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  <info>
    <author>Biancani, Giuseppe</author>
    <title>Aristotelis loca mathematica</title>
    <date>1615</date>
    <place>Bologna</place>
    <translator></translator>
    <lang>la</lang>
    <cvs_file>bianc_locam_009_la_1615</cvs_file>
    <cvs_version></cvs_version>
    <locator>009.xml</locator>
  </info>
  <text>
    <front>
      <pb xlink:href="009/01/001.jpg"></pb>
      <section>
        <p type="head">
          <s id="s.000001">ARISTOTELIS <lb></lb>LOCA MATHEMATICA <lb></lb>
Ex vniuerſis ipſius Operibus collecta, <lb></lb>
&amp; explicata.</s>
        </p>
        <p type="head">
          <s id="s.000002"><emph type="italics"></emph>Aristotelicæ videlicet expoſitionis complementum <lb></lb>
hactenus deſideratum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000003">Acceſſere de Natura Mathematicarum ſcientiarum Tractatio; <lb></lb>
<expan abbr="atq;">atque</expan> Clarorum Mathematicorum Chronologia.</s>
        </p>
        <p type="head">
          <s id="s.000004"><emph type="italics"></emph>Authore IOSEPHO BLANCANO Bononienſi è Societate Ieſu, <lb></lb>
Mathematicarum in Gymnaſio Parmenſi Profeſſore.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000005">Ad Illuſtriſſimum, ac Nobiliſſimum <lb></lb>PETRVM FRANCISCVM MALASPINAM <lb></lb>
Aedificiorum Marchionem, apud Cæſ. Maieſtatem <lb></lb>
pro Sereniſs. Parmenſium Duce Legatum.</s>
        </p>
        <p type="head">
          <s id="s.000006">BONONIÆ M. D C. X V.</s>
        </p>
        <p type="head">
          <s id="s.000007">Apud Bartholomæum Cochium. </s>
          <s id="s.000008">Superiorum permiſſu.</s>
        </p>
        <p type="head">
          <s id="s.000009">Sumptibus Hieronymi Tamburini.</s>
        </p>
        <pb xlink:href="009/01/002.jpg"></pb>
        <pb pagenum="3" xlink:href="009/01/003.jpg"></pb>
      </section>
      <section>
        <p type="head">
          <s id="s.000010">ILLVSTRISSIMO <lb></lb>AC NOBILISSIMO <lb></lb>PETRO FRANCISCO <lb></lb>MALASPINAE <lb></lb>
ÆDIFICIORVM MARCHIONI.</s>
        </p>
        <p type="main">
          <s id="s.000011"><emph type="italics"></emph>En tandem Illustriß. Marchio opus no­<lb></lb>
strum de Locis Mathematicis apud Ari­<lb></lb>
stotelem, vnà cum Tractatione de natura <lb></lb>
ſcientiarum Mathematicarum, necnon <lb></lb>
Clarorum <expan abbr="Mathematicorũ">Mathematicorum</expan> Chronologia; <lb></lb>
quod tibi Mecœnati meo munificentiſsimo iure meritò <lb></lb>
dicare, ac ſub clarißimi tui nominis patrocinio in lucem <lb></lb>
dare conſtitui. </s>
          <s id="s.000012">primùm quidem, vt mei perpetui erga te <lb></lb>
amoris, &amp; obſeruantiæ hoc vnum ſaltem specimen exta­<lb></lb>
ret: tùm vt idoneum, æquumque propoſitæ rei iudicem <lb></lb>
nanciſcerer. </s>
          <s id="s.000013">cùm enim ad iuſtum <expan abbr="arbitrũ">arbitrum</expan> duo potißimùm <lb></lb>
requirantur, rerum ſcilicet cognitio, atque prudentia, quem <lb></lb>
te rei, de qua agitur peritiorem, quemuè prudentiorem <lb></lb>
inuenire potuerim? </s>
          <s id="s.000014">tu enim cùm Phyſiologiæ, ac Mathe­<lb></lb>
maticarum omnium Encyclopædiam mirum in modum<emph.end type="italics"></emph.end> <pb pagenum="4" xlink:href="009/01/004.jpg"></pb><emph type="italics"></emph>excolueris, adintima Mathematicarum penetralia ita <lb></lb>
perſuaſiſti, vt Archimedis, &amp; Apollonij admirandis, ac <lb></lb>
ſubtilißimis Demonſtrationibus detinearis. </s>
          <s id="s.000015">Quanta por­<lb></lb>
rò in rebus agendis prudentia valeas, toti penè Europæ <lb></lb>
innotuit, cùm pro noſtris Sereniß. Ducibus, non ſolùm ad <lb></lb>
omnes ferè Italiæ, atque Germaniæ Principes, verùm etiam <lb></lb>
ad Cæſaream Maieſtatem rebus fœliciter geſtis Legatus <lb></lb>
decimùm extiteris; ac demùm à Sereniß. Duce Ranutio <lb></lb>
inter primarios de Rep. </s>
          <s id="s.000016">Conſiliorum Authores adſcitus <lb></lb>
fueris. </s>
          <s id="s.000017">Cæterùm in Clarorum Mathematicorum Chro­<lb></lb>
nologia perlegenda, ſæpißimè tibi nobilißimi æquè, ac do­<lb></lb>
ctißimi Viri, tui omnino perſimiles occurrent, quod tibi <lb></lb>
nonniſi gratißimum accidere poſſe arbitror. </s>
          <s id="s.000018">Complectere <lb></lb>
igitur ea benignitate, atque clementia, qua ſoles noſtra stu­<lb></lb>
dia promouere, mea hæc quantulacumque munuſcula. <lb></lb>
</s>
          <s id="s.000019">quæ ſi tibi accepta eſſe intellexero, iam tandem ma­<lb></lb>
ximorum munerum loco habenda eſſe cenſe­<lb></lb>
bo. </s>
          <s id="s.000020">incolumem tibi, ac fœlicem D. Opt. <lb></lb>
Max. longæuitatem tueatur. <lb></lb>
</s>
          <s id="s.000021">Vale.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000022"><emph type="italics"></emph>Parmæ Idibus Maij M. DC. XIIII.<emph.end type="italics"></emph.end><lb></lb>
</s>
        </p>
      </section>
      <pb pagenum="5" xlink:href="009/01/005.jpg"></pb>
      <section>
        <p type="head">
          <s id="s.000023">Liber de ſe ipſo.</s>
        </p>
        <p type="head">
          <s id="s.000024"><emph type="italics"></emph>Nec diſcet Lector me ſolo interprete totum, <lb></lb>
Nec ſine me totum diſcet Aristotelem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000025">Ego Iordanus Caſſini Præpoſitus Prouincialis Prouinciæ Venetæ Societatis <lb></lb>
Ieſu, ex auctoritate Adm. Reuer. P. noſtro Præpoſiti Generalis P. Claudij <lb></lb>
Aquæuiuæ, facultatem concedo, vt hoc opus P. Ioſephi Blancani eiuſdem <lb></lb>
Societatis, quod inſcribitur, Ariſt. Loca Mathematica ex vniuerſis ipſius <lb></lb>
operibus collecta, &amp; explicata, à deputatis Patribus recognitum, &amp; ap­<lb></lb>
probatum typis mandari poſſit. </s>
          <s id="s.000026">Parmæ die 15. Ianuarij 1615.</s>
        </p>
        <p type="main">
          <s id="s.000027"><emph type="italics"></emph>Iordanus Caſſini P.<emph.end type="italics"></emph.end><lb></lb>
Don Marcellus Baldaſſinus pro Illuſtriſs. </s>
          <s id="s.000028">&amp; Reuerendiſs. Archiepiſc. Bonon.</s>
        </p>
        <p type="main">
          <s id="s.000029">Imprimatur</s>
          <s id="s.000030">Fr. Hieronymus Onuphrius pro Reuerendiſs. P. Inquiſitore Bonon.</s>
        </p>
      </section>
      <pb pagenum="6" xlink:href="009/01/006.jpg"></pb>
      <section>
        <p type="head">
          <s id="s.000031">LECTORI.</s>
        </p>
        <p type="main">
          <s id="s.000032">Qvod priſcis olim temporibus (humaniſſime Lector) ſum­<lb></lb>
mi duo Philoſophi, Philippus Mendeus, ac Theon Smyr­<lb></lb>
næus in Platonis Dialogis egregiè perfecerunt, vt videli­<lb></lb>
cet quæ paſſim ſummus hic Philoſophus de Mathemati­<lb></lb>
cis ſcripta reliquit, eadem ipſa ab illis ſelecta, &amp; in vnum <lb></lb>
quaſi corpus redacta lucubrationibus illuſtrarent: idem ego quoque in <lb></lb>
Ariſtotelis operibus efficere ſum conatus, vt quæ de Mathematicis re­<lb></lb>
bus in vniuerſis eiuſdem monumentis ſparſa leguntur, eadem in vnum <lb></lb>
à me collecta, &amp; explicata ijs Philoſophiæ ſtudioſis maxime ſeruirent, <lb></lb>
qui priſca illa conſuetudine relicta, Mathematicarum omnium ignari <lb></lb>
non ſine graui ſtudiorum ſuorum detrimento Philoſophiæ curriculum <lb></lb>
aggrediuntur. </s>
          <s id="s.000033">Vt autem huius operis neceſſitas, <expan abbr="variæq́">variæque</expan>; vtilitates pla­<lb></lb>
nius cognoſcantur operæpretium erit initio illius cauſas exponere; quæ <lb></lb>
me potiſſimum ad illud conſcribendum compulerunt, quarum</s>
        </p>
        <p type="main">
          <s id="s.000034">Prima ſit, quod hæc Ariſt. loca Mathematica, quæ quidem ferè 408. <lb></lb>
numerantur, peſſimè latinis literis conſignata ſunt vſque adeò, vt Ari­<lb></lb>
ſtotelem ipſum, vel inuitum (quod poſtea multis in locis planum fiet) in <lb></lb>
abſurdiſſima errata ſæpiſſimè compellant.</s>
        </p>
        <p type="main">
          <s id="s.000035">Secunda, quòd plurima huiuſmodi loca à nemine, quod ſciam, adhuc <lb></lb>
declarata in tenebris magno noſtrorum malo deliteſcunt: cuiuſmodi <lb></lb>
ſunt ad ſexaginta problemata, libellus de lineis in ſecabilibus, libellus <lb></lb>
de mundo, ſi tamen Ariſtotelis eſt, &amp; Mechanicæ quæſtiones, quamuis <lb></lb>
enim Picolomineus in eas paraphraſim ediderit, loca tamen earum dif­<lb></lb>
ficiliora non ſatis illuſtrauit. </s>
          <s id="s.000036">Vt autem dixi 408. in vniuerſum loca mi­<lb></lb>
nimùm numerantur, quibus illud Platonis inſcriptum eſt <foreign lang="grc">αγαιομέτρητος <lb></lb>
υδείς εισίτο</foreign>; &amp; in quibus Mathematicæ diſciplinæ rudes, &amp; imperiti, quem <lb></lb>
ſequuntur ducem Ariſt. eum ſæpe deſerere non ſine turpi dedecoris no­<lb></lb>
ta coguntur; quo fit vt exempla illa Mathematica lucem rebus aliquan­<lb></lb>
do allatura, tenebras cimmerijs, vt aiunt vmbris craſſiores ijſdem <lb></lb>
obducant.</s>
        </p>
        <p type="main">
          <s id="s.000037">Tertia, quia Græci eorumdem locorum commentatores breuiter, &amp; <lb></lb>
obſcurè admodum ea, quæ ad Mathematicum ſpectant, attingunt, hoc <lb></lb>
enim ab ipſis <expan abbr="certũ">certum</expan> ponitur, Lectorem eſſe, vt moris tunc erat, omnium <lb></lb>
Philo ſophorum, Mathematicis imbutum; at verò noſtra ætate magna <lb></lb>
cum Philoſophiæ iactura, quamplurimi earumdem diſciplinarum deſti­<lb></lb>
tuti præſidijs, ne Græcorum quidem Interpretum explanationes, ne­<lb></lb>
dum Ariſt. obſcurè dicta intelligunt.</s>
        </p>
        <pb pagenum="7" xlink:href="009/01/007.jpg"></pb>
        <p type="main">
          <s id="s.000038">Quarta. </s>
          <s id="s.000039">Adde, quod etiam ſi quis leuiter ſit erudito illo Mathemati­<lb></lb>
corum puluere conſperſus, adeò tamen peruerſa eſt eorumdem Græco­<lb></lb>
rum in Latinum tranſlatio, <expan abbr="tantaq́">tantaque</expan>; figurarum, quæ neceſſariæ erant <lb></lb>
confuſio, &amp; deprauatio, vt nec abeo, qui ſit Mathematicarum ſcientia <lb></lb>
excultus, ſine magno labore percipi poſſint. </s>
          <s id="s.000040">Quin etiam figuræ illæ, quæ <lb></lb>
omnino neceſſariæ ſunt ob Scriptorum, &amp; Typographorum inſcitiam, <lb></lb>
aut inertiam pluribus in locis deſiderantur. </s>
          <s id="s.000041">Latini verò multo minus, <lb></lb>
quàm Græci Mathematicæ periti, qua ratione eadem loca pertractaue­<lb></lb>
rint, facilius eſt conijcere, quàm vt dici oporteat.</s>
        </p>
        <p type="main">
          <s id="s.000042">Quinta. </s>
          <s id="s.000043">Ex his omnibus in aliud incommodum, vel maximum Phi­<lb></lb>
loſophi quidam incidebant; aut enim horum locorum expoſitionem ta­<lb></lb>
citi declinabant: aut eam minime neceſſariam ad Ariſt. percipiendam <lb></lb>
ſententiam aſſerebant; quo quid abſurdius, quid ſtudioſorum progreſ­<lb></lb>
ſibus pernicioſius excogitari poteſt? </s>
          <s id="s.000044">Eorum verò nonnulli eorumdem <lb></lb>
locorum expoſitionem audacter nimis aggrediebantur, <expan abbr="atq;">atque</expan> hinc pueri­<lb></lb>
les illæ, ac ridiculæ expoſitiones paſſim auditæ, cuiuſmodi eſt illa, quan­<lb></lb>
do Ariſtoteles ait, quod illi frequentiſſimum eſt, omnis triangulus ha­<lb></lb>
bet tres; nihil aliud ſignificari volunt, quàm omnem triangulum habe­<lb></lb>
re tres angulos. </s>
          <s id="s.000045">quod ſi dicat, omnis triangulus habet tres æquales duo­<lb></lb>
bus rectis: hic hærent, hinc anguntur: <expan abbr="cumq;">cumque</expan> ex his anguſtijs, ac tricis <lb></lb>
ſe minimè expedire valeant, aurea verba illa, quibus ingentes ſapientiæ <lb></lb>
theſauri continentur, alto ſilentij velo contegere Mathematicarum eos <lb></lb>
cogit inſcitia: vnde illud, quod Græcæ linguæ imperitis mutata oratio­<lb></lb>
ne acclamandum illis foret, Mathematicum eſt, non legitur. </s>
          <s id="s.000046">Nec mi­<lb></lb>
nus elegans illa altera expoſitio; Diametrum eſſe incommenſurabilem <lb></lb>
coſtæ; quod ſæpe apud Ariſt. legentibus occurrit, nihil aliud ſibi velle, <lb></lb>
quam Diametrum eſſe longiorem coſta, quam quidem aſymetriæ huius <lb></lb>
ignorantiam Plato de legibus dial. 7. non hominum, ſed ſuum, <expan abbr="peco-rumq;">peco­<lb></lb>
rumque</expan> appellare non dubitauit. </s>
          <s id="s.000047">Quid illa? </s>
          <s id="s.000048">cum Ariſt. ait duo cubi, cu­<lb></lb>
bus, ipſum loqui putant de duplatione Geometrici cubi, nondum in­<lb></lb>
uenta; non intelligentes, eum ibi de numeris cubis ſermonem habere. <lb></lb>
</s>
          <s id="s.000049">Auerroes ipſe tantus vir 5. Phyſ. commen. 15. quàm ſe Mathematicis, <lb></lb>
reliquiſque Philoſophis irridendum præbet dum à permutata propor­<lb></lb>
tione putat ſerectè in hunc modum pluribus apud ipſum verbis explica­<lb></lb>
tum, argumentari,</s>
        </p>
        <p type="main">
          <s id="s.000050"><emph type="italics"></emph>Vt ſe habet voluntas noua ad effectum nouum, <lb></lb>
It a voluntas antiqua ad effectum antiquum. <lb></lb>
</s>
          <s id="s.000051">Ergo permutatim, vt ſe habebit voluntas noua ad effectum <lb></lb>
antiquum, ita voluntas antiqua ad effectum nouum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000052"><expan abbr="Spectatũ">Spectatum</expan> admiſſi riſum teneatis amici? </s>
          <s id="s.000053">nego, ait; quiſpiam conſequen­


<pb pagenum="8" xlink:href="009/01/008.jpg"></pb>tiam, non enim hoc eſt argumentari à permutata ratione, deberet enim <lb></lb>
inferre, ſic, ergo ita ſe habebit voluntas noua ad antiquam, quemad­<lb></lb>
modum effectus nouus ad antiquum. </s>
          <s id="s.000054">quæ vitioſa argumentatio quan­<lb></lb>
tumuis læuis ſit, &amp; manifeſta, quoſdam tamen magni nominis philoſo­<lb></lb>
phantes adeò torſit, vt adhuc torqueat.</s>
        </p>
        <p type="main">
          <s id="s.000055">Quanta autem miſeratione digni, qui publicè aliquando apud ſuos <lb></lb>
auditores totam Perſpectiuam, qua nihil iucundius eſt, de medio tolle­<lb></lb>
re conati ſunt, propterea quod illæ viſuales lineæ, illi anguli, illæ pyra­<lb></lb>
mides, aut coni, quibus viſio perficitur nullibi extarent, ſed eſſent vana <lb></lb>
quædam opticorum figmenta. </s>
          <s id="s.000056">Quì verò fieri potuit, vt non aduerterint <lb></lb>
iſti ſe Ariſtoteli ſuo manifeſte repugnare, qui ſæpius de lineis viſualibus <lb></lb>
perſpectiuum pertractare aſſerit, <expan abbr="diſcrimenq́">diſcrimenque</expan>; inter lineam phyſicam, &amp; <lb></lb>
opticam aſſignat, <expan abbr="ipſiusq́">ipſiusque</expan>; optices tanquam veræ ſcientiæ mentionem <lb></lb>
ſæpius facit.</s>
        </p>
        <p type="main">
          <s id="s.000057">Alij ex altera parte contra Aſtronomos in ſurgunt, eccentricos, <expan abbr="atq;">atque</expan> <lb></lb>
epiciclos omnes de cœlo detrahere cupientes. </s>
          <s id="s.000058">Verum id iſti nulla ex­<lb></lb>
preſſa nedum probabili ratione faciunt, falsò exiſtimantes Aſtronomos <lb></lb>
admirandam illam Cœlorum fabricam aſſerere, non autem ſupponere: <lb></lb>
ſed aſtronomi illam ſupponunt, <expan abbr="eamq́">eamque</expan>; propterea hypotheſim <expan abbr="appellãt">appellant</expan>, <lb></lb>
non aſſerunt. </s>
          <s id="s.000059">Quod ſi aliqua probabili ratione id facerent, vti nonnulli <lb></lb>
ex recentioribus, quorum Ticho Coripheus eſt, laudandi potius, quam <lb></lb>
vituperandi eſſent. </s>
          <s id="s.000060">Impugnant <expan abbr="itaq;">itaque</expan> aſtronomachi iſti hypotheſim pro <lb></lb>
aſſertione; <expan abbr="talesq́">talesque</expan>; ſæpè hi ſunt, vt non ſatis intelligant, quid ſit Aequa­<lb></lb>
tor, aut Zodiacus, ne dum quid Epiciclus, aut Eccentricus. </s>
          <s id="s.000061">Nec defuit <lb></lb>
qui viginti duo argumenta excogitarit, <expan abbr="atq;">atque</expan> in medium protulerit, qui­<lb></lb>
bus contra Aſtronomos probare conatus eſt, nullo modo Solem, aut <lb></lb>
Lunam moueri poſſe motibus contrarijs, ideſt, ab oriente in <expan abbr="occidẽtem">occidentem</expan> <lb></lb>
motu diurno, &amp; proprio ab occidente in orientem. </s>
          <s id="s.000062">Sed exiſtimandum <lb></lb>
eſt <expan abbr="iſtũ">iſtum</expan> Lunam nouam à Sole quotidie magis, ac magis verſus orientem <lb></lb>
recedere, nunquam animaduertiſſe; ab ea enim hanc motuum concor­<lb></lb>
diam didiciſſet.</s>
        </p>
        <p type="main">
          <s id="s.000063">Quid tandem <expan abbr="dicẽdum">dicendum</expan> de quodam magni nominis Philoſopho, om­<lb></lb>
nium tamen <expan abbr="Mathematicarũ">Mathematicarum</expan> experte, qui in publica diſputatione axio­<lb></lb>
ma illud Mathematicum, omne totum eſt maius ſua parte, in ſenſu in <lb></lb>
quo à Mathematicis effertur negare non erubuit, eò, quod in infinito, <lb></lb>
vt aiebat non concederetur ab omnibus. </s>
          <s id="s.000064">ſcilicet non intelligebat ma­<lb></lb>
thematicum tantummodo tractare de Quantitate finita, ac terminata, <lb></lb>
in qua axioma prædictum ab omnibus conceditur. </s>
          <s id="s.000065">Neque vero hic <lb></lb>
nonnullorum infenſus in Mathematicas animus quieuit, verum etiam <lb></lb>
eò progreſſus eſt, vt eas omnes omnino conuellere, atque ex albo ſcien­


<pb pagenum="9" xlink:href="009/01/009.jpg"></pb>tiarum, quamuis non Ariſtotele tantum, ſed ipſa etiam veritate repu­<lb></lb>
gnante, expungere conati ſint; <expan abbr="idq;">idque</expan> neſcio an vlla alia de cauſa egerint, <lb></lb>
quàm quod eas non ſatis calerent; non ſecus <expan abbr="atq;">atque</expan> Aeſopica illa Vulpes, <lb></lb>
quæ cum cauda mutilata eſſet, caudarum mutilationem reliquis vulpi­<lb></lb>
bus vafrè perſuadere conabatur. </s>
          <s id="s.000066">Verum enim verò optimè ſcio, ea, <lb></lb>
quę hactenus dicta ſunt non in omnes noſtri temporis Philoſophos qua­<lb></lb>
drare, cum non pauci hodie quoque ſint, qui more antiquorum Mathe­<lb></lb>
maticis ſuffulti, optimè ſuis ſtudijs conſulentes, reliquam Philoſophiam <lb></lb>
non ſine magno compendio aggrediuntur. </s>
          <s id="s.000067">Quo fit, vt cæteros ageo­<lb></lb>
metretos ita antecellant, vt eorum Magiſtri appellari poſſint, &amp; <expan abbr="debeãt">debeant</expan>; <lb></lb>
tales fuerunt in primis Vicomercatus, Cardanus, Zabarella, Toletus, <lb></lb>
Bonamicus, &amp; alij, quibus paulo antiquiores fuerunt D. Tho. &amp; Scotus, <lb></lb>Hi omnes Mathematicarum auxilio, quantum inter reliquos philoſo­<lb></lb>
phantes excelluerint, nemo eſt qui non nouerit. </s>
          <s id="s.000068">Illud hoc loco minimè <lb></lb>
tacendum, Iacobum Zabarellam in ſuis logicæ commentarijs teſtari ſe <lb></lb>
bis ſumma diligentia totum Euclidem perlegiſſe, vt perfectè Ariſt. de <lb></lb>
demonſtratione ſententiam aſſequi poſſet.</s>
        </p>
        <p type="main">
          <s id="s.000069">Hi ridiculas illas, ac pueriles expoſitiones ſuperius allatas minimè <lb></lb>
effutierunt, neque reliquis ſupra recenſitis incommodis obnoxij fue­<lb></lb>
runt, quibus magno etiam cum dedecore alij Mathematicarum ope ca­<lb></lb>
rentes afficiuntur.</s>
        </p>
        <p type="main">
          <s id="s.000070">In horum igitur gratiam operam diligenter dedi, vt quantum in me <lb></lb>
eſſet damna à me ſupra enarrata aliqua ex parte reſarcirem. </s>
          <s id="s.000071">Quaprop­<lb></lb>
ter loca hæc mathematica num rectè eſſent è græco in latinum tranſlata <lb></lb>
diligenter prius expendi. </s>
          <s id="s.000072">Deinde claritate, quàm potui maxima eadem <lb></lb>
loca interpretatus ſum, &amp; in horum, de quibus dixi gratiam, quædam <lb></lb>
fanè tenuia proſequutus ſum, quæ alioquin libenter omiſiſſem. </s>
          <s id="s.000073">Tum fi­<lb></lb>
guras omnes, aut correxi, aut reſtitui, aut nouas appoſui. </s>
          <s id="s.000074">Hoc igitur <lb></lb>
noſtro qualicunque labore poterit quiſque omnia illa facile intelligere, <lb></lb>
<expan abbr="atq;">atque</expan> enumerata incommoda euitare, vnum tantummodo à Lectore ma­<lb></lb>
thematicarum experte requiram, vt principia ſaltem illa, ſcilicet defini­<lb></lb>
tiones, poſtulata, &amp; axiomata, quæ primò Euclideis libro præponuntur, <lb></lb>
diligenter prius perlegat cum illa ſua perſpicuitate omnibus ſint obuia; <lb></lb>
cætera ego explicanda recipio. </s>
          <s id="s.000075">Obiter etiam auctaria nonnulla partim <lb></lb>
mathematica, partim naturalia inſerui, quæ ob nouitatem, ac pulchri­<lb></lb>
tudinem grata Lectori, atque iucunda fore exiſtimaui.</s>
        </p>
        <p type="main">
          <s id="s.000076">Sciat præterea Lector noſtrum inſtitutum eſſe loca hæc mathemati­<lb></lb>
ca, quatenus mathematica ſunt declarare, ſiue ea ſupplere, quæ ex ma­<lb></lb>
thematicis petenda eſſent: reliqua autem me tantum attingere, quan­<lb></lb>
tum harum rerum cum illis connexio poſtulat.</s>
        </p>
        <pb pagenum="10" xlink:href="009/01/010.jpg"></pb>
        <p type="main">
          <s id="s.000077">His omnibus placuit appendices opportune nonnullas addere, qua­<lb></lb>
rum prima de natura mathematicarum ſcientiarum: altera, qua omnes <lb></lb>
demonſtrationes primi libri Euclidis breuiter ad Logicam normam ex­<lb></lb>
penduntur, vt pateat, quonam demonſtrationis genere <expan abbr="cẽſeri">cenſeri</expan> <expan abbr="vnaquęq;">vnaquęque</expan> <lb></lb>
debeat, &amp; ex illis de cæteris iudicium fiat. </s>
          <s id="s.000078">Tandem in gratiam etiam <lb></lb>
Mathematicorum tertiam appendicem appendi, qua omnia loca Ariſt. <lb></lb>
Geometrica ad Euclidis ordinem referuntur; vnde &amp; ipſi ad ſuas pręle­<lb></lb>
ctiones exornandas aliquid ſubinde depromere queant.</s>
        </p>
        <p type="main">
          <s id="s.000079">Fruere igitur amice Lector hoc noſtro qua liquali labore, quo ad ple­<lb></lb>
nam totius Ariſt. intelligentiam, cui adhuc mathematicarum ignoratio <lb></lb>
obſtitit peruenire tandem poſſis: <expan abbr="illudq́">illudque</expan>; experiaris, quod optimus qui­<lb></lb>
dam Philoſophus, cum totum hunc librum perlegiſſet, effatus eſt, vide­<lb></lb>
licet, opus hoc <emph type="italics"></emph>Aristotelicæ expoſitionis complementum ad hanc vſque <lb></lb>
diem deſideratum<emph.end type="italics"></emph.end> iure ac meritò nuncupari poſſe.</s>
        </p>
        <p type="main">
          <s id="s.000080">Illud demum tanquam parergon addam, quod ego his elucubran­<lb></lb>
dis experientia didici, ad veram ſcilicet, ac perfectam to­<lb></lb>
tius Ariſtotelis intelligentiam linguæ in primis <lb></lb>
græcæ, necnon mathematicarum om­<lb></lb>
nium diſciplinarum haud medio­<lb></lb>
crem cognitionem ne­<lb></lb>
ceſſariam eſſe. <lb></lb>
</s>
          <s id="s.000081">Vale.</s>
        </p>
      </section>
      <pb pagenum="11" xlink:href="009/01/011.jpg"></pb>
      <section>
        <p type="head">
          <s id="s.000082">Pręcipua quędam, aut noua, aut reſtaurata, <lb></lb>
quæ obiter pertractantur.<lb></lb>
<arrow.to.target n="table1"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table1"></table.target>
          <row>
            <cell><emph type="italics"></emph>1<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De reſolutione. numero marginali.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>4<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>2<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De figuris vacuum replentibus, vbi Aristotelis, &amp; expo-ſitorum erratum aperitur. num.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>121<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell></cell>
            <cell><emph type="italics"></emph>Inibi, Apum mirabilis quædam in cellis ſuis hexagonis   constituendis induſtria detegitur. num.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>120<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>3<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De ijs, quæ aquæ inſident, vnà cum noua demonstratione problema-tis illius Archimedis, quo metallorum mixtionem indiſſoluta Co-rona, explorauit. in additione. ante num.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>124<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>4<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De Cometa, recentiorum ſententia. num.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>136<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>5<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De altitudine montis Caucaſi.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>148<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>6<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De Terræ rotunditate, ac mundi duratione.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>151<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>7<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De Iride. in additione.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>181<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>8<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Scytala quid.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>250<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>9<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Securis antiqua quæ, &amp; qua ratione fieret.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>258<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>10<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Statera antiqua quæ,<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>259<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>11<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De Aeſtu Maris.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>272<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>12<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Araneorum induſtria nuper patefacta: vbi Democritus contra Ariſt.   defenditur.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>293<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>13<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De Lucis figuratione, &amp; rerum ſimulacris in obſcuro loco.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>345<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>14<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De Pupilla oculi.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>408<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>15<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>De Mathematicarum natura. propè finem operis.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>16<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Clarorum Mathematicorum Chronologia, in fine operis.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
        </table>
      </section>
      <pb pagenum="12" xlink:href="009/01/012.jpg"></pb>
      <section>
        <p type="head">
          <s id="s.000083"><emph type="italics"></emph>PRIMVS INDEX LOCORVM ARIST.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000084"><emph type="italics"></emph>Quæ in hoc opere explicantur, iuxta ordine librorum <lb></lb>
ipſius ex vulgata editione Lugdunenſi.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000085">In Prædicamentis.</s>
        </p>
        <p type="main">
          <s id="s.000086"><emph type="italics"></emph>Capite s. </s>
          <s id="s.000087">de Relatione, vbi de Quadratura circuli.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000088"><emph type="italics"></emph>Cap. de Priori, vbi de Principijs Mathematicarum,<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000089"><emph type="italics"></emph>Cap. de Motu, vbi de Gnomone.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000090">In Primo Priorum Reſolutoriorum.</s>
        </p>
        <p type="main">
          <s id="s.000091"><emph type="italics"></emph>Ad titulum libri de Reſolutione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000092"><emph type="italics"></emph>Cap. 23. ſect 1. libri 1. de Incommenſurabilibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000093"><emph type="italics"></emph>Cap. 24. ſecti 1. lib. 1. de Deſcriptionibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000094"><emph type="italics"></emph>Cap. 2. ſect 2. lib. 1. de Deſcriptionibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000095"><emph type="italics"></emph>Cap. 3. ſecti 2. lib. 1. de Incommenſurabili.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000096"><emph type="italics"></emph>Cap. 1. ſecti 3. lib. 1. de eo, quod est, omnis triangulus habet tres angulos æquales <lb></lb>
æquales duobus rectis: Aequalitas Geometrica, quæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000097"><emph type="italics"></emph>Cap. eodem, de exemplis, quibus vtuntur Geometræ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000098">In ſecundo Priorum Reſol.<emph type="italics"></emph>Cap. 21. de lineis Paralellis, ſeu Coalternis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000099"><emph type="italics"></emph>Cap. eodem. </s>
          <s id="s.000100">de Paralellis, &amp; de triangulo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000101"><emph type="italics"></emph>Cap. 26. Quod omnis triangulus habet tres, &amp; c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000102"><emph type="italics"></emph>Cap. 31. de Abductione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000103"><emph type="italics"></emph>Cap. codem, de circuli Quadratura, ſecundum Hippocratem Chium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000104">In primo Poſteriorum.</s>
        </p>
        <p type="main">
          <s id="s.000105"><emph type="italics"></emph>Textu primo, De Præcognitis Mathematicarum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000106"><emph type="italics"></emph>T. 2. Omnis triangulus habet tres, &amp; c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000107"><emph type="italics"></emph>T. 5. De Diametro incommenſurabili. </s>
          <s id="s.000108">Item De Mathematicarum Principijs.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000109"><emph type="italics"></emph>T. eodem, De Indiuiſibilitate vnitatis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000110"><emph type="italics"></emph>T. 9. De Puncto, &amp; linea. </s>
          <s id="s.000111">Item de recto, &amp; circulari. </s>
          <s id="s.000112">Item de numero pari, impari; <lb></lb>
primo, &amp; compoſito; æquilatero, &amp; altera parte longiore.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000113"><emph type="italics"></emph>T. 11. Lineæ punctum inest per ſe, &amp; c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000114"><emph type="italics"></emph>T. 13. De Parallelis. </s>
          <s id="s.000115">De Iſoſcele. </s>
          <s id="s.000116">De Alterna Proportione, <lb></lb>
Item quod omnis triangulus habet tres, &amp; c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000117"><emph type="italics"></emph>T. 14. De ijſdem cum præcedentibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000118"><emph type="italics"></emph>T. 20. Magnitudines euadunt numeri. </s>
          <s id="s.000119">Item, quod non duo cubi cubus. </s>
          <s id="s.000120">Item de <lb></lb>
Mathematicis ſubalternatis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000121"><emph type="italics"></emph>T. 23. Quadratura circuli ſecundum Bryſonem. </s>
          <s id="s.000122">Item perſectam illam eſſe Demon­<lb></lb>
ſtrationem, qua Geometræ oſtendunt, quod Omnis triangulus habet tres, &amp; c. <lb></lb>
</s>
          <s id="s.000123">Perſpectiuam, &amp; Mechanicam ſubalternari Geometriæ, Muſicam, Arithmeticæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000124"><emph type="italics"></emph>T. 24. De numero pari, impari, quadrangulo, cubo. </s>
          <s id="s.000125">In Geometria quid irrationale, <lb></lb>
refrangi, concurrere. </s>
          <s id="s.000126">Quid Astronomia conſideret.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000127"><emph type="italics"></emph>T. 25. Geometram non mentiri in ſuis exemplis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000128"><emph type="italics"></emph>T. 28. De Parallelis.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="13" xlink:href="009/01/013.jpg"></pb>
        <p type="main">
          <s id="s.000129"><emph type="italics"></emph>T. 29. Cur in Mathematicis non ſit Paralogiſmus. </s>
          <s id="s.000130">Item quid multiplicata propor­<lb></lb>
tio. </s>
          <s id="s.000131">Quid Cæneus dixerit. </s>
          <s id="s.000132">Cur Affectiones <expan abbr="Mathematicorũ">Mathematicorũ</expan> maximè <expan abbr="conuertãtur">conuertantur</expan>.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000133"><emph type="italics"></emph>T. 30. De Lunæ ſphæricitate. </s>
          <s id="s.000134">Quid ſtereometria. </s>
          <s id="s.000135">&amp; De ſubalternatione, &amp;c. </s>
          <s id="s.000136">&amp; Ma­<lb></lb>
thematicorum eſt ſcire Propter quid: ſenſitiuorum verò ſcire Quod.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000137"><emph type="italics"></emph>T. 37. Iſoſceles, &amp; Scalenum habere tres æquales, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000138"><emph type="italics"></emph>T. 38. Quid Mina, quid Dieſis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000139"><emph type="italics"></emph>T. 39. Habere tres angulos æquales, &amp;c. </s>
          <s id="s.000140">Item, quod omnis figura habet ſuos angu­<lb></lb>
los externos æquales quatuor tantum rectis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000141"><emph type="italics"></emph>T. 43. Triangulum tres æquales, &amp;c. </s>
          <s id="s.000142">De Eclypſi.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000143"><emph type="italics"></emph>De combuſtione per refractionem ex ſphæra vitrea. </s>
          <s id="s.000144">De principijs ſcientiarum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000145"><emph type="italics"></emph>T. 44. Diameter incommenſurabilis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000146">In 2. Poſteriorum.</s>
        </p>
        <p type="main">
          <s id="s.000147"><emph type="italics"></emph>T. 1. Aequalitas, &amp; inæqualitas. </s>
          <s id="s.000148">Terram eſſe in medio mundi ab Aſtronomis per­<lb></lb>
fectè demonſtratur. </s>
          <s id="s.000149">Item Quid conſonantia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000150"><emph type="italics"></emph>T. 2. Omnis triangulus habet tres, &amp;c. </s>
          <s id="s.000151">Item de Definitionibus Mathematicarum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000152"><emph type="italics"></emph>T. 7. Geometra, quædam accipit, quædam demonſtrat.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000153"><emph type="italics"></emph>T. 11. Angulum in ſemicirculo rectum eſſe probari à Geometra per cauſam materia­<lb></lb>
lem. </s>
          <s id="s.000154">Zabarella correctus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000155"><emph type="italics"></emph>T. 24. Echo, Imago è ſpeculo, Iris.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000156"><emph type="italics"></emph>T. 25. Permutatim proportionale quid; exemplum in triangulis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000157">In primo lib. Topicorum.</s>
        </p>
        <p type="main">
          <s id="s.000158"><emph type="italics"></emph>Cap. 13. Diameter est incommenſurabilis. </s>
          <s id="s.000159">Vox acuta velox, cur. </s>
          <s id="s.000160">&amp;c. </s>
          <s id="s.000161">Colores in <lb></lb>
Muſica, qui. </s>
          <s id="s.000162">tria genera veteris Muſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000163">In 4. libro.</s>
        </p>
        <p type="main">
          <s id="s.000164"><emph type="italics"></emph>Cap. 1. loco 1. lineæ inſecabiles.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000165">In 6. libro.</s>
        </p>
        <p type="main">
          <s id="s.000166"><emph type="italics"></emph>Cap. 2. loco 32. Definitio lineæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000167">In 8. libro.</s>
        </p>
        <p type="main">
          <s id="s.000168"><emph type="italics"></emph>Cap. 2. loco 41. Vſus Definitionum in Mathematicis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000169"><emph type="italics"></emph>Cap. 4. loco 86. Elementa geometrica: Numeri capitales.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000170">In Elenchorum lib. 1.</s>
        </p>
        <p type="main">
          <s id="s.000171"><emph type="italics"></emph>Cap. 10. Quid Pſeudographia. </s>
          <s id="s.000172">Quadratura rurſus Hippocratis, &amp; Bryſonis. </s>
          <s id="s.000173">Mathe­<lb></lb>
maticæ non contentioſæ. </s>
          <s id="s.000174">Quadratio Antiphontis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000175">Ex 1. Phyſic.</s>
        </p>
        <p type="main">
          <s id="s.000176"><emph type="italics"></emph>T. 11. De Quadraturis circuli Hippocratis, &amp; Antiphontis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000177">Ex 2. Phyſic.</s>
        </p>
        <p type="main">
          <s id="s.000178"><emph type="italics"></emph>T. 20. Quatenus Perſpectiuus conſideret lineam.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000179"><emph type="italics"></emph>T. 28. Quid conſonantia Diapaſon.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000180"><emph type="italics"></emph>T. 68. Mathematicas Demonſtrationes habere cauſam, quæ reducitur ad defini­<lb></lb>
tionem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000181"><emph type="italics"></emph>T. 8. De neceſſario, quod eſt in Mathematicis. </s>
          <s id="s.000182">&amp; omnis triangulus habet tres an­<lb></lb>
gulos, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000183">Ex 3. Phyſic.</s>
        </p>
        <p type="main">
          <s id="s.000184"><emph type="italics"></emph>T. 76. Quinam numeri dicantur Gnomones.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000185"><emph type="italics"></emph>T. 31. Quonam infinito vtantur Mathematici.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000186"><emph type="italics"></emph>T. 71. De infinito Mathematica.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="14" xlink:href="009/01/014.jpg"></pb>
        <p type="head">
          <s id="s.000187">Ex 4. Phyſic.</s>
        </p>
        <p type="main">
          <s id="s.000188"><emph type="italics"></emph>T. 120. De commenſurab. </s>
          <s id="s.000189">&amp; incommenſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000190">Ex 5. Phyſic.</s>
        </p>
        <p type="main">
          <s id="s.000191"><emph type="italics"></emph>T. 6. De chordis, graui, acuta; media, &amp; vltima.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000192">Ex 8. Phyſic.</s>
        </p>
        <p type="main">
          <s id="s.000193"><emph type="italics"></emph>T. 15. Omnis triangulus habet tres æquales, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000194">Ex 1. de Cœlo.</s>
        </p>
        <p type="main">
          <s id="s.000195"><emph type="italics"></emph>T. 33. De minimo indiuiſibili.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000196"><emph type="italics"></emph>T. 36. Ratione vtitur lineari; vt probet mundum eſſe finitum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000197"><emph type="italics"></emph>T. 48. Commenſurab. &amp; incommenſurab. </s>
          <s id="s.000198">quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000199"><emph type="italics"></emph>T. 119. Omnis triangulus habet tres, &amp;c. </s>
          <s id="s.000200">Item de commenſurabili.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000201">Ex 2. de Cœlo.</s>
        </p>
        <p type="main">
          <s id="s.000202"><emph type="italics"></emph>T. 24. Plato ex planis ſolida componebat, quì.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000203"><emph type="italics"></emph>T. 25. Ordo figurarum planarum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000204"><emph type="italics"></emph>T. 31. Aquæ ſuperficiem eſſe ſphæricam.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000205"><emph type="italics"></emph>T. 46. Maiorem circulum velocius moueri. </s>
          <s id="s.000206">Recentiorum obſeruationes.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000207"><emph type="italics"></emph>T. 57. De ordine Cœlorum ex ſententia Aſtronomorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000208"><emph type="italics"></emph>T. 59. De rotunditate Lunæ, bis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000209"><emph type="italics"></emph>T. 107. Centrum duplex grauit: &amp; molis. </s>
          <s id="s.000210">Qua ratione grauia ad mundi centrum <lb></lb>
aptarentur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000211"><emph type="italics"></emph>T. 109. Terram eſſe rotundam. </s>
          <s id="s.000212">alio item modo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000213"><emph type="italics"></emph>T. 110. Terram eſſe paruam reſpectu Cœli.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000214"><emph type="italics"></emph>T. 111. Mare occidentale coniungi indico.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000215"><emph type="italics"></emph>T. 112. De quantitate Terræ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000216">Ex 3. de Cœlo.</s>
        </p>
        <p type="main">
          <s id="s.000217"><emph type="italics"></emph>T. 40. Vt componatur ſphæra.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000218"><emph type="italics"></emph>T. 66. Omne corpus diuiſibile.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000219"><emph type="italics"></emph>T. 66. Quænam planarum figurarum totum ſpatium repleant. </s>
          <s id="s.000220">Hinc de admirabili <lb></lb>Apum ingenio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000221"><emph type="italics"></emph>T. eodem. </s>
          <s id="s.000222">Num plures Pyramides locum replere valeant, vbi Ariſtoteles, &amp; omnes <lb></lb>
expoſitores erraſſe oſtenduntur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000223"><emph type="italics"></emph>T. 71. Terram eſſe cubum, cur dictum ſit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000224">Ex 4. de Cœlo.</s>
        </p>
        <p type="main">
          <s id="s.000225"><emph type="italics"></emph>T. 33. Extare centrum Mundi, quò grauia tendunt, à quo leuia abeant.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000226"><emph type="italics"></emph>T. 44. &amp; <expan abbr="ſeq.">ſeque</expan> Cur quædam grauiora quàm aqua, ſupernatent.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000227">Ex 2. de Generatione, &amp; Corruptione.</s>
        </p>
        <p type="main">
          <s id="s.000228"><emph type="italics"></emph>Tex. 56. Cur Planetæ duobus motibus moueri dicantur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000229">Ex 1. Meteororum.</s>
        </p>
        <p type="main">
          <s id="s.000230"><emph type="italics"></emph>Summa prima cap. 3. De magnitudine Terræ ad aſtra, &amp; ſolem collata.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000231"><emph type="italics"></emph>Cap. eodem. </s>
          <s id="s.000232">De magnitudine Aſtrorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000233"><emph type="italics"></emph>Cap. 4. De ordine Luminarium Solis, &amp; Lunæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000234"><emph type="italics"></emph>Summa 2. cap. 3. de Mercurij stella. </s>
          <s id="s.000235">Item de Cometa: eſſe in Cœlo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000236"><emph type="italics"></emph>Cap. 5. De Magnitudine Solis, &amp; de vmbra Terræ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000237"><emph type="italics"></emph>Cap. 5. De Glaxia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000238"><emph type="italics"></emph>Cap. 6. Sententia Ariſtotelis de Glaxia, partim defenditur: vera, deinde <lb></lb>
aperitur.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="15" xlink:href="009/01/015.jpg"></pb>
        <p type="main">
          <s id="s.000239"><emph type="italics"></emph>Summa 4. cap 1. De Monte Parnaſſo, dubia. </s>
          <s id="s.000240">Mare extraneum, quod. </s>
          <s id="s.000241">Errata quæ­<lb></lb>
dam veterum Geographorum, &amp; Ariſt. corriguntur. </s>
          <s id="s.000242">Altitudo montis Caucaſi.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000243"><emph type="italics"></emph>Cap. 2. De permutatione Aquarum, &amp; continentis. </s>
          <s id="s.000244">Noua obſeruatio de rotundi­<lb></lb>
tate Terræ, <expan abbr="atq;">atque</expan> Mundi duratione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000245">Ex 2. Meteororum.</s>
        </p>
        <p type="main">
          <s id="s.000246"><emph type="italics"></emph>Summa 1. cap. 1. De Mari rubro.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000247"><emph type="italics"></emph>Summa 2. cap. 2. De ortu stellarum fixarum: Item de occaſu earumdem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000248"><emph type="italics"></emph>Cap. eodem, De Canicula. </s>
          <s id="s.000249">De Zonis temperatis. </s>
          <s id="s.000250">Corona Ariadnæ. </s>
          <s id="s.000251">Zonam torridam <lb></lb>
falsò putabant inhoſpitalem. </s>
          <s id="s.000252">cur habitabilis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000253"><emph type="italics"></emph>Cap. 3. De Ventorum ſitu.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000254">Ex 3. Meteor.</s>
        </p>
        <p type="main">
          <s id="s.000255"><emph type="italics"></emph>Summa 2. cap. 2. De Halone, ſeu Area, ſeu Corona, Mathematica demonſtratio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000256"><emph type="italics"></emph>Cap. 4. De Iridis figura Mathematica demonſtratio, ſed deficiens. </s>
          <s id="s.000257">Noua de eadem <lb></lb>
tractatio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000258"><emph type="italics"></emph>Cap. 5. De Parelio. </s>
          <s id="s.000259">Rationes Ariſtotelis refelluntur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000260">Ex 1. De Anima.</s>
        </p>
        <p type="main">
          <s id="s.000261"><emph type="italics"></emph>Tex. 11. Quid rectum, quid obliquum. </s>
          <s id="s.000262">&amp; omnis triangulus habet tres, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000263"><emph type="italics"></emph>T. 13. Sphæra planum tangit in puncto.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000264">Ex 2. De Anima.</s>
        </p>
        <p type="main">
          <s id="s.000265"><emph type="italics"></emph>T. 12. Definitionem formalem, &amp; cauſalem explicat exemplo Quadrationis Geo­<lb></lb>
metricæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000266"><emph type="italics"></emph>T. 86. Acutum, &amp; Graue, vt differant.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000267"><emph type="italics"></emph>T. 159. De Solis magnitudine ad terram.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000268">Ex 3. De Anima.</s>
        </p>
        <p type="main">
          <s id="s.000269"><emph type="italics"></emph>T. 21. Incommenſurabile.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000270"><emph type="italics"></emph>T. 25. Indiuiſibilia eſſe priuationes.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000271"><emph type="italics"></emph>T. 32. Permutata proportio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000272">Ex lib. De Senſu.</s>
        </p>
        <p type="main">
          <s id="s.000273"><emph type="italics"></emph>Capite 6. Dieſis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000274"><emph type="italics"></emph>Cap. 8. Nete. </s>
          <s id="s.000275">Diapaſon. </s>
          <s id="s.000276">Diapenſe.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000277">Ex lib. De Memoria, &amp; Rem.</s>
        </p>
        <p type="main">
          <s id="s.000278"><emph type="italics"></emph>Cap. 1. Omnis triangulus habet tres, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000279"><emph type="italics"></emph>Cap. 3. Mathemata facile reminiſcibilia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000280">Ex lib. De Somnijs.</s>
        </p>
        <p type="main">
          <s id="s.000281"><emph type="italics"></emph>Cap. 2. Terra, cur nauigantibus moueri videatur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000282"><emph type="italics"></emph>Cap. 3. Cur Oculus digito dimotus res geminatas videat.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000283">Ex 1. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000284"><emph type="italics"></emph>Cap. 1. Initium Mathematicarum ab Aegyptiorum Sacerdotibus. </s>
          <s id="s.000285">Item, Automata, <lb></lb>
quæ ſolstitia. </s>
          <s id="s.000286">Diameter incommenſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000287"><emph type="italics"></emph>Summa 2. cap. 3. Pythagorei Mathematicas cæteris præferebant.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000288"><emph type="italics"></emph>T. 47. Geometria habet ſuas præcognitiones.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000289">Ex 2. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000290"><emph type="italics"></emph>T. 14. Leges apud Muſicos quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000291">Ex 3. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000292"><emph type="italics"></emph>Tex. </s>
          <s id="s.000293">Mathematicas puras carere cauſis efficiente, &amp; finali. </s>
          <s id="s.000294">Ariſtippus, vt Mathe­<lb></lb>
maticas ſugillaret. </s>
          <s id="s.000295">Tetragoniſmus est inuentio mediæ.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="16" xlink:href="009/01/016.jpg"></pb>
        <p type="main">
          <s id="s.000296"><emph type="italics"></emph>Tex. 8. Geodæſia quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000297">Ex 4. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000298"><emph type="italics"></emph>T. 4. Quæ ſint primæ, &amp; quæ ſecundæ inter Mathematicas.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000299"><emph type="italics"></emph>T. 28. Diameter, commenſurabilis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000300">Ex 5. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000301"><emph type="italics"></emph>T. 2. Exemplum cauſæ formalis ex Diapaſon. </s>
          <s id="s.000302">Quæ ſint proportiones Muſicales.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000303"><emph type="italics"></emph>T. 3. Quæ ſit Materia in Mathematicis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000304"><emph type="italics"></emph>T. 4. Quidnam ſint elementa apud Geometras.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000305"><emph type="italics"></emph>T. 12. Dieſis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000306"><emph type="italics"></emph>T. 17. Diameter incommenſurab. </s>
          <s id="s.000307">Quid potentia vnius lineæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000308"><emph type="italics"></emph>T. 34. Diameter incommenſurabilis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000309">Ex 6. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000310"><emph type="italics"></emph>T. 1. Principia, elementa, &amp; cauſæ in Mathem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000311"><emph type="italics"></emph>T. 8. Diameter. </s>
          <s id="s.000312">commenſurab.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000313"><emph type="italics"></emph>T. 20. Deſcriptiones. </s>
          <s id="s.000314">Omnis triangulus habet tres, &amp;c. </s>
          <s id="s.000315">Cur Angulus in ſemicir­<lb></lb>
culo rectus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000316"><emph type="italics"></emph>T. 22. Omnis triangulus habet tres, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000317">Ex 10. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000318"><emph type="italics"></emph>T. 4. Motum diurnum menſuram reliquorum. </s>
          <s id="s.000319">Dieſis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000320"><emph type="italics"></emph>T. 11. Similes figuræ quæ. </s>
          <s id="s.000321">Diuerſum in Math. quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000322">Ex 11. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000323"><emph type="italics"></emph>Cap. 2. Ortus punctorum, linearum, ſuperficierum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000324">Ex 12. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000325"><emph type="italics"></emph>T. 44. Peculiariſſimam Philoſophiam, Mathematicorum videlicet, Aſtronomiam <lb></lb>
pluralitatem Cœlorum docere.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000326"><emph type="italics"></emph>T. 45. Numerus orbium cœleſtium ſecundum Eudoxum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000327"><emph type="italics"></emph>T. 46. Itidem ex Eudoxo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000328"><emph type="italics"></emph>T. 47. Orbium cœleſtium numerus, &amp; fabrica ex Calippo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000329">Ex 13. Methaphyſ.</s>
        </p>
        <p type="main">
          <s id="s.000330"><emph type="italics"></emph>Cap. 3. Qua ratione Mathematici tractant de Bono.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000331">In Mechanicas Quæſtiones.</s>
        </p>
        <p type="main">
          <s id="s.000332"><emph type="italics"></emph>Cap. 1. Quæ ſit Mechanica facultas.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000333"><emph type="italics"></emph>Cap. 2. De Admirandis circuli.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000334"><emph type="italics"></emph>Quæſtio 1. De Libra. </s>
          <s id="s.000335">cur maior, exactior. </s>
          <s id="s.000336">inibi Ariſt. lapſus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000337"><emph type="italics"></emph>Quæſt. 2. Duplex Libra. </s>
          <s id="s.000338">Piccolomineus reiectus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000339"><emph type="italics"></emph>Quæſt. 3. De Vecte.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000340"><emph type="italics"></emph>Quæſt. 4. De Remo; Petri Nonÿ in Arist. correctio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000341"><emph type="italics"></emph>Quæſt. 5. De Temone Nauis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000342"><emph type="italics"></emph>Quæſt. 6. De Antenna.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000343"><emph type="italics"></emph>Quæſt. 8 De Rota.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000344"><emph type="italics"></emph>Quæſt. 9. De Trochlea, &amp; Scytali. figura antiquæ ſcytalis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000345"><emph type="italics"></emph>Quæſt. 10. De Libra vacua.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000346"><emph type="italics"></emph>Qùæſt. 11. De Curru, &amp; ſcytala.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000347"><emph type="italics"></emph>Quæſt. 13. De lugo. </s>
          <s id="s.000348">De Succula.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000349"><emph type="italics"></emph>Quæſt. 15. De Vmbelicis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000350"><emph type="italics"></emph>Quæſt. 16. De ligni oblongi, ac breuis flexura.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="17" xlink:href="009/01/017.jpg"></pb>
        <p type="main">
          <s id="s.000351"><emph type="italics"></emph>Quæſt. 17. De Cuneo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000352"><emph type="italics"></emph>Quæſt. 18. De Trochlea; error Piccolominei.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000353"><emph type="italics"></emph>Quæſt. 19. De Securi. </s>
          <s id="s.000354">Securis veteris figura, &amp; conſtructio; vnà cum affectione <lb></lb>
eius mirabili.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000355"><emph type="italics"></emph>Quæſt. 20. De Statera. </s>
          <s id="s.000356">Veteris stateræ figura restaurata.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000357"><emph type="italics"></emph>Quæſt. 21. De Dentiforcipe.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000358"><emph type="italics"></emph>Quæſt. 22. De Nucifrago.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000359"><emph type="italics"></emph>Quæſt. 23. De Motibus in Rhombo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000360"><emph type="italics"></emph>Quæſt. 24. De duobus circulis concentricis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000361"><emph type="italics"></emph>Quæſt. 25. De funibus lectulorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000362"><emph type="italics"></emph>Quæſt. 26. De ligno humeris gestato.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000363"><emph type="italics"></emph>Quæſt. 27. De ponderibus humero geſtatis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000364"><emph type="italics"></emph>Quæſt. 28. De Tollenone.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000365"><emph type="italics"></emph>Quæſt. 29. De onere à duobus phalanga geſtato.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000366"><emph type="italics"></emph>Quæſt. 30. De ſurgente à ſeſſione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000367">In libello De Mundo ad Alex.<emph type="italics"></emph>Cap. 2. Ordo Planetarum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000368"><emph type="italics"></emph>Cap. 3. De Cometis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000369"><emph type="italics"></emph>Cap. 5. De fluxu maris. </s>
          <s id="s.000370">noua de maris æſtu ſententia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000371">In libro De Admirandis audit.</s>
        </p>
        <p type="main">
          <s id="s.000372"><emph type="italics"></emph>Num. 8. De nouo orbe.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000373"><emph type="italics"></emph>Nu. </s>
          <s id="s.000374">100. De Iſtro, error Ariſt. &amp; veterum Geographorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000375">In libello De lineis inſecabilibus.</s>
        </p>
        <p type="main">
          <s id="s.000376"><emph type="italics"></emph>Primus locus. </s>
          <s id="s.000377">De commenſurabili, &amp; incommenſurabili.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000378"><emph type="italics"></emph>2. locus. </s>
          <s id="s.000379">De figuris incommenſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000380"><emph type="italics"></emph>3. locus. </s>
          <s id="s.000381">Quæ linea rationalis, quæ irrationalis. </s>
          <s id="s.000382">Binomio, Apotome.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000383"><emph type="italics"></emph>4. locus. </s>
          <s id="s.000384">De communi menſura.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000385"><emph type="italics"></emph>5. locus. </s>
          <s id="s.000386">Lineæ rectæ motus in ſemicirculum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000387"><emph type="italics"></emph>6. locus. </s>
          <s id="s.000388">Circulorum æqualium ab inuicem motus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000389"><emph type="italics"></emph>7. locus. </s>
          <s id="s.000390">Multum Mathematicis demonſtrationibus tribuitur ab Ariſtotele.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000391"><emph type="italics"></emph>8. locus. </s>
          <s id="s.000392">Si extarent indiuidua, omnes lineæ eſſent commenſ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000393"><emph type="italics"></emph>9. locus. </s>
          <s id="s.000394">Idem probat aliteŕ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000395"><emph type="italics"></emph>10. locus. </s>
          <s id="s.000396">Idem ex triangulo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000397"><emph type="italics"></emph>11. locus. </s>
          <s id="s.000398">Idem ex quadrato.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000399"><emph type="italics"></emph>12. Ex diuiſione lineæ idem confirmatur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000400"><emph type="italics"></emph>13. Idem eodem ferè modo cum præcedenti.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000401"><emph type="italics"></emph>14. A quadrato cuiuſuis lineæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000402"><emph type="italics"></emph>15. Idem probat ex ſuperficie, &amp; ex corpore.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000403"><emph type="italics"></emph>16. Idem ex contactu circuli cum linea recta.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000404">Ex lib. 9. Hiſtoriæ Animalium.</s>
        </p>
        <p type="main">
          <s id="s.000405"><emph type="italics"></emph>Cap. 39. error Ariſt. &amp; noua obſeruatio de admiranda quadam Aranearum induſtria.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000406">De Inceſſu Animal.</s>
        </p>
        <p type="main">
          <s id="s.000407"><emph type="italics"></emph>Cap. 7. qua ratione in greſſu fiat hypotenuſa. </s>
          <s id="s.000408">&amp; ea quid ſit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000409">De Motu Animal.</s>
        </p>
        <p type="main">
          <s id="s.000410"><emph type="italics"></emph>Cap. 1. in flexuris animalium eſſe centrum, &amp; circulum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000411"><emph type="italics"></emph>Cap. 3. Automata.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="18" xlink:href="009/01/018.jpg"></pb>
        <p type="head">
          <s id="s.000412">De Generatione Animal.</s>
        </p>
        <p type="main">
          <s id="s.000413"><emph type="italics"></emph>Lib. 2. cap. 1. Automata.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000414"><emph type="italics"></emph>Lib. 2. cap. 4. Omnis triangulus habet tres, &amp;c. </s>
          <s id="s.000415">Ibidem Diametrum eſſe incommen­<lb></lb>
ſurabilem coſtæ, habet cauſam, &amp; demonſtrationem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000416">In Ethicis ad Nicom.</s>
        </p>
        <p type="main">
          <s id="s.000417"><emph type="italics"></emph>Lib. 1. cap. 7. Faber, &amp; Geometra diuersè conſiderant angulum rectum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000418"><emph type="italics"></emph>Lib. 2. cap. 6. De Arithmetica proportione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000419"><emph type="italics"></emph>cap. 9. Centrum circuli reperire.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000420"><emph type="italics"></emph>Lib. 3. cap. 3. Diameter, &amp; latus incommenſurabilis: Item quid reſolutio Geome­<lb></lb>
trica: Quid deſignatio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000421"><emph type="italics"></emph>Lib. 5. cap. 3. Vnitarius numerus. </s>
          <s id="s.000422">Quid Proportionalitas. </s>
          <s id="s.000423">Eam in 4. terminis con­<lb></lb>
ſiſtere. </s>
          <s id="s.000424">Item quid Permutata proportio. </s>
          <s id="s.000425">Item quid Geometrica proportio. </s>
          <s id="s.000426">Propor­<lb></lb>
tio continuata, &amp; diſiuncta quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000427"><emph type="italics"></emph>cap. 6. Proportio Geometrica, &amp; Arithmetica.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000428"><emph type="italics"></emph>Lib. 6. cap. 5. Omnis triangulus, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000429"><emph type="italics"></emph>cap. 8. Principia Mathematica non pendere ab experientia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000430"><emph type="italics"></emph>Lib. 7. cap. 8. De principijs Mathem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000431">Ex 1. Magnorum Moralium.</s>
        </p>
        <p type="main">
          <s id="s.000432"><emph type="italics"></emph>Cap. 1. Numerus pariter par.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000433"><emph type="italics"></emph>Cap. 2. Omnis triangulus habet, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000434"><emph type="italics"></emph>Cap. 10 Omnis triangulus habet, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000435"><emph type="italics"></emph>Cap. 16. Quadratum quatuor rectis æquales habere.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000436"><emph type="italics"></emph>Cap. 30. Proportionale in quatuor terminis conſiſtit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000437">Ex 1. lib. Moralium Eudemiorum.</s>
        </p>
        <p type="main">
          <s id="s.000438"><emph type="italics"></emph>Cap. 5. Duplum inter multiplices rationes primum tenet locum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000439">Ex 1. lib. Mor. Eudemiorum.</s>
        </p>
        <p type="main">
          <s id="s.000440"><emph type="italics"></emph>Cap. Omnis triangulus habet tres, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000441"><emph type="italics"></emph>Cap. 10. Diametrum commenſ. </s>
          <s id="s.000442">eſſe. </s>
          <s id="s.000443">Circuli quadratio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000444"><emph type="italics"></emph>Cap. 12. Triangulus habet tres, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000445">Ex 7. lib. Mor. Eudemiorum.</s>
        </p>
        <p type="main">
          <s id="s.000446"><emph type="italics"></emph>Cap. 12. Diametralis oppoſitio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000447">Ex 3. lib. Politicorum.</s>
        </p>
        <p type="main">
          <s id="s.000448"><emph type="italics"></emph>Cap. 2. Modi Dorius, &amp; Phrygius apud Muſicos, quì.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000449">Ex 4. lib. Polit.</s>
        </p>
        <p type="main">
          <s id="s.000450"><emph type="italics"></emph>Cap. 3. Modus Doricus, &amp; Phrygius.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000451">Ex 5. lib. Polit.</s>
        </p>
        <p type="main">
          <s id="s.000452"><emph type="italics"></emph>Cap. 1. Aequitas Arithmetica, &amp; quæ ſecundum dignitatem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000453">Ex 8. Polit.</s>
        </p>
        <p type="main">
          <s id="s.000454"><emph type="italics"></emph>Cap. 5. Muſica nuda, &amp; cum melodia. </s>
          <s id="s.000455">Rithmus quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000456"><emph type="italics"></emph>Harmonia lydia. </s>
          <s id="s.000457">Rithmus quid ſit dicetur in Problematibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000458"><emph type="italics"></emph>Cap. 7. Harmoniæ, &amp; Rithmi, vt in præcedenti.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000459">Ex Problematibus.</s>
        </p>
        <p type="main">
          <s id="s.000460"><emph type="italics"></emph>Sectione 1. num. </s>
          <s id="s.000461">3. De ortu ſyderum inerrantium: Succulæ, Hypades, Atlantides, <lb></lb>
Virgiliæ, Pleiades. </s>
          <s id="s.000462">num. </s>
          <s id="s.000463">17. De occaſu affixarum ſtellarum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000464"><emph type="italics"></emph>Sectione 15. num. </s>
          <s id="s.000465">1. Diametri ethymon.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000466"><emph type="italics"></emph>num. </s>
          <s id="s.000467">2. Iterum Diametri ethymologia.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="19" xlink:href="009/01/019.jpg"></pb>
        <p type="main">
          <s id="s.000468"><emph type="italics"></emph>num. </s>
          <s id="s.000469">3. Denarius numerus cur perfectus. </s>
          <s id="s.000470">eius dignitates. </s>
          <s id="s.000471">Petri Apponenſis deceptio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000472"><emph type="italics"></emph>4. De inæquali ſolis vmbrarum incremento.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000473"><emph type="italics"></emph>5. Cur Solis illuminationes ſemper rotundæ, quamuis per anguloſa foramina ingre­<lb></lb>
diantur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000474"><emph type="italics"></emph>6. Cur Luna ſemiplena videtur linea recta terminari? </s>
          <s id="s.000475">vbi de illuminatione Lunæ, <lb></lb>
quæ experientia docetur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000476"><emph type="italics"></emph>7. Cur Sol, &amp; Luna videantur plana?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000477"><emph type="italics"></emph>8. De vmbris Solis orientis, occidentis, meridiantis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000478"><emph type="italics"></emph>9. Cur Lunæ, quàm Solis minores vmbræ?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000479"><emph type="italics"></emph>10. Cur in defectu Solis etiam illuminationes ipſius defectiuæ ſunt? </s>
          <s id="s.000480">modus commodè <lb></lb>
videndi eclypſim Solis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000481"><emph type="italics"></emph>Sect. 16. nu. </s>
          <s id="s.000482">1. Cur baſes bullarum in aquis ſunt albæ?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000483"><emph type="italics"></emph>3. Opplumbati tali.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000484"><emph type="italics"></emph>4. De reſultu cadentium in terram.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000485"><emph type="italics"></emph>5. Cur conus, &amp; cylindrus diuersè moueantur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000486"><emph type="italics"></emph>6. De voluminum ſectione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000487"><emph type="italics"></emph>12. Idem cum præcedenti 3.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000488"><emph type="italics"></emph>13. Idem cum 4 ſuperiori. </s>
          <s id="s.000489">reflexio radiorŭm pulchrè comparatur corporŭm reſultationi.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000490">Ex ſectione 19. De Muſica.</s>
        </p>
        <p type="main">
          <s id="s.000491"><emph type="italics"></emph>num. </s>
          <s id="s.000492">2. Lineæ duplæ quadratum quadruplum. </s>
          <s id="s.000493">Hoc loco ſequentium probl. </s>
          <s id="s.000494">cauſa, <lb></lb>
præmittitur totius Muſicæ ortus breuis tractatio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000495"><emph type="italics"></emph>3. Vox tam in hypate, quam in nete cantando rumpitur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000496"><emph type="italics"></emph>4. Cur facilius hypate, quam nete canitur?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000497"><emph type="italics"></emph>5. Cur ſuauius notam cantilenam audimus?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000498"><emph type="italics"></emph>7. Cur veteres hypatem omittebant.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000499"><emph type="italics"></emph>8. Cur grauis ſonum potest acutæ?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000500"><emph type="italics"></emph>9. Cur cantus ad tibiam vnam, aut lyram ſuauior?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000501"><emph type="italics"></emph>10. Teretizare, quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000502"><emph type="italics"></emph>11. Vox deſinens acutior fit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000503"><emph type="italics"></emph>12. Grauior è fidibus cantilenam ſuſcipit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000504"><emph type="italics"></emph>13. In Diapaſon graue eſt acuti Antiphonum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000505"><emph type="italics"></emph>14. Cur Diapaſon vnica vox videtur. </s>
          <s id="s.000506">Punicum quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000507"><emph type="italics"></emph>15. Leges Muſicæ, quæ. </s>
          <s id="s.000508">Genera, Diatonicum, Chromaticum, Encharmonium. <lb></lb>
</s>
          <s id="s.000509">Tetrachorda quæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000510"><emph type="italics"></emph>16. Antiphonum ſuauius est ſymphono, cur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000511"><emph type="italics"></emph>17. Cur ſola Diapaſon canitur. </s>
          <s id="s.000512">Magadis quid. </s>
          <s id="s.000513">Magadare.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000514"><emph type="italics"></emph>18. De Antiphonis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000515"><emph type="italics"></emph>19. Cur Diapente, &amp; Diabeſſacon non canunt in Antiphonis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000516"><emph type="italics"></emph>20. Meſe ſola diſſonante, totum deſſonat pſalterium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000517"><emph type="italics"></emph>21. Vocum grauium errores manifestiores, cur?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000518"><emph type="italics"></emph>23. Cur nete duplo acutior, quam hypate?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000519"><emph type="italics"></emph>24. Nete interpellata, hypate reſonare videtur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000520"><emph type="italics"></emph>25. Cur Meſe ſic appellata eſt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000521"><emph type="italics"></emph>27. Cur ſola audibilia mores obtinent. </s>
          <s id="s.000522">Rithmus quid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000523"><emph type="italics"></emph>28. Cur cantilenæ quædam leges decebantur?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000524"><emph type="italics"></emph>30. De Harmonijs, ſeu Modis, ſeu Tonis priſcorum.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="20" xlink:href="009/01/020.jpg"></pb>
        <p type="main">
          <s id="s.000525"><emph type="italics"></emph>31. Vetustiores fuiſſe magis Melopæos.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000526"><emph type="italics"></emph>32. De ipſius Diapaſon ethymo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000527"><emph type="italics"></emph>33. Cur aptè de acuto in graue, non è contra canitur?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000528"><emph type="italics"></emph>34. Cur biſdiapente, aut biſdiateſſaron conſonantia non eſt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000529"><emph type="italics"></emph>35. Cur diapaſon omnium pulcherrima eſt conſonantia?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000530"><emph type="italics"></emph>36. Meſe ſola diſſonante, tota perit harmonia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000531"><emph type="italics"></emph>37. Cur difficilius acutum canere, quam graue?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000532"><emph type="italics"></emph>38. Cur Rythmo, &amp; harmonij omnes gaudent?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000533"><emph type="italics"></emph>39. Cur ſuauius eſt ſymphonum vniſono?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000534"><emph type="italics"></emph>40. Cur ſolam Diapaſon magadari ſolent?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000535"><emph type="italics"></emph>41. Idem cum 5.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000536"><emph type="italics"></emph>42. Idem cum 34.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000537"><emph type="italics"></emph>43. Idem cum 24. Iugum in lyra veteri quodnam fuerit, vna cum figura vete­<lb></lb>
ris lyræ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000538"><emph type="italics"></emph>44. Cur ſuauius ad tibiam, quam ad lyram cantatur?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000539"><emph type="italics"></emph>45. Idem cum 25. ſuperiori.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000540"><emph type="italics"></emph>46. Idem cum 22.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000541"><emph type="italics"></emph>47. Idem cum 26.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000542"><emph type="italics"></emph>48. Idem cum 7. quid Grauidenſum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000543"><emph type="italics"></emph>49. Idem cum 30. In choris tragœdiarum, nec ſubdorius, nec ſubphrygius modus <lb></lb>
erat in vſu.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000544"><emph type="italics"></emph>50. Cur grauior Melodia eſt etiam mollior?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000545"><emph type="italics"></emph>51. Dolia duo æqualia, quorum alterum plenum ſit, alterum dimidium, Diapaſon <lb></lb>
reſonant.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000546">Ex ſectione 23.</s>
        </p>
        <p type="main">
          <s id="s.000547"><emph type="italics"></emph>De immerſione Nauigij.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000548">Ex ſectione 30.</s>
        </p>
        <p type="main">
          <s id="s.000549"><emph type="italics"></emph>6. Omnis triangulus habet tres æquales, &amp;c.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000550">Ex ſectione 31.</s>
        </p>
        <p type="main">
          <s id="s.000551"><emph type="italics"></emph>7. Cur o culos, <expan abbr="abſq;">abſque</expan> vlla vi, ab inuicem diſſociari nequimus?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000552"><emph type="italics"></emph>Cur duobus oculis res vna tantum videatur. </s>
          <s id="s.000553">Cur aliquando rei viſæ gemina­<lb></lb>
tio accidat.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000554"><emph type="italics"></emph>11. Cur diſtractis oculis res vna duæ apparent?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000555"><emph type="italics"></emph>17. Oculo in latera contorto, cur non fit geminatio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000556"><emph type="italics"></emph>21. Cur ſolam rectitudinem vnico oculo inſpiciamus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000557">Auctarium De Oculi Pupilla.</s>
        </p>
        <p type="main">
          <s id="s.000558"><emph type="italics"></emph>Oculi fabrica præmittitur, colores oculi vbi ſint: vnde qui noctu vident.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000559"><emph type="italics"></emph>Primo. </s>
          <s id="s.000560">De pupillæ voce.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000561"><emph type="italics"></emph>2. Cur in oculo appareat.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000562"><emph type="italics"></emph>3. Cur non in tota cornea.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000563"><emph type="italics"></emph>4. Pupillæ definitio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000564"><emph type="italics"></emph>5. Cur nigra in omnibus hominibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000565"><emph type="italics"></emph>6. Cur in Sole euaneſcat.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000566"><emph type="italics"></emph>7. Quantitas ipſius num videatur?<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000567"><emph type="italics"></emph>8. Cur modo maior, modo minor videatur, &amp; cuiſdam lepida deceptio.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="21" xlink:href="009/01/021.jpg"></pb>
        <p type="main">
          <s id="s.000568">Additamentum de natura Mathematicarum diſciplinarum.</s>
        </p>
        <p type="main">
          <s id="s.000569"><emph type="italics"></emph>Primo. </s>
          <s id="s.000570">De ſubiecto Mathem. ſeu de materia intelligibili: vbi oſtenduntur definitio­<lb></lb>
nes Mathematicæ eſſe perfectiſſimæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000571">2. <emph type="italics"></emph>Demonſtrationes Mathematicas eſſe perfectiſſimas.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000572">3. <emph type="italics"></emph>Obiectiones: <expan abbr="atq;">atque</expan> etiam calumniæ diluuntur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000573">4. <emph type="italics"></emph>De præstantia cognitionis, quam Geometria, &amp; Arithmetica gignunt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000574">5. <emph type="italics"></emph>De 4. Mathematicis medijs: Aſtronomia, Perſpectiua, Muſica, Mechanica.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000575">6. <emph type="italics"></emph>Appendix de reſolutione omnium demonstrationum primi Euclidis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000576">7. <emph type="italics"></emph>Clarorum Mathematicorum Chronicon.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000577"><emph type="bold"></emph>Finis Primi Indicis.<emph.end type="bold"></emph.end></s>
        </p>
      </section>
      <pb pagenum="22" xlink:href="009/01/022.jpg"></pb>
      <section>
        <p type="head">
          <s id="s.000578"><emph type="italics"></emph>ALTER INDEX<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000579"><emph type="italics"></emph>Quo loca Aristotelis Geometrica, in hoc Opere explicata, <lb></lb>
ad Euclidem, ſecundum propoſitionum ordinem refe­<lb></lb>
runtur; vt Mathematicarum Profeſſores habeant, <lb></lb>
vnde ſuas prælectiones aliquando valeant locupletare.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000580"><emph type="italics"></emph>In Primo Elem. Euclidis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000581">Ad verbum ipſum <emph type="italics"></emph>(Elementum Euclidis)<emph.end type="italics"></emph.end> vide infra tex. 4. quinti <lb></lb>
Methaph.</s>
        </p>
        <p type="main">
          <s id="s.000582">Ad principia primi elementorum, vide infra tex. 5. pri. Poſter.</s>
        </p>
        <p type="main">
          <s id="s.000583">Ad definitionem 10. pri. pro angulo recto, vide 30. quæſt. </s>
          <s id="s.000584">Mecha­<lb></lb>
nic. &amp; cap. 7. lib. 1. Eth.</s>
        </p>
        <p type="main">
          <s id="s.000585">Ad axioma 10. quamuis Ariſtoteles nihil hac de re dicat; ſcias tamen velim <lb></lb>
hoc vno axiomate quęſtionem <expan abbr="quãdam">quandam</expan> inter Philoſophos valdè difficilem, <lb></lb>
facile diſſolui. </s>
          <s id="s.000586">ea eſt, vtrum marmor, aut adamas, aliudue quidpiam infle­<lb></lb>
xibile ſucceſſiuè findi, &amp; aperiri poſſit. </s>
          <s id="s.000587">qui enim aiunt, ſic refelluntur, quia <lb></lb>
nimirum ſequeretur, duas rectas lineas habere ſegmentum commune: in­<lb></lb>
telligantur enim duæ lineæ, vna in vna ſuperficie, altera vero in altera, quæ <lb></lb>
antequam incipiat apertio, congruant; Quoniam igitur tota illa apertio <lb></lb>
non fit in inſtanti, ſed ſucceſſiuè, facta iam aliqua apertionis parte conſi­<lb></lb>
derentur prædictæ lineæ, erit igitur earum pars aliqua ab inuicem ſepara­<lb></lb>
ta, altera verò adhuc alteri congruens, ergo ſequetur, duas lineas habere <lb></lb>
ſegmentum commune, quod eſt impoſſibile, quia contra 10. axioma.</s>
        </p>
        <p type="main">
          <s id="s.000588">Ad Calcem axiomatum primi accommodetur tex. 1. primi Poſter.</s>
        </p>
        <p type="main">
          <s id="s.000589">Ad primam primi, poſt ipſius explicationem, commodè declarari poteſt, cur <lb></lb>
Ariſt. Demonſtrationes Geometricas appellet Deſcriptiones, &amp; Deſigna­<lb></lb>
tiones, vide cap. de Priori, &amp; cap. 24. ſecti primi, libri primi Priorum, &amp; <lb></lb>
tex. 4. quinti Methaph. &amp; tex. 20. ſexti Methaph. &amp; cap. 3. lib. 3. Ethic. <lb></lb>
</s>
          <s id="s.000590">Item ad primam primi, vide tex. 7. ſecundi Poſter. loco 2.</s>
        </p>
        <p type="main">
          <s id="s.000591">Ad 5. primi, vide cap. 24. ſecti 1 lib. 1. Priorum.</s>
        </p>
        <p type="main">
          <s id="s.000592">Ad 21. primi, vide tex. 20. primi Poſter. loco 2.</s>
        </p>
        <p type="main">
          <s id="s.000593">Ad 22. primi, vide locum 10. de lineis inſecabilibus.</s>
        </p>
        <p type="main">
          <s id="s.000594">Ad 28. primi, vide cap. 21. &amp; cap. 22. ſecundi <expan abbr="Priorũ">Priorum</expan>, &amp; tex. 13. primi Poſter.</s>
        </p>
        <p type="main">
          <s id="s.000595">Ad 32. primi, vide cap. 1. ſecti 3. lib. 1. Prior. &amp; cap. 26. ſecundi <expan abbr="Priorũ">Priorum</expan>, &amp; tex. 2. <lb></lb>
primi Poſter. loco 4. &amp; tex. 23. primi Poſter. vbi ait hanc eſſe potiſſimam <lb></lb>
<expan abbr="demõſtrationem">demonſtrationem</expan>. </s>
          <s id="s.000596">&amp; tex. 37. primi Poſter. &amp; tex. 39. primi Poſter. </s>
          <s id="s.000597">Ibidem <lb></lb>
loco 4. &amp; tex. 43. primi Poſter. &amp; tex. 2. ſecundi Poſter. bis. </s>
          <s id="s.000598">&amp; tex. 89. ſe­<lb></lb>
cundi Phyſ. &amp; tex. 15. octaui Phyſ. &amp; tex. 119. primi de Cœlo. </s>
          <s id="s.000599">&amp; tex. 25. <lb></lb>
ſecundi de Cœlo. </s>
          <s id="s.000600">tex 11. primi de Anima. &amp; cap. 1. de mem. </s>
          <s id="s.000601">&amp; reminiſc. <lb></lb>
</s>
          <s id="s.000602">&amp; tex. 35. quinti Methaphyſ. &amp; tex. 20. ſexti Methaphyſ. &amp; tex. 22. ſexti <lb></lb>
Methaphyſ. &amp; cap. 4. lib. 2. de Generat. animal. </s>
          <s id="s.000603">&amp; cap. 5. lib. 6. Ethic. &amp; <lb></lb>
cap. 2. Magnorum Moral. &amp; cap. 10. Mag. Moral. &amp; cap. 16. Mag. Moral. <lb></lb>
&amp; cap. 7. ſecundi Eudem. &amp; cap. 12. ſecundi Eudem. &amp; problema 6. ſectio­


<pb pagenum="23" xlink:href="009/01/023.jpg"></pb>nis 30. tot Ariſtotelis loca illuſtrat vnica hæc Euclidis Demonſtratio.</s>
        </p>
        <p type="main">
          <s id="s.000604">Ad ſcholion præcedentis 32. primi, vide tex. 39. primi Poſter. loco 3. Item <lb></lb>
tex. 25. ſecundi Poſter. loco vlt.</s>
        </p>
        <p type="main">
          <s id="s.000605">Ad 45. primi, vide locum 9. de lineis inſecabilibus.</s>
        </p>
        <p type="main">
          <s id="s.000606">Ad 46. primi, vide locum 11. de lineis inſecabilibus.</s>
        </p>
        <p type="main">
          <s id="s.000607">Ad 47. primi, vide locum 11. de lineis inſecab. </s>
          <s id="s.000608">Item locum 14. de ijſdem.</s>
        </p>
        <p type="head">
          <s id="s.000609"><emph type="italics"></emph>In ſecundo Elem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000610">Ad 2. definitionem 2. Gnomonis, vide cap. de Motu in Poſtprædicam. </s>
          <s id="s.000611">Qua­<lb></lb>
dratum augetur Gnomone circumpoſito.</s>
        </p>
        <p type="main">
          <s id="s.000612">Ad 14. propoſ. </s>
          <s id="s.000613">2. opportunum eſt Auditores de Quadratura circuli erudire, <lb></lb>
vide igitur cap. de relatione in prædicam. </s>
          <s id="s.000614">&amp; cap. 31. ſecundi Priorum, &amp; <lb></lb>
tex. 23. primi Poſter. &amp; finem 1. cap. primi Elenchorum. </s>
          <s id="s.000615">lege primam Ar­<lb></lb>
chimedis de dimenſione circuli.</s>
        </p>
        <p type="head">
          <s id="s.000616"><emph type="italics"></emph>In tertio Elem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000617">Ad primam 3. vide cap. 9. lib. 2. Ethycorum.</s>
        </p>
        <p type="main">
          <s id="s.000618">Ad 2. tertij, vide tex. 13. lib. 1. de Anima. &amp; locum 16. de lineis inſecab.</s>
        </p>
        <p type="main">
          <s id="s.000619">Ad 31. tertij, vide tex. 11. ſecundi Poſter. &amp; tex. 20. ſexti Methaph. loco 2.</s>
        </p>
        <p type="head">
          <s id="s.000620"><emph type="italics"></emph>In quarto.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000621">Ad commentarium P. Clauij extremum lib. 4. elementorum. </s>
          <s id="s.000622">lege tex. 66. <lb></lb>
tertij de Cœlo.</s>
        </p>
        <p type="head">
          <s id="s.000623"><emph type="italics"></emph>In quinto.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000624">Ad 4. definitionem 5. vide cap. 3. lib. 2. Ethyc.</s>
        </p>
        <p type="main">
          <s id="s.000625">Ad 9. definitionem 5. vide cap. 3. lib. 5. Ethyc. loco 4. &amp; cap. 31. primi Ma­<lb></lb>
gnorum Moralium.</s>
        </p>
        <p type="main">
          <s id="s.000626">Ad 10. definitionem 5. vide tex. 29. primi Poſter. loco 2.</s>
        </p>
        <p type="main">
          <s id="s.000627">Ad 12. definitionem 5. vide tex. 13. primi Poſter. loco 3. &amp; tex. 25. ſecundi <lb></lb>
Poſter. &amp; tex. 32. tertij de Anima. &amp; cap. 3. lib. 5. Ethyc. loco 4.</s>
        </p>
        <p type="main">
          <s id="s.000628">Ad 16. propoſ. </s>
          <s id="s.000629">5. vide tex. 25. ſecundi Poſter. loco 2. ex hac Euclidis demon­<lb></lb>
ſtratione patet, vitioſam eſſe illam Auerrois argumentationem, 8. Phyſ. <lb></lb>
comm. 15. ſcilicet.</s>
        </p>
        <p type="main">
          <s id="s.000630">Vt ſe habet voluntas antiqua ad antiquum effectum, <lb></lb>
Ita ſe habet etiam voluntas noua ad effectum nouum: <lb></lb>
Ergo permutando, ita ſe habebit voluntas antiqua ad effectum nouum. <lb></lb>
</s>
          <s id="s.000631">Quemadmodum voluntas noua ad effectum antiquum.</s>
        </p>
        <p type="main">
          <s id="s.000632">Non enim in permutando confert antecedentem ad antecedentem, &amp; con­<lb></lb>
ſequentem ad conſequentem, vt par erat, ſed confert antecedentem ad <lb></lb>
conſequentem, quod non licet.</s>
        </p>
        <p type="head">
          <s id="s.000633"><emph type="italics"></emph>In ſexto.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000634">Ad 2. propoſit. </s>
          <s id="s.000635">6. vide cap. 2. lib. 8. Topicorum loco 41.</s>
        </p>
        <p type="main">
          <s id="s.000636">Ad 13. ſexti, vide tex. 12. ſecundi de Anima, &amp; tex. 3. tertij Methaphyſ.<emph type="italics"></emph>In ſeptimo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000637">Ad primam definitionem 7. vide tex. 5. primi Poſter.</s>
        </p>
        <p type="main">
          <s id="s.000638">Ad 8. definitionem 7. vide cap. 1. lib. 1. Magnorum Moral.</s>
        </p>
        <p type="head">
          <s id="s.000639"><emph type="italics"></emph>In octauo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000640">Ad 4. propoſ. </s>
          <s id="s.000641">9. vide tex. 20. primi Poſter. loco 2.</s>
        </p>
        <p type="main">
          <s id="s.000642">Ad 8. propoſ. </s>
          <s id="s.000643">9. vide problem. </s>
          <s id="s.000644">3. ſectionis 15. loco 4.</s>
        </p>
        <pb pagenum="24" xlink:href="009/01/024.jpg"></pb>
        <p type="head">
          <s id="s.000645"><emph type="italics"></emph>In decimo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000646">Ad primam definitionem 10. vide cap. 23. ſecti 1. primi Priorum. </s>
          <s id="s.000647">&amp; tex. 48. <lb></lb>
primi de Cœlo.</s>
        </p>
        <p type="main">
          <s id="s.000648">Ad 118. decimi, vide cap. 23. ſecti 1. libri 1. Priorum. </s>
          <s id="s.000649">&amp; ſecto 2. cap. 23. li­<lb></lb>
bri 1. Priorum. </s>
          <s id="s.000650">&amp; cap. 22. lib. 2. Priorum. </s>
          <s id="s.000651">&amp; tex. 5. primi Poſter. &amp; tex. 44. <lb></lb>
primi Poſter. &amp; cap. 15. primi Poſter. &amp; tex. 119. primi de Cœlo. </s>
          <s id="s.000652">&amp; tex. <lb></lb>
120. quarti Phyſ. &amp; tex. 21. tertij de Anima. &amp; cap. 1. primi Methaphyſ. <lb></lb>
&amp; tex. 28. quarti Met. &amp; tex. 34. quinti Met. &amp; tex. 8. ſexti Met. &amp; cap. 4. <lb></lb>
lib. 2. de Generat. animal. </s>
          <s id="s.000653">&amp; lib. 3. cap. 3. Ethyc. &amp; cap. 10. ſecundi Eu­<lb></lb>
dem. </s>
          <s id="s.000654">tot Ariſt. loca ab hac vna Euclidis Demonſtratione illuſtrantur.</s>
        </p>
        <p type="head">
          <s id="s.000655"><emph type="italics"></emph>In decimotertio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000656">Ad primam propoſ. </s>
          <s id="s.000657">13. ſecundum editionem Commandini, aut Zamberti. <lb></lb>
</s>
          <s id="s.000658">vide initio Priorum, in verbum (Reſolutio)</s>
        </p>
        <p type="main">
          <s id="s.000659">Atque hæc ſunt, quæ ex Elementorum opere Ariſtoteles paſſim vſurpauit, <lb></lb>
quæque nos infra explicabimus.</s>
        </p>
        <p type="head">
          <s id="s.000660"><emph type="bold"></emph>Finis Secundi Indicis.<emph.end type="bold"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000661">Quæ verò ad alias Mathematicas, Muſicam ſcilicet, Perſpe­<lb></lb>
ctiuam, Mechanicam, &amp; Aſtronomiam pertinens, facilè <lb></lb>
poterunt ex primo Indice ad vnamquamque earum ſeor­<lb></lb>
ſum cum libuerit, ſecerni.</s>
        </p>
      </section>
      <pb pagenum="25" xlink:href="009/01/025.jpg"></pb>
      <section>
        <p type="head">
          <s id="s.000662"><emph type="italics"></emph>TERTIVS INDEX ALPHABETICVS,<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.000663"><emph type="italics"></emph>cuius numeri reſpondent numeris marginalibus Operis.<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table2"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table2"></table.target>
          <row>
            <cell><emph type="italics"></emph>A<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Abductio quid. eius inuentor. numero 16. marginali.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Acuti denſum quid.<emph.end type="italics"></emph.end></cell>
            <cell>399</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Aequalitas mathematica, quæ.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Aeſtus maris natura.<emph.end type="italics"></emph.end></cell>
            <cell>272</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Aequitas arithmetica, &amp; æquitas ſecundum dignitatem.<emph.end type="italics"></emph.end></cell>
            <cell>330</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Agathirſi populi.<emph.end type="italics"></emph.end></cell>
            <cell>382</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Angulus quid. vt nominari debeat 10. angulum in ſemicirculo eſſe rectum oſtendi   per cauſam materialem.<emph.end type="italics"></emph.end></cell>
            <cell>71</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Angulus rectus variè conſideratur à Geometra, &amp; à Fabro.<emph.end type="italics"></emph.end></cell>
            <cell>301</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Antiphontis quadratura circuli.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Antennæ nauis problema.<emph.end type="italics"></emph.end></cell>
            <cell>248</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Antiphonæ voces.<emph.end type="italics"></emph.end> 358. 363. 370. 371.</cell>
            <cell>373</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Apum mirabilis induſtria.<emph.end type="italics"></emph.end></cell>
            <cell>120</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Apotome linea, quæ.<emph.end type="italics"></emph.end></cell>
            <cell>279</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Aquæ ſuperficiem eſſe ſphæricam ratione mathematica.<emph.end type="italics"></emph.end></cell>
            <cell>107</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Arithmetica proportio.<emph.end type="italics"></emph.end></cell>
            <cell>302</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Aranei industria patefacta, qua ad res inacaſſas tranſeat.<emph.end type="italics"></emph.end></cell>
            <cell>293</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>filum emittit ex ſeceſſu contra Ariſtotelem pro Democrito.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>ibidem.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Astronomiæ principia duo, Apparentia, &amp; Obſeruatio.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Automata, quæ.<emph.end type="italics"></emph.end> 199. 298.</cell>
            <cell><emph type="italics"></emph>a. b.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>B<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Baſis ballarum in aqua, cur ſit alba, &amp; cur non faciat vmbram.<emph.end type="italics"></emph.end></cell>
            <cell>351</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Binomium linea, quæ.<emph.end type="italics"></emph.end></cell>
            <cell>279</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Braſilienſes, qua ratione numerare ſoliti.<emph.end type="italics"></emph.end></cell>
            <cell>340</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Bryſonis quadratura circuli.<emph.end type="italics"></emph.end></cell>
            <cell>35</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>C<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Calippi opinio de numero Cœlorum.<emph.end type="italics"></emph.end></cell>
            <cell>236</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cantilenam notam ſuauius, quam ignotam audimus.<emph.end type="italics"></emph.end></cell>
            <cell>362</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Centrum circuli reperire. 303. Centrum mundi mathematicè oſtenditur. 123. Cen-trum grauitatis, &amp; molis.<emph.end type="italics"></emph.end> 38.</cell>
            <cell>112</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Chordarum veterum nomina.<emph.end type="italics"></emph.end></cell>
            <cell>359</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Circuli quadratura quid. an poſſibilis. num. 1. Circulorum concentricorum maiores   velocius moueri. 108. Circuli admiranda. 239. De circulorum concentricorum   volutatione.<emph.end type="italics"></emph.end></cell>
            <cell>263</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Coalternæ lineæ, quæ.<emph.end type="italics"></emph.end> 12. 14.</cell>
            <cell>44</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cœlorum ordinem petendum ex Aſtronomis 109. item numerum<emph.end type="italics"></emph.end></cell>
            <cell>233</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Colores in muſica 78. Colores oculorum vnde.<emph.end type="italics"></emph.end></cell>
            <cell>408</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cometarum tractatiuncula ex recentioribus, qui eas in Cœlo collocant. 136. eſſe ſu-pra aerem longiſſimo ſaltem interuallo oſtenditur mathematicè. 129. in additione.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Conſonantia, quid. 64. quibus numeris conſonantiæ conſtent.<emph.end type="italics"></emph.end></cell>
            <cell>210</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Conus, &amp; cylindrus, cur variè mouentur.<emph.end type="italics"></emph.end></cell>
            <cell>355</cell>
          </row>
          <pb pagenum="26" xlink:href="009/01/026.jpg"></pb>
          <row>
            <cell><emph type="italics"></emph>Cubus numerus. duo cubi cubus, quid ſignificet.<emph.end type="italics"></emph.end></cell>
            <cell>33</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Curru problema.<emph.end type="italics"></emph.end></cell>
            <cell>252</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cunei problema.<emph.end type="italics"></emph.end></cell>
            <cell>256</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cylindri, &amp; coni motus comparatio problematica.<emph.end type="italics"></emph.end></cell>
            <cell>335</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>D<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Definitiones mathematicæ eſſe eſſentiales, &amp; perfeſſimas. cap. 1. de nat. Math.   definitionum vſus in Mathematicis.<emph.end type="italics"></emph.end></cell>
            <cell>81</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Deſcriptio, &amp; deſcribere, quid.<emph.end type="italics"></emph.end> 2. 6. 7.</cell>
            <cell>205</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Deſignatio pro demonſtratione mathematica.<emph.end type="italics"></emph.end></cell>
            <cell>305</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Demonſtrationis perfectæ exemplum. 36. demonſtrationum mathematicarum præ-ſtantia. cap. 4. de nat. Mathem.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Dentiforcipis problema.<emph.end type="italics"></emph.end></cell>
            <cell>260</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Denarij numeri perfectio. 339. cur <expan abbr="vſq;">vſque</expan> ad denariŭ omnes <expan abbr="gẽtes">gentes</expan> <expan abbr="numerẽt">numerent</expan>.<emph.end type="italics"></emph.end></cell>
            <cell>339. 8. <emph type="italics"></emph>&amp;c.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diameter incommenſurabilis costæ. 5. diametri etymon.<emph.end type="italics"></emph.end></cell>
            <cell>337</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diapaſon quid. 90. 350. omnium conſonantiarum pulcherrima.<emph.end type="italics"></emph.end></cell>
            <cell>388</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diapaſon diapente.<emph.end type="italics"></emph.end></cell>
            <cell>359</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diapente conſonantia, quæ.<emph.end type="italics"></emph.end></cell>
            <cell>359</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diateſſaron conſonantia, quæ.<emph.end type="italics"></emph.end></cell>
            <cell>359</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diſd apaſon conſonantia, quæ.<emph.end type="italics"></emph.end></cell>
            <cell>359</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Dieſis, quid.<emph.end type="italics"></emph.end> 53.</cell>
            <cell>226</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Dolia duo, quomodo aliquando Diapaſon reſonent.<emph.end type="italics"></emph.end></cell>
            <cell>402</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Duplum inter multiplicia primum eſt.<emph.end type="italics"></emph.end></cell>
            <cell>322</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>E<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Elementa mundi non componi ex figuris geometricis.<emph.end type="italics"></emph.end></cell>
            <cell>120</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Elementa geometrica, quæ.<emph.end type="italics"></emph.end> 82.</cell>
            <cell>213</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Eudoxi opinio de numero Cœlorum.<emph.end type="italics"></emph.end></cell>
            <cell>234</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Exempla mathematicorum, qualia. 11. non eſſe falſa.<emph.end type="italics"></emph.end></cell>
            <cell>43</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Exemplorum veritas, &amp; conformitas, quatenus requirantur.<emph.end type="italics"></emph.end></cell>
            <cell>36</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>F<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Figuram omnem planam habere ſuos angulos externos <expan abbr="quotcŭmq;">quotcŭmque</expan> æquales quatuor   rectis angulis, quæ eſt mira proprietas.<emph.end type="italics"></emph.end></cell>
            <cell>59</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Figuræ ſimiles, quæ.<emph.end type="italics"></emph.end></cell>
            <cell>70</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Figurarum planarum ordo 88. quæ nam totum locum repleant.<emph.end type="italics"></emph.end></cell>
            <cell>96</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Figurarum ſolidarum, quænam totum locum repleant: vbi Ariſt. &amp; omnium expoſi-torum ratum aperitur.<emph.end type="italics"></emph.end></cell>
            <cell>121</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Figuratio lucis.<emph.end type="italics"></emph.end></cell>
            <cell>345</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Figurationes pro demonſtrationibus Mathem.<emph.end type="italics"></emph.end></cell>
            <cell>194</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Filum Araneorum, ex qua parte corporis exeat, &amp; ex qua materia.<emph.end type="italics"></emph.end></cell>
            <cell>293</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Fluxus, ac refluxus maris.<emph.end type="italics"></emph.end></cell>
            <cell>272</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Funium lectorum problema.<emph.end type="italics"></emph.end></cell>
            <cell>264</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>G<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Galaxia quid. 131. Ariſtoteles defenſus.<emph.end type="italics"></emph.end> 132.</cell>
            <cell>140</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Galibei recens obſeruatio.<emph.end type="italics"></emph.end></cell>
            <cell>141</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Generatria Muſicæ veteris. 78. Fusè explicantur.<emph.end type="italics"></emph.end></cell>
            <cell>371</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Geodæſia.<emph.end type="italics"></emph.end></cell>
            <cell>207</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Geographiæ veteris plura errata,<emph.end type="italics"></emph.end> 145. 146. 147. 148.</cell>
            <cell>149</cell>
          </row>
          <pb pagenum="27" xlink:href="009/01/027.jpg"></pb>
          <row>
            <cell><emph type="italics"></emph>Gnomon, quid. 3. &amp;<emph.end type="italics"></emph.end></cell>
            <cell>331</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Gnomones numeri.<emph.end type="italics"></emph.end></cell>
            <cell>93</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Graue qua ratione ad centrum mundi deſcenderet, eiqué aptaretur.<emph.end type="italics"></emph.end></cell>
            <cell>112</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Grauidenſum, quid.<emph.end type="italics"></emph.end></cell>
            <cell>399</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>H<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Halonis demonſtratio.<emph.end type="italics"></emph.end></cell>
            <cell>161</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hippocratis chij quadratura circuli. 17. <expan abbr="eiuſdẽ">eiuſdem</expan> quadratura lunulæ optima.<emph.end type="italics"></emph.end></cell>
            <cell>17</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hyades, Atlantides, &amp; Succulæ.<emph.end type="italics"></emph.end></cell>
            <cell>335</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hypate, quid.<emph.end type="italics"></emph.end></cell>
            <cell>360</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hypotenuſa in inceſſu animalium.<emph.end type="italics"></emph.end></cell>
            <cell>294</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>I<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Illuminationes Solis deficientis per foramina tranſeuntes, eur ſint defectiuæ.<emph.end type="italics"></emph.end></cell>
            <cell>350</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>modus videndi eclypſim facilis, ac iucundus.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>ibidem.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Inceſſus animalium lineis explicatur.<emph.end type="italics"></emph.end></cell>
            <cell>294. <emph type="italics"></emph>&amp; <expan abbr="ſeqq.">ſeqque</expan><emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Incommenſurabilia, quæ, &amp; eorum inuentores.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Indiuiſibilia mathematica eſſe priuationes. 189. oriri ex diuiſione. 231. eorum duo   genera.<emph.end type="italics"></emph.end></cell>
            <cell>276</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Infinito, qua ratione vtantur Mathematici.<emph.end type="italics"></emph.end> 94.</cell>
            <cell>96</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iridis demonſtratio ſecundum Ariſt. 163. &amp; ſequentibus. Item noua de Iride tra-ctatio. ibidem in additione.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iugum in lyra quid, &amp; eius figura.<emph.end type="italics"></emph.end></cell>
            <cell>396</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>L<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Leges muſicales.<emph.end type="italics"></emph.end></cell>
            <cell>204</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Libra maior, cur exactior. initio Mechanicarum quæſt.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Linea, quæ terminus est illuminationis in Luna, cur modo recta, modo curua vi-deatur.<emph.end type="italics"></emph.end></cell>
            <cell>346</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Lineæ rationales, &amp; irrationales, &amp;c.<emph.end type="italics"></emph.end></cell>
            <cell>279</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Lumen oculorum noctu videntium, in qua oculi parte manet. in addit. de Pupilla.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Lumen Solis, cur ſit circulare, quamuis per foramina anguloſa ingrediatur.<emph.end type="italics"></emph.end></cell>
            <cell>345</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Luna plana, cur appareat, cum ſit ſphærica. 347. cur in eadem altitudine cum Sole   ſupra horizontem, maiorem vmbram efficiat.<emph.end type="italics"></emph.end></cell>
            <cell>349</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Lunam eſſe ſphæricam. 48. illuminari ſphæricè quid: ibidem &amp; de illuminatione Lu-næ. iterum eſſe ſphæricam ab eclypſibus.<emph.end type="italics"></emph.end></cell>
            <cell>111</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Luminarium Solis, &amp; Lunæ ordo.<emph.end type="italics"></emph.end></cell>
            <cell>133</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Lychanos, quid.<emph.end type="italics"></emph.end></cell>
            <cell>360</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Lyræ veteris figura.<emph.end type="italics"></emph.end></cell>
            <cell>396</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>M<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Magalis, ſeu magas, &amp; magadiſſare.<emph.end type="italics"></emph.end> 373.</cell>
            <cell>393</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Materia intelligibilis. fusè verò. explicatur in tract. de nat. Mathem.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mathematicæ mediæ, ſeu ſubalternatæ habent propter quid ſuarum demonſiratio-nem.<emph.end type="italics"></emph.end></cell>
            <cell>50</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mathematici negant reperiri quantitatem indiuiſibilem, ſeu minimam.<emph.end type="italics"></emph.end></cell>
            <cell>100</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mathematicæ non ſunt contentioſæ. 83. ostendunt per cauſam formalcm.<emph.end type="italics"></emph.end></cell>
            <cell>91</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mathematicas inuenerunt Aegyptij Sacerdotes.<emph.end type="italics"></emph.end></cell>
            <cell>198</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mathematicæ oſtendunt per cauſam materialem, &amp; formalem. 205. earum vtilitas.   ibidem. De earum natura. in proprio tractatu.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <pb pagenum="28" xlink:href="009/01/028.jpg"></pb>
          <row>
            <cell><emph type="italics"></emph>Mathematicæ maximè tractant de Bono, &amp; Pulcro, qua ratione.<emph.end type="italics"></emph.end></cell>
            <cell>237</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mechanica facultas, quæ.<emph.end type="italics"></emph.end></cell>
            <cell>238</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Melodia.<emph.end type="italics"></emph.end></cell>
            <cell>331</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Melopeia quid.<emph.end type="italics"></emph.end></cell>
            <cell>384</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Medium Demonstrationum Mathem. in earum tractatu.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Meſe quid.<emph.end type="italics"></emph.end></cell>
            <cell>360</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mina in menſuris quid.<emph.end type="italics"></emph.end></cell>
            <cell>53</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Monochordium.<emph.end type="italics"></emph.end></cell>
            <cell>359</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Motus nauigij, &amp; remi comparatio. 247. Pulchra P. Nonij in id annotatio con-tra Ariſt.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Modi muſici.<emph.end type="italics"></emph.end></cell>
            <cell>383</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Modorum antiquorum ordo, numerus, &amp;c.<emph.end type="italics"></emph.end></cell>
            <cell>383</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Motus primi mobilis, ſeu diurnus eſt menſura cœlestium motuum.<emph.end type="italics"></emph.end></cell>
            <cell>225</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Muſici recentiores reprehenſi. 331. &amp; in fine Chronologiæ.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Muſicæ totius elementa.<emph.end type="italics"></emph.end></cell>
            <cell>359</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Muſica nuda, &amp; cum melodia.<emph.end type="italics"></emph.end></cell>
            <cell>331</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>N<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Nete quid.<emph.end type="italics"></emph.end></cell>
            <cell>360</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Nucifragi inſtrumenti problema.<emph.end type="italics"></emph.end></cell>
            <cell>261</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Numerus, par, impar, primus, &amp; compoſitus, quadratus, ſeu æquilaterus, altera   parte longior. 24. Cubus num.<emph.end type="italics"></emph.end></cell>
            <cell>33</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Numeri capitales, qui.<emph.end type="italics"></emph.end></cell>
            <cell>82</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Numerum parem eſſe cauſam infiniti: imparem verò finiti.<emph.end type="italics"></emph.end></cell>
            <cell>93</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Numerorum parium alij ſunt primi, alij non.<emph.end type="italics"></emph.end></cell>
            <cell>224</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Numerus vnitarius.<emph.end type="italics"></emph.end></cell>
            <cell>307</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>O<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Oculi cur moueantur conſimiliter.<emph.end type="italics"></emph.end></cell>
            <cell>405</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Oculi anathome.<emph.end type="italics"></emph.end></cell>
            <cell>408. <emph type="italics"></emph>&amp;c.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Omophonæ voces.<emph.end type="italics"></emph.end> 372.</cell>
            <cell>392</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Oppoſitio diametralis eſt omnium maxima.<emph.end type="italics"></emph.end></cell>
            <cell>327</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ortus, &amp; occaſus ſyderum, quid, &amp; quotuplex: vbide Orione, &amp; Canicula.<emph.end type="italics"></emph.end></cell>
            <cell>153</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>P<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Paranete quæ voces, aut chordæ.<emph.end type="italics"></emph.end></cell>
            <cell>360</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Parameſe<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Parhypate<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Parelia, cur appareant nondum ſatis explicari.<emph.end type="italics"></emph.end></cell>
            <cell>182</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Parnaſſus mons, vbinam ſit. Item paropameſſus.<emph.end type="italics"></emph.end></cell>
            <cell>145</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Partes quantitatis ſunt materia illius.<emph.end type="italics"></emph.end></cell>
            <cell>211</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Perſpectiuus, quatenus conſideret lineam.<emph.end type="italics"></emph.end></cell>
            <cell>89</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Paſſiones Mathematicorum cum ſubiecto conuertuntur.<emph.end type="italics"></emph.end></cell>
            <cell>47</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Pila chriſtallina, vel vitrea, qua ratione comburat.<emph.end type="italics"></emph.end></cell>
            <cell>60</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Planetæ, qua ratione moueantur duplici motu.<emph.end type="italics"></emph.end></cell>
            <cell>130</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Plato ſolida ex planis componebat. 105. cur Elementis figuras Geometricas attri-bueret.<emph.end type="italics"></emph.end></cell>
            <cell>122</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Planetarum ordo.<emph.end type="italics"></emph.end></cell>
            <cell>271</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Principia Mathematicorum.<emph.end type="italics"></emph.end> 2.</cell>
            <cell>118</cell>
          </row>
          <pb pagenum="29" xlink:href="009/01/029.jpg"></pb>
          <row>
            <cell><emph type="italics"></emph>Principia ſcientiarum duplicia. Ex quibus, &amp; circa quod.<emph.end type="italics"></emph.end></cell>
            <cell>61</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Principia Mathematica non pendere ab experientia.<emph.end type="italics"></emph.end></cell>
            <cell>315</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Proportio alterna. 28. multiplicata, ſeu multiplex ſecundum Cæneum.<emph.end type="italics"></emph.end></cell>
            <cell>46</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Pſeudographia quid.<emph.end type="italics"></emph.end></cell>
            <cell>83</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Proportionalitas quid.<emph.end type="italics"></emph.end></cell>
            <cell>308</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Proportio continuata, &amp; diſiuncta quid. 310. alterna, ſeu permutata quid.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>inibi.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Proportio Geometrica. 311. Arithmetica.<emph.end type="italics"></emph.end></cell>
            <cell>302</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Proportio ſecundum dignitatem, eſt Geometrica.<emph.end type="italics"></emph.end></cell>
            <cell>330</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Problemata muſicalia varia à 360. <expan abbr="vſq;">vſque</expan> ad finem ſectionis 19. problematum.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Punicum, muſicum inſtrumentum.<emph.end type="italics"></emph.end></cell>
            <cell>370</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Pupillæ oculi etymon., &amp; natura. 408. cur in oculo noſtro imago pupillæ appareat.   problem.<emph.end type="italics"></emph.end> 2.</cell>
            <cell><emph type="italics"></emph>ibidem.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cur nigra in omnibus hominibus. probl.<emph.end type="italics"></emph.end> 5.</cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cur in Sole euaneſcat. probl.<emph.end type="italics"></emph.end> 6.</cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cur modo maior, modo minor appareat.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Pythagorici primi Mathematicis <expan abbr="operã">operam</expan> dedere, easqué ceteris ſcientijs præponebat.<emph.end type="italics"></emph.end></cell>
            <cell>202</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Q<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Qvadratura circuli. vide circulus. de quadratura figurarum rectangularum, &amp;   quadraturæ duplex definitio cauſalis, &amp; formalis.<emph.end type="italics"></emph.end></cell>
            <cell>185</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Quantitas an conſtet ex indiuiſibili. toto libello de lineis inſecabilibus argumentis   mathematicis.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>R<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Remi problema.<emph.end type="italics"></emph.end></cell>
            <cell>245</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Reſolutio logica, &amp; mathematica, vt conueniant. 4. &amp;<emph.end type="italics"></emph.end></cell>
            <cell>305</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Reſultus cadentium in terram, quibus angulis fiat.<emph.end type="italics"></emph.end></cell>
            <cell>354</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Rythmus fusè explicatur.<emph.end type="italics"></emph.end></cell>
            <cell>381</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Rubrum mare duplex.<emph.end type="italics"></emph.end></cell>
            <cell>152</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>S<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Scythala quid, &amp; eius figura 250. &amp;<emph.end type="italics"></emph.end></cell>
            <cell>252</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Securis problema, vbi de antiquæ ſecuris figura, &amp; angulo pulchra demonſtran-tur.<emph.end type="italics"></emph.end></cell>
            <cell>258</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Semitonium, quid.<emph.end type="italics"></emph.end></cell>
            <cell>360</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Solem eſſe terra multo maiorem: probatur.<emph.end type="italics"></emph.end></cell>
            <cell>131</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Sphæram planum tangit in puncto. demonſtratur.<emph.end type="italics"></emph.end></cell>
            <cell>184</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Statera antiqua, quæ: eius figura, &amp; problema.<emph.end type="italics"></emph.end></cell>
            <cell>259</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Stereomatria, vt differat à Geometria.<emph.end type="italics"></emph.end></cell>
            <cell>49</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Succula.<emph.end type="italics"></emph.end></cell>
            <cell>253</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Symphonæ voces.<emph.end type="italics"></emph.end> 372.</cell>
            <cell>392</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Symphonia.<emph.end type="italics"></emph.end></cell>
            <cell>391</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>T<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Temonis nauis problema.<emph.end type="italics"></emph.end></cell>
            <cell>246</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Terram eſſe rotundam ex eclypſi. 114. Item aliter. 115. eſſe reſpectu Cœli   paruam valde. 115. eſſe cubum cur Plato voluerit.<emph.end type="italics"></emph.end></cell>
            <cell>122</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Terræ quantitas.<emph.end type="italics"></emph.end></cell>
            <cell>115</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Terram paulatim reduci ad pefféctam rotunditatem.<emph.end type="italics"></emph.end></cell>
            <cell>151</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Tetragoniſmus. vide Quadratura.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <pb pagenum="30" xlink:href="009/01/030.jpg"></pb>
          <row>
            <cell><emph type="italics"></emph>Teretizare, quid.<emph.end type="italics"></emph.end></cell>
            <cell>366</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Tetrachordon, quid.<emph.end type="italics"></emph.end></cell>
            <cell>386</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Tollenonis problema.<emph.end type="italics"></emph.end></cell>
            <cell>267</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Tonus muſicus, qui; vnde oriatur.<emph.end type="italics"></emph.end></cell>
            <cell>360</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Trochleæ problemata.<emph.end type="italics"></emph.end> 249. 250.</cell>
            <cell>251</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Tunicæ oculi. 408. in tractatu de Pupilla.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>V<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ventorum nomina, &amp; ſitus.<emph.end type="italics"></emph.end></cell>
            <cell>160. <emph type="italics"></emph>a.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vectis quotuplex, &amp; c.<emph.end type="italics"></emph.end></cell>
            <cell>244</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Veteres canere ſolitos non ſolum in choris, ſed etiam in ſcenis.<emph.end type="italics"></emph.end> 371. 384.</cell>
            <cell>400</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Virgiliæ, Pleiades.<emph.end type="italics"></emph.end></cell>
            <cell>335</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Viſæ res gemmantur diſtractis oculis.<emph.end type="italics"></emph.end></cell>
            <cell>406</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Viſæ rei geminatio non fit altero oculo in latera torto, cur.<emph.end type="italics"></emph.end></cell>
            <cell>407</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Viſus res viſas, cur non duplicet, etiam ſi duos oculos habeamus.<emph.end type="italics"></emph.end></cell>
            <cell>405</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vmbelici litoralis problema.<emph.end type="italics"></emph.end></cell>
            <cell>254</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vmbram terræ parum ſupra Lunam tranſcendere.<emph.end type="italics"></emph.end></cell>
            <cell>137</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vmbrarum incrementa, &amp; decrementa, cur inæqualia.<emph.end type="italics"></emph.end> 344.</cell>
            <cell>348</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Viſus res geminat, ſi alter oculorum digito pellatur, cur.<emph.end type="italics"></emph.end></cell>
            <cell>197</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vocum muſicalium antiquæ appellationes.<emph.end type="italics"></emph.end></cell>
            <cell>360</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vox acuta velocior, grauis verò tarda, cur.<emph.end type="italics"></emph.end></cell>
            <cell>77</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Voluminum ſectio modo rectam lineam, modo curuam refert, cur.<emph.end type="italics"></emph.end></cell>
            <cell>356</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vnitas, cur indiuiſibilis.<emph.end type="italics"></emph.end></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Z<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Zonas terræ, vt Arist. deſignet: &amp; quæ ſecundum ipſum ſint habitabiles.<emph.end type="italics"></emph.end></cell>
            <cell>156</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Zonam torridam quatuor reddunt habitabilem.<emph.end type="italics"></emph.end></cell>
            <cell>159</cell>
          </row>
        </table>
        <p type="head">
          <s id="s.000664">Finis Tertij Indicis.</s>
        </p>
        <pb pagenum="31" xlink:href="009/01/031.jpg"></pb>
      </section>
      <section>
        <p type="main">
          <s id="s.000665">Viſum eſt etiam opportunum Lectori fore, ea ſimul in vnum <lb></lb>
loca colligere, in quibus Ariſtoteles mihi viſus eſt in Ma­<lb></lb>
thematicis ſcopum non attigiſſe, vt alij pręſertim Peripa­<lb></lb>
tetici facilius ea inuenire, <expan abbr="atq;">atque</expan> de ijſdem iudicium ferre <lb></lb>
poſſint.<lb></lb>
<arrow.to.target n="table3"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table3"></table.target>
          <row>
            <cell><emph type="italics"></emph>121<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Nvmero marginali: vbi ait plura Octaedra, ſeu Pyramides re-plere locum: in quo omnes pariter expoſitores lapſi ſunt.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>124<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Latitudinem figura, ait, cauſam eſſe ſupernatationis. &amp; aquam   reſiſtere ſimpliciter diuiſioni.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>136<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Cometas in ſuprema aeris regione collocat; cuius contrarium ibi   line ari demonſtratione ostenditur.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>147<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Ait Tanaim, &amp; Indum oriri ex monte Paropamiſſo. &amp; c.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>148<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Ait, tertia parte noctis Caucaſi verticem illuminari à Sole.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>149<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Ait Danubium ex Pyreneo monte defluere.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>150<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Ait fluuium quendam non minorem Rhodano in Liguria abſorberi,   &amp; iterum egredi.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>152<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Ait Rubrum mare parum Atlantico Oceano commiſceri.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>159<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Zonam torridam inhabitabilem exiſtimat.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>164<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Putat Iridis angulos non poſſe vnum ſupra alterum collocari, ſed   tantummodo in orbem.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>182<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Rationes, quas in Parelij dubitationibus affert, videntur inanes.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>236<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>In ſubducendo cœleſtium orbium numero, memoria labitur.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>243<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Ait lineam O L, ſuperare lineam L R, quantitate P L, vt in figura.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>245<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Remum ad vectem primi generis reducit.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>246<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Temonem nauis reducit ad vectem primi generis.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>247<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>In motu Remi collato cum motu Nauigij, à Nonio erroris manifestè   arguitur. eodem modo in motu Temonis.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>250<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Ait maioribus trochleis, aut rotulis facilius onera ſubleuari.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>256<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Reducit cuneum ad vectem primi generis.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>270<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Cur res in vorticibus ad medium ferantur, veram cauſam aſsigna-re non videtur.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>275<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Ait Danubium altero ramo in Mediterraneum, altero in Pontum   effluere.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>293<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Negat Araneum filum ab intrinſeco emittere, Democritum iniuria   refellens. quamuis hoc vltimum ad Phyſicum pertineat.<emph.end type="italics"></emph.end></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>403<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>Problem. 2. ſect. 2 3. non benè videtur aſsignare cauſam variæ im-merſionis nauigij.<emph.end type="italics"></emph.end></cell>
          </row>
        </table>
        <pb xlink:href="009/01/032.jpg"></pb>
        
      </section>
    </front>
    <body>
      <chap>
        <pb pagenum="33" xlink:href="009/01/033.jpg"></pb>
        <p type="head">
          <s id="s.000666">LOCA <lb></lb>MATHEMATICA <lb></lb>EX LIBRO <lb></lb>PRÆDICAMENTORVM <lb></lb>
Per ordinem declarata.</s>
        </p>
        <p type="main">
          <s id="s.000667"><arrow.to.target n="marg1"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000668"><margin.target id="marg1"></margin.target>1</s>
        </p>
        <p type="main">
          <s id="s.000669">Ex c. 3. De his, quæ ad aliquid. </s>
          <s id="s.000670">Porrò quemadmodum vnus angulus vni angulo æqualis eſt, ita <lb></lb>
<expan abbr="aliquãdo">aliquando</expan> duo anguli ſunt vni angulo æquales, vt patet, ſi vnus angulus, v.g. <lb></lb>
angulus B A C, vbi ait <emph type="italics"></emph>(Scientia verò ſi non ſit, <lb></lb>
nihil probibet eſſe ſcibile, vt circuli quadratura, ſi eſt ſcibilis, <lb></lb>
ſcientia quidem eius nondum eſt)<emph.end type="italics"></emph.end> Cum velit Ariſt. oſtendere, <lb></lb>
nó omnia correlatiua ſimul eſſe natura, id de ſcibili, &amp; ſcien­<lb></lb>
tia variè probat, præſertim verò, quia multa ſint ſcibilia, <lb></lb>
quæ tamen nondum ſciantur, vt patet, inquit, in Quadratu­<lb></lb>
ra circuli, &amp; ſcientia ipſius, quia quamuis ipſa circuli quadratura ſit ſcibi­<lb></lb>
lis, nondum tamen ſimul cum ipſa, ſcientia illius extat. </s>
          <s id="s.000671">Quæ vt perfectè <lb></lb>
intelligantur, ſciendum eſt, quadraturam circuli, quæ à Græcis tetrago­<lb></lb>
niſmus dicitur, nihil aliud eſſe, quàm propoſito cuilibet circulo exhibere <lb></lb>
quadratum æquale. </s>
          <s id="s.000672">Quæ æqualitas debet intelligi de areis, ſeu ſpatijs, ita <lb></lb>
vt area circuli, ſeu ſpatium illud, ſiue ſuperficies illa circularis, ſit æqualis <lb></lb>
areæ, ſeu ſuperficiei quadratæ. </s>
          <s id="s.000673">Qua in re plurimi decipiuntur exiſtimantes <lb></lb>
per quadraturam cir culi inquiri æqualitatem linearum, ita vt circumferen­<lb></lb>
tia circuli debeat eſſe æqualis ambitui, ſeu quatuor lateribus quadrati: <lb></lb>
quod omnino falſum eſt.</s>
        </p>
        <p type="main">
          <s id="s.000674">Quadratio porrò circuli dupliciter proponi poteſt, vel tanquam Theo­<lb></lb>
rema, vel tanquam Problema <emph type="italics"></emph>(theorema autem eſt propoſitio, in qua nihil fa­<lb></lb>
ciendum proponitur; problema verò aliquid fieri expoſcit)<emph.end type="italics"></emph.end> neutrum autem tem­<lb></lb>
pore Ariſt. erat adinuentum nam theorema inuentum eſt poſt ipſum ducen­<lb></lb>
tis circiter annis ab Archimede: problema verò nondum à quoquam per­<lb></lb>
fectè potuit reperiri. </s>
          <s id="s.000675">qua diſtinctione ſaluari poſſunt nonnulli, vt Boetius <lb></lb>
hoc loco, qui aiunt, ſe vidiſſe Demonſtrationem quadraturæ huius, ſi nimi­<lb></lb>
rum intelligant theorema. </s>
          <s id="s.000676">&amp; alij etiam verum aſſerunt, dum negant hacte­<lb></lb>
nus repertam eſſe, ſi nimirum de problemate loquantur, theorema Archi­


<pb pagenum="34" xlink:href="009/01/034.jpg"></pb>medis eſt propoſitio prima acutiſſimi libelli de Dimenſione circuli; eſt au­<lb></lb>
tem huiuſmodi. </s>
          <s id="s.000677">Quilibet circulus æqualis eſt triangulo rectangulo, cuius <lb></lb>
quidem ſemidiameter vni laterum, quæ circa rectum angulum ſunt, ambi­<lb></lb>
tus verò baſi eius eſt æqualis.</s>
        </p>
        <figure id="id.009.01.034.1.jpg" place="text" xlink:href="009/01/034/1.jpg"></figure>
        <p type="main">
          <s id="s.000678">Sit, v.g. datus circulus, cuius ſemidiameter A B; &amp; fit triangulum rectangu­<lb></lb>
lum A B C, cuius angulus B, ſit rectus, &amp; latus B A, <expan abbr="conſtituẽs">conſtituens</expan> angulum re­<lb></lb>
ctum B, cum baſi B C, ſit æquale ſemidiametro A B; baſis verò B C, ſit æqua­<lb></lb>
lis peripheriæ eiuſdem circuli dati. </s>
          <s id="s.000679">demonſtrat iam ibi Archimedes acuta <lb></lb>
æquè, ac euidenti demonſtratione triangulum iſtud æquale eſſe circulo illi. <lb></lb>
</s>
          <s id="s.000680">quod perinde eſt, ac ſi oſtendiſſet cuinam quadrato ſit æqualis, cum per vl­<lb></lb>
timam 2. Eucl. poſſimus triangulo huic quadratum æquale conſtruere, quod <lb></lb>
conſequenter dato circulo æquale erit. </s>
          <s id="s.000681">Quod ſi in modum Problematis ita <lb></lb>
proponatur: Dato circulo æquale quadratum conſtruere, nondum inuenta <lb></lb>
eſt ratio, quæ demonſtratione confirmetur, qua id geometricè penitus, hoc <lb></lb>
eſt ad æqualitatem mathematicam, ſeu exactiſſimam effici poſſit, <expan abbr="totaq́">totaque</expan>; dif­<lb></lb>
ficultas poſita eſſe videtur in inueſtigando, quonam modo exhibeamus li­<lb></lb>
neam rectam B C, æqualem peripheriæ circuli dati. </s>
          <s id="s.000682">quam nullus hactenus <lb></lb>
geometricè illi æqualem potuit exhibere, <expan abbr="atq;">atque</expan> exhibita <expan abbr="euidẽti">euidenti</expan> demonſtra­<lb></lb>
tione comprobare; Quamuis Archimedes acumine ſanè mirabili in lib. de <lb></lb>
lineis ſpiralibus, eam <expan abbr="quoq;">quoque</expan> theorematicè, non tamen problematicè inue­<lb></lb>
ſtigauit. </s>
          <s id="s.000683">nam propoſitione 18. illius <expan abbr="admirãdi">admirandi</expan> operis inuenit lineam rectam <lb></lb>
æqualem circumferentiæ primi circuli ſpiralis lineæ; propoſ verò 19. repe­<lb></lb>
rit aliam rectam æqualem circumferentiæ ſecundi circuli. </s>
          <s id="s.000684">tu ipſum conſule, <lb></lb>
ſi admirandarum rerum contemplatione delectaris. </s>
          <s id="s.000685">Multa hac de re Pap­<lb></lb>
pus Alexandrinus lib. 4. Math. coll. </s>
          <s id="s.000686">&amp; Ioannes Buteo vnico volumine om­<lb></lb>
nes quadraturas tain priſcorum, quam recentiorum <expan abbr="cõprehenſus">comprehenſus</expan> eſt. </s>
          <s id="s.000687">Qua­<lb></lb>
re qui plura cupit, eos adeat; nos tamen infra ſuis locis explicabimus tres <lb></lb>
illas celebres antiquorum Antiphontis, Briſſonis, &amp; Hippocratis quadra­<lb></lb>
turas, quamuis falſas, <expan abbr="quarũ">quarum</expan> ſæpe meminit Ariſt. &amp; alij. </s>
          <s id="s.000688">ſolet autem à non­<lb></lb>
nullis diſputari, vtrum quadratura iſta problematica ſit poſſibilis, nec ne, <lb></lb>
cum videant eam à nemine, quamuis diu magno labore perquiſitam, hacte­<lb></lb>
nus adinuentam eſſe. </s>
          <s id="s.000689">ego quidem eſſe poſſibilem exiſtimo, quis enim dubi­<lb></lb>
tare poteſt, poſſe exiſtere quadratum æquale circulo propoſito? </s>
          <s id="s.000690">Quod ſi po­<lb></lb>
teſt fieri, quare non etiam demonſtrari? </s>
          <s id="s.000691">pręfertim cum videamus ab Archi­<lb></lb>
mede iam inuentam eſſe, quatenus Theorema eſt. </s>
          <s id="s.000692">&amp; præterea conſtet, Hip­<lb></lb>
pocratem quadraſſe lunulam, vt ſuo loco dicemus, &amp; Archimedem in libel­


