml:space="preserve">Fig. 72.</note> +</div> +</div> +<div xml:id="echoid-div263" type="section" level="1" n="198"> +<head xml:id="echoid-head255" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s7602" xml:space="preserve">Sit altitudo ſuperficiei aqueæ A B ſupra orificium o = a; </s> + <s xml:id="echoid-s7603" xml:space="preserve">erit velocitas<unsure/> <lb/>aquæ in o effluentis, ſi prima fluxus momenta excipias, uniformis cenſenda <lb/>& </s> + <s xml:id="echoid-s7604" xml:space="preserve">= √a, quia vas conſtanter plenum conſervari aſſumimus; </s> + <s xml:id="echoid-s7605" xml:space="preserve">poſitaque ratio-<lb/>ne amplitudinum tubi ejusque foraminis = {n/1}, erit velocitas aquæ in tu-<lb/>bo = {√a/n}: </s> + <s xml:id="echoid-s7606" xml:space="preserve">Si vero omne fundum F D abeſſet, foret velocitas ultima aquæ in <lb/>eodem tubo = √a, quæ major eſt quam a; </s> + <s xml:id="echoid-s7607" xml:space="preserve">Igitur aqua in tubo tendit ad ma-<lb/>jorem motum, niſus autem ejus ab appoſito fundo F D impeditur: </s> + <s xml:id="echoid-s7608" xml:space="preserve">Ab hoc <lb/>niſu & </s> + <s xml:id="echoid-s7609" xml:space="preserve">reniſu comprimitur aqua, quæ ipſa compreſſio coërcetur à lateribus <lb/>tubi, hæcque proinde ſimilem preſſionem ſuſtinent. </s> + <s xml:id="echoid-s7610" xml:space="preserve">Apparet ſic preſſionem +<pb o="259" file="0273" n="273" rhead="SECTIO DUODECIMA."/> +laterum proportionalem eſſe accelerationi ſeu incremento velocitatis, quod <lb/>aqua ſit acceptura, ſi in inſtanti omne obſtaculum motus evaneſcat, ſic ut <lb/>immediate in aërem ejiciatur.</s> + <s xml:id="echoid-s7611" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7612" xml:space="preserve">Res igitur jam eo perducta eſt, ut ſi durante fluxu aquæ per o, tubus <lb/>E D in temporis puncto abrumpatur in c d, quæratur quantam acceleratio-<lb/>nem guttula a c b d inde ſit perceptura: </s> + <s xml:id="echoid-s7613" xml:space="preserve">tantam enim preſſionem ſentiet par-<lb/>ticula a c in lateribus tubi ſumta à præterfluente aqua: </s> + <s xml:id="echoid-s7614" xml:space="preserve">Hunc in finem con-<lb/>ſiderandum eſt vas A B E c d C, atque pro eo invenienda acceleratio particu-<lb/>læ aqueæ effluxui proximæ, ſi hæc habuerit velocitatem {√a/n}: </s> + <s xml:id="echoid-s7615" xml:space="preserve">Iſtud nego-<lb/>tium fecimus generaliſſime in paragrapho tertio ſect. </s> + <s xml:id="echoid-s7616" xml:space="preserve">V. </s> + <s xml:id="echoid-s7617" xml:space="preserve">Attamen quia in hoc <lb/>caſu particulari brevis eſt calculus, motum in vaſe decurtato A B E c d C hic <lb/>iterum calculo ſubducemus.</s> + <s xml:id="echoid-s7618" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7619" xml:space="preserve">Sit velocitas aquæ in tubo Ed, quæ nunc ut variabilis conſideranda eſt, <lb/>= v: </s> + <s xml:id="echoid-s7620" xml:space="preserve">amplitudo tubi ut antea = n, longitudo E c = c: </s> + <s xml:id="echoid-s7621" xml:space="preserve">indicetur longi-<lb/>tudo a c particulæ aqueæ infinite parvæ & </s> + <s xml:id="echoid-s7622" xml:space="preserve">effluxui proxime per d x: </s> + <s xml:id="echoid-s7623" xml:space="preserve">Erit <lb/>guttula æqualis in E tubum ingreſſura eodem temporis puncto quo altera <lb/>a c d b ejicitur: </s> + <s xml:id="echoid-s7624" xml:space="preserve">dum autem guttula in E, cujus maſſa = n d x, tubum in-<lb/>greditur acquirit velocitatem v, atque vim vivam n v v d x, quæ vis viva tota <lb/>fuit de novo generata; </s> + <s xml:id="echoid-s7625" xml:space="preserve">nullum enim, ob amplitudinem vaſis A E infinitam, <lb/>motum guttula in E habuit tubum nondum ingreſſa: </s> + <s xml:id="echoid-s7626" xml:space="preserve">huic vi vivæ n v v d x <lb/>addendum eſt incrementum vis vivæ, quod aqua in Eb accipit, dum gut-<lb/>tula a d effluit, nempe 2 n c v d v: </s> + <s xml:id="echoid-s7627" xml:space="preserve">aggregatum debetur deſcenſui actuali guttu-<lb/>læ n d x per altitudinem B E ſeu a: </s> + <s xml:id="echoid-s7628" xml:space="preserve">habetur igitur nvvdx + 2ncvdv = nadx <lb/>ſive {vdv/dx} = {a - vv/2c}.</s> + <s xml:id="echoid-s7629" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7630" xml:space="preserve">In omni autem motu eſt incrementum velocitatis d v proportionale <lb/>preſſioni ductæ in tempuſculum quod hic eſt {d x/v}: </s> + <s xml:id="echoid-s7631" xml:space="preserve">igitur in noſtro caſu eſt <lb/>preſſio, quam guttula ad patitur, proportionalis quantitati {vdv/dx}, id eſt, quan-<lb/>titati {a - vv/2c}.</s> + <s xml:id="echoid-s7632" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7633" xml:space="preserve">Eſt vero in eo temporis puncto, quo tubus abrumpitur, v = {√α/n} <lb/>vel vv = {a/u<unsure/>n}, hic igitur valor @ſubſtituendus eſt in expreſſione {a - vv/2c}, quæ +<pb o="260" file="0274" n="274" rhead="HYDRODYNAMICÆ"/> +ſic abit in hanc alteram {nn - 1/2 nnc}a. </s> + <s xml:id="echoid-s7634" xml:space="preserve">Et hæc eſt quantitas, cui preſſio aquæ con-<lb/>tra particulam tubi a c proportionalis eſt, quamcunque amplitudinem tubus <lb/>habuerit, aut quocunque foramine ipſius fundum perforatum fuerit. </s> + <s xml:id="echoid-s7635" xml:space="preserve">Igitur <lb/>ſi in unico caſu preſſio aquæ cognita fuerit, innoteſcet ſimul in omnibus re-<lb/>liquis: </s> + <s xml:id="echoid-s7636" xml:space="preserve">talem autem habemus, nempe cum foramen eſt infinite parvum aut <lb/>n infinite magna ratione unitatis: </s> + <s xml:id="echoid-s7637" xml:space="preserve">tunc enim ex ſe patet, aquam exercere <lb/>integram ſuam preſſionem, quæ toti altitudini a convenit, hancque preſſio-<lb/>nem deſignabimus per a: </s> + <s xml:id="echoid-s7638" xml:space="preserve">ſed quando n eſt infinita, evaneſcit unitas præ nu-<lb/>mero nn, fitque quantitas cui preſſio eſt proportionalis = {a/2c}: </s> + <s xml:id="echoid-s7639" xml:space="preserve">Ergo ſi ge-<lb/>neraliter ſcire velimus, quanta ſit preſſio cum n eſt numerus qualiscunque, <lb/>talis inſtituenda eſt analogia. </s> + <s xml:id="echoid-s7640" xml:space="preserve">Si quantitati {a/2c} convenit preſſio a, quænam <lb/>erit preſſio pro quantitate {nn - 1/2 nnc} a: </s> + <s xml:id="echoid-s7641" xml:space="preserve">Et ſic invenitur preſſio quæſita = {nn - 1/nn} a. <lb/></s> + <s xml:id="echoid-s7642" xml:space="preserve">Q. </s> + <s xml:id="echoid-s7643" xml:space="preserve">E. </s> + <s xml:id="echoid-s7644" xml:space="preserve">I.</s> + <s xml:id="echoid-s7645" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div264" type="section" level="1" n="199"> +<head xml:id="echoid-head256" xml:space="preserve">Corollarium 1.</head> +<p> + <s xml:id="echoid-s7646" xml:space="preserve">§. </s> + <s xml:id="echoid-s7647" xml:space="preserve">6. </s> + <s xml:id="echoid-s7648" xml:space="preserve">Quia litera c ex calculo abiit, ſequitur omnes partes tubi, tam <lb/>eæ quæ ſunt vaſi A G propiores, quam quæ remotiores, æqualiter ab aqua <lb/>præterfluente premi, & </s> + <s xml:id="echoid-s7649" xml:space="preserve">quidem minus quam partes fundi C G: </s> + <s xml:id="echoid-s7650" xml:space="preserve">differen-<lb/>tiamque eo majorem eſſe, quo majus ſit foramen o: </s> + <s xml:id="echoid-s7651" xml:space="preserve">nullamque amplius <lb/>preſſionem ſuſtinere latera tubi, ſi in hoc omnis obex F D abſit, ſic ut ple-<lb/>no orificio aquæ effluant.</s> + <s xml:id="echoid-s7652" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div265" type="section" level="1" n="200"> +<head xml:id="echoid-head257" xml:space="preserve">Corollarium 2.</head> +<p> + <s xml:id="echoid-s7653" xml:space="preserve">§. </s> + <s xml:id="echoid-s7654" xml:space="preserve">7. </s> + <s xml:id="echoid-s7655" xml:space="preserve">Si alicubi foraminulo minimo, & </s> + <s xml:id="echoid-s7656" xml:space="preserve">quidem tali ratione foraminis <lb/>o, perforetur tubus, exiliet aqua velocitate, qua ad altitudinem {nna - a/nn} aſcen-<lb/>dere poſſit, ſi modo impedimenta aliena nihil obſtent: </s> + <s xml:id="echoid-s7657" xml:space="preserve">Erit nempe altitudo <lb/>jactus, in figura 73, ſeu ln = {nna - a/nn}. </s> + <s xml:id="echoid-s7658" xml:space="preserve">Si vero tubulus adſit verticalis, aut <lb/>etiam utcunque inclinatus g m, communicans cum tubo horizontali, ſed ita <lb/>tamen, ut extremitas tubuli inſerti non promineat intra cavitatem tubi hori-<lb/>zontalis, ne aqua præterfluens illidat in illam extremitatem, erit altitudo aquæ +<pb o="261" file="0275" n="275" rhead="SECTIO DUODECIMA."/> +verticalis gh in tubo inſerto hærentis pariter æqualis {nna - a/nn}: </s> + <s xml:id="echoid-s7659" xml:space="preserve">neque neceſſe <lb/>eſt in hoc poſteriori caſu, ut tubulus g m ſit admodum ſtrictus.</s> + <s xml:id="echoid-s7660" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div266" type="section" level="1" n="201"> +<head xml:id="echoid-head258" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s7661" xml:space="preserve">§. </s> + <s xml:id="echoid-s7662" xml:space="preserve">8. </s> + <s xml:id="echoid-s7663" xml:space="preserve">Poterit ergo hæc theoria experimento confirmari facillimo, eo <lb/>majoris futuro momenti, quod nemo adhuc hujusmodi æquilibria, quorum <lb/>uſus latiſſime patet, definiverit: </s> + <s xml:id="echoid-s7664" xml:space="preserve">quod eadem methodo niſus aquarum per ca-<lb/>nales fluentium generaliſſime obtineri poſſit pro aquæ ductibus utcunque in-<lb/>clinatis, incurvatis, amplitudinisque variatæ ac velocitate aquarum quali-<lb/>cunque; </s> + <s xml:id="echoid-s7665" xml:space="preserve">tum etiam, quod nonſolum hæcce preſſionum, ſed tota inſuper <lb/>motuum theoria, quam ſupra dedimus, hujusmodi experimentis confirme-<lb/>tur, quia arguunt, recte à nobis definitas fuiſſe accelerationes aquarum. </s> + <s xml:id="echoid-s7666" xml:space="preserve">Cu-<lb/>randum autem eſt in experimento, ut tubus horizontalis ſit interius bene <lb/>politus, perfecte cylindricus atque horizontalis: </s> + <s xml:id="echoid-s7667" xml:space="preserve">ſitque ſatis amplus, ut ab <lb/>adhæſione aquæ ad latera tubi notabile motus decrementum oriri non poſſit: <lb/></s> + <s xml:id="echoid-s7668" xml:space="preserve">vas ipſum ſit ampliſſimum atque continue plenum conſervetur. </s> + <s xml:id="echoid-s7669" xml:space="preserve">Obſervan-<lb/>dum quoque eſt, quanta ſit virtus tubulo vitreo g m aquas ſtagnantes elevan-<lb/>di, quæ virtus omnibus tubis capillaribus aut admodum ſtrictis ineſt: </s> + <s xml:id="echoid-s7670" xml:space="preserve">hæc <lb/>enim elevatio ab altitudine g h eſt ſubtrahenda: </s> + <s xml:id="echoid-s7671" xml:space="preserve">aut potius aſſumendus eſt tu-<lb/>bus æqualis craſſitiei & </s> + <s xml:id="echoid-s7672" xml:space="preserve">obturato orificio o, notandum eſt punctum m, tum-<lb/>que fluxu aquis conceſſo notandum quoque eſt punctum h: </s> + <s xml:id="echoid-s7673" xml:space="preserve">erit autem ſe-<lb/>cundum theoriam deſcenſus m h = {1/nn} X a = {1/nn} X E B.</s> + <s xml:id="echoid-s7674" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7675" xml:space="preserve">Tandem etiam attendendum eſt ad venam aquæ in o effluentis; </s> + <s xml:id="echoid-s7676" xml:space="preserve">hujus enim <lb/>contractio etiam facit, ut aqua in tubo horizontali minori transfluat velocita-<lb/>te, quam {√a/n}. </s> + <s xml:id="echoid-s7677" xml:space="preserve">De iſta contractione eamque præveniendi modo egi in Sect. </s> + <s xml:id="echoid-s7678" xml:space="preserve">IV. <lb/></s> + <s xml:id="echoid-s7679" xml:space="preserve">His autem quamvis ita occurri poſſit incommodis, ut error ſenſibilis in ex-<lb/>perimento ſupereſſe nequeat, tamen ſi majorem adhibere velimus accuratio-<lb/>nem, experimento indaganda erit quantitas aquæ dato tempore effluentis, <lb/>quæ cum amplitudine tubi comparata rectiſſime dabit velocitatem aquæ intra <lb/>tubum fluentis, quam in calculo poſuimus = {√a/n}: </s> + <s xml:id="echoid-s7680" xml:space="preserve">Si vero experimento mi-<lb/>nor inventa fuerit, talis nempe, quæ debeatur altitudini b, tunc erit proxi-<lb/>me preſſio aquæ præterfluentis = a - b.</s> + <s xml:id="echoid-s7681" xml:space="preserve"/> +</p> +<pb o="262" file="0276" n="276" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div267" type="section" level="1" n="202"> +<head xml:id="echoid-head259" xml:space="preserve">Corollarium 3.</head> +<p> + <s xml:id="echoid-s7682" xml:space="preserve">§. </s> + <s xml:id="echoid-s7683" xml:space="preserve">9. </s> + <s xml:id="echoid-s7684" xml:space="preserve">Si orificium in o prius digito obturetur, poſteaque fluxus aqui@ <lb/>concedatur, mutatur à primo fluxus momento preſſio a in preſſionem {nna - a/nn}: <lb/></s> + <s xml:id="echoid-s7685" xml:space="preserve">iſta vero preſſionum mutatio non fit in inſtanti; </s> + <s xml:id="echoid-s7686" xml:space="preserve">imo ſi accurate loquendum <lb/>eſt, fit demum poſt tempus infinitum, quia, ut vidimus in ſectione quinta, <lb/>omnis aquarum velocitas, quanta in calculo à nobis poſita fuit integræ altitu-<lb/>dini a reſpondens, nunquam accurate adeſt: </s> + <s xml:id="echoid-s7687" xml:space="preserve">attamen incredibili acceleratio-<lb/>ne ſtatim poſt primas ejectas guttulas ad hanc velocitatem tendunt, ita ut <lb/>totam, quantum ſenſibus dijudicari poteſt, ſine mora ulla ſenſibili acquiſiviſſe <lb/>videantur, niſi prælongi ſint aquæ ductus, tum enim aquarum acceleratio-<lb/>nes oculis diſtincte dijudicari poſſunt, cujus rei exemplum dedi in Sect. </s> + <s xml:id="echoid-s7688" xml:space="preserve">V. </s> + <s xml:id="echoid-s7689" xml:space="preserve"><lb/>§. </s> + <s xml:id="echoid-s7690" xml:space="preserve">13. </s> + <s xml:id="echoid-s7691" xml:space="preserve">In his igitur canalibus aquas ex caſtello longiſſime ſito ad fontem ſa-<lb/>lientem ducentibus, ſi preſſiones alicubi experimento explorentur eo quo ſu-<lb/>pra dixi modo, invenietur preſſionem celeriter quidem, nec tamen in in-<lb/>ſtanti diminui, preſſionumque intervalla dignoſcere licebit.</s> + <s xml:id="echoid-s7692" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7693" xml:space="preserve">Ut vero generaliter niſus aquarum definiatur, ponenda eſt, pro v ea ve-<lb/>locitas, quam aqua eo ipſo in loco temporisque puncto, quibus niſus deſi-<lb/>deratur, habet, ſique ea velocitas convenire intelligatur altitudini b, erit niſus <lb/>aquarum = a - b. </s> + <s xml:id="echoid-s7694" xml:space="preserve">Unde collatis cum præſentibus his quæ in ſectione quinta <lb/>tradita fuerunt, definire licebit quanta ſingulis momentis preſſio futura ſit.</s> + <s xml:id="echoid-s7695" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7696" xml:space="preserve">Ex his non obſcurum eſt prævidere leges hujuſce hydraulico-ſtaticæ, ſi & </s> + <s xml:id="echoid-s7697" xml:space="preserve"><lb/>figura vaſis & </s> + <s xml:id="echoid-s7698" xml:space="preserve">aquarum per canales transfluentium velocitas pro lubitu fingan-<lb/>tur qualeſcunque. </s> + <s xml:id="echoid-s7699" xml:space="preserve">Erit nempe preſſio aquarum conſtanter = a - b, ubi per <lb/>a intelligitur altitudo debita velocitati, quacum aqua abrupto canali vaſeque <lb/>conſtanter pleno conſervato poſt tempus infinitum effluxura ſit, & </s> + <s xml:id="echoid-s7700" xml:space="preserve">per b al-<lb/>titudo debita velocitati, qua cum aqua actu transfluit. </s> + <s xml:id="echoid-s7701" xml:space="preserve">Mirum ſane eſt ſim-<lb/>pliciſſimam hanc regulam, quam natura affectat, adhuc latere potuiſſe. </s> + <s xml:id="echoid-s7702" xml:space="preserve">Ja@@ <lb/>igitur illam demonſtrabo expreſſius.</s> + <s xml:id="echoid-s7703" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div268" type="section" level="1" n="203"> +<head xml:id="echoid-head260" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s7704" xml:space="preserve">§. </s> + <s xml:id="echoid-s7705" xml:space="preserve">10. </s> + <s xml:id="echoid-s7706" xml:space="preserve">Invenire preſſionem aquæ, per canalem utcunque formatum at-<lb/>que inclinatum, velocitate quacunque fluentis uniformi.</s> + <s xml:id="echoid-s7707" xml:space="preserve"/> +</p> +<pb o="263" file="0277" n="277" rhead="SECTIO DUODECIMA."/> +</div> +<div xml:id="echoid-div269" type="section" level="1" n="204"> +<head xml:id="echoid-head261" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s7708" xml:space="preserve">Sit canalis A C D (Fig. </s> + <s xml:id="echoid-s7709" xml:space="preserve">74.) </s> + <s xml:id="echoid-s7710" xml:space="preserve">per cujus foramen o transfluere ponantur <lb/> +<anchor type="note" xlink:label="note-0277-01a" xlink:href="note-0277-01"/> +aquæ velocitate uniformi & </s> + <s xml:id="echoid-s7711" xml:space="preserve">tali quæ debeatur altitudini verticali o S: </s> + <s xml:id="echoid-s7712" xml:space="preserve">ducatur <lb/>S N & </s> + <s xml:id="echoid-s7713" xml:space="preserve">fingatur vas infinite amplum N M Q Paquis plenum usque in N P, ex <lb/>quo canalis aquas ſuas perpetuo & </s> + <s xml:id="echoid-s7714" xml:space="preserve">æquabiliter hauriat: </s> + <s xml:id="echoid-s7715" xml:space="preserve">hæc ideo ſic fingo, ut <lb/>cauſa adſit ſeu vis propellens uniformis, quæ aquas data velocitate propellat <lb/>ſeu fluxum aquarum conſervet æquabilem: </s> + <s xml:id="echoid-s7716" xml:space="preserve">Et ſine hac hypothef<unsure/>i problema <lb/>noſtrum foret indeterminatum, quia velocitas eadem in eodem canali infini-<lb/>tis modis ad temporis punctum generari poteſt & </s> + <s xml:id="echoid-s7717" xml:space="preserve">propterea, ut habeatur <lb/>menſura cauſæ aquas propellentis, fingenda eſt uniformitas in motu aquarum.</s> + <s xml:id="echoid-s7718" xml:space="preserve"/> +</p> +<div xml:id="echoid-div269" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0277-01" xlink:href="note-0277-01a" xml:space="preserve">Fig. 74.</note> +</div> +<p> + <s xml:id="echoid-s7719" xml:space="preserve">Fuerit nunc aquarum preſſio definienda in C F (aut c f): </s> + <s xml:id="echoid-s7720" xml:space="preserve">huncque in <lb/>finem putabimus rurſus abrumpi canalem in C E (aut c e) ſectione ad cana-<lb/>lem perpendiculari examinaturi, quamnam accelerationem retardationemve <lb/>guttula C E G F (vel c e g f) poſt primum rupturæ momentum receptura ſit: <lb/></s> + <s xml:id="echoid-s7721" xml:space="preserve">quâ de cauſa generaliter motum momentaneum per vas decurtatum N M E C A <lb/>Q P (vel N M c e A Q P) definiendum habemus. </s> + <s xml:id="echoid-s7722" xml:space="preserve">Sitigitur velocitas guttulæ in-<lb/>finite parvæ CEGF (ſeu c e g f) ipſo decurtationis puncto = v: </s> + <s xml:id="echoid-s7723" xml:space="preserve">maſſa ejus <lb/>= dx: </s> + <s xml:id="echoid-s7724" xml:space="preserve">erit vis viva aquæ in vaſe decurtato motæ proportionalis quantitati <lb/>v v, eamque proinde faciemus = α v v, intelligendo per litteram a quantita-<lb/>tem quamcunque conſtantem, quæ pendet ab amplitudinibus canalis abrupti; </s> + <s xml:id="echoid-s7725" xml:space="preserve"><lb/>præciſa autem ejus determinatio hic non requiritur. </s> + <s xml:id="echoid-s7726" xml:space="preserve">Notetur vim vivam aquæ <lb/>in vaſe ficto N M QP negligi ob infinitam ejus amplitudinem: </s> + <s xml:id="echoid-s7727" xml:space="preserve">nulla tamen ſi <lb/>vel infinitæ non eſſet amplitudinis inde in calculo oritura fuiſſet variatio. </s> + <s xml:id="echoid-s7728" xml:space="preserve">Ha. </s> + <s xml:id="echoid-s7729" xml:space="preserve"><lb/>bemus jam incrementum vis vivæ aquæ in vaſe decurtato motæ = 2avdv, cui <lb/>ſi addatur vis viva ſimul genita in guttula ejecta, oritur 2avdv + vvdx, quod <lb/>eſt incrementum vis vivæ totale, debitum deſcenſui actuali guttulæ dx per alti-<lb/>tudinem verticalem aquæ ſupra punctum C (vel c,) quam deſignabimus per a: </s> + <s xml:id="echoid-s7730" xml:space="preserve"><lb/>hinc igitur iſtud incrementum vis vivæ totale faciendum eſt æquale adx, ſic <lb/>ut ſit <lb/>2avdv + vvdx = adx vel <lb/>{vdv/dx} = {a - vv/2a}.</s> + <s xml:id="echoid-s7731" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7732" xml:space="preserve">Reliqua ſi fiant, ut in paragrapho quinto & </s> + <s xml:id="echoid-s7733" xml:space="preserve">ponatur velocitas v talis <lb/>quæ debeatur altitudini b, invenietur preſſionem aquæ in C F (aut cf) tantam +<pb o="264" file="0278" n="278" rhead="HYDRODYNAMICÆ"/> +eſſe, quanta in aqua ſtagnante ad altitudinem a - b. </s> + <s xml:id="echoid-s7734" xml:space="preserve">Ubi notari poteſt eſſe al-<lb/>titudinem b ad altitudinem o S, ſi nulla motus impedimenta aliena ſint, vena-<lb/>que effluens in o non contrahatur, in ratione quadrata foraminis o & </s> + <s xml:id="echoid-s7735" xml:space="preserve">ſectionis <lb/>CE (aut c e).</s> + <s xml:id="echoid-s7736" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div271" type="section" level="1" n="205"> +<head xml:id="echoid-head262" xml:space="preserve">Corollarium.</head> +<p> + <s xml:id="echoid-s7737" xml:space="preserve">§. </s> + <s xml:id="echoid-s7738" xml:space="preserve">11. </s> + <s xml:id="echoid-s7739" xml:space="preserve">Cùm b major eſt quam a, fit quantitas a - b negativa atque ſic <lb/>preſſio in ſuctionem mutatur, id eſt, latera canalis introrſum premuntur: </s> + <s xml:id="echoid-s7740" xml:space="preserve">tunc <lb/>autem res ita conſideranda eſt, ac ſi loco columnæ aqueæ CT ſuperincumben-<lb/>tis & </s> + <s xml:id="echoid-s7741" xml:space="preserve">in æquilibrio poſitæ cum aqua præterfluente, ſit columna aquea appen-<lb/>ſa e t, cujus niſus deſcendendi impediatur ab attractione aquæ præterfluentis: <lb/></s> + <s xml:id="echoid-s7742" xml:space="preserve">veluti ſi v. </s> + <s xml:id="echoid-s7743" xml:space="preserve">gr. </s> + <s xml:id="echoid-s7744" xml:space="preserve">amplitudo canalis c e æqualis ſit orificio o, tunc erit b = o S, nul-<lb/>la habita ratione motus impedimentorum accidentalium: </s> + <s xml:id="echoid-s7745" xml:space="preserve">hinc ſi tubulus ex ca-<lb/>nali deſcendat c r, hicque ſit aqua plenus à ſua origine c uſque in punctum t <lb/>cum orificio o ad libellam poſitum, manebit aqua c t ſuſpenſa ſine motu: </s> + <s xml:id="echoid-s7746" xml:space="preserve">ſi verò <lb/>punctum t infra o poſitum ſit, deſcendet aqua per tubulum cr, & </s> + <s xml:id="echoid-s7747" xml:space="preserve">effluet perpe-<lb/>tuo in r, neque tamen ut facile quis exiſtimare potuiſſet nondum hâc viſa theo-<lb/>ria, velocitas aquæ in r effluentis talis erit, quæ debeatur altitudini N P ſu-<lb/>pra r, etiamſi omnia impedimen@a auferantur, reſpondebit potius hæc velo-<lb/>citas, ſi modo tubulus admodum ſtrictus ſit ratione canalis, altitudini t r. </s> + <s xml:id="echoid-s7748" xml:space="preserve"><lb/>Si punctum t altius poſitum ſit puncto o, aqua ſua ſponte aſcendet & </s> + <s xml:id="echoid-s7749" xml:space="preserve">cum <lb/>omnis canalem ingreſſa erit, aër per tubulum attrahetur, moxque vena aquea <lb/>in o effluens ab admixto aëre turbabitur pelluciditate atque ſoliditate orbata. </s> + <s xml:id="echoid-s7750" xml:space="preserve">Ap-<lb/>paret igitur, quando preſſio futura ſit affirmativa & </s> + <s xml:id="echoid-s7751" xml:space="preserve">quando negativa: </s> + <s xml:id="echoid-s7752" xml:space="preserve">nempe <lb/>eo major eſt in tubo preſſio, quo amplior eſt & </s> + <s xml:id="echoid-s7753" xml:space="preserve">quo humilius poſitus: </s> + <s xml:id="echoid-s7754" xml:space="preserve">Al-<lb/>titudo b eſt quidem in theoria = {1/nn} X oS, ſi {1/n} denotet rationem inter am-<lb/>plitudinem orificii & </s> + <s xml:id="echoid-s7755" xml:space="preserve">ejus tubi ſectionis, pro qua preſſio eſt definienda. </s> + <s xml:id="echoid-s7756" xml:space="preserve">Cum vero <lb/>obſtacula notabiliter diminuunt motum, conveniet potius in æſtimandis preſ-<lb/>ſionibus, ut'velocitas aquæ, qualis actu eſt, experimento cognoſcatur & </s> + <s xml:id="echoid-s7757" xml:space="preserve">alti-<lb/>tudo illi velocitati debita pro b ſubſtituatur: </s> + <s xml:id="echoid-s7758" xml:space="preserve">ſimiliter accuratius æſtimabitur <lb/>preſſio, ſi pro a non tam ponatur altitudo ſuperficiei aqueæ N P ſupra <lb/>effluxus locum, quam altitudo velocitatis, quacum aquæ actu effluant <lb/>ex canali eodem in loco abrupto: </s> + <s xml:id="echoid-s7759" xml:space="preserve">Hæ tamen correctiones non ſemper locum <lb/>habent: </s> + <s xml:id="echoid-s7760" xml:space="preserve">Iſtam vero theoriam generalem jam exemplis quibuſdam illuſtrabo.