<pb pagenum="35" xlink:href="009/01/035.jpg"></pb>lo de quadratura Paraboles, quadraſſe ipſam Parabolem, quæ tamen duæ fi­<lb></lb>
guræ, lunula ſcilicet, &amp; parabola ſunt curuilineæ.</s>
        </p>
        <p type="main">
          <s id="s.000693"><arrow.to.target n="marg2"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000694"><margin.target id="marg2"></margin.target>2</s>
        </p>
        <p type="main">
          <s id="s.000695">Ex cap. de Priori <emph type="italics"></emph>(in ſcientijs demonſtratiuis eſt prius, &amp; poſterius ordine, <lb></lb>
elementa enim priora ſunt ijs, quæ deſcribuntur, nam principia prior a ſunt theore­<lb></lb>
matibus ordine)<emph.end type="italics"></emph.end> verba illa, nam principia, &amp;c. </s>
          <s id="s.000696">quæ non ſunt in antiqua tran­<lb></lb>
ſlatione deſumpſimus ex caſtigatiſſimo græco codice editionis Francfor­<lb></lb>
dienſis, propterea quod totum hunc locum declarant; ſunt autem iſta, <lb></lb>
<foreign lang="grc">αί γαρ αρχαί πρότεραι τῶν θεωρημάτων τῃ τάξη. </foreign> per ſcientias autem demonſtra­<lb></lb>
tiuas intelligendas eſſe hoc loco ipſas Mathematicas ex eo patet, quod illis <lb></lb>
aſſignet Ariſt. Deſcriptiones; nam hoc verbo, Deſcriptiones, ſeu figuratio­<lb></lb>
nes, ſolet ipſe Mathematicas Demonſtrationes innuere, quod in ipſis figu­<lb></lb>
rationes, &amp; Deſcriptiones adhibeantur, vt alijs locis patebit: idcirco ver­<lb></lb>
ba illa à nobis addita ex græco, optimè <expan abbr="præcedẽtia">præcedentia</expan> exponunt, cum per ele­<lb></lb>
menta intelligantur principia, qualia ſunt initio Euclidis, &amp; per deſcriptio­<lb></lb>
nes exponant theoremata. </s>
          <s id="s.000697">quod autem principia illa ordine priora ſint de­<lb></lb>
monſtrationibus, ſiue ipſas præcedant, ex ipſa primi Euclidis inſpectione <lb></lb>
patere poteſt.</s>
        </p>
        <p type="main">
          <s id="s.000698"><arrow.to.target n="marg3"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000699"><margin.target id="marg3"></margin.target>3</s>
        </p>
        <p type="main">
          <s id="s.000700">Ex cap. de motu <emph type="italics"></emph>(Quadratum augetur Gnomone circumpoſito)<emph.end type="italics"></emph.end> Gnomon vox <lb></lb>
græca inter alia ſignificat inſtrumentum illud, quod Latini tum amuſſim, <lb></lb>
<figure id="id.009.01.035.1.jpg" place="text" xlink:href="009/01/035/1.jpg"></figure><lb></lb>
tum normam appellant, Itali verò, Squadra, ad <lb></lb>
cuius ſimilitudinem Geometræ denominarunt fi­<lb></lb>
guram quandam, ſeu portionem cuiuſuis paralle­<lb></lb>
logrammi, vt videre eſt in definitione ſecunda <lb></lb>
2. elem. </s>
          <s id="s.000701">&amp; in præſenti figura, in qua quadratum <lb></lb>
A B C D, circumpoſito gnomone E F G, augetur, <lb></lb>
&amp; fit maius quadratum H B I L.</s>
        </p>
        <p type="main">
          <s id="s.000702">Idem etiam verum eſt in quadrato arithmeti­<lb></lb>
co, ſiue in numero quadrato: is enim pariter ad­<lb></lb>
dito Gnomone augetur. </s>
          <s id="s.000703">i. </s>
          <s id="s.000704">addito numero impari. <lb></lb>
</s>
          <s id="s.000705">quemadmodum infra 3. Phyſ. tex. 26. fusè explicabimus.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.000706"><emph type="italics"></emph>Ex Primo Priorum reſolutoriorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000707"><arrow.to.target n="marg4"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000708"><margin.target id="marg4"></margin.target>4</s>
        </p>
        <p type="main">
          <s id="s.000709">Aliquorum opinio eſt, Ariſtotelem hoſce libros appellaſſe reſolu­<lb></lb>
torios, quod per illos doceat ſyllogiſmum, ac demonſtrationem <lb></lb>
iam factam in ſua immediata principia reſoluere, quam opinio­<lb></lb>
nem meum non eſt, nunc refellere. </s>
          <s id="s.000710">perſuaſum tamen mihi eſt, rem <lb></lb>
multo aliter ſe habere, veram rationem huius tituli petendam eſſe ex peni­<lb></lb>
tiori Mathematicorum eruditione. </s>
          <s id="s.000711">Sciendum <expan abbr="itaq;">itaque</expan> id, quod tradit Pappus <lb></lb>
Alex. initio ſeptimi Mathem. collect. </s>
          <s id="s.000712">antiquiſſimos videlicet Geometras, <lb></lb>
Euclidem, Apollonium Pergæum, &amp; Ariſtęum ſcripſiſſe libros de reſolutio­<lb></lb>
ne, in quibus ars tradebatur, qua propoſito quouis theoremate, aut proble­<lb></lb>
mate poſſent facile ex eo, tanquam vero accepto inueſtigare aliquam veri­<lb></lb>
tatem, per quam deinde componerent illius, quod quærebatur, Demonſtra­<lb></lb>
tionem; inueſtigationem illam appellabant reſolutionem: compoſitionem <lb></lb>
verò nominabant diſcurſum <expan abbr="illũ">illum</expan>, quo ex vero illo per reſolutionem inuento,


<pb pagenum="36" xlink:href="009/01/036.jpg"></pb>oſtendebant concluſionem. </s>
          <s id="s.000713">Porrò Diogenes Laert. huius reſolutionis in­<lb></lb>
uentorem facit Platonem: à quo eam Leodamas Thaſius didicit, cuius be­<lb></lb>
neficio, pluries deinde Geometricas demonſtrationes adinuenit. </s>
          <s id="s.000714">definitio <lb></lb>
<expan abbr="vtriuſq;">vtriuſque</expan> eſt apud Euclidem ad primam propoſ. 13. Elem. iuxta tranſlatio­<lb></lb>
nem Zamberti, &amp; Commandini; vbi etiam <expan abbr="quinq;">quinque</expan> priora theoremata, pri­<lb></lb>
mò per reſolutionem, deinde per compoſitionem demonſtrantur, quæ tan­<lb></lb>
quam perſpicua exempla rei propoſitæ inſeruire poſſunt. </s>
          <s id="s.000715">ſunt præterea fre­<lb></lb>
quentes huiuſmodi reſolutiones in operibus Archimedis, Apollonij, &amp; Pap­<lb></lb>
pi. </s>
          <s id="s.000716">extat adhuc liber Datorum Euclidis, qui geometricis reſolutionibus in­<lb></lb>
ſeruiebat. </s>
          <s id="s.000717">vtinam extarent etiam alij de reſolutione, quorum auxilio non <lb></lb>
tantopere recentiores Mathematici in inueniendis Demoſtrationibus la­<lb></lb>
borarent; hanc reſolutionem, ſic Pappus fuſius, quam Euclides explicat; <lb></lb>
reſolutio eſt via à quæſito tanquam conceſſo per ea, quæ ex ipſo conſequun­<lb></lb>
tur ad aliquod certum, &amp; conceſſum: in reſolutione enim id, quod quæritur <lb></lb>
tanquam factum, &amp; verum ſupponentes, quid ex hoc ſequatur, conſidera­<lb></lb>
mus, quouſque incidamus in aliquod iam cognitum, vel quod ſit è numero <lb></lb>
principiorum. </s>
          <s id="s.000718">Quod quidem erat ſignum euidens, quæſitum quoque verum <lb></lb>
eſſe. </s>
          <s id="s.000719">eadem omnino habet Proclus in comm. ad ſextam primi elem. </s>
          <s id="s.000720">Quod <lb></lb>
porrò Ariſt. ipſe hanc reſolutionem Mathematicam cognouerit eſſe medij <lb></lb>
inquiſitionem manifeſtum eſt ex cap. 3. lib.  3. Ethyc. vbi ſic ait <emph type="italics"></emph>(Qui enim <lb></lb>
conſultat, quærere videtur, &amp; reſoluere prædicto modo, quemadmodum deſigna­<lb></lb>
tiones)<emph.end type="italics"></emph.end> vbi per deſignationes intelligit Geometricas demonſtrationes, vt <lb></lb>
ſupra innuimus, &amp; infra probabimus; cum ergo conſultatio nihil aliud ſit, <lb></lb>
quam medij idonei ad finem in rebus agendis inquiſitio, eamque dicat eſſe <lb></lb>
ſimilem reſolutioni Geometricæ, manifeſtum eſt, ipſam <expan abbr="quoq;">quoque</expan> reſolutionem <lb></lb>
eſſe medij in rebus ſpeculatiuis idonei perueſtigationem. </s>
          <s id="s.000721">Exiſtimo igitur <lb></lb>
cum doctiſſimis Zabarella, Burana, Toleto, &amp; alijs, Ariſtotilem non ſolum <lb></lb>
hanc ſuam logicam ad mathematicarum ſcientiarum typum compegiſſe, <lb></lb>
verum potius imitatum eſſe opus illud Euclidis de reſolutione, atque ex eo <lb></lb>
non ſolum plurima exempla Geometrica, verum etiam titulum deſumpſiſſe, <lb></lb>præſertim cum argumentum eſſet ferè idem vtrobique, ſed Ariſt. intentio <lb></lb>
fuerit accommodare reſolutionem omnibus <expan abbr="ſciẽtijs">ſcientijs</expan>; Euclidis verò, &amp; alio­<lb></lb>
rum Geometriæ ſoli. </s>
          <s id="s.000722">hinc patere poteſt, cur hi libri reſolutorij inſcribantur, <lb></lb>
quod ſcilicet tradunt methodum, qua valeamus quæſitum quoduis reſolue­<lb></lb>
re, ideſt, ex quæſito tanquam vero inueſtare aliquam veritatem, per quam <lb></lb>
deinde propoſitæ quæſtionis rationem methodo compoſitiua reddamus. </s>
          <s id="s.000723">Et <lb></lb>
verò cum reliquas appellationes Problematis, Theorematis, Propoſitionis, <lb></lb>
definitionum, poſtulatorum, axiomatum, &amp; alia huiuſmodi ex Geometri­<lb></lb>
cis ad omnes ſcientias tranſtulerit, quid ni etiam reſolutionem? </s>
          <s id="s.000724">maximè <lb></lb>
verò, quia ſi horum lib. intentio eſſet docere iam factum ſyllogiſmum in ſua <lb></lb>
principia reſoluere, parum eſſet vtilis; imò nec vtilis, ſed ſuperfluum quid. <lb></lb>
</s>
          <s id="s.000725">at verò vbinam docuit hanc reſolutionem? </s>
          <s id="s.000726">profecto nullibi. </s>
          <s id="s.000727">quid opus eſt <lb></lb>
iam factum ſyllogiſmum reſoluere? </s>
          <s id="s.000728">at verò propoſitam quæſtionem reſol­<lb></lb>
uere veterum mathematicorum more, hoc opus, hic labor eſt.</s>
        </p>
        <p type="main">
          <s id="s.000729">Hanc porrò reſolutionem attendendam eſſe primò penes formam, quam <lb></lb>
docet primis duobus analyticis; ſecundò penes materiam, quam tradit duo­


<pb pagenum="37" xlink:href="009/01/037.jpg"></pb>bus vltimis, non prætereundum. </s>
          <s id="s.000730">reliquas duas logicæ partes, Topicam ſci­<lb></lb>
licet, &amp; Elenchos, quæ ſyllogiſmos probabilem, &amp; apparentem docent, no­<lb></lb>
luit appellare reſolutorios, quamuis inuentionem mediorum doceant, quia <lb></lb>
iam mos iſte inoleuerat apud Philoſophos, &amp; Mathematicos, vt illa ſola <lb></lb>
pars, quæ ex materia neceſſaria doceret ſyllogiſmum demonſtratiuum con­<lb></lb>
ſtruere, diceretur reſolutio: cum Mathematici, qui primi de reſolutione <lb></lb>
ſcripſerunt, talem materiam ſolum conſiderent.</s>
        </p>
        <p type="main">
          <s id="s.000731"><arrow.to.target n="marg5"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000732"><margin.target id="marg5"></margin.target>5</s>
        </p>
        <p type="main">
          <s id="s.000733">Ex cap. 23. ſecti primi lib. 1. <emph type="italics"></emph>(Vt quod diameter incommenſurabilis eo, quod <lb></lb>
imparia æqualia paribus fiant, ſi fuerit poſita commenſurabilis. </s>
          <s id="s.000734">æqualia igitur fieri <lb></lb>
imparia paribus ratiocinantur, diametrum vtrò incommenſurabilem eſſe ex ſuppo­<lb></lb>
ſitione <expan abbr="monſtrãt">monſtrant</expan>, quoniam falſum accidit propter contradictionem)<emph.end type="italics"></emph.end> Euclides pri­<lb></lb>
mis duabus definitionibus 10. elem. </s>
          <s id="s.000735">definit, quæ nam ſint magnitudines <lb></lb>
commenſ. </s>
          <s id="s.000736">&amp; quæ incommenſ. </s>
          <s id="s.000737">ſic; commenſ. </s>
          <s id="s.000738">magnitudines dicuntur, quas <lb></lb>
<figure id="id.009.01.037.1.jpg" place="text" xlink:href="009/01/037/1.jpg"></figure><lb></lb>
eadem menſura metitur, vt ſi fuerint duæ magnitu­<lb></lb>
dines, A, &amp; B, quas eadem menſura C, ideſt quan­<lb></lb>
titas C, metiatur, ideſt <expan abbr="quãtitas">quantitas</expan> C, applicata quan­<lb></lb>
titati A, &amp; per ipſam aliquoties replicata ipſam ad­<lb></lb>
æquatè abſumat, vt ſi linea C, quinquies ſuper li­<lb></lb>
neam A, replicata eam præcisè, &amp; perfectè omninò <lb></lb>
adæquaret: &amp; eadem linea C, applicata lineæ B, &amp; ſuper illam ter, v.g. re­<lb></lb>
petita ipſam conſumeret, diceretur <expan abbr="vtranq;">vtranque</expan> metiri, &amp; proinde duas lineas <lb></lb>
A, &amp; B, eſſe comm. definit poſtea <expan abbr="incommẽſ">incommenſ.</expan> hoc modo, incomm. autem, qua­<lb></lb>
rum nullam contingit communem menſuram reperiri; vt ſi duarum linea­<lb></lb>
<figure id="id.009.01.037.2.jpg" place="text" xlink:href="009/01/037/2.jpg"></figure><lb></lb>
rum, A, B, nunquam poſſet reperiri aliqua menſu­<lb></lb>
ra, quæ <expan abbr="vtranq;">vtranque</expan> adæquatè metiretur, v. g. ſi linea <lb></lb>
C, menſuraret A, quater ſumpta, ter autem ſumpta <lb></lb>
non adæquaret omnino <expan abbr="lineã">lineam</expan> B, ſed deficeret, vel ex­<lb></lb>
cederet aliquantulum, <expan abbr="atq;">atque</expan> hoc fieret in quauis alia <lb></lb>
menſura, loco ipſius C, aſſumpta, ſiue maior, ſiue <lb></lb>
minor ipſa C, vt <expan abbr="vtranq;">vtranque</expan> nunquam perfectè metiretur, eſſent duæ illæ lineæ <lb></lb>
incommenſ. </s>
          <s id="s.000739">Extare porrò tales lineas, &amp; ſuperficies, &amp; corpora, <expan abbr="eaq́">eaque</expan>; quam­<lb></lb>
plurima, ac penè infinita ex 10. Elem. manifeſtum eſt. </s>
          <s id="s.000740">inuentum autem hu­<lb></lb>
ius aſymmetriæ, quod Pythagoricis veteres attribuunt, mihi ſemper viſum <lb></lb>
eſt omni maius admiratione, cum nulla experientia, <expan abbr="nullusq́">nullusque</expan>; effectus in ip­<lb></lb>
ſius cognitionem potuerit priſcos illos Geometras inducere. </s>
          <s id="s.000741">Quapropter <lb></lb>
non immeritò diuinus ille Plato lib. 7. de legib. </s>
          <s id="s.000742">huius aſymmetriæ ignora­<lb></lb>
tionem, adeo deteſtatus eſt, vt eam non hominum, ſed ſuum, pecorumque <lb></lb>
ignorantiam cenſuerit. </s>
          <s id="s.000743">inter lineas incommenſ. ſunt diameter, &amp; latus eiuſ­<lb></lb>
dem quadrati, quia nulla poteſt reperiri menſura quantumuis exigua, vti <lb></lb>
<figure id="id.009.01.037.3.jpg" place="text" xlink:href="009/01/037/3.jpg"></figure><lb></lb>
eſt lineola E, in præſenti quadrato, etiamſi illam in <lb></lb>
infinitum ſubdiuidas, quæ <expan abbr="vtranq;">vtranque</expan> lineam, diame­<lb></lb>
trum ſcilicet A C, &amp; latus quoduis ex quatuor, v.g. <lb></lb>
latus B C, præcisè omnino metiatur. </s>
          <s id="s.000744">theorema <lb></lb>
iſtud demonſtratur in vltima 10. Elem. eodem me­<lb></lb>
dio, quod ab Ariſtotele hic innuitur; Euclides ex <lb></lb>ſuppoſitione alterius partis contradictionis ipſius <pb pagenum="38" xlink:href="009/01/038.jpg"></pb>propoſitionis, quæ falſa eſt, nimirum ſuppoſito prædictas lineas eſſe comm. <lb></lb>
deducit ad impoſſibile, ſiue, vt ait hic Ariſt. falſum ratiocinatur, quod ſci­<lb></lb>
licet idem numerus eſſet par, &amp; impar, quod Ariſt. ſignificat, quando ait, <lb></lb>
imparia æqualia paribus fiunt. </s>
          <s id="s.000745">ex quo abſurdo deducitur falſam eſſe prædi­<lb></lb>
ctam ſuppoſitionem, quæ aſtruebat eſſe comm. &amp; proinde altera pars con­<lb></lb>
tradictionis, quæ eſt, eſſe incomm. vera aſtruitur. </s>
          <s id="s.000746">ex quibus ſatis videtur ex­<lb></lb>
plicari hic locus. </s>
          <s id="s.000747">videas igitur, quàm leuiter nonnulli noſtræ tempeſtatis <lb></lb>
ageometreti iſtud exponant, dicentes diametrum eſſe incomm. coſtæ, nihil <lb></lb>
aliud ſignificare, quam diametrum eſſe longiorem coſta, qua expoſitione <lb></lb>
nihil ineptius. </s>
          <s id="s.000748">Aduerte tandem figuram vulgatæ editionis eſſe ineptam, <lb></lb>
cum habeat duo quadrata alterum ſuper diametro alterius, quorum maius <lb></lb>
ſuperuacaneum eſt.</s>
        </p>
        <p type="main">
          <s id="s.000749"><arrow.to.target n="marg6"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000750"><margin.target id="marg6"></margin.target>6</s>
        </p>
        <p type="main">
          <s id="s.000751">Et cap. 24. ſecti primi libri primi <emph type="italics"></emph>(Sed magis efficitur manifeſtum in deſcri­<lb></lb>
ptionibus, vt quod æquicruris, qui ad baſim æquales ſint, ad centrum ductæ A B, <lb></lb>A C, ſi igitur æqualem accipiat A G, angulum ipſi A B D, non omnino exiſtimans <lb></lb>
æquales, qui ſemicirculorum, &amp; rurſus G, ipſi D, non omnem aſſumens eum, qui ſe­<lb></lb>
cti. </s>
          <s id="s.000752">amplius ab æqualibus existentibus totis angulis, &amp; ablatorum æquales eſſe re­<lb></lb>
liquos E, F, quod ex principio petet, niſi acceperit ab æqualibus demptis æqualia <lb></lb>
derelinqui.)<emph.end type="italics"></emph.end> Primum ſcias characteres vulgatæ editionis, vna cum figura ip­<lb></lb>
ſis reſpondente, eſſe mendoſos; propterea ex textu græco vtrunque corri­<lb></lb>
gendum putaui in hunc, quem vidiſti modum. </s>
          <s id="s.000753">Secundo, per deſcriptiones <lb></lb>
Ariſt. intelligere <expan abbr="demõſtrationes">demonſtrationes</expan> Geometricas ſupra diximus, quod ex hoc <lb></lb>
loco euidenter confirmatur, vbi manifeſtè loco deſcriptionis ſupponit li­<lb></lb>
nearem demonſtrationem. </s>
          <s id="s.000754">In hoc <expan abbr="itaq;">itaque</expan> exemplo vult Ariſt. illud demon­<lb></lb>
ſtrare, quod Euclides in 5. primi oſtendit, alio tamen modo, ſcilicet Iſoſce­<lb></lb>
lium triangulorum, qui ad baſim ſunt anguli, inter ſe ſunt æquales. </s>
          <s id="s.000755">eſt au­<lb></lb>
tem figura in omnibus textibus deprauata, quam ſic puto <expan abbr="rèſtītuendam">rèſtituendam</expan> eſſe <lb></lb>
ex quodam græco codice, qui characteres hoc modo appoſuerat. </s>
          <s id="s.000756">ſit Iſoſce­<lb></lb>
<figure id="id.009.01.038.1.jpg" place="text" xlink:href="009/01/038/1.jpg"></figure><lb></lb>
les C A B, cuius baſis C B, Dico angulos ſupra baſim, <lb></lb>
in quibus literæ E F, eſſe inuicem æquales. </s>
          <s id="s.000757">facto centro <lb></lb>
in A, deſcribatur circulus A B C, tranſiens per puncta <lb></lb>
C B, iam ſic. </s>
          <s id="s.000758">omnes anguli ſemicirculi ſunt æquales in­<lb></lb>
ter ſe, ergo anguli A C G, A B D, ſunt æquales. </s>
          <s id="s.000759">Præte­<lb></lb>
rea cùm anguli eiuſdem ſectionis ſint æquales ad inui­<lb></lb>
cem, erunt anguli ſectionis C B D G, nimirum anguli, <lb></lb>
in quibus ſunt G, &amp; D, inter ſe æquales: <expan abbr="cumq́">cumque</expan>; hi duo <lb></lb>
anguli ſectionis ſint partes <expan abbr="angulorũ">angulorum</expan> ſemicirculi A C G, <lb></lb>
A B D, ſi illi ab his auferantur, auferuntur æquales anguli ab æqualibus an­<lb></lb>
gulis, ergo anguli, qui remanent, ſcilicet E, &amp; F, erunt æquales, quod erat <lb></lb>
demonſtrandum. </s>
          <s id="s.000760">hinc Ariſt. infert manifeſtum eſſe oportere in omni ſyllo­<lb></lb>
giſmo, reperiri vniuerſales, &amp; affirmatiuas propoſitiones, vt Factum eſt in <lb></lb>
præcedenti aliter eſſet petitio principij. </s>
          <s id="s.000761">Quænam vero ſit æqualitas, quam <lb></lb>
Geometræ conſiderant, infra cap. 1. ſecti 3. explicabitur.</s>
        </p>
        <p type="main">
          <s id="s.000762"><arrow.to.target n="marg7"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000763"><margin.target id="marg7"></margin.target>7</s>
        </p>
        <p type="main">
          <s id="s.000764">Ex cap. 2. ſecti 2. lib. 1. <emph type="italics"></emph>(Secundum veritatem quidem ex ijs, quæ ſecundum <lb></lb>
veritatem deſcribuntur ineſſe, ad dialecticos autem ſyllogiſmos ex propoſitionibus <lb></lb>
ſecundum opinionem)<emph.end type="italics"></emph.end> verba illa; ex ijs, quæ <expan abbr="ſecundũ">ſecundum</expan> veritatem deſcribuntur <pb pagenum="39" xlink:href="009/01/039.jpg"></pb>ineſſe; ſic græcè, <foreign lang="grc">έκ τῶν κατὰ αληθείαν διαγεγραμμένον </foreign> vbi manifeſtè vtitur <lb></lb>
verbo, Deſcribere, per quod ſuperius annotauimus apud Ariſt. ſignificari <lb></lb>
Geometricas demonſtrationes, nam eas opponit dialecticis ſyllogiſmis, ſe­<lb></lb>
quentibus verbis, cum dixit (ad dialecticos autem ſyllogiſmos ex propoſi­<lb></lb>
tionibus ſecundum opinionem) hac adhibita conſideratione, quam inter­<lb></lb>
pres non videtur adhibuiſſe, ſenſus huius loci non erit obſcurus.</s>
        </p>
        <p type="main">
          <s id="s.000765"><arrow.to.target n="marg8"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000766"><margin.target id="marg8"></margin.target>8</s>
        </p>
        <p type="main">
          <s id="s.000767">Ex eodem loco paulo poſt <emph type="italics"></emph>(Quare principia quidem, quæ ſecundum <expan abbr="vnum-quodq;">vnum­<lb></lb>
quodque</expan> ſunt experimenti est tradere: dico autem, vt aſtrologicam experientiam <lb></lb>
aſtrologicæ ſcientiæ: acceptis enim apparentibus <expan abbr="ſufficiẽter">ſufficienter</expan>, ita inuentæ ſunt aſtro­<lb></lb>
logicæ demonstrationes)<emph.end type="italics"></emph.end> Cum rationem tradat inueniendorum mediorum ad <lb></lb>quodlibet problema demonſtrandum; nunc docet, non omnia in ſcientijs <lb></lb>poſſe probari, aut demoνſtrari: principia enim ſcientiarum <expan abbr="nõ">non</expan> demonſtran­<lb></lb>
tur, ſed ſola experientia manifeſta ſunt; vt patet in Aſtronomia, quæ ab ex­<lb></lb>
perientia ſua ſolet ſtabilire principia: principijs autem <expan abbr="experimẽto">experimento</expan> conſti­<lb></lb>
tutis ex ipſis reliqua problemata <expan abbr="demonſtrãtur">demonſtrantur</expan>. </s>
          <s id="s.000768">duo autem ſunt apud aſtro­<lb></lb>
nomos genera experimenti, primum dicitur Phænomena, ideſt, <expan abbr="apparẽtiæ">apparentiæ</expan>; <lb></lb>
&amp; ſunt ea, quæ vulgo omnibus patent, vt Solem oriri, &amp; occidere; aſtra fer­<lb></lb>
ri circulariter, diem augeri modo, modo minui: &amp; his ſimilia. </s>
          <s id="s.000769">alterum ge­<lb></lb>
nus dicitur obſeruationes, quæ tantummodo aſtronomiæ peritis per obſer­<lb></lb>
uationem innoteſcunt, vt Solem inæqualiter ferri proprio motu per Zodia­<lb></lb>
cum; aliquando maiorem, aliquando minorem videri; plures dies immo­<lb></lb>
rari citra æquatiorem in parte Zodiaci boreali, quam in altera vltra æqua­<lb></lb>
torem auſtrali. </s>
          <s id="s.000770">dies naturales eſſe inuicem inæquales, &amp;c. </s>
          <s id="s.000771">ex quibus deinde <lb></lb>
ponunt eccentricos, &amp; augem, ad ſaluandas tum apparentias, tum obſerua­<lb></lb>
tiones; &amp; hac ratione aſtrologica ſcientia paulatim reperta eſt, ac in dies <lb></lb>
reperitur.</s>
        </p>
        <p type="main">
          <s id="s.000772"><arrow.to.target n="marg9"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000773"><margin.target id="marg9"></margin.target>9</s>
        </p>
        <p type="main">
          <s id="s.000774">Ex cap. 3. ſecti 2. lib. 1. <emph type="italics"></emph>(Vt an ne diameter incomm.)<emph.end type="italics"></emph.end> loquitur de aſymme­<lb></lb>
tria diametri, &amp; coſtæ eiuſdem quadrati, de qua fusè egimus ſuperius in <lb></lb>
cap. 23. ſecti 1. huius libri; quæ ſi repetantur, optimè hunc <expan abbr="locũ">locum</expan> declarant.</s>
        </p>
        <p type="main">
          <s id="s.000775"><arrow.to.target n="marg10"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000776"><margin.target id="marg10"></margin.target>10</s>
        </p>
        <p type="main">
          <s id="s.000777">Ex cap. 1. ſecti 3. lib. 1. <emph type="italics"></emph>(Sit A, duo recti, in quo B, triangulus, in quo C, <lb></lb>
æquicrus, ipſi <expan abbr="itaq;">itaque</expan> C, ineſt A. per B; ipſi vero B, non amplius per aliud, per ſe <lb></lb>
namque triangulus habet duos rectos)<emph.end type="italics"></emph.end> nullum aliud exemplum tam frequenter <lb></lb>
vſurpat Philoſophus, quam iſtud ex Mathematicis deſumptum de triangu­<lb></lb>
lo, ſcilicet, omnis triangulus habet tres angulos æquales duobus rectis an­<lb></lb>
gulis, cuius Demonſtratio eſt in 32. primi Elem. quod, vt probè intelliga­<lb></lb>
tur, explicandum eſt penes quid attendenda ſit æqualitas inter angulum, &amp; <lb></lb>
angulum, quod facile aſſequemur, ſi meminerimus angulum eſſe inclinatio­<lb></lb>
nem illam, quam duæ lineæ non in directum poſitæ faciunt: ſiue etiam (vt <lb></lb>
melius percipiamus) angulum eſſe acumen illud, ſiue mucronem <expan abbr="illũ">illum</expan>, quem <lb></lb>
duæ lineæ non in directum conſtitutæ faciunt, vt duarum linearum A B, A C, <lb></lb>
<figure id="id.009.01.039.1.jpg" place="text" xlink:href="009/01/039/1.jpg"></figure><lb></lb>
inclinatio in puncto A, ſiue acumen illud, ſiue mucro, <lb></lb>
eſt ratio anguli. </s>
          <s id="s.000778">ſolum igitur duo anguli erunt æqua­<lb></lb>
les, <expan abbr="quãdo">quando</expan> vnius acumen æquale erit acumini alterius; <lb></lb>
etiam ſi lineæ conſtituentes vnum angulum ſint lon­<lb></lb>
giores lineis alterum angulum conſtituentibus, quia <lb></lb>quantitas anguli non attenditur penes longitudinem <pb pagenum="40" xlink:href="009/01/040.jpg"></pb><expan abbr="linearũ">linearum</expan>, ſed penes inclinationem, &amp; mucronem, quem faciunt: vnde etiamſi <lb></lb>
duæ lineæ prædictæ A B, A C, productæ, ſiue etiam decurtatæ fuerint, dum­<lb></lb>
modo ſitus, ſiue poſitio ipſarum, quam ad inuicem habent, non varietur, <lb></lb>
erit ſemper eadem quantitas anguli A. </s>
          <s id="s.000779">Aduertendum præterea rationem <lb></lb>
anguli non poſſe ſaluari in ſolo puncto A, in quo lineæ concurrunt, ſed ne­<lb></lb>
ceſſariam eſſe aliquam quantitatem, quamuis exiguam, linearum A B, A C. <lb></lb>
</s>
          <s id="s.000780">Notandum etiam, quod in nominatione angulorum, quæ fit per tres lite­<lb></lb>
ras, ſemper literam illam eſſe medio loco proferendam, quæ ad acumen ip­<lb></lb>
ſum poſita eſt, vt in ſuperiori, litera A, debet ſemper media proferri, dicen­<lb></lb>
do angulum B A C, ſiue C A B, <expan abbr="nũquam">nunquam</expan> tamen licet dicere angulum A C B, <lb></lb>
vel C B A. </s>
          <s id="s.000781">Porrò quemadmodum vnus angulus vni angulo æqualis eſt, ita <lb></lb>
<expan abbr="aliquãdo">aliquando</expan> duo anguli ſunt vni angulo æquales, vt patet, ſi vnus angulus, v.g. <lb></lb>
angulus B A C, diuidatur in duos angulos à linea A D. tunc enim duo angu­<lb></lb>
<figure id="id.009.01.040.1.jpg" place="text" xlink:href="009/01/040/1.jpg"></figure><lb></lb>
li partiales B A D, D A C, erunt æquales totali angulo <lb></lb>
B A C, cum partes omnes ſimul ſumptæ ſint ſuo toti æqua­<lb></lb>
les. </s>
          <s id="s.000782">pariter tres anguli poſſunt æquari &amp; vni, &amp; duobus <lb></lb>
alijs angulis, quando nimirum a cumina, ſiue mucrones il­<lb></lb>
li ſimul ad vnum punctum conſtituti <expan abbr="adæquarẽtur">adæquarentur</expan> mucro­<lb></lb>
ni illi, quem conſtituerent alij duo anguli, quibus illi tres <lb></lb>
ſunt pares, v.g. ſint tres anguli trianguli A B C, <expan abbr="ſintq́">ſintque</expan>; alij duo anguli recti, <lb></lb>
<figure id="id.009.01.040.2.jpg" place="text" xlink:href="009/01/040/2.jpg"></figure><lb></lb>
quos linea perpendicularis D E, facit cum li­<lb></lb>
nea F G; ſit <expan abbr="inquã">inquam</expan> anguli recti D E F, D E G, <lb></lb>
tunc tres anguli illius <expan abbr="triãguli">trianguli</expan> <expan abbr="dicẽtur">dicentur</expan> æqua­<lb></lb>
les duobus hiſce rectis, ſi tres illi mucrones <lb></lb>
trianguli ſimul ſumpti, &amp; vniti ad punctum <lb></lb>
E, ad quod duo <expan abbr="quoq;">quoque</expan> mucrones angulorum <lb></lb>
<figure id="id.009.01.040.3.jpg" place="text" xlink:href="009/01/040/3.jpg"></figure><lb></lb>
rectorum coeunt, congruent omnino duobus <lb></lb>
prædictis angulis rectis, ſiue duobus illis mu­<lb></lb>
cronibus angulorum rectorum, ſiue conſti­<lb></lb>
tuent lineam rectam F E G, ſicuti faciunt <lb></lb>
etiam duo illi anguli recti; ſiue etiam dica­<lb></lb>
mus, occupabunt idem ſpatium omninò, &amp; <lb></lb>
præcisè, quod occupant duo recti: v.g. ſi mucro B, ibi poneretur, faceret <lb></lb>
angulum F E H, &amp; ſi ibi iuxta ipſum apponeretur mucro A, faceret angulum <lb></lb>
H E I. quem ſi deinceps ſubſequetur reliquus angulus C, conſtitueret <expan abbr="reli-quũ">reli­<lb></lb>
quum</expan> angulum I E G. iam, vt vides, illi tres anguli ad E, tranſlati, ſunt æqua­<lb></lb>
les duobus rectis ad E, pariter conſtitutis, cum illi tres fiant partes duorum <lb></lb>rectorúm, vel quia occupant idem ſpatium, vel eandem lineam rectam F E G, <lb></lb>
conſtituant. </s>
          <s id="s.000783">habet igitur omne triangulum ſiue ęquilaterum, ſiue ſcalenum, <lb></lb>
ſiue Iſoſceles mirabilem hanc proprietatem, vt tres anguli, cuiuſuis trian­<lb></lb>
guli ſint æquales duobus rectis angulis. </s>
          <s id="s.000784">Quam demonſtrationem primi om­<lb></lb>
nium Pythagorici perfecerunt, vt refert Proclus ad 32. primi Elem. Eucli­<lb></lb>
des deinde ibidem aliter, quam Pythagorici idem demonſtrauit. </s>
          <s id="s.000785">Quod ſi <lb></lb>
quis huius rei <expan abbr="experiẽtiam">experientiam</expan> aliquam velit; etiamſi non exactam (cum æqua­<lb></lb>
litas mathematica non cadat ſub ſenſum, ſed ſola intelligentia percipiatur, <lb></lb>quippe quæ in materia intelligibili, non autem ſenſibili verſatur, &amp; cuius <pb pagenum="41" xlink:href="009/01/041.jpg"></pb>æqualitas nullum diſcrimen, quantumuis minimum admittat, quod ſenſui <lb></lb>
vitare ob ſui imperfectionem non licet: vnde inter eæ, quæ mathematicè <lb></lb>
ſunt æqualia, nullus intellectus aliquam valeat reperire differentiam) ſumat <lb></lb>
inquam triangulum quodpiam materiale, vt ex charta, quantum fieri po­<lb></lb>teſt perfectum, deinde ducat lineam vnam perpendicularem ſuper aliam, <lb></lb>
quæ ſcilicet faciat, cum illa duos angulos rectos. </s>
          <s id="s.000786">poſtea abſcindat tres an­<lb></lb>
gulos trianguli materialis, <expan abbr="eosq́">eosque</expan>; ita ſimul componat, vt mucrones illorum <lb></lb>
ſint vniti, &amp; contigui ad punctum lineæ perpendicularis cum altera, vti eſt <lb></lb>in ſuperiori figura punctum E; &amp; illicò apparebit tres illos angulos mate­<lb></lb>
riales obtegere adæquatè totum illud ſpatium duorum rectorum, quos per­<lb></lb>
pendicularis conſtituit. </s>
          <s id="s.000787">Hoc autem experiri poteris in diuerſis admodum <lb></lb>
triangulis Scalenis, Rectangulis, Iſoſcelibus, Aequilateris, &amp;c. </s>
          <s id="s.000788">non ſine de­<lb></lb>
lectatione, atque hic eſt ſenſus illorum verborum, omnis triangulus habet <lb></lb>
tres ęquales duobus rectis. </s>
          <s id="s.000789">Abſtineo à demonſtrationibus geometricis, quo­<lb></lb>
niam ij, qui Mathematicis ſunt imbuti, noſtra hac opera parum indigent. <lb></lb>
</s>
          <s id="s.000790">ſi quis tamen volet, conſulat 32. primi Elem. </s>
          <s id="s.000791">Ex hac igitur declaratione <lb></lb>
licet cognoſcere nonnullos ageometretos locum hunc, &amp; ſimiles ſubſequen­<lb></lb>
tes non ſatis intelligere, dicentes, nihil aliud verba illa Ariſt. velle ſignifi­<lb></lb>
care, quàm omnem triangulum habere tres angulos, quod inquiunt, notiſ­<lb></lb>
ſimum eſt. </s>
          <s id="s.000792">Sed ſi incidant in ſequentia; æquales duobus rectis, tunc, cum <lb></lb>
hæc non intelligant, abſtinent etiam à priorum declaratione, quibus præ­<lb></lb>
miſſis facile eſt Ariſt. textum percipere. </s>
          <s id="s.000793">ſit A, duo recti, ideſt, duo anguli <lb></lb>
recti ſint paſſio demonſtranda, in quo B, triangulus, in quo C, æquicrus. </s>
          <s id="s.000794">ipſi <lb></lb>
itaque C, ideſt triangulo æquicruſi, ineſt A, ſcilicet duo recti, hoc eſt, ineſt <lb></lb>
æquicruſi hæc, paſſio habere tres angulos æquales duobus rectis per B, ideſt <lb></lb>
per <expan abbr="triangulũ">triangulum</expan> vniuerſale, quia hæc proprietas eſt trianguli propria, &amp; <expan abbr="cõpe-tit">compe­<lb></lb>
tit</expan> æquicruſi, non vt æquicrus eſt, ſed, vt triangulum eſt; quare B, non erit <lb></lb>
medium ipſius A, quia prædicta paſſio. </s>
          <s id="s.000795">A, non competit triangulo B, per <lb></lb>
aliud, ſed per ſe, de eo enim primo, &amp; per ſe demonſtratur in 32. primi Elem. <lb></lb>
optimè Aegydius, &amp; Niphus in hunc locum.</s>
        </p>
        <p type="main">
          <s id="s.000796"><arrow.to.target n="marg11"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000797"><margin.target id="marg11"></margin.target>11</s>
        </p>
        <p type="main">
          <s id="s.000798">Ex eodem cap. <emph type="italics"></emph>(Non oportet autem exiſtimare penes id, quod exponimus, ali­<lb></lb>
quid accidere abſurdum, nihil enim vtimur eo, quod eſt hoc aliquid eſſe. </s>
          <s id="s.000799">ſed ſicut <lb></lb>
Geometra pedalem, &amp; rectam hanc, &amp; ſine latitudine dicit, quæ non ſunt. </s>
          <s id="s.000800">verum <lb></lb>
non ſic vtitur, tanquam ex his ratiocinans)<emph.end type="italics"></emph.end> Quoniam Ariſt. in exemplis affert <lb></lb>
pro rebus characteres, A, B, C, poſſet quiſpiam ſuſpicari aliquod propterea <lb></lb>
abſurdum accidere: cui ſuſpicioni Ariſt. reſpondet, dicens, nihil inde abſur­<lb></lb>
di accidere poſſe, quoniam ipſe vtitur hiſce literis, <expan abbr="nõ">non</expan> quatenus literæ ſunt, <lb></lb>
ſed quatenus rerum vicem, pro quibus exponuntur, gerunt: quemadmodum <lb></lb>
etiam Geometræ faciunt, qui lineam, quæ pedalis non eſt, pedalem, &amp; quæ <lb></lb>
non eſt recta, rectam; &amp; quæ lata eſt, non latam, ſupponunt, &amp; tamen nihil <lb></lb>
inde abſurdi contingit. </s>
          <s id="s.000801">Ex quibus intelligimus per lineas illas ſenſibiles, &amp; <lb></lb>
phyſicas, quas Geometræ in ſuis figuris ducunt, intelligendas eſſe lineas ve­<lb></lb>
rè Mathematicas omni latitudine carentes; vtitur enim inquit Ariſt. Geo­<lb></lb>
metra lineis phyſicis, non tanquam phyſicis, nec de eis tanquam de phyſicis <lb></lb>
lineis ratiocinatur, ſed ijs vtitur tanquam verè mathematicis. </s>
          <s id="s.000802">idem dicen­<lb></lb>dum eſt de ſuperficiebus, necnon de corporibus, quæ ijdem Geometræ de­<lb></lb>
ſcribunt, vt per ea, de verè mathematicis diſcurrant.</s>
        </p>
      </chap>
      <pb pagenum="42" xlink:href="009/01/042.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.000803"><emph type="italics"></emph>Ex Libro ſecundo Priorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000804"><arrow.to.target n="marg12"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000805"><margin.target id="marg12"></margin.target>12</s>
        </p>
        <p type="main">
          <s id="s.000806">Ex cap. 21. <emph type="italics"></emph>(Quod faciunt, qui coalternas putant ſcribere, latent enim ipſe<lb></lb>ſe ipſos talia accipientes, quæ non est poſſibile monstrare uon exiſtentibus <lb></lb>
coalternis)<emph.end type="italics"></emph.end> Vult Ariſt. exemplo mathematico explicare, quid ſit pe­<lb></lb>
titio principij. </s>
          <s id="s.000807">vbi per coalternas intelligit parallelas lineas, vox <lb></lb>
enim græca <foreign lang="grc">παραλληλος,</foreign> idem ſignificat, ac mutuus, &amp; coalternus. </s>
          <s id="s.000808">quoad <lb></lb>
exempli explicationem vtor figura textibus apponi ſolita, quæ eſt præſens. <lb></lb>
<figure id="id.009.01.042.1.jpg" place="text" xlink:href="009/01/042/1.jpg"></figure><lb></lb>
probat Euclides in 28. primi Elem. quod ſi <lb></lb>
linea recta quædam, vti E F, cadens ſuper <lb></lb>
duas rectas, vti ſunt A B, C D, fecerit angu­<lb></lb>
los alternos ęquales, angulos <expan abbr="nimirũ">nimirum</expan> A G H, <lb></lb>
G H D, ij enim dicuntur alterni; ſiue alios <lb></lb>
duos, nimirum B G H, G H C, hi enim ſunt <lb></lb>
<expan abbr="quoq;">quoque</expan> alterni; probat inquam has duas li­<lb></lb>
neas A B, C D, eſſe inuicem parallelas. </s>
          <s id="s.000809">Iam ſi quis vellet probare, ſe duas <lb></lb>
parallelas duxiſſe, hac ratione, quia ſcilicet faciunt prædictos angulos al­<lb></lb>
ternos æquales; &amp; probaret facere angulos alternos æquales, quia ſunt pa­<lb></lb>
rallelæ, hic peteret principium, ideſt, illud, quod principio <expan abbr="probandũ">probandum</expan> erat, <lb></lb>
afferret pro ratione, &amp; cauſa, quod dicitur peti principium, quia tunc pe­<lb></lb>
timus, vt concedatur nobis, id, quod principio, &amp; primo omnium demon­<lb></lb>
ſtrare propoſueramus. </s>
          <s id="s.000810">aduerte, quod characteres, qui ſunt in ſequentibus <lb></lb>
verbis huius loci, non appellant characteres figuræ appoſitæ; in quo quidam <lb></lb>
decepti, nullo pacto poterant locum hunc intelligere.</s>
        </p>
        <p type="main">
          <s id="s.000811"><arrow.to.target n="marg13"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000812"><margin.target id="marg13"></margin.target>13</s>
        </p>
        <p type="main">
          <s id="s.000813">Ex cap. 22. lib. 2. Priorum <emph type="italics"></emph>(Vt ſi volens monſtrare, quod diameter eſt incom­<lb></lb>
menſ. argueret Zenonis rationem, quod non eſt moueri)<emph.end type="italics"></emph.end> ſuperius ſecto 3. lib. 1. <lb></lb>
fusè explicauimus hanc aſymmetriam, quam ſi quis vellet demonſtrare ea­<lb></lb>
dem illa ratione, qua Zeno motum impugnabat, quia ſcilicet <expan abbr="mẽſura">menſura</expan> com­<lb></lb>munis, quæ debet vtramque, quantitatem menſurare, debet in menſurando <lb></lb>infinitas partes pertranſire, nimirum medietates medietatum in infinitum, <lb></lb>
eſt autem impoſſibile pertranſire infinitas huiuſmodi partes, &amp; propterea <lb></lb>
non poterit metiri, <expan abbr="neq;">neque</expan> vnam, <expan abbr="neq;">neque</expan> alteram ex <expan abbr="quãtitatibus">quantitatibus</expan>, quæ putaban­<lb></lb>
tur commenſurabiles, afferret hic, inquit Ariſt. non cauſam pro cauſa.</s>
        </p>
        <p type="main">
          <s id="s.000814"><arrow.to.target n="marg14"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000815"><margin.target id="marg14"></margin.target>14</s>
        </p>
        <p type="main">
          <s id="s.000816">Ex eodem cap. <emph type="italics"></emph>(Quoniam idem <expan abbr="vtiq;">vtique</expan> falſum per plures petitiones accidere <lb></lb>
nihil fortaſſe inconueniens, veluti coalternas coincidere; &amp; ſi maior eſt extrinſecus <lb></lb>
angulus intrinſeco; &amp; ſi triangulus habet plures rectos duobus)<emph.end type="italics"></emph.end> per plures poſi­<lb></lb>
tiones ſubaudi falſas. </s>
          <s id="s.000817">per coalternas intellige lineas æquidiſtantes, ſeu pa­<lb></lb>
rallelas, vt in ſuperiori cap. monuimus. </s>
          <s id="s.000818">Cæterum Euclides propoſ. </s>
          <s id="s.000819">28. pri­<lb></lb>
mi Elem. oſtendit, quod ſi fuerint duæ parallelæ veluti in præcedenti figura, <lb></lb>
A B, C D, ſuper quas alia recta E F, incidat, neceſſario faciet angulum ex­<lb></lb>
trinſecum E G B, v. g. æqualem interno, &amp; oppoſito, &amp; ad eaſdem partes, <lb></lb>
angulo videlicet G H D. ſi ergo inquit Ariſt ſupponamus iſtud falſum, an­<lb></lb>
gulum ſcilicet E G B, externum eſſe maiorem angulo interno G H D, ſequi­<lb></lb>
tur etiam falſum, videlicet lineas <expan abbr="æquidiſtãtes">æquidiſtantes</expan> A B, C D, concurrere. </s>
          <s id="s.000820">&amp; pro­<lb></lb>batur conſequentia hoc modo, quia ſi angulus E G B, maior eſt angulo <pb pagenum="43" xlink:href="009/01/043.jpg"></pb>G H D, appoſito <expan abbr="vtiq;">vtique</expan> communi angulo B G H, erant primum, duo anguli <lb></lb>
E G B, B G H, maiores, quam ſint duo B G H, G H D, quia ſi inæqualibus <lb></lb>
æqualia addantur, tota erunt inæqualia, vt prius per 4, axioma: hoc loco <lb></lb>
communis angulus additur ſemel maiori angulo, &amp; ſemel minori; &amp; ideo <lb></lb>
totum illud, in quo eſt maior angulus, adhuc maius eſt altero toto, in quo <lb></lb>
minor angulus continetur. </s>
          <s id="s.000821">at illi duo E G B, B G H, per 13. primi, ſunt <lb></lb>
æquales duobus rectis angulis, ergo duo <expan abbr="quoq;">quoque</expan> recti erunt maiores duobus <lb></lb>
internis B G H, D H G, ſiue hi duo interni erunt minores duobus rectis. <lb></lb>
</s>
          <s id="s.000822">At quando hi duo interni ſunt minores duobus rectis, tunc lineæ A B, C D, <lb></lb>
ſunt concurrentes, ſi protrahantur ad partes prædictorum <expan abbr="angulorũ">angulorum</expan>. </s>
          <s id="s.000823">quod <lb></lb>
P. Clauius luculenti, &amp; hactenus deſiderata demonſtratione ad 28. primi <lb></lb>
demonſtrauit. </s>
          <s id="s.000824"><expan abbr="Atq;">Atque</expan> hoc pacto ex prima falſa ſuppoſitione, nimirum angu­<lb></lb>
lum illum externum eſſe maiorem interno, &amp; oppoſito; ſequitur falſum, ni­<lb></lb>
mirum lineas parallelas concurrere.</s>
        </p>
        <p type="main">
          <s id="s.000825">Præterea ſi ſupponamus aliam falſitatem, ſcilicet triangulum habere tres <lb></lb>
angulos maiores duobus rectis, ſequetur eadem iterum falſitas, ſcilicet pa­<lb></lb>
<figure id="id.009.01.043.1.jpg" place="text" xlink:href="009/01/043/1.jpg"></figure><lb></lb>
rallelas coincidere, &amp; probatur ſic; ſint enim <lb></lb>
<expan abbr="triãguli">trianguli</expan> A B C, tres anguli maiores, quam duo <lb></lb>
recti anguli, &amp; per punctum C, ducta ſit recta <lb></lb>
C D, parallela lateri B A. quia ergo angulus <lb></lb>
A, æqualis eſt angulo ſibi alterno A C D, per <lb></lb>
29. primi, &amp; quia totalis angulus B C D, æqua­<lb></lb>
lis eſt duobus angulis B C A, A C D, quos tanquam ſuas partes adæquatas <lb></lb>
continet, quorum alter, ſcilicet A C D, eſt æqualis angulo A. erit idem to­<lb></lb>
talis angulus B C D, æqualis duobus angulis A, &amp; A C B, trianguli propoſi­<lb></lb>
ti. </s>
          <s id="s.000826">ergo totus iſte angulus B C D, ſimul cum reliquo <expan abbr="triãguli">trianguli</expan> angulo B. con­<lb></lb>
ſtabit compoſitionem ex tribus angulis trianguli dati: &amp; conſequenter ta­<lb></lb>
lis compoſitio trium angulorum erit maior, quam ſint duo anguli recti. </s>
          <s id="s.000827">ex <lb></lb>
quo ſequitur duas rectas B A, C D, ſuper quas cadit linea B C, faciens duos <lb></lb>
angulos internos, &amp; ad eaſdem partes, ſcilicet A B D, maiores duobus re­<lb></lb>
ctis non eſſe parallelas, ſed concurrentes (vt patet ex nuper citata demon­<lb></lb>
ſtratione P. Clauij) quod falſum eſt. </s>
          <s id="s.000828">&amp; ſequitur ex ſecunda falſa ſuppoſitio­<lb></lb>
ne. </s>
          <s id="s.000829">ex quibus textus Ariſt. videtur ſatis clarus.</s>
        </p>
        <p type="main">
          <s id="s.000830"><arrow.to.target n="marg15"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000831"><margin.target id="marg15"></margin.target>15</s>
        </p>
        <p type="main">
          <s id="s.000832">Ex cap. 26. <emph type="italics"></emph>(Vt ſi A, duo recti, in quo autem P., triangulus, in quo vero C, <lb></lb>ſenſibilis triangulus, ſuſpicari <expan abbr="namq;">namque</expan> poſſet aliquis non eſſe C, ſciens, quod omnis <lb></lb>
triangulus habet duos rectos: quare ſimul noſcet, &amp; ignorabit idem. </s>
          <s id="s.000833">noſce enim <lb></lb>
omnem triangulum, quod duobus rectis, non ſimplex eſt: ſed hoc quidem eo, quod <lb></lb>
vniuerſalem habet ſcientiam: illud autem eo, quod ſingularem. </s>
          <s id="s.000834">ſic igitur, vt vni­<lb></lb>
uerſale nouit C, quod duo recti; vt autem ſingulare non nouit, quare non habebit <lb></lb>
contrarias)<emph.end type="italics"></emph.end> vide, quæ diximus lib. 1. ſecto 3. cap. 1. ex quibus quidquid Ma­<lb></lb>
thematicum eſt hic, clarum redditur. </s>
          <s id="s.000835">reliqua verò, quæ ad Logicum ſpe­<lb></lb>
ctant, huius loci commentatores proſequuntur.</s>
        </p>
        <p type="main">
          <s id="s.000836">In cap. 31. de Abductione.</s>
        </p>
        <p type="main">
          <s id="s.000837"><arrow.to.target n="marg16"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000838"><margin.target id="marg16"></margin.target>16</s>
        </p>
        <p type="main">
          <s id="s.000839">Notandum hic cum eruditiſſimo Burana, Abductionem hanc, de qua in hoc <lb></lb>
cap. agitur eſſe vocem mathematicam, <expan abbr="camq́">eamque</expan>; Ariſt. quemadmodum multa <lb></lb>
alia à Mathematicis mutuatum ad omnes alias ſcientias tranſtuliſſe. </s>
          <s id="s.000840">eſſe <pb pagenum="44" xlink:href="009/01/044.jpg"></pb>autem terminum mathematicum colligitur manifeſtè ex Proelo, qui lib. 3. <lb></lb>
in comm. Elem. Euclidis ad primam propoſitionem primi Elementi, pag. <lb></lb>
</s>
          <s id="s.000841">121. ſic ait, Abductio verò eſt tranſitus à propoſito problemate, vel theo­<lb></lb>
remate ad aliud, quo cognito, aut comparato Propoſitum quoque perſpi­<lb></lb>
cuum eſt. </s>
          <s id="s.000842">Exempli cauſa, cum cubi duplicatio propoſita eſſet ad inueſti­<lb></lb>
gandam quæſtionem in aliud tranſtulere, quod illud propoſitum conſequi­<lb></lb>
tur, ad duarum nempe mediarum linearum inuentionem tranſlata eſt quæ­<lb></lb>
ſtio, &amp; ſic quærebant deinceps, quonam modo datis duabus rectis lineis, <lb></lb>
duæ mediæ proportionales reperirentur. </s>
          <s id="s.000843">Primum autem dicunt Hippocra­<lb></lb>
tem Chium prędictorum titulorum, Abductionem feciſſe, qui &amp; lunulæ qua­<lb></lb>
dratum fecit æquale, &amp; alia multa in Geometria inuenit. </s>
          <s id="s.000844">hæc Proclus. </s>
          <s id="s.000845">vbi <lb></lb>
non diſſimulandum nos reſtituiſſe verbum, Abductionem, cuius loco inter­<lb></lb>
pres Procli vtitur inductionis voce, ſequuti &amp; rationem, &amp; græcum textum, <lb></lb>
qui noſtram hanc expoſitionem euidenter poſtulat, <foreign lang="grc">απαγωγὴ</foreign> enim valet &amp; <lb></lb>
inductionem, &amp; abductionem, ſed abductio omnino rei propoſitæ quadrat.</s>
        </p>
        <p type="main">
          <s id="s.000846">Notandum præterea Hippocratem Chium fuiſſe auctorem huius Abdu­<lb></lb>
ctionis, <expan abbr="eumq́">eumque</expan>; feciſſe Abductionem à propoſito Problemate quadrandi cir­<lb></lb>
culi, vnde manifeſtè apparet, Ariſtotelem ex Mathematicis hunc terminum <lb></lb>
mutuò accepiſſe, quandoquidem ex ijſdem accepit etiam exemplum Abdu­<lb></lb>ctionis Mathematicæ, imò etiam exemplum ipſius authoris Abductionis <lb></lb>
Mathematicæ. </s>
          <s id="s.000847">ſyllogiſmus autem Hippocratis, quo oſtendebat circuli qua­<lb></lb>
draturam reducebatur ad has propoſitiones, omnis rectilinea figura qua­<lb></lb>
dratur, ſed circulus reducitur ad figuram rectilineam, ergo circulus qua­<lb></lb>
dratur. </s>
          <s id="s.000848">in probatione minoris facta eſt Abductio, cum enim ipſe vellet re­<lb></lb>
ctificare circumferentiam circuli per lunulas, nec valeret, alij per lineam, <lb></lb>
quandam quadratricem, vt eſt apud Pappum <expan abbr="Alexandrinũ">Alexandrinum</expan>, &amp; apud P. Cla­<lb></lb>
uium in fine ſexti Elem. &amp; alij aliter fruſtra conarentur, facta eſt Abductio <lb></lb>circa probationem minoris, in qua adhuc Mathematici verſantur; quæ pro­<lb></lb>
batio, ſi tandem inueniri poſſet, mox ſequeretur principale <expan abbr="propoſitũ">propoſitum</expan> pro­<lb></lb>
blema, nimirum circulus quadraretur; vide quæ ſcripſimus in cap. 3. Præ­<lb></lb>
dicam. de hac re, quia plurimum hunc conferunt. </s>
          <s id="s.000849">ſed iam ad textus expli­<lb></lb>
cationem veniamus.</s>
        </p>
        <p type="main">
          <s id="s.000850"><arrow.to.target n="marg17"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000851"><margin.target id="marg17"></margin.target>17</s>
        </p>
        <p type="main">
          <s id="s.000852">Ex eodem cap. <emph type="italics"></emph>(Veluti ſi K, eſſet quadrari, in quo autem E, rectilineum, in <lb></lb>
quo verò F, circulus, ſi ipſius E F, vnum ſolum eſſet medium, hoc, quod eſt, cum <lb></lb>
lunulis æqualem fieri circulum rectilineo, eſſe poſſet propè ipſum cognoſcere, cum <lb></lb>
vero B C, neque credibilius ſit, quam A C, <expan abbr="neq;">neque</expan> pauca media, non dico Abductio­<lb></lb>
nem: <expan abbr="neq;">neque</expan> quando B C, ſit immediatum, tale enim ſcientia est)<emph.end type="italics"></emph.end> Aduerte figuram <lb></lb>
vulgatæ editionis eſſe mendoſam, &amp; propterea reſtituendam eſſe, qualis pri­<lb></lb>
ma ſequens ex Simplicio ad tex. 11. primi Phyſic. hoc modo Hippocrates <lb></lb>
Chius conabatur circulum ad quadrum redigere; fit circulus A B G C, qua­<lb></lb>
drandus; conſtituatur <expan abbr="itaq;">itaque</expan> ſuper diametro eius B C, quadratum B C D F, <lb></lb>
cuius diameter B D, ſecatur bifariam in G, à circumferentia circuli dati, <lb></lb>
quod patet ducta ſemidiametro H G, perpendiculari ex B C, quæ ſuo extre­<lb></lb>
mo puncto G, ſecat bifariam, &amp; <expan abbr="diametrũ">diametrum</expan> B D, &amp; circumferentiam B G C. <lb></lb>
facto ergo centro G, deſcribatur alter circulus per puncta B C D F, <expan abbr="conne-ctaturq́">conne­<lb></lb>
ctaturque</expan>; recta G C. in triangulo orthogonio B C D, latus B D, ſubtenditur <pb pagenum="45" xlink:href="009/01/045.jpg"></pb><figure id="id.009.01.045.1.jpg" place="text" xlink:href="009/01/045/1.jpg"></figure><lb></lb>
angulo recto C, ergo quadratum eius ex corol­<lb></lb>
lario 47. primi, duplum erit quadrati B C, quare <lb></lb>
etiam circulus B C D F, duplus erit circuli A B­<lb></lb>
G C, per 2. duodecimi, &amp; ſemicirculus B C D, <lb></lb>
duplus erit ſemicirculi B A C: &amp; quadrans B E­<lb></lb>
C G, æqualis erit ſemicirculo B A C: ablato igi­<lb></lb>
tur communi ſegmento B E C H, remanet lunu­<lb></lb>
la B A C E, æqualis triangulo B C G, quod trian­<lb></lb>
gulum ſi per vltimam ſecundi quadretur, erit lu­<lb></lb>
nula B A C, conſequenter quadrata. </s>
          <s id="s.000853"><expan abbr="hucuſq;">hucuſque</expan> be­<lb></lb>
nè procedit Hippocrates. </s>
          <s id="s.000854">ſed vt reliquum circu­<lb></lb>
li quadret, ſic pergit, ponatur recta L M, dupla <lb></lb>
ipſius B C, ſupra quam ſemicirculus deſcribatur <lb></lb>
<figure id="id.009.01.045.2.jpg" place="text" xlink:href="009/01/045/2.jpg"></figure><lb></lb>
L O M, cui inſcribatur hexagoni <lb></lb>
æquilateri dimidium L Q S M, &amp; ſu­<lb></lb>
per tribus hexagoni lateribus, ſint <lb></lb>
tres ſemicirculi, vt in figura. </s>
          <s id="s.000855">&amp; <expan abbr="quo-niã">quo­<lb></lb>
niam</expan> diameter L M, dupla eſt <expan abbr="vniuſ-cuiuſq;">vniuſ­<lb></lb>
cuiuſque</expan> <expan abbr="diametrorũ">diametrorum</expan> B C, L Q, Q S, <lb></lb>
S M, erit ſemicirculus L O M, ęqua­<lb></lb>
lis quatuor ſemicirculis prædictis <lb></lb>
per 2. duodecimi, &amp; per 4. ſecundi <lb></lb>
ablatis igitur tribus <expan abbr="ſegmẽtis">ſegmentis</expan> com­<lb></lb>
munibus L N Q, Q O S, S P M, relinquetur trapezium L Q S M, æquale ſe­<lb></lb>
micirculo B A C, &amp; tribus lunulis L H Q N, Q R S O, S X M P, abſcindan­<lb></lb>
tur <expan abbr="itaq;">itaque</expan> de trapezio tria triangula æqualia tribus lunulis, eo modo, quo ſu­<lb></lb>
pra in prima figura factum eſt, &amp; quod relinquetur æquale erit ſemicirculo <lb></lb>
B A C. quod deinde quadretur per vlt. </s>
          <s id="s.000856">ſecundi, ſed aduerte, quod quando <lb></lb>
ait, abſcindantur de trapezio tria triangula æqualia lunulis, eo modo, quo <lb></lb>
ſupra, committit deceptionem, quia eodem modo, quo ſupra minimè id fa­<lb></lb>
cere poſſumus, quia in ſuperiori figura triangula erant conſtituta ſuper la­<lb></lb>
tus B C, quadrati B C D F, intra circulum deſcripti, qui circulus facit cum <lb></lb>
B C, maius <expan abbr="ſegmentũ">ſegmentum</expan>, quam faciat ſemicirculus L O M, cum lateribus L Q, <lb></lb>
Q S, S M. &amp; propterea ſemicirculus iſte non habet eandem proportionem <lb></lb>
ad vnamquamque lunularum ſuarum, quam habet ſemicirculus ſuperior <lb></lb>
B C D, ad lunulam B A C E. <expan abbr="atq;">atque</expan> hæc eſt fallacia, quam authorem ſuum mi­<lb></lb>
nimè latuiſſe putandum, cuius Ariſt. ſæpius mentionem in ſequentibus fa­<lb></lb>
ciet : quì enim fieri poteſt, vt tam acutus inuentor, adeo manifeſtum erro­<lb></lb>
rem non vidiſſet, verum propter adinuenti excellentiam, authori ſuo pla­<lb></lb>
cuit paralogyſmus. </s>
          <s id="s.000857">mirabilis tamen ſemper habita eſt illa ſuperior lunulæ <lb></lb>
quadratio. </s>
          <s id="s.000858">Ex quibus ſatis clara eſſe poſſunt ea, quæ ad <expan abbr="Mathematicũ">Mathematicum</expan> per­<lb></lb>
tinent, ad locum hunc de Abductione declarandum. </s>
          <s id="s.000859">facta eſt igitur abdu­<lb></lb>
ctio ab Hippocrate in quadratione trium poſteriorum lunularum, in qua­<lb></lb>
rum quadratione diu immoratus, nunquam niſi cum paralogyſmo quadra­<lb></lb>
re valuit. </s>
          <s id="s.000860">Hæc pluribus, vt ſequentibus etiam textibus, in quibus huius te­<lb></lb>
tragoniſmi fit mentio ſatisfacere poſſimus. </s>
          <s id="s.000861">Hippocrates iſte Chius eſt alter <pb pagenum="46" xlink:href="009/01/046.jpg"></pb>ab illo Hippocrate Coo medicorum Magiſtro, vt colligitur ex Alexandre <lb></lb>
Aphrod. in Primum Meteororum de Cometis.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.000862"><emph type="italics"></emph>Ex Primo Posteriorum reſolutoriorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000863"><arrow.to.target n="marg18"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000864"><margin.target id="marg18"></margin.target>18</s>
        </p>
        <p type="main">
          <s id="s.000865">Textu primo <emph type="italics"></emph>(Omnis doctrina, &amp; omnis diſciplina diſcurſiua ex præexi­<lb></lb>
ſtenti fit cognitione. </s>
          <s id="s.000866">manifeſtum autem hoc ſpeculantibus in omnibus, <lb></lb>
Mathematicæ <expan abbr="namq;">namque</expan> ſcientiarum per hunc modum accedunt)<emph.end type="italics"></emph.end> quo mo­<lb></lb>
do Mathematicæ fiant ex præcedenti cognitione, ſcilicet Princi­<lb></lb>
piorum perſpicuè quilibet videbit, qui ſaltem primum <expan abbr="Elemẽtorum">Elementorum</expan> Eucli­<lb></lb>
dis, vel è ianuis inſpexerit; pręcedunt enim primo principiorum tria gene­<lb></lb>
ra, quorum primum continet definitiones ſubiecti Geometriæ, vt definitio­<lb></lb>
nes lineæ, ſuperficiei, trianguli, &amp;c: Secundum continet Poſtulata. </s>
          <s id="s.000867">Tertium <lb></lb>
Axiomata, ſeu communes omnium conceptiones, &amp; ſententias, ex quibus <lb></lb>tanquam ex vberrimis, &amp; chriſtallinis fontibus Demonſtrationes Geome­<lb></lb>
tricæ deriuantur. </s>
          <s id="s.000868">Idem vìdere licet in operibus aliorum Geometrarum, <lb></lb>
Archimedis, Apollonij, Pappi, &amp; cæterorum. </s>
          <s id="s.000869">Aliæ ſimiliter mathematicæ, <lb></lb>
vt Arithmetica, Perſpectiua, Muſica, Mechanica, Aſtronomia, non niſt ex <lb></lb>
præmiſſis, ac manifeſtiſsimis principijs ſuas demonſtrationes deducunt. <lb></lb>
</s>
          <s id="s.000870">Nulla porrò alia ſcientia tam diſtinctè ſua præmittit principia, <expan abbr="tamq́">tamque</expan>; per­<lb></lb>
ſpicua, ſicuti Mathematicæ, vt non immeritò Philoſophus eas, tamquam <lb></lb>
veræ ſcientiæ <expan abbr="typũ">typum</expan>, <expan abbr="eumq́">eumque</expan>; omnibus numeris abſolutum ſibi ob oculos pro­<lb></lb>
poſuerit, ex quo veræ ſcientiæ deſcriptionem hiſce libris complecteretur.</s>
        </p>
        <p type="main">
          <s id="s.000871"><arrow.to.target n="marg19"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000872"><margin.target id="marg19"></margin.target>19</s>
        </p>
        <p type="main">
          <s id="s.000873">Tex. 2. <emph type="italics"></emph>(Quod enim omne triangulum habet duobus rectis æquales, præſciuit: <lb></lb>
quod autem hoc, quod eſt in ſemicirculo triangulum eſt, ſimul inducens cognouit)<emph.end type="italics"></emph.end><lb></lb>
vide primo, quæ ſupra libro 1. Prior. ſecto 3. cap. 1. explicaui de angulis <lb></lb>
trianguli. </s>
          <s id="s.000874">deinde ſcias, quod quando Ariſt. ait, hoc, quod eſt in ſemicircu­<lb></lb>
lo triangulum, &amp;c. </s>
          <s id="s.000875">alludit ad demonſtrationem quandam, quam ipſe infe­<lb></lb>
rius in exemplum adducet, &amp; quæ eſt in 3. Elem. Euclidis 31. in qua talis fi­<lb></lb>
gura proponitur qualis eſt præſens, in qua vides triangulum A B C. in ſe­<lb></lb>
<figure id="id.009.01.046.1.jpg" place="text" xlink:href="009/01/046/1.jpg"></figure><lb></lb>
micirculo. </s>
          <s id="s.000876">tunc autem dicitur triangulum in <lb></lb>
ſemicirculo, quando baſis ipſius eſt diameter <lb></lb>
ſemicirculi, &amp; reliqua duo latera ita concur­<lb></lb>runt ſimul in angulum B, vt ipſum pariter in <lb></lb>
circumferentia conſtituant, quibus pręmiſsis <lb></lb>
ſic textum explicaueris: quod enim omne <lb></lb>
triangulum habet tres angulos æquales duo­<lb></lb>
bus rectis angulis præſciuit vniuerſaliter per <lb></lb>
32. primi; quod autem hoc particulare triangulum A B C, quod eſt in ſe­<lb></lb>micirculo habeat eandem proprietatem, ſimul, ac quiſpiam animaduertit <lb></lb>
illud eſſe triangulum cognoſcit, <expan abbr="abſq;">abſque</expan> vlla demonſtratione, ſed ſolum virtu­<lb></lb>
te illius maioris propoſitionis; omne triangulum habet tres, &amp;c.</s>
        </p>
        <p type="main">
          <s id="s.000877"><arrow.to.target n="marg20"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000878"><margin.target id="marg20"></margin.target>20</s>
        </p>
        <p type="main">
          <s id="s.000879">Tex. 5. <emph type="italics"></emph>(Vera quidem igitur oportet eſſe, quoniam non eſt non ens ſcire, vt quod <lb></lb>diameter ſit commenſurabilis)<emph.end type="italics"></emph.end> conſule ea, quæ ſcripſimus ad cap. 23. primi <lb></lb>
Priorum, ſecto 1. ſine quibus locus hic ſatis intelligi nequit; ijs autem per­<lb></lb>
ceptis ſic <expan abbr="locũ">locum</expan> hunc explicare poſſumus, cum diameter quadrati ſit incom­


<pb pagenum="47" xlink:href="009/01/047.jpg"></pb>menſurabilis lateri ſui quadrati, falſum erit dicere diametrum eſſe com­<lb></lb>
menſurabilem prædicto lateri, quod autem falſum eſt, illud non eſt; igitur <lb></lb>
impoſsibile eſt ſcire diametrum eſſe commenſurabile.</s>
        </p>
        <p type="main">
          <s id="s.000880"><arrow.to.target n="marg21"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000881"><margin.target id="marg21"></margin.target>21</s>
        </p>
        <p type="main">
          <s id="s.000882">Hoc eodem cap. plura dicuntur de Principijs Demonſtrationis, ſiue ſcien­<lb></lb>
tiæ, vt ſunt Dignitates, Poſitiones, Definitiones, &amp; ſimilia, quæ quo modo <lb></lb>
ſe habeant, &amp; quo modo illis Demonſtrationes innitantur, optimè ex con­<lb></lb>
templatione primi libri Elem. Euclidis percipi poteſt. </s>
          <s id="s.000883">vt propterea benè ij <lb></lb>
ſentiant, inter quos præcipui ſunt Toletus, &amp; Zabarella, qui aſſerunt, Ariſt. <lb></lb>
Mathematicas ſcientias tamquam typum perfectiſsimarum ſcientiarum <lb></lb>
ſibi ob oculos propoſuiſſe; ex quo typo veræ ſcientiæ deſcriptionem his li­<lb></lb>
bris complectaretur.</s>
        </p>
        <p type="main">
          <s id="s.000884"><arrow.to.target n="marg22"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000885"><margin.target id="marg22"></margin.target>22</s>
        </p>
        <p type="main">
          <s id="s.000886">Eodem tex. 5. <emph type="italics"></emph>(Ponit enim Arithmeticus vnitatem indiuiſibilem eſſe ſecun­<lb></lb>
dum quantum)<emph.end type="italics"></emph.end> hoc quamquam non ponatur ab Arithmeticis expreſsè, præ­<lb></lb>
ſupponitur tamen ab eis: nuſquam enim Euclides in totis tribus Arithme­<lb></lb>
ticis libris, infra vnitatem deſcendit, vt propterea appareat, ipſam in quan­<lb></lb>
titate diſcreta eſſe minimum, &amp; indiuiſibile. </s>
          <s id="s.000887">Verum dubitabit fortè quiſ­<lb></lb>
piam hoc modo, ſi vnitas minimum, <expan abbr="atq;">atque</expan> indiuiſibile eſt in quanto diſcreto, <lb></lb>
qua igitur ratione Arithmetici practici eam diuidunt in dimidium, in trien­<lb></lb>
tem, in quadrantem, &amp; alijs ſimiliter modis, vnde numeri illi, qui fractio­<lb></lb>
nes appellantur, exurgunt? </s>
          <s id="s.000888">Reſpondemus, <expan abbr="quotieſeunq;">quotieſcunque</expan> vnitas diuiditur ab <lb></lb>
Arithmeticis, tunc ipſi eam accipiunt tanquam totum quoddam <expan abbr="cõtinuum">continuum</expan> <lb></lb>
in plures partes diuiſibile: ſiue tanquam aggregatum quoddam vnitatum, <lb></lb>
quæ vnitates ſunt partes illius, vt quando dicunt, vnum horæ quadrantem, <lb></lb>
vel duos horæ quadrantes, vel tres horæ quadrantes, accipiunt horam tan­<lb></lb>
quam aggregatum quatuor quadrantum, &amp; propterea numeri illi 1/4. 2/4. 3/4. <lb></lb>
&amp; ſimiles fractiones, nihil aliud ſunt, quam numeri partium vnius horæ: ex <lb></lb>
quo patet huiuſmodi fractiones omnes reduci ad numeros integros, qui <lb></lb>
enim dicit tres quadrantes 3/4. dicit tres partes alicuius totius, quod intel­<lb></lb>
ligitur diuiſum eſſe in 4. æquales partes, ex quibus illæ tres tantummodo <lb></lb>
numerat.</s>
        </p>
        <p type="main">
          <s id="s.000889"><arrow.to.target n="marg23"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000890"><margin.target id="marg23"></margin.target>23</s>
        </p>
        <p type="main">
          <s id="s.000891">Tex. 9. <emph type="italics"></emph>(Per ſe autem, <expan abbr="quæcunq;">quæcunque</expan> &amp; inſunt in eo, quod quid eſt, vt triangulo li­<lb></lb>
nea, &amp; lineæ punctum; ſubſtantia <expan abbr="namq;">namque</expan> ipſorum ex his eſt, &amp; in oratione dicen­<lb></lb>
te, quid eſt, inſunt)<emph.end type="italics"></emph.end> aggreditur explicare quænam ſint ea, quæ per ſe dicun­<lb></lb>
tur: <expan abbr="quotq́">quotque</expan>; modis dicatur aliquid per ſe. </s>
          <s id="s.000892">quorum primus eſt, ea ſcilicet, <lb></lb>
per ſe de aliquo ſubiecto dici, <expan abbr="quæcunq;">quæcunque</expan> in definitione illius ponuntur, cu­<lb></lb>
iuſmodi ſunt linea, &amp; punctum, quæ per ſe prædicantur, illa de triangulo, <lb></lb>iſtud de linea; in definitione enim trianguli ponitur linea recta, quia linea <lb></lb>
recta dum terminat illam ſuperficiem, quæ dicitur triangulus illi trianguli <lb></lb>
naturam impertitur, &amp; ideo triangulus definitur ſic, triangulus eſt figura <lb></lb>
tribus lineis rectis terminata. </s>
          <s id="s.000893">ſimiliter in definitione lineæ, non infinitæ, <lb></lb>
ſed finitæ, &amp; terminatæ ponitur punctum, quia duo puncta, quæ ſunt extre­<lb></lb>
ma illius, faciunt, vt ea ſit line a finita, &amp; definitur ſic, linea finita eſt lon­<lb></lb>gitudo, cuius extrema ſunt puncta. </s>
          <s id="s.000894">quamuis autem hæc definitio apud Eu­<lb></lb>
clidem expreſſa non habeatur, tamen ex definitionibus ipſius præſertim ſe­<lb></lb>
cunda, tertia, &amp; quarta elici poteſt.</s>
        </p>
        <p type="main">
          <s id="s.000895"><arrow.to.target n="marg24"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000896"><margin.target id="marg24"></margin.target>24</s>
        </p>
        <p type="main">
          <s id="s.000897">Eodem tex. 9. <emph type="italics"></emph>(Et <expan abbr="quibuſcunq;">quibuſcunque</expan> inexiſtentium ipſis, ipſæ ſunt in oratione, quid<emph.end type="italics"></emph.end> <pb pagenum="48" xlink:href="009/01/048.jpg"></pb><emph type="italics"></emph>est declarante, quemadmodum rectum ineſt lineæ, &amp; circulare: &amp; impar, &amp; par <lb></lb>
numero, &amp; primum, &amp; compoſitum, &amp; æquilaterum, &amp; altera parte longius. &amp; <lb></lb>
<expan abbr="oĩbus">omnibus</expan> bis inſunt in oratione, quid eſt <expan abbr="declarãte">declarante</expan>, ibi quidem linea, hic vero numerus)<emph.end type="italics"></emph.end><lb></lb>
quia locus hic benè exponitur à Toleto, &amp; melius etiam à Conymbr. </s>
          <s id="s.000898">addam <lb></lb>
tantummodo quædam, quæ ad perfectam eius intelligentiam deſiderantur. <lb></lb>
</s>
          <s id="s.000899">Sciendum igitur primò, nuſquam ab Euclide definiri rectum, circulare, <lb></lb>
impar, par, primum, compoſitum, æquilaterum, nec altera parte longius: <lb></lb>
<expan abbr="verũ">verum</expan> ab ipſo in definitionibus primi definiri lineam rectam, non tamen cir­<lb></lb>
cularem expreſsè. </s>
          <s id="s.000900">in definitionibus deinde ſeptimi definiri <expan abbr="numerũ">numerum</expan> parem, <lb></lb>
&amp; imparem, item numerum primum, &amp; compoſitum, &amp; æquilaterum, &amp; al­<lb></lb>
tera parte longiorem. </s>
          <s id="s.000901">ex quibus definitionibus poſſunt erui definitiones re­<lb></lb>
cti, circularis, imparis, &amp; cæterorum, quorum hic Ariſtoteles meminit. <lb></lb>
</s>
          <s id="s.000902">Cæterum Euclides definitione 11. ſeptimi, ſic definit numerum primum: <lb></lb>
primus numerus eſt, quem vnitas ſola metitur. </s>
          <s id="s.000903">numerus autem, vel vnitas <lb></lb>
metiri dicitur alium numerum, quando ſæpius repetita ipſum omnino ad­<lb></lb>
æquat, vt ternarius metitur nouenarium, quia ter repetitus ipſum ad vn­<lb></lb>
guem explet. </s>
          <s id="s.000904">illi igitur numeri dicuntur ab Arithmeticis primi, qui à nullo <lb></lb>
alio, præterquam ab vnitate menſurantur, quales ſunt, 2. 3. 5. 7. &amp;c. </s>
          <s id="s.000905">Defi­<lb></lb>
nitione verò 13. definit numerum compoſitum ſic; compoſitus numerus eſt, <lb></lb>
quem numerus quiſpiam metitur, vt ſenarius erit compoſitus, quia ipſum <lb></lb>
binarius metitur, nam ter repetitus, ipſi perfectè adæquatur.</s>
        </p>
        <p type="main">
          <s id="s.000906">Per æquilaterum, intelligit quadratum, quadratus autem numerus defi­<lb></lb>
nitione 18. ſeptimi ſic explicatur: Quadratus numerus eſt, qui ſub duobus <lb></lb>
æqualibus numeris continetur, ideſt, qui fit ex ductu vnius numeri in ſe ip­<lb></lb>
<figure id="id.009.01.048.1.jpg" place="text" xlink:href="009/01/048/1.jpg"></figure><lb></lb>
ſum, vt ſi ducantur 3. in 3. fient 9. qui continetur ſub duobus <lb></lb>
ternarijs; omnes autem ternarij ſunt æquales. </s>
          <s id="s.000907">is autem nu­<lb></lb>
merus dicetur quadratus, quia, vt apparet in figura, nouem <lb></lb>
ipſius vnitates poſſunt in plano ita ad inuicem collocari, vt <lb></lb>
referant quadratum; &amp; ſicuti quadratum geometricum ha­<lb></lb>
bet latera æqualia, ita etiam quadratum arithmeticum: ſi­<lb></lb>
ue numerus quadratus, habet ſua latera æqualia, quot enim vnitates ſunt <lb></lb>
in vno, tot etiam ſunt in reliquis, vt in præſenti ſunt tres vnitates in ſingulis <lb></lb>
lateribus. </s>
          <s id="s.000908">pręterea quemadmodum quadratum geometricum reſolui poteſt <lb></lb>
in plura quadrata, ita etiam arithmeticum, vt præſens, qui reſoluitur in <lb></lb>
quatuor quadrata arithmetica. </s>
          <s id="s.000909"><expan abbr="Neq;">Neque</expan> enim poteſt quilibet numerus, vt opi­<lb></lb>
nantur ageometreti, in hunc modum diſponi, ſed ſolum ij, qui producuntur <lb></lb>
ex multiplicatione numeri alicuius in ſe ipſum.</s>
        </p>
        <p type="main">
          <s id="s.000910">Per altera parte longius, intelligit numerum, qui producitur à duobus <lb></lb>
<figure id="id.009.01.048.2.jpg" place="text" xlink:href="009/01/048/2.jpg"></figure><lb></lb>
numeris inæqualibus inuicem multiplicatis, qualis eſt <lb></lb>
duodenarius, qui ex ductu trium in quatuor produci­<lb></lb>
tur, &amp; refert figuram altera parte longiorem, ſiue, vt <lb></lb>
ait Boetius longilateram, cuius vnum latus eſt maius <lb></lb>
altero, vt in appoſita figura videre licet. </s>
          <s id="s.000911">atque hæc <lb></lb>
ſunt, quæ ex Mathematicis petenda erant, ad huius <lb></lb>
loci intelligentiam.</s>
        </p>
        <p type="main">
          <s id="s.000912"><arrow.to.target n="marg25"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000913"><margin.target id="marg25"></margin.target>25</s>
        </p>
        <p type="main">
          <s id="s.000914">Tex. 11. <emph type="italics"></emph>(Per ſe autem, &amp; ſecundum quod ipſum, idem, vt per ſe lineæ inest<emph.end type="italics"></emph.end> <pb pagenum="49" xlink:href="009/01/049.jpg"></pb><emph type="italics"></emph>punctum, &amp; rectum; etenim ſecundum quod linea, &amp; triangulo, ſecundum quod <lb></lb>
triangulum duo recti: etenim per ſe triangulum duobus rectis æquale. </s>
          <s id="s.000915">Vniuerſale <lb></lb>
autem eſt tunc, quando in quolibet, &amp; primo monſtratur, vt duos rectos habere, <lb></lb>
<expan abbr="neq;">neque</expan> figuræ eſt vniuerſale, quamuis eſt monſtrare de figura, quod duos rectos habet, <lb></lb>
ſed non de qualibet figura, <expan abbr="neq;">neque</expan> vtitur qualibet figura monstrans, quadrangulum <lb></lb>
enim figura a quidem est, non habet autem duobus rectis æquales. </s>
          <s id="s.000916">Aequicrus verò <lb></lb>habet quidem <expan abbr="quodcunq;">quodcunque</expan> duobus rectis æquales, ſed non primò, ſed triangulum <lb></lb>
prius. </s>
          <s id="s.000917">quod igitur quoduis primum monſtratur duos rectos habens, aut <expan abbr="quodcunq;">quodcunque</expan> <lb></lb>
aliud, huic primo ineſt vniuerſale, &amp; demonstratio de hoc vniuerſaliter eſt, de alijs <lb></lb>
verò quodammodo, non per ſe, <expan abbr="neq;">neque</expan> de æquicrure eſt vniuerſaliter, ſed in plus)<emph.end type="italics"></emph.end> pro <lb></lb>
quorum intelligentia neceſſaria ſunt ea, quæ primo Priorum ſecto 3. cap. 1. <lb></lb>
ſcripſimus. </s>
          <s id="s.000918">deinde memineris figuram vniuerſaliorem eſſe triangulo, &amp; tri­<lb></lb>
angulum vniuerſalius æquicrure. </s>
          <s id="s.000919">quando ait (vt duos rectos habere) vult <lb></lb>
dicere, habere duos angulos rectos non actu, ſed potentia; quæ affectio eſt <lb></lb>
trianguli, quia, vt ſuperius diximus, habet tres angulos æquales duobus <lb></lb>
rectis angulis: quæ proprietas vniuerſaliter, &amp; primò competit triangulo. <lb></lb>
</s>
          <s id="s.000920">non autem figuræ, quia figura eſt vniuerſalior. </s>
          <s id="s.000921"><expan abbr="neq;">neque</expan> iſoſceli, quia iſoſceles eſt <lb></lb>
reſtrictius triangulo. </s>
          <s id="s.000922">omittimus reliqua ſingillatim exponere, tum quia ſa­<lb></lb>
tis clara ſunt, tum quia ab interpretibus benè explicantur.</s>
        </p>
        <p type="main">
          <s id="s.000923"><arrow.to.target n="marg26"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000924"><margin.target id="marg26"></margin.target>26</s>
        </p>
        <p type="main">
          <s id="s.000925">Tex. 13. <emph type="italics"></emph>(Si quis igitur monſtrauerit, quod rectæ <expan abbr="nõ">non</expan> coincidunt, videbitur <expan abbr="vtiq;">vtique</expan> <lb></lb>huius eſſe demonstratio, eo quod in omnibus eſt rectis; non eſt autem: ſi quidem <lb></lb>
non quoniam ſic æquales, fit hoc, ſed ſecundum quod <expan abbr="quomodocunq;">quomodocunque</expan> æquales)<emph.end type="italics"></emph.end> pro­<lb></lb>
ponit tres errores, qui circa demonſtrationem de vniuerſali contingunt, <lb></lb>
quos omnes Geometricis exemplis illuſtrat; affert autem primo pro tertio <lb></lb>
errore duo exempla, quorum primum in præmiſſis verbis continetur, <expan abbr="atq;">atque</expan> <lb></lb>
ex 28. primi Elem. deſumitur, quam propterea primo loco exponendam <lb></lb>
<figure id="id.009.01.049.1.jpg" place="text" xlink:href="009/01/049/1.jpg"></figure><lb></lb>
cenſui. </s>
          <s id="s.000926">Quando igitur duæ rectæ conſtitu­<lb></lb>
tæ fuerint, vt A B, C D, in quas alia recta, <lb></lb>
vt G F, incidens, faciat duos angulos in­<lb></lb>
ternos, reſpectu rectarum A B, C D, &amp; ad <lb></lb>
eaſdem partes rectæ E F, vt ſunt ex parte <lb></lb>
ſiniſtra anguli A G H, C H G; exparte ve­<lb></lb>
rò dextra B G H, D H G; ſi <expan abbr="inquã">inquam</expan> linea E F, <lb></lb>
fecerit duos illos angulos ex parte ſiniſtra ſimul ſumptos, æquales duobus <lb></lb>
rectis angulis, vel duos ex parte dextra pariter æquales duobus rectis, pro­<lb></lb>
bat Euclides rectas A B, C D, non concurrere, ſiue parallelas eſſe. </s>
          <s id="s.000927">Verum, <lb></lb>
quia linea E F, poteſt facere aliquando prædictos angulos non <expan abbr="tantũ">tantum</expan> æqua­<lb></lb>
les duobus rectis, verum etiam rectos, quo etiam modo <expan abbr="probarẽtur">probarentur</expan> cædem <lb></lb>
lineæ eſſe parallelæ, vt in ſequenti figura, cum ſint anguli A G I, C I G, re­<lb></lb>
<figure id="id.009.01.049.2.jpg" place="text" xlink:href="009/01/049/2.jpg"></figure><lb></lb>
cti, probabitur de rectis A B, C D, æquidiſtan­<lb></lb>
tia. </s>
          <s id="s.000928">Ex his facile textum in hunc modum expo­<lb></lb>
nemus; ſi quis igitur monſtrauerit, quod rectæ <lb></lb>A B, C D, nunquam coincidunt, etiamſi in infi­<lb></lb>
nitum producantur, ſeu quod ſunt æquidiſtantes, <lb></lb>
quando anguli prædicti interni ſunt duo recti, <lb></lb>
videbitur <expan abbr="vtiq;">vtique</expan> huius eſſe demonſtratio de vniuerſali per ſe, &amp; de primo ſu­


<pb pagenum="50" xlink:href="009/01/050.jpg"></pb>biecto, vel ſecundum quod ipſum, eò quod probatur vniuerſaliter de lineis <lb></lb>
omnibus habentibus prædictos angulos rectos. </s>
          <s id="s.000929">non autem de omni, ſecun­<lb></lb>
dum quod ipſum, ſi quidem non competit affectio hæc, eſſe parallelas, li­<lb></lb>
neis habentibus illos angulos rectos actu; ſed primò, &amp; vniuerſaliter, &amp; ſe­<lb></lb>
cundum quod ipſum competit lineis habentibus illos angulos æquales duo­<lb></lb>
bus rectis, <expan abbr="quomodocunq;">quomodocunque</expan> æquales ſint duobus rectis, ſiue ambo ſint recti, <lb></lb>
ſiue vnus acutus, alter obtuſus, ſed tamen ambo ſimul æquentur duobus re­<lb></lb>
ctis, quales ſunt lineæ primæ figuræ. </s>
          <s id="s.000930">In tertio igitur errore, vniuerſale exi­<lb></lb>
ſtit quidem, &amp; habet nomen, ſed tamen prætermittetur, ſeu ſtrictius ſume­<lb></lb>
tur, quam oportet. </s>
          <s id="s.000931">alij latini, quos quidem viderim, præter Zabarellana <lb></lb>
perperam omnino ob mathematicarum ignorantiam, exemplum iſtud in­<lb></lb>
terpretantur.</s>
        </p>
        <p type="main">
          <s id="s.000932"><arrow.to.target n="marg27"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000933"><margin.target id="marg27"></margin.target>27</s>
        </p>
        <p type="main">
          <s id="s.000934">Ibidem <emph type="italics"></emph>(Et ſi triangulum non eſſet aliud, quam Iſoſceles, ſecundum quod Iſo­<lb></lb>
ſceles videretur <expan abbr="vtiq;">vtique</expan> ineſſe)<emph.end type="italics"></emph.end> iſtud eſt ſecundum exemplum tertij erroris. </s>
          <s id="s.000935">Por­<lb></lb>
rò cum tres ſint ſpecies triangulorum, æquilaterum, Iſoſceles, Scalenum, ſi <lb></lb>
accideret, vt ex illis tribus vna tantum ſpecies, v. g. Iſoſceles in mundo re­<lb></lb>
periretur; <expan abbr="tuncq́">tuncque</expan>; quiſpiam de Iſoſcele oſtenderet affectionem quampiam, <lb></lb>
putans ſe <expan abbr="oſtẽdiſſe">oſtendiſſe</expan> paſſionem de proprio ſubiecto, &amp; primo, falleretur, quia <lb></lb>
aifectio illa competeret Iſoſceli, non vt huic ſpeciei Iſoſcelis, ſed quatenus <lb></lb>
eſt triangulum, cui primo, &amp; per ſe, &amp; ſecundum quod ipſum conuenit. </s>
          <s id="s.000936">hoc <lb></lb>
loco diſceſſimus à Zabarella, qui putat iſtud eſſe exemplum primi erroris, <lb></lb>
cum verba textus adeo clara ſint, vt expoſitionem illius nullo modo admit­<lb></lb>
tant. </s>
          <s id="s.000937">ſunt autem hæc textus verba <emph type="italics"></emph>(Et ſi triangulum non eſſet aliud, quam Iſo­<lb></lb>
ſceles, ſecundum quod Iſoſceles videretur <expan abbr="vtiq;">vtique</expan> ineſſe)<emph.end type="italics"></emph.end> quibus verbis manifeſtè <lb></lb>
apparet Ariſt. accipere pro ſubiecto vniuerſali non indiuiduum vnum, vt in <lb></lb>
primo errore contingit, ſed ſpeciem loco generis, ſcilicet Iſoſceles, quod <lb></lb>
eſt ſpecies trianguli pro genere ipſo, nimirum pro Triangulo. </s>
          <s id="s.000938">ait enim, ſi <lb></lb>
non eſſet aliud, quam Iſoſceles, <expan abbr="ſecundũ">ſecundum</expan> quod Iſoſceles: quibus verbis cla­<lb></lb>rè ſpeciem, non indiuiduum, ſignificat, ex his duobus exemplis manifeſtus <lb></lb>
eſt tertius error, qui erat, quando erat <emph type="italics"></emph>(vt in parte totum)<emph.end type="italics"></emph.end> <expan abbr="quodq́">quodque</expan>; illis verbis <lb></lb>
expoſuerat <emph type="italics"></emph>(vel contingit etiam, vt in parte totum, in quo monſtratur: ijs enim, <lb></lb>
quæ ſunt in parte inerit quidem demonſtratio, &amp; erit de omni, ſed tamen non erit <lb></lb>
huius primi vniuerſaliter demonſtratio. </s>
          <s id="s.000939">dico autem huius primi, ſecundum quod <lb></lb>
huius demonstrationem, quando ſit primi vniuerſaliter)<emph.end type="italics"></emph.end> ideſt, quando vniuerſale <lb></lb>
ſubiectum exiſtit quidem, ſed tamen non de ipſo ſit demonſtratio, ſed de ali­<lb></lb>
qua parte ipſius, v. g. de ſpecie aliqua demonſtratur aliquid, quod deberet <lb></lb>
oſtendi primò de ipſo vniuerſali, cum illi primò competat.</s>
        </p>
        <p type="main">
          <s id="s.000940"><arrow.to.target n="marg28"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000941"><margin.target id="marg28"></margin.target>28</s>
        </p>
        <p type="main">
          <s id="s.000942">Ibidem <emph type="italics"></emph>(Et proportionale, quod alternatim, ſecundum quod numeri, &amp; ſecun­<lb></lb>
dum quod lineæ, &amp; ſecundum quod ſolida, &amp; ſecundum quod tempora: quemad­<lb></lb>
modum &amp; monſtrabatur aliquando ſeorſum, contingens <expan abbr="vtiq;">vtique</expan> de omnibus vnica <lb></lb>demonſtratione monſtrari; ſed quia non ſunt nominatum quidam omnia hæc vnum, <lb></lb>
numeri, longitudines, tempora ſolida, &amp; ſpecie differunt à ſeinuicem ſeorſum <expan abbr="ac-cipiebãtur">ac­<lb></lb>
cipiebantur</expan>. </s>
          <s id="s.000943">nunc autem vniuer aliter monſtratur, <expan abbr="neq;">neque</expan> enim ſecundum quod lineæ, <lb></lb>aut ſecundum quod numeri, inerat; ſed ſecundum quod hoc, quod vniuerſale ſup­<lb></lb>
ponunt eſſe)<emph.end type="italics"></emph.end> affert exemplum ſecundi erroris, qui accidit, quando vniuerſa­<lb></lb>
le exiſtit quidem, ſed tamen eſt innominatum, pro cuius explicatione ſcien­


<pb pagenum="51" xlink:href="009/01/051.jpg"></pb>dum quid ſit alterna proportio. </s>
          <s id="s.000944">Alternam igitur proportionem definit Eu­<lb></lb>
clides definitione 12. quinti, ſic, eſt ſumptio antecedentis ad <expan abbr="antecedẽtem">antecedentem</expan>, <lb></lb>
<figure id="id.009.01.051.1.jpg" place="text" xlink:href="009/01/051/1.jpg"></figure><lb></lb>
&amp; conſequentis ad conſequentem. </s>
          <s id="s.000945">Explico, exponantur qua­<lb></lb>
tuor quantitates proportionales, v.g. vt 6. ad 3. ita ſint 4. ad <lb></lb>
2. ſi igitur argumentemur ſic, vt 6. ad 3. ita 4. ad 2. ergo al­<lb></lb>
ternatim erit, vt 6. ad 4. ita 3. ad 2. ſiue dixerimus, vt pri­<lb></lb>
mum ad ſecundum, ita tertium ad quartum, igitur alterna­<lb></lb>
tim erit, vt primum ad tertium, ita ſecundum ad quartum: valebit conſe­<lb></lb>
quentia; quæ quidem probatur deinde propoſitione 16. quinti de magnitu­<lb></lb>
dinibus, hoc eſt in vniuerſum de lineis, ſuperficiebus, &amp; ſolidis. </s>
          <s id="s.000946">quando igi­<lb></lb>
tur Ariſt. ait, monſtramus proportionale, ideſt, quaſuis quatuor quantita­<lb></lb>
tes proportionales, habere hanc proprietatem, vt ſint etiam alternatim <lb></lb>
proportionales, &amp; non monſtramus vnica demonſtratione de omni quouis <lb></lb>
proportionali, ſed ſeparatim de magnitudinibus in 16. quinti, de numeris <lb></lb>
in 13. ſeptimi, &amp; ſeorſum de temporibus in aſtronomia, vel phyſica; hoc <lb></lb>
modo non oſtendimus vniuerſaliter de primo ſubiecto, quia talis affectio <lb></lb>
conuenit ſingulis, non vt numeri, aut magnitudines, aut tempora ſunt, ſed <lb></lb>
ſecundum quandam naturam illis omnibus communem, cui primò illa paſ­<lb></lb>
ſio debetur; quæ quidem natura communis nomine caret, &amp; propterea eſt <lb></lb>
cauſa erroris.</s>
        </p>
        <p type="main">
          <s id="s.000947"><arrow.to.target n="marg29"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000948"><margin.target id="marg29"></margin.target>29</s>
        </p>
        <p type="main">
          <s id="s.000949"><emph type="italics"></emph>Nunc autem vniuerſaliter demonſtratur)<emph.end type="italics"></emph.end> nuſquam apud Mathematicos in­<lb></lb>
uenio hanc demonſtrationem vniuerſalem de illo communi omnibus præ­<lb></lb>
dictis, quare dicendum cum Zabarella, illud, nunc, eſſe intelligendum ſic, <lb></lb>
nunc autem, ideſt, in præſentia autem deberet vniuerſaliter demonſtrari, <lb></lb>
quod tamen cum non fiat, contingit nos decipi putantes vniuerſaliter de­<lb></lb>
monſtraſſe. </s>
          <s id="s.000950">vel dicendum iſtud verificari tantum de lineis, ſuperficiebus, &amp; <lb></lb>
ſolidis, de quibus ſimul in vnica natura communi, quæ eſt magnitudo, de­<lb></lb>
monſtratur in 16. quinti vniuerſaliter. </s>
          <s id="s.000951"><expan abbr="atq;">atque</expan> hoc modo explicatum eſt exem­<lb></lb>
plum ſecundi erroris, qui verbis illis <emph type="italics"></emph>(Vel ſit quidem, ſed innominatum ſit in <lb></lb>
rebus ſpecie differentibus)<emph.end type="italics"></emph.end> continebatur.</s>
        </p>
        <p type="main">
          <s id="s.000952"><arrow.to.target n="marg30"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000953"><margin.target id="marg30"></margin.target>30</s>
        </p>
        <p type="main">
          <s id="s.000954">Ibidem <emph type="italics"></emph>(Propter hoc ſi quis monſtrauerit ſingulum triangulum. </s>
          <s id="s.000955">demonſtratio­<lb></lb>
ne aut vna, aut altera, quod duos rectos habet vnumquodque, <expan abbr="æquilateiũ">æquilaterum</expan> ſeorſum, <lb></lb>
&amp; ſcalenum, &amp; æquicrus: nondum nouit triangulum, quod duobus rectis, niſi ſo­<lb></lb>
phiſtico modo, <expan abbr="neq;">neque</expan> vniuerſaliter triangulum, <expan abbr="neq;">neque</expan> ſi vllum eſt præter prædicta <lb></lb>
triangulum alterum. </s>
          <s id="s.000956">non enim ſecundum quod triangulum, <expan abbr="neq;">neque</expan> omne triangulum, <lb></lb>
niſi ſecundum numerum, ſecundum ſpeciem autem non omne; &amp; ſi nullum eſt, quod <lb></lb>
non nouit)<emph.end type="italics"></emph.end> vltimo loco ponit exemplum primi erroris, quem ſupra verbis il­<lb></lb>
lis <emph type="italics"></emph>(Quando vel nihil ſit accipere ſuperius, præter ſingulare)<emph.end type="italics"></emph.end> expreſſerat, quod, <lb></lb>
vt benè intelligamus, opus eſt ea, legere, quæ libro primo Priorum ſecto 3. <lb></lb>
cap. 1. ſcripſimus de proprietate illa trianguli, quod ſcilicet habet tres an­<lb></lb>
gulos æquales duobus rectis angulis, quibus præmiſſis, ſic deinde locum <lb></lb>
hunc interpretaberis; Propter hoc, quod præcedenti textu dictum eſt; no­<lb></lb>
tandum in primo errore vniuerſale, tanquam ſi non eſſet vniuerſale oſten­<lb></lb>
ditur de ſingulari, ſi quis igitur monſtrauerit ſingillatim de <expan abbr="vnoquoq;">vnoquoque</expan> trian­<lb></lb>
gulo in ſingulari, ſcilicet de vno æquilatero, tantum, &amp; de vno Scaleno, &amp; <lb></lb>de vno Iſoſcele, ſeparatim, vtens aut eadem demonſtratione dum de <expan abbr="vnoq́">vnoque</expan>; <pb pagenum="52" xlink:href="009/01/052.jpg"></pb>ſeparatim oſtendit, aut vtens diuerſis demonſtrationibus, vna pro æquila­<lb></lb>
tero, altera pro Iſoſcele, tertia pro Scaleno, oſtendens, quod <expan abbr="vnumquodq;">vnumquodque</expan> <lb></lb>
illorum habet tres angules æquales duobus rectis angulis; iſte nondum no­<lb></lb>
uit triangulum omne habere talem affectionem, niſi modo ſophiſtico, quia <lb></lb>
non cognoſcit hanc affectionem illis <expan abbr="cõpetere">competere</expan> propter naturam illam com­<lb></lb>
munem trianguli, cui primo, &amp; per ſe competit; &amp; neque vniuerſaliter co­<lb></lb>
gnoſcit triangulum omne eſſe tale, etiam ſi nullum aliud reperiatur trian­<lb></lb>
gulum, præter illud æquilaterum, vel illud Iſoſceles, vel illud Scalenum, de <lb></lb>
quibus ſeparatim <expan abbr="demõſtrauit">demonſtrauit</expan>, &amp; ſecundum numerum, ideſt de vnoquoque, <lb></lb>
quatenus eſt vnum numero. </s>
          <s id="s.000957">non nouit autem ſecundum ſpeciem, idest fecun­<lb></lb>
dum naturam, &amp; formam communem illis tribus indiuiduis, quæ eſt natu­<lb></lb>
ra trianguli. </s>
          <s id="s.000958">hoc autem eſſe exemplum primi erroris manifeſtè conuincitur, <lb></lb>
tum ex verbis illis, quando nihil ſit ſuperius, præter ſingulare, tum ex hu­<lb></lb>
ius textus verbis illis <emph type="italics"></emph>(Singulum triangulum)<emph.end type="italics"></emph.end> &amp; ex illis <emph type="italics"></emph>(Niſi ſecundum nume­<lb></lb>
rum)<emph.end type="italics"></emph.end> ideſt, niſi de vno, quod ſit vnum numero. </s>
          <s id="s.000959">propterea nos de ſingulari <lb></lb>
triangulo omiſſa Zabarellæ ſententia explicauimus tandem in confirma­<lb></lb>tionem noſtræ expoſitionis in hæc tria errata illud non omittendum, ſatius <lb></lb>
eſſe dicere, Ariſt. attuliſſe pro tribus erratis tria exempla ordine retrogra­<lb></lb>
do, quàm, quod facit Zabarella, primum eſſe pro tertio, ſecundum pro pri­<lb></lb>
mo, tertium verò pro ſecundo; eo enim modo, Ariſt. confuſionem nulla ra­<lb></lb>
tione, imò contra omnem rationem imponimus.</s>
        </p>
        <p type="main">
          <s id="s.000960"><arrow.to.target n="marg31"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000961"><margin.target id="marg31"></margin.target>31</s>
        </p>
        <p type="main">
          <s id="s.000962">Textu 14. continet quidem quædam mathematica, ſed ferè eadem cum <lb></lb>
ſuperioribus, quæ quia tum ex prædictis facile intelligi poſſunt, tum quia <lb></lb>
benè ab expoſitoribus explicantur, ne actum agamus, prætermittimus.</s>
        </p>
        <p type="main">
          <s id="s.000963"><arrow.to.target n="marg32"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000964"><margin.target id="marg32"></margin.target>32</s>
        </p>
        <p type="main">
          <s id="s.000965">Tex. 20. <emph type="italics"></emph>(Niſi magnitudines numeri ſint)<emph.end type="italics"></emph.end> hoc eſt, niſi magnitudines ſint di­<lb></lb>
feretæ, ita vt cadant ſub numerum, vt ſi linea quæpiam diuidatur in partes <lb></lb>
decem, vel duodecim, tunc euadit quantitas diſcreta, ſiue numerus. </s>
          <s id="s.000966">&amp; tunc <lb></lb>
linea numerus eſt. </s>
          <s id="s.000967">idem de ſuperficie, ac ſolido intelligendum.</s>
        </p>
        <p type="main">
          <s id="s.000968"><arrow.to.target n="marg33"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000969"><margin.target id="marg33"></margin.target>33</s>
        </p>
        <p type="main">
          <s id="s.000970">Ibidem <emph type="italics"></emph>(Propter hoc Geometriæ non licet monſtrare, quod contrariorum vna <lb></lb>
eſe ſcientia, ſed neque quod duo cubi cubus)<emph.end type="italics"></emph.end> quo ad verba illa, duo cubi cubus, <lb></lb>
quæ ad nos pertinent, vult Ariſt. docere, quod non debet Geometra oſten­<lb></lb>dere numerorum affectiones (per cubos enim intelligit numeros quoſdam <lb></lb>
ſic dictos, vt paulo poſt oſtendam) vt ſi quis vellet geometricè oſtendere id, <lb></lb>
quod oſtenditur in 4. noni Elem. ſcilicet, ſi cubus numerus cubum numerum <lb></lb>
multiplicauerit, productus numerus erit pariter cubus. </s>
          <s id="s.000971">nonnulli latinorum <lb></lb>
perperam textum hunc expoſuerunt putantes reperiri ſolummodo cubos <lb></lb>
geometricos, at Euclides definit. </s>
          <s id="s.000972">19. ſeptimi, ſic arithmeticum cubum de­<lb></lb>
finit, cubus numerus eſt, qui ſub tribus numeris æqualibus continetur, qua­<lb></lb>
lis eſt. </s>
          <s id="s.000973">8. qui eſt ad inſtar cubi geometrici, &amp; continetur ſub tribus binarijs <lb></lb>
multiplicatis inuicem, quæ multiplicatio ſic inſtituitur, exponuntur tres bi­<lb></lb>
<figure id="id.009.01.052.1.jpg" place="text" xlink:href="009/01/052/1.jpg"></figure><lb></lb>
narij, 2, 2, 2, primus ducitur in ſecundum, &amp; producitur. <lb></lb>
</s>
          <s id="s.000974">4. qui eſt numerus quadratus huius figuræ, <figure id="id.009.01.052.2.jpg" place="text" xlink:href="009/01/052/2.jpg"></figure>, deinde <lb></lb>
tertius binarius ducitur in prædictum quadratum 4. &amp; pro­<lb></lb>
ducitur 8. qui dicitur cubus, quia ſi intelligantur duo qua­<lb></lb>
ternarij, vnus ſupra alterum, vt in præſenti figura refe­<lb></lb>
runt cubicam figuram, cuius tam longitudo, quam latitudo,


<pb pagenum="53" xlink:href="009/01/053.jpg"></pb>&amp; altitudo, eſt 2. Similiter cubus numerus eſt 27. quia ſit ex tribus terna­<lb></lb>
rijs inuicem modo prædicto multiplicatis, 3. 3. 3. nam 3. in 3. ductis ſit 9. <lb></lb>
<figure id="id.009.01.053.1.jpg" place="text" xlink:href="009/01/053/1.jpg"></figure><lb></lb>
qui eſt quadratus. </s>
          <s id="s.000975">quo deinde ducto in tertium ter­<lb></lb>
narium, producitur 27. qui eſt cubus, &amp; refert figu­<lb></lb>
ram cubicam hanc. </s>
          <s id="s.000976">Iam verò ſi cubus 8. multipli­<lb></lb>
cet cubum 27. procreabitur 216. qui pariter cubus <lb></lb>
eſt. </s>
          <s id="s.000977"><expan abbr="atq;">atque</expan> hoc ſibi volunt verba illa, ſi duo cubi cubus, <lb></lb>
ideſt, ſi duo numeri cubi multiplicentur mutuò, cu­<lb></lb>
bus alter producetur; ex quibus videas, quam in­<lb></lb>
eptè illi <expan abbr="interpretẽtur">interpretentur</expan> hunc locum, qui dicunt, Ari­<lb></lb>
ſtotilem velle dicere non pertinere ad Geometram <lb></lb>
probare duos cubos geometricos ſibi additos face­<lb></lb>
re alium cubum, quod erat problema Delphicum de <lb></lb>
duplatione cubi, nondum inuentum; bis enim iſti peccant, primo in Logi­<lb></lb>
cam, quia ſic non tranſiret Geometra de genere in genus, ipſius enim eſt <lb></lb>
agere de duplatione cubi; ſecundò in Mathematicas, cum nondum noue­<lb></lb>rint arithmeticos cubos; &amp; præterea ignorent duos cubos ſibi additos, non <lb></lb>
facere alium cubum. </s>
          <s id="s.000978">Quod præterea hoc loco intelligendi ſint cubi arith­<lb></lb>
metici certò certius conſtat, ex ſequenti 24. textu, vbi ſic dicitur <emph type="italics"></emph>(Veluti <lb></lb>
Arithmetica quidem, quid impar, aut par, aut quadrangulum, aut cubus.)<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.000979"><arrow.to.target n="marg34"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000980"><margin.target id="marg34"></margin.target>34</s>
        </p>
        <p type="main">
          <s id="s.000981">Ibidem <emph type="italics"></emph>(<expan abbr="Neq;">Neque</expan> alij ſcientiæ quod alterius, niſi <expan abbr="quæcunq;">quæcunque</expan> ita ſe habent inter ſe, <lb></lb>
vt ſit alterum ſub altero, vt perſpectiua ad Geometriam, &amp; harmonica ad Arith­<lb></lb>
meticam)<emph.end type="italics"></emph.end> excipit ab illa regula (qua prohibetur, quamuis ſcientiam in alie­<lb></lb>
nam falcem immittere) ſcientias ſubalternatas, quæ propriè in Mathemati­<lb></lb>
cis reperiuntur, Perſpectiua enim propriè ſubalternatur Geometriæ, quia <lb></lb>
vtitur Demonſtrationibus linearibus, quas applicat lineis viſualibus, &amp; Mu­<lb></lb>
ſica ſubalternatur Arithmeticæ, quia ab ipſa mutuatur <expan abbr="demõſtrationes">demonſtrationes</expan> nu­<lb></lb>
merorum, quas applicat numeris ſonoris. </s>
          <s id="s.000982">v.g. Perſpectiua dicit, ea, quæ vi­<lb></lb>
dentur eminus videri minora, quam quæ videntur cominus, quia illa viden­<lb></lb>
tur ſub angulo minori, hæc verò ſub angulo maiori, quod verò remotiora <lb></lb>
videantur ſub angulo minori, quam propinquiora cæteris paribus probat <lb></lb>
<figure id="id.009.01.053.2.jpg" place="text" xlink:href="009/01/053/2.jpg"></figure><lb></lb>
per 21. primi Elem. ſit enim ma­<lb></lb>
gnitudo viſa A B, remotior ab o­<lb></lb>
culo in C, poſito, &amp; viſa propin­<lb></lb>
quior ab oculo in D. ductis lineis <lb></lb>
viſualibus C A, C B: D A, D B; ab <lb></lb>
oculis C, &amp; D, ad extremitates <lb></lb>
ſpectatæ magnitudinis, erit remo­<lb></lb>
tioris viſionis angulus C, minor <lb></lb>
angulo D, propinquioris, vt ex præallegata Demonſtratione pater. </s>
          <s id="s.000983">Hine <lb></lb>
perſpicuè vides, qua ratione Perſpectiua Geometriæ ſubalternetur, ſiue <lb></lb>
quid ſit ipſa ſubalternatio, vbi medium eſt Geometricum, concluſio autem <lb></lb>
optica. </s>
          <s id="s.000984">Exemplum ſubalternationis Muſicæ ſit, <expan abbr="conſonãtia">conſonantia</expan> Diapaſon, quam <lb></lb>
vulgò octauam appellant in data chorda collocare, hoc eſt, vocem grauio­<lb></lb>
rem facere duplam vocis acutioris ſumatur chorda A B, &amp; diuidatur bifa­<lb></lb>riam, ſine in æqualia in C; tota igitur chorda A B, ad dimidium A C, habet <pb pagenum="54" xlink:href="009/01/054.jpg"></pb><figure id="id.009.01.054.1.jpg" place="text" xlink:href="009/01/054/1.jpg"></figure><lb></lb>
proportionem, quam 2. ad 1. <lb></lb>
ſiue duplam, ergo etiam ſo­<lb></lb>
nus totius chordæ A B, ad <expan abbr="ſo-nũ">ſo­<lb></lb>
num</expan> chordæ dimidiæ A C, ha­<lb></lb>
bebit eandem rationem, <expan abbr="nimirũ">nimirum</expan> quam 2. ad 1. ſiue duplam. </s>
          <s id="s.000985">ſed ſonus chor­<lb></lb>
dæ A B, ad ſonum chordæ A C, conſonat diapaſon, ſeu octauam, ergo in <lb></lb>
data chorda collocata eſt conſonantia diapaſon, quod oportebat. </s>
          <s id="s.000986">vides me­<lb></lb>
dium eſſe arithmeticam, concluſionem verò harmonicam. </s>
          <s id="s.000987">Aliud exemplum <lb></lb>
Tonus, quod eſt <expan abbr="interuallũ">interuallum</expan> primæ vocis, Vt, ad ſecundam, Rè, in duo æqua­<lb></lb>
lia ſemitonia diuidi nequit, ratio eſt Arithmetica, quia proportio ſuper­<lb></lb>
particularis in duo æqualia arithmeticè ſecari nequit; at Tonus conſiſtit in <lb></lb>
ratione ſuperparticulari, nempè in ſeſquioctaua, ergo Tonus bifariam diui­<lb></lb>
di nequit. </s>
          <s id="s.000988">deſumptum eſt ex Boetio.</s>
        </p>
        <p type="main">
          <s id="s.000989"><arrow.to.target n="marg35"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000990"><margin.target id="marg35"></margin.target>35</s>
        </p>
        <p type="main">
          <s id="s.000991">Tex. 23. <emph type="italics"></emph>(Est autem ſic monſtrare, quemadmodum Bryſo quadraturam, ſecun­<lb></lb>
dum enim commune monſtrant tales rationes)<emph.end type="italics"></emph.end> cum velit oſtendere veram de­<lb></lb>
monſtrationem conſtare debere ex proprijs, non autem ex communibus; <lb></lb>
primum affert exemplum demonſtrationis cuiuſdam Bryſonis, quæ ex com­<lb></lb>
munibus procedat, vt autem benè intelligamus, qualeſnam ſint huiuſmodi <lb></lb>
demonſtrationes, quæ per communia oſtendunt, legenda prius ea ſunt, quæ <lb></lb>
ſcripſimus de quadratura circuli in prędicamento relationis. </s>
          <s id="s.000992">Bryſo itaque, <lb></lb>
vt tradit Alexander, in hunc modum conabatur quadrare <expan abbr="circulũ">circulum</expan>. </s>
          <s id="s.000993">ſit qua­<lb></lb>drandus circulus A B C D, cui circumſcribatur quadratum E F G H. per <lb></lb>
7 quarti, &amp; alterum quadratum I L M N, eidem inſcribatur per 6. quarti, <lb></lb>
quid autem ſit circumſcribere, &amp; inſcribere figuram circulo, ex definitione <lb></lb>
<figure id="id.009.01.054.2.jpg" place="text" xlink:href="009/01/054/2.jpg"></figure><lb></lb>
3. &amp; 4. eiuſdem libri petatur, quamuis <lb></lb>
ex inſpectione figuræ <expan abbr="pręsẽtis">pręsentis</expan> ſatis per­<lb></lb>
cipi poſſit; deinde aliud <expan abbr="quadratũ">quadratum</expan> me­<lb></lb>
dium inter prædicta duo conſtituatur, <lb></lb>
<expan abbr="ſitq́">ſitque</expan>; O P Q R. </s>
          <s id="s.000994">Iam ſic oſtendebat iſtud <lb></lb>
medium quadratum eſſe æquale circu­<lb></lb>
lo propoſito. </s>
          <s id="s.000995"><expan abbr="Quæcunq;">Quæcunque</expan> ſunt, ſimul ma­<lb></lb>
iora eodem, &amp; minora eodem, ſunt in­<lb></lb>
uicem æqualia, ſed circulus, &amp; quadra­<lb></lb>
tum medium, ſunt ambo maiora qua­<lb></lb>
drato inſcripto, &amp; ambo minora qua­<lb></lb>
drato circumſcripto, ergo circulus, &amp; <lb></lb>
quadratum medium, ſunt æqualia. </s>
          <s id="s.000996">vte­<lb></lb>
batur, inquit Ariſt prędicto principio, <lb></lb>
etiam numeris, lineis, temporibus, &amp; <lb></lb>
qualitatibus communi, <expan abbr="neq;">neque</expan> deducto ex natura circuli, aut quadrati, de qui­<lb></lb>
bus erat demonſtratio. </s>
          <s id="s.000997">præterea aduertendum eſt, illud eſſe falſum, nam ſex, <lb></lb>
&amp; quinque, ambo ſunt maiores, quam quatuor, &amp; minores, quam ſeptem, <lb></lb>
&amp; tamen non ſunt æquales.</s>
        </p>
        <p type="main">
          <s id="s.000998"><arrow.to.target n="marg36"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.000999"><margin.target id="marg36"></margin.target>36</s>
        </p>
        <p type="main">
          <s id="s.001000">In codem textu <emph type="italics"></emph>(<expan abbr="Vnumquodq;">Vnumquodque</expan> autem ſcimus, non ſecundum accidens, quando <lb></lb>
ſecundum illud cognoſcamus, ſecundum quod ineſt ex principijs illius, ſecundam <lb></lb>
quod illud; vt duobus rectis æquales, habere, cui ineſt per ſe, quod dictum eſt ex<emph.end type="italics"></emph.end> <pb pagenum="55" xlink:href="009/01/055.jpg"></pb><emph type="italics"></emph>principijs huius)<emph.end type="italics"></emph.end> affert nunc exemplum alterius demonſtrationis, quæ non <lb></lb>
ex communibus, vt præcedens Bryſonis, ſed ex proprijs principijs oſtendit <lb></lb>
affectionem de ſubiecto proprio. </s>
          <s id="s.001001">Eſt autem illud exemplum toties decan­<lb></lb>
tatum de triangulo habente tres angulos æquales duobus rectis angulis; id­<lb></lb>
circo operæpretium eſſe puto explicare demonſtrationem, 32. primi Eucli­<lb></lb>
dis, quæ iſtud ex proprijs principijs demonſtrat, &amp; quam hoc loco Ariſto­<lb></lb>
teles innuit, hoc enim modo ipſius Ariſt. mentem probè penetrare poteri­<lb></lb>
<figure id="id.009.01.055.1.jpg" place="text" xlink:href="009/01/055/1.jpg"></figure><lb></lb>
mus. </s>
          <s id="s.001002">ſit ergo <expan abbr="triãgulum">triangulum</expan> A B C. </s>
          <s id="s.001003">Dico ag­<lb></lb>
gregatum <expan abbr="triũ">trium</expan> ipſius angulorum A, B, C, <lb></lb>
eſſe æquale aggregato ex duobus angu­<lb></lb>
lis rectis (vt autem melius intelligas, quæ <lb></lb>
ſequuntur, lege prius ea, quæ dicta ſunt <lb></lb>
in lib. 1. Priorum ſecto 3. cap. 1.) produ­<lb></lb>
catur latus B C, <expan abbr="vſq;">vſque</expan> in D, vt fiat angulus <lb></lb>
externus A C D; Iam ſic, quoniam <expan abbr="pro-batũ">pro­<lb></lb>
batum</expan> eſt in 13. primi, duos angulos, quos <lb></lb>
facit linea A C, cum linea B D, ſcilicet angulos A C B, A C D, eſſe pares <lb></lb>
duobus rectis: &amp; quia pariter in prima parte huius propoſ. </s>
          <s id="s.001004">32. probatum <lb></lb>
eſt ab Euclide duos angulos A B, eſſe æquales externo angulo A C D: ſi ter­<lb></lb>
tius angulus reliquus A C B, ſumatur bis, ſemel cum duobus angulis A, B, <lb></lb>
&amp; ſemel cum externo A C D, <expan abbr="addẽtur">addentur</expan> æqualia æqualibus, &amp; propterea tres <lb></lb>
anguli A, B, A C B, ſimul ſumpti, erunt æquales duobus A C D, A C B, ſimul <lb></lb>
ſumptis; ſed his duobus ſunt æquales duo recti, ergo cum quæ ſunt æqualia <lb></lb>
vni tertio, ſint etiam æqualia inuicem, erit aggregatum trium angulorum <lb></lb>
A, B, A C B, æquale aggregato duorum rectorum; quod erat demonſtran­<lb></lb>
dum. </s>
          <s id="s.001005">Medium <expan abbr="itaq;">itaque</expan> huius demonſtrationis, ſi res ad trutinam Logicam ex­<lb></lb>
pendatur, eſt, quod partes aggregati <expan abbr="triũ">trium</expan> <expan abbr="angulorũ">angulorum</expan> A, B, A C B, ſunt æqua­<lb></lb>
les partibus aggregati <expan abbr="duorũ">duorum</expan>, &amp; ideo <expan abbr="aggregatũ">aggregatum</expan>, aggregato æqua­<lb></lb>
le eſt. </s>
          <s id="s.001006">quod medium eſt in genere cauſæ materialis. </s>
          <s id="s.001007">quod verò partes illius <lb></lb>
ſint æquales partibus huius, probatur, per dignitatem <expan abbr="illã">illam</expan>, quæ ſunt æqualia <lb></lb>
vni tertio, ſunt etiam inter ſe. </s>
          <s id="s.001008">partes porrò aggregati trium angulorum <lb></lb>
erant hæ, anguli A, B, vna; altera verò angulus A C B; partes verò aggre­<lb></lb>
gati duorum rectorum erant A C B, A C D, quibus partibus, illæ ſunt æqua­<lb></lb>
les, &amp; ideo totum toti æquale. </s>
          <s id="s.001009">quod medium eſt omnino intrinſecum, &amp; ex <lb></lb>
proprijs ipſius trianguli, ſiue ex proprijs angulorum ipſius, cum ſint ipſius <lb></lb>
partes. </s>
          <s id="s.001010">quod pariter medium ex parte paſſionis, quæ demonſtratur, eſt ex <lb></lb>
proprijs, cum ſint partes illius materiales. </s>
          <s id="s.001011">per materiam autem oportet <lb></lb>
hoc loco intelligere materiam intelligibilem, ideſt quantitatem à qualita­<lb></lb>tibus abſtractam, &amp; terminatam, de qua pluribus agemus infra in tractatu <lb></lb>
de natura mathematicarum. </s>
          <s id="s.001012">Hinc videas eos magnopere decipi, qui pu­<lb></lb>
tant, hanc demonſtrationem eſſe per extrinſeca, eò quod ad demonſtran­<lb></lb>
dum producatur linea B C, in D, putantes lineam illam productam C D, <lb></lb>
eſſe demonſtrationis medium; lineæ <expan abbr="namq;">namque</expan> huiuſmodi, quæ in demonſtra­<lb></lb>
tionibus geometricis conſtruuntur, nunquam ſunt media propria demon­<lb></lb>
ſtrationum, ſed tantummodo aſſumuntur ad probandum medium iam ex­<lb></lb>
cogitatum eſſe veram cauſam concluſionis. </s>
          <s id="s.001013">Hinc etiam manifeſtè colligas <pb pagenum="56" xlink:href="009/01/056.jpg"></pb>Mathematicas facultates habere demonſtrationes perfectiſſimas, quod <lb></lb>
ageometreti negare ſolent, ſed audacter aiunt exempla Ariſt. non eſſe vera: <lb></lb>
<expan abbr="neq;">neque</expan> requiri veritatem exemplorum; in <expan abbr="quorũ">quorum</expan> <expan abbr="vtroq;">vtroque</expan> peccant, nam dictum <lb></lb>
illud vſurpari ſolet, &amp; debet de exemplis moralibus. </s>
          <s id="s.001014">at vero requiri confor­<lb></lb>
mitatem exemplorum cum regulis traditis, nemo ſanæ mentis dubitabit. <lb></lb>
</s>
          <s id="s.001015">Verum iſti confundunt conformitatem cum veritate. </s>
          <s id="s.001016">Veritas exemplo tunc <lb></lb>
ineſt, quando illud, quod in exemplo narratur, verè extitit, vt ſi quis in <lb></lb>
exemplum pudicitiæ afferret hiſtoriam Ioſephi, <expan abbr="verũ">verum</expan> iſtud eſſet exemplum. <lb></lb>
</s>
          <s id="s.001017">quæ veritas in exemplis moralibus non ſemper eſt neceſſaria, talia exempla <lb></lb>
ſunt ſæpè parabolæ, &amp; fabulæ, quæ nunquam extiterunt, v. g. narratur ab <lb></lb>
Ariſt. de quodam filio, qui patrem crudeliter traxerat, qui poſtea grandior <lb></lb>
factus, cum filium procreaſſet, ab eodem pariter raptatus eſt ipſe, vſque ad <lb></lb>
eundem locum, quo ipſe patrem ſuum impiè raptauerat. </s>
          <s id="s.001018">non eſt neceſſe, ta­<lb></lb>
lem extitiſſe filium, <expan abbr="neq;">neque</expan> patrem. </s>
          <s id="s.001019">Verumtamen ſemper conformitas exem­<lb></lb>
pli cum regulis, &amp; præceptis, quæ traduntur neceſſaria eſt, alioquin exem­<lb></lb>
pla deſtruerent id, quod præceptio conſtruit, <expan abbr="illiq́">illique</expan> contraria eſſet, quod om­<lb></lb>
nino abſurdum foret. </s>
          <s id="s.001020">non ſecus, ac ſi quis vellet alium docere characteres <lb></lb>
latinos, <expan abbr="illiq́">illique</expan>; barbaros, quos Gothicos vocant in exemplum proponeret. </s>
          <s id="s.001021">re­<lb></lb>
quiritur igitur ſemper in omni exemplo conformitas cum eo, quod doce­<lb></lb>
tur; in moralibus tamen non ſemper requiritur veritas, vti diximus; Alij <lb></lb>
verò dicunt non requiri in exemplis determinatam veritatem, ſed ſatis eſſe, <lb></lb>
ſi exemplum verum ſit ſecundum opinionem aliquorum: <expan abbr="quorũ">quorum</expan> ſententiam <lb></lb>
non improbamus. </s>
          <s id="s.001022">Exempla igitur ab Ariſt. paſſim ex mathematicis allata, <lb></lb>
congrua, <expan abbr="conformiaq́">conformiaque</expan>; omninò ſunt ipſius doctrinæ, aliter ipſum perpetuò <lb></lb>
mentientem facimus. </s>
          <s id="s.001023">Poſtremò illud etiam eſt aduertendum, fortè Ariſt. in <lb></lb>
præſenti textu ſpectaſſe <expan abbr="nõ">non</expan> ad hanc Euclidianam demonſtrationem, ſed po­<lb></lb>
tius ad Pithagoricam. </s>
          <s id="s.001024">Pithagorei enim eam aliter, quamuis per idem me­<lb></lb>
dium, ſcilicet à cauſa materiali, demonſtrabant; conſtruebant enim aliter, <lb></lb>
<expan abbr="neq;">neque</expan> vlla vtebantur diuiſione. </s>
          <s id="s.001025">quod dictum velim propter nonnullos, qui ab <lb></lb>
huiuſmodi diuiſionibus abhorrent, <expan abbr="timentq́">timentque</expan>; ne demonſtrationis perfectio­<lb></lb>
ni per eas plurimum derogetur. </s>
          <s id="s.001026">Pithagoreorum demonſtrationem vide <lb></lb>
apud Clauium in ſcholio 32. primi Euclidis, quam ex Eudemo etiam Pro­<lb></lb>
clus in comm. eiuſdem recitat.</s>
        </p>
        <p type="main">
          <s id="s.001027"><arrow.to.target n="marg37"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001028"><margin.target id="marg37"></margin.target>37</s>
        </p>
        <p type="main">
          <s id="s.001029">Ibidem <emph type="italics"></emph>(Sed quemadmodŭm harmonica per Arithmeticam)<emph.end type="italics"></emph.end> vide ſupra tex. 20.</s>
        </p>
        <p type="main">
          <s id="s.001030"><arrow.to.target n="marg38"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001031"><margin.target id="marg38"></margin.target>38</s>
        </p>
        <p type="main">
          <s id="s.001032">Ibidem <emph type="italics"></emph>(Demonſtratio autem non computatur in aliud genus; niſi, vt dictum <lb></lb>eſt geometricæ demonſtrationes in Perſpectiuas, aut Mechanicas, &amp; arithmeticæ in <lb></lb>
harmonicas)<emph.end type="italics"></emph.end> exempla ſubalternationis Perſpectiuæ, &amp; Muſicæ in tex. 20. at­<lb></lb>
tulimus; nunc Mechanicæ ſubalternationis, quam hic Ariſt. inſinuat, exem­<lb></lb>
plum ſit illud, quod Archimedes prop. 14. primi Aequep. demonſtrat, ni­<lb></lb>
mirum centrum grauitatis omnis trianguli eſſe punctum illud, in quo rectæ <lb></lb>
lineæ ab angulis trianguli ad dimidia latera oppoſita ductæ concurrunt. </s>
          <s id="s.001033">ſit <lb></lb>
triangulum A B C, à cuius angulis A, &amp; B, ducantur duæ rectæ A D, B E, ita <lb></lb>
vt bifariam ſecent latera A C, B C, in punctis D, &amp; E, &amp; concurrant in F. <lb></lb>
</s>
          <s id="s.001034">Dico F, eſſe centrum grauitatis propoſiti trianguli. </s>
          <s id="s.001035">Quoniam enim in 13. <lb></lb>
Aequep. probauit centrum grauitatis eſſe in ea linea, quæ ducta ab angulo <lb></lb>
quouis ſecat oppoſitum latus bifariam, crit in linea A D, <expan abbr="centrũ">centrum</expan> grauitatis.


<pb pagenum="57" xlink:href="009/01/057.jpg"></pb><figure id="id.009.01.057.1.jpg" place="text" xlink:href="009/01/057/1.jpg"></figure><lb></lb>
ſed eadem ratione erit etiam in linea B E, er­<lb></lb>
go non niſi in puncto F, quod <expan abbr="ſolũ">ſolum</expan> eſt in vtra­<lb></lb>
que, quod erat demonſtrandum. </s>
          <s id="s.001036">ex quibus ap­<lb></lb>
paret, qua ratione mechanica concluſio Geo­<lb></lb>
metriæ ſubiaceat, dum lineari diſcurſu ipſa <lb></lb>
demonſtratio perficitur. </s>
          <s id="s.001037">Scias præterea cen­<lb></lb>
trum grauitatis eſſe tale punctum, ex quo ſi ſu­<lb></lb>
ſpendatur corpus triangulare vniformis craſ­<lb></lb>
ſitici, manet ſemper horizonti parallelum, ſi <lb></lb>
tamen antequam ſuſpenderetur, iacebat plano horizontis, æquidiſtans; <lb></lb>
<expan abbr="neq;">neque</expan> ſi ſuſpenſum feratur huc illud nutat, ſed ſemper in <expan abbr="codẽ">codem</expan> ſitu perſeuerat.</s>
        </p>
        <p type="main">
          <s id="s.001038"><arrow.to.target n="marg39"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001039"><margin.target id="marg39"></margin.target>39</s>
        </p>
        <p type="main">
          <s id="s.001040">Tex. 24. <emph type="italics"></emph>(Veluti Arithmetica quidem quid impar, aut par; aut quadrangu­<lb></lb>
lum, aut cubus)<emph.end type="italics"></emph.end> cognoſcas hinc certò certius quadrangulum, &amp; cubum eſſe <lb></lb>
ſpecies numerorum, ſicuti ſupra tex. 9. &amp; 20. explicauimus, quò nunc te vi­<lb></lb>
ciſſim, vt præſentem locum intelligas, remittimus.</s>
        </p>
        <p type="main">
          <s id="s.001041"><arrow.to.target n="marg40"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001042"><margin.target id="marg40"></margin.target>40</s>
        </p>
        <p type="main">
          <s id="s.001043">Ibidem <emph type="italics"></emph>(Geometrica verò quid irrationale, aut refrangi, aut concurrere)<emph.end type="italics"></emph.end> per <lb></lb>
verbum, irrationale, non videtur Ariſt. intellexiſſe proprietatem illam duo­<lb></lb>
rum linearum incommenſurabilium longitudine, &amp; potentia, quia vſus fuiſ­<lb></lb>
ſet verbo, <foreign lang="grc">άορητον.</foreign> quod apud Geometras vſurpari ſolet in illa ſignificatio­<lb></lb>
ne, ſed vſus eſt verbo, <foreign lang="grc">ὰλογον,</foreign> quod latinè redditur improportionale.</s>
        </p>
        <p type="main">
          <s id="s.001044"><arrow.to.target n="marg41"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001045"><margin.target id="marg41"></margin.target>41</s>
        </p>
        <p type="main">
          <s id="s.001046">Per verbum <emph type="italics"></emph>(Refrangi)<emph.end type="italics"></emph.end> ſeu frangi, intelligit lineam aliquam rectam, non <lb></lb>in directum tendere, ſed in aliquo puncto frangi, ſeu declinari à rectitudine, <lb></lb>
ita vt conſtituat angulum.</s>
        </p>
        <p type="main">
          <s id="s.001047">Per verbum <emph type="italics"></emph>(Concurrere)<emph.end type="italics"></emph.end> intelligit, non eſſe parallelas, ſed ad idem ali­<lb></lb>
quod punctum coire, ſi protrahantur.</s>
        </p>
        <p type="main">
          <s id="s.001048"><arrow.to.target n="marg42"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001049"><margin.target id="marg42"></margin.target>42</s>
        </p>
        <p type="main">
          <s id="s.001050">Ibidem <emph type="italics"></emph>(Et Astrologia ſimiliter)<emph.end type="italics"></emph.end> per Aſtrologiam intelligit Ariſt. non iu­<lb></lb>
diciariam, quamuis à recentioribus hoc nomine vocetur, ſed quam hodie <lb></lb>
dicunt Aſtronomiam, <expan abbr="aitq́">aitque</expan>; ipſam conſiderare quantitatem, figuram, mo­<lb></lb>
tum, &amp; locum totius Mundi, ac partium ipſius integrantium, vt ſunt Cœli, <lb></lb>
&amp; Elementa.</s>
        </p>
        <p type="main">
          <s id="s.001051"><arrow.to.target n="marg43"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001052"><margin.target id="marg43"></margin.target>43</s>
        </p>
        <p type="main">
          <s id="s.001053">Tex. 25. <emph type="italics"></emph>(<expan abbr="Neq;">Neque</expan> Geometra falſa ſupponit, quemadmodum quidam aſſeruere di­<lb></lb>
centes, quod non oportet falſo vti: Geometram verò mentiri dicentem pedalem, non <lb></lb>
pedalem, aut rectam deſcriptam, non rectam <expan abbr="existẽtem">existentem</expan>: Geometra verò nihil con­<lb></lb>cludit eò, quod hæc eſt linea, ſed quæ per hæc oſtenduntur)<emph.end type="italics"></emph.end> innuit his verbis eam <lb></lb>
materiam intelligibilem, quæ eſt ſubiectum Geometriæ: eam ſcilicet, quæ <lb></lb>
ſub figuris Geometricis ſenſibilibus, &amp; <expan abbr="plerunq;">plerunque</expan> falſis latet; nam ſæpè Geo­<lb></lb>
metra vtitur linea quadam ſenſibili pro recta, quæ verè nec eſt linea mathe­<lb></lb>
matica, nec recta; ſupponit aliquando talem lineam eſſe pedalem, quæ ve­<lb></lb>
rè non eſt pedalis: Verumtamen non mentitur, quia reſpicit ad veram li­<lb></lb>
neam mathematicam, quæ ſub illa intelligitur, &amp; quæ recta concipitur; &amp; <lb></lb>
quidem hæc omnia verè concipiuntur, quoniam ita eſſe re vera poſſunt.</s>
        </p>
        <p type="main">
          <s id="s.001054"><arrow.to.target n="marg44"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001055"><margin.target id="marg44"></margin.target>44</s>
        </p>
        <p type="main">
          <s id="s.001056">Tex. 28. <emph type="italics"></emph>(Coaltern as verò coincidere)<emph.end type="italics"></emph.end> per coalternas intelligendas eſſe pa­<lb></lb>
rallelas lineas, alias, &amp; nunc <expan abbr="quoq;">quoque</expan> monemus.</s>
        </p>
        <p type="main">
          <s id="s.001057"><arrow.to.target n="marg45"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001058"><margin.target id="marg45"></margin.target>45</s>
        </p>
        <p type="main">
          <s id="s.001059">Tex. 29. <emph type="italics"></emph>(In Mathematicis verò non est ſimiliter paralogiſmus, quoniam me­<lb></lb>
diŭ eſt ſemper, quod duplex, de hoc enim omni, &amp; hoc rurſus de alio dicitur omni)<emph.end type="italics"></emph.end><lb></lb>
aduerte, quod quamuis nonnulli codices habeant pro, in mathematicis, in


<pb pagenum="58" xlink:href="009/01/058.jpg"></pb>diſciplinis, idem tamen apud græcos <foreign lang="grc">μαθηματα</foreign> ſunt, ac apud latinos diſci­<lb></lb>
plinæ; verbum autem <foreign lang="grc">μαθηματα</foreign> vſurpat hoc loco Ariſtoteles. </s>
          <s id="s.001060">Porrò non <lb></lb>
eſt in mathematicis, ſicut in alijs paralogiſmus, quia in omni demonſtra­<lb></lb>
tione maius extremum dicitur de omni medio, &amp; rurſus medium dicitur de <lb></lb>
omni minori extremo, ac ſi diceret mathematicæ demonſtrationes ſunt in <lb></lb>
primo modo, qui barbarè à latinis recentioribus Barbara appellatur. </s>
          <s id="s.001061">Hæc <lb></lb>
eſt autem pulcherrima mathematicarum commendatio, quippe præclarum <lb></lb>
eſt à laudato laudari. </s>
          <s id="s.001062">In mathematicis, inquit, non accidit ſimiliter para­<lb></lb>
logiſmus, ideſt, tam frequenter, quemadmodum in ſyllogiſmis dialecticis, <lb></lb>
quia modus argumentandi mathematicarum eſt perfectiſſimus, quippe in <lb></lb>
primo modo primæ figuræ.</s>
        </p>
        <p type="main">
          <s id="s.001063"><arrow.to.target n="marg46"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001064"><margin.target id="marg46"></margin.target>46</s>
        </p>
        <p type="main">
          <s id="s.001065">Eodem tex. <emph type="italics"></emph>(Contingit autem quoſdam non ſyllogiſticè dicere, &amp; quod ex vtriſ­<lb></lb>
que conſequentia accipiunt, quemadmodum &amp; Cæneus facit, quod ignis in multi­<lb></lb>
plici proportione: etenim ignis celeriter gignitur, vt ait: &amp; hæc est proportio. </s>
          <s id="s.001066">ſic <lb></lb>autem non eſt ſyllogiſmus, niſi celerrimam proportio ſequatur multiplex: &amp; ignem <lb></lb>
celerrima in motu proportio)<emph.end type="italics"></emph.end> verba illa (in multiplici proportione) græcè ſic <lb></lb>
ſe habent, <foreign lang="grc">εν τῃ πολλαπλασιονι αναλογιᾳ,</foreign> quod melius redditur latinè in mul­<lb></lb>
tiplici proportione, quemadmodum fecimus, quam in multiplicata, quem­<lb></lb>
admodum in vulgata editione. </s>
          <s id="s.001067">porrò quid inter multiplicem, &amp; multipli­<lb></lb>
catam rationem interſit, optimè declarat noſter Clauius ad 4. definit. </s>
          <s id="s.001068">lib. 5. <lb></lb>
Elem. ex quo etiam loco pauca decerpam, quæ huic loco declarando con­<lb></lb>
ducunt. </s>
          <s id="s.001069">Proportio igitur multiplex eſt habitudo inter duas quantitates in­<lb></lb>
æquales, quarum maior continet minorem, bis, vel ter, vel quater, &amp;c. </s>
          <s id="s.001070">vn­<lb></lb>
de proportio multiplex habet ſub ſe genera infinita, quando enim maior <lb></lb>
continet minorem bis, dicitur Dupla: quando ter, Tripla: quando quater, <lb></lb>
Quadrupla: &amp; ſic in infinitum: v. g. 2. ad 1. eſt proportio dupla; 3. ad 1. tri­<lb></lb>
pla; 4. ad 1. quadrupla, &amp;c. </s>
          <s id="s.001071">omnes tamen continentur ſub genere multipli­<lb></lb>
cis rationis. </s>
          <s id="s.001072">porrò ſi quępiam proportio ex genere multiplici progrediatur <lb></lb>
per plures terminos, v. g. proportio quadrupla progrediatur hoc modo, <lb></lb>
1. 4. 16. 64. 256. &amp;c. </s>
          <s id="s.001073">fit, vt ſubſequentes termini mirum in modum augean­<lb></lb>
tur. </s>
          <s id="s.001074">hic vides primum ipſam quadruplam rationem in diſpoſitis terminis <lb></lb>
progredi, quia quilibet ſequens terminus ad præcedentem eſt quadruplus. <lb></lb>
</s>
          <s id="s.001075">cernis etiam in paucis terminis, quinque ſcilicet magnum factum eſſe incre­<lb></lb>
mentum, cum <expan abbr="vſq;">vſque</expan> ad 256. excreuerint. </s>
          <s id="s.001076">Cæneus igitur dicens ignem augeri <lb></lb>
ſecundum multiplicem rationem, vnam ex prædictis intelligebat aliquam, <lb></lb>
quia quælibet illarum magnopere creſcit, ſi propagetur, vt ad 10. quinti <lb></lb>
definit. </s>
          <s id="s.001077">traditur: &amp; vt paulo ante exemplo licuit perſpicere. </s>
          <s id="s.001078">argumentaba­<lb></lb>
tur igitur Cæneus in hunc modum; quod in multiplici ratione augetur, ce­<lb></lb>
lerrimè augetur: ignis celerrimè augetur, ergo ignis in multiplici ratione <lb></lb>
augetur, quæ argumentatio vitioſa eſt, ex duabus quippe affirmatiuis in ſe­<lb></lb>
cunda figura procedens, vt colligitur ex verbis illis tex. <emph type="italics"></emph>(Ex viriſque conſe­<lb></lb>
quentia accipiunt<emph.end type="italics"></emph.end>) ex his mathematica huius locis patere ſatis poſſunt.</s>
        </p>
        <p type="main">
          <s id="s.001079"><arrow.to.target n="marg47"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001080"><margin.target id="marg47"></margin.target>47</s>
        </p>
        <p type="main">
          <s id="s.001081">Ibidem (<emph type="italics"></emph>Conuertuntur autem magis, quæſunt in mathematicis, quoniam nul­<lb></lb>
lum accidens accipiunt (in quo quidem ijs præſtăt, quæ diſputationibus traduntur) <lb></lb>
ſed definitiones<emph.end type="italics"></emph.end>) Hæc eſt altera mathematicarum laus, vnde earum quoque <lb></lb>
præſtantia elucet, quia ſcilicet mathematicæ pro medijs vtuntur definitio­


<pb pagenum="59" xlink:href="009/01/059.jpg"></pb>nibus ſubiecti, aut paſſionis, quæ nullo modo ſunt accidentalia concluſioni, <lb></lb>
v. g. in prima Euclidis demonſtratione per definitionem ſubiecti probantur <lb></lb>
tres lineæ eſſe æquales, quia nimirum ſint ſemidiametri circulorum æqua­<lb></lb>
lium, quæ eſt ipſarum definitio. </s>
          <s id="s.001082">&amp; in 4. primi probantur baſis, &amp; anguli <lb></lb>
vnius trianguli æquales eſſe baſi, &amp; angulis alterius trianguli per formalem <lb></lb>
definitionem paſſionis, videlicet æqualitatis, quæ traditur in octauo axio­<lb></lb>
mate ſic, quæ ſibi mutuo congruunt, ea inter ſe ſunt æqualia. </s>
          <s id="s.001083">probat igitur <lb></lb>
Euclides in quarta baſim, &amp; angulos vnius trianguli eſſe æqualia baſi, &amp; an­<lb></lb>
gulis alterius trianguli, quia oſtendit, quod, ſi baſis illa huic baſi, &amp; illi an­<lb></lb>
guli hiſce angulis ſuperponantur, congruunt; ex qua congruentia mutua, <lb></lb>
quæ eſt æqualitatis definitio, infert æqualitatem ipſarum baſium, necnon <lb></lb>
angulorum. </s>
          <s id="s.001084">eadem deinde æqualitatis definitione totam demonſtrationem <lb></lb>
concludit, ſcilicet totum triangulum toti triangulo æquale eſſe, quia vnum <lb></lb>
alteri congruat. </s>
          <s id="s.001085">Aſtronomi <expan abbr="quoq;">quoque</expan> demonſtrant eclypſim de Luna, per in­<lb></lb>
terpoſitionem terræ inter Lunam, &amp; Solem, quæ interpoſitio eſt definitio <lb></lb>
cauſalis ipſius eclypſis, ſcilicet paſſionis. </s>
          <s id="s.001086">huiuſmodi <expan abbr="ſexcẽtas">ſexcentas</expan> reperies apud <lb></lb>
Geometras, Arithmeticos, Aſtronomos, <expan abbr="cæterosq́">cæterosque</expan>; Mathematicas demon­<lb></lb>
ſtrationes: ita vt meritò dixerit Ariſt. Mathematicas alias omnes natura­<lb></lb>
les ſcientias, quæ diſputabilibus rationibus traduntur ex hac parte antecel­<lb></lb>
lere. </s>
          <s id="s.001087">aſſumunt igitur terminos conuertibiles, quia adhibent ſæpè definitio­<lb></lb>
nes ad demonſtrandum. </s>
          <s id="s.001088">Reliqua logici expoſitores declarant.</s>
        </p>
        <p type="main">
          <s id="s.001089"><arrow.to.target n="marg48"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001090"><margin.target id="marg48"></margin.target>48</s>
        </p>
        <p type="main">
          <s id="s.001091">Tex. 30. (<emph type="italics"></emph>Rurſus quemadmodum monſtrant Lunam, quod ſphærica ſit per aug­<lb></lb>menta: ſi enim quod ita augetur, eſt ſphæricum; augetur autem Luna; planŭm quod <lb></lb>
ſphærica<emph.end type="italics"></emph.end>) Illius demonſtrationis, quæ ab effectu procedit, affert exemplum <lb></lb>
ex aſtronomia; Aſtronomi enim demonſtrant Lunam eſſe ſphæricam ab ef­<lb></lb>
fectu ipſius ſphæricitatis, qui eſt illuminatio ſphærica: ſic enim ratiocinan­<lb></lb>
tur: ea, quæ ſphæricè illuminantur ſunt ſphærica, Luna ſphæricè illumina­<lb></lb>
tur, ergo ſphærica eſt: quæ argumentatio fuſius explicanda eſt; quod ait, <lb></lb>
quod ita augetur, ideſt, ſphæricè, eſt ſphæricum, ideſt, quia lumen nouæ Lu­<lb></lb>
næ augetur ſphæricè, hoc eſt, ad eum modum, quo quæuis ſphæra obiecta <lb></lb>
corpori luminoſo ſolet illuminari. </s>
          <s id="s.001092">illuminatio porrò Lunæ in ſe ſemper eſt <lb></lb>
eadem, quia ſemper dimidium Lunæ quod Solem aſpicit, illuminatur; dici­<lb></lb>
tur tamen augeri reſpectu oculi noſtri, quia ſcilicet initio facto à nouilunio <lb></lb>
pars illuminata incipit quotidie magis vergere ad oculum noſtrum, ita vt <lb></lb>
in dies maiorem, ac maiorem illuminationem videamus, donec opponatur <lb></lb>
Soli, in qua oppoſitione totum ferè Lunæ <expan abbr="illuminatũ">illuminatum</expan> conſpicitur. </s>
          <s id="s.001093">Vt autem <lb></lb>
huius illuminationis non iniucundam facias experientiam; cape ſphæram <lb></lb>
quampiam ſolidam manu, cum qua recede ad medium cubiculi, &amp; pone lu­<lb></lb>
men ſeorſum ad partem aliquam: deinde brachio extenſo oppone ſphæram <lb></lb>
lumini, quo ſitu nihil de illuminatione videbis, quamuis dimidium ferè il­<lb></lb>
lius illuminetur. </s>
          <s id="s.001094">poſtea conuerte te ipſum ibidem paulatim, ita vt aliquid <lb></lb>
illuminationis oculo tuo appareat; &amp; videbis partem illam illuminationis, <lb></lb>
falcatæ, ſeu nouæ Lunæ ſimilem. </s>
          <s id="s.001095">Deinde adhuc magis te conuerte, &amp; cer­<lb></lb>
nes illuminationem dimidiatæ Lunæ ſimilem: verte adhuc te ipſum donec <lb></lb>
ſit ſphæra ita lumini oppoſita, vt inter ipſam, &amp; lumen oculus tuus ſit me­<lb></lb>
dius; apparebit tunc tota illuminatio, quæ erit inſtar plenilunij. </s>
          <s id="s.001096">perge ad­


<pb pagenum="60" xlink:href="009/01/060.jpg"></pb>huc te ipſum conuertere, &amp; videbis paulatim lumen oculo tuo decreſcere <lb></lb>
non aliter ac in Luna ſeneſcente. </s>
          <s id="s.001097"><expan abbr="atq;">atque</expan> hoc eſt ſphæricè illuminari, fierique <lb></lb>
ſphærica illuminationis augmenta. </s>
          <s id="s.001098">cum ergo videamus Lunam eo modo lu­<lb></lb>
mine augeri, quo ſphæra, hinc ipſam <expan abbr="quoq;">quoque</expan> ſphæricam eſſe argumentamur.</s>
        </p>
        <p type="main">
          <s id="s.001099"><arrow.to.target n="marg49"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001100"><margin.target id="marg49"></margin.target>49</s>
        </p>
        <p type="main">
          <s id="s.001101">Poſt nonnulla (<emph type="italics"></emph>Vt Perſpectiua ad Geometriam, &amp; Mechanica ad Stereome­<lb></lb>
tricam, &amp; Harmonica ad Arithmeticam, vt Apparentia ad Aſtrologicam<emph.end type="italics"></emph.end>) ſupra <lb></lb>
tex. 20. exempla ſubalternationum Perſpectiuæ, &amp; Mechanicæ cum Geo­<lb></lb>
metria ſunt allata. </s>
          <s id="s.001102">hic primo notandum Stereometriam non eſſe ſcientiam <lb></lb>
diſtinctam à Geometria, niſi ſicuti partem à toto: nam cum Geometria <lb></lb>
conſideret quantitatem, ſecundum tres dimenſiones, longitudinem, latitu­<lb></lb>dinem, &amp; profunditatem, oritur triplex illius diuiſio, de lineis, de ſuperfi­<lb></lb>
ciebus, de ſolidis. </s>
          <s id="s.001103">pars igitur, quæ de ſolidis tractat, <expan abbr="partimq́">partimque</expan>; continetur <lb></lb>
11. 12. 13. 14. &amp; 15. Euclidis, partim aliorum Geometrarum libris, vt li­<lb></lb>
bro Archim. de Sphæra, &amp; Cyl. &amp; ſimilibus, dicitur Stereometria à græco <lb></lb>
<foreign lang="grc">στερεον,</foreign> ideſt ſolidum. </s>
          <s id="s.001104">Porrò cur malit Ariſt. Mechanicam ſubalternari Ste­<lb></lb>
reometriæ, quam toti Geometriæ, qua tamen, vt videre eſt apud Archime­<lb></lb>
dem, innititur, fortè ea ratio eſt, quia Mechanica præcipuè conſiderat ma­<lb></lb>
chinas, quæ corpora ſunt, &amp; propterea præcipuè, &amp; primò debet Stereome­<lb></lb>
triæ, quæ corpora pariter contemplatur, ſubalternari. </s>
          <s id="s.001105">Quod ait Apparen­<lb></lb>tia ad Aſtrol. </s>
          <s id="s.001106">intelligit per Apparentia vulgarem quandam Nautarum, &amp; <lb></lb>
Agricolarum aſtronomiam, quæ quodammodo ſubalternatur, &amp; pendet ex <lb></lb>
ſcientia Aſtrologiæ; indiget enim cognitione ortus, &amp; motus aſtrorum, <lb></lb>
præſertim Lunæ, Hyadum, Pleiadum, &amp; Canis. </s>
          <s id="s.001107">Reliqua <expan abbr="vſq;">vſque</expan> ad finem ca­<lb></lb>
pitis optimè à Zabarella explicantur, <expan abbr="neq;">neque</expan> ad nos pertinet, cum de Mathe­<lb></lb>
maticis agant, quatenus ad Logicum ſpectant.</s>
        </p>
        <p type="main">
          <s id="s.001108"><arrow.to.target n="marg50"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001109"><margin.target id="marg50"></margin.target>50</s>
        </p>
        <p type="main">
          <s id="s.001110">Poſt nonnulla (<emph type="italics"></emph>Hic enim ipſum quidem quod ſenſitiuorum eſt ſcire, ipſum ve­<lb></lb>
rò Propter quid Mathematicorum; hi <expan abbr="namq;">namque</expan> habent cauſarum demonſtrationes, <lb></lb>
&amp;c.<emph.end type="italics"></emph.end>) ſenſus eſt in ſubalternatis, &amp; dependentibus diſciplinis, quas ſenſitiuas <lb></lb>
appellat, quia de rebus ſenſibilibus ſunt, vt in Perſpectiua de obiectis viſibi­<lb></lb>
libus, &amp; in Muſica de ſonis cognoſcitur Quod, ideſt effectus: cuius effectus <lb></lb>
cauſa, ſeu Propter quid ſcitur auxilio Mathematicarum, ideſt, traditur à <lb></lb>
ſcientijs ſubalternantibus. </s>
          <s id="s.001111">v. g. alicuius effectus in Perſpectiua cauſa inqui­<lb></lb>ritur, &amp; inuenitur ope Geometriæ, cui illa ſubiacet. </s>
          <s id="s.001112">Hic obiter notandum, <lb></lb>
Ariſt. fateri manifeſtè Mathematicas ſubalternatas, ſeu medias oſtendere <lb></lb>
per cauſas, quas ſubalternantium ope perueſtigant.</s>
        </p>
        <p type="main">
          <s id="s.001113"><arrow.to.target n="marg51"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001114"><margin.target id="marg51"></margin.target>51</s>
        </p>
        <p type="main">
          <s id="s.001115">Et poſtea (<emph type="italics"></emph>Se habet autem &amp; ad Perſpectiuam, vt hæc ad Geometriam, alia ad <lb></lb>hanc, vt quod eſt de Iride ipſum enim quod Naturalis eſt ſcire, ipſum verò Prop­<lb></lb>
ter quid Perſpectiui<emph.end type="italics"></emph.end>) ſicut ſe habet, inquit, <expan abbr="ſciẽtià">ſcientià</expan> Naturalis de Iride ad Per­<lb></lb>
ſpectiuam, ita Perſpectiua ad Geometriam. </s>
          <s id="s.001116">qua verò ratione cauſa Iridis <lb></lb>
pertineat ad opticam, <expan abbr="atq;">atque</expan> hine tandem ad Geometriam, optimè patebit <lb></lb>
in Meteoris, cum ipſius demonſtrationem afferemus.</s>
        </p>
        <p type="main">
          <s id="s.001117"><arrow.to.target n="marg52"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001118"><margin.target id="marg52"></margin.target>52</s>
        </p>
        <p type="main">
          <s id="s.001119">Tex. 37. (<emph type="italics"></emph>Vt æquicruri, &amp; Scaleno hoc, quod eſt duobus rectis æquales habere <lb></lb>
ſecandum commune aliquod ineſt<emph.end type="italics"></emph.end>) quid ſit habere tres æquales duobus rectis <lb></lb>
ſatis explicatum eſt lib. r. </s>
          <s id="s.001120">Priorum ſecto 3. cap. r. </s>
          <s id="s.001121">nunc igitur paraphraſim <lb></lb>
ſolum huius loci dabo. </s>
          <s id="s.001122">Triangulo Iſoſceli, &amp; Scaleno convenit paſſio illa,<lb></lb>
habere tres angulos æquales duobus rectis angulis ſecundum aliquod com­


<pb pagenum="61" xlink:href="009/01/061.jpg"></pb>mune, quia illis competit, quatenus ambo ſunt figura quædam, ideſt, qua­<lb></lb>
tenus <expan abbr="vtrumq;">vtrumque</expan> illorum triangulum eſt; triangulo <expan abbr="namq;">namque</expan> omni primo com­<lb></lb>
petit habere tres angulos æquales duobus rectis.</s>
        </p>
        <p type="main">
          <s id="s.001123"><arrow.to.target n="marg53"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001124"><margin.target id="marg53"></margin.target>53</s>
        </p>
        <p type="main">
          <s id="s.001125">Tex. 38. (<emph type="italics"></emph>Et quemadmodum in alijs principium ſimplex, hoc autem non idem <lb></lb>
vbique, ſed in pondere quidem mina, in cătu verò dieſis<emph.end type="italics"></emph.end>) Dieſis apud Muſicos eſt <lb></lb>
pars Toni. </s>
          <s id="s.001126">Tonus autem eſt interuallum duarum vocum, quale eſt inter pri­<lb></lb>
mam vocem, Vt, &amp; ſecundam Rè, vt modo loquuntur. </s>
          <s id="s.001127">iſtud interuallum <lb></lb>
diuidunt Muſici primum in ſemitonia, non tamen æqualia, ſed vnum maius <lb></lb>
altero. </s>
          <s id="s.001128">minus iterum in duas partes æquales ſubdiuidunt, quarum <expan abbr="vtramq;">vtramque</expan> <lb></lb>
veteres harmonici dieſim dixerunt. </s>
          <s id="s.001129">&amp; hęc dieſis eſt minima vox ab eis con­<lb></lb>
ſiderata; &amp; quæ prima cadit ſub ſenſum; &amp; propterea veluti ſimplex prin­<lb></lb>cipium, &amp; elementum, ex quo alia maiora interualla conſtent; &amp; in quod <lb></lb>
reſoluuntur. <foreign lang="grc">διέοις</foreign> porrò græcè valet inter alia, diuiſionem. </s>
          <s id="s.001130">igitur interual­<lb></lb>
lum iſtud minimum dictum eſt dieſis, quod ſit quædam diuiſio, ſeu ſegmen­<lb></lb>tum Toni (<emph type="italics"></emph>Quemadmodum in pondere mina<emph.end type="italics"></emph.end>) qui de ponderibus antiquis tra­<lb></lb>
ctant, aſſerunt, Minam fuiſſe maiorem libra per ſemunciam, æquipondera­<lb></lb>
bat enim centum drachmis: quæ refragantur huic loco. </s>
          <s id="s.001131">ſed fortè <expan abbr="dicẽdum">dicendum</expan>, <lb></lb>
Ariſt. conſideraſſe, Minam reſpectu Talenti, reſpectu enim illius dici poteſt <lb></lb>
principium, cum ſex millia minarum in Attico talento continerentur.</s>
        </p>
        <figure id="id.009.01.061.1.jpg" place="text" xlink:href="009/01/061/1.jpg"></figure>
        <p type="main">
          <s id="s.001132"><arrow.to.target n="marg54"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001133"><margin.target id="marg54"></margin.target>54</s>
        </p>
        <p type="main">
          <s id="s.001134">Tex. 39. <emph type="italics"></emph>(Si enim quod duobus rectis ineſt, non in <lb></lb>
quantum æquicrus, ſed in quantum triangulus, no­<lb></lb>
ſcens, &amp;c.)<emph.end type="italics"></emph.end> ideſt, ſi enim qui cognoſcit, quod ha­<lb></lb>
bere tres angulos æquales duobus rectis conuenit <lb></lb>
æquicruri, non quatenus æquicrus eſt, ſed quate­<lb></lb>
nus triangulus eſt, &amp;c. </s>
          <s id="s.001135">quid ſit habere tres æqua­<lb></lb>
les duobus rectis, &amp;c. </s>
          <s id="s.001136">fusè explicatum eſt in lib. 1. <lb></lb>
Priorum ſecto 3. cap. 1. quò te nunc mitto.</s>
        </p>
        <p type="main">
          <s id="s.001137"><arrow.to.target n="marg55"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001138"><margin.target id="marg55"></margin.target>55</s>
        </p>
        <p type="main">
          <s id="s.001139">Poſt pauca <emph type="italics"></emph>(Ineſt omni triangulo hoc quod est <lb></lb>
duos, &amp;c.)<emph.end type="italics"></emph.end> ideſt, hæc proprietas, quæ eſt habere <lb></lb>
duos angulos rectos non actu, ſed per æquiualen­<lb></lb>
tiam trium angulorum trianguli. </s>
          <s id="s.001140">Vide quæ im­<lb></lb>mediatè ſupra de hac re dixi, &amp; quò te remiſi.</s>
        </p>
        <p type="main">
          <s id="s.001141"><arrow.to.target n="marg56"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001142"><margin.target id="marg56"></margin.target>56</s>
        </p>
        <p type="main">
          <s id="s.001143">Eodem tex <emph type="italics"></emph>(Quando igitur cognoſcimus, quod <lb></lb>
quatuor exteriores ſunt æquales, quoniam Iſoſceles, <lb></lb>adhuc deficit, propier quid Iſoſceles? </s>
          <s id="s.001144">quoniam trian­<lb></lb>
gulus: &amp; hoc quoniam figura rectilinea, &amp;c.)<emph.end type="italics"></emph.end> exem­<lb></lb>
plo geometrico vult oſtendere demonſtrationem <lb></lb>
vniuerſalem eſſe particulari præſtantiorem: eſt <lb></lb>
autem exemplum de pulcherrima, <expan abbr="atq;">atque</expan> admira­<lb></lb>
bili proprietate, quæ omnibus figuris rectilineis <lb></lb>
conuenit, eſt <expan abbr="q́">que</expan>; huiuſmodi: Omnis figuræ rectili­<lb></lb>neæ anguli externi omnes ſimul ſumpti, ſunt æqu<lb></lb>
les quatuor rectis angulis, quæ affectio demon­<lb></lb>
ſtratur in ſcholio 32. primi Elem. dicuntur autem <lb></lb>
anguli externi, qui productis lateribus fiunt, vt in <lb></lb>triangulo præſenti anguli externi ſunt, B D C,<pb pagenum="62" xlink:href="009/01/062.jpg"></pb>D F E, F B A, ita vt quælibet figura tot angulos externos ſortiatur, quot <lb></lb>
habet latera; cum exproductis lateribus oriantur. </s>
          <s id="s.001145">Vt autem propoſitio ve­<lb></lb>
rificetur, ſingula latera ordinatim ſunt producenda, hoc eſt, verſus eandem <lb></lb>
partem, vt in figuris appoſitis vides. </s>
          <s id="s.001146">Quæuis igitur figura rectilinea, ſiue <lb></lb>
trilatera ſit, ſiue quadrilatera, vel etiam millelatera, &amp; proinde mille quo­<lb></lb>
que angulos externos habeat, hanc tamen mirabilem proprietatem (quod <lb></lb>
vix credi poteſt) poſſidet, vt omnes illi anguli externi ſimul ſint æquales <lb></lb>
quatuor rectis angulis. </s>
          <s id="s.001147">vnde tres externi anguli trianguli, &amp; quatuor exter­<lb></lb>
ni quadranguli, &amp; quinque externi <expan abbr="pẽtagoni">pentagoni</expan>, &amp;c. </s>
          <s id="s.001148">ſunt æquales quatuor tan­<lb></lb>
tum rectis, nec aliter res ſe habet in figura millelatera. </s>
          <s id="s.001149">Ex quo fit, vt an­<lb></lb>guli externi cuiuſuis figuræ ſint æquales angulis omnibus externis alterius <lb></lb>
cuiuſlibet figuræ. </s>
          <s id="s.001150">Ariſt. igitur inquit, quando cognoſcimus, quod quatuor <lb></lb>
angulis rectis ſunt æquales exteriores omnes anguli alicuius figuræ, quo­<lb></lb>
niam figura illa eſt triangulum ſcalenum, adhuc talis cognitio eſt defecti­<lb></lb>
ua, quia non illi competit illa paſſio, quia ſit triangulum ſcalenum, neque <lb></lb>
competit ſcaleno, quia ſit triangulum; ſed his omnibus competit, quia ſunt <lb></lb>
figuræ rectilineæ, cui hæc proprietas ineſt primo, &amp; vniuerſaliter: qui igi­<lb></lb>
tur ſcit, ſcalenum habere prædictam affectionem, ex eo, quod ſit figura re­<lb></lb>
ctilinea, perfectius ſcit, quia nihil amplius quæri poteſt, quia illa figura re­<lb></lb>
ctilinea illud vniuerſale eſt, cui primo competit; reliquis autem per illam. <lb></lb>
</s>
          <s id="s.001151">qui igitur vniuerſale ſcit, perfectius ſcit; quod volebat Ariſt. demonſtrare.</s>
        </p>
        <p type="main">
          <s id="s.001152"><arrow.to.target n="marg57"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001153"><margin.target id="marg57"></margin.target>57</s>
        </p>
        <p type="main">
          <s id="s.001154">Eodem tex. <emph type="italics"></emph>(Vt ſi quis nouit, quod omnis triangulus habet tres duobus rectis <lb></lb>
æquales)<emph.end type="italics"></emph.end> nihil frequentius. </s>
          <s id="s.001155">vide ſupra lib. 1. Priorum ſecto 3. cap. 1.</s>
        </p>
        <p type="main">
          <s id="s.001156"><arrow.to.target n="marg58"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001157"><margin.target id="marg58"></margin.target>58</s>
        </p>
        <p type="main">
          <s id="s.001158">Tex. 43. <emph type="italics"></emph>(Sed planum, quod etſi eſſet ſentire triangulum, quod duobus rectis <lb></lb>
æquales habet angulos)<emph.end type="italics"></emph.end> vide ſupra lib. 1. Priorum ſecto 3. cap. 1.</s>
        </p>
        <p type="main">
          <s id="s.001159"><arrow.to.target n="marg59"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001160"><margin.target id="marg59"></margin.target>59</s>
        </p>
        <p type="main">
          <s id="s.001161">Poſt pauca <emph type="italics"></emph>(Quare &amp; ſi ſupra Lunam eſſemus, &amp; videremus obiectam terram, <lb></lb>
non <expan abbr="vtiq;">vtique</expan> ſciremus cauſam eclypſis)<emph.end type="italics"></emph.end> loquitur de defectu Lunæ, qui fit, quando <lb></lb>
terra inter Lunam, &amp; Solem poſita, impedit, ne lumen Solis feratur in Lu­<lb></lb>
nam, ſed efficit, vt vmbra ipſius terræ eam contegat.</s>
        </p>
        <p type="main">
          <s id="s.001162"><arrow.to.target n="marg60"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001163"><margin.target id="marg60"></margin.target>60</s>
        </p>
        <figure id="id.009.01.062.1.jpg" place="text" xlink:href="009/01/062/1.jpg"></figure>
        <p type="main">
          <s id="s.001164">Et paulo poſt <emph type="italics"></emph>(Quemadmodŭm ſi vi­<lb></lb>
trum perforatum videremus, &amp; lumen <lb></lb>
permeans, planum vtique eſſet propter <lb></lb>
quid comburit)<emph.end type="italics"></emph.end> Ioquitur de ea com­<lb></lb>
buſtione, cuæ fit per refractionem <lb></lb>
media ſphæra vitrea. </s>
          <s id="s.001165">de qua Vitel­<lb></lb>
lio propoſ. </s>
          <s id="s.001166">48. decimi libri; non au­<lb></lb>
tem de ea, quæ fit per reflexionem <lb></lb>
ex ſpeculo concauo quando combu­<lb></lb>
ſtio fit per refractionem, cauſatur à <lb></lb>
radijs Solis vitrum permeantibus, <lb></lb>
in quo ita franguntur, vt egredien­<lb></lb>
tes è vitro ſimul vniantur, ex qua <lb></lb>
vnione ita calor intenditur, vt ibi <lb></lb>
comburat. </s>
          <s id="s.001167">vt in appoſita figura cer­<lb></lb>
nere facile eſt; in qua radij à Sole <lb></lb>
manentes, ſphæram vitream perua­


<pb pagenum="63" xlink:href="009/01/063.jpg"></pb>dunt, <expan abbr="atq;">atque</expan> in exitu ita refraguntur, vt ad A, punctum coaceruati, ibi poſ­<lb></lb>
ſint, ſi quid combuſtibile occurrat, comburere. </s>
          <s id="s.001168">Si igitur, inquit Ariſt. vide­<lb></lb>
remus illos radios ſic permeare, &amp; refrangi, planum <expan abbr="vtiq;">vtique</expan> nobis eſſet pro­<lb></lb>
pter quid incendant.<lb></lb>
<arrow.to.target n="marg61"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001169"><margin.target id="marg61"></margin.target>61</s>
        </p>
        <p type="main">
          <s id="s.001170">Ad finem tex. 43. <emph type="italics"></emph>(Principia enim duplicia ſunt, ex quibus, &amp; circa quod: <lb></lb>
quæ quidem igitur, ex quibus, communia ſunt: quæ autem circa quod propria, vt <lb></lb>
numerus, magnitudo)<emph.end type="italics"></emph.end> nonnulli codices corruptè legunt (vt numerus magni­<lb></lb>
tudine) ſed ex græco tex. corrigendi ſunt, vti fecimus. </s>
          <s id="s.001171">Cæterum per prin­<lb></lb>
cipia, ex quibus intelligit Dignitates, quia ex illis diſcurrimus. </s>
          <s id="s.001172">per princi­<lb></lb>
pia verò circa quod, intelligit Definitiones, quibus, vt apparet apud Eucli­<lb></lb>
dem, explicatur ſubiectum, circa quod ſcientia verſatur; vt in definitioni­<lb></lb>
bus primi Elem. docemur, quid ſit linea, quid triangulum, quid circulus, <lb></lb>
quid magnitudines reliquæ, quæ ſunt materia, circa quam Geometria ſpe­<lb></lb>
culatur. </s>
          <s id="s.001173">In ſeptimo verò traduntur definitiones numerorum, quid ſit nu­<lb></lb>
merus, quid impar, quid compoſitus, quadratus, cubus, &amp; reliquæ nume­<lb></lb>
rorum ſpecies, quæ ſunt materia ſeptimi, octaui, &amp; noni, in quibus de Arith­<lb></lb>
metica tractatur.</s>
        </p>
        <p type="main">
          <s id="s.001174"><arrow.to.target n="marg62"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001175"><margin.target id="marg62"></margin.target>62</s>
        </p>
        <p type="main">
          <s id="s.001176">Tex. 44. <emph type="italics"></emph>(Commenſurabilem namq; eſſe diametrum verè opinari, abſurdum eſt)<emph.end type="italics"></emph.end><lb></lb>
vide, quæ de <expan abbr="commẽſurabilitate">commenſurabilitate</expan> diametri quadrati cum latere expoſuimus <lb></lb>
lib. 1. Priorum ſecto 1. cap. 23. ait igitur Ariſt. abſurdum eſſe opinari dia­<lb></lb>
metrum eſſe commenſurabilem coſtæ, ſeu lateri eiuſdem quadrati, reli­<lb></lb>
qua ſunt Logica.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001177"><emph type="italics"></emph>Ex Secundo Posteriorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001178"><arrow.to.target n="marg63"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001179"><margin.target id="marg63"></margin.target>63</s>
        </p>
        <p type="main">
          <s id="s.001180">Tex. 1. <emph type="italics"></emph>(Dico autem ſimpliciter quidem ſubiectum, vt Lunam, aut ter­<lb></lb>
ram, aut Solem, aut triangulum; aliquid verò defectum, æqualitatem, <lb></lb>
inæqualitatem. </s>
          <s id="s.001181">ſi in medio, aut non)<emph.end type="italics"></emph.end> Zabarella locum hunc, etiam <lb></lb>
quatenus ad Mathematicum attinet, optimè declarat. </s>
          <s id="s.001182">In quæ­<lb></lb>
ſtionibus, &amp; demonſtrationibus duo ſunt, ſubiectum, &amp; <expan abbr="prædicatũ">prædicatum</expan>, <expan abbr="vtriuſq;">vtriuſque</expan> <lb></lb>
cauſæ exiſtunt, &amp; quæruntur: v. g. Luna, terra, Sol, &amp; triangulum ſunt ſu­<lb></lb>
biectum in demonſtratione, quorum prædicata ſunt, Lunæ quidem, &amp; So­<lb></lb>
lis, eclypſis. </s>
          <s id="s.001183">terræ autem eſſe in medio mundi, quod ab Aſtronomis ratione <lb></lb>
ab eclypſibus deſumpta, euidentius, quam ab alio quoquam demonſtratur, <lb></lb>vt patet ex tractatu de ſphæra. </s>
          <s id="s.001184">in quo Zabarella non probatur, qui ſolum <lb></lb>
ait, terram eſſe in medio mundi, à Phyſicis demonſtrari. </s>
          <s id="s.001185"><expan abbr="triãgulum">triangulum</expan> autem, <lb></lb>
ſeu angulorum ipſius <expan abbr="prædicatũ">prædicatum</expan> eſt æqualitas, &amp; inæqualitas: vt cum in 32. <lb></lb>
primi Elem. demonſtrat Euclides, omne triangulum habere tres angulos <lb></lb>
æquales duobus rectis.</s>
        </p>
        <p type="main">
          <s id="s.001186"><arrow.to.target n="marg64"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001187"><margin.target id="marg64"></margin.target>64</s>
        </p>
        <p type="main">
          <s id="s.001188">Ibidem <emph type="italics"></emph>(Quid eſt conſonantia? </s>
          <s id="s.001189">ratio numerorum in acuto, &amp; graui, &amp;c)<emph.end type="italics"></emph.end> tan­<lb></lb>
git breuiter Ariſt. cauſam formalem conſonantiæ, &amp; conſequenter defini­<lb></lb>
tionem ipſius. </s>
          <s id="s.001190">definiunt igitur Muſici conſonantiam hoc modo; Conſonan­<lb></lb>
tia eſt compoſitio ſoni grauis, &amp; acuti, quæ ſuauiter auribus accidit; &amp; quo­<lb></lb>
rum ſonorum proportio ad inuicem ſit ſicuti proportio numerorum, qui <lb></lb>
quaternario includuntur: vt eſt proportio 2. ad 1. vel 3. ad 1. vel 4. ad 1. <lb></lb>
vel 3. ad 2. vel 4. ad 3. <expan abbr="Quotieſeunq;">Quotieſcunque</expan> igitur duo ſoni habuerin quampiam <pb pagenum="64" xlink:href="009/01/064.jpg"></pb>ex <expan abbr="quinq;">quinque</expan> prædictis proportionibus, ſi ſimul coaluerint, ita vt ex eis vnus <lb></lb>
tantum ſonus efficiatur; ſonus ille erit concordans, &amp; auribus gratus. </s>
          <s id="s.001191"><expan abbr="atq;">atque</expan> <lb></lb>
hæc eſt ſententia priſcorum præſertim Pythagoreorum, qui propterea di­<lb></lb>
cebant non licere Muſico vltra quaternarium pertranſire, eò quod ſolæ pro­<lb></lb>
portiones, vt diximus, numerorum quaternario contentorum, concordem, <lb></lb>
ac conſonantem concentum efficere poterant: quod vt adhuc melius per­<lb></lb>
<figure id="id.009.01.064.1.jpg" place="text" xlink:href="009/01/064/1.jpg"></figure><lb></lb>
cipiamus, accipe exemplum. </s>
          <s id="s.001192">Sint duæ chordæ <lb></lb>
A, &amp; B, æqualis craſſitici, &amp; æquè tenſæ. </s>
          <s id="s.001193">qua­<lb></lb>
rum A, dupla ſit ipſius B, quia igitur corpora <lb></lb>
ſonantia ſunt in dupla proportione, erunt pa­<lb></lb>
riter eorum ſoni in ratione dupla (vt patet ex <lb></lb>
principijs harmonicæ) hoc eſt, <expan abbr="eorũ">eorum</expan> ſoni erunt, <lb></lb>
vt 2. ad 1. quia ſcilicet ſonus maioris chordæ A, erit duplus ad ſonum mi­<lb></lb>
noris chordæ B. hoc eſt, erit, vt 2. ad 1. &amp; propterea, ſi ſimul ambæ chordæ <lb></lb>
pulſentur, ſonus, quem ex duobus mixtum edent, conſonans, <expan abbr="atq;">atque</expan> gratiſſi­<lb></lb>
mus auribus noſtris perueniet. </s>
          <s id="s.001194">huiuſmodi porrò conſonantia, quæ eſt in <lb></lb>
proportione dupla, <expan abbr="quæq́">quæque</expan> omnium ſuauiſſima eſt, à græcis dicebatur Dia­<lb></lb>paſon. </s>
          <s id="s.001195"><expan abbr="atq;">atque</expan> hæc in præſentia ſufficiant, cum plura de his ad ſectionem pro­<lb></lb>
blematum 19. quæ tota eſt de Muſica, dicenda ſint.</s>
        </p>
        <p type="main">
          <s id="s.001196"><arrow.to.target n="marg65"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001197"><margin.target id="marg65"></margin.target>65</s>
        </p>
        <p type="main">
          <s id="s.001198">Tex. 2. <emph type="italics"></emph>(Vt quod omnis triangulus duobus rectis æquales habet)<emph.end type="italics"></emph.end> vide anno­<lb></lb>
tata lib. 1. Priorum ſecto 3. cap. 1.</s>
        </p>
        <p type="main">
          <s id="s.001199"><arrow.to.target n="marg66"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001200"><margin.target id="marg66"></margin.target>66</s>
        </p>
        <p type="main">
          <s id="s.001201">Eodem tex. <emph type="italics"></emph>(Definitiones verò apparent omnes ſupponentes, &amp; accipientes <lb></lb>
ipſum quid eſt, vt Mathematicæ, quid vnitas, quid par, &amp; impar)<emph.end type="italics"></emph.end> alludit ad de­<lb></lb>
finitiones 7. Elem. vbi agitur de numeris. </s>
          <s id="s.001202">Quæ verò hoc loco de principijs <lb></lb>
dicuntur, luculentiſſimè patent conſideranti definitiones, &amp; axiomata, quæ <lb></lb>
Mathematicis demonſtrationibus in omnibus ferè libris præmittuntur; ex <lb></lb>
quibus ſtatim demonſtrationes deriuantur.</s>
        </p>
        <p type="main">
          <s id="s.001203"><arrow.to.target n="marg67"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001204"><margin.target id="marg67"></margin.target>67</s>
        </p>
        <p type="main">
          <s id="s.001205">Et paulo poſt <emph type="italics"></emph>(<expan abbr="Neq;">Neque</expan> <expan abbr="vtiq;">vtique</expan> de plano figura, non enim eſt planum figura, <expan abbr="neq;">neque</expan> fi­<lb></lb>
gura planum)<emph.end type="italics"></emph.end> alludit ad definitiones planarum figurarum, qualis eſt circu­<lb></lb>
lus, cuius definitio eſt inter definitiones primi Elem. 15. &amp; eſt huiuſmodi: <lb></lb>
circulus eſt figura plana, ſub vnica linea comprehenſa, quæ periphæria ap­<lb></lb>
pellatur, ad quam ab vno puncto eorum, quæ intra figuram ſunt poſita, ca­<lb></lb>
dentes omnes rectæ lineæ inter ſe ſunt æquales: in qua quidem definitione <lb></lb>
non prædicatur planum de figura, nec figura de plano: <expan abbr="neq;">neque</expan> enim planum, <lb></lb>
ſeu plana ſuperficies eſt figura ſecundum ſe, niſi terminetur; <expan abbr="neq;">neque</expan> figura eſt <lb></lb>
plana ſuperficies, cum plurimæ ſint figuræ curuæ, &amp; præterea ſolidæ quam­<lb></lb>
plurimæ.</s>
        </p>
        <p type="main">
          <s id="s.001206"><arrow.to.target n="marg68"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001207"><margin.target id="marg68"></margin.target>68</s>
        </p>
        <p type="main">
          <s id="s.001208">Ibidem <emph type="italics"></emph>(Quoniam monſtratum eſt Iſoſceles habere tres angulos æquales duo­<lb></lb>
bus rectis, ſi id de omni triangulo monſtratum ſit)<emph.end type="italics"></emph.end> ex dictis lib. 1. Priorum ſecto <lb></lb>
3. cap. 1. petatur huius loci declaratio.</s>
        </p>
        <p type="main">
          <s id="s.001209"><arrow.to.target n="marg69"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001210"><margin.target id="marg69"></margin.target>69</s>
        </p>
        <p type="main">
          <s id="s.001211">Tex. 7. <emph type="italics"></emph>(Quid enim ſignificat triangulum, accipit Geometra)<emph.end type="italics"></emph.end> vt manifeſtum <lb></lb>
eſt in 20. definitione primi Elem.</s>
        </p>
        <p type="main">
          <s id="s.001212"><arrow.to.target n="marg70"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001213"><margin.target id="marg70"></margin.target>70</s>
        </p>
        <p type="main">
          <s id="s.001214">Ibidem <emph type="italics"></emph>(Quod autem ſit, monstrat)<emph.end type="italics"></emph.end> vt perſpicuum eſt in prima <expan abbr="demõſtra-tione">demonſtra­<lb></lb>
tione</expan> primi Elem. vbi triangulum æquilaterum conſtruit, &amp; poſtea probat <lb></lb>
illud eſſe triangulum æquilaterum. </s>
          <s id="s.001215">Certum tamen eſt, Geometram ſuppo­<lb></lb>
nere triangulum in communi, cum inter definitiones ipſius contineatur,


<pb pagenum="65" xlink:href="009/01/065.jpg"></pb>quod tamen non obſtat, quominus probare poſſit, aliquando poſſe <expan abbr="cõſtrni">conſtrui</expan>, <lb></lb>
&amp; eſſe aliquod particulare triangulum, vt fit in prædicta demonſtratione, <lb></lb>
Euclidis.</s>
        </p>
        <p type="main">
          <s id="s.001216"><arrow.to.target n="marg71"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001217"><margin.target id="marg71"></margin.target>71</s>
        </p>
        <p type="main">
          <s id="s.001218">Tex. 11. <emph type="italics"></emph>(Manifeſtum autem, &amp; ſic, propter quid eſt rectus in ſemicirculo)<emph.end type="italics"></emph.end><lb></lb>
affert exemplum demonſtrationis per cauſam materialem, <expan abbr="idq́">idque</expan>; vti ſolet ex <lb></lb>
Mathematicis petitum, eſt enim apud Euclidem 31. demonſtratio 3. Elem. <lb></lb>
vbi ipſe oſtendit angulum in ſemicirculo eſſe rectum. </s>
          <s id="s.001219">Vbi aduertendum eſt <lb></lb>
propoſitionem hanc 31. ab Euclide demonſtrari duobus modis; ex quibus <lb></lb>
ſecundum innuit hoc loco Ariſt. cui aſcripta eſt figura ſimilis huic noſtræ; <lb></lb>
in editione Clauiana. </s>
          <s id="s.001220">quod fortè non benè aduertens Iacobus Zabarella, <lb></lb>
alioquin in his ſatis oculatus incidit in errorem, dicens, ſe nullo pacto vi­<lb></lb>
dere medium Euclidianæ demonſtrationis eſſe cauſam materialem; quod <lb></lb>
tamen nos mox aperiemus. </s>
          <s id="s.001221">per angulum in ſemicirculo intelligas eum, qui <lb></lb>
fit à lineis ductis ab extremitatibus diametri, &amp; ſimul in quoduis punctum <lb></lb>
<figure id="id.009.01.065.1.jpg" place="text" xlink:href="009/01/065/1.jpg"></figure><lb></lb>
circumferentiæ coeuntibus, vt in figura <lb></lb>
præſenti vides lineas A C, B C, ad C, pun­<lb></lb>
ctum conuenire, <expan abbr="ibiq́">ibique</expan>; facere angulum, <lb></lb>
A C B, qui dicitur angulus in ſemicircu­<lb></lb>
lo, quia deſcriptus eſt in ſemicirculo A­<lb></lb>
C B. <expan abbr="eſtq́">eſtque</expan>; ſanè mirabilis hæc ſemicirculi <lb></lb>
proprietas, cum <expan abbr="vbicunq;">vbicunque</expan> punctum C, in <lb></lb>
periphæria ſumptum fuerit, ſemper ta­<lb></lb>
men angulus A C B, fiat rectus. </s>
          <s id="s.001222">quod Euclides eodem prorſus medio, quod <lb></lb>
Ariſt. hic innuit, hoc modo demonſtrat. </s>
          <s id="s.001223">ducta enim recta D C, à centro D, <lb></lb>
ad punctum C, exurgunt duo lſoſcelia triangula A D C, C D B, ergo per <lb></lb>
5. primi, anguli D C A, D A C, ſunt æquales: pariter anguli D C B, D B C, <lb></lb>
æquales ſunt. </s>
          <s id="s.001224">&amp; quia per 32. primi, anguli D A C, D C A, ſimul ſunt æqua­<lb></lb>
les angulo externo C D B, &amp; inter ſe æquales, erit angulus A C D, dimidium <lb></lb>
anguli C D B. eadem ratione probatur angulus D C B, eſſe dimidium an­<lb></lb>
guli C D A. ergo totus angulus A C B, dimidium erit duorum angulorum <lb></lb>
A D C, C D B, qui per 13. primi, ſunt vel recti, vel duobus rectis <expan abbr="æquiualẽt">æquiualent</expan>. <lb></lb>
</s>
          <s id="s.001225">Sequitur igitur, angulum A C B, in ſemicirculo eſſe dimidium duorum re­<lb></lb>
ctorum; &amp; quia omnes recti ſunt æquales, ſequitur dimidium duorum re­<lb></lb>
ctorum, nihil aliud eſſe, quam vnum rectum angulum, ergo angulus in ſe­<lb></lb>
micirculo, cum ſit ſemiſſis duorum <expan abbr="rectorũ">rectorum</expan>, erit vnus rectus angules; quod <lb></lb>
erat probandum. </s>
          <s id="s.001226">ex quibus vides medium illud, quod Ariſt. aſſumpſit, eſſe <lb></lb>
omnino idem cum eo, quo Euclides vtitur, ſcilicet, eſſe dimidium duorum <lb></lb>
rectorum, &amp; propterea eſſe rectum: quod etiam medium in toto demon­<lb></lb>
ſtrationis decurſu eſt vltimum, &amp; principale, quod proximè concluſionem <lb></lb>
attingit, &amp; propterea dici meretur eſſe medium huius demonſtrationis. <lb></lb>
</s>
          <s id="s.001227">Cæterum, quod medium iſtud ſit in genere cauſæ materialis, patet ex eo, <lb></lb>
quod eſt, eſſe dimidium; nam eſſe dimidium, vel eſſe tertiam partem, &amp; ſi­<lb></lb>
milia, nihil aliud eſt, quam eſſe partem; eſſe autem partem eſt eſſe materiam <lb></lb>
totius, etiam ex ſententia ipſius Ariſt. ex hac præterea materia conflatur <lb></lb>
definitio minoris extremi, vel ſubiecti; dum dicitur, angulus in ſemicircu­<lb></lb>
lo eſt dimidium duorum rectorum. </s>
          <s id="s.001228">ſyllogiſmus enim reducitur tandem ad <pb pagenum="66" xlink:href="009/01/066.jpg"></pb>hanc formam, dimidium duorum rectorum eſt rectus, angulus in ſemicir­<lb></lb>
culo eſt dimidium duorum <expan abbr="rectorũ">rectorum</expan>, ergo angulus in ſemicirculo eſt rectus. <lb></lb>
</s>
          <s id="s.001229">vides in minori propoſitione contineri definitionem ſubiecti materialem? <lb></lb>
</s>
          <s id="s.001230">adeò vt hæc ſit demonſtratio omnibus numeris abſoluta per cauſam mate­<lb></lb>
rialem, vt benè ſentit Ariſt. </s>
          <s id="s.001231">Reliqua ad logicum pertinent, etiamſi per cha­<lb></lb>
racteres more mathematicorum exponantur.</s>
        </p>
        <p type="main">
          <s id="s.001232"><arrow.to.target n="marg72"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001233"><margin.target id="marg72"></margin.target>72</s>
        </p>
        <p type="main">
          <s id="s.001234">Tex. 24. <emph type="italics"></emph>(Vt propter quid reſonat? </s>
          <s id="s.001235">aut propter quid apparet? </s>
          <s id="s.001236">aut propter quid <lb></lb>
Iris? </s>
          <s id="s.001237">omnia enim hæc idem problemata ſunt genere, omnia enim ſunt refractio, ſed <lb></lb>
ſpecie altera)<emph.end type="italics"></emph.end> propter quid reſonat? </s>
          <s id="s.001238">ſcilicet echo; propter quid apparet? <lb></lb>
</s>
          <s id="s.001239">ſcilicet imago in ſpeculo. </s>
          <s id="s.001240">dicit cauſam echo, imaginis in ſpeculo, &amp; iridis <lb></lb>
in nubibus eſſe eandem; nimirum refractionem; quamuis tres illæ refractio­<lb></lb>
nes, ſeu; vt melius loquamur, reflexiones differant ſpecie ab inuicem, illa <lb></lb>
enim eſt repercuſſio vocis; hæc reflexio ſpeciei viſibilis ex corpore terſo; <lb></lb>
iſta <expan abbr="deniq;">denique</expan> radiorum Solis ex nube rorida in ſtato angulo repercuſſus. </s>
          <s id="s.001241">qua <lb></lb>
ratione autem iſta omnia fiant, longum eſſet exponere, &amp; ab intelligentia <lb></lb>
huius loci fortè alienum. </s>
          <s id="s.001242">Illud tamen non prætereundum, quod ſi propriè <lb></lb>
cum Perſpectiuis loqui velimus, dicendum eſſe, omnia illa eſſe reflexionem, <lb></lb>
non refractionem. </s>
          <s id="s.001243">nam reflexio eſt, quando linea viſualis, per quam fertur <lb></lb>ſpecies in aliquod corpus terſum, impingit, ex quo deinde ad oculos refle­<lb></lb>
ctitur. </s>
          <s id="s.001244">refractio tunc eſt, quando ſpecies obiecti viſibilis tranſit per media <lb></lb>
diuerſæ craſſitiei., vt quando ſpecies lapilli per aquam primùm, deinde per <lb></lb>aerem means ad oculum peruenit; tunc enim linea, per quam ſpecies pro­<lb></lb>
greditur, frangitur in confinio aquæ, &amp; aeris, ita vt ſpecies non per vnicam <lb></lb>
lineam rectam, ſed per fractam, ſeu refractam in confinio illo, oculis tan­<lb></lb>
dem accidat.</s>
        </p>
        <p type="main">
          <s id="s.001245">In fine textus <emph type="italics"></emph>(Quoniam Luna deficit)<emph.end type="italics"></emph.end> non intelligit defectum illum, qui <lb></lb>
eclypſis appellatur, ſed ilium, quo paulatim lumen Lunæ minus oculis no­<lb></lb>
ſtris apparet: decreſcente enim Luna ſolent humida augeri.</s>
        </p>
        <p type="main">
          <s id="s.001246"><arrow.to.target n="marg73"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001247"><margin.target id="marg73"></margin.target>73</s>
        </p>
        <p type="main">
          <s id="s.001248">Tex. 25. <emph type="italics"></emph>(Vt propter quid, &amp; permutatim proportionale? </s>
          <s id="s.001249">&amp; c.<emph.end type="italics"></emph.end>) quod quan­<lb></lb>
titates, quæ ſunt proportionales, ſint etiam alternatim, ſeu permutatim <lb></lb>
proportionales explicatum eſt ad tex. 13. primi Poſter. quæ etiam neceſſa­<lb></lb>
ria ſunt ad hunc locum benè intelligendum. </s>
          <s id="s.001250">Illud autem commune propter <lb></lb>
quod ea, quæ ſunt proportionalia, ſint etiam permutatim proportionalia, <lb></lb>
eſt quoddam innominatum, de quo ibi dictum eſt, quod cum conueniat li­<lb></lb>
neis, &amp; numeris, &amp; tamen ſeparatim de vtriſque illa paſſio demonſtretur, <lb></lb>
quærit cuinam primò, &amp; per ſe conueniat hæc paſſio, eſſe permutatim pro­<lb></lb>
portionale; ſcilicet quidnam ſit illud innominatum; in quo deinde commu­<lb></lb>
nicent lineæ, &amp; numeri, vt inde habeant eſſe etiam permutatim propor­<lb></lb>
tionalia.</s>
        </p>
        <p type="main">
          <s id="s.001251"><arrow.to.target n="marg74"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001252"><margin.target id="marg74"></margin.target>74</s>
        </p>
        <p type="main">
          <s id="s.001253">Ibidem (<emph type="italics"></emph>Hic quidem fortaſſe proportionaliter habere latera, &amp; angulos<emph.end type="italics"></emph.end>) vult <lb></lb>indicare, in quonam conſiſtat ſimilitudo inter duas figuras rectilineas geo­<lb></lb>
metricas, quam ſimilitudinem Euclides definit. </s>
          <s id="s.001254">1. ſexti, ſic explicat: ſimi­<lb></lb>
les figuræ rectilíneæ ſunt; quæ &amp; angulos ſingulos, ſingulis angulis æquales <lb></lb>
habent, <expan abbr="atq;">atque</expan> etiam latera, quæ circa angulos æquales ſunt proportionalia. <lb></lb>
</s>
          <s id="s.001255">vt ſi duo triangula appoſita habeant angulos æquales, <expan abbr="angulũ">angulum</expan> A, angulo D: <lb></lb>angulum B, angulo E. angulum C, angulo F. &amp; præterea latera, quæ ſunt <pb pagenum="67" xlink:href="009/01/067.jpg"></pb><figure id="id.009.01.067.1.jpg" place="text" xlink:href="009/01/067/1.jpg"></figure><lb></lb>
circa angulos æquales, v. g. circa an­<lb></lb>
gulos A, &amp; D, habeant proportiona­<lb></lb>
lia, hoc eſt, vt latus A B, ad latus A C; <lb></lb>
ita ſit latus D E, ad latus D F; &amp; ſic de <lb></lb>
lateribus alijs circa reliquos angulos <lb></lb>
æquales; erunt tunc prædicta duo tri­<lb></lb>
angula ſimilia.</s>
        </p>
        <p type="main">
          <s id="s.001256"><arrow.to.target n="marg75"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001257"><margin.target id="marg75"></margin.target>75</s>
        </p>
        <p type="main">
          <s id="s.001258">Ibidem (<emph type="italics"></emph>Vt extrinſecos æquales eſſe<emph.end type="italics"></emph.end>) ideſt extrinſecos angulos cuiuſuis fi­<lb></lb>
guræ rectilineæ æquales eſſe quatuor rectis angulis: vide quæ ſcripſimus de <lb></lb>
hac re ad tex. 39. ſecundi Poſter. quæ huic pariter loco ſatisfaciunt.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001259"><emph type="italics"></emph>EX TOPICIS.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.001260"><emph type="italics"></emph>Ex Primo Libro.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001261"><arrow.to.target n="marg76"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001262"><margin.target id="marg76"></margin.target>76</s>
        </p>
        <p type="main">
          <s id="s.001263">Cap. 13. (<emph type="italics"></emph>Conſiderare, quod diameter est coſtæ incommenſurabilis<emph.end type="italics"></emph.end>) vide <lb></lb>
quæ de hac re ſcripſi lib. 1. Priorum ſecto 1. cap. 23.</s>
        </p>
        <p type="main">
          <s id="s.001264"><arrow.to.target n="marg77"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001265"><margin.target id="marg77"></margin.target>77</s>
        </p>
        <p type="main">
          <s id="s.001266">Eodem cap. (<emph type="italics"></emph>Similiter autem &amp; acutum; non enim idem ſimpliciter <lb></lb>
in omnibus dicitur: nam vox acuta quidem velox (ſicut dicunt, qui ſe­<lb></lb>cundum numeros harmonici ſunt) angulus autem acutus, qui minor eſt recto; gla­<lb></lb>
dius verò, qui eſt anguli acuti<emph.end type="italics"></emph.end>) affert tres ſpecies acuti, aliud dicens eſſe acu­<lb></lb>
tum, quod eſt in voce acuta; aliud, quod eſt in angulo acuto: aliud denique, <lb></lb>
quod eſt in gladio acuto horum enim trium acumen diuerſo modo ſe habet. <lb></lb>
</s>
          <s id="s.001267">nam acumen vocis, &amp; ſoni ex celeritate motus, qua aer percuſſus impelli­<lb></lb>
tur; grauitatem autem ex tarditate oriri tradiderunt antiqui Muſici om­<lb></lb>
nes: quamuis non ex ſola celeritate, &amp; tarditate, ſed ex alijs etiam cauſis <lb></lb>
oriri poſſe voluerint. </s>
          <s id="s.001268">Primus <expan abbr="omniũ">omnium</expan> Architas Tarentinus, vt eſt apud Por­<lb></lb>
phirium in harmonicis Ptolæmei, &amp; Zarlinum pag. </s>
          <s id="s.001269">58. complem. </s>
          <s id="s.001270">muſica­<lb></lb>
lium, ait, ſi virga celerius feriat aerem, gigni motum celeriorem in aere, <lb></lb>
<expan abbr="atq;">atque</expan> hinc ſonum acutiorem reddi, experientia conſtat: ſi autem eadem vir­<lb></lb>
ga tardius aerem feriat, gigni motum in aere tardiorem, ex quo etiam ſo­<lb></lb>
num grauem, vt experientia docet. </s>
          <s id="s.001271">Ptolæmeus deinde lib. 1. cap. 3. Harm. <lb></lb>
cum ex alijs, tum ex celeritate oriri ſonum acutum, grauem verò ex tardi­<lb></lb>
tate aſſerit; vt ſi chorda eadem parum intenſa pulſetur, tardius aerem ver­<lb></lb>
berat, &amp; ideo grauiorem ſonum efficit: ſi autem magis intendatur, validius <lb></lb>
aerem pulſabit, &amp; proinde citiorem motum illi imprimet, &amp; propterea <lb></lb>
acutiorem ſonum reddet. </s>
          <s id="s.001272">hæc ille. </s>
          <s id="s.001273">videmus etiam, quod cannæ organo­<lb></lb>
rum maiores cum plus aeris moueant, &amp; idcirco tardius, ſonum grauiorem <lb></lb>
emittunt, quàm cannæ graciliores, quæ quia parum aeris cient, &amp; ideo ce­<lb></lb>
lerius, ſonum acutum edunt. </s>
          <s id="s.001274">ab hac ſententia poſteriores Muſici non recelſ<lb></lb>ſerunt, vt videre eſt apud Zarlinum.</s>
        </p>
        <p type="main">
          <s id="s.001275">In quo poſtea conſiſtat ratio acuti anguli, explicat <expan abbr="inducẽs">inducens</expan> definitionem <lb></lb>
ipſius, quæ eſt inter definitiones primi Elem. huiuſmodi, Angulus acutus <lb></lb>
eſt, qui minor recto eſt. </s>
          <s id="s.001276">Demum explicat, cur nam gladius dicatur acutus, <lb></lb>
quia nimirum habet angulum acutum ſuperficialem, ideſt, quem duæ ſuper­<lb></lb>
ficies ſimul in acie gladij concurrentes efficiunt.</s>
        </p>
        <pb pagenum="68" xlink:href="009/01/068.jpg"></pb>
        <p type="main">
          <s id="s.001277"><arrow.to.target n="marg78"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001278"><margin.target id="marg78"></margin.target>78</s>
        </p>
        <p type="main">
          <s id="s.001279">Eodem cap. (<emph type="italics"></emph>Rurſum ſi eorundem; quæ ſunt ſub eodem nomine diuerſæ diffe­<lb></lb>
rentiæ ſunt; vt coloris, qui eſt in corporibus, &amp; in melodijs<emph.end type="italics"></emph.end>) veteres Muſici can­<lb></lb>
tilenas omnes ad tria genera reuocarunt, videlicet Enharmonicum, Chro­<lb></lb>maticum, &amp; Diatonicum; quæ diſtinguebantur inuicem ex varia diuiſione <lb></lb>interuallorum, ex quibus ipſorum Monochordia conſtabant: ſiue ex varijs <lb></lb>
vocum interuallis, v. g. quia in vno continebantur plures toni, vt in Diato­<lb></lb>
nico; in alio plures dieſes, vt in Enharmonico; in tertio verò plura ſemito­<lb></lb>
nia, vt in Chromatico: quæ vox deducitur à chroma, græco, quod latinis <lb></lb>
eſt color; quare Chromaticum latinè redditur coloratum. </s>
          <s id="s.001280">Hic eſt igitur <lb></lb>
color ille, quem hic Ariſt. innuit. </s>
          <s id="s.001281">quod genus forſitan à calore denomina­<lb></lb>
batur, quòd ipſius notæ muſicales eſſent coloratæ, vt hoc modo ab alijs ge­<lb></lb>
neribus dignoſceretur. quam conſuetudinem exiſtimat Zarlinus cap. 46. ſe­<lb></lb>
cundæ partis, etiam noſtra tempeſtate aliquo modo perſeuerare, cum vi­<lb></lb>
deamus in organis, &amp; alijs huiuſmodi inſtrumentis, quæ pinnas, vulgò ta­<lb></lb>ſtos, habent; illas inquam pinnas, quæ chromaticis interuallis deputatæ <lb></lb>
ſunt, colore nigro tinctas eſſe.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001282"><emph type="italics"></emph>Libro Quarto.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001283"><arrow.to.target n="marg79"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001284"><margin.target id="marg79"></margin.target>79</s>
        </p>
        <p type="main">
          <s id="s.001285">Cap. 1. loco 10. (<emph type="italics"></emph>Si quis inſecabiles ponens lineas<emph.end type="italics"></emph.end>) nonnulli antiquorum <lb></lb>
Philoſophorum putarunt omnia ex indiuiſibilibus componi, vt Demo­<lb></lb>
critus, &amp; Leucippus, &amp; propterea dixerunt, etiam lineas conſtare ex lineis <lb></lb>quibuſdam adeò paruis, quæ omnino eſient inſecabiles, ſeu indiuiſibiles: de <lb></lb>
quibus plura in libello de line is inſecabilibus.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001286"><emph type="italics"></emph>Libro Sexto.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001287"><arrow.to.target n="marg80"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001288"><margin.target id="marg80"></margin.target>80</s>
        </p>
        <p type="main">
          <s id="s.001289">Cap. 2. loco 32. (<emph type="italics"></emph>Vt qui lineam definiunt longitudinem ſine latitudine eſſe<emph.end type="italics"></emph.end>) <lb></lb>ſupponimus lectorem intellexiſſe definitiones ſaltem primi Elem. in­<lb></lb>
ter quas definitio lineæ eſt ſecunda, <expan abbr="cademq́">eademque</expan>; cum hac Ariſtotelis.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001290"><emph type="italics"></emph>Libro Octauo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001291"><arrow.to.target n="marg81"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001292"><margin.target id="marg81"></margin.target>81</s>
        </p>
        <p type="main">
          <s id="s.001293">Cap. 2. loco 41. (<emph type="italics"></emph>Videntur autem in diſciplinis, ſeu Mathematicis quædam <lb></lb>ob definitionis defectum non facile deſcribi; vt &amp; quoniam, quæ ad latus ſe­<lb></lb>
cat planum linea, ſimiliter diuidit &amp; lineam, &amp; locum: definitione autem dicta, <lb></lb>
ſtatim manifeſtum eſt, quod dicitur, nam eandem ablationem habent loca, &amp; linea, <lb></lb>
ſive latus planæ figuræ, est autem definitio eiuſdem proportionis hæc<emph.end type="italics"></emph.end>) mendosè <lb></lb>legitur à nonnullis (<emph type="italics"></emph>Eſt distem definitio eiuſdem orationis hæc<emph.end type="italics"></emph.end>) quos puto de­<lb></lb>
ceptos ab æquiuoco <foreign lang="grc">λσγους</foreign> quod &amp; orationem, &amp; rationem, ſiue proportio­<lb></lb>
nem ſignificat: hic autem ſignificare proportionem res ſubrecta ſatis mani­<lb></lb>
feſtat. </s>
          <s id="s.001294">Notandum poſtea cum Alexandro (quod &amp; ſuperius alias commo­<lb></lb>nui in cap. de Priori, &amp; alibi) per verbum (Deſcribi) ſignificari hoc loco <lb></lb>
geometricè demonſtrare, quoniam Geometræ <expan abbr="nõ">non</expan> niſi adhibitis deſcriptio­<lb></lb>nibus, ſeu figuris demonſtrant. </s>
          <s id="s.001295">Vult autem Ariſt. exemplo mathematico <lb></lb>
oſtendere, difficile eſſe diſputare, aut <expan abbr="argumẽtari">argumentari</expan>, niſi prius rectè aſſignetur <pb pagenum="69" xlink:href="009/01/069.jpg"></pb>definitio illius rei, de qua diſſeritur. </s>
          <s id="s.001296">Porrò exemplum mathematicum hic <lb></lb>
allatum ſic videtur explicandum: Conetur aliquis demonſtrare hanc pro­<lb></lb>
poſitionem; ſi linea ducta fuerit æquidiſtans lateri vnius plani trianguli, ſe­<lb></lb>cabit &amp; latera, &amp; locum, ideſt ſuperficiem illam triangularem ſimiliter, ideſt <lb></lb>
<figure id="id.009.01.069.1.jpg" place="text" xlink:href="009/01/069/1.jpg"></figure><lb></lb>
in eadem proportione, vt in triangulo A B C, <lb></lb>
linea D E, parallela baſi B C, ſecat latera A B, <lb></lb>
&amp; A C, in punctis D, &amp; E, in eadem ratione, <lb></lb>
in qua etiam fecat totum triangulum, ita vt <lb></lb>
eadem ſit proportio lineæ A D, ad D B, &amp; lineæ <lb></lb>
A E, ad E C, quæ eſt partium totalis trianguli <lb></lb>A B C, ſcilicet quæ eſt partis A D E, ad partem <lb></lb>
E D C, fiue ad partem D E B. quod conſtat ex <lb></lb>
ſecunda 6. Elem. </s>
          <s id="s.001297">Inquit ergo Ariſt. </s>
          <s id="s.001298">Si quis <lb></lb>
vellet hoc demonſtrare nondum præmiſſa defi­<lb></lb>
nitione eorum, quæ habent eandem rationem, ſiue nondum definitione al­<lb></lb>
lata quantitatum proportionalium, hic difficile id valeret oſtendere: at ve­<lb></lb>rò allata prius definitione quantitatum proportionalium facile demonſtra­<lb></lb>
bit. </s>
          <s id="s.001299">Subdit verò Ariſt. dictam definitionem, dicens, tunc quantitates eſſe <lb></lb>
proportionales, quando habent eandem ablationem, ideſt, eandem diuiſio­<lb></lb>
nem, ideſt, eadem diuiſio ne tantum proportionaliter de vna, quantum de <lb></lb>
altera magnitudine reſecatur: Quemadmodum etiam Euclides loco cita­<lb></lb>
to probat, latera illius trianguli, &amp; ſuperficiem eſſe ſimiliter diuiſa, ex quo <lb></lb>
ſequitur eſſe proportionalia. </s>
          <s id="s.001300">Porrò Euclides definit. </s>
          <s id="s.001301">ſeptima 5. paulo ali­<lb></lb>
ter definit quantitates proportionales eſſe illas, quæ eandem habent ratio­<lb></lb>
nem, v. g. ſi ſit, vt prima ad ſecundam, ita tertia ad quartam. </s>
          <s id="s.001302">ex quibus <lb></lb>
quoad Mathematicas ſpectat, huic loco ſatisfactum ſit.</s>
        </p>
        <p type="main">
          <s id="s.001303"><arrow.to.target n="marg82"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001304"><margin.target id="marg82"></margin.target>82</s>
        </p>
        <p type="main">
          <s id="s.001305">Cap. 4. loco 86. <emph type="italics"></emph>(Tentandum autem, &amp; ea, in quæ ſæpiſſimè incidunt diſputa­<lb></lb>
tiones, tenere, nam quemadmodum in Geometria ante opus eſt circa elementa exer­<lb></lb>
citatum eſſe, &amp; in numeris circa capitales promptè ſe habere, &amp; multum refert ad <lb></lb>hoc, &amp; alium numerum cognoſcere multiplicatum)<emph.end type="italics"></emph.end> Elementa vocabant antiqui <lb></lb>
demonſtrationes faciliores, &amp; ſimpliciores, quales propriè ſunt omnes, quæ <lb></lb>
ſex prioribus libris Euclidianis continentur: ex illis enim tanquam ex ele­<lb></lb>
mentis abſtruſiores, &amp; difficiliores demonſtrationes deducebant. </s>
          <s id="s.001306"><expan abbr="atq;">atque</expan> hæc <lb></lb>
eſt ratio, cur Euclides ſuos libros elementa nuncupauerit. </s>
          <s id="s.001307">ait igitur curan­<lb></lb>
dum eſſe horum elementorum cognitionem in promptu habere, quia fre­<lb></lb>
quens de ipſis incidit diſputatio. </s>
          <s id="s.001308">Per capitales numeros intelligo ſimplices <lb></lb>
ab vnitate, <expan abbr="vſq;">vſque</expan> ad nouem incluſiuè. </s>
          <s id="s.001309">&amp; quando ait, alium numerum cogno­<lb></lb>
ſcere multiplicatum, ſignificat vtile valdè eſſe ad quotidianum vſum <lb></lb>
cognoſcere, quemnam numerum producant numeri capitales, <lb></lb>
ſi ad inuicem multiplicentur, quamuis huiuſmodi co­<lb></lb>
gnitio facilis, ac leuis ſit: qua de cauſa vide­<lb></lb>
mus vſque in hanc diem pueros diu in <lb></lb>
Abaco memoriter perdiſcen­<lb></lb>
do detineri.</s>
        </p>
      </chap>
      <chap>
        <pb pagenum="70" xlink:href="009/01/070.jpg"></pb>
        <p type="head">
          <s id="s.001310"><emph type="italics"></emph>Ex Primo Elenchorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001311"><arrow.to.target n="marg83"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001312"><margin.target id="marg83"></margin.target>83</s>
        </p>
        <p type="main">
          <s id="s.001313">Cap. 10. <emph type="italics"></emph>(Nam pſeudographiæ non contentioſæ (ſecundum enim ea, quæ <lb></lb>
ſub arte ſunt, captioſæ ſunt ratiocinationes) <expan abbr="neq;">neque</expan> ſi aliqua eſt pſeudogra­<lb></lb>
phia circa verum, vt Hippocratis quadratura, quæ per lunulas, ſed, vt <lb></lb>
Bryſſo quadrauit circulum; &amp; tametſi quadretur circulus, quia tamen <lb></lb>
non ſecundum rem, ideo ſophiſticus)<emph.end type="italics"></emph.end> qua ratione Hippocrates orbi quadrum <lb></lb>
exhibere æquale tentauerit, explicatum eſt abundè in 2. Priorum cap. 31. <lb></lb>
&amp; quo itidem modo Bryſſo lib. 1. Poſter. tex. 23. <expan abbr="ſolũmodo">ſolummodo</expan> id hoc loco no­<lb></lb>
tandum per pſeudographiam intelligere, vt apertè etiam inferius explicat, <lb></lb>
Geometricam demonſtrationem fallacem, eò quod demonſtrationes geo­<lb></lb>metricæ fiant adhibitis deſcriptionibus, ſeu figurationibus: pſeudographia <lb></lb>
autem latinè idem eſt, ac falſa deſcriptio; quemadmodum è contrariò, ſi­<lb></lb>
cuti ſupra in Topicis, &amp; alibi obſeruaui, per deſcribere intelligit geometri­<lb></lb>
cè demonſtrare, &amp; per deſcriptiones intelligit demonſtrationes geometri­<lb></lb>
cas. </s>
          <s id="s.001314">Qua ratione item Hippocrates ex ijs, quæ ſub arte Geometriæ ſunt, <lb></lb>
procederet ibi dictum eſt, propter quod non eſt contentioſa, quamuis fallax <lb></lb>
ipſius demonſtratio: appellat enim Ariſt illas demonſtrationes contentio­<lb></lb>
ſas, quæ non procedunt ex proprijs illius ſcientiæ, in qua fiunt, ſed ex com­<lb></lb>
munibus alijs ſcientijs: captioſas verò, &amp; ſophiſticas, quæ ex proprijs ſcien­<lb></lb>
tiæ, in qua fiunt, decipiunt. </s>
          <s id="s.001315">At verò demonſtratio, ſeu pſeudographia Bryſ­<lb></lb>
ſonis erat contentioſa, quia ex communibus, &amp; extra Geometriam petitis <lb></lb>
argumentabatur: quemadmodum ibi explicatum eſt.</s>
        </p>
        <p type="main">
          <s id="s.001316"><arrow.to.target n="marg84"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001317"><margin.target id="marg84"></margin.target>84</s>
        </p>
        <p type="main">
          <s id="s.001318">Eodem cap. <emph type="italics"></emph>(Quadratura per lunulas non contentioſa)<emph.end type="italics"></emph.end> inquit Hippocratis <lb></lb>
tetragoniſmum, de quo in 2. Priorum, quæ non contentioſa dicitur, quia ex <lb></lb>
proprijs Geometriæ deducebatur.</s>
        </p>
        <p type="main">
          <s id="s.001319"><arrow.to.target n="marg85"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001320"><margin.target id="marg85"></margin.target>85</s>
        </p>
        <p type="main">
          <s id="s.001321">Ibidem <emph type="italics"></emph>(Bryſſonis autem contentioſa: &amp; illam quidem non eſt transferre, niſi <lb></lb>
ad Geometriam ſolum; eo quod ex proprijs ſit principijs)<emph.end type="italics"></emph.end> <expan abbr="quãdo">quando</expan> ait <emph type="italics"></emph>(&amp; illam qui­<lb></lb>
dem)<emph.end type="italics"></emph.end> intelligit quadrationem Hippocratis. </s>
          <s id="s.001322">vide 2. Prior cap. 31. &amp; quæ pau­<lb></lb>
lo ante in præcedentibus locis diximus.</s>
        </p>
        <p type="main">
          <s id="s.001323"><arrow.to.target n="marg86"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001324"><margin.target id="marg86"></margin.target>86</s>
        </p>
        <p type="main">
          <s id="s.001325">Ibidem <emph type="italics"></emph>(Hanc autem ad plures)<emph.end type="italics"></emph.end> intelligit tetragoniſmum Bryſſonis, qui <lb></lb>
per communia deducebatur. </s>
          <s id="s.001326">lege ſuperius dicta in præcedentibus locis hu­<lb></lb>
ius capituli.</s>
        </p>
        <p type="main">
          <s id="s.001327"><arrow.to.target n="marg87"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001328"><margin.target id="marg87"></margin.target>87</s>
        </p>
        <figure id="id.009.01.070.1.jpg" place="text" xlink:href="009/01/070/1.jpg"></figure>
        <p type="main">
          <s id="s.001329">Ad finem cap. <emph type="italics"></emph>(Aut vt Antiphon quadra­<lb></lb>
uit)<emph.end type="italics"></emph.end> ſimile peccatum peccaſſe Antiphon­<lb></lb>
tem in orbe quadrando, ac Hippocratem, <lb></lb>
Ariſt. his verbis videtur ſignificare, ideſt, <lb></lb>
ipſum, quamuis ex proprijs Geometriæ, <lb></lb>
falſis tamen ratiocinatum eſſe. </s>
          <s id="s.001330">Cæterum <lb></lb>
Antiphontem in hunc modum orbem ad <lb></lb>
quadrum redigere tentaſſe, tradit Simpli­<lb></lb>
cius. </s>
          <s id="s.001331">circulo quadrando inſcribebat pri­<lb></lb>
mò quadratum A B C D. deinde in ſingu­<lb></lb>
lis quatuor ſegmentis inſcribebat totidem <lb></lb>trigona æquilatera, vt patet in adſcripta <pb pagenum="71" xlink:href="009/01/071.jpg"></pb>figura. </s>
          <s id="s.001332">poſtea ſuper ſingula latera horum triangulorum in reliquis ſegmen­<lb></lb>
tis inſcribebat adhuc triangula ſimilia triangulo A I E. alia inſuper trigona <lb></lb>
ſuper latera iſtorum conſtituebat, donec ambitus figuræ illius multilateræ <lb></lb>in circulo delineatæ, circumferentiæ circuli aptaretur. </s>
          <s id="s.001333">quod fieri poſſe ille <lb></lb>
falsò contra Geometriæ principia aſſumebat; eſt enim principium Geome­<lb></lb>
tricum continuum eſſe diuiſibile in infinitum, <expan abbr="neq;">neque</expan> per diuiſionem abſumi <lb></lb>poſſe; cui principio aduerſatur, dum putat ſe conſumpturum totum circu­<lb></lb>
lum, diuidendo illud in triangula ſemper minora; vel quia putat, lineam <lb></lb>
curuam conſtare ex minimis lineis rectis. </s>
          <s id="s.001334">Similiter igitur <expan abbr="atq;">atque</expan> Hippocra­<lb></lb>tes errauit, qúi pariter in Geometria fallebatur: Antiphon quidem contra <lb></lb>
principia illius: Hippocrates verò aſſumens falſi quidpiam in Geometria. <lb></lb>
</s>
          <s id="s.001335">At Bryſſo, eo quod per communia alijs ſcientijs deduceret ratiocinatio­<lb></lb>
nem propterea pſeudographia Antiphontis non litigioſa quidem, ſed <lb></lb>
tamen fallax extitit, non enim per communia alijs ſcientijs <lb></lb>
procedat; vnde nec transferri poterat ipſius falſa de­<lb></lb>
ſcriptio, ſeu demonſtratio extra Geometriæ li­<lb></lb>
mites, quod cauſa eſt contentionis.</s>
        </p>
        <p type="head">
          <s id="s.001336"><emph type="italics"></emph>Logicorum locorum finis.<emph.end type="italics"></emph.end></s>
        </p>
      </chap>
      <pb pagenum="72" xlink:href="009/01/072.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.001337">EX PRIMO LIBRO <lb></lb>
PHYSICORVM.<lb></lb>
<arrow.to.target n="marg88"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001338"><margin.target id="marg88"></margin.target>88</s>
        </p>
        <p type="main">
          <s id="s.001339">Tex. 11. <emph type="italics"></emph>(Simul autem <expan abbr="neq;">neque</expan> conuenit omnia ſoluere&#039;, ſed <expan abbr="quæcunq;">quæcunque</expan> ex <lb></lb>
principijs aliquis demonſtrans <expan abbr="mẽtitur">mentitur</expan>; <expan abbr="quæcunq;">quæcunque</expan> verò non, minimè: <lb></lb>
vt tetragoniſmum, eum quidem, qui per ſectiones Geometrici est diſ­<lb></lb>
ſoluere: illum autem, qui Antiphontis non Geometrici eſt<emph.end type="italics"></emph.end>) Tetrago­<lb></lb>
niſmum, ſeu circuli quadraturam per ſectiones, eſſe illam Hip­<lb></lb>
pocratis Chij exiſtimant græci expoſitores, qui per lunulas, quas Ariſt. ſe­<lb></lb>
ctiones appellat, orbem quadrare tentabat. </s>
          <s id="s.001340">Eius demonſtrationem expli­<lb></lb>
caui ad cap. 31. de Abductione in 2. Priorum, quam inibi videas. </s>
          <s id="s.001341">hoc ſolum <lb></lb>
hic notandum pertinere ad Geometram, ipſam refellere, quia ex falſa qua­<lb></lb>
dam præmiſſa ex Geometria deſumpta, ratiocinabatur, idcirco debet (in­<lb></lb>
quit Ariſt.) Geometra illius deceptionem inuenire. </s>
          <s id="s.001342">Tetragoniſmum autem <lb></lb>
Antiphontis non eſt Geometræ <expan abbr="cõfutare">confutare</expan>, quia aduerſabatur principijs Geo­<lb></lb>
metriæ, ſupponebat enim circuli circumferentiam ex indiuiduis, <expan abbr="minimisq́">minimisque</expan>; <lb></lb>
lineis rectis componi: cuius falſam demonſtrationem explicatam inuenies <lb></lb>
ad cap. 10. primi Elench. poſſumus addere tertiam rationem quia ſcilicet <lb></lb>
Hippocrates non procedebat per communia alijs ſcientijs, vt videre eſt ad <lb></lb>
tex. 23. primi Poſter. cap. 8. vbi ipſius pſeudographiam expoſui. Quemad­<lb></lb>
modum igitur Geometra diſſoluit falſas tantummodo rationes eas, quæ ſer­<lb></lb>
uatis Geometricis principijs procedunt; non autem eas, quæ Geometriæ <lb></lb>
principia conuellunt: ita Phyſico non incumbit <expan abbr="cõtra">contra</expan> Parmenidem, ac Me­<lb></lb>
liſſum naturæ principia deſtruentes diſceptare, aut fallaces eorum rationes <lb></lb>
coarguere. </s>
          <s id="s.001343">Hoc volebat Ariſtoteles inferre.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001344"><emph type="italics"></emph>Ex Secundo Phyſicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001345"><arrow.to.target n="marg89"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001346"><margin.target id="marg89"></margin.target>89</s>
        </p>
        <p type="main">
          <s id="s.001347">Tex. 20. (<emph type="italics"></emph>Geometria enim de phyſica linea conſiderat, ſed non quatenus <lb></lb>
eſt phyſici: Perſpectiua autem mathematicam quidem lineam, ſed non <lb></lb>
quatenus phyſica eſt<emph.end type="italics"></emph.end>) quamuis textus hic non pertineat ad Mathe­<lb></lb>
maticum, libuit tamen illum in ordinem noſtrum recenſere, ope­<lb></lb>
ræpretium etenim eſt ea, quæ in ipſo continentur à nonnullis recentioribus <lb></lb>
rectè intelligi, vt ab his moniti, ab inani quadam optices impugnatione ab­<lb></lb>
ſtineant, ac tandem ex Ariſt. lineas illas viſuales quas ipſi de medio tollunt, <lb></lb>
perſpicuè videant. </s>
          <s id="s.001348">cætera, quæ in præcedentibus locis Ariſt. de Natura Ma­<lb></lb>
thematicarum habet, ſunt præter noſtrum inſtitutum.</s>
        </p>
        <p type="main">
          <s id="s.001349"><arrow.to.target n="marg90"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001350"><margin.target id="marg90"></margin.target>90</s>
        </p>
        <p type="main">
          <s id="s.001351">Tex. 28. (<emph type="italics"></emph>Alio autem modo, forma, &amp; exemplum: hæc autem eſt ratio ipſius, <lb></lb>
quod quid erat eſſe, &amp; huius genera, vt ipſius diapaſon duo ad vnum, &amp; omnino <lb></lb>
numerus, &amp; partes, quæ in ratione ſunt<emph.end type="italics"></emph.end>) vt benè intelligas, quod in præſenti <lb></lb>
textu <expan abbr="mathematicũ">mathematicum</expan> eſt, conſule prius, quæ ſcripſi ad tex. 1. cap. primi 2. Po­<lb></lb>
ſter. ſuper verba illa (<emph type="italics"></emph>Quid eſt conſonantia?<emph.end type="italics"></emph.end>) vbi perſpicuè videbis, cur <expan abbr="con-ſonãtiæ">con­<lb></lb>
ſonantiæ</expan>, quæ dicitur Diapaſon, eſſentia, &amp; definitio ſit ipſa proportio dupla, <lb></lb>
quæ ſub his num. </s>
          <s id="s.001352">2.1. continetur: quibus perſpectis facilis erit phyſico totius <lb></lb>
loci intelligentia.</s>
        </p>
        <pb pagenum="73" xlink:href="009/01/073.jpg"></pb>
        <p type="main">
          <s id="s.001353"><arrow.to.target n="marg91"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001354"><margin.target id="marg91"></margin.target>91</s>
        </p>
        <p type="main">
          <s id="s.001355">Tex. 68. (<emph type="italics"></emph>Aut enim ad ipſum quid eſt, reducitur ipſum propter quid in immo­<lb></lb>bilibus, vt in Mathematicis, ad definitionem enim recti, aut commenſurabilis, aut <lb></lb>
alius cuiuſpiam reducitur vltimum<emph.end type="italics"></emph.end>) ex his manifeſtè videas Mathematicas <expan abbr="de-mõſtrare">de­<lb></lb>
monſtrare</expan> per cauſam formalem, cum cauſam ipſam ad ipſum quid eſt, ideſt, <lb></lb>
ad definitionem reducant. </s>
          <s id="s.001356">quorum exempla in logicis ex Mathematicis at­<lb></lb>
tuli: ſed etiam ſequentis loci exemplum de triangulo idem apertè manife­<lb></lb>
ſtat; in quo probat duos angulos A C B, A C D, eſſe rectos, ex definitione <lb></lb>
ipſorum, ſiue ex definitione lineæ perpendicularis A C, quod idem eſt.</s>
        </p>
        <p type="main">
          <s id="s.001357"><arrow.to.target n="marg92"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001358"><margin.target id="marg92"></margin.target>92</s>
        </p>
        <p type="main">
          <s id="s.001359">Tex 89. (<emph type="italics"></emph>Eſt autem neceſſarium in Mathematicis, &amp; in his, quæ ſecundum <lb></lb>naturam fiunt quaſi eodem modo; quoniam enim hoc rectum eſt, neceſſe eſt, trian­<lb></lb>gulum tres angulos habere æquales duobus rectis; ſed non, ſi hoc, illud; ſed ſi hoc <lb></lb>
non eſt, <expan abbr="neq;">neque</expan> rectum eſt.<emph.end type="italics"></emph.end>) cum animaduerterim non parum eſſe diſſenſionis, &amp; <lb></lb>
difficultatis in exemplo hoc mathematico explicando, ita vt recentiores <lb></lb>
quidam textum <expan abbr="hũc">hunc</expan> pro arbitratu ſuo perperam latinè verterint: ideò pri­<lb></lb>
mum ex græcis codicibus interpretationem hanc veram attuli. </s>
          <s id="s.001360">deinde, quia <lb></lb>
etiam græci in exemplo mathematico enodando, vel malè, vt Simplicius; <lb></lb>
vel obſcurè nimis, vt reliqui; Latini verò vel nihil, vel peius multò loquun­<lb></lb>
tur, ideò ſic ego exponendum cenſui. </s>
          <s id="s.001361">cum velit Ariſt. oſtendere neceſſita­<lb></lb>tem, quæ in ſcientijs inter præmiſſas, ſeu medium, &amp; concluſionem reperi­<lb></lb>
tur, affert exemplum illud mathematicum ſibi familiare, demonſtrationem <lb></lb>
ſcilicet illam, qua oſtenditur, omne triangulum habere tres angulos æqua­<lb></lb>
les duobus rectis angulis, cuius fuſiſſimam explicationem inuenies ſupra in <lb></lb>
primo Priorum, ſecto 3. cap. 1. quam neceſſe eſt, conſulas. </s>
          <s id="s.001362">pro medio autem <lb></lb>
huius paſſionis accipit lineam perpendicularem, quam innuit verbis illis <lb></lb>
<emph type="italics"></emph>(quoniam enim hoc rectum eſt<emph.end type="italics"></emph.end>) vt in figura ſit triangulum A B C, <expan abbr="ſitq́">ſitque</expan>; vt latus <lb></lb>
<figure id="id.009.01.073.1.jpg" place="text" xlink:href="009/01/073/1.jpg"></figure><lb></lb>
A C, ſit perpendiculare <expan abbr="cũ">cum</expan> latere B C, &amp; pro­<lb></lb>
ducatur B C, in D; tunc triangulum A B C, <lb></lb>
habere tres angulos, A, B, &amp; A C B, æquales <lb></lb>
duobus rectis planum erit: nam <expan abbr="cũ">cum</expan> latus A C, <lb></lb>
ſit perpendiculare (quod Ariſt. dicit, cum <expan abbr="re-ctũ">re­<lb></lb>
ctum</expan> hoc ſit) erunt duo anguli deinceps A C B, <lb></lb>
A C D, recti, ex definitione lineæ perpendicu­<lb></lb>
laris, cum ergo duo anguli A, &amp; B, externo, <expan abbr="rectoq́">rectoque</expan>; A C D, ſint æquales per <lb></lb>
32. primi, &amp; reliquus angulus A C B, communis, ideſt, ſit angulus triangu­<lb></lb>
li, &amp; angulus vnus lineæ perpendicularis, &amp; ideò rectus; manifeſtè apparet, <lb></lb>
tres angulos A, B, A C B, eſſe æquales neceſſariò duobus rectis, ex poſitio­<lb></lb>
ne illius recti, ſiue lateris perpendicularis, quia ex verò, verum neceſſariò <lb></lb>
ſequitur; non tamen poſita hac paſſione, ſiue concluſione, habere ſcilicet <lb></lb>
tres angulos æquales duobus rectis, neceſſariò ſequitur illud eſſe rectum, <lb></lb>ideſt latus illud A C, eſſe perpendiculare ad latus B C, quia verum <lb></lb>
ſequi poteſt ex verò, &amp; falsò. </s>
          <s id="s.001363">valebit tamen hæc conſequen­<lb></lb>
tia, ſi triangulum non habet hanc proprietatem, ne­<lb></lb>
que illud rectum eſt, ideſt, <expan abbr="neq;">neque</expan> latus prædi­<lb></lb>ctum erit <expan abbr="perpẽdiculare">perpendiculare</expan>, quia falſum <lb></lb>non, niſi ex falſo ſequitur.</s>
        </p>
      </chap>
      <pb pagenum="74" xlink:href="009/01/074.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.001364"><emph type="italics"></emph>Ex Tertio Phyſicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001365"><arrow.to.target n="marg93"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001366"><margin.target id="marg93"></margin.target>93</s>
        </p>
        <p type="main">
          <s id="s.001367">Tex. 26. <emph type="italics"></emph>(Et hi quidem infinitum eſſe par; hoc enim compræhenſura, &amp; <lb></lb>
ab impari terminatum tribuit ijs, quæ ſunt, infinitatem. </s>
          <s id="s.001368">ſignum autem <lb></lb>huius id eſſe, quod contingit in numeris, circumpoſitis enim Gnomoni­<lb></lb>bus circa vnum, &amp; ſeorſum, aliquando quidem ſemper aliam fieri ſpe­<lb></lb>
ciem, aliquando autem vnam)<emph.end type="italics"></emph.end> vt melius percipiantur ea, quæ ſequuntur, lege <lb></lb>
prius, quæ in cap. de Motu in poſt prædicamentis ſcripſi de Gnomone, ad <lb></lb>
ſimilitudinem enim Gnomonis illius Geometrici, inueniuntur etiam in nu­<lb></lb>
meris Gnomones Arithmetici. </s>
          <s id="s.001369">Pythagorici enim (à quibus iſta mutuatus <lb></lb>
eſt Ariſt. numeros impares ſolos appellabant Gnomones, eò quod in for­<lb></lb>
mam normæ æquilateræ, ſiue Gnomonis conſtitui poſſint, vt patet in his <lb></lb>
<figure id="id.009.01.074.1.jpg" place="text" xlink:href="009/01/074/1.jpg"></figure><lb></lb>
nimirum in ternario, quinario, ſeptenario, &amp; ſic de <lb></lb>
reliquis imparibus. </s>
          <s id="s.001370">pares autem numeri, quia ne­<lb></lb>
queunt in figuram normæ æquilateræ diſponi, cum <lb></lb>
non habeant vnitatem pro angulo, &amp; paria poſtea la­<lb></lb>
tera, vt oportet, non merentur appellari Gnomones, vt quaternarius, ſi di­<lb></lb>
ſponatur ſic <figure id="id.009.01.074.2.jpg" place="text" xlink:href="009/01/074/2.jpg"></figure> non refert Gnomonem, quia lateribus inęqualibus con­<lb></lb>
ſtat; <expan abbr="neq;">neque</expan> ſi hoc modo <figure id="id.009.01.074.3.jpg" place="text" xlink:href="009/01/074/3.jpg"></figure> quia deeſt huic figuræ angularis vnitas, quæ <lb></lb>illi neceſſaria eſt. </s>
          <s id="s.001371">Pythagorici igitur dicebant, numerum parem ideò eſſe <lb></lb>
infinitum ipſum, quia videbant ipſum eſſe cauſam perpetuæ diuiſionis, cum <lb></lb>
quælibet res quanta ſit diuiſibilis bifariam, ideſt in duo ſecundum numerum <lb></lb>
parem, &amp; ſubdiuiſibilis poſtea bifariam, &amp; ſic in infinitum, vt de linea pro­<lb></lb>
blematicè probatur in 10. primi Elem. quamuis theorematicè ſit axioma. <lb></lb>
</s>
          <s id="s.001372">hunc porrò numerum parem dicebant terminatum eſſe ab impari, quia ori­<lb></lb>
tur ex diuiſione cuiuſuis rei, quæ vna ſit, ſumentes vnitatem pro impari. <lb></lb>
</s>
          <s id="s.001373">ſignum præterea huius finitatis ab impari, &amp; infinitatis à pari numero pro­<lb></lb>
cedentis, aiunt eſſe Gnomones, numeros ſcilicet impares: Gnomones enim, <lb></lb>
ideſt impares numeri vnitati additi, producunt eandem perpetuò numero­<lb></lb>
rum formam, videlicet quadratum: at verò è contrariò numeri pares vni­<lb></lb>
tati additi, conflant perpetuò varias numerorum formas: quapropter vi­<lb></lb>dentur numeri impares eſſe finitatis cauſa; ſicut pares ex aduersò infinitatis <lb></lb>
principium. </s>
          <s id="s.001374">quæ vt melius intelligas, declaranda eſt 26. propoſ. </s>
          <s id="s.001375">7. Arith­<lb></lb>
metices lordani, vbi iſtud idem demonſtrat, quæ eſt hæc. </s>
          <s id="s.001376">ſit vnitas, &amp; ſuo or­<lb></lb>
dine ſequantur impares, vt in ſequenti hac ſerie apparet 1. 3. 5. 7. 9. &amp; c. <lb></lb>
<figure id="id.009.01.074.4.jpg" place="text" xlink:href="009/01/074/4.jpg"></figure><lb></lb>
ſi igitur vnitati addatur ternarius in Gnomo­<lb></lb>
nis modum, vt vides in prima figura, produ­<lb></lb>
cetur quaternarius numerus, qui eſt numerus <lb></lb>
quadratus (quid ſit quadratus numerus expli­<lb></lb>
caui in Logicis tex. 9. primi Poſter.) etſi huic <lb></lb>
quaternario addatur ſequens impar, qui eſt <lb></lb>quinarius in modum Gnomonis, vt in ſecunda <lb></lb>
figura, ſit numerus nouenarius, qui pariter eſt quadratus. </s>
          <s id="s.001377">etſi huic ſimiliter <lb></lb>addatur ſequens impar, nimirum ſeptenarius, conflabitur ſedenarius, qui <lb></lb>
numerus pariter quadratus eſt, vt in tertia figura, &amp; hoc modo, ſi in infini­


<pb pagenum="75" xlink:href="009/01/075.jpg"></pb>tum procedatur, numeri ſemper quadrati progignentur. </s>
          <s id="s.001378">Vides igitur, qui <lb></lb>
ratione Gnomonum, ſiue imparium additione fiat ſemper eadem ſpecies, <lb></lb>
ſcilicet quadratus numerus, quod ſignum eſt, inquiunt, imparem numerum <lb></lb>
non infinitatis, ſed finitatis eſſe auctorem. </s>
          <s id="s.001379">Poſt prædictam 26. propoſitio­<lb></lb>nem Iordani, ſunt aliquot propoſitiones, quarum ſumma hæc eſt: ſi pares <lb></lb>
numeri ab vnitate coaceruentur; coaceruati erunt ſemper variæ formæ nu­<lb></lb>
merorum. </s>
          <s id="s.001380">quæ ſic explicantur: ſint ab vnitate pares diſpoſiti ordinatim <lb></lb>
hoc modo, 1. 2. 4. 6. &amp;c. </s>
          <s id="s.001381">ſi igitur vnitati binarius coaceruetur, fit numerus <lb></lb>
<figure id="id.009.01.075.1.jpg" place="text" xlink:href="009/01/075/1.jpg"></figure><lb></lb>
triangularis, vt in prima figura. </s>
          <s id="s.001382">ſi huic ternario <lb></lb>
coaceruetur ſequens par, fiet altera ſpecies, ni­<lb></lb>
mirum hexagonus numerus, vt in ſecunda figu­<lb></lb>
ra. </s>
          <s id="s.001383">cui ſi ſequens addatur par, ſcilicet ſenarius, <lb></lb>
fiet iterum noua numeri forma, v. g. </s>
          <s id="s.001384">dodecago­<lb></lb>
nus, vt in tertia figura. </s>
          <s id="s.001385">&amp; ſic ſemper in infinitum nouæ ac variæ numerorum <lb></lb>
formæ ex hac additione parium prouenient, quod argumento eſt numerum <lb></lb>
parem infiniti naturam ſapere. </s>
          <s id="s.001386">Porrò reperiri numeros triangulares, pen­<lb></lb>
tagonos, &amp; ſimiles, conſtat ex Arithmetica Nicomachi, Boetij, &amp; Iordani, <lb></lb>citati in definitionibus 7. ſuæ Arithmeticæ, atque ex tractatu Diophantis <lb></lb>
Alex. de numeris rectangulis. </s>
          <s id="s.001387"><expan abbr="atq;">atque</expan> ex his locus hic ſatis clarus redditur.</s>
        </p>
        <p type="main">
          <s id="s.001388"><arrow.to.target n="marg94"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001389"><margin.target id="marg94"></margin.target>94</s>
        </p>
        <p type="main">
          <s id="s.001390">Tex. 31. <emph type="italics"></emph>(Vtuntur etiam Mathematici infinito)<emph.end type="italics"></emph.end> <expan abbr="aliquãdo">aliquando</expan> Mathematici du­<lb></lb>
cunt lineas quantumuis longas, ſeu indefinitæ longitudinis, quas etiam in­<lb></lb>
finitas appellant: &amp; hoc modo vtuntur infinito, vt infra tex. 71. ipſe Ariſt. <lb></lb>
exponit. </s>
          <s id="s.001391">alio præterea modo vtuntur infinito, vt quando ſupponunt data <lb></lb>
quauis quantitate poſſe ſumi maiorem, vel etiam minorem in infinitum, vt <lb></lb>
patet ex 6. poſtulato primi Elem. editionis Clauianæ. </s>
          <s id="s.001392">numerum <expan abbr="quoq;">quoque</expan> au­<lb></lb>
geri poſſe in infinitum, eſt ſecundum poſtulatum libri 7. Elem. vel demum <lb></lb>
quando probant quamlibet lineam poſſe diuidi bifariam, quia hinc ſequitur <lb></lb>
poſſe ſub diuidi in <expan abbr="infinitũ">infinitum</expan>; his igitur modis Mathematicis <expan abbr="infinitũ">infinitum</expan> in vſu eſt.</s>
        </p>
        <p type="main">
          <s id="s.001393"><arrow.to.target n="marg95"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001394"><margin.target id="marg95"></margin.target>95</s>
        </p>
        <p type="main">
          <s id="s.001395">Tex. 68. &amp; 69. plura de magnitudine, &amp; numero continent; ſed quæ non <lb></lb>
indigeant opera noſtra.</s>
        </p>
        <p type="main">
          <s id="s.001396"><arrow.to.target n="marg96"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001397"><margin.target id="marg96"></margin.target>96</s>
        </p>
        <p type="main">
          <s id="s.001398">Tex. 71. <emph type="italics"></emph>(Non remouet autem ratio Mathematicos à contemplatione auferens <lb></lb>
ſic eſſe infinitum, vt actu ſit verſus augmentum, vt intranſibile, <expan abbr="ncq;">neque</expan> enim nunc in­<lb></lb>
digent infinito, <expan abbr="neq;">neque</expan> vtuntur, ſed ſolum eſſe <expan abbr="quantumcunqu;">quantumcunque</expan> velint finitam)<emph.end type="italics"></emph.end> ratio <lb></lb>
phyſica tollens infinitum actu, non eſt Mathematicis impedimento, quia ipſi <lb></lb>
non vtuntur infinito actu; quam enim ipſi ducunt lineam infinitam, non eſt <lb></lb>
verè infinita, ſed indefinita, eam enim quantumlibet magnam producunt, vt <lb></lb>
poſſit ad demonſtrandum ſufficere.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001399"><emph type="italics"></emph>Ex Quarto Phyſicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001400"><arrow.to.target n="marg97"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001401"><margin.target id="marg97"></margin.target>97</s>
        </p>
        <p type="main">
          <s id="s.001402">Tex. 120. ter in hoc textu meminit commenſurabilitatis, &amp; incommen­<lb></lb>
ſurabilitatis, quæ eſt diametri ad coſtam: cuius explicationem vide <lb></lb>
primo Priorum, ſecto primo, cap. 23.</s>
        </p>
      </chap>
      <pb pagenum="76" xlink:href="009/01/076.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.001403"><emph type="italics"></emph>Ex Quinto Phyſicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001404"><arrow.to.target n="marg98"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001405"><margin.target id="marg98"></margin.target>98</s>
        </p>
        <p type="main">
          <s id="s.001406">Tex. 6. <emph type="italics"></emph>(Vt media grauis ad vltimam, &amp; acuta ad primam)<emph.end type="italics"></emph.end> alludit ad or­<lb></lb>
dinem chordarum in muſicis inſtrumentis, vbi media chorda edit ſo­<lb></lb>
num, reſpectu quidem vltimæ, &amp; ſupremæ chordæ grauem: reſpectu verò <lb></lb>
primæ, &amp; infimæ acutum.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001407"><emph type="italics"></emph>Ex Octauo Phyſicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001408"><arrow.to.target n="marg99"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001409"><margin.target id="marg99"></margin.target>99</s>
        </p>
        <p type="main">
          <s id="s.001410">Tex. 15. <emph type="italics"></emph>(Etenim triangulus habet tres angulos æquales duobus rectis angulis)<emph.end type="italics"></emph.end><lb></lb>
lib. 1. Priorum, ſecto 3. cap. 1. huius rei explicationem reperies.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001411"><emph type="italics"></emph>EX PRIMO DE COELO.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001412"><arrow.to.target n="marg100"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001413"><margin.target id="marg100"></margin.target>100</s>
        </p>
        <p type="main">
          <s id="s.001414">Tex. 33. <emph type="italics"></emph>(Vt ſi quis minimam quădam eſſe dicat magnitudinem, hic enim <lb></lb>
minimum introducens, maxima <expan abbr="vbiq;">vbique</expan> amoueret mathematicorŭm)<emph.end type="italics"></emph.end> ideſt, <lb></lb>
ſi quis, vt Democritus poſuerit in magnitudinibus eſſe minima, <lb></lb>
ſeu indiuiſibilia, ex quibus entia mathematica componerentur, <lb></lb>
hic euerteret maxima mathematicorum, ideſt maxime ipſorum demonſtra­<lb></lb>
tiones, atque etiam effata euerterentur: v. g. 10. primi Elem. quæ docet <lb></lb>
quamlibet lineam poſſe diuidi bifariam nulla eſſet, quia linea illa, quæ con­<lb></lb>
ſtaret ex tribus Democriti atomis, nulla ratione bifariam ſecari poſſet. </s>
          <s id="s.001415">pa­<lb></lb>
riter totus ferè decimus liber Elem. deceptiuus, &amp; nullus eſſet, ſi enim da­<lb></lb>
rentur illæ atomi, ex quibus <expan abbr="quãtitas">quantitas</expan> conflaretur, nullæ eſſent lineæ incom­<lb></lb>
menſurabiles, quandoquidem omnes communi illa, ac indiuidua, commen­<lb></lb>
ſurarentur. </s>
          <s id="s.001416">poſtulatum <expan abbr="quoq;">quoque</expan> illud, qualibet data magnitudine ſumi poſſe <lb></lb>
minorem prorſus irritum redderetur, quia data atomo, illa minor accipi <lb></lb>
non poſſet.</s>
        </p>
        <p type="main">
          <s id="s.001417"><arrow.to.target n="marg101"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001418"><margin.target id="marg101"></margin.target>101</s>
        </p>
        <p type="main">
          <s id="s.001419">Tex. 36. <emph type="italics"></emph>(Sit <expan abbr="itaq;">itaque</expan> linea, in qua A G E, infinita ad partes E; &amp; alia vtrinque <lb></lb>
infinita, in qua <foreign lang="grc">β</foreign> B; ſi <expan abbr="itaq;">itaque</expan> deſcribat circulum linea A G E, circa centrum G, fe-<emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.076.1.jpg" place="text" xlink:href="009/01/076/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>retur circulariter linea A G E, ſecans ali­<lb></lb>
quando lineam <foreign lang="grc">β</foreign> B, tempore finito; totum <lb></lb>
enim tempus, in quo circulariter latum <lb></lb>
eſt Cœlum finitum eſt, &amp; ablatum igitur, <lb></lb>quo ſecans ferebatur; erit igitur aliquod <lb></lb>principium, quo primum linea A G E, li­<lb></lb>
neam <foreign lang="grc">β</foreign> B, ſecuit. </s>
          <s id="s.001420">ſed impoſſibile est; non <lb></lb>
est igitur circulariter verti <expan abbr="infinitũ">infinitum</expan>, quare <lb></lb>
<expan abbr="neq;">neque</expan> mundum, ſi eſſet infinitus)<emph.end type="italics"></emph.end> quamuis <lb></lb>
textus hic parum ſit mathematicus, <lb></lb>
quia tamen ſupponit figuram mathe­<lb></lb>
maticam, quæ in codicibus pariter, ac <lb></lb>
commentarijs deſideratur, illam pla­<lb></lb>
cuit apponere. </s>
          <s id="s.001421">in qua quidem, quamuis duæ lineæ infinitæ ſupponantur, vna <lb></lb>
ad alteram <expan abbr="tãtum">tantum</expan> partem in qua E: altera verò ad <expan abbr="vtramq;">vtramque</expan> partem <foreign lang="grc">β,</foreign> &amp; B,


<pb pagenum="77" xlink:href="009/01/077.jpg"></pb>non potuerunt tamen deſcribi, niſi finitæ; appoſitæ idcircò ſunt ad partes <lb></lb>
illas, ad quas deberent eſſe infinitæ lineolæ quædam infinitatem indicantes. <lb></lb>
</s>
          <s id="s.001422">debemus poſtea, vt mentem Ariſt. percipiamus concipere lineam A G E, <lb></lb>
moueri circulariter facto centro in G. quæ quia infinita ſupponitur ad par­<lb></lb>
tem E, ſecabit neceſſariò alteram <expan abbr="vtrinq;">vtrinque</expan> infinitam <foreign lang="grc">β</foreign> B, <expan abbr="illamq́">illamque</expan>; neceſſariò <lb></lb>
finito tempore percurret, finito enim tempore tota mundi circulatio per­<lb></lb>
agitur, ſpatio videlicet viginti quatuor horarum. </s>
          <s id="s.001423">ex quo Ariſt. infert mun­<lb></lb>
dum non poſſe eſſe infinitæ magnitudinis; quia ſi mundus eſſet infinitus; &amp;. <lb></lb>
</s>
          <s id="s.001424">duæ lineæ infinitæ, quales ſunt prædictæ in ipſo, <expan abbr="atq;">atque</expan> cum ipſo moueri alte­<lb></lb>
ra earum A E, intelligatur, alteram <foreign lang="grc">β</foreign> B, manentem in tempore finito, ideſt, <lb></lb>
in diurna conuerſione pertranſibit: fieri autem nequit, vt infinita magni­<lb></lb>
tudo finito tempore percurratur; quare dicendum eſt, mundum eſſe finita <lb></lb>
magnitudine præditum.</s>
        </p>
        <p type="main">
          <s id="s.001425"><arrow.to.target n="marg102"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001426"><margin.target id="marg102"></margin.target>102</s>
        </p>
        <p type="main">
          <s id="s.001427">Tex. 48. <emph type="italics"></emph>(Nihil autem refert grauitates, commenſurabiles ſint, an incommen­<lb></lb>ſurabiles)<emph.end type="italics"></emph.end> quidnam ſit commenſurabilitas, &amp; incommenſurabilitas, expli­<lb></lb>
catum eſt lib. 1. Priorum, ſecto 1. cap. 23.</s>
        </p>
        <p type="main">
          <s id="s.001428"><arrow.to.target n="marg103"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001429"><margin.target id="marg103"></margin.target>103</s>
        </p>
        <p type="main">
          <s id="s.001430">Tex. 119. <emph type="italics"></emph>(Est autem impoſſibile, &amp; poſſibile; falſum, &amp; verum, ex ſuppoſitio­<lb></lb>
ne quidem, dico autem, vt triangulum impoſſibile eſt duos rectos habere, ſi hæc)<emph.end type="italics"></emph.end><lb></lb>
ideſt, ſi ſupponantur falſa quædam, quæ ſupponi poſſunt, ſequetur impoſſi­<lb></lb>
bile eſſe triangulum habere tres angulos æquales duobus rectis angulis, vi­<lb></lb>
de, quæ ſcripſi lib. 1. Priorum, ſecto 3. cap. 1. de hoc, quod eſt, habere tres <lb></lb>
angulos æquales duobus rectis. </s>
          <s id="s.001431">v. g. ſi in triangulo pag. </s>
          <s id="s.001432">73. producto late­<lb></lb>
re A C, in D. ſi ſupponatur externus angulus B C D, non eſſe æqualis duobus <lb></lb>internis, &amp; oppoſitis A, &amp; B, nunquam poterimus eo modo, quo Euclides, <lb></lb>
demonſtrare paſſionem prædictam de triangulo A B C. huiuſmodi impoſſi­<lb></lb>
bile, cuius oppoſitum non ſolum poſſibile, ſed etiam neceſſarium eſt, vocat <lb></lb>
Ariſt. impoſſibile ex ſuppoſitione, quia ſcilicet impoſſibile euadit ex quo­<lb></lb>
dam falſo ſuo ſuppoſito, vt in allato exemplo, triangulum habere tres an­<lb></lb>
gulos æquales duobus rectis, quamuis neceſſarium ſit, tamen ex falſa ſup­<lb></lb>poſitione, impoſſibile euaſit.</s>
        </p>
        <p type="main">
          <s id="s.001433"><arrow.to.target n="marg104"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001434"><margin.target id="marg104"></margin.target>104</s>
        </p>
        <p type="main">
          <s id="s.001435">Ibidem <emph type="italics"></emph>(Et diameter commenſurabilis est coſtæ, ſi hæc)<emph.end type="italics"></emph.end> vide primo Priorum, <lb></lb>
ſecto 3. cap. 23. hoc ſolum nunc addendum <emph type="italics"></emph>(Si hæc)<emph.end type="italics"></emph.end> v. g. ſi ſupponamus li­<lb></lb>
neas eſſe compoſitas ex indiuiſibilibus, conſectarium erit diametrum eſſe <lb></lb>
commenſurabilem coſtæ, quia indiuiſibile illud, ex quo vtraque linea con­<lb></lb>
ſtat, erit <expan abbr="vtriuſq;">vtriuſque</expan> menſura communis.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001436"><emph type="italics"></emph>Ex Secundo de Cælo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001437"><arrow.to.target n="marg105"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001438"><margin.target id="marg105"></margin.target>105</s>
        </p>
        <p type="main">
          <s id="s.001439">Tex. 24. <emph type="italics"></emph>(Amplius qui ſolida diuidunt in plana, <expan abbr="atq;">atque</expan> ex planis corpora <lb></lb>
generant, his teſtes fuiſſe videntur: ſolam enim figurarum ſolidarum <lb></lb>
ſphæram non diuidunt, vt non plures ſuperficies. </s>
          <s id="s.001440">quam vnam <expan abbr="habẽum">habentem</expan>. <lb></lb>
</s>
          <s id="s.001441">diuiſio enim in plana non perinde efficitur, vt quiſpiam <expan abbr="diuidẽs">diuidens</expan> in par­<lb></lb>tes diuidat totum, ſed vt in ſpecie diuerſa: patet igitur ſphæram eſſe ſolidarum <lb></lb>
primam)<emph.end type="italics"></emph.end> qui ſolida diuidunt in plana, ea diuidunt <expan abbr="ſecũdum">ſecundum</expan> numerum ſuper­<lb></lb>ficierum, quibus ambiuntur, v. g. diuidunt cubum in ſex ſuperficies, quia <lb></lb>cubus ſex quadratis planis ſuperficiebus continetur: qua ratione nequeunt <pb pagenum="78" xlink:href="009/01/078.jpg"></pb>ſphæram in plana vlla reſoluere, <expan abbr="neq;">neque</expan> in alias plures ſuperficies, quia ſphæ­<lb></lb>
ra ambitur vnica tantum ſuperficie ſphærica. </s>
          <s id="s.001442">quando verò ex planis corpo­<lb></lb>
ra generant, vt facit Plato in Timæo, accipíunt primò triangulum æquila­<lb></lb>
terum, &amp; ex quatuor triangulis æquilateris ſimul compactis conficiunt py­<lb></lb>
ramidem; &amp; hoc modo alia ſolida à pluribus ſuperficiebus ambita conſti­<lb></lb>
tuunt: verum hac ratione nullo modo poſſunt ſphæram componere, quia <lb></lb>
vnica tantum, <expan abbr="eaq́">eaque</expan>; ſphærica ſuperficie compræhenditur: <expan abbr="atq;">atque</expan> hoc pacto iſti <lb></lb>
diuidentes, &amp; componentes corpora fidem faciunt, ſphæram, cum ex nullis <lb></lb>
componatur, ſolidorum eſſe primam.</s>
        </p>
        <p type="main">
          <s id="s.001443"><arrow.to.target n="marg106"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001444"><margin.target id="marg106"></margin.target>106</s>
        </p>
        <p type="main">
          <s id="s.001445">Tex. 25. <emph type="italics"></emph>(Est autem, &amp; ſecundum numerorum ordinem aſſignantibus, ſic po­<lb></lb>
nentibus rationabiliſſimam, circulum quidem ſecundum vnum; triangulum autem <lb></lb>
ſecundum dualitatem, quoniam duo recti. </s>
          <s id="s.001446">ſi autem ſecundum triangulum, vnum. <lb></lb>
</s>
          <s id="s.001447">circulus non erit figura)<emph.end type="italics"></emph.end> In ordine figurarum conueniens eſt, inquit, primam <lb></lb>
facere circulum propter ſimpliciſsimam ipſius naturam, cum vnica, ac per­<lb></lb>
fecta circulari linea comprehendatur: <expan abbr="Triangulũ">Triangulum</expan> verò ſecundam, quoniam <lb></lb>
duo anguli recti, ideſt, quia triangulum habet tres angulos æquales duobus <lb></lb>rectis angulis; quod fusè explicatum eſt lib. 1. Priorum, ſecto 3. cap. 1. De­<lb></lb>
mum ſi primum locum dederimus triangulo, nullus alius remanet pro cir­<lb></lb>
culo, quod eſt inconueniens, ergo circulus prima figura erit.</s>
        </p>
        <p type="main">
          <s id="s.001448"><arrow.to.target n="marg107"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001449"><margin.target id="marg107"></margin.target>107</s>
        </p>
        <p type="main">
          <s id="s.001450">Tex. 31. <emph type="italics"></emph>(At verò, quod aquæ ſuperficies talis ſit, manifeſtum eſt hac ſuppoſi­<lb></lb>
tione ſumpta, quod apta natura eſt ſemper confluere aqua ad magis concauum: ma­<lb></lb>
gis autem concauum eſt, quod centro propinquius est. </s>
          <s id="s.001451">ducantur ergo ex centro A,<emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.078.1.jpg" place="text" xlink:href="009/01/078/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>linea A B, &amp; linea A C, &amp; producatur, in qua B C, <lb></lb>
ducta igitur ad baſim linea, in qua A D, minor eſt eis, <lb></lb>
quæ ex centro. </s>
          <s id="s.001452">magis igitur concauus locus eſt, quare <lb></lb>
influet aqua, donec <expan abbr="vtiq;">vtique</expan> æquetur. </s>
          <s id="s.001453">æqualis eſt autem eis, <lb></lb>
quæ ex centro linea A E, quare neceſſe eſt apud eas, quæ <lb></lb>
ex centro, eſſe aquam, tunc enim quieſcet. </s>
          <s id="s.001454">linea autem, <lb></lb>
quæ eas, quæ ex centro tangit, circularis eſt, ſphærica <lb></lb>
igitur aquæ ſuperficies eſt, in qua B E C.)<emph.end type="italics"></emph.end> toto hoc <lb></lb>
textu lineari demonſtratione probat aquæ manen­<lb></lb>
tis ſuperficiem eſſe ſphæricam: quæ demonſtratio <lb></lb>
perſpicua euadit, ſi figura, quæ in codicibus tam <lb></lb>
græcis, quam latinis, <expan abbr="atq;">atque</expan> etiam in commentarijs deſideratur, quemadmo­<lb></lb>
dum fecimus, reſtituatur. </s>
          <s id="s.001455">ſit igitur in præcedenti figura A, centrum mundi, <lb></lb>
ex quo educantur duæ rectæ lineæ æquales A B, A C, quæ deinde alia recta <lb></lb>
B C, coniungantur. </s>
          <s id="s.001456">educatur <expan abbr="quoq;">quoque</expan> recta alia ex centro A, quæ pertingat <lb></lb>
ad B C, quæ baſis eſt trianguli B A C, &amp; producatur vlterius quantumlibet <lb></lb>
in E. intelligatur demum circumferentia tranſire per puncta B, &amp; C, quia <lb></lb>
illæ duæ lineæ A B, A C, ſunt æquales, quæ circumferentia alteram A D, quæ <lb></lb>
fuit protracta, ſecet in E. </s>
          <s id="s.001457">Iam ſic argumentatur: aqua natura ſua ſemper <lb></lb>defluit ad locum magis concauum, ideſt, ad loca centro A, terræ propin­<lb></lb>
quiora, quale eſſet in figura locus D, reſpectu locorum B, &amp; C, quia A D, <lb></lb>
linea minor eſt ijs, quæ ex centro eductæ ſunt A B, A C. quapropter aqua <lb></lb>debet defluere ex B, ad D, vel ex C, ad idem D, donec pertingat ad E. qui <lb></lb>
locus non eſt decliuior punctis B, &amp; C. quare cum loca B, E, C, quæ ſunt ex­