</s> + <s xml:id="echoid-s7761" xml:space="preserve"/> +</p> +<pb o="265" file="0279" n="279" rhead="SECTIO DUODECIMA."/> +</div> +<div xml:id="echoid-div272" type="section" level="1" n="206"> +<head xml:id="echoid-head263" xml:space="preserve">Exemplum 1.</head> +<p> + <s xml:id="echoid-s7762" xml:space="preserve">§. </s> + <s xml:id="echoid-s7763" xml:space="preserve">12. </s> + <s xml:id="echoid-s7764" xml:space="preserve">Sit vas A B F G (Fig. </s> + <s xml:id="echoid-s7765" xml:space="preserve">75.) </s> + <s xml:id="echoid-s7766" xml:space="preserve">ex cujus fundi medio deſcendit tubus <lb/> +<anchor type="note" xlink:label="note-0279-01a" xlink:href="note-0279-01"/> +D E formam habens coni truncati inferiora verſus divergentis: </s> + <s xml:id="echoid-s7767" xml:space="preserve">Affundantur <lb/>perpetuo aquæ in A G, ita ut ſic vas plenum conſervetur.</s> + <s xml:id="echoid-s7768" xml:space="preserve"/> +</p> +<div xml:id="echoid-div272" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0279-01" xlink:href="note-0279-01a" xml:space="preserve">Fig. 75.</note> +</div> +<p> + <s xml:id="echoid-s7769" xml:space="preserve">Sit autem altitudo ſuperficiei aqueæ ſupra orificium E = a, & </s> + <s xml:id="echoid-s7770" xml:space="preserve">ſupra D <lb/>(qui locus eſt pro quo preſſio aquæ quæritur) = c: </s> + <s xml:id="echoid-s7771" xml:space="preserve">amplitudo orificii in E = m; <lb/></s> + <s xml:id="echoid-s7772" xml:space="preserve">& </s> + <s xml:id="echoid-s7773" xml:space="preserve">amplitudo ſeu ſectio horizontalis in D = n. </s> + <s xml:id="echoid-s7774" xml:space="preserve">Erit preſſio aquæ in D = <lb/>c - {mm/nn} a, quæ quantitas vi hypotheſium eſt negativa, ſic ut latera canalis <lb/>introrſum premantur à columna aquea altitudinis {mm/nn} a - c.</s> + <s xml:id="echoid-s7775" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7776" xml:space="preserve">Igitur ſi concipiatur tubus incurvus D L N alteri D E inſertus, erit aqua <lb/>præterfluens in D in æquilibrio cum aqua D L N, quando altitudo D ſupra N <lb/>eſt = {mm/nn} a - c. </s> + <s xml:id="echoid-s7777" xml:space="preserve">Si altitudo hæc minor eſt, ſua ſponte aqua aſcendet nec aſ-<lb/>cendere deſinet, quamdiu aquis orificium N ſubmerſum eſt, ita ut ſic aquæ <lb/>ex loco humiliori in ſublimiorem ſine ulla vi externa elevari poſſint, ſi in A G <lb/>aquæ ſufficiente copia affluant. </s> + <s xml:id="echoid-s7778" xml:space="preserve">At vero cum altitudo verticalis D ſupra N ma-<lb/>jor eſt quam {mm/nn} a - c, aſcendet aqua in crure L N, donec illi fuerit æqualis.</s> + <s xml:id="echoid-s7779" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7780" xml:space="preserve">Cæterum hic in memoriam revoc<unsure/>andum eſt, quod paſſim monui ex-<lb/>perientiam docere, nempe multum abeſſe quominus aquæ per tubos à vaſe, <lb/>cui implantati ſunt, divergentes tota ſua velocitate, quam vi theoriæ obtinere <lb/>deberent, effluant; </s> + <s xml:id="echoid-s7781" xml:space="preserve">cujus rei rationes indicavi paragrapho 26. </s> + <s xml:id="echoid-s7782" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s7783" xml:space="preserve">3.</s> + <s xml:id="echoid-s7784" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7785" xml:space="preserve">Fit inde ut altitudo D ſupra N admodum minor ſit, quam vi theoriæ <lb/>expoſita eſſe deberet: </s> + <s xml:id="echoid-s7786" xml:space="preserve">Error corrigetur ſi loco {mm/nn} a ponatur altitudo velo-<lb/>citatis, quam aqua in D habet; </s> + <s xml:id="echoid-s7787" xml:space="preserve">quæ altitudo per experimentum de quantitate <lb/>aquæ dato tempore effluentis ſumtum obtinetur.</s> + <s xml:id="echoid-s7788" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div274" type="section" level="1" n="207"> +<head xml:id="echoid-head264" xml:space="preserve">Exemplum 2.</head> +<p> + <s xml:id="echoid-s7789" xml:space="preserve">§. </s> + <s xml:id="echoid-s7790" xml:space="preserve">13. </s> + <s xml:id="echoid-s7791" xml:space="preserve">Si ſimili vaſi appenſus ſit tubus verticalis, qualis repræſentatur <lb/>in Fig. </s> + <s xml:id="echoid-s7792" xml:space="preserve">76. </s> + <s xml:id="echoid-s7793" xml:space="preserve">per C E, in quo amplitudines ubique rationem habeant inverſam <lb/> +<anchor type="note" xlink:label="note-0279-02a" xlink:href="note-0279-02"/> +<pb o="266" file="0280" n="280" rhead="HYDRODYNAMICÆ"/> +ſubduplicatam altitudinum aquæ ſuperincumbentis, tubus iſte nihil afficitur ab <lb/>aqua præterfluente, neque ullibi vel preſſionem ſive ſuctionem ſuſtinet.</s> + <s xml:id="echoid-s7794" xml:space="preserve"/> +</p> +<div xml:id="echoid-div274" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0279-02" xlink:href="note-0279-02a" xml:space="preserve">Fig. 76.</note> +</div> +<p> + <s xml:id="echoid-s7795" xml:space="preserve">Sequitur inde figuram naturalem fili aquei verticalis, quamdiu hoc'con-<lb/>tiguum eſt, eandem eſſe, quæ tubi C F E, quod & </s> + <s xml:id="echoid-s7796" xml:space="preserve">ratio & </s> + <s xml:id="echoid-s7797" xml:space="preserve">experientia con-<lb/>firmat: </s> + <s xml:id="echoid-s7798" xml:space="preserve">filum autem eo citius attenuabitur quo minor eſt altitudo ſuperficiei <lb/>aqueæ ſupra orificium C, ſeu quo tardius effluunt aquæ: </s> + <s xml:id="echoid-s7799" xml:space="preserve">apparet filum aqueum <lb/>ejus eſſe indolis, ut eadem aquæ quantitas per ſingulas ſectiones transfluat, nec <lb/>velocitas ullibi mutetur, ubicunque filum abrumpatur, quæ eadem proprietas <lb/>etiam in tubum C F E cadit, adeo ut rectiſſime hæc inter ſe conveniant.</s> + <s xml:id="echoid-s7800" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div276" type="section" level="1" n="208"> +<head xml:id="echoid-head265" xml:space="preserve">Exemplum 3.</head> +<p> + <s xml:id="echoid-s7801" xml:space="preserve">§. </s> + <s xml:id="echoid-s7802" xml:space="preserve">14. </s> + <s xml:id="echoid-s7803" xml:space="preserve">Devehantur aquæ e caſtello per canalem, in cujus fundum fo-<lb/>ramen ſit per quod aquæ veluti in fonte ſaliente verticaliter exiliant, dico preſ-<lb/>ſionem aquæ in ſingula canalis puncta ubique æqualem fore, ſi amplitudines <lb/>ejus ſint reſpective ut √{a/x - b}, ubi a exprimit altitudinem aquæ in caſtello ſu-<lb/>pra orificium effluxus; </s> + <s xml:id="echoid-s7804" xml:space="preserve">x altitudinem ejuſdem aquæ ſupra locum ad libitum in <lb/>canali ſumtum & </s> + <s xml:id="echoid-s7805" xml:space="preserve">b altitudinem arbitrariam conſtantem, & </s> + <s xml:id="echoid-s7806" xml:space="preserve">tunc fore ubique <lb/>preſſionem aquæ fluentis ad preſſionem aquæ ſtagnantis ut b ad a. </s> + <s xml:id="echoid-s7807" xml:space="preserve">Quia vero cæ-<lb/>teris paribus canales ampliores minus rupturæ reſiſtunt quam ſtrictiores, & </s> + <s xml:id="echoid-s7808" xml:space="preserve">id <lb/>quidem in ratione radiorum ſeu quia conatus aquæ ad canalem rumpendum cæ-<lb/>teris paribus rationem ſequitur ſubduplicatam amplitudinum, patet canalem <lb/>idem rupturæ periculum in ſingulis locis ſubiturum eſſe, ſi amplitudo (y) ratio-<lb/>ne orificii aquas ejicientis (1) ubique ſequatur legem hujus æquationis <lb/>(x - {a/yy}) √y = b vel <lb/>xxy<emph style="super">4</emph> - bby<emph style="super">3</emph> - 2axyy + aa = o.</s> + <s xml:id="echoid-s7809" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7810" xml:space="preserve">In canali per totum ſuum tractum æquabilis amplitudinis aquarum niſus <lb/>ad rumpendum canalem ubique proportionalis erit firmitati canalis, ſi craſſities <lb/>laterum canalis rationem ſequatur ut x - {a/mm}, intellecta per m amplitudine ca-<lb/>nalis ratione orificii (1).</s> + <s xml:id="echoid-s7811" xml:space="preserve"/> +</p> +<pb o="267" file="0281" n="281" rhead="SECTIO DUODECIMA."/> +</div> +<div xml:id="echoid-div277" type="section" level="1" n="209"> +<head xml:id="echoid-head266" xml:space="preserve">Exemplum 4.</head> +<p> + <s xml:id="echoid-s7812" xml:space="preserve">§ 15. </s> + <s xml:id="echoid-s7813" xml:space="preserve">Fieri poteſt, ut altitudo ſuperficiei aqueæ ratione loci, pro quo <lb/>preſſio indaganda eſt, ſit negativa, veluti in ſiphonibus recurvis aquas ex vaſe <lb/>uno in aliud humilius poſitum ducentibus: </s> + <s xml:id="echoid-s7814" xml:space="preserve">Tuncque preſſio fit duplici titulo <lb/>negativa, nempe = - a - b, denotante a altitudinem loci ſupra ſuperficiem <lb/>aquæ & </s> + <s xml:id="echoid-s7815" xml:space="preserve">b altitudinem velocitati aquæ in illo loco debitam.</s> + <s xml:id="echoid-s7816" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7817" xml:space="preserve">Iſta vero ſufficient, ut puto, ad recte intelligendam fluidorum moto-<lb/>rum ſtaticam: </s> + <s xml:id="echoid-s7818" xml:space="preserve">Venio jam ad alia quædam phænomena, quorum ſolutio ab <lb/>iſtis, quas dedimus modo, regulis pendet.</s> + <s xml:id="echoid-s7819" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7820" xml:space="preserve">§. </s> + <s xml:id="echoid-s7821" xml:space="preserve">16. </s> + <s xml:id="echoid-s7822" xml:space="preserve">In Sectione tertia §. </s> + <s xml:id="echoid-s7823" xml:space="preserve">25. </s> + <s xml:id="echoid-s7824" xml:space="preserve">mentionem feci cohæſionis aquæ per <lb/>tubos fluentis: </s> + <s xml:id="echoid-s7825" xml:space="preserve">veras autem iſtius cohæſionis menſuras ubique definire res eſt, <lb/>quæ ſine iſta præmiſſa hydraulico-ſtatica expediri nequit: </s> + <s xml:id="echoid-s7826" xml:space="preserve">neque enim altitudi-<lb/>nes conſideraſſe verticales ſupra orificium effluxus ſufficit, ut vulgo putatur, ſed <lb/>oportet etiam noſſe velocitates aquis convenientes, hæque cognoſcuntur ex <lb/>amplitudinibus. </s> + <s xml:id="echoid-s7827" xml:space="preserve">Ut vero ſtatim appareat lex generalis in definienda vi cohæ-<lb/>ſionis ſeu conatu, quo fluida ad mutuam ſeparationem ſolicitantur, dico il-<lb/>lam vim cohæſionis æqualem eſſe vi, qua latera canalis introrſum premuntur, <lb/>quam definivimus §. </s> + <s xml:id="echoid-s7828" xml:space="preserve">10. </s> + <s xml:id="echoid-s7829" xml:space="preserve">Propoſitio hæc alia demonſtratione egere mihi non <lb/>videtur; </s> + <s xml:id="echoid-s7830" xml:space="preserve">prouti enim compreſſio aquæ, ſeu vis quâ ejus partes ad ſe invicem <lb/>comprimuntur, æqualis eſt ſuperincumbenti columnæ aqueæ ſtagnanti, ita <lb/>viciſſim conatus fluida ſeparandi æqualis cenſendus eſt appenſæ columnæ ver-<lb/>ticali aqueæ ſtagnanti, quæ cum aquis præterfluentibus in æquilibrio ſit. </s> + <s xml:id="echoid-s7831" xml:space="preserve">Exem-<lb/>plorum loco eadem accipiemus, quibus ſupra pro indicandis aquarum preſſio-<lb/>nibus negativis uſi ſumus.</s> + <s xml:id="echoid-s7832" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7833" xml:space="preserve">(I) In Figura ſeptuageſima quinta §. </s> + <s xml:id="echoid-s7834" xml:space="preserve">12. </s> + <s xml:id="echoid-s7835" xml:space="preserve">explicata, ſi in tubulo D L N <lb/>altitudo D ſupra N talis ſit, ut aqua in eo ſtagnans cum aquis in D præterflu-<lb/>entibus in æquilibrio ſit, tanta debet eſſe vis cohæſionis in D, ne aqua ibi-<lb/>dem diſcerpatur, quantam habet pondus columnæ aqueæ ſimilis baſis & </s> + <s xml:id="echoid-s7836" xml:space="preserve">alti-<lb/>tudinis verticalis D N. </s> + <s xml:id="echoid-s7837" xml:space="preserve">Inde intelligitur quod dixi §. </s> + <s xml:id="echoid-s7838" xml:space="preserve">25. </s> + <s xml:id="echoid-s7839" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s7840" xml:space="preserve">3. </s> + <s xml:id="echoid-s7841" xml:space="preserve">poſſe longi-<lb/>tudinem tubi ita augeri, ut tandem aquæ deſinant eſſe continuæ in tubo, quin <lb/>poti{us} in column{as} dividantur, idque fieri in tubis cylindricis cum infra tri- +<pb o="268" file="0282" n="282" rhead="HYDRODYNAMICÆ"/> +ginta duos pedes deſcendant; </s> + <s xml:id="echoid-s7842" xml:space="preserve">in tubis divergentib{us} autem minorem deſcenſum <lb/>requiri: </s> + <s xml:id="echoid-s7843" xml:space="preserve">ita ſi v. </s> + <s xml:id="echoid-s7844" xml:space="preserve">gr. </s> + <s xml:id="echoid-s7845" xml:space="preserve">orificium inferius duplo majus fuerit orificio ſuperiori in <lb/>caſtellum hiante non poſſe tubos infra octo pedes deſcendere, quin periculum <lb/>adſit aquarum diſſolutionis. </s> + <s xml:id="echoid-s7846" xml:space="preserve">In his tamen exemplis theoretice conſideratis <lb/>aquæ omni ſua velocitate ſine diminutione motus effluere ponuntur.</s> + <s xml:id="echoid-s7847" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7848" xml:space="preserve">(II.) </s> + <s xml:id="echoid-s7849" xml:space="preserve">Ex eadem ratione patet, ſi tubi inferiora verſus convergant, tunc <lb/>illos majorem quam 32. </s> + <s xml:id="echoid-s7850" xml:space="preserve">ped. </s> + <s xml:id="echoid-s7851" xml:space="preserve">admittere deſcenſum: </s> + <s xml:id="echoid-s7852" xml:space="preserve">imo ſine fine tubum con-<lb/>tinuari poſſe in caſu Figuræ 76. </s> + <s xml:id="echoid-s7853" xml:space="preserve">§. </s> + <s xml:id="echoid-s7854" xml:space="preserve">13. </s> + <s xml:id="echoid-s7855" xml:space="preserve">explicatæ, ut & </s> + <s xml:id="echoid-s7856" xml:space="preserve">infinitis aliis modis.</s> + <s xml:id="echoid-s7857" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7858" xml:space="preserve">(III) Si vero altitudo ſuperficiei aqueæ in caſtello ratione loci propo-<lb/>ſiti negativa fuerit, veluti fit, cum aquæ trans montem vehendæ ſunt, nun-<lb/>quam poterit quomodocunque res inſtituatur, altitudo excedere triginta duos <lb/>pedes, quod patet ex § 15. </s> + <s xml:id="echoid-s7859" xml:space="preserve">Si enim aquæ vel plane infinitè parva transfluant <lb/>velocitate, vis cohæſionis jam requiritur, quæ ſit æqualis toti columnæ aqueæ, <lb/>atque major vis requiritur, ſi notabili velocitate transfluxerint. </s> + <s xml:id="echoid-s7860" xml:space="preserve">Hinc remedia <lb/>ab aliquibus Scriptoribus allata vana puto: </s> + <s xml:id="echoid-s7861" xml:space="preserve">ſcio quidem ſine alio artificio aquas <lb/>ſæpe ſuſpenſas hærere ultra altitudinem 32. </s> + <s xml:id="echoid-s7862" xml:space="preserve">pedum, & </s> + <s xml:id="echoid-s7863" xml:space="preserve">Mercurium ultra 30. <lb/></s> + <s xml:id="echoid-s7864" xml:space="preserve">pollices; </s> + <s xml:id="echoid-s7865" xml:space="preserve">ſed is effectus incertus eſt nec ſibi conſtans. </s> + <s xml:id="echoid-s7866" xml:space="preserve">Quidam etiam affirmant <lb/>fluxum aquarum per ſiphones recurvos fieri in vacuo: </s> + <s xml:id="echoid-s7867" xml:space="preserve">an vero vacuum tale <lb/>fuerit, ut ne ſexageſima quidem aëris pars in recipiente remanſerit, & </s> + <s xml:id="echoid-s7868" xml:space="preserve">num al-<lb/>titudo tubi plus quam dimidio pede ſuperficiem aquæ hauriendæ exceſſerit <lb/>ignoro. </s> + <s xml:id="echoid-s7869" xml:space="preserve">Sic igitur, quæ de ſubſecutura aquarum ſolutione dixi, non aliter <lb/>quam hypothetice dicta velim conſiderentur. </s> + <s xml:id="echoid-s7870" xml:space="preserve">Sufficiet quod accurate determi-<lb/>naverim quanta vi aquæ ad ſeparationem mutuam urgeantur.</s> + <s xml:id="echoid-s7871" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7872" xml:space="preserve">§. </s> + <s xml:id="echoid-s7873" xml:space="preserve">17. </s> + <s xml:id="echoid-s7874" xml:space="preserve">Sunt porro alia naturæ phænomena, quorum vera explicatio <lb/>ab iſta theoria hydraulico-ſtatica pendet: </s> + <s xml:id="echoid-s7875" xml:space="preserve">veluti quod fumus per caminum aſ-<lb/>cendens aërem per foramen in camino factum magno poſt ſe trahat impe u: <lb/></s> + <s xml:id="echoid-s7876" xml:space="preserve">quod ventus ex loco anguſtiori in apertiorem flans aliquid de ſua elaſticitate <lb/>perdat, prouti id colligitur ex eo, quod feneſtræ apertæ ab aëre, è camera <lb/>egreſſum ob majorem ſuam elaſticitatem, tentante claudantur; </s> + <s xml:id="echoid-s7877" xml:space="preserve">& </s> + <s xml:id="echoid-s7878" xml:space="preserve">hujusmodi <lb/>alia, quæ examinare ſingula non licet.</s> + <s xml:id="echoid-s7879" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7880" xml:space="preserve">Poſſunt fluidorum motorum preſſiones quidem infinitis variari modis; <lb/></s> + <s xml:id="echoid-s7881" xml:space="preserve">puto tamen omnia ad principia noſtra reduci poſſe: </s> + <s xml:id="echoid-s7882" xml:space="preserve">duas iſtius theoriæ exami- +<pb o="269" file="0283" n="283" rhead="SECTIO DUODECIMA."/> +navimus ſpecies; </s> + <s xml:id="echoid-s7883" xml:space="preserve">primam deduxi ex cognito motu, quem fluidum habiturum <lb/>ſit, ſi in loco determinandæ preſſionis foraminulo infinite parvo vas perfore-<lb/>tur: </s> + <s xml:id="echoid-s7884" xml:space="preserve">alteram à priori, ut dicunt, ex theoria noſtra generali deduxi; </s> + <s xml:id="echoid-s7885" xml:space="preserve">fæpe utra-<lb/>que ſimul locum obtinet, ut altera alterius opem requirat, & </s> + <s xml:id="echoid-s7886" xml:space="preserve">tunc alia ori-<lb/>tur preſſionum æſtimatio, quam unico indicabo exemplo.</s> + <s xml:id="echoid-s7887" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7888" xml:space="preserve">§. </s> + <s xml:id="echoid-s7889" xml:space="preserve">18. </s> + <s xml:id="echoid-s7890" xml:space="preserve">Putemus in vaſe, quod figura 72. </s> + <s xml:id="echoid-s7891" xml:space="preserve">ſiſtit, tubum horizontalem <lb/>nonſolum in extremitate, ſed & </s> + <s xml:id="echoid-s7892" xml:space="preserve">in ſua inſertione E G laminam habere in pla-<lb/>no verticali in medio perforatam, manentibus cæteris poſitionibus § 5. </s> + <s xml:id="echoid-s7893" xml:space="preserve">in-<lb/>dicatis: </s> + <s xml:id="echoid-s7894" xml:space="preserve">aliam patientur preſſionem latera tubi E D à transfluente aquâ, quam <lb/>nulla appoſita lamina E G & </s> + <s xml:id="echoid-s7895" xml:space="preserve">quidem minorem, quamvis minori velocitate <lb/>transfluant. </s> + <s xml:id="echoid-s7896" xml:space="preserve">Ut preſſio hæc accurate definiatur, via calcanda eſt eadem, <lb/>quæ in citato paragrapho quinto: </s> + <s xml:id="echoid-s7897" xml:space="preserve">nempe ante omnia quærenda eſt velocitas, <lb/>quâ aquæ in tubo E D transfluunt, poſtquam hæc jam uniformis facta eſt. <lb/></s> + <s xml:id="echoid-s7898" xml:space="preserve">Deinde etiam inquirendum eſt in valorem {vdv/dx}, ſi tubus alicubi abrumpi <lb/>ponatur.</s> + <s xml:id="echoid-s7899" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7900" xml:space="preserve">Quomodo autem hoc inveniri poſſit, res eſt quæ potiſſimum pertinet <lb/>ad ſectionem octavam, adhibitis ſimul cautelis §. </s> + <s xml:id="echoid-s7901" xml:space="preserve">14. </s> + <s xml:id="echoid-s7902" xml:space="preserve">ſectionis ſeptimæ: </s> + <s xml:id="echoid-s7903" xml:space="preserve">In ſe-<lb/>ctione octava generaliter oſtenditur motus fluidorum per plura foramina <lb/>transfluentium & </s> + <s xml:id="echoid-s7904" xml:space="preserve">in §. </s> + <s xml:id="echoid-s7905" xml:space="preserve">14. </s> + <s xml:id="echoid-s7906" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s7907" xml:space="preserve">7. </s> + <s xml:id="echoid-s7908" xml:space="preserve">in ſpecie monſtratur, quomodo æſtiman-<lb/>dus ſit aſcenſus potentialis, qui in guttulis generatur, quando hæ per foramen, <lb/>non in aquam veluti ſtagnantem, ſed in aquam motu, qui negligi nequit, <lb/>latam influit.</s> + <s xml:id="echoid-s7909" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7910" xml:space="preserve">Si recte indicatis hiſce inſiſtas veſtigiis, reperies velocitatem, quacum <lb/>aqua uniformiter per tubum E D transfluit, convenire huic altitudini <lb/>{mmppa/mmnn + nnpp - mmpp}, <lb/>ubi per m, p, & </s> + <s xml:id="echoid-s7911" xml:space="preserve">n indicantur reſpective amplitudines foraminum in laminis <lb/>E G & </s> + <s xml:id="echoid-s7912" xml:space="preserve">F D factorum ut & </s> + <s xml:id="echoid-s7913" xml:space="preserve">tubi E D: </s> + <s xml:id="echoid-s7914" xml:space="preserve">per a autem intelligitur altitudo aquæ <lb/>ſupra tubum E D horizontaliter poſitum.</s> + <s xml:id="echoid-s7915" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7916" xml:space="preserve">Si porro tubum abrumpi ponas in cd, guttulamque ad velocitate mo-<lb/>veri v ſeu altitudinem huic velocitati debitam = vv, ſimulque lo gitudi-<lb/>nem E c indices per c, longitudinem minimam ac per dx: </s> + <s xml:id="echoid-s7917" xml:space="preserve">æquationem in-<lb/>venies hanc +<pb o="270" file="0284" n="284" rhead="HYDRODYNAMICÆ"/> +2cvdv + {nn/mm} vvdx = adx, ſive <lb/>{vdv/dx} = {mma - nnvv/2mmc}; <lb/></s> + <s xml:id="echoid-s7918" xml:space="preserve">ſtubſtituatur nunc pro vv valor modo indicatus {mmppa/mmnn + nnpp - mmpp}, <lb/>& </s> + <s xml:id="echoid-s7919" xml:space="preserve">erit <lb/>{vdv/dx} = {mmnn - mmpp/2c(mmnn + nnpp - mmpp)}a, <lb/>cui preſſio quæſita eſt proportionalis. </s> + <s xml:id="echoid-s7920" xml:space="preserve">Sed ſi amplitudo orificii extremi in-<lb/>dicata per p eſt veluti infinite parva, preſſio fit = a; </s> + <s xml:id="echoid-s7921" xml:space="preserve">Igitur eſt generaliter <lb/>preſſio quæſita vi paragraphi quinti æqualis <lb/>{mmnn - mmpp/mmnn + nnpp - mmpp} a.</s> + <s xml:id="echoid-s7922" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7923" xml:space="preserve">§. </s> + <s xml:id="echoid-s7924" xml:space="preserve">19. </s> + <s xml:id="echoid-s7925" xml:space="preserve">Si amplitudo tubi n eſt veluti infinita ratione amplitudinum <lb/>in laminarum foraminibus, fit preſſio = {mma/mm + pp}: </s> + <s xml:id="echoid-s7926" xml:space="preserve">& </s> + <s xml:id="echoid-s7927" xml:space="preserve">tanta etiam eſt altitudo, <lb/>ad quam aqua in o effluens velocitate ſua aſcendere poteſt: </s> + <s xml:id="echoid-s7928" xml:space="preserve">id igitur con-<lb/>forme cum paragrapho quarto ſectionis octavæ, quia figura vaſis ceu ubique in-<lb/>finitæ amplitudinis non differre facit velocitatem aquæ exilientis.</s> + <s xml:id="echoid-s7929" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7930" xml:space="preserve">Cum nulla eſt lamina in F, fit p = n, totaque preſſio evaneſcit. </s> + <s xml:id="echoid-s7931" xml:space="preserve">No-<lb/>tari id meretur, quia rationem oſtendit, cur in tubis divergentibus ſuctio <lb/>tanta non ſit, quanta vi hypotheſeos, qua omnis vis viva conſervari ponitur, <lb/>eſſe deberet: </s> + <s xml:id="echoid-s7932" xml:space="preserve">In præſenti enim caſu rationem habuimus illius vis vivæ, quæ <lb/>continue abſumitur. </s> + <s xml:id="echoid-s7933" xml:space="preserve">Ita quoque nullam preſſionem patiuntur latera tubi, <lb/>cum lamina quæ eſt in E G foramen veluti infinite minus, illo, quod eſt in <lb/>F D, habet. </s> + <s xml:id="echoid-s7934" xml:space="preserve">Denique notari id quoque meretur, quod quamvis fluida per <lb/>canales nullis laminis inſtructos mota generaliter affectent preſſionem, quæ <lb/>reſpondeat differentiæ altitudinum illis velocitatibus debitarum, qua flui-<lb/>dum effluat poſt tempus infinitum per canalem abruptum & </s> + <s xml:id="echoid-s7935" xml:space="preserve">qua actu transfluit <lb/>per canalem non abruptum, hanc legem tamen in præſenti caſu minime valere, <lb/>ad quod animum attendere velim hos, qui viſa theoria noſtra hydraulico-ſtatica, <lb/>propoſitionem generalem §. </s> + <s xml:id="echoid-s7936" xml:space="preserve">10. </s> + <s xml:id="echoid-s7937" xml:space="preserve">ſynthetice demonſtrare volent. </s> + <s xml:id="echoid-s7938" xml:space="preserve">Erunt enim for-<lb/>taſſe, quibus res hæcita per ſe obvia videbitur, ut vix demonſtranda ſit: </s> + <s xml:id="echoid-s7939" xml:space="preserve">hos autem, <lb/>ſi qui futuri ſint, ex falſa quadam veriſimilitudine ſibimet imponere, oſten-<lb/>dunt hujus modi leges particulares, quæ in hydraulico-ſtatica occurrunt.