<pb pagenum="79" xlink:href="009/01/079.jpg"></pb>trema linearum, ſint æquè decliuia, neceſſe eſt aquæ ſuperficiem apud ipſa <lb></lb>
conſiſtere, tunc enim debet quieſcere, aliter nunquam quieſceret; ſed vide­<lb></lb>
mus aquam manentem, &amp; quietam, ergo quieſcit circa puncta B, E, C, à <lb></lb>
centro terræ æquidiſtantia, per quæ tranſit linea circularis coniungens illa; <lb></lb>etſi ſuperficies per eiuſmodi loca pertranſiret, eſſet ſphærica: ſed ſuperfi­<lb></lb>
cies aquæ tranſit per talia loca, ergo ſphærica eſt. </s>
          <s id="s.001458">Huius etiam habes acu­<lb></lb>
tiſſimam Archimedis demonſtrationem initio libelli de ijs, quæ vehuntur <lb></lb>
in aqua, quam in ſuam ſphæram retulit Clauius.</s>
        </p>
        <p type="main">
          <s id="s.001459"><arrow.to.target n="marg108"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001460"><margin.target id="marg108"></margin.target>108</s>
        </p>
        <p type="main">
          <s id="s.001461">Tex. 46. <emph type="italics"></emph>(Reliquum eſt orbes quidem moueri, stellas verò quieſcere, &amp; infixas <lb></lb>
ipſis orbibus ferri; ſolum enim ſic nullum abſurdum accidit. </s>
          <s id="s.001462">celeriorem enim eſſe <lb></lb>
maioris circuli velocitatem, rationabile eſt circa idem centrum infixis: vt enim in <lb></lb>
alijs maius corpus velocius fertur propria latione, ſic, &amp; in circularibus: maius <lb></lb>
enim eſt eorum, quæ auferuntur ab eis, quæ ex centro, maioris circuli ſegmentum)<emph.end type="italics"></emph.end><lb></lb>
ex intellectione vltimæ periodi textus totius intelligentia pendet: ſit igitur <lb></lb>
<figure id="id.009.01.079.1.jpg" place="text" xlink:href="009/01/079/1.jpg"></figure><lb></lb>
figura præſens, in qua cum ſint duo circuli concen­<lb></lb>
trici, vnus altero maior, <expan abbr="eductæq́">eductæque</expan>; ſint ex <expan abbr="cẽtro">centro</expan> duæ <lb></lb>
ſemidiametri A D, A E, quæ <expan abbr="vtrunq;">vtrunque</expan> circulum ſe­<lb></lb>
cant, apparet maius eſſe <expan abbr="ſegmentũ">ſegmentum</expan> D E, quod è ma­<lb></lb>
iori circulo ſemidiametri ex <expan abbr="cẽtro">centro</expan> eductæ auferunt, <lb></lb>
quam ſegmentum B C, minoris circuli, quod eiſdem <lb></lb>ſemidiametris intercipitur. </s>
          <s id="s.001463">Verumtamen ſi circuli <lb></lb>
ambo ſimul moueantur, maior circulus æquali tem­<lb></lb>
pore maius illud ſpatium D E, &amp; minor minus B C, <lb></lb>
pertranſibit: idem igitur de cœleſtibus orbibus di­<lb></lb>
cendum, qui quamuis omnes diurnum ſimul motum <lb></lb>
abſoluunt, maiores tamen celerius conuertuntur: quo fit, vt ſtellæ maiori­<lb></lb>
bus circulis infixæ, <expan abbr="atq;">atque</expan> delatæ, maiori celeritate ſuos curſus peragant, ne­<lb></lb>
que oportet eas, dum mouentur cœlum diſſecare, quod accideret, ſi pro­<lb></lb>prio motu veluti piſces per aquam progrederentur.</s>
        </p>
        <p type="main">
          <s id="s.001464">Hæc quidem Ariſt. conſentanea obſeruationibus veterum Aſtronomo­<lb></lb>
rum; at verò illis noſtræ ætatis obſeruationes repugnant; præſertim illæ, <lb></lb>
quæ fiunt circa ſtellas errantes: ex quibus fatendum eſſe videtur, Cœlum, <lb></lb>
qua parte Planetas continet, liquidum eſſe, ac per illud Planetas proprio <lb></lb>
motu, ceu piſces in aqua progredi. </s>
          <s id="s.001465">Tycho <expan abbr="namq;">namque</expan> Brahe, <expan abbr="alijq́">alijque</expan>; plures exactè <lb></lb>
demonſtrant Cometas in regione Planetarum eſſe, <expan abbr="eosq́">eosque</expan>; motu quodam in <lb></lb>
tranſuerſum moueri, quo neceſſario Cęlú deberent perforare; ijdem oſten­<lb></lb>
dunt nonnullos Planetas, Martem præſertim, ac Venerem modo ſupra So­<lb></lb>
lem, modo infra aſcendere, &amp; deſcendere. </s>
          <s id="s.001466">Idem patet ex obſeruatione no­<lb></lb>
ua per nouum Teleſcopij i <expan abbr="ſtrumẽtum">ſtrumentum</expan> in Venere facta, quæ lunulata <expan abbr="vtrinq;">vtrinque</expan> <lb></lb>
à Sole apparet: quando nimirum eſt in imo epicyclo. </s>
          <s id="s.001467"><expan abbr="iterumq́">iterumque</expan>; rotunda ve­<lb></lb>
luti Luna plena, cum in ſummo epicyclo verſatur: quæ minimè apparerent, <lb></lb>
niſi ſupra, ac infra Solem circumiret. </s>
          <s id="s.001468">His rationibus conantur ipſi proba­<lb></lb>
re Cœlum eſſe liquidum; <expan abbr="atq;">atque</expan> in eo Planetas, veluti aues in aere, permeare: <lb></lb>quarum ſolutio mihi nulla occurrit, alijs fortaſſis occurret.</s>
        </p>
        <p type="main">
          <s id="s.001469"><arrow.to.target n="marg109"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001470"><margin.target id="marg109"></margin.target>109</s>
        </p>
        <p type="main">
          <s id="s.001471">Tex. 57. <emph type="italics"></emph>(De ordine autem ipſorum, quo quidem modo ſingula diſponantur, vt <lb></lb>
quædam ſint priora, quædam posteriora, &amp; quomodo ſpatijs ſe ă<expan abbr="habeãt">habeant</expan> ad inuicem,<emph.end type="italics"></emph.end>


<pb pagenum="80" xlink:href="009/01/080.jpg"></pb><emph type="italics"></emph>ex ijs circa Aſtrologiam, conſideretur: dicitur enim ſufficienter)<emph.end type="italics"></emph.end> ſumit hoc loco <lb></lb>
Aſtrologiam, pro Aſtronomia, ſi iuxta recentiores loqui velimus. </s>
          <s id="s.001472">Dicit igi­<lb></lb>
tur ordinem cœlorum, ac ſyderum, item ſitum, &amp; proportiones magnitu­<lb></lb>dinum eorundem, cum per naturalis ſcientiæ principia ſciri nequeant, ex <lb></lb>
rationibus Aſtronomorum petenda eſſe, apud quos iſta ſufficienter <expan abbr="demon-ſtrẽtur">demon­<lb></lb>
ſtrentur</expan>. </s>
          <s id="s.001473">&amp; meritò quidem hæc dicuntur; poſteriores enim ab Ariſt. ordines, <lb></lb>
ſitus, ac magnitudines tam cœlorum, quam ſyderum firmis rationibus, <expan abbr="atq;">atque</expan> <lb></lb>
inuentu peracutis demonſtrarunt. </s>
          <s id="s.001474">quorum princeps fuit ptolæmeus; noſtra <lb></lb>
tamen ætate Tycho Brahe, qui certis obſeruationibus, quas maximo labo­<lb></lb>
re, ac ſumptu exantlauit, in nonnullis à Ptolæmeo, ac reliquis diſſentjt: ſtan­<lb></lb>
dum autem eſſe recentioribus obſeruationibus apud Aſtronomiæ peritos in <lb></lb>
confeſſo eſt.</s>
        </p>
        <p type="main">
          <s id="s.001475"><arrow.to.target n="marg110"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001476"><margin.target id="marg110"></margin.target>110</s>
        </p>
        <p type="main">
          <s id="s.001477">Tex. <emph type="italics"></emph>(Luna autem oſtenditur per ea, quæ circa viſum, quod ſphærica ſit: non <lb></lb>
enim <expan abbr="vtiq;">vtique</expan> fieret accreſcens, &amp; decreſcens, plurimŭm quidem alter a ex parte curua, <lb></lb>
altera concaua, aut <expan abbr="vtrmq;">vtrinque</expan> curua, ſemel autem bipartita)<emph.end type="italics"></emph.end> ait per ea, quæ circa <lb></lb>
viſum, ideſt per opticem probari Lunam eſſe ſphæricam: ſed conſule, quæ <lb></lb>
primo Poſter. tex. 3. de hac re ſcripſi, &amp; plenam etiam huius loci intelligen­<lb></lb>
tiam aſſequeris, præſertim ſi experimentum ibi traditum inieris.</s>
        </p>
        <p type="main">
          <s id="s.001478"><arrow.to.target n="marg111"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001479"><margin.target id="marg111"></margin.target>111</s>
        </p>
        <p type="main">
          <s id="s.001480">Ibidem <emph type="italics"></emph>(Et rurſus per Astrologica, quia <expan abbr="vtiq;">vtique</expan> non eſſent ſolis eclypſes lunulæ <lb></lb>ſpeciem præfeferentes. </s>
          <s id="s.001481">Quare ſi vnum est tale, palam eſt, quod &amp; alia <expan abbr="vtiq;">vtique</expan> erunt <lb></lb>
talia)<emph.end type="italics"></emph.end> ſicuti <expan abbr="præcedẽs">præcedens</expan> ſphæricitatis Lunæ ratio ex Perſpectiua deſumpta eſt, <lb></lb>
ita præſens ex Aſtronomia, ex eò enim, quod eclypſis Solis habeat figuram <lb></lb>
lunulæ, ideſt, ſi inſtar Lunæ falcatæ, probant Aſtronomi Lunam eſſe ſphæri­<lb></lb>
cam. </s>
          <s id="s.001482">intellige tamen partem illam Solis, quæ non eclypſatur, habere figu­<lb></lb>
ram lunulæ, pars enim à Luna obumbrata non videtur, etſi videretur oua­<lb></lb>lem quandam ſpeciem, præſeferret: pars igitur, illa eſt corniculata, quia <lb></lb>
<figure id="id.009.01.080.1.jpg" place="text" xlink:href="009/01/080/1.jpg"></figure><lb></lb>
cum Solis defectio ex interpoſitione Lunæ inter nos, &amp; <lb></lb>
Solem contingat, &amp; Luna ſit ſphærica, neceſſariò ſphæ­<lb></lb>
ricè, &amp; circulariter Solem obumbrabit; quare pars illa <lb></lb>
non obumbrata remanet falcata, &amp; corniculata, vt in <lb></lb>præſenti figura videre eſt; vbi cernis, Lunam Solem or­<lb></lb>
biculariter offuſcare in linea A D C, partem Solis de­<lb></lb>
tectam <expan abbr="contentã">contentam</expan> lineis curuis A B C D, eſſe lunularem, <lb></lb>
&amp; falcatam; cum ergo in hunc modum fiat Solis deli­<lb></lb>
quium, ſignum certum eſt, Lunam eſſe ſphæricam.</s>
        </p>
        <p type="main">
          <s id="s.001483"><arrow.to.target n="marg112"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001484"><margin.target id="marg112"></margin.target>112</s>
        </p>
        <p type="main">
          <s id="s.001485">Tex. 107. <emph type="italics"></emph>(Quod autem dubitatur, hoc eſt; videre autem non eſt difficile, ſi pa­<lb></lb>
rum conſiderauerimus, &amp; diſtinxerimus, quonam modo cenſeamus quantamuis ma­<lb></lb>
gnitudinem grauem ad medium ferri. </s>
          <s id="s.001486">manifeſtum enim eſt, quod non quouſque ex­<lb></lb>
tremum tangat ipſum centrum; ſed maior pars vincat, oportet, <expan abbr="quouſq;">quouſque</expan> ſuo medio <lb></lb>
ipſum medium compræhendat; <expan abbr="hucnſq;">hucuſque</expan> enim habet propenſionem)<emph.end type="italics"></emph.end> ſenſus Ariſto­<lb></lb>
telis eſt, debere nos exiſtimare, quod ſi quæpiam grauis magnitudo deſcen­<lb></lb>dat ad centrum mundi, eam non permanſuram, ſtatim ac ipſius extremum <lb></lb>centrum mundi attigent; ſed eò <expan abbr="vſq;">vſque</expan> deſcenſuram, <expan abbr="quouſq;">quouſque</expan> ipſius medium, <lb></lb>mundi medium, ſiue centrum aſſequutum ſit; maior enim ipſius pars, in qua <lb></lb>
ſcilicet medium eſt, minorem partem propellit, donec vtrinque à centro <lb></lb>
mundi æquè emineat; omne enim graue <expan abbr="hucuſq;">hucuſque</expan> habet propenſionem, ſiue <pb pagenum="81" xlink:href="009/01/081.jpg"></pb><expan abbr="hucuſq;">hucuſque</expan> grauitat, v. g. ſi lapis illuc deſcenderet, non quieſceret ſtatim ac <lb></lb>
prima ipſius pars ad mundi centrum pertingeret, ſed reliquæ ipſius partes <lb></lb>
adhuc grauitarent, <expan abbr="ſicq́">ſicque</expan>; vlterius primam partem impellerent, donec lapi­<lb></lb>dis medium, mundi medio congrueret: quo facto lapis quieſceret. </s>
          <s id="s.001487">quæ num <lb></lb>
vera ſint, vt intelligamus, oportet prius præmittere, iuxta Mathematicos <lb></lb>
duplex eſſe medium, ſiue centrum cuiuſuis magnitudinis: aliud enim eſt <lb></lb>
centrum molis, aliud eſt centrum grauitatis. </s>
          <s id="s.001488">centrum molis eſt illud pun­<lb></lb>
ctum, à quo extrema æquidiſtant: centrum grauitatis eſt punctum illud, à <lb></lb>
quo extrema æque ponderant, ſiue à quo graue ſuſpenſum æquè ponderat, <lb></lb>
ſiue in æquilibrio manet. </s>
          <s id="s.001489">Porrò in corporibus regularibus, ſi vniformia ſint <lb></lb>
idem, &amp; vnum ſunt centrum molis, ac centrum grauitatis: vt in ſphæra <lb></lb>plumbea, idem erit <expan abbr="vtrumq;">vtrumque</expan> centrum: ſi verò difformia ſint in grauitate, <lb></lb>
vt in ſphæra partim plumbea, partim lignea, diuerſum erit centrum molis, <lb></lb>
à centro grauitatis; illud enim erit in medio ſphæræ; centrum verò graui­<lb></lb>
tatis in parte plumbea exiſtet. </s>
          <s id="s.001490">In corporibus deinde irregularibus, etiamſi <lb></lb>
ſint vniformis ponderis, aliud tamen eſſe poteſt centrum molis à <expan abbr="cẽtro">centro</expan> gra­<lb></lb>
uitatis, vt in corpore oblongo, cuius alterum extremum ſit reliquis parti­<lb></lb>
bus multò maius, vti eſt claua: vbi centrum molis erit in medio longitudi­<lb></lb>
nis clauæ; centrum verò grauitatis, erit propinquius capiti clauæ. </s>
          <s id="s.001491">quando <lb></lb>
igitur Ariſt. ait, graue deſcenſurum, donec ipſius medium, ſiue centrum, <lb></lb>
mundi centrum attingat; benè dicit, ſi de medio grauitatis intelligat; ma­<lb></lb>
lè autem ſi de medio molis. </s>
          <s id="s.001492">quia grauia omnia ratione centri grauitatis <lb></lb>
ponderant, <expan abbr="neq;">neque</expan> manent; niſi ipſum maneat: quare niſi ipſum <expan abbr="attingãt">attingant</expan> cen­<lb></lb>
trum mundi ſemper grauitabunt, &amp; mouebuntur. </s>
          <s id="s.001493">Verum enim verò ex an­<lb></lb>
tiquorum monumentis manifeſtum eſt, Archimedem, qui multò poſt Ari­<lb></lb>
ſtotelem floruit, primum omnium de centro grauitatis eſſe philoſophatum, <lb></lb>
qua ratione dicendum eſſet, Ariſtotelem de centro, molis loquutum eſſe, <lb></lb>
&amp; perinde non <expan abbr="vſquequaq;">vſquequaque</expan> verè.</s>
        </p>
        <p type="main">
          <s id="s.001494"><arrow.to.target n="marg113"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001495"><margin.target id="marg113"></margin.target>113</s>
        </p>
        <p type="main">
          <s id="s.001496">Tex. 109. <emph type="italics"></emph>(Præterea <expan abbr="quoq;">quoque</expan> &amp; per ea, quæ apparent ſecundum ſenſum, neque <lb></lb>
enim Lunæ eclypſes tales <expan abbr="haberẽt">haberent</expan> deciſiones; nunc enim in ijs, quæ ſecundum men­<lb></lb>
ſem fiunt, figurationibus, omnes accipit diuiſiones: etenim recta fit, &amp; vtrinque <lb></lb>
curua, &amp; concaua)<emph.end type="italics"></emph.end> probat terram eſſe ſphæricam ratione aſtronomica, ex <lb></lb>
Lunæ eclypſibus deſumpta: nam niſi terra eſſet rotunda, nunquam Luna in <lb></lb>
eclypſi haberet tales deciſiones, ideſt non haberet falcatas, aut lunulatas <lb></lb>partes illas, quæ in eclypſi obſcurantur, &amp; quaſi à Luna reſecantur. </s>
          <s id="s.001497">quam­<lb></lb>
uis enim ſingulis menſibus Luna terminetur modo linea concaua, vt quan­<lb></lb>
do noua eſt; modo recta, vt quando diuidua eſt: modo vtrinque curua, vt <lb></lb>cum à diuidua ad plenilunium tendit. </s>
          <s id="s.001498">quod fuſius primo Poſter. tex. 30. ex­<lb></lb>
poſui. </s>
          <s id="s.001499">in eclypſibus tamen ſemper curuam habet lineam illam, quæ partem <lb></lb>eclypſatam deſinit; vt paulo poſt explicabo. </s>
          <s id="s.001500">Vide precedentem textum 59. <lb></lb>
&amp; ca, quæ ibi annotaui, <expan abbr="quæq;">quæque</expan> tibi propoſui, &amp; plenam huius loci intelligen­<lb></lb>
tiam aſſequeris. </s>
          <s id="s.001501">vide etiam, quæ mox ſubdam circa huius loci reliquum.</s>
        </p>
        <p type="main">
          <s id="s.001502"><arrow.to.target n="marg114"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001503"><margin.target id="marg114"></margin.target>114</s>
        </p>
        <p type="main">
          <s id="s.001504">Ibidem <emph type="italics"></emph>(Circa autem eclypſes, ſemper curuam habet terminătem lineam: qua­<lb></lb>re quoniam eclypſim patitur propter terræ obiectionem, terræ <expan abbr="circumferẽtia">circumferentia</expan> ſphæ­<lb></lb>
rica exiſtens, figuræ cauſa erit)<emph.end type="italics"></emph.end> probat rotunditatem terræ ab eclypſi lunari, <lb></lb>
ex eo, quod Luna ſphæricè eclypſetur, quod innuitur illis verbis, ſemper


<pb pagenum="82" xlink:href="009/01/082.jpg"></pb>curuam habet terminantem lineam, linea ſcilicet, quæ terminat partem <lb></lb>
eclypſatam à non eclypſata, ſemper apparet circularis; cum autem hæc li­<lb></lb>
nea ſit terminus vmbræ terræ, quæ lumen obumbrat, ſignum <expan abbr="manifeſtũ">manifeſtum</expan> eſt <lb></lb>
vmbram ipſam eſſe rotundam; nam cum Luna deficiat propter terræ obie­<lb></lb>
ctionem inter ipſam, &amp; Solem, ita, vt vmbra terræ protendatur <expan abbr="vſq;">vſque</expan> ad Lu­<lb></lb>
nam, <expan abbr="eamq́">eamque</expan>; in omni eclypſatione, ſiue eclypſis ſit ſupra terram, ſiue infra, <lb></lb>
ad quamlibet <expan abbr="deniq;">denique</expan> partem terræ fiat, orbiculariter eam contegit, ſignum <lb></lb>
perſpicuum eſt terram proijcere quoquouerſus vmbram rotundam, quæ vt <lb></lb>
in ſphæra oſtenditur, eſt rotunda ad modum coni; cum ergo vmbra terræ <lb></lb>
ex quauis parte proijciatur, ſit rotunda, certò certius colligitur, <expan abbr="terramq́">terramque</expan>; <lb></lb>
<expan abbr="quoq;">quoque</expan> ipſam rotunda figura præditam eſſe. </s>
          <s id="s.001505">hanc eandem rationem, ſi libue­<lb></lb>
rit, fuſius pertractatam videre poteris apud P. Clauium in ſphæra.</s>
        </p>
        <p type="main">
          <s id="s.001506"><arrow.to.target n="marg115"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001507"><margin.target id="marg115"></margin.target>115</s>
        </p>
        <p type="main">
          <s id="s.001508">Tex. <emph type="italics"></emph>(Præterea per astrorum apparentiam, non ſolum manifeſtum eſt, quod re<lb></lb>tunda, ſed &amp; quod magnitudine non magna ſit; paruo enim facto nobis tranſitu ad <lb></lb>meridiem, &amp; Vrſam, manifeſtè fit alter horizon circulus, ita vt aſtra, quæ ſuper <lb></lb>caput, magnam habeant mutationem, &amp; non eadem appareant, &amp; ad Vrſam, &amp; ad <lb></lb>meridiem tranſeuntibus, quædam enim in Aegypto quidem stellæ <expan abbr="vidẽtur">videntur</expan>, &amp; cir­<lb></lb>
ca Cyprum, in ijs autem, quæ ad Vrſam vergunt regionibus, non <expan abbr="viaẽtur">videntur</expan>. </s>
          <s id="s.001509">&amp; aſtro­<lb></lb>
rum ea, quæ ſemper in ijs, quæ ad Vrſam vergunt, apparent, in illis locis occidunt. <lb></lb>
</s>
          <s id="s.001510">Quare non ſolum ex his manifeſtum eſt rotundam eſſe figuram terræ, ſed &amp; ſphæræ <lb></lb>
non magnæ: non enim tam celeriter inſigne quippiam faceret, tranſlatis nobis adeò <lb></lb>
parum)<emph.end type="italics"></emph.end> hic textus ei, qui ſphæram mundi audiuerit perfacilis eſt: propte­<lb></lb>
rea eum breuiter ſic paraphraſticè exponam. </s>
          <s id="s.001511">Terram eſſe rotundam, <expan abbr="atq;">atque</expan> <lb></lb>
reſpectu cœleſtium corporum non magnam, ſignum eſt, quod facto à nobis <lb></lb>
paruo itinere ſiue ad meridionalem plagam, ſiue ad <expan abbr="ſeptẽtrionalem">ſeptentrionalem</expan> (quam <lb></lb>
Vrſam dicit) magnopere mutatur horizon: quod apparet primo ex varia­<lb></lb>tione aſtrorum, nam quæ in primo loco ſupra noſtrum verticem <expan abbr="trãſibant">tranſibant</expan>, <lb></lb>
in ſecundo loco non amplius, ſed alia, <expan abbr="atq;">atque</expan> alia valde ab inuicem ſeiuncta <lb></lb>
<figure id="id.009.01.082.1.jpg" place="text" xlink:href="009/01/082/1.jpg"></figure><lb></lb>ex facto quamuis paruo itinere tranſeunt. </s>
          <s id="s.001512">ſit in <lb></lb>
præſenti figura terra, vbi A, in qua facta parua <lb></lb>
mutatione ex loco F, in locum G, fieret magna <lb></lb>
mutatio <expan abbr="aſtrorũ">aſtrorum</expan> verſicalium B, in C, quæ mul­<lb></lb>
tum ab inuicem diſtant. </s>
          <s id="s.001513">ſi autem terra eſſet <lb></lb>
maior, v. g. circulus medius, tunc facta maio­<lb></lb>
ri mutatione ex D, in E, fieret eadem aſtrorum <lb></lb>
variatio ex B, in C; ſed cum nos experiamur <lb></lb>fieri magnam aſtrorum mutationem, ex parua <lb></lb>
locorum intercapedine, ſignum eſt magnope­<lb></lb>
re mutari horizontem, ac proinde terram eſſe <lb></lb>
rotundam, ac reſpectu cœleſtium corporum <lb></lb>
paruam. </s>
          <s id="s.001514">aliud præterea ſignum huius horizontis permutationis eſt, quod <lb></lb>
ſtellæ, quæ in priori loco ſupra horizontem apparebant, mutato paululum <lb></lb>
loco ad alterutram plagam, ſtatim abſconduntur; aliæ verò nouæ <expan abbr="apparẽt">apparent</expan> <lb></lb>vt in Aegypto, &amp; Cypro, ſtella, quæ dicitur Canobus ſupra horizontem <lb></lb>
aſcendit; quæ ſi paululum Vrſam, ſeu ſeptentrionem ambulaueris, ſtatim <lb></lb>
latitabit. </s>
          <s id="s.001515">Demum eiuſdem citæ mutationis finitoris indicium etiam ſit,<pb pagenum="83" xlink:href="009/01/083.jpg"></pb>quod regiones ſeptentrionales incolentibus plurima ſunt aſtra, quæ nun­<lb></lb>quam occidunt, quamuis horizontem leuiter perſtringant, quæ tamen Cy­<lb></lb>
prijs, <expan abbr="atq;">atque</expan> Aegyptijs oriuntur, <expan abbr="atq;">atque</expan> occidunt. </s>
          <s id="s.001516">ex quibus &amp; rotunditas, &amp; <lb></lb>
paruitas terræ colligi poteſt. </s>
          <s id="s.001517">has eaſdem rationes fuſius explicatas repe­<lb></lb>
ries apud P. Clauium in ſphæra.</s>
        </p>
        <p type="main">
          <s id="s.001518"><arrow.to.target n="marg116"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001519"><margin.target id="marg116"></margin.target>116</s>
        </p>
        <p type="main">
          <s id="s.001520">Tex. 111. <emph type="italics"></emph>(Quapropter existimantes eum, qui circa Herculeas columnas eſt lo­<lb></lb>cum coniungi ei, qui circa Indiam, &amp; hoc modo mare vnum eſſe, non admodum <lb></lb>
incredibilia exiſtimare videntur &amp;c.)<emph.end type="italics"></emph.end> exiſtimatores hoſce non perperam exi­<lb></lb>
ſtimaſſe apertè <expan abbr="cõuincunt">conuincunt</expan> Chriſtophori Columbi, Argonautarum principis <lb></lb>
nauigationes; quibus nouus orbis repertus eſt, qui inter columnas Hercu­<lb></lb>
lis, <expan abbr="atq;">atque</expan> orientalem Indiam totus vna <expan abbr="cũ">cum</expan> mari Oceano Atlantico interiacet.</s>
        </p>
        <p type="main">
          <s id="s.001521"><arrow.to.target n="marg117"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001522"><margin.target id="marg117"></margin.target>117</s>
        </p>
        <p type="main">
          <s id="s.001523">Tex. 112. <emph type="italics"></emph>(Mathematicorum etiam, qui circumferentiæ magnitudinem ratio­<lb></lb>cinari tentant, ad 400. dicunt ſtadiorum millia, &amp;c.)<emph.end type="italics"></emph.end> quam ſubtilibus rationi­<lb></lb>
bus inueſtigauerint Aſtronomi quantitatem terræ, optimè, ac dilucidè ex­<lb></lb>
ponitur à P. Clauio in ſphæra: quem ſi libet, conſule, ne inani labore opu­<lb></lb>
ſculum iſtud exereſcat.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001524"><emph type="italics"></emph>Ex Tertio de Cœlo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001525"><arrow.to.target n="marg118"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001526"><margin.target id="marg118"></margin.target>118</s>
        </p>
        <p type="main">
          <s id="s.001527">Tex. 40. <emph type="italics"></emph>(Figuræ autem omnes componuntur ex pyramidibus: rectilinea <lb></lb>quidem ex rectilineis: ſphæra verò ex octo partibus componitur)<emph.end type="italics"></emph.end> Ale­<lb></lb>xander exiſtimat, Ariſtotelem dicere ſphæram conſtare ex octo <lb></lb>
partibus illis, quæ deſignantur per tres circulos, quorum duo ſe­<lb></lb>cant ſe mutuò ad angulos rectos, vt in ſphæra mundi faciunt duo coluri; <lb></lb>
tertius verò medios illos diuidit æquidiſtanter à ſectionibus <expan abbr="illorũ">illorum</expan> mutuis, <lb></lb>
quemadmodum æquator in ſphæra mundi ſecat duos coluros. </s>
          <s id="s.001528">ex quibus ſe­<lb></lb>
ctionibus tota ſphæra in octo partes diuiditur, quibus ſphæram componi <lb></lb>
vult Ariſtoteles. </s>
          <s id="s.001529">aduerte tamen hanc ſphæræ compoſitionem nullo modo <lb></lb>
habere partes actu, cum ſphæra ſit vnica ſimplici ſuperficie terminata; ſed <lb></lb>
quæ tantum ſint à prædictis imaginatis circulis deſignatæ: at verò aliæ fi­<lb></lb>
guræ, quæ pluribus planis terminantur, vt cubus, octaedrum, &amp; ſimilia, quæ <lb></lb>Ariſt. vocat rectilineas, quia terminantur ſuperficiebus rectilineis actu di­<lb></lb>
ſtinctis ab inuicem ex natura ſua, non per noſtram deſignationem, ideò re­<lb></lb>
ctè dicuntur componi ex pyramidibus, v. g. dicimus cubum componi ex ſex <lb></lb>
pyramidibus, quia cum habeat ſex baſes, cogitamus ſupra <expan abbr="vnamquamq;">vnamquamque</expan> il­<lb></lb>
larum ſingulas pyramides erigi, quarum omnium vertices ad idem punctum <lb></lb>
medium intra cubum imaginatum coeant. </s>
          <s id="s.001530">&amp; ſic de reliquis ſolidis. </s>
          <s id="s.001531">quæ qua <lb></lb>
ratione reſoluantur in plures pyramides, conſtat ex 10. 11. 12. &amp; 13. Ele­<lb></lb>
mentorum Euclidis, at verò in ſphæra nullum reale compoſitionis, aut di­<lb></lb>
uiſionis fundamentum reperitur.</s>
        </p>
        <p type="main">
          <s id="s.001532"><arrow.to.target n="marg119"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001533"><margin.target id="marg119"></margin.target>119</s>
        </p>
        <p type="main">
          <s id="s.001534">Tex. <emph type="italics"></emph>(Ad hæc neceſſe eſt non omne corpus eſſe diuiſibile dicere, ſed repugnare <lb></lb>certiſſimis ſcientijs; nam Mathematicæ ipſum quidem intelligibile, accipiunt diui­<lb></lb>
ſibile)<emph.end type="italics"></emph.end> ipſum intelligibile, ideſt, quantitatem abſtractam tam continuam, <lb></lb>quam diſcretam, quam ſtatuunt Philoſophi eſſe ſubiectam materiam ma­<lb></lb>
thematicarum. </s>
          <s id="s.001535">quam ideo appellant intelligibilem, quia cum ſit abſtracta <lb></lb>
per intellectum à ſenſibilibus affectionibus, reſtat vt ſit tantummodo intel­


<pb pagenum="84" xlink:href="009/01/084.jpg"></pb>lectu perceptibilis. </s>
          <s id="s.001536">Hanc eandem ſupponunt eſſe diuiſibilem in infinitum, <lb></lb>
vt ſupra 3. Phyſ. textu 31. dictum eſt.</s>
        </p>
        <p type="main">
          <s id="s.001537"><arrow.to.target n="marg120"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001538"><margin.target id="marg120"></margin.target>120</s>
        </p>
        <p type="main">
          <s id="s.001539">Tex. 66. <emph type="italics"></emph>(Omninò autem eniti ſimplicibus corporibus figuras tribuere irratio­<lb></lb>
nabile eſt. </s>
          <s id="s.001540">primò quidem, quia accidit non repleri totum; nam in planis tres figuræ <lb></lb>
videntur implere locum, Triangulus, Quadratum, &amp; Sexangulus)<emph.end type="italics"></emph.end> per ſimplicia <lb></lb>
corpora intelligit quatuor elementa. </s>
          <s id="s.001541">Vult enim probare quatuor elemen­<lb></lb>
ta non habere figuras illas mathematicas, quas illis Plato tribuebat, vt au­<lb></lb>
tem Ariſt. rationem probè percipiamus, ſciendum, quod implere totum, <lb></lb>
ſiue locum, illæ figuræ dicuntur, quæ ſimul ſuis angulis in plano quopiam ad <lb></lb>
vnum, <expan abbr="atq;">atque</expan> idem punctum vnitæ locum illum totum, qui circa punctum il­<lb></lb>
lud conſiſtit, <expan abbr="cõtegunt">contegunt</expan>, ita vt nihil vacui inter ipſas relinquatur. </s>
          <s id="s.001542">tales ſunt, <lb></lb>
quibus fieri poſſunt pauimenta, oportet enim, vt ſimul vnitæ nihil vacui in <lb></lb>
pauimento relinquant. </s>
          <s id="s.001543">huiuſmodi ſunt triangula æquilatera (de his enim <lb></lb>
intelligendus eſt textus) quadrata, &amp; hexagona, ſiue ſexilatera regularia; <lb></lb>
<figure id="id.009.01.084.1.jpg" place="text" xlink:href="009/01/084/1.jpg"></figure><lb></lb>
nam ſex triangula æquilatera ſimul iuncta in plano paui­<lb></lb>
re poſſunt, vt patet in figura præſenti; ratio huius eſt, <lb></lb>
quia omnes anguli circa idem punctum (y. </s>
          <s id="s.001544">g. A, in hac <lb></lb>
figura) in plano, quotquot fuerint conſtituti, ſunt æqua­<lb></lb>
les quatuor rectis, ex coroll. </s>
          <s id="s.001545">ſecundo 15. primi Elemen­<lb></lb>
ti: cum igitur ſex anguli, trianguli æquilateri <expan abbr="æquiualeãt">æquiualeant</expan> <lb></lb>
quatuor rectis angulis, conſtituti omnes circa punctum <lb></lb>
A, totum locum circa illud implere poſſunt. </s>
          <s id="s.001546">Quadratum etiam replere lo­<lb></lb>
<figure id="id.009.01.084.2.jpg" place="text" xlink:href="009/01/084/2.jpg"></figure><lb></lb>cum manifeſtum eſt, cum enim ipſius anguli ſint recti, ſi <lb></lb>
quatuor quadrata ad idem punctum A, copulentur, vt in <lb></lb>
figura apparet, replebunt eadem de cauſa vacuum.</s>
        </p>
        <p type="main">
          <s id="s.001547">Hexagonum quoque regulare, ideſt æquilaterum, &amp; <lb></lb>
æquiangulum idem præſtare poteſt; cum enim tres angu­<lb></lb>
li ipſius æquiualeant quatuor rectis, ſi tria hexagona ad <lb></lb>
idem punctum A, vt in figura adaptentur, neceſſariò ni­<lb></lb>
hil vacui inter ipſa relinquetur, vt in figura hac oſtenditur. </s>
          <s id="s.001548">præter has tres <lb></lb>
<figure id="id.009.01.084.3.jpg" place="text" xlink:href="009/01/084/3.jpg"></figure><lb></lb>figuras, nulla alia reperitur, quæ iſtud efficere poſ­<lb></lb>
ſit. </s>
          <s id="s.001549">cuius demonſtrationem perfectam videre pote­<lb></lb>
ris in fine commentarij P. Clauij ſuper 4. Elem. nos <lb></lb>
ea tantum attingimus, quæ percipi poſſint ab homi­<lb></lb>
ne vix mathematicis tincto: ſed tamen, quæ ſenſum <lb></lb>
Ariſtotelis patefaciunt. </s>
          <s id="s.001550">Aliæ porrò figuræ replen­<lb></lb>
tes locum planum, quibus aliquando Architectores <lb></lb>
vtuntur, vel ſunt irregulares, vel ad prædictas redu­<lb></lb>
ci poſſunt. </s>
          <s id="s.001551">cum igitur tres tantum ex figuris planis <lb></lb>
totum repleant, hæ ſolæ poterunt elementis attri­<lb></lb>
bui, ac propterea non ſufficient, niſi pro tribus elementis. </s>
          <s id="s.001552">quare quartum <lb></lb>
<expan abbr="abſq;">abſque</expan> figura relinquetur; quod eſt abſurdum.</s>
        </p>
        <pb pagenum="85" xlink:href="009/01/085.jpg"></pb>
        <p type="head">
          <s id="s.001553"><emph type="italics"></emph>Admirabilis quædam Ap̀um industria.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001554">Cæterum occaſione harum figurarum illud hoc loco apponere vi­<lb></lb>
ſum eſt, quod Pappus <expan abbr="Alexãdrinus">Alexandrinus</expan> initio quinti libri collectionum <lb></lb>
mathematicarum ſcribit, De admirabili Apum induſtria, atque <lb></lb>
prudentia in conſtruendo ſuas cellulas figura hexagona regulari. <lb></lb>
</s>
          <s id="s.001555">cum enim vellent omne vacuum excludere, &amp; præterea capaciſſimam <expan abbr="om-niũ">om­<lb></lb>
nium</expan> figuram habere, hexagonam accepere, quæ inter prædictas tres vtrum­<lb></lb>
que præſtat, nam &amp; inane omne excludit, &amp; illarum trium capaciſſima eſt, <lb></lb>
cum magis ad circularem figuram accedat: vt patet ex tractatu de figuris <lb></lb>Iſoperimetris, qui eſt apud Clauium in ſphæra, necnon in Geometria pra­<lb></lb>
ctica. </s>
          <s id="s.001556">hoc ideò libentius recenſui, quia animaduerti naturales hiſtoriogra­<lb></lb>
phos omnes latere, vel ipſum Aldobrandum noſtrum, qui quamuis indu­<lb></lb>
ſtrioſæ Apis inſtar omnia delibauerit, iſtud tamen de Apibus artificium tan­<lb></lb>
ta ſapientia plenum, neſcio quo modo prætermiſit.</s>
        </p>
        <p type="main">
          <s id="s.001557"><arrow.to.target n="marg121"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001558"><margin.target id="marg121"></margin.target>121</s>
        </p>
        <p type="main">
          <s id="s.001559">Ibidem <emph type="italics"></emph>(In ſolidis verò duæ ſolum pyramis, &amp; cubus)<emph.end type="italics"></emph.end> ideſt replent locum <lb></lb>
ſolidum. </s>
          <s id="s.001560">nullum reperi, qui in hoc loco explicando non errauerit; nam Græ­<lb></lb>
ci, qui alioqui ſolent mathematica probè intelligere, hic omnes lapſi ſunt, <lb></lb>
<expan abbr="ſecumq́">ſecumque</expan>; &amp; Arabes, &amp; Latinos in <expan abbr="eãdem">eandem</expan> foueam ſupra ſe miſerè traxerunt. <lb></lb>
</s>
          <s id="s.001561">communis ferè error omnium fuit, pyramides plures ſimul compactas poſ­<lb></lb>
ſe replere ſolidum locum. </s>
          <s id="s.001562">quod vt melius intelligamus, ſciendum eſt, reple­<lb></lb>
re locum <expan abbr="ſolidũ">ſolidum</expan> nihil aliud eſſe, quam ſi plura corpora ſolida ſimul ad idem <lb></lb>
punctum coaptata, ita conſtipentur, vt totum ſpatium, quod eſt circa pun­<lb></lb>
ctum illud omninò occupent, hoc eſt, nihil vacui inter ipſa relinquatur: ſi­<lb></lb>
cut enim prædictæ tres figuræ planæ, de quibus paulò ante, replent locum <lb></lb>planum, ideſt ſuperficiem; ita cubi replent ſolidum, ideſt ſoliditatem ſimul <lb></lb>
vniti conſtituunt, ita vt ſi octo cubi ſimul ad idem punctum <expan abbr="coaptẽtur">coaptentur</expan>, con­<lb></lb>ſtituant corpus ſolidum ex octo illius conſtatum, <expan abbr="nihilq́">nihilque</expan>; inane inter ipſos <lb></lb>
cubos relinquatur. </s>
          <s id="s.001563">&amp; ſicuti planæ illæ figuræ erant conficiendis pauimentis <lb></lb>
aptæ, ita ſolidæ hæ muris, qui corpora ſunt ſolida, <expan abbr="conſtruẽdis">conſtruendis</expan> idonea ſunt. <lb></lb>
</s>
          <s id="s.001564"><expan abbr="Notã">Notam</expan> dum præterea, quod per pyramidem debemus intelligere pyramidem <lb></lb>
regularem, quæ dicitur etiam Tetraedrum, <expan abbr="eſtq́">eſtque</expan>; ſecunda inter <expan abbr="quinq;">quinque</expan> cor­<lb></lb>
pora regularia rectilinea, quæ alias Platonica corpora dicuntur. </s>
          <s id="s.001565"><expan abbr="eorumq́">eorumque</expan>; <lb></lb>definitiones ſunt in 11. Elem. </s>
          <s id="s.001566">Tetraedrum autem ſic definitur, eſt figura ſo­<lb></lb>
lida ſub quatuor triangulis æquilateris, <expan abbr="atq;">atque</expan> inuicem æqualibus contenta: <lb></lb>
de hac inquam eſt ſermo. </s>
          <s id="s.001567">quia ſi liceret intelligere de irregularibus figuris, <lb></lb>infinitæ reperirentur figuræ tam planæ, quam ſolidæ, quæ vtrumque locum <lb></lb>
complerent. </s>
          <s id="s.001568">Aduertendum tandem Ariſt. videri loqui de repletione loci <lb></lb>
ſolidi, quia tranſit à planïs figuris ad ſolidas. </s>
          <s id="s.001569">&amp; quia ſi hæ duæ pyramis, &amp; <lb></lb>cubus replent locum ſolummodo ſecundum ſuas ſuperficies, quæ ſunt trian­<lb></lb>
gulum, &amp; quadratum, iam de his cum proximè ante dixiſſet, quid opus fuiſ­<lb></lb>
ſet idem poſt modum repetere. </s>
          <s id="s.001570">ad hæc ſi in medium ſolida hæc duo profert, <lb></lb>
<expan abbr="aitq́">aitque</expan>; ipſa replere locum, intelligens, planum, profectò non loquitur forma­<lb></lb>liter, ideſt de ipſis, vt ſolida ſunt. </s>
          <s id="s.001571">Quare Ariſt. videretur ſibi non conſtare, <lb></lb>
vel perperam exiſtimaſſe plura Tetraedra complere ſoliditatem. </s>
          <s id="s.001572">deceptus <pb pagenum="86" xlink:href="009/01/086.jpg"></pb>fortè fuit Ariſt. eò quod videret Icoſaedrum conſtare ex viginti pyramidi­<lb></lb>
bus, verùm illæ non ſunt regulares, ideſt <expan abbr="nõ">non</expan> ſunt Tetraedra, vt poſtea oſten­<lb></lb>
dam. </s>
          <s id="s.001573">Verum quidem eſt octo cubos ſimul adactos ſoliditatem conficere, <lb></lb>
quia ad id neceſſarij ſunt octo anguli ſolidi, quos octo cubi præbere poſſunt, <lb></lb>
cum anguli ipſorum ſint recti, &amp; ſolidi. </s>
          <s id="s.001574">Verum enim verò plures pyramides <lb></lb>
regulares, ſiue plura Tetraedra non poſſe replere vacuum, <expan abbr="ſolidumq́">ſolidumque</expan>; con­<lb></lb>
ſtituere, ex eo patet, quia ſi id præſtarent, conflarent neceſſariò, vel vnum <lb></lb>
ex <expan abbr="quinq;">quinque</expan> corporibus regularibus, de quibus in 13. Elemen. vel aliud quod­<lb></lb>
piam; non aliud, nam, vt patet ex ſcholio 13. Elem. non dantur, niſi illa. <lb></lb>
</s>
          <s id="s.001575">quinque; <expan abbr="neq;">neque</expan> vllum ex illis, quia diameter huiuſmodi corporis, quod com­<lb></lb>
poneretur ex illis pyramidibus, eſſet dupla lateris eiuſdem, vt patet, quia <lb></lb>
pyramides illæ omnes concurrerent ad centrum ſphæræ illas omnes com­<lb></lb>
plectentis, quare latus vnius pyramidis à ſuperficie ſphæræ incipiens deſi­<lb></lb>
neret in centrum, ergo latus iſtud eſſet ſemidiameter, quapropter tota dia­<lb></lb>meter illius ſphęræ, &amp; conſequenter huius corporis in illa inſcripti, eſſet du­<lb></lb>
pla lateris eiuſdem figuræ ſolidæ inſcriptæ, ſed nullo talis proportio diame­<lb></lb>
tri alicuius ex illis <expan abbr="quinq;">quinque</expan> ſolidis regularibus ad latus eiuſdem reperitur, <lb></lb>
quæ ſit nimirum dupla, vt patet ex vltimis demonſtrationibus 13. Elem. ini­<lb></lb>
tio facto à 13. demonſtratione, in quibus nulla reperitur proportio dupla <lb></lb>
inter diametrum, &amp; latus eiuſdem alicuius ex illis ſolidis; ex quibus mani­<lb></lb>
feſtum eſt, plures regulares pyramides quouis pacto ſimul vnitas nullo mo­<lb></lb>
do replere locum ſolidum. </s>
          <s id="s.001576">cum igitur animaduerterem, ſenſum Ariſt. nullo <lb></lb>
modo poſſe verificari de repletione ſolidi per plura Tetraedra, &amp; omnes <lb></lb>
tamen commentatores auctoritate Ariſt. decepti pro ipſo ſtarent, dubius, <lb></lb>
<expan abbr="ancepsq́">ancepsque</expan>; diu hæſi, neque quid quam mea Minerua aſſerere auſus ſum, ſed P. <lb></lb>
Clauium præceptorem meum per literas conſului, qui in hunc modum hu­<lb></lb>
maniſſimè reſpondit; cubus implet locum quater ſumptus, ad idem enim <lb></lb>
punctum quatuor cubi coaptantur: ſic etiam pyramis ſexies ſumpta, ſeu ſex <lb></lb>pyramides ad idem punctum iunctæ ratione ſubſtantium <expan abbr="triangulorũ">triangulorum</expan> æqui­<lb></lb>
laterorum. </s>
          <s id="s.001577">Verum hac ratione non videntur implere locum ſolidum, fa­<lb></lb>teor; ſed tamen Ariſt. in eo tex. non loquitur de repletione loci ſolidi. </s>
          <s id="s.001578">hæc <lb></lb>
ipſe. </s>
          <s id="s.001579">ſi igitur libeat Ariſtotelem, quod fortè Clauius intendebat defendere, <lb></lb>
dicendum eſt cum eo Ariſt non loqui de repletione loci ſolidi: <expan abbr="neq;">neque</expan> loqui <lb></lb>
de cubo, &amp; Tetraedro, quatenus ſunt corpora, ſed quatenus habent ſuper­<lb></lb>
ficies, cubus quidem ſex quadratas, Tetraedrum autem quatuor æquilate­<lb></lb>
ras ſuperficies, quæ duæ figuræ, vt ſupra in hoc textu vidimus, replent lo­<lb></lb>
cum: <expan abbr="atq;">atque</expan> hoc modo facimus Ariſtotelem non formaliter loquentem. </s>
          <s id="s.001580">ex­ <lb></lb>
aduersò ne videamur magis Ariſt. quam veritatem ſequi, videtur dicen­<lb></lb>
dum, Ariſtotilem formaliter locutum eſſe, &amp; vt patet ex rationibus ſupra <lb></lb>
allatis de repletione ſolidi eſſe intelligendum, vt etiam intellexerunt omnes <lb></lb>
huius loci expoſitores; Verumtamen ipſum erraſſe, dum plures pyramides <lb></lb>
replere ſolidum exiſtimauit. </s>
          <s id="s.001581">Vtrumuis dixerimus, non tamen Ariſt. ab om­<lb></lb>ni errore vindicabimus. </s>
          <s id="s.001582">Hoc tamen certum eſt, ex prædictis, Græcos om­<lb></lb>
nes pariter, ac Latinos, illos ſequentes, lapos eſſe, aſſerentes duodecim py­<lb></lb>
ramides complere ſolidum locum, <expan abbr="atq;">atque</expan> Dodecaedrum conſtituere; nam py­<lb></lb>
ramides Dodecaedron conſtituentes non ſunt regulares, ideſt, non ſunt Te­


<pb pagenum="87" xlink:href="009/01/087.jpg"></pb>traedra (de quibus tamen Ariſt. loquitur) vt patet ex ſupra dictis. </s>
          <s id="s.001583">Indul­<lb></lb>
geas Lector, ſi hoc loco neceſſe fuit in Geometriæ penetralia ingredi: ope­<lb></lb>
ræpretium enim eſt aliquando ipſis Mathematicis ſatisfacere. </s>
          <s id="s.001584">tu verò, ſi <lb></lb>
adeo es mathematicis imbutus, conſule poſtremas demonſtra. </s>
          <s id="s.001585">13. Elem. &amp; <lb></lb>
præcipuè ſcholium vltimum, vbi plura de his corporibus ſcitu digniſſima, <lb></lb>
<expan abbr="atq;">atque</expan> huc ſpectantia reperies ex his omnibus Mathematica, quæ noſtræ ſunt <lb></lb>
partes, perſpicuè ſatis expoſuimus.</s>
        </p>
        <p type="main">
          <s id="s.001586">Multo poſt tempore, quàm hæc ſcripſeram incidi fortè in cap. 38. ſpecu­<lb></lb>
lationem 10. Benedicti de placitis Ariſt. <expan abbr="reperiq́">reperique</expan>; ab eo vno Ariſt. hoc loco <lb></lb>
erroris notari, dum aſſeruit duodecim pyramides replere <expan abbr="locũ">locum</expan> corporeum, <lb></lb>
ideſt, vt exponit ipſe, ſex pyramides ſuper hexagonam aliquam figuram <lb></lb>
ſuperficialem, &amp; ſex ſub eadem, id præſtarent, cum potius maius vacuum <lb></lb>
remaneat ad quamlibet partium ſupra, &amp; infra, quam plenum. </s>
          <s id="s.001587">hæc ipſe. </s>
          <s id="s.001588">ſed <lb></lb>
expoſitio iſta puerili, ne dum Ariſt. ingenio prorſus indigna eſt: vt propte­<lb></lb>
rea exiſtimem caſu potius eum Ariſt. rectè reprehendiſſe, quam ex certa <lb></lb>
ſcientia, cum illius erratum maiori errato conetur corrigere. </s>
          <s id="s.001589">Incidi po­<lb></lb>
ſtremò in Indicem librorum, quem Maurolyius ſuæ Coſmographiæ præpo­<lb></lb>
nit, vbi ſic ait: Demonſtramus autem in libello de figuris planis, <expan abbr="ſolidisq́">ſolidisque</expan>; <lb></lb>locum replentibus, cubos per ſe, pyramides verò cum octaedris compactas <lb></lb>
dumtaxat implere locum, qua in re Auerroem erraſſe pueriliter manifeſtum <lb></lb>
erit. </s>
          <s id="s.001590">Vides igitur tanti viri auctoritate confirmari noſtram ſententiam, py­<lb></lb>
ramides videlicet per ſe, non replere vacuum. </s>
          <s id="s.001591">cum igitur conſtet vnam tan­<lb></lb>
tum ex figuris ſolidis, ſiue etiam dicas, vt perperam Ariſt. &amp; alij plures exi­<lb></lb>
ſtimarunt, replere totum ſolidum; nulla ratione poterunt <expan abbr="elemẽta">elementa</expan> quatuor, <lb></lb>
quatuor diuerſis figuris indui, ſed vnum tantummodo, quare reliqua <expan abbr="abſq;">abſque</expan> <lb></lb>
figura remanere neceſſe eſſet: quod eſt omnino inconueniens.</s>
        </p>
        <p type="main">
          <s id="s.001592"><arrow.to.target n="marg122"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001593"><margin.target id="marg122"></margin.target>122</s>
        </p>
        <p type="main">
          <s id="s.001594">Tex. 71 <emph type="italics"></emph>(Deinde ſi terra eſt cubus &amp;c.)<emph.end type="italics"></emph.end> lege definitiones 11. Elem. quæ ſunt <lb></lb>admodum faciles, ibi reperies definitiones quinque corporum regularium, <lb></lb>
quorum figuras Plato elementis tribuebat: qua verò id ratione faceret, ha­<lb></lb>
bes in ſphæra Clau. </s>
          <s id="s.001595">Simpl. etiam hoc loco ſatisfacit.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001596"><emph type="italics"></emph>Ex Quarto de Cœlo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001597"><arrow.to.target n="marg123"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001598"><margin.target id="marg123"></margin.target>123</s>
        </p>
        <p type="main">
          <s id="s.001599">Tex. 33. <emph type="italics"></emph>(Deinde ad ſimiles videtur angulos ignis quidem ſurſum ferri, <lb></lb>terra autem deorſum, &amp; omninò quod grauitatem habet, quare neceſſe <lb></lb>
est ferri ad medium. </s>
          <s id="s.001600">hoc autem vtrum accidit ad ipſum terræ medium, <lb></lb>
an ad vniuerſi, quoniam idem ipſorum ſit, alius ſermo eſt)<emph.end type="italics"></emph.end> cum vellet <lb></lb>
<figure id="id.009.01.087.1.jpg" place="text" xlink:href="009/01/087/1.jpg"></figure><lb></lb>
probare Ariſtoteles dari <expan abbr="pũctum">punctum</expan> quoddam in medio <lb></lb>
mundi, ad quod grauia deſcendant, &amp; concurrent: <lb></lb>
&amp; à quo leuia aſcendat; vtitur, præter alias, etiam <lb></lb>
ratione aliqua ex parte mathematica; quæ eſt huiuſ­<lb></lb>
modi. </s>
          <s id="s.001601">videmus ignem, &amp; cætera lęuia aſcendere à <lb></lb>
terra ſurſum ad angulos æquales; ſimiliter videmus <lb></lb>terram, &amp; cętera grauia deſcendere ad terram deor­<lb></lb>
ſum ad angulos æquales, quod ſignum eſt omnia iſta <lb></lb>
idem mundi medium reſpicere: v.g. ſit terra in figu­<lb></lb>
ra præſenti circulus E C D, cuius medium, ſine cen­


<pb pagenum="88" xlink:href="009/01/088.jpg"></pb>trum A. via, qua aſcendit ignis ſit in linea A C B, quæ facit angulos in ſu­<lb></lb>
perficie terræ æquales, nimirum angulos B C D, B C E. ſimiliter terra per <lb></lb>eandem lineam <expan abbr="faciẽs">faciens</expan> eoſdem angulos æquales deſcendit. </s>
          <s id="s.001602">linea autem, quæ <lb></lb>
facit tales angulos tendit ad centrum ſphæræ A, vt patet ad ſenſum in figu­<lb></lb>
ra, &amp; probari poteſt geometricè ex primis tertij Elem. ex quibus patet tam <lb></lb>
læuia, quam grauia, quæ per talem lineam ferantur, reſpicere centrum A, <lb></lb>
ſphæræ. </s>
          <s id="s.001603">Vtrum autem iſtud centrum ſit idem cum <expan abbr="cẽtro">centro</expan> totius mundi, alius, <lb></lb>
inquit, eſt ſermo, hoc eſt, ad aſtronomum pertinet. </s>
          <s id="s.001604">vide igitur hac de re <lb></lb>
pulchram deſſertationem apud Clauium in ſphæra: qui probat euidenter <lb></lb>
eſſe vnum, &amp; idem.</s>
        </p>
        <p type="main">
          <s id="s.001605"><arrow.to.target n="marg124"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001606"><margin.target id="marg124"></margin.target>124</s>
        </p>
        <p type="main">
          <s id="s.001607"><emph type="italics"></emph>Hoc loco deſideratur commentarius in cap. vlt. </s>
          <s id="s.001608">de Cœlo. </s>
          <s id="s.001609">cuius loco ìn-<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001610"><arrow.to.target n="marg125"></arrow.to.target><lb></lb>
<arrow.to.target n="marg126"></arrow.to.target><lb></lb>
<emph type="italics"></emph>terim Lector adeat Diſcurſum Italicum Galilæi Galilæi, de his,<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="marg127"></arrow.to.target><lb></lb>
<emph type="italics"></emph>quæ in aqua mouentur, ac natant: ubi propè finem, plura in hu-<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="marg128"></arrow.to.target><lb></lb>
<emph type="italics"></emph>ius capitis explicationem affert.<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="marg129"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001611"><margin.target id="marg125"></margin.target>125</s>
        </p>
        <p type="margin">
          <s id="s.001612"><margin.target id="marg126"></margin.target>126</s>
        </p>
        <p type="margin">
          <s id="s.001613"><margin.target id="marg127"></margin.target>127</s>
        </p>
        <p type="margin">
          <s id="s.001614"><margin.target id="marg128"></margin.target>128</s>
        </p>
        <p type="margin">
          <s id="s.001615"><margin.target id="marg129"></margin.target>129</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001616"><emph type="italics"></emph>Ex Lib. 2. de Generatione, &amp; Corruptione.<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="marg130"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001617"><margin.target id="marg130"></margin.target>130</s>
        </p>
        <p type="main">
          <s id="s.001618">Tex. 56. <emph type="italics"></emph>(ldeoqué non prima latio cauſa Generationis, &amp; Corruptionis eſt, <lb></lb>
ſed quæ ſecundum obliquum circulum, in hac enim &amp; continuum vnum <lb></lb>
eſt &amp; moueri duobus motibus)<emph.end type="italics"></emph.end> per primam lationem intelligit mo­<lb></lb>tum primi mobilis, qui ſit ſuper polis mundi, quo Stellæ omnes <lb></lb>
ab oriente in occidentem rectà feruntur. </s>
          <s id="s.001619">per obliquum verò circulum in­<lb></lb>
telligit Zodiacum, qui obliquus eſt, quia poli eius ſunt alij à polis mundi, &amp; <lb></lb>
quia non tendit rectà ab ortu ad occaſum, ſed in ſphæra mundi tranſuer­<lb></lb>
ſus eſt, &amp; deflectit à ſeptentrione in meridiem, quamuis non rectà, vt in <lb></lb>
ſphæra explicari ſolet. </s>
          <s id="s.001620">motus ergo Planetarum, qui fit ſecundum hunc cir­<lb></lb>
culum, &amp; ipſe obliquus, &amp; tranſuerſus codem modo erit; ferrentur que per <lb></lb>
eum à Borea ad Auſtrum, &amp; è conuerſo; ex quo acceſſu, &amp; receſſu efficiunt <lb></lb>
æſtatem, &amp; hyemem, item generationes, &amp; corruptiones. </s>
          <s id="s.001621">Sol porrò, &amp; pla­<lb></lb>
netæ, qui motibus proprijs hunc circulum peragunt, dicuntur moueri duo­<lb></lb>
bus motibus, &amp; quidem contrarijs: quoniam dum Sol. </s>
          <s id="s.001622">v. g. per Zodiacum <lb></lb>
graditur motu proprio, interim etiam à primo mobili fertur ab ortu in oc­<lb></lb>
caſum: ex quibus duobus motibus fit vnus tantum Solis motus ſpiralis, qui <lb></lb>
mixtus eſt, ideſt, qui fit à duobus motoribus; vnde re vera Sol non mouetur <lb></lb>
duobus motibus contrarijs re ipſa diſtinctis; hoc enim impoſſibile eſt: ſed <lb></lb>
motu mixto ex duobus, qui ſpiralis eſt, circa mundum deſcribens ſpiras ab <lb></lb>
vno tropico ad alterum: qui, vt dixi, cauſatur à duobus motoribus, qui ſunt <lb></lb>
Sol ipſe, mouens ſe ipſum per Zodiacum: &amp; primum mobile mouens inſu­<lb></lb>
per ipſum Solem, &amp; Zodiacum ab ortu in occaſum circa mundum.</s>
        </p>
      </chap>
      <pb pagenum="89" xlink:href="009/01/089.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.001623"><emph type="italics"></emph>EX PRIMO METEORORVM.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001624"><arrow.to.target n="marg131"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001625"><margin.target id="marg131"></margin.target>131</s>
        </p>
        <p type="main">
          <s id="s.001626">Svmma 1. cap. 3. <emph type="italics"></emph>(Moles autem terræ quanta ſit ad ambientes magnitudi­<lb></lb>nes, non immanifestum, iam enim viſum est per aſtrologica theoremata, <lb></lb>
quod multò etiam quibuſdam aſtris est minor)<emph.end type="italics"></emph.end> Quantitas terræ non ſo­<lb></lb>
lum abſolutè conſiderata, ab Aſtronomis explorata habetur, vt vi­<lb></lb>
dere eſt in ſphæra Clauij; ſed etiam reſpectiuè conſiderata, ideſt reſpectu <lb></lb>
aliorum elementorum, &amp; ipſorum etiam aſtrorum; cuius demonſtrationes <lb></lb>
ſunt partim in libello Ariſtarchi Samij, de magnitudine, &amp; diſtantia Solis, <lb></lb>
&amp; Lunæ, partim apud Ptolæmeum in magna Syntaxi, ſiue Almageſto: par­<lb></lb>
tim apud Albategnium de ſcientia ſtellarum: partim demum apud Ticho­<lb></lb>
nem Brahe. </s>
          <s id="s.001627">Porrò facile eſt demonſtrare Solem eſſe terra multò maiorem, <lb></lb>
terram verò maiorem Luna, <expan abbr="idq́">idque</expan>; ex eclypſi lunari, cuius imaginem habes <lb></lb>
in figura ſequenti; vbi vmbra terræ eſt D B E, in quam Luna nigricans im­<lb></lb>
mergitur, ac lumine deficit, reliqua cognitu ſunt facilia: quia igitur Aſtro­<lb></lb>
nomi obſeruarunt vmbram terræ paulò ſupra Lunam pertingere, cum ſupe­<lb></lb>
riora aſtra non adeat, hinc collegerunt eam neceſſariò eſſe acuminatam, ſeu <lb></lb>
conicam, vt figura refert. </s>
          <s id="s.001628">Cum ergo terra vmbram proijciat turbinatam, <lb></lb>
neceſſariò corpus Solis, quod ipſam illuminat, eadem maior erit: quoti­<lb></lb>
diana enim experientia docemur, corpore illuminante exiſtente maiore <lb></lb>
quà ſit illuminatum, vmbram proijci faſtigiatam: cum deinde Solem val­<lb></lb>
de a terra diſtare certum ſit, optimè infertur, eum reſpectu terræ eſſe maxi­<lb></lb>
mum: quanto enim duæ lineæ, ſiue radij B A, B C. à terra ad partes Solis <lb></lb>
<figure id="id.009.01.089.1.jpg" place="text" xlink:href="009/01/089/1.jpg"></figure><lb></lb>
magis elongantur, tan­<lb></lb>
to maius corpus <expan abbr="illu-minãs">illu­<lb></lb>
minans</expan> intercipiunt. </s>
          <s id="s.001629">ha­<lb></lb>
ctenus de magnitudine <lb></lb>
terræ ad Solem. </s>
          <s id="s.001630">Cum <lb></lb>
verò Luna eclypſatio­<lb></lb>
nis tempore, aliquan­<lb></lb>
do non ſolum tota in <lb></lb>
vmbræ vertice lateat, <lb></lb>
verùm etiam <expan abbr="aliquãdo">aliquando</expan> <lb></lb>
moram trahat, euidens <lb></lb>
eſt, eam eſſe multò mi­<lb></lb>
norem illa vmbræ par­<lb></lb>
te, in quam immergi­<lb></lb>
tur; quæ pars cum ſit <lb></lb>
conicæ vmbræ media, <lb></lb>erit multò gracilior <lb></lb>
quàm ſit ipſa terra. <lb></lb>
</s>
          <s id="s.001631">Ex quo manifeſtè apparet, Lunam, quæ illa vmbra minor eſt, eſſe à fortio­<lb></lb>
ri multò minorem ipſa terreſtri mole. </s>
          <s id="s.001632">Atque hæc de comparatione terræ <lb></lb>
ad Lunam. </s>
          <s id="s.001633">harum rerum demonſtrationes exactiores pertractare non eſt <lb></lb>
huius loci.</s>
        </p>
        <p type="main">
          <s id="s.001634"><arrow.to.target n="marg132"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001635"><margin.target id="marg132"></margin.target>132</s>
        </p>
        <p type="main">
          <s id="s.001636">Eodem cap. <emph type="italics"></emph>(Conſiderantes vtique, quæ nunc oſtenduntur per Mathematica<emph.end type="italics"></emph.end> <pb pagenum="90" xlink:href="009/01/090.jpg"></pb><emph type="italics"></emph>ſufficienter, fortè vtique deſisterent ab hac puerili opinione; valde enim ſimplex <lb></lb>
eſt putare <expan abbr="vnumquodq;">vnumquodque</expan> eorum quæ feruntur eſſe paruum magnitudinibus, quia vi­<lb></lb>detur aſpicientibus, hinc nobis ſic)<emph.end type="italics"></emph.end> vtinam iſta, necnon alia his ſimilia, quæ <lb></lb>
paſſim apud Ariſt. occurrunt, <expan abbr="pleriq;">plerique</expan> noſtræ ætatis conſiderarent, qui nulla <lb></lb>
ratione probari poſſe exiſtimant, Solem, v. g. terra eſſe centies ſexagies ſe­<lb></lb>
xies maiorem; ſed etiam, quod peius eſt, negant eſſe maiorem; ad demon­<lb></lb>
ſtrationes autem aſtronomicas dicunt ſe exiſtimare eas eſſe fallaces; at que <lb></lb>impoſſibile eſſe nos res adeo à nobis diſtantes ſufficienter perueſtigare: <lb></lb>
quanto ſapientius, ac prudentius eorum Magiſter Ariſt. alibi ſæpius, ſed hoc <lb></lb>
præcipuè loco; quippe qui Mathematicis ſufficienter excultus erat; quibus <lb></lb>
iſti deſtituti, nullo vnquam modo veſtigia præceptoris aſſequi poterunt.</s>
        </p>
        <p type="main">
          <s id="s.001637"><arrow.to.target n="marg133"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001638"><margin.target id="marg133"></margin.target>133</s>
        </p>
        <p type="main">
          <s id="s.001639">Summa 1. cap. 4. <emph type="italics"></emph>(Quæ igitur astrorum eſt, velox quidem; longè autem: quæ <lb></lb>verò Lunæ deorſum quidem, tarda autem: quæ autem Solis ambo hæc habet ſuffi­<lb></lb>
cienter)<emph.end type="italics"></emph.end> quæ igitur aſtrorum, ideſt latio aſtrorum eſt velox, ſed procul à ter­<lb></lb>
ra; Lunæ verò latio terræ quidem proxima, tarda tamen: at verò Solis la­<lb></lb>
tio medio modo ſe habet inter vtrumque, ideſt, quia <expan abbr="neq;">neque</expan> nimis vt aſtra di­<lb></lb>
ſtat, <expan abbr="neq;">neque</expan> tardè ſicut Luna circunfertur. </s>
          <s id="s.001640">exiſtimo Ariſt. loqui de motu diur­<lb></lb>
no, quia ſecundum hunc aſtra inerrantia ſunt Sole citatiora, Sol verò ipſa <lb></lb>
Luna citior. </s>
          <s id="s.001641">Verumenimuerò illud non prætereundum, quod plurium inua­<lb></lb>
luerit opinio exiſtimantium Ariſt. his verbis, Solem ſupra Lunam proximè <lb></lb>
collocaſſe; quod tamen ex ipſis nullo pacto deduci poteſt; ſed ſolummodo <lb></lb>
ipſum ſupra Lunam collocaſſe. </s>
          <s id="s.001642">quod ſi ita ſenſiſſet venia dignus haberetur, <lb></lb>
cum tunc temporis nondum fortè adinuentæ eſſent demonſtrationes illæ <lb></lb>
aſtronomicæ, quibus ordo Planetarum certiſſimè conſtat, <expan abbr="Solq́">Solque</expan>; medius in­<lb></lb>
ter Planetas collocatur. </s>
          <s id="s.001643">At verò nulla ratione ferendi ſunt <expan abbr="quicunq;">quicunque</expan> noſtra <lb></lb>
hac tempeſtate non ſolum Ariſt. ita ſenſiſſe, ſed etiam contra firmiſſimas <lb></lb>
aftronomorum demonſtrationes, quibus adeò Ariſt. deferebat, vnica, vt pu­<lb></lb>
tant ipſius auctoritate fulti, Solem ſecundum à Luna locum occupare om­<lb></lb>
ni ope defendunt.</s>
        </p>
        <p type="main">
          <s id="s.001644"><arrow.to.target n="marg134"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001645"><margin.target id="marg134"></margin.target>134</s>
        </p>
        <p type="main">
          <s id="s.001646">Summa 2. cap. 3. <emph type="italics"></emph>(Quod accidit circa Mercurij stellam, quia enim modicum <lb></lb>
<expan abbr="ſuperaſcẽdis">ſuperaſcendis</expan>, ſæpè non apparet, it a vt poſt tempus multum appareat)<emph.end type="italics"></emph.end> quod Mer­<lb></lb>
curius non niſi rarò conſpici poſſit, cauſa eſt, quia parum à Sole elongatur, <lb></lb>
ſiue ipſum antecedat, ſiue ſubſequatur. </s>
          <s id="s.001647">ex quo fit, vt diu ferè ſimul cum So­<lb></lb>le circumferatur, &amp; propterea ſiue oriatur, ſiue occidat, parum ſupra ho­<lb></lb>
rizontem eleuatus apparere poteſt, quod Ariſt. ait modicum <expan abbr="ſuperaſcẽdit">ſuperaſcendit</expan>. <lb></lb>
</s>
          <s id="s.001648">vnde fit tum propter nimiam Solis vicinitatem, cuius lumine tegitur; tum <lb></lb>
propter vapores, qui horizonti vt plurimum incumbunt, vt rarò, &amp; poſt ma­<lb></lb>
gna temporis interualla conſpiciatur. </s>
          <s id="s.001649">non me fugit hæc omnia ab aſtrono­<lb></lb>
mis per epiciclum excuſari; ſed ego mediocritati eorum, in quorum gra­<lb></lb>
tiam hæc ſcribo, conſultum volo.</s>
        </p>
        <p type="main">
          <s id="s.001650"><arrow.to.target n="marg135"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001651"><margin.target id="marg135"></margin.target>135</s>
        </p>
        <p type="main">
          <s id="s.001652">Eodem cap. <emph type="italics"></emph>(Ad auſtrum autem quando feratur, copiam quidem habere talís <lb></lb>humiditatis, ſed quia parua eſt ſectio circuli, quæ ſuper terram, quæ autem deor­<lb></lb>
ſum multiplex, non poſſe viſum hominum fractum ferri ad Solem, <expan abbr="neq;">neque</expan> ipſi tropico <lb></lb>
auſtrino appropinquanti; <expan abbr="neq;">neque</expan> in æſtiuis verſionibus exiſtente Sole. </s>
          <s id="s.001653">quapropter in <lb></lb>
lis quidem locis neque fieri cometem ipſum. </s>
          <s id="s.001654">quando verò ad Boream ſubdefecerit, <lb></lb>accipere comam, quia magna eſt circunferentia, quæ eſt ſupra horizontem; quæ au-<emph.end type="italics"></emph.end>


<pb pagenum="91" xlink:href="009/01/091.jpg"></pb><emph type="italics"></emph>tem eſt ſubtus, pars circuli parua; facilè enim viſum hominum pertingere tunc ad <lb></lb>
Solem)<emph.end type="italics"></emph.end> cur cometa in regione auſtrali vltra Solis, <expan abbr="anniq́">annique</expan>; vias conſtitutus <lb></lb>non appareret, cauſam referebat Hippocrates paruitatem circuli, quem <lb></lb>
motu diurno cometa deſcribebat, ob quam adeò parum ſupra horizontem <lb></lb>
attolleretur, vt <expan abbr="nõ">non</expan> poſſet viſus noſter ab ipſo ad Solem reflecti; quod ſecun­<lb></lb>
dum ipſum erat neceſſarium ad cometarum apparitionem. </s>
          <s id="s.001655">loquitur igitur <lb></lb>Hippocrates de circulis, quos diurna conuerſione cometes circumducit, qui <lb></lb>
omninò ſimiles ſunt ijs, quos etiam Sol, <expan abbr="reliquaq́">reliquaque</expan>; aſtra eodem motu de ſi­<lb></lb>
gnant. </s>
          <s id="s.001656">qui quidem omnes in noſtra ſphæra obliqua ita ſe habent, vt ij, qui <lb></lb>ſunt vltra æquatorem ad Capricorni tropicum, minus ſupra horizontem <lb></lb>extent, quàm infra deprimantur, &amp; tanto minus, quanto magis ab æquato­<lb></lb>re in auſtrum recedunt: contra verò faciunt, qui citra æquatorem ad Can­<lb></lb>cri conuerſionem collocantur, quanto enim magis ab æquatore in boream <lb></lb>
remouentur, tantò eorum ſectio, quæ eſt ſupra horizontem, maior eſt ea, <lb></lb>
quæ infra horizontem latet. </s>
          <s id="s.001657">quæ quidem omnia clara ſunt adhibita ſphæra <lb></lb>
materiali, quam ſi ad tuam poli eleuationem accommodaueris, illicò vi­<lb></lb>
debis tropici, Cancri ſectionem, quæ eſt ſupra horizontem multo maiorem <lb></lb>
ea, quæ eſt infra. </s>
          <s id="s.001658">oppoſitum verò in altero Capricorni tropico, cuius mini­<lb></lb>
mam portionem ſupra, maximam verò infra horizontem exiſtere videbis. <lb></lb>
</s>
          <s id="s.001659">Idem proportionaliter imaginari debes de circulis, quos cometa tam vltra <lb></lb>
Capricornum, quàm citra Cancrum delineat; nam eorum, qui ſunt vltra <lb></lb>
Capricornum ad auſtrum minores adhuc ſectiones ſupra horizontem exi­<lb></lb>
ſterent, quàm opus ſit ad cometen ſpectandum. </s>
          <s id="s.001660"><expan abbr="Atq́ue">Atque</expan> hæc cauſa eſt ex ſen­<lb></lb>
tentia Hippocr. cur in illa auſtrali plaga <expan abbr="nũquam">nunquam</expan> cometes effulgeat. </s>
          <s id="s.001661">è con­<lb></lb>
trario autem, quia ad boream ſectiones illæ maximæ ſunt, <expan abbr="aptæq́">aptæque</expan>; ad refra­<lb></lb>
ctionem viſus noſtri vſque ad Solem, idcircò in hac mundi parte cometas <lb></lb>
conſpicere ſolemus. </s>
          <s id="s.001662">Reliqua Vicomercatus, <expan abbr="atq;">atque</expan> Alexand. optimè expli­<lb></lb>
cant, quos tu conſule, ne actum agatur.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001663"><emph type="italics"></emph>In cap. 4. ſummæ 2. lib. 1. Meteor. de Cometis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001664"><arrow.to.target n="marg136"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001665"><margin.target id="marg136"></margin.target>136</s>
        </p>
        <p type="main">
          <s id="s.001666">In præſenti cap. Ariſt. ſuam de Cometis ſententiam exponit: Come­<lb></lb>
tam nimirum infra Lunam in elementari mundo procreari, &amp; ignitum <lb></lb>
quoddam Meteoron, ex lenta, pingui, <expan abbr="ſiccaq́">ſiccaque</expan>; materia à terra in ſu­<lb></lb>
premam aeris regionem attracta, exiſtere; <expan abbr="ibiq́">ibique</expan>; rapti aeris calore, <lb></lb>
vel elementi ignis (quod illic eſſe putat) vicinitate, vel etiam vi aſtrorum <lb></lb>
incendi, <expan abbr="atq;">atque</expan> impelli. </s>
          <s id="s.001667">Hanć porrò opinionem &amp; ſi probabilibus tantum ra­<lb></lb>
tionibus confirmatam vulgò tamen <expan abbr="vſq;">vſque</expan> ad hanc diem receptam, cum fal­<lb></lb>
ſam eſſe aſtronomi exiſtiment, non erit abs re rationes eas ex ſecundo pro­<lb></lb>
gymn. </s>
          <s id="s.001668">Tichonis volumine, deſumptas hic breuiter referre, quibus aſtrono­<lb></lb>
mus ille eos ſupra Lunam in ætherea regione collocauit: quas quidem ra­<lb></lb>
tiones ille ex diuturnis obſeruationibus per exquiſita organa factis adinue­<lb></lb>
nit: eaſque Mathematicis linearum, ac numerorum demonſtrationibus <lb></lb>
explicauit.</s>
        </p>
        <p type="main">
          <s id="s.001669">Prima. </s>
          <s id="s.001670">ſed vt ab auctoritate, in quam obiter incidimus <expan abbr="initiũ">initium</expan> faciamus, <lb></lb>non eſt exiſtimandum nonnullos ſolum ex recentioribus id conſtanter aſſe­<pb pagenum="92" xlink:href="009/01/092.jpg"></pb>ueraſſe, ſed ſuperiori etiam ætate id ipſum Hieron. </s>
          <s id="s.001671">Cardan. libro de ſubtili­<lb></lb>
tate conatus eſt, <expan abbr="neq;">neque</expan> irrito conatu, demonſtrare; qui præterea idem cum <lb></lb>ſe ipſo ſenſiſſe ait Albumazar. </s>
          <s id="s.001672">quibus etiam ex antiquis Seneca annumeran­<lb></lb>
dus eſt. </s>
          <s id="s.001673">prędicti autem recentiores omnes varijs demonſtrationibus ex ac­<lb></lb>curata obſeruatione erutis illud certò certius confirmare contendunt: <expan abbr="idq́">idque</expan>; <lb></lb>
non in vno dumtaxat, ſed in <expan abbr="quinq;">quinque</expan> cometis; quorum demonſtrationes apud <lb></lb>
Tychonem partim in progymn. </s>
          <s id="s.001674">partim in epiſt. fuſius explicatas reperies.</s>
        </p>
        <p type="main">
          <s id="s.001675">2. Quarum potiſſima illa eſt, quæ ex parallaxi, ſeu aſpectus diuerſitate <lb></lb>
deſumitur, certiſſimum enim eſt lumen illud eſſe altero ſublimius, quod mi­<lb></lb>
norem exhibet parallaxim: expertos autem ſe eſſe hi omnes, affirmant ho­<lb></lb>
ſce quinque cometas multò minorem pati parallaxim, quam Lunam; imò <lb></lb>
quempiam minorem, quàm Sol ipſe patiatur, quo poſito manifeſtè conuin­<lb></lb>
ceretur eos omnes ſupra Lunam in ætherea regione effulſiſſe.</s>
        </p>
        <p type="main">
          <s id="s.001676">3. Ratio, qua etiam ante nouas obſeruationes vti ſolebant, deſumitur <lb></lb>
ex motu cometæ diurno, quo ſcilicet oritur, &amp; occidit, quemadmodum cæ­<lb></lb>
tera ſydera, hoc eſt ſpatio 24. horarum diurnam conuerſionem circa totam <lb></lb>
terram abſoluit. </s>
          <s id="s.001677">ſi igitur comete eſſet in ſublimiori aeris regione, vbi cæte­<lb></lb>
ra ignita meteora collocantur, <expan abbr="mouereturq́">mouereturque</expan>; diurno motu circa terram, ſe­<lb></lb>
queretur neceſſariò eum tanta velocitate videri à nobis circumferri, vt po­<lb></lb>
tius fulgor quidam, ſeu radius pertranſiens ab oriente in occidentem appa­<lb></lb>
reret, quam ſtella quędam: <expan abbr="idq́">idque</expan>; propter propinquitatem; aſtra enim ob ni­<lb></lb>
miam diſtantiam videntur tardè moueri, quamuis velociſſimè moueantur.</s>
        </p>
        <figure id="id.009.01.092.1.jpg" place="text" xlink:href="009/01/092/1.jpg"></figure>
        <p type="main">
          <s id="s.001678">Quod melius ex ſequenti figura <lb></lb>
<expan abbr="cõuincitur">conuincitur</expan>, vbi circulus interior eſt <lb></lb>
terra, cuius ſemidiameter A B. cir­<lb></lb>
culus verò exterior eſt cometæ gy­<lb></lb>
rus, quem ipſe ſpatio 24. horarum <lb></lb>
percurrit, qui ſecundum veram pro­<lb></lb>
portionem deberet adhuc ipſi terræ <lb></lb>
propinquior, ac proinde minor eſſe, <lb></lb>
iuxta aeris ſupremam partem. </s>
          <s id="s.001679">hori­<lb></lb>
zon eſt recta D C, tangens terram in <lb></lb>
B, vbi eſt oculus noſter, qui nihil in­<lb></lb>
fra ipſam D C, videre poteſt; quare <lb></lb>
ſi cometa 24. horarum totum gyrum <lb></lb>
D C E, percurrit, non videbitur, niſi <lb></lb>
quando percurret portionem D C, <lb></lb>
ſupra horizontem; quæ quidem por­<lb></lb>
tio, <expan abbr="neq;">neque</expan> ſemihoræ reſponderet, ſi fi­<lb></lb>
gura iuxta veram proportionem conſtrueretur. </s>
          <s id="s.001680">experientia tamen conſtat, <lb></lb>
cometas videri ſupra horizontem tot horis, quot ſtellæ fixæ, ſub quibus mo­<lb></lb>
uentur: non ergo eſt in ſupremo aere. </s>
          <s id="s.001681">Quod ſi fiat figura, in qua exterior <lb></lb>
cometæ ambitus adeò magnus ſit, vt ipſius portio D C, ſupra horizontem <lb></lb>
exiſtens, reſpondeat tempori, quo cometa ſupra noſtrum pariter horizon­<lb></lb>
tem ſpectatur, ea figura terræ ſemidiametrum A B. toties multiplicabit, vt <lb></lb>
ipſi Lunæ circuitui proximè accedat.</s>
        </p>
        <pb pagenum="93" xlink:href="009/01/093.jpg"></pb>
        <p type="main">
          <s id="s.001682">Præterea aiunt, quis ſanæ mentis dixerit, Meteoron vllum ex materia <lb></lb>
vaga, ac fluxa conſtans, poſſe tanta pernicitate moueri, vt diurnam con­<lb></lb>
uerſionem abſoluat? </s>
          <s id="s.001683">vnde illi motus iſte? </s>
          <s id="s.001684">præſertim cum videamus cætera <lb></lb>
ignita meteora eſſe ad modum temporanea, <expan abbr="atq;">atque</expan> euanida.</s>
        </p>
        <p type="main">
          <s id="s.001685">4. Comprobationem nobis ſuppeditant ex via, ſeu ductus circuli, quem <lb></lb>
toto durationis tempore proprio curſu deſignarunt: prædicti <expan abbr="namq;">namque</expan> quin­<lb></lb>
que cometæ motu ſibi proprio, quo ab occidente non omninò orientem <lb></lb>
verſus, ſed ad aquilonem deflectentes ab initio ſuæ apparitionis, <expan abbr="vſq;">vſque</expan> ad vl­<lb></lb>timum finem exquiſitiſſimè portionem circuli maximi in cęlo deſignarunt; <lb></lb>non aliter quàm Sol proprio motu per eclypticam in cœlo mundi ſphæram <lb></lb>
in duo æqualia diuidentem deſcribit. </s>
          <s id="s.001686">necnon aliter ac Luna ſuum iter per <lb></lb>
circulum maximum cœlum bifariam diuidentem perficit. </s>
          <s id="s.001687">quapropter co­<lb></lb>
metas hoſce <expan abbr="nõ">non</expan> minus quam Sol, vel Luna in ipſo æthere ſpatiatos eſſe con­<lb></lb>
tendunt. </s>
          <s id="s.001688">qui enim, aiunt, fieri potuiſſet, ſi in mundo elementari flagraſſent, <lb></lb>
vt tam regulari, <expan abbr="atq;">atque</expan> conſtanti ductu circuli maximi portionem tam exactè <lb></lb>
delineaſſent, quam quidem inter elementa vagum, <expan abbr="atq;">atque</expan> inſtabilem pro ma­<lb></lb>
teriæ inſtabilitate exercere debuiſſent?</s>
        </p>
        <p type="main">
          <s id="s.001689">5. Adde, quod in maximo hoc circulo deſcribendo, etiam ſi inæquali ve­<lb></lb>
locitate viſi ſint moueri, inæqualitatem tamen illam regularem <expan abbr="vbiq;">vbique</expan> ſem­<lb></lb>
per ſeruauerunt, in principio quidem velociores, deinde ſucceſſiuè, &amp; pro­<lb></lb>
portionaliter velocitatem illam ſimili analogia ſemper ſeruata <expan abbr="inhibuerũt">inhibuerunt</expan>, <lb></lb>
nullo igitur pacto inordinatam inæqualitatem, qua à tardiore motu ſubito <lb></lb>
in celeriorem, &amp; rurſus ſtatim ab hoc in <expan abbr="illũ">illum</expan> proſilirent exhibuerunt: prout <lb></lb>
omnia Meteora, quæ in mundi parte elementari ex flammanti materia ge­<lb></lb>
nerantur, talem diſparem, <expan abbr="atq;">atque</expan> inconſtantem motum obtinere cernuntur.</s>
        </p>
        <p type="main">
          <s id="s.001690">6. Argumento præterea eſt cometas hoſce minimè elementares fuiſſe, <lb></lb>quod hic eorum proprius motus, quo maximo illo tramite ferebantur, nun­<lb></lb>
quam tantus fuit, vt proprium Lunæ motum, vel tardiſſimum adæquauerit, <lb></lb>
quæ quidem cum lentiſſima eſt plus denis gradibus vna die promouetur; <lb></lb>
cum tamen cometæ initio cum velociſſimi ſunt non multum vltra quinos <lb></lb>
gradus diurno motu progreſſi ſint, vt ob id longè ſupra Lunam curſum ſuum <lb></lb>
abſoluiſſe manifeſtè comprobari poſſit: quo enim ſydera magis à terra at­<lb></lb>
tolluntur, <expan abbr="octauæq́">octauæque</expan>; ſphæræ propius accedunt, eò tardioribus proprijs la­<lb></lb>tionibus proferuntur: ita vt ſtellæ iſtæ cœlo adſcititiæ ſupra Lunam admo­<lb></lb>
dum euehendæ videantur. </s>
          <s id="s.001691">Quod ſi in ſuprema aeris regione conflagrarent, <lb></lb>
qua nam ratione vnà cum toto cœlo diurnam conuerſionem abſoluiſſent: <lb></lb>
<expan abbr="neq;">neque</expan> enim putandum eſt ſupremum hunc aeris limbum eadem perne citate, <lb></lb>
qua cœleſtes orbes, verum minori admodum imò tardiſſimè à diurno mo­<lb></lb>
tu, ſi tamen eo rapitur circumduci.</s>
        </p>
        <p type="main">
          <s id="s.001692">7. Tandem argumentum ex ipſorum duratione deſumatur. </s>
          <s id="s.001693">cætera nam­<lb></lb>
que meteora ſtatim <expan abbr="atq;">atque</expan> apparuerint, veluti temporanea prorſus, <expan abbr="atq;">atque</expan> eua­<lb></lb>
nida extinguuntur: At verò cometæ ad menſem aliquando integrum per­<lb></lb>
ſeuerant. </s>
          <s id="s.001694">quì igitur fieri potuerit, vt in hac corruptibili <expan abbr="mũdi">mundi</expan> parte ex ma­<lb></lb>teria adeò fluxa, &amp; vaga, quam illis Ariſtoteles ſupponit, tandiu perdura­<lb></lb>
re potuiſſent.</s>
        </p>
        <p type="main">
          <s id="s.001695"><expan abbr="Atq;">Atque</expan> hæ ſunt rationes, quibus plurimi aſtronomorum recentiorum, co­


<pb pagenum="94" xlink:href="009/01/094.jpg"></pb>metas hoſce motum æthereæ regioni conformem, contrà quam Ariſt. opi­<lb></lb>natus eſt, obtinuiſſe, manifeſtum eſſe volunt; ac proinde eorum locum, &amp; <lb></lb>
curſum in cœleſti mundi parte extitiſſe, ſe comprobaſſe exiſtimant: qua de <lb></lb>
re prudentis Lectoris eſto iudicium: <expan abbr="neq;">neque</expan> enim, vt ille cecinit, noſtrum eſt, <lb></lb>
tantas componere lites.</s>
        </p>
        <p type="main">
          <s id="s.001696">Verumenimuerò Peripatetica omnis ſchola reclamat; Cœlum eſt inge­<lb></lb>nerabile, &amp; incorruptibile, nihil igitur noui cœlo poteſt accidere. </s>
          <s id="s.001697">ſed age <lb></lb>
reſpondent, nonne omnium aſtronomorum conſenſu ſtellæ tres nouæ noſtro <lb></lb>
hoc ſæculo in cœlo toti mundo conſpicuæ illuxerunt? </s>
          <s id="s.001698"><expan abbr="easq́">easque</expan>; in octaua ſphæ­<lb></lb>ra reſediſſe conſtans eſt omnium aſſertio? </s>
          <s id="s.001699">quarum prior anno 1572. in con­<lb></lb>
ſtellatione Caſſiopeæ apparuit. </s>
          <s id="s.001700">Secunda anno 1600. in Cygno, quæ nec dum <lb></lb>
extinguitur. </s>
          <s id="s.001701">Tertia anno 1604. inter Sagittarij ſtellas viſa eſt, de quibus vi­<lb></lb>
de P. Clauium in ſphæra breuiter de illis tractantem: aut ſi mauis, &amp; vacat, <lb></lb>
vide quoad primam primum volumen progymnaſmatum Tychonis Brahe, <lb></lb>
vbi etiam aliorum aſtronomorum de eadem certiſſimas commentationes <lb></lb>
reperies. </s>
          <s id="s.001702">conſule etiam de reliquis duabus Ioannis Kepleri Cæſareæ Maie­<lb></lb>
ſtatis Mathematici commentaria; &amp; coactus libenter fateberis noui ali­<lb></lb>
quid cœlo aduenire poſſe.</s>
        </p>
        <p type="main">
          <s id="s.001703">Poſtremò tandem poſſet quiſpiam in hunc <expan abbr="modũ">modum</expan> opponere: etiam ſi con­<lb></lb>
ſtet <expan abbr="quinq;">quinque</expan> cometas cęlo oberraſſe, non propterea dicemus reliquos omnes <lb></lb>
eſſe pariter cœleſtes, <expan abbr="nullumq́">nullumque</expan>; proinde ſublunarem. </s>
          <s id="s.001704">Huic memorati Aſtro­<lb></lb>
nomi ſic reſponderent; id quidem mathematica, &amp; infallibili ratione non <lb></lb>
colligi, imò aliquot parum infra Lunam extitiſſe, non omninò negandum <lb></lb>
videri: at verò in ſuperiori aeris plaga, in tam fluxa, ac inſtabili mundi par­<lb></lb>
te, cometas vnquam effulſiſſe, nemo ſibi ob allatas rationes meritò perſua­<lb></lb>
dere poſſe.</s>
        </p>
        <p type="main">
          <s id="s.001705"><arrow.to.target n="marg137"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001706"><margin.target id="marg137"></margin.target>137</s>
        </p>
        <p type="main">
          <s id="s.001707">Summæ 2. cap. 5. <emph type="italics"></emph>(Ad hæc autem ſi quemadmodum oſtenditur in ijs, quæ cir­<lb></lb>ca Astrologiam ſpeculationibus, Solis magnitudo maior eſt quàm terræ; &amp; diſtan­<lb></lb>
tia multò maior aſtrorum ad terram quàm So is; ſicut Solis ad terram quàm Lu­<lb></lb>
næ; non <expan abbr="vtiq;">vtique</expan> longè alicubi à terra conus, qui à Sole, conijciet radios, <expan abbr="neq;">neque</expan> vtique <lb></lb>vmbra terræ, quæ vocatur nox, erit apud astra; ſed neceſſe Solem omnia aſtra cir­<lb></lb>
cunſpicere, &amp; nulli ipſorum terram obſistere)<emph.end type="italics"></emph.end> ex dictis ſumma 1. cap. 3. huius, <lb></lb>
&amp; ex figura ibi deſcripta, facilè eſt intelligere præſentem locum; nam cum <lb></lb>
Sol ſit multò maior terra, vt ibi probatur, ac minus diſter à terra quàm fixæ <lb></lb>
ſtellæ, magis tamen quàm Luna, vt patet ex ſolari eclypſi, ſequitur neceſſa­<lb></lb>
riò vmbram terræ, quæ nox eſt ipſa, effici turbinatam, &amp; valdè procul à ter­<lb></lb>
ra acumen coni vmbræ aſcendet, ſed paulò ſupra Lunam conus hic vmbræ <lb></lb>
permittet radios Solis ſe ipſum ambientes iterum ſimul committi, quod il­<lb></lb>
lis verbis <emph type="italics"></emph>(Conijciet radios)<emph.end type="italics"></emph.end> ideſt committet radios expreſſit Ariſt. cum igi­<lb></lb>
tur vmbra apud Lunam ſit ſatis gracilis, breui ſupra Lunam deſinet, neque <lb></lb>
vllo pacto ad affixa ſydera protendetur, <expan abbr="neq;">neque</expan> illis tenebras offundet. </s>
          <s id="s.001708">quod <lb></lb>
etiam experientia confirmat, cum nunquam aſtra illa, quæ Soli opponuntur, <lb></lb>
<expan abbr="quæq́">quæque</expan>; vertex vmbræ collimat, vllam <expan abbr="patiãtur">patiantur</expan> eclypſim. </s>
          <s id="s.001709">quare ſine vllo ter­<lb></lb>ræ impedimento Sol poteſt affixa omnia ſydera per luſtrare. </s>
          <s id="s.001710">Exactiores ha­<lb></lb>
rum rerum demonſtrationes ſunt alterius loci.</s>
        </p>
        <p type="main">
          <s id="s.001711"><arrow.to.target n="marg138"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001712"><margin.target id="marg138"></margin.target>138</s>
        </p>
        <p type="main">
          <s id="s.001713">Eodem cap. <emph type="italics"></emph>(Amplius autem eſt tertia quædam opinio de ipſo, dicunt enim<emph.end type="italics"></emph.end> <pb pagenum="95" xlink:href="009/01/095.jpg"></pb><emph type="italics"></emph>quidam lac eſſe reflexionem noſtri viſus ad Solem; ſicut &amp; ſtellam comatam; im­<lb></lb>
poſſibile autem eſt &amp; hoc, ſi enim videns quieuerit &amp; ſpeculum, &amp; quod videtur <lb></lb>
omne in eodem puncto ſpeculi eadem apparebit <expan abbr="vtiq;">vtique</expan> pars imaginis, ſi autem mo­<lb></lb>
ueatur ſpeculum, &amp; quod videtur, in eadem quidem diſtantia ad videns, &amp; quie­<lb></lb>
ſcens; ad inuicem autem <expan abbr="neq;">neque</expan> æquè velociter, <expan abbr="neq;">neque</expan> in eadem ſemper diſtantia im­<lb></lb>
poſſibile eandem imaginem in eadem eſſe parte ſpeculi. </s>
          <s id="s.001714">Quæ autem in lactis circu­<lb></lb>
lo feruntur aſtra, &amp; Sol, ad quem fit reflexio, mouentur manentibus nobis, &amp; ſi­<lb></lb>
militer, &amp; æqualiter ad nos diſtantia; à ſe ipſis autem non æqualiter: aliquando <lb></lb>
enim medijs noctibus Delphin oritur, aliquando verò diluculo. </s>
          <s id="s.001715">partes autem lactis <lb></lb>
eædem manent in vnoquoque; atqui non oportebat, ſi erat imago, ſed non in eiſdem <lb></lb>
adhuc eſſet hæc paſſio locis)<emph.end type="italics"></emph.end> in his Ariſt. confutat opinionem dicentium Gala­<lb></lb>
xiam apparere per quandam reflexionem viſus noſtri ab illa parte cęli, ceu, <lb></lb>ex quodam ſpeculo ad Solem: probat autem hoc eſſe impoſſibile ratione <lb></lb>
deſumpta ex parte Optices, quæ dicitur Catoptrica, ſiue ſpecularia, quia <lb></lb>
tractat de viſione reflexa, quæ fit mediante ſpeculo, quam quidem rationem <lb></lb>
ſi vellem mathematicè explicare, longa nimis, ac præter inſtitutum fieret <lb></lb>
tractatio. </s>
          <s id="s.001716">Pauca tamen addam, quæ Ariſtotelis <expan abbr="ſentẽtiam">ſententiam</expan> ſatis perſpicuam <lb></lb>
reddant. </s>
          <s id="s.001717">ſi igitur inquit, Galaxia nihil aliud eſſet quàm reflexio noſtri viſus <lb></lb>
ex illa cœli parte, in qua ipſa apparet tanquam ex ſpeculo ad Solem, ita vt <lb></lb>nihil aliud ipſa eſſet, quàm Sol viſus per reflexionem ex illa cœli parte tan­<lb></lb>
quam ſpeculo; ſequeretur eam non ſemper in eadem cœli parte apparere, <lb></lb>
ſed modo in vna, modo in alia, ita vt ſpatio vnius anni totum cœlum perua­<lb></lb>
garetur: quod tamen non accidit. </s>
          <s id="s.001718">quod autem illud conſequatur manife­<lb></lb>
ſtum eſſe poteſt ex obſeruatione eorum, quæ ex ſpeculis videntur: tunc enim <lb></lb>
res per ſpe culum viſa in eadem ſpeculi parte apparet, quando &amp; videns, &amp; <lb></lb>
ſpeculum, &amp; obiectum immota manent: quod ſi &amp; ſpeculum, &amp; obiectum ad <lb></lb>
inuicem accedant, vel recedant, ſeruata tamen eadem ab inſpectore diſtan­<lb></lb>
tia, nullo modo fieri poteſt, vt eadem imago, in eadem ſpeculi parte ſpe­<lb></lb>
ctanti videatur, niſi obiectum ſpeculo per eandem lineam accedat, ſecun­<lb></lb>
dum quam illi incidebat. </s>
          <s id="s.001719">At verò partibus illis lactei circuli, ſiue aſtris, quæ <lb></lb>
in eo fulgent, Sol perpetuò accedit, vel recedit, <expan abbr="neq;">neque</expan> per lineam incidentiæ <lb></lb>
<expan abbr="eãdem">eandem</expan>, ſeruata tamen eadem à nobis diſtantia, quod quidem inde patet, quia <lb></lb>
Delphini conſtellatio, qui in ipſo ferè lacte exiſtit, <expan abbr="aliquãdo">aliquando</expan> medijs noctibus, <lb></lb>
aliquando verò mane, aliquando etiam veſperi oritur; quod inde accidit, <lb></lb>
quia illi Sol modò appropinquat, modò coniungitur, modò ab eo recedit, <lb></lb>
quare neceſſe eſſet, vt lacteus orbis, non ſemper in ijſdem locis, ſed perpe­<lb></lb>
tuò in alijs, <expan abbr="atq;">atque</expan> alijs cerneretur, cuius tamen contrarium videmus. </s>
          <s id="s.001720">ex qui­<lb></lb>
bus conſtat falſam omninò eſſe eorum ſententiam, qui Galaxiam per huiuſ­<lb></lb>modi reflexionem fieri opinabantur. </s>
          <s id="s.001721">Quæ dicta ſunt de ſpeculo, &amp; obiecto <lb></lb>
ſatius eſt aſſumpto aliquo ſpeculo experiri, quàm ea pluribus obſcurare: qua <lb></lb>etiam experientia Ariſt. ratio confirmabitur.</s>
        </p>
        <p type="main">
          <s id="s.001722"><arrow.to.target n="marg139"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001723"><margin.target id="marg139"></margin.target>139</s>
        </p>
        <p type="main">
          <s id="s.001724">Ibidem <emph type="italics"></emph>(Quæ autem in lactis cir culo feruntur astra, &amp; Sol, ad quem fit refle­<lb></lb>xio, mouentur manentibus nobis, &amp; ſimiliter, &amp; æqualiter ad nos diſtantia à ſe <lb></lb>
ipſis autem non æqualiter)<emph.end type="italics"></emph.end> quæ hic ab Ariſtotele dicuntur <expan abbr="nõ">non</expan> ſunt vſque quaque <lb></lb>vera propter apogæum, ac perigæum Solis, quæ quidem duo ab omnibus <lb></lb>
aſtronomis aſſeruatur: quando igitur Sol eſt in apogæo, maiori multo in­


<pb pagenum="96" xlink:href="009/01/096.jpg"></pb>teruallo diſtat à nobis, quàm quando eſt in perigæo, interuallum enim illud <lb></lb>
conſtat diametris terræ duobus, &amp; quadraginta, hoc eſt milliarijs 208000. <lb></lb>
ferè, ideſt octonis millibus ſupra ducenta millia. </s>
          <s id="s.001725">quæ differentia facit vt Sol <lb></lb>
manifeſtè appareat nobis minor apogæus, quàm perigæus. </s>
          <s id="s.001726">Sol præterea ſi­<lb></lb>
militer ipſis inerrantibus ſtellis fit tantumdem modo remotior, modo pro­<lb></lb>
pinquior: ſed fortè Ariſt. iſta non occurrerunt, vel tunc temporis nondum <lb></lb>
perſpecta erant.</s>
        </p>
        <p type="main">
          <s id="s.001727"><arrow.to.target n="marg140"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001728"><margin.target id="marg140"></margin.target>140</s>
        </p>
        <p type="main">
          <s id="s.001729">Ibidem <emph type="italics"></emph>(Aliquando enim medijs noctibus Delphin oritur)<emph.end type="italics"></emph.end> vt probet, Gala­<lb></lb>
xiam non ſemper ſeruare à Sole diſtantiam eandem, accipit tanquam huius <lb></lb>
rei ſignum, manifeſtum, quod Delphini conſtellatio aliquando medijs no­<lb></lb>
ctibus oriatur ſupra horizontem, aliquando verò diluculo; non ideò tamen <lb></lb>
putes hanc rationem ſupponere Delphinum eſſe in ipſo lacteo circulo, quod <lb></lb>
tamen verum non eſt, non enim eſt in Galaxia, ſed tamen illi proximus, vt <lb></lb>
noctu videre eſt in cœlo, vel etiam ſi mauis in globo aſtronomico: non ta­<lb></lb>
men ob id Ariſt. ratio minus valida redditur, cum Delphinus ſemper Gala­<lb></lb>
xiæ eodem modo ſit proximus, <expan abbr="eoq́">eoque</expan>; moto, ipſa pariter moueatur.</s>
        </p>
        <p type="main">
          <s id="s.001730"><arrow.to.target n="marg141"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001731"><margin.target id="marg141"></margin.target>141</s>
        </p>
        <p type="main">
          <s id="s.001732">Summæ 2. cap. 6. Sunt qui velint Ariſt. Galaxiam nihil aliud eſſe, quàm <lb></lb>
quandam refractionem lucis ſtellarum illarum, quæ ſunt in ætherea Gala­<lb></lb>
xia, quæ inquam refractio fiat circa ſupremam aeris regionem ex occurſu <lb></lb>
exhalationum, quæ ibi perpetuò conſeruantur, &amp; vi earumdem ſtellarum <lb></lb>
ſurſum ſemper attrahuntur, quæ refractio fiat ad eum modum, quo halo cir­<lb></lb>
ca Solem, &amp; Lunam. </s>
          <s id="s.001733">&amp; quemadmodum halo, ſiue area omnibus <expan abbr="vndecunq;">vndecunque</expan> <lb></lb>
aſpicientibus ſemper videntur in eodem cœli loco, hoc eſt è regione Solis, <lb></lb>
vel Lunæ; ſimiliter Galaxia in aere omnibus <expan abbr="vndecunq;">vndecunque</expan> intuentibus appa­<lb></lb>
reat in eadem cœli parte, ideſt ex aduersò eorumdem ſyderum, quæ cœle­<lb></lb>
ſtem lacteam viam conficiunt. </s>
          <s id="s.001734">Porrò qui ſic mentem Ariſt. exponunt, nul­<lb></lb>
lo modo poſſunt à Mathematicis redargui per rationem deſumptam à di­<lb></lb>
uerſitate aſpectus (quam poſtea explicabo) quamuis phyſicis rationibus re­<lb></lb>
fellantur. </s>
          <s id="s.001735">Alij ſunt, quorum ſententia magis videtur <expan abbr="improbãda">improbanda</expan>, eò quod <lb></lb>
Ariſt. ſummum Philoſophum pueriliter in aſtronomia lapſum fateri cogan­<lb></lb>
tur. </s>
          <s id="s.001736">Exiſtimant hi Galaxiam hanc Ariſtotelicam nihil aliud eſſe, quàm ip­<lb></lb>
ſas tenues exhalationes in aere ſubuectas, directèque infra ſtellas illas la­<lb></lb>
cteum circulum in cœlo conſtituentes nobis obiectas. </s>
          <s id="s.001737">qui præter innumera, <lb></lb>
ac magna abſurda è naturali Philoſophia petita, vnum maximum ex Aſtro­<lb></lb>
nomia, nempè ex diuerſitate aſpectus deſumptum, nullo modo vitare poſ­<lb></lb>
ſunt; <expan abbr="eſtq́">eſtque</expan>; huiuſmodi, quia ſi lacteus hic circulus eſſet in aere, non ab om­<lb></lb>
nibus, <expan abbr="neq;">neque</expan> ex omni terræ loco per eadem ſydera commeare cerneretur, ſed <lb></lb>
è diuerſis, &amp; præcipuè ab inuicem valde diſſitis, circa diuerſa aſtra ſe ſe ocu­<lb></lb>
lis noſtris obijceret: at teſtimonio ſenſus conſtat, Galaxiam ſemper in eo­<lb></lb>
dem loco; <expan abbr="eademq́">eademque</expan>; à ſyderibus fixis diſtantia albicare, ergò nullo modo <lb></lb>
viam hanc in aere quaſi pendulam fabricare debemus. </s>
          <s id="s.001738">rationem hanc di­<lb></lb>
uerſitatis aſpectus aſtronomicè magis explicatam reperies apud Clauium <lb></lb>
in ſphæra. </s>
          <s id="s.001739">Porrò hæc ratio quamuis adeo certa, ac noſtra tempeſtate vul­<lb></lb>
gata, parum tamen à nonnullis de rebus Meteorologicis commentaria con­<lb></lb>
farcinantibus intellecta, minimè eos abſterrere potuit, quin prædictam opi­<lb></lb>
nionem, non ſolum Ariſtoteli imponerent, verum etiam ipſi <expan abbr="tãquam">tanquam</expan> veram


<pb pagenum="97" xlink:href="009/01/097.jpg"></pb>aſtruerent: huiuſmodi patiuntur incommoda, qui <expan abbr="abſq;">abſque</expan> Mathematicarum <lb></lb>
auxilio Philoſophiam aggrediuntur.</s>
        </p>
        <p type="main">
          <s id="s.001740"><arrow.to.target n="marg142"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001741"><margin.target id="marg142"></margin.target>142</s>
        </p>
        <p type="main">
          <s id="s.001742">Eodem cap. <emph type="italics"></emph>(Ad hæc autem locus plenus eſt aſtris maximis, &amp; fulgidiſſimis, <lb></lb>
&amp; adhuc ſparſis vocatis)<emph.end type="italics"></emph.end> non ſolum viam hanc lacteam aſtris plurimis refer­<lb></lb>
tiſſimam eſſe videmus, ſed præterea eandem ſtellarum admodum feracem <lb></lb>
appellare licebit, ſi quidem ſtellæ omnes illæ nouæ, quæ noſtra tempeſtate <lb></lb>
apparuerunt, omnes in hac via exortæ ſunt. </s>
          <s id="s.001743">prima enim anno 1572. effulſit <lb></lb>
in Caſſiopea; altera anno 1600. in Cygno. </s>
          <s id="s.001744">tertia demum anno 1604. in Sa­<lb></lb>
gittario, quæ omnes conſtellationes intra lacteum circulum continentur. <lb></lb>
</s>
          <s id="s.001745">Veriſſimum præterea eſſe hoc idem confirmatur inſtrumenti illius mirabi­<lb></lb>
lis auxilio, quod ſuperiori anno in Belgio excogitatum, &amp; poſtea in Italia <lb></lb>
à Galilæo perfectius <expan abbr="redditũ">redditum</expan> eſt, <expan abbr="quodq́">quodque</expan>; ipſe primum Italicè Cannocchiale, <lb></lb>
Latinè verò, &amp; quidem aptè à Græcis mutuato vocabulo alius Teleſcopium <lb></lb>
appellauit: hoc inquam ſpecillo adhibito perſpicuum ſtatim fit non ſolum <lb></lb>
in via lactea innumeras ſtellas contineri, verum quid ipſa ſit, certò certius <lb></lb>
conſtat; ſed ſatius eſt ipſius Galilæi verba ex Nuncio ſydereo referre: Quod <lb></lb>
tertio inquit, loco à nobis fuit obſeruatum eſt ipſiuſmet lactei circuli eſſen­<lb></lb>
tia, ſen materies, quam Teleſcopij beneficio adeò ad ſenſum licet intueri, <lb></lb>
vt &amp; altercationes omnes, quæ per tot ſæcula Philoſophos excruciarunt ab <lb></lb>
oculata certitudine <expan abbr="dirimãtur">dirimantur</expan>, <expan abbr="nosq́">nosque</expan>; à verboſis diſputationibus liberemur: <lb></lb>
eſt enim Galaxia nihil aliud, quàm innumerarum ſtellarum coaceruatim <lb></lb>
conſitarum congeries, in <expan abbr="quãcunq;">quancunque</expan> enim regionem illius ſpecillum dirigas, <lb></lb>ſtatim ſtellarum ingens frequentia ſe ſe in conſpectum profert, <expan abbr="quarũ">quarum</expan> com­<lb></lb>
plures ſatis magnæ, ac valdè conſpicuæ videntur; ſed exiguarum multitudo <lb></lb>
prorſus inexplorabilis eſt. </s>
          <s id="s.001746">hæc ille.</s>
        </p>
        <p type="main">
          <s id="s.001747"><arrow.to.target n="marg143"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001748"><margin.target id="marg143"></margin.target>143</s>
        </p>
        <p type="main">
          <s id="s.001749">Eodem cap. <emph type="italics"></emph>(Conſideretur autem &amp; circulus, &amp; quæ ſunt in ipſo aſtra ex de­<lb></lb>
ſcriptione)<emph.end type="italics"></emph.end> id eſt, conſideretur Galaxia, &amp; aſtra ipſius inſpiciantur diligenter <lb></lb>
ex deſcriptione alicuius Globi aſtronomici, in quo ſolent Aſtronomi omnes <lb></lb>
conſtellationes, ac ſtellas ſuis locis reddere, <expan abbr="atq;">atque</expan> etiam lacteum ipſum cir­<lb></lb>
culum graphicè effingere. </s>
          <s id="s.001750">huiuſmodi globum veteres ſphęram Aratæam di­<lb></lb>
cebant ab Arato Poeta græco, qui <expan abbr="cõſtellationes">conſtellationes</expan> omnes carmine proſequu­<lb></lb>
tus eſt, ac proinde globum hunc ordine expoſuit:</s>
        </p>
        <p type="main">
          <s id="s.001751"><arrow.to.target n="marg144"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001752"><margin.target id="marg144"></margin.target>144</s>
        </p>
        <p type="main">
          <s id="s.001753">Eodem cap. <emph type="italics"></emph>(Sparſa autem vocata)<emph.end type="italics"></emph.end> putò ſparſa hæc ſydera illa eſſe, quæ <lb></lb>
recentiores informia appellant, eò quod ad aliorum aſteriſmorum formas <lb></lb>
minimè reuocentur.</s>
        </p>
        <p type="main">
          <s id="s.001754"><arrow.to.target n="marg145"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001755"><margin.target id="marg145"></margin.target>145</s>
        </p>
        <p type="main">
          <s id="s.001756">Summa 4. cap. 1. <emph type="italics"></emph>(In Aſia igitur plurimi ex Parnaſſo vocato monte videntur <lb></lb>fluentes)<emph.end type="italics"></emph.end> rectè dubitat Alexander, qua ratione mons Parnaſſus ab Ariſt. po­<lb></lb>
natur in Aſia, cum certò certius conſtet, ipſum in Græcia Europæ regione <lb></lb>
ſitum eſſe. </s>
          <s id="s.001757">fortè legendum eſt, vt vult Vicomercatus, ex Paropameſſo, non <lb></lb>
autem ex Parnaſſo, quamuis Græci codices aduerſentur; Paropameſſum <lb></lb>
<expan abbr="namq;">namque</expan> Plinius, &amp; Strabo in Aſia collocant, <expan abbr="voluntq́">voluntque</expan>; ipſum eſſe iugum quod­<lb></lb>dam montis Caucaſi: Caucaſum autem ſupra Pontum oriri, &amp; <expan abbr="vſq;">vſque</expan> ad Hir­<lb></lb>
canum, &amp; vltra mare per totam Aſiam ſe proferre, tradunt veteres Geo­<lb></lb>
graphi. </s>
          <s id="s.001758">vide Theſaurum geographicum Abrahami Ortelij. </s>
          <s id="s.001759">Strabo lib. 15. <lb></lb>
ſic: Indiam à ſeptentrione Tauri extrema terminant, ab Ariana vſque in <lb></lb>
orientale mare, quæ extrema indigenæ particulatim nominant Poropamiſ­


<pb pagenum="98" xlink:href="009/01/098.jpg"></pb>ſum, Emodum, Imauum, &amp; alijs nominibus: Macedones verò Caucaſum <lb></lb>
vocant.</s>
        </p>
        <p type="main">
          <s id="s.001760"><arrow.to.target n="marg146"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001761"><margin.target id="marg146"></margin.target>146</s>
        </p>
        <p type="main">
          <s id="s.001762">Ibidem <emph type="italics"></emph>(Apparet mare, quod eſt extra)<emph.end type="italics"></emph.end> intelligit illud mare <expan abbr="Oceanũ">Oceanum</expan>, quod <lb></lb>
Arabiam, ac Perſiam alluit, <expan abbr="Indicoq́">Indicoque</expan>; Oceano committitur: <expan abbr="quodq́">quodque</expan>; à pri­<lb></lb>
ſcis Geographis Rubrum mare appellatur, cuius alterum Rubrum mare, <lb></lb>
quod inter Africam, &amp; Arabiam ſe inſinuat, eſt quidam ſinus, quem nunc <lb></lb>
communiter omnes Rubrum mare appellant. </s>
          <s id="s.001763">de illo inquam meritò intel­<lb></lb>
ligit Alexander, non de hoc Aegyptiaco, cum ex aſpectu illius à monte Pa­<lb></lb>
ropameſſo, ſequatur ipſum eſſe editiſſimum, quod non ſequeretur ex altero <lb></lb>
ob illius propinquitatem. </s>
          <s id="s.001764">Dixit autem mare, quod eſt extra, ideſt extra <lb></lb>
terram habitatam, ad diſtinctionem maris Mediterranei, quod eſt intra <lb></lb>
terram habitatam, ac propterea Mediterraneum dictum eſt.</s>
        </p>
        <p type="main">
          <s id="s.001765"><arrow.to.target n="marg147"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001766"><margin.target id="marg147"></margin.target>147</s>
        </p>
        <p type="main">
          <s id="s.001767">Ibidem <emph type="italics"></emph>(Ex hoc igitur fluunt &amp; alij fluuij, &amp; Bactrus, &amp; Choaſpes, &amp; Ara­<lb></lb>
xes. </s>
          <s id="s.001768">ab hoc autem abſcinditur Tanais pars exiſtens in Meotidem paludem fluit au­<lb></lb>
tem, &amp; Indus ex ipſo, omnium fluuiorum fluxio maxima)<emph.end type="italics"></emph.end> hæc omnia ſunt falſa, <lb></lb>
&amp; impoſſibilia; nam cum Bactrus Bactrianam regionem irriget, quæ eſt vl­<lb></lb>
tra Perſiam, Choaſpes verò Perſiam ipſam, Indus <expan abbr="deniq;">denique</expan> in India oriatur: <lb></lb>
quì fieri poteſt, vt in Regionibus adeò inuicem diſſitis orti fluuij ab eodem <lb></lb>
<expan abbr="quoq;">quoque</expan> Paropameſſo monte ortum ducant. </s>
          <s id="s.001769">nec minus falſum eſt illud de Ta­<lb></lb>
nai, quod ſit quaſi ipſius Araxis ramus quidam, Tanais enim ex Riphæis <lb></lb>
<expan abbr="mõtibus">montibus</expan> Scythiæ delabitur in Meotidem paludem longè longius ab Araxi. <lb></lb>
</s>
          <s id="s.001770"><expan abbr="eumq́">eumque</expan>; terminum inter Europam, &amp; Aſiam Geographi conſtituunt, vnde <lb></lb>
Dionyſius Afer ſic cecinit:</s>
        </p>
        <p type="main">
          <s id="s.001771"><emph type="italics"></emph>Europam, <expan abbr="atq;">atque</expan> Aſiam Tanais diſterminat amnis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001772">verùm huiuſmodi errata Ariſt. <expan abbr="atq;">atque</expan> adeò Geographis illius temporis con­<lb></lb>
donanda ſunt, cum nondum Geographia ſatis exculta eſſet.</s>
        </p>
        <p type="head">
          <s id="s.001773"><emph type="italics"></emph>De altitudine montis Caucaſi.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001774"><arrow.to.target n="marg148"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001775"><margin.target id="marg148"></margin.target>148</s>
        </p>
        <p type="main">
          <s id="s.001776">Eod. cap. <emph type="italics"></emph>(Caucaſus autem maximus mons eſt eorum qui ad ori<expan abbr="ẽtem">entem</expan> æſtiua­<lb></lb>
lem, &amp; multitudine, &amp; altitudine ſigna autem altitudinis quidem, quia <lb></lb>
videtur &amp; à vocatis Profundis, &amp; à nauigantibus in Stagnum inſuper il­<lb></lb>
luſtrantur à Sole ipſius ſummitates, vſque ad tertiam partem nocte, &amp; ab <lb></lb>
aurora, &amp; iterum a veſpera)<emph.end type="italics"></emph.end> Caucaſus mons ſitus eſt inter mare Euxinum, &amp; <lb></lb>
Caſpium, ſupra Cholchidem, &amp; Iberiam regiones, vbi polus eleuatur 47. <lb></lb>
circiter grad. ac reſpectu Græciæ, &amp; maris Euxini vergit ad eam mundi pla­<lb></lb>
gam, vnde illis æſtiuo tempore Sol oritur. </s>
          <s id="s.001777">ait Ariſt. eum eſſe omnium mon­<lb></lb>
tium illius plagæ altiſſimum, quod probat primò, quia admodum à longè <lb></lb>
cernitur, <expan abbr="nimirũ">nimirum</expan> ab illo Euxini loco, qui Profunda vocatur, eò quòd à Nau­<lb></lb>
tis nuſquam ibi fundus reperiatur. </s>
          <s id="s.001778">&amp; præterea à Nauigantibus in Stagnum, <lb></lb>
ſiue in Meotidem paludem, quæ quidem loca minimùm diſtant a Caucaſo <lb></lb>
560. milliaribus. </s>
          <s id="s.001779">Secundò, probat il ius altitudinem ex eo, quòd ſummi­<lb></lb>
tates ipſius <expan abbr="vſq;">vſque</expan> ad tertiam partem nocte, &amp; veſperi à Sole illuſtrentur. </s>
          <s id="s.001780">Lo­<lb></lb>
cum hunc fusè pertractat eruditiſſimus Iacobus Mazonius ſectione 3. &amp; 4. <lb></lb>
de Comparatione Platonis, &amp; Ariſt. quo in opere plurima habet ex Mathe­<lb></lb>
maticis deſumpta, quibus naturalem Philoſophiam mirificè illuſtrat, <expan abbr="mani-feſtumq́">mani­<lb></lb>
feſtumque</expan>; reddit, quàm neceſſariæ ſint Mathematicæ ad philoſophicæ veri­


<pb pagenum="99" xlink:href="009/01/099.jpg"></pb>tatis inſpectionem. </s>
          <s id="s.001781">Is igitur ſect. </s>
          <s id="s.001782">3. cap. 5. de hoc Ariſt. loco ſie loquitur: <lb></lb>hic locus diligenter expendendus videtur tum quia difficillimus eſt, <expan abbr="tũ">tum</expan> quia <lb></lb>
multis anſam dedit reprehendendi Ariſt. tanquam puerilia effutientem. </s>
          <s id="s.001783">tex­<lb></lb>
tus <expan abbr="itaq;">itaque</expan> Ariſt. duplicem habet ſenſum; alter à quo non abhorret <expan abbr="Alexãder">Alexander</expan>; <lb></lb>
vt tertia illa pars ad montem referatur, quaſi dicat, quod antequam Sol ima <lb></lb>
montis illuſtret, illuminat illius cacumen <expan abbr="vſq;">vſque</expan> ad tertiam montis partem: <lb></lb>
ſed hæc Mazonij expoſitio nulla eſt, cuiuſlibet enim montis etiam medio­<lb></lb>
cris altitudinis Sol illuſtrat non ſolum tertiam partem, ſed &amp; dimidium, &amp; <lb></lb>
duas tertias, &amp; ferè totum, antequam ad planam illius baſim deſcendat. <lb></lb>
</s>
          <s id="s.001784">Ego ſic exponendum cenſeo, vt Ariſt. dicat, mane, ideſt initio Crepuſculi <lb></lb>
matutini, &amp; veſpere, ideſt, in fine Crepuſculi veſpertini ipſius tertiam par­<lb></lb>
tem illuminatam conſpici ab ijs, quorum horizonti tunc incipit, vel deſinit <lb></lb>
Crepuſculum; ex quibus illi neceſſariò reſpectu Caucaſi ſunt occidentales, <lb></lb>
quì manè hoc vident, vti ſunt ij, qui in Euxino, ſeu Ponto, &amp; Meotide naui­<lb></lb>
gant, vel loca proxima inhabitant: illi verò, qui in fine Crepuſculi veſper­<lb></lb>
tini hoc cernunt, neceſſariò reſpectu Caucaſi erunt orientales. </s>
          <s id="s.001785">Alter huius <lb></lb>
loci ſenſus eſt, ait Mazonius, vt non de tertia montis parte, ſed de tertia <lb></lb>
noctis portione loquatur, ita vt manè. </s>
          <s id="s.001786">v. g. initio tertiæ, &amp; vltimæ noctis <lb></lb>
parte, cacumen Caucaſi illuminetur. </s>
          <s id="s.001787">hæc ille. </s>
          <s id="s.001788">vbi animaduertendum expo­<lb></lb>
ſitionem <expan abbr="hãc">hanc</expan> parùm differre à noſtra modò allata, cùm <expan abbr="vtraq;">vtraque</expan> in idem tem­<lb></lb>
pus recidat; nam ſi dixerimus initio Crepuſculi matutini illuminari ter­<lb></lb>
tiam partem Caucaſi, tempus hoc coincidit cum initio tertiæ partis noctis, <lb></lb>
quantitas enim Crepuſculi in poli eleuatione 47. grad. qualem habet Cau­<lb></lb>
caſus, per totam æſtatem tres horas plus minus continet, vt patet ex tabu­<lb></lb>
la quantitatis Crepuſculi, quæ eſt apud Nonium, &amp; apud Clauium in ſphæ­<lb></lb>
ra vltimæ editionis; quæ quantitas reperiri geometrico calculo poteſt, vt <lb></lb>
docent Nonius, Clauius, &amp; Maginus lib. 10. primi mob. </s>
          <s id="s.001789">quod quidem trium <lb></lb>
circiter horarum tempus eſt tertia ferè noctis pars in ijs regionibus, quibus <lb></lb>
polus eleuatur 47. grad. ſiue ergo dicamus id contingere initio Crepuſculi, <lb></lb>
ſiue initio tertiæ partis noctis, erit idem tempus, trium ſcilicet horarum. <lb></lb>
</s>
          <s id="s.001790">ſi ergo, inquit Mazonius, ſequamur priorem declarationem, neceſſarium <lb></lb>
eſt dicere, quod ea tertia pars montis, quæ initio auroræ Solis lumine per­<lb></lb>
funditur, ſit ea montis altitudo, qua ipſe exuperat illam aeris regionem, <lb></lb>
vnde Crepuſculum incipit apparere. </s>
          <s id="s.001791">quo poſito aptè, ac ſagaciter altitudi­<lb></lb>
nem Caucaſi inueſtigat hoc pacto. </s>
          <s id="s.001792">præmittit autem ſeptem propoſitiones <lb></lb>
apud Mathematicos manifeſtas, quas ego miſſas facio cum non mihi neceſ­<lb></lb>
ſariæ videantur. </s>
          <s id="s.001793">poſtea ſic diſcurrit; His ergo ita ſe habentibus, dico nos in­<lb></lb>
uenire poſſe viam, qua ſaltem rudi Minerua, montis altitudinem comper­<lb></lb>
tam habeamus. </s>
          <s id="s.001794">ſi enim in principio Crepuſculi v. g. matutini (ita enim, vt <lb></lb>
ſupra annotaui intelligendus eſt Ariſt.) illuminatur tertia pars, neceſſarium <lb></lb>videtur tertiam illam partem ſupra eam regionem collocari, ex qua Cre­<lb></lb>
puſculum in planitie apparere incipit, ſed illa regio ex Alhazino, &amp; Vitell. <lb></lb>
de Crepuſculis milliaribus 52. à terra recedit, ergo duæ tertiæ montis par­<lb></lb>
tes, quæ Solem initio auroræ non vident, ſunt 52. milliaria ad perpendicu­<lb></lb>
lum, &amp; tertia alia pars illuminata eſt ad perpendiculum 26. milliaria: ita <lb></lb>
vt totius montis altitudo perpendicularis ſit 78. mill. ſed papè in quos acu­


<pb pagenum="100" xlink:href="009/01/100.jpg"></pb>leos imprudens me conieci? </s>
          <s id="s.001795">rident enim hoc Ariſt. dictum Mathematici, <lb></lb>
putant enim eum pueriliter lapſum eſſe. </s>
          <s id="s.001796">Cæterum ego pro præceptoris tu­<lb></lb>tela, dico eum ſequutum eſſe famam. </s>
          <s id="s.001797">hæc Mazonius, quorum nonnulla in­<lb></lb>
digent conſideratione cuiuſmodi, ſunt illa, quando dicit, neceſſarium vi­<lb></lb>
detur, quod ea pars ſupra eam regionem attollatur, vnde Crepuſculum in <lb></lb>planitie apparere incipit. </s>
          <s id="s.001798">videtur enim his verbis velle dicere, quod quan­<lb></lb>
do habitantibus planitiem, quæ eſt ad pedem montis Caucaſi, vel horizon­<lb></lb>
tem eiuſdem, incipit Crepuſculum, ijſdem etiam tunc tertia montis pars <lb></lb>
appareat illuminata; in quo ſenſu errat poſtea in colligenda montis altitu­<lb></lb>
dine, quamuis enim verum eſſet partem illuminatam eminere totam ſupra <lb></lb>
52. milliaria, non tamen ſequitur ipſam ſolam eminere, ſed alia etiam pars <lb></lb>
eminere poteſt, quod ſic geometricè demonſtrabo. </s>
          <s id="s.001799">deſcribatur enim figura <lb></lb>
<figure id="id.009.01.100.1.jpg" place="text" xlink:href="009/01/100/1.jpg"></figure>