</s> + <s xml:id="echoid-s7940" xml:space="preserve"/> +</p> +<pb o="271" file="0285" n="285" rhead="SECTIO DUODECIMA."/> +<p> + <s xml:id="echoid-s7941" xml:space="preserve">§. </s> + <s xml:id="echoid-s7942" xml:space="preserve">20. </s> + <s xml:id="echoid-s7943" xml:space="preserve">E re erit de his quoque, quæ §. </s> + <s xml:id="echoid-s7944" xml:space="preserve">18. </s> + <s xml:id="echoid-s7945" xml:space="preserve">dicta ſunt, experimenta ſumere, <lb/>tum pro velocitate aquarum in o effluentium, tum pro preſſione; </s> + <s xml:id="echoid-s7946" xml:space="preserve">inde enim <lb/>præter preſſionum leges confirmabitur etiam illa accelerationum theoria, quæ <lb/>obtinet, cum continue pars quædam vis vivæ inutiliter abſumitur, quod ar-<lb/>gumentum in ſectione octava præſertim pertractavimus; </s> + <s xml:id="echoid-s7947" xml:space="preserve">In experimento au-<lb/>tem ſumendo evitentur, quantum fieri poteſt, impedimenta, quorum jam <lb/>ſæpe mentionem fecimus.</s> + <s xml:id="echoid-s7948" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7949" xml:space="preserve">§. </s> + <s xml:id="echoid-s7950" xml:space="preserve">21. </s> + <s xml:id="echoid-s7951" xml:space="preserve">Adjiciam hic quæſtionem quæ quidem non ad ſtaticam fluidorum <lb/>pertinet, ſed ad hydraulicam ſeu motum fluidorum, quæ vero ſine iſtis <lb/>præmiſſis regulis hydraulico-ſtaticis ſolvi nequit. </s> + <s xml:id="echoid-s7952" xml:space="preserve">Quæritur in figura ſeptuage-<lb/>ſima ſecunda (nullam hic-amplius in E G laminam conſidero) ſi tubus fora-<lb/>mine in ac perforetur finitam rationem habente tum ad amplitudinem tubi <lb/>tum ad amplitudinem foraminis o, motusque aquarum jam uniformis factus <lb/>fuerit, quæritur, inquam, quanta velocitate aquæ per utramque aperturam <lb/>erupturæ ſint.</s> + <s xml:id="echoid-s7953" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7954" xml:space="preserve">Sit jam rurſus altitudo B E = a, amplitudo tubi = n amplitudo ori-<lb/>ficii in o = p: </s> + <s xml:id="echoid-s7955" xml:space="preserve">amplitudo foraminis ac = m: </s> + <s xml:id="echoid-s7956" xml:space="preserve">velocitas aquæ per o effluen-<lb/>tis = v: </s> + <s xml:id="echoid-s7957" xml:space="preserve">Erit velocitas aquæ quæ foramen ac præterfluit = {p/n} v. </s> + <s xml:id="echoid-s7958" xml:space="preserve">Igitur <lb/>ibidem in latera tubi exercet preſſionem, quæ eſt = a - {ppvv/nn} (per§. </s> + <s xml:id="echoid-s7959" xml:space="preserve">5.)</s> + <s xml:id="echoid-s7960" xml:space="preserve">& </s> + <s xml:id="echoid-s7961" xml:space="preserve"><lb/>propterea ſuppono proxime fore tantam quoque altitudinem, quæ generare <lb/>poſſit velocitatem, qua aqua per foramen ac exilit: </s> + <s xml:id="echoid-s7962" xml:space="preserve">ipſam vero hanc velocitatem <lb/>eſſe = √(a - {ppvv/nn}). </s> + <s xml:id="echoid-s7963" xml:space="preserve">Hoc poſito erunt velocitates in foraminibus o & </s> + <s xml:id="echoid-s7964" xml:space="preserve">ac <lb/>ut v ad √(a - {ppvv/nn}): </s> + <s xml:id="echoid-s7965" xml:space="preserve">ſicque quælibet guttula tubum in G E ingre-<lb/>diens, cum pervenit ad regionem primi foraminis, in duas diſpeſcitur par-<lb/>tes, quarum altera per ac, altera per o effluit: </s> + <s xml:id="echoid-s7966" xml:space="preserve">ſuntque hæ partes reſpective, <lb/>ut velocitates, quibus fit effluxus utrobique ductæ in amplitudines forami-<lb/>num. </s> + <s xml:id="echoid-s7967" xml:space="preserve">Igitur ſi maſſa guttulæ integræ G E dicatur g, erit pars ejus per ac <lb/>effluens æqualis <lb/>gm √(a - {ppvv/nn}):</s> + <s xml:id="echoid-s7968" xml:space="preserve">[pv + m√(a - {ppvv/nn})] <lb/>& </s> + <s xml:id="echoid-s7969" xml:space="preserve">pars altera per o effluens = +<pb o="272" file="0286" n="286" rhead="HYDRODYNAMICÆ"/> +gpv:</s> + <s xml:id="echoid-s7970" xml:space="preserve">[pv + m√(a - {ppvv/nn})].</s> + <s xml:id="echoid-s7971" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7972" xml:space="preserve">Si hæ partes multiplicentur reſpective per quadrata ſuarum velocitatum, <lb/>habebuntur earundem vires vivæ, quarum aggregatum æquandum eſt <lb/>cum g X a, id eſt, cum deſcenſu actuali guttulæ g per altitudinem a. </s> + <s xml:id="echoid-s7973" xml:space="preserve">Sic ob-<lb/>tinetur talis æquatio, ſi reducatur <lb/>n<emph style="super">3</emph>vv - n<emph style="super">3</emph>a = mpv√(nna - ppvv) ſive <lb/>vv = {2n<emph style="super">6</emph> + mmnnpp + nnmp√4n<emph style="super">4</emph> + mmpp - 4nnpp)/2n<emph style="super">6</emph> + 2mmp<emph style="super">4</emph>.</s> + <s xml:id="echoid-s7974" xml:space="preserve">}a, <lb/>hæcque quantitas exprimit altitudinem pro velocitate aquæ in o effluentis, qua <lb/>cognita habetur quoque altitudo ſimilis pro altero foramine ac, quæ nempe <lb/>eſt = a - {ppvv/nn}.</s> + <s xml:id="echoid-s7975" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7976" xml:space="preserve">§. </s> + <s xml:id="echoid-s7977" xml:space="preserve">22. </s> + <s xml:id="echoid-s7978" xml:space="preserve">Si p = n, fit vv = a; </s> + <s xml:id="echoid-s7979" xml:space="preserve">ergo tunc aquæ tota velocitate exiliunt <lb/>ſolita per foramen o, & </s> + <s xml:id="echoid-s7980" xml:space="preserve">per alterum foramen a c nihil effluit. </s> + <s xml:id="echoid-s7981" xml:space="preserve">In utroque <lb/>porro foramine velocitas reſpondet integræ altitudini a, ſi p eſt veluti infini-<lb/>te parva: </s> + <s xml:id="echoid-s7982" xml:space="preserve">Si vero m eſt infinite parva, fit quidem v v = a, ſed altitudo ve-<lb/>locitatis pro foraminulo ac eſt = a - {pp/nn}a, ut §. </s> + <s xml:id="echoid-s7983" xml:space="preserve">7. </s> + <s xml:id="echoid-s7984" xml:space="preserve">jam indicatum fuit: <lb/></s> + <s xml:id="echoid-s7985" xml:space="preserve">Si m = p, fit vv = {n<emph style="super">4</emph>a/n<emph style="super">4</emph> - nnpp + p<emph style="super">4</emph>}; </s> + <s xml:id="echoid-s7986" xml:space="preserve">& </s> + <s xml:id="echoid-s7987" xml:space="preserve">a - {ppvv/nn} = {(nn - pp)<emph style="super">2</emph>a/n<emph style="super">4</emph> - nnpp + p<emph style="super">4</emph>}.</s> + <s xml:id="echoid-s7988" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7989" xml:space="preserve">Denique obſervari poteſt, aquas per foramen o ſemper majori velo-<lb/>citate ejici, quam quæ altitudini a reſpondet, quod utique fit, quia aquæ <lb/>in E d veluti impetum faciunt in aquas d F.</s> + <s xml:id="echoid-s7990" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s7991" xml:space="preserve">Interim quamvis omnia hæc Corollaria egregie cum indole argumenti <lb/>conſentiunt, non poteſt tamen ſolutio iſtius problematis aliter quam proxi-<lb/>me vera cenſeri.</s> + <s xml:id="echoid-s7992" xml:space="preserve"/> +</p> +<pb o="273" file="0287" n="287" rhead="SECTIO DUODECIMA."/> +</div> +<div xml:id="echoid-div278" type="section" level="1" n="210"> +<head xml:id="echoid-head267" xml:space="preserve">EXPERIMENTA</head> +<head xml:id="echoid-head268" style="it" xml:space="preserve">Hydraulico - ſtatica pro Sectione XII.</head> +<head xml:id="echoid-head269" xml:space="preserve">Ad §. §. 3. & 4.</head> +<p> + <s xml:id="echoid-s7993" xml:space="preserve">PReſſiones, quæ dictis expoſitæ fuerunt paragraphis, facili experimen-<lb/>to confirmari poterunt, ſi vas, quale figura quadrageſima tertia ſi-<lb/>ſtit, quodque §. </s> + <s xml:id="echoid-s7994" xml:space="preserve">30. </s> + <s xml:id="echoid-s7995" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s7996" xml:space="preserve">8. </s> + <s xml:id="echoid-s7997" xml:space="preserve">deſcribitur, confici curetur, ejusdemque <lb/>laminæ L Q tubus vitreus verticaliter implantetur, cujus orificium utrum-<lb/>que apertum ſit: </s> + <s xml:id="echoid-s7998" xml:space="preserve">obſervabitur ſic obturatis foraminibus H & </s> + <s xml:id="echoid-s7999" xml:space="preserve">N totoque ſy-<lb/>ſtemate aquis repleto, aquam in tubo vitreo ad libellam A B aſcendere, aut illam <lb/>propter naturam tubulorum capillarium tranſcendere. </s> + <s xml:id="echoid-s8000" xml:space="preserve">Dein autem ſi digitus ab <lb/>orificio N removeatur, obſervabitur, aquam in tubo vitreo deſcendere & </s> + <s xml:id="echoid-s8001" xml:space="preserve"><lb/>captis menſuris, invenietur, ni fallor, altitudinem aquæ in tubo vitreo re-<lb/>ſiduam (detracta altitudine virtuti tuborum capillarium debita) eſſe = <lb/>{αα x LB - γγ x NQ/αα + γγ}, uti dictum eſt §. </s> + <s xml:id="echoid-s8002" xml:space="preserve">3. </s> + <s xml:id="echoid-s8003" xml:space="preserve">ubi denominationes harum litte-<lb/>rarum explicantur.</s> + <s xml:id="echoid-s8004" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8005" xml:space="preserve">Si porro ab utroque orificio H & </s> + <s xml:id="echoid-s8006" xml:space="preserve">N digitus removeatur, tunc erit ea-<lb/>dem altitudo aquæ in tubo vitreo reſidua talis, quæ §. </s> + <s xml:id="echoid-s8007" xml:space="preserve">4. </s> + <s xml:id="echoid-s8008" xml:space="preserve">indicatur. </s> + <s xml:id="echoid-s8009" xml:space="preserve">Simi-<lb/>liter poteſt tubus vitreus laminæ Q N inſeri, isque deinde inflecti, ut cognoſ-<lb/>ci poſſit, an preſſiones quoque in lamina Q N recte definitæ fuerint.</s> + <s xml:id="echoid-s8010" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8011" xml:space="preserve">Experimenta vero quæ ad preſſiones aquarum per tubos latarum per-<lb/>tinent ipſemet coram Societate noſtra inſtitui & </s> + <s xml:id="echoid-s8012" xml:space="preserve">deſcripta ſunt in tom. </s> + <s xml:id="echoid-s8013" xml:space="preserve">IV. <lb/></s> + <s xml:id="echoid-s8014" xml:space="preserve">Commentariorum pag. </s> + <s xml:id="echoid-s8015" xml:space="preserve">194. </s> + <s xml:id="echoid-s8016" xml:space="preserve">Illa igitur, ut ibi deſcripta ſunt, hic allegabo.</s> + <s xml:id="echoid-s8017" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8018" xml:space="preserve">„Uſus ſum arca lignea, cujus latitudo erat unius pedis, longitudo trium <lb/>pedum, altitudo 14. </s> + <s xml:id="echoid-s8019" xml:space="preserve">pollicum. </s> + <s xml:id="echoid-s8020" xml:space="preserve">Hanc aqua implevi ejuſque parti infimæ <lb/>fiſtulam accurate cylindricam ex ferro fabricatam infixi horizontaliter. </s> + <s xml:id="echoid-s8021" xml:space="preserve">Ita <lb/>autem factus erat tubus iſte ferreus: </s> + <s xml:id="echoid-s8022" xml:space="preserve">longitudinem nempe habuit A B <lb/>(Fig. </s> + <s xml:id="echoid-s8023" xml:space="preserve">77.) </s> + <s xml:id="echoid-s8024" xml:space="preserve">4. </s> + <s xml:id="echoid-s8025" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8026" xml:space="preserve">2. </s> + <s xml:id="echoid-s8027" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8028" xml:space="preserve">Angl. </s> + <s xml:id="echoid-s8029" xml:space="preserve">diametrum B C 7. </s> + <s xml:id="echoid-s8030" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8031" xml:space="preserve">in medio tubus <lb/> +<anchor type="note" xlink:label="note-0287-01a" xlink:href="note-0287-01"/> +foraminulo m erat perforatus, ibidemque tubulus D E pariter ferreus ſex li-<lb/>neas longus acſesquilineam in diametro habens afferruminatus erat, ita ut +<pb o="274" file="0288" n="288" rhead="HYDRODYNAMICÆ"/> +foraminulum m in medio baſis foveret: </s> + <s xml:id="echoid-s8032" xml:space="preserve">Huic poſtmodum tubulo impoſui <lb/>tubum vitreum aquabilis amplitudinis, ut apparet in figura 79. </s> + <s xml:id="echoid-s8033" xml:space="preserve">quæ modum <lb/>totius experimenti indicat. </s> + <s xml:id="echoid-s8034" xml:space="preserve">Porro tria opercula confieri curavi tubo fer-<lb/>reo adaptata, foramine diverſæ magnitudinis pertuſa: </s> + <s xml:id="echoid-s8035" xml:space="preserve">tale operculum repræ-<lb/>ſentatur Figura 78.</s> + <s xml:id="echoid-s8036" xml:space="preserve"/> +</p> +<div xml:id="echoid-div278" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0287-01" xlink:href="note-0287-01a" xml:space="preserve">Fig. 77. <lb/>78 & 79.</note> +</div> +<p> + <s xml:id="echoid-s8037" xml:space="preserve">„Hiſce omnibus conjunctis eum in modum, quem oſtendit figura 79, <lb/>factoque, ne aqua per alias rimas, quam per aperturam in B C efflueret, ob-<lb/>turavi orificium in B C, tumque obſervavi in tubo vitreo verticaciter poſi-<lb/>to punctum n, ad quod aquæ aſcendebant, idque filo ſericeo circumvolu-<lb/>to notavi: </s> + <s xml:id="echoid-s8038" xml:space="preserve">prius autem exploraveram virtutem capillarem iſtius tubi vitrei, <lb/>hancque inveneram quinque linearum, ita ut tubo aquæ verticaliter immiſ-<lb/>ſo differentia inter utramque aquæ ſuperficiem eſſet quinque linearum: <lb/></s> + <s xml:id="echoid-s8039" xml:space="preserve">propterea punctum n ſupra ſuperficiem E F elevatum fuit totidem lineis, <lb/>hincque in calculo quævis altitudo D n, D g, quinque lineis diminuta cen-<lb/>ſenda eſt.</s> + <s xml:id="echoid-s8040" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8041" xml:space="preserve">„In ſingulis experimentis arca aquis ita plena conſervata fuit, ut alti-<lb/>tudo A F eſſet 9. </s> + <s xml:id="echoid-s8042" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8043" xml:space="preserve">7. </s> + <s xml:id="echoid-s8044" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8045" xml:space="preserve">altitudo autem D n 10. </s> + <s xml:id="echoid-s8046" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8047" xml:space="preserve">His omnibus ita <lb/>ad experimentum præparatis, tunc aperto orificio in B C aquis effluxus <lb/>concedebatur & </s> + <s xml:id="echoid-s8048" xml:space="preserve">protinus deſcendit aqua in tubo vitreo, veluti ex n in g, quem <lb/>locum g rurſus alio filo ſericeo antea tubo circumvoluto notavi. </s> + <s xml:id="echoid-s8049" xml:space="preserve">Et ſic de-<lb/>nique talia cepimus experimenta quæ reſpondent §. </s> + <s xml:id="echoid-s8050" xml:space="preserve">5. </s> + <s xml:id="echoid-s8051" xml:space="preserve">& </s> + <s xml:id="echoid-s8052" xml:space="preserve">ſeqq.</s> + <s xml:id="echoid-s8053" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div280" type="section" level="1" n="211"> +<head xml:id="echoid-head270" xml:space="preserve">Experimentum 1.</head> +<p> + <s xml:id="echoid-s8054" xml:space="preserve">„Cum diameter foraminis in operculo B C eſſet 2 {1/5} lin. </s> + <s xml:id="echoid-s8055" xml:space="preserve">fuit deſcenſus <lb/>n g tantillo major una linea, ita ut nulla differentia inter theoriam & </s> + <s xml:id="echoid-s8056" xml:space="preserve">ſucceſ-<lb/>ſum experimenti obſervari potuerit.</s> + <s xml:id="echoid-s8057" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div281" type="section" level="1" n="212"> +<head xml:id="echoid-head271" xml:space="preserve">Experimentum 2.</head> +<p> + <s xml:id="echoid-s8058" xml:space="preserve">„Aſſumto alio operculo, in quo diameter foraminis erat 3 {2/5} lin. </s> + <s xml:id="echoid-s8059" xml:space="preserve">aut poul-<lb/>lulum major, deſcenſus n g obſervatus fuit ſex linearum cum duabus ter-<lb/>tiis, plane rurſus, ut theoria indicat.</s> + <s xml:id="echoid-s8060" xml:space="preserve"/> +</p> +<pb o="275" file="0289" n="289" rhead="SECTIO DUODECIMA."/> +</div> +<div xml:id="echoid-div282" type="section" level="1" n="213"> +<head xml:id="echoid-head272" xml:space="preserve">Experimentum 3.</head> +<p> + <s xml:id="echoid-s8061" xml:space="preserve">„Adhibito tertio operculo, in quo diameter foraminis erat 5. </s> + <s xml:id="echoid-s8062" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8063" xml:space="preserve">aut <lb/>aliquantulum minor, deſcenſum n g obſervavimus 28. </s> + <s xml:id="echoid-s8064" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8065" xml:space="preserve">Vi theoriæ de-<lb/>bebat eſſe circiter 29. </s> + <s xml:id="echoid-s8066" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8067" xml:space="preserve">nec enim foramen omnino quinque lineas in dia-<lb/>metro habere viſum fuit. </s> + <s xml:id="echoid-s8068" xml:space="preserve">Differentia parvula tribuenda eſt impedimentis, <lb/>quæ aqua in transfluxu per fiſtulam patitur, majoribus quam in præceden-<lb/>tibus experimentis, ob auctum motum intra fiſtulam.</s> + <s xml:id="echoid-s8069" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div283" type="section" level="1" n="214"> +<head xml:id="echoid-head273" xml:space="preserve">Experimentum 4.</head> +<p> + <s xml:id="echoid-s8070" xml:space="preserve">„Denique nullo appoſito operculo aquas pleno orificio effluere ſivi-<lb/>mus, tuncque omnis fere aqua è tubo vitreo egreſſa fuit: </s> + <s xml:id="echoid-s8071" xml:space="preserve">pars tamen ali-<lb/>qua remanſit, quam deprehendimus octo lineas altam: </s> + <s xml:id="echoid-s8072" xml:space="preserve">Earum autem quin-<lb/>que tribuendæ ſunt virtuti tubi capillaris, tres reliquæ debentur impedimen-<lb/>tis, quæ aqua in transfluxu à D uſque ad B offendit.</s> + <s xml:id="echoid-s8073" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8074" xml:space="preserve">„Sic igitur experimenta ad amuſſim cum theoria conveniunt: </s> + <s xml:id="echoid-s8075" xml:space="preserve">Inde <lb/>autem non difficile eſt prævidere, fieri poſſe, ut latera fiſtulæ non ſolum <lb/>non premantur verſus exteriora, ſed & </s> + <s xml:id="echoid-s8076" xml:space="preserve">ut verſus axem fiſtulæ introrſum <lb/>comprimantur (confer. </s> + <s xml:id="echoid-s8077" xml:space="preserve">§. </s> + <s xml:id="echoid-s8078" xml:space="preserve">11.)</s> + <s xml:id="echoid-s8079" xml:space="preserve">. </s> + <s xml:id="echoid-s8080" xml:space="preserve">Id autem edoctus ſum hoc alio experi-<lb/>mento.</s> + <s xml:id="echoid-s8081" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div284" type="section" level="1" n="215"> +<head xml:id="echoid-head274" xml:space="preserve">Experimentum 5.</head> +<p> + <s xml:id="echoid-s8082" xml:space="preserve">„Loco tubi cylindrici A B adhibui conicum, cujus orificium exter-<lb/>num erat majus orificio interno, ſimulque uſus ſum tubo vitreo incurvato, <lb/>qualem oſtendit Figura 80. </s> + <s xml:id="echoid-s8083" xml:space="preserve">Et cum ante fluxum aqua hæſit in tubo vitreo <lb/> +<anchor type="note" xlink:label="note-0289-01a" xlink:href="note-0289-01"/> +in n, deſcendit in eodem tubo aqua uſque in g, cum aquæ effluerent per tu-<lb/>bum conicum: </s> + <s xml:id="echoid-s8084" xml:space="preserve">fuitque punctum g infra D, indicio compreſſum fuiſſe du-<lb/>rante fluxu tubum conicum. </s> + <s xml:id="echoid-s8085" xml:space="preserve">In his autem caſibus impedimenta motus ſunt <lb/>inſignia, quæ faciunt ut velocitates aquæ in orificio externo admodum <lb/>minores ſint, quam quæ reſpondent altitudini aquæ: </s> + <s xml:id="echoid-s8086" xml:space="preserve">hancque ob rationem <lb/>altitudo puncti D ſupra g tanta non fuit, quanta alias futura fuiſſet, fuit ta-<lb/>men aliqua. </s> + <s xml:id="echoid-s8087" xml:space="preserve">Similem effectum alio obtinui modo, ſed admodum notabi-<lb/>liorem (confer. </s> + <s xml:id="echoid-s8088" xml:space="preserve">§. </s> + <s xml:id="echoid-s8089" xml:space="preserve">12.)</s> + <s xml:id="echoid-s8090" xml:space="preserve">. </s> + <s xml:id="echoid-s8091" xml:space="preserve">Experimentum hoc alterum ſubſequente anno coram <lb/>Academicis inſtitui, præſente Sereniſſimo Portugaliæ Principe Emanuele.</s> + <s xml:id="echoid-s8092" xml:space="preserve"/> +</p> +<div xml:id="echoid-div284" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0289-01" xlink:href="note-0289-01a" xml:space="preserve">Fig. 80.</note> +</div> +<pb o="276" file="0290" n="290" rhead="HYDRODYNAMICÆ"/> +</div> +<div xml:id="echoid-div286" type="section" level="1" n="216"> +<head xml:id="echoid-head275" xml:space="preserve">Experimentum 6.</head> +<p> + <s xml:id="echoid-s8093" xml:space="preserve">„In Figura 81. </s> + <s xml:id="echoid-s8094" xml:space="preserve">repræſentat A C F B cylindrum, in cujus fundo im-<lb/> +<anchor type="note" xlink:label="note-0290-01a" xlink:href="note-0290-01"/> +plantatus erat tubus conicus D G H E; </s> + <s xml:id="echoid-s8095" xml:space="preserve">hicque ad latus habuit parvulum tu-<lb/>bulum in l, qui reciperet extremitatem tubi vitrei incurvati l m n; </s> + <s xml:id="echoid-s8096" xml:space="preserve">altitudo <lb/>C A erat 3. </s> + <s xml:id="echoid-s8097" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8098" xml:space="preserve">10. </s> + <s xml:id="echoid-s8099" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8100" xml:space="preserve">E l 4. </s> + <s xml:id="echoid-s8101" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8102" xml:space="preserve">l H 2. </s> + <s xml:id="echoid-s8103" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8104" xml:space="preserve">9 {1/2}. </s> + <s xml:id="echoid-s8105" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8106" xml:space="preserve">amplitudo tubi conici in l <lb/>erat ad amplitudinem orificii G H ut 10. </s> + <s xml:id="echoid-s8107" xml:space="preserve">ad 16. </s> + <s xml:id="echoid-s8108" xml:space="preserve">ln erat 5. </s> + <s xml:id="echoid-s8109" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8110" xml:space="preserve">6. </s> + <s xml:id="echoid-s8111" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8112" xml:space="preserve">ejuſque <lb/>orificium n erat aquæ in vaſculo M ſubmerſum.</s> + <s xml:id="echoid-s8113" xml:space="preserve"/> +</p> +<div xml:id="echoid-div286" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0290-01" xlink:href="note-0290-01a" xml:space="preserve">Fig. 81.</note> +</div> +<p> + <s xml:id="echoid-s8114" xml:space="preserve">„Appoſito digito orificio G H impletoque vaſe ſtillabat aqua per tu-<lb/>bum vitreum l m n in vas M: </s> + <s xml:id="echoid-s8115" xml:space="preserve">remoto autem digito & </s> + <s xml:id="echoid-s8116" xml:space="preserve">effluentibus jam aquis <lb/>per G H, motu reciproco aqua ſponte ex vaſculo M aſcendit per tubum <lb/>n m l, & </s> + <s xml:id="echoid-s8117" xml:space="preserve">una cum reliquis effluxit per G H, donec totum vaſculum M eva-<lb/>cuatum eſſet. </s> + <s xml:id="echoid-s8118" xml:space="preserve">Affundebantur autem ſuperius continue aquæ, ut vas plenum <lb/>conſervaretur. </s> + <s xml:id="echoid-s8119" xml:space="preserve">Si digito pars orificii G H obtegebatur, facile erat efficere ut <lb/>pro lubitu aquæ in tubo vitreo l m n ſurſum deorſumve moverentur.</s> + <s xml:id="echoid-s8120" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8121" xml:space="preserve">Si quis etiam experimentis explorare voluerit, num theoria cum pro-<lb/>blemate §. </s> + <s xml:id="echoid-s8122" xml:space="preserve">18. </s> + <s xml:id="echoid-s8123" xml:space="preserve">conveniat, non male operam ſuam collocaverit, quandoqui-<lb/>dem non ſolum ſic novam hanc noſtram hydraulico-ſtaticam, ſed & </s> + <s xml:id="echoid-s8124" xml:space="preserve">theoriam <lb/>Sect. </s> + <s xml:id="echoid-s8125" xml:space="preserve">8. </s> + <s xml:id="echoid-s8126" xml:space="preserve">novam pariter & </s> + <s xml:id="echoid-s8127" xml:space="preserve">à nemine tractatam egregio exemplo eoque facil-<lb/>limo illuſtraverit.</s> + <s xml:id="echoid-s8128" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8129" xml:space="preserve">Hiſce jam in chartam conjectis ipſe experimenta ſumſi, quorum mo-<lb/>do mentionem feci: </s> + <s xml:id="echoid-s8130" xml:space="preserve">Machina ad id uſus ſum eadem, quam modo deſcrip-<lb/>ſi, quæque Figura 79. </s> + <s xml:id="echoid-s8131" xml:space="preserve">repræſentatur: </s> + <s xml:id="echoid-s8132" xml:space="preserve">ſed inſuper, ut natura rei poſtulat, <lb/>in A tubo aliud operculum impoſui: </s> + <s xml:id="echoid-s8133" xml:space="preserve">eratque altitudo aquæ A F 8. </s> + <s xml:id="echoid-s8134" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8135" xml:space="preserve">Lond. <lb/></s> + <s xml:id="echoid-s8136" xml:space="preserve">diameter tubi ferrei A C rurſus 7. </s> + <s xml:id="echoid-s8137" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8138" xml:space="preserve">Operculis quoque iiſdem uſus ſum, qui-<lb/>bus ante: </s> + <s xml:id="echoid-s8139" xml:space="preserve">In quovis autem experimento deſcenſum obſervavi, quem ſuper-<lb/>ficies n fecit, cum digitus ab operculo B C removeretur: </s> + <s xml:id="echoid-s8140" xml:space="preserve">ſimul autem men-<lb/>ſura capta altitudinis verticalis orificii C ſupra pavimentum obſervavi diſtan-<lb/>tiam iſtius lineæ verticalis à loco, in quem vena aquea incidebat. </s> + <s xml:id="echoid-s8141" xml:space="preserve">Hanc diſtan-<lb/>tiam vocabo amplitudinem jactus: </s> + <s xml:id="echoid-s8142" xml:space="preserve">altitudo autem hæc verticalis erat in ſingulis <lb/>experimentis 19. </s> + <s xml:id="echoid-s8143" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8144" xml:space="preserve">His ita præparatis experimenta feci talia.</s> + <s xml:id="echoid-s8145" xml:space="preserve"/> +</p> +<pb o="277" file="0291" n="291" rhead="SECTIO DUODECIMA."/> +</div> +<div xml:id="echoid-div288" type="section" level="1" n="217"> +<head xml:id="echoid-head276" xml:space="preserve">Experimentum 7.</head> +<p> + <s xml:id="echoid-s8146" xml:space="preserve">Cum diameter orificii interioris operculi eſſet 2 {1/5}. </s> + <s xml:id="echoid-s8147" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8148" xml:space="preserve">& </s> + <s xml:id="echoid-s8149" xml:space="preserve">diameter orifi-<lb/>cii exterioris orificii 3 {2/5}. </s> + <s xml:id="echoid-s8150" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8151" xml:space="preserve">fuit deſcenſus n g paullo minor, quam 7. </s> + <s xml:id="echoid-s8152" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8153" xml:space="preserve">ampli-<lb/>tudo jactus 9. </s> + <s xml:id="echoid-s8154" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8155" xml:space="preserve">In theoria autem §. </s> + <s xml:id="echoid-s8156" xml:space="preserve">18. </s> + <s xml:id="echoid-s8157" xml:space="preserve">expoſita, indicatur deſcenſus ng <lb/>6. </s> + <s xml:id="echoid-s8158" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8159" xml:space="preserve">10. </s> + <s xml:id="echoid-s8160" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8161" xml:space="preserve">& </s> + <s xml:id="echoid-s8162" xml:space="preserve">amplitudo jactus 9 {1/2}. </s> + <s xml:id="echoid-s8163" xml:space="preserve">poll.</s> + <s xml:id="echoid-s8164" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div289" type="section" level="1" n="218"> +<head xml:id="echoid-head277" xml:space="preserve">Experimentum 8.</head> +<p> + <s xml:id="echoid-s8165" xml:space="preserve">Deinde fuit diameter orificii interni 5. </s> + <s xml:id="echoid-s8166" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8167" xml:space="preserve">& </s> + <s xml:id="echoid-s8168" xml:space="preserve">diameter alterius orifi-<lb/>cii 3 {2/5}. </s> + <s xml:id="echoid-s8169" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8170" xml:space="preserve">fuit deſcenſus n g fere 17. </s> + <s xml:id="echoid-s8171" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8172" xml:space="preserve">& </s> + <s xml:id="echoid-s8173" xml:space="preserve">amplitudo jactus 24. </s> + <s xml:id="echoid-s8174" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8175" xml:space="preserve">In theoria <lb/>eſt ng 17 {3/4}. </s> + <s xml:id="echoid-s8176" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8177" xml:space="preserve">& </s> + <s xml:id="echoid-s8178" xml:space="preserve">amplitudo jactus 23. </s> + <s xml:id="echoid-s8179" xml:space="preserve">poll.</s> + <s xml:id="echoid-s8180" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div290" type="section" level="1" n="219"> +<head xml:id="echoid-head278" xml:space="preserve">Experimentum 9.</head> +<p> + <s xml:id="echoid-s8181" xml:space="preserve">Porro cum eſſet diameter orificii interni 3 {2/5}. </s> + <s xml:id="echoid-s8182" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8183" xml:space="preserve">& </s> + <s xml:id="echoid-s8184" xml:space="preserve">diameter orificii ex-<lb/>terioris 5. </s> + <s xml:id="echoid-s8185" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8186" xml:space="preserve">fuit deſcenſus n g fere idem, qui in experimento 7. </s> + <s xml:id="echoid-s8187" xml:space="preserve">nempe circi-<lb/>ter 7. </s> + <s xml:id="echoid-s8188" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8189" xml:space="preserve">Verum amplitudo jactus fuit major, ſcilicet 11. </s> + <s xml:id="echoid-s8190" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8191" xml:space="preserve">In theoria eſt n g <lb/>6. </s> + <s xml:id="echoid-s8192" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8193" xml:space="preserve">11. </s> + <s xml:id="echoid-s8194" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8195" xml:space="preserve">& </s> + <s xml:id="echoid-s8196" xml:space="preserve">amplitudo jactus fere 11. </s> + <s xml:id="echoid-s8197" xml:space="preserve">poll.</s> + <s xml:id="echoid-s8198" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div291" type="section" level="1" n="220"> +<head xml:id="echoid-head279" xml:space="preserve">Experimentum 10.</head> +<p> + <s xml:id="echoid-s8199" xml:space="preserve">Denique exiſtente diametro orificii interioris 3 {2/5}. </s> + <s xml:id="echoid-s8200" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8201" xml:space="preserve">& </s> + <s xml:id="echoid-s8202" xml:space="preserve">diametro ’ori-<lb/>ficii exterioris 2 {1/5}. </s> + <s xml:id="echoid-s8203" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8204" xml:space="preserve">fuit deſcenſus n g circiter unius pollicis atque amplitudo <lb/>jactus 23. </s> + <s xml:id="echoid-s8205" xml:space="preserve">poll. </s> + <s xml:id="echoid-s8206" xml:space="preserve">In theoria eſt ng = 14. </s> + <s xml:id="echoid-s8207" xml:space="preserve">lin. </s> + <s xml:id="echoid-s8208" xml:space="preserve">& </s> + <s xml:id="echoid-s8209" xml:space="preserve">amplitudo jactus = 22 {1/2}. </s> + <s xml:id="echoid-s8210" xml:space="preserve">poll.</s> + <s xml:id="echoid-s8211" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8212" xml:space="preserve">Omnia profecto hæc experimenta egregiè cum theoria conveniunt; <lb/></s> + <s xml:id="echoid-s8213" xml:space="preserve">fortaſſe major conſenſus futurus fuiſſet, ſi majori accuratione foraminum men-<lb/>furas accipere licuiſſet; </s> + <s xml:id="echoid-s8214" xml:space="preserve">nemo tamen, ut puto, minimis iſtis numerorum dif-<lb/>ferentiis offendetur. </s> + <s xml:id="echoid-s8215" xml:space="preserve">Oriuntur autem maximè à compreſſione aquæ in A C, <lb/>quæ producitur, dum guttulæ per orificium interius canalem ingredientes <lb/>partem motus amittunt, hinc amplitudo jactus tantillo major & </s> + <s xml:id="echoid-s8216" xml:space="preserve">deſcenſus <lb/>n g minor funt in theoria quam in experimentis, nolui hujus rei menſuram <lb/>adjicere, quamvis id in poteſtate fuiſſet, ne calculus fierit intricatior.</s> + <s xml:id="echoid-s8217" xml:space="preserve"/> +</p> +<pb o="278" file="0292" n="292" rhead="(278)"/> +</div> +<div xml:id="echoid-div292" type="section" level="1" n="221"> +<head xml:id="echoid-head280" xml:space="preserve"><emph style="bf">HYDRODYNAMICÆ</emph></head> +<head xml:id="echoid-head281" xml:space="preserve">SECTIO DECIMA TERTIA.</head> +<head xml:id="echoid-head282" style="it" xml:space="preserve">De reactione fluidorum ex vaſis efflluentium eo-<lb/>rundemque, poſtquam effluxerunt, impetu in <lb/>plana quibus occurrunt.</head> +<head xml:id="echoid-head283" xml:space="preserve">§. 1.</head> +<p> + <s xml:id="echoid-s8218" xml:space="preserve">AQuæ dum ex vaſe ejiciuntur ſimili agunt modo in vas, ex quo effluunt, <lb/>quo globus in tormentum bellicum aut ſclopetum, ex quo explodi-<lb/>tur: </s> + <s xml:id="echoid-s8219" xml:space="preserve">vas nempe retropellunt: </s> + <s xml:id="echoid-s8220" xml:space="preserve">& </s> + <s xml:id="echoid-s8221" xml:space="preserve">id quidem jam annotavit Newtonus <lb/>in princ. </s> + <s xml:id="echoid-s8222" xml:space="preserve">Math. </s> + <s xml:id="echoid-s8223" xml:space="preserve">phil. </s> + <s xml:id="echoid-s8224" xml:space="preserve">nat. </s> + <s xml:id="echoid-s8225" xml:space="preserve">edit. </s> + <s xml:id="echoid-s8226" xml:space="preserve">prim. </s> + <s xml:id="echoid-s8227" xml:space="preserve">p. </s> + <s xml:id="echoid-s8228" xml:space="preserve">332. </s> + <s xml:id="echoid-s8229" xml:space="preserve">recteque inde deducit aſ-<lb/>cenſum pilarum, quæ pulvere pyrio, carbone temperato implentur; </s> + <s xml:id="echoid-s8230" xml:space="preserve">materia <lb/>enim inflammata, dum per foramen paullatim expirat, pilas in altum projicit.</s> + <s xml:id="echoid-s8231" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8232" xml:space="preserve">Sed nec ſatis generaliter pro rei momento argumentum pertractavit ci-<lb/>tatus auctor (cum id ex ipſius inſtituto non erat) nec veram rei menſuram de-<lb/>dit. </s> + <s xml:id="echoid-s8233" xml:space="preserve">Imo in duabus editionibus poſterioribus id prorſus ſilentio præteriit: </s> + <s xml:id="echoid-s8234" xml:space="preserve">pu-<lb/>tavit autem vim illam repulſionis eſſe æqualem ponderi cylindri aquei, cujusbaſis <lb/>ſit orificium aquas tranſmittens & </s> + <s xml:id="echoid-s8235" xml:space="preserve">cujus altitudo ſit æqualis altitudini ſuperfi-<lb/>ciei aqueæ ſupra foramen. </s> + <s xml:id="echoid-s8236" xml:space="preserve">Recte quidem hæc menſura deducitur ex opinio-<lb/>ne, quam tunc temporis fovebat Newtonus, circa velocitatem aquæ ex vaſe <lb/>effluentis, dum ſtatueret aquam ad dimidiam ſuperficiei altitudinem ſua velo-<lb/>citate aſcendere poſſe.</s> + <s xml:id="echoid-s8237" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8238" xml:space="preserve">Prouti autem hujus propoſitionis falſitas nemini amplius nunc ignota <lb/>eſt, ita & </s> + <s xml:id="echoid-s8239" xml:space="preserve">alterius defectum inde quivis facile colliget, quamvis prima fronte <lb/>ſatis veriſimilis.</s> + <s xml:id="echoid-s8240" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8241" xml:space="preserve">§. </s> + <s xml:id="echoid-s8242" xml:space="preserve">2. </s> + <s xml:id="echoid-s8243" xml:space="preserve">Conſiderabimus primo rem in caſu ſimpliciſſimo, quo nempe <lb/>aquas ex vaſe infinitæ amplitudinis horizontaliter effluere ponemus. </s> + <s xml:id="echoid-s8244" xml:space="preserve">Habeo <lb/>autem demonſtratum repulſionis vim non ſtatim à fluxus initio totam adeſſe, <lb/>niſi quatenus & </s> + <s xml:id="echoid-s8245" xml:space="preserve">ipſa velocitas in aquis effluentibus tota adſit, ita ut ſi vas non +<pb o="279" file="0293" n="293" rhead="SECTIO DECIMA TERTIA."/> +ſit infinitæ amplitudinis, vis repulſionis una cum velocitate aquarum effluen-<lb/>tium ſenſim ſenſimque creſcat, aut etiam decreſcat pro circumſtantiarum na-<lb/> +<anchor type="note" xlink:label="note-0293-01a" xlink:href="note-0293-01"/> +tura: </s> + <s xml:id="echoid-s8246" xml:space="preserve">Ab his autem mutationibus momentaneis animum primo abſtrahemus, <lb/>fluxum ex vaſe infinito fieri æquabilem ponendo. </s> + <s xml:id="echoid-s8247" xml:space="preserve">Atque ſic optime definietur <lb/>vis repulſionis, ſi inquiratur, quænam ſit vis ad motum producendum re-<lb/>quiſita: </s> + <s xml:id="echoid-s8248" xml:space="preserve">Hunc vero in finem non ſolum ad velocitatem aquæ effluentis, ſed & </s> + <s xml:id="echoid-s8249" xml:space="preserve"><lb/>ad illius quantitatem erit reſpiciendum; </s> + <s xml:id="echoid-s8250" xml:space="preserve">quantitas autem pendet partim à ma-<lb/>gnitudine orificii, partim à contractione venæ, quæ poſterior variabilis eſt: <lb/></s> + <s xml:id="echoid-s8251" xml:space="preserve">Vidimus quidem in Sect. </s> + <s xml:id="echoid-s8252" xml:space="preserve">IV. </s> + <s xml:id="echoid-s8253" xml:space="preserve">poſſe totam evitari; </s> + <s xml:id="echoid-s8254" xml:space="preserve">ſi tamen quædam ſit, erit <lb/>Sectio venæ maxime contractæ ſive attenuatæ ceu orificium conſiderandum & </s> + <s xml:id="echoid-s8255" xml:space="preserve"><lb/>tunc dico fore vim repulſionis æqualem ponderi cylindri aquei, cujus baſis ſit <lb/>orificium aquas tranſmittens (id eſt, Sectio venæ horizontalis maxime con-<lb/>tractæ) & </s> + <s xml:id="echoid-s8256" xml:space="preserve">cujus altitudo ſit æqualis duplæ altitudini ſuperficiei aqueæ ſuprafo-<lb/>ramen vel accuratius, duplæ altitudini, velocitati aquæ effluentis debitæ. </s> + <s xml:id="echoid-s8257" xml:space="preserve"><lb/>Igitur ſi nulla ſit venæ contractio, prouti nulla eſt, cum per tubulum brevem <lb/>aquæ effluant, repulſio duplo aut fere duplo major erit, quam à Newtono de-<lb/>finita fuit.</s> + <s xml:id="echoid-s8258" xml:space="preserve"/> +</p> +<div xml:id="echoid-div292" type="float" level="2" n="1"> +<note position="right" xlink:label="note-0293-01" xlink:href="note-0293-01a" xml:space="preserve">Fig. 74.</note> +</div> +<p> + <s xml:id="echoid-s8259" xml:space="preserve">§. </s> + <s xml:id="echoid-s8260" xml:space="preserve">3. </s> + <s xml:id="echoid-s8261" xml:space="preserve">Ut hanc propoſitionem demonſtremus, conſiderandum hic erit <lb/>principium aliquod Mechanicum cujus uſum in aliis etiam quæſtionibus ſol-<lb/>vendis ſæpe expertus ſum: </s> + <s xml:id="echoid-s8262" xml:space="preserve">principium hoc eſt:</s> + <s xml:id="echoid-s8263" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8264" xml:space="preserve">Si corpus à quiete velocitatem eandem per preßiones motrices directas <lb/>utcunque variabiles acquiſiverit, at que ſingulæ preßiones in tempuſcula ſua mul-<lb/>tiplicentur, erit ſumma omnium productorum ſemper eadem, id eſt, ſi preßio <lb/>fit = p, tempuſculum = dt, erit ſ p d t conſtans. </s> + <s xml:id="echoid-s8265" xml:space="preserve">Hanc rem clarius expoſui <lb/>in Comment. </s> + <s xml:id="echoid-s8266" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s8267" xml:space="preserve">Imp. </s> + <s xml:id="echoid-s8268" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s8269" xml:space="preserve">tom. </s> + <s xml:id="echoid-s8270" xml:space="preserve">1. </s> + <s xml:id="echoid-s8271" xml:space="preserve">pag. </s> + <s xml:id="echoid-s8272" xml:space="preserve">132.</s> + <s xml:id="echoid-s8273" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8274" xml:space="preserve">§. </s> + <s xml:id="echoid-s8275" xml:space="preserve">4. </s> + <s xml:id="echoid-s8276" xml:space="preserve">Ponamus jam cylindrum infinitæ veluti amplitudinis, ex quo <lb/>aquæ horizontaliter effluant velocitate uniformi, abſtrahendo ab actione, quam <lb/>gravitas exerit in particulas, poſtquam jam effluxerunt, ita ut ſingulæ hori-<lb/>zontaliter & </s> + <s xml:id="echoid-s8277" xml:space="preserve">uniformiter moveri pergant; </s> + <s xml:id="echoid-s8278" xml:space="preserve">particulæ autem accelerantur preſ-<lb/>ſionemque patiuntur, quamdiu maximus velocitatis gradus nondum adeſt, <lb/>huncque obtinent cum ad locum venæ maxime contractæ pervenerunt; </s> + <s xml:id="echoid-s8279" xml:space="preserve">hæc <lb/>eſt ratio, quod ſectionem venæ ibidem conceptam ceu orificium effluxus con- +<pb o="280" file="0294" n="294" rhead="HYDRODYNAMICÆ"/> +ſiderandum eſſe dixi. </s> + <s xml:id="echoid-s8280" xml:space="preserve">Sit amplitudo iſtius Sectionis = 1, habeantque ibi aquæ <lb/>velocitatem quæ debeatur altitudini A: </s> + <s xml:id="echoid-s8281" xml:space="preserve">ponatur, cylindrum aquæ effluxiſſe, <lb/>qui pro baſe habeat 1 & </s> + <s xml:id="echoid-s8282" xml:space="preserve">pro longitudine L: </s> + <s xml:id="echoid-s8283" xml:space="preserve">ſi tempus exprimatur per ſpa-<lb/>tium diviſum per velocitatem, erit velocitas altitudini A debita exprimenda <lb/>per √ 2 A, tempuſque fluxus per {L/√2A}. </s> + <s xml:id="echoid-s8284" xml:space="preserve">His præmiſſis indagabimus in preſ-<lb/>ſionem motricem, quæ poſſit tempore ({L/√2a}) cylindro L communicare ve-<lb/>locitatem √ 2 A: </s> + <s xml:id="echoid-s8285" xml:space="preserve">ſit illa preſſio = p: </s> + <s xml:id="echoid-s8286" xml:space="preserve">putetur brevioris calculi ergo egiſſe <lb/>tempore t cylindroque dediſſe velocitatem v; </s> + <s xml:id="echoid-s8287" xml:space="preserve">erit d v = {pdt/L} & </s> + <s xml:id="echoid-s8288" xml:space="preserve">v = {pt/L}, <lb/>hinc p = {Lv/t}; </s> + <s xml:id="echoid-s8289" xml:space="preserve">ponatur jam √ 2 A pro v & </s> + <s xml:id="echoid-s8290" xml:space="preserve">{L/√2A} pro t atque erit p = <lb/>(L √2A): </s> + <s xml:id="echoid-s8291" xml:space="preserve">(L/√2A} = 2 A. </s> + <s xml:id="echoid-s8292" xml:space="preserve">Eſt igitur preſſio aquam ad effluxum conſtanter <lb/>ſollicitans æqualis ponderi cylindri aquei, cujus baſis ſit orificium aquas tranſ-<lb/>mittens ſupra definitum & </s> + <s xml:id="echoid-s8293" xml:space="preserve">cujus altitudo ſit æqualis duplæ altitudini velocitati <lb/>aquæ effluentis debitæ: </s> + <s xml:id="echoid-s8294" xml:space="preserve">& </s> + <s xml:id="echoid-s8295" xml:space="preserve">tanta quoque eſt reactio, quæ vas repellit. </s> + <s xml:id="echoid-s8296" xml:space="preserve">Q.</s> + <s xml:id="echoid-s8297" xml:space="preserve">E.</s> + <s xml:id="echoid-s8298" xml:space="preserve">D.</s> + <s xml:id="echoid-s8299" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8300" xml:space="preserve">§. </s> + <s xml:id="echoid-s8301" xml:space="preserve">5. </s> + <s xml:id="echoid-s8302" xml:space="preserve">Eadem eſt demonſtratio ſi aquæ non per orificium ſed per tubum <lb/>horizontalem cylindricum velocitate uniformi effluant, aut etiam per tubum <lb/>utcunque inæqualiter amplum: </s> + <s xml:id="echoid-s8303" xml:space="preserve">poſterius id directe demonſtrari etiam poteſt, <lb/>ſi bene exprimatur preſſio requiſita in ſingulis guttis, ut hæ debita velocita-<lb/>tum incrementa aut decrementa ſuſcipiant.</s> + <s xml:id="echoid-s8304" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8305" xml:space="preserve">§. </s> + <s xml:id="echoid-s8306" xml:space="preserve">6. </s> + <s xml:id="echoid-s8307" xml:space="preserve">Altitudo, quam vocavimus A, parum quidem differt in experi-<lb/>mentis ab altitudine aquæ ſupra orificium effluxus, præſertim ſi aquæ ex vaſe <lb/>valde amplo per orificium ſimplex, idque non admodum parvum effluant: <lb/></s> + <s xml:id="echoid-s8308" xml:space="preserve">differt autem ſæpius notabiliter orificium effluxus à ſectione minima venæ, quam <lb/>nos ceu orificium aquas tranſmittens conſideramus; </s> + <s xml:id="echoid-s8309" xml:space="preserve">id quantitas aquæ dato <lb/>tempore effluentis cum velocitate ſua comparata in experimentis indicat.</s> + <s xml:id="echoid-s8310" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8311" xml:space="preserve">Hinc fit ut propoſitio noſtra §. </s> + <s xml:id="echoid-s8312" xml:space="preserve">3. </s> + <s xml:id="echoid-s8313" xml:space="preserve">ad experientiam vocata ordinario <lb/>non multum diſcrepet ab propoſitione Newtoni §. </s> + <s xml:id="echoid-s8314" xml:space="preserve">1. </s> + <s xml:id="echoid-s8315" xml:space="preserve">expoſita; </s> + <s xml:id="echoid-s8316" xml:space="preserve">ſi vero omnia <lb/>ſollicite evitentur, quæ contractionem venæ producere & </s> + <s xml:id="echoid-s8317" xml:space="preserve">quæ velocitatem <lb/>diminuere poſſunt, vis repellens ſecundum theoriam noſtram fiet tantum non <lb/>duplo major, quam quæ à Newtono fuit definita & </s> + <s xml:id="echoid-s8318" xml:space="preserve">tunc talis etiam experi-<lb/>mentis confirmatur.</s> + <s xml:id="echoid-s8319" xml:space="preserve"/> +</p> +<pb o="281" file="0295" n="295" rhead="SECTIO DECIMA TERTIA."/> +<p> + <s xml:id="echoid-s8320" xml:space="preserve">At ut rem plane in apricum ponamus, eam generalius nunc proſe-<lb/>quemur, idque tentabimus, ut vim repellentem à fluxus initio, dum veloci-<lb/>tates continue mutantur, determinemus: </s> + <s xml:id="echoid-s8321" xml:space="preserve">neque enim primum noſtrum theo-<lb/>rema aliter quam cum velocitas invariata manet locum habet. </s> + <s xml:id="echoid-s8322" xml:space="preserve">Ut in quæſtio-<lb/>ne hâc paullo intricatiore pertractanda eo intelligibiliores ſimus, hîc quædam <lb/>generaliora præmonuiſſe juvabit.</s> + <s xml:id="echoid-s8323" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8324" xml:space="preserve">§. </s> + <s xml:id="echoid-s8325" xml:space="preserve">7. </s> + <s xml:id="echoid-s8326" xml:space="preserve">Quantit{as} mot{us} eſt factum ex velocitate in maſſam: </s> + <s xml:id="echoid-s8327" xml:space="preserve">ſi velocitates <lb/>ſint inæquales, habebitur quantitas mot{us} abſoluta, ſi ſingulæ particulæ per ſuam <lb/>reſpective velocitatem multiplicentur productorumque fumma accipiatur. <lb/></s> + <s xml:id="echoid-s8328" xml:space="preserve">Quantitas mot{us} generatur à preſſionibus motricibus dato tempore urgentibus & </s> + <s xml:id="echoid-s8329" xml:space="preserve"><lb/>effectus cauſæ eſt æqualis cenſendus: </s> + <s xml:id="echoid-s8330" xml:space="preserve">Igitur ſumma preſſionum motricium per <lb/>ſua tempuſcula multiplicatorum æſtimanda eſt ex genita quantitate motus. </s> + <s xml:id="echoid-s8331" xml:space="preserve">Et <lb/>quia quælibet preſſio motrix reagit in vas, ex quo aquæ effluunt, erit tota vis re-<lb/>pellens pro quovis momento æqualis novæ quantitati motus diviſæ per tempuſ-<lb/>culum, quo generatur. </s> + <s xml:id="echoid-s8332" xml:space="preserve">His præmonitis ad quæſtionem ipſam progredior.</s> + <s xml:id="echoid-s8333" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8334" xml:space="preserve">§. </s> + <s xml:id="echoid-s8335" xml:space="preserve">8. </s> + <s xml:id="echoid-s8336" xml:space="preserve">Sit igitur vas infinitæ amplitudinis A C D B (Fig. </s> + <s xml:id="echoid-s8337" xml:space="preserve">82.) </s> + <s xml:id="echoid-s8338" xml:space="preserve">eique ho-<lb/> +<anchor type="note" xlink:label="note-0295-01a" xlink:href="note-0295-01"/> +rizontaliter infixa fiſtula E H I D, cujus amplitudines utcunque inæquales po-<lb/>nuntur: </s> + <s xml:id="echoid-s8339" xml:space="preserve">amplitudo orificii H I fuerit = 1, longitudo fiftulæ = m; </s> + <s xml:id="echoid-s8340" xml:space="preserve">velocitas <lb/>utcunque variabilis in H I = √ 2 v, ſeu talis, quæ debeatur altitudini v: </s> + <s xml:id="echoid-s8341" xml:space="preserve">dico <lb/>primo, fore quantitatem motus abſolutam aquæ in fiſtula contentæ æqualem <lb/>m√2v, id eſt, talem ac ſi fiſtula eſſet cylindrica ſuaque amplitudine orificium <lb/>H I exæquaret, quia nempe cujuslibet ſtrati F G gf velocitas eſt maſſæ reci-<lb/>proce proportionalis.</s> + <s xml:id="echoid-s8342" xml:space="preserve"/> +</p> +<div xml:id="echoid-div293" type="float" level="2" n="2"> +<note position="right" xlink:label="note-0295-01" xlink:href="note-0295-01a" xml:space="preserve">Fig. 82.