<pb pagenum="101" xlink:href="009/01/101.jpg"></pb>illa, qua ad vaporum altitudines indagandas vtuntur Alhazenus, Vitellio, <lb></lb>
&amp; Clauius, in qua terræ globus eſt F L G E, regiò vaporum, &amp; exhalatio­<lb></lb>
num M X N T. horizon aſtronomicus O P. phyſicus Q R, tangens terram <lb></lb>
in puncto F, vbi etiam ponendus eſt huius horizontis habitator, vnà cum. <lb></lb>
</s>
          <s id="s.001800">Caucaſo F V. </s>
          <s id="s.001801">Sol A B C, qui initio Crepuſculi infra horizontem O P, depri­<lb></lb>
mitur gr. 18. vti ab Aſtronomis compertum eſt, hoc eſt, arcum D P, eſſe <lb></lb>
grad. 18. radius autem C I K, tangens terram, incipit illuminare halitus, <lb></lb>
qui ſunt ad K, in extremo horizonte ſenſibili F K. quique poſſunt videri ab <lb></lb>
oculo in F, ideſt ab huius horizontis habitatore. </s>
          <s id="s.001802">Cæterùm prædicti autho­<lb></lb>
res poſt longam ratiocinationem ex calculo planorum <expan abbr="triangulorũ">triangulorum</expan> tandem <lb></lb>
oſtendunt in triangulo H F K, latus H K, continere milliaria 3631. ex quo <lb></lb>detracta H L, ſemidiametro terræ, quæ eſt milliar. 3579. reliqua L K, ſum­<lb></lb>ma halituum eleuatio relinquatur 52. milliar. </s>
          <s id="s.001803">quibus ab ipſis demonſtra­<lb></lb>
tis, ſi H F, terræ ſemidiameter, quæ continet milliar. 3579. ponatur ſinus <lb></lb>
totus 100000. &amp; latus F K, ponatur tangens anguli ad H, quem prędicti au­<lb></lb>
thores probant eſſe grad. 8. 54. erit F K, tangens partium 15659. fiat igi­<lb></lb>
tur per 2. pro. </s>
          <s id="s.001804">trjang. </s>
          <s id="s.001805">rectil. </s>
          <s id="s.001806">Clauij;<lb></lb>
<arrow.to.target n="table4"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table4"></table.target>
          <row>
            <cell>vt H F, ſinus totus,</cell>
            <cell>ad milliar.</cell>
            <cell>ita tangens F K,</cell>
            <cell>ad milliar.</cell>
          </row>
          <row>
            <cell>100000.</cell>
            <cell>3579.</cell>
            <cell>15659.</cell>
            <cell>560.</cell>
          </row>
        </table>
        <p type="main">
          <s id="s.001807">&amp; inueniemus per auream regulam latus F K, continere milliar. </s>
          <s id="s.001808">560. quan­<lb></lb>
ta ſcilicet eſt diſtantia ab oculo noſtro ad exhalationes Crepuſculi initium <lb></lb>
efficientes. </s>
          <s id="s.001809">Conſideremus iam triangulum F K V, vt ipſius latus F V, quæ <lb></lb>
eſt Caucaſi altitudo, in milliaribus innoteſcat. </s>
          <s id="s.001810">iam ipſius latus F K, inno­<lb></lb>
tuit, angulus verò ad F, eſt rectus; at angulus ad K, ſic manifeſtabitur; in <lb></lb>
quadrilatero F K I H, quatuor anguli ſunt æquales 4. rectis ex 32. primi. </s>
          <s id="s.001811">duo <lb></lb>
autem F, &amp; I, ſunt recti ex 18. 3. ergo reliqui duo H, &amp; K, æquales erunt duo­<lb></lb>
bus rectis, quorum alter H, eſt gr. 17. 48. vt præditi Mathematici <expan abbr="oſtẽdunt">oſtendunt</expan>, <lb></lb>
reliquus igitur ad K, erit gr. 162. 12. vt compleat duos rectos. </s>
          <s id="s.001812">qui ſi detra­<lb></lb>
hatur à duobus rectis, qui ſunt deinceps ad lineam F K, reliquus angulus <lb></lb>
F K V, erit gr. 17. 48. ſi ergo latus F K, notum ponatur ſinus totus 100000. <lb></lb>
latus verò F V, tangens anguli noti, erit ipſa 32100. fiat igitur,<lb></lb>
<arrow.to.target n="table5"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table5"></table.target>
          <row>
            <cell>vt F K, ſinus totus,</cell>
            <cell>ad milliar.</cell>
            <cell>ita F V, tangens</cell>
            <cell>ad milliar.</cell>
          </row>
          <row>
            <cell>100000.</cell>
            <cell>560.</cell>
            <cell>32100.</cell>
            <cell>180.</cell>
          </row>
        </table>
        <p type="main">
          <s id="s.001813"><expan abbr="inueniemusq́">inueniemusque</expan>; latus F V, continere milliar. </s>
          <s id="s.001814">180. cuius pars F X, quæ eſt in­<lb></lb>
fra habituum altitudinem continet milliar. </s>
          <s id="s.001815">52. quibus detractis ex 180. re­<lb></lb>
manent 128. pro tota X V, quæ tota eſt ſupra vapores, nondum tamen illu­<lb></lb>
minata. </s>
          <s id="s.001816">vnde patet Mazonium erraſſe in colligenda hoc modo Caucaſi al­<lb></lb>
titudine, ex prima Crepuſculi illuminatione in horizonte Caucaſi facta, <lb></lb>
cum ex præmiſſo calculo conſtet partem montis F V, totam tunc temporis <lb></lb>
eſſe tenebroſam, quamuis ſuperet multò regionem vaporum, contrà quàm <lb></lb>
ipſe putabat, ſuperat enim eam milliar. </s>
          <s id="s.001817">128. quare duæ tertiæ montis erunt <lb></lb>
non 52. mill. vt ipſe ait, ſed mill. 180. &amp; proinde tota altitudo erit mill. 270. <lb></lb>
quod ſanè ridiculum eſt, cum nullius montis altitudo ſeſquimilliare tran­<lb></lb>
ſcendat. </s>
          <s id="s.001818">Quod ſi ſequamur alteram expoſitionem, vt nimirum Ariſtot. lo­<lb></lb>
quatur non de tertia montis parte, ſed noctis, ita vt dicat, circa initium <lb></lb>
tertiæ partis noctis apicem montis illuſtrari, altitudo eius erit tantum­


<pb pagenum="102" xlink:href="009/01/102.jpg"></pb>modo 180. quot continet latus F V. vt vidimus, quæ quamuis illa minor ſit, <lb></lb>
adhuc tamen abſurda eſt.</s>
        </p>
        <p type="main">
          <s id="s.001819">Si verò dixerimus Ariſt. intelligere hæc omnia, non reſpectu horizontis <lb></lb>
Caucaſi, ſed alterius, cuius habitator in principio ſui Crepuſculi tertiam <lb></lb>
Caucaſi partem iam illuſtratam videat, vti accideret ſi Caucaſus ſtatuere­<lb></lb>
tur in L K, vbi incipit Crepuſculum habitanti in F. tunc eſſet altitudo tanta, <lb></lb>
quanta colligit Mazonius, ſi tamen Ariſt. intelligatur de tertia montis par­<lb></lb>
te; eſt enim L K, altitudo habituum 52. mill. &amp; duæ tertiæ montis, quare <lb></lb>
totus mons erit 78. ſi autem intelligatur circa tertiam noctis partem, mon­<lb></lb>tis apicem illuminatum videri ab habitatore F, ſic altitudo eius erit tan­<lb></lb>
tummodo 52. mill. quæ tamen adhuc omnem veritatem nimium ſuperat. <lb></lb>
</s>
          <s id="s.001820">Cum ergo hinc inde ſequantur abſurda, putat Mazonium excuſandum eſſe <lb></lb>
Ariſtot. dicendo eum ſequutum eſſe famam, <expan abbr="loquutumq́">loquutumque</expan>; eſſe populariter. <lb></lb>
</s>
          <s id="s.001821">Verumenimuerò ſapientiores iudicent num rectè philoſophus, cuius eſt re­<lb></lb>
condita, <expan abbr="atq;">atque</expan> abdita docere, excuſetur, ſi dicatur, eum, popularem famam <lb></lb>
ſequutum eſſe.</s>
        </p>
        <p type="main">
          <s id="s.001822">Tandem monendus mihi Lector eſt, in demonſtratione Magini, quæ eſt <lb></lb>
apud Mazonium ſect. </s>
          <s id="s.001823">4. citati operis; aſſumi radium Solis tangentem terræ <lb></lb>
globum, qui cum horizonte faciat angulum gr. 18. quod falſum eſt, ſolus <lb></lb>
enim radius centralis, qui à centro Solis ad centrum terræ ducitur talem <lb></lb>
facit angulum, <expan abbr="atq;">atque</expan> hac de cauſa ipſe colligit altitudinem noſtra maiorem; <lb></lb>
noſtra eſt 270. mill. ſua verò 276. vbi etiam, ſicut &amp; nos aſſumit horizon­<lb></lb>
tem Caucaſi.</s>
        </p>
        <p type="main">
          <s id="s.001824">Aduertendum tandem Mazonium admodum aduerſantia loquutum eſſe, <lb></lb>
ſect. </s>
          <s id="s.001825">enim 3. demonſtratinè concludit altitudinem 76. mill. ſect. </s>
          <s id="s.001826">verò 4. ſi­<lb></lb>
mul <expan abbr="cũ">cum</expan> Magino demonſtratiuè pariter colligit altitudinem eiuſdem 276. m. <lb></lb>
</s>
          <s id="s.001827">quæ nimis ab inuicem diſcrepant, cum tamen <expan abbr="vtrobiq;">vtrobique</expan> demonſtret, &amp; ve­<lb></lb>
ritas ſit vna. </s>
          <s id="s.001828">At verò cauſa huius diſcrepantiæ eſt, quòd ſect. </s>
          <s id="s.001829">3. accipit Cre­<lb></lb>
puſculum non horizontis Caucaſi, ſed illius, in cuius extremitate orientali, <lb></lb>
vbi incipit Crepuſculum, Caucaſus ſitus ſit, <expan abbr="diſtetq́">diſtetque</expan>; ab habitatore 560. m. <lb></lb>
</s>
          <s id="s.001830">vt ſupra oſtendimus. </s>
          <s id="s.001831">ſect. </s>
          <s id="s.001832">verò 4. accipit horizontem ipſius Caucaſi, vt ex <lb></lb>
figura illic deſcripta videre eſt. </s>
          <s id="s.001833">ex hac igitur horizontum varia ſuppoſitio­<lb></lb>
ne, varia etiam altitudo colligitur, quamuis <expan abbr="vtrobiq;">vtrobique</expan> ex <expan abbr="vtraq;">vtraque</expan> ſuppoſitio­<lb></lb>
ne <expan abbr="vtramq;">vtramque</expan> altitudinem rectè concludat. </s>
          <s id="s.001834"><expan abbr="Atq;">Atque</expan> hæc de Caucaſo ſufficiant.</s>
        </p>
        <p type="main">
          <s id="s.001835"><arrow.to.target n="marg149"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001836"><margin.target id="marg149"></margin.target>149</s>
        </p>
        <p type="main">
          <s id="s.001837">Eodem cap. <emph type="italics"></emph>(Ex Pyreneo autem, hic autem est mons ad occidentem æquino­<lb></lb>ctialem in Gallia, fluunt iſter, &amp; Tarteſſus, iste quidem extra columnas, Iſter au­<lb></lb>tem per totam Europam in Pontum Euxinum)<emph.end type="italics"></emph.end> Ariſt. fortè ſequutus eſt Herodo­<lb></lb>tum, qui falsò tradit Iſtrum, ſine Dannbium ex Pyreneis defluere, nam Iu­<lb></lb>
ce clarius conſtat ipſum ex ijs Alpibus, quæ Heluetiorum montes dicuntur, <lb></lb>
propè Baſileam ex Adula monte ortum ducere. </s>
          <s id="s.001838"><expan abbr="neq;">neque</expan> verum eſt Tarteſſum, <lb></lb>
quem &amp; Bœtim alij nominant ex Pyreneis deſcendere. </s>
          <s id="s.001839">Tarteſſum hunc Ma­<lb></lb>
ginus putat eſſe Tagum, cui fauet vocabulorum qualiſcunque ſimilitudo. <lb></lb>
</s>
          <s id="s.001840">extra tamen columnas Herculis quiſquis ſit in Oceanum occidentale illa­<lb></lb>
bitur. </s>
          <s id="s.001841">Ignoſcenda ſunt iſta Ariſt. tunc enim Geographia <expan abbr="nondũ">nondum</expan> adoleuerat.</s>
        </p>
        <p type="main">
          <s id="s.001842"><arrow.to.target n="marg150"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001843"><margin.target id="marg150"></margin.target>150</s>
        </p>
        <p type="main">
          <s id="s.001844">Ad finem eiuſdem cap. <emph type="italics"></emph>(Et circa Liguſticam non minor Rhodano abſorbetur <lb></lb>
quidam fluuius, &amp; iterum egreditur ſecundum alium locum)<emph.end type="italics"></emph.end> <expan abbr="incompertũ">incompertum</expan> &amp; hoc <pb pagenum="103" xlink:href="009/01/103.jpg"></pb>Ariſt. vt ſuperiora, ob Geographiæ illius ſeculi imperfectionem, nuſquam <lb></lb>
enim in tota Liguria quidpiam tale reperitur.</s>
        </p>
        <p type="head">
          <s id="s.001845"><emph type="italics"></emph>De Terræ rotunditate.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001846"><arrow.to.target n="marg151"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001847"><margin.target id="marg151"></margin.target>151</s>
        </p>
        <p type="main">
          <s id="s.001848">Svmma 4. cap. 2. quod eſt de permutatione, &amp; viciſſitudine aquarum, <lb></lb>
&amp; continentis. </s>
          <s id="s.001849">Pergratum Lectori fore exiſtimaui, nec alienum ab <lb></lb>
inſtituto, ſi occaſione huius permutationis maris, ac terræ, rem ex­<lb></lb>poſuero ſcitu digniſſimam, quam pridem obſeruare cœpi, ac in dies <lb></lb>
obſeruo, præſertim cum nullus præteritorum ſcriptorum, quod ſciam, eam <lb></lb>
literis mandauerit: Terræ ſcilicet totius molem paulatim reduci ad perfe­<lb></lb>
ctam ſphæricitatem, ita vt aliquando neceſſe ſit futurum ipſam à mari inun­<lb></lb>
dari, <expan abbr="atq;">atque</expan> omninò inhabitabilem reddi. </s>
          <s id="s.001850">Primum igitur illud ex ſacris lite­<lb></lb>
ris ſtatuendum, orbem terræ in ſuo primordio fuiſſe ab opifice rerum om­<lb></lb>
nium, figura ſphærica donatum, hoc eſt <expan abbr="abſq;">abſque</expan> montium eminentijs, atque <lb></lb>
vallium depreſſionibus. </s>
          <s id="s.001851">quod patet ex eo, quia <expan abbr="tũc">tunc</expan> tota Mari obtegebatur, <lb></lb>
ita vt minimè apta eſſet animantibus ad inhabitandum. </s>
          <s id="s.001852">redditam verò ha­<lb></lb>
bitabilem, cum ipſius conditor <expan abbr="quãdam">quandam</expan> ipſius partem humiliorem, &amp; quan­<lb></lb>
dam eminentiorem effeciſſet; transferendo nimirum maximam terræ por­<lb></lb>
tionem ex vno loco in alium, vnde illic maris concauitas, iſtic verò mon­<lb></lb>
tium ſublimitas emerſit. </s>
          <s id="s.001853">quo facto aquæ omnes in loca illa decliuiora ſua <lb></lb>
ſpontè receſſerunt, quæ aquarum congregatio Mare appellatum eſt. </s>
          <s id="s.001854">Hine <lb></lb>
nonnulli auctores grauiſſimi aſſerere non dubitarunt, montes <expan abbr="cõflatos">conflatos</expan> fuiſ­<lb></lb>
ſe ex terra illa, quæ locum illum occupabat, quem poſtea maria inuaſerunt. <lb></lb>
</s>
          <s id="s.001855">quæ cum ita ſint. </s>
          <s id="s.001856">ſequitur terram <expan abbr="nũc">nunc</expan> eſſe extra naturalem ſuam figuram, &amp; <lb></lb>
propterea in quodam ſtatu violento, <expan abbr="violẽtum">violentum</expan> autem <expan abbr="nullũ">nullum</expan> <expan abbr="perpetuũ">perpetuum</expan>. </s>
          <s id="s.001857">præ­<lb></lb>
terea cum terra ſit grauior quàm aqua, nulla ratione deberent terræ partes <lb></lb>
ſuperiores a quæ ſuperficiem ſuperare, cuius tamen <expan abbr="contrariũ">contrarium</expan> accidit, nam <lb></lb>
ſuperficies ipſa terræ, &amp; multò magis <expan abbr="mõtana">montana</expan> loca ſuperficiem maris cuiuſ­<lb></lb>
uis non parum ſuperant; quæ altera violentia terræ, &amp; aquæ ineſt, &amp; ideò <lb></lb>
minimè mirum eſt, imò <expan abbr="vtriuſq;">vtriuſque</expan> naturæ valdè conueniens terram redire ad <lb></lb>
priſtinam, ac primigeniam figuram, ex qua conſectarium erit aquam <expan abbr="quoq;">quoque</expan> <lb></lb>
ſuam pariter illam ſibi primæuam recuperaturam eſſe figuram. </s>
          <s id="s.001858">cauſam au­<lb></lb>
tem reſtauratricem huius terrenæ <expan abbr="rotũditatis">rotunditatis</expan> eſſe aquas tum pluuiales, tum <lb></lb>
fluuiales iamdiù obſeruauimus, vt ex ſequentibus obſeruationibus patebit.</s>
        </p>
        <p type="main">
          <s id="s.001859">Primò, videmus flumina quotidie montium radices corrodere, ac quaſi <lb></lb>
ſuffodere, ita vt paſſim ex hoc, vel illo monte magnas faciant ruinas, ac prę­<lb></lb>
cipitia, <expan abbr="atq;">atque</expan> hinc inde prærupti appareant montes, vt meritò legamus apud <lb></lb>Iob cap. 14. alluuione paulatim terra conſumitur. </s>
          <s id="s.001860">humum porrò illam ex <lb></lb>
montibus delapſam ſemper ad loca humiliora fluuij ſecum detrahunt. </s>
          <s id="s.001861">Ex <lb></lb>
continua etiam hac inter montes corroſione facta manifeſtè apparet, flumi­<lb></lb>
num alueos in montanis modò eſſe humiliores quàm olim, quamuis contra­<lb></lb>rium accidat alueis fluuiorum per plana decurrentium, qui modò altiores <lb></lb>
ſunt <expan abbr="quã">quam</expan> exordio mundi, vt paulò poſt <expan abbr="oſtẽdam">oſtendam</expan>. </s>
          <s id="s.001862">Illud autem liquidò apparet <lb></lb>
ex ſignis, ſeu ſymbolis, ſeu ex ſimilitudine terræ, aut lapidis, quæ in altiſſimis <lb></lb>
fluminum ripis hinc inde paſſim <expan abbr="vidẽtur">videntur</expan>, quæ indicio ſunt montes illos iam <pb pagenum="104" xlink:href="009/01/104.jpg"></pb>olim fuiſſe continuos, <expan abbr="atq;">atque</expan> vnam, <expan abbr="eandemq́">eandemque</expan>; terram <expan abbr="continẽtem">continentem</expan>, antequam <lb></lb>
flumen eos ab inuicem ſepararet; <expan abbr="flumenq́">flumenque</expan>; ipſum olim altius, vbi ſunt ſigna <lb></lb>
illa ambulaſſe; quemadmodum in Pyramo Ciliciæ amne obſeruauit Strabo, <lb></lb>
dum libro 12. de illius ripis hæc tradit, mira præterea eſt montis cæſura, <lb></lb>
per quam alueus ducitur; nam quemadmodum in petris per medium ſciſſis <lb></lb>
contingit, alterius partis depreſſioribus ita conuenire alterius partis emi­<lb></lb>
nentias, vt coniungi poſſint: ſic videre eſt imminentes flumini petras vtrin­<lb></lb>
que ferè <expan abbr="vſq;">vſque</expan> ad montis ſumma pertendentes duorum, triumuè iugerum <lb></lb>
ſpatio concauitates quaſdam eminentijs oppoſitas habere. </s>
          <s id="s.001863">hæc Strabo de <lb></lb>
vno, quod nos in pluribus obſeruauimus. </s>
          <s id="s.001864">Pręterea videmus quotidie pluuias <lb></lb>
aquas, idem quantum poſſunt efficere, ſuperficies montium, eorum maxi­<lb></lb>
mè, qui coluntur, perpetuò abſumentes, <expan abbr="atq;">atque</expan> ad loca conuallium deducen­<lb></lb>
tes. </s>
          <s id="s.001865">hinc videre eſt, montes cæteris duriores, vt ſunt lapidoſi, cæteris altio­<lb></lb>
res remanſiſſe; quippe qui magis &amp; pluuijs, &amp; fluuialibus aquis ſua duritie <lb></lb>
obſtiterunt. </s>
          <s id="s.001866">idem montani incolæ omnes confirmant, qui omnes aiunt ſibi <lb></lb>
hanc montium demolitionem iampridem innotuiſſe, ex eo quod nonnulli <lb></lb>
montes olim ſibi impedimento erant, ne arcem, turremuè in vlteriore mon­<lb></lb>
te ſitam conſpicerent, quam deinde plures poſt annos intermedio monte <lb></lb>
depreſſo, commodè videbant. </s>
          <s id="s.001867">Ad hæc; antiqua in montium verticibus con­<lb></lb>
ſtituta ædeficia, propterea intercidunt, quia terra hinc, &amp; inde ab aquis <lb></lb>paulatim conſumpta, <expan abbr="deorſumq́">deorſumque</expan>; delapſa, fundamenta ipſorum nuda primò <lb></lb>
relinquit; deindé terra etiam ipſa, qua fundamenta innitebatur ſenſim de­<lb></lb>
lapſa, ipſa <expan abbr="quoq;">quoque</expan> fundamenta vnà cum toto ædeficio neceſſe eſt collabi, hu­<lb></lb>
ius ſigna infinita propemodum videri poſſunt; vnum tamen, quod toti orbi <lb></lb>conſpicuum eſt, non ommittam; Capitolium videlicet Romanum, cuius <lb></lb>
modo fundamenta tota extant, quæ olim altè ſub terram deſcendebant. </s>
          <s id="s.001868">vi­<lb></lb>
de pulcherrimam hac de re tractationem apud Georgium Agricolam lib. 3. <lb></lb>
cap. 1. qui amplius addit illud, quod &amp; mihi maximè probatur; flumina ni­<lb></lb>
mirum producere montes, <expan abbr="collesq́">collesque</expan>; hoc modo; vult enim initio mundi non <lb></lb>
extitiſſe tot particulares montes ab inuicem diſcretos, ſed fuiſſe perpetua <lb></lb>
quædam terræ iuga eminentia quidem, ſed non tot vallibus diſſecta: v. g. <lb></lb>
mons noſter Apenninus erat iugum, ſiue dorſum quoddam terræ eminens <lb></lb>
quidem, ſed nullis vallibus in tot particulares colles, aut montes diſſectum; <lb></lb>
ſed poſtquam flumina à ſummitate ipſius deorſum fluere cœperunt; paula­<lb></lb>
tim corrodentes humum in dies magis, ac magis effecerunt valles, <expan abbr="atq;">atque</expan> hac <lb></lb>
ratione in colles, <expan abbr="montesq́">montesque</expan>; plurimos totus Apenninus diuiſus eſt. </s>
          <s id="s.001869">hæc de <lb></lb>
montibus ſufficiant, nunc ad plana deſcendamus.</s>
        </p>
        <p type="main">
          <s id="s.001870">Contrarium igitur omninò accidere videmus in planis, quoniam eædem <lb></lb>
aquæ, quæ ex montibus quotidie terram ſecum deducunt, eam ad humilio­<lb></lb>
ra loca, vt ſunt plana, &amp; campeſtria, ſiue ibi ſint maria, ſiue arida, compor­<lb></lb>
tant, <expan abbr="eamq́">eamque</expan>; ibidem deponunt. </s>
          <s id="s.001871">hinc videmus antiqua ædeficia in planis locis <lb></lb>
exſtructa, eſſe iam penè tota ſepulta, contra quam in montanis, cuius exem­<lb></lb>
plum habes etiam Romæ propè ipſum Capitolium, in Arcu triumphali Sep­<lb></lb>
timij, qui iam ferè totus ruinoſa vndique terra obruitur. </s>
          <s id="s.001872">ſic Pantheon. </s>
          <s id="s.001873">ſic <lb></lb>
etiam templa Epiſcopalia, quæ <expan abbr="plerunq;">plerunque</expan> ſatis peruetuſta ſunt, admodum <lb></lb>
infra terram conſpiciuntur. </s>
          <s id="s.001874">Idem affirmant cœmentarij, &amp; architectores <pb pagenum="105" xlink:href="009/01/105.jpg"></pb>omnes, quibus <expan abbr="vbiq;">vbique</expan> terrarum, dum in planis ædeficiorum fundamenta ex­<lb></lb>
canant, occurrit primò terra quædam, quam ipſi motam appellant, quæ li­<lb></lb>
gnis, ruderibus, ferramentis, numiſmatis, ſepulturis, <expan abbr="varijsq́">varijsque</expan>; rebus per­<lb></lb>
mixta eſt; qua eruta, reperitur terra alia, quam nunquam fuiſſe motam, ap­<lb></lb>
paret, ex eo quod ſolida, ac benè compacta ſit, neque vllis externis rebus, <lb></lb>
præſertim artificiatis admixta, terra illa, quam motam dicunt, variam va­<lb></lb>
rijs in locis ſortita eſt altitudinem, prout aquæ plurimum, vel minimum <lb></lb>
montanæ terræ huc, vel illuc comportarunt: alicubi vt hic Parmæ erit ſex <lb></lb>
vlnarum, alibi viginti, vt Mutinæ; alibi triginta, vt Romæ, nonnullis in lo­<lb></lb>
cis. </s>
          <s id="s.001875">Comprobatur tandem hæc noſtra obſeruatio ex arte illa, qua per eaſ­<lb></lb>
dem fluuiales aquas ſolent, tam loca depreſſiora per aggerationem paula­<lb></lb>
tim replere, <expan abbr="atq;">atque</expan> eleuare: quàm etiam altiora per aquarum earumdem cor­<lb></lb>
roſionem deprimere. </s>
          <s id="s.001876">qua in arte exercitatiſſimum P. Auguſtinum Spernac­<lb></lb>
ciatum noſtræ Societatis videmus modo de mandato Summi Pontificis Pa­<lb></lb>
dum, ac Renum Bononienſem ob aggerationem ſtagnantes in mari emitte­<lb></lb>
re; cui totus hic noſter diſcurſus maximè probatur. </s>
          <s id="s.001877">Ex quibus omnibus ſe­<lb></lb>
quitur ſuperficiem terræ tam montium, quam planorum quotidie variari. <lb></lb>
</s>
          <s id="s.001878">illam nimirum deprimi, hanc attolli. </s>
          <s id="s.001879">vnde aliud maximum notandum ſe­<lb></lb>
quitur, videlicet hac tempeſtate non eſſe eandem agrorum ſuperficiem, quæ <lb></lb>
erat antiquitus, cum in montanis agris ſit multò humilior, in campeſtribus <lb></lb>
verò altior, quàm antiqua illa, ac primigenia; quapropter mirum videri <lb></lb>
non debet, ſi quorumdam locorum adeò immutata natura eſt, vt quæ olim <lb></lb>
generoſa vina ferebant, vel quouis alio eſſent prædita munere, adeò dege­<lb></lb>
nerauerint, vt &amp; vina, &amp; alia nullius modò valoris, vel in parua copia pro­<lb></lb>
ferant. </s>
          <s id="s.001880">Quod verò ad marium aggerationem ſpectat, dicimus ijſdem aquis <lb></lb>magnam arenarum copiam perpetuò importantibus, fieri aggerationem, <lb></lb>
hoc eſt littora quotidie magis creſcere, ſeu in mare ingredi, &amp; conſequen­<lb></lb>
ter mare recedere. </s>
          <s id="s.001881">quod primò Ariſt. teſtimonio in hoc cap. comprobatur, <lb></lb>
cum quo pariter ſentiunt veteres Geographi, &amp; Hiſtorici omnes. </s>
          <s id="s.001882">Ariſt. igi­<lb></lb>
tur in comprobationem huius adducit primò magnam Aegypti aggeratio­<lb></lb>
nem; pars enim illa Aegypti, quæ Delta, <expan abbr="Niliq́">Nilique</expan>; donum appellatur ab He­<lb></lb>
rodoto, ex arenis, &amp; limo, ex Aethyopiæ montibus ſimul cum Nilo in mare <lb></lb>
delabentibus, eſt conflata, <expan abbr="atq;">atque</expan> antiquo littori addita, cui locum paulatim <lb></lb>
mare ceſſit; <expan abbr="eſtq́">eſtque</expan>; propterea donum Nili appellata, quod ab ipſo illuc are­<lb></lb>
nas importante ſit facta. </s>
          <s id="s.001883">ſecundum, Ariſt. exemplum eſt Ammonia Regio, <lb></lb>
cuius humiliora loca. </s>
          <s id="s.001884">f. </s>
          <s id="s.001885">maritima, palam eſt, inquit, quod aggeratione facta, <lb></lb>
fiunt ſtagna, &amp; continens: ſuccedente autem tempore, ſtagnans aqua ob <lb></lb>
nouam aggerationem deſiccata eſt, &amp; iam annihilata. </s>
          <s id="s.001886">tertium eſt Meotidis <lb></lb>
Paludis; At verò, ait, &amp; quæ ſunt circa Meotidem Paludem creuerunt allu­<lb></lb>
uione fluuiorum tantum, vt multò minores magnitudine naues, nunc innare <lb></lb>
poſſint, quàm anno ab hinc ſexageſimo. </s>
          <s id="s.001887">quare ex hoc facilè eſt ratiocinari, <lb></lb>
quod &amp; primò, vt multa ſtagnorum, ita &amp; hoc opus eſt fluuiorum, &amp; tan­<lb></lb>
dem neceſſe eſt totum fieri ſiccum. </s>
          <s id="s.001888"><expan abbr="quartũ">quartum</expan> eſt illi Boſphorus Tracius; quod <lb></lb>
vnà cum præcedentibus ſatius eſt apud ipſum, vel potius apud eius expoſi­<lb></lb>
torem Vicomercatum videre, vt breuitati conſulatur. </s>
          <s id="s.001889">Accedit &amp; Plinij te­<lb></lb>ſtimonium, qui tradit multas terras naſci, non ſolum fluminum inuectu, ſed <pb pagenum="106" xlink:href="009/01/106.jpg"></pb>etiam marium receſſu; ſic mare ab Ambraciæ portu 10. millia paſſuum; ab <lb></lb>
Athenarum verò <expan abbr="quinq;">quinque</expan> millia, &amp; alijs in locis plus minuſuè receſſiſſe ſcri­<lb></lb>
bit. </s>
          <s id="s.001890">Huc facit locus quidam Strabonis ex lib. 12. de Pyramo Ciliciæ fluuio: <lb></lb>
ſic; montes verò egreſſus tantum limum in mare deducit, partim ex Ca­<lb></lb>taonia, partim ex Ciliciæ campis, vt huiuſmodi de eo oraculum feratur;</s>
        </p>
        <p type="main">
          <s id="s.001891"><emph type="italics"></emph>Tempus erit rapidis olim cum Pyramus vndis <lb></lb>In ſacram veniet congeſto litore, Cyprum:<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001892">hic enim fluuius è regione Cypri inſulæ in mari influit, &amp;c. </s>
          <s id="s.001893">hæc Strabo.</s>
        </p>
        <p type="main">
          <s id="s.001894">Verùm recentiora non deſunt exempla. </s>
          <s id="s.001895">Rauenna olim erat in extremo <lb></lb>
littore ſita, nunc paulatim aggeratione aucto litore, mare multum ab ea <lb></lb>
receſſit. </s>
          <s id="s.001896">Patauium pariter, vt fertur mare alluebat, quod modo 25. paſſuum <lb></lb>
millibus ab eo diſtat. </s>
          <s id="s.001897">Aæſtuarium ipſum Venetum, ob arenas à varijs flu­<lb></lb>
minibus in ipſum immiſſas adeò fundum extulit, vt vix amplius nauigatio­<lb></lb>
ni ſit aptum, <expan abbr="periculamq́">periculamque</expan>; ſit ne Venetiarum mirabilis locus, ex maritimo <lb></lb>
fiat terreſtris. </s>
          <s id="s.001898">demum exemplum ſit Bononienſium Renus, qui quamuis exi­<lb></lb>
guus ſit torrens, paucis tamen annis Padum ipſum, in quem immiſſus fue­<lb></lb>
rat arena ita repleuit, vt &amp; ſibi, &amp; Pado magno vicinorum agrorum damno <lb></lb>
viam in mare obſtruxerit. </s>
          <s id="s.001899">Cum igitur mare ob hanc ad aggerationem co­<lb></lb>
gatur ſe quotidie magis recipere, <expan abbr="fiatq́">fiatque</expan>; propterea alueus ipſius anguſtior, <lb></lb>
<expan abbr="atq;">atque</expan> elatior, neceſſe eſt etiam ipſam quoque maris aquam quotidie magis <lb></lb>
coanguſtari, <expan abbr="atq;">atque</expan> attolli, &amp; aliquando futurum, vt exundare incipiat. </s>
          <s id="s.001900">quod <lb></lb>
iam <expan abbr="pleriſq;">pleriſque</expan> in locis accidit, vt in littore Baltico, Danico, &amp; Hollandico, <lb></lb>
quibus in locis ſunt hac tempeſtate extructi prælongi, ac præalti aggeres <lb></lb>
contra maritimas innundationes: quibus antiquitus minimè fuiſſe opus hi­<lb></lb>
ſtoricorum, ac <expan abbr="Geographorũ">Geographorum</expan> ſilentium comprobat. </s>
          <s id="s.001901">Hoc igitur modo ter­<lb></lb>
ra, qua montes, <expan abbr="collesq́">collesque</expan>; conſtant paulatim ab aquis in maris concauitates <lb></lb>
deportata, cauſa eſt, vt mare ſenſim modo hac, modo illac, terræ ſuperfi­<lb></lb>
ciei ſuperfundatur, <expan abbr="terraq́">terraque</expan>; iterum, quemadmodum exordio mundi inhabi­<lb></lb>tabilis reddatur: quod tunc maximè accidet cum aquæ tam fluuiales, quàm <lb></lb>
pluuiæ, ſuper faciem terræ perpetuò diſcurrentes, totam illam montanam <lb></lb>
terram in priſtinum locum, vbi ab initio fuerat, <expan abbr="vndeq́">vndeque</expan>; ſublata fuit, reſti­<lb></lb>
tuerint; tunc terra erit iterum rotunda, &amp; ſphærica, hoc eſt ſuæ primigeniæ <lb></lb>
iterum figuræ reſtituetur: quapropter mare etiam rurſus ſicut initio mundi <lb></lb>
totam terræ faciem <expan abbr="circumquaq;">circumquaque</expan> innundabit, quod probare volebam.</s>
        </p>
        <p type="main">
          <s id="s.001902"><emph type="italics"></emph>Tantum æui mutare potest longæua vetuſtas.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001903">Hinc nonnulla colligi poſſunt non minus notatu, ac ſcitu, quàm præceden­<lb></lb>
tia digniſſima, quibus Ethnicorum Philoſophorum error redarguatur, ſides <lb></lb>
verò noſtra magis roboretur: mundum nimirum ab æterno neutiquam ex­<lb></lb>
titiſſe, vel ſaltem terram ab æterno non fuiſſe hac figura præditam, qua nunc <lb></lb>videmus, nec mundum perpetuò duraturum. </s>
          <s id="s.001904">nam ſi hæc montuoſa illi figu­<lb></lb>
ra ab æterno ineſſet, iampridem tota illa montium tuberoſitas fuiſſet ab <lb></lb>
aquis exæſa, &amp; conſumpta: <expan abbr="neq;">neque</expan> æterna erit, quia ſucceſſu temporis, vt pro­<lb></lb>
bauimus, reducetur ad rotunditatem, <expan abbr="atq;">atque</expan> à mari innundabitur, &amp; idcircò <lb></lb>
inhabitabilis, vnde neceſſariò mortalium genus interibit. </s>
          <s id="s.001905">Quapropter niſi <lb></lb>
igne illo, quem ſacræ literæ innuunt cataclyſmus ille præueniatur, aqua <lb></lb>
mundus interiturus eſſet. </s>
          <s id="s.001906">ſed de his hactenus.</s>
        </p>
        <pb pagenum="107" xlink:href="009/01/107.jpg"></pb>
        <p type="main">
          <s id="s.001907">Quoad magnum illud Diluuium, quod Ariſt. hoc capite exiſtimat poſt <lb></lb>
multa ſecula reuolui, hoc veritati eſſe conſentaneum argumento ſunt, ac <lb></lb>
pariter admirationi varia <expan abbr="cõchiliorum">conchiliorum</expan> genera, quæ tùm in Apennino mon­<lb></lb>
te, tùm in Alpibus obſeruaui; <expan abbr="Ìdemq́">Ìdemque</expan>; in alijs mundi partibus inueniri pu­<lb></lb>
to; præſertim in tam immenſa copia, <expan abbr="atq;">atque</expan> intra viſcera montium colloca­<lb></lb>ta, quæ nulla vis humana illuc contuliſſet, niſi temporibus cataclyſmi ebul­<lb></lb>
lientibus aquis maris ſuper terram facta fuiſſet hæc varia rerum maritima­<lb></lb>
rum cum terreſtribus commixtio: quæ quidem optimè ex Pomponio Mela <lb></lb>
comprobantur, qui libro 1. de Numidia ſic narrat: interius, &amp; longè ſatis <lb></lb>
à litore, ſi fides res capit, mirum admodum, ſpinæ piſcium, <expan abbr="Muricũ">Muricum</expan>, <expan abbr="Oſtreo-rumq́">Oſtreo­<lb></lb>
rumque</expan>; fragmenta, ſaxi atritu, vti ſolent fluctibus, &amp; non differentia mari­<lb></lb>
nis, infixæ cautibus anchoræ, <expan abbr="aliaq́">aliaque</expan>; huiuſmodi ſigna, &amp; veſtigia effuſi olim <lb></lb>
<expan abbr="vſq;">vſque</expan> ad ea loca pelagi, in campis nihil alentibus eſſe inuenirique narrantur. <lb></lb>
</s>
          <s id="s.001908">neque locus ille Ouid. Met. 15. extra rem:</s>
        </p>
        <p type="main">
          <s id="s.001909"><emph type="italics"></emph>Vidi ego, quod fuerat olim ſolidiſſima petra <lb></lb>Eſſe fretum, vidi factas ex æquore terras: <lb></lb>
Et procul à Pelago conchæ iacuere marinæ, <lb></lb>
Et vetus inuenta eſt in montibus anchora ſummis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001910">Nos autem Chriſtiani ad Noemi Diluuium iſta referre debemus.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.001911"><emph type="italics"></emph>Ex Secundo Meteororum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001912"><arrow.to.target n="marg152"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001913"><margin.target id="marg152"></margin.target>152</s>
        </p>
        <p type="main">
          <s id="s.001914">Cap. 1. ait multa eſſe maria, quæ ad inuicem non communicant. <lb></lb>
</s>
          <s id="s.001915">Eorum rubrum mare vnum eſſe; quod cum Oceano <expan abbr="Atlãtico">Atlantico</expan>, qui <lb></lb>
eſt extra Herculeum fretum ad occidentem parum videtur com­<lb></lb>
miſceri ſiue Ariſt. pro Rubro mari intelligat Oceanum illum, qui <lb></lb>
Arabiam, ac Perſiam alluit, ſiue illius ſinum, qui Arabiam, <expan abbr="atq;">atque</expan> Aethiopiam <lb></lb>
interluit, falſum eſt ipſum parum communicare cum occidentali Oceano, <lb></lb>
vt quotidianis Luſitanorum nauigationibus ad Indos patet. </s>
          <s id="s.001916">ſed meritò hoc <lb></lb>
Ariſtot. condonandum, cum tunc temporis nondum tota Africa eſſet certò <lb></lb>
circumluſtrata, <expan abbr="neq;">neque</expan> iter ab Hiſpania ad Indos maritimum, adeo nunc fre­<lb></lb>
quens, patefactum eſſet.</s>
        </p>
        <p type="main">
          <s id="s.001917"><arrow.to.target n="marg153"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001918"><margin.target id="marg153"></margin.target>153</s>
        </p>
        <p type="main">
          <s id="s.001919">Summæ 2. cap. 2. <emph type="italics"></emph>(Quapropter &amp; circa Orionis ortum maximè fit tranquilli­<lb></lb>
tas)<emph.end type="italics"></emph.end> quando Medici, Philoſophi, Poetæ, ac reliqui auctores loquuntur de <lb></lb>
ortu aſtrorum fixorum, aut conſtellationum, quæ ſunt in firmamento, vti <lb></lb>
eſt Orion (&amp; Canis, de quo poſtea) intelligunt ſemper de ortu ipſorum, qui <lb></lb>
fit matutino tempore, quando ſcilicet vel ſimul cum Sole, vel paulò ante <lb></lb>
Solem emergunt, ita vt videantur à nobis; qui ortus dicitur Coſmicus, tunc <lb></lb>
propriè, quando ſimul aſtrum cum Sole oritur; quando autem incipit appa­<lb></lb>rere mane ante Solem, dicitur ortus Heliacus. </s>
          <s id="s.001920">i. </s>
          <s id="s.001921">ſolaris, quia oritur quodam­<lb></lb>
modo ex radijs Solis, ſub quibus antea latebat. </s>
          <s id="s.001922">Aſtra verò inerrantia, &amp; <lb></lb>
planetæ Sole tardiores oriuntur <expan abbr="vtroq;">vtroque</expan> modo. </s>
          <s id="s.001923">nam cùm ipſa Sol, quippe il­<lb></lb>
lis velocior primum aſſequitur, ea ſuo lumine obtegit, <expan abbr="eſtq́">eſtque</expan>; hic occaſus eo­<lb></lb>
rum heliacus: cum verò eadem præterierit, ac poſt ſe reliquerit fit, vt mo­<lb></lb>
tu diurno toto cœlo conuerſo, mane ante Solem effulgeant, ſiue heliacè <lb></lb>oriantur: &amp; cum quotidie magis Sol ab illis recedat, ipſaque magis à Sole <pb pagenum="108" xlink:href="009/01/108.jpg"></pb>elongentur, fit, vt quotidie magis ortum Solis anticipent, &amp; citius mane au­<lb></lb>
te Solem videantur. </s>
          <s id="s.001924"><expan abbr="ſicq́">ſicque</expan>; tanto in dies citius, vt deinde media etiam nocte <lb></lb>
oriantur; tum ante mediam noctem poſtea paulò ante occaſum Solis. </s>
          <s id="s.001925">de­<lb></lb>
mum cum fuerint Soli oppoſita, occidente Sole oriantur, qui ortus dicitur <lb></lb>
Veſpertinus, vel Acronicus. </s>
          <s id="s.001926">poſtea oriuntur ſemper in die ante Solis occa­<lb></lb>
ſum, donec Sol ipſa iterum aſſequatur, <expan abbr="eaq́">eaque</expan>; radijs ſuis offuſcet, quod eſt he­<lb></lb>
liacè occidere; &amp; mox cum ipſo Sole occumbant, quod Acronicè eſt occi­<lb></lb>
dere. </s>
          <s id="s.001927">Totum porrò illud tempus, quo per diem oriuntur, non eorum ortui, <lb></lb>
ſed occaſui deputatur, eò quod non cernuntur oriri, vt ſequenti loco expli­<lb></lb>
cabitur. </s>
          <s id="s.001928">Quæ omnia adhibito Globo aſtronomico, in quo conſtellationes <lb></lb>
omnes depictæ ſunt, <expan abbr="eoq́">eoque</expan>; ad tui poli eleuationem conſtituto, appoſitoque <lb></lb>
Sole ſuo loco in Zodiaco, qui paulatim per Zodiacum orientem verſus gra­<lb></lb>diatur, &amp; interim diurno motu globus conuertatur, ad ſenſum manifeſta <lb></lb>
apparebunt. </s>
          <s id="s.001929">In ſumma auctores intelligunt de ortu, qui mane fit ante So­<lb></lb>
lem, quia tunc primum poſt diuturnas latebras incipit apparere. </s>
          <s id="s.001930"><expan abbr="nõ">non</expan> autem <lb></lb>
intelligunt de ortu Acronico, quia ante hunc ortum videbatur noctu, <expan abbr="itaq;">itaque</expan> <lb></lb>
ortu Acronico non fit noua apparitio; ideo de hoc non intelligunt. </s>
          <s id="s.001931">fit au­<lb></lb>
tem ortus hic Orionis, heliacus, &amp; matutinus, de quo Ariſt. hoc loco, &amp; alij <lb></lb>
auctores, noſtra hac tempeſtate paulò ante Solis ingreſſum in Cancrum, ſi­<lb></lb>
ue ante ſolſtitium æſtiuum circa 22. Iunij.<arrow.to.target n="marg154"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001932"><margin.target id="marg154"></margin.target>154</s>
        </p>
        <p type="main">
          <s id="s.001933">Eodem cap. <emph type="italics"></emph>(Incertus autem, &amp; moleſtus Orion eſſe videtur &amp; occumbens, &amp; <lb></lb>
oriens, quia in tranſmutatione temporis accidit occaſus, &amp; ortus, aſtate, aut hye­<lb></lb>
me, &amp; propter magnitudinem aſtri dierum ſit aliqua pluralitas)<emph.end type="italics"></emph.end> hoc loco Vico­<lb></lb>
mercatus ex ſententia aſtronomorum occaſum Orionis fieri autumni tem­<lb></lb>
pore, Sole Scorpionem obſidente docet, quod &amp; verba Ariſt. clarè ſignifi­<lb></lb>
cant, cum dicat ortum ipſius fieri æſtate; in tranſmutatione verò temporis, <lb></lb>
videlicet in autumno fieri occaſum. </s>
          <s id="s.001934">Porrò occaſus hic fieri incipit primum <lb></lb>
mane oriente Sole, <expan abbr="diciturq́">diciturque</expan>; occaſus coſmicus, quia dum Sol eſt in oriente, <lb></lb>
Orion eſt in occidente, &amp; infra orizontem cadit: deinde paulò ante Solis or­<lb></lb>
tum, ſed tamen nocturno tempore, ita vt occaſus eius videri poſſit, donec <lb></lb>
occidat parum poſt Solis occaſum, &amp; tandem cum Sole ipſo heliacè euane­<lb></lb>
ſcat. </s>
          <s id="s.001935">Scriptores autem ferè ſemper cum loquuntur de occaſu inerrantium <lb></lb>
ſyderum, de eo, qui noctu videatur, intelligunt: ſicuti ortum intelligunt <lb></lb>
eum, qui noctu fit, <expan abbr="noctuq́">noctuque</expan>; videtur. </s>
          <s id="s.001936">affixa <expan abbr="namq;">namque</expan> ſydera per ſex fermè men­<lb></lb>
ſes noctu oriuntur, <expan abbr="oririq́">oririque</expan>; ea conſpicimus, &amp; propterea totum illud tem­<lb></lb>
pus, ortui ipſorum deputamus: Reliquum verò <expan abbr="tẽpus">tempus</expan>, quo per diem oriun­<lb></lb>
tur, &amp; idcircò ortus illorum minimè apparet, nulla ratione ortui debuit <lb></lb>
aſcribi: totum verò tempus, quo noctu occidunt, &amp; occidere cernuntur, oc­<lb></lb>
caſui illorum meritò attribuitur. </s>
          <s id="s.001937">&amp; <expan abbr="quemadmodũ">quemadmodum</expan> temporis illius initium, <lb></lb>
quo primo de nocte apparere incipiunt, dicitur abſolutè ortus cuiuſuis ſy­<lb></lb>
deris; ſic etiam initium temporis illius, quo primum per noctem ea occide­<lb></lb>
re videmus, ſimpliciter occaſum appellamus.</s>
        </p>
        <p type="main">
          <s id="s.001938"><arrow.to.target n="marg155"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001939"><margin.target id="marg155"></margin.target>155</s>
        </p>
        <p type="main">
          <s id="s.001940">Eodem cap. <emph type="italics"></emph>(Eteſiæ autem flant post verſiones, &amp; Canis ortum)<emph.end type="italics"></emph.end> per verſio­<lb></lb>
nes intelligit tropicos, quod &amp; tropici etymon confirmat, <expan abbr="cũ">cum</expan> tropicus idem <lb></lb>
valeat, ac conuerſiuus. </s>
          <s id="s.001941">circa Canis ortum eadem ſunt notanda, quæ ſupra <lb></lb>de ortu Orionis annotaui; intelligit enim eum Canis ortum, qui mane fiat <pb pagenum="109" xlink:href="009/01/109.jpg"></pb>primum paulò ante Solis ortum, cum ſcilicet incipit apparere.</s>
        </p>
        <p type="main">
          <s id="s.001942">Cum porrò in cęlo ſit Canis maior, &amp; Canis minor, qui &amp; Procyon, ideſt <lb></lb>
Anticanis dicitur, exiſtimo Canem maiorem eſſe eum, qui vulgò Canicula <lb></lb>
nominatur, <expan abbr="ſoletq́">ſoletque</expan>; vehementes, ac noxios calores excitare. </s>
          <s id="s.001943">de quo etiam <lb></lb>
putò Ariſt. intelligere. </s>
          <s id="s.001944">eius porrò ortus in noſtra poli eleuatione quadra­<lb></lb>
ginta quinque graduum, circa diem tertium Auguſti contingit, Sole autem <lb></lb>
10. gradum Leonis occupante. </s>
          <s id="s.001945">Ex Magini tabulis ante ephemerides.</s>
        </p>
        <p type="main">
          <s id="s.001946"><arrow.to.target n="marg156"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001947"><margin.target id="marg156"></margin.target>156</s>
        </p>
        <p type="main">
          <s id="s.001948">Eodem cap. <emph type="italics"></emph>(Duobus enim exiſtentibus ſegmentis habitabilis regionis: vno <lb></lb>
quidem ad ſuperiorem polum, qui noſter eſt; altero ad alterum, &amp; ad meridiem: <lb></lb>
<expan abbr="eaq́">eaque</expan>, tympani ſpeciem habeant, talem enim figuram terræ excidunt ex centro ipſius <lb></lb>ductæ lineæ, &amp; faciunt duos conos, hunc quidem habentem baſim tropicum, alte­<lb></lb>
rum autem habentem baſim circulum ſemper manifestum, verticem autem in me­<lb></lb>
dio terræ. </s>
          <s id="s.001949">eodem autem modo ad inferiorem polum alij duo coni terræ ſegmenta fa­<lb></lb>
ciunt)<emph.end type="italics"></emph.end> vt benè duas haſce terræ portiones, quas ſolas habitabiles putat Ari­<lb></lb>
ſtot. concipias, <expan abbr="reliquaq;">reliquaque</expan> huius loci intelligas, inſpice ſequentem figuram. <lb></lb>
<figure id="id.009.01.109.1.jpg" place="text" xlink:href="009/01/109/1.jpg"></figure><lb></lb>
Maior circulus ſit cœlum, in quo polus L, articus; M, antarticus, ille eleua­<lb></lb>
tus ſupra noſtrum horizontem S N, 45. gradibus, iſte verò totidem infra <lb></lb>
depreſſus. </s>
          <s id="s.001950"><expan abbr="ſintq́">ſintque</expan>; diametri circuli ſemper apparentium maximi S R, necnon <pb pagenum="110" xlink:href="009/01/110.jpg"></pb>diametri ſemper occultorum maximi Y N: tropicorum item T Q, Cancri, <lb></lb>
X O, Capricorni, vt vides in figura. </s>
          <s id="s.001951">Terra ſit A B C H G F E D Z K. à cu­<lb></lb>
ius centro Z, educantur primo duæ lineæ rectæ Z R, Z S. ad circulum ſem­<lb></lb>
per apparentium maximum, quæ in terra tranſeant per puncta B, K. &amp; iun­<lb></lb>
gatur linea B K: iam vides conum S R Z, cuius baſis eſt circulus ſemper ap­<lb></lb>
parens S R, vertex autem Z, in centro terræ, vt ait Ariſtot. </s>
          <s id="s.001952">educantur nunc <lb></lb>
duæ aliæ rectæ ad tropicum Cancri Z T, Z Q, quæ in terra faciant puncta <lb></lb>
I, C, <expan abbr="iungaturq́">iungaturque</expan>; recta I C; hic pariter vides conum alterum T Q Z, cuius ba­<lb></lb>
ſis eſt circulus Cancri, vertex verò centrum terræ Z. conſidera iam figuram <lb></lb>
B K I C, inter duas rectas B K, I C, &amp; duos circuli terræ arcus contentam; <lb></lb>
hanc Ariſt. appellat tympanum vnum terræ habitabile, quod eſt ad Vrſam, <lb></lb>
ideſt in ſeptentrionali plaga, in qua ſumus nos: quæ quidem portio ſi conſi­<lb></lb>
deretur vt ſolida, &amp; à reliqua terra præciſa, erit corpus rotundum, <expan abbr="vtrinq;">vtrinque</expan> <lb></lb>
tamen duobus planis circulis ad inſtar tympani terminatum: Ductis dein­<lb></lb>
de ſimiliter alijs quattuor lineis à centro Z, verſus polum antarticum fit al­<lb></lb>
terum tympanum H D E G, auſtralis terræ habitabilis, vt in figura manife­<lb></lb>
ſtum eſt. </s>
          <s id="s.001953">fuiſſe autem huiuſmodi habitabilis terræ ſegmenta figuræ tympa­<lb></lb>
ni ſimilia, optimè declarant veteres figuræ geographicæ Ptolęmei, &amp; patet <lb></lb>
etiam ex longitudine, &amp; latitudine, vt benè ait Ariſt. quas Geographi por­<lb></lb>
tioni terræ habitabili attribuebant, longitudinem enim dixerunt eius di­<lb></lb>
menſionem ab occaſu ad ortum: latitudinem autem à ſeptentrione in meri­<lb></lb>
diem, eò quòd illa multò hac longior eſſet. </s>
          <s id="s.001954">Ex quibus apparet habitatam <lb></lb>
fuiſſe veluti Zonam, terram ab occaſu ad ortum præcingentem. </s>
          <s id="s.001955">quæ Zona <lb></lb>
ſi ſumatur cum ſoliditate, quam ambit, ab Ariſt. tympano aſſimilatur.</s>
        </p>
        <p type="main">
          <s id="s.001956"><arrow.to.target n="marg157"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001957"><margin.target id="marg157"></margin.target>157</s>
        </p>
        <p type="main">
          <s id="s.001958">Eodem cap. <emph type="italics"></emph>(Hæ autem habitari ſolæ poſſibiles: &amp; <expan abbr="neq;">neque</expan> vltra verſiones; vm­<lb></lb>
bra enim non <expan abbr="vtiq;">vtique</expan> eſſet ad Vrſam: nunc autem inhabitabilia prius fiunt loca, quàm <lb></lb>
ſubdeficiat, aut permutetur vmbra ad meridiem. </s>
          <s id="s.001959">Quæ autem ſub Vrſa, è frigore <lb></lb>
inhabitabilia)<emph.end type="italics"></emph.end> quod ait vltra verſiones, ideſt intra tropicos in ipſa ſcilicet <lb></lb>
Zona torrida, non poſſe habitari, falſum eſſe oſtendunt plurimæ regiones <lb></lb>
tam veteris, quam noui orbis, ſuperiori ſeculo patefactæ, in quibus magna <lb></lb>
in amœnitate, ac fertilitate, <expan abbr="ſummisq́">ſummisque</expan>; delicijs viuitur. </s>
          <s id="s.001960">Quoad vmbram il­<lb></lb>
lam, intellige meridianam. </s>
          <s id="s.001961">i. </s>
          <s id="s.001962">quam Sole circa meridiem exiſtente, nos qui <lb></lb>
Boreales ſumus, ſemper ad <expan abbr="ſeptẽtrionem">ſeptentrionem</expan> proijcimus. </s>
          <s id="s.001963">Quod ſi ad meridiem <lb></lb>
perrexerimus, occurret inhabitabilis (vt falsò putat) terra, prius quam. <lb></lb>
</s>
          <s id="s.001964">vmbra meridiana in Boream vergens deficiat. </s>
          <s id="s.001965">quæ ſigna ſunt noſtram habi­<lb></lb>
tationem eſſe citra Zonam torridam, in Boreali parte. </s>
          <s id="s.001966">Quæ autem ſub Vr­<lb></lb>
ſa, ideſt ſub polo arctico, ob nimium frigus inhoſpita omninò habetur, nam</s>
        </p>
        <p type="main">
          <s id="s.001967"><emph type="italics"></emph>Quod latus mundi nebulæ, <expan abbr="malusq́">malusque</expan>; <lb></lb>
Iupiter vrget.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.001968">Verumtamen, quæ ſub <expan abbr="vtroq;">vtroque</expan> polo partes ſunt adhuc incognitæ manent.</s>
        </p>
        <p type="main">
          <s id="s.001969"><arrow.to.target n="marg158"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001970"><margin.target id="marg158"></margin.target>158</s>
        </p>
        <p type="main">
          <s id="s.001971">Eodem cap: <emph type="italics"></emph>(Fertur autem, &amp; corona ſecundam hunc locum, videtur enim ſu­<lb></lb>per caput eſſe nobis, cum fuerit ſecundum meridianum)<emph.end type="italics"></emph.end> conſtellatio videlicet, <lb></lb>
quæ corona Ariadnæ dicitur, hæc cum in cœlo manifeſtè ſit Borealis, <expan abbr="no-ſtroq́">no­<lb></lb>
ſtroque</expan>; vertici noctu, quando meridianum pertranſit, incumbat: clarè indi­<lb></lb>
cat nos <expan abbr="quoq;">quoque</expan> eſſe Boreales.</s>
        </p>
        <p type="main">
          <s id="s.001972"><arrow.to.target n="marg159"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001973"><margin.target id="marg159"></margin.target>159</s>
        </p>
        <p type="main">
          <s id="s.001974">Eodem cap. <emph type="italics"></emph>(Et quidem ad latitudinem <expan abbr="vſq;">vſque</expan> ad inhabitabilia ſcimus habita-<emph.end type="italics"></emph.end>


<pb pagenum="111" xlink:href="009/01/111.jpg"></pb><emph type="italics"></emph>tam, hic enim propter frigus non amplius habitant, illic autem propter æſtum)<emph.end type="italics"></emph.end><lb></lb>
illic autem, ideſt ſub Zona torrida, compertum autem eſt nunc totam ferè <lb></lb>
torridam Zonam, &amp; quidem alicubi percommodè habitari, cuius cauſæ ſunt <lb></lb>
quatuor, quæ ipſum latuerunt. </s>
          <s id="s.001975">prima <expan abbr="eaq́">eaque</expan>; toti Zonæ torridæ communis, <lb></lb>
eſt perpetuum æquinoctium, quo Sol tantum ſupra, quantum infra terram <lb></lb>
immoratur. </s>
          <s id="s.001976">accedit, quòd Sol nocturno tempore maximè ad imum cœli fe­<lb></lb>
ratur, <expan abbr="plurimumq́">plurimumque</expan>; ab horizonte, <expan abbr="ſuperoq́">ſuperoque</expan>; hemiſpherio recedat. </s>
          <s id="s.001977">atque ob <lb></lb>
hanc ſolam rationem Campanus in ſua ſphæra Zonam hanc putat maximè <lb></lb>
eſſe habitabilem: quamuis hæc ſola cauſa, vt quotidiana docet experientia, <lb></lb>
non ſufficiat. </s>
          <s id="s.001978">ſecunda ſunt pluuiæ, quæ alicubi quotidie ſtata hora decidunt. <lb></lb>
</s>
          <s id="s.001979">tertia venti, qui veluti flabella quædam aerem agitant. </s>
          <s id="s.001980">quarta præalti mon­<lb></lb>
tes perpetuis niuibus obſiti. </s>
          <s id="s.001981">quæ quatuor torridam hanc paſſim refrigerant, <lb></lb>
atque habitabilem reddunt.</s>
        </p>
        <p type="main">
          <s id="s.001982"><arrow.to.target n="marg160"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.001983"><margin.target id="marg160"></margin.target>160.a</s>
        </p>
        <p type="main">
          <s id="s.001984">Summæ 2. cap. 3. de <expan abbr="vẽtis">ventis</expan> <emph type="italics"></emph>(Oportet autem de ſitu ſimul rationes ex deſcriptio <lb></lb>
ne conſiderare)<emph.end type="italics"></emph.end> ideſt rationes ventorum ex deſcriptione, ideſt in figura ali­<lb></lb>
qua, vt in ſequenti conſiderare; ſolet enim Ariſt. figuras, imò demonſtratio­<lb></lb>
nes ipſas Mathematicorum, deſcriptiones appellare, vt ſæpius in Logicis <lb></lb>
monuimus.</s>
        </p>
        <p type="main">
          <s id="s.001985"><emph type="italics"></emph>Deſcriptus ſit igitur, vt clarior res euadat horizontis circulus quapropter, &amp; <lb></lb>
rotundus)<emph.end type="italics"></emph.end> vt in ſequenti figura circulus A G B H, deſcriptus horizontem <lb></lb>
referret,</s>
        </p>
        <figure id="id.009.01.111.1.jpg" place="text" xlink:href="009/01/111/1.jpg"></figure>
        <pb pagenum="112" xlink:href="009/01/112.jpg"></pb>
        <p type="main">
          <s id="s.001986"><emph type="italics"></emph>Oportet autem ipſius alteram portionem intelligere, quæ nobis habitatur; quæ <lb></lb>
eodem modo diuidi poterit)<emph.end type="italics"></emph.end> ideſt oportet intelligere ipſius horizontis, vel ter­<lb></lb>
ræ habitatæ partem, quæ quamuis rotunda non ſit, poterit tamen, ac ſi ro­<lb></lb>
tunda eſſet in figura circulari repreſentari, <expan abbr="atq;">atque</expan> in plures partes eo modo, <lb></lb>
quo circulus ſecatur, ſecari.</s>
        </p>
        <p type="main">
          <s id="s.001987"><emph type="italics"></emph>Supponatur autem primò contraria ſecundum locum, eſſe plurimum diſtantia <lb></lb>
ſecundum locum; ſicut ſecundum ſpeciem contraria, plurimum diſtant ſecundum <lb></lb>
ſpeciem. </s>
          <s id="s.001988">plurimum autem diſtant ſecundum locum, quæ per diametrum opponuntur, <lb></lb>
ſit igitur vbi A, occidens æquinoctionalis, contrarius autem huic locus vltimus B, <lb></lb>
ortus æquinoctionalis)<emph.end type="italics"></emph.end> ideſt in ſequenti figura ducta diametro B A. in altera <lb></lb>
ipſius extremitate vbi A. ſit occaſus æquinoctialis, qui fit Sole exiſtente in <lb></lb>
alterutro æquinoctio; huic igitur per diametrum opponatur ortus æquino­<lb></lb>
ctialis in B. qui pariter contingit tempore æquinoctiorum: linea autem B A, <lb></lb>
refert ipſum æquatorem.</s>
        </p>
        <p type="main">
          <s id="s.001989"><emph type="italics"></emph>Alia autem diameter hanc perpendiculariter ſecet, cuius punctum illud, in quo <lb></lb>
G, ſit Vrſa: huic autem contrarium ex oppoſito illud, in quo H, meridies)<emph.end type="italics"></emph.end> hæc dia­<lb></lb>
meter erit ipſa linea meridiana. </s>
          <s id="s.001990">pro Vrſa verò intelligit ſeptentrionem, <lb></lb>
quod ibi ſit Vrſæ conſtellatio.</s>
        </p>
        <p type="main">
          <s id="s.001991"><emph type="italics"></emph>Id autem, in quo F, ortus æſtiualis; in quo verò E, occidens æſtiualis)<emph.end type="italics"></emph.end> quæ duo <lb></lb>
puncta iunguntur linea F E, quæ refert ſectionem tropici, Cancri cum ho­<lb></lb>
rizonte: ortus enim, &amp; occaſus æſtiualis contingunt Sole Cancri tropicum <lb></lb>
percurrente.</s>
        </p>
        <p type="main">
          <s id="s.001992"><emph type="italics"></emph>Id autem, in quo D, oriens hyemalis; vbi verò C, occidens hyemalis)<emph.end type="italics"></emph.end> linea au­<lb></lb>
tem D C, erit ſectio tropici Capricorni, &amp; horizontis; Sole enim hunc tro­<lb></lb>
picum attingente ortus, &amp; occaſus hybernus fiunt.</s>
        </p>
        <p type="main">
          <s id="s.001993"><emph type="italics"></emph>Ab F, autem ducatur diameter ad C, &amp; à D, ad E. quoniam igitur plurimum <lb></lb>
diſtantia ſecundum locum, contraria ſunt ſecundum locum: plurimum autem di­<lb></lb>
stantia, quæ ſecundum diametrum; neceſſarium eſt, &amp; flatuum hos inuicem con­<lb></lb>
trarios eſſe, <expan abbr="quicunq;">quicunque</expan> ſecundum diametrum exiſtunt. </s>
          <s id="s.001994">vocantur autem ſecundum po­<lb></lb>
ſitionem locorum venti ſic; Zephyrus quidem ab A, hoc enim eſt occidens æquino­<lb></lb>
ctialis. </s>
          <s id="s.001995">Boreas autem, &amp; Aparetias à G. hic enim Vrſa, contrarius autem huic <lb></lb>
Notus ab H. </s>
          <s id="s.001996">Meridies enim eſt hic, à quo flat, &amp; H, ipſi G, contrarium eſt; ſecun­<lb></lb>
dum enim diametrum ſunt. </s>
          <s id="s.001997">Ab F, autem Cæcias; hic enim oriens æſtiuus eſt; cui <lb></lb>
contrarius est, non qui flat ab E, ſed qui à C. Libs, iſte enim ab occidente hyemali <lb></lb>
flat; <expan abbr="estq́">estque</expan>, illi contrarius, quia ſecundum diametrum illi opponitur. </s>
          <s id="s.001998">Qui verò à D, <lb></lb>
Eurus, iſte enim ab horiente hyberno flat, vicinus existens Noto, vnde &amp; ſæpè Eu­<lb></lb>
ronoti flare dicuntur: <expan abbr="cõtrarius">contrarius</expan> autem huic, non qui à C. Libs, ſed qui ab E, quem <lb></lb>
vocant, hi quidem Argeſten, hi autem Olympium, alij verò Scironem; iste enim ab <lb></lb>
occidente æſtiuo flat, &amp; ſecundum diametrum ipſi ſolus opponitur. </s>
          <s id="s.001999">Venti igitur, qui <lb></lb>
ſecundum diametrum poſiti ſunt, &amp; quibus alij aduerſantur, ij ſunt. </s>
          <s id="s.002000">Alij autem <lb></lb>
ſunt, ſecundum quos non ſunt contrarij venti, ab I, quem vocant Traſciam, qui me­<lb></lb>dius eſt inter Argesten, &amp; Apparitiam, à K, autem, quem vocant Meſen, Medius <lb></lb>
enim eſt Cæciæ, &amp; Aparetiæ. </s>
          <s id="s.002001">Diameter autem K I, iuxta circulum ſemper conſpi­<lb></lb>
cuum eſſe ſolet, non tamen exactè)<emph.end type="italics"></emph.end> ideſt linea K I, ſolet in horizonte referre <lb></lb>
diametrum circuli omnium ſemper apparentium maximi, eo quod ſit ferè <lb></lb>ſub diametro illius, in qualibet enim ſphæra obliqua, ideſt, in qua polus ele­<pb pagenum="113" xlink:href="009/01/113.jpg"></pb>uatur, intelligunt Aſtronomi circulum quendam ſemper apparentium ma­<lb></lb>ximum, quem deſcribunt ex ipſo polo, tanquam centro, &amp; interuallo vſque <lb></lb>
ad horizontem, circa ipſum polum: hunc appellant ſemper apparentium, <lb></lb>
maximum, quia intra hunc alios quamplurimos concipiunt circa eundem <lb></lb>
polum, quorum minores ſemper ſunt polo propinquiores. </s>
          <s id="s.002002">huius igitur dia­<lb></lb>
metrum vult Ariſt. per lineam, quæ à K, in I, duceretur (quamuis non exa­<lb></lb>
ctè) repreſentari.</s>
        </p>
        <p type="main">
          <s id="s.002003"><emph type="italics"></emph>Contrarij autem non ſunt his ſtatibus, <expan abbr="neq;">neque</expan> ipſi Meſe, ſpiraret enim <expan abbr="vtiq;">vtique</expan> aliquis <lb></lb>
ab eo, in quo M. hoc enim illi eſt ſecundum diametrum; <expan abbr="neq;">neque</expan> Trafciæ ab N, enim, <lb></lb>
quod punctum per diametrum aduerſum illi eſt, ſpiraret. </s>
          <s id="s.002004">Niſi ab eo veniat, qui ta­<lb></lb>
men non longè progreditur ventus quidam, quem accolæ Phæniciam vocant. </s>
          <s id="s.002005">maxi­<lb></lb>
mè igitur præcipui, &amp; definiti venti hi ſunt: <expan abbr="hocq́">hocque</expan>, modo diſpoſiti)<emph.end type="italics"></emph.end> ſupradicta por­<lb></lb>
rò omnia ex ſequenti figura optimè poterunt intelligi, quam diligenti ope­<lb></lb>
ra ad mentem Ariſt. ex græcis codicibus reſtituere conatus ſum, cum ani­<lb></lb>maduerterem figuras valdè deprauatas paſſim apud <expan abbr="cõmentatores">commentatores</expan> reperiri. <lb></lb>
</s>
          <s id="s.002006">Porrò ad literam M, in figura ſcripſi ventum Libonotum, quem Ariſt. qui­<lb></lb>
dem non ponit propter ipſius paruitatem; imò apertè dicit Heleſpontum <lb></lb>
non habere contrarium: ſed feci, vt completum ventorum numerum, quem <lb></lb>
alij tradunt, haberemus.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002007"><emph type="italics"></emph>Ex Tertio Meteororum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002008"><arrow.to.target n="marg161"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002009"><margin.target id="marg161"></margin.target>160.b</s>
        </p>
        <p type="main">
          <s id="s.002010">Antequam textuum explicationem aggrediar, illud animaduerten-<lb></lb>
dum eſt, <expan abbr="vbicunq;">vbicunque</expan> interpretatio antiqua vtitur verbis, refractio, <lb></lb>
&amp; refrangere; ibi Vicomercatum in ſua interpretatione meritò, <lb></lb>
&amp; propriè vſum eſſe verbis; reflexio, &amp; reflecti: differunt enim <lb></lb>
valdè apud Opticos refractio, &amp; reflexio, vt etiam refrangere, &amp; reflectere. <lb></lb>
</s>
          <s id="s.002011">propterea optimè hoc loco Olympiodorus diſtinguit inter <foreign lang="grc">ανακλασιν, και <lb></lb>διακλασιν,</foreign> reflexionem, &amp; refractionem. </s>
          <s id="s.002012">Reflexio enim fit ex repercuſſo, vt <lb></lb>
quando lumen Solis incidens in aliquod ſpeculum, inde reſilit in oppoſitum <lb></lb>
parietem, illud reſilire eſt propriè perſpectiuis reflecti, vnde reflexio. </s>
          <s id="s.002013">Re­<lb></lb>
fractio autem fit ex tranſpectu: vt quando lapis, qui eſt in aqua, emittit <lb></lb>
fuam ſpeciem ad oculum, qui eſt in aere, tunc enim, quia ſpecies lapidis re­<lb></lb>
preſentatiua non tendit recta ad oculum, ſed in confinio aquæ, &amp; aeris fran­<lb></lb>
gitur, dicitur fieri refractio, &amp; refrangi, in refractione igitur requiruntur <lb></lb>duo media, per quæ fiat viſio, quæ ſint diuerſæ denſitatis, vt ſunt aqua, &amp; <lb></lb>
aer: vapor, exhalatio, &amp; aer: vitrum, &amp; aer, &amp;c. </s>
          <s id="s.002014">quando igitur videmus <lb></lb>
Solem, aut Lunam per vapores, aut exhalationes fit refractio, quia denſior <lb></lb>
eſt vapor, &amp; exhalatio, quam aer.</s>
        </p>
        <p type="main">
          <s id="s.002015">Notandum etiam Aream, de qua mox dicam explicari poſſe tam per re­<lb></lb>
flexionem, quàm per refractionem: per reflexionem, quia ſupponunt Philo­<lb></lb>ſophi eſſe in acre rorido innumera ſpecula parua inuicem valdè proxima, <lb></lb>ideſt guttulas, per quas reflectatur ad oculum noſtrum ſpecies ſyderis. </s>
          <s id="s.002016">per <lb></lb>refractionem verò, vt vult Vitellio, quia ſumit totum illum aerem humi­<lb></lb>
dum magis denſum eſſe aere paro, qui eſt circa oculos noſtros, &amp; hoc modo <lb></lb>conſtituit diuerſa media in denſitate, per quam fiat viſio; corpus inquam <pb pagenum="114" xlink:href="009/01/114.jpg"></pb>illud humidum denſius, &amp; aerem deinde circa oculum rarius. </s>
          <s id="s.002017">Vicomerca­<lb></lb>
tus igitur quamuis vtatur voce reflexionis in Halone, non tamen ex prædi­<lb></lb>ctis videtur reprehendendus.</s>
        </p>
        <p type="main">
          <s id="s.002018"><arrow.to.target n="marg162"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002019"><margin.target id="marg162"></margin.target>161</s>
        </p>
        <p type="main">
          <s id="s.002020">Summæ 2. cap. 2. De Areæ figura <emph type="italics"></emph>(Refrangitur autem à conſiſtente caligine <lb></lb>
circa Solem, aut Lunam viſus; quapropter non ex oppoſito ſicut iris, apparet. </s>
          <s id="s.002021"><expan abbr="Vn-diq;">Vn­<lb></lb>
dique</expan> autem ſimiliter refracto, neceſſe eſt circulum eſſe, aut circuli partem. </s>
          <s id="s.002022">ab co­<lb></lb>
dem enim ſigno ad idem ſignum æquales frangentur ſuper circuli lineam ſemper. </s>
          <s id="s.002023">ſit <emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.114.1.jpg" place="text" xlink:href="009/01/114/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>enim à puncto, in quo A, ad B, fracta, &amp; ea, quæ est <lb></lb>
A C B, &amp; quæ A F B, &amp; quæ A D B, æquales autem <lb></lb>&amp; hæ A C, A F, A D, inuicem. </s>
          <s id="s.002024">&amp; quæ ad B, inui­<lb></lb>
cem ſcilicet C B, E B, D B. &amp; protrahatur A E B, <lb></lb>
quare trianguli æquales, etenim ſuper æqualem, quæ <lb></lb>
eſt A E B, ducantur autem <expan abbr="perpẽdiculares">perpendiculares</expan> ad A E B, <lb></lb>
ex angulis; à C, quidem, quæ eſt C E; ab F, autem, <lb></lb>
quæ eſt F E; à D, autem, quæ eſt D E, æquales itaque <lb></lb>hæ, in æqualibus enim triăngulis, &amp; in vno plano om­<lb></lb>nes, ad rectum enim omnes ei, quæ eſt A E B. &amp; ad <lb></lb>
vnum punctum E, copulantur, circulus igitur erit <lb></lb>
deſcripta, centrum autem E. ſit autem B, quidem Sol, <lb></lb>
A, autem viſus, quæ autem eſt circa C D F, circun­<lb></lb>
ferentia nubes, à qua refrangitur viſus ad Solem)<emph.end type="italics"></emph.end><lb></lb>
quia ſuppono Aream, ſiue Halonem fieri per re­<lb></lb>
fractionem, vt vult etiam Vitellio, propterea <lb></lb>
<expan abbr="præmittẽdum">præmittendum</expan> eſt principium quoddam, quo tra­<lb></lb>
ctatio de refractione innititur; eſt autem huiuſ­<lb></lb>
modi; ea, quæ <expan abbr="vidẽtur">videntur</expan> per refractionem, ſiue ſub <lb></lb>
aliquo refractionis angulo, manentibus nobis &amp; <lb></lb>
aſtro, &amp; medio ijſdem in locis, non poſſunt vide­<lb></lb>
ri ſub diuerſo angulo à priori, nec per conſe<expan abbr="quẽs">quens</expan> <lb></lb>
alibi apparere. </s>
          <s id="s.002025">v. g. Sol (vt in præſenti figura) <lb></lb>
videatur ab oculo A, media nube C D F, ſub an­<lb></lb>
gulo refractionis B C A, vel B F A, &amp; alijs ſimilibus angulis in eadem nube; <lb></lb>
manente igitur oculo A, &amp; aſtro B, necnon nube C D E. eodem in loco, im­<lb></lb>
poſſibile eſt Solem videri ab eodem oculo ſub diuerſo angulo à priori, nec <lb></lb>
conſequenter alibi apparere, quam in B. </s>
          <s id="s.002026">Nunc ad textus declarationem, in <lb></lb>
quo continetur Geometrica demonſtratio rotunditatis Areæ, quam ſic bre­<lb></lb>
uiter prius veteres excogitarunt: Viderunt primò Solem in Area apparere <lb></lb>
in orbem, &amp; conſimiliter: hinc intulerunt neceſſe eſſe apparere etiam per <lb></lb>
conſimiles, ſiue æquales refractionis angulos; quia diuerſi anguli, diuerſam <lb></lb>
etiam <expan abbr="apparẽtiam">apparentiam</expan> efficiunt: atqui conſimiles, ſiue æquales refractionis an­<lb></lb>
gulos neceſſe eſt in circulum <expan abbr="cõſtitui">conſtitui</expan>, vt mox conſtabit; cauſa igitur rotun­<lb></lb>
ditatis huius, eſt angulorum refractionis æqualitas. </s>
          <s id="s.002027">Sed iam textum Ariſt. <lb></lb>qui geometricam huius rei continet demonſtrationem, explicemus. </s>
          <s id="s.002028">Suppo­<lb></lb>
nit igitur primò Ariſt. lineas viſuales à ſydere B, ad oculos noſtros A, per <lb></lb>
nubem roridam C D F, procedentes, in nube conſimiliter refrangi, ideſt <expan abbr="vn-diq;">vn­<lb></lb>
dique</expan> circa Solem, Lunamuè facere angulos refractionis æquales. </s>
          <s id="s.002029">quod etiam <pb pagenum="115" xlink:href="009/01/115.jpg"></pb>patet ex 48. 10. Vitellionis; vt in figura, in qua ſydus B, oculus A, nubes <lb></lb>
C D F, radij viſuales tres refracti in nube ſint B C A, B D A, B E A, facien­<lb></lb>
tes conſimilem refractionem, ideſt angulos refractos B C A, B D A, B E A, <lb></lb>
æquales in punctis C, D, F: <expan abbr="atq;">atque</expan> hoc eſt conſimilem facere refractionem. <lb></lb>
</s>
          <s id="s.002030">Supponit ſecundò lineas à ſydere ad nubem, vſque extenſas eſſe æquales, vt <lb></lb>
ſunt B C, B D, B F: ſimiliter reliquas tres à nube ad viſum A. pares eſſe C A, <lb></lb>
D A, F A. his ſuppoſitis, ſi deinde protrahatur recta A B, ab oculo ad ſydus, <lb></lb>exurgunt tria triangula omninò æqualia, &amp; ſimilia, cum duo latera vnius <lb></lb>
ſint æqualia duobus alterius <expan abbr="vtrunq;">vtrunque</expan> vtrique, &amp; angulus angulo, &amp; præterea <lb></lb>
baſis ſit communis; ideò per quartam primi ſunt omninò æqualia. </s>
          <s id="s.002031">ducan­<lb></lb>
tur nunc ex angulis C, D, F, tres perpendiculares ad rectam A B, quæ ſint <lb></lb>
C E, D E, F E, in figura; quæ tres neceſſariò erunt æquales, cum ſint ductæ <lb></lb>
ab angulis æqualibus æqualium triangulorum ad communem baſim, &amp; di­<lb></lb>
uident neceſſariò baſim in eodem puncto E, cum diuidant triangula æqua­<lb></lb>
lia proportionaliter; <expan abbr="eruntq́">eruntque</expan>; propterea hæ tres rectæ in eodem plano, quod <lb></lb>
in nube concipitur ex 5. 11. Quare ſi concipiamus ſuperficiem, ſiue planum <lb></lb>
delineari circa E, ad interuallum linearum æqualium C E, D E, F E, de­<lb></lb>
ſcriptus erit circulus per 9. tertij, cuius circumferentia C D F. </s>
          <s id="s.002032">Ex quibus <lb></lb>
patet tria illa puncta C, D, E, per quæ Sol tranſparet eſſe in orbem diſpoſi­<lb></lb>
ta. </s>
          <s id="s.002033">cauſa igitur rotunditatis Areæ, eſt ſimilitudo angulorum refractionis, <lb></lb>
quibus Sol tranſparet: vel ideo rotunda eſt, quia ſimiles anguli neceſſariò <lb></lb>
in orbem conſtituuntur, vt oſtenſum eſt. </s>
          <s id="s.002034">Eadem ratione omnia alia puncta <lb></lb>
eiuſdem <expan abbr="circũferentiæ">circunferentiæ</expan> ſunt puncta, per quæ Sol videtur refractè; &amp; hoc mo­<lb></lb>
do ad ſimilitudinem trium linearum A C B, A D B, A F B, refractarum, in­<lb></lb>
finitæ <expan abbr="vndiq;">vndique</expan> intelligendæ ſunt, quarum aliæ refrangantur in circunferentia <lb></lb>
prædicta, aliæ verò in alia periphæria maiori, aliæ etiam in minori, ita vt <lb></lb>
ex tota nube fiant refractiones circulares plurimæ, ex quibus in nube area <lb></lb>
conſtituatur. </s>
          <s id="s.002035"><expan abbr="Atq;">Atque</expan> hæc cur Halonis figura orbicularis videatur, rationem <lb></lb>
reddunt, <expan abbr="vnaq́">vnaque</expan>; textui lucem afferunt.</s>
        </p>
        <p type="head">
          <s id="s.002036"><emph type="italics"></emph>Summæ 2. cap. 4. De Iridis figura.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002037"><arrow.to.target n="marg163"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002038"><margin.target id="marg163"></margin.target>162</s>
        </p>
        <p type="main">
          <s id="s.002039"><emph type="italics"></emph>Qvod autem <expan abbr="neq;">neque</expan> circulum poſſibile ſit fieri Iridis, <expan abbr="neq;">neque</expan> maiorem ſemicir­<lb></lb>
culo portionem, &amp; de alijs accidentibus circa ipſam, ex deſcriptione <lb></lb>
erit conſiderantibus manifeſtum)<emph.end type="italics"></emph.end> In Logicis ſæpius monui Ariſt. per <lb></lb>
deſcriptiones intelligere geometricas demonſtrationes, quod <lb></lb>
etiam hoc loco confirmatur, vbi Geometrica demonſtratione quam deſcri­<lb></lb>
ptionem appellat, Iridis figuræ accidentia oſtendit; nimirum cur ſit quidem <lb></lb>
circularis, nunquam tamen circulus integer, imò <expan abbr="neq;">neque</expan> ſemicirculo vnquam <lb></lb>
maior, ſed tamen ſemicirculo minor.</s>
        </p>
        <p type="main">
          <s id="s.002040"><arrow.to.target n="marg164"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002041"><margin.target id="marg164"></margin.target>163</s>
        </p>
        <p type="main">
          <s id="s.002042">Ibidem <emph type="italics"></emph>(Hemiſphærio enim exiſtente ſuper horizontis circulum in quo A. cen­<lb></lb>
tro autem K, alio autem quodam oriente puncto, in quo G, ſi lineæ, quæ à K, ſecun­<lb></lb>
dum conum excidentes faciant velut axem lineam in qua G K, &amp; à K. ad M, co­<lb></lb>
pulatæ refrangantur ab hemiſphærio ad G, ſuper maiorem angulum, circuli circun­<lb></lb>
ferentiam incident lineæ, quæ à K, &amp; ſi quidem in ortu, aut in occaſu aſtri reflexio <lb></lb>
fiat, ſemicirculus ab <expan abbr="horizõte">horizonte</expan> aſſumetur ſuper terram factus. </s>
          <s id="s.002043">ſi autem ſupra, minor<emph.end type="italics"></emph.end> <pb pagenum="116" xlink:href="009/01/116.jpg"></pb><figure id="id.009.01.116.1.jpg" place="text" xlink:href="009/01/116/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>ſemper ſemicirculo, minus autem, <lb></lb>
cum in meridie fuerit aſtrum)<emph.end type="italics"></emph.end> quod <lb></lb>
ſupra monui, iterum moneo, <expan abbr="re-tinẽdam">re­<lb></lb>
tinendam</expan> vocem reflexionis, <expan abbr="quã-uis">quam­<lb></lb>
uis</expan> in antiqua tranſlatione lega­<lb></lb>
tur refractio, eſt enim apud om­<lb></lb>
nes in confeſſo Iridem fieri per <lb></lb>
reflexionem. </s>
          <s id="s.002044">Eſt igitur in ſupe­<lb></lb>
riori figura, quam textui, vt par <lb></lb>
erat reſtitui, horizon G K O. cuius centrum K. in quo eſt viſus noſter, <expan abbr="ſitq́">ſitque</expan>; <lb></lb>
hemiſphærium noſtrum in arcu G A M O, repræſentatum, <expan abbr="ſitq́">ſitque</expan>; nubes rori­<lb></lb>
da, in qua Iris appareat, vbi M, quod punctum M, nubem referens, in figu­<lb></lb>
ra ponitur in hemiſphærij ambitu, quod cœlum repræſentat, cum tamen <lb></lb>nubes parum à terra ſubuehatur; id enim ad demonſtrationem ferè perinde <lb></lb>
eſt. </s>
          <s id="s.002045">in oriente G, ſit aſtrum. </s>
          <s id="s.002046">ſi ergò lineæ viſuales à K, ad M, nubem tenden­<lb></lb>
tes reflectantur ſuper maiorem angulum M K G, ad G, erit reflexarum vna <lb></lb>
veluti M G. </s>
          <s id="s.002047">Porro omnes lineæ viſuales, quæ ad nubem M, incidunt, neceſ­<lb></lb>
ſariò, vt probabo, cadent in ambitum circularem. </s>
          <s id="s.002048">debemus enim innume­<lb></lb>ras lineas imaginari à K, in coni figuram excidentes, cuius vertex ſit in K, <lb></lb>
&amp; axis G K O, quas omnes repræſentat vna K M, <expan abbr="meliusq́">meliusque</expan>; repræſentabit, fi <lb></lb>
cogitemus axem G K O, circa polos G, O, manentes circumuolui, <expan abbr="ſecumq́">ſecumque</expan>; <lb></lb>
lineam K M, circumducere. </s>
          <s id="s.002049">in hac etiam giratione linea K M, tranſibit per <lb></lb>
omnes illas lineas, quas imaginabamur; <expan abbr="deſcribetq́">deſcribetque</expan>; conum, quem illæ con­<lb></lb>
formare debebant. </s>
          <s id="s.002050">In prædicta autem axis volutatione, extremum M, li­<lb></lb>
neæ K M, neceſſariò deſcribit circulum, qui eſt circulus Iridis, &amp; eſt baſis <lb></lb>
memorati coni.</s>
        </p>
        <p type="main">
          <s id="s.002051">Si igitur oriente, vel occidente aſtro fiat iris, Iris erit ſemicirculus, ideſt <lb></lb>
illa ſemiſſis circuli prędicti (quem horizon bifariam diuidit) quæ ſupra ter­<lb></lb>
ram extabit. </s>
          <s id="s.002052">ſi autem aſtrum eleuatum ſupra horizontem fuerit, quando fit <lb></lb>
iris, erit ſemper arcus Iridis ſemicirculo minor; <expan abbr="tuncq́">tuncque</expan>; minimus <expan abbr="cũ">cum</expan> aſtrum <lb></lb>
<expan abbr="meridianũ">meridianum</expan> <expan abbr="circulũ">circulum</expan> occupauerit. </s>
          <s id="s.002053">hęc tria ſunt, quæ deinceps <expan abbr="probãda">probanda</expan> recipit.</s>
        </p>
        <p type="main">
          <s id="s.002054"><arrow.to.target n="marg165"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002055"><margin.target id="marg165"></margin.target>264</s>
        </p>
        <figure id="id.009.01.116.2.jpg" place="text" xlink:href="009/01/116/2.jpg"></figure>
        <p type="main">
          <s id="s.002056">Ibidem <emph type="italics"></emph>(Sit enim in <expan abbr="oriẽte">oriente</expan> pri­<lb></lb>
mum vbi G, &amp; refracta ſit K M, <lb></lb>
ad G, &amp; planum erectum ſit in quo <lb></lb>
A, à triangulo in quo G K M, cir­<lb></lb>
culus igitur erit ſectio ſphæræ, qui <lb></lb>
maximus ſit in quo A, differet enim <lb></lb>nihil ſi quod<expan abbr="cŭq;">cŭque</expan> eorum, quæ ſuper <lb></lb>
G K, ſecundum triangulŭ K M G, <lb></lb>
erectum fuerit planum. </s>
          <s id="s.002057">lineæ igitur <lb></lb>ab ijs, quæ G, K, ductæ in hac ratio­<lb></lb>ne non conſtituentur ad aliud, &amp; <lb></lb>
aliud punctum, quàm ſemicirculi <lb></lb>
in quo A. </s>
          <s id="s.002058">Quoniam enim puncta <lb></lb>
G, K, data ſunt, &amp; quæ K M, vtique data erit; &amp; quæ M G, ad M K; datam igi­<lb></lb>
tur circunferentiam tanget M, fit <expan abbr="itaq;">itaque</expan> hæc in qua M N, quare ſectio circunferen-<emph.end type="italics"></emph.end>


<pb pagenum="117" xlink:href="009/01/117.jpg"></pb><emph type="italics"></emph>tiarum data eſt. </s>
          <s id="s.002059">apud autem aliud punctum, quam ipſius M N, circunferentiæ, ab <lb></lb>
ijſdem punctis, eadem ratio in eodem plano non conſiſtit)<emph.end type="italics"></emph.end> eorum omnium, quæ <lb></lb>
demonſtranda ſunt, præmittenda ſunt duo neceſſaria fundamenta. </s>
          <s id="s.002060">Primum <lb></lb>
eſt; ea, quæ videmus per reflexionem ſub quopiam angulo, manentibus no­<lb></lb>
bis ſpeculo, &amp; obiecto ijſdem in locis, non poſſunt videri ſub alio diuerſo <lb></lb>
angulo, nec alibi conſequenter apparere. </s>
          <s id="s.002061">v. g. in ſuperiori figura, quam <lb></lb>
textui reſtituimus exiſtente Sole in G, oculo in K, &amp; nube in M. ex qua ra­<lb></lb>dius Solis G M, reflectatur ad viſum in K, per <expan abbr="lineã">lineam</expan> M K, ſub angulo G M K, <lb></lb>
impoſſibile eſt manentibus illis, vt dixi, videri Solem in nube M, ſub diuer­<lb></lb>
ſo angulo à priori, nec alibi apparere. </s>
          <s id="s.002062">Alterum eſt apud Opticos vulga­<lb></lb>
tum; ea ſcilicet, quæ per reflexionem (de quorum numero eſt Iris) viden­<lb></lb>
tur, videri, tunc ſolum, quando angulus incidentiæ fuerit æqualis angulo <lb></lb>
reflexionis, quia tunc breuiſſimis lineis fit viſio; quibus ſoli, natura (ſi fieri <lb></lb>
<figure id="id.009.01.117.1.jpg" place="text" xlink:href="009/01/117/1.jpg"></figure><lb></lb>
poteſt) vtitur. </s>
          <s id="s.002063">v. g. in figura præſenti ſit ſpe­<lb></lb>
culum C D E, obiectum A, oculus B, linea in­<lb></lb>
cidentiæ eſt A D, &amp; angulus pariter inciden­<lb></lb>
tiæ eſt A D C. linea verò D B, eſt linea refle­<lb></lb>
xionis, &amp; angulus pariter reflexionis eſt B D­<lb></lb>
E, qui duo anguli niſi fuerint æquales, nun­<lb></lb>
quam videbitur obiectum A, ab oculo B, hinc <lb></lb>
eſt, quod aliquando poſito ſpeculo, obiectum <lb></lb>
quamuis illi aduerſum, à nobis pariter ante <lb></lb>
ſpeculum conſtitutis, videri nequit, quia ſci­<lb></lb>
licet in tali poſitione ſpeculi, obiecti, &amp; noſtri, nulla linea incidentiæ, ideſt, <lb></lb>
quæ ab obiecto in ſpeculum tendit, facere poteſt angulum cum ſpeculo, qui <lb></lb>
dicitur angulus incidentiæ, æqualem angulo illi, quem facit linea eadem re­<lb></lb>flexa à ſpeculo ad oculum, quem dicunt angulum reflexionis. </s>
          <s id="s.002064">Cum ergo in <lb></lb>
Iride videamus colorem Solis per reflexionem, tunc ſolum apparebit Iris, <lb></lb>
quando Sol, nubes, &amp; oculus fuerint in ea conſtitutione, qua radius <expan abbr="incidẽs">incidens</expan> <lb></lb>
nubi, &amp; radius à nube repercuſſus faciant pares angulos. </s>
          <s id="s.002065">Et quia quando <lb></lb>nubes roſcida perpendiculariter opponitur Soli, &amp; nobis, poſſunt fieri præ­<lb></lb>
dicti anguli æquales non in vno loco nubis, ſed in pluribus, conſtitutis ta­<lb></lb>
men in circuli periphæria, hinc fit, quod Solis color reflectatur ex pluribus <lb></lb>
locis in orbem conſtitutis, quæ reflexio eſt ipſius Iridis arcus. </s>
          <s id="s.002066">ex Vitellion <lb></lb>
63. 10. Totam autem figuræ Iridis demonſtrationem ſic breuiter puto ad­<lb></lb>
inuentam eſſe. </s>
          <s id="s.002067">cum Sol in Iride videatur in orbem, <expan abbr="atq;">atque</expan> conſimiliter, ne ceſ­<lb></lb>
ſe eſt id prouenire ex angulis reflexionum conſimilibus, ſiue æqualibus: diſ­<lb></lb>
ſimiles enim anguli, diſſimilem <expan abbr="vtiq;">vtique</expan> efficiunt Solis <expan abbr="apparẽtiam">apparentiam</expan>. </s>
          <s id="s.002068">atqui con­<lb></lb>
ſimiles anguli, ſiue æquales, non niſi in orbem poſſunt conſtitui; igitur an­<lb></lb>
gulorum æqualitas cauſa erit rotundationis arcus. </s>
          <s id="s.002069">hęc eſt ſumma totius di­<lb></lb>
ſcurſus, quem pluribus, &amp; nimis obſcurè Ariſt. explicat.</s>
        </p>
        <p type="main">
          <s id="s.002070">Inquit igitur Ariſt. ſit enim in oriente, &amp;c. </s>
          <s id="s.002071">vbi aggreditur probare vnum <lb></lb>
ex tribus illis, quæ ſupra propoſuit, nimirum tunc Iridem eſſe ſemicircu­<lb></lb>
lum, quando aſtrum fuerit in oriente, ſiue in horizonte, vbi G. ſi igitur per <lb></lb>
triangulum G M K, intelligamus <expan abbr="planũ">planum</expan> extenſum, in quo A, in figura, adeo <lb></lb>
magnum, vt totum ſecet hemiſphærium, faciet in ſuperficie hemiſphærij ſe­


<pb pagenum="118" xlink:href="009/01/118.jpg"></pb>ctionem, quæ erit portio maximi circuli, per 6. Theodoſij, cum planum ſe­<lb></lb>
cans hemiſphærium, tranſeat per <expan abbr="centrũ">centrum</expan> ipſius, quæ ſectio, ſiue circuli por­<lb></lb>tio repræſentatur in figura, per ſemicirculum in quo A, ſiue in quo G A M­ <lb></lb>
R O. nihil autem refert quodcunque intelligas planum ſuper axem G K O, <lb></lb>
tranſiens ſiue per triangulum G K M, ſiue per aliud illi ſimile. </s>
          <s id="s.002072">Præmitten­<lb></lb>
dum præterea non poſſe in ſemicirculo ſuperiori, quod eſt planum, &amp; ſectio <lb></lb>
trianguli G K M, poni alias duas lineas. </s>
          <s id="s.002073">v. g. G R, K R, ad aliud punctum, <lb></lb>
vti eſt R, quæ habeant eandem inuicem proportionem, quam habent prio­<lb></lb>
res duæ G M, K M, quod probatur, quia ſi ſint vt G M, ad K M, ita G R, ad <lb></lb>
K R, cum G R, ſit centro K, propinquior quam G M, erit etiam eadem G R, <lb></lb>
longior ipſa G M, per 15. 3. &amp; tamen deberet eſſe æqualis illi; quemadmo­<lb></lb>
dum K M, eſt æqualis alteri K R; nequeunt autem duæ lineæ inæquales inui­<lb></lb>
cem, habere eandem rationem ad duas inuicem æquales: ergo non habent <lb></lb>
eandem rationem G M, &amp; K M, quam habent G R, &amp; K R. quod ſi punctum <lb></lb>
R, ſumatur ſupra M, erit ſimilis <expan abbr="demõſtratio">demonſtratio</expan>, ſi literæ M, &amp; R, loca permu­<lb></lb>
tent. </s>
          <s id="s.002074">his poſitis, ait <emph type="italics"></emph>(Quoniam enim G, K, puncta data ſunt, &amp; c.)<emph.end type="italics"></emph.end> ideſt data <lb></lb>
ſunt poſitione, cum notum ſit vbi ſint. </s>
          <s id="s.002075">G, enim eſt in ortu. </s>
          <s id="s.002076">K, verò in centro <lb></lb>horizontis, ſequitur, quod etiam linea G K, cuius ipſa ſunt extrema, data <lb></lb>
ſit, &amp; poſitione, &amp; magnitudine, per 26. Datorum Euclidis. </s>
          <s id="s.002077">eadem quoque <lb></lb>
ratione data erit K M, linea; ſiue quia eſt æqualis ipſi G K, ſiue quia per <lb></lb>
aſtrolabium poſſumus ipſius longitudinem, &amp; poſitionem inueſtigare; qua­<lb></lb>
re &amp; punctum M, datum erit per 27. Datorum, quare &amp; linea G M, data <lb></lb>
erit quoad ſitum, &amp; magnitudinem per 26. Datorum. </s>
          <s id="s.002078">Quare per primam <lb></lb>
Datorum erit data proportio linearum G M, M K, punctum <expan abbr="itaq;">itaque</expan> M, tanget <lb></lb>
ambitum datum, qui baſis eſt coni, quem linea K M, deſcribit in reuolutio­<lb></lb>
ne axis G K O, ſuper polis G, O. cum enim data ſit K M, poſitu, &amp; magni­<lb></lb>
tudine, <expan abbr="eaq́">eaque</expan>; ſit latus prædicti coni, ſequitur periphæriam, vel ambitum ba­<lb></lb>
ſis coni eſſe datum per ſimilem definitionem 5. definitioni Datorum. </s>
          <s id="s.002079">ſit <expan abbr="au-tẽ">au­<lb></lb>
tem</expan> ambitus ille in figura ſequenti notatus literis L M N. qui ambitus L M N, <lb></lb>
non eſt <expan abbr="concipiẽdus">concipiendus</expan> in eodem plano ſemicirculi G A N O, quemadmodum <lb></lb>
falsò pingitur in figura; ſed debemus ipſum concipere tanquam erectum ad <lb></lb>
angulos rectos cum prædicto ſemicirculo, necnon cum horizonte G K O. <lb></lb>
</s>
          <s id="s.002080">Iam ſi <expan abbr="triãgulum">triangulum</expan> G M K, prioris figuræ circumuoluatur circa axem G K O, <lb></lb>
punctum ipſius M, deſcribit prædictum ambitum L M N. hunc ambitum <lb></lb>inquit Ariſtot. linea K M, attinget, <expan abbr="eritq́">eritque</expan>; hic ambitus datus, vt dictum eſt. <lb></lb>
<figure id="id.009.01.118.1.jpg" place="text" xlink:href="009/01/118/1.jpg"></figure><lb></lb>
Erit præterea ſectio circunferentiarum ho­<lb></lb>rizontis, &amp; huius ambitus data, cuius extre­<lb></lb>
ma puncta eſſent L, &amp; N. ſi enim <expan abbr="cõcipiamus">concipiamus</expan> <lb></lb>
in figura non ſolum horizontis diametrum <lb></lb>
G K O, ſed etiam circunferentiam (in qua <lb></lb>
circunferentia eſſent duo illa puncta L, &amp; N, <lb></lb>vt in præſenti deſcriptione melius intellige­<lb></lb>
tur, in qua horizon G N O L, &amp; ambitus <lb></lb>
prædictus eſt L M N, qui debet intelligi ele­<lb></lb>
uatus ſupra horizontem perpendiculariter) <lb></lb>tunc ſectio ipſius mutua cum horizonte eſſet <pb pagenum="119" xlink:href="009/01/119.jpg"></pb>linea N P L, cuius extrema puncta ſunt L, N, quæ data erunt, cum ſint ex­<lb></lb>
trema lineæ K M, circumlatæ; &amp; quemadmodum dabatur ſuperius punctum <lb></lb>
M. eadem ratione ex Datis, dabitur punctum N, &amp; L. quare etiam ſectio <lb></lb>
N P L, quæ inter data puncta continetur, data erit ex 26. Datorum.</s>
        </p>
        <p type="main">
          <s id="s.002081">Illud nunc in memoriam <expan abbr="reuocãdum">reuocandum</expan>, quod paulò ante probaui, nimirum <lb></lb>
proportionem linearum G M, K M, non poſſe ſeruari in alijs lineis, quæ ſint <lb></lb>
in eodem plano trianguli G M K, ſi ducantur ab ijſdem punctis G, K. poteſt <lb></lb>
tamen ſeruari in alijs duabus, quæ cadant in prædictum ambitum, ſiue <expan abbr="cir-cunferẽtiam">cir­<lb></lb>
cunferentiam</expan> L M N, <expan abbr="quæq́">quæque</expan>; ſint in alio plano, <expan abbr="quã">quam</expan> in plano trianguli G M K, <lb></lb>
quod tamen tranſeat per axem G K O, <expan abbr="ſitq́">ſitque</expan>; vnum ex planis illis, de quibus <lb></lb>
ſupra dictum eſt. </s>
          <s id="s.002082">Verumenimuerò ad quid probatio hæc? </s>
          <s id="s.002083">non poſſe duas <lb></lb>
alias lineas in eodem plano, &amp;c.? exiſtimo Ariſt. idcircò hoc probaſſe, quia <lb></lb>
ſi aliæ duæ lineæ habentes eandem rationem, poſſent collocari in eodem <lb></lb>
plano; eſſent <expan abbr="permutãdo">permutando</expan> illæ duæ (in priori figura) G R, R K. <expan abbr="vtraq;">vtraque</expan> <expan abbr="vtriq;">vtrique</expan> <lb></lb>
æquales prioribus G M, M K, per quas videtur Iris, cum enim K R, ſit æqua­<lb></lb>
lis ipſi K M, erit, &amp; G M, æqualis ipſi G R, per 7. 5. &amp; in eius ſcholio. </s>
          <s id="s.002084">qua­<lb></lb>
re natura ageret tam per lineas breuiſſimas <expan abbr="agẽdo">agendo</expan> per has, quam per illas, <lb></lb>
<expan abbr="hocq́">hocque</expan>; pacto per has etiam Iris videri poſſet. </s>
          <s id="s.002085">cum ergò conſtet non poſſe has <lb></lb>
eſſe prioribus proportionales, ſed maiorem, vel minorem, alteram illarum, <lb></lb>
quàm ſit G M, ſequitur, quod non faciunt angulum æqualem angulo G M K, <lb></lb>
ſub quo videtur Iris, <expan abbr="nimirũ">nimirum</expan> angulum G R K, qui ſit æqualis angulo G M K; <lb></lb>
habet enim Iris hunc angulum determinatum, ita vt ſub maiori, vel mino­<lb></lb>
ri videri nequeat; ex 10. Baptiſta Porta. </s>
          <s id="s.002086">ſi autem punctum R, eſſet infra M, <lb></lb>
angulus G R K, eſſet minor angulo Iridis G M K, ſi verò ſupra eſſet maior <lb></lb>
eodem, quod vel ad ſenſum patere poteſt in quouis circulo, <expan abbr="idq́">idque</expan>; ſufficiat, ne <lb></lb>
longior euadat hæc tractatio. </s>
          <s id="s.002087">Fortè etiam addi poteſt, quod alibi exiſten­<lb></lb>
te puncto R, quàm in M, non poſſent anguli incidentiæ, &amp; reflexionis eſſe <lb></lb>
æquales, quæ cauſa eſſet cur ſub alio angulo, quam prædicto G M K, Iris <lb></lb>
non appareret.</s>
        </p>
        <p type="main">
          <s id="s.002088">Prædicta omnia ſunt ſecundum Ariſtot. diſcurſum, &amp; figurationem dicta, <lb></lb>
nam ſecundum veritatem poſſunt in eadem nube conſtitui plures anguli <lb></lb>
æquales, nec tamen in eodem orbe, ſed vnus ſupra <expan abbr="alterũ">alterum</expan>; vt in figura præ­<lb></lb>
<figure id="id.009.01.119.1.jpg" place="text" xlink:href="009/01/119/1.jpg"></figure><lb></lb>
ſenti, ſi nubes eſſet vbi B D. <lb></lb>
oculus in C, Sol in A. eſſent <lb></lb>
duo anguli A B C, A D C, æ­<lb></lb>
quales per 33. 3. qui tamen <lb></lb>
non ſunt in gyrum conſtituti, <lb></lb>
poſſet igitur, per <expan abbr="illorũ">illorum</expan> vtrun­<lb></lb>
que Sol Iridem efficere. </s>
          <s id="s.002089">atque <lb></lb>
animaduerſio hęc videtur ma­<lb></lb>
gni <expan abbr="momẽti">momenti</expan> eſſe, ad Iridis <expan abbr="de-monſtrationẽ">de­<lb></lb>
monſtrationem</expan> conſtituendam: <lb></lb>
cum hinc vſitatæ demonſtra­<lb></lb>
tiones infringatur. </s>
          <s id="s.002090">Fortè confugiendum eſt ad illud, quod Maurolycus, &amp; <lb></lb>
10. Baptiſta Porta obſeruarunt; debere <expan abbr="nimirũ">nimirum</expan> diſtantiam ab oculo ad cen­<lb></lb>
trum Iridis eſſe æqualem altitudini, ſiue ſemidiametro Iridis. </s>
          <s id="s.002091">Ita vt non ſo­


<pb pagenum="120" xlink:href="009/01/120.jpg"></pb>lum requiratur idem angulus, ſed etiam tanta Iridis altitudo, <expan abbr="quãta">quanta</expan> requi­<lb></lb>
ritur vt angulus in orbem conſtituatur, ex quo Iris poſſit apparere. </s>
          <s id="s.002092">hæc à <lb></lb>
nemine hactenus animaduerſa placuit addere, vt ex ijs demonſtratio Iridis <lb></lb>
omnibus numeris aliquando abſolui poſſit, quod infra (ni fallor, fauente <lb></lb>
Deo) præſtabimus.<lb></lb>
<arrow.to.target n="marg166"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002093"><margin.target id="marg166"></margin.target>165</s>
        </p>
        <p type="main">
          <s id="s.002094">Ibidem <emph type="italics"></emph>(Extraponatur igitur quædam linea, quæ D B, &amp; ſeindatur vt M G, ad <lb></lb>
M K, ſic quæ D, ad B, maior autem quæ M G, ea quàm M K, quoniam ſuper ma­<lb></lb>
iorem angulum reflexio coni, maiori enim angulo ſubtenditur trianguli M K G. <lb></lb>
</s>
          <s id="s.002095">Maior igitur eſt &amp; ipſa D, ipſa B. addatur igitur ad eam, quæ B, ea in qua F, vt <lb></lb>
ſit quod D, ad B, quæ B F, ad D. </s>
          <s id="s.002096">Deinde quod F, ad K G, quæ B, ad aliam fiat, <lb></lb>
quæ K P. &amp; à P, ad M, copuletur quæ P M, erit igitur P. polus circuli, ad quem <lb></lb>
lineæ, quæ à K, incidunt)<emph.end type="italics"></emph.end> <expan abbr="hucuſq;">hucuſque</expan> oſtendit lineas viſuales cadere ad M, pun­<lb></lb>
ctum in Iridis periphæriam, pergit deinceps inueſtigare polum, &amp; poſtea <lb></lb>
centrum eiuſdem ambitus, vtraque autem exiſtere in horizonte reperit, vt <lb></lb>
hinc inferat Iridis portionem illam, quæ oriente Sole ſupra horizontem ap­<lb></lb>
paret, eſſe ſemicirculum, vt propoſuerat. </s>
          <s id="s.002097">Differt autem polus circuli à cen­<lb></lb>
tro eiuſdem circuli. </s>
          <s id="s.002098">polus eſt punctum extra planum circuli, ex quo tamen <lb></lb>
vt <expan abbr="cẽtro">centro</expan> adhibito circino circuli periphæria deſcribi poteſt; ſic polus æqua­<lb></lb>
toris eſt idem, qui polus mundi: <expan abbr="centrũ">centrum</expan> verò eſt in plano ſui cir culi, ſic cen­<lb></lb>
trum æquatoris eſt idem cum centro mundi, cum æquator per illud incedat.</s>
        </p>
        <p type="main">
          <s id="s.002099">Dicit <expan abbr="itaq;">itaque</expan> Ariſt. cum data ſit proportio linearum K M, &amp; M G, in ſupe­<lb></lb>
riori ſecunda figura numeri 164. quam nunc iterum inſpicere opertet; ex­<lb></lb>
<figure id="id.009.01.120.1.jpg" place="text" xlink:href="009/01/120/1.jpg"></figure><lb></lb>
ponatur alia linea recta B D. quæ diui­<lb></lb>
datur in partes B, &amp; D. proportionales <lb></lb>
cum lineis K M, G M, per 10. 6. cum <lb></lb>
ergo K M, ſit minor quàm G M, per 19. <lb></lb>
primi, quia in triangulo G M K, oppo­<lb></lb>
nitur minori angulo, erit <expan abbr="quoq;">quoque</expan> B, minor quàm D, addatur iam ipſi B. linea <lb></lb>
nea F, ita vt ſit tota F B, tertia proportionalis ad duas B, &amp; D, per 11. 6. <lb></lb>
hoc ordine, vt F B, ad D. ita D, ad B. </s>
          <s id="s.002100">Deinde vt ſe habet F, ad K G. ita ſit <lb></lb>
B, ad aliam, quæ ſit K P, in eadem figura per 12. 6. &amp; à puncto P, ad M, iun­<lb></lb>
gatur recta P M. </s>
          <s id="s.002101">Dico P, eſſe polum circuli, quem dixi Iridis, &amp; in quem li­<lb></lb>
neæ à K, procedentes turbinis formam effingunt, probatur autem ab Ariſt. <lb></lb>
in ſequentibus.</s>
        </p>
        <p type="main">
          <s id="s.002102"><arrow.to.target n="marg167"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002103"><margin.target id="marg167"></margin.target>166</s>
        </p>
        <p type="main">
          <s id="s.002104">Ibidem <emph type="italics"></emph>(Erit etiam, quod quæ F, ad K G. &amp; quæ B, ad K P. &amp; quæ D, ad P M. <lb></lb>non enim ſit, ſed aut ad minorem, aut ad maiorem ea, quæ P M, nihil enim differet. <lb></lb>
</s>
          <s id="s.002105">ſit enim ad P R. eandem ergo rationem G K, &amp; K P, &amp; P R, inuicem habebunt, <lb></lb>
quam quæ F, B, D: quæ autem F, B, D, proportionales crant, quod quidem D, ad <lb></lb>
B. quæ F B, ad D: quare quod quæ P G, ad P R, quæ P R, ad eam, quæ P K. ſi igi­<lb></lb>
tur ab ijs, quæ K G, quæ G R, &amp; K R, ad R, coniungantur, coniunctæ hæ eandem <lb></lb>
habebunt rationem, quam quæ G P, ad eam, quæ P R, circa eundem enim angulum <lb></lb>
P, proportion aliter, &amp; quæ trianguli G P R, &amp; eius, qui K R P. quare &amp; quæ G R, <lb></lb>
ad eam quæ K R, eandem rationem habebit, quam &amp; quæ G P, ad eam quæ P R, <lb></lb>
habet autem &amp; quæ M G, ad M K, eam rationem, quam quæ D, ad eam quæ B, <lb></lb>quare ambæ à punctis G K, non ſolum ad circunferentiam M N, conſtituentur ean­<lb></lb>dem habentes rationem, ſed &amp; alibi, quod quidem impoſſibile)<emph.end type="italics"></emph.end> incipit, vt dixi,


<pb pagenum="121" xlink:href="009/01/121.jpg"></pb>probare P, eſſe polum prædicti ambitus, ſic. </s>
          <s id="s.002106">Primò enim ſciendum in præ­<lb></lb>
miſſa conſtructione eſſe, vt F, ad G K, &amp; B, ad K P, ita D, ad P M. nam ſi non <lb></lb>
ſit eadem ratio D, ad P M, cum alijs prædictis, erit eadem ratio eiuſdem D, <lb></lb>ad aliam maiorem, vel minorem ipſa P M. ſit ad minorem P R. nihil enim <lb></lb>
refert ſiue dixeris habere eandem rationem ad minorem, ſiue ad maiorem, <lb></lb>
ergo permutando erunt G K, K P, P R, proportionales cum F, B, D. ſed li­<lb></lb>
neæ F, B, D, erant proportionales <expan abbr="componẽdo">componendo</expan> hoc modo, vt F B, ad D, ita <lb></lb>
D, ad B: quare ſimiliter erunt vt G P, ad P R, ita P R, ad P K. per 18. 5. ſi igi­<lb></lb>
tur à punctis G, &amp; K, figuræ nu. </s>
          <s id="s.002107">164. <expan abbr="iungãtur">iungantur</expan> lineæ ad R, quæ ſint G R, K R, <lb></lb>
erit vt G R, ad K R, ita G P, ad P R. quia orta <expan abbr="sũt">sunt</expan> duo <expan abbr="triãgula">triangula</expan> G P R, K P R, <lb></lb>
quæ habent eundem angulum ad P. &amp; latera proportionalia circa dictum <lb></lb>
angulum. </s>
          <s id="s.002108">eſt etiam vt G P, ad P R, in maiori triangulo, ita P R, ad K P, in <lb></lb>
minori, ex conſtructione, quare per 6. 6. erunt illa duo triangula æquian­<lb></lb>
gula; ergò per 4. 6. erunt latera circum æquales angulos proportionalia; <lb></lb>
quare erit vt G P, ad P R. ita G R, ad R K: erat autem vt K M, ad G M, ita <lb></lb>
B, ad D. &amp; ita etiam G P, ad P R; ergò per 11. 5. vt K M, ad M G. ita K R, <lb></lb>
ad R G, intra eandem circunferentiam, &amp; in eodem plano: quod eſſe im­<lb></lb>
poſſibile ſupra oſtendimus, hoc autem impoſſibile, ſequitur ſi neges eſſe vt <lb></lb>
F, ad G K; &amp; B, ad K P, ita D, ad P M.</s>
        </p>
        <p type="main">
          <s id="s.002109"><arrow.to.target n="marg168"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002110"><margin.target id="marg168"></margin.target>167</s>
        </p>
        <p type="main">
          <s id="s.002111">Ibidem <emph type="italics"></emph>(Quoniăm igitur quæ D, <expan abbr="neq;">neque</expan> ad minorem ea, quæ P M, <expan abbr="neq;">neque</expan> ad maiorem <lb></lb>(ſimiliter enim demonſtrabimus) palam eſt, quod ad ipſam <expan abbr="vtiq;">vtique</expan> erit, in qua P M, <lb></lb>
quare erit, quod quæ M P, ad P K, quæ P G, ad M P. </s>
          <s id="s.002112">Si igitur eo in quo P, polo <lb></lb>vtens, diſtantia autem ea, in qua P M, circulus deſcribatur, omnes angulos attin­<lb></lb>
get, quos reflexæ faciunt, quæ à K, G. ſi autem non, ſimiliter oſtendentur eandem <lb></lb>
babere rationem, quæ alibi, quam in ſemicirculo conſtituuntur; quod quidem erat <lb></lb>
impoſſibile)<emph.end type="italics"></emph.end> quoniam igitur, inquit, linea D, <expan abbr="neq;">neque</expan> ad minorem, <expan abbr="neq;">neque</expan> ad ma­<lb></lb>
iorem quam P M, habet eam rationem, quæ eſt ipſius F, ad G K, aut ipſius <lb></lb>
B, ad K P. ſimiliter enim demonſtratur abſurdum ſequi. </s>
          <s id="s.002113">palàm eſt, quoniam <lb></lb>
erit D, ad P M, vt prædictæ ad prædictas: quare componendo, &amp; permu­<lb></lb>
tando, erunt tandem vt G P, ad P M, ita P M, ad P K, &amp; ita G M, ad M K, <lb></lb>
aſſumpſimus enim in conſtructione eſſe G M, ad M K, ita F B, ad D, &amp; D, ad <lb></lb>
B. quare cum ſit vt G M, ad M K, ita F B, ad D. &amp; G P, ad P M. &amp; P M, ad <lb></lb>
K P; erunt per 11. 5. vt G M, ad M K. ita G P, ad P M. &amp; P M, ad P K. ſi quis <lb></lb>
igitur vtens puncto P, tanquam polo, &amp; interuallo P M, circulum deſcribat, <lb></lb>
omnes angulos reflexionis attinget, quos faciunt lineæ productæ à K, &amp; re­<lb></lb>
flexæ ab M, ad G. harum enim infinitam multitudinem debemus imaginari <lb></lb>
à K, ad infinita puncta M, produci in ambitu illo conſtituta, <expan abbr="reſlectiq́">reflectique</expan>; ad G. <lb></lb>
ſi enim non attingat omnes illos angulos, ſequitur, vt ſupra, in eodem ſemi­<lb></lb>
circulo <expan abbr="cõſtitui">conſtitui</expan> poſſe duas alias rectas proportionales prioribus G M, M K, <lb></lb>
quod eſt impoſſibile. </s>
          <s id="s.002114">Porrò ſub angulo G M K, linearum G M, M K, Iris <lb></lb>
apparet: quare apparebit etiam ſub alijs omnibus, quæ à punctis G K, duci <lb></lb>
poſſunt ad extremum lineæ P M, quia erunt in eadem ratione cum illis; cum <lb></lb>
non deſinant in eundem <expan abbr="ſemicirculũ">ſemicirculum</expan>, ſed in ambitum Iridis M N, in quo M, <lb></lb>
punctum imaginamur circumduci. </s>
          <s id="s.002115">Ex quibus pater P, eſſe polum Iridis, ex <lb></lb>
quo per puncta M, vbi ſit reflexio, deſcribitur arcus attingens omnes Iridis <lb></lb>
reflexiones.</s>
        </p>
        <pb pagenum="122" xlink:href="009/01/122.jpg"></pb>
        <p type="main">
          <s id="s.002116"><arrow.to.target n="marg169"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002117"><margin.target id="marg169"></margin.target>168</s>
        </p>
        <p type="main">
          <s id="s.002118">Ibidem <emph type="italics"></emph>(Si igitur circumducas ſemicirculŭm, in quo A, circa diametrum in qua <lb></lb>
G K P, que à G, K, reflexæ ad id in quo M; in omnibus planis ſimiliter ſe habebunt, <lb></lb>&amp; æqualem facient angulum, qui K M G, &amp; quem etiam facient angulum, quæ <lb></lb>
K P, &amp; P M, ſuper eam, quæ G P, ſemper æqualis erit. </s>
          <s id="s.002119">Trianguli igitur ſuper eam, <lb></lb>
quæ G P, æquales ei, qui G M P. conſiſtunt. </s>
          <s id="s.002120">horum autem perpendiculares ad idem <lb></lb>
ſignum cadent eius, quæ G P, &amp; æquales erunt, cadunt ad <foreign lang="grc">ω,</foreign> centrum ergò circuli <lb></lb>
<foreign lang="grc">ω</foreign> ſemicirculus autem, qui circa M N, abſectus eſt ab horizonte)<emph.end type="italics"></emph.end> hac vltima <lb></lb>
textus parte concludit Iridis portionem ſupra horizontem aſtro <expan abbr="oriẽte">oriente</expan> exi­<lb></lb>
ſtentem eſſe ſemicirculum, hoc modo; ſi igitur imaginatione circumducas <lb></lb>
ſemicirculum, in quo A, circa diametrum horizontis G K P, in hac circum­<lb></lb>
uolutione duæ lineæ G M, M K, in omnibus planis conſtitui poſſibilibus cir­<lb></lb>
ca prædictam diametrum, quæ ſupra etiam fieri à triangulis infinitis dixi­<lb></lb>
mus, ſucceſſiuè erunt; ſiue percurrent ſimiliter omnia illa plana, &amp; facient <lb></lb>
vbique angulum Iridis K M G, eundem: pariter duæ lineæ K P, P M, facient <lb></lb>
vndique eundem angulum K P M. quare omnia triangula in predictis planis <lb></lb>
imaginata, &amp; <expan abbr="cõſtituta">conſtituta</expan> ſuper linea G P, ſimilia ipſi G M P, &amp; æqualia erunt; <lb></lb>
ſi igitur ab angulis ipſorum, in quibus M, ductæ ſint perpendiculares ad la­<lb></lb>
tus G P, omnes cadent in idem punctum <foreign lang="grc">ω,</foreign> vt in figura; <expan abbr="quarũ">quarum</expan> vna erit M <foreign lang="grc">ω,</foreign><lb></lb>
quæ tamen cæteras omnes repreſentabit, <expan abbr="eisq́">eisque</expan>; omnibus in volutatione axis <lb></lb>
G K <foreign lang="grc">ω,</foreign> coincidit; erunt autem omnes æquales, quandoquidem ſunt trian­<lb></lb>
gulorum æqualium. </s>
          <s id="s.002121"><expan abbr="eruntq́">eruntque</expan>; in eodem eiuſdem circuli plano, &amp; punctum <foreign lang="grc">ω,</foreign><lb></lb>
erit centrum ipſius. </s>
          <s id="s.002122">ſimilia dicta ſunt in Halone. </s>
          <s id="s.002123">Cum ergò ipſius centrum <lb></lb>
<foreign lang="grc">ω</foreign>, ſit in diametro horizontis G K <foreign lang="grc">ω</foreign> P O, manifeſtum fit portionem eius, quæ <lb></lb>
ſupra horizontem eminet, eſſe ſemicirculum, qui in figura notatur lineis <lb></lb>
L M N. </s>
          <s id="s.002124">Atque hoc accidit Sole, vel Luna in horizonte exiſtentibus; quod <lb></lb>
erat primo loco demonſtrandum.</s>
        </p>
        <p type="main">
          <s id="s.002125">Porrò ſciendum poſſe nos breuius polum prædictum inuenire, ſi nimirum <lb></lb>
<figure id="id.009.01.122.1.jpg" place="text" xlink:href="009/01/122/1.jpg"></figure><lb></lb>
ad M, ducatur M P, faciens angulum K P M, æqua­<lb></lb>
lem angulo G M K, per 23. primi, erunt enim duo <lb></lb>
triangula <expan abbr="æquiãgula">æquiangula</expan> G P M, K P M, angulus enim <lb></lb>
P, eſt communis, angulus verò M K P, eſt æqualis <lb></lb>
duobus G, &amp; G M K, per 32. primi, ergo etiam <lb></lb>
duobus ad M, ſiue toti G M P, &amp; reliquus K M P, <lb></lb>
reliquo, quare per 4.6. latera circa angulos æqua­<lb></lb>
les proportionalia erunt, &amp; omologa G M, ad M K, ita G P, ad P M, quæ <lb></lb>æqualibus angulis ſubtenduntur. </s>
          <s id="s.002126"><expan abbr="eaſdẽ">eaſdem</expan> autem proprietates habebant etiam <lb></lb>
triangula Ariſt. in figura, de qua paulò ante dicebam. </s>
          <s id="s.002127">Verba illa <emph type="italics"></emph>(Quæ ali­<lb></lb>bi quam in ſemicirculo constituuntur)<emph.end type="italics"></emph.end> ſunt perperam in antiqua tranſlatione <lb></lb>
tranſlata, nam Græcè ſic, <foreign lang="grc">αι αλλοθι τοῡ ημικοκλνού συνισταμεναι,</foreign> transferenda <lb></lb>
eſſent, quæ in alio circuli loco concurrunt.</s>
        </p>
        <p type="main">
          <s id="s.002128"><arrow.to.target n="marg170"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002129"><margin.target id="marg170"></margin.target>169</s>
        </p>
        <p type="main">
          <s id="s.002130">Ibidem <emph type="italics"></emph>(Iterum ſit horizon quidem in quo A C. oriatur autem ſupra hunc G, <lb></lb>
axis autem ſit nunc in quo G P. </s>
          <s id="s.002131">Alia igitur omnia ſimiliter oſtendentur vt &amp; prius. <lb></lb>
</s>
          <s id="s.002132">Polus autem circuli, in quo P, erit ſub horizonte eo, in quo A C, eleuato puncto, <lb></lb>
in quo G. in eadem autem &amp; polus, &amp; centrum circuli, &amp; terminantis nunc ortum, <lb></lb>
eſt enim iſte, in quo G P. </s>
          <s id="s.002133">Quoniam autem ſupra diametrum, quæ A C, quod K G, <lb></lb>
centrum vtique erit ſub horizonte priori eius, in quo A C, in linea K P, in quo <foreign lang="grc">ω,</foreign><emph.end type="italics"></emph.end>