</note> +</div> +<p> + <s xml:id="echoid-s8343" xml:space="preserve">Jam vero fingamus dato tempuſculo infinite parvo exilire per orificium <lb/>H I columellam H L M I, cujus longitudinem H L vel I M ponemus = a: <lb/></s> + <s xml:id="echoid-s8344" xml:space="preserve">erit maſſa hujus columellæ = a, habebitque quantitatem motus = a√2v: </s> + <s xml:id="echoid-s8345" xml:space="preserve"><lb/>fed eodem tempore maſſa aquæ in fiſtula contentæ acquiſivit quantitatem mo-<lb/>tus {mdv/√2v} (habuit enim m√2v); </s> + <s xml:id="echoid-s8346" xml:space="preserve">eſt igitur quantitas motus abſoluta dato tem-<lb/>puſculo genita = a√2v + {mdv/√2v}; </s> + <s xml:id="echoid-s8347" xml:space="preserve">hæc vero ſi dividatur per idem tempuſ-<lb/>culum (quod exprimendum eſt per {a/√2v}) habebitur, ut vidimus §. </s> + <s xml:id="echoid-s8348" xml:space="preserve">7. </s> + <s xml:id="echoid-s8349" xml:space="preserve">preſſio <lb/>quæſita vas repellens, quæ proinde ſi vocetur p, erit +<pb o="282" file="0296" n="296" rhead="HYDRODYNAMICÆ"/> +p = (α√2v + {mdv/√2v}): </s> + <s xml:id="echoid-s8350" xml:space="preserve">{a/√2v}, ſive <lb/>p = 2v + {mdv/a}.</s> + <s xml:id="echoid-s8351" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8352" xml:space="preserve">(α) Apparet inde ultimam definitionem quæſtionis pendere à ratione <lb/>quæ intercedit inter d v & </s> + <s xml:id="echoid-s8353" xml:space="preserve">α; </s> + <s xml:id="echoid-s8354" xml:space="preserve">hanc vero in Sectione tertia generaliter defini-<lb/>vimus, nulla tamen impedimentorum, quæ debentur caſui, facta attentione. <lb/></s> + <s xml:id="echoid-s8355" xml:space="preserve">Igitur & </s> + <s xml:id="echoid-s8356" xml:space="preserve">figura fiſtulæ hic aliquid confert.</s> + <s xml:id="echoid-s8357" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8358" xml:space="preserve">(β) Sequitur porro, ſi fluxus uniformis factus ponatur, eſſe p con-<lb/>ſtanter = 2v, quia tunc dv = o: </s> + <s xml:id="echoid-s8359" xml:space="preserve">Id vero conforme eſt cum eo, quod <lb/>demonſtravimus §. </s> + <s xml:id="echoid-s8360" xml:space="preserve">5. </s> + <s xml:id="echoid-s8361" xml:space="preserve">Donec vero fluxus incrementa accipit (quod qui-<lb/>dem facit notabiliter, idque diu ſatis, ſi canalis E I longior fuerit) vas aliam <lb/>atque aliam patitur vim repellentem.</s> + <s xml:id="echoid-s8362" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8363" xml:space="preserve">(γ) Habet dv ad α ſemper rationem realem: </s> + <s xml:id="echoid-s8364" xml:space="preserve">ergo vis repellens nun-<lb/>quam eſt nulla, ſic ut à primo fluxus tempore vas repellatur, etiamſi tunc <lb/>aquæ fere nullæ effluant ob exiguam earundem velocitatem. </s> + <s xml:id="echoid-s8365" xml:space="preserve">Verum, ut <lb/>uſus regulæ noſtræ generalis unicuique pateat, eam nunc ad caſum ſpecia-<lb/>l<unsure/>em applicabimus, tribuendo fiſtulæ EHID figuram cylindricam amplitu-<lb/>dinis 1.</s> + <s xml:id="echoid-s8366" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8367" xml:space="preserve">§. </s> + <s xml:id="echoid-s8368" xml:space="preserve">9. </s> + <s xml:id="echoid-s8369" xml:space="preserve">Si igitur fiſtula ponatur cylindrica tota aperta in H I retentis <lb/>cæteris poſitionibus & </s> + <s xml:id="echoid-s8370" xml:space="preserve">denominationibus, erit vis viva aquæ in fiſtula con-<lb/>tentæ = mv; </s> + <s xml:id="echoid-s8371" xml:space="preserve">hujus incrementum = mdv, cui addenda vis viva columel-<lb/>læ H L M I ſeu a v, eorumque ſumma æqualis facienda facto ex altitudine, <lb/>quam habet ſuperficies aquæ A B ſupra orificium H I, quamque vocabimus <lb/>a, & </s> + <s xml:id="echoid-s8372" xml:space="preserve">ex maſſula α. </s> + <s xml:id="echoid-s8373" xml:space="preserve">Eſt igitur mdv + αv = αa, unde hic fit {dv/α} = {a - v/m}. <lb/></s> + <s xml:id="echoid-s8374" xml:space="preserve">lſto autem valore ſubſtituto in æquatione ſuperioris paragrahi fit <lb/>p = a + v. </s> + <s xml:id="echoid-s8375" xml:space="preserve"><lb/>unde talia deduco conſectaria.</s> + <s xml:id="echoid-s8376" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8377" xml:space="preserve">(α) Longitudo fiſtulæ nihil ad vim repellentem, quam vas ſuſtinet, <lb/>tribuit, ſi velocitas eadem ponatur, quia littera m è calculo evanuit, facit <lb/>autem hæc longitudo (ſicuti in ſuperioribus ſatis ſuperque demonſtravimus) <lb/>ut velocitates citiora aut lentiora incrementa capiant; </s> + <s xml:id="echoid-s8378" xml:space="preserve">quo longior enim fue- +<pb o="283" file="0297" n="297" rhead="SECTIO DECIMA TERTIA."/> +rit fiſtula, eo tardius accelerantur aquæ & </s> + <s xml:id="echoid-s8379" xml:space="preserve">viciſſim, ſic ut in inſtanti à quiete <lb/>maximum ſuum celeritatis gradum acquirant, ſi longitudo fiſtulæ nulla fue-<lb/>rit; </s> + <s xml:id="echoid-s8380" xml:space="preserve">at ſi infinitæ fuerit eadem hæc fiſtula longitudinis, aquæ nonniſi poſt <lb/>tempus infinitum notab lem celeritatis gradum acquirere poſſunt.</s> + <s xml:id="echoid-s8381" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8382" xml:space="preserve">(β) Fieri igitur poteſt non mutata aquarum altitudine, ut diſpendio <lb/>aquarum quantumvis parvo, vis repellens notabilis ſit, eaque pro lubitu <lb/>duret; </s> + <s xml:id="echoid-s8383" xml:space="preserve">& </s> + <s xml:id="echoid-s8384" xml:space="preserve">id quidem duplici obtineri poteſt modo, tum prolongando fiſtu-<lb/>lam, tum etiam obturando ſæpius orificium, antequam aquæ notabilem ve-<lb/>locitatis gradum attigerint; </s> + <s xml:id="echoid-s8385" xml:space="preserve">prior tamen modus liberum aquarum fluxum <lb/>per fiſtulam ponit: </s> + <s xml:id="echoid-s8386" xml:space="preserve">retardato enim ab impedimentis extrinſecis, in prælon-<lb/>gis fiſtulis nunquam vitabilibus, aquarum fluxu, diminuitur quoque vis <lb/>repellens.</s> + <s xml:id="echoid-s8387" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8388" xml:space="preserve">(γ) Liceat hic paucis attingere verbis propoſitionem aliquam ex princ. <lb/></s> + <s xml:id="echoid-s8389" xml:space="preserve">math. </s> + <s xml:id="echoid-s8390" xml:space="preserve">phil. </s> + <s xml:id="echoid-s8391" xml:space="preserve">nat. </s> + <s xml:id="echoid-s8392" xml:space="preserve">edit. </s> + <s xml:id="echoid-s8393" xml:space="preserve">2. </s> + <s xml:id="echoid-s8394" xml:space="preserve">Newtoni: </s> + <s xml:id="echoid-s8395" xml:space="preserve">Auctor hic poſtquam ſententiam ſuam de ve-<lb/>locitate aquarum ex vaſe effluentium in prima citati operis editione exhibi-<lb/>tam mutaſſet, easque, ſi verticaliter ſurſum ejiciantur, ad integram ſuper-<lb/>ficiei aquæ altitudinem aſcendere agnoviſſet in editione ſecunda, talia ſubje-<lb/>cit verba in libro ſecundo propos. </s> + <s xml:id="echoid-s8396" xml:space="preserve">36. </s> + <s xml:id="echoid-s8397" xml:space="preserve">coroll. </s> + <s xml:id="echoid-s8398" xml:space="preserve">2. </s> + <s xml:id="echoid-s8399" xml:space="preserve">Vis qua totus aquæ exilientis <lb/>motus generari poteſt, æqualis eſt ponderi cylindricæ columellæ aquæ, cujus <lb/>baſis eſt for amen E F (vid. </s> + <s xml:id="echoid-s8400" xml:space="preserve">fig. </s> + <s xml:id="echoid-s8401" xml:space="preserve">Nevvt.) </s> + <s xml:id="echoid-s8402" xml:space="preserve">& </s> + <s xml:id="echoid-s8403" xml:space="preserve">cujus altitudo eſt 2 G I vel 2 C K. </s> + <s xml:id="echoid-s8404" xml:space="preserve"><lb/>Iſta ſententia à me olim & </s> + <s xml:id="echoid-s8405" xml:space="preserve">ab aliis fuit impugnata, ab aliis rurſus confirma-<lb/>ta. </s> + <s xml:id="echoid-s8406" xml:space="preserve">Nunc autem poſtquam hanc aquarum motarum theoriam medita-<lb/>tus ſum, lis ita dirimenda mihi videtur, ut cum aquæ ad motum unifor-<lb/>mem pervenerint, quæ quidem hypotheſis eſt Newtoni, tunc recte altitu-<lb/>dine 2 G I vis illa definiatur, ſed ab initio fluxus, ubi velocitas adhuc nulla <lb/>eſt, vis ſimplici altitudini G I reſpondeat, moxque creſcente velocitate ſi-<lb/>mul vis aquam ad effluxum animans creſcat, & </s> + <s xml:id="echoid-s8407" xml:space="preserve">tandem ad eam magnitudi-<lb/>nem exſurgat, quam Newtonus aſſignavit. </s> + <s xml:id="echoid-s8408" xml:space="preserve">Hæc nunc ſunt unicuique ob-<lb/>via, quia vis motum aquæ generans, de qua Newtonus loquitur, non po-<lb/>teſt non eſſe æqualis vi repellenti, quam vidimus eſſe æqualem a + v. </s> + <s xml:id="echoid-s8409" xml:space="preserve">Re-<lb/>cte etiam Jll. </s> + <s xml:id="echoid-s8410" xml:space="preserve">Riccatus, cum quo mihi de hoc argumento res erat interro-<lb/>gatus, unde vis illa duplæ aquarum altitudini conveniens oriri po{ſsi}t, cum +<pb o="284" file="0298" n="298" rhead="HYDRODYNAMICÆ"/> +obturato orificio gutta eidem imminens vi ſimplicis altitudinis urgeri manife-<lb/>ſte appareat, reſpondit, diſtinguendum eſſe ſtatum quietis à ſtatu motus.</s> + <s xml:id="echoid-s8411" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8412" xml:space="preserve">§. </s> + <s xml:id="echoid-s8413" xml:space="preserve">10. </s> + <s xml:id="echoid-s8414" xml:space="preserve">Si fiſtula vaſi implantata non ſit cylindrica, calculus ita erit po-<lb/>nendus.</s> + <s xml:id="echoid-s8415" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8416" xml:space="preserve">Sit amplitudo canalis in F G vel fg = y; </s> + <s xml:id="echoid-s8417" xml:space="preserve">diſtantia ſtrati F G gf ab ori-<lb/>ficio E D = x, retineanturque cæteræ denominationes: </s> + <s xml:id="echoid-s8418" xml:space="preserve">erit vis viva aquæ in <lb/>fiſtula contentæ = vſ {dx/y}, ejusque incrementum = dvſ {dx/y}, cui ut in §. <lb/></s> + <s xml:id="echoid-s8419" xml:space="preserve">præcedente factum eſt, addatur vis viva columellæ H L M I ſeu a v, eritque <lb/>d vſ {dx/y} + αv = αa; </s> + <s xml:id="echoid-s8420" xml:space="preserve">unde ſic oritur <lb/>{d v/α} = (a - v): </s> + <s xml:id="echoid-s8421" xml:space="preserve">ſ {dx/y}, <lb/>quo valore ſubſtituto in æquatione §. </s> + <s xml:id="echoid-s8422" xml:space="preserve">8. </s> + <s xml:id="echoid-s8423" xml:space="preserve">fit <lb/>p = 2v + m (a - v): </s> + <s xml:id="echoid-s8424" xml:space="preserve">ſ{dx/y}.</s> + <s xml:id="echoid-s8425" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8426" xml:space="preserve">Igitur cum in fluxu aquarum uniformi ſit v = a, erit tunc rurſus <lb/>p = 2a. </s> + <s xml:id="echoid-s8427" xml:space="preserve">Cæterum quamdiu aquarum fluxus acceleratur, motus aquæ in <lb/>vaſe A C D B orificio D E proximæ, à quo in toto hoc opere animum ab-<lb/>ſtraximus, hic non eſt negligendus: </s> + <s xml:id="echoid-s8428" xml:space="preserve">determinari autem recte motus iſte non <lb/>poteſt, nec igitur accurate quadrat expreſſio quam dedi pro vi repellente ſi <lb/>aquæ nondum uniformiter fluere ceperint, ſed cum æquabiliter fluunt aquæ <lb/>valet expreſſio accuratiſſime.</s> + <s xml:id="echoid-s8429" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8430" xml:space="preserve">§. </s> + <s xml:id="echoid-s8431" xml:space="preserve">11. </s> + <s xml:id="echoid-s8432" xml:space="preserve">Poſtquam ſic demonſtravimus pro effluxu aquarum uniformi, <lb/>vim repellentem ſemper eſſe æqualem ponderi cylindri aquei foramini ſuper-<lb/>inſtructi & </s> + <s xml:id="echoid-s8433" xml:space="preserve">ad duplam aquæ altitudinem exſurgentis, lubet id etiam indire-<lb/>cte demonſtrare per deductionem ad abſurdum, ut & </s> + <s xml:id="echoid-s8434" xml:space="preserve">regularum mechanicarum <lb/>ignari propoſitionis hujus ſatis paradoxæ veritatem perſpiciant.</s> + <s xml:id="echoid-s8435" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8436" xml:space="preserve">Hunc in finem conſiderabimus aquas verticaliter defluentes ex cylindro, <lb/>abſtrahendo animum ab impedimentis velocitati aquarum aliquid deroganti-<lb/>bus & </s> + <s xml:id="echoid-s8437" xml:space="preserve">ab illa contractione venæ, quæ vitari poteſt. </s> + <s xml:id="echoid-s8438" xml:space="preserve">Foramini reſpondeat tu-<lb/>bus verticalis, qualis conſpicitur Fig. </s> + <s xml:id="echoid-s8439" xml:space="preserve">76. </s> + <s xml:id="echoid-s8440" xml:space="preserve">habeant ſe omnia, ut in Sect. </s> + <s xml:id="echoid-s8441" xml:space="preserve">XII. <lb/></s> + <s xml:id="echoid-s8442" xml:space="preserve">§. </s> + <s xml:id="echoid-s8443" xml:space="preserve">13. </s> + <s xml:id="echoid-s8444" xml:space="preserve">dictum fuit: </s> + <s xml:id="echoid-s8445" xml:space="preserve">aquæ habeant fluxum æquabilem: </s> + <s xml:id="echoid-s8446" xml:space="preserve">latera vaſis & </s> + <s xml:id="echoid-s8447" xml:space="preserve">canalis +<pb o="285" file="0299" n="299" rhead="SECTIO DECIMA TERTIA."/> +gravitate carere intelligantur, altitudo cylindri ponatur = a, & </s> + <s xml:id="echoid-s8448" xml:space="preserve">altitudo fiſtu-<lb/>læ = b, altitudo c F = x, amplitudo in E = 1; </s> + <s xml:id="echoid-s8449" xml:space="preserve">erit amplitudo in F = {√(a + b)/√(a + x)} <lb/>& </s> + <s xml:id="echoid-s8450" xml:space="preserve">in C = {√(a + b)/√a}: </s> + <s xml:id="echoid-s8451" xml:space="preserve">Denique ponatur amplitudo cylindri = M. </s> + <s xml:id="echoid-s8452" xml:space="preserve">His poſitis <lb/>quæremus pondus omnis aquæ A B C E: </s> + <s xml:id="echoid-s8453" xml:space="preserve">exprimemus pondus aquæ A B C per <lb/>M a & </s> + <s xml:id="echoid-s8454" xml:space="preserve">ſic erit pondus aquæ C E = 2a + 2b - 2√(aa + ab); </s> + <s xml:id="echoid-s8455" xml:space="preserve">ergo pon-<lb/>dus omnis aquæ A B C E = Ma + 2a + 2b - 2√(aa + ab): </s> + <s xml:id="echoid-s8456" xml:space="preserve">Sic igitur <lb/>poſito aquas ſtagnare in vaſe & </s> + <s xml:id="echoid-s8457" xml:space="preserve">fiſtula, vis requiſita ad ſuſpendendam aquam <lb/>eſt = Ma + 2a + 2b - 2√(aa + ab).</s> + <s xml:id="echoid-s8458" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8459" xml:space="preserve">Jam vero indagabimus vim ſimilem cum aquæ per E tota ſua velocitate <lb/>(quâ nempe ad altitudinem a + b aſcendere poſſunt) effluunt: </s> + <s xml:id="echoid-s8460" xml:space="preserve">hæc autem ha-<lb/>bebitur, ſi à priori vi ſubtrahatur vis repellens: </s> + <s xml:id="echoid-s8461" xml:space="preserve">Si proinde hæc vis repellens <lb/>ponatur, ut nos ſtatuimus, = 2a + 2b, erit vis aquas, durante fluxu ſuſpen-<lb/>dens = Ma - 2√(aa + ab).</s> + <s xml:id="echoid-s8462" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8463" xml:space="preserve">At vero finge abeſſe tubum C E, & </s> + <s xml:id="echoid-s8464" xml:space="preserve">erit per eaſdem noſtras regulas vis ſu-<lb/>ſpenſoria, dumaquæ per orificium C erumpunt, rurſus = Ma - 2√(aa + ab). <lb/></s> + <s xml:id="echoid-s8465" xml:space="preserve">ideo, quia pondus aquæ A B C eſt Ma & </s> + <s xml:id="echoid-s8466" xml:space="preserve">quia amplitudo foraminis C eſt {√a + b/√a}, <lb/>quæ multiplicata per duplam altitudinem a dat 2√(aa + ab). </s> + <s xml:id="echoid-s8467" xml:space="preserve">Monſtrat igi-<lb/>tur noſtra virium repellentium æſtimatio, vim ſuſpenſoriam durante aquarum <lb/>effluxu eandem eſſe, ſive nulla ſit fiſtula, ſive adſit & </s> + <s xml:id="echoid-s8468" xml:space="preserve">quamcunque habeat lon-<lb/>gitudinem, modo fiſtula figuram habeat §. </s> + <s xml:id="echoid-s8469" xml:space="preserve">13. </s> + <s xml:id="echoid-s8470" xml:space="preserve">Sect. </s> + <s xml:id="echoid-s8471" xml:space="preserve">XII. </s> + <s xml:id="echoid-s8472" xml:space="preserve">deſcriptam: </s> + <s xml:id="echoid-s8473" xml:space="preserve">atque hujus <lb/>conſenſus & </s> + <s xml:id="echoid-s8474" xml:space="preserve">identitatis neceſſitas apparet quoque ſine calculo ex ipſa rei natu-<lb/>ra, quando fiſtula ita formata nullam in aquis transfluentibus facit mutationem, <lb/>cum vena aquæ ſua ſponte eandem figuram induit, quam habet fiſtula, quam-<lb/>diu aquæ cohærent. </s> + <s xml:id="echoid-s8475" xml:space="preserve">Sed ſi<unsure/> aliter vim repellentem æſtimemus, nunquam con-<lb/>ſenſum illum inter vires ſuſpenſorias generaliter obtinebimus: </s> + <s xml:id="echoid-s8476" xml:space="preserve">Ita v. </s> + <s xml:id="echoid-s8477" xml:space="preserve">gr. </s> + <s xml:id="echoid-s8478" xml:space="preserve">ſi ſe-<lb/>cundum ſententiam communem dicamus vim repellentem eſſe æqualem pon-<lb/>deri ſimplicis cylindri ſæpe nominati, erit vis repellens, dum aquæ per cana-<lb/>lem C E ex vaſe A C B effluere finguntur = a + b; </s> + <s xml:id="echoid-s8479" xml:space="preserve">& </s> + <s xml:id="echoid-s8480" xml:space="preserve">hæc vis ſi ſubtrahatur à <lb/>pondere totius aquæ A B C E ſeu Ma + 2a + 2b - 2√(aa + ab), relinqui-<lb/>tur Ma + a + b - 2√(aa + ab) quæ eſt vis requiſita ad ſuſpendendum ſy-<lb/>ſtema A B C E, dum aquæ fluunt: </s> + <s xml:id="echoid-s8481" xml:space="preserve">Vidimus autem hanc vim eandem eſſe debe- +<pb o="286" file="0300" n="300" rhead="HYDRODYNAMICÆ"/> +re, ſi canalis C E abſit: </s> + <s xml:id="echoid-s8482" xml:space="preserve">Sed tunc eſt vis ſuſpenſorra = Ma - √(aa + ab), <lb/>quia pondus aquæ A B C eſt = Ma & </s> + <s xml:id="echoid-s8483" xml:space="preserve">vis repellens per hypotheſin eſt ſimplex <lb/>cylindrus foramini C ad altitudinem a ſuperinſtructus. </s> + <s xml:id="echoid-s8484" xml:space="preserve">Deberet igitur in hâc <lb/>hypotheſi ſemper eſſe Ma + a + b - 2√(aa + ab) = Ma - √(aa + ab) <lb/>ſeu a + b = √(aa + ab), quod eſt abſurdum. </s> + <s xml:id="echoid-s8485" xml:space="preserve">Similis abſurditas demonſtra-<lb/>ri poſſet, ſi vena ſurſum verticaliter aſcendere putetur: </s> + <s xml:id="echoid-s8486" xml:space="preserve">& </s> + <s xml:id="echoid-s8487" xml:space="preserve">fruſtra hic excipe-<lb/>retur pro communi ſententia firmanda, venam effluentem C E fingi non poſſe <lb/>tanquam continuam, niſi aliqua aquæ tenacitas fingatur ſimul (aliàs enim ve-<lb/>nam mox præ orificio in guttulas abruptum iri) & </s> + <s xml:id="echoid-s8488" xml:space="preserve">tenacitatem rei ſtatum per-<lb/>mutare: </s> + <s xml:id="echoid-s8489" xml:space="preserve">nam profecto nec velocitates aquæ à cohæſione mutua aquæ in C E <lb/>mutantur nec latera canalis C E preſſionem ullam ſentiunt, ſicut demonſtravi <lb/>Sect. </s> + <s xml:id="echoid-s8490" xml:space="preserve">XII. </s> + <s xml:id="echoid-s8491" xml:space="preserve">§. </s> + <s xml:id="echoid-s8492" xml:space="preserve">13. </s> + <s xml:id="echoid-s8493" xml:space="preserve">ut taceam cohæſionem aquæ non oriri à tenacitate ſed ab ali-<lb/>qua virtute magnetica ſeu à mutua attractione, à qua virtute centrum gravita-<lb/>tis in nullo ſyſtemate nec majorem nec minorem velocitatem acquirere poteſt. <lb/></s> + <s xml:id="echoid-s8494" xml:space="preserve">Sed hæc porro adverſariorum exceptio in venis verticaliter aſcendentibus nul-<lb/>lum plane locum habet, cum aquæ ibi continuè maneant, ſi vel nulla aquisin-<lb/>ſit tenacitas aut mutua attractio.</s> + <s xml:id="echoid-s8495" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8496" xml:space="preserve">At poſſem infinitis aliis modis & </s> + <s xml:id="echoid-s8497" xml:space="preserve">exemplis particularibus ſententiam <lb/>noſtram confirmare, ſi hiſce diutius inſiſtere vellem. </s> + <s xml:id="echoid-s8498" xml:space="preserve">Ita v. </s> + <s xml:id="echoid-s8499" xml:space="preserve">gr. </s> + <s xml:id="echoid-s8500" xml:space="preserve">in Fig. </s> + <s xml:id="echoid-s8501" xml:space="preserve">29. </s> + <s xml:id="echoid-s8502" xml:space="preserve">Sect. <lb/></s> + <s xml:id="echoid-s8503" xml:space="preserve">V. </s> + <s xml:id="echoid-s8504" xml:space="preserve">§. </s> + <s xml:id="echoid-s8505" xml:space="preserve">4. </s> + <s xml:id="echoid-s8506" xml:space="preserve">deſcripta, ſi ſit altitudo N S = 1, orificium L M = 1, & </s> + <s xml:id="echoid-s8507" xml:space="preserve">orificium <lb/>R S = 2, erit P B = {1/3}, vis repellens, quæ oritur ab effluxu aquæ per R S <lb/>= 2 X {2/3} = {4/3}, & </s> + <s xml:id="echoid-s8508" xml:space="preserve">demonſtrare poſſum vim repellentem, quæ prodit ab ef-<lb/>fluxu aquæ ex ſimplici cylindro R N per L M eſſe etiam = {4/3}, & </s> + <s xml:id="echoid-s8509" xml:space="preserve">ſic vim re-<lb/>pellentem totalem eſſe = {8/3}, quæ præciſe facit duplum cylindrum aqueum fo-<lb/>ramini L M ad altitudinem N S + P B inſiſtentem. </s> + <s xml:id="echoid-s8510" xml:space="preserve">Talis autem conſenſus ex <lb/>aliis theoriis falſo receptis minime prodit, ita ut de noſtra amplius non poſ-<lb/>ſint dubitare, niſi harum rerum penitus ignari: </s> + <s xml:id="echoid-s8511" xml:space="preserve">Id vero, quod dixi, vim re-<lb/>pellentem aquæ ex ſimplici cylindro R N per L M effluentis eſſe = {4/3}, ſi de-<lb/>monſtrare velim, poſtulat ut vis repellens definiatur, cum aquæ ex vaſe non <lb/>infinito data velocitate quacunque non variata fluunt: </s> + <s xml:id="echoid-s8512" xml:space="preserve">Ne vero prolixior ſim <lb/>in hâc re, id aliis efficiendum relinquo; </s> + <s xml:id="echoid-s8513" xml:space="preserve">neque id nunc amplius magnam fa-<lb/>ceſſet operam; </s> + <s xml:id="echoid-s8514" xml:space="preserve">Pergo ad alia.</s> + <s xml:id="echoid-s8515" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8516" xml:space="preserve">§. </s> + <s xml:id="echoid-s8517" xml:space="preserve">12. </s> + <s xml:id="echoid-s8518" xml:space="preserve">Demonſtrationes quas adhuc dedimus non valent niſi pro fiſtu- +<pb o="287" file="0301" n="301" rhead="SECTIO DECIMA TERTIA."/> +lis rectis, in quibus nempe uniuscujuſque guttulæ vis motrix, indeque ori-<lb/>unda vis repellens, inter ſe ſingulæ conſpirant, communemque habent dire-<lb/>ctionem: </s> + <s xml:id="echoid-s8519" xml:space="preserve">at cum fiſtulæ vaſi implantatæ, per quas aquæ effluunt, ſunt incur-<lb/>vatæ, alius adhibendus eſt demonſtrandi modus: </s> + <s xml:id="echoid-s8520" xml:space="preserve">Ut nihil in iſto argumento <lb/>prorſus novo omittamus, hunc quoque caſum docebimus: </s> + <s xml:id="echoid-s8521" xml:space="preserve">nec erit, quod <lb/>laboris pœniteat, cum inde veræ preſſionum leges, quas natura non ſolum in <lb/>his caſibus, ſed & </s> + <s xml:id="echoid-s8522" xml:space="preserve">multis aliis ſequatur, apparebunt.</s> + <s xml:id="echoid-s8523" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8524" xml:space="preserve">§. </s> + <s xml:id="echoid-s8525" xml:space="preserve">13. </s> + <s xml:id="echoid-s8526" xml:space="preserve">Concipiamus itaque vaſi infinito fiſtulam implantatam eſſe uni-<lb/>formis quidem amplitudinis, ſed incurvatam ſecundum curvaturam qualem-<lb/>cunque A S (Fig. </s> + <s xml:id="echoid-s8527" xml:space="preserve">83.) </s> + <s xml:id="echoid-s8528" xml:space="preserve">ita ut A locus ſit inſertionis, S locus effluxus: </s> + <s xml:id="echoid-s8529" xml:space="preserve">Du-<lb/> +<anchor type="note" xlink:label="note-0301-01a" xlink:href="note-0301-01"/> +cantur tangentes in A & </s> + <s xml:id="echoid-s8530" xml:space="preserve">S, nempe A R & </s> + <s xml:id="echoid-s8531" xml:space="preserve">S B, ſitque A B ad S B perpendi-<lb/>cularis: </s> + <s xml:id="echoid-s8532" xml:space="preserve">fuerit velocitas aquæ per fiſtulam transfluentis uniformis & </s> + <s xml:id="echoid-s8533" xml:space="preserve">talis, <lb/>quæ debeatur altitudini A; </s> + <s xml:id="echoid-s8534" xml:space="preserve">amplitudo fiſtulæ ubique = 1: </s> + <s xml:id="echoid-s8535" xml:space="preserve">Dico totam vim <lb/>repellentem in directione S B ſumtam fore rurſus = 2 A, hancque ſolam adfore.</s> + <s xml:id="echoid-s8536" xml:space="preserve"/> +</p> +<div xml:id="echoid-div294" type="float" level="2" n="3"> +<note position="right" xlink:label="note-0301-01" xlink:href="note-0301-01a" xml:space="preserve">Fig. 83.</note> +</div> +<p> + <s xml:id="echoid-s8537" xml:space="preserve">Demonſtrationis gratia ducantur infinite propinquæ nq, ep ad S B per-<lb/>pendiculares; </s> + <s xml:id="echoid-s8538" xml:space="preserve">n m parallela eidem S B; </s> + <s xml:id="echoid-s8539" xml:space="preserve">ſit S q = x, qp = dx; </s> + <s xml:id="echoid-s8540" xml:space="preserve">qn = y; <lb/></s> + <s xml:id="echoid-s8541" xml:space="preserve">e m = dy: </s> + <s xml:id="echoid-s8542" xml:space="preserve">erit radius oſculi in e n = {- dsdy/ddx}, ſumtis elementis en quæ <lb/>vocabo ds pro conſtantibus; </s> + <s xml:id="echoid-s8543" xml:space="preserve">habet autem columella aquæ intercepta inter e & </s> + <s xml:id="echoid-s8544" xml:space="preserve">n <lb/>vim centrifugam, ſic determinandam: </s> + <s xml:id="echoid-s8545" xml:space="preserve">gravitas columellæ eſt = ds (quia <lb/>baſis ejus = 1 & </s> + <s xml:id="echoid-s8546" xml:space="preserve">altitudo = ds) atque ſi radius oſculi foret = 2 A, ha-<lb/>beretur per theorema Hugenianum vis centrifuga particulæ æqualis ejusdem <lb/>gravitati, & </s> + <s xml:id="echoid-s8547" xml:space="preserve">ſunt vires centrifugæ cæteris paribus in reciproca ratione radio-<lb/>rum: </s> + <s xml:id="echoid-s8548" xml:space="preserve">eſt igitur vis centrifuga columellæ = {- 2 Addx/dy}: </s> + <s xml:id="echoid-s8549" xml:space="preserve">exprimatur hæc vis <lb/>centrifuga per ec ad curvam perpendicularem, ducaturque co ipfi B S paral-<lb/>lela: </s> + <s xml:id="echoid-s8550" xml:space="preserve">reſolvatur vis e c in oc & </s> + <s xml:id="echoid-s8551" xml:space="preserve">eo; </s> + <s xml:id="echoid-s8552" xml:space="preserve">erit (ob ſimilitudinem triangulorum eoc <lb/>& </s> + <s xml:id="echoid-s8553" xml:space="preserve">nme) vis oc = {- 2 Addx/ds}, vis eo = {- 2 Adxddx/dyds} = (ob d s conſtans) <lb/>{2 Addy/ds}.</s> + <s xml:id="echoid-s8554" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8555" xml:space="preserve">Sed vis elementaris oc agit ſola in directione S B, dum altera e o pro <lb/>hac directione eſt negligenda: </s> + <s xml:id="echoid-s8556" xml:space="preserve">ſumatur integrale vis elementaris oc cum con-<lb/>ſtanti tali, ut integrale una cum abſciſſa evaneſcat: </s> + <s xml:id="echoid-s8557" xml:space="preserve">integrale hoc eſt = 2A +<pb o="288" file="0302" n="302" rhead="HYDRODYNAMICÆ"/> +- {2 Adx/ds}, quia in S eſt dx = d s: </s> + <s xml:id="echoid-s8558" xml:space="preserve">Nunc ut habeatur vis in directione <lb/>tangentis S B pro tota fiſtula, ponendum eſt {RB/RA} pro {dx/ds}, ergo tota vis ſe-<lb/>cundum tangentem SB = 2A - {2A x RB/RA}. </s> + <s xml:id="echoid-s8559" xml:space="preserve">Hæc vero oritur à vi centrifu-<lb/>ga cujusvis guttulæ: </s> + <s xml:id="echoid-s8560" xml:space="preserve">ſed alia vis ſupereſt conſideranda; </s> + <s xml:id="echoid-s8561" xml:space="preserve">nempe dum aqua <lb/>ex vaſe infinite amplo continue in fiſtulam influit velocitate uniformi reſpon-<lb/>dente altitudini A, vas repellitur ſecundum directionem R A vi 2 A (per §. </s> + <s xml:id="echoid-s8562" xml:space="preserve">4.) <lb/></s> + <s xml:id="echoid-s8563" xml:space="preserve">quâ reſoluta in tangentialem ſecundum S B eique perpendicularem ſecundum <lb/>B A, prior {2A x RB/RA} erit ſola conſideranda; </s> + <s xml:id="echoid-s8564" xml:space="preserve">& </s> + <s xml:id="echoid-s8565" xml:space="preserve">quia habet directionem com-<lb/>munem cum vi 2A - {2A x RB/RA} à vi centrifuga oriunda & </s> + <s xml:id="echoid-s8566" xml:space="preserve">modo definita, erit <lb/>eidem addenda: </s> + <s xml:id="echoid-s8567" xml:space="preserve">ſicque ſumma 2A - {2A x RB/RA} + {2A x RB/RA} vel 2A expri-<lb/>met vim repellentem ſecundum directionem S B.</s> + <s xml:id="echoid-s8568" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8569" xml:space="preserve">Ut porro demonſtremus ſub nulla alia directione vas repelli, recurre-<lb/>mus ad vim elementarem eo, quam vidimus = {2Addy/ds}, cujus integrale = <lb/>{2A x AB/AR}, quæ præciſe annihilatur à vi 2A vas repellente ſecundum directio-<lb/>nem R A, poſtquam hæc debite reſoluta fuit. </s> + <s xml:id="echoid-s8570" xml:space="preserve">Q. </s> + <s xml:id="echoid-s8571" xml:space="preserve">E. </s> + <s xml:id="echoid-s8572" xml:space="preserve">D.</s> + <s xml:id="echoid-s8573" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8574" xml:space="preserve">§. </s> + <s xml:id="echoid-s8575" xml:space="preserve">14. </s> + <s xml:id="echoid-s8576" xml:space="preserve">Hæc theorematis generaliſſimi ſimplicitas, quâ nempe vis re-<lb/>pellens in directione aquis uniformiter effluentibus contraria indicatur con-<lb/>ſtanter = 2A, argumentum eſſe poteſt, quod dicitur ad hominem pro ejus-<lb/>dem bonitate, iis qui ratiocinium noſtrum aut non aſſequentur aut exami-<lb/>nare non cupient ſufficienti attentione. </s> + <s xml:id="echoid-s8577" xml:space="preserve">Si vero vim repellentem aquæ ex <lb/>vaſe infinito in fiſtulam influentis ſub directione A R ſtatuas = A, vides ſy-<lb/>ſtema repelli in directione S B vi quæ ſit = 2A - {A x RB/RA}, quod abſurdum <lb/>eſſe vel ipſa mihi indicare videtur formula. </s> + <s xml:id="echoid-s8578" xml:space="preserve">Neque in hâc opinione nulla eſ-<lb/>ſet vis in directione ad priorem perpendiculari: </s> + <s xml:id="echoid-s8579" xml:space="preserve">Nam vas deberet reprimi ſe-<lb/>cundum directionem B A vi {A x AB/AR}, quod iterum mihi eſt abſurdum & </s> + <s xml:id="echoid-s8580" xml:space="preserve">cu-<lb/>jus falſitatem experimento cognovi, in caſu quo angulus A R S erat rectus & </s> + <s xml:id="echoid-s8581" xml:space="preserve"><lb/>A B = A R.</s> + <s xml:id="echoid-s8582" xml:space="preserve"/> +</p> +<pb o="289" file="0303" n="303" rhead="SECTIO DECIMA TERTIA."/> +<p> + <s xml:id="echoid-s8583" xml:space="preserve">Multa alia theoremata pro hoc argumento in tota ſua, quam habere <lb/>poteſt, extenſione, ſumto erui & </s> + <s xml:id="echoid-s8584" xml:space="preserve">demonſtrari poterunt, pro fluxu aquarum <lb/>nondum uniformi eoque per fiſtulam utcunque inæqualem, modo ſimul at-<lb/>tendatur ad ea, quæ §. </s> + <s xml:id="echoid-s8585" xml:space="preserve">8. </s> + <s xml:id="echoid-s8586" xml:space="preserve">monita fuerunt. </s> + <s xml:id="echoid-s8587" xml:space="preserve">Quia vero per ſingula ire non va-<lb/>cat, ad aliam progredior vim examinandam priori ſub directione contraria <lb/>æqualem, illam nempe quam vena effluens in planum exerit, dum in illud <lb/>perpendiculariter impingit.</s> + <s xml:id="echoid-s8588" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8589" xml:space="preserve">§. </s> + <s xml:id="echoid-s8590" xml:space="preserve">15. </s> + <s xml:id="echoid-s8591" xml:space="preserve">De impetu venæ aqueæ in planum impingentis multi commen-<lb/>tati ſunt, plurimaque ſumſere experimenta. </s> + <s xml:id="echoid-s8592" xml:space="preserve">Ego quoque hâc de re quæ-<lb/>dam dedi in Comm. </s> + <s xml:id="echoid-s8593" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s8594" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s8595" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s8596" xml:space="preserve">tom. </s> + <s xml:id="echoid-s8597" xml:space="preserve">2. </s> + <s xml:id="echoid-s8598" xml:space="preserve">Experimenta extant apud Mariot-<lb/>tum in tract. </s> + <s xml:id="echoid-s8599" xml:space="preserve">de mot aquarum, in hiſt. </s> + <s xml:id="echoid-s8600" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s8601" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s8602" xml:space="preserve">conſcripta a D. </s> + <s xml:id="echoid-s8603" xml:space="preserve">du Hamel. </s> + <s xml:id="echoid-s8604" xml:space="preserve">p. </s> + <s xml:id="echoid-s8605" xml:space="preserve">48. <lb/></s> + <s xml:id="echoid-s8606" xml:space="preserve">& </s> + <s xml:id="echoid-s8607" xml:space="preserve">alibi. </s> + <s xml:id="echoid-s8608" xml:space="preserve">Equidem non admodum conveniunt, plurima tamen indicare prima <lb/>fronte videntur niſum venæ aqueæ uniformiter fluentis æqualem eſſe ponderi <lb/>cylindri aquei, cujus baſis ſit foramen, per quod aquæ effluunt & </s> + <s xml:id="echoid-s8609" xml:space="preserve">cujus al-<lb/>titudo ſit æqualis altitudini aquæ ſupra foramen: </s> + <s xml:id="echoid-s8610" xml:space="preserve">Huic ſententiæ plerique imo <lb/>omnes, adhæſerunt & </s> + <s xml:id="echoid-s8611" xml:space="preserve">adhuc adhærent, quia cum aliis quoque experimentis, <lb/>præſertim quæ de globis in medio reſiſtente motis ſumi ſolent, mire conve-<lb/>nit: </s> + <s xml:id="echoid-s8612" xml:space="preserve">Eandem igitur ipſemet ſecutus ſum, quamvis plura animum ſuſpende-<lb/>bant, in cit. </s> + <s xml:id="echoid-s8613" xml:space="preserve">Comm. </s> + <s xml:id="echoid-s8614" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s8615" xml:space="preserve">nec hæſitavi in ipſo hoc opere, quod ſub mani-<lb/>bus habeo, Sectione nempe IX. </s> + <s xml:id="echoid-s8616" xml:space="preserve">§. </s> + <s xml:id="echoid-s8617" xml:space="preserve">§. </s> + <s xml:id="echoid-s8618" xml:space="preserve">31. </s> + <s xml:id="echoid-s8619" xml:space="preserve">32. </s> + <s xml:id="echoid-s8620" xml:space="preserve">illa inſtar exempli uti. </s> + <s xml:id="echoid-s8621" xml:space="preserve">Ve-<lb/>rum enim vero re attentius perpenſa, novisque adhibitis principiis, ſimulque <lb/>aliis novi generis experimentis inſtitutis, clare tandem vidi communem iſtam <lb/>opinionem de impetu venæ aqueæ eodem modo mutandam eſſe, ſicuti New-<lb/>toni de vi repellente, ſcilicet ut loco orificii conſideretur ſectio venæ contra-<lb/>ctæ & </s> + <s xml:id="echoid-s8622" xml:space="preserve">loco altitudinis aquæ adhibeatur dupla altitudo velocitati aquarum reali <lb/>reſpondens: </s> + <s xml:id="echoid-s8623" xml:space="preserve">Demonſtratum enim habeo, vim repulſionis §. </s> + <s xml:id="echoid-s8624" xml:space="preserve">2.</s> + <s xml:id="echoid-s8625" xml:space="preserve">, expoſitam <lb/>omnino æqualem eſſe impetui venæ, ſi hæc tota in planum perpendiculariter <lb/>incidat: </s> + <s xml:id="echoid-s8626" xml:space="preserve">ſequitur inde impetum venæ majorem eſſe, quo minor fuerit venæ <lb/>contractio, hâcque plane evanescente, & </s> + <s xml:id="echoid-s8627" xml:space="preserve">aquis ſimul tota ſua velocitate, quam <lb/>in theoria habere poſſunt, erumpentibus, tum impetum duplo majorem eſſe, <lb/>quam vulgo ſtatuitur: </s> + <s xml:id="echoid-s8628" xml:space="preserve">quia vero ſemper & </s> + <s xml:id="echoid-s8629" xml:space="preserve">velocitati aliquid decedit & </s> + <s xml:id="echoid-s8630" xml:space="preserve">vena non <lb/>raro ad dimidium fere contrahitur, factum eſt ut experimenta pleraque ſimplam <lb/>in Cylindro altitudinem arguere viſa fuerint in impetu illo æſtimando. </s> + <s xml:id="echoid-s8631" xml:space="preserve">Velim <lb/>autem probe notetur, de venis ſolitariis tantum mihi hic ſermonem eſſe, quas +<pb o="290" file="0304" n="304" rhead="HYDRODYNAMICÆ"/> +plana totas excipiant, non autem de fluidis corpora ambientibus in eademque <lb/>impetum facientibus, veluti de Ventis aut fluminibus: </s> + <s xml:id="echoid-s8632" xml:space="preserve">dico enim hos dupli-<lb/>cis generis impetus quos auctores adhuc confuderunt, probe à ſe invicem di-<lb/>ſtinguendos eſſe, ob rationes infra breviter exponendas.</s> + <s xml:id="echoid-s8633" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8634" xml:space="preserve">§. </s> + <s xml:id="echoid-s8635" xml:space="preserve">16. </s> + <s xml:id="echoid-s8636" xml:space="preserve">Ratione venæ aqueæ ſic cenſeo: </s> + <s xml:id="echoid-s8637" xml:space="preserve">aquas velocitate uniformi ex <lb/>cylindro infinite amplo verticali A B M (Fig. </s> + <s xml:id="echoid-s8638" xml:space="preserve">84.) </s> + <s xml:id="echoid-s8639" xml:space="preserve">per foramen laterale C M <lb/> +<anchor type="note" xlink:label="note-0304-01a" xlink:href="note-0304-01"/> +horizontaliter effluere pono, venamque perpendiculariter impingere in lami-<lb/>nam E F: </s> + <s xml:id="echoid-s8640" xml:space="preserve">ita facile video, quia particulæ inſequentes priores impediunt ne <lb/>reſilire poſſint, fore ut ſingulæ ad latera deflectantur, idque motu laminæ E F <lb/>(ſi modo hæc ſat magna fuerit, ut vena tota quamvis diſperſa excipiatur) pa-<lb/>rallelo vel tantum non tali: </s> + <s xml:id="echoid-s8641" xml:space="preserve">Et quia omnia ſunt in ſtatu permanentiæ, fingere <lb/>licet laminam E F vaſi eſſe affirmatam, venamque lateribus C H D G L M <lb/>circumdatam, ita, ut aquæ per hiatum circularem D E G F effluere ex vaſe <lb/>A B C H D E F G L M cenſeri poſſint. </s> + <s xml:id="echoid-s8642" xml:space="preserve">Hoc ſi ita fuerit, demonſtravimus <lb/>§. </s> + <s xml:id="echoid-s8643" xml:space="preserve">13. </s> + <s xml:id="echoid-s8644" xml:space="preserve">guttulas in D E effluentes vim repellentem quidem producere ſecun-<lb/>dum E F; </s> + <s xml:id="echoid-s8645" xml:space="preserve">ſed ſimul apparet vim repellentem eſſe in G F priori contrariam <lb/>ita ut ad hanc virium repellentium claſſem hic non ſit attendendum. </s> + <s xml:id="echoid-s8646" xml:space="preserve">Quod <lb/>vero ad directionem, laminæ E F vel cylindro B C, perpendicularem atti-<lb/>net, demonſtravimus in fine ejuſdem §. </s> + <s xml:id="echoid-s8647" xml:space="preserve">13. </s> + <s xml:id="echoid-s8648" xml:space="preserve">ſub ea directione plane nullam <lb/>fieri repulſionem: </s> + <s xml:id="echoid-s8649" xml:space="preserve">Igitur tantum lamina E F propellitur, quantum cylindrus <lb/>repellitur. </s> + <s xml:id="echoid-s8650" xml:space="preserve">Idque eſt quod demonſtrare volui: </s> + <s xml:id="echoid-s8651" xml:space="preserve">Atque inde jam ſequitur, preſ-<lb/>ſionem venæ aqueæ, quæ tota in laminam incurrit, tantam eſſe quanta pon-<lb/>dus cylindri aquei, qui pro baſe habeat ſectionem venæ (poſtquam hæc unifor-<lb/>mem acquiſivit amplitudinem) & </s> + <s xml:id="echoid-s8652" xml:space="preserve">pro altitudine duplam altitudinem velocita-<lb/>ti aquarum (poſtquam hæc ſimiliter uniformis facta eſt) debitam.</s> + <s xml:id="echoid-s8653" xml:space="preserve"/> +</p> +<div xml:id="echoid-div295" type="float" level="2" n="4"> +<note position="left" xlink:label="note-0304-01" xlink:href="note-0304-01a" xml:space="preserve">Fig. 84.</note> +</div> +<p> + <s xml:id="echoid-s8654" xml:space="preserve">§. </s> + <s xml:id="echoid-s8655" xml:space="preserve">17. </s> + <s xml:id="echoid-s8656" xml:space="preserve">Non dubito multos fore, quibus propoſitio hæc plane nova <lb/>ſuſpecta videatur atque experimentis contraria: </s> + <s xml:id="echoid-s8657" xml:space="preserve">Hos vero perpendere velim, <lb/>experimenta hactenus ſumta nequaquam regulæ communi accurate reſponde-<lb/>re, & </s> + <s xml:id="echoid-s8658" xml:space="preserve">in pleriſque caſibus noſtram Regulam parum differre à communi, quam-<lb/>vis in theoria maxime ſint diverſæ: </s> + <s xml:id="echoid-s8659" xml:space="preserve">tum etiam eos in anteceſſum monitos cu-<lb/>pio, alia me inſtituiſſe experimenta, quæ ſingula meam ſententiam exactiſſime <lb/>confirmant, veteremque plane refellunt! experimenta a me ſumpta in fine Se-<lb/>ctionis recenſebo. </s> + <s xml:id="echoid-s8660" xml:space="preserve">Demonſtrandi modus quo uſus fui, fortaſſe etiam parum +<pb o="291" file="0305" n="305" rhead="SECTIO DECIMA TERTIA."/> +videbitur quibusdam accuratus, habeo autem aliam demonſtrationem directam, <lb/>quæ nova proprietate innititur Mechanica mihi aliquando obſervata, quam-<lb/>que hic communicabo, tum quia dictam demonſtrationem facillime quivis in-<lb/>de deducere, tum etiam quia ad alios uſus eandem impendere poterit: </s> + <s xml:id="echoid-s8661" xml:space="preserve">Ita au-<lb/>tem ſe habet.</s> + <s xml:id="echoid-s8662" xml:space="preserve"/> +</p> +<p style="it"> + <s xml:id="echoid-s8663" xml:space="preserve">Si cerpus movetur velocitate uniformi, directiones autem ſuas con-<lb/>t<unsure/>inue mutat à cauſis quibuſcunque & </s> + <s xml:id="echoid-s8664" xml:space="preserve">utcunque agentibus, donec directionem <lb/>acquiſiverit perpendicularem ad primam directionem, ſique ſingulæ preßiones <lb/>corpus deflectentes reſolvantur in duas claſſes, alteram parallelam primæ dire-<lb/>ctioni, alteram perpendicularem; </s> + <s xml:id="echoid-s8665" xml:space="preserve">Denique ſi preßiones ſingulæ parallelæ multi-<lb/>plicantur per ſuatempora; </s> + <s xml:id="echoid-s8666" xml:space="preserve">dico fore ſummam productorum conſtanter eandem, <lb/>& </s> + <s xml:id="echoid-s8667" xml:space="preserve">quidem æqualem ei, quæ totum motum à quiete generare aut generatum <lb/>totum abſorbere valet.</s> + <s xml:id="echoid-s8668" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8669" xml:space="preserve">Hâc affectione Dynamica, cum utimur in præſenti noſtro negotio, con-<lb/>ſideranda eſt lamina E F, quæ ſua in aquas reactione, earundem directionem <lb/>mutat, uſque dum perpendicularis ad primam facta fuerit: </s> + <s xml:id="echoid-s8670" xml:space="preserve">Ergo propoſitio <lb/>præcedentis paragraphi ope hujus affectionis eodem modo demonſtrabitur, <lb/>quo uſi ſumus §. </s> + <s xml:id="echoid-s8671" xml:space="preserve">4. </s> + <s xml:id="echoid-s8672" xml:space="preserve">ad determinandam vim repellentem ope principii §. </s> + <s xml:id="echoid-s8673" xml:space="preserve">3. </s> + <s xml:id="echoid-s8674" xml:space="preserve">ex-<lb/>poſiti. </s> + <s xml:id="echoid-s8675" xml:space="preserve">Hæc igitur vera idea videtur, quam de impetu aquarum mente conci-<lb/>pere debemus: </s> + <s xml:id="echoid-s8676" xml:space="preserve">ponit autem guttas aquæ ſingulas ſecundum directionem lami-<lb/>næ ad latera reſilire, à quâ indole aquas non recedere ſemper obſervavi: </s> + <s xml:id="echoid-s8677" xml:space="preserve">vidi <lb/>tamen etiam guttulas aliquas ſed paucas retrorſum reſilire; </s> + <s xml:id="echoid-s8678" xml:space="preserve">hæ autem majorem <lb/>preſſionem producunt, quam quæ ad latera deflectuntur: </s> + <s xml:id="echoid-s8679" xml:space="preserve">Et eo ipſo inducor, ut <lb/>firmiter credam, ſi vena aquea magno impetu oblique contra planum impin-<lb/>gat, v. </s> + <s xml:id="echoid-s8680" xml:space="preserve">gr. </s> + <s xml:id="echoid-s8681" xml:space="preserve">ſub angulo triginta graduum, preſſionem inde orituram pluſquam <lb/>dimidiam ejus, quæ à vena eadem directe impingente oritur, cum ſecundum <lb/>regulas ordinarias exactè dimidiam vim exerere deberet: </s> + <s xml:id="echoid-s8682" xml:space="preserve">ratio ejus rei eſt, quod <lb/>in impulſu obliquo plures particulæ reſilire poſſint, quam in directo, imo fe-<lb/>re omnes, ſi magna fuerit velocitas.</s> + <s xml:id="echoid-s8683" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8684" xml:space="preserve">Si autem omnes ita reſilire ponantur, ut angulus incidentiæ angulo <lb/>reflexionis æqualis ſit, tunc uterque impulſus idem cenſendus erit. </s> + <s xml:id="echoid-s8685" xml:space="preserve">Optimus <lb/>hic aquarum preſſiones æſtimandi modus eſt, qui ratiocinio à poſteriori inni-<lb/>titur.</s> + <s xml:id="echoid-s8686" xml:space="preserve"/> +</p> +<pb o="292" file="0306" n="306" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s8687" xml:space="preserve">§. </s> + <s xml:id="echoid-s8688" xml:space="preserve">18. </s> + <s xml:id="echoid-s8689" xml:space="preserve">Sequitur porro ex præfata affectione probe intellecta, eundem <lb/>oriri à preſſionibus effectum ſive lamina aquas ad latera deflectat, ſive cauſa <lb/>fingatur motum omnem, quem particulæ aqueæ cylindrum egreſſæ acquiſive-<lb/>runt, abſorbens: </s> + <s xml:id="echoid-s8690" xml:space="preserve">Inde intelligitur quid futurum ſit, ſi orificium C M (Fig. </s> + <s xml:id="echoid-s8691" xml:space="preserve">85.) <lb/></s> + <s xml:id="echoid-s8692" xml:space="preserve"> +<anchor type="note" xlink:label="note-0306-01a" xlink:href="note-0306-01"/> +per quod aquæ ex cylindro A B M effluunt, aliis aquis in vaſe P Q F E ſtagnan-<lb/>tibus ſubmerſum fuerit: </s> + <s xml:id="echoid-s8693" xml:space="preserve">repelletur nempe cylindrus A B M verſus P Q intra vas <lb/>P Q F E, ſi hoc cum cylindro non cohæreat; </s> + <s xml:id="echoid-s8694" xml:space="preserve">At ſi vaſa inter ſe fuerint firmata, <lb/>nullam patietur ſyſtema preſſionem prævalentem; </s> + <s xml:id="echoid-s8695" xml:space="preserve">quanta enim eſt preſſio ver-<lb/>ſus P Q ab effluentibus aquis, tanta quoque oritur preſſio contraria verſus E F <lb/>à continua deſtructione motus, quem particulæ cylindrum egreſſæ acquiſivere.</s> + <s xml:id="echoid-s8696" xml:space="preserve"/> +</p> +<div xml:id="echoid-div296" type="float" level="2" n="5"> +<note position="left" xlink:label="note-0306-01" xlink:href="note-0306-01a" xml:space="preserve">Fig. 85.</note> +</div> +<p> + <s xml:id="echoid-s8697" xml:space="preserve">§. </s> + <s xml:id="echoid-s8698" xml:space="preserve">19. </s> + <s xml:id="echoid-s8699" xml:space="preserve">Dixi de preſſione venæ, quam lamina totam etiamſi expanſam <lb/>excipit: </s> + <s xml:id="echoid-s8700" xml:space="preserve">Venio ad alteram ſpeciem impetus aquarum, quem ſcilicet ſuſtinent <lb/>laminæ fluido undique ſubmerſæ: </s> + <s xml:id="echoid-s8701" xml:space="preserve">puto autem hanc non poſſe abſolute defini-<lb/>ri, quia ſingulæ particulæ in laminam impingentes aliter deflectuntur. </s> + <s xml:id="echoid-s8702" xml:space="preserve">Si vero <lb/>cujuslibet particulæ deviatio cognita ponatur, non difficilis erit amplius quæ-<lb/>ſtionis ſolutio, mutato paullum theoremate, quo §. </s> + <s xml:id="echoid-s8703" xml:space="preserve">17. </s> + <s xml:id="echoid-s8704" xml:space="preserve">uſi ſumus eoque ge-<lb/>neraliori reddito, nempe tali: </s> + <s xml:id="echoid-s8705" xml:space="preserve">ſi angulus mutatæ in corpore moto directionis non <lb/>fuerit rectus, ſed recto minor, tunc quoque minor erit ſumma productorum (de <lb/>quâ antea ſermo fuit) in ratione ut ſinus verſus mutatæ directionis ad ſinum <lb/>totum.</s> + <s xml:id="echoid-s8706" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8707" xml:space="preserve">Igitur pro quâvis guttula indagandum eſſet, quantum directionem mo-<lb/>tus ab obice, ſeu lamina curſui oppoſita mutare cogatur. </s> + <s xml:id="echoid-s8708" xml:space="preserve">At in theoria hu-<lb/>juſmodi definitiones exhiberi accurate vix poſſunt; </s> + <s xml:id="echoid-s8709" xml:space="preserve">nec experientia probattheo-<lb/>remata hanc in rem exhiberi ſolita; </s> + <s xml:id="echoid-s8710" xml:space="preserve">veluti quod conatus fluminis directe con-<lb/>tra circulum impingentis duplo ſit major conatu ejuſdem fluminis contra ſphæ-<lb/>ram ejuſdem diametri, & </s> + <s xml:id="echoid-s8711" xml:space="preserve">quæ ſunt ſimilia: </s> + <s xml:id="echoid-s8712" xml:space="preserve">quod autem quantitas preſſionis <lb/>pro ſphæra, qualis dari ſolet ab auctoribus, cum experimentis à Newtono <lb/>aliiſque inſtitutis & </s> + <s xml:id="echoid-s8713" xml:space="preserve">in princ. </s> + <s xml:id="echoid-s8714" xml:space="preserve">math. </s> + <s xml:id="echoid-s8715" xml:space="preserve">phil. </s> + <s xml:id="echoid-s8716" xml:space="preserve">nat. </s> + <s xml:id="echoid-s8717" xml:space="preserve">recenſitis, ſatis accurate conveniat, <lb/>id omnibus bene perpenſis caſui fortuito tribuendum eſſe cenſeo.</s> + <s xml:id="echoid-s8718" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8719" xml:space="preserve">Theoremata quæ ad motum in mediis reſiſtentibus theoretice conſideratum <lb/>faciunt, tum etiam varias obſervationes phyſicas dedi in tom. </s> + <s xml:id="echoid-s8720" xml:space="preserve">II. </s> + <s xml:id="echoid-s8721" xml:space="preserve">Comm. </s> + <s xml:id="echoid-s8722" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s8723" xml:space="preserve">Sc. <lb/></s> + <s xml:id="echoid-s8724" xml:space="preserve">Petrop. </s> + <s xml:id="echoid-s8725" xml:space="preserve">& </s> + <s xml:id="echoid-s8726" xml:space="preserve">ſeqq. </s> + <s xml:id="echoid-s8727" xml:space="preserve">Neque proinde ea hic repetam, quamvis ad inſtitutum no-<lb/>ſtrum pertineant; </s> + <s xml:id="echoid-s8728" xml:space="preserve">diutius meditationibus hiſce hydrodynamicis immorari non +<pb o="293" file="0307" n="307" rhead="SECTIO DECIMA TERTIA."