<pb pagenum="123" xlink:href="009/01/123.jpg"></pb><figure id="id.009.01.123.1.jpg" place="text" xlink:href="009/01/123/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>Quare minor erit ſuperior ſectio ſemicir­<lb></lb>
culo, in qua S T, (nam Q S T, ſemicir­<lb></lb>
culus est, nunc autem interſectus eſt ab <lb></lb>
horizonte A C; <expan abbr="itaq;">itaque</expan> Q S, diſparens erit) <lb></lb>
eleuato ipſo Sole)<emph.end type="italics"></emph.end> demonſtrat propoſi­<lb></lb>
tionem ſecundam nimirum Sole ſupra <lb></lb>horizontem eleuato, ambitum Iridis <lb></lb>
eſſe minorem circuli portionem, ſiue <lb></lb>
ſemicirculo minorem. </s>
          <s id="s.002134">ſit igitur in fi­<lb></lb>
gura ſuperiori, quam textui <expan abbr="cõgruen-tem">congruen­<lb></lb>
tem</expan> reſtituimus, linea A C, horizon­<lb></lb>
talis, ſupra quam Sol ſit eleuatus in <lb></lb>
circulo altitudinis in loco G, axis au­<lb></lb>
rem coni, quem reflexè faciunt ſit <lb></lb>
G K <foreign lang="grc">ω</foreign> P. alia igitur omnia, quæ ſupra exiſtente in ortu aſtro oſtenſa ſunt, hic <lb></lb>
pariter oſtendi poſſunt, ſcilicet Iridem fieri tantum per lineas proportiona­<lb></lb>
les, &amp; æquales lineis G M, M K, quia Iris videri nequit, niſi in tali, ac deter­<lb></lb>
minata reflexione, &amp; angulo, vt initio ſuppoſui; &amp; quia lineæ illis propor­<lb></lb>
tionales non poſſunt alibi conſtitui, quam in ambitu circulari, &amp; in diuerſis <lb></lb>
planis, ſequitur, vt ſupra Iridem eſſe circularem M N L; <expan abbr="eiusq́">eiusque</expan>; polum P, &amp; <lb></lb>
centrum <foreign lang="grc">ω,</foreign> inueniemus ſimiliter in axe G K <foreign lang="grc">ω</foreign> P, &amp; quia axis hic ſecat hori­<lb></lb>
zontem in K, in hac vltima figura propter eleuationem Solis ſupra A C, in <lb></lb>
G, ſequitur partem axis, in qua <foreign lang="grc">ω,</foreign> &amp; P, exiſtunt, infra horizontem deprimi. <lb></lb>
</s>
          <s id="s.002135">&amp; quia (vt pater ex 64. 10. Vitell.) &amp; P, polus, &amp; centrum <foreign lang="grc">ω,</foreign> Iridis, &amp; cen­<lb></lb>trum K, circuli horizontis, cuius ſcilicet diameter eſſet A K S, &amp; Sol, ſunt <lb></lb>
in eadem linea G K <foreign lang="grc">ω</foreign> P, ſi centrum Iridis <foreign lang="grc">ω,</foreign> ſit infra horizontem, patet mi­<lb></lb>
norem circuli portionem, quam ſit ſemicirculus ſupra horizontem eminere, <lb></lb>
in qua poſui literas S L T, nam Q S L T R, eſt ſemicirculus, cuius pars con­<lb></lb>
tenta inter duos arcus Q S, &amp; T R, eſt infra horizontem. </s>
          <s id="s.002136">debemus autem <lb></lb>
hunc ſemicirculum, &amp; hanc portionem ipſius S L T, extantem ſupra hori­<lb></lb>
zontem imaginari erectam eſſe, vt planum ipſius circuli faciat angulos re­<lb></lb>
ctos ſiue ſit perpendiculare cum axe G K P; &amp; <expan abbr="circulũ">circulum</expan> altitudinis A G M N, <lb></lb>
modo fungi vice horizontis. </s>
          <s id="s.002137">ſic enim ſola portio S L T, appareret nobis, <expan abbr="eſ-ſetq́">eſ­<lb></lb>
ſetque</expan>; rationabiliter conſtituta. </s>
          <s id="s.002138">Ex quibus 2. Ariſt. propoſitio manifeſta eſt.</s>
        </p>
        <p type="main">
          <s id="s.002139"><arrow.to.target n="marg171"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002140"><margin.target id="marg171"></margin.target>180</s>
        </p>
        <p type="main">
          <s id="s.002141">Ibidem <emph type="italics"></emph>(Minima autem cum in meridie, quanto enim ſuperius G, tanto infe­<lb></lb>
rius &amp; polus, &amp; centrum circuli erit)<emph.end type="italics"></emph.end> probat tertiam propoſitionem, nimi­<lb></lb>
rum Sole exiſtente in meridie minimam <expan abbr="omniũ">omnium</expan> eſſe Iridis arcus portionem: <lb></lb>
ratio autem eſt, quia tunc G, ſiue Sol, eſt altiſſimus ſupra horizontem, &amp; <lb></lb>
conſequenter <foreign lang="grc">ω;</foreign> centrum Iridis eſt depreſsiſſimum, quare tunc maxima cir­<lb></lb>
culi Iridis portio abſcondetur, &amp; proinde minima apparebit, quod erat vl­<lb></lb>
timo <expan abbr="demõſtrandum">demonſtrandum</expan>. </s>
          <s id="s.002142">Non me latet has Ariſt. figurationes eſſe apud Olym­<lb></lb>
piodorum nonnullis obiectionibus obnoxias, ſed cum facilè dilui poſſint, &amp; <lb></lb>
etiam ſi non diluantur, ſaluetur tamen veritas Ariſtotelicæ demonſtratio­<lb></lb>
nis, breuitati ſtudens, conſultò eas prætermitto.</s>
        </p>
        <p type="main">
          <s id="s.002143">Aduertendum præterea Vicomercatum inordinatè citare librum Dato­<lb></lb>
rum Euclidis, &amp; <expan abbr="quandoq;">quandoque</expan> etiam malè citare Euclidem ipſum. </s>
          <s id="s.002144">peius verò <pb pagenum="124" xlink:href="009/01/124.jpg"></pb>faciunt ij, qui has demonſtrationes <expan abbr="abſq;">abſque</expan> vlla libri Datorum mentione ex­<lb></lb>
plicare conantur, cum manifeſtè illo innitantur.</s>
        </p>
        <p type="main">
          <s id="s.002145">Cæterum ſi quis breues, ac dilucidas harum rerum demonſtrationes re­<lb></lb>
quirat, is legat 74. 75. 76. propoſitiones 10. Vitell. vel ſequentem noſtram <lb></lb>
de Iride additionem. </s>
          <s id="s.002146">ego enim longiorem hanc, <expan abbr="atq;">atque</expan> impeditam Ariſt. tra­<lb></lb>
ctationem in gratiam textus illius, vt inſtituti mei ratio poſtulabat, perſe­<lb></lb>
quutus ſum.</s>
        </p>
        <p type="main">
          <s id="s.002147"><arrow.to.target n="marg172"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002148"><margin.target id="marg172"></margin.target>181</s>
        </p>
        <p type="main">
          <s id="s.002149">Ibidem <emph type="italics"></emph>(Quod autem in minoribus quidem diebus ijs, qui poſt æquinoctium au­<lb></lb>
tumnale <expan abbr="cõtingit">contingit</expan> ſemper fieri Iridem: in longioribus aurem diebus ijs qui ab æqui­<lb></lb>noctio altero, ad æquinoctium alterum circa meridiem non fit Iris, cauſa est, quia <lb></lb>
quæ ad Vrſam ſectiones omnes maiores ſunt ſemicirculo, &amp; ſemper ad maiores quod <lb></lb>
autem eſt occultum, paruum: quæ autem ad æquatoris meridiem ſectiones, quæ qui­<lb></lb>
dem ſupra ſectio, parua; quæ autem ſub terra magna, &amp; ſemper maiores, quæ lon­<lb></lb>
gius. </s>
          <s id="s.002150">quare in ijs, qui ad æſtiuas verſiones diebus propter magnitudinem ſectionis, <lb></lb>
antequam veniat G, ad medium ſectionis, infra iam pœnitus fit P; propterea quod <lb></lb>
longè diſtat à terra meridies propter magnitudinem ſectionis. </s>
          <s id="s.002151">In ijs autem diebus, <lb></lb>qui ad hyemales verſiones, quia non multŭμ ſunt ſupra terram ſectiones circulorum, <lb></lb>
contrarium neceſſarium fieri, modicum enim eleuato in quo G, in meridie fit Sol)<emph.end type="italics"></emph.end><lb></lb>
quærit cur poſt æquinoctium autumnale vſque ad vernum, hoc eſt hyemali <lb></lb>
tempore, Iris appareat etiam Sole meridiem occupante: reliquo autem <lb></lb>
tempore æſtiuo, quod eſt ab æquinoctio verno ad autumnale appareat tan­<lb></lb>
tum Sole vel in ortu, aut occaſu exiſtente, vel parum ſupra terram ſublato. <lb></lb>
</s>
          <s id="s.002152">cauſa autem huius refert in ſectiones parallelorum circulorum, quos Sol <lb></lb>
diurno motu inter <expan abbr="vtrunq;">vtrunque</expan> <expan abbr="tropicũ">tropicum</expan> deſcribit: nam ſectiones parallelorum, <lb></lb>
qui ſunt ad Vrſam, ideſt in parte ſphæræ Boreali, qui omnes ſunt inter æqua­<lb></lb>
torem, &amp; tropicum Cancri; ſectiones inquam horum circulorum, quæ ſunt <lb></lb>
ſupra horizontem, maiores ſunt ſectionibus infra horizontem depreſſis, &amp; <lb></lb>ſemper eò maiores, quò propiores ſunt Cancro, ita vt magna valdè ſit ea <lb></lb>
portio, quæ eſt ſupra terram, exigua verò admodum, quæ infra (intelligan­<lb></lb>
tur hæc in ſphæra obliqua, cuius polus eleuetur grad. 45. circiter) quare <lb></lb>
quando aſtrum G, conſcenderit meridiem, adeò P, polus Iridis, &amp; etiam <foreign lang="grc">ω,</foreign><lb></lb>
centrum eius infra terram deprimitur, vt aut nihil, aut inſenſibile quid de <lb></lb>
Iridis ambitu ſupra terram eleuari poſſit, contrarium accidit in parallelis <lb></lb>
meridionalibus, quia eorum ſectiones ſuperiores ſunt ſemper inferioribus <lb></lb>
minores, quapropter etiam ſi aſtrum ad meridiem eleuetur, parum tamen <lb></lb>
attollitur, &amp; conſequenter centrum <foreign lang="grc">ω,</foreign> Iridis parum infra horizontem <lb></lb>
deſcendit, ac propterea etiam in meridie pars ipſius ſatis ma­<lb></lb>
gna conſpicitur. </s>
          <s id="s.002153">quæ omnia adhibita ſphæra materia­<lb></lb>
li, eaque aſtronomicè ad ſuam eleuationem <lb></lb>
accommodata, nullo negotio li­<lb></lb>
cebit intueri.</s>
        </p>
        <pb pagenum="125" xlink:href="009/01/125.jpg"></pb>
        <p type="head">
          <s id="s.002154"><emph type="italics"></emph>Additio de Iride.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002155">Cvm ſuperior Ariſtot. de Iride tractatio obſcura, ac tricis pluribus <lb></lb>
impedita euaſerit, <expan abbr="cumq́">cumque</expan>; aliorum etiam demonſtrationes aliqua <lb></lb>ex parte vacillent, viſum eſt breuiter expeditam, <expan abbr="atq;">atque</expan> abſolutam <lb></lb>
ipſius apponere demonſtrationem. </s>
          <s id="s.002156">Cum igitur in cœleſti arcu <lb></lb>
duo potiſſimum ſint, quæ ſui admiratione <expan abbr="Philoſophorũ">Philoſophorum</expan> animos in ſui con­<lb></lb>
templationem alliciant, colores, ſcilicet, &amp; figura: nos mirabilem illam co­<lb></lb>
lorum triadem, tanquam alienam, phyſicis relinquentes, de figura ipſius iu­<lb></lb>
re mathematico diſſeremus: rotunditatis ſcilicet Iridis cauſam opticis ra­<lb></lb>
tionibus venabimur, cur aliquando ſemicirculus, aliquando ſemicirculo mi­<lb></lb>
nor appareat. </s>
          <s id="s.002157">vt igitur ordine procedamus.</s>
        </p>
        <p type="main">
          <s id="s.002158">Primo loco aduertendum eſt tria ad Iridis viſionem eſſe neceſſaria, So­<lb></lb>
lem, oculum, &amp; nubem tenuem, ac roſcidam, quæ ſcilicet minutis guttulis <lb></lb>
iam ſcateat; hac enim ratione guttulæ illæ innumera erunt veluti parua <lb></lb>
ſpecula, quæ lumen Solis ob paruitatem imperfecto quodam modo repre­<lb></lb>
ſentare poſſint, ex tali enim repreſentatione Iris apparet. </s>
          <s id="s.002159">quæ tria debent <lb></lb>
eſſe ita diſpoſita, vt Sol, oculus, &amp; centrum Iridis ſint in eadem recta linea <lb></lb>
conſtituta, <expan abbr="oculusq́">oculusque</expan>; medium locum, inter Solem, &amp; Iridis <expan abbr="cẽtrum">centrum</expan> obtineat, <lb></lb>
vt in prima figura videre eſt, in qua Sol vbi A, oculus in C. nubes verò <lb></lb>
G H L E, in qua apparet Iris in arcu E B F, quem debemus concipere eſſe <lb></lb>
in receſſu, vt pictores aiunt, depictum. </s>
          <s id="s.002160">i. </s>
          <s id="s.002161">non in hoc ſitu, &amp; ouali figura, ſed <lb></lb>
<figure id="id.009.01.125.1.jpg" place="text" xlink:href="009/01/125/1.jpg"></figure><lb></lb>
eſſe perfectè ſemicircularem, <expan abbr="habereq́">habereque</expan>; talem poſitionem, vt pars ipſius B F, <lb></lb>
ſit citra chartam eleuata, <expan abbr="ipſiq́">ipſique</expan>; perpendicularis, pars verò E B, vltra pagi­


<pb pagenum="126" xlink:href="009/01/126.jpg"></pb>nam rectà recedat, <expan abbr="ſicq́">ſicque</expan>; diameter Iridis E F, faciat angulos rectos cum <lb></lb>
linea horizontali A C L, in quo ſitu oculo C, totus ex oppoſito directè ſpe­<lb></lb>
ctaretur, non aliter ac Iridem ipſam conſpicere ſolemus. </s>
          <s id="s.002162">Quod autem ne­<lb></lb>
ceſſaria ſit nubes roſcida, pulcherrima hac experientia <expan abbr="cõprobatur">comprobatur</expan>: ſi enim <lb></lb>
in Sole poſiti ore aquam efflantes leui aſpergine aerem Soli, ac nobis ad­<lb></lb>
uerſum irroremus, actutum Iridis arcum guttulis illis, quamuis volitanti­<lb></lb>bus inhærentem ſumma voluptate ſpectabimus. </s>
          <s id="s.002163">Quod præterea oculus no­<lb></lb>
ſter, cum Iridem videmus, medius ſit inter Solem, &amp; Iridis centrum, expe­<lb></lb>
rimento diuturno, manifeſtum eſt.</s>
        </p>
        <p type="main">
          <s id="s.002164">Secundò, notandum eſt, arcum per reflexionem fieri: quod quidem pri­<lb></lb>
mo eadem experientia, qua præcedens concluſio confirmatur: deinde, quia <lb></lb>
Iridem ſemper in oppoſita Soli, ac nobis parte <expan abbr="cõſpicimus">conſpicimus</expan>; quemadmodum <lb></lb>
in eadem figura oſtenditur, quod aliter quàm per reflexionem fieri nequit.</s>
        </p>
        <p type="main">
          <s id="s.002165">Tertiò, ſciendum eſt ex Maurolyco, &amp; 10. Baptiſta Porta, tantam eſſe di­<lb></lb>
ſtantiam C D, ab oculo ad centrum arcus, quanta eſt altitudo, ſeu ſemidia­<lb></lb>
meter D B, obſeruarunt enim ipſi angulos D C B, &amp; C B D, eſſe ſemirectos, <lb></lb>
&amp; proinde æquales, &amp; conſequenter duo latera C D, D B, trianguli C D B, <lb></lb>
per 6. 1. æqualia ſunt.</s>
        </p>
        <p type="main">
          <s id="s.002166">Quartò, conſiderandum eſt lineas A B, A D, ob maximam Solis ab Iride <lb></lb>
diſtantiam inſenſibiliter differre; &amp; ideò ſupponi poſſunt æquidiſtantes, <lb></lb>
quare angulus A B C, qui æqualis eſt alterno B C D, ſumi poteſt abſque vllo <lb></lb>
errore pro ſemirecto. </s>
          <s id="s.002167">hic autem angulus A B C, dicitur angulus reflexionis <lb></lb>
Iridis, ſub tali enim reflexione lumen Solis occurrens nubi in B, reflectitur <lb></lb>
ad oculum C.</s>
        </p>
        <p type="main">
          <s id="s.002168">Quintò, ſequitur ex prædictis arcum videri ſemper ſub ſtato, ac determi­<lb></lb>
nato reflexionis angulo, ſcilicet ſub ſemirecto, <expan abbr="neq;">neque</expan> poſſe per alium videri. <lb></lb>
</s>
          <s id="s.002169">quod etiam probari poteſt ex Ariſt. quia nimirum videmus arcum apparere <lb></lb>
conſimiliter in ambitu circulari, ergò neceſſariò apparebit <expan abbr="vbiq;">vbique</expan> in toto il­<lb></lb>
lo ambitu per conſimilem reflexionem, ſiue per æquales reflexionis angulos, <lb></lb>
pro quibus omnibus vnus cernitur in figura angulus A B C.</s>
        </p>
        <p type="main">
          <s id="s.002170">Sextò, ad Iridis viſionem, præter ea, requiri aeris rorantis multiplica­<lb></lb>
tionem; ſicuti enim nebulam videre nequimus, niſi aer exhalatione illa in­<lb></lb>
fectus multus ſit ante oculum noſtrum: ſic etiam exiſtimo ad Iridis appari­<lb></lb>
tionem, opus eſſe plurima nube roreſcente, vt ex multiplicatione guttula­<lb></lb>
rum, quarum aliæ poſt alias ſint, totus tandem Iris appareat. </s>
          <s id="s.002171">quia paucæ <lb></lb>
guttulæ, etiam ſi quælibet illarum aliquid Iridis efficeret, ob paruitatem <lb></lb>
tamen illarum, nulla arcus figura ſpectaretur. </s>
          <s id="s.002172">Quod ſi ante oculum pluri­<lb></lb>
mæ ſint in toto aere aliæ poſt alias, tunc ſe mutuò iuuantes, obiectum ſatis <lb></lb>ſenſibile, quod Iris eſt, efficere poſſunt. </s>
          <s id="s.002173">Adde, quod etiam ex tali guttula­<lb></lb>
rum multiplicatione, aer opacatur, quæ opacatio plurimum iuuat ad Iri­<lb></lb>
dem ſpectandam.</s>
        </p>
        <p type="main">
          <s id="s.002174">Septimò, Iridis rotundationis cauſam ex præmiſſis conſtare potiſſimum <lb></lb>
ex duabus. </s>
          <s id="s.002175">primò, ex angulo reflexionis determinato, qui videlicet ſit ferè <lb></lb>
ſemirectus. </s>
          <s id="s.002176">ſecundò, ex paribus diſtantijs C D, D B, huiuſmodi enim plures <lb></lb>
anguli, qui ad Iridem ſunt neceſſarij (debent enim ſingulæ Iridis partes ſub <lb></lb>
huiuſmodi angulo repreſentari) non poſſunt aliter quàm in gyrum <expan abbr="cõſtitui">conſtitui</expan>,


<pb pagenum="127" xlink:href="009/01/127.jpg"></pb>quem gyrum optimè concipiemus, ſi imaginemur triangulum A B C, cir­<lb></lb>
cumuerti circa lineam horizontalem A C L, fixam, tanquam circa axem. </s>
          <s id="s.002177">in <lb></lb>
hac enim conuerſione angulus Iridis B, deſcribet circulum, qui erit Iris, &amp; <lb></lb>pertranſibit omnes angulos, qui in tali Solis, oculi, ac nubis ſitu, arcum ef­<lb></lb>
ficere ſunt idonei.</s>
        </p>
        <p type="main">
          <s id="s.002178">Sed contra prædicta de angulo Iridis determinato eadem nobis obijcies, <lb></lb>
quæ nos ſupra ad finem numeri 164. Ariſt. &amp; alijs obiecimus, plures <expan abbr="nimi-rũ">nimi­<lb></lb>
rum</expan> poſſe conſtitui angulos æquales angulo Iridis B, in plano trianguli A B C, <lb></lb>
qui non ſint in eodem orbe conſtituti, in quo ſunt omnes anguli B. </s>
          <s id="s.002179">Iridem <lb></lb>
reflectentes, <expan abbr="quiq;">quique</expan> reflexionem faciant ad eundem oculum C, vnde ſequitur <lb></lb>
prædictam Iridis altitudinem non eſſe, vti diximus, determinatam, cum <lb></lb>
poſſit angulus B, alios ſibi æquales tam ſupra, quàm infra habere, qua ra­<lb></lb>
tione deberet etiam Iris, &amp; altius, &amp; inferius apparere.</s>
        </p>
        <p type="main">
          <s id="s.002180">Huic dubitationi reſpondeo, quod quamuis huiuſmodi plures anguli <lb></lb>
æquales fiant, non tamen Iridis generationi obſtant, quinimò ad eam valdè <lb></lb>
neceſſarij ſunt; <expan abbr="cũ">cum</expan> enim omnes ſint in <expan abbr="circunferẽtia">circunferentia</expan> circuli A C D B, quar­<lb></lb>
tæ figuræ num. </s>
          <s id="s.002181">164. quæ modo inſpicienda eſt, vt ſunt in ea anguli A D C, <lb></lb>
A B C; quæ circunferentia ob ſui circuli immenſitatem ad ſenſum eſt inſtar <lb></lb>
lineæ rectæ, fit vt omnes illi anguli tàm qui ſupra B, quàm qui infra ſunt, <lb></lb>
ſint quoad ſenſum in eadem recta C D B, ante viſum protenſa, <expan abbr="ſicq́">ſicque</expan>; Iris, qui <lb></lb>
apparet in D, &amp; in B, &amp;c. </s>
          <s id="s.002182">ob medij rorantis multiplicationem vnam <expan abbr="tãtùm">tantùm</expan> <lb></lb>
oculo Iridem repreſentet. </s>
          <s id="s.002183">locus tamen, in quo apparet, &amp; vbi eſt angulus <lb></lb>
B, qui propriè Iridis appellatur, eſt in tanta diſtantia à centro arcus, quan­<lb></lb>
ta eſt ab eodem centro ad oculum, vt ſupra dictum eſt.</s>
        </p>
        <p type="main">
          <s id="s.002184">Quod verò alibi extra circunferentiam illius circuli, poni nequeat angu­<lb></lb>lus æqualis angulo B, præſentis figuræ, qui reflectat ad C. patet ſic, ſit enim <lb></lb>
angulus A N O, ſemirectas, &amp; ideò æqualis angulo B, erunt ergo B C, N O, <lb></lb>
parallelæ, quare non concurrent ambæ ad C, ſed altera ad E, altera verò ad <lb></lb>
O, quæ propterea oculo in O, poſito Iridem efficeret, non autem oculo C: <lb></lb>
<expan abbr="ſicq́">ſicque</expan>; oculus C, &amp; oculus O, viderent diuerſos arcus. </s>
          <s id="s.002185">eodem modo oſtendi <lb></lb>
poteſt, <expan abbr="neq;">neque</expan> in ſuperiori parte nubis vbi P, conſtitui poſſe angulum æqualem <lb></lb>
angulo B, qui oculo C, Iridem valeat oſtendere. </s>
          <s id="s.002186">Ex quibus ſatis patefacta <lb></lb>
eſt cauſa rotunditatis arcus, angulus ſcilicet determinatus cum diſtantia­<lb></lb>
rum C D, D B, paritate, necnon cum medij rorantis ſufficienti multiplica­<lb></lb>
tione. </s>
          <s id="s.002187">Ex his etiam Iridis definitio in hunc modum concinnari poteſt, Iris <lb></lb>
eſt arcus multicolor in nube rorida, ex radiorum Solis, aut Lunæ reflexio­<lb></lb>ne ſub ſtatuto angulo effulgens.</s>
        </p>
        <p type="main">
          <s id="s.002188">Octauo loco Problemata nonnulla reſoluemus.</s>
        </p>
        <pb pagenum="128" xlink:href="009/01/128.jpg"></pb>
        <p type="head">
          <s id="s.002189"><emph type="italics"></emph>Problema Primum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002190">Cur oriente, aut occumbente Sole, Iris ſemicirculus eſt?</s>
        </p>
        <p type="main">
          <s id="s.002191">Cauſa huius hæc eſt; ſupra enim dictum eſt, in omni Iridis appari­<lb></lb>
tione tria hæc, Solem, oculum, &amp; Iridis centrum eſſe in eadem re­<lb></lb>
cta linea, v. g. in linea A C D, præcedentis figuræ, cum igitur Sol <lb></lb>
tam oriens, quam occidens ſit in horizonte, v. g. in A, horizontis <lb></lb>
puncto, ſimiliter oculus ſit in C, horizontis centro, conſectarium eſt, cen­<lb></lb>
trum etiam Iridis D, eſſe pariter in horizontis ſuperficie, quare ſecabitur <lb></lb>
ab horizonte per centrum, vnde etiam ſequitur ipſius Iridis portionem <lb></lb>
E B F, quæ ſupra horizontem extat eſſe ſemicirculum. </s>
          <s id="s.002192">Quod ſi horizon non <lb></lb>
obſtaret, <expan abbr="integrũ">integrum</expan> Iris compleret orbem, <expan abbr="cernereturq́">cernereturque</expan>; toto ambitu B F M E.</s>
        </p>
        <p type="head">
          <s id="s.002193">An <expan abbr="quandoq́">quandoque</expan>; maior ſemicirculo appareat?</s>
        </p>
        <p type="head">
          <s id="s.002194"><emph type="italics"></emph>Problema Secundum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002195">Maior quidem, imò etiam integer circulus, ſed ab oculo in ſummitate <lb></lb>
montis conſtituto, <expan abbr="Soleq́">Soleque</expan>; iam multum eleuato videri poteſt, vt in <lb></lb>
hac ſecunda figura cernitur, vbi euecto Sole ad locum E, ſupra horizontem <lb></lb>
<figure id="id.009.01.128.1.jpg" place="text" xlink:href="009/01/128/1.jpg"></figure><lb></lb>
A B, poterit oculus in vertice montis C, poſitus Iridem F G H I, comple­<lb></lb>
tam videre, quia infra lineam E C D, in qua exiſtunt Sol, oculus, &amp; Iridis <lb></lb>
centrum, nihil eſt ad partes D, vbi nubes irrorat, quod Iridis apparitioni <lb></lb>
ſit impedimento.</s>
        </p>
        <pb pagenum="129" xlink:href="009/01/129.jpg"></pb>
        <p type="head">
          <s id="s.002196">Cur quanto Sol altior eſt, tanto inferior, <expan abbr="tantoq́">tantoque</expan>; ſemicir­<lb></lb>
culo minor appareat Iris?</s>
        </p>
        <p type="head">
          <s id="s.002197"><emph type="italics"></emph>Problema Tertium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002198">Qvia eleuato Sole ad E, vt in hac tertia figura, neceſſario centrum Iri­<lb></lb>
dis D, infra horizontem A B, deprimetur, cum in eadem recta E C D. <lb></lb>
<figure id="id.009.01.129.1.jpg" place="text" xlink:href="009/01/129/1.jpg"></figure><lb></lb>
Sol E, oculus C, <expan abbr="centrũq́">centrumque</expan>; Iridis D, exiſtant: vnde neceſſariò ſequitur Iridis <lb></lb>
portionem F G H, ſupra horizontem extantem, ſemicirculo minorem eſſe.</s>
        </p>
        <p type="head">
          <s id="s.002199">Cur Iris inſequentes fugit, fugientes verò inſequitur?</s>
        </p>
        <p type="head">
          <s id="s.002200"><emph type="italics"></emph>Problema Quartum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002201">Pvlcherrimum iſtud phænomenon primus omnium Philippus Mendæus <lb></lb>
Platonis diſcipulus, obſeruauit; Cuius ratio eſt, quia arcus non niſi ſub <lb></lb>determinato angulo, diſtantijs etiam illis paribus, ac tandem idonea aſper­<lb></lb>
ginoſæ nubis multiplicatione ſpectatur; quapropter ſi quis per aerem to­<lb></lb>
tum <expan abbr="vndiq;">vndique</expan> roſcidum inambulet, <expan abbr="vbicunq;">vbicunque</expan> illi anguli, <expan abbr="illæq́">illæque</expan>; conditiones af­<lb></lb>
fuerint Iris apparebit: quod ſi in aperta planitie obequitans arcu conſpe­<lb></lb>
cto, additis equo calcaribus citatum curſum ad eum direxerit, fugientem <lb></lb>
ante ſe Iridem ſumma cum iucunditate mirabitur.</s>
        </p>
        <p type="main">
          <s id="s.002202">Ex dictis prętere a patet, ſimpliciter nimis eos hallucinari, qui exiſtimant <lb></lb>
in plana, aut concaua nubis ſuperficie Iridem tantummodo apparere poſſe.</s>
        </p>
        <pb pagenum="130" xlink:href="009/01/130.jpg"></pb>
        <p type="head">
          <s id="s.002203">Cur lunares Irides fiunt rariores?</s>
        </p>
        <p type="head">
          <s id="s.002204"><emph type="italics"></emph>Problema Quintum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002205">Qvoniam iuxta plenilunia tantum, cum ſcilicet Luna plurimo lumine <lb></lb>
abundat, quod Iridem efficere debet, contingunt: præterea quia cum <lb></lb>
lunare lumen debile ſit, niſi aliæ cauſæ perfectæ admodum concur­<lb></lb>
rant, quod rarò accidit, Iris nullo modo effulgere valet. </s>
          <s id="s.002206">Hactenus de Iri­<lb></lb>
dis figura ſit ſatis.</s>
        </p>
        <p type="head">
          <s id="s.002207"><emph type="italics"></emph>Summa 2. cap. 5. De Parelio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002208"><arrow.to.target n="marg173"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002209"><margin.target id="marg173"></margin.target>182</s>
        </p>
        <p type="main">
          <s id="s.002210">Textus <emph type="italics"></emph>(Fiunt autem vt diximus, &amp; Virgæ, &amp; Parelia in ortu, &amp; oc­<lb></lb>
caſu, &amp; nec ſupra Solem, nec infra, ſed ex lateribus, nec propè admo­<lb></lb>
dum, nec procul omninò. </s>
          <s id="s.002211">propinquam enim concretionem Sol diſſoluit: <lb></lb>
ſi autem procul abſit, aſpectus non reflectetur, ſi enim à paruo ſpeculo <lb></lb>
procul protenditur imbecillus fit. </s>
          <s id="s.002212">quare, &amp; Coronæ è regione Solis non fiunt. </s>
          <s id="s.002213">ſi igi­<lb></lb>
tur ſupra fuerit, &amp; proxima; eam Sol diſſoluet: ſi verò procul aſpectus minor <lb></lb>quam vt reflecti poſſit in Solem non incidet; à latere autem fieri poteſt, vt ſpecu­<lb></lb>
lum ita distet à Sole, vt non ſoluatur, &amp; aſpectus totus ad eum perueniat, eo quod <lb></lb>
ad terram dum fertur, quaſi per immenſum feratur, peruenire nequeat. </s>
          <s id="s.002214">ſub Sole <lb></lb>
verò non fit, quia cum ad terram propius acceſſerit à Sole diſſoluitur, cum medium <lb></lb>
cœli tenuerit aſpectus diſtrahitur. </s>
          <s id="s.002215">omninò ne à latere quidem, Sole medium cœli <lb></lb>tenente, efficitur, quia aſpectus ſub terram non fertur, quare exiguus ad ſpeculum <lb></lb>producitur, &amp; qui reflectitur prorſus imbecillis redditur)<emph.end type="italics"></emph.end> ibi <emph type="italics"></emph>(propinquam enim <lb></lb>
concretionem Sol diſſoluit)<emph.end type="italics"></emph.end> rationes, quas affert circa Parelia videntur (auda­<lb></lb>
cter loquar) admodum debiles. </s>
          <s id="s.002216">præſens ea eſt, vt Parelium non fiat propè <lb></lb>
Solem, quia illa nubis concretio, quæ Parelio neceſſaria eſt, nequit adeo So­<lb></lb>
li propinqua eſſe, quia nimirum Sol ob propinquitatem eam diſſolueret; ſed <lb></lb>
quis non videt eam nubem, quam vulgò exiſtimamus eſſe Soli propinquam, <lb></lb>
ſeu quaſi inter nos, &amp; Solem tantum, imò etiam minus aliquando à Sole ve­<lb></lb>
rè diſtare, quàm alia, quàm vulgò remotiorem à Sole putabimus? </s>
          <s id="s.002217">præte­<lb></lb>
rea omnes nubes noſtri horizontis re vera æquidiſtare à Sole certum eſt, ob <lb></lb>
maximam enim Solis diſtantiam totus noſter horizon phyſicus eſt inſenſi­<lb></lb>
bilis quantitatis ad Solem, &amp; vnius puncti vicem gerit.</s>
        </p>
        <p type="main">
          <s id="s.002218">Ibi verò <emph type="italics"></emph>(Si autem procul abſit, &amp;c.)<emph.end type="italics"></emph.end> reddit rationem, cur parelium non <lb></lb>
appareat in nube à Sole valde remota ſecundum vulgarem æſtimationem, <lb></lb>
vnde vulgarem etiam rationem affert, ait enim, nubem illam eſſe veluti ſpe­<lb></lb>
culum Solis repreſentatiuum, ſpeculum autem tàm longè à Sole poſitum, <lb></lb>
reddi debile, &amp; proptereá non poſſe Solis imaginem referre: Verùm ratio <lb></lb>
hæc nulla eſſe videtur, quis enim ignorat non propterea eſſe remotius à So­<lb></lb>
le, quamuis maiorem habere videatur à Sole lateralem diſtantiam, vt pau­<lb></lb>
lò ante dixi? </s>
          <s id="s.002219">Eandem rationem illi dubitationi accommodat, cur <expan abbr="neq;">neque</expan> vi­<lb></lb>
deatur ſupra Solem, quamuis non ei quadret, poteſt enim aliqua nubes vi­


<pb pagenum="131" xlink:href="009/01/131.jpg"></pb>deri ſupra Solem, quæ tamen remotior ſit à Sole, quam illa, in qua Parelium <lb></lb>
gignitur. </s>
          <s id="s.002220">Ait poſtea <emph type="italics"></emph>(A latere autem, &amp;c.)<emph.end type="italics"></emph.end> cur appareat in nube fatis Soli <lb></lb>
à latere vicina, in diſtantiam à Sole refert: ſed quæ dudum dicta ſunt, iſtud <lb></lb>
<expan abbr="quoq;">quoque</expan> refellunt. </s>
          <s id="s.002221">Verba illa <emph type="italics"></emph>(Eo quod ad terram dum fertur quaſi per immenſum <lb></lb>
feratur, peruenire nequeat)<emph.end type="italics"></emph.end> videntur alieno loco dicta; ſimilia præcedentibus <lb></lb>
ſunt reliqua, præſertim quæ ibi <emph type="italics"></emph>(Sub Sole verò non fit, quia cum ad terram pro­<lb></lb>
pius acceſſerit)<emph.end type="italics"></emph.end> cur non videatur infra Solem, rationem quandam, quæ fortè <lb></lb>
inanis eſt reddit; nunquid enim non poſſumus tam infra Solem, quàm ſupra <lb></lb>
ita ſpeculum accommodare, vt Solem noſtris viſibus remittat? </s>
          <s id="s.002222">huic certè <lb></lb>
Optice tota repugnat. </s>
          <s id="s.002223">Cum igitur Mathematica ratione hæ rationes non <lb></lb>
conſiſtant, alias alij excogitent. </s>
          <s id="s.002224">Mirum tamen eſt, omnes, quos viderim <lb></lb>
commentatores, eas tanquam optimas admittere.</s>
        </p>
        <p type="head">
          <s id="s.002225"><emph type="italics"></emph>In quarto Meteororum nihil Mathematicum occurrit.<emph.end type="italics"></emph.end></s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002226"><emph type="italics"></emph>EX LIB. PRIMO DE ANIMA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002227"><arrow.to.target n="marg174"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002228"><margin.target id="marg174"></margin.target>183</s>
        </p>
        <p type="main">
          <s id="s.002229">Tex. 11. <emph type="italics"></emph>(Videtur autem non ſolum ipſum quid eſt cognoſcere vtile eſſe <lb></lb>
ad cognoſcendas cauſas accidentium ſubſtantijs: ſicut in Mathemati­<lb></lb>cis quid rectum, &amp; quid obliquum, aut quid linea, &amp; planum, ad co­<lb></lb>gnoſcendum quot rectis, trianguli anguli ſunt æquales)<emph.end type="italics"></emph.end> quid ſit <expan abbr="vnum-quodq;">vnum­<lb></lb>
quodque</expan> ex prædictis patet tum ex definitionibus primi Elem. tum ex com­<lb></lb>
mentarijs ipſarum; quamuis autem ibi non definiatur <expan abbr="rectũ">rectum</expan>, nec obliquum <lb></lb>
in genere, definitur tamen linea recta, &amp; obliqua, &amp; plana ſuperficies, ſiue <lb></lb>
planum, ex quibus facilè definitio recti, &amp; obliqui colligi poteſt: quæ defi­<lb></lb>
nitiones neceſſariæ ſunt ad cognoſcendum quot rectis angulis æquales ſint <lb></lb>tres anguli cuiuſuis trianguli. </s>
          <s id="s.002230">vide quæ de hac æqualitate ſcripſi lib, primo <lb></lb>
Priorum, ſecto 3. cap. 1.</s>
        </p>
        <p type="main">
          <s id="s.002231"><arrow.to.target n="marg175"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002232"><margin.target id="marg175"></margin.target>184</s>
        </p>
        <p type="main">
          <s id="s.002233">Tex. 13. <emph type="italics"></emph>(Si igitur eſt aliqua animæ operatio, aut paſſio propria, continget vti­<lb></lb>
que ipſam ſeparari: ſi verò nulla eſt propria ipſius non vtique erit ſeparabilis. </s>
          <s id="s.002234">ſed <lb></lb>
ſicut recto in quantum rectum multa accidunt, vt tangere æneam ſphæram ſecun­<lb></lb>
dum punctum, non tamen tanget hoc, rectum ipſum ſeparatum: inſeparabile enim, <lb></lb>
ſi quidem cum corpore quodam ſemper eſt)<emph.end type="italics"></emph.end> Propoſitio 2. tertij Elem. ṕrobat li­<lb></lb>
<figure id="id.009.01.131.1.jpg" place="text" xlink:href="009/01/131/1.jpg"></figure><lb></lb>
neam rectam, duo quælibet puncta <expan abbr="quãtumuis">quantumuis</expan> pro­<lb></lb>
pinqua in circuli ambitu aſſumpta coniungentem <lb></lb>
cadere intra circulum. </s>
          <s id="s.002235">v. g. puncta A B, quantum­<lb></lb>uis ſibi inuicem propinqua unerint, attamen ſi line a <lb></lb>
A B, ea coniungat, ipſa cadet intra circulum, &amp; <lb></lb>
veluti chorda ſubtendet arcum A B, quantulum­<lb></lb>
cunque. </s>
          <s id="s.002236">ex qua demonſtratione colligitur in corol­<lb></lb>
lario eius lineam rectam tangentem circulum ip­<lb></lb>
ſum in vnico puncto tangere. </s>
          <s id="s.002237">v. g. rectam C D, tan­<lb></lb>
gere circulum in puncto E. ſi enim dixeris tangere <lb></lb>
in duobus admodum propinquis, vt in E F, tunc non erit amplius tangens, <lb></lb>
ſed ſecans, quia vt modo dixi, pars lineæ rectæ, quæ <expan abbr="cõiungeret">coniungeret</expan> puncta E F,


<pb pagenum="132" xlink:href="009/01/132.jpg"></pb>intra circulum per ſecundam præallegatam caderet, quod eſt abſurdum, <lb></lb>
quia contra hypotheſim, cum ſupponamus illam ſolùm tangere, non autem <lb></lb>
ſecare circulum. </s>
          <s id="s.002238">Ex hac Euclidis doctrina Theodoſius primo ſphæricorum, <lb></lb>
propoſitione 3. probat planum, ſiue ſuperficiem planam tangere ſphæram <lb></lb>
in vnico puncto, vt hoc loco innuit Philoſophus. </s>
          <s id="s.002239">probat autem hac ferè ra­<lb></lb>
<figure id="id.009.01.132.1.jpg" place="text" xlink:href="009/01/132/1.jpg"></figure><lb></lb>
tione. </s>
          <s id="s.002240">ſit ſphæra A B C, quæ tangat quodpiam planum <lb></lb>
in duobus punctis A, B, ſi fieri poteſt. </s>
          <s id="s.002241">per quæ duo pun­<lb></lb>
cta intelligatur ducta recta linea A B, intelligatur <expan abbr="etiã">etiam</expan> <lb></lb>
circulus A B C, qui ſecet ſphæram per centrum C. &amp; <lb></lb>
per puncta A, B, ergo ex demonſtratis ab Euclide li­<lb></lb>
nea A B, quæ coniungit puncta A B, cadet intra prædi­<lb></lb>
ctum circulum; ſed linea hæc eſt in plano tangente ex <lb></lb>
ſuppoſitione, circulus verò in ſphæra; ergò cum linea <lb></lb>
cadat intra circulum, cadet etiam neceſſariò planum <lb></lb>
in quo eſt linea, &amp; cum linea cadat intra circulum, cadet etiam neceſſariò <lb></lb>
intra ſphæram; <expan abbr="idemq́">idemque</expan>; faciet planum, quod eam neceſſariò ſequatur, ergò <lb></lb>
planum ſecat ſphæram, non autem tangit, quod eſt abſurdum, quia contra <lb></lb>
hypotheſim, ſupponunt autem Mathematici, entia hæc mathematica eſſe <lb></lb>
perfecta, qualia in ſublunaribus fortè non reperiuntur; ænea enim ſphæra <lb></lb>
nulla erit perfectè rotunda, vel planum aliquod perfectè complanatum, vt <lb></lb>
ipſi ſupponunt, eò quod materiæ imperfectio, ac ruditas id nequaquam pa­<lb></lb>
tiatur. </s>
          <s id="s.002242">quare cum huiuſmodi entia non reperiantur abſtracta ab impura hac <lb></lb>
materia, nullum erit inquit Ariſt. abſtractum planum, quod poſſit mathe­<lb></lb>
maticè, <expan abbr="atq;">atque</expan> adeò in vnico puncto mathematico ſphæram tangere. </s>
          <s id="s.002243"><expan abbr="hucuſq;">hucuſque</expan> <lb></lb>
neceſſaria ſunt mathematica ad huius loci <expan abbr="intelligentiã">intelligentiam</expan>. </s>
          <s id="s.002244">ex quibus ea etiam, <lb></lb>
quæ ad phyſicum ſpectant manifeſta fiunt, nimirum ſicut entia mathemati­<lb></lb>
ca à materia non exiſtunt ſeparata, quia ſic nullam haberent operationem; <lb></lb>
ita etiam anima, ſi nullam habet propriam operationem non exiſtet à cor­<lb></lb>
pore ſeparata.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002245"><emph type="italics"></emph>Ex Secundo de Anima.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002246"><arrow.to.target n="marg176"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002247"><margin.target id="marg176"></margin.target>185</s>
        </p>
        <p type="main">
          <s id="s.002248">Tex. 12. <emph type="italics"></emph>(Non enim ſolum ipſum, quod ſit, oportet definitiuam rationem <lb></lb>oſtendere, ſicut plures definitionum dicunt, ſed &amp; cauſam ineſſe, &amp; ap­<lb></lb>
parere. </s>
          <s id="s.002249">nunc autem, vt concluſiones rationes definitionum ſunt, vt quid <lb></lb>
tetragoniſmus? </s>
          <s id="s.002250">æquale altera parte longiori rectangulum æquilaterum <lb></lb>
eſſe, talis autem definitio ratio concluſionis. </s>
          <s id="s.002251">dicens autem, quod tetragoniſmus eſt <lb></lb>
medij inuentio rei cauſam dicit)<emph.end type="italics"></emph.end> aggreſſurus Ariſt. animæ definitionem præ­<lb></lb>
mittit duplicem eſſe definitionem, alteram ſcilicet, quæ explicat ſolum rei <lb></lb>
eſſentiam, quam dicunt formalem definitionem; alteram verò, quæ præte­<lb></lb>
rea explicat etiam rei cauſam, quam dicunt cauſalem definitionem: vtram­<lb></lb>
que autem exemplo Geometrico explicat.</s>
        </p>
        <p type="main">
          <s id="s.002252">In cap. igitur de relatione plura ſcripſi de tetragoniſmo, ſeu quadratio­<lb></lb>
ne circuli, quæ huc ſpectant. </s>
          <s id="s.002253">propterea nunc tantum propria huius loci <expan abbr="de-clarãda">de­<lb></lb>
claranda</expan> reſtant. </s>
          <s id="s.002254">loquitur igitur hic Philoſophus non de quadratione circuli,<pb pagenum="133" xlink:href="009/01/133.jpg"></pb>ſed figuræ rectilineæ illius, quæ dicitur Altera parte longior, qualis eſt præ­<lb></lb>
ſens figura A B C D, cuius quadrandæ ratio eſt huiuſmodi. </s>
          <s id="s.002255">per 13. 6. inue­<lb></lb>
<figure id="id.009.01.133.1.jpg" place="text" xlink:href="009/01/133/1.jpg"></figure><lb></lb>
niatur recta linea media proportionalis inter <lb></lb>
duo latera figuræ A B, B C, <expan abbr="eaq́">eaque</expan>; ſit B D, in ſe­<lb></lb>
quenti figura. </s>
          <s id="s.002256">eſſe autem mediam proportio­<lb></lb>
nalem nihil aliud eſt quam ita eſſe A B, ad B D, <lb></lb>
ſicut B D, ad B C. <expan abbr="diciturq́">diciturque</expan>; media proportio­<lb></lb>
nalis, quia in hac habitudine medium locum obtinet. </s>
          <s id="s.002257">quadratum autem li­<lb></lb>
neæ B D, æquale eſt rectangulo dato A B C D, per 17.6. Inuentio porrò hu­<lb></lb>
ius mediæ proportionalis, quia facilis eſt, &amp; ſcitu iucunda, eam ſic habeto. <lb></lb>
<figure id="id.009.01.133.2.jpg" place="text" xlink:href="009/01/133/2.jpg"></figure><lb></lb>
accipe duo latera A B, &amp; B C, <expan abbr="quadrãdi">quadrandi</expan> rectan­<lb></lb>
guli, <expan abbr="eaq́">eaque</expan>; in directum conſtitue, vt vnicam re­<lb></lb>
ctam conſtituant A C, vt apparet in figura; de­<lb></lb>
inde diuiſa tota A C, bifariam in E, facto cen­<lb></lb>
tro in E, deſcribe ſemicirculum ſuper lineam <lb></lb>
A C, demum à puncto B, in quo duo latera con­<lb></lb>
iunguntur, erigatur linea perpendicularis <expan abbr="vſq;">vſque</expan> <lb></lb>
ad periphæriam, quæ ſit B D. hæc enim B D, eſt media proportionalis inter <lb></lb>
latera A B, B D, quam nimirum habitudinem habet A B, ad B D, eam quo­<lb></lb>
que obtinet B D, ad B C. </s>
          <s id="s.002258">Quadratum igitur huius B D, hoc eſt quadratum, <lb></lb>cuius quatuor latera ſint æqualia lineæ B D, quale eſt præſens, æquale erit <lb></lb>
<figure id="id.009.01.133.3.jpg" place="text" xlink:href="009/01/133/3.jpg"></figure><lb></lb>
dato ſuperiori rectangulo A B C D, <expan abbr="atq;">atque</expan> hoc modo per­<lb></lb>
acta erit quadratio, ſeu tetragoniſmus dati quadrilateri <lb></lb>
A B C D. </s>
          <s id="s.002259">Vides igitur, qua ratione quadratum conſti­<lb></lb>
tuatur æquale dato quadrilatero; &amp; qua rationem inuen­<lb></lb>
tio illius mediæ proportionalis ſit cauſa quadraturæ re­<lb></lb>
ctanguli, &amp; proinde ſi quis dicat quadrationem hanc eſſe <lb></lb>
effectionem rectanguli æquilateri, ideſt quadrati, æqualis dato quadrilate­<lb></lb>
ro, hic definitionem formalem ſolum afferet: quæ definitio, vt dixit in Lo­<lb></lb>
gicis, eſt inſtar concluſionis. </s>
          <s id="s.002260">ſi quis verò dicat tetragoniſmum hunc quadri­<lb></lb>
lateri dati eſſe mediæ prædictæ inuentionem cauſalem afferet definitionem, <lb></lb>
cum rei cauſam dicat. </s>
          <s id="s.002261">Aduerte 10. Grammaticum immeritò accuſare Ale­<lb></lb>
xandrum, quod dicat quadrationem hanc per inuentionem mediæ propor­<lb></lb>
tionalis tradi in 2. Elem. nam verè in 14. 2. traditur talis inuentio, quam­<lb></lb>uis enim ibi nulla fiat expreſſa mentio huiuſmodi mediæ, in ipſa tamen ea <lb></lb>
reperitur, ac per eam figuræ rectilineæ quadrantur: quod patet ex figura <lb></lb>
14. prædictæ, quæ eadem eſt cum figura 13. 6. qua docemur prædictam in­<lb></lb>
uentionem.</s>
        </p>
        <p type="main">
          <s id="s.002262"><arrow.to.target n="marg177"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002263"><margin.target id="marg177"></margin.target>186</s>
        </p>
        <p type="main">
          <s id="s.002264">Tex. 86. <emph type="italics"></emph>(Acutum mouet ſenſum in tempore pauco multùm: graue autem in <lb></lb>multo parùm; non igitur velox eſt acutum, graue autem tardum, ed ſit illius qui­<lb></lb>
dem propter velocitatem motus huiuſmodi, huius autem propter tarditatem)<emph.end type="italics"></emph.end> vide <lb></lb>
quæ de hac re primo topic. </s>
          <s id="s.002265">cap. 13. dicta ſunt, illa enim omnia in hunc lo­<lb></lb>
cum quadrant. </s>
          <s id="s.002266">Verum occurrit illa dubitatio; quod cum Ariſt. ibi dicat <lb></lb>
<emph type="italics"></emph>(Vox acuta quidem velox)<emph.end type="italics"></emph.end> hic autem <emph type="italics"></emph>(Non igitur velox eſt acutum<emph.end type="italics"></emph.end>) repugnan­<lb></lb>
tia dicere videtur. </s>
          <s id="s.002267">cui dubitationi ſic occurrendum; vt dicamus ibi Philo­<lb></lb>ſophum dicere vocem acutam eſſe velocem, quatenus acumen vocis oritur <pb pagenum="134" xlink:href="009/01/134.jpg"></pb>ex velocitate motus aerem impellentis. </s>
          <s id="s.002268">hic verò diſtinguere acutum à ve­<lb></lb>
loci, tanquam effectum à cauſa.</s>
        </p>
        <p type="main">
          <s id="s.002269"><arrow.to.target n="marg178"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002270"><margin.target id="marg178"></margin.target>187</s>
        </p>
        <p type="main">
          <s id="s.002271">Tex. 159. <emph type="italics"></emph>(Apparent autem, &amp; falſa, de quibus ſimul exiſtimationem veram <lb></lb>habet, vt apparet ſol vnius pedis, perſuaſum autem eſt, eum maiorem eſſe habitata)<emph.end type="italics"></emph.end><lb></lb>
habitata, ideſt terra habitata. </s>
          <s id="s.002272">Vide, quæ cap. 3. ſummæ 1. primi Meteor. <lb></lb>
</s>
          <s id="s.002273">Item capite 5. ſummæ 2. de Solis magnitudine ſcripſi, ea enim huic loco <lb></lb>
abundè ſatisfaciunt.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002274"><emph type="italics"></emph>Ex Tertio de Anima.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002275"><arrow.to.target n="marg179"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002276"><margin.target id="marg179"></margin.target>188</s>
        </p>
        <p type="main">
          <s id="s.002277">Tex. 21. <emph type="italics"></emph>(Vt incommenſurabile, &amp; diameter)<emph.end type="italics"></emph.end> vide, quæ de incom­<lb></lb>
menſuratione diametri, &amp; coſtæ ſcripta ſunt lib. 1. Priorum, cap. <lb></lb>
23. vnde ſatis huic loco fieri poteſt.</s>
        </p>
        <p type="main">
          <s id="s.002278"><arrow.to.target n="marg180"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002279"><margin.target id="marg180"></margin.target>189</s>
        </p>
        <p type="main">
          <s id="s.002280">Tex 25. <emph type="italics"></emph>(Punctum autem, &amp; omnis diuiſio, &amp; ſic indiuiſibile mon­<lb></lb>ſtratur ſicut priuatio)<emph.end type="italics"></emph.end> punctum enim cum ſit terminus lineæ, eſt negatio vl­<lb></lb>
terioris lineæ <emph type="italics"></emph>(Et omnis diuiſio)<emph.end type="italics"></emph.end> innuit his verbis præter punctum, lineam <lb></lb>
etiam, &amp; ſuperficiem, nam quemadmodum punctus oritur ex diuiſione li­<lb></lb>
neæ, ita linea ex diuiſione ſuperficiei, &amp; ſuperficies ex diuiſione corporis. <lb></lb>
</s>
          <s id="s.002281">&amp; quamuis punctum, linea, ſuperficies, ſint indiuiſibilia, monſtrantur ta­<lb></lb>
men quatenus ſunt priuationes, ſeu negationes, illud vlterioris lineæ, iſta <lb></lb>
vlterioris ſuperficiei, hæc tandem vlterioris corporis.</s>
        </p>
        <p type="main">
          <s id="s.002282"><arrow.to.target n="marg181"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002283"><margin.target id="marg181"></margin.target>190</s>
        </p>
        <p type="main">
          <s id="s.002284">Tex. 32. <emph type="italics"></emph>(Sit igitur vt A, quidem album, ad B, quod nigrum; ſic C, ad D; qua­<lb></lb>
re &amp; permutatim)<emph.end type="italics"></emph.end> ideſt, quare &amp; permutando (vt aiunt Geometræ) erit vt <lb></lb>
A, ad C, ita B, ad D, hunc argumentandi modum à permutata proportio­<lb></lb>
ne explicaui in primo Poſter. cap. 5. tex. 13. dicitur etiam alterna ratio; <lb></lb>
&amp; definitur ab Euclide definitione 12, 5.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002285"><emph type="italics"></emph>Ex Libro de Senſu.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002286"><arrow.to.target n="marg182"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002287"><margin.target id="marg182"></margin.target>191</s>
        </p>
        <p type="main">
          <s id="s.002288">Cap, 6. <emph type="italics"></emph>(Et qui in Dieſi ſonus latet, quamuis continuum exiſtentem audit <lb></lb>omnem cantum, diſtantia enim eius ad extremos ſonos latet)<emph.end type="italics"></emph.end> quid ſit <lb></lb>
Dieſis apud Muſicos explicatum eſt primo Poſter. tex. 38. cum <lb></lb>
autem Dieſis ſit minima diſtantia, ſeu vt loquuntur Muſici, mini­<lb></lb>
mum <expan abbr="interuallũ">interuallum</expan> inter duas voces, hinc fit vt hæc minima diſtantia inter ex­<lb></lb>
tremos ſonos non exaudiatur, quemadmodum nec minima particula alicu­<lb></lb>
ius magni corporis à longè viſi <expan abbr="nõ">non</expan> percipitur, ſed latet inter extrema illius.</s>
        </p>
        <p type="main">
          <s id="s.002289"><arrow.to.target n="marg183"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002290"><margin.target id="marg183"></margin.target>192</s>
        </p>
        <p type="main">
          <s id="s.002291">Cap. 8. <emph type="italics"></emph>(<expan abbr="Vnumquodq;">Vnumquodque</expan> magis eſt ſentire ſimplex exiſtens, quàm mixtum, velut <lb></lb>
vinum non temperatum, quàm temperatum; &amp; mel, &amp; colorem, &amp; neten ſolam. <lb></lb>
</s>
          <s id="s.002292">quàm in diapaſon, quia obſcurant ſe inuicem)<emph.end type="italics"></emph.end> nete apud veteres muſicos erat <lb></lb>
in muſicis inſtrumentis omnium chordarum acutiſſima, cuiuſmodi apud <lb></lb>
nos eſt, quam vulgò canto appellant. </s>
          <s id="s.002293">Hypate verò erat chorda omnium <lb></lb>
grauiſſima, qualis eſt ea, quam modo Baſſo vocant. </s>
          <s id="s.002294">hæ duæ ſimul pulſatæ <lb></lb>edebant conſonantiam, quæ Diapaſon dicitur, &amp; vulgò octaua. </s>
          <s id="s.002295">ex quibus <lb></lb>
ſenſus verberum Ariſt. manifeſtus eſt.</s>
        </p>
        <pb pagenum="135" xlink:href="009/01/135.jpg"></pb>
        <p type="main">
          <s id="s.002296"><arrow.to.target n="marg184"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002297"><margin.target id="marg184"></margin.target>193</s>
        </p>
        <p type="main">
          <s id="s.002298">Eodem cap. <emph type="italics"></emph>(Velut Diapaſon, &amp; Diapente)<emph.end type="italics"></emph.end> quid ſit conſonantia Diapa­<lb></lb>
ſon, explicaui in primo Poſter. tex. 1. Diapente verò eſt conſonantia ex duo­<lb></lb>
<figure id="id.009.01.135.1.jpg" place="text" xlink:href="009/01/135/1.jpg"></figure><lb></lb>
bus ſonis coaleſcens, quorum proportio ſit vt <lb></lb>
3. ad 2. quæ dicitur ſeſquialtera. </s>
          <s id="s.002299">v. g. ſint duæ <lb></lb>chordæ æqualis craſſitiei, <expan abbr="atq;">atque</expan> æquè tenſæ: vna <lb></lb>
tamen habeat ad alteram proportionem ſeſ­<lb></lb>
quialteram, vt in figura apparet; ſi ſimul pul­<lb></lb>
ſentur, edent conſonantiam Diapente. </s>
          <s id="s.002300">vulgò autem quinta.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002301"><emph type="italics"></emph>Ex Libro de Memoria, &amp; reminiſcentia.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002302"><arrow.to.target n="marg185"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002303"><margin.target id="marg185"></margin.target>194</s>
        </p>
        <p type="main">
          <s id="s.002304">Cap. 1. <emph type="italics"></emph>(Sic meminit eos, qui trianguli, quod duobus rectis æquales)<emph.end type="italics"></emph.end> ideſt <lb></lb>
ſic meminit tres angulos cuiuſuis trianguli ſimul ſumptos æqua­<lb></lb>
les eſſe duobus angulis rectis ſimul ſumptis. </s>
          <s id="s.002305">lege annotata primo <lb></lb>
Poſter. ſecto 3. cap. 1.</s>
        </p>
        <p type="main">
          <s id="s.002306"><arrow.to.target n="marg186"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002307"><margin.target id="marg186"></margin.target>195</s>
        </p>
        <p type="main">
          <s id="s.002308">Cap. 3. <emph type="italics"></emph>(Sunt facilè reminiſcibilia, <expan abbr="quæcunq;">quæcunque</expan> habent ordinationem aliquam, <lb></lb>
vt mathemata)<emph.end type="italics"></emph.end> hęc Philoſophus dicens ſpectabat ad mirabilem illam, ac per­<lb></lb>
petuam de monſtrationum connexionem, qua Geometræ omnes, &amp; præci­<lb></lb>
puè Euclides opera ſua ab initio ad finem vſque, diuino planè ingenij acu­<lb></lb>
mine deduxerunt.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002309"><emph type="italics"></emph>Ex Libro de Somnijs.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002310"><arrow.to.target n="marg187"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002311"><margin.target id="marg187"></margin.target>196</s>
        </p>
        <p type="main">
          <s id="s.002312">Cap. 2. <emph type="italics"></emph>(Cur autem fallimur, cauſa eſt, quoniam non ſolum cum ſenſibile <lb></lb>
mouetur apparent quælibet, ſed etiam cum ſenſus ipſe mouetur, ſi eodem <lb></lb>
modo moueatur, quemadmodum à ſenſibili. </s>
          <s id="s.002313">dico autem velut terra vi­<lb></lb>
detur nauigantibus moueri, dummodo viſus ab alio)<emph.end type="italics"></emph.end> reddit rationem, <lb></lb>
cur nauigantibus videatur terra ipſa moueri, ac retrocedere, non autem <lb></lb>ipſi nauigantes, quin potius ipſi ſibi ſtare videantur. </s>
          <s id="s.002314">cauſam igitur eam eſ­<lb></lb>
ſe ait, quia ex motu nauis, terra ipſa manente, accidit, vt eodem modo im­<lb></lb>
mutetur ſenſus viſus, ac ſi terra ipſa moueretur, viſus verò quieſceret. <lb></lb>
</s>
          <s id="s.002315">At cur eodem modo afficitur ſenſus? </s>
          <s id="s.002316">Perſpectiui rationem eſſe dicunt, quia <lb></lb>
ea, quæ circa oculum ſunt, vt nauis, &amp; ea, quæ in naui ſunt, non mutant ſi­<lb></lb>
tum reſpectu oculi, quemadmodum facerent, ſi nos ipſi ſine naui progrede­<lb></lb>
remur. </s>
          <s id="s.002317">arbores autem, &amp; reliqua, quæ in terra ſunt, variant ſitum reſpectu <lb></lb>
oculi, non ſecus, ac ſi ipſæ arbores retro deferrentur. </s>
          <s id="s.002318">propterea igitur viſus <lb></lb>
tunc arbores remeare iudicat, quia quæ circa oculum ſunt reſpectu ipſius <lb></lb>
oculi non mouentur, ſiue non variant ſitum ad ipſum; ex variatione enim <lb></lb>
ſitus rei reſpectu oculi, percipimus cuiuſuis rei localem motum.</s>
        </p>
        <p type="main">
          <s id="s.002319"><arrow.to.target n="marg188"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002320"><margin.target id="marg188"></margin.target>197</s>
        </p>
        <p type="main">
          <s id="s.002321">Cap. 3. <emph type="italics"></emph>(Quemadmodum igitur, ſi quem lateat ſuppoſitus oculo digitus, non <lb></lb>ſolum apparebit, ſed etiam putabitur duo, quod eſt vnum. </s>
          <s id="s.002322">Si verò non lateat appa­<lb></lb>
rebit quidem, non putabitur tamen)<emph.end type="italics"></emph.end> eſt hæc optica deceptio, quæ tunc accidit, <lb></lb>
cum aliquod obiectum intuentes, interim digito alterum oculum ſurſum <lb></lb>
pellimus, ita vt oculi propterea varient ſitum reſpectu obiecti, ſiue non eo­


<pb pagenum="136" xlink:href="009/01/136.jpg"></pb>dem ſitu vterque obiectum intueatur, hoc eſt, vt optici aiunt, axes viſuales <lb></lb>
non amplius concurrunt ſimul in rem viſam. </s>
          <s id="s.002323">Vnde ſequitur ſpeciem rei in­<lb></lb>
tentionalem oculis vario ſitu affectis imprimi, ac proinde eam eundem ſi­<lb></lb>
tum in vtroque oculo minimè obtinere, ſed ea, quæ oculo à ſuo naturali <lb></lb>ſtatu dimoto accidit ab altera alterius oculi differt; quapropter vario <lb></lb>etiam modo, duplici nimirum, obiectum repreſentant. </s>
          <s id="s.002324">atque hæc <lb></lb>
ipſa cauſa eſt, cur illud, quod vnum tantum eſt, duo tamen <lb></lb>
emoto oculorum altero, videatur. </s>
          <s id="s.002325">Vide Alhaze­<lb></lb>
num lib. 3. propoſit. </s>
          <s id="s.002326">11. &amp; 12. &amp; infra <lb></lb>
Problem. 7. ſectionis 31.</s>
        </p>
      </chap>
      <pb pagenum="137" xlink:href="009/01/137.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.002327">EX PRIMO <lb></lb>
METAPHYSICAE.</s>
        </p>
        <p type="main">
          <s id="s.002328"><arrow.to.target n="marg189"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002329"><margin.target id="marg189"></margin.target>198</s>
        </p>
        <p type="main">
          <s id="s.002330">Capite 1. <emph type="italics"></emph>(Circa Aegyptum Mathematicæ artes constitutæ ſunt; illic <lb></lb>
enim gens Sacerdotum vacare permittitur)<emph.end type="italics"></emph.end> Notanda maximè no­<lb></lb>
bilis Mathematicarum origo, cum ab Aegyptiorum Sacerdoti­<lb></lb>
bus teſte Philoſopho fuerint adinuentæ, quibus occaſionem præ­<lb></lb>
buit anniuerſaria agrorum ob Nili innundationem, diuiſio: cum enim iam <lb></lb>
perplures dimetiendorum agrorum rationes repertæ fuiſſent, Sacerdotes <lb></lb>ipſi, quibus per otium licebat, illarum praxium demonſtrationes cœperunt <lb></lb>
perueſtigare, <expan abbr="ſicq́">ſicque</expan>; paulatim poſtea Geometria amplius exculta adoleuit; <lb></lb>
quæ deinde ijſdem ad res aſtronomicas perſcrutandas <expan abbr="adiumẽto">adiumento</expan> fuit, <expan abbr="hacq́">hacque</expan>; <lb></lb>
ratione reliquas etiam in mathematicas inciderunt.</s>
        </p>
        <p type="main">
          <s id="s.002331"><arrow.to.target n="marg190"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002332"><margin.target id="marg190"></margin.target>199</s>
        </p>
        <p type="main">
          <s id="s.002333">Cap. 2. <emph type="italics"></emph>(Sicut de præſtigioſis, quæ per ſe mouentur, illi qui nondum ſpeculati <lb></lb>
ſunt cauſam<emph.end type="italics"></emph.end>) verbis illis (<emph type="italics"></emph>Quæ per ſemouentur<emph.end type="italics"></emph.end>) vnica dictio <expan abbr="Græcareſpõdet">Græca reſpondet</expan>, <lb></lb>
Automata. </s>
          <s id="s.002334">erant autem Automata apud veteres Gręcos machinæ quędam, <lb></lb>
quæ à Mathematicis Mechanicæ artis occultis quibuſdam ingenijs, ea arte <lb></lb>
conſtruebantur, vt à ſeipſis de loco ad locum, ac ſi viuæ eſſent ſpontè pro­<lb></lb>grederentur; vnde, &amp; automata, quaſi ſpontanea dicebantur. </s>
          <s id="s.002335">Extat adhuc <lb></lb>
de huiuſmodi machinis liber Heronis Alexandrini, quem nuper ex græco <lb></lb>
latinum reddidit doctiſſimus Abbas Guaſtallenfis. </s>
          <s id="s.002336">de huiuſmodi artificioſis <lb></lb>
operibus, quibus ſæpè priſci ita admirationi fuere, vt præſtigia quædam ar­<lb></lb>
tificium ignorantibus, viderentur, intelligit hoc loco Ariſt.</s>
        </p>
        <p type="main">
          <s id="s.002337"><arrow.to.target n="marg191"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002338"><margin.target id="marg191"></margin.target>200</s>
        </p>
        <p type="main">
          <s id="s.002339">Cap. 3. (<emph type="italics"></emph>Aut de ſolſtitijs<emph.end type="italics"></emph.end>) quid ſolſtitium, cur dicatur ſolſtitium, &amp; cur Sol <lb></lb>
in <expan abbr="vtroq;">vtroque</expan> topico, quoad dierum incrementum, ac decrementum, &amp; quoad <lb></lb>
eleuationem eius, aut depreſſionem meridianam, videatur moras trahere, <lb></lb>
quamuis noſtrum ſit explicare, ob rei tamen facilitatem omittantur. </s>
          <s id="s.002340">Hoc <lb></lb>
tantum ſcias velim ſolſtitiorum cauſam eſſe Zodiaci ad Tropicos longio­<lb></lb>
rem adhæſionem, ideſt, quòd Zodiacus propè contactum tropicorum ab ijs <lb></lb>
parum recedat, cum ergo Sol motu proprio ſemper per Zodiacum inam­<lb></lb>
bulet, fit vt ipſe <expan abbr="quoq;">quoque</expan> pariter modicum à tropicis remoueatur, imò pluri­<lb></lb>
mum ſecus illos incedat, ita vt eo tempore, quo ad eos paulatim accedit, <lb></lb>
aut ab eis paulatim recedit, quaſi ſtare, ſiue quieſcere apud eoſdem videa­<lb></lb>
tur: <expan abbr="atq;">atque</expan> hinc etiam quantitas <expan abbr="dierũ">dierum</expan>, ac noctium videatur ferè nihil variari; <lb></lb>&amp; noua eleuatio, aut depreſſio Solis ſupra horizontem nulla ferè appareat.</s>
        </p>
        <p type="main">
          <s id="s.002341"><arrow.to.target n="marg192"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002342"><margin.target id="marg192"></margin.target>201</s>
        </p>
        <p type="main">
          <s id="s.002343">Ibidem (<emph type="italics"></emph>Aut de diametri incommenſurabilitate, admirabile enim omnibus vi­<lb></lb>
detur, ſi quid, cum non ſit minimum non menſuretur, decet autem in contrarium, <lb></lb>&amp; in melius ſecundum prouerbium conſumare, quemadmodŭm in his fit, cum diſcant, <lb></lb>nihil enim magis vir Geometricus admiraretur, quàm ſi diameter commenſurabi­<lb></lb>lis fieret<emph.end type="italics"></emph.end>) vide quæ de hac commenſurabilitate ſcripſi lib. 1. Priorum, ſect. </s>
          <s id="s.002344">1. <lb></lb>
cap. 1. Videtur inquit mirum à principio Geometriam aggredienti diame­<lb></lb>
trum, &amp; latus eiuſdem quadrati non commenſurari, cum in neutro eorum <lb></lb>
detur minimum, ſeu indiuiſibile, videtur enim omne diuiſibile poſſe menſu­


<pb pagenum="138" xlink:href="009/01/138.jpg"></pb>rari. </s>
          <s id="s.002345">poſtea tamen cum in Geometria verſatus fuerit, maximè admirare­<lb></lb>
tur, ſi audiret diametrum eſſe lateri commenſurabilem.</s>
        </p>
        <p type="main">
          <s id="s.002346"><arrow.to.target n="marg193"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002347"><margin.target id="marg193"></margin.target>202</s>
        </p>
        <p type="main">
          <s id="s.002348">Summa 2. cap. 3. (<emph type="italics"></emph>Pythagorici primi Mathematicis operam dedere, hæc præpo­<lb></lb>
nebant, &amp; in cis enutriti, eorum principia, entium <expan abbr="quoq;">quoque</expan> cunctorum putant eſſe <lb></lb>
principia<emph.end type="italics"></emph.end>) vtinam noſtrates Philoſophi Pythagoricos imitarentur; enimue­<lb></lb>
rò multò melius &amp; ſibi, &amp; Philoſophiæ conſulerent. </s>
          <s id="s.002349">At verò non ſine ma­<lb></lb>
gno artium, <expan abbr="atq;">atque</expan> diſciplinarum omnium diſpendio à plurimis hac tempe­<lb></lb>
ſtate deſpectui habentur; ſed quid mirum cum quas ſcientiarum omnium <lb></lb>alumni Pythagorei omnibus ſcientijs anteferebant; eas noſtri ſeculi quam­ <lb></lb>
plures omnibus alijs facultatibus poſthabeant.</s>
        </p>
        <p type="main">
          <s id="s.002350"><arrow.to.target n="marg194"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002351"><margin.target id="marg194"></margin.target>203</s>
        </p>
        <p type="main">
          <s id="s.002352">Tex. 47. (<emph type="italics"></emph>Qui Geometriam diſcit aliqua præſcire contingit<emph.end type="italics"></emph.end>) ideſt definitio­<lb></lb>
nes, poſtulata, axiomata, quæ ſunt tria principiorum genera, ex quibus to­<lb></lb>
ta Geometria deducitur.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002353"><emph type="italics"></emph>Ex Secundo Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002354"><arrow.to.target n="marg195"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002355"><margin.target id="marg195"></margin.target>204</s>
        </p>
        <p type="main">
          <s id="s.002356">Tex. 14. (<emph type="italics"></emph>Quantam verò vim conſuetudo habeat, leges declarant, in qui­<lb></lb>
bus fabuloſa, ac puerilia plus poſſunt propter conſuetudinem, quàm ſi <lb></lb>
ea cognoſceremus<emph.end type="italics"></emph.end>) per leges intelligit cantilenas illas, quas vete­<lb></lb>
res Muſici leges appellabant, eò quòd eas ſolas, cæteris abroga­<lb></lb>
tis liceret lata lege decantari. </s>
          <s id="s.002357">Vide declarationem problematis 15. &amp; 28. <lb></lb>
ſect. </s>
          <s id="s.002358">19. <expan abbr="problematũ">problematum</expan> vbi <expan abbr="tanquã">tanquam</expan> in proprio loco iſta fuſius pertractabuntur.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002359"><emph type="italics"></emph>Ex Tertio Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002360"><arrow.to.target n="marg196"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002361"><margin.target id="marg196"></margin.target>205</s>
        </p>
        <p type="main">
          <s id="s.002362">Tex. 3. Verba huius textus, cum ſatis perſpicua ſint, ac parum ma­<lb></lb>
thematicis indigeant, omittenda duxi. </s>
          <s id="s.002363">Quod ad mathematicas <lb></lb>
attinet, ait, eas non demonſtrare, nec per cauſam finalem, nec <lb></lb>
per efficientem (quod intelligendum eſt de Mathematicis puris, <lb></lb>
&amp; ſpeculatiuis nam mathematicæ practicæ reliquas etiam cauſas, efficien­<lb></lb>
tem, &amp; finalem neceſſariò habere debent, quapropter ſophiſta quidam no­<lb></lb>
mine Ariſtippus, eas irridebat, <expan abbr="atq;">atque</expan> adeo illiberalibus, ac ſedentarijs arti­<lb></lb>
bus poſthabebat, quæ cauſam efficientem, quia ſcilicet operantur, &amp; fina­<lb></lb>
lem ſcilicet quæſtum ſibi proponunt. </s>
          <s id="s.002364">fuit autem iſte ex Plutarcho, &amp; Laer­<lb></lb>
tio primus, qui pacto pretio doceret, <expan abbr="philoſophiamq́">philoſophiamque</expan>; faceret quæſtuoſam: <lb></lb>
<expan abbr="ideoq́">ideoque</expan>; mathematicas paruipendebat, quòd neglecta cauſa efficiente, nihil <lb></lb>
efficerent; &amp; finali, nihil lucrarentur. </s>
          <s id="s.002365">videas igitur quales ſint pulcherrima­<lb></lb>
rum facultatum contemptores, ij nimirum, qui philoſophiæ, aut lucri, aut <lb></lb>
ambitionis cauſa dant operam. </s>
          <s id="s.002366">Quod autem Mathematicæ nihil efficiant, <lb></lb>
<expan abbr="nihilq́">nihilque</expan>; lucrentur, ne videamur vtile paruifacere, eſt omninò falſum: ſunt <lb></lb>
enim plures mathematicæ practicæ, quæ innumera, <expan abbr="atq;">atque</expan> <expan abbr="admirãda">admiranda</expan> efficiunt <lb></lb>
opera, <expan abbr="quæq́">quæque</expan>; magnos quæſtus quotidie faciunt. </s>
          <s id="s.002367">huiuſmodi ſunt Geometria <lb></lb>
practica, qua menſurationes omnes vel ſolo viſu perficiuntur. </s>
          <s id="s.002368">Arithmeti­<lb></lb>
ca, cuius vſus quàm latè patet? </s>
          <s id="s.002369">Muſica practica, qua quotidie ipſi oblecta­


<pb pagenum="139" xlink:href="009/01/139.jpg"></pb>mur; <expan abbr="Deoq́">Deoque</expan>; Optimo Maximo laudes debitas concinimus. </s>
          <s id="s.002370">Mechanica pra­<lb></lb>
ctica, cuius ope ingentia pondera, vel exigua vi, <expan abbr="inuitaq́">inuitaque</expan>; natura ſuſq; <expan abbr="deq́">deque</expan>; <lb></lb>
commouentur. </s>
          <s id="s.002371">Perſpectiua, quæ Pictoribus, &amp; Architectoribus adeo inſer­<lb></lb>
uit, vt <expan abbr="abſq;">abſque</expan> ea nihil fermè audeant. </s>
          <s id="s.002372">Aſtronomia tandem, ſi in praxim de­<lb></lb>
ducatur, ex vna ſolum eclypſium prædictione, quantam vniuerſo orbi ad­<lb></lb>
mirationem parit? </s>
          <s id="s.002373">mitto hanc ſolam dierum, menſium, &amp; annorum diſtri­<lb></lb>
butionem, ac temporum emendationem exhibere, rem adeò Reipublicæ <lb></lb>
Chriſtianæ neceſſariam.</s>
        </p>
        <p type="main">
          <s id="s.002374"><arrow.to.target n="marg197"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002375"><margin.target id="marg197"></margin.target>206</s>
        </p>
        <p type="main">
          <s id="s.002376">Eodem tex. 3. (<emph type="italics"></emph>Item &amp; in cæteris tunc ſcire vnumquodque arbitramur torum, <lb></lb>
quorum ſunt demonstrationes, cum quid eſt ſciamus, vt puta quid tetragoniſmus, <lb></lb>
quòd inuentio mediæ<emph.end type="italics"></emph.end>) eadem reperies ſuperius in ſecundo de Anima, tex. 12. <lb></lb>
fuſius explicata.</s>
        </p>
        <p type="main">
          <s id="s.002377"><arrow.to.target n="marg198"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002378"><margin.target id="marg198"></margin.target>207</s>
        </p>
        <p type="main">
          <s id="s.002379">Tex. 8. (<emph type="italics"></emph>Si enim in hoc differret ſolum Geometria à Geodæſia, quod hæc quidem <lb></lb>
eorum eſt, quæ ſentimus, illa verò non ſenſibilium eſt<emph.end type="italics"></emph.end>) Geodæſia eſt pars Geo­<lb></lb>
metriæ practicæ, ea ſcilicet, quæ circa diuiſionem ſuperficierum verſatur. <lb></lb>
</s>
          <s id="s.002380">audi Pediaſmum de menſuratione: Terræ inquit menſuratio in duas partes <lb></lb>
diuiditur, Geometriam ſcilicet, &amp; Geodæſiam: Areæ <expan abbr="namq;">namque</expan> ſecundum ar­<lb></lb>
tem menſuratio, &amp; terræ menſuratio eſt, &amp; meritò Geometria vocatur. <lb></lb>
</s>
          <s id="s.002381">Vnius verò, &amp; eiuſdem areæ, ſeu loci diuiſio inter diuerſas perſonas, parti­<lb></lb>
tio quædam eſt terræ, &amp; iure optimo Geodæſia appellatur. </s>
          <s id="s.002382">hæc ille. </s>
          <s id="s.002383">dicitur <lb></lb>
autem Geodæſia à <foreign lang="grc">γεα</foreign>, terra, &amp; <foreign lang="grc">δάιω</foreign>, diuido. </s>
          <s id="s.002384">Vocabulum tamen iſtud Geo­<lb></lb>
dæſiæ fuit poſtea ad latiorem tranſlatum ſignificationem: extat enim Geo­<lb></lb>
dæſia Heronis Mechanici antiqui ſcriptoris, quampridem Baroccius lati­<lb></lb>
nitate donauit, quæ quidem ars eſt eadem cum Geometria practica, cum <lb></lb>non ſolum diuiſiones, ſed menſurationes omnes etiam per dioptricam fa­<lb></lb>
cultatem, ſeu per lineas viſuales doceat inueſtigare.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002385"><emph type="italics"></emph>Ex Quarto Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002386"><arrow.to.target n="marg199"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002387"><margin.target id="marg199"></margin.target>208</s>
        </p>
        <p type="main">
          <s id="s.002388">Tex. 4. <emph type="italics"></emph>(Philoſophus <expan abbr="namq;">namque</expan> eſt, vt ille, qui Mathematicus dicitur, &amp; <lb></lb>hæc enim habet partes: ac prima quædam, &amp; ſecunda ſcientia eſt: cæ <lb></lb>
teræ <expan abbr="quoq;">quoque</expan> conſequenter in mathematibus<emph.end type="italics"></emph.end>) inter mathematicas pri­<lb></lb>
mæ ſcientiæ ſunt Geometria, &amp; Arithmetica, quia ipſæ à cæteris <lb></lb>
nulla ratione dependent; imò cæteræ ipſis innituntur, quæ ſecundæ hoc lo­<lb></lb>
co appellantur, hæ ſunt Perſpectiua, Muſica, Mechanica, Aſtronomia. </s>
          <s id="s.002389">illas <lb></lb>
duas recentiores ſubalternantes, has verò ſecundas ſubalternatas vocant. <lb></lb>
</s>
          <s id="s.002390">Exempla ſubalternationum varia attuli in Logicis tex. 20. &amp; 23. primi Po­<lb></lb>
ſter. vbi clarè licet intueri quid ſit ſubalternatio, vnde etiam præſens lo­<lb></lb>
cus illuſtratur.</s>
        </p>
        <p type="main">
          <s id="s.002391"><arrow.to.target n="marg200"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002392"><margin.target id="marg200"></margin.target>209</s>
        </p>
        <p type="main">
          <s id="s.002393">Tex. 28. (<emph type="italics"></emph>Vti diametrum commenſurabilem eſſe<emph.end type="italics"></emph.end>) legenda ſunt ea, quæ libro <lb></lb>
primo Priorum, ſecto 1. cap. 23. de hac commenſurabilitate, &amp; incommen­<lb></lb>
ſurabilitate tractata ſunt.</s>
        </p>
      </chap>
      <pb pagenum="140" xlink:href="009/01/140.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.002394"><emph type="italics"></emph>Ex Quinto Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002395"><arrow.to.target n="marg201"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002396"><margin.target id="marg201"></margin.target>210</s>
        </p>
        <p type="main">
          <s id="s.002397">Tex. 2. (<emph type="italics"></emph>Alia verò cauſa eſt forma, &amp; exemplar: hæc autem eſt ratio ip­<lb></lb>ſius quid erat eſſe, &amp; horum genera, vt ipſius Diapaſon duo ad vnum, <lb></lb>&amp; ſimpliciter numerus, &amp; partes, quæ in ratione ſunt<emph.end type="italics"></emph.end>) affert exem­<lb></lb>
plum cauſæ formalis ex Muſica petitum; <expan abbr="aitq́">aitque</expan>; cauſam formalem <lb></lb>
illius conſonantiæ, quæ Diapaſon dicitur, <expan abbr="eſtq́">eſtque</expan>; omnium perfectiſſima, eſſe <lb></lb>
duplam proportionem, ideſt, quæ eſt inter duo, &amp; vnum, id, quod omnes <lb></lb>
Muſici <expan abbr="fatẽtur">fatentur</expan>. </s>
          <s id="s.002398">quod vt inelius intelligas, repete, quæ in 2. Poſter. ad tex. 1. <lb></lb>ſcripta ſunt: necnon quæ in libro de Senſu in cap. 8. Amplius inquit cauſam <lb></lb>
formalem genericam eiuſdem Diapaſon eſſe numerum, &amp; partes numeri, <lb></lb>
ſub numero enim continentur &amp; duo, &amp; vnum. </s>
          <s id="s.002399">Occurrit hoc loco vnum <lb></lb>
magnopere notandum, videlicet tam conſonantias, quam diſſonantias ha­<lb></lb>
bere proportiones numerorum, hoc tamen diſcrimine, quod conſonantiæ <lb></lb>
habent ſolùm proportiones numerorum eorum, qui quaternario continen­<lb></lb>
tur, ex veterum præſertim Pythagoreorum ſententia, qui propterea vltra <lb></lb>
quaternarium progredi vetabant. </s>
          <s id="s.002400">Recentiores tamen uſque ad ſenarium <lb></lb>
procedunt, quippe, qui omnes vocum conſonantias admittunt, quæ pro­<lb></lb>
portionibus numerorum ſenario contentorum præditæ ſint. </s>
          <s id="s.002401">Diſſonantiæ <lb></lb>verò ſecundum priſcos habent proportiones numerorum extra quaterna­<lb></lb>
rium progredientium, iuxta noſtros autem extra ſenarium. </s>
          <s id="s.002402">qua de re pluri­<lb></lb>
bus Zarlinus colloquio 2. definit. </s>
          <s id="s.002403">3.</s>
        </p>
        <p type="main">
          <s id="s.002404"><arrow.to.target n="marg202"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002405"><margin.target id="marg202"></margin.target>211</s>
        </p>
        <p type="main">
          <s id="s.002406">Tex. 3. <emph type="italics"></emph>(Partes <expan abbr="quoq;">quoque</expan> totius<emph.end type="italics"></emph.end>) ideſt ſunt materia; loquitur enim de cauſa <lb></lb>
materiali. </s>
          <s id="s.002407">libuit locum hunc annotare in gratiam Geometricarum demon­<lb></lb>
ſtrationum, quorum media ſæpè ſunt ex cauſa materiali ſumpta, quod ta­<lb></lb>
men non ita ab omnibus obſeruatur, <expan abbr="quotieſcunq;">quotieſcunque</expan> enim probant affectio­<lb></lb>
nem quampiam de aliquo ſubiecto, ex eo, quod ſubiectum illud ſit, vel di­<lb></lb>
midium alicuius, vel duplum, vel reliquum, vel tertia pars, &amp; his ſimilia, <lb></lb>
erit talis ratio in genere cauſæ materialis. </s>
          <s id="s.002408"><expan abbr="neq;">neque</expan> eſt cur recentiores quidam, <lb></lb>naturalibus ſcientijs aſſueti, negent huiuſmodi materiam veram eſſe mate­<lb></lb>riam, ac proinde neque, Geometricas demonſtrationes veras eſſe demonſtra­<lb></lb>
tiones; dicendum enim talem quidem materiam non eſſe veram materiam <lb></lb>
phyſicam, &amp; proinde illas demonſtrationes <expan abbr="nõ">non</expan> eſſe veras naturales demon­<lb></lb>ſtrationes, eſſe tamen veram materiam intelligibilem, quæ Geometriæ ſu­<lb></lb>
bijcitur, &amp; proinde demonſtrationes illas veras eſſe demonſtrationes Geo­<lb></lb>
metricas; id quod Ariſt. ſæpius in libris Poſter, apertè ſignificat, tum aſſer­<lb></lb>
tionibus, tum exemplis quamplurimis. </s>
          <s id="s.002409">Quapropter cauendum eſt illis, ne <lb></lb>
ingrati animi notam incurrant, dum pulcherrimam artem reſolutoriam, <lb></lb>
quam Ariſt. à Mathematicis acceptam omnibus ſcientijs accommodauit <lb></lb>
(vt initio Priorum oſtenſum eſt) eam ipſi ita alijs facultatibus adaptent, vt <lb></lb>
Mathematicis ipſis, ex quibus orta, &amp; ſub quibus adoleuit, pulla ratione <lb></lb>conuenire poſſit. </s>
          <s id="s.002410">De hac materia fuſius infra in additamento de natura Ma­<lb></lb>
thematicarum.</s>
        </p>
        <p type="main">
          <s id="s.002411"><arrow.to.target n="marg203"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002412"><margin.target id="marg203"></margin.target>212</s>
        </p>
        <p type="main">
          <s id="s.002413">Tex. 3. (<emph type="italics"></emph>Et ipſius Diapaſon duplum, &amp; numerus<emph.end type="italics"></emph.end>) ſcilicet cauſæ formales <lb></lb>
ſunt, quemadmodum ſupra tex. 2. huius cap. explicatum eſt.</s>
        </p>
        <pb pagenum="141" xlink:href="009/01/141.jpg"></pb>
        <p type="main">
          <s id="s.002414"><arrow.to.target n="marg204"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002415"><margin.target id="marg204"></margin.target>213</s>
        </p>
        <p type="main">
          <s id="s.002416">Tex. 4. (<emph type="italics"></emph>Similiter autem figurationum <expan abbr="quoq;">quoque</expan> elementa dicuntur, ac ſimpliciter <lb></lb>demonſtrationum primæ enim demonſtrationes, quæ in pluribus demonstrationibus <lb></lb>
inſunt, hæc elementa demonſtrationum dicuntur<emph.end type="italics"></emph.end>) verbo (<emph type="italics"></emph>Figurationum<emph.end type="italics"></emph.end>) ſiue <expan abbr="de-ſcriptionũ">de­<lb></lb>
ſcriptionum</expan>, Ariſt, intelligere demonſtrationes Geometricas, ſæpius dictum <lb></lb>
eſt, præſertim in Logicis, &amp; ex hoc loco pariter confirmatur. </s>
          <s id="s.002417">Ex hoc por­<lb></lb>
rò loco illud innoteſcit dignum, quod præcipuè à Mathematico non igno­<lb></lb>
retur, quæ nam ſint demonſtrationes illæ, quæ nomine <expan abbr="elementorũ">elementorum</expan> debeant <lb></lb>appellari, necnon cauſa cur Euclides ſuum opus elementa nuncupauerit, <lb></lb>
ſunt enim illæ, quæ in pluribus demonſtrationibus inſunt, ideſt, quæ ſæpius <lb></lb>
in alijs demonſtrationibus citantur, vti ſunt præcipuè ſex priores libri Eu­<lb></lb>
clidis: <expan abbr="atq;">atque</expan> hac ratione elementa appellantur.</s>
        </p>
        <p type="main">
          <s id="s.002418"><arrow.to.target n="marg205"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002419"><margin.target id="marg205"></margin.target>214</s>
        </p>
        <p type="main">
          <s id="s.002420">Tex. 12. <emph type="italics"></emph>(Principium <expan abbr="itaq;">itaque</expan> ſcibilis, circa <expan abbr="vnumquodq;">vnumquodque</expan> ipſum vnum, non eſt au­<lb></lb>tem idem in cunctis generibus vnum, ſed hic quidem dieſis, hic verò vocalis, aut <lb></lb>
muta)<emph.end type="italics"></emph.end> ideſt, in Muſica quidem principium omnium, &amp; elementum eſt die­<lb></lb>
ſis, quæ eſt minima vox, aut ſonus, qui ſub Muſici conſiderationem cadat. <lb></lb>
</s>
          <s id="s.002421">Porrò ad tex. 38. primi Poſter. de dieſi plura ſunt dicta.</s>
        </p>
        <p type="main">
          <s id="s.002422"><arrow.to.target n="marg206"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002423"><margin.target id="marg206"></margin.target>215</s>
        </p>
        <p type="main">
          <s id="s.002424">Tex. 17. <emph type="italics"></emph>(Veluti diametrum commenſurabilem eſſe impoſſibile est)<emph.end type="italics"></emph.end> huius expo­<lb></lb>
ſitionem inuenies 1. Priorum, ſecto 1. cap. 23.</s>
        </p>
        <figure id="id.009.01.141.1.jpg" place="text" xlink:href="009/01/141/1.jpg"></figure>
        <p type="main">
          <s id="s.002425"><arrow.to.target n="marg207"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002426"><margin.target id="marg207"></margin.target>216</s>
        </p>
        <p type="main">
          <s id="s.002427">Tex. eodem <emph type="italics"></emph>(Metaphoricè autem, quæ in Geometria po­<lb></lb>
tentia dicitur)<emph.end type="italics"></emph.end> potentiam vnius lineæ appellant Geometræ <lb></lb>
quadratum illius, ideſt quadratum ſuper ipſam conſtru­<lb></lb>
ctum. </s>
          <s id="s.002428">v. g. quadratum in quo C, dicitur potentia lineæ <lb></lb>
D B, quia ſuper illam conſtructum eſt.</s>
        </p>
        <p type="main">
          <s id="s.002429"><arrow.to.target n="marg208"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002430"><margin.target id="marg208"></margin.target>217</s>
        </p>
        <p type="main">
          <s id="s.002431">Tex. 34. (<emph type="italics"></emph>Quemadmodum dicitur diametrum eſſe commenſurabilem<emph.end type="italics"></emph.end>) vide an­<lb></lb>notata 1. Priorum, ſecto 1. cap. 23.</s>
        </p>
        <p type="main">
          <s id="s.002432"><arrow.to.target n="marg209"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002433"><margin.target id="marg209"></margin.target>218</s>
        </p>
        <p type="main">
          <s id="s.002434">Tex. 35. (<emph type="italics"></emph>Vt triangulo duos rectos habere<emph.end type="italics"></emph.end>) ideſt affectio trianguli eſt habe­<lb></lb>
re tres angulos æquales duobus rectis angulis. </s>
          <s id="s.002435">Vide declarationem huius <lb></lb>lib. primo Priorum, ſecto 3. cap. 1.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002436"><emph type="italics"></emph>Ex Sexto Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002437"><arrow.to.target n="marg210"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002438"><margin.target id="marg210"></margin.target>219</s>
        </p>
        <p type="main">
          <s id="s.002439">Tex. 1. (<emph type="italics"></emph>Mathematicorum quoque principia, elementa, &amp; cauſæ ſunt<emph.end type="italics"></emph.end>) <lb></lb>
notanda ſunt hæc aduerſus quoſdam, qui negant in Mathemati­<lb></lb>
cis cauſas reperiri, vt hinc <expan abbr="quoq;">quoque</expan> illis ſcientiam auferant. </s>
          <s id="s.002440">enim­<lb></lb>
uerò apertè patet eos falli ex toto hoc Ariſt. diſcurſu.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002441"><emph type="italics"></emph>Ex Nono Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002442"><arrow.to.target n="marg211"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002443"><margin.target id="marg211"></margin.target>220</s>
        </p>
        <p type="main">
          <s id="s.002444"><emph type="italics"></emph>Vt ſi quis dicat diametrum poſſe commenſurari, non tamen commenſu­<lb></lb>
rabitur<emph.end type="italics"></emph.end>) &amp; paulò infra (<emph type="italics"></emph>Commenſurari enim impoſſibile eſt<emph.end type="italics"></emph.end>) expoſi­<lb></lb>
tionem horum reperies 1. Priorum, ſecto 1. cap. 23.</s>
        </p>
        <p type="main">
          <s id="s.002445"><arrow.to.target n="marg212"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002446"><margin.target id="marg212"></margin.target>221</s>
        </p>
        <p type="main">
          <s id="s.002447">Tex. 20. (<emph type="italics"></emph>Deſcriptiones <expan abbr="quoq;">quoque</expan> actu inueniuntur, diuidentes nanque <lb></lb>inuenirent, quod ſi diuiſæ eſſent, manifeſtè eſſent, nunc autem inſunt potentia, cur <lb></lb>
triangulus duo recti? </s>
          <s id="s.002448">quia qui circa vnum punctum anguli duobus rectis æquales<emph.end type="italics"></emph.end> <pb pagenum="142" xlink:href="009/01/142.jpg"></pb><emph type="italics"></emph>ſunt, ſi igitur quæ ad latus educeretur, videnti mox eſſet manifeſtum<emph.end type="italics"></emph.end>) per deſcri­<lb></lb>
ptiones, vel figurationes, vel deſignationes intelligendas eſſe demonſtra­<lb></lb>
tiones Geometricas ſæpius ſupra dictum eſt, &amp; pariter ex hoc loco com­<lb></lb>
probatur. </s>
          <s id="s.002449">Dicit igitur, quod demonſtrationes ſuas Geometræ inueniunt, <lb></lb>
reducendo ad actum ea, quæ erant in potentia, diuidentes enim educunt in <lb></lb>actum, figuras, angulos, lineas, &amp; cætera huiuſmodi, quæ prius ſolùm erat <lb></lb>
in potentia, ex quibus poſtea ſuas demonſtrationes perficiunt (<emph type="italics"></emph>Cur triangu­<lb></lb>
lus duo recti<emph.end type="italics"></emph.end>) affert exemplum eius, quod proximè dixerat, ſcilicet Geome­<lb></lb>
tras demonſtrare producendo ad actum entia quædam Mathematica, quod <lb></lb>
exemplum, vt intelligas ijs opus habes, quæ primo Priorum, ſecto 3. cap. 1. <lb></lb>
conſcripta ſunt (<emph type="italics"></emph>Cur triangulus duo recti?<emph.end type="italics"></emph.end>) ideſt, cur triangulus habet tres <lb></lb>
angulos æquales duobus rectis angulis (<emph type="italics"></emph>Quia qui circa vnum punctum anguli <lb></lb>
duobus rectis angulis æquales ſunt<emph.end type="italics"></emph.end>) niſi hoc dictum ad bonum trahatur ſenſum, <lb></lb>
<figure id="id.009.01.142.1.jpg" place="text" xlink:href="009/01/142/1.jpg"></figure><lb></lb>
falſum eſt, nam omnes anguli, qui circa vnum <lb></lb>
punctum, v. g. A, ſunt conſtituti, æquales ſunt <lb></lb>
non duobus, vt eſt in textu, ſed quatuor rectis, <lb></lb>
vt patet ex corollario 2. 15. primi Elem. quot­<lb></lb>
quot enim anguli conſtituantur ad punctum A, <lb></lb>
omnes ſimul erunt æquales quatuor rectis, quos <lb></lb>
faciunt præſentes lineæ B C, D E. vniuerſi enim <lb></lb>
illi congruent his quatuor rectis: ſed Ariſt. ſen­<lb></lb>
ſus eſt omnes angulos ad eaſdem partes conſti­<lb></lb>
tutos, v. g. ad partes ſuperiores lineæ B C, eſſe <lb></lb>
æquales duobus rectis B A D, D A C, vt oſtenditur in 13. primi, necnon <lb></lb>
etiam patere poteſt ex corollario 2. 15. eiuſdem. </s>
          <s id="s.002450">tales ſunt quatuor anguli <lb></lb>
ad ſuperiores partes lineæ B C, &amp; ad punctum A, conſtituti, qui, vt patet, <lb></lb>
<figure id="id.009.01.142.2.jpg" place="text" xlink:href="009/01/142/2.jpg"></figure><lb></lb>
ſunt æquales duobus rectis B A D, D A C, <lb></lb>
tales etiam ſunt in hac ſecunda figura tres <lb></lb>
anguli B C A, A C D, D C E, qui quidem <lb></lb>
æquales ſunt duobus rectis angulis. </s>
          <s id="s.002451">hoc <lb></lb>
ſenſiſſe Ariſt. patet ex demonſtratione 32. <lb></lb>
primi, quæ demonſtrat <expan abbr="memoratã">memoratam</expan> ab Ari­<lb></lb>
ſtot. trianguli affectionem, &amp; ad quam <lb></lb>
propterea ipſe ſpectabat, cuius figura eſt <lb></lb>
eadem cum hac ſecunda, in qua Euclides oſtendit prædictos tres angulos <lb></lb>
æquari duobus rectis. </s>
          <s id="s.002452">ſubdit poſtea, ſi igitur linea C D, quæ ad latus A B, <lb></lb>
parallela eſt in potentia, educeretur in actum, videnti mox eſſet manifeſtum <lb></lb>
tres angulos trianguli A B C, eſſe pares duobus rectis. </s>
          <s id="s.002453">ducta enim C D, pa­<lb></lb>
rallela lateri B A, apparet ſtatim angulus A, æqualis angulo A C D, &amp; an­<lb></lb>
gulus B, angulo D C E; cum reliquus verò <expan abbr="triãguli">trianguli</expan> angulus B C A, ſit apud <lb></lb>prædictos duos ad idem punctum C, conſtitutus; <expan abbr="atq;">atque</expan> omnes hi tres duobus <lb></lb>rectis æquentur, mox inſpicienti talem figurationem manifeſtum fit tres an­<lb></lb>gulos illius trianguli eſſe duobus rectis æquales.</s>
        </p>
        <p type="main">
          <s id="s.002454"><arrow.to.target n="marg213"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002455"><margin.target id="marg213"></margin.target>222</s>
        </p>
        <p type="main">
          <s id="s.002456">Ibidem (<emph type="italics"></emph>Cur in ſemicirculo vniuerſaliter rectus? </s>
          <s id="s.002457">quia ſi tres æquales, &amp; quæ <lb></lb>
baſis eſt duo, &amp; quæ ex medio ſupra stat recta, videnti manifestum erit ei, qui illud <lb></lb>
ſciat<emph.end type="italics"></emph.end>) In 2. Poſter. tex. 11. inuenies huius loci expoſitionem. </s>
          <s id="s.002458">nunc ſolùm <pb pagenum="143" xlink:href="009/01/143.jpg"></pb><figure id="id.009.01.143.1.jpg" place="text" xlink:href="009/01/143/1.jpg"></figure><lb></lb>
hæc addenda ſunt. </s>
          <s id="s.002459">Reſpondet Ariſt. quæ­<lb></lb>
ſito pręcedenti, cur ſcilicet angulus in ſe­<lb></lb>
micirculo ſit rectus, qualis eſt in figura <lb></lb>
angulus A C B, <expan abbr="dicitq́">dicitque</expan>; cauſam eſſe, quia <lb></lb>
in figura tres lineæ ſunt æquales, duæ ni­<lb></lb>
mirum, in quas baſis B A, diuiditur, quæ <lb></lb>
ſunt B D, D A, &amp; tertia, quæ ex medio <lb></lb>
baſis erigitur, <expan abbr="eſtq́">eſtque</expan>; D C, cum omnes ſint <lb></lb>ſemidiametri eiuſdem circuli. </s>
          <s id="s.002460">educta <expan abbr="itaq;">itaque</expan> linea D C, de potentia in actum, <lb></lb>
ſi cuipiam trium harum linearum æqualitas innoteſcat, continuò ei etiam <lb></lb>
manifeſtum erit angulum A C B, in ſemicirculo, eſſe rectum. </s>
          <s id="s.002461">quia ſtatim ap­<lb></lb>
parent duo iſoſcelia B D C, A D C, quorum anguli ad baſes B C, A C, ſunt <lb></lb>
æquales inuicem; &amp; anguli duo ad D, ſunt dupli duorum <expan abbr="angulorũ">angulorum</expan> A C D, <lb></lb>
D C B, ex quibus conflatur totus angulus A C B, ergo duo anguli ad D, ſunt <lb></lb>
dupli anguli B C A, ſed duo anguli ad D, ſunt æquales duobus rectis, ergo <lb></lb>
duo recti ſunt dupli anguli A C B, ergo angulus B C A, eſt dimidium duo­<lb></lb>
rum rectorum. </s>
          <s id="s.002462">cum autem omnes recti ſint æquales, conſectarium eſt dimi­<lb></lb>
dium duorum rectorum eſſe angulum rectum. </s>
          <s id="s.002463">patet igitur, qua ratione ex <lb></lb>
ductu linearum prædictarum actu, manifeſtum fiat angulum in ſemicirculo <lb></lb>
A C B, eſſe rectum. </s>
          <s id="s.002464">ne mireris ſi vulgatam tranſlationem antiquam non <lb></lb>
ſum ſequutus, indigebat enim correctione, quam iuxta græcum exem­<lb></lb>
plar adhibui.</s>
        </p>
        <p type="main">
          <s id="s.002465"><arrow.to.target n="marg214"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002466"><margin.target id="marg214"></margin.target>223</s>
        </p>
        <p type="main">
          <s id="s.002467">Tex. 22. (<emph type="italics"></emph>Vt puta ſi triangulum non putet mutari, non opinabitur modo duos <lb></lb>
rectos habere, modo non, mutaretur enim<emph.end type="italics"></emph.end>) quia nimirum huius habemus ſcien­<lb></lb>
tiam per demonſtrationem 32. primi Elementorum. </s>
          <s id="s.002468">quomodo autem tri­<lb></lb>
angulus habeat duos rectos, ideſt tres angulos æquales duobus rectis angu­<lb></lb>
lis, explicatum eſt primo Priorum, ſecto 3. cap. 1.</s>
        </p>
        <p type="main">
          <s id="s.002469"><arrow.to.target n="marg215"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002470"><margin.target id="marg215"></margin.target>224</s>
        </p>
        <p type="main">
          <s id="s.002471">Ibidem (<emph type="italics"></emph>Verum aliquid quidem, aliquid verò non, vt puta parem numerum <lb></lb>
primum nullum eſſe; aut quoſdam quidem, quoſdam verò non<emph.end type="italics"></emph.end>) definitione 11. <lb></lb>
7. Elem. ſic numerus ille, qui à Mathematicis dicitur primus, definitur, pri­<lb></lb>
mus numerus eſt, quem vnitas ſola metitur, vnde patet inter numeros pa­<lb></lb>
res ſolum binarium eſſe primum, cum ipſum ſola vnitas bis replicata men­<lb></lb>
ſuraret. </s>
          <s id="s.002472">quaternarium autem, ſenarium, &amp;c. </s>
          <s id="s.002473">pares, non eſſe primos, cum <lb></lb>
eos non ſola vnitas, ſed alius numerus metiatur: quaternarium enim bina­<lb></lb>
rius bis replicatus menſurat: ſenarium menſurat &amp; binarius, &amp; ternarius: <lb></lb>
quare verum erit exiſtimare inter pares numeros aliquos eſſe primos, ideſt <lb></lb>
binarium, aliquos verò non, ideſt cæteros pares vltra binarium.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002474"><emph type="italics"></emph>Ex Decimo Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002475"><arrow.to.target n="marg216"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002476"><margin.target id="marg216"></margin.target>225</s>
        </p>
        <p type="main">
          <s id="s.002477">Tex. 4. (<emph type="italics"></emph>Ac etiam motum ſimplici, &amp; velociſſimo motu menſurant, mi­<lb></lb>
nimum enim tempus hic habet. </s>
          <s id="s.002478">quapropter in Aſtrologia tale vnŭm prin­<lb></lb>
cipium, &amp; menſura eſt. </s>
          <s id="s.002479">motum enim æqualem, &amp; velociſſimŭm cœli ſup­<lb></lb>ponunt, ad quem cæteros iudicant<emph.end type="italics"></emph.end>) intelligit motum diurnum, quam <lb></lb>primo cœlo, ſeu mobili aſcribunt, hic enim velociſſimus eſt omnium reli­<pb pagenum="144" xlink:href="009/01/144.jpg"></pb>quorum cœleſtium motuum, ac ſimpliciſſimus, &amp; valdè vniformis, ac regu­<lb></lb>
laris, &amp; propterea minimum habet tempus, ideſt tempus vnius diei natura­<lb></lb>
lis, quo tempore totum primum mobile circulationem integram perficit. <lb></lb>
</s>
          <s id="s.002480">per minimum tempus, poſſunt etiam intelligi partes diei, quæ ſunt horæ, &amp; <lb></lb>
horarum partes. </s>
          <s id="s.002481">conſiderant hunc motum in circulo æquàtoris, quia æqua­<lb></lb>tor motu primi mobilis, ſeu diurno vniformiter, ae maximè regulariter <lb></lb>
mouetur: hac de cauſa hunc motum tanquam reliquorum menſuram, ac <lb></lb>
normam meritò aſſumpſerunt.</s>
        </p>
        <p type="main">
          <s id="s.002482"><arrow.to.target n="marg217"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002483"><margin.target id="marg217"></margin.target>226</s>
        </p>
        <p type="main">
          <s id="s.002484">Ibidem <emph type="italics"></emph>(Et in Muſica Dieſis primus ſenſibilis ſonus, quia minimum)<emph.end type="italics"></emph.end> ideſt mi­<lb></lb>
nimum interuallum, quod à Muſicis conſideretur, eſt menſura maiorum in­<lb></lb>
teruallorum. </s>
          <s id="s.002485">ad tex. 38. primi Poſter. ſatis dictum eſt de Dieſi, quæ videas.</s>
        </p>
        <p type="main">
          <s id="s.002486"><arrow.to.target n="marg218"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002487"><margin.target id="marg218"></margin.target>227</s>
        </p>
        <p type="main">
          <s id="s.002488">Eodem tex. ſed cap. 3. <emph type="italics"></emph>(Nox ſemper autem menſura numero vnum eſt, verum <lb></lb>aliquando plura, vt puta dieſes duæ, non quidem ſecundum auditum, ſed in ratio­<lb></lb>
nibus, &amp; voces plures, quibus menſuramus, &amp; diameter duobus menſuratur, &amp; la­<lb></lb>
tus, &amp; omnes magnitudines)<emph.end type="italics"></emph.end> ita corrigenda eſt antiqua tranſlatio. </s>
          <s id="s.002489">quid dieſis <lb></lb>
dictum ſit ad tex. 38. primi Poſter. quando autem ait <emph type="italics"></emph>(Vt puta duæ dieſes)<emph.end type="italics"></emph.end><lb></lb>
ideſt duæ dieſes ſunt menſura vnius interualli muſici, qui tonus appellatur: <lb></lb>
quæ quidem duæ dieſes non ſunt menſura ſenſibilis, quæ ſcilicet auribus per­<lb></lb>
cipiatur, ſed tantummodò exiſtunt in numerorum proportionibus, ibi per <lb></lb>
intellectum excogitatis, quando ait <emph type="italics"></emph>(Et voces plures quibus menſuramus)<emph.end type="italics"></emph.end><lb></lb>
quando vtimur eodem interuallo, ſiue eadem voce ad cantus menſuram, <lb></lb>
tunc ſunt plures menſuræ numero, quamuis vna tantum ſpecie. Ait <emph type="italics"></emph>(Et dia­<lb></lb>
meter duobus menſuratur)<emph.end type="italics"></emph.end> v. g. duobus ſemidiametris: vel duobus pedibus. <lb></lb>
</s>
          <s id="s.002490">&amp; latus pariter quadrati, duobus. </s>
          <s id="s.002491">v. g. pedibus mensuratur; eode<expan abbr="mq́">mque</expan>; mo­<lb></lb>do reliquæ omnes magnitudines poſſunt ab eadem menſura ſæpius replica­<lb></lb>
ta menſurari.</s>
        </p>
        <p type="main">
          <s id="s.002492"><arrow.to.target n="marg219"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002493"><margin.target id="marg219"></margin.target>228</s>
        </p>
        <p type="main">
          <s id="s.002494">Eodem tex. <emph type="italics"></emph>(Semper autem menſura eiuſdem generis eſt, magnitudinum nam­<lb></lb>que magnitudo, &amp; ſecundum vnumquodque, longitudinis longitudo<emph.end type="italics"></emph.end>) ex his ratio <lb></lb>
manifeſta apparet, cur Geometræ practici menſurent longitudines per ali­<lb></lb>
quam longitudinem, vt puta per vlnam, digitum, vnciam, &amp;c. </s>
          <s id="s.002495">ſuperficies <lb></lb>
etiam per aliquam ſuperficiem, ſed quæ ſit quadrata, vt puta per vlnam qua­<lb></lb>
dratam, palmum quadratum, &amp;c. </s>
          <s id="s.002496">corpora <expan abbr="quoq;">quoque</expan> per corpus, quod tamen <lb></lb>
ſit cubus, vt per vlnam cubicam, palmum cubicum, vnciam cubicam, &amp;c.</s>
        </p>
        <p type="main">
          <s id="s.002497"><arrow.to.target n="marg220"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002498"><margin.target id="marg220"></margin.target>229</s>
        </p>
        <p type="main">
          <s id="s.002499">Tex. 11. <emph type="italics"></emph>(Similia verò ſi cum non ſint eadem ſimpliciter, nec ſecundum ſubſtan­<lb></lb>tiam ſubiectam indifferentia ſecundum formam eadem ſit: quemadmodum quadra­<lb></lb>tum maius minori ſimile eſt, &amp; lineæ inæquales, hæ enim ſimiles quidem, verŭm non <lb></lb>
cædem ſimpliciter ſunt)<emph.end type="italics"></emph.end> Prima definitio ſexti definit ſimiles figuras eas eſſe, <lb></lb>
quæ ſunt æquiangulæ inuicem, &amp; quæ habent latera proportionalia circa <lb></lb>
æquales angulos. </s>
          <s id="s.002500">cum ergò quadratum maius, &amp; minus ſint æquiangula, <lb></lb>
quia habent omnes angulos rectos; &amp; præterea habeant latera circa æqua­<lb></lb>
les angulos proportionalia, ſicut enim latera maioris quadrati circa vnum <lb></lb>
angulum rectum ſunt in proportione æqualitatis; ita <expan abbr="quoq;">quoque</expan> latera minoris <lb></lb>
circa vnum angulum rectum ſunt illis proportionalia, cum ſint inuicem pa­<lb></lb>
riter in proportione æqualitatis, erunt neceſſariò ſimilia hæc duo quadrata. <lb></lb>
</s>
          <s id="s.002501">duæ etiam, exempli gratia, lineæ rectæ ſunt inuicem ſimiles, quamuis vna <lb></lb>
ſit maior altera.</s>
        </p>
        <pb pagenum="145" xlink:href="009/01/145.jpg"></pb>
        <p type="main">
          <s id="s.002502"><arrow.to.target n="marg221"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002503"><margin.target id="marg221"></margin.target>230</s>
        </p>
        <p type="main">
          <s id="s.002504">Eodem tex. <emph type="italics"></emph>(Tertium ſicut illa, quæ in Mathematicis)<emph.end type="italics"></emph.end> tertium ſcilicet mo­<lb></lb>
dum diuerſi, ponit in entibus Mathematicis, ſicut enim poſuit idem eſſe in <lb></lb>
Mathematicis, quando duæ figuræ ſunt ſimiles, &amp; æquales: ita ex oppoſito <lb></lb>
diuerſum erit in Mathematicis, quando duæ figuræ fuerint diſſimiles, &amp; in­<lb></lb>
æquales, <expan abbr="dicenturq́">dicenturque</expan>; diuerſæ, in quo conſiſtat ſimilitudo figurarum dictum <lb></lb>
eſt in præcedenti expoſitione.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002505"><emph type="italics"></emph>Ex Vndecimo Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002506"><arrow.to.target n="marg222"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002507"><margin.target id="marg222"></margin.target>231</s>
        </p>
        <p type="main">
          <s id="s.002508">Svmma r. </s>
          <s id="s.002509">cap. 2. <emph type="italics"></emph>(Si quis verò lineas, aut quæ has ſequuntur, dico autem <lb></lb>
primas ſuperficies principia eſſe ponat. </s>
          <s id="s.002510">hæc non ſunt ſubſtantiæ ſeparabiles, <lb></lb>
verùm ſectiones, &amp; diuiſiones, illæ quidem in ſuperficierum, hæc verò cor­<lb></lb>
porum, puncta verò linearum ſunt, &amp; etiam ipſarum earumdem termini; <lb></lb>
hæc autem omnia in alijs ſunt, &amp; nihil ſeparabile eſt)<emph.end type="italics"></emph.end> ait puncta oriri ex ſectio­<lb></lb>
ne lineæ, quamuis ſint etiam termini illius; lineas verò oriri ex diuiſione <lb></lb>
ſuperficierum, quamuis ſint etiam termini illarum. </s>
          <s id="s.002511">ſuperficies <expan abbr="quoq;">quoque</expan> oriri <lb></lb>
ex diuiſione corporum, quamuis ſint etiam termini, illorum. </s>
          <s id="s.002512">Hæc placuit <lb></lb>
annotare propter <expan abbr="ipſorũ">ipſorum</expan> conuenientiam <expan abbr="cũ">cum</expan> ijs, quæ à Geometris traduntur.</s>
        </p>
        <p type="main">
          <s id="s.002513"><arrow.to.target n="marg223"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002514"><margin.target id="marg223"></margin.target>232</s>
        </p>
        <p type="main">
          <s id="s.002515">Summa 3. cap. 2. <emph type="italics"></emph>(Vt puta ſub Cane fiat frigus)<emph.end type="italics"></emph.end> ideſt ſub ortum Canis cœ­<lb></lb>
læſtis, ſeu Caniculæ. </s>
          <s id="s.002516">Vide quæ libro ſecundo Meteororum, ſumma 2. cap. 2. <lb></lb>
de hac ſtella ſcripſimus.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002517"><emph type="italics"></emph>Ex Duodecimo Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002518"><arrow.to.target n="marg224"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002519"><margin.target id="marg224"></margin.target>233</s>
        </p>
        <p type="main">
          <s id="s.002520">Tex. 44. <emph type="italics"></emph>(Pluralitatem verò lationum ex peculiariſſima Philoſophia <lb></lb>
Mathematicarum ſcientiarum, videlicet ex Aſtronomia conſiderandum <lb></lb>
est: hæc enim de ſubſtantia ſenſibili quidem, ac ſempiterna ſpeculatur)<emph.end type="italics"></emph.end><lb></lb>
pluralitatem nimirum cœleſtium motuum petendam eſſe aſſerit <lb></lb>
ex præcipua totius Philoſophiæ parte, quam ait eſſe Aſtronomiam. </s>
          <s id="s.002521">dignum <lb></lb>
porrò conſideratione eſt, quanti faciat Ariſt. Mathematicas diſciplinas, ac <lb></lb>
præcipuè ſyderalem ſcientiam.</s>
        </p>
        <p type="main">
          <s id="s.002522"><arrow.to.target n="marg225"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002523"><margin.target id="marg225"></margin.target>234</s>
        </p>
        <p type="main">
          <s id="s.002524">Tex. 45. <emph type="italics"></emph>(Eudoxus igitur Solis, &amp; Lunæ lationem poſuit fieri à tribus orbibus, <lb></lb>quorum primus quidem eſſet, qui inerrantium ſtellarum; ſecundus verò ſecundum <lb></lb>
id, quod per medium Zodiacum; tertius tandem, ſecundum quem qui in latitudine <lb></lb>
Zodiaci obliquatur. </s>
          <s id="s.002525">in maiori autem latitudine obliquari eum ſecundum quem Lu­<lb></lb>na, quàm eum ſecundum quem Sol fertur)<emph.end type="italics"></emph.end> Eudoxi tempore nondum ſatis ex­<lb></lb>
culta fuerat Aſtronomia, vt propterea minimè mirandum ſit, eum hoc lo­<lb></lb>
co imperfecta admodum circa cęleſtia tradere. </s>
          <s id="s.002526">omittit enim in Sole orbem <lb></lb>
motum augis conficientem; necnon duos eccentricos, qui ſolis anomaliam, <lb></lb>
<expan abbr="atq;">atque</expan> eccentricitatis variationem excuſant. </s>
          <s id="s.002527">attribuit præterea Soli motum <lb></lb>
quendam in latitudinem, quod falſum eſt omninò, cum Sol perpetuò directè <lb></lb>
ſub eclyptica incedat. </s>
          <s id="s.002528">In Luna pariter plures neceſſarios illi orbes ad motus <lb></lb>
ipſius ſaluandos prætermittit. </s>
          <s id="s.002529">Ex ſententia tamen Tychonis Brahe hos or­<lb></lb>
bes, ac circulos tanquam ab inuicem diſtinctos abrogare debemus.</s>
        </p>
        <pb pagenum="146" xlink:href="009/01/146.jpg"></pb>
        <p type="main">
          <s id="s.002530"><arrow.to.target n="marg226"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002531"><margin.target id="marg226"></margin.target>235</s>
        </p>
        <p type="main">
          <s id="s.002532">Tex. 46. <emph type="italics"></emph>(Errantium verò ſtellarum <expan abbr="vniuſcuiuſq;">vniuſcuiuſque</expan> in quatuor ſphæris, quarum <lb></lb>primam quidem, &amp; ſecundam eandem illis eſſe: etenim, quæ fixarum eſt eam illam <lb></lb>eſſe, quæ omnes fert: at eam, quæ ſub ipſa ordinata eſt, ac quæ ſecundum Zodiacum <lb></lb>
lationem habet, communem omnibus eſſe. </s>
          <s id="s.002533">Tertiæ verò omnium polos in eo, quod <lb></lb>
per medium Zodiacum eſt. </s>
          <s id="s.002534">Quartæ autem lationem ſecundum eum, qui obliquatus <lb></lb>ad medium eius eſt; eſſe verò tertiæ ſphæræ polos aliarum quidem proprios, Veneris <lb></lb>
autem, &amp; Mercurij eoſdem)<emph.end type="italics"></emph.end> pergit tradere theoriam reliquorum errantium <lb></lb>
<expan abbr="quinq;">quinque</expan> ſyderum, ſecundum mentem Eudoxi, qui propriè Planetæ dicuntur: <lb></lb>
Sol autem, &amp; Luna hoc nomine non eſt complexus, eo quod ipſa mereantur <lb></lb>
potius duo mundi luminaria appellari, quàm cum cęteris ſtellis in ordinem <lb></lb>
redigi. </s>
          <s id="s.002535">Reliquis igitur <expan abbr="quinq;">quinque</expan> erronibus ſingulis quatuor ſphæris attribue­<lb></lb>
bat, quarum prima, &amp; ſecunda eodem modo ſe habebant, ac in Sole, &amp; Lu­<lb></lb>
na, etenim octaua ſphæra, ſeu firmamentum, quod affixa ſibi ſydera differt <lb></lb>
communicabat, ſecundum ipſum reliquis inferioribus ſphæris motum ſuum <lb></lb>
peculiarem, videlicet diurnum, quo ab oriente in occidentem tota cęli ma­<lb></lb>
china conuertebatur. </s>
          <s id="s.002536">fecundam eam facit, quæ Planetas omnes ſecundum <lb></lb>
Zodiaci longitudinem ab occidente in <expan abbr="oriẽtem">orientem</expan> vehebat, quæ pariter eodem <lb></lb>
modo ſe habet in ſingulis. </s>
          <s id="s.002537">Tertiam verò eam confinxit, cuius poli eſſent in <lb></lb>
eclyptica, in quibus cita, ab eclyptica vltrò, <expan abbr="citroq́">citroque</expan>; dilataretur. </s>
          <s id="s.002538">Quartam <lb></lb>
demum poſuit, quæ tertiam bifariam ſecaret, <expan abbr="eamq́">eamque</expan>; tali motu cieret, ne ab <lb></lb>
eclyptica plus iuſto verſus mundi polos exorbitaret. </s>
          <s id="s.002539">porrò in reliquis vo­<lb></lb>
luit polos tertij orbis eſſe peculiares, Veneri autem, &amp; Mercurio eoſdem <lb></lb>
eſſe, ideſt eſſe in eadem linea. </s>
          <s id="s.002540">Ex mente igitur Eudoxi cœleſtes orbes in <lb></lb>
vniuerſum 27. numerantur, in Sole ſimul, ac Luna 6. in reliquis quinque er­<lb></lb>
rantibus 20. <expan abbr="atq;">atque</expan> octauæ ſphæræ 1. Non me later, has Eudoxi poſitiones, <lb></lb>
ob ratas poſteriorum aſtronomorum obſeruationes non ſubſiſtere. </s>
          <s id="s.002541">at verò <lb></lb>
hic non ipſius placita, ſed præcipuè textus intelligentiam perſequor.</s>
        </p>
        <p type="main">
          <s id="s.002542"><arrow.to.target n="marg227"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002543"><margin.target id="marg227"></margin.target>236</s>
        </p>
        <p type="main">
          <s id="s.002544">Tex. 47. <emph type="italics"></emph>(At Calippus ſitum quidem ſphærarum eundem Eudoxo ponebat, hoc <lb></lb>
eſt diſtantiarum ordinem. </s>
          <s id="s.002545">pluralitatem autem ſtellæ quidem Iouis, ac Saturni ean­<lb></lb>
dem illi attribuebat. </s>
          <s id="s.002546">Solis verò, &amp; Lunæ duas adhuc putabat ſphæras addendas <lb></lb>eſſe, ſi quis eorum, quæ ſenſibiliter apparent, cauſas aſſignare debeat. </s>
          <s id="s.002547">Cæteris ve­<lb></lb>
rò errantium vnicuique vnam. </s>
          <s id="s.002548">neceſſe verò eſſe, ſi debent omnes ſimul poſitæ, quæ <lb></lb>
apparent reddere, ſecundam <expan abbr="vnamquamq;">vnamquamque</expan> errantium alteras ſphæras vna paucio­<lb></lb>
res eſſe, quæ reuoluant, &amp; ad idem poſitione ſemper primam eius astri ſphæram, <lb></lb>
quod inferius ordinatum eſt, conſtituant. </s>
          <s id="s.002549">hoc enim modo ſolùm contingit errantium <lb></lb>
lationem omnia facere. </s>
          <s id="s.002550">Cùm igitur, in quibus ipſa quidem feruntur ſphæris, hæ <lb></lb>quidem octo, bæ verò <expan abbr="vigintiquinq;">vigintiquinque</expan> ſint. </s>
          <s id="s.002551">horum ſane non oportet illas ſolas reuo­<lb></lb>lui, in quibus fertur, quod infimè ordinatum eſt. </s>
          <s id="s.002552">quæ quidem duarum ſphærarum <lb></lb>
primas reuoluant, ſex erunt. </s>
          <s id="s.002553">quæ verò poſteriorum quatuor, ſexdecim. </s>
          <s id="s.002554">cunctarum <lb></lb>
verò numerus, tùm earum quæ ferunt, tùm quæ reuoluunt eas, quinquaginta quin­<lb></lb>
que. </s>
          <s id="s.002555">quòd ſi Lunæ, &amp; Soli, non addat aliquis quos diximus motus, omnes ſphæræ <lb></lb>
erunt ſeptem, &amp; quadraginta. </s>
          <s id="s.002556">pluralitas <expan abbr="itaq;">itaque</expan> ſphærarum tanta ſit)<emph.end type="italics"></emph.end> textum hunc <lb></lb>
per paraphraſim ſic explico; Calippus igitur eundem quidem ordinem, at­<lb></lb>
que diſtantiam ſphærarum cum Eudoxo ponebat: <expan abbr="eandemq́">eandemque</expan>; pluralitatem <lb></lb>orbium mouentium Saturnum, ac Iouem; quatuor <expan abbr="nimirũ">nimirum</expan> <expan abbr="vnicuiq;">vnicuique</expan> eorum. <lb></lb>
</s>
          <s id="s.002557">ſed putabat ſoli duas addendas, ac Lunæ ſimiliter, ſi quis eorum <expan abbr="apparẽtias">apparentias</expan> <pb pagenum="147" xlink:href="009/01/147.jpg"></pb>ſaluare vellet. </s>
          <s id="s.002558">cæteris verò errantium, Marti, Veneri, &amp; Mercurio <expan abbr="vnicuiq;">vnicuique</expan> <lb></lb>
vnam. </s>
          <s id="s.002559">neceſſe præterea exiſtimabat eſſe, vt prædictæ omnes ſphæræ ſimul <lb></lb>
apparentias omnes excuſarent, addendas eſſe alias ſingulis planetis toti­<lb></lb>
dem ſphæras vna minus, quas Reuoluentes appellabat; ita vt qui quatuor <lb></lb>Mouentes ſphæras habuiſſet, tribus præterea reuoluentibus opus haberet: <lb></lb>
quæ ſphæræ reuoluentes id præſtabant, vt quaſi priores Mouentes ita in of­<lb></lb>
ficio continerent, vt priori poſitioni aſtrum, quod interiori orbi affigebur <lb></lb>
ſuo tempore reſtituerent, vt Alexander exponit. </s>
          <s id="s.002560">hoc enim ſolummodo poſ­<lb></lb>
ſibile putabat omnes errantium lationes nos imitari poſſe. </s>
          <s id="s.002561">Cum igitur mo­<lb></lb>
uentes ſphæræ illæ quidem Saturni, ac Iouis ſint octo; reliquorum verò vi­<lb></lb>
gintiquinque, nam reliqui Planetæ <expan abbr="quinq;">quinque</expan> ſinguli ſphæras <expan abbr="quinq;">quinque</expan> mouentes <lb></lb>
habent, quæ omnes ſimul numerum <expan abbr="vigintiquinq;">vigintiquinque</expan> explent: quarum omnium <lb></lb>
ſolæ inferiores, quibus aſtrum affixum volebat, non indigebant reuoluente, <lb></lb>
ſequitur duorum ſuperiorum Saturni, &amp; Iouis, quorum octo erant mouen­<lb></lb>
tes, ſex debere eſſe reuoluentes. </s>
          <s id="s.002562">Inferiorum verò quatuor planetarum re­<lb></lb>
uoluentes erunt ſexdecim: ſed hoc loco Ariſt. memoria fallit, deberet enim <lb></lb>
dicere, reliquorum <expan abbr="quinq;">quinque</expan> planetarum reuoluentes erunt vigintì, ſunt enim <lb></lb>
planetæ ſeptem, quorum Saturno, ac Ioui ſupremis ſex reuoluentes attri­<lb></lb>buit habita ratione ſphærarum mouentium; reliquis igitur <expan abbr="quinq;">quinque</expan> planetis <lb></lb>
habita ratione ſuorum orbium mouentium, 25. cum ſinguli habeant <expan abbr="quinq;">quinque</expan> <lb></lb>mouentes, habebunt ex præſcripto Calippi ſinguli 4. reuoluentes; ac pro­<lb></lb>
inde 20. in vniuerſum erunt reuoluentes. </s>
          <s id="s.002563">Omnium igitur ſphærarum tam <lb></lb>
mouentium, quàm reuoluentium ſummam ait, ſed perperam, eſſe quinqua­<lb></lb>ginta quinque; cum enim mouentes Saturni, &amp; Iouis ſint 8. reliquorum au­<lb></lb>
tem 25. reuoluentes verò Saturni, &amp; Iouis ſint 6. reliquorum autem, vt ip­<lb></lb>
ſe memoria falſus ponit, ſexdecim, conflant quidem ſummam prædictam, <lb></lb>
ſed illi in memoria reuocandus eſt, planeta ille, quem oblitus eſt, cuius ſunt <lb></lb>
quatuor reuoluentes, qui prioribus additi ſphærarum errantium numerum <lb></lb>
quinquaginta nouem conſtituent: quibus etiam addenda eſt octaua ſphæra, <lb></lb>
ſeu firmamentum, quod inerrantium ſedes eſt, non enim ſolum errantium, <lb></lb>
ſed omnium cœleſtium orbium numerum inueſtigare volebat, <expan abbr="ſicq́">ſicque</expan>; eſſent <lb></lb>
omnes ſecundum Calippum ſphęræ ſexaginta. </s>
          <s id="s.002564">Quod ſi Lunæ, &amp; Soli non ad­<lb></lb>
dantur ſingulis duo mouentes, vt facit Calippus, <expan abbr="neq;">neque</expan> conſequenter quatuor <lb></lb>
illis debiti reuoluentes non erunt omnes, 55. verùm, detractis octo prædi­<lb></lb>
ctis, erunt tantum 47. ſeu vt melius loquatur non erunt in vniuerſum, 60. ſed <lb></lb>
52. tantum. </s>
          <s id="s.002565">Hactenus de numero cœlorum.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.002566"><emph type="italics"></emph>Ex Decimotertio Metaphyſicæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002567"><arrow.to.target n="marg228"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002568"><margin.target id="marg228"></margin.target>237</s>
        </p>
        <p type="main">
          <s id="s.002569">Svmma 1. cap. 3. <emph type="italics"></emph>(Qui dicunt Mathematicas ſcientias nihil de bono, vel <lb></lb>
pulchro dicere, falſum dicunt. </s>
          <s id="s.002570">dicunt. </s>
          <s id="s.002571">n. </s>
          <s id="s.002572">&amp; maximè <expan abbr="oſtẽdunt">oſtendunt</expan>. </s>
          <s id="s.002573">nam &amp; ſi non <lb></lb>
nominant, quia tamen opera, &amp; rationes ostendunt, non ne dicunt de eis? <lb></lb>
</s>
          <s id="s.002574">pulchra <expan abbr="namq;">namque</expan> maximè ſpecies ſunt, ordo, commenſuratio, &amp; definitŭ, quæ <lb></lb>
maximè à Mathematicis ſcientijs oſtenduntur, &amp;c.)<emph.end type="italics"></emph.end> placuit hæc in Mathemati­<lb></lb>
carum commendationem, ac defenſionem apponere, cum non deſint hac <lb></lb>noſtra tempeſtate ageometreti complures, qui eas libenter ſugillare ſolent.</s>
        </p>
      </chap>
      <pb pagenum="148" xlink:href="009/01/148.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.002575"><emph type="italics"></emph>IN MECHANICAS QVÆSTIONES.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002576">Qvidquid Mathematicum in his quæſtionibus occurret, illud, vt <lb></lb>
plurimum per paraphraſim exponemus, ita tamen, vt tex. Ariſt. <lb></lb>
&amp; figuræ textui reſpondentes per eam, quantum fieri poterit re­<lb></lb>
ſtituantur, &amp; ſi quæ ſe offerent difficilia, pro viribus ſoluantur. <lb></lb>
</s>
          <s id="s.002577">Eſt autem nonnullis in locis textus tam græcus, quàm latinus adeò corrup­<lb></lb>
tus, ac deprauatus, vt nullo modo emendari queat.</s>
        </p>
        <p type="head">
          <s id="s.002578"><emph type="italics"></emph>Caput Primum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002579">Quæ ſit artis Mechanicæ facultas.</s>
        </p>
        <p type="main">
          <s id="s.002580"><arrow.to.target n="marg229"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002581"><margin.target id="marg229"></margin.target>238</s>
        </p>
        <p type="main">
          <s id="s.002582">Eorum, quæ miraculo ſunt, alia quidem natura contingunt, <expan abbr="ſuntq́">ſuntque</expan>; ea, <lb></lb>
quorum ignorantur cauſæ: alia verò ſunt, quæ præter naturam per <lb></lb>
artificium aliquod ad hominum vtilitatem perficiuntur, in multis <lb></lb>
<expan abbr="namq;">namque</expan> natura ei, quod nobis vſui eſſe poteſt, contrarium facit, quod <lb></lb>
inde oritur, quia natura eundem ſemper, ac ſimplicem ſeruat modum: quod <lb></lb>
autem nobis vtile eſt, plurimas ſubit varietates. </s>
          <s id="s.002583">quando igitur quippiam <lb></lb>
præter naturam facere opportuerit, illud, quod faciendum eſt, difficultate <lb></lb>
ſua nos remoratur, <expan abbr="arteq́">arteque</expan>; propterea indigemus. </s>
          <s id="s.002584">quamobrem eam artis <lb></lb>
partem, quæ huiuſmodi ſuccurrit difficultatibus, Mechanicam appellamus. <lb></lb>
</s>
          <s id="s.002585">Cæterùm optimè Antiphon Poeta in hunc modum cecinit;</s>
        </p>
        <p type="head">
          <s id="s.002586"><emph type="italics"></emph>Arte ſuperamus ea, in quibus à natura vincimur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002587">Quemadmodum accidit, cum minora ſuperant maiora, &amp; quæcunque exi­<lb></lb>
guam vim habentia, magna tamen mouent pondera, &amp; omnia ferè illa, quæ <lb></lb>
ſub ea cadunt problemata, quæ mechanica nuncupari <expan abbr="ſolẽt">ſolent</expan>. </s>
          <s id="s.002588">ſunt autem hæc <lb></lb>
<expan abbr="neq;">neque</expan> naturalibus omninò quæſtionibus eadem, <expan abbr="neq;">neque</expan> ſeiugata valde: verùm <lb></lb>
mathematicarum contemplationum, <expan abbr="naturaliumq;">naturaliumque</expan> communia. </s>
          <s id="s.002589">Poſtea in <lb></lb>
græcis codicibus hæc ſequuntur (<foreign lang="grc">τὸ μεν γὰρ ῶ δία των μαθηματικῶν δηλον· το<lb></lb>δε περὶ ὅ, δὶα τῶν φυσικῶν</foreign>) ideſt, ſi quidem quomodo ſint, ſeu qua ratione <lb></lb>
exiſtant, manifeſtum eſt per Mathematica: illud verò circa quod verſantur, <lb></lb>
hoc eſt obiectum, de quo pertractant Mechanicæ quæſtiones per ſcientias <lb></lb>
phyſicas habetur, ideſt res naturalis eſt; eſt enim pondus, &amp; vis, aut poten­<lb></lb>
tia pondus ipſum mouens, quatenus quanta ſunt; ſiue dixeris eſt quantitas <lb></lb>
ponderum, <expan abbr="atq;">atque</expan> potentiarum. </s>
          <s id="s.002590">Mathematicæ enim mediæ, de quorum nu­<lb></lb>
mero eſt facultas Mechanica, conſiderant quantitatem rei alicuius <lb></lb>
determinatæ, ſic Aſtronomia circa cœleſtium corporum, <expan abbr="mo-tuumq́">mo­<lb></lb>
tuumque</expan>; quantitates, Perſpectiua circa linearum viſua­<lb></lb>
lium; Muſica circa ſonorum quantitates ver­<lb></lb>
ſantur. </s>
          <s id="s.002591">quæ placuit annotare, vt ſcien­<lb></lb>
tiæ huius naturam perſpectam <lb></lb>
haberemus.</s>
        </p>
        <pb pagenum="149" xlink:href="009/01/149.jpg"></pb>
        <p type="head">
          <s id="s.002592">De dignitatibus, <expan abbr="admirandisq́">admirandisque</expan>; Circuli proprietatibus.</s>
        </p>
        <p type="head">
          <s id="s.002593"><emph type="italics"></emph>Cap. Secundum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002594"><arrow.to.target n="marg230"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002595"><margin.target id="marg230"></margin.target>239</s>
        </p>
        <p type="main">
          <s id="s.002596">Cvm vellet Ariſt. mirabilium effectuum, quos in Mechanicis admi­<lb></lb>
ramur, cauſam referre in circulum: meritò ante omnia de admi­<lb></lb>
randa ipſius circuli natura diſſerit, quo minus mirum deinde vi­<lb></lb>deatur prædictas mirabiles operationes ex ipſo procedere. </s>
          <s id="s.002597">quan­<lb></lb>
doquidem exadmiranda cauſa admirabiles effectus prodire debeant. </s>
          <s id="s.002598">qua­<lb></lb>lia ſunt ea, quæ circa vectem, cum magna <expan abbr="omniũ">omnium</expan> admiratione contingunt. <lb></lb>
</s>
          <s id="s.002599">videmus enim exiguam prorſus vim ingens pondus, quod <expan abbr="abſq;">abſque</expan> vecte mini­<lb></lb>
mè mouere poſſet, addito etiam ipſius vectis pondere, facilè <expan abbr="quocunq;">quocunque</expan> vo­<lb></lb>
luerit propellere. </s>
          <s id="s.002600">quod quidem auditu abſurdum foret, niſi viſu conſtaret. <lb></lb>
</s>
          <s id="s.002601">omnium autem huiuſmodi cauſæ principium circulus obtinet: &amp; hoc qui­<lb></lb>
dem meritò, ex admirabili enim, quippiam mirandum accidere rationi <lb></lb>omninò conſentaneum eſt.</s>
        </p>
        <p type="main">
          <s id="s.002602">Primò igitur maximè admirandum eſt contraria ſimul fieri, aut exiſtere:<lb></lb>
circulus tamen ex contrarijs eſt conſtitutus, oritur enim circulus ex com­<lb></lb>moto, &amp; manente, quæ quidem naturaliter ſunt inuicem contraria. </s>
          <s id="s.002603">ſit au­<lb></lb>
tem circulus ex commoto, &amp; manente, quia oritur ex circumuolutione <lb></lb>
vnius rectæ lineæ, cuius alterum extremum fixum manet, alterum verò cir­<lb></lb>
cumagitur; quamobrem iſthæc cernentes minus admirari <expan abbr="cõuenit">conuenit</expan> reliquas, <lb></lb>
quæ in ipſo ſunt contrarietates. </s>
          <s id="s.002604">cuiuſmodi eſt hæc, quod cum linea, quæ cir­<lb></lb>
culi orbem complectitur, <expan abbr="quæq́">quæque</expan>; circunferentia appellatur, nullam habeat <lb></lb>
latitudinem, ei tamen contraria quodammodo inſunt, concauum ſcilicet, <lb></lb>
&amp; curuum; quæ quidem eo modo ſunt contraria, quo etiam magnum, &amp; pa­<lb></lb>
ruum, horum enim medium eſt æquale; illorum verò rectum. </s>
          <s id="s.002605">&amp; ſicuti quan­<lb></lb>
do magnum, &amp; paruum inuicem commutantur, ita vt quod magnum eſt fiat <lb></lb>
paruum, quod verò paruum fiat magnum, neceſſe eſt, vt perueniant ad <lb></lb>
æquale priuſquam ad extremum alterutrum; ita linea curua antequam fiat <lb></lb>
concaua, debet prius fieri recta: &amp; ex concaua, vt tranſeat ad conuexam, <lb></lb>
&amp; circularem, debet ſimiliter prius eſſe recta.</s>
        </p>
        <p type="main">
          <s id="s.002606">Alterum contrarium, quod circulo ineſt, eſt ſimul <expan abbr="cõtrarijs">contrarijs</expan> motibus mo­<lb></lb>neri: ſimul enim ad anteriorem mouetur locum, &amp; ad poſteriorem. </s>
          <s id="s.002607">&amp; eo­<lb></lb>
dem modo linea illa, quæ ex vno extremo manens, ex altero verò circum­<lb></lb>lata circulum deſcribit, ſe habet; contraria enim ſimul continet, primum <lb></lb>
ſcilicet, &amp; extremum. </s>
          <s id="s.002608">Ex quo enim primo loco circumagi incipit ad eun­<lb></lb>
dem rurſus poſtremò reuertitur, ita, vt primum ipſius, &amp; poſtremum idem <lb></lb>
ſint; quapropter, vt prius dicebamus non eſt inconueniens, ipſum circulum <lb></lb>
miraculorum omnium eſſe principium. </s>
          <s id="s.002609">Admiranda igitur ea, quæ circa li­<lb></lb>bram fiunt, ad circulum <expan abbr="tãquam">tanquam</expan> cauſam referuntur, quæ verò circa vectem <lb></lb>
ad ipſam libram: alia autem ferè omnia, quæ circa mechanicas contingunt <lb></lb>
motiones, ad vectem reducuntur.</s>
        </p>
        <p type="main">
          <s id="s.002610">Præter prædicta aliud tandem mirum ipſi ineſt, quia nimirum cum innu­<lb></lb>
mera ſint puncta in vna <expan abbr="eademq́">eademque</expan>; linea, quæ ſemidiameter eſt, omnia tamen <pb pagenum="150" xlink:href="009/01/150.jpg"></pb>quando ſemidiameter circa centrum mouetur, quamuis cum ipſa mouean­<lb></lb>tur, inæquali velocitate mouentur; Nam punctum illud ſemper velocius <lb></lb>
mouetur, quod remotius eſt à centro circuli, ſeu à manente ſemidiametri <lb></lb>
termino, &amp; proinde illud tardius, quod centro proximius eſt. </s>
          <s id="s.002611"><expan abbr="Atq;">Atque</expan> ex hac <lb></lb>
mira circuli proprietate, <expan abbr="pleraq;">pleraque</expan> miraculorum accidunt circuli motioni­<lb></lb>
bus, vt in ſequentibus quæſtionibus manifeſtum erit.</s>
        </p>
        <p type="main">
          <s id="s.002612">Quoniam autem ſecundum contrarias ſimul motiones mouetur circulus, <lb></lb>
&amp; alterum quidem diametri extremum vbi A, in figura præſenti antrorſum <lb></lb>
<figure id="id.009.01.150.1.jpg" place="text" xlink:href="009/01/150/1.jpg"></figure><lb></lb>
mouetur; alterum verò vbi B, retror­<lb></lb>
ſum, efficiunt nonnulli, vt ab vnica mo­<lb></lb>
tione multi contrariò ſimul mouean­<lb></lb>
tur denticulati circuli: vt ſunt ij, quos <lb></lb>
in locis proponunt ſacris, quorum alij <lb></lb>
ſunt ænei, alij ferrei. </s>
          <s id="s.002613">ſi enim circulus <lb></lb>
A B, alterum circulum C D, contige­<lb></lb>
rit, mota diametro A B, ita vt A, an­<lb></lb>
trorſum eat, commouebit alteram dia­<lb></lb>
metrum C D, ita vt C, retrorſum, hoc eſt in contrarium ipſi A, veniat, in <lb></lb>
contrarium igitur mouebitur ſecundus circulus C D, ad circulum A B, &amp; <lb></lb>
rurſus circulus E F. in contrarium ipſi C D, commouebitur ab ipſo C D, ob <lb></lb>
eandem rationem. </s>
          <s id="s.002614">eodem etiam modo ſi plures fuerint, idem facient vno <lb></lb>
ſolo tanquam primo motore <expan abbr="cõmoto">commoto</expan>. </s>
          <s id="s.002615">hanc igitur circuli naturam animad­<lb></lb>
uertentes Architecti, inſtrumentum artificiosè <expan abbr="fabricãt">fabricant</expan>, motus principium <lb></lb>
occultantes, vt machinæ ſolù manifeſtum ſit illud, quod admirationem <lb></lb>
parit, cauſa verò lateat: quod genus machinarum Automata dicebantur, <lb></lb>
quia ſpontè à ſe ipſis mouebantur.</s>
        </p>
        <p type="main">
          <s id="s.002616">In primis igitur, quæ circa libram accidunt, dubitare faciunt, quamnam <lb></lb>
ob cauſam maiores libræ minoribus ſint exactiores: huius autem rei prin­<lb></lb>
cipium eſt illud, quod ſupra innuimus, quod ſcilicet, quæ à centro plus di­<lb></lb>
ſtat linea, ſiue quæ longior eſt, eadem vi commota citius fertur, quam illa, <lb></lb>
quæ minus à centro diſtat, ſeu quæ minor eſt. </s>
          <s id="s.002617">Porrò citius bifariam dicitur; <lb></lb>
ſiue enim in minori tempore æquale pertranſit ſpatium: ſiue in æquali tem­<lb></lb>
pore, maius conficit interuallum; citius feciſſe dicitur. </s>
          <s id="s.002618">ſi autem duæ lineæ <lb></lb>
circa idem centrum moueantur vna maior, &amp; altera minor in æquali tem­<lb></lb>
pore; maior maiorem circulum deſcribet, quam minor; quia circulus à ma­<lb></lb>iori deſcriptus, alterum à minori delineatum circumpleρetur, <expan abbr="atq;">atque</expan> intra ſe <lb></lb>
continebit; maius autem eſt continens, quàm <expan abbr="contẽtum">contentum</expan>. </s>
          <s id="s.002619">horum autem cau­<lb></lb>
ſa, quoniam quæ circulum deſcribit linea, duabus fertur lationibus, quæ nul­<lb></lb>
lam inuicem obtinent analogiam: quod antequam probemus, ſciendum <lb></lb>
eſt, quod, quidquid duobus motibus inuicem proportionatis, mouetur, ne­<lb></lb>ceſſe eſt, quod motu ex illis mixto progrediatur per lineam rectam, quæ dia­<lb></lb>
meter eſt quadrilateri, cuius latera habeant illam proportionem, quam <lb></lb>
duo illi motus. </s>
          <s id="s.002620">ſit enim in figura proportio lateris A B, ad latus A C, quam <lb></lb>
ctiam habent duo motus, ſecundum quos latum quodpiam feratur, <expan abbr="ſitq́">ſitque</expan>; la­<lb></lb>
tum illud A, &amp; feratur motu vno verſus B, per lineam A B, altero verò mo­<lb></lb>
tu feratur verſus C. quod fiet ſi cogitemus latus A B, <expan abbr="deſcẽdere">deſcendere</expan> verſus M C,