/> +vacat: </s> + <s xml:id="echoid-s8729" xml:space="preserve">Igitur ad finem propero. </s> + <s xml:id="echoid-s8730" xml:space="preserve">Novam hanc circa reactionem & </s> + <s xml:id="echoid-s8731" xml:space="preserve">impetum <lb/>fluidorum theoriam, quæ receptam ab omnibus adhuc auctoribus opinionem <lb/>evertit in re magni momenti, ſingulari Diſſertatione proſecutus ſum, quæ ſuo <lb/>tempore Commentar. </s> + <s xml:id="echoid-s8732" xml:space="preserve">Academ. </s> + <s xml:id="echoid-s8733" xml:space="preserve">Scient. </s> + <s xml:id="echoid-s8734" xml:space="preserve">Imp. </s> + <s xml:id="echoid-s8735" xml:space="preserve">Petropol. </s> + <s xml:id="echoid-s8736" xml:space="preserve">inſeretur eandemque indubita-<lb/>tis confirmavi experimentis. </s> + <s xml:id="echoid-s8737" xml:space="preserve">Venio nunc ad argumentum aliud, Geome-<lb/>trarum attentione minime indignum.</s> + <s xml:id="echoid-s8738" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8739" xml:space="preserve">§. </s> + <s xml:id="echoid-s8740" xml:space="preserve">20. </s> + <s xml:id="echoid-s8741" xml:space="preserve">Mentem aliquando ſubiit, poſſe ea quæ de vi repellente flui-<lb/>dorum, dum ejiciuntur, meditatus fueram, quæque hic maximam partem <lb/>expoſui, utiliter applicari ad novum inſtituendum navigationis modum: </s> + <s xml:id="echoid-s8742" xml:space="preserve">ne-<lb/>que enim video, quid obſtet, quo minus maximæ naves ſine velis remiſque eo <lb/>modo promoveri poſſint, ut aquæ continue in altum eleventur effluxuræ per <lb/>foramina in ima navis parte, faciendo ut directio aquarum effluentium verſus <lb/>puppim ſpectet. </s> + <s xml:id="echoid-s8743" xml:space="preserve">Ne quis vero opinionem hanc in ipſo limine rideat, ceu ni-<lb/>mis inſulſam, è re erit noſtra argumentum iſtud accuratius excutere & </s> + <s xml:id="echoid-s8744" xml:space="preserve">ad cal-<lb/>culum revocare: </s> + <s xml:id="echoid-s8745" xml:space="preserve">utile enim eſſe poteſt multisque diſquiſitionibus geometri-<lb/>cis eſt fertiliſſimum.</s> + <s xml:id="echoid-s8746" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8747" xml:space="preserve">Incipiam ab eo, ex quo deinde apparebit, ſub quibus circumſtantiis <lb/>maximus ſucceſſus à nova iſta navigatione expectari debeat.</s> + <s xml:id="echoid-s8748" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8749" xml:space="preserve">§. </s> + <s xml:id="echoid-s8750" xml:space="preserve">21. </s> + <s xml:id="echoid-s8751" xml:space="preserve">Notandum igitur eſt, navem ab hauſtis aquis continue retarda-<lb/>ri ob inertiam earundem, quando illis eadem velocitas communicatur qua-<lb/>cum navis fertur & </s> + <s xml:id="echoid-s8752" xml:space="preserve">dum communicatur, navis à reactione aquarum retrorſum <lb/>urgetur, ſimul ac ab earundem effluxu antrorſum premitur. </s> + <s xml:id="echoid-s8753" xml:space="preserve">Iſte actionum con-<lb/>trariarum concurſus limites ponit vi naves propellenti à data potentia abſoluta <lb/>obtinendæ: </s> + <s xml:id="echoid-s8754" xml:space="preserve">niſi enim actio prior adeſſet (de qua ut verum fatear diu non co-<lb/>gitavi) poſſet labore hominum quantumvis parvo vis naves propellens utcun{q́ue<unsure/>} <lb/>magna obtineri, quod ſic demonſtro.</s> + <s xml:id="echoid-s8755" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8756" xml:space="preserve">In ſectione nona (vide præſertim §. </s> + <s xml:id="echoid-s8757" xml:space="preserve">26.) </s> + <s xml:id="echoid-s8758" xml:space="preserve">oſtendi, laborem hominum <lb/>in elevandis aquis impenſum, quem voce potentiæ abſolutæ deſigno, æſtiman-<lb/>dum eſſe ex producto quantitatis aquarum in altitudinem elevationis, ita ut <lb/>verbi gratia labore ſecundum omnes menſuras eodem poſſint & </s> + <s xml:id="echoid-s8759" xml:space="preserve">quatuor pedes <lb/>cubici ad altitudinem ſedecim pedum & </s> + <s xml:id="echoid-s8760" xml:space="preserve">ſedecim pedes cubici ad alitudinem <lb/>quatuor pedum elevari: </s> + <s xml:id="echoid-s8761" xml:space="preserve">Dico nunc porro preſſionem uniformem, naves <lb/>antrorſum propellentem adeſſe, quamdiu fluida velocitate æquali effluunt, <lb/>quæ preſſio æſtimanda ſit ex quantitate aquarum effluentium & </s> + <s xml:id="echoid-s8762" xml:space="preserve">ex radice al- +<pb o="294" file="0308" n="308" rhead="HYDRODYNAMICÆ"/> +titudinis aquarum in vaſe ſupra foramen poſitarum: </s> + <s xml:id="echoid-s8763" xml:space="preserve">fuerit enim quantitas <lb/>aquarum dato tempore effluentium = Q; </s> + <s xml:id="echoid-s8764" xml:space="preserve">altitudo earum = A, erit ma-<lb/>gnitudo foraminis aquas eructantis proportionalis cenſenda quantitati {Q/√ A} <lb/>pro eodem tempore: </s> + <s xml:id="echoid-s8765" xml:space="preserve">at vero vis repellens, quæ hic navem promovet, æqualis <lb/>eſt magnitudini foraminis ductæ in duplam altitudinem aquarum (per §. </s> + <s xml:id="echoid-s8766" xml:space="preserve">4.) <lb/></s> + <s xml:id="echoid-s8767" xml:space="preserve">id eſt, æqualis quantitati {Q/√ A} X <emph style="bf">2</emph> A ſeu 2 Q √ A. </s> + <s xml:id="echoid-s8768" xml:space="preserve">Ex comparatione utrius-<lb/>que propoſitionis ſequitur laborem hominum in elevandis aquis exantlatum <lb/>eſſe ad vim naves propellentem inde obtinendam, ut Q A ad 2 Q √ A ſive ut <lb/>√ A ad quantitatem aliquam conſtantem: </s> + <s xml:id="echoid-s8769" xml:space="preserve">igitur quo minor eſt altitudo ad <lb/>quam aquæ elevantur, eò major vis naves promovens ab eodem labore obti-<lb/>netur, ita ut labore hominum quantumvis parvo vis naves propellens utcun-<lb/>que magna obtineri poßit. </s> + <s xml:id="echoid-s8770" xml:space="preserve">Verum etiam inertia aquarum, quæ hauriuntur, <lb/>(de qua ab initio hujus paragraphi diximus) naves retardans eo majorem <lb/>obtinet rationem ad vim naves propellentem, quo minor aſſumitur altitudo <lb/>A, ad quod animus hic probe eſt advertendus.</s> + <s xml:id="echoid-s8771" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8772" xml:space="preserve">§. </s> + <s xml:id="echoid-s8773" xml:space="preserve">22. </s> + <s xml:id="echoid-s8774" xml:space="preserve">Perſpicuum eſt ex præcedente paragrapho, altitudinem ad quam <lb/>aquæ ſunt elevandæ eſſe ex earum claſſe, quæ alicubi maximæ ſunt. </s> + <s xml:id="echoid-s8775" xml:space="preserve">Ut ve-<lb/>ro altitudo maxime ad propoſitum proficua determinetur, aliæ nobis ſe of-<lb/>ferunt quæſtiones prius examinandæ.</s> + <s xml:id="echoid-s8776" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div298" type="section" level="1" n="222"> +<head xml:id="echoid-head284" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s8777" xml:space="preserve">Ponatur navis uniformi progredi velocitate, quæ generatur lapſu li-<lb/>bero per altitudinem B, fingaturque aquas continue affluere in navem, ve-<lb/>luti ſub forma pluviarum, & </s> + <s xml:id="echoid-s8778" xml:space="preserve">quidem tanta quantitate, quantam remotis om-<lb/>nibus impedimentis alienis ſuppeditaret cylindrus conſtanter plenus ad alti-<lb/>tudinem A per orificium magnitudinis M. </s> + <s xml:id="echoid-s8779" xml:space="preserve">Quæritur quantam reſiſtentiam <lb/>navis ab iſto perpetuo & </s> + <s xml:id="echoid-s8780" xml:space="preserve">uniformi aquarum affluxu earundemque inertia pa-<lb/>tiatur.</s> + <s xml:id="echoid-s8781" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div299" type="section" level="1" n="223"> +<head xml:id="echoid-head285" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s8782" xml:space="preserve">Aſſumatur tempus quodcunque t, quod ſi æſtimetur ex ſpatio, quod <lb/>fluidum affluens ſua velocitate percurrit, diviſo per eandem velocitatem, tunc +<pb o="295" file="0309" n="309" rhead="SECTIO DECIMA TERTIA."/> +velocitas eſt exprimenda per √ 2 A & </s> + <s xml:id="echoid-s8783" xml:space="preserve">erit quantitas aquæ tempore t affluens <lb/>æqualis cylindro ſuper baſi M conſtructo longitudinis t √ 2 A: </s> + <s xml:id="echoid-s8784" xml:space="preserve">iſta vero <lb/>quantitas tempore t, dum à nave aufertur, accipit velocitatem debitam alti-<lb/>tudini B & </s> + <s xml:id="echoid-s8785" xml:space="preserve">exprimendam per √ 2 B: </s> + <s xml:id="echoid-s8786" xml:space="preserve">quærenda itaque eſt vis uniformis, quæ <lb/>poſſit tempore t, cylindro aqueo M t √ 2 A communicare velocitatem √2 B <lb/>& </s> + <s xml:id="echoid-s8787" xml:space="preserve">erit iſta vis ob reactionem, quæ in navem reagit, æqualis cenſenda reſi-<lb/>ſtentiæ quæſitæ. </s> + <s xml:id="echoid-s8788" xml:space="preserve">Sit præfata vis = p, puteturque dediſſe tempore θ veloci-<lb/>tatem v cylindro aqueo M t √ 2 A & </s> + <s xml:id="echoid-s8789" xml:space="preserve">erit d v = {pdθ/Mt √ 2A}, atque v = {pθ/Mt √2A}: <lb/></s> + <s xml:id="echoid-s8790" xml:space="preserve">ponatur jam √ 2 B pro v & </s> + <s xml:id="echoid-s8791" xml:space="preserve">t pro θ eritque √ 2B = {p/M √2A} ſivè p = 2M √A B.</s> + <s xml:id="echoid-s8792" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8793" xml:space="preserve">Eſt igitur reſiſtentia quæſita æqualis ponderi cylindri aquei, cujus ba-<lb/>ſis eſſet æqualis orificio M & </s> + <s xml:id="echoid-s8794" xml:space="preserve">cujus longitudo æqualis duplæ mediæ propor-<lb/>tionali inter altitudines A & </s> + <s xml:id="echoid-s8795" xml:space="preserve">B.</s> + <s xml:id="echoid-s8796" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div300" type="section" level="1" n="224"> +<head xml:id="echoid-head286" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s8797" xml:space="preserve">§. </s> + <s xml:id="echoid-s8798" xml:space="preserve">23. </s> + <s xml:id="echoid-s8799" xml:space="preserve">Sit in navi cylindrus altitudinis ſupra ſuperficiem maris A, per <lb/>cujus orificium in eadem ſuperficie poſitum amplitudinis M aquæ verſus pup-<lb/>pim effluant ſine ullo impedimento, conſerveturque cylindrus aqua conſtan-<lb/>ter plenus, determinare potentiam navem continue propellentem.</s> + <s xml:id="echoid-s8800" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div301" type="section" level="1" n="225"> +<head xml:id="echoid-head287" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s8801" xml:space="preserve">Potentia navem propellens eſt æqualis reactioni aquarum dum effluunt, <lb/>ſeu vi repellenti diminutæ potentia in præcedente paragrapho definita ab in-<lb/>ertia aquarum, quæ continue hauriuntur, oriunda. </s> + <s xml:id="echoid-s8802" xml:space="preserve">Vis repellens eſt æqua-<lb/>lis, per paragraphum hujus ſectionis quartum, 2 M A & </s> + <s xml:id="echoid-s8803" xml:space="preserve">hæc navem pro-<lb/>movet: </s> + <s xml:id="echoid-s8804" xml:space="preserve">vis altera quæ navem retardat eſt per præcedentem paragraphum <lb/>= 2 M √ A B. </s> + <s xml:id="echoid-s8805" xml:space="preserve">Eſt igitur potentia abſoluta navem promovens = 2 M -<lb/>2 M √A B.</s> + <s xml:id="echoid-s8806" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div302" type="section" level="1" n="226"> +<head xml:id="echoid-head288" xml:space="preserve">Corollarium.</head> +<p> + <s xml:id="echoid-s8807" xml:space="preserve">§. </s> + <s xml:id="echoid-s8808" xml:space="preserve">24. </s> + <s xml:id="echoid-s8809" xml:space="preserve">Si navis nullam habeat velocitatem, erit vis navem urgens = <lb/>2 M A; </s> + <s xml:id="echoid-s8810" xml:space="preserve">atque ſi navis eadem velocitate movetur qua aquæ in plagam contra-<lb/>riam effluunt, fit B = A & </s> + <s xml:id="echoid-s8811" xml:space="preserve">tunc navis nulla vi propellitur. </s> + <s xml:id="echoid-s8812" xml:space="preserve">Si proinde na- +<pb o="296" file="0310" n="310" rhead="HYDRODYNAMICÆ"/> +vis vel liberrime moveretur ſuper mari, non acquireret tamen ab actione <lb/>aquarum, quæ continue hauriuntur inferiusque effluunt, majorem velocita-<lb/>tem quam eam, qua aquæ effluunt, non quod aquæ ex vaſe uniformiter mo-<lb/>to effluentes vas minori vi quam ex vaſe immoto repellant, ſed quod tunc in-<lb/>ertia aquarum reſiſtentiam producat vi repellenti æqualem.</s> + <s xml:id="echoid-s8813" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div303" type="section" level="1" n="227"> +<head xml:id="echoid-head289" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s8814" xml:space="preserve">§. </s> + <s xml:id="echoid-s8815" xml:space="preserve">25. </s> + <s xml:id="echoid-s8816" xml:space="preserve">Data potentia operariorum, qui aquas elevant, & </s> + <s xml:id="echoid-s8817" xml:space="preserve">data altitudi-<lb/>ne ad quam aquæ elevantur, invenire amplitudinem foraminis effluxus & </s> + <s xml:id="echoid-s8818" xml:space="preserve">vim <lb/>repellentem.</s> + <s xml:id="echoid-s8819" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div304" type="section" level="1" n="228"> +<head xml:id="echoid-head290" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s8820" xml:space="preserve">Sit potentia talis, qua ſingulis minutis ſecundis numerus pedum cu-<lb/>bicorum aquæ N poſſit ad altitudinem unius pedis elevari, quam potentiam <lb/>vi experimenti ſecundi ſectioni nonæſubjuncti exerere poteſt operariorum nu-<lb/>merus deſignandus per {5/4} N. </s> + <s xml:id="echoid-s8821" xml:space="preserve">Sit altitudo ad quam aquæ continue elevantur <lb/>= A in pedilus expreſſa: </s> + <s xml:id="echoid-s8822" xml:space="preserve">amplitudo orificii in pedibus quadratis = M; </s> + <s xml:id="echoid-s8823" xml:space="preserve">erit <lb/>numerus pedum cubicorum aquæ, quem operarii data potentia ad altitudi-<lb/>nem A ſingulis minutis ſecundis elevare poſſunt, = {N/A} (per §. </s> + <s xml:id="echoid-s8824" xml:space="preserve">22. </s> + <s xml:id="echoid-s8825" xml:space="preserve">ſect. </s> + <s xml:id="echoid-s8826" xml:space="preserve">9.) <lb/></s> + <s xml:id="echoid-s8827" xml:space="preserve">erit igitur orificium ejus amplitudinis conſtruendum, ut ſingulis minutis <lb/>ſecundis numerus iſte pedum cubicorum aquæ per id effluere poſſit, ſi liber-<lb/>rime effluant. </s> + <s xml:id="echoid-s8828" xml:space="preserve">Sumamus autem loco minutorum ſecundorum tempus, quod <lb/>corpus inſumit, dum libere cadit per altitudinem A: </s> + <s xml:id="echoid-s8829" xml:space="preserve">tempus id eſt hic expri-<lb/>mendum {1/4} √ A, (poſito concinnioris calculi gratia corpus à quiete libere <lb/>cadens intra minutum ſec. </s> + <s xml:id="echoid-s8830" xml:space="preserve">abſolvere 16. </s> + <s xml:id="echoid-s8831" xml:space="preserve">ped.) </s> + <s xml:id="echoid-s8832" xml:space="preserve">& </s> + <s xml:id="echoid-s8833" xml:space="preserve">hoc tempore debet effluere <lb/>numerus pedum cubicorum aquæ deſignandus per {N/A} X {1/4} √ A ſeu {N/4 √ A}: </s> + <s xml:id="echoid-s8834" xml:space="preserve"><lb/>effluit autem revera 2 M A, nempe cylindrus aqueus cujus baſis eſt M & </s> + <s xml:id="echoid-s8835" xml:space="preserve">cu-<lb/>jus longitudo facit duplicem altitudinem A: </s> + <s xml:id="echoid-s8836" xml:space="preserve">eſt igitur {N/4 √ A} = 2MA; </s> + <s xml:id="echoid-s8837" xml:space="preserve">unde <lb/>amplitudo orificii ſeu</s> +</p> +<p> + <s xml:id="echoid-s8838" xml:space="preserve">M = {N/8A √ A}.</s> + <s xml:id="echoid-s8839" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8840" xml:space="preserve">Vis autem repellens fit æqualis 2 M A ſeu = {N/4 √ A}.</s> + <s xml:id="echoid-s8841" xml:space="preserve"/> +</p> +<pb o="297" file="0311" n="311" rhead="SECTIO DECIMA TERTIA."/> +</div> +<div xml:id="echoid-div305" type="section" level="1" n="229"> +<head xml:id="echoid-head291" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s8842" xml:space="preserve">§. </s> + <s xml:id="echoid-s8843" xml:space="preserve">26. </s> + <s xml:id="echoid-s8844" xml:space="preserve">In quavis nave aquæ ad aliam atque aliam altitudinem ſunt ele-<lb/>vandæ, ut eadem potentia, quæ in hauriendis aquis inſumitur, vis navem <lb/>promovens maxima obtineatur & </s> + <s xml:id="echoid-s8845" xml:space="preserve">duo requiruntur ad altitudinem illam uti-<lb/>liſſimam definiendam pro certo operariorum numero. </s> + <s xml:id="echoid-s8846" xml:space="preserve">Primo ut cognitum <lb/>ſit quamnam velocitatem propoſita navis à data potentia acquirat: </s> + <s xml:id="echoid-s8847" xml:space="preserve">ratione <lb/>hujus poſtulati, ponemus navem à preſſione, quæ ſit æqualis ponderi unius <lb/>pedis cubici aquæ ſeu circiter 72 librarum acquirere velocitatem, quæ gene-<lb/>retur lapſu libero per altitudinem C, & </s> + <s xml:id="echoid-s8848" xml:space="preserve">quia deinceps ſemper in pedibus <lb/>menſuras omnes exprimemus, erit pondus unius pedis cubici aquæ expri-<lb/>mendum per unitatem. </s> + <s xml:id="echoid-s8849" xml:space="preserve">Secundo pro cognita aſſumenda eſt relatio inter ce-<lb/>leritates navis & </s> + <s xml:id="echoid-s8850" xml:space="preserve">potentias navem propellents: </s> + <s xml:id="echoid-s8851" xml:space="preserve">ſtatuitur hic vulgo velocita-<lb/>tes habere rationem ſubduplicatam virium propellentium, experimenta qui-<lb/>dem hanc hypotheſin non exactè confirmant in motibus lentis; </s> + <s xml:id="echoid-s8852" xml:space="preserve">interim ta-<lb/>men eam reliquis omnibus præferendam cenſemus. </s> + <s xml:id="echoid-s8853" xml:space="preserve">Si quis velit rem ſub alia <lb/>hypotheſi explorare, is poterit eodem modo, quo nunc utemur, calculum <lb/>inſtituere.</s> + <s xml:id="echoid-s8854" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div306" type="section" level="1" n="230"> +<head xml:id="echoid-head292" xml:space="preserve">Problema.</head> +<p> + <s xml:id="echoid-s8855" xml:space="preserve">§. </s> + <s xml:id="echoid-s8856" xml:space="preserve">27. </s> + <s xml:id="echoid-s8857" xml:space="preserve">Invenire altitudinem, ad quam aquæ continue elevandæ ſunt, <lb/>inſtituto utiliſſimam, nempe talem, ut eadem potentia in elevandis aquis <lb/>adhibenda vis navem promovens maxima oriatur.</s> + <s xml:id="echoid-s8858" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div307" type="section" level="1" n="231"> +<head xml:id="echoid-head293" xml:space="preserve">Solutio.</head> +<p> + <s xml:id="echoid-s8859" xml:space="preserve">Serventur denominationes omnes in hoc argumento adhibitæ: </s> + <s xml:id="echoid-s8860" xml:space="preserve">erit an-<lb/>te omnia inquirenda velocitas navis ſeu altitudo huic velocitati debita quam <lb/>vocavimus B. </s> + <s xml:id="echoid-s8861" xml:space="preserve">Quia vero velocitates navis ponuntur proportionales radici-<lb/>bus potentiarum navem urgentium, erunt altitudines velocitatum ipſis po-<lb/>tentiis proportionales. </s> + <s xml:id="echoid-s8862" xml:space="preserve">Erit igitur talis analogia inſtituenda.</s> + <s xml:id="echoid-s8863" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8864" xml:space="preserve">Sicuti pondus unius pedis cubici ad altitudinem C (conf. </s> + <s xml:id="echoid-s8865" xml:space="preserve">§. </s> + <s xml:id="echoid-s8866" xml:space="preserve">26.) </s> + <s xml:id="echoid-s8867" xml:space="preserve">ita <lb/>preſſio navem urgens ſeu 2MA - 2M√AB (vid. </s> + <s xml:id="echoid-s8868" xml:space="preserve">§. </s> + <s xml:id="echoid-s8869" xml:space="preserve">23.) </s> + <s xml:id="echoid-s8870" xml:space="preserve">ad altitudinem ve-<lb/>locitati navis reſpondentem, quæ proinde erit 2MC X (A - √AB): </s> + <s xml:id="echoid-s8871" xml:space="preserve">Hanc <lb/>vero altitudinem vocavimus B: </s> + <s xml:id="echoid-s8872" xml:space="preserve">Eſt itaque</s> +</p> +<pb o="298" file="0312" n="312" rhead="HYDRODYNAMICÆ"/> +<p> + <s xml:id="echoid-s8873" xml:space="preserve">B = 2 MC X (A - √AB).</s> + <s xml:id="echoid-s8874" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8875" xml:space="preserve">Exinde fit preſſio navem urgens = {B/C}, atque adeo’ proportionalis <lb/>altitudini B, quia C eſt quantitas conſtans: </s> + <s xml:id="echoid-s8876" xml:space="preserve">ergo & </s> + <s xml:id="echoid-s8877" xml:space="preserve">preſſio navem promo-<lb/>vens & </s> + <s xml:id="echoid-s8878" xml:space="preserve">altitudo navis velocitati reſpondens ſimul fiunt maximæ: </s> + <s xml:id="echoid-s8879" xml:space="preserve">Igitur ſi pro <lb/>præſenti inſtituto differentietur quantitas 2MA - 2M√AB, quæ preſſionem <lb/>navem propellentem exprimit, poterit poni d B = o. </s> + <s xml:id="echoid-s8880" xml:space="preserve">Prius vero quam dif-<lb/>ferentiatio inſtituatur oportet pro M ſubſtituere valorem ejus §. </s> + <s xml:id="echoid-s8881" xml:space="preserve">25. </s> + <s xml:id="echoid-s8882" xml:space="preserve">& </s> + <s xml:id="echoid-s8883" xml:space="preserve">tunc <lb/>fit preſſio navem promovens = {N/4√A} - {N√B/4A}, in qua littera N eſt con-<lb/>ſtans, litteræ vero B & </s> + <s xml:id="echoid-s8884" xml:space="preserve">A variabiles. </s> + <s xml:id="echoid-s8885" xml:space="preserve">Sumatur nunc ejus differentiale, facien-<lb/>do d B = o, idque fiat = o; </s> + <s xml:id="echoid-s8886" xml:space="preserve">atque ſic reperietur A = 4B.</s> + <s xml:id="echoid-s8887" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8888" xml:space="preserve">Eſt igitur vis navem promovens maxima cum altitudo, ad quam aquæ <lb/>elevantur, eſt quadrupla altitudinis velocitati navis debitæ.</s> + <s xml:id="echoid-s8889" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8890" xml:space="preserve">Ponatur in æquatione B = 2 M C X (A - √AB) ſuperius inventa <lb/>A = 4B & </s> + <s xml:id="echoid-s8891" xml:space="preserve">reperietur <lb/>M = {1/4C}, <lb/>& </s> + <s xml:id="echoid-s8892" xml:space="preserve">quia (per §. </s> + <s xml:id="echoid-s8893" xml:space="preserve">25.) </s> + <s xml:id="echoid-s8894" xml:space="preserve">eſt M = {N/8A√A}, fit tunc <lb/>A = ({1/2} NC)<emph style="super">{2/3}</emph>, atque <lb/>B = {1/4}({1/2} NC)<emph style="super">{2/3}</emph>.</s> + <s xml:id="echoid-s8895" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div308" type="section" level="1" n="232"> +<head xml:id="echoid-head294" xml:space="preserve">Corollarium.</head> +<p> + <s xml:id="echoid-s8896" xml:space="preserve">§. </s> + <s xml:id="echoid-s8897" xml:space="preserve">28. </s> + <s xml:id="echoid-s8898" xml:space="preserve">Si ad præceptum præcedentis paragraphi orificio, per quod <lb/>aquæ inferius ex canali verſus puppim effluunt, concilietur amplitudo {1/4C}, <lb/>id eſt, talis, quæ ſe habeat ad amplitudinem unius pedis quadrati, ſicuti men-<lb/>ſura unius pedis, ad altitudinem quadruplam velocitati navis, vi 72. </s> + <s xml:id="echoid-s8899" xml:space="preserve">libra-<lb/>rum animatæ, debitam, fiet tunc ut navis dimidia velocitate feratur ejus qua <lb/>aquæ effluunt & </s> + <s xml:id="echoid-s8900" xml:space="preserve">erit vis repellens aquarum effluentium <lb/>2MA = {1/2C} X ({1/2} NC)<emph style="super">{2/3}</emph>, +<pb o="299" file="0313" n="313" rhead="SECTIO DECIMA TERTIA."/> +vis vero navem promovens hujus erit dimidia, adeo ut dimidius effectus <lb/>perdatur ab inertia earundem, quæ continue hauriuntur, aquarum.</s> + <s xml:id="echoid-s8901" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div309" type="section" level="1" n="233"> +<head xml:id="echoid-head295" xml:space="preserve">Scholium.</head> +<p> + <s xml:id="echoid-s8902" xml:space="preserve">§. </s> + <s xml:id="echoid-s8903" xml:space="preserve">29. </s> + <s xml:id="echoid-s8904" xml:space="preserve">Poſtquam ſic demonſtravimus, quomodo utiliſſime maximo-<lb/>que cum ſucceſſu iſte navigandi modus ſit inſtituendus, nunc porro rem iſtam <lb/>exemplo illuſtrandam eſſe puto tali, quod cum ipſa rei natura non male con-<lb/>venire crediderim ut ſimul appareat, qualis præterpropter eventus futurus <lb/>ſit.