<pb pagenum="151" xlink:href="009/01/151.jpg"></pb><figure id="id.009.01.151.1.jpg" place="text" xlink:href="009/01/151/1.jpg"></figure><lb></lb>ipſi æquidiſtanter, dum punctum A, mouetur <lb></lb>
ad B. his duabus lationibus A, latum. </s>
          <s id="s.002621">neceſſa­<lb></lb>riò motu mixto progredietur per diametrum <lb></lb>
A M, quod ſic probari poteſt; ſit iam A, mo­<lb></lb>
tum primo motu <expan abbr="vſq;">vſque</expan> ad D, linea verò ex ſe­<lb></lb>
cundo motu ſit in G F E, quo motu punctum <lb></lb>
A, quod erat in D, <expan abbr="tractũ">tractum</expan> erit in F. quod pun­<lb></lb>
ctum eſt in diametro A M, quoniam enim mo­<lb></lb>
uetur duobus motibus, cum lineis A B, A C, proportionalibus, motus au­<lb></lb>
tem <expan abbr="hucuſq;">hucuſque</expan> ſunt A D, A E, quæ debent eſſe proportionales, cum A B, A C <lb></lb>
compleatur rectangulum A D F E, erunt ſimiliter proportionalia F E, D E, <lb></lb>
cum ſint æqualia duobus D A, A E, quare per 26. 6. cum quadrilaterum <lb></lb>
paruum A D F E, ſit ſimile toti A B M C, erit A M, <expan abbr="vtriuſq;">vtriuſque</expan> diameter, ergò <lb></lb>
punctum F, in quo eſt A, eſt in diametro A M. eodem modo, de quouis pun­<lb></lb>
cto in linea A B, ad quod A, perueniat, probabitur ab altero motu deſcen­<lb></lb>
diſſe vſque ad diametrum. </s>
          <s id="s.002622">ſemper ergò latum A, per rectam A M, diame­<lb></lb>
trum quadrilateri, cum illis motibus proportionalibus progreditur, quod <lb></lb>
probandum erat. </s>
          <s id="s.002623">è conuersò manifeſtum etiam eſt, quod ſi quid ſecundum <lb></lb>
diametrum duabus fertur lationibus, eas lationes eſſe proportionales late­<lb></lb>
ribus quadrilateri, cuius eſt illa diameter, ſi enim illæ lationes non ſunt la­<lb></lb>
teribus proportionales, latum illud non feretur ſecundum diametrum il­<lb></lb>
lam, ſed ſecundum aliam alterius quadrilateri.</s>
        </p>
        <p type="main">
          <s id="s.002624">Quod ſi quid duabus lationibus nullam habentibus proportionem per­<lb></lb>
petuò ferratur, impoſſibile eſt ipſum motu mixto lineam rectam deſcribere. <lb></lb>
</s>
          <s id="s.002625">ſi enim dixeris illud poſſe deſcribere rectam lineam, tunc circa rectam il­<lb></lb>
lam tanquam diametrum deſcribam quadrilaterum, &amp; poſtea oſtendam, vt <lb></lb>
proximè oſtenſum eſt, illud latum eſſe ſecundum laterum illius proportio­<lb></lb>
nem, quare impoſſibile eſt id, quod mouetur duabus lationibus nullam in­<lb></lb>
uicem rationem habentibus, ferri per lineam rectam: quapropter <expan abbr="dicẽdum">dicendum</expan> <lb></lb>
eſt hoc modo <expan abbr="latũ">latum</expan>, neceſſariò ferri circulariter, ſiue per lineam circularem. <lb></lb>
</s>
          <s id="s.002626">Quod autem ea, quæ deſcribit circulum linea, dum altero eius manente <lb></lb>
extremo circumagitur, duabus ſimul feratur lationibus, ex quibus motus <lb></lb>
orbicularis oriatur, manifeſtum eſt ex ſuperioribus, quia &amp; antrorſum, &amp; <lb></lb>
retrorſum impellitur; tùm etiam, quia ſi rectà tenderet recta <expan abbr="deſcribẽs">deſcribens</expan> cir­<lb></lb>
<figure id="id.009.01.151.2.jpg" place="text" xlink:href="009/01/151/2.jpg"></figure><lb></lb>
culum, nunquam ad diametri perpendiculum <lb></lb>
perueniret, ſed tamen peruenit, ita vt ſit ipſa <lb></lb>
à centro perpendicularis diametro. </s>
          <s id="s.002627">ſit circuli <lb></lb>
figura A B C D, in qua extremum diametri <lb></lb>
B, feratur ad alterum extremum vbi D, per <lb></lb>
ipſius diametri B D, circumuolutionem circa <lb></lb>
centrum F, neceſſe eſt aliquando B, perueniat <lb></lb>
ad C. ſi igitur B, feretur duabus lationibus <lb></lb>
aliquo modo proportionatis, v. g. vt eſt pro­<lb></lb>
portio lateris B E, ad E C, latus, ſequeretur <lb></lb>
ex demonſtratis ipſum B, ferri per <expan abbr="rectã">rectam</expan> B C, <lb></lb>quæ diameter eſſet quadrilateri B E C G. ſed <pb pagenum="152" xlink:href="009/01/152.jpg"></pb>quia in nulla proportione fertur, propterea per circularem lineam B E C, <lb></lb>
progreditur ad C, ita vt ipſa diameter B D, in poſitione A C, fiat perpendi­<lb></lb>
cularis priori diametro B D. ex quibus ſequitur eam moueri duobus moti­<lb></lb>
bus nullam rationem habentibus; quod erat intentum.</s>
        </p>
        <p type="main">
          <s id="s.002628">Hoc modo Ariſt. probare conatur, lineam circulum deſcribentem, dua­<lb></lb>bus ferri lationibus, quæ nullam habeant analogiam: Verùm, vt liberè fa­<lb></lb>
tear nullo modo mihi videtur intentum aſſequi, nam <expan abbr="neq;">neque</expan> ex dictis pater, <lb></lb>
ipſam duobus motibus ferri, quibus opus eſſet: neque patet eos (quamuis <lb></lb>
concedantur) nullam inuicem habere analogiam: qui enim fieri poteſt, vt <lb></lb>
duo motus reperiantur, quì nulla ſe mutuò habitudine reſpiciant? </s>
          <s id="s.002629">Præte­<lb></lb>
rea ſi B, ferretur illis motibus, non ſequitur debere moueri per lineam cir­<lb></lb>
cularem, cum præter lineam rectam ſint plures curuæ, quæ tamen non ſunt <lb></lb>
circulares, vt ſunt ſectiones parabolicæ, &amp; lineæ ſpirales. </s>
          <s id="s.002630">Deinde pergit.</s>
        </p>
        <p type="main">
          <s id="s.002631"><arrow.to.target n="marg231"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002632"><margin.target id="marg231"></margin.target>241</s>
        </p>
        <p type="main">
          <s id="s.002633">Vt autem cauſa appareat, cur ea, quæ à centro longior eſt linea velocius <lb></lb>
moueatur, ſiue quod in eadem ſemidiametro remotiora puncta à <expan abbr="cẽtro">centro</expan> ve­<lb></lb>
locius moueantur, vt ſupra dictum eſt, ſciendum eſt, Quod ſi duo mouean­<lb></lb>
tur ab eadem potentia, quorum vnum à quopiam alio mouente plus repel­<lb></lb>
latur à motu priori, alterum verò minus, rationi <expan abbr="cõſentaneum">conſentaneum</expan> eſſe, tardius <lb></lb>
moueri id, quod plus, eo quod minus impeditur; quod videtur accidere <lb></lb>
maiori, &amp; minori illarum, quæ à centro egreſſæ circulos delineant. </s>
          <s id="s.002634">quoniam <lb></lb>
enim propius eſt manenti eius, quæ minor eſt extremum, quàm extremum <lb></lb>
maioris, propterea plus à centro, cui propius eſt, retrahitur à priori mo­<lb></lb>
tu, <expan abbr="hincq́">hincque</expan>; motus eius tardior redditur, ideſt, quia centro propius eſt; hinc <lb></lb>
fit, vt extremum illud deſcribat lineam circularem quidem, ſed tamen <lb></lb>
curuiorem quam deſcribat extremum longioris lineæ, quæ circulum minus <lb></lb>
curuum, ſeu magis ad rectam lineam accedentem delineat. </s>
          <s id="s.002635">omni quidem <lb></lb>
igitur lineæ circulum deſcribenti iſtud accidit, vt duobus feratur motioni­<lb></lb>
bus; vna quidem, quæ illi naturalis, ac ſecundum circunferentiam, qua re­<lb></lb>
ctà tenderet niſi impediretur: altera verò, quæ illi innaturalis, qua in tranſ­<lb></lb>
uerſum agitur, ſeu ſecus centrum, ob quam cogitur in gyrum duci, minor <lb></lb>
autem linea ſecundum hanc motionem innaturalem plus fertur, quàm ma­<lb></lb>
ior, ideſt plus ipſius progreſſio inflectitur in orbem; quia enim eſt centro <lb></lb>
<figure id="id.009.01.152.1.jpg" place="text" xlink:href="009/01/152/1.jpg"></figure><lb></lb>
vicinior, quod quodammodo retra­<lb></lb>
hit à motu naturali, propterea ma­<lb></lb>
gis vincitur, quàm remotior. </s>
          <s id="s.002636">Quod <lb></lb>
ex his erit <expan abbr="manifeſtũ">manifeſtum</expan>. </s>
          <s id="s.002637">ſit circulus vbi <lb></lb>
B C E D, &amp; alter in eo minor, vbi <lb></lb>
N M O P, circa idem centrum A. &amp; <lb></lb>
proijciantur diametri in magno qui­<lb></lb>
dem C D, B E, in minori verò M O, <lb></lb>
N P. &amp; altera parte longius quadri­<lb></lb>
laterum compleatur D K R C. ſi igi­<lb></lb>
tur ſemidiameter A B, circumacta <lb></lb>
deſcribit circulum maiorem, reuer­<lb></lb>
titur tandem ad locum B A, vnde di­<lb></lb>
greſſa eſt. </s>
          <s id="s.002638">ſimiliter M A, circumuoluta <pb pagenum="153" xlink:href="009/01/153.jpg"></pb>redibit ad priorem poſitionem in M A. </s>
          <s id="s.002639">Tardius autem fertur M A, quàm <lb></lb>
B A, vt dictum eſt, quia maior illi fit retractio à recta progreſſione. </s>
          <s id="s.002640">Sit igi­<lb></lb>
tur linea A B, mota vſque ad locum A L F, &amp; à puncto L, ducatur L Q, per­<lb></lb>
pendicularis ipſi A B, in minori circulo. </s>
          <s id="s.002641">&amp; rurſus ducatur L S, parallela ei­<lb></lb>
dem A B, &amp; à puncto S, in maiori circulo ducatur S T, perpendicularis ei­<lb></lb>
dem B A, necnon F X. erunt igitur S T, L Q, latera rectanguli T L, æqualia <lb></lb>
per 34. primi. </s>
          <s id="s.002642">erit poſtea B T, minor quam M Q, quia æquales rectæ S T, <lb></lb>
L Q, ductæ à circunferentia ad diametrum perpendiculares in circulis in­<lb></lb>
æqualibus, ea quæ eſt in maiori circulo minorem reſecat diametri portio­<lb></lb>
nem, quàm quæ in minori.</s>
        </p>
        <p type="main">
          <s id="s.002643">In quanto autem tempore ipſa A L, lata eſt per circunferentiam M L, in <lb></lb>
tanto temporis ſpatio in maiori circulo B, extremum ipſius B A, latum erit <lb></lb>per maiorem arcum quàm ſit B S; iam conſiderandum eſt motus vtriuſque <lb></lb>
lineæ in hoc caſu æquales eſſe, ſunt enim deſcripti per lineas æquales T S, <lb></lb>
Q L, quæ ſunt rectæ; tam enim linea B A, quàm M A, naturali motu recta <lb></lb>
tenderet, vt dictum eſt, <expan abbr="peragraſſetq́">peragraſſetque</expan>; illa rectam T S: hæc verò rectam Q L. <lb></lb>
</s>
          <s id="s.002644">Verum lationes innaturales ſunt impares, latio enim B T, breuior eſt M <expan abbr="q.">que</expan> <lb></lb>
quantitate autem B T, retracta eſt B A, à motu ſibi naturali, &amp; recto: quan­<lb></lb>
titate verò M Q, retracta eſt M A, vnde apparet motu hoc violento magis <lb></lb>
retractam eſſe minorem M A, quàm maiorem B A, quod erat primo de­<lb></lb>
clarandum.</s>
        </p>
        <p type="main">
          <s id="s.002645">Quod autem ob id A B, maior cęlerius mota ſit motu naturali, quàm mi­<lb></lb>
nor M A, palàm fiet. </s>
          <s id="s.002646">quia enim oportet <expan abbr="vtramq;">vtramque</expan> lineam maiorem, &amp; mi­<lb></lb>
norem eadem vi motam, confeciſſe binos illos motus proportionales, ideſt <lb></lb>
ita ſe debet habere motus naturalis maioris ad motum innaturalem eiuſ­<lb></lb>
dem, quemadmodum ſe habet motus naturalis minoris ad motum innatu­<lb></lb>
ralem eiuſdem: Oportet ergo, vt ſi A B, &amp; A M, ſunt eadem vi commotæ, <lb></lb>
vt ſit eadem ratio T S, ad Q L, quæ eſt B T, ad M Q, non eſt autem, vt oſten­<lb></lb>
ſum eſt; ergo linea A B, eadem vi commota, ac M A, conficit pluſquam <lb></lb>
B S, ſed neceſſariò peruenit ad F. hoc enim in puncto erunt prædicti motus <lb></lb>
proportionales, vt oportet, erit enim motus naturalis in maiori perpendi­<lb></lb>
cularis F X, &amp; innaturalis B X, in minori verò naturalis L Q, innaturalis <lb></lb>
M <expan abbr="q.">que</expan> quod ſi ducantur rectè B F, M L, apparebunt duo triangula æquian­<lb></lb>
gula B X F, M Q L, &amp; erunt per 4. 6. vt F X, ad B X. ita L Q, ad M Q, &amp; <lb></lb>
permutando erunt etiam vt F X, ad L Q, ita B X, ad M <expan abbr="q.">que</expan> ideſt, vt motus <lb></lb>
naturalis ad naturalem, ita innaturalis ad innaturalem. </s>
          <s id="s.002647">In alio autem lo­<lb></lb>
co præter F, non erunt eædem proportiones.</s>
        </p>
        <p type="main">
          <s id="s.002648">Ex quibus patere ſatis poteſt, cur A B, longior à centro velocius mouea­<lb></lb>
tur quàm minor M A, ſeu cur puncta eiuſdem B A, velocius vertuntur, quo <lb></lb>
longius abſunt à centro A, ideſt maiorem arcum B F, peractum eſſe à B, <lb></lb>
quàm ſit arcus M L, peractus ab M, quod erat oſtendendum.</s>
        </p>
        <p type="main">
          <s id="s.002649">Atque hic eſt diſcurſus ille Ariſt. quo putat ſe cauſam aperuiſſe, cur lon­<lb></lb>
gior ſemidiameter velocius moueatur: quod num rectè attigerit, non puto <lb></lb>
operæpretium eſſe hoc loco diſcutere, præſertim cum ad naturalem Philo­<lb></lb>
ſophum ſpectet.</s>
        </p>
        <p type="main">
          <s id="s.002650">Mihi tamen maximè conſiderandum videtur hoc ipſum quod aſſeruit, &amp; <pb pagenum="154" xlink:href="009/01/154.jpg"></pb>ex ſe patet, remotiores ſcilicet partes diametrorum à centro velocius mo­<lb></lb>
ueri, quàm viciniores; ex hac enim maiori velocitate ſequitur maiore etiam <lb></lb>
vi moueri, vnde &amp; potentiæ mouenti in extremo eius vis augebitur, &amp; plus <lb></lb>
poterit quam ſola ſine vecte, eſt enim vectis duæ ſemidiametri altera alte­<lb></lb>
ram longior; ex quibus fortè apparet vnde vectis vires oriantur.</s>
        </p>
        <p type="main">
          <s id="s.002651">His igitur tanquam huius Mechanicæ facultatis principijs poſitis, ad va­<lb></lb>
rias Quæſtiones diſcutiendas accedit.</s>
        </p>
        <p type="head">
          <s id="s.002652"><emph type="italics"></emph>QVÆSTIO PRIMA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002653"><emph type="italics"></emph>De Libra.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002654"><arrow.to.target n="marg232"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002655"><margin.target id="marg232"></margin.target>242</s>
        </p>
        <p type="main">
          <s id="s.002656">Cvr autem maiores libræ minoribus ſint exactiores, palàm eſt ex <lb></lb>
præmiſſis principijs. </s>
          <s id="s.002657">conſiderare enim oportet, quod in motu li­<lb></lb>
bræ deſcribitur quidam circulus, cuius diameter ſunt ipſa libræ <lb></lb>brachia, centrum verò eſt ſpartum, ſiue trutina; hoc enim pun­<lb></lb>
ctum in motu libræ manet: duo verò brachia ſunt veluti duæ ſemidiametri <lb></lb>
<figure id="id.009.01.154.1.jpg" place="text" xlink:href="009/01/154/1.jpg"></figure><lb></lb>
à centro exeuntes, vt in figura cerne­<lb></lb>
re eſt, in qua centrum, ſiue ſpartum <lb></lb>
eſt vbi C, reliqua ſunt manifeſta. </s>
          <s id="s.002658">In <lb></lb>
eadem porrò figura libra maior ſit <lb></lb>
A B. minor verò circa idem ſpartum <lb></lb>
C, ſit F G. </s>
          <s id="s.002659">Iam vt præmiſſum eſt, ea­<lb></lb>
dem vi, vel eodem onere in lance B, <lb></lb>
poſito, mouebitur velocius brachium <lb></lb>
libræ maioris, quàm minoris ſit ma­<lb></lb>
ior tranſlata ad <expan abbr="locũ">locum</expan> D E, ergò com­<lb></lb>
mota eſt per arcum B E, vel A D. </s>
          <s id="s.002660">Minor autem libra acta eſſet per mino­<lb></lb>
rem arcum G I, vel F H, melius autem apparet arcus B E, maior, quam mi­<lb></lb>
nor G I, <expan abbr="atq;">atque</expan> hoc eſt, quòd maiores libras exactiores facit. </s>
          <s id="s.002661"><expan abbr="hincq́">hincque</expan>; etiam eſt, <lb></lb>
quòd nonnulla pondera in minimis libris adeò paruam brachiorum aper­<lb></lb>
tionem faciant, vt ægrè percipi poſſit; in magnis verò propter brachiorum <lb></lb>
longitudinem valdè ſenſibilem efficiant. </s>
          <s id="s.002662">quædam verò benè, &amp; in magnis, <lb></lb>
&amp; in paruis apparent, ſed tamen ſemper melius in magnis ob dictam ratio­<lb></lb>
nem. </s>
          <s id="s.002663">Quamobrem machinantur ij, qui purpuram vendunt, vt pendendo <lb></lb>
defraudent, tum in medio libræ non ponentes ſpartum, vt hoc modo bra­<lb></lb>
chium ex vna parte longius factum facilius moueatur, &amp; proinde à minori <lb></lb>
purpuræ pondere; tum etiam <expan abbr="plumbũ">plumbum</expan> in lancem illam infundentes inquam <lb></lb>
merces imponitur, vel partem illam lancis, quam magis grauitare cu­<lb></lb>
piunt ex ligno radici proximo, vel ex nodoſo facientes: lignum <lb></lb>
enim, quod radici proximum eſt, graue admodum eſt, <lb></lb>
quemadmodum etiam nodus; quia nodus eſt, <lb></lb>
quædam radix. </s>
          <s id="s.002664"><expan abbr="Atq;">Atque</expan> hæc eſt huius pri­<lb></lb>
mæ quæſtionis paraphraſis.</s>
        </p>
        <pb pagenum="155" xlink:href="009/01/155.jpg"></pb>
        <p type="head">
          <s id="s.002665"><emph type="italics"></emph>QVÆSTIO SECVNDA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002666"><emph type="italics"></emph>De Libra<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002667"><arrow.to.target n="marg233"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002668"><margin.target id="marg233"></margin.target>243</s>
        </p>
        <p type="main">
          <s id="s.002669"><emph type="italics"></emph>Cvr ſi quidem ſurſum fuerit ſpartum, quando deorſum lato pondere, quiſ­<lb></lb>
piam id amouet, rurſum aſcendit libra? </s>
          <s id="s.002670">Si autem deorſum constitutum <lb></lb>
fuerit, non aſcendit, ſed manet? </s>
          <s id="s.002671">An quia ſurſum ſparto quidem exiſten­<lb></lb>
te plus libræ extra perpendiculum fit, ſpartum enim eſt perpendiculum, <lb></lb>
quare neceſſe est deorſum ferri id, quod plus est, donec aſcendat, quæ bifariam li­<lb></lb>bram diuidit ad ipſum perpendiculŭm, cum onus incumbat ad libræ partem tractam.<emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.155.1.jpg" place="text" xlink:href="009/01/155/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>ſit libra vbi recta B C, ſpartum autem <lb></lb>
A D: hoc igitur ſurſum erecto, perpendi­<lb></lb>
culum erit vbi A D M. ſi igitur in ipſo B, <lb></lb>
ponatur onus, B, quidem deſcendet vbi E; <lb></lb>
C, autem aſcendat vbi H, quamobrem ea, <lb></lb>
quæ bifariam libram ſecat, primò quidem <lb></lb>
erit D M, ipſius perpendiculi; incumben­<lb></lb>
te autem onere erit D G, quare libræ ip­<lb></lb>
ſius vbi E H, quod extra A M, perpendi­<lb></lb>
culum eſt, vbi eſt D D H, maius eſt dimidio. </s>
          <s id="s.002672">ſi igitur amoueatur onus ab ipſo E, ne­<lb></lb>
ceſſe eſt H, deorſum ferri, minus enim eſt ipſum E D. ſi quidem igitur ſurſum ha­<lb></lb>
buerit ſpartum, propter hoc aſcendit libra. </s>
          <s id="s.002673">ſi autem deorſum fuerit, id quod ſub­<lb></lb>
stat, contrarium facit; plus enim dimidio fit libræ, quæ deorſum eſt, pars, quàm <lb></lb>
quod perpendiculum ſecet; quapropter non aſcendit, pars enim eleuata leuior eſt.<emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.155.2.jpg" place="text" xlink:href="009/01/155/2.jpg"></figure><lb></lb>
<emph type="italics"></emph>ſit libra vbi N G, perpendiculum autem <lb></lb>
K L M, bifariam igitur ſecatur N G. im­<lb></lb>
poſito autem onere in ipſo N, erit quidem <lb></lb>
N, vbi O, ipſum autem G, vbi R; K L, au­<lb></lb>
tem vbi K P, quare maius eſt L P O, quàm <lb></lb>
L R, ipſo P L. </s>
          <s id="s.002674">Ablato igitur onere, ne­<lb></lb>
ceſſe eſt manere; incumbit enim, ceu onus <lb></lb>
exceſſus medietatis in quo P L.)<emph.end type="italics"></emph.end> Aduer­<lb></lb>
te textum græcum eſſe mendoſum, la­<lb></lb>
tinum vero mendoſiſſimum. </s>
          <s id="s.002675">Ego partim ex certa rei intelligentia, vti vi­<lb></lb>
des reſtitui. </s>
          <s id="s.002676"><expan abbr="Idemq́">Idemque</expan>; circa figuras præſtiti. </s>
          <s id="s.002677">Porrò quoniam Piccolomineus, <lb></lb>
&amp; ſi plurimum, vt ipſe fatetur, inſudauerit, non tamen ſolutionem huius <lb></lb>
quæſtionis eſt aſſecutus, eam tibi ex Mechanicis Guidibaldi tradam. </s>
          <s id="s.002678">Ariſt. <lb></lb>
igitur ponit duas libræ ſpecies, ſiue potius duas eiuſdem libræ poſitiones, <lb></lb>vnam, quæ habet ſpartum, ſiue perpendiculum ſupra; alteram, quæ infra. <lb></lb>
<figure id="id.009.01.155.3.jpg" place="text" xlink:href="009/01/155/3.jpg"></figure><lb></lb>
vt in præſenti figura, ſit libra B C, cuius <lb></lb>
ſpartum, ſiue perpendiculum A D, ſit ſur­<lb></lb>
ſum, ita vt in puncto A, ſit affixum perpen­<lb></lb>
diculum, &amp; circa idem punctum A, tan­<lb></lb>
quam circa centrum tota libra circum­<lb></lb>
uertatur. </s>
          <s id="s.002679">hæc eſt prima libræ collocatio. </s>
          <s id="s.002680">ſit deinde libra B C, cuius ſpartum, <lb></lb>
ſiue perpendiculum A D, ſit deorſum, vt in altera figura, <expan abbr="ſitq́">ſitque</expan>; circa pun­


<pb pagenum="156" xlink:href="009/01/156.jpg"></pb><figure id="id.009.01.156.1.jpg" place="text" xlink:href="009/01/156/1.jpg"></figure><lb></lb>
ctum A, tanquam circa <expan abbr="centrũ">centrum</expan>, aut axem <lb></lb>
ita fixum, vt ipſi libræ conuerſio innita­<lb></lb>
tur, quæ eſt altera libræ poſitio. </s>
          <s id="s.002681">Quærit <lb></lb>
igitur, cur ſi in libra ſurſum <expan abbr="habẽte">habente</expan> per­<lb></lb>
pendiculum, &amp; centrum, ponatur ex vna <lb></lb>
parte onus quodpiam, v. g. in parte B, vt in prima textus figura factum eſt, <lb></lb>
libra de primo ſitu B C, mouetur ad ſitum E H, ſed tamen ablato pondere <lb></lb>
reuertitur ſua ſpontè ad priſtinum ſitum B C. ſi autem in libra, cuius per­<lb></lb>
pendiculum, ac centrum deorſum ſit, vt in ſecunda figura textus, pondus <lb></lb>
imponatur, ipſa quidem à ſitu B C, ad ſitum O R, transferretur; verumta­<lb></lb>
men ablato onere, <expan abbr="nõ">non</expan> amplius ad priorem poſitionem, vti prior, reuertitur.</s>
        </p>
        <p type="main">
          <s id="s.002682">Huic quæſtioni, vt reſpondeat, tacitè ſupponit omne graue tendere de­<lb></lb>
orſum, hoc pacto, vt centrum grauitatis ipſius tendat per lineam rectam <lb></lb>
ad mundi centrum ab ipſo grauitatis centro protractam, quam lineam Di­<lb></lb>
rectionis Recentiores appellant. </s>
          <s id="s.002683">ſciendum autem centrum grauitatis eſſe <lb></lb>
punctum quoddam in quolibet graui, ex quo ſi graue illud ſuſpendatur, ſem­<lb></lb>
per manet in æquilibrio, nec vnquam poſitionem reſpectu ſuarum partium <lb></lb>
mutat, quamuis ita ſuſpenſum huc illuc transferatur. </s>
          <s id="s.002684">Ita Pappus Alexan­<lb></lb>
drinus initio octaui libri Mathematicarum collectionum. </s>
          <s id="s.002685">Totius igitur li­<lb></lb>
bræ abſque onere centrum grauitatis eſſet circa punctum D, quod eſſet di­<lb></lb>
ſtinctum à centro circumuolutionis A. quod grauitatis centrum, ſemper <lb></lb>
quantum fieri poteſt, ſi nihil obſtet, centro mundi appropinquat; &amp; propte­<lb></lb>
rea facit, vt prior libra ſine onere ſuſpenſa in A, in æquilibrio, atque hori­<lb></lb>
zonti parallela permaneat, ſtante enim D, centro mundi maximè propin­<lb></lb>
quo, ſiue in loco humillimo, erit inter punctum A, &amp; centrum mundi, ac <lb></lb>
conſequenter in linea directionis. </s>
          <s id="s.002686">quæ linea directionis in prima figura <lb></lb>
textus eſſet eadem cum perpendiculo A D M, manente libra ſine pondere <lb></lb>
horizonti parallela; in <expan abbr="ſecũda">ſecunda</expan> autem figura textus coincideret pariter cum <lb></lb>
perpendiculo K L M, antequam libra ob impoſitum onus ab æquilibrio di­<lb></lb>
moueretur. </s>
          <s id="s.002687">per hanc enim lineam centrum grauitatis libræ, quod eſt propè <lb></lb>
puncta D, &amp; L, tenderet ad mundi centrum, ſi libra liberè ad centrum mun­<lb></lb>
di dilaberetur. </s>
          <s id="s.002688">his præmiſſis ſic quæſtioni ſatisfacit, &amp; primò primæ parti, <lb></lb>
quando nimirum ſpartum ſupernè collocatum eſt. </s>
          <s id="s.002689">Ratio igitur, cur tunc li­<lb></lb>
bra amoto pondere ad horizontis æquilibrium reuertatur eſt, quia pondus <lb></lb>
libræ impoſitum in altera tantum libræ parte, grauitando impellit libram <lb></lb>
ad alium ſitum E H, ita vt maior pars libræ conſtituatur ex altera parte li­<lb></lb>neæ directionis prioris A D M, in qua etiam parte exiſtit centrum grauita­<lb></lb>
tis libræ ipſius, eſt enim circa D, quod centrum vi ponderis incumbentis in <lb></lb>
E, cogitur paulùm aſcendere, <expan abbr="atq;">atque</expan> contra ipſius naturalem inclinationem à <lb></lb>
mundi centro recedere, vt ſi in libra B C, appendatur onus in B, vt in pri­<lb></lb>
ma textus figura; B, deſcendet ad E, &amp; C, aſcendet ad H, &amp; centrum graui­<lb></lb>tatis D, paulùm aſcendet à centro mundi, &amp; linea A D M, quæ libram bi­<lb></lb>
fariam ſecabat modo tranſlato perpendiculo in A D G, non amplius cam <lb></lb>
bifariam ſecabit; ſed libræ E H, maior pars erit vltra perpendiculum A D­<lb></lb>
M, quæ maior pars eſt D D H.</s>
        </p>
        <p type="main">
          <s id="s.002690">Si igitur nunc onus amoueatur libræ E H, centrum grauitatis, quod eſt <pb pagenum="157" xlink:href="009/01/157.jpg"></pb>ad D, remanet vltra priorem Directionis lineam; &amp; quia pondus non am­<lb></lb>
plius illi æ que ponderat, grauitabit, &amp; quia libra cùm affixa ſit ad A, nequit <lb></lb>
deorſum recta tendere, circumferretur circa A, trahente ipſam grauitatis <lb></lb>
centro, cum nihil obſit, donec iterum perpendiculum A D G, priori ſitui <lb></lb>
A D M, congruat: hac enim ratione centrum grauitatis, quantum poteſt, <lb></lb>
iuxta naturam ſuam deſcendet, <expan abbr="libraq́">libraque</expan>; ad priſtinum æquilibrij B C, ſitum <lb></lb>
reſtituetur. </s>
          <s id="s.002691">Si autem deorſum fuerit ſpartum in ſecunda figura textus, im­<lb></lb>
poſito pondere contrarium accidit, quia maior pars libræ, &amp; in qua cen­<lb></lb>
trum grauitatis eſt, in tali motu deſcendit: altera autem pars minor, ac læ­<lb></lb>
uior ſurſum tollitur. </s>
          <s id="s.002692">&amp; quia graue natura ſua nequit aſcendere, propterea <lb></lb>
ablato pondere non reuertitur ad æquilibrium B C, cum centrum grauita­<lb></lb>
tis aſcendere ne queat, quod tunc oporteret.</s>
        </p>
        <p type="main">
          <s id="s.002693">Sit libra N G, in ſecunda figura, cuius perpendiculum, <expan abbr="ſimulq́">ſimulque</expan>; directio­<lb></lb>nis linea ſit K L M, quæ libram in prima poſitione diuidit bifariam; impoſi­<lb></lb>
to autem onere in N. N, trahetur ad O, &amp; G, ad R, &amp; K L, vbi K P. quare <lb></lb>
maior eſt O L, in quo <expan abbr="centrũ">centrum</expan> grauitatis, &amp; propterea grauior quàm ſit L R: <lb></lb>
ſuperat enim O L, ipſam L R, exceſſu duplæ P L, quod facilè apparet ſi po­<lb></lb>
natur tota O R, 10. &amp; dimidia O P, &amp; O R, 5. &amp; P L, ponatur 2. erit enim <lb></lb>
tunc O L, 7. &amp; L R, 3. quæ hanc ſuperat 4. duplo ſcilicet ipſius P L, 2. qua­<lb></lb>
re neſcio cur Ariſt. dicat, ipſam O L, ſuperare ipſam L R, ſolùm quantitate <lb></lb>
P L. </s>
          <s id="s.002694">Quapropter etiam ſi onus auferatur, neceſſe eſt ibi libram manere, <lb></lb>
quia maior, &amp; grauior ipſius pars deorſum eſt, nec poteſt natura ſua læui­<lb></lb>
tare, vel aſcendere, vt oporteret, ſi ad priſtinum ſitum N G, reſtitui debe­<lb></lb>
ret. </s>
          <s id="s.002695">remanebit igitur in O R.</s>
        </p>
        <p type="main">
          <s id="s.002696">Ex his, quæſtionis ſolutionem, textus explicationem, ac reſtitutio­<lb></lb>
nem habeto.</s>
        </p>
        <p type="main">
          <s id="s.002697">Aduertendum quoad ſecundam libram, ne ſimul cum 10. Baptiſta Bene­<lb></lb>
dicto in libro ſpeculationum immeritò Ariſt. erroris arguamus: ipſe enim, <lb></lb>
quia libram hanc non agnouit, auſus eſt affirmare, Ariſtot. hoc loco falſum <lb></lb>prorſus dixiſſe, cum dixit libram ſparto infimè collocato, non redire ad <lb></lb>
priſtinam poſitionem.</s>
        </p>
        <p type="head">
          <s id="s.002698"><emph type="italics"></emph>QVÆSTIO TERTIA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002699"><emph type="italics"></emph>De Vecte.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002700"><arrow.to.target n="marg234"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002701"><margin.target id="marg234"></margin.target>244</s>
        </p>
        <p type="main">
          <s id="s.002702">Cvm textus tam græci, quàm latini mendis ſcateant, <expan abbr="neq;">neque</expan> hi textus <lb></lb>
maioris ſint momenti, eos per paraphraſim explicabo, in qua ta­<lb></lb>
men totus textus continebitur, <expan abbr="atq;">atque</expan> emendabitur. </s>
          <s id="s.002703">Cur exiguæ <lb></lb>
vires (quemadmodum à principio dictum eſt) adhibito vecte, ma­<lb></lb>
iora mouent pondera, quam <expan abbr="abſq;">abſque</expan> vecte? </s>
          <s id="s.002704">contrarium enim videtur debere <lb></lb>
fieri, nam mouenti additur grauitas vectis, &amp; ideò pondus augetur, ergò <lb></lb>
difficilius ipſum cum vecte, quàm ſine eo mouere deberet.</s>
        </p>
        <p type="main">
          <s id="s.002705">Vectis porrò eſt inſtrumentum oblongum, quo ad ſubleuandum graue <lb></lb>
quodpiam vtuntur opifices, quod innititur cuidam fulcimento, quod græcè <lb></lb>hypomoclion dicitur: hypomoclion autem oneri leuando, quantum fieri <pb pagenum="158" xlink:href="009/01/158.jpg"></pb>poteſt proximum eſſe debet, vt vectis pars longior ſit ad partes potentiæ <lb></lb>
mouentis. </s>
          <s id="s.002706">vt plurimum verò fulcimentum eſt inter pondus, &amp; potentiam: <lb></lb>
aliquando etiam eſt ex altero vectis extremo, ita vt onus ſit inter fulturam, <lb></lb>
&amp; potentiam; aliquando potentia eſt inter vtrunque, vnde tres vectis ſpe­<lb></lb>
cies exiſtunt. </s>
          <s id="s.002707">vt in ſubiectis figuris apparet. </s>
          <s id="s.002708">In prima, vectis eſt A B, fultu­<lb></lb>
<figure id="id.009.01.158.1.jpg" place="text" xlink:href="009/01/158/1.jpg"></figure><lb></lb>
ra E, onus C. potentia autem ſeu vis, <lb></lb>
ſeu aliud pondus <expan abbr="mouẽs">mouens</expan> ſit vbi D. quæ <lb></lb>
deorſum in D, præmens eleuabit ſur­<lb></lb>
ſum ex altera parte onus C. &amp; vectis <lb></lb>
circa fulturam E, tanquam centrum <lb></lb>
conuertetur. </s>
          <s id="s.002709">In altera figura pondus <lb></lb>
eſt inter fulturam, &amp; potentiam, ful­<lb></lb>
tura autem in altera extremitate, vt <lb></lb>
patet in figura, hic autem potentia <lb></lb>
non præmit deorſum in D: ſed ſurſum <lb></lb>
vectem eleuando pondus C, attollitur. <lb></lb>
</s>
          <s id="s.002710">In tertia tandem figura potentia, eſt <lb></lb>
inter vtrunque, eſt enim in D, ibique <lb></lb>
ſurſum vrget. </s>
          <s id="s.002711">verum tamen eſt hunc vectem artificibus eſſe inutilem, quip­<lb></lb>
pe qui nullo modo iuuet potentiam, imò verò pondus ipſum grauius reddit: <lb></lb>
<expan abbr="neq;">neque</expan> hoc genere in his Mechanicis indigemus.</s>
        </p>
        <p type="main">
          <s id="s.002712">Reſpondet igitur dubitationi, dicens rationem huius incrementi poten­<lb></lb>
tiæ motricis, quod fit aſſumpto vecte fortè inde oriri, quod vectis ſit quæ­<lb></lb>
dam libra, cuius alterum brachium ſit altero longius; in prima autem quæ­<lb></lb>
ſtione explicatum eſt, cur libra maior, maiorem vim habeat, eam ad cir­<lb></lb>
culum reducendo; vectis autem fit libra, hypomoclion enim eſt loco ſparti, <lb></lb>
tam enim ſpartum, quam hypomoclion veluti centra manent. </s>
          <s id="s.002713">quoniam ve­<lb></lb>
rò ab eodem pondere, cęlerius, ſiue maiori vi mouetur linea, quantò lon­<lb></lb>
gior à centro fuerit, vt dictum eſt de admiranda circuli natura; hinc fit, vt <lb></lb>
cum duæ ſint in vecte potentiæ, ſiue duo pondera, mouens, &amp; motum, illud <lb></lb>
facilius ac maiore vi moueat, ſiue vires ex vecte acquirat, quod longiorem <lb></lb>
vectis partem preſſerit. </s>
          <s id="s.002714">quemadmodum igitur pars vectis longior, quæ ſpe­<lb></lb>
ctabat ad mouentem potentiam, ſuperat minorem partem, in qua eſt mo­<lb></lb>
tum; ita etiam maius eſt pondus <expan abbr="motũ">motum</expan>, quàm mouens. </s>
          <s id="s.002715">ſemper autem quan­<lb></lb>
to ab hypomoclio magis diſtabit potentia, tantò facilius mouebit, cuius <lb></lb>
cauſa ſupra reddita eſt, quoniam nimirum, quæ plus à centro elongatur ma­<lb></lb>
iorem deſcribit circulum, qui magis ad lineam rectam accedit: quare ab <lb></lb>
eadem potentia adhibito vecte, tantò facilius pars vectis mouens dimoue­<lb></lb>
bitur, quantò magis à fulcimento diſtabit. </s>
          <s id="s.002716">Exempli gratia ſit in ſuperiori <lb></lb>
prima figura vectis A B, pondus C, mouens D, hypomoclion E, in qua præ­<lb></lb>
dicta poteris contemplari. </s>
          <s id="s.002717">vltima illa textus verba <emph type="italics"></emph>(Quod autem vbi D, mo­<lb></lb>
uens, vbi F, motum autem vbi C, pondus in G,)<emph.end type="italics"></emph.end> videntur ſuperuacanea, atque <lb></lb>
mendosè addita.</s>
        </p>
        <p type="main">
          <s id="s.002718">In hac quæſtione reſpexit Ariſt. ſolùm ad primam vectis ſpeciem. </s>
          <s id="s.002719">Illud <lb></lb>
demum, quod dixit eandem habere rationem potentiam ad pondus, quàm <lb></lb>partes vectis inuicem demonſtratum eſt poſtea acutiſſimè ab Archimede <pb pagenum="159" xlink:href="009/01/159.jpg"></pb>propoſitione 6. &amp; 7. de æqueponderantibus: &amp; noſtra <expan abbr="tẽpeſtate">tempeſtate</expan> alio quam­<lb></lb>
uis modo, &amp; vnica demonſtratione à Guido Vbaldo in ſuis Mechanicis pro­<lb></lb>
poſitione 1. de Vecte, quæ eſt huiuſmodi; Potentia ſuſtinens pondus vecti <lb></lb>
appenſum, eandem ad ipſum pondus proportionem habet, quam vectis di­<lb></lb>
ſtantia inter fulcimentum, ac ponderis ſuſpenſionem, ad diſtantiam, à fulci­<lb></lb>
mento ad potentiam interiectam. </s>
          <s id="s.002720">quod de omni vecte ab eo demonſtratur, <lb></lb>
cuius propoſitionis ſenſus eſt hic; in ſuperiori prima figura ſi pars vectis <lb></lb>
E B, fuerit, v.g. quadrupla partis A E; etiam pondus C, erit quadruplo ma­<lb></lb>
ius pondere, ſeu vi in D, quæ ip ſum C, ope vectis ſuſtinet. </s>
          <s id="s.002721">quod etiam trans­<lb></lb>
ferre debes ad ſecundam figuram.</s>
        </p>
        <p type="head">
          <s id="s.002722"><emph type="italics"></emph>QVÆSTIO QVARTA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002723"><emph type="italics"></emph>De Remo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002724"><arrow.to.target n="marg235"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002725"><margin.target id="marg235"></margin.target>245</s>
        </p>
        <p type="main">
          <s id="s.002726">EI, qui ſuperiora intellexerit ſatis clara videtur. </s>
          <s id="s.002727">Illud tamen non <lb></lb>
omittendum, ſcilicet dicendum potius Remum eſſe vectem ſecundi <lb></lb>
generis, quàm primi, quod fortè Ariſt. non animaduertit, nec Pic­<lb></lb>
colomineus, nam mare eſt hypomoclion, reſpectu enim nauis non <lb></lb>
mouetur, ſed manet, ſcalmus autem ſimul cum tota naui eſt pondus motum; <lb></lb>
verè enim nauis ipſa mouetur. </s>
          <s id="s.002728">mouens eſt ipſe remex. </s>
          <s id="s.002729">Reliqua in textu <lb></lb>
ſunt clara.</s>
        </p>
        <p type="head">
          <s id="s.002730"><emph type="italics"></emph>QVÆSTIO QVINTA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002731"><emph type="italics"></emph>De Temone Nauis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002732"><arrow.to.target n="marg236"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002733"><margin.target id="marg236"></margin.target>246</s>
        </p>
        <p type="main">
          <s id="s.002734">Qvemadmodum in præcedenti quæſtione Ariſt. vectem ſecundi ge­<lb></lb>
neris ad ſolutionem non adhibuit, vt par erat, &amp; propterea obſcu­<lb></lb>rior euaſit, ita etiam in præſenti, quęſtionem ad vectem primi ge­<lb></lb>
neris reducit, quæ ad alterum reducendà erat: <expan abbr="atq;">atque</expan> hinc obſcuri­<lb></lb>
tas, atque prolixitas ſolutionis manauit. </s>
          <s id="s.002735">Eſt enim propriè Temo, ſiue gu­<lb></lb>
bernaculum nauis, vectis ſecundi generis, vt mox explicabo, eſt enim temo <lb></lb>
<figure id="id.009.01.159.1.jpg" place="text" xlink:href="009/01/159/1.jpg"></figure><lb></lb>
inſtrumentum in extrema nanis par­<lb></lb>te, ſeu puppi affixum, vt in figura pre­<lb></lb>
ſenti vides tabellam, in qua B C D, <lb></lb>
cuius manubrium A B, intra nauim <lb></lb>
recipitur, quæ tabella, ſeu temo in <lb></lb>
duobus cardinibus, vbi C, &amp; D, cir­<lb></lb>
cumuertitur à Nauis gubernatore, <lb></lb>
manubrium vbi A, tractante; ex qua <lb></lb>
conuerſione nauigium, quò vult ipſe <lb></lb>
gubernator facilè dirigit, ipſumque <lb></lb>
nauigium huc illuc quamuis adeò magnum ipſe ſolus impellit, &amp; agitat. </s>
          <s id="s.002736">eſt <lb></lb>
enim temo vectis, cuius auxilio vires mirum in modum augentur, nam to­<lb></lb>
ta A B, eſt ipſa Vectis longitudo, cuius hypomoclion eſt mare, cui contra­


<pb pagenum="160" xlink:href="009/01/160.jpg"></pb>nititur tabella B E; onus autem eſt puppis, quod onus præſertim in cardini­<lb></lb>
bus C D, mouenti reſiſtit, &amp; quod præcipuè mouere gubernator intendit. <lb></lb>
</s>
          <s id="s.002737">cum igitur motum onus ſit intra vectis extrema, hypomoclion in extremo <lb></lb>
ad B E, vbi in motu temonis tabella mare vrget, quod minimè cedit, <expan abbr="ipſaq́">ipſaque</expan>; <lb></lb>
in hoc motu ferè maneat, &amp; fiat quaſi centrum, circa quod totus temo cir­<lb></lb>
cumducitur, patet temonem eſſe vectem ſecundæ ſpeciei, vt dicebam. </s>
          <s id="s.002738">quod <lb></lb>
etiam hinc patere poteſt, quia temo eſt veluti remus, cuius ſcalmus ſint car­<lb></lb>
dines C, D. ſicut ergo remus eſt vectis ſecundi generis, cuius pondus eſt <lb></lb>ſcalmus, &amp; mare hypomoclion; ita temo erit vectis eiuſdem generis, cuius <lb></lb>
pondus erit vbi cardines, fultura verò mare.</s>
        </p>
        <p type="main">
          <s id="s.002739">Quærit igitur Ariſt. vnde nam tantas vires paruus nauis temo guberna­<lb></lb>
tori ſuggerat, <expan abbr="reſpondetq́">reſpondetque</expan>; propterea id contingere, quod temo vectis na­<lb></lb>
turam obtineat, cuius inquit onus eſt mare, melius autem, vt dixi, dixiſſet <lb></lb>
onus eſſe nauim, mare autem hypomoclion, mouens autem eſt gubernator. <lb></lb>
</s>
          <s id="s.002740">Differunt autem remus, &amp; temo, quamuis <expan abbr="vterq;">vterque</expan> ſit vectis, quoniam remus <lb></lb>
ſecundum latitudinem nauis, ſeu ad latera nauis mari obnititur. </s>
          <s id="s.002741">temo au­<lb></lb>
tem in directum ferè nauigij conſtitutus mare ſcindit. </s>
          <s id="s.002742">hinc fit, vt remus ad <lb></lb>
nauem antrorſum rectà agitandam, gubernaculum verò ad eam in latera, <lb></lb>&amp; obliquè contorquendam idoneum ſit. </s>
          <s id="s.002743">quoniam enim mare eſt hypomo­<lb></lb>
clion, fit vt dum gubernator mouet anſam temonis in A, ſeu ad dextram, <lb></lb>
ſeu ad ſiniſtram ſecum ad eandem partem trahat nauigium, quod temoni <lb></lb>
eſt connexum; ad <expan abbr="cõtrariam">contrariam</expan> tamen partem trahit ei, ſecundum quam mare <lb></lb>
impingit. </s>
          <s id="s.002744"><expan abbr="atq;">atque</expan> hoc pacto remus antrorſum, temo verò obliquè nauim agit.</s>
        </p>
        <p type="main">
          <s id="s.002745">Poſthæc ſequuntur huiuſmodi verba <emph type="italics"></emph>(In extremo autem, &amp; non in medio <lb></lb>
iacet, quoniam <expan abbr="mouẽti">mouenti</expan> facillimum est ab extremo motum mouere: prima enim pars <lb></lb>celerrimè fertur, quoniam quemadmodum in ijs, quæ feruntur in fine deficit latio, <lb></lb>
ſic ipſius continui in fine imbeciliſſima eſt latio, imbeciliſſima autem ad <expan abbr="expellẽdum">expellendum</expan> <lb></lb>est facilis, propter hæc igitur in puppi gubernaculum ponitur)<emph.end type="italics"></emph.end> quorum ſenſus <lb></lb>
videtur difficilis, <expan abbr="neq;">neque</expan> græcus textus excuſandus eſt, benè enim tranſlata <lb></lb>
ſunt. </s>
          <s id="s.002746">Piccolominæus quidem plura quàm Ariſt. fatur, ſed non clariora. </s>
          <s id="s.002747">dif­<lb></lb>
ficultas eſt in verbis illis <emph type="italics"></emph>(Prima enim pars celerrimè fertur)<emph.end type="italics"></emph.end> &amp; in illis <emph type="italics"></emph>(Sic ip­<lb></lb>ſius continui in fine imbeciliſſima eſt latio)<emph.end type="italics"></emph.end> videtur velle dicere, quod quando <lb></lb>
continuum aliquod proiectum fertur per aera, pars ipſius anterior ea eſt, <lb></lb>
quæ præ cæteris partibus principaliter mouetur, &amp; ad cuius motum reliquæ <lb></lb>poſteriores tanquam ſubſequentes moueantur; quaſi dicat tota vis lationis <lb></lb>
eſt in anteriori parte: ſiue ipſi impetus maior ineſt: videmus enim proiecta, <lb></lb>
quorum vna pars eſt cæteris grauior, quia ei parti melius imprimitur mo­<lb></lb>
tus, eam etiam fieri anteriorem in latione, quamuis initio fuerit poſterior. <lb></lb>
</s>
          <s id="s.002748">ſic etiam quando graue fertur deorſum, dicimus ipſum ferri ſecundum cen­<lb></lb>
trum grauitatis ipſius, <expan abbr="ibiq́">ibique</expan>; maiorem vim grauitandi exiſtere, ſic in proie­<lb></lb>
ctis partem anteriorem dicere poſſumus eſſe, ſecundum quam totum conti­<lb></lb>
nuum fertur: <expan abbr="ibiq́">ibique</expan>; totum eſſe impetum lationis, &amp; propterea etiam maio­<lb></lb>
ri impetu, <expan abbr="atq;">atque</expan> celerrimè ferri: &amp; <expan abbr="conſequẽter">conſequenter</expan> partem poſteriorem, quam­<lb></lb>
uis priorem æqua velocitate conſequatur, non tamen tanto impetu, cum ip­<lb></lb>ſa ad alterius impetum moueatur, &amp; propterea latio ipſius eſt admodum <lb></lb>
imbecillis.</s>
        </p>
        <pb pagenum="161" xlink:href="009/01/161.jpg"></pb>
        <p type="main">
          <s id="s.002749">Si quis ſagittam per aerem latam à ſuo motu vellet deflectere, eam faci­<lb></lb>
lius in poſteriore parte à ſuo curſu deuiaret, quàm in anteriore. </s>
          <s id="s.002750">hunc con­<lb></lb>
cinui corporis motum continuo proiectorum motui aſſimilat: quemadmo­<lb></lb>
dum enim motus proiectorum in fine debilior lenteſcit: ſic totum conti­<lb></lb>
nuum in poſtrema parte ſegnius impellitur. </s>
          <s id="s.002751">Quia igitur nauis eſt <expan abbr="cõtinuum">continuum</expan>, <lb></lb>
quod vi remorum recta antrorſum fertur, &amp; propterea maiore vi prora, <lb></lb>
quàm puppis, facilius eſt à ſuo directo curſu nauem deflectere, eam in pup­<lb></lb>
pi, quàm in prora commouendo. </s>
          <s id="s.002752">hac igitur de cauſa, gubernaculum puppi <lb></lb>
affigitur. </s>
          <s id="s.002753">quæ quidem ratio, &amp; quantum valeat, &amp; an naui quadret, &amp; num <lb></lb>
benè ſit explicata, phyſicorum eſt iudicare.</s>
        </p>
        <p type="main">
          <s id="s.002754">Ego tamen aliam huius rationem video, quia nimirum ſi temo in priori <lb></lb>
parte eſſet, quando à rectitudine ipſius nauis ad dextram, aut ad ſiniſtram <lb></lb>eſſet inclinandus, tunc quia aqua in vnam tantum ipſius partem, ſeu faciem <lb></lb>
tota impingeret, in eam ſcilicet, quæ antrorſum reſpiceret, eam aqua re­<lb></lb>
trorſum ſimul cum tota naui auerteret, <expan abbr="ſicq́">ſicque</expan>; totam nauim inuerteret, ita <lb></lb>
vt prora, cui adhæreret temo extrema fieret. </s>
          <s id="s.002755">impetus igitur aquæ, &amp; naui­<lb></lb>
gij temonati, cogit temonem eſſe poſtremum non primum, nec medium. <lb></lb>
</s>
          <s id="s.002756"><expan abbr="atq;">atque</expan> hinc oritur neceſſitas <expan abbr="eũ">eum</expan> poſteriori parti affigendi. </s>
          <s id="s.002757">ſubdit poſtea aliam <lb></lb>
eiuſdem rationem, quia nimirum parua motione facta in puppi multo ma­<lb></lb>
ius interuallum cogitur mutare prora; nam idem angulus, quo eius lineæ <lb></lb>
ſunt longiores, eò maiorem ſubtenſam ſibi lineam reſpicit, quod facilè in <lb></lb>
<figure id="id.009.01.161.1.jpg" place="text" xlink:href="009/01/161/1.jpg"></figure><lb></lb>
adſcripta figura intueri licet; in qua duæ <lb></lb>
lineæ A B, A C, continent angulum A, cui <lb></lb>
angulo ſubtenduntur tres lineæ parallelæ <lb></lb>
F G, D E, B C, quarum B C, maxima eſt, <lb></lb>
quia ibi maiores, ſiue remotiores ſunt ab <lb></lb>
angulo A, duæ rectæ A B, A C, ipſum con­<lb></lb>
tinentes, quod Geometricè per 4. 6. pro­<lb></lb>
bari poteſt. </s>
          <s id="s.002758">ſic etiam facta motione, vel <lb></lb>
parua in puppi, tota nauis transfertur ad <lb></lb>
alium ſitum, ita vt prora multum aliò transferatur, quod non accideret, ſi <lb></lb>
eadem motio fieret ad medium nauigij. </s>
          <s id="s.002759">propterea igitur aptiſſimè puppi <lb></lb>
gubernaculum connectitur.</s>
        </p>
        <p type="main">
          <s id="s.002760"><arrow.to.target n="marg237"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002761"><margin.target id="marg237"></margin.target>247</s>
        </p>
        <p type="main">
          <s id="s.002762">Ex ijſdem etiam rationibus mathematicis patet, cur magis antrorſum <lb></lb>
procedit nauigium, quàm remi ipſius palmula retrorſum: eadem enim ma­<lb></lb>
gnitudo, ijſdem mota viribus in aere plus, quàm in aqua progreditur. <lb></lb>
</s>
          <s id="s.002763">Sit igitur A B, remus, G, verò ſcalmus. </s>
          <s id="s.002764">A, autem in nauigio ſit remi initium. <lb></lb>
</s>
          <s id="s.002765">B, verò in mari palmula. </s>
          <s id="s.002766">ſi igitur A, vbi D, transferatur, per totum ſpa­<lb></lb>
tium A D, non permeabit tantumdem ſpatij B, <expan abbr="vſq;">vſque</expan> ad E. </s>
          <s id="s.002767">B E, enim ponitur <lb></lb>
æqualis ipſi A D, ſed minus interuallum propter reſiſtentiam aquæ ex ſup­<lb></lb>
poſitione percurret, quale eſt B F, quod minus eſt quàm A D, quare etiam li­<lb></lb>
nea B G, abbreuiabitur, <expan abbr="eritq́">eritque</expan>; veluti F Y, quæ etiam erit minor ipſa D G, <lb></lb>quæ facta eſt D Y, propter duo ſimilia triangula D Y A, B Y F, ſimilia au­<lb></lb>
tem triangula ſunt ea, quorum anguli vnius ſunt æquales angulis alterius, <lb></lb>
quo poſito ſunt etiam latera vnius proportionalia lateribus alterius, vt pa­<lb></lb>
tet ex prima definitione 6. necnon ex quarta eiuſdem demonſtratione. </s>
          <s id="s.002768">hæc <pb pagenum="162" xlink:href="009/01/162.jpg"></pb><figure id="id.009.01.162.1.jpg" place="text" xlink:href="009/01/162/1.jpg"></figure><lb></lb>
quidem duo triangula ſunt ſimi­<lb></lb>
lia, &amp; rectè concluditur F Y, mi­<lb></lb>
nus eſſe quàm D Y, ſed tamen <lb></lb>
non videntur iſta propoſitum <lb></lb>
oſtendere, quod erat, plus nauim <lb></lb>
procedere, quàm palmulam re­<lb></lb>
trocedere. </s>
          <s id="s.002769">Fateor quidem tex­<lb></lb>
tum hunc eſſe obſcuriſſimum, <lb></lb>
<expan abbr="idq́">idque</expan>; propterea fortè quia eſt admodum corruptus, præſertim circa chara­<lb></lb>
cteres, qui corrigendi ſunt vti nos facimus. </s>
          <s id="s.002770">neſcio qua ratione Piccolomi­<lb></lb>
neus videatur ſibi locum hunc explicaſſe. </s>
          <s id="s.002771">Forſitan addenda ſunt nonnulla <lb></lb>
hoc pacto; cum initio remigationis ponamus remum in ſitu A B, in fine ve­<lb></lb>
rò primæ impulſionis in D F, ſcalmum verò circa medium remi in G, pri­<lb></lb>
mo; vltimo erit etiam circa medium D F, vbi H, quare ſcalmus tranſlatus <lb></lb>
eſt à G, ad H, <expan abbr="totaq́">totaque</expan>; G H, perficit, quam deberet Ariſtot. vt ſibi conſtaret <lb></lb>
probare eſſe maiorem ipſa B F, quam palmula obiuit, &amp; conſequenter pro­<lb></lb>
baſſet nauigium plus proceſſiſſe, quàm palmula receſſerit: quod propoſue­<lb></lb>
rat. </s>
          <s id="s.002772">Verum hoc non demonſtrat; <expan abbr="neq;">neque</expan> ex præmiſſis deduci poteſt. </s>
          <s id="s.002773">poſtea <lb></lb>
ſubdit <emph type="italics"></emph>(Stans autem erit medium vbi eſt G, in contrarium enim ipſi, qu<foreign lang="grc"></foreign>od in mari <lb></lb>eſt, extremo B, procedit, vbi extremum in nauigio eſt A, non procederet autem <lb></lb>vbi est D, niſi commoueretur nauigiŭm, &amp; eò transferretur vbi eſt remi principium)<emph.end type="italics"></emph.end><lb></lb>
vbi in textu mendosè legitur C, pro G.</s>
        </p>
        <p type="main">
          <s id="s.002774">Senſus porrò horum verborum eſt hic; ſi remus circa ſcalmum G, verte­<lb></lb>
retur, &amp; tamen nauis ab eo non propelleretur, ſed ſtaret, tunc medium na­<lb></lb>
uis maneret vbi G, per motum enim remi impellitur in contrarias partes <lb></lb>
ipſi palmulæ B, quæ eſt in mari, quia ſequitur motum alterius extremi A, <lb></lb>
manubrij ſcilicet remi, qui eſt in naui: quod autem nauigium à remo mo­<lb></lb>
neatur, ſignum eſt, quia manubrium A, non procederet vbi eſt D, niſi pari­<lb></lb>
ter cum remo nauigium illorſum conſequeretur. </s>
          <s id="s.002775">Hæc quidem Ariſt. circa <lb></lb>
motum nauigij imperfectè admodum niſi textus corruptionem cauſetur, di­<lb></lb>
xiſſe videatur. </s>
          <s id="s.002776">Quapropter operæpretium me facturum exiſtimo, ſi Petri <lb></lb>
Nonij acutiſſimi Mathematici, ſubtiliſſimas, <expan abbr="ſcituq́">ſcituque</expan>; digniſſimas in præſens <lb></lb>problema annotationes hoc loco deſcripſero, ex quibus perfectè, ac ma­<lb></lb>
thematicè toti huic quæſtioni fit ſatis, quæ ſic ſe habent.</s>
        </p>
        <p type="head">
          <s id="s.002777"><emph type="italics"></emph>In Problema Mechanicum Arist. de motu Nauigij <lb></lb>
ex remis, annotatio Petri Nonij.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002778">Cvm olim diſcipulis noſtris mechanicas Ariſt. quæſtiones interpre­<lb></lb>
taremur, nonnulla circa problema illud annotauimus, cur magis <lb></lb>
procedat nauigium, quam remi palmula in contrarium. </s>
          <s id="s.002779">Ariſtot. <lb></lb>
enim ratiocinatio obſcura eſt; quam nos tamen, vt aliquid lucis <lb></lb>
haberet, ad hunc modum explicauimus; &amp; propter materiæ ſimilitudinem <lb></lb>
hiſce noſtris libris de nauigandi ratione adiunximus. </s>
          <s id="s.002780">Supponit autem ipſe <lb></lb>
auctor remi palmulam retrocedere, quoties nauigium in anteriora progre­


<pb pagenum="163" xlink:href="009/01/163.jpg"></pb>ditur, <expan abbr="locumq́">locumque</expan>; ſcalmi, ſuper quo circulari motu remus vertitur, in medio <lb></lb>
ipſius remi poſitum eſſe, vt ſcilicet tantum diſtet à manubrio, quantum à <lb></lb>
palmula. </s>
          <s id="s.002781">Duæ <expan abbr="itaq;">itaque</expan> rectæ lineæ ponantur æquales A B, &amp; D E, quæ quidem <lb></lb>
in C, puncto medio ſe inuicem ſecent, &amp; connectantur A B, &amp; D E: remus <lb></lb>
autem in initio vnius remigationis poſitionem habeat rectam lineam A B, <lb></lb>
<expan abbr="ſitq́">ſitque</expan>; A, manubrium; B, palmula; C, verò ſcalmus. </s>
          <s id="s.002782">Cum igitur A, remi ca­<lb></lb>
put in fine ipſius remigationis eò tranſlatum fuerit D, non erit B, vbi E; ſi <lb></lb>
<figure id="id.009.01.163.1.jpg" place="text" xlink:href="009/01/163/1.jpg"></figure><lb></lb>
enim ibi fuerit; remus igitur poſitionem <lb></lb>
habebit rectam lineam D E; &amp; quoniam <lb></lb>
contrapoſiti anguli, qui ad C, æquales ſunt, <lb></lb>
&amp; duo latera A C, &amp; D C, trianguli A D C, <lb></lb>
duobus lateribus B C, &amp; C E, trianguli B­<lb></lb>
E C, æqualia etiam ſunt: reliqui igitur an­<lb></lb>
guli, <expan abbr="atq;">atque</expan> baſes ipſorum <expan abbr="triãgulorum">triangulorum</expan> æqua­<lb></lb>
les erunt per 4. propoſitionem primi libri <lb></lb>
Euclidis, &amp; propterea tantum ſpatium per­<lb></lb>
curret B, quantum A: ſcalmus verò C, im­<lb></lb>
motus omninò erit: &amp; nauigium idcircò, in <lb></lb>
quo ipſe ſcalmus, immotum etiam erit con­<lb></lb>
tra hypotheſim. </s>
          <s id="s.002783">ſupponitur enim in queſtio­<lb></lb>
ne, quod nauigium illa remigatione in anteriora moueatur, remi verò pal­<lb></lb>
mula retrocedat. </s>
          <s id="s.002784">Scalmus porrò quamquam circularis remi motus expers <lb></lb>
ſit; motu tamen nauigij commouetur. </s>
          <s id="s.002785">Remus igitur poſitionem habeat in <lb></lb>fine ipſius remigationis rectam lineam D Z, quæ quidem rectam A B, ſecet <lb></lb>
in T, inter B, &amp; C; rectam verò B E, in Z. </s>
          <s id="s.002786">Et quoniam duo coalterni anguli <lb></lb>
C A D, &amp; C B E, æquales <expan abbr="oſtẽſi">oſtenſi</expan> ſunt, &amp; angulus A T D, contrapoſito B T Z, <lb></lb>
æqualis eſt: duo igitur triangula A T D, &amp; B Z T, æquiangula erunt per 32. <lb></lb>
primi, &amp; communem ſententiam. </s>
          <s id="s.002787">Similia <expan abbr="itaq;">itaque</expan> erunt ipſa triangula, <expan abbr="late-raq́">late­<lb></lb>
raque</expan>; habebunt proportionalia per 4. 6. ſicut A T, ad B T, ita D A, ad B Z. <lb></lb>
</s>
          <s id="s.002788">Maior eſt autem A T, quàm B T: maior igitur D A, quàm B Z, quod etiam <lb></lb>
per <expan abbr="cõmunem">communem</expan> <expan abbr="ſentẽtiam">ſententiam</expan> neglecta <expan abbr="triangulorũ">triangulorum</expan> ſimilitudine concludi poteſt.</s>
        </p>
        <p type="main">
          <s id="s.002789">Maius <expan abbr="itaq;">itaque</expan> ſpatium decurrit manubrium, quàm remi palmula, <expan abbr="atq;">atque</expan> illuc <lb></lb>
tranſuehetur nauigium, quò remi capulus deportatus fuerit: nauigium igi­<lb></lb>
tur in diuerſa procedens, plus ſpatij, quàm remi palmula tranſmittet. </s>
          <s id="s.002790">Vti­<lb></lb>
mur aurem tralatione, <expan abbr="atq;">atque</expan> demonſtrationis figura Victoris Fauſti. </s>
          <s id="s.002791">Aduer­<lb></lb>
tendum eſt tamen, quod cum remus poſitionem habuerit D Z, remi palmu­<lb></lb>
la erit infra Z. </s>
          <s id="s.002792">Nam quoniam <expan abbr="triãguli">trianguli</expan> A D C, duo latera A C, &amp; D C, æqua­<lb></lb>lia poſita ſunt: duo igitur anguli, qui ad D, &amp; A, æquales erunt: angulus <lb></lb>
igitur A D T, angulo D A T, maior erit: &amp; idcircò latus A T, trianguli A­<lb></lb>
T D, latere D T, maius erit per 19. primi. </s>
          <s id="s.002793">Aæqualis porrò oſtenſus eſt an­<lb></lb>
guius B Z T, angulo A D T, præterea angulus D A T; angulo T B Z, æqua­<lb></lb>
lis: angulus igitur B Z T, angulo T B Z, maior erit, &amp; propterea latus B T, <lb></lb>
trianguli B T Z, latere T Z, maius erit: tota igitur recta linea A B, tota <lb></lb>
D Z, maior erit: &amp; idcircò cum remus poſitionem habuerit rectam lineam <lb></lb>
D Z palmula erit vltra Z. </s>
          <s id="s.002794">Eſto igitur in K, &amp; connectantur rectæ lineæ B D, <lb></lb>&amp; B K: ſpatium igitur decurſum ab ipſa palmula non erit B Z, ſed B K: quod <pb pagenum="164" xlink:href="009/01/164.jpg"></pb>quidem minus etiam oſtendemus eſſe ipſo D A. </s>
          <s id="s.002795">Nam quoniam duo latera <lb></lb>
B D, &amp; D K, trianguli B D K, duobus lateribus B D, &amp; D E, <expan abbr="triãguli">trianguli</expan> B E D, <lb></lb>æqualia ſunt, ſed minor eſt angulus B D K, angulo B D E: minor igitur erit <lb></lb>
baſis B K, baſe B E, per 24. primi, quod demonſtrandum erat</s>
        </p>
        <p type="main">
          <s id="s.002796">Præterea, quod Ariſt. ratiocinando ſumit tantum ſpatium conficere na­<lb></lb>
uigium, quantum remi manubrium, ambiguum eſt. </s>
          <s id="s.002797">Nam remi manubrium <lb></lb>
duabus fertur motionibus: vna propria, <expan abbr="circulariq́">circularique</expan>; ſuper ſcalmo: altera <lb></lb>
verò, qua vnà fertur cum ipſo nauigio. </s>
          <s id="s.002798">ſpatium igitur, quod omninò decur­<lb></lb>ſum eſt à remi manubrio, eo quod à nauigio confectum eſt, maius erit. </s>
          <s id="s.002799">At <lb></lb>
ſi paria ſpatia decurſa eſſe intelligat à remi manubrio motu proprio, &amp; à <lb></lb>
nauigio, <expan abbr="neq;">neque</expan> hoc difficultate caret. </s>
          <s id="s.002800">Nam nauigium interdum maius ſpa­<lb></lb>
tium percurret, interdum minus, iuxta remigum vires, &amp; prout mari remi <lb></lb>
palmula immerſa fuerit: remi verò manubrium tametſi ab exiguis viribus <lb></lb>
moueatur haud minorem tamen ambitum deſcribet, quàm ſi à multo ma­<lb></lb>
iore virtute moueretur. </s>
          <s id="s.002801">Quapropter, vt huiuſmodi Ariſt. ſententiam exa­<lb></lb>minaremus, Theoremata, quæ ſequuntur, demonſtrauimus.</s>
        </p>
        <p type="head">
          <s id="s.002802"><emph type="italics"></emph>PROPOSITIO PRIMA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002803">Si Remiges nauigium mouere poſſunt, maius ſemper ſpa­<lb></lb>
tium remi manubrium percurrit, quàm nauigium.</s>
        </p>
        <p type="main">
          <s id="s.002804">Sit enim remus A C, manubrium A, ſcalmus B, qui propter nauigij <lb></lb>motum ſpatium percurrat à B, in D, in quo loco ipſe remus A C, ſi­<lb></lb>
<figure id="id.009.01.164.1.jpg" place="text" xlink:href="009/01/164/1.jpg"></figure><lb></lb>
tum rectitudinis habeat E F. </s>
          <s id="s.002805">Spatium <lb></lb>itaque, quod A, conficit, curua linea <lb></lb>
ſit A E, cui recta linea reſpondeat A Z, in re­<lb></lb>ctam E F, perpendicularis. </s>
          <s id="s.002806">Nauigium verò <lb></lb>idem ſpatium conficiet, quod ſcalmus B: aio <lb></lb>
igitur ipſam A Z, rectam lineam, recta B D, <lb></lb>
maiorem eſſe. </s>
          <s id="s.002807">ſecet enim recta A C, rectam <lb></lb>
E F, in G: æquiangula ſunt igitur bina trian­<lb></lb>
gula A G Z, &amp; B G D, quapropter ſicut A G, <lb></lb>
ad B G, ſie A Z, ad B D, per. </s>
          <s id="s.002808">4. 6. libri Eucli­<lb></lb>dis: maior eſt autem A G, ipſa B G, &amp; maior <lb></lb>
igitur erit A Z, quam B D. &amp; proinde maius <lb></lb>
ſpatium remi manubrium percurrit, quam <lb></lb>
nauigium, quod demonſtrandum erat.</s>
        </p>
        <p type="main">
          <s id="s.002809">Quod ſi à puncto B, rectam lineam vtrinque <lb></lb>
ducamus H K, ad remi menſuram, rectos facientem angulos cum B D, <expan abbr="re-ctamq́">re­<lb></lb>
ctamque</expan>; A Z, ſecantem in I, manifeſtè intelligemus ipſam rectam A Z, con­<lb></lb>
ſtare ex A I, &amp; I Z, quarum prior reſpondet curuæ A H, quæ motu proprio <lb></lb>
manubrij deſcripta eſt; poſterior verò æqualis eſt rectæ B D, quæ motu na­<lb></lb>
uigij decurſa eſt.</s>
        </p>
        <pb pagenum="165" xlink:href="009/01/165.jpg"></pb>
        <p type="head">
          <s id="s.002810"><emph type="italics"></emph>PROPOSITIO SECVNDA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002811">Si remi manubrium motu proprio, &amp; nauigium, æqualia <lb></lb>
ſpatia pertranſierint, fieri non poterit, vt palmula mo­<lb></lb>
ueatur: ſed veluti centrum immota manebit.</s>
        </p>
        <p type="main">
          <s id="s.002812">Esto iterum remus A C, manubrium A, ſcalmus B: tantum autem ſpa­<lb></lb>
tium conficiat nauigium; quantum motu proprio A. Dico, quod C, <lb></lb>
remi palmula immota manebit. </s>
          <s id="s.002813">Nam ſi a loco ſuo dimota fuerit: <lb></lb>
ſpatium igitur permeet C D, ad poſteriora: quo quidem decurſo, <lb></lb>remus A C, poſitionem rectitudinis habeat F D, ſcalmus <expan abbr="itaq;">itaque</expan> B, tranſlatus <lb></lb>
erit in G. </s>
          <s id="s.002814">Excitetur autem à puncto B, in <expan abbr="vtramq;">vtramque</expan> partem linea E B R, ad <lb></lb>
<figure id="id.009.01.165.1.jpg" place="text" xlink:href="009/01/165/1.jpg"></figure><lb></lb>
rectos angulos ſuper B G, &amp; à <expan abbr="pũcto">puncto</expan> A, recta A H, <lb></lb>
ſuper D F: itemque à puncto E, recta C E, ſuper <lb></lb>
E R; ipſarum verò rectarum linearum E R, &amp; <lb></lb>
A H, ſectio ſit in K, ſed C F., &amp; D F, ſit in Z, &amp; quo­<lb></lb>
niam A K, id ſpatium eſt, quod motu proprio re­<lb></lb>
mi manubrium permeauit, curuilineo enim re­<lb></lb>
ſpondeat A R, recta autem B G, id ſpatium eſt, <lb></lb>
quod nauigium confecit: ipſæ igitur rectæ lineæ <lb></lb>
H K, &amp; B G, æquales erunt. </s>
          <s id="s.002815">Atqui in duobus æqui­<lb></lb>
angulis triangulis E B C, &amp; B A K, vel per 26. <lb></lb>
propoſitionem primi Euclidis, vel 4. 6. æquales <lb></lb>
eſſe concludes A K, &amp; E C, rectas lineas: quapro­<lb></lb>
pter æqualis erit E C, rectæ B G, per communem <lb></lb>
ſententiam: eidem autem B G, æqualis eſt E Z, <lb></lb>
in parallelogrammo, per 34. propoſitionem ip­<lb></lb>
ſius primi libri: æqualis igitur erit recta E Z, re­<lb></lb>
ctæ E C, pars toti, quod eſt impoſſibile. </s>
          <s id="s.002816">Et pro­<lb></lb>
pterea immota manebit palmula C, quod erat à <lb></lb>
nobis oſtendendum.</s>
        </p>
        <p type="head">
          <s id="s.002817"><emph type="italics"></emph>PROPOSITIO TERTIA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002818">Si remi manubrium motu proprio duplum confecerit ſpa­<lb></lb>
tium, quàm nauigium, tantum prouehetur ea remiga­<lb></lb>
tione nauigium, quantum palmula retroceſſerit.</s>
        </p>
        <p type="main">
          <s id="s.002819">Remus enim incipiente motu poſitionem habeat A C, deſinente <lb></lb>
verò rectitudinis ſitum F G. ſcalmus igitur B, propter nauigij <lb></lb>motum, ſpatium conficiet B D. </s>
          <s id="s.002820">Excitetur à puncto B, in <expan abbr="vtramq;">vtramque</expan> <lb></lb>
partem perpendicularis E Z, in quam veniant a punctis A, &amp; C, <lb></lb>ad rectos angulos rectæ lineæ A E, &amp; C Z: ſpatium autem A E, à manubrio <pb pagenum="166" xlink:href="009/01/166.jpg"></pb><figure id="id.009.01.166.1.jpg" place="text" xlink:href="009/01/166/1.jpg"></figure><lb></lb>
decurſum motu proprio ſpatij B D, duplum <lb></lb>
ſit: recta verò linea C H, curuæ reſpondeat <lb></lb>
C G, quæ à remi palmula deſcripta eſt. </s>
          <s id="s.002821">Di­<lb></lb>
co ipſas rectas lineas B D, &amp; C H, æquales <lb></lb>
eſſe. </s>
          <s id="s.002822">Nam in duobus triangulis B A E, &amp; <lb></lb>
C B Z, duæ rectæ lineæ A E, &amp; C Z, æqua­<lb></lb>
les ſunt. </s>
          <s id="s.002823">In parallelogrammo autem B H, <lb></lb>
duæ B D, &amp; H Z, æquales, atqui recta A E, <lb></lb>
dupla eſt rectæ B D, per hypotheſim; dupla <lb></lb>
eſt igitur, &amp; C Z, rectæ H Z, quapropter <lb></lb>
C H, &amp; H Z, æquales erunt, Duæ igitur <lb></lb>
C H, &amp; B D, æquales per communem ſen­<lb></lb>
tentiam.</s>
        </p>
        <p type="main">
          <s id="s.002824">Et quia nauigium tantum ſpatium de­<lb></lb>
currit ſemper, quantum ſcalmus: ſi igitur <lb></lb>
remi manubrium motu proprio duplum <lb></lb>
confecerit ſpatium, quàm nauigium, tan­<lb></lb>
tum prouehetur nauigium, quantum pal­<lb></lb>
mula retroceſſerit, quod demonſtrandum <lb></lb>
erat.</s>
        </p>
        <p type="head">
          <s id="s.002825"><emph type="italics"></emph>PROPOSITIO QVARTA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002826">Si nauigium minus ſpatium decurrat, quàm remi manu­<lb></lb>
brium, ſed ſupra dimidium, magis prouehetur, quàm pal­<lb></lb>
mula retrocedat; ſi verò citra dimidium, minus.</s>
        </p>
        <p type="main">
          <s id="s.002827">In deſcripta enim figura ponatur B D, minor quam A E, ſed eius dimi­<lb></lb>
dio maior. </s>
          <s id="s.002828">Dico, quod ipſa B D, maior eſt quàm C H. </s>
          <s id="s.002829">Nam B D, &amp; <lb></lb>
H Z, æquales ſunt: Ad hæc A E, &amp; C Z, æquales ſunt rectæ lineæ; ma­<lb></lb>
ior igitur erit H Z, dimidio ipſius A E: quapropter reliqua C H, mi­<lb></lb>
nor dimidio erit eiuſdem A E, &amp; minor igitur erit C H, quàm B D. </s>
          <s id="s.002830">Spa­<lb></lb>
tium autem B D, id eſt, quod nauigium conficit, ſpatium verò C H, remi <lb></lb>
palmula in contrarium decurrit; idcircò prior pars Theorematis vera eſt. <lb></lb>
</s>
          <s id="s.002831">Poſterior autem ſimiliter oſtendetur. </s>
          <s id="s.002832">ſi enim B D, minor eſt dimidio ipſius <lb></lb>
A E: minor igitur erit, &amp; H Z, dimidio eiuſdem A E; &amp; quoniam A E, &amp; <lb></lb>
C Z, æquales ſunt: reliqua igitur C H, dimidio eiuſdem A E, maior erit: &amp; <lb></lb>
proinde minor erit B D, quàm C H. </s>
          <s id="s.002833">Nauigium igitur minus ſpatium de­<lb></lb>
curret in anteriora, quam remi palmula in contrarium, quod demonſtran­<lb></lb>
dum ſuſcepimus.</s>
        </p>
        <p type="head">
          <s id="s.002834"><emph type="italics"></emph>Corollarium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002835">Ex hac, &amp; præcedenti infertur, quod ſi remi manubrium motu proprio <lb></lb>
maius ſpatium decurrat, quàm nauigium, ſiue id ſit duplum, ſiue mi­


<pb pagenum="167" xlink:href="009/01/167.jpg"></pb>nus duplo, ſiue maius duplo, ſpatium, quod nauigium interim decurrit ad <lb></lb>
anteriora, &amp; quod palmula remi in contrarium ſimul iuncta, ei quod ipſum <lb></lb>
remi manubrium motu proprio conficit, æqualia erunt. </s>
          <s id="s.002836">ſemper enim B D, <lb></lb>
æqualis eſt H Z: tota verò C Z, quæ æqualis eſt A E, ex ſuis partibus C H, <lb></lb>
&amp; H Z, conſtabit.</s>
        </p>
        <p type="head">
          <s id="s.002837"><emph type="italics"></emph>Propoſitionis conuerſio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002838">Si nauigium longius progrediatur, quàm remi palmula re­<lb></lb>
trocedat, ſpatium conficiet pluſquam dimidium eius, <lb></lb>
quod motu proprio remi manubrium decurrit: <lb></lb>
ſi minus, citra dimidium.</s>
        </p>
        <p type="head">
          <s id="s.002839"><emph type="italics"></emph>Huius demonſtratio ex ſupradictis facilè colligi poterit.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002840"><emph type="italics"></emph>PROPOSITIO QVINTA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002841">Si celerius feratur nauigium, quàm remi manubrium, mo­<lb></lb>
uebitur palmula in vlteriora, <expan abbr="nilq́">nilque</expan>; vnquam retroce­<lb></lb>
det, <expan abbr="idq́">idque</expan>; ſpatium decurret, quo nauigij motus <lb></lb>
motum manubrij ſuperat.</s>
        </p>
        <p type="main">
          <s id="s.002842">Habeat enim remus incipiente motu poſitionem A C: deſinente <lb></lb>
<figure id="id.009.01.167.1.jpg" place="text" xlink:href="009/01/167/1.jpg"></figure><lb></lb>
verò <expan abbr="ſitũ">ſitum</expan> rectitudinis F G. ſcal­<lb></lb>
mus igitur B, propter nauigij <lb></lb>
motum tranſlatus, erit in D, ſit <lb></lb>
<expan abbr="itaq;">itaque</expan> ſpatium B D, maius quàm A H, à re­<lb></lb>
mi manubrio motu proprio decurſum: ſic <lb></lb>
enim celerius dicetur ferri <expan abbr="nauigiũ">nauigium</expan>, quàm <lb></lb>
manubrium. </s>
          <s id="s.002843">Dico, quòd palmula C, in <lb></lb>
vlteriora mouebitur. </s>
          <s id="s.002844">Nam cum ſcalmus <lb></lb>
B, prouectus fuerit in D: tranſlata erit ip­<lb></lb>ſa palmula C, vbi G, in rectitudinis ſitu, <lb></lb>
<expan abbr="ſpatiumq́">ſpatiumque</expan>; conficiet C G, curuilineum, cui <lb></lb>
reſpondet C K: mouebitur igitur palmula <lb></lb>
in vlteriora. </s>
          <s id="s.002845">Nihil autem vnquam retro­<lb></lb>
cedere, oſtendetur in hunc modum. </s>
          <s id="s.002846">eadem <lb></lb>
enim celeritate mouentur A, in H, &amp; C, <lb></lb>
verſus I, circa ſcalmum. </s>
          <s id="s.002847">Atqui per hypo­<lb></lb>
theſim celerius fertur nauigium, quam A. <lb></lb>
in H, celerius igitur ipſum nauigium fer­<lb></lb>tur, quàm C, verſus I. ſed mouetur idem <pb pagenum="168" xlink:href="009/01/168.jpg"></pb>C. ipſa nauigij celeritate verſus K; celerius igitur ferretur C, ad K, quam <lb></lb>
ad I, quapropter nihil vnquam retrocedet ipſum C, imò verò in vlteriora <lb></lb>
progredietur, <expan abbr="ſpatiumq́">ſpatiumque</expan>; decurret C K, quod quidem relinquitur detracto <lb></lb>
I C, ex I K. ſi enim remi palmula tota ipſa nauigij celeritate moueretur, vl­<lb></lb>
tra K, progrederetur, cum B, perueniret ad D: ſed retrahitur interim, pro­<lb></lb>
pter eum motum, qui fit circa B. </s>
          <s id="s.002848">Sic igitur palmulæ celeritate, quæ à mo­<lb></lb>
tu nauigij prouenit retardata, decurſum ſpatium erit C K. </s>
          <s id="s.002849">Videtur autem <lb></lb>
ſolo remorum impulſu hoc fieri non poſſe, ſed alia inſuper virtute impel­<lb></lb>
lente opus eſſe, vt venti, vel aquæ.</s>
        </p>
        <p type="main">
          <s id="s.002850">Ex his Theorematis liquet, inquit Nonius, quàm incerta interroget Ari­<lb></lb>
ſtoteles, &amp; quàm inſcitè reſpondeat. </s>
          <s id="s.002851">Nam non continuò ſi nauigium in an­<lb></lb>teriora mouetur, remi palmula retrocedet; neque etiam ſi retrocedat, mi­<lb></lb>
nus ſpatìum tranſmittit in contrarium, quàm nauigium progrediatur. </s>
          <s id="s.002852">De­<lb></lb>
monſtrant hoc ſecunda, &amp; tertia propoſitio. </s>
          <s id="s.002853">Remi verò manubrium motu <lb></lb>
proprio, qui circa ſcalmum fit, &amp; vnà cum nauigij motu maius ſpatium con­<lb></lb>
ficit quàm nauigium. </s>
          <s id="s.002854">ſolo autem proprio motu, ſi contingat tantum ſpa­<lb></lb>
tium conficere, quantum nauigium, fieri non poterit, vt palmula mouea­<lb></lb>
tur. </s>
          <s id="s.002855">fruſtra igitur conatur in vniuerſum demonſtrare remi manubrium ma­<lb></lb>
ius ſpatium decurrere, quàm palmulam in contrarium. </s>
          <s id="s.002856">Præterea quando <lb></lb>
nauigium <expan abbr="lõgius">longius</expan> progreditur, quàm remi palmula regrediatur, minus ſpa­<lb></lb>
tium decurrit, quam manubrium: igitur hon æquale. </s>
          <s id="s.002857">Et proinde conſtat <lb></lb>
neque veritatem in propoſito, neque demonſtrationem in ijs, quæ conge­<lb></lb>
rit, reperiri.</s>
        </p>
        <p type="main">
          <s id="s.002858">Huiuſque Petrus Nonius:</s>
        </p>
        <p type="main">
          <s id="s.002859">Reliqua huius textus vtinam quemadmodum ſunt clara, ita etiam vera <lb></lb>
eſſent: ſed quia quæ modo dixit de remo, eadem temoni applicat propte­<lb></lb>
rea ijſdem etiam obnoxia ſunt difficultatibus.</s>
        </p>
        <p type="head">
          <s id="s.002860"><emph type="italics"></emph>QVÆSTIO SEXTA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002861"><emph type="italics"></emph>De Antenna.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002862"><arrow.to.target n="marg238"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002863"><margin.target id="marg238"></margin.target>248</s>
        </p>
        <p type="main">
          <s id="s.002864">Qværit cur quanto Antenna ſublimior fuerit, ijſdem velis, &amp; vento <lb></lb>
eodem celerius ferantur nauigia. </s>
          <s id="s.002865">Reſpondet inde id prouenire, <lb></lb>
quia malus, ſiue arbor nauis in huiuſmodi ventorum impulſu ve­<lb></lb>
ctis euadit, cuius auxilio idem ventus, qui mouens eſt, maiorem <lb></lb>
vim acquirit, quanto longior fuerit pars vectis, quæ inter hypomoclion, &amp; <lb></lb>
vim mouentem intercipitur: quando autem altior fuerit antenna, tunc ea <lb></lb>
vectis pars longior euadit, &amp; propterea accidit, vt vires ventorum augean­<lb></lb>
tur. </s>
          <s id="s.002866">ſed iſta melius in figura inſpiciamus. </s>
          <s id="s.002867">ſit nauis A B, cuius arbor C D E, <lb></lb>
antenna F C G, velum F G H, vectis eſt arbor, cuius fultura eſt in E, extre­<lb></lb>
mo mali in fundo nauis, onus autem in D, vbi malus exit è carina. </s>
          <s id="s.002868">mouens <lb></lb>
potentia eſt ventus, qui mouet in antenna F C G. quanto igitur ſublimior <lb></lb>
eſt antenna, tanto longior euadit vectis E C, <expan abbr="tantoq́">tantoque</expan>; maiores fiunt venti <lb></lb>
vires. </s>
          <s id="s.002869">dixi autem onus eſſe in D, quia ſi nauis vento obſiſteret, ipſa inuerte­<lb></lb>retur hac ratione, vt puppis A, eleuata, prora B, demergeretur, manente <pb pagenum="169" xlink:href="009/01/169.jpg"></pb><figure id="id.009.01.169.1.jpg" place="text" xlink:href="009/01/169/1.jpg"></figure><lb></lb>
veluti centro parte E. quia ve­<lb></lb>
rò ob maris liquiditatem na­<lb></lb>uis minimè obſiſtit, ſed facilè <lb></lb>
cedens à ventis vrgetur, hinc <lb></lb>
fit, vt meritò dixerim pondus <lb></lb>
nauis eſſe ad D, fulcimentum <lb></lb>
verò ad E.</s>
        </p>
        <p type="main">
          <s id="s.002870">Quæſtio ſeptima, &amp; ſatis per <lb></lb>
ſe clara eſt; <expan abbr="neq;">neque</expan> Mathemati­<lb></lb>
ci eſt eam exponere.</s>
        </p>
        <p type="head">
          <s id="s.002871"><emph type="italics"></emph>QVÆSTIO OCTAVA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002872"><emph type="italics"></emph>De Rota.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002873">Cur ex figurarum genere quæcunque rotundæ ſunt, &amp; cir­<lb></lb>
culares facilius mouentur?</s>
        </p>
        <p type="main">
          <s id="s.002874"><arrow.to.target n="marg239"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002875"><margin.target id="marg239"></margin.target>249</s>
        </p>
        <p type="main">
          <s id="s.002876">Tribus autem modis circulum rotari contingit; aut enim ſecun­<lb></lb>
dum apſidem, ſiue curuaturam centro ſimul moto, quemadmo­<lb></lb>
dum plauſtrorum rotæ vertuntur: aut circa manentem axem, <lb></lb>
tanquam centrum veluti rotulæ illæ, ex quibus trochlea compo­<lb></lb>
nitur; vel quibus ad puteos vtimur, quæ quidem rectæ ad horizontem ſo­<lb></lb>
lent conſtitui. </s>
          <s id="s.002877">aut quem ad modum rota figuli, quæ pariter circa <expan abbr="manẽs">manens</expan> cen­<lb></lb>
trum gyratur, ſed quaſi proſtrata horizonti æquidiſtans collocata eſt. </s>
          <s id="s.002878">Quæ <lb></lb>
igitur primo modo mouentur, fortè facilius quam figuræ rectilineæ, vt ſunt <lb></lb>
triangulares, quadratæ, pentagonæ, &amp;c. </s>
          <s id="s.002879">mouentur, quia circulares figuræ <lb></lb>
parua ſui parte, &amp; quaſi in puncto planum, ſeu pauimentum contingunt, vn­<lb></lb>
de fit, vt <expan abbr="neq;">neque</expan> offenſent, <expan abbr="neq;">neque</expan> impingant; cuius cauſa eſt, quia à terra ſemo­<lb></lb>
tus eſt angulus, ideſt tali angulo planum contingunt, vt ab eo ſtatim rotæ <lb></lb>
curuatura à terra eleuari incipiat, &amp; propterea parum terræ hæreat: in fi­<lb></lb>
guris verò rectilineis, in quadrata. </s>
          <s id="s.002880">v. g. ſecus accidit, quia ab angulo ad an­<lb></lb>
gulum linea recta tenditur, vnde in ipſius volutatione poſt contactum vnius <lb></lb>
anguli tota recta linea ſequens, plano adaptabitur, &amp; non ſemouebitur ſta­<lb></lb>
tim in altum, &amp; ideò multum offenſabit, &amp; impinget, <expan abbr="tardeq́">tardeque</expan>; idcircò mo­<lb></lb>
uebitur. </s>
          <s id="s.002881">Præterea circulares etiam, ſi cui obuiam fiunt corpori, illud ſimi­<lb></lb>liter ſecundum puſillum tangunt: rectilineæ verò figuræ, rectitudine ſua <lb></lb>
plani multum contingerent. </s>
          <s id="s.002882">Ad hæc motor mouens huiuſmodi rotas, eas <lb></lb>
mouet, quò nutant: nam quando rota erecta eſt ſuper pauimentum, dia­<lb></lb>meter ipſius, quæ à contactu pauimenti ad angulos rectos, ad ſupremum <pb pagenum="170" xlink:href="009/01/170.jpg"></pb>rotæ perducitur totum rotæ pondus in duas æquas partes diuidit, ita vt ta­<lb></lb>
le pondus in æquilibrio conſtituatur, cum ex vna parte tantum ſit, quantum <lb></lb>
ex altera; ex quo fit, vt vel exigua vis ipſam impellere valeat: quando enim <lb></lb>
duo æqualia pondera ſunt in æquilibrio, quelibet vis poteſt ea ab æquilibrio <lb></lb>
dimouere. </s>
          <s id="s.002883">quando poſtea rota eſt in motu, vel cum primum ei motus fuerit <lb></lb>
à motore inditus, ſemper nutat ad partes illas, ad quas primum fuit incita­<lb></lb>
ta per impreſſam motionem, quapropter nullo negotio ad eaſdem partes, <lb></lb>
ſeu antrorſum mouetur; quò enim <expan abbr="vnumquodq;">vnumquodque</expan> vergit, illuc facillimè fer­<lb></lb>
tur: quemadmodum è contrario difficillimum eſt in contrariam nutus ſui <lb></lb>
partem vnumquodque pellere. </s>
          <s id="s.002884">Huc etiam pertinet, quod nonnulli dicunt, <lb></lb>
circuli nimirum periphæriam perenni verſari motu, <expan abbr="atq;">atque</expan> hinc facilius mo­<lb></lb>
ueri. </s>
          <s id="s.002885">ſicuti etiam dicunt, quod manentia propterea manent, quia contrani­<lb></lb>
tuntur, &amp; obſiſtunt mouenti: quod fortè dicebant propter maximam circu­<lb></lb>li ad motum aptitudinem. </s>
          <s id="s.002886">&amp; quia ſicut diameter ad diametrum, ita maio­<lb></lb>
ris circuli periphæria ad minoris periphæriam (vt poſtea oſtendam) &amp; quia <lb></lb>
quo <expan abbr="lõgior">longior</expan> diameter eſt, eò facilius, vt initio probaui, mouetur, fit vt etiam <lb></lb>
periphæria maioris facilius, quàm minoris moueatur, ſiue dixeris, quod an­<lb></lb>
gulus maioris circuli ad angulum minoris nutum quendam habet; &amp; quia <lb></lb>
facilius mouetur angulus maioris, quàm minoris, fit, vt maior rota adhi­<lb></lb>
beatur ad minorem mouendam: &amp; quia intra maiorem infinitæ circa idem <lb></lb>
centrum concipi poſſunt, hinc fit, vt rotæ maiores facilius moueantur, &amp; <lb></lb>
motæ moueant cæteras intra ſe contentas. </s>
          <s id="s.002887">quod dictum eſt de nutu anguli <lb></lb>
maioris circuli ad angulum minoris ex appoſita figura facilè patebit, vbi <lb></lb>
<figure id="id.009.01.170.1.jpg" place="text" xlink:href="009/01/170/1.jpg"></figure><lb></lb>
pro minore angulo intelligendus eſt arcus C B, <lb></lb>
pro maiore autem arcus D E, quorum <expan abbr="vterq;">vterque</expan> vo­<lb></lb>
catur angulus, quoniam angulo A, qui eſt in cen­<lb></lb>
tro opponuntur. </s>
          <s id="s.002888">Atque hæc ſufficiant de ijs, quæ <lb></lb>primo modo mouentur.</s>
        </p>
        <p type="main">
          <s id="s.002889">Nunc ad ea, quæ reliquis duobus modis cieri <lb></lb>
ſolent, quæ ſcilicet non mouentur ſecundum apſi­<lb></lb>
dem, ſed aut iuxta planitiem, ideſt, quæ æquidi­<lb></lb>ſtanter pauimento collo<expan abbr="cãtur">cantur</expan>, vt rotæ figulorum, <lb></lb>
aut quæ in loco à terra eleuato, vt troclearum or­<lb></lb>
biculi. </s>
          <s id="s.002890">rotæ hæ facilius ipſæ, &amp; ea etiam, quæ ipſis annectuntur commouen­<lb></lb>
tur, quam ſi rectilinea figura conſtarent; non quia parua ſui portione vel <lb></lb>
tangant planum, vel offenſent, ſed ob aliam inclinationem, de qua initio <lb></lb>
huius operis ante quæſtiones dictum eſt, vbi diximus circulum duas incli­<lb></lb>
nationes ad motum obtinere, ſecundum quas à motore mouetur; vna eſt, <lb></lb>
quam diximus naturalem, qua ſolet cieri ſecundum periphæriam, motor <lb></lb>
enim ſemper mouet circulum in periphæria, &amp; ſecundum hanc inclinatio­<lb></lb>
nem extremum diametri rectà, non circulariter moueretur: hanc inclina­<lb></lb>
tionem fortè habet à materia grauitante, &amp; in ipſo circulo conſtituta in <lb></lb>
æquilibrio: quæ autem in æquilibrio, facillimè cedunt; &amp; qui talia mouent, <lb></lb>
quaſi prius mota mouent, &amp; ideò facillimè. </s>
          <s id="s.002891">Secundum igitur inclinatio­<lb></lb>
nem hanc, quæ in obliquum eſt, ideſt, quæ ſecundum circunferentiam ſit, <lb></lb>
ipſam rotam mouens facillimè mouet. </s>
          <s id="s.002892">altera latio eſt, ſecundum quam cir­


<pb pagenum="171" xlink:href="009/01/171.jpg"></pb>culus à ſeipſo ſecundum diametrum mouetur, ideſt circa ſuum centrum re­<lb></lb>
trahit continuò extrema diametri; ne recta ſecundum naturalem lationem <lb></lb>
ferantur, ſed in orbem circulariter circa centrum gyrentur. </s>
          <s id="s.002893">hæc Ariſt. </s>
          <s id="s.002894">Re­<lb></lb>
ſtat vt ſatisfaciam promiſſis.</s>
        </p>
        <p type="main">
          <s id="s.002895">Dictum eſt ab Ariſt. in textu <emph type="italics"></emph>(Sicut diameter ad diametrum, ita maior circu­<lb></lb>
lus ad maiorem)<emph.end type="italics"></emph.end> quæ verba intelligenda eſſe non de circulis, ſed de periphæ­<lb></lb>
rijs, vti expoſui, manifeſtum eſt ex 11. propoſit. </s>
          <s id="s.002896">5. Pappi Alexandrini, quæ <lb></lb>
talis eſt: Circulorum circunferentiæ inter ſe ſunt vt diametri. </s>
          <s id="s.002897">quam etiam <lb></lb>
Pater Clauius demonſtrat propoſ. </s>
          <s id="s.002898">2. lib. 8. &amp; propoſ. </s>
          <s id="s.002899">1. lib. 4. Geom. pract. <lb></lb>
ſi autem de ipſis circulis intelligerentur falſa eſſent, non enim eſt circulus <lb></lb>
ad circulum, vt diameter ad diametrum; ſed circuli ſunt inter ſe, quemad­<lb></lb>
modum à diametris ipſorum quadrata per ſecundam 12. Elem. quadrata <lb></lb>
autem ſunt inter ſe in duplicata ratione laterum per 20. 6. <expan abbr="eiusq;">eiusque</expan> corolla­<lb></lb>
rium; hoc eſt ſi fiat, vt latus maioris quadrati ad latus minoris, ita latus mi­<lb></lb>
noris ad aliam tertiam lineam, erit quadratum maius ad minus, vt latus <lb></lb>
ipſius ad tertiam illam lineam; non autem vt ad latus minoris. </s>
          <s id="s.002900">cum ergo <lb></lb>
circulus ſit ad circulum, vt quadratum diametri ad quadratum diametri, <lb></lb>
&amp; quadrata non <expan abbr="habeãt">habeant</expan> rationem laterum, ſeu diametrorum prædictorum, <lb></lb>
ſed illorum duplicatam, <expan abbr="neq;">neque</expan> circuli inuicem illam habere poterunt.</s>
        </p>
        <p type="main">
          <s id="s.002901">Illud demum non ignorandum, quod Guidus Vbaldus propoſit. </s>
          <s id="s.002902">1. de Tro­<lb></lb>
chlea, demonſtrat, quod nimirum potentia ſuſtinens pondus per rotulam, <lb></lb>
cui funis ſupernæ fuerit circumductus, qualis ea eſt, qua ad hauriendam ex <lb></lb>
puteis aquam vtimur, talis inquam potentia eſt æqualis ponderi; cuius ra­<lb></lb>
tio eſt, quia tunc trochlea fit vectis, cuius fulcimentum eſt in medio vectis, <lb></lb>
pondus verò, &amp; potentia in extremitatibus ſunt, &amp; æquidiſtant ab hypomo­<lb></lb>
clio, &amp; propterea cum ſit eadem proportio ponderis ad potentiam, quæ di­<lb></lb>
ſtantiæ ad diſtantiam, vt ſupra quęſt. </s>
          <s id="s.002903">3. probatum eſt ex Archimede, &amp; Gui­<lb></lb>
do Vbaldo, diſtantiæ autem ſint æquales, erunt etiam pondus, &amp; potentia <lb></lb>
æqualia, ideſt, ſi pondus eſſet vnius libræ, ſuſtineretur à tanta vi, <expan abbr="quãta">quanta</expan> opus <lb></lb>
eſt ad libram vnam ſuſtinendam, &amp; non amplius. </s>
          <s id="s.002904">vt autem clarè appareat <lb></lb>
vectis in trochlea, &amp; hypomoclion, &amp; æquales diſtantiæ, ſit figura, in qua <lb></lb>
<figure id="id.009.01.171.1.jpg" place="text" xlink:href="009/01/171/1.jpg"></figure><lb></lb>
pondus D, ductario funi D C B E, alligatum. </s>
          <s id="s.002905">poten­<lb></lb>
tia <expan abbr="ſuſtinẽs">ſuſtinens</expan> E. axis autem erit diameter rotulæ B A C, <lb></lb>
nam potentia premit rotulam in B, &amp; pondus in C, &amp; <lb></lb>
cum rotula ſuſtineatur in A, à ſuſpenſorio F A. erit <lb></lb>
punctum A, hypomoclion, quia in motu vectis eua­<lb></lb>
dit centrum, <expan abbr="eſtq́">eſtque</expan>; punctum manens. </s>
          <s id="s.002906">æquales autem <lb></lb>
diſtantiæ <expan abbr="vtrinq;">vtrinque</expan> ab hypomoclio ſunt B A, A C, ſunt <lb></lb>
enim ex centro eodem. </s>
          <s id="s.002907">ex quibus manifeſtum eſt hu­<lb></lb>
iuſmodi rotulam nullam vim mouenti addere, ſed ſo­<lb></lb>
lum illud præſtat, vt omne tollat impedimentum, <lb></lb>
quemadmodum ait Ariſt. manifeſtum etiam eſt ma­<lb></lb>
iorem vim quamlibet, quam ſit ea, quæ ſuſtinet, poſſe <lb></lb>
idem pondus ſurſum mouere. </s>
          <s id="s.002908">hæc &amp; præſenti loco, &amp; <lb></lb>
ſequentibus lucem afferre poſſunt.</s>
        </p>
        <pb pagenum="172" xlink:href="009/01/172.jpg"></pb>
        <p type="head">
          <s id="s.002909"><emph type="italics"></emph>QVÆSTIO NONA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002910"><emph type="italics"></emph>De Trochleis, &amp; Scytalis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002911"><arrow.to.target n="marg240"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002912"><margin.target id="marg240"></margin.target>250</s>
        </p>
        <p type="main">
          <s id="s.002913"><emph type="italics"></emph>Cvr ea, quæ per maiores circulos tollantur, &amp; trahuntur facilius, &amp; ci­<lb></lb>
tius mouentur? </s>
          <s id="s.002914">veluti per maiores trochleas, quàm per minores, &amp; ſcy­<lb></lb>
talas ſimiliter? </s>
          <s id="s.002915">An quanto maior fuerit illa, quæ à centro eſt, in æquali <lb></lb>
temporis ſpatio maius ſpatium conficit? </s>
          <s id="s.002916">quamobrem æqualì inexiſtente <lb></lb>
onere, idem faciet, ſicuti diximus maiores libras minoribus exactiores eſſe; ſpar­<lb></lb>
tum enim in illis centrum eſt: partes verò libræ vtrinque à ſparto ſunt veluti lineæ <lb></lb>
ex centro)<emph.end type="italics"></emph.end> Cum textus huius quæſtionis fatis clarus ſit, præſertim ſi prius <lb></lb>
legantur, quæ dicta ſunt de libra in prima quæſt. </s>
          <s id="s.002917">&amp; quæ de rota, &amp; trochlea <lb></lb>
in proxima præcedenti, à paraphraſi ipſius ſuperſedebo. </s>
          <s id="s.002918">Illud tamen, quod <lb></lb>
magis neceſſarium eſt, non omittam, vt ſcilicet difficultatibus quibuſdam <lb></lb>
occurram. </s>
          <s id="s.002919">Et primo, quod Ariſt. ait, ea quæ per maiores circulos veluti <lb></lb>
trochleas, ſeu rotulas trahuntur, facilius trahi, quàm ea, quæ per minores, <lb></lb>
non videtur ex omni parte verom. </s>
          <s id="s.002920">nam ſicuti in <expan abbr="præcedẽti">præcedenti</expan> quæſtione oſten­<lb></lb>
ſum eſt ex Guido Vbaldo, trochlea ſimplex, ſiue rotula illa ſtriata, cui funis <lb></lb>
ſupernè inditur, vt in ſuperiori figura; nullas addit vires potentiæ, quia re­<lb></lb>
ducitur ad vectem, cuius fultura ſit in medio ipſius. </s>
          <s id="s.002921">ſiue igitur rotula illa <lb></lb>
magna fuerit, ſiue parua, ſemper in talem vectem reſoluetur, &amp; propterea, <lb></lb>
vt etiam experientia conſtat eodem labore aquam hauriunt, ſiue rotula illa <lb></lb>
magna fuerit, ſine parua. </s>
          <s id="s.002922">nec minus vera videtur reſponſio, cum ait <emph type="italics"></emph>(An quia <lb></lb>
quanto maior fuerit illa, quæ à centro eſt, in æquali <expan abbr="tẽpore">tempore</expan> maius mouetur ſpatium)<emph.end type="italics"></emph.end><lb></lb>
quæ quidem vera ſunt, ſi intelligantur hoc modo, nimirum, quod quando <lb></lb>
plures <expan abbr="circulſconcẽtrici">circuli concentrici</expan>, <expan abbr="atq;">atque</expan> inuicem connexi fuerint, ita vt vnus ſine alijs <lb></lb>
moueri nequeat, tunc quanto maior fuerit diameter, &amp; conſequenter cir­<lb></lb>
cunferentia, tanto velocius mouebitur. </s>
          <s id="s.002923">ſi autem intelligantur de duobus <lb></lb>
circulis ab inuicem ſeparatis, quorum vnus <expan abbr="abſq;">abſque</expan> altero moueri poteſt, vt ſie <lb></lb>
quando vtimur modo rotula magna, modo parua ad aquam hauriendam <lb></lb>
non videntur vera, in quo ſenſu manifeſtè loquitur Ariſt. </s>
          <s id="s.002924">Quapropter vt ſin­<lb></lb>
cerè loquar, nunc neſcio, qua ratione Ariſt ab errore excuſare valeam, alijs <lb></lb>
fortè occurret.</s>
        </p>
        <p type="main">
          <s id="s.002925">Secundo loco videndum quid ſint ſcyntalæ. </s>
          <s id="s.002926">Vt autem conſtat ex ſequenti <lb></lb>
quæſtione 11. ſcyntala erat inſtrumentum quoddam vectorium, quod ro­<lb></lb>
tas, ſicut currus, aliter tamen factas, habebat, porrò <foreign lang="grc">σκυταλὴ,</foreign> ideſt ſcytala <lb></lb>
inter alia ſignificat <expan abbr="baculũ">baculum</expan>, ſiue lignum oblongum, ac teres, qualia ea ſunt, <lb></lb>
quibus vtimur in ſucculis, vulgò Naſpe; &amp; in axe in peritrochio, vt videre <lb></lb>
eſt apud <expan abbr="Guidũ">Guidum</expan> Vbaldum. </s>
          <s id="s.002927">hinc factum eſt, vt apud Lacædemonios ſcytala <lb></lb>
ſignificaret quoddam genus epiſtolæ, quam ſcytalem laconicam dicebant, <lb></lb>
quia in charta inſtar zonæ oblonga, &amp; circa ſcytalam, hoc eſt circa bacillum <lb></lb>quendam ſpiratim circumuoluta exarabatur; ita vt verſus ſcripturæ ſecun­<lb></lb>dum ſurculi longitudinem ducerentur, ex quo fiebat, vt per iuncturas mem­<lb></lb>
branæ, literæ, ac verba procederent, membranam hanc ex ſcytala reuolu­<lb></lb>
tam, &amp; aliter complicatam Imperatori mittebant, reſolutio autem mem­


<pb pagenum="173" xlink:href="009/01/173.jpg"></pb>branæ literas truncas, atque mutilas reddebat; cum partim continerentur <lb></lb>
citra iuncturas, partim vltra: eæquè partes, quæ ſimul fuerant ſcriptæ, &amp; <lb></lb>continuatæ, poſt reſolutionem erant ab inuicem valde diſſitæ. </s>
          <s id="s.002928">quapropter <lb></lb>
Imperator commenti totius conſcius, eandem membranam ſcytali alteri <lb></lb>
priori omninò ſimili, <expan abbr="æqualiq́">æqualique</expan>; eodem modo, quo prius circumponebat, <expan abbr="ſicq́">ſicque</expan>; <lb></lb>
iuncturæ priores redibant, quæ literas, ac verba mutila, &amp; imperfecta in <lb></lb>
integrum reſtituebant, vt facilè legi poſſent. </s>
          <s id="s.002929">hoc illi vtebantur ſecreto, cum <lb></lb>
literas ad Imperatores ſuos miſſas, hoſtibus occultas eſſe volebant.</s>
        </p>
        <p type="main">
          <s id="s.002930">Ex quibus conijcere licet ſcytalam fuiſſe lignum oblongum, &amp; teres, ſiue <lb></lb>
vt Geometræ dicunt, Cylindrum; in cuius tamen extremitatibus eſſent <lb></lb>
margines duo aliquantulum prominentes, ceu binæ rotæ, cum ipſo tamen <lb></lb>
continuæ, &amp; connexæ, vt cum ipſo ſimul conuoluerentur; non tamen tan­<lb></lb>
<figure id="id.009.01.173.1.jpg" place="text" xlink:href="009/01/173/1.jpg"></figure><lb></lb>
quam circa axem. </s>
          <s id="s.002931">cuius hanc accipe fi­<lb></lb>
guram. </s>
          <s id="s.002932">Quærit igitur Ariſt. cur huiuſ­<lb></lb>
modi ſcytalæ facilius moueantur, quo <lb></lb>
maiores ipſarum ſunt rotæ. </s>
          <s id="s.002933">Cui quæ­<lb></lb>
ſtioni ſimul, ijſdemque verbis, quibus <lb></lb>
quæſtioni de maioribus rotulis reſpondet, ſed non ſatisfacit ob eandem ra­<lb></lb>
tionem, quam ibi attuli. </s>
          <s id="s.002934">Crediderim tamen maiores ſcytalas, &amp; maiores <lb></lb>
curruum rotas, &amp; alia id generis, quæ volutantur, ita vt motu progreſſiuo <lb></lb>
mutent locum, facilius moueri, ſed ob aliam cauſam, quia nimirum maio­<lb></lb>
res rotæ minus ſi quid obuiam fiat, offenſant, quia ſua magnitudine quem­<lb></lb>
libet obicem facilè ſuperare poſſunt; cuius cauſa eſt angulus <expan abbr="acutiſſimũs">acutiſſimus</expan>, <lb></lb>
quem cum terra facit; at verò exiguæ rotæ, ſi cui maiori obſtaculo obuia­<lb></lb>
rint, ipſum nequeunt ſuperare, aut ſuperaſcendere, quia angulum cum ter­<lb></lb>
ra faciunt inſto maiorem, vnde facilè ipſorum curſus inhibetur, <expan abbr="ipſæq́">ipſæque</expan>; pro­<lb></lb>
pterea præ maioribus tardiores euadunt. </s>
          <s id="s.002935">Atque hæc in hanc quæſtionem <lb></lb>
dicta ſufficiant.</s>
        </p>
        <p type="head">
          <s id="s.002936"><emph type="italics"></emph>QVÆSTIO DECIMA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002937"><emph type="italics"></emph>De libra vacua, &amp; alijs ſimilibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002938"><arrow.to.target n="marg241"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002939"><margin.target id="marg241"></margin.target>251</s>
        </p>
        <p type="main">
          <s id="s.002940">Cvr libræ, quæ omni incumbente pondere ſunt vacuæ ab impoſito <lb></lb>
pondere facilius mouentur, quàm ſi quopiam inexiſtente pondere <lb></lb>
aliud rurſus onus ſuperaddatur. </s>
          <s id="s.002941">ſimiliter etiam rota, &amp; huiuſmodi <lb></lb>
quippiam, quod grauius quidem eſt, difficilius commouetur quàm <lb></lb>
læue, v. g. rota ferrea difficilius, quàm lignea. </s>
          <s id="s.002942">ſimiliter quæ maiora ſunt, <lb></lb>
etiam ſi ex eadem materia conſtent difficilius mouentur quàm minora, vt <lb></lb>
rota maior ferrea, quàm minor etiam ferrea. </s>
          <s id="s.002943">Habet hæc quæſtio tres par­<lb></lb>
tes, quibus Ariſt. reſpondet dicens, quod graue eſt ægrè moneri non ſolum <lb></lb>contra nutum ſuum, ideſt ſurſum, ſed etiam in obliquum, ſeu ad latera, quia <lb></lb>
grauia deorſum <expan abbr="nutãt">nutant</expan>, non ſurſum, nec in tranſuerſum: ideo libræ cum one­<lb></lb>re, quia ſunt grauiores, &amp; rota ferrea quàm lignea, &amp; ferrea etiam maior, <lb></lb>
quàm minor grauior eſt, ideò difficilius agitatur.</s>
        </p>
        <p type="main">
          <s id="s.002944">Contra quam reſponſionem ſic fortè obijcies; in præcedenti enim quæ­