</s> + <s xml:id="echoid-s8905" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8906" xml:space="preserve">Conſideremus triremem, vulgo galeram, cum 260 remigibus: </s> + <s xml:id="echoid-s8907" xml:space="preserve">pona-<lb/>mus hanc galeram pondere unius pedis cubici aquæ ſeu 72. </s> + <s xml:id="echoid-s8908" xml:space="preserve">librarum tractam <lb/>perficere ſingulis minutis ſecundis ſpatium duorum pedum, cujus velocita-<lb/>tis altitudo genitrix indicata per C eſt = {1/16}, poſito corpus grave libere à quie-<lb/>te decidens primo minuto ſecundo perficere 16. </s> + <s xml:id="echoid-s8909" xml:space="preserve">ped. </s> + <s xml:id="echoid-s8910" xml:space="preserve">Quia porro 260. </s> + <s xml:id="echoid-s8911" xml:space="preserve">ope-<lb/>rarii adhibentur, quorum quivis vi experimenti ſecundi ad Sect. </s> + <s xml:id="echoid-s8912" xml:space="preserve">9. </s> + <s xml:id="echoid-s8913" xml:space="preserve">pertinentis <lb/>poteſt ſingulis minutis ſecundis quatuor quintas partes pedis cubici ad altitu-<lb/>dinem unius pedis elevare, erit N = {4/5} X 260 = 208. </s> + <s xml:id="echoid-s8914" xml:space="preserve">Fiat igitur orificium, <lb/>per quod aquæ effluant, amplitudinis 4 pedum quadratorum: </s> + <s xml:id="echoid-s8915" xml:space="preserve">poteruntque <lb/>operarii aquam in canali ſupra orificium elevatam conſervare ad altitudinem <lb/>proxime 3 {1/2} ped. </s> + <s xml:id="echoid-s8916" xml:space="preserve">quæ indicatur litera A, & </s> + <s xml:id="echoid-s8917" xml:space="preserve">ſi ſumas hujus altitudinis quar-<lb/>tam partem habebis B = {7/8} ped. </s> + <s xml:id="echoid-s8918" xml:space="preserve">adeo ut navis tali velocitate ſit iſta navigatione <lb/>progreſſura, quam grave acquirit lapſu libero per altitudinem {7/8} ped. </s> + <s xml:id="echoid-s8919" xml:space="preserve">ſic ergo <lb/>navis ſingulis minutis ſecundis ſpatium 7 {1/2} ped. </s> + <s xml:id="echoid-s8920" xml:space="preserve">perficiet & </s> + <s xml:id="echoid-s8921" xml:space="preserve">ſingulis horis 27000 <lb/>ped. </s> + <s xml:id="echoid-s8922" xml:space="preserve">id eſt, plus duobus milliaribus gallicis: </s> + <s xml:id="echoid-s8923" xml:space="preserve">tanta navis velocitas remigatio-<lb/>ne vix ac ne vix quidem obtineri poteſt.</s> + <s xml:id="echoid-s8924" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8925" xml:space="preserve">Jam vero calculum alia hypotheſi ſuperſtruam, quam rei nauticæ intel-<lb/>ligentes non admodum improbaturos eſſe, confido: </s> + <s xml:id="echoid-s8926" xml:space="preserve">quadrat enim cum mul-<lb/>tis, quos ipſe ſuper mari feci, obſervationibus: </s> + <s xml:id="echoid-s8927" xml:space="preserve">ſupponam vela triremis perpen-<lb/>diculariter ad carinam expanſa ſuperficiem habere 1600 pedum quadratorum, <lb/>hæcque ventum excipere directe impingentem, qui ſingulis minutis ſecundis <lb/>ſpatium percurrat 18. </s> + <s xml:id="echoid-s8928" xml:space="preserve">ped. </s> + <s xml:id="echoid-s8929" xml:space="preserve">navem vero in eadem directione ſic ſingulis minu-<lb/>tis ſecundis ſpatium perficere 6 pedum. </s> + <s xml:id="echoid-s8930" xml:space="preserve">Ita ventus in vela incurret velocita- +<pb o="300" file="0314" n="314" rhead="HYDRODYNAMICÆ"/> +te reſpectiva 12 pedum: </s> + <s xml:id="echoid-s8931" xml:space="preserve">vim iſtius venti æſtimo = ponderi {9 x 1600/850} ped. </s> + <s xml:id="echoid-s8932" xml:space="preserve">cub. <lb/></s> + <s xml:id="echoid-s8933" xml:space="preserve">aquæ, ſeu fere 17. </s> + <s xml:id="echoid-s8934" xml:space="preserve">ped. </s> + <s xml:id="echoid-s8935" xml:space="preserve">cub. </s> + <s xml:id="echoid-s8936" xml:space="preserve">aquæ.</s> + <s xml:id="echoid-s8937" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8938" xml:space="preserve">Hæc ſi ita ſint, ſequitur navem ab elevatione aquarum 260. </s> + <s xml:id="echoid-s8939" xml:space="preserve">operario-<lb/>rum poſſe ea velocitate propelli, qua ſingulis minutis ſecundis ſpatium per-<lb/>currat 6 {1/2}. </s> + <s xml:id="echoid-s8940" xml:space="preserve">pedum.</s> + <s xml:id="echoid-s8941" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s8942" xml:space="preserve">Æſtimatio non admodum diverſa ſequitur ex iis, quæ D. </s> + <s xml:id="echoid-s8943" xml:space="preserve">Chazelles <lb/>habet in Comm. </s> + <s xml:id="echoid-s8944" xml:space="preserve">Acad. </s> + <s xml:id="echoid-s8945" xml:space="preserve">Reg. </s> + <s xml:id="echoid-s8946" xml:space="preserve">Sc. </s> + <s xml:id="echoid-s8947" xml:space="preserve">Pariſ. </s> + <s xml:id="echoid-s8948" xml:space="preserve">ad ann. </s> + <s xml:id="echoid-s8949" xml:space="preserve">1702. </s> + <s xml:id="echoid-s8950" xml:space="preserve">p. </s> + <s xml:id="echoid-s8951" xml:space="preserve">98. </s> + <s xml:id="echoid-s8952" xml:space="preserve">edit Pariſ. </s> + <s xml:id="echoid-s8953" xml:space="preserve">Ut vero recte <lb/>ad inſtitutum noſtrum applicari poſſint, notandum erit in remigatione, vim <lb/>triremem propellentem non eſſe æſtimandam ex preſſione remigum in remos, <lb/>ſed ex preſſione, quam remorum extremitates aquis ſubmerſæ contra aquas <lb/>exerunt. </s> + <s xml:id="echoid-s8954" xml:space="preserve">Ut hanc proxime definiamus, hæc prius erunt obſervanda. </s> + <s xml:id="echoid-s8955" xml:space="preserve">Remiges <lb/>fuere adhibiti 260. </s> + <s xml:id="echoid-s8956" xml:space="preserve">totis viribus remigantes: </s> + <s xml:id="echoid-s8957" xml:space="preserve">ſingulis minutis primis remorum <lb/>impulſus (gallicè palades) facti ſunt 24: </s> + <s xml:id="echoid-s8958" xml:space="preserve">integra remorum agitatio tribus abſol-<lb/>vitur motibus, quos ejuſdem durationis ponam, eorumque unus ſolus <lb/>triremem promovet: </s> + <s xml:id="echoid-s8959" xml:space="preserve">hoc modo triremis velocitate fuit provecta, qua <lb/>ſingulis minutis ſecundis ſpatium 7 {1/5}. </s> + <s xml:id="echoid-s8960" xml:space="preserve">ped. </s> + <s xml:id="echoid-s8961" xml:space="preserve">abſolvebat, pars remi intra navem <lb/>fuit 6. </s> + <s xml:id="echoid-s8962" xml:space="preserve">pedum & </s> + <s xml:id="echoid-s8963" xml:space="preserve">extra navem 12. </s> + <s xml:id="echoid-s8964" xml:space="preserve">pedum: </s> + <s xml:id="echoid-s8965" xml:space="preserve">ſuperficies autem (gallicè les pales) <lb/>omnium remorum, quæ contra aquas impelluntur, in unam collectæ D. <lb/></s> + <s xml:id="echoid-s8966" xml:space="preserve">Chazelles facit 130. </s> + <s xml:id="echoid-s8967" xml:space="preserve">pedum quadratorum: </s> + <s xml:id="echoid-s8968" xml:space="preserve">notavit porro extremitatem inter-<lb/>nam remi ſingulis agitationibus ſpatium deſcribere ſex pedum: </s> + <s xml:id="echoid-s8969" xml:space="preserve">& </s> + <s xml:id="echoid-s8970" xml:space="preserve">quia quævis <lb/>agitatio tempore {60/24}. </s> + <s xml:id="echoid-s8971" xml:space="preserve">unius minuti ſecundi abſolvitur ſimulque ex tribus con-<lb/>ſtat motibus, quos pono tautochronos, apparet quamvis remi retractionem <lb/>fieri tempore {20/24}. </s> + <s xml:id="echoid-s8972" xml:space="preserve">ſeu {5/6}. </s> + <s xml:id="echoid-s8973" xml:space="preserve">unius minuti ſecundi & </s> + <s xml:id="echoid-s8974" xml:space="preserve">hoc tempore extremitas remi <lb/>interna abſolvit ſpatium 6. </s> + <s xml:id="echoid-s8975" xml:space="preserve">pedum. </s> + <s xml:id="echoid-s8976" xml:space="preserve">Porro ob longitudinem ſuperficiei remo-<lb/>rum, quæ contra aquas impellitur, non tota eſt ad diſtantiam 12. </s> + <s xml:id="echoid-s8977" xml:space="preserve">pedum <lb/>cenſenda: </s> + <s xml:id="echoid-s8978" xml:space="preserve">illam igitur diſtare ponam 10. </s> + <s xml:id="echoid-s8979" xml:space="preserve">pedibus, quaſi nempe pars remi ex-<lb/>tra navem promineret 10. </s> + <s xml:id="echoid-s8980" xml:space="preserve">pedes longa: </s> + <s xml:id="echoid-s8981" xml:space="preserve">hujus partis extremitas deſcribet 10. </s> + <s xml:id="echoid-s8982" xml:space="preserve"><lb/>pedes tempore {5/6} unius minuti ſecundi: </s> + <s xml:id="echoid-s8983" xml:space="preserve">quia vero ipſ<unsure/>a triremis velocitatem <lb/>habet, qua eodem tempore ſex pedes abſolvit, cenſendum eſt, remorum ex-<lb/>tremitates contra aquam impelli velocitate reſpectiva, qua tempore {5/6}. </s> + <s xml:id="echoid-s8984" xml:space="preserve">min. </s> + <s xml:id="echoid-s8985" xml:space="preserve"><lb/>ſec. </s> + <s xml:id="echoid-s8986" xml:space="preserve">4. </s> + <s xml:id="echoid-s8987" xml:space="preserve">pedes deſcribat: </s> + <s xml:id="echoid-s8988" xml:space="preserve">igitur vis triremem propellens eſt æqualis vi, quam <lb/>aqua contra ſuperficiem 130. </s> + <s xml:id="echoid-s8989" xml:space="preserve">pedum quadratorum exereret, ſi velocitate in illam <lb/>incurreret, qua tempore {5/6} min. </s> + <s xml:id="echoid-s8990" xml:space="preserve">ſec. </s> + <s xml:id="echoid-s8991" xml:space="preserve">4. </s> + <s xml:id="echoid-s8992" xml:space="preserve">ped. </s> + <s xml:id="echoid-s8993" xml:space="preserve">abſolvat: </s> + <s xml:id="echoid-s8994" xml:space="preserve">hanc vim ſecundum vul- +<pb o="301" file="0315" n="315" rhead="SECTIO DECIMA TERTIA."/> +garem æſtimationem invenio præter propter æqualem ponderi 40. </s> + <s xml:id="echoid-s8995" xml:space="preserve">ped. </s> + <s xml:id="echoid-s8996" xml:space="preserve">cub. <lb/></s> + <s xml:id="echoid-s8997" xml:space="preserve">aquæ; </s> + <s xml:id="echoid-s8998" xml:space="preserve">iſta vero vis non continue applicatur, ſed tantum eo tempore quo re-<lb/>mi retrahuntur: </s> + <s xml:id="echoid-s8999" xml:space="preserve">ſunt igitur duo trientes iſtius vis auferendi, ita ut vis quæ tri-<lb/>remem continue propellat, cenſenda denique ſit æqualis ponderi 13 {1/3}. </s> + <s xml:id="echoid-s9000" xml:space="preserve">ped. </s> + <s xml:id="echoid-s9001" xml:space="preserve">cub. </s> + <s xml:id="echoid-s9002" xml:space="preserve"><lb/>aquæ.</s> + <s xml:id="echoid-s9003" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s9004" xml:space="preserve">Exinde ſequitur, ſi velocitates navis rationem ſequi ſubduplicatam vi-<lb/>rium propellentium ponantur, quod eadem hæc triremis pondere unius pedis <lb/>cubici aquæ impulſa velocitatem habitura fuiſſet, qua poſſit ſingulis minutis <lb/>ſecundis perficere proxime duos pedes; </s> + <s xml:id="echoid-s9005" xml:space="preserve">quæ hypotheſis eadem eſt, cum illa <lb/>quam primo loco adhibuimus, ita ut rurſus exinde ſequatur triremem velo-<lb/>citatem ab iſta navigatione acquiſituram eſſe, qua poſſit perficere ſingulis mi-<lb/>nutis ſecundis 7 {1/2} pedes, quæ velocitas tantillo major eſt illa, quæ triremi re-<lb/>migatione fortiſſima 260. </s> + <s xml:id="echoid-s9006" xml:space="preserve">remigum data fuit.</s> + <s xml:id="echoid-s9007" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s9008" xml:space="preserve">Rebus bene perpenſis hæſito, utrum navigationis genus ſit præferen-<lb/>dum, an remigatio, an aquarum elevatio, ſucceſſum fere æqualem credide-<lb/>rim utriuſque, & </s> + <s xml:id="echoid-s9009" xml:space="preserve">pro certo affirmare audeo, ſi minus promoveatur navis ab <lb/>aquarum elevatione, defectum parvum fore: </s> + <s xml:id="echoid-s9010" xml:space="preserve">fortaſſe autem promovebitur <lb/>magis. </s> + <s xml:id="echoid-s9011" xml:space="preserve">Interim non dubito, quin nova iſta navigationis idea harum rerum <lb/>ignaris vana & </s> + <s xml:id="echoid-s9012" xml:space="preserve">ridicula appareat. </s> + <s xml:id="echoid-s9013" xml:space="preserve">Ego vero aliter ſentio velimque ut animus <lb/>porro ad ſequentia advertatur.</s> + <s xml:id="echoid-s9014" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s9015" xml:space="preserve">Primo. </s> + <s xml:id="echoid-s9016" xml:space="preserve">Quod aquæ in omni navium genere, ubi remi plane adhiberi <lb/>nequeunt, commode elevari poſſunt, ita ut nova iſta navigatione naves etiam <lb/>bellicæ prægraves, quibus in pugnis navalibus utuntur, deficiente omni vento, <lb/>quo lubet agi poſſint.</s> + <s xml:id="echoid-s9017" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s9018" xml:space="preserve">Secundo. </s> + <s xml:id="echoid-s9019" xml:space="preserve">Quod ſic in theoria exemplum habetur, dari vires motrices <lb/>ſive propellentes, quæ dici poſſunt intrinſecæ: </s> + <s xml:id="echoid-s9020" xml:space="preserve">Excitabuntur iſto exemplo <lb/>ingenia ad excogitanda hujuſmodi alia motus principia eaque magis perficien-<lb/>da & </s> + <s xml:id="echoid-s9021" xml:space="preserve">ad navigationis uſum adhibenda.</s> + <s xml:id="echoid-s9022" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s9023" xml:space="preserve">Tertio. </s> + <s xml:id="echoid-s9024" xml:space="preserve">Quod multis modis ſublevari poteſt labor hominum in elevan-<lb/>dis aquis ſecus atque fieri poteſt in remorum uſu: </s> + <s xml:id="echoid-s9025" xml:space="preserve">ſunt nempe res naturales <lb/>inſigni & </s> + <s xml:id="echoid-s9026" xml:space="preserve">fere incredibili virtute præditæ eæque mediocri pretio comparandæ, <lb/>quibus idem quod labore hominum effici poteſt: </s> + <s xml:id="echoid-s9027" xml:space="preserve">harum uſus præſertim bre- +<pb o="302" file="0316" n="316" rhead="HYDRODYNAMICÆ"/> +vibus trajectibus ſerena & </s> + <s xml:id="echoid-s9028" xml:space="preserve">tranquilla tempeſtate inſtituendis inſervire poſſet. <lb/></s> + <s xml:id="echoid-s9029" xml:space="preserve">De virtute iſtiuſmodi rebus naturalibus inſita, de effectibus inde obtinendis <lb/>horumque menſuris egi in Sect. </s> + <s xml:id="echoid-s9030" xml:space="preserve">X. </s> + <s xml:id="echoid-s9031" xml:space="preserve">§. </s> + <s xml:id="echoid-s9032" xml:space="preserve">40. </s> + <s xml:id="echoid-s9033" xml:space="preserve">& </s> + <s xml:id="echoid-s9034" xml:space="preserve">ſequentibus: </s> + <s xml:id="echoid-s9035" xml:space="preserve">imprimis autem ve-<lb/>lim ut attendatur ad §. </s> + <s xml:id="echoid-s9036" xml:space="preserve">43. </s> + <s xml:id="echoid-s9037" xml:space="preserve">quo omnes quibus ingenium à natura datum fuit <lb/>felix ad machinas excogitandas, excitari deberent ad rei iſtius perfectionem <lb/>tentandam.</s> + <s xml:id="echoid-s9038" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s9039" xml:space="preserve">Quarto. </s> + <s xml:id="echoid-s9040" xml:space="preserve">Quod nonnulla alia compendia purè mechanica adhiberi poſ-<lb/>ſint ſimilia illi quod §. </s> + <s xml:id="echoid-s9041" xml:space="preserve">27. </s> + <s xml:id="echoid-s9042" xml:space="preserve">datum fuit, quorum nempe ope ab eodem labo-<lb/>re effectus in promovendis navibus non parum creſcit: </s> + <s xml:id="echoid-s9043" xml:space="preserve">Verum non licet jam <lb/>ſecundum veram rei indolem omnia pertractare.</s> + <s xml:id="echoid-s9044" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div310" type="section" level="1" n="234"> +<head xml:id="echoid-head296" xml:space="preserve">EXPERIMENTA</head> +<head xml:id="echoid-head297" style="it" xml:space="preserve">In Sectionem decimam tertiam.</head> +<p> + <s xml:id="echoid-s9045" xml:space="preserve">UT vim repelle@tem experimento recte cognoſcere liceat, adhiberi pote-<lb/>rit vas quod habeat formam parallelopipedi ejuſque pondus ſumi tam <lb/>vacui quam aqua pleni, poſteaque indagari ratio inter amplitudinem <lb/>vaſis & </s> + <s xml:id="echoid-s9046" xml:space="preserve">amplitudinem foraminis, quod in latere vaſis eſſe debet, ſicut & </s> + <s xml:id="echoid-s9047" xml:space="preserve">ratio in-<lb/>ter altitudines aquæ ſupra foramen & </s> + <s xml:id="echoid-s9048" xml:space="preserve">ſupra baſin: </s> + <s xml:id="echoid-s9049" xml:space="preserve">Inde deducere licebit ratio-<lb/>nem inter pondus vaſis aqua pleni & </s> + <s xml:id="echoid-s9050" xml:space="preserve">cylindri aquei foramini verticaliter ſuper-<lb/>incumbentis. </s> + <s xml:id="echoid-s9051" xml:space="preserve">Porro ex obſervata amplitudine jactus habebitur velocitas aquæ: <lb/></s> + <s xml:id="echoid-s9052" xml:space="preserve">ex hac, ſi ſimul jungas quantitatem aquæ dato tempore effluentem pariter <lb/>obſervandam, colliges amplitudinem venæ contractæ, quam comparare pote-<lb/>ris cum amplitudine orificii.</s> + <s xml:id="echoid-s9053" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s9054" xml:space="preserve">His omnibus exploratis ſuſpendatur vas ex filo prælongo adhibita ſi-<lb/>mul cura, ut alium motum habere non poſſit, quam qui ſit directioni aqua-<lb/>rum effluentium contrarius. </s> + <s xml:id="echoid-s9055" xml:space="preserve">Tum demum aquis effluxus concedatur & </s> + <s xml:id="echoid-s9056" xml:space="preserve">ob-<lb/>ſervabitur filum ſitum verticalem deſerere & </s> + <s xml:id="echoid-s9057" xml:space="preserve">ex angulo declinationis cognoſ-<lb/>cetur vis repellens eaque cum menſuris, quas indicavimus, comparari poterit.</s> + <s xml:id="echoid-s9058" xml:space="preserve"/> +</p> +<pb o="303" file="0317" n="317" rhead="SECTIO DECIMA TERTIA."/> +</div> +<div xml:id="echoid-div311" type="section" level="1" n="235"> +<head xml:id="echoid-head298" xml:space="preserve">Experimentum 1.</head> +<p> + <s xml:id="echoid-s9059" xml:space="preserve">Feci ipſe aliquando omnia, ut nunc monui, viſumque fuit regulam <lb/>noſtram §. </s> + <s xml:id="echoid-s9060" xml:space="preserve">2. </s> + <s xml:id="echoid-s9061" xml:space="preserve">recte confirmari: </s> + <s xml:id="echoid-s9062" xml:space="preserve">non potui tamen tum temporis fufficiente <lb/>accuratione experimentum inſtituere, nec illud poſtea repetii.</s> + <s xml:id="echoid-s9063" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div312" type="section" level="1" n="236"> +<head xml:id="echoid-head299" xml:space="preserve">Experimentum 2.</head> +<p> + <s xml:id="echoid-s9064" xml:space="preserve">Alio tempore rem aliter tentavi: </s> + <s xml:id="echoid-s9065" xml:space="preserve">vas nempe de quo omnes menſuras <lb/>requiſitas ſumſeram aqua plenum naviculæ impoſui in puppi: </s> + <s xml:id="echoid-s9066" xml:space="preserve">navicula aquis <lb/>in alveo innatabat: </s> + <s xml:id="echoid-s9067" xml:space="preserve">Deinde aquis ex vaſe effluentibus (ita tamen ut in navicu-<lb/>lam non illiderent) navicula in plagam contrariam progreſſa eſt: </s> + <s xml:id="echoid-s9068" xml:space="preserve">velocitatem <lb/>naviculæ ex ſpatio dato tempore percurſo rectiſſime exploravi. </s> + <s xml:id="echoid-s9069" xml:space="preserve">Deinde in-<lb/>quiſivi quantum ponduſculum naviculæ eſſet appendendum, ut illo pondere <lb/>ſollicitata eandem velocitatem acquireret. </s> + <s xml:id="echoid-s9070" xml:space="preserve">Inſtituta deinde comparatione iſtius <lb/>ponderis cum pondere cylindri aquei datæ diametri, inde rectiſſime theoriam <lb/>noſtram confirmari vidi.</s> + <s xml:id="echoid-s9071" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div313" type="section" level="1" n="237"> +<head xml:id="echoid-head300" xml:space="preserve">Experimentum 3.</head> +<p> + <s xml:id="echoid-s9072" xml:space="preserve">Effluentibus aquis ex vaſe naviculæ ſuperimpoſito in naviculam, hæc <lb/>omnino immota permanſit: </s> + <s xml:id="echoid-s9073" xml:space="preserve">Id indicat impetum venæ aqueæ æqualem eſſe vi <lb/>repellenti, ut demonſtravi §. </s> + <s xml:id="echoid-s9074" xml:space="preserve">§. </s> + <s xml:id="echoid-s9075" xml:space="preserve">16. </s> + <s xml:id="echoid-s9076" xml:space="preserve">& </s> + <s xml:id="echoid-s9077" xml:space="preserve">17. </s> + <s xml:id="echoid-s9078" xml:space="preserve">Tum etiam ſi vena aquea directe <lb/>impingebat in planum naviculæ affixum, hæc fimiliter immota ſtetit, quod <lb/>rurſus æqualitatem impetus & </s> + <s xml:id="echoid-s9079" xml:space="preserve">vis repellentis probat: </s> + <s xml:id="echoid-s9080" xml:space="preserve">at ſi vena oblique in pla-<lb/>num incidebat, navicula quidem motum obtinuit ſed lentiorem.</s> + <s xml:id="echoid-s9081" xml:space="preserve"/> +</p> +<p> + <s xml:id="echoid-s9082" xml:space="preserve">Denique ſi aquæ effluentes à navicula excipiebantur, ita ut orificium <lb/>aquis in navicula ſtagnantibus eſſet ſubmerſum, ſimiliter abſque motu perſte-<lb/>tit navicula, documento, quod eadem preſſio à vena oriatur, ſive fiat ut om-<lb/>nis ejus motus cohibeatur, ſive ut ad angulum rectum declinetur, prouti de-<lb/>monſtratum fuit §. </s> + <s xml:id="echoid-s9083" xml:space="preserve">18. </s> + <s xml:id="echoid-s9084" xml:space="preserve">æqualitatem inter vim repellentem & </s> + <s xml:id="echoid-s9085" xml:space="preserve">vim venæ aqueæ <lb/>perpendiculariter in planum incidentis plurimis aliis modis exactiſſime confir-<lb/>mavi. </s> + <s xml:id="echoid-s9086" xml:space="preserve">Hanc autem vim theoriæ noſtræ conformem opinionique omnibus adhuc +<pb o="304" file="0318" n="318" rhead="HYDRODYNAMICÆ"/> +communi contrariam experimento omni exceptione majori confirmavi, quod <lb/>præſentibus D. </s> + <s xml:id="echoid-s9087" xml:space="preserve">Emanuele Kœnig, Patrueli meo Nicolao Bernoullio atque <lb/>Patre meo in ædibus meis inſtitui tanta cum fiducia, ut acceptis omnibus <lb/>menſuris, preſſionem venæ aqueæ, quanta futura eſſet, etſi nunquam antea <lb/>à me capto experimento, omni præciſione prædixerim. </s> + <s xml:id="echoid-s9088" xml:space="preserve">Hæc omnia novis <lb/>principiis mechanicis eruta communicavi cum Academia Scientiarum Petro-<lb/>politana, cujus Commentariis a liquando inſerentur.</s> + <s xml:id="echoid-s9089" xml:space="preserve"/> +</p> +</div> +<div xml:id="echoid-div314" type="section" level="1" n="238"> +<head xml:id="echoid-head301" xml:space="preserve">Experimentum 4.</head> +<p> + <s xml:id="echoid-s9090" xml:space="preserve">Ut etiam oſtenderem falſitatem regulæ receptæ tum de vi repellente <lb/>tum de impetu aquarum, adhibui vas quale oſtendit Figura 86. </s> + <s xml:id="echoid-s9091" xml:space="preserve">inſtructum ca-<lb/> +<anchor type="note" xlink:label="note-0318-01a" xlink:href="note-0318-01"/> +nali A B uniformis amplitudinis & </s> + <s xml:id="echoid-s9092" xml:space="preserve">incurvato, cujus directio in A erat horizon-<lb/>talis, in B verticalis: </s> + <s xml:id="echoid-s9093" xml:space="preserve">vidi vas plane non repelli horizontaliter; </s> + <s xml:id="echoid-s9094" xml:space="preserve">ergo per §. </s> + <s xml:id="echoid-s9095" xml:space="preserve">14. <lb/></s> + <s xml:id="echoid-s9096" xml:space="preserve">falſa eſt regula, quæ ſimplici cylindro ibidem definito adhæret.</s> + <s xml:id="echoid-s9097" xml:space="preserve"/> +</p> +<div xml:id="echoid-div314" type="float" level="2" n="1"> +<note position="left" xlink:label="note-0318-01" xlink:href="note-0318-01a" xml:space="preserve">Fig. 86.</note> +</div> +</div> +<div xml:id="echoid-div316" type="section" level="1" n="239"> +<head xml:id="echoid-head302" xml:space="preserve">FINIS.</head> + <figure> + <image file="0318-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0318-01"/> + </figure> +<pb file="0319" n="319"/> + <figure> + <image file="0319-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0319-01"/> + </figure> +<pb file="0319a" n="320"/> + <figure> + <image file="0319a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0319a-01"/> + </figure> +<pb file="0320" n="321"/> +<pb file="0321" n="322"/> + <figure> + <image file="0321-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0321-01"/> + </figure> +<pb file="0321a" n="323"/> + <figure> + <image file="0321a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0321a-01"/> + </figure> +<pb file="0322" n="324"/> +<pb file="0323" n="325"/> + <figure> + <image file="0323-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0323-01"/> + </figure> +<pb file="0323a" n="326"/> + <figure> + <image file="0323a-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/xxxxxxxx/figures/0323a-01"/> + </figure> +<pb file="0324" n="327"/> +<pb file="0325" n="328"/> + <figure> + <image file="0325-01" 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