<pb pagenum="174" xlink:href="009/01/174.jpg"></pb>ſtione dictum eſt ab Ariſt. maiores trochleas, &amp; ſcytalas, minoribus facilius <lb></lb>
commoueri, hic autem dicit maiorem rotam difficilius quàm minorem mo­<lb></lb>
ueri. </s>
          <s id="s.002945">Hanc obiectionem Piccolomineus diſſimulaſſe videtur, cui ego, inge­<lb></lb>
nuè fateor, me ſatisfacere neſcire, vt enim in præcedenti annotaui, nulla <lb></lb>
mihi ratio Ariſt. excuſandi occurrit, alijs fortè occurret. </s>
          <s id="s.002946">In præſenti au­<lb></lb>
tem benè quidem reſpondet, ſed tamen intimam rei cauſam non attingit.</s>
        </p>
        <p type="main">
          <s id="s.002947">Sciendum igitur eſt id, quod Guidus Vbaldus in tractatu de libra pluri­<lb></lb>
bus demonſtrauit: quod ſi quoduis graue ſuſpendatur prorſus in <expan abbr="cẽtro">centro</expan> gra­<lb></lb>
uitatis, ita vt in perfecto ſit æquilibrio, tunc ſiue magnum, ſiue paruum, <lb></lb>
ſiue graue, grauiuſuè fuerit, à quauis exigua vi poterit ab æquilibrio dimo­<lb></lb>
ueri. </s>
          <s id="s.002948">cur ergo in libris, &amp; rotis grauioribus, aut maioribus <expan abbr="experiẽtia">experientia</expan> con­<lb></lb>
trarium oſtendit? </s>
          <s id="s.002949">ratio eſt, quia hæc omnia communiter non collocantur, <lb></lb>
ita vt circa centrum ſuum, quod etiam centrum grauitatis eſt, conuerti <lb></lb>
poſſint: verum aptantur circa axem, &amp; quidem iuſto maiorem, laxiuſque <lb></lb>
circa ipſum conuertuntur, vnde fit, vt ipſa ob inſitam grauitatem premant <lb></lb>
axem in ſuperiori parte, vnde quando ab aliquo gyrantur, non propriè gy­<lb></lb>
rant, ſed in ſuperiori axis parte hærentes ipſum atterunt; ex qua attritione <lb></lb>
fit, vt retardentur, <expan abbr="idq́">idque</expan>; eò magis, quo grauiora magis premunt; hærent, <lb></lb>
<expan abbr="difficiliusq́">difficiliusque</expan>; propterea raptantur potius, quàm gyrentur.</s>
        </p>
        <p type="main">
          <s id="s.002950">Ex his, &amp; textus, &amp; ratio Ariſtotelis ſatis clara redduntur.</s>
        </p>
        <p type="head">
          <s id="s.002951"><emph type="italics"></emph>QVÆSTIO VNDECIMA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002952"><emph type="italics"></emph>De Scytala, &amp; Curru.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002953"><arrow.to.target n="marg242"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002954"><margin.target id="marg242"></margin.target>252</s>
        </p>
        <p type="main">
          <s id="s.002955">Cvr ſuper ſcytalas facilius portantur onera quàm ſuper currus, cum <lb></lb>
tamen currus magnas habeant rotas, ſcytalæ verò puſillas?</s>
        </p>
        <p type="main">
          <s id="s.002956">Quidnam ſcytala eſſet explicatum eſt in 9. quæſt. </s>
          <s id="s.002957">Quo autem <lb></lb>
modo per ſcytalas onera <expan abbr="portẽtur">portentur</expan>, ſic, accipe: exiſtimo binas ſcy­<lb></lb>
talas inuicem æquidiſtantes, &amp; aliquantulum ſemotas inuicem ſic diſponi, <lb></lb>
vt efficiant inſtrumentum vectorium currus inſtar, &amp; fortè veteres vteban­<lb></lb>
tur his ſcytalis eo modo, quo nunc architectores vtuntur duobus illis lignis <lb></lb>
longis, ac rotundis, quæ vulgò dicuntur Ruccioli.</s>
        </p>
        <p type="main">
          <s id="s.002958">Reſpondet igitur id accidere, quia rotæ ſcytalarum ſimul ſunt cum ſuo <lb></lb>axe compactæ, ita vt ſimul cum ipſo rotentur: rotæ autem curruum, quia <lb></lb>
ſeiunctæ ſunt ab earum axe, ita vt ſine illius rotatione ipſæ voluantur, fit vt <lb></lb>illæ firmius incedant, nec huc, <expan abbr="illucq́">illucque</expan>; nutent, veluti rotæ plauſtri: <expan abbr="neq;">neque</expan> illæ <lb></lb>
ad ipſum axem offenſent, quemadmodum iſtæ. </s>
          <s id="s.002959">addit aliam rationem, quia <lb></lb>
currus nimia oneris grauitate premens rotas ipſas ferè ſiſtit, quod ſcytalis <lb></lb>
non accidit, cum rotæ ipſarum vnum, &amp; idem cum ſuo ſint axe. </s>
          <s id="s.002960">quæ ratio <lb></lb>
quantum valeat, neſcio, nam quamuis rotæ ſcytalarum non premantur ab <lb></lb>
axe, premitur tamen axis ipſarum ab onere, à quo ſimiliter ſiſti debe­<lb></lb>
rent ſcytalæ.</s>
        </p>
        <p type="main">
          <s id="s.002961">Crediderim ego facilius portari magna onera per ſcytalas, propter ipſa­<lb></lb>
rum firmitatem, currus enim <expan abbr="ipſorumq;">ipſorumque</expan> rotæ ſunt multò debiliores, neque <lb></lb>
maioribus oneribus ſufficiunt. </s>
          <s id="s.002962">Concludit poſtea quæſtionem dicens, quia <pb pagenum="175" xlink:href="009/01/175.jpg"></pb>igitur ſcytalæ ab ipſo onere non ita premuntur quin moueri melius poſſint <lb></lb>
quàm currus, imò ab ipſo onere iam commoto, ipſæ quoque incitentur, &amp; <lb></lb>
præterea à potentia per planum infernè, benè ſubſtratum, &amp; complanatum <lb></lb>
trahantur, fit, vt quaſi in duobus locis ipſarum rotæ impellantur ab onere <lb></lb>
ſupra, &amp; à potentia infra; ſicque facilius quam currus ingentia præſertim <lb></lb>
onera vehunt.</s>
        </p>
        <p type="head">
          <s id="s.002963"><emph type="italics"></emph>De Funda.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002964"><emph type="italics"></emph>QVÆSTIO DVODECIMA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002965">Non videtur declaratione indigere.</s>
        </p>
        <p type="head">
          <s id="s.002966"><emph type="italics"></emph>QVÆSTIO DECIMATERTIA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002967"><emph type="italics"></emph>De Iugo, &amp; Succula.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002968"><arrow.to.target n="marg243"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002969"><margin.target id="marg243"></margin.target>253</s>
        </p>
        <p type="main">
          <s id="s.002970">Declarandum prius quid ſit hoc loco iugum: eſt igitur iugum li­<lb></lb>
gnum illud cylindricum, quod vulgò dicitur Subbio. </s>
          <s id="s.002971">quorum bi­<lb></lb>
na ponuntur in ea machina textoria, quam vulgò dicunt Telaio, <lb></lb>
quaſi telarium, eo quod in ipſa telæ texantur. </s>
          <s id="s.002972">alteri autem iugo <lb></lb>
conuoluitur ſtamen: alteri verò contexta iam tela ſubinde cum opus eſt cir­<lb></lb>
cumponitur: quæ duo textores faciunt ipſa iuga conuertendo. </s>
          <s id="s.002973">quæ vt faci­<lb></lb>
lius conuertant, iugis vtrinque inſerunt per bina foramina binos collopes. <lb></lb>
</s>
          <s id="s.002974">qui collopes ſunt duo ligna oblonga ſatis gracilia vnius vlnæ ferè in longi­<lb></lb>
tudinem; quibus <expan abbr="appræhẽſis">appræhenſis</expan>, <expan abbr="motisq́">motisque</expan>; iugum facilè verſatur. </s>
          <s id="s.002975">quanto autem <lb></lb>
collopes ſunt longiores, facilius iugum circumagitur. </s>
          <s id="s.002976">cuius cauſa eſt, quia <lb></lb>
collops ad vectem reducitur, cuius fultura eſt circa medium iugi, pondus <lb></lb>
verò eſt extima iugi ſuperficies è qua telæ, aut ſtaminis pondus pendet: in <lb></lb>
altera verò extremitate collopis, quæ extra iugum multum prominet, eſt <lb></lb>
potentia: ibi enim textoris manus premit, vel trahit. </s>
          <s id="s.002977">quando ergò longior <lb></lb>
eſt collops, ea pars, quæ eſt inter fulturam, &amp; vim, augetur; altera non mu­<lb></lb>
tata; quia ſemper inter fulturam, ſeu centrum iugi, &amp; vltimam iugi ſuper­<lb></lb>ficiem continetur; quanto autem illa hanc ſuperat, tantum virium po­<lb></lb>
tentiæ addi.</s>
        </p>
        <p type="main">
          <s id="s.002978">Secundò, videndum quid ſit ſuccula: hanc vulgò Naſpa appellant, ni fal­<lb></lb>
lor à verbo græco <foreign lang="grc">ἀγασπάω,</foreign> oriunda, quod ſurſum extrahere ſignificat. </s>
          <s id="s.002979">cum <lb></lb>
quo, &amp; voce, &amp; ſignificatione conuenit; eſt enim inſtrumentum, quo ſæpius <lb></lb>
architectores in extrahendis ſurſum ruderibus effoſſis vtuntur. </s>
          <s id="s.002980">eſt autem <lb></lb>compago quædam cylindrica non admodum longa, cui ex vna parte potiſ­<lb></lb>
ſimum prominent plures collopes non mobiles, vt in iugo, verum ſtabiles, <lb></lb>
ac cum ipſa ſuccula compacti, quibus manu appræhenſis ſuccula ſupra bi­<lb></lb>
nos polos verſatur, <expan abbr="eiq́">eique</expan>; interim ductarius funis circumuoluitur, ſecumque <lb></lb>
ſurſum pondus educit. </s>
          <s id="s.002981">cuius imaginem <expan abbr="qualcmcunq;">qualemcunque</expan> inſpice. </s>
          <s id="s.002982">quærit igitur, <lb></lb>cur quanto gracilius fuerit corpus ſucculæ A B, tanto facilius vertitur. <lb></lb>
</s>
          <s id="s.002983">Ratio eſt, quia collops, quemadmodum etiam iugum, reducitur ad vectem,


<pb pagenum="176" xlink:href="009/01/176.jpg"></pb><figure id="id.009.01.176.1.jpg" place="text" xlink:href="009/01/176/1.jpg"></figure><lb></lb>
cuius hypomoclion eſt in medio <lb></lb>
ſucculæ, ſiue in axe ipſius ſucculæ; <lb></lb>
potentia verò eſt in ſummitatibus <lb></lb>
collopum, vt in C, E, F, D, pon­<lb></lb>
dus verò eſt vbi funis ductarius <lb></lb>
cum onere pendet è ſuccula in ſu­<lb></lb>
perficie nimirum, vt vbi L, quare <lb></lb>
pars vectis inter axim, &amp; ſuperfi­<lb></lb>
ciem ſucculæ eadem eſt, quæ inter <lb></lb>
hypomoclium, &amp; pondus. </s>
          <s id="s.002984">quanto <lb></lb>
igitur ſucculæ corpus gracilius fuerit, tanto hæc pars minuetur; &amp; conſe­<lb></lb>
quenter altera inter hypomoclium, &amp; potentiam productior euadet: eaque <lb></lb>
propter facilius à motore verſabitur.</s>
        </p>
        <p type="head">
          <s id="s.002985"><emph type="italics"></emph>De ligno ad genu fracto.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002986"><emph type="italics"></emph>QVÆSTIO DECIMAQVARTA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002987">Satis perſe clara videtur.</s>
        </p>
        <p type="head">
          <s id="s.002988"><emph type="italics"></emph>QVÆSTIO DECIMAQVINTA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002989"><emph type="italics"></emph>De Vmbilicis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002990"><arrow.to.target n="marg244"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002991"><margin.target id="marg244"></margin.target>254</s>
        </p>
        <p type="main">
          <s id="s.002992">Notandum primò, quæ Græcis <foreign lang="grc">Κροκαι,</foreign> ideſt Crocæ dicuntur, Latinis <lb></lb>
Vmbilicos appellari; de his enim loquitur Cic. 2. de Oratore, vbi <lb></lb>
ſic, non audeo dicere de talibus viris, ſed tamen ita narrare ſole­<lb></lb>
bat Sceuola, conchas, eos, &amp; vmbilicos ad Caietam, &amp; ad Lucri­<lb></lb>
num legere conſueuiſſe. </s>
          <s id="s.002993">hos autem vmbilicos exponunt Grammatici eſſe <lb></lb>
lapillos paruos, acrotundos, politoſque, de quibus etiam Ariſt. loquitur. <lb></lb>
</s>
          <s id="s.002994">Quare decipitur Piccolomineus dum negat, nos harum crocarum latinum <lb></lb>
nomen habere. </s>
          <s id="s.002995">Cæterùm, &amp; quæſtio, &amp; reſponſio, ex ſuperioribus ſatis <lb></lb>
perſpicua eſſe videntur.</s>
        </p>
        <p type="head">
          <s id="s.002996"><emph type="italics"></emph>QVÆSTIO DECIMASEXTA<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.002997"><emph type="italics"></emph>De ligno oblongo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.002998"><arrow.to.target n="marg245"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.002999"><margin.target id="marg245"></margin.target>255</s>
        </p>
        <p type="main">
          <s id="s.003000">Ex appoſita figura totus huius problematis textus, alioquin ſatis cla­<lb></lb>
rus patebit. </s>
          <s id="s.003001">ſint duo ligna oblonga, vnum altero longius, &amp; craſſius. <lb></lb>
</s>
          <s id="s.003002">in eleuatione maioris, fulcimentum eſt in B, vbi manus altera ferè <lb></lb>
manens appræhendit; in C, verò, vbi altera manus mouens premit <lb></lb>
eſt potentia, ſiue maius onus. </s>
          <s id="s.003003">in A, verò onus ipſius ligni, deorſum tendens <lb></lb>
premit, quod nunc eſt inſtar potentiæ motricis, quare A, &amp; C, ſunt ſibi in­<lb></lb>
uicem, &amp; potentiæ, &amp; pondera. </s>
          <s id="s.003004">In minori autem ligno, onus ligni in D,


<pb pagenum="177" xlink:href="009/01/177.jpg"></pb><figure id="id.009.01.177.1.jpg" place="text" xlink:href="009/01/177/1.jpg"></figure><lb></lb>
fultura manus in E, potentia alterius ma­<lb></lb>nus in F. iam inquit Ariſt. maius lignum <lb></lb>
A B C, magis flectitur, quamuis craſſius <lb></lb>ſit, quàm lignum D E F, quod eſt tenuius, <lb></lb>
ſed multò breuius; quia in maiori onus <lb></lb>
ipſius ligni, quod circa A, deorſum pre­<lb></lb>
mit <expan abbr="lõgius">longius</expan> diſtat ab hypomoclio B, quàm <lb></lb>
in minori ligno. </s>
          <s id="s.003005">Ex quo ſequitur iuxta <lb></lb>
ipſius principia, vt onus A, facilius lignum mouere, aut inflectere <lb></lb>
poſſit.</s>
        </p>
        <p type="main">
          <s id="s.003006">Cæterùm exiſtimo, quod ſi maioris ligni longitudo ad eiuſdem <lb></lb>
craſſitiem haberet <expan abbr="eãdem">eandem</expan> proportionem, quàm minoris longitudo ad eiuſ­<lb></lb>
dem craſſitiem, <expan abbr="ſicq́">ſicque</expan>; <expan abbr="vtrumq;">vtrumque</expan> eſſet ab hypomoclio in eadem ratione diui­<lb></lb>
ſum, fore, vt <expan abbr="vtrunq;">vtrunque</expan> eodem modo inflecteretur, quia haberent pondera <lb></lb>
eandem rationem ad diſtantias ab hypomoclio, oportet igitur vt ſint non <lb></lb>
analoga, ſed aloga, vt eis præſens problema Ariſtotelis vnà cum eiuſdem <lb></lb>
ſolutione competat.</s>
        </p>
        <p type="head">
          <s id="s.003007"><emph type="italics"></emph>QVÆSTIO XVII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003008"><emph type="italics"></emph>De Cuneo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003009"><arrow.to.target n="marg246"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003010"><margin.target id="marg246"></margin.target>256</s>
        </p>
        <p type="main">
          <s id="s.003011">Cvr paruo cuneo magna finduntur onera, &amp; corporum moles, <expan abbr="adeoq;">adeoque</expan> <lb></lb>
valida fit impreſſio? </s>
          <s id="s.003012">fortè, quia cuneus duobus vectibus ſibi inui­<lb></lb>
cem oppoſitis conſtat; quorum vterque, &amp; potentiam mouentem, <lb></lb>
&amp; hypomoclion, &amp; <expan abbr="põdus">pondus</expan> habet. </s>
          <s id="s.003013">hypomoclion autem illud ipſum <lb></lb>
eſſe ait, quod cuneo diuellitur; hoc autem dicit Ariſtot. quia non agnouit <lb></lb>
alium, præter primi generis vectem, vt ſupra etiam dixi.</s>
        </p>
        <p type="main">
          <s id="s.003014">Verum ſatius eſt cum Guido Vbaldo reducere cuneum ad duos ſecundi <lb></lb>
generis vectes, quorum fultura ſit in cunei apice extremo, pondus verò in­<lb></lb>
tra vectem, ea nimirum pars ligni, que à cuneo vrgetur, ac diuellitur. </s>
          <s id="s.003015">cuneo <lb></lb>
præterea vires adduntur ex valida mallei percuſſione; malleus autem ipſe <lb></lb>
magna vi percutit, quia motus mouet, ſeu quia mouens malleum, mouet <lb></lb>
ipſum etiam dum eſt in ipſa latione, vnde ipſa lationis celeritate malleus <lb></lb>
fit valentior: <expan abbr="hocq́">hocque</expan>; modo paruos cunei vectes maiores conſequuntur vires, <lb></lb>
<figure id="id.009.01.177.2.jpg" place="text" xlink:href="009/01/177/2.jpg"></figure><lb></lb>
quàm ipſa vectium magnitudo poſtulet. <lb></lb>
</s>
          <s id="s.003016">ſit cuneus A B C. lignum autem ſcinden­<lb></lb>
dum D E F G, <expan abbr="vectesq́">vectesque</expan> duo ſint A C, &amp; <lb></lb>
B C, quorum commune hypomoclion eſt <lb></lb>
in C, onus autem vectis B C, eſt pars li­<lb></lb>
gni G, hæc enim ipſi contranititur, <expan abbr="atq;">atque</expan> <lb></lb>
ab eo expellitur. </s>
          <s id="s.003017">potentia verò mouens <lb></lb>
vectem eſt in malleo, dum ſuperius latus <lb></lb>
cunei A B, percutit. </s>
          <s id="s.003018">alter huic auerſus <lb></lb>
vectis eſt latus A C, cuius fultura eſt C, <lb></lb>eadem cum priori, onus propulſatum D,<pb pagenum="178" xlink:href="009/01/178.jpg"></pb>potentia cum altero communis eſt in latere A B, à malleo validè percuſſo. <lb></lb>
</s>
          <s id="s.003019">cunei igitur virtus partim ex vectibus, partim ex percuſſione conſtat.</s>
        </p>
        <p type="head">
          <s id="s.003020"><emph type="italics"></emph>QVÆSTIO XVIII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003021"><emph type="italics"></emph>De Trochlea.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003022"><arrow.to.target n="marg247"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003023"><margin.target id="marg247"></margin.target>257</s>
        </p>
        <p type="main">
          <s id="s.003024">Hvius quæſtionis ſenſus, ac verba optimè intelligentur ex ſequen­<lb></lb>
tibus. </s>
          <s id="s.003025">Trochlea, vt patet ex ſuperioribus Ariſt. eſt orbiculus in <lb></lb>
periphæria ſtriatus, vna cum toto loculumento, cui inſeritur: <lb></lb>
cuius imaginem ad 8. quæſt. </s>
          <s id="s.003026">exhibui. </s>
          <s id="s.003027">Apud Architectores verò <lb></lb>
trochlea conſtat ſaltem ex duobus prædictis loculamentis, in quibus ſunt <lb></lb>
orbiculi; &amp; vnus orbiculus eſt ſupernè collocatus, alter verò infernè, vt pa­<lb></lb>
tebit in ſequenti figuratione: quod <expan abbr="inſtrumentũ">inſtrumentum</expan> nunc vulgò dicitur Taglia, <lb></lb>
à nonnullis dicitur etiam Rechamo. </s>
          <s id="s.003028">Auxilio huius inſtrumenti machinato­<lb></lb>
res parua vi attollunt ingentia pondera. </s>
          <s id="s.003029">communiter autem conſtat ex plu­<lb></lb>
<figure id="id.009.01.178.1.jpg" place="text" xlink:href="009/01/178/1.jpg"></figure><lb></lb>
ribus orbiculis, qui partim ſuperiori loculamento, <lb></lb>partim inferiori inditi ſunt, per quos orbiculos cer­<lb></lb>
ta lege circumductus eſt ductarius funis, qui deinde <lb></lb>
in ſui poſtrema parte à potentia tractus omnes illos <lb></lb>
orbiculos, per quos tranſit circumuoluens inferius <lb></lb>
loculamentum, cui appenſum eſt pondus, vnà cum <lb></lb>pondere attollit. </s>
          <s id="s.003030">figuram ſimplicis trochleæ, con­<lb></lb>
ſtantis ſcilicet ex duobus tantum orbiculis, facilita­<lb></lb>
tis cauſa exhibebo, in hac enim melius apparebit, <lb></lb>
qua ratione trochlea ad vectem reducatur. </s>
          <s id="s.003031">vnde, &amp; <lb></lb>
Ariſt. ſenſum, quamuis obſcuriſſimum, ac proinde <lb></lb>
problematis ſolutionem optimè percipere licebit. <lb></lb>
</s>
          <s id="s.003032">Sit igitur orbiculus ſuperior A, qui in pegmate I K­<lb></lb>
L D, voluatur circa axem G, <expan abbr="ſitq́">ſitque</expan>; pegma iſtud ſupe­<lb></lb>
rius fixum, &amp; immobile à clauo H, pendens. </s>
          <s id="s.003033">Infe­<lb></lb>
rior orbiculus B, in loculamento O P Q R, circa <lb></lb>
axem B, conuoluatur: ſitque funis ductarius circa <lb></lb>
hos orbiculos hoc modo circumductus. </s>
          <s id="s.003034">primo ca­<lb></lb>
put funis religetur clauo D, in ſuperiori pegmate <lb></lb>
infixo, hinc demiſſus ſubtus inferiorem rotulam per <lb></lb>
ipſius ſtriam deſcendat per puncta L S, aſcendatque <lb></lb>
poſtea per M E N, ad ſuperiorem rotulam, ſupra <lb></lb>
quam aſcendat per punctum T, <expan abbr="deſcendatq́">deſcendatque</expan>; ad V, &amp; <lb></lb>
inde demittatur ad <expan abbr="potẽtiam">potentiam</expan> F. Iam ſi quepiam po­<lb></lb>
tentia in F, traxerit funem F V, deorſum, interim <lb></lb>
partes T, N, E, M, ſurſum attrahentur, &amp; locula­<lb></lb>
mentum inferius ſimul cum appenſo pondere eleua­<lb></lb>
bitur, manente tamen interim fune prope D, vbi <lb></lb>
clauo D, eſt religatus, &amp; immobilis. </s>
          <s id="s.003035">ſed vbinam hic <lb></lb>
vectis? </s>
          <s id="s.003036">conſidera diametrum M L, inferioris orbiculi, hæc enim ea eſt, quæ <pb pagenum="179" xlink:href="009/01/179.jpg"></pb>vectem gerit. </s>
          <s id="s.003037">huius enim extrema L M, à fune tanguntur, &amp; ab eius medio <lb></lb>B, onus pendet, &amp; grauitat; &amp; quia funis in M, ſurſum trahitur, <expan abbr="ſecumq́">ſecumque</expan>; ex <lb></lb>parte illa ſurſum elenat diametrum L M, erit potentia mouens, &amp; eleuans <lb></lb>
in M. pondus verò intra vectem ad B, medium vectis; quare fulcimentum <lb></lb>
erit in reliquo extremo L, vbi funis ſuſtinet loculamentum, &amp; vbi diameter, <lb></lb>
ſeu vectis innititur. </s>
          <s id="s.003038">quare diameter hæc eſt vectis ſecundi generis expoſiti. <lb></lb>
</s>
          <s id="s.003039">aduerte præterea vectem hunc eſſe mobilem, ſimul cum <expan abbr="fulcimẽto">fulcimento</expan>, quia dum <lb></lb>
ex parte M, ſurſum tollitur ſimul cum toto orbiculo, ac loculamento, ſub­<lb></lb>
ſequitur etiam alterum extremum L, quod fune fulcitur, &amp; in ipſo fune ſur­<lb></lb>ſum verſus D, aſcendit; &amp; hoc modo inferius tignum cum onere tandem ad <lb></lb>
ſuperius tignum ſublatum erit. </s>
          <s id="s.003040">hinc verum dixiſſe Ariſt. conſtat, trochleam <lb></lb>
ſcilicet idem eſſe, ac vectem. </s>
          <s id="s.003041">quod tamen de ſolo inferiori orbiculo intelli­<lb></lb>
gi debet, ſuperior enim rotula quamuis vectis fiat, non tamen vires vllas <lb></lb>
potentiæ tribuit, cum eius hypomoclion ſit in medio, quemadmodum ſupra <lb></lb>
ad 8. quæſt. </s>
          <s id="s.003042">expoſui. </s>
          <s id="s.003043">Inferior igitur ille eſt, qui mouenti maximo eſt adiu­<lb></lb>
mento. </s>
          <s id="s.003044">quod ſi ſcire aueas quantum iuuet, reſpondeo ipſum vires potentiæ <lb></lb>
duplicare; adeo vt ſi quatuor. </s>
          <s id="s.003045">v. g. homines erant neceſſarij ad pondus tol­<lb></lb>
lendum, auxilio huius ſimplicis trochleæ duo tantum ſufficiant. </s>
          <s id="s.003046">quod ſi ad­<lb></lb>
dantur duo alij orbiculi, vnus ſuperior, alter inferior, rurſus vires duplica­<lb></lb>
buntur, <expan abbr="eritq́">eritque</expan>; vnus <expan abbr="tãtum">tantum</expan> homo neceſſarius. </s>
          <s id="s.003047">quod ſi plures aliæ rotulæ tam <lb></lb>ſupernè, quàm infernè addantur, vt ſolet in maioribus trochleis, quas ve­<lb></lb>
teres Polyſpaſtos, ideſt multum trahentes dixerunt, augebuntur vires in in­<lb></lb>
finitum. </s>
          <s id="s.003048">quod dixi de virium duplicatione conſtat ex 6. &amp; 7. propoſitione <lb></lb>
Archimedis de Aequip. </s>
          <s id="s.003049">quia enim in vecte noſtro L M, dupla eſt proportio <lb></lb>
inter L M, &amp; L B, eadem etiam proportio erit inter pondus, &amp; potentiam, <lb></lb>
quare pondus C, duplum erit potentiæ in M, hoc eſt à minore potentia ſibi <lb></lb>
ſubdupla ſuſtinebitur: &amp; à quauis adhuc <expan abbr="quantumcunq;">quantumcunque</expan> maiore eleuabitur.</s>
        </p>
        <p type="main">
          <s id="s.003050">Qui plura de trochlea deſiderat, adeat Guidi Vbaldi, Mechanica, cuius <lb></lb>
auxilio fateor me verum ſenſum harum Mechanicarum Ariſt. &amp; præſertim <lb></lb>
huius loci enucleaſſe. </s>
          <s id="s.003051">quæ ſi cum Piccolominei expoſitione contuleris, vide­<lb></lb>
bis eum nequaquam cognouiſſe, vbi nam vectis in trochlea lateret, eumque <lb></lb>
tam ſuperiorem, quàm inferiorem <expan abbr="rotulã">rotulam</expan> æquè vectem facere; in quo etiam <lb></lb>
Io. Baptiſta Benedictus pariter erraſſe videtur in ſuis ſpeculationibus, cum <lb></lb>
inferiores tantummodo vice vectium fungantur, vt probatum eſt.</s>
        </p>
        <p type="main">
          <s id="s.003052"><expan abbr="Atq;">Atque</expan> ex his ſatis mihi videtur textus, ac ſenſus Ariſt. illuſtrari.</s>
        </p>
        <p type="head">
          <s id="s.003053"><emph type="italics"></emph>QVÆSTIO XVIIII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003054"><emph type="italics"></emph>De Securi.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003055"><arrow.to.target n="marg248"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003056"><margin.target id="marg248"></margin.target>258</s>
        </p>
        <p type="main">
          <s id="s.003057">Partim ex ſe, partim ex dictis in 17. quæſt. </s>
          <s id="s.003058">ſatis clara eſt. </s>
          <s id="s.003059">placet au­<lb></lb>
tem his, quæ de cuneo, &amp; ſecuri dicta ſunt, nonnulla ex Guido Vbal­<lb></lb>
do loco corollarij adijcere, videlicet. </s>
          <s id="s.003060">Ad huiuſmodi facultatis in­<lb></lb>ſtrumentum ea <expan abbr="quoq;">quoque</expan> omnia commodè referri poſſunt, quæ percuſ­<lb></lb>
ſione, ſiue impulſu incidunt, diuidunt, perforant, <expan abbr="huiuſmodiq́">huiuſmodique</expan>; alia obeunt <lb></lb>
munera; vt enſes, gladij, mucrones, ſecures, terebræ, &amp; ſimilia: ſerra <expan abbr="quoq;">quoque</expan> <lb></lb>
ad hoc reducitur, dentes enim percutiunt, <expan abbr="cuneiq́">cuneique</expan>; inſtar exiſtunt.</s>
        </p>
        <pb pagenum="180" xlink:href="009/01/180.jpg"></pb>
        <p type="head">
          <s id="s.003061"><emph type="italics"></emph>Additio de veteri Securi, &amp; Bipenne.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003062">Libet etiam huic tractationi de ſecuri nonnulla addere, quæ olim oc­<lb></lb>
caſione ex Proclo accepta in tenebris diu deliteſcentia in lucem re­<lb></lb>
ſtituimus, ſunt autem hæc. </s>
          <s id="s.003063">Primò, antiquæ ſecuris, necnon bipen­<lb></lb>
nis figuram reſtituam. </s>
          <s id="s.003064">Secundò, oſtendam angulum ſecuris, qui <lb></lb>
curuilineus eſt, æqualem eſſe angulo trianguli æquilateri, qui rectilineus eſt. <lb></lb>
</s>
          <s id="s.003065">Proclus igitur in comm. 23. primi Euclidis, ſic ait: oſtenſum fuit ab anti­<lb></lb>
quis, ſcilicet Geometris, quod angulus figuræ illius, quæ ſecuri ſimilis eſt, <lb></lb>
æqualis eſt angulo rectilineo, quippe qui duabus tertijs anguli recti æqualis <lb></lb>
eſt. </s>
          <s id="s.003066">hanc anguli ſecuris affectionem, cum nec ille, nec alij, quod ſciam de­<lb></lb>
monſtrent, ego paulò poſt demonſtrabo. </s>
          <s id="s.003067">deinde ſubdit; fit autem huiuſmo­<lb></lb>
di ſecuralis figura, quæ pelecoides vocatur duobus circulis per centra ſe <lb></lb>
mutuò ſecantibus. </s>
          <s id="s.003068">hæc Proclus. </s>
          <s id="s.003069">Ex his autem poſtremis verbis deſcriptio­<lb></lb>
nem antiquæ ſecuris, ſic puto eruendam. </s>
          <s id="s.003070">Ducatur primo recta A C, quæ <lb></lb>
<figure id="id.009.01.180.1.jpg" place="text" xlink:href="009/01/180/1.jpg"></figure><lb></lb>
erit inſtar manubrij ſecuris. </s>
          <s id="s.003071">de­<lb></lb>
inde ex centro C, interuallo. </s>
          <s id="s.003072">v. g. <lb></lb>
C B, deſcribatur circulus B F; ſi­<lb></lb>
militer eodem interuallo B D, ex <lb></lb>
centro D, deſcribatur circulus <lb></lb>
B E; tandem ex B, centro, atque <lb></lb>
eodem interuallo ducatur alius <lb></lb>
circulus D E F C, qui priores duos ſecabit in punctis E F. <expan abbr="cõſideremus">conſideremus</expan> iam, <lb></lb>
reliquis circulorum partibus ommiſſis, curuilineam figuram B E F, quam <lb></lb>
eſſe veteris ſecuris formam ex <expan abbr="ſentẽtia">ſententia</expan> Proclinon eſt dubitandum, cum cir­<lb></lb>
culis ſe mutuò per centra ſecantibus conſtituatur, vt vult ipſe, &amp; præterea <lb></lb>
habeat angulos E F, tantos, quantos ipſe tradit, vt mox patebit; linea au­<lb></lb>
tem A B C, ſecuris manubrium refert.</s>
        </p>
        <p type="main">
          <s id="s.003073">Quod autem tam angulus E, quàm angulus F, ſint æquales duabus tertijs <lb></lb>vnius anguli recti, ſiue quod idem eſt angulo trianguli æquilateri, manife­<lb></lb>ſtum erit hoc modo. </s>
          <s id="s.003074">Deſcribatur iterum ſecuralis figura prædicto modo, <lb></lb>
ſ<expan abbr="itq;">itque</expan> ea A B C. ducantur præterea ad ſingulos angulos tres rectæ A B, B C, <lb></lb>C A, quæ conſtituunt triangulum æquilaterum A B C, tria enim ipſius late­<lb></lb>
<figure id="id.009.01.180.2.jpg" place="text" xlink:href="009/01/180/2.jpg"></figure><lb></lb>
ra ſubtendunt tres arcus æquales A B, B C, C A, <lb></lb>
ſunt enim tres ſextantes æqualium circulorum, <lb></lb>
ut facilè colligi poteſt ex 15. 4. ex quo etiam ſe­<lb></lb>quitur tres illas circulorum portiones, quas re­<lb></lb>
ctè cum ſuis arcubus conſtituunt eſſe inuicem <lb></lb>æquales, &amp; ſimiles portiones nimirum A B E, <lb></lb>
B C D, C A F. hinc pręterea ſequitur angulos ip­<lb></lb>
ſarum eſſe inuicem æquales, angulos, v.g. A B E, <lb></lb>C B D, mixtos eſſe æquales, quod facilè eſt per imaginariam ſuperpoſitio­<lb></lb>
nem demonſtrare. </s>
          <s id="s.003075">cum igitur prædicti duo anguli ſint æquales, ſitque inter <lb></lb>eos medius alius angulus E B C, qui pariter mixtus eſt, ſi ipſe addatur tam <lb></lb>angulo C B D, quàm angulo A B E, inuicem æqualibus, erunt duo anguli <pb pagenum="181" xlink:href="009/01/181.jpg"></pb>A B C, rectilineus, &amp; E B D, curuilineus æquales. </s>
          <s id="s.003076">ille autem eſt angulus <lb></lb>
æquilateri, qui æqualis eſt duabus tertijs vnius recti ex corollario 32. primi. <lb></lb>
</s>
          <s id="s.003077">hic verò eſt angulus ſecuris. </s>
          <s id="s.003078">eſt igitur angulus ſecuris æqualis duabus ter­<lb></lb>
tijs vnius recti, vt ait Proclus, quod demonſtrandum erat. </s>
          <s id="s.003079">quod etiam ma­<lb></lb>nifeſtum ſignum eſt ſecuris figuram a me reſtitutam eſſe illam veterem, de <lb></lb>
qua idem Proclus loquitur.</s>
        </p>
        <p type="main">
          <s id="s.003080">Reſtat, vt de antiquæ bipennis etiam figura diſſeramus; quæ nihil aliud <lb></lb>
erat, quàm duplex ſecuris, ſiue ſecuris anceps, qualis eſt præſens figura, vt <lb></lb>
<figure id="id.009.01.181.1.jpg" place="text" xlink:href="009/01/181/1.jpg"></figure><lb></lb>
propterea etiam ſæpius <expan abbr="bipẽnis">bipennis</expan> ip­<lb></lb>
ſa ſecuris appelletur. </s>
          <s id="s.003081">dicitur enim <lb></lb>
bipennis, quaſi binis pinnis, quæ ſe­<lb></lb>
cures erant, conſtet, vt &amp; Græcis <lb></lb>
<foreign lang="grc">διπτερος</foreign> dicebatur. </s>
          <s id="s.003082">teſte etiam No­<lb></lb>
nio, illud bipenne eſt, quod <expan abbr="vtrinq;">vtrinque</expan> <lb></lb>
acutum eſt. </s>
          <s id="s.003083">collegi autem <expan abbr="vtcunq;">vtcunque</expan> <lb></lb>
hanc bipennis figuram ex Simmiæ <lb></lb>
peruetufti poetæ græci <expan abbr="epigrãmate">epigrammate</expan>, quod Simmiæ ſecuris appellatur. </s>
          <s id="s.003084">quod <lb></lb>
epigramma carminibus loco linearum conſtat, quæ in ſecuris formam con­<lb></lb>
ſtituta ſunt.</s>
        </p>
        <p type="main">
          <s id="s.003085">Sciendum namque eſt Simmiam, poeticam hanc ſecurim concinnaſſe in <lb></lb>
gratiam Epei illius, qui equum Troianum ligneum fuerat architectatus, vt <lb></lb>
eſt apud Virg. </s>
          <s id="s.003086">Et ipſe doli fabricator Epeus. </s>
          <s id="s.003087">qui cum ſoluendi voti cauſa <lb></lb>
vellet ſecurim, ſiue bipennem, qua in equi Durij molitione vſus fuerat, Mi­<lb></lb>
neruæ Deæ, quod ſibi in eo opere faciendo auxilio fuiſſet, dedicare, <expan abbr="eamq́">eamque</expan>; <lb></lb>
vt Ariſt. in libello de admirandis audit. </s>
          <s id="s.003088">num. </s>
          <s id="s.003089">104. narrat, in templo græ­<lb></lb>
cæ Mineruæ, quod erat in Gargaria Italiæ Regione propè Metapontum, <lb></lb>
ſuſpendere, a præfato Simmia quæſiuit, vt epigrammate aliquo dedicatio­<lb></lb>
nem hanc ſuam complecteretur. </s>
          <s id="s.003090">qui vt illi morem gereret ingenioſæ illius <lb></lb>
bipennis dedicationem, vt melius imitaretur, ſecuri hac carminum com­<lb></lb>
plexus eſt. </s>
          <s id="s.003091">quæ dedicatio, ſiue epigramma, quod adhuc extat, deinceps ſe­<lb></lb>
curis Simmiæ vocitata eſt; ex qua figura bipennis illius, equi Durij fabrica­<lb></lb>
tricis nobis adhuc magna cum voluptate innotuit. </s>
          <s id="s.003092">Porrò gratum, <expan abbr="atq;">atque</expan> ad <lb></lb>
ea, quæ diximus intelligenda vtile Lectori fore arbitrati ſumus, ipſam Sim­<lb></lb>
miæ bipennem ex operibus Theocriti, quibus addi ſolet, huc referre; quam <lb></lb>
P. Ricardus Eſius de noſtra Societate linguæ græcæ peritiſſimus, in hunc <lb></lb>
modum tranſtulit. </s>
          <s id="s.003093">hoc autem ordine legenda eſt: lectio à manubrio <lb></lb>
incipiat, deinde legatur carmen; fortiſſimæ Deæ, quod ſubſe­<lb></lb>
quatur; dedit Epeus, &amp; ſic in orbem lectio, <expan abbr="vſq;">vſque</expan> ad me­<lb></lb>
dium circumducatur. </s>
          <s id="s.003094">hæc ſunt, quæ præſertim <lb></lb>
in gratiam eorum, qui ſuauiſſimo an­<lb></lb>
tiquitatis ſtudio tenentur, la­<lb></lb>
tere nolui.</s>
        </p>
        <pb pagenum="182" xlink:href="009/01/182.jpg"></pb>
        <p type="head">
          <s id="s.003095"><emph type="italics"></emph>Simmiæ Rhodij <lb></lb>
Bipennis.<emph.end type="italics"></emph.end></s>
        </p>
        <figure id="id.009.01.182.1.jpg" place="text" xlink:href="009/01/182/1.jpg"></figure>
        <pb pagenum="183" xlink:href="009/01/183.jpg"></pb>
        <p type="head">
          <s id="s.003096"><emph type="italics"></emph>QVÆSTIO XX.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003097"><emph type="italics"></emph>De Statera.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003098"><arrow.to.target n="marg249"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003099"><margin.target id="marg249"></margin.target>259</s>
        </p>
        <p type="main">
          <s id="s.003100">Antequam ad textus explicationem accedamus, conſultius eſſe iu­<lb></lb>
dico veteris ſtateræ figuram, atque deſcriptionem præmittere, <lb></lb>
quàm ex hoc Ariſt. loco, magna mihi licuit cum delectatione col­<lb></lb>
ligere: quod etiam antiquitatis ſtudioſis pergratum fore non du­<lb></lb>
bito: <expan abbr="atq;">atque</expan> hinc etiam ineptas, <expan abbr="atq;">atque</expan> ad ſcititias textus huius figuras tanquam <lb></lb>
adulterinas reijcere; <expan abbr="inq́">inque</expan>; earum locum veras reſtituere licebit. </s>
          <s id="s.003101">erat igitur <lb></lb>
<figure id="id.009.01.183.1.jpg" place="text" xlink:href="009/01/183/1.jpg"></figure><lb></lb>
ſtatera, quantum ex Ariſt. conijcio <lb></lb>
primum haſta oblonga, qualis eſt in <lb></lb>
præſenti figura A B, ex cuius altero <lb></lb>
extremo B, pendebat appendicu­<lb></lb>
lum, quod propriè æquipondium <lb></lb>
dicitur: ex altera verò extremitate <lb></lb>
A, lanx vna pendebat; in qua carnes, aliæuè merces ponderabantur: in me­<lb></lb>
dia <expan abbr="deniq;">denique</expan> haſta paribus interuallis plures trutinæ, ex quibus ſingulis modo <lb></lb>
hac, modo illa, prout pondus emptoris poſtulabat ſuſpendebatur, <expan abbr="atq;">atque</expan> in­<lb></lb>
terim tantum mercis lanci imponebatur, donec æquipondio præpondera­<lb></lb>
ret in æquilibrio. </s>
          <s id="s.003102">ſingulæ autem trutinæ ad aliquod determinatum pondus <lb></lb>
trutinandum, erant conſtitutæ, v. g. vna ad ſex libras, altera ad octo, &amp;c. <lb></lb>
</s>
          <s id="s.003103">quam diuiſionem, ac fabricam ſtateræ non eſt difficilè exhibere, cum ex Ar­<lb></lb>
chimede propoſ. </s>
          <s id="s.003104">6. &amp; 7. de æquip. </s>
          <s id="s.003105">eadem ſit proportio inter pondus mer­<lb></lb>
cis, &amp; pondus æquipondij, quæ eſt permutatim inter diſtantias vtrinque ab <lb></lb>
aſſumpta trutina, quæ in trutinando hypomoclij vicem gerit: nam ſtatera <lb></lb>
reducitur ad vectem; pondus erit æquipondium; &amp; merces in lance erit po­<lb></lb>
tentia mouens: ſunt autem in tota ſtateræ haſta trutinæ plures, hoc enim <lb></lb>
modo tota fit vniformis quoad pondus. </s>
          <s id="s.003106">æquipondium præterea debet ha­<lb></lb>
bere tantum pondus, quantum eſt in nuda lance, vt ſic tota ſtatera ſit per ſe <lb></lb>
ſola æquilibrabilis: &amp; præterea debet habere pondus ſtatum, a c legitimum, <lb></lb>
v. g. vnius libræ, aut duarum, aut trium, prout magis <expan abbr="trutinãdæ">trutinandæ</expan> merci ido­<lb></lb>
neum erit, &amp; hoc erit proprium æquipondij pondus. </s>
          <s id="s.003107">vt autem ex ſingulis <lb></lb>
trutinis ſingula pondera ponderentur. </s>
          <s id="s.003108">ſingulis nota aliqua ſculpenda eſt, vt <lb></lb>
facilè mercatores merces ponderent, quod hac ratione fieri poteſt. </s>
          <s id="s.003109">pona­<lb></lb>
mus æquipondium eſſe 12. librarum. </s>
          <s id="s.003110">dico, quod trutina C, dabit in lance <lb></lb>
pondus mercis 12. librarum, ſi ex ea fiat æquilibrium, eſt enim vt A C, ad <lb></lb>
C B, ita permutatim æquipondium 12. ad mercem; ſed A C, ipſi C B, eſt <lb></lb>
æqualis, ergò etiam æquipondium 12. erit merci æquale, hoc eſt vtrunque <lb></lb>
erit, 12. librarum.</s>
        </p>
        <p type="main">
          <s id="s.003111">Similiter ſi fieret æquilibrium ex trutina D, eſſet vt A D, 3. ad B D, 9. <lb></lb>
ita 12. ad 36. tandem trutina E, æquilibrante, eſſet vt A E, 9. ad E B, 3. ita <lb></lb>
12. ad 4. Si igitur trutina C, notetur 12. numero, trutina D, num. </s>
          <s id="s.003112">36. tru­<lb></lb>
tina E, num. </s>
          <s id="s.003113">4. &amp; idem de cæteris: ſtatim facilè erit quodlibet pondus per <lb></lb>
huiuſmodi ſtateram exhibere. </s>
          <s id="s.003114">Vnde videas contrario ab illis modo in no­


<pb pagenum="184" xlink:href="009/01/184.jpg"></pb>ſtris ſtateris æquipondium totam haſtam percurrere; in illis verò manentè <lb></lb>
æquipondio trutinam quodammodo per haſtam moueri.</s>
        </p>
        <p type="main">
          <s id="s.003115">His præmiſſis ad textus paraphraſim veniamus.</s>
        </p>
        <p type="main">
          <s id="s.003116">Cur ſtatera, qua carnes ponderantur, paruo appendiculo magna truti­<lb></lb>
nat onera, cum alioquin tota ſtatera nihil aliud ſit, quàm dimidiata libra, <lb></lb>
vbi enim onus mercis imponitur vna lanx pendet, quam vnicam ſtatera ha­<lb></lb>
bet; in altera autem parte, vbi libra habet alteram lancem, ſtatera nullam <lb></lb>
habet, ſed ſola ſine lance eſt. </s>
          <s id="s.003117">Cauſa igitur eſt, quia ſtatera ſimul, &amp; libra eſt, <lb></lb>
&amp; vectis. </s>
          <s id="s.003118">libra eſt, quia ſpartorum, ſiue trutinarum quælibet fit veluti cen­<lb></lb>
trum libræ, <expan abbr="inq́">inque</expan>; altera parte eſt lanx; in altera verò loco lancis ipſum æqui­<lb></lb>
pondium, quod libræ incumbit, <expan abbr="fungiturq́">fungiturque</expan>; vice alterius lancis, cui ſit onus <lb></lb>
impoſitum; manifeſtum enim eſt, quod æquipondium ſtateræ tantumdem <lb></lb>
trahit oneris, quantum eſt illud, quod in altera lance eſt. </s>
          <s id="s.003119">eapropter ſtatera <lb></lb>
quodammodo tot libras in ſe continet, quot trutinas: quarum vna quæque <lb></lb>
cum ſit intra appendiculum, &amp; lancem, apta eſt eſſe medium, ſeu centrum <lb></lb>
ſtateræ, <expan abbr="atq;">atque</expan> adeo etiam libræ; quæ vnam quidem lancem habeat ex vna <lb></lb>
parte, ex altera verò pro lance æquipondium. </s>
          <s id="s.003120">ſtatera verò dicitur, quate­<lb></lb>
nus ex vna parte habet non lancem, ſed perpendiculum. </s>
          <s id="s.003121">ſed hoc nihil eſt <lb></lb>
aliud quàm eſſe plures in vna libras; Cur autem ſparta, quæ lanci, ſiue ap­<lb></lb>
penſo oneri proximiora ſunt, maiora ſubleuent onera, cauſa eſt vectis natu­<lb></lb>
ra, quæ ſtateræ ineſt. </s>
          <s id="s.003122">eſt enim ſtatera vectis, quamuis quodammodo inuer­<lb></lb>ſus, eſt enim ipſius fulcimentum trutina ipſa ſupernè collocata, pondus ve­<lb></lb>rò leuandum eſt ipſa merx, potentia verò appendiculum. </s>
          <s id="s.003123">quantò autem pro­<lb></lb>
ductior fuerit pars vectis à fulcimento ad potentiam, tanto facilius poten­<lb></lb>
tia mouet, vt in præſentia accidit. </s>
          <s id="s.003124">mouet autem <expan abbr="vſq;">vſque</expan> ad æquilibrium; <expan abbr="hocq́">hocque</expan>; <lb></lb>
modo pars illa productior ſtateræ, quæ vergit ad æquipondium, facit, vt <lb></lb>
onus ſtateræ impoſitum facilè trutinetur.</s>
        </p>
        <p type="head">
          <s id="s.003125"><emph type="italics"></emph>QVÆSTIO XXI.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003126"><emph type="italics"></emph>De Dentiforcipe.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003127"><arrow.to.target n="marg250"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003128"><margin.target id="marg250"></margin.target>260</s>
        </p>
        <p type="main">
          <s id="s.003129">Cvr Medici facilius dentes extrahunt dentiforcipis onere adiecto, <lb></lb>
quàm ſi ſola manu vtantur? </s>
          <s id="s.003130">fortè, quia ex manu facilius dens ela­<lb></lb>
bitur propter ſui ipſius lubricitatem, quàm ex forcipe. </s>
          <s id="s.003131">Vel etiam, <lb></lb>
quia digiti propter carnis mollitiem cedentem nequeunt dentem <lb></lb>firmiter circumplecti; ferrum verò, cum vndique durum æque ſit, nec ce­<lb></lb>dens, melius dentem comprehendit. </s>
          <s id="s.003132">Aut tandem, quia forceps hæc duos <lb></lb>
in ſe continet contrarios vectes; quorum, vnum tantum eſt hypomoclion, <lb></lb>
<figure id="id.009.01.184.1.jpg" place="text" xlink:href="009/01/184/1.jpg"></figure><lb></lb>eorum ſcilicet connexio; Virtute igitur <lb></lb>
vectis arctius dentem perſtringunt, <expan abbr="atq;">atque</expan> <lb></lb>
adeò obtinent, <expan abbr="atq;">atque</expan> hinc etiam facilius <lb></lb>
commouent. </s>
          <s id="s.003133">ſit dentiforcipis figura, ex­<lb></lb>
poſita, cuius alterum extremum, vbi ſunt <lb></lb>
A, B, eſt illud, quod binis ſemicirculis <lb></lb>
concurrentibus dentem arctè <expan abbr="cõſtringit">conſtringit</expan>,


<pb pagenum="185" xlink:href="009/01/185.jpg"></pb>&amp; commouet. </s>
          <s id="s.003134">Vectis vnus eſt A G D, alter B G C, communis fultura eſt G, <lb></lb>
vbi eſt ipſorum decuſſata connexio; dens loco ponderis eſt; vtroque igitur <lb></lb>
C, &amp; D, tanquam manubrijs vectium dentem Medici compræhendentes ip­<lb></lb>
ſum facilè commouent: quando autem commotus fuerit, facilius manu, <lb></lb>
quàm inſtrumento extrahitur.</s>
        </p>
        <p type="head">
          <s id="s.003135"><emph type="italics"></emph>QVÆSTIO XXII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003136"><emph type="italics"></emph>De Instrumento Nucifrago.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003137"><arrow.to.target n="marg251"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003138"><margin.target id="marg251"></margin.target>261</s>
        </p>
        <p type="main">
          <s id="s.003139">Tempore Ariſt. vt colligitur ex hac quæſtione, ad frangendas nu­<lb></lb>
ces peculiare inſtrumentum ligneum adhibeant, quod erat inſtar <lb></lb>
forcipis, ita tamen concinnatum, vt non ad ſcindendum, nec ad <lb></lb>
extrahendum, ſed ad frangendum per <expan abbr="cõpreſſionem">compreſſionem</expan> eſſet aptum. <lb></lb>
</s>
          <s id="s.003140">cuius hanc qualemcumque figuram inſpice. </s>
          <s id="s.003141">cuius latus inferius A D, fortè <lb></lb>
alicui fulcimento in plano horizontis, fixum hærebat: alterum verò A C, <lb></lb>
manu tractabatur, vt ſic expeditæ nucium plurima quantitas breui poſſet <lb></lb>
confringi. </s>
          <s id="s.003142">Credibile eſt nucifragam hanc ad capita F E, habuiſſe aliquod <lb></lb>
impedimentum, ne omninò conſtringeretur, vt nuces <expan abbr="frangerẽtur">frangerentur</expan> quidem, <lb></lb>
non autem comminuerentur. </s>
          <s id="s.003143">Cur igitur nuces <expan abbr="abſq;">abſque</expan> ictu facilè confringun­<lb></lb>
tur hiſce inſtrumentis, quæ ad eum fiunt vſum? </s>
          <s id="s.003144">contrarium <expan abbr="namq;">namque</expan> accidere <lb></lb>
deberet, vtentes enim prædictis inſtrumentis, omnibus illis viribus deſti­<lb></lb>
tuuntur, quas motio, ac violentia percuſſionis afferre ſolent. </s>
          <s id="s.003145">præterea cur <lb></lb>
ligneo vtuntur, ac proinde leui? </s>
          <s id="s.003146">non ne aptius eſſet durum, <expan abbr="atq;">atque</expan> pondero­<lb></lb>
ſum veluti ferreum?</s>
        </p>
        <p type="main">
          <s id="s.003147">His reſpondendum eſt, nucifragum iſtud inſtrumentum reduci ad binos <lb></lb>
vectes, quemadmodum etiam dentiforcipem. </s>
          <s id="s.003148">nux igitur hoc modo duplici <lb></lb>
vecte comprimitur. </s>
          <s id="s.003149">vecte autem facilè onera quælibet <expan abbr="obuiãtia">obuiantia</expan> diuelluntur. <lb></lb>
</s>
          <s id="s.003150">qui duo vectes vnicum habent hypomoclion ipſam ſcilicet connexionem <lb></lb>
<figure id="id.009.01.185.1.jpg" place="text" xlink:href="009/01/185/1.jpg"></figure><lb></lb>
A. vectes ſunt binæ inſtrumenti haſtæ, F A D, <lb></lb>
E A C. <expan abbr="dilatãdo">dilatando</expan> igitur extrema C D, deducun­<lb></lb>
tur etiam alia extrema F, E, &amp; impoſita nuce in <lb></lb>
hiatu K, quæuis potentia conſtringendo C, D, <lb></lb>
conſtringet ſimul F, E, <expan abbr="ipſamq́">ipſamque</expan>; nucem confrin­<lb></lb>
get. </s>
          <s id="s.003151">quod igitur cum percuſſione feciſſet pon­<lb></lb>
dus mallei, id valentiori vectium virtute efficiunt F A D, E A C. quanto au­<lb></lb>
tem locus nucis K, propinquior fuerit hypomoclio A, tanto celerius <lb></lb>
confringitur, quia partes vectium A C, A D, tunc à centro <lb></lb>
A, productiores fiunt, ideſt multò maiores fiunt, <lb></lb>
quàm ſint diſtantiæ inter nucem, &amp; cen­<lb></lb>
trum A, quod maximè poten­<lb></lb>
tiam iuuat.</s>
        </p>
        <p type="main">
          <s id="s.003152">Ex quibus præſenti quæſtioni ſatisfactum videtur.</s>
        </p>
        <pb pagenum="186" xlink:href="009/01/186.jpg"></pb>
        <p type="head">
          <s id="s.003153"><emph type="italics"></emph>QVÆSTIO XXIII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003154"><emph type="italics"></emph>De Rhombo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003155"><arrow.to.target n="marg252"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003156"><margin.target id="marg252"></margin.target>262</s>
        </p>
        <p type="main">
          <s id="s.003157">Rhombus ex definitione 23. primi Elem. eſt figura æquilatera qui­<lb></lb>
<figure id="id.009.01.186.1.jpg" place="text" xlink:href="009/01/186/1.jpg"></figure><lb></lb>
dem, ſed non æquiangula, habet enim <lb></lb>
binos oppoſitos angulos acutos, &amp; alies <lb></lb>
binos oppoſitos obtuſos, talis eſt præ­<lb></lb>
ſens figura A B D C. </s>
          <s id="s.003158">In præſenti porrò quæſtione <lb></lb>
ſupponitur punctum A, quod eſt vnum extremum <lb></lb>
in rhombo moueri ſuper latus A B, verſus B, &amp; ſi­<lb></lb>
militer interim æqua velocitate moueri alterum <lb></lb>extremum B, ſuper idem latus A B, verſus A, &amp; in­<lb></lb>
terim dum hæc duo puncta hoc modo ſibi obuiam <lb></lb>
procedunt, moueri latus totum A B, eadem ve­<lb></lb>
locitate, verſus latus C D, ita vt ſemper ipſi C D, <lb></lb>æquidiſter, <expan abbr="deſcendatq́">deſcendatque</expan>; per latera A C, B D, quo­<lb></lb>
uſque ipſi C D, congruat.</s>
        </p>
        <p type="main">
          <s id="s.003159">Horum igitur trium motuum quemadmodum <lb></lb>
æquæ ſunt celeritates, ita etiam ſpatia, quibus peraguntur, nam puncta duo <lb></lb>
mouentur in latere A B, ipſum verò A B, mouetur in lateribus A C, &amp; B D, <lb></lb>
quæ cum priori A B, ſunt æqualia.</s>
        </p>
        <p type="main">
          <s id="s.003160">Aduertendum præterea, quod hac ratione duo puncta A, &amp; B, duabus la­<lb></lb>
tionibus mouebuntur, ſi quidem proprio motu <expan abbr="mouẽtur">mouentur</expan> in ipſo latere A B, <lb></lb>
&amp; quia latus A B, per quod ipſa incedunt eodem tempore mouetur verſus <lb></lb>
C D, ſequitur, quod etiam ipſa hoc eodem motu ferantur. </s>
          <s id="s.003161">erit igitur ipſo­<lb></lb>
rum motus ex his duobus mixtus; &amp; quidem ipſius A, latio erit per longio­<lb></lb>
rem diametrum A D; ipſius verò B, per breuiorem B C. </s>
          <s id="s.003162">Quare cum pun­<lb></lb>
ctum A, peruenerit ad D, etiam punctum B, eadem cęleritate acceſſerit ad <lb></lb>
C. maius autem eſt ſpatium A D, quod confecit A, quam ſpatium B C, con­<lb></lb>
fectum a C. </s>
          <s id="s.003163">Quærit igitur primò, cur cùm A, &amp; B, mota ſint æquali celeri­<lb></lb>tate in vtra que latione, vnum tamen maiorem lineam, quàm alterum per­<lb></lb>
tranſiuit? </s>
          <s id="s.003164">Quærit ſecundò, cur punctum B, confecit lineam B C, quæ mi­<lb></lb>
nor eſt quam ipſum latus A C, quod in ſuo motu conficit latus A B, quando <lb></lb>
ad D C, acceſſit. </s>
          <s id="s.003165">&amp; tamen B, duplici fertur latione; A B, verò vnica; vtrun­<lb></lb>
que autem in æquali velocitate? </s>
          <s id="s.003166">Quod autem punctus A, motu illo deſcri­<lb></lb>
bat lineam A D, punctus verò B. lineam B C, manifeſtum erit hoc modo. </s>
          <s id="s.003167">ſit <lb></lb>
v. g. punctum A, motu proprio delatum, <expan abbr="vſq;">vſque</expan> ad punctum E, medium late­<lb></lb>
ris A B, erit interim totum latus A B, tranſlatum vbi eſt F G, hoc eſt, ad ſui <lb></lb>
itineris dimidium, quia horum motus ponuntur æquales: hoc autem motu <lb></lb>
ipſum punctum A, erit neceſſariò in K, hoc eſt in linea A D, vt dicebamus. <lb></lb>
</s>
          <s id="s.003168">Similiter in fine <expan abbr="vtriuſq;">vtriuſque</expan> motus, A, erit in B, proprio motu, ſed alieno in D, <lb></lb>
extremo ſcilicet lineæ A D. ſimili ratione oſtendi poteſt de ipſo B, qui cum <lb></lb>
æqua velocitate moueatur, ac punctum A, quando A erit in E; B, pariter <lb></lb>
illi occurret in E, proprio motu: ſed alieno à latere B A, proueniente erit


<pb pagenum="187" xlink:href="009/01/187.jpg"></pb>in K, vbi etiam ob alterum motum erit A: erit igitur B, in linea B C, vt vo­<lb></lb>
lebamus. </s>
          <s id="s.003169">à quo poſtea diſcedens verſus C, motu pariter compoſito ſiſtitur <lb></lb>
tandem in C, extremo lineæ pariter B C. eodem ergo tempore duo rhombi <lb></lb>
extrema puncta æquè velocia, ſecundum <expan abbr="vtramq;">vtramque</expan> lationem mota, interual­<lb></lb>
la nequaquam æqualia confecerunt, ſed A, maius, nimirum A D; B, verò <lb></lb>
minus nimirum B C.</s>
        </p>
        <p type="main">
          <s id="s.003170">Ex quibus etiam ſecundæ quæſtionis explicatio, &amp; dubitandi ratio pate­<lb></lb>
bit: nam cum in rhombo duo ſint obtuſi anguli B, &amp; C, &amp; duo acuti A, &amp; D, <lb></lb>
punctus ille, qui ab obtuſo angulo B, recedit, fertur duabus lationibus inui­<lb></lb>
cem ferè contrarijs, propria enim tendit ſurſum ad A, aliena verò deorſum <lb></lb>
trahitur verſus D; cauſa huius contrarietatis ſunt lineæ D B, B A, obtuſum <lb></lb>
angulum continentes, quæ à prædicto angulo in contrarias partes ſeparan­<lb></lb>
tur: per has autem lineas fiunt prædicti motus, vnde ipſi quoque contrarij <lb></lb>
ſint neceſſe eſt: &amp; propterea ſe mutuò impediunt: <expan abbr="atq;">atque</expan> hinc neceſſe eſt pun­<lb></lb>ctum B, motu compoſito hinc inhibito minus interuallum B C, pertranſire. <lb></lb>
</s>
          <s id="s.003171">At verò punctum A, quia ab acuto angulo deſcendit, <expan abbr="vtraq;">vtraque</expan> latione fertur <lb></lb>
deorſum, quæ lationes ſe mutuò iuuant, <expan abbr="faciuntq́">faciuntque</expan>; vt A, maius, quamuis eo­<lb></lb>
dem tempore, &amp; eadem celeritate peragret ſpatium A D. nam punctum A, <lb></lb>
ſua ſpontè <expan abbr="deſcẽdit">deſcendit</expan> per latus A B, &amp; ab ipſo latere A B, quod fertur ad C D, <lb></lb>
pariter deorſum vehitur. </s>
          <s id="s.003172">nihil igitur mirum fit, ſi A, maius <expan abbr="interuallũ">interuallum</expan> A D, <lb></lb>
quam B C, percurrat. </s>
          <s id="s.003173">cauſa verò huius motuum concordiæ eſt angulus acu­<lb></lb>
tus A, ob quem latera rhombi magis inuicem approximantur, redduntque <lb></lb>
longiorem A D, quàm B C: è contrariò autem, quo obtuſiores ſunt anguli <lb></lb>
B, C, minorem faciunt ipſam B C, latera enim ſemper magis ad rectam li­<lb></lb>
neam accedunt; donec tandem omni angulo euaneſcente in directum con­<lb></lb>
ſtituantur; quo caſu congruerent cum linea A D, <expan abbr="rhombusq́">rhombusque</expan>; ipſe amplius <lb></lb>
nullus eſſet.</s>
        </p>
        <p type="main">
          <s id="s.003174">Ex his igitur ſequitur, quod punctum A, ab angulo A, acuto diſcedens, <lb></lb>
duobus feratur motibus ſimilibus ad eandem partem tendentibus, &amp; quò <lb></lb>
acutiores ſunt anguli, eò magis tendent ad eandem partem; &amp; melius ſe <lb></lb>mutuò iuuabunt. </s>
          <s id="s.003175">B, autem vice verſa, quoniam quanto obtuſior eſt angulus <lb></lb>
B, tanto magis latera illius diuaricantur; duæ etiam motiones, quibus B, <lb></lb>
progreditur in diuerſas partes tendent; fiunt enim per illa latera; &amp; tanto <lb></lb>
etiam magis ſibi contrariæ erunt; <expan abbr="magisq́">magisque</expan>; ſibi mutuò impedimento erunt. <lb></lb>
</s>
          <s id="s.003176">&amp; propterea punctum B, minus interuallum, quale eſt B C, percurret, quan­<lb></lb>
do A, maius A D, percurrit.</s>
        </p>
        <p type="main">
          <s id="s.003177">Ad ſecundam verò quæſtionis partem, reſpondeo conſiderandum eſſe <lb></lb>
latus B A, moueri vnico motu ad D C, quare à nullo impedi­<lb></lb>
tur, vnde nihil mirum videri debet, quòd ipſum vnica <lb></lb>
latione maius conficiat ſpacium quàm B, quod <lb></lb>
quamuis duplici pellatur motu, vnus <lb></lb>
tamen ab altero inhibetur.</s>
        </p>
        <pb pagenum="188" xlink:href="009/01/188.jpg"></pb>
        <p type="head">
          <s id="s.003178"><emph type="italics"></emph>QVÆSTIO XXIIII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003179"><emph type="italics"></emph>De duobus circulis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003180"><arrow.to.target n="marg253"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003181"><margin.target id="marg253"></margin.target>263</s>
        </p>
        <p type="main">
          <s id="s.003182">Vnde eſt, quod ſi duo circuli, vnus altero maior, circa idem cen­<lb></lb>
trum poſiti, volutentur, ita vt etiam centrum feratur, eo ſcilicet <lb></lb>
modo, quo plauſtrorum rotæ ſolent, ſecundum æqualem lineam <lb></lb>
conuoluuntur, ſiue æquale ſpatium conficiunt: ſi verò ſeorſum <lb></lb>
ſeparati quilibet eodem modo volutetur, non æquale <expan abbr="ſpatiũ">ſpatium</expan> pertranſibunt, <lb></lb>
ſed maior maiorem lineam, quàm minor; <expan abbr="idq́">idque</expan>; ea proportione, quam inui­<lb></lb>
cem eorum circunferentiæ obtinent, cum in hac veluti rotæ conuolutione, <lb></lb>
circunferentia tota ſucceſſiuè decurſo ſpatio adaptetur, ita vt tanta ſit de­<lb></lb>
curſa linea, quanta eſt rotæ circunferentia? </s>
          <s id="s.003183">Quin etiam eodem exiſtente <lb></lb>
<expan abbr="vtriuſq;">vtriuſque</expan> centro, aliquando confectum ſpatium ab <expan abbr="vtroq;">vtroque</expan> tantum eſt, quan­<lb></lb>
tum minor circulus ſolus, ſecundum ſuam periphæriam reuolutus perfeciſ­<lb></lb>
ſet; <expan abbr="quandoq́">quandoque</expan>; verò quantum maior ſolus abſoluiſſet. </s>
          <s id="s.003184">Quod autem maior <lb></lb>
ſolus in ſua reuolutione maiorem lineam deſcribat, manifeſtum eſt hinc, <lb></lb>
quia ſenſu patet maiorem circunferentiam in maiori circulo ſubtendere <lb></lb>
angulum, qui fit à diametris in centro; minorem verò circunferentiam <lb></lb>
ſubtendere eundem angulum in minori orbe, vt etiam in 8. quæſt. </s>
          <s id="s.003185"><expan abbr="dictũ">dictum</expan> eſt: <lb></lb>
eandem igitur, vt proximè dixi habebunt etiam proportionem illæ lineæ, <lb></lb>
quæ à ſingulis ſeorſum orbibus reuolutis deſignabuntur. </s>
          <s id="s.003186">Quod præterea ſe­<lb></lb>
cundum æqualem conuoluuntur, quando circa idem poſiti fuerint centrum, <lb></lb>
manifeſtum eſt, ita tamen, vt aliquando ambæ æquales ſint ei, ſecundum <lb></lb>
quam ſolus maior conuolueretur; aliquando verò ſecundum quam minor. <lb></lb>
<figure id="id.009.01.188.1.jpg" place="text" xlink:href="009/01/188/1.jpg"></figure><lb></lb>
ſit enim circulus maior quidem vbi <lb></lb>
D F C, minor verò vbi E G B, <expan abbr="vtriq;">vtrique</expan> <lb></lb>
autem centrum A, linea, ſecundum <lb></lb>
quam quadrans F C, maioris per ſe <lb></lb>
rotaretur, ſit F L. linea verò, ſecun­<lb></lb>
dum quam <expan abbr="quadrãs">quadrans</expan> G B, minoris ſe­<lb></lb>
iuncti à maiori, volutaretur ſit G K, <lb></lb>
quæ æqualis eſt dicto quadranti G B, <lb></lb>
ſicut etiam F I, æqualis eſt quadran­<lb></lb>
ti F C. ſi quis igitur impellat mino­<lb></lb>
rem orbem mouens ſimul commune <lb></lb>
centrum A, cui maior eſt circumpo­<lb></lb>
ſitus, donec diameter A B, perpendicularis ſit lineæ G K, in puncto K. tunc <lb></lb>
pariter diameter maioris A C, erit perpendicularis lineæ F L, in puncto L. <lb></lb>
</s>
          <s id="s.003187">G K, autem, &amp; F L, neceſſariò erunt æquales per 34. primi, æquales igitur <lb></lb>
lineas hoc modo peragrarunt inæquales circunferentiæ, ſiue quadrantes <lb></lb>
G B, F C. ſi autem quadrantes hoc præſtant, manifeſtum eſt, quod &amp; toti <lb></lb>
ambitus idem efficiunt, quare quando tota periphæria G B E G, fuerit re­<lb></lb>
uoluta etiam tota F C D F, ſuum orbem <expan abbr="completũ">completum</expan> habebit. </s>
          <s id="s.003188">ſimiliter ſi ma­<lb></lb>iorem quis mouerit, cui minor ſit annexus eodem exiſtente centro, ſimul ac <pb pagenum="189" xlink:href="009/01/189.jpg"></pb>diameter A C, erit perpendicularis ad F I, in puncto I, erit etiam A B, per­<lb></lb>
pendicularis ipſi G M, in M; ſunt autem G M, &amp; F I, æquales, quare quan­<lb></lb>
do F C, quadrans maioris pertranſiuerit rectam F C, etiam C B, quadrans <lb></lb>
minoris tranſactam habebit illi parem G M. hoc autem accidit nulla inter­<lb></lb>
cedente mora in vllo ipſorum: quando enim mouetur maior, nihil ceſſat <lb></lb>
minor: &amp; quando minor agitur, maior nunquam quieſcit. </s>
          <s id="s.003189">quod ſi hoc acci­<lb></lb>
dit quartæ parti circulorum, idem, &amp; totis accidit periphærijs. </s>
          <s id="s.003190">vbi inſuper <lb></lb>
illud etiam mirum, centrum nimirum ipſorum eadem celeritate motum, <lb></lb>ac vnica ſemper exiſtenti latione, modo maius, modo minus ſpatium per­<lb></lb>
ficere; idem verò eadem velocitate latum, æquale ſemper deberet interual­<lb></lb>
lum tranſilire. </s>
          <s id="s.003191">&amp; tamen in præſentia vtrouis modo moueas eadem pernici­<lb></lb>
tate, modò maius, modò minus ſpatium pertranſibit.</s>
        </p>
        <p type="main">
          <s id="s.003192">Huius quæſtionis enodandæ cauſa, ſupponendum primò eſt, quod eadem, <lb></lb>
ſeu æqualis potentia, hanc quidem magnitudinem tardius, illam verò citius <lb></lb>
mouere poteſt. </s>
          <s id="s.003193">ſi enim fuerit quippiam, quod à ſeipſo moueri minimè ap­<lb></lb>
tum ſit; &amp; aliud, quod à ſe ipſo moueri aptum ſit; qui hoc ſimul cum illo <lb></lb>
coniunctum mouerit, tardius mouebit, quàm ſi ipſum ſolum moueret. </s>
          <s id="s.003194">&amp; ſi <lb></lb>
quid moueatur, quod aptum ſit ex ſe moueri, verumtamen in eo motu nihil <lb></lb>
ex ſe moueatur, perinde eſt, ac ſi minimè aptum ſit ad motum, &amp; proinde <lb></lb>
tardius mouebitur; nec fieri poterit, vt pluſquam mouens moueatur, cum <lb></lb>
nihil innata motione vtatur. </s>
          <s id="s.003195">Si quis igitur minorem circulum, quem mo­<lb></lb>
do B, appello, mouerit ſupra ſuam circunferentiam, cui annexus ſit maior, <lb></lb>
quem modo appello A, ſic quidem maior mouebitur, non autem ex ſe, ſed <lb></lb>
ſolum quatenus à minori feretur, vnde tantum pertranſibit de recta F L, <lb></lb>
quantum à minori fuerit impulſus; tantum autem eſt impulſus, quantum <lb></lb>
minor eſt motus; quare æqualem cum illo viam confecit. </s>
          <s id="s.003196">ſi igitur minor fe­<lb></lb>
cit pedalem G K, maior confecit etiam pedalem F L, quia maior nihil de <lb></lb>
proprio motu addidit, ſed ſolum motione minoris eſt tranſlatus. </s>
          <s id="s.003197">ſimiliter <lb></lb>
ſi quis rotet maiorem ſupra ſuam circunferentiam annexo minori, tantum <lb></lb>
minor mouebitur, quantum à maiori deportabitur, quia nihil ex ſe impel­<lb></lb>
litur. </s>
          <s id="s.003198">Verum ſi ſeorſum ambo ex ſe ſecundum ſuos ambitus moueantur, ſiue <lb></lb>
citò, ſiue tardè, eadem etiam velocitate perficiant integram ſuæ periphæ­<lb></lb>
riæ volutationem, maior maius, minor verò minus conficiet ſpatium.</s>
        </p>
        <p type="main">
          <s id="s.003199">Sed fortè augebitur difficultas conſideranti, quod prædicti circuli ſunt <lb></lb>
circa idem centrum, &amp; circa illud mouentur. </s>
          <s id="s.003200">moueri autem circulum cir­<lb></lb>
ca ſuum centrum, eſt moueri ſecundum ſuum naturalem motum, ad quem <lb></lb>
circuli ex ſe ſunt apti. </s>
          <s id="s.003201">ſi verò vnus moueretur circa ſuum centrum, alter ve­<lb></lb>
rò non, vt quando alter alteri non eſt circa idem centrum compactus, &amp; ab <lb></lb>
altero mouetur, vbi manifeſtè apparet, quod fertur omninò ab illo, &amp; in il­<lb></lb>
la latione non circumuertitur circa proprium centrum, quare tunc minimè <lb></lb>
mirum eſt, ſi <expan abbr="neq;">neque</expan> plus, <expan abbr="neq;">neque</expan> minus ſpatium conficiat, quàm ab altero de­<lb></lb>
portetur, cui quoquo modo adiacet, aut appenſus eſt extra illius centrum.</s>
        </p>
        <p type="main">
          <s id="s.003202">Huic obiectioni <expan abbr="reſpondẽdum">reſpondendum</expan> eſt, quod quamuis prædicti orbes ſint con­<lb></lb>
centrici, nihilominus non mouentur ambo ſuamet motione, ſed ille, qui ab <lb></lb>
alio fertur mouetur ſecundum motionem illam, tanquam ſi nullam ad eam <lb></lb>haberet aptitudinem; quamuis enim poſſit moueri circa centrum illud A,<pb pagenum="190" xlink:href="009/01/190.jpg"></pb>propria natura, in præſenti tamen caſu minimè vtitur illa aptitudine; &amp; <lb></lb>
propterea motus debet moueri, quantum mouens, nec plus, nec minus.</s>
        </p>
        <p type="main">
          <s id="s.003203">Quòd autem ſpectat ad id, quod initio dicebatur de eodem centro, &amp; de <lb></lb>
mouente eadem velocitate, &amp; de æquali ab inæqualibus orbibus pertranſi­<lb></lb>ta linea, ſubeſt huic dubitationi paralogiſmus: quamuis enim ſit idem am­<lb></lb>
borum centrum, eſt tamen vnius centrum per ſe in motione, alteri verò per <lb></lb>
accidens, veluti per accidens eſt eundem virum eſſe muſicum, &amp; album. </s>
          <s id="s.003204">ille <lb></lb>
enim circulus, qui mouet alterum, obtinet illud centrum per ſe, &amp; ex natu­<lb></lb>
ra ſua; alter verò, qui mouetur, habet illud idem per accidens, quia non <lb></lb>
vtitur illo tanquam centro. </s>
          <s id="s.003205">non igitur circa idem ſimpliciter centrum fit <lb></lb>
horum motus, ſed alio modo vnus, alio modo alter, vnde &amp; reliquis dubi­<lb></lb>
tationibus facilè ſatisfiet.</s>
        </p>
        <p type="head">
          <s id="s.003206"><emph type="italics"></emph>QVÆSTIO XXV.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003207"><emph type="italics"></emph>De Lecto<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003208"><arrow.to.target n="marg254"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003209"><margin.target id="marg254"></margin.target>264</s>
        </p>
        <p type="main">
          <s id="s.003210">Cvr lectulorum ſpondas faciunt ſecundum duplam proportionem, <lb></lb>
hoc eſt longiorem ſpondam duplo longiorem, quàm ſit altera: il­<lb></lb>
lam enim ſex pedum, vel paulò plus, hanc verò trium? </s>
          <s id="s.003211">præterea <lb></lb>
cur reſtes, quibus culcitræ ſuſtinentur non extendunt per diame­<lb></lb>
trum, ſed per tranſuerſum?</s>
        </p>
        <p type="main">
          <s id="s.003212">Ad primum reſpondetur ideò facere ſpondas in dupla ratione, vt ſint hu­<lb></lb>
mano corpori proportionatæ, ſic enim lecti longitudinem habebunt qua­<lb></lb>
tuor cubitorum, latitudinem verò duorum, in tali enim ſpatio commo­<lb></lb>
dè cubamus.</s>
        </p>
        <p type="main">
          <s id="s.003213">Ad ſecundum verò dicendum extendi illos funes non per diametrum, ſed <lb></lb>
ex oppoſito, quia hoc modo ligna ipſius lecti minus diſtrahuntur: facilè <lb></lb>
enim ex natura ſua ligna hæc ab inuicem ſecundum longum ſeparantur; ar­<lb></lb>
ctius autem ductis funibus per tranſuerſum, quàm per diametrum inuicem <lb></lb>
conſtringuntur: præterea, quia ſic etiam funes minus laborant, cum ſint eo­<lb></lb>rum ductus breuiores; &amp; quia debent ſuſtinere onus ſtragulorum, <expan abbr="atq;">atque</expan> cul­<lb></lb>
cìtrarum, ſic certè ex hoc onere minus laborabunt ſi tranſuerſim, quàm ſi <lb></lb>
diametraliter ſubtendantur.</s>
        </p>
        <p type="main">
          <s id="s.003214">Tertia demum ratio eſt, quia hac ratione minus reſtium abſumitur, quæ <lb></lb>
<figure id="id.009.01.190.1.jpg" place="text" xlink:href="009/01/190/1.jpg"></figure><lb></lb>vt benè intelligatur, deſcriba­<lb></lb>
tur lectuli figura A F G K, &amp; <lb></lb>
bifariam diuidatur latus F G, <lb></lb>
in B. &amp; quia tota F G, dupla <lb></lb>
eſt ipſius A F, erit dimidium <lb></lb>
F B, æquale ipſi A F. &amp; propte­<lb></lb>
rea tot erunt foramina, quibus <lb></lb>
funes immittuntur in F B, quot <lb></lb>
in A F. <expan abbr="extẽdunt">extendunt</expan> autem funem <lb></lb>
hoc modo incipiunt ab A, &amp; <lb></lb>
ducunt ad B, poſtea per C, re­


<pb pagenum="191" xlink:href="009/01/191.jpg"></pb>uertuntur ad D; hinc flectunt per H, vſque ad E, &amp; per G, angulum iterum <lb></lb>
deſcendunt ad M, à quo recta tendunt in F, hinc per 2. deducunt ad 3. à quo <lb></lb>
foramine, per foramen 4. reflexum faciunt ad 5. à quo iterum per B, deſcen­<lb></lb>
dunt ad angulum K, <expan abbr="ibiq́">ibique</expan>; alterum funis extremum deſinit: <expan abbr="hocq́">hocque</expan>; modo duo <lb></lb>
anguli A, &amp; K, reſtis habent capita, &amp; reſtes extenſæ ſunt non diametrali­<lb></lb>
ter, ſed tranſuerſim.</s>
        </p>
        <p type="main">
          <s id="s.003215">Notandum autem, quod reſtes æquales ſunt cum ſuis curuaturis. </s>
          <s id="s.003216">v. g. re­<lb></lb>
ſtis A B, cum ſua curuatura B C, æqualis eſt reſti C D, vnà cum eius curua­<lb></lb>
tura D H, &amp; aliæ eodem modo ſe habent, quia eadem demonſtratio omni­<lb></lb>
bus accommodari poteſt: quia enim figura A B G M, parallelogrammum <lb></lb>
eſt, æqualia enim ſunt latera B G, A M, &amp; quot foramina ſunt in vno, tot <lb></lb>
etiam ſunt in altero, <expan abbr="eaq́">eaque</expan>; inuicem æquidiſtant, ſequitur omnes reſtes eſſe <lb></lb>
parallelas, &amp; æquales, per 33, primi. </s>
          <s id="s.003217">ex qua etiam ſequitur prædictas cu­<lb></lb>
ruaturas, B C, D H, E G, eſſe æquales. </s>
          <s id="s.003218">quare manifeſtum eſt in dimidio le­<lb></lb>
ctulo tot eſſe reſtes æquales reſti A B, quot ſunt foramina in dimidio latere <lb></lb>
B G, vel in dimidio F B, hoc eſt eſſe quatuor. </s>
          <s id="s.003219">porrò oportet quantitatem <lb></lb>
harum omnium reſtium perſcrutari, vt eam cum quantitate reſtium diame­<lb></lb>
traliter extenſarum conferamus, quod geometricè hoc modo aſſeque mur: <lb></lb>
triangulum enim B G K, rectangulum eſt, ergò per 47. primi, quadrata la­<lb></lb>terum B G, G K, æqualia ſunt quadrato lineæ B K: latus B G, eſt trium pe­<lb></lb>
dum, quemadmodum etiam latus G K quadratus autem numerus ternarij <lb></lb>
eſt 9. ergo duo quadrati numeri 9. ſiue 18. æquales ſunt quadrato lineæ B K, <lb></lb>
ergò linea B K, eſt radix quadrata numeri 18. quæ radix non poteſt exactè <lb></lb>
in numeris repræſentari, eſt enim, vt aiunt, radix ſurda. </s>
          <s id="s.003220">verumtamen per <lb></lb>
radicum extractionem, <expan abbr="atq;">atque</expan> approximationem ea poni poteſt eſſe 41/4. ideſt <lb></lb>
quatuor pedum cum vna quarta. </s>
          <s id="s.003221">cum igitur in toto lecto ſint huiuſmodi <lb></lb>
octo reſtes, erit omnium ſumma pedum 34. ferè. </s>
          <s id="s.003222">ſi autem ſeeundum diame­<lb></lb>
trum extendantur reſtes, vti factum eſt in lectulo A B C D, neutiquam re­<lb></lb>
ſtes omnes ſimul ſuperiori quantitati adæquabuntur, ſed illam longè ſupe­<lb></lb>
<figure id="id.009.01.191.1.jpg" place="text" xlink:href="009/01/191/1.jpg"></figure><lb></lb>
rabunt. </s>
          <s id="s.003223">Sit igitur lectus A B­<lb></lb>
C D, in quo diametraliter du­<lb></lb>
ctæ ſint reſtes B D, E H, &amp; re­<lb></lb>
liquæ, vt in figura. </s>
          <s id="s.003224">harûm quan­<lb></lb>
titas ſi per 47. primi, &amp; per ra­<lb></lb>
dicis quadratæ extractionem <lb></lb>
inueniatur, erit ſumma earum <lb></lb>
pedum quadraginta cum dimi­<lb></lb>
dio; quæ quantitas præcedenti <lb></lb>
maior eſt ſex pedibus cum di­<lb></lb>
midio.</s>
        </p>
        <p type="main">
          <s id="s.003225"><expan abbr="Atq;">Atque</expan> hic eſt ſenſus Ariſt. quamuis tex. ipſius propter nimiam tam in græ­<lb></lb>
cis, quàm in latinis codicibus corruptionem, totus reſtitui nequiuerit.</s>
        </p>
        <pb pagenum="192" xlink:href="009/01/192.jpg"></pb>
        <p type="head">
          <s id="s.003226"><emph type="italics"></emph>QVÆSTIO XXVI.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003227"><emph type="italics"></emph>De ligno humeris gestato.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003228"><arrow.to.target n="marg255"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003229"><margin.target id="marg255"></margin.target>265</s>
        </p>
        <p type="main">
          <s id="s.003230">Cvr difficilius eſt <expan abbr="lõga">longa</expan> ligna ab extremo ſuper humeros ferre, quàm <lb></lb>
ſecundum medium, cùm tamen <expan abbr="vtroq;">vtroque</expan> modo ſit ſemper idem pon­<lb></lb>
dus? </s>
          <s id="s.003231">An quia dum fertur lignum ſuper humeros ab altero extre­<lb></lb>mo, alterum extremum vibratur, &amp; agitatur, quæ agitatio ipſius <lb></lb>
lationem impedit? </s>
          <s id="s.003232">An quia licet nihil inflectatur ob agitationem, <expan abbr="neq;">neque</expan> ma­<lb></lb>
gnam habeat longitudinem, difficilius tamen ab extremo fertur, quoniam <lb></lb>
facilius ex medio eleuatur, quàm ab extremo, &amp; quia latio eſt quaſi quæ­<lb></lb>
dam continua eleuatio, propterea etiam difficilius ſic portatur? </s>
          <s id="s.003233">cauſa au­<lb></lb>
tem cur facilius ex medio eleuetur eſt, quia hoc modo totum lignum fit ve­<lb></lb>
ctis, cuius hypomoclion eſt in medio, vbi is, qui eleuat, tenet aut fert: ex­<lb></lb>
trema autem ſibi mutuò <expan abbr="æqueponderãt">æqueponderant</expan>, ita vt <expan abbr="abſq;">abſque</expan> vllo alio auxilio, â tan­<lb></lb>
ta vi, quantum eſt totum ligni pondus ſuſtineatur; quod ſi ab extremo ele­<lb></lb>
uetur non ſufficit amplius prædicta vis, ſed opus erit maiori, quia non ſo­<lb></lb>lum oportebit illud eleuare, ſed præterea etiam illud in æquilibrio conſti­<lb></lb>
tuere, &amp; conſeruare. </s>
          <s id="s.003234">pondus enim totius ligni vergit ferè ad alteram ligni <lb></lb>medietatem, quæ ab hypomoclio productior euadit, quapropter ad onus <lb></lb>
iſtud æquilibrandum, opus eſt alia potentia in altero extremo. </s>
          <s id="s.003235">ſit lignum <lb></lb>
<figure id="id.009.01.192.1.jpg" place="text" xlink:href="009/01/192/1.jpg"></figure><lb></lb>
A B, ſuſpenſum ex medio C. <lb></lb>
hoc modo lignum ponderi­<lb></lb>
bus libratum ſuis manet in <lb></lb>
æquilibrio, poteſtque à ſola <lb></lb>
potentia illud eleuante etiam deferri: quia A, &amp; B, extrema ſe mutuò ſuſti­<lb></lb>
<figure id="id.009.01.192.2.jpg" place="text" xlink:href="009/01/192/2.jpg"></figure><lb></lb>
nent. </s>
          <s id="s.003236">quod ſi non ex medio eleuaretur, <lb></lb>
ſed ab extremo, vt in ſecunda figura, <lb></lb>
eleuans potentia ex C, æqualis oportet, <lb></lb>
vt ſit præcedenti; ſed præterea opus eſt <lb></lb>
alia vi, quæ in B, æquiponderet alteri <lb></lb>
extremo A, quod magis grauitat, quo ab C, longius fuerit; &amp; hoc modo in <lb></lb>
æquilibrio conſtitutum, &amp; conſeruatum poterit non ſolum eleuari, ſed <lb></lb>
etiam circumferri.</s>
        </p>
        <p type="head">
          <s id="s.003237"><emph type="italics"></emph>QVÆSTIO XXVII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003238"><emph type="italics"></emph>De Gestatis ſuper humerum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003239"><arrow.to.target n="marg256"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003240"><margin.target id="marg256"></margin.target>266</s>
        </p>
        <p type="main">
          <s id="s.003241">Cvr ſi valdè procerum ſuerit idem pondus difficilius ſuper humeros <lb></lb>
geſtatur, etiam ſi ex medio illud feratur, quàm ſi breuius ſit? </s>
          <s id="s.003242">quod <lb></lb>
enim dudum dictum eſt cauſa non eſt, ſed vibratio, &amp; ſuccuſſatio <lb></lb>
ligni nunc eſt: quando enim ab humero productius fuerit, magis <lb></lb>
vibrantur extrema, quam ob rem contingit portantem difficilius geſtare. <lb></lb>
</s>
          <s id="s.003243">vibrationis autem cauſa eſt, quoniam ab eadem vi mouente magis extrema <pb pagenum="193" xlink:href="009/01/193.jpg"></pb>huc illuc transferuntur, quanto procerius fuerit lignum, quia tunc maior <lb></lb>
fit diſtantià à centro, ſeu hypomoclio, quod modo eſt humerus ipſe. </s>
          <s id="s.003244">ſit vt <lb></lb>
in prima præcedentis quæſtionis figura, humerus vbi A. diſtantiæ autem ab <lb></lb>
ipſo centro ſunt A B, A C, quod autem maior diſtantia; faciliorem reddat <lb></lb>
motum oſtenſum eſt initio huius operis.</s>
        </p>
        <p type="head">
          <s id="s.003245"><emph type="italics"></emph>QVÆSTIO XXVIII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003246"><emph type="italics"></emph>De Tollenone.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003247"><arrow.to.target n="marg257"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003248"><margin.target id="marg257"></margin.target>267</s>
        </p>
        <p type="main">
          <s id="s.003249">Inſtrumentum iſtud, quod græca voce Leonicus interpres Celonia vo­<lb></lb>
cat, latinis dicitur Tolleno, à tollendo; quod etiam manifeſtum eſt <lb></lb>
ex Feſto, qui ait, Tolleno eſt genus machinæ, quo hauritur aqua in al­<lb></lb>
teram partem prægrauante pondere; quæ tollenonis deſcriptio om­<lb></lb>
ninò machinæ præſentis quæſtionis competit. </s>
          <s id="s.003250">Hiſpani Telonam fortè a tol­<lb></lb>
lenone nuncupant. </s>
          <s id="s.003251">Eſt autem tolleno <expan abbr="inſtrumentũ">inſtrumentum</expan> hauriendæ è puteo aquæ <lb></lb>
idoneum, quo ruſtici paſſim vtuntur: <expan abbr="idq́">idque</expan>; iuxta puteos ſtabile, ac firmum <lb></lb>
conſtruunt, quale à figura ſequenti refertur. </s>
          <s id="s.003252">vbi puteus F, tolleno conſtat <lb></lb>
<figure id="id.009.01.193.1.jpg" place="text" xlink:href="009/01/193/1.jpg"></figure><lb></lb>
erecto tigno D C, &amp; tranſ­<lb></lb>
uerſa haſta A C B, vnà cum <lb></lb>
fune B E, &amp; hydria E. ap­<lb></lb>
ponitur præterea onus ſa­<lb></lb>
tis graue ad <expan abbr="partẽ">partem</expan> A, quale <lb></lb>
eſt G. haſta porrò A B, ve­<lb></lb>
luti vectis circa <expan abbr="pũctum">punctum</expan> C, <lb></lb>
tanquam hypomoclion, <lb></lb>
<expan abbr="ſusq́">ſusque</expan>; <expan abbr="deq́">deque</expan>; agitur, à poten­<lb></lb>
tia funem B E, trahente. <lb></lb>
</s>
          <s id="s.003253">ſed iam textus exponatur.</s>
        </p>
        <p type="main">
          <s id="s.003254">Cur iuxta puteos tolle­<lb></lb>
nones faciunt eo, quo vi­<lb></lb>
ſuntur modo, ligno enim <lb></lb>
tranſuerſo A B, adiungunt <lb></lb>
onus plumbi G, cum alio­<lb></lb>
quin vas ipſum E, &amp; vacuum, &amp; plenum pondus habeat: cur inquam, vt fa­<lb></lb>
cilius moueant tollenonem, tollenonis oneri onus addunt G? </s>
          <s id="s.003255">An quoniam <lb></lb>
cùm opus hauriendi diuidatur in duo, in intingendi nimirum, &amp; ſurſum tra­<lb></lb>
hendi tempora: accidit quidem <expan abbr="abſq;">abſque</expan> plumbi onere facilius intingere, quia <lb></lb>
tunc vas eſt vacuum: at verò ſurſum vas deinde plenum trahere, laborio­<lb></lb>
ſius erit. </s>
          <s id="s.003256">ſi verò addatur onus G, tunc quidem paulò difficilius intingemus, <lb></lb>
ſed tamen vas plenum poſtea multò facilius, quod opus, &amp; labor eſt, ſurſum <lb></lb>
educemus: operæpretium igitur eſt, onus illud plumbi, aut lapidis adiun­<lb></lb>
gere in extremo A, quia ſic pondus illud tanquam quædam potentia vecte <lb></lb>
A B, vtens ſurſum hydriam plenam rapiet, <expan abbr="hacq́">hacque</expan>; ratione nos labore leua­<lb></lb>
bit, <expan abbr="totumq́">totumque</expan>; hauriendi opus demiſſione, <expan abbr="atq;">atque</expan> eleuatione <expan abbr="conſtãs">conſtans</expan>, alleuabit,</s>
        </p>
        <pb pagenum="194" xlink:href="009/01/194.jpg"></pb>
        <p type="head">
          <s id="s.003257"><emph type="italics"></emph>QVÆSTIO XXVIIII.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003258"><emph type="italics"></emph>De onere phalanga gestato.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003259"><arrow.to.target n="marg258"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003260"><margin.target id="marg258"></margin.target>268</s>
        </p>
        <p type="main">
          <s id="s.003261">Cvr quando ſuper ligno, aut huiuſmodi quopiam duo portauerint <lb></lb>
homines æquale pondus, non ſimiliter grauantur, niſi quando pon­<lb></lb>
dus in medio eorum fuerit; ſed magis ille premitur, cui onus vici­<lb></lb>
nius fuerit? </s>
          <s id="s.003262">An quia lignum illud vectis efficitur, cuius hypomo­<lb></lb>
clion eſt vbi pondus geſtatum ſuſpenditur; geſtantium autem oneri proxi­<lb></lb>
mior gerit vicem illius, quod vecte mouetur, remotior verò eſt potentia <lb></lb>
vecte mouens. </s>
          <s id="s.003263">quanto igitur plus diſtat ab hypomoclio, ſeu geſtato ponde­<lb></lb>
re, tanto facilius mouet, hoc eſt, alterum magis deorſum premit, contra­<lb></lb>
nitente nimirum geſtato onere <expan abbr="tãquam">tanquam</expan> hypomoclio. </s>
          <s id="s.003264">ſi autem in medio fue­<lb></lb>
rit pondus, nihilo magis alter geſtantium fit id, quod vecte mouetur, quàm <lb></lb>
alter; <expan abbr="neq;">neque</expan> magis mouet: ſed eodem modo alter alteri fit pondus.</s>
        </p>
        <p type="main">
          <s id="s.003265">Cæterum ſciendum huiuſmodi lignum, quo tranſuerſo onera <expan abbr="deportãtur">deportantur</expan> <lb></lb>
dici à latinis phalangam, vnde etiam verbum phalangare deducitur, quod <lb></lb>
huiuſmodi geſtationem ſignificat; <expan abbr="eſtq́">eſtque</expan>; Vitruuio vſitatum, &amp; Afranio, qui <lb></lb>
ait, capream vnam ſemilaceram quaterni ſimul phalangabant.</s>
        </p>
        <p type="head">
          <s id="s.003266"><emph type="italics"></emph>QVÆSTIO XXX.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003267"><emph type="italics"></emph>De ſurgente à ſeßione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003268"><arrow.to.target n="marg259"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003269"><margin.target id="marg259"></margin.target>269</s>
        </p>
        <p type="main">
          <s id="s.003270">Cvm ſedemus, præcipuè ſi commodè ſedeamus, ſolemus duos angu­<lb></lb>
los rectos facere, vnum quidem, quem facit thorax cum femore; <lb></lb>
alterum quem facit femur cum crure, vt in figura thorax ſit A B, <lb></lb>
<figure id="id.009.01.194.1.jpg" place="text" xlink:href="009/01/194/1.jpg"></figure><lb></lb>
femur B C, crus C D, anguli duo recti ſunt B, <lb></lb>
&amp; C. </s>
          <s id="s.003271">Quærit igitur, cur quando ſurgere volumus angu­<lb></lb>
los hoſce rectos in acutos commutamus, nam crus re­<lb></lb>
trahimus ſub femur ad acutum angulum, v. g. ad poſitio­<lb></lb>
nem C F. <expan abbr="fitq́">fitque</expan>; acutus angulus B C F. ſimiliter thoracem <lb></lb>
femori aptamus ad acutum angulum E B C, alioquin ſur­<lb></lb>
gere non valemus? </s>
          <s id="s.003272">An quia id, quod æquale eſt, quietis <lb></lb>
<expan abbr="vbiq;">vbique</expan> eſt cauſa, rectus autem angulus eſt angulus æquali­<lb></lb>
tatis, <expan abbr="atq;">atque</expan> ſtationis? </s>
          <s id="s.003273"><expan abbr="quæcunq;">quæcunque</expan> enim angulis rectis con­<lb></lb>
ſtant, vt quadratum, vt cubus, quieti, ac ſtationi ſunt <lb></lb>idonea, vt propterea Pytagorei dicerent terram eſſe cubicam, propter ip­<lb></lb>
ſius immobilitatem. </s>
          <s id="s.003274">eſt autem angulus rectus, angulus æqualitatis, quia <lb></lb>
omnes anguli recti ſunt inuicem æquales, vel quia linea illa, quæ angulum <lb></lb>
rectum facit eſt perpendicularis alteri lineæ, cui incumbit, <expan abbr="æqualiterq́">æqualiterque</expan>; in <lb></lb>
<expan abbr="vtramq;">vtramque</expan> partem inclinata eſt: quapropter fit, vt quæcunque conſtituta ſint <lb></lb>ſuper ſuperficiem terræ ad angulos rectos non cadant, ſed recta maneant. <lb></lb>
</s>
          <s id="s.003275">pariter <expan abbr="quæcunq;">quæcunque</expan> ad angulos rectos pauimento incumbunt, non ſolum, quia <lb></lb>cum illo faciant angulos rectos, ſed etiam, quia ſimul faciunt cum ſuperficie <pb pagenum="195" xlink:href="009/01/195.jpg"></pb>terræ perpendiculum. </s>
          <s id="s.003276">An quia qui ſurgit fit rectus; rectus autem manens, <lb></lb>oportet, vt ſit ſuperficiei terræ perpendicularis. </s>
          <s id="s.003277">debet igitur eſſe ſecundum <lb></lb>
eandem rectitudinem, ideſt caput ſupra thoracem, thorax verò ſupra femo­<lb></lb>
ra, femora verò ſupra crura in eadem rectitudine, quæ horizonti perpendi­<lb></lb>
culariter inſiſtat: quando autem ſedemus thorax, &amp; crura, non ſunt in ea­<lb></lb>
dem linea horizonti perpendiculariter erecta, quapropter neceſſe eſt pedes <lb></lb>
retrahere, caput autem reclinare, vt ſic in eadem recta linea horizonti per­<lb></lb>
pendiculariter conſtituantur, <expan abbr="hocq́">hocque</expan>; modo aſſurgere erit poſſibile.</s>
        </p>
        <p type="main">
          <s id="s.003278"><arrow.to.target n="marg260"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003279"><margin.target id="marg260"></margin.target>270</s>
        </p>
        <p type="main">
          <s id="s.003280">Reliquæ quæſtiones ad Phyſicum ſpectant. </s>
          <s id="s.003281">In 33. aperit propriam ſen­<lb></lb>
tentiam de motu proiectorum.</s>
        </p>
        <p type="main">
          <s id="s.003282">In 35. &amp; vltima de vortice quamuis videatur mathematicam ſapere, eſt <lb></lb>
tamen phyſica. </s>
          <s id="s.003283">Eius autem reſolutiones tres ab Ariſt. allatas, falſas eſſe <lb></lb>
ſuſpicor; experientia enim docet, quod ſi quippiam ponatur ſupra rotam <lb></lb>
figuli, id non ad centrum, ſed extra rotam proijcitur. </s>
          <s id="s.003284">ſed cauſa eſt, quia in <lb></lb>vortice aqua ipſa ſpiratim circumcurrens tandem in centrum, vbi demer­<lb></lb>
gitur deſcendit; neceſſe igitur eſt, vt etiam ea, quæ in ipſa ſunt, ſimul cum <lb></lb>
illa ad centrum per plures conuolutiones deducantur. </s>
          <s id="s.003285">Cæterum ſi quis ve­<lb></lb>
lit Mechanicam facultatem ſeriò aggredi, nequaquam paucis his ab Ariſt. <lb></lb>
traditis, <expan abbr="eisq́">eisque</expan>; leui brachio pertractatis, contentus ſit: verùm Archimedem <lb></lb>
de Aquæponderantibus, Commandinum, ac Lucam Valerium de centro <lb></lb>
grauitatis ſolidorum, ac tandem Guidi Vbaldi Mechanica adeat, vbi hu­<lb></lb>
ius ſcientiæ admiranda plurima, <expan abbr="eaq́">eaque</expan>; firmiſſimè demonſtrata reperiet.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003286"><emph type="italics"></emph>IN LIBELLVM DE MVNDO <lb></lb>
AD ALEXANDRVM.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003287">Cvm plures libellum hunc Ariſt. attribuant, cogor loca ipſius ma­<lb></lb>
thematica ex inſtituto exponere.</s>
        </p>
        <p type="main">
          <s id="s.003288"><arrow.to.target n="marg261"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003289"><margin.target id="marg261"></margin.target>271</s>
        </p>
        <p type="main">
          <s id="s.003290">In 2. cap. recenſet Planetarum ordinem, iuxta antiquiſſimorum <lb></lb>
Aſtronomorum traditiones, qui ob paucas, <expan abbr="easq́">easque</expan>; imperfectas ob­<lb></lb>ſeruationes multa ignorarunt, <expan abbr="atq;">atque</expan> in multis, &amp; præcipuè in ordine Plane­<lb></lb>
tarum ſtatuendo, falſi ſunt: Aſtronomi enim poſteriores, &amp; maximè Ptolæ­<lb></lb>
meus, vnà cum recentioribus noſtri ſeculi alium ordinem exactioribus ob­<lb></lb>
ſeruationibus, <expan abbr="atq;">atque</expan> demonſtrationibus aſtruentes vetuſtiſſimorum illorum <lb></lb>
errores patefecerunt. </s>
          <s id="s.003291">Eſt autem verus ordo, vt Luna ſit omnium terris pro­<lb></lb>
xima, deinde Mercurius, tùm Venus, poſtea Sol, Mars, Iupiter, <expan abbr="Saturnusq́">Saturnusque</expan>; <lb></lb>
à terris altiſſimus, quos omnes ſtellarum affixarum ſphæra, quæ etiam fir­<lb></lb>
mamentum dicitur, complectitur. </s>
          <s id="s.003292">non me latet huius noſtri ſeculi di­<lb></lb>
ligentiſſimos aſtronomos nouam mundani ſyſtematis hy­<lb></lb>
potheſim inducere; ſed ea prædicto Planctarum <lb></lb>
ordini parum, aut nihil repugnat.</s>
        </p>
        <pb pagenum="196" xlink:href="009/01/196.jpg"></pb>
        <p type="head">
          <s id="s.003293"><emph type="italics"></emph>De æstu Maris.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003294"><arrow.to.target n="marg262"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003295"><margin.target id="marg262"></margin.target>272</s>
        </p>
        <p type="main">
          <s id="s.003296">In 3. cap. <emph type="italics"></emph>(Aiunt etiam multos æstus vndarumqué ſublationes fiatis quibuſ­<lb></lb>dam temporibus cum Luna circumagi)<emph.end type="italics"></emph.end> Perpaucis maris fluxum, &amp; reflu­<lb></lb>xum attingit, qui quia ex motu præcipuè Lunæ pendet, non videtur <lb></lb>
alienum hoc loco eum fuſius explicare, <expan abbr="atq;">atque</expan> nonnullis difficultatibus <lb></lb>occurrere, quibus recentiores nonnulli nimis implicantur. </s>
          <s id="s.003297">Aeſtus maris <lb></lb>
eſt quædam maris ebullitio, ob quam vt ſolet in ebullientibus aquis, mare <lb></lb>
intumeſcit: fiunt autem in toto mundo duobus tantum in locis ex hoc æſtu <lb></lb>
tumores duo, quorum vnus ſemper directè Lunæ ſubiacet, alter verò in <lb></lb>
auerſa terræ parte, ſiue huic antipoda, &amp; diametraliter oppoſita.</s>
        </p>
        <p type="main">
          <s id="s.003298">Ex his Marium tumoribus fit vt aquæ, quæ naturæ ſua decliuiora petunt, <lb></lb>quaſi exundantes ad littora fluant. </s>
          <s id="s.003299">atque hic aquarum curſus fluxus maris <lb></lb>
appellatur. </s>
          <s id="s.003300">decreſcente deinde maris æſtu, &amp; tumore ex receſſu Lunæ, aquæ <lb></lb>
iterum ad medium mare refluunt: <expan abbr="atq;">atque</expan> hic maris refluxus dicitur. </s>
          <s id="s.003301">Cum au­<lb></lb>
tem in toto die ſint 24. horæ &amp; ſemper ſint ſimul in mundo duo æſtus, &amp; tu­<lb></lb>
mores, fit vt ſint pariter ſemper in mundo duo fluxus, qui tumores illos co­<lb></lb>
mitantur; necnon duo refluxus, qui eoſdem ſubſequantur; hinc fit vt <expan abbr="vni-cuiq;">vni­<lb></lb>
cuique</expan> illorum ſex heræ <expan abbr="conueniãt">conueniant</expan>, ſex fluxui, ſex refluxui, qui ſub Luna fiunt; <lb></lb>
ſex verò fluxui, &amp; ſex tandem refluxui Lunæ auerſis, quæ totam Lunæ circa <lb></lb>
mundum periodum 25. horarum expleant. </s>
          <s id="s.003302">Cauſam autem cur mare hoc <lb></lb>
modo ſtatis horis, paulò tamen ſerius ob Lunæ tardiorem ortum ſemper <lb></lb>
creſcat, &amp; decreſcat antiqui omnes in Lunam retulerunt, vt primus omnium <lb></lb>
Ariſt. hoc loco, deinde Strabo, Pomponius Mela, Plinius, Solinus, &amp; alij <lb></lb>
plures idem ſenſerunt. </s>
          <s id="s.003303">Lunam ſcilicet eam habere vim in mare, vt pars il­<lb></lb>la, quæ Lunæ ſubiacet, ſiue quam Luna radijs ferit, æſtuet, &amp; turgeat; non <lb></lb>
aliter pars maris huic antipoda, &amp; auerſa, quamuis tota terræ moles inter <lb></lb>
<figure id="id.009.01.196.1.jpg" place="text" xlink:href="009/01/196/1.jpg"></figure><lb></lb>
ipſam, &amp; Lunam interpona­<lb></lb>
tur, æſtuat, fluxumque, ac re­<lb></lb>
fluxum quamuis priori mi­<lb></lb>
norem, efficit. </s>
          <s id="s.003304">quæ omnia <lb></lb>
melius in figura cernentur; <lb></lb>
vbi infra Lunam vides tumo­<lb></lb>
rem A, ex quo fluxus deriua­<lb></lb>
tur. </s>
          <s id="s.003305">&amp; in parte huic auerſa <lb></lb>
tumorem B, ex quo alter flu­<lb></lb>
xus deriuatur. </s>
          <s id="s.003306">&amp; quia in alijs <lb></lb>
duobus mundi lateribus non <lb></lb>
<expan abbr="fiũt">fiunt</expan> huiuſmodi tumores, imò <lb></lb>
mare ob refrigerationem <lb></lb>
ſubſidet, ibi fiunt duo reflu­<lb></lb>
xus C, &amp; D, ita vt ſemper ſint <lb></lb>
in mari præſertim Oceano <lb></lb>
quatuor prædicti effectus, qui <lb></lb>
ſimul, vt ait hic Ariſt. &amp; ex­


<pb pagenum="197" xlink:href="009/01/197.jpg"></pb>perientia teſtatur, ſimul cum Luna circa mundum circumaguntur. </s>
          <s id="s.003307">hoc eſt <lb></lb>
ſi Luna, quæ modo eſt in ſuperiori parte meridionali, venerit ad locum E, <lb></lb>occidentalem, eam fluxus A, ſubſequitur, <expan abbr="vergitq́">vergitque</expan>; tumorem ſuum ad occi­<lb></lb>
dentem E, vnde, &amp; fluxus B, promouebitur ad orientem, ita vt punctum F, <lb></lb>
orientalem aſpiciat.</s>
        </p>
        <p type="main">
          <s id="s.003308">Alij præterea duo refluxus eadem proportione promoti erunt, vbi prius <lb></lb>
erant fluxus: quæ <expan abbr="conſequẽtia">conſequentia</expan> ad Lunam perpetua, manifeſtum eſt, ſignum, <lb></lb>hoſce fluxus, ac refluxus non aliunde quàm à Luna manare. </s>
          <s id="s.003309">quod adhuc ma­<lb></lb>
nifeſtius erit, ſi conſideremus, quod quanto tardius quotidie Luna oritur, <lb></lb>
tanto etiam maris æſtus tardius incipit. </s>
          <s id="s.003310">Porrò vt appareat hanc eſſe vete­<lb></lb>
rum ſententiam libet hic attexere quædam ex lib. 3. Strabonis, quæ ipſe ex <lb></lb>
Poſſidonio acceperat. </s>
          <s id="s.003311">ſic. </s>
          <s id="s.003312">Oceani verò motum ait, ſcilicet Poſſidonius, ſy­<lb></lb>
deris ſubire circuitum, quendam quidem diurnum, quendam menſtruum, <lb></lb>
quendam annuum, vt Lunæ etiam contingit. </s>
          <s id="s.003313">quo etiam tempore iſta ſuper <lb></lb>
horizontem aſcenderit, mare terram aſcendere incipit, ſenſu teſte, <expan abbr="quouſq;">quouſque</expan> <lb></lb>
ad cœli medium Luna conſcenderit. </s>
          <s id="s.003314">Vbi verò declinare ſydus ipſum cœ­<lb></lb>
perit, ſenſim rurſus à terra pelagus ad medium mare reuertitur, donec ad <lb></lb>occidentis punctum Luna deſcenderit. </s>
          <s id="s.003315">deinde tanto eadem inconſtantia <lb></lb>tempore manet, quanto Luna ad ipſum occaſum coniungitur, &amp; adhuc tan­<lb></lb>to magis, quanto ſub terram mota, ſignum ab horizonte diſtet. </s>
          <s id="s.003316">poſtea rur­<lb></lb>ſus mare aſcendere, quouſque ſub tellurem in medio cœli ſit Luna, deinde <lb></lb>
mare à littore regredi quoad iterum Luna in orientem procedat, ac ſupra <lb></lb>horizontem eleuetur, conſiſtere verò vſque quo ſignum ſupra terram eleue­<lb></lb>
tur, &amp; rurſus terras mare aſcendere. </s>
          <s id="s.003317">Hanc diurnam eſſe circuitionem aſſe­<lb></lb>
rit Poſſidonius, menſtruam verò, &amp;c. </s>
          <s id="s.003318">vbi pergit explicare, qua ratione, ma­<lb></lb>ria etiam alijs motibus menſtruo. </s>
          <s id="s.003319">ſcilicet, &amp; annuo cieantur, iuxta Lunæ <lb></lb>
periodos menſtruam, &amp; annuam. </s>
          <s id="s.003320">Eadem omninò habet Plinius, &amp; alij ve­<lb></lb>
teres omnes, quos tu conſulere poteris vnde mirum videri debeat, cur re­<lb></lb>
centiores plurimi, <expan abbr="neq;">neque</expan> veterum auctoritate, <expan abbr="neq;">neque</expan> ratione, aut experientia <lb></lb>
nixi, hanc maris affectionem, à Luna effici negarint.</s>
        </p>
        <p type="main">
          <s id="s.003321">Verum ipſi duabus potiſſimum rationibus id negant.</s>
        </p>
        <p type="main">
          <s id="s.003322">Prima eſt, quod vario admodum tempore, &amp; modo in diuerſis fiant ma­<lb></lb>
ribus, &amp; in nonnullis nihil horum æſtuum appareat.</s>
        </p>
        <p type="main">
          <s id="s.003323">Huic reſpondendum eſt, id ex varia marium diſpoſitione, tum etiam va­<lb></lb>rio ſitu, quo Lunam aſpiciunt prouenire. </s>
          <s id="s.003324">hoc modo videmus vario tempo­<lb></lb>
re, &amp; modo, in toto orbe effici dies, ac noctes, æſtatem, &amp; hyemem; &amp; ta­<lb></lb>
men certum eſt Solem iſta omnia efficere. </s>
          <s id="s.003325">Sed melius etiam huic dubita­<lb></lb>
tioni occurremus certa quadam, <expan abbr="atq;">atque</expan> omninò explorata experientia ex ar­<lb></lb>
te Nautica deſumpta. </s>
          <s id="s.003326">libri enim nautici abſque vlla dubitatione Lunæ hæc <lb></lb>omnia verè aſcribunt, dum quaſdam regulas tradunt, eas tamen pro varijs <lb></lb>maribus varias, quibus per ætatem Lunæ, &amp; ſitum ipſius ſupra horizontem <lb></lb>
illius maris certò certius horam fluxus; &amp; refluxus, imò eorum etiam ma­<lb></lb>gnitudinem prænoſcunt, ac prædicunt. </s>
          <s id="s.003327">huiuſmodi librum vidi ego Parmæ, <lb></lb>
manu ſcriptum, auctore Auguſtino Cæſareo, quem ille olim Sereniſs. Duci <lb></lb>
Octauio dono dederat. </s>
          <s id="s.003328">quod ſi hi æſtus à Luna minimè penderent, nulla ra­<lb></lb>tione regulæ illæ effici potuiſſent, quibus per ætatem ipſius, ac ſitum ſupra <lb></lb>
horizontem eos prædicere tuto valerent.</s>
        </p>
        <pb pagenum="198" xlink:href="009/01/198.jpg"></pb>
        <p type="main">
          <s id="s.003329">Secunda verò ratio, quæ maximè eos torquet eſt quanam ratione à Luna <lb></lb>
effici poſſit ſecundus refluxus B, primò oppoſitus, cum tota terræ moles in­<lb></lb>
teriecta obſtare videatur.</s>
        </p>
        <p type="main">
          <s id="s.003330">Verum huic difficultati optimè ex opticis ſatisfacere poſſumus, fi dixe­<lb></lb>
rimus, æſtum illum effici quidem à Luna, &amp; Sole, ſed tamen per lumen ex <lb></lb>
ſyderibus ad partem illam auerſam reflexum; quod vt melius explicetur, &amp; <lb></lb>
confirmetur. </s>
          <s id="s.003331">Illud primò ſciendum non ſolam Lunam, verum etiam Solem <lb></lb>ad æſtum maris ciendum concurrere, quamuis primas in hoc Lunæ conce­<lb></lb>
dat; experientia enim conſtat maiorem fieri fluxum, quando Sol, &amp; Luna <lb></lb>
ſimul ſunt coniuncta, vt in nouilunio accidit, quia lumina, &amp; eorum virtu­<lb></lb>
tes vnitæ fortius eandem maris partem directis radijs percellunt. </s>
          <s id="s.003332">ſimiliter <lb></lb>
maior fit, quando luminaria ſunt oppoſita, vt in plenilunio contingit, quia <lb></lb>
tunc radij vnius directi, aſſociantur cum reflexis alterius radijs, <expan abbr="hocq́">hocque</expan>; mo­<lb></lb>
do duplicati eaſdem terræ partes, &amp; directè, &amp; reflexè feriunt, vt melius in <lb></lb>
ſequenti figura patebit.</s>
        </p>
        <p type="main">
          <s id="s.003333">Secundò præmittendum eſt, lumen Solis, &amp; Lunæ reflecti ex denſis, ac per­<lb></lb>politis corporibus, vti ſunt omnia ſydera.</s>
        </p>
        <p type="main">
          <s id="s.003334">Tertiò, ex opticis aſſumendum, ſi corpora plurima ſphærica lumen re­<lb></lb>flectentia fuerint in circulari ambitu conſtituta, quemadmodum ſunt ſtellæ <lb></lb>
affixæ in ambitu firmamenti collocatæ, reflectere <expan abbr="plurimũ">plurimum</expan> lumen ad vnum, <lb></lb>
&amp; idem punctum, quod ſit inter lumen, &amp; ambitum illum; quod aſſumptum <lb></lb>
manifeſtum eſt ex Iride, vbi ex plurimis ſphæricis guttulis lumen Solis re­<lb></lb>
flectitur ad oculum; quamuis geometricè, &amp; quidem facilè à Perſpectiuo <lb></lb>
demonſtrari poſſit.</s>
        </p>
        <p type="main">
          <s id="s.003335">Quartò, ex opticis, dato corpore luminoſo, &amp; ſphærico reflectente, &amp; <lb></lb>
puncto quouis, ad quod poſſit reflecti lumen, poteſt inueniri in ſphæra refle­<lb></lb>
ctente punctum reflexionis.</s>
        </p>
        <p type="main">
          <s id="s.003336">Quintò, quanto radij perpendiculariores incidunt, tanto maiorem <lb></lb>
vim habere.</s>
        </p>
        <p type="main">
          <s id="s.003337">Sit ergò Sol, &amp; Luna ſimul, vt in figura <expan abbr="ſitq́">ſitque</expan>; octauæ ſphæræ portio A B C, <lb></lb>
cum innumeris in ea affixis ſyderibus. </s>
          <s id="s.003338">eſſe autem totum cœlum ſtellis penè <lb></lb>
infinitis, ac conſtipatis refertum ſenſui palam fit, adhibito nouo illo, ac mi­<lb></lb>
rabili Teleſcopij inuento.</s>
        </p>
        <p type="main">
          <s id="s.003339">Iam, vt patet ex 39.5. Alhazeni, ex ſingulis ſtellis Solis, ac Lunæ lumen <lb></lb>
reflecti poteſt (niſi quid obſtet) ad partem terræ D, luminaribus auerſam, <lb></lb>
vt quarto loco ſuppoſui. </s>
          <s id="s.003340">&amp; præterea ex ſtellis circa B, poſitis radij Solis re­<lb></lb>
percuti poſſunt ad eandem terræ partem D, perpendiculares, qui præ cæte­<lb></lb>
ris maximam vim obtinent. </s>
          <s id="s.003341">quemadmodum lineæ in figura reflexæ <expan abbr="vtcunq;">vtcunque</expan> <lb></lb>
oſtendunt, ideò aſſerendum eſt eos, æſtum D, excitare præcipuè poſſe, <expan abbr="neq;">neque</expan> <lb></lb>
terræ quantitas Solis luci obeſt, cum conſtet vmbram terræ parum ſupra <lb></lb>
Lunæ cœlum produci. </s>
          <s id="s.003342">poteſt tamen Lunæ eſſe impedimento quoad hos ra­<lb></lb>
dios perpendiculares; ſed tamen alios minus perpendiculares, ſeu parum <lb></lb>
obliquos nullo modo impedire poteſt, quo minus ad D, reſiliant. </s>
          <s id="s.003343">qui quam­<lb></lb>
uis ſint minus quàm perpendiculares efficaces, obtinent tamen non modi­<lb></lb>
cam vim. </s>
          <s id="s.003344">Ex ſtellis igitur circa A, &amp; C, reflecti poteſt ex quarto fundamen­<lb></lb>
to lumen <expan abbr="vtriuſq;">vtriuſque</expan> luminaris ad D, quod ſatis eſt efficax, cùm ferè perpendi­


<pb pagenum="199" xlink:href="009/01/199.jpg"></pb><figure id="id.009.01.199.1.jpg" place="text" xlink:href="009/01/199/1.jpg"></figure><lb></lb>
culariter terræ D, incidat. </s>
          <s id="s.003345">quamuis autem ex ſtellis F, E, lumen aliquod ad <lb></lb>
D, tranſmittatur, tamen cum obliquè admodum illi accidat, nihil penè ef­<lb></lb>
ficere valet. </s>
          <s id="s.003346">Verumenimuerò quiſpiam in hunc modum obijciet: hac ra­<lb></lb>
tione deberet fieri etiam æſtus in terræ lateribus H, I, quando quidem etiam <lb></lb>
illuc lumen ex quarto fundamento reflecti poteſt.</s>
        </p>
        <p type="main">
          <s id="s.003347">Cui ſic reſpondendum, poſſe quidem aliquod lumen illuc reſilire, ſed ta­<lb></lb>
men exiguum admodum, &amp; proinde nullius penè roboris, quod experientia <lb></lb>
deſumpta ex illuminatione Lunæ comprobari poteſt; videmus enim, quod <lb></lb>
quanto Luna magis Soli opponitur, &amp; proinde ſuam illuminationem magis <lb></lb>
verſus terram obuertit, vt in plenilunio, tanto maiorem eam vim habere <lb></lb>
æſtus excitandi. </s>
          <s id="s.003348">multo verò minorem, quando eſt in aſpectu Solis quadrato, <lb></lb>
quia dimidiam tantum ſui illuminationem nobis reflectit. </s>
          <s id="s.003349">Idem proportio­<lb></lb>
naliter de ſtellis dicendum, quæ enim luminari maximè opponuntur, vt quæ <lb></lb>
ſunt circa B, illæ totam illuminationem terræ oſtendunt, vnde, &amp; efficacio­<lb></lb>
res ſunt. </s>
          <s id="s.003350">cæteræ, quo magis ab illis diſtant minus de ſua illuminatione ter­<lb></lb>
ræ, ſeu mari <expan abbr="obuertũt">obuertunt</expan>, &amp; proinde minus efficiunt. </s>
          <s id="s.003351">vnde fit, vt quamuis non­<lb></lb>
nulli radij etiam perpendiculares ad terræ latera H, I, referri poſſint, tamen <lb></lb>
quia pauciores ſunt, quàm alibi, propterea nullam ibi æſtus <expan abbr="prouocãdi">prouocandi</expan> vim <lb></lb>
obtinent. </s>
          <s id="s.003352">ſydera porrò illa, quæ ſupra Solem exiſtunt, etiam ſi ipſorum illu­


<pb pagenum="200" xlink:href="009/01/200.jpg"></pb>minatio tota ad terras vergat, tamen in lateribus terræ prædictis nihil ef­<lb></lb>
ficiunt, quia in illa vel obliquè admodum radij incidunt, vel ea tantummo­<lb></lb>
do tangunt. </s>
          <s id="s.003353">Verum illuminatione ſua eaſdem maris partes, quæ ſunt ad G, <lb></lb>
vnà cum Sole, ac Luna percellunt.</s>
        </p>
        <p type="main">
          <s id="s.003354">Ex quibus apparet duas tantum orbis terræ partes totis, ac plenis aſtro­<lb></lb>rum luminibus impeti, in quibus ſcilicet duo oppoſiti æſtus ebulliunt.</s>
        </p>
        <p type="main">
          <s id="s.003355">Idem poſſumus hoc modo confirmare, quia cum totum firmamentum ſit <lb></lb>
innumeris penè ſyderibus ſtipatum, loco concaui, ac ſphæriçi ſpeculi ha­<lb></lb>
beri poteſt, &amp; proinde illius inſtar amborum luminarium lumen reflectere; <lb></lb>
qua ratione patet omnem ferè ad partes prædictas D, emitti reflexionem.</s>
        </p>
        <p type="main">
          <s id="s.003356">His rationibus manifeſtum eſſe patet prædictum æſtus tumorem lumina­<lb></lb>
ribus auerſum, <expan abbr="atq;">atque</expan> antipodum ex prædicta reflexione exurgere.</s>
        </p>
        <p type="main">
          <s id="s.003357">Poſſet etiam quiſpiam ſic opponere, ſi illuc prædicta luminum reflexio <lb></lb>
pertineret, non ſolum illam aquarum ebullitionem efficeret, verum etiam <lb></lb>
lucem aliquam eòdem afferret, quod tamen ſenſu minimè apparet. </s>
          <s id="s.003358">cui ſic <lb></lb>
reſpondendum videtur, neceſſarium non eſſe, vt reflexio illa, quæ hoc modo <lb></lb>
mare afficit tanta ſit, vt etiam illud luce ſolito maiori afficiat; quod <expan abbr="expe-riẽtia">expe­<lb></lb>
rientia</expan> conſtat in alijs cęli influxibus: quàm ſæpè enim Luna nubiloſo etiam <lb></lb>tempore, fluxum, ac refluxum priorem parit, cum tamen nullam tunc lu­<lb></lb>
cem nobis afferat? </s>
          <s id="s.003359">quamuis enim lumen ſtellarum ſuperficiem maris non <lb></lb>
attingat, attingit tamen ſuperficiem vaporum, exhalationum, ac nubium, <lb></lb>
quæ terram in ſphæræ modum ambiunt, ac parum à terra <expan abbr="circumquaq;">circumquaque</expan> at­<lb></lb>
tolluntur: quem exhalationum ambitum deinde luminarium virtus facilè <lb></lb>
penetrare poteſt. </s>
          <s id="s.003360">Nullum præterea lumen apparet, quia lumen reflexum <lb></lb>
præſertim ex conuexis corporibus, vt ſunt ſtellæ, valde debile eſt, quia <expan abbr="con-uexũ">con­<lb></lb>
uexum</expan> illud reflectendo non vnit, ſed diſgregat, contra quam facit concauum.</s>
        </p>
        <p type="main">
          <s id="s.003361">Tandem quærere quis poſſet, cur æſtus hic ſecundus minor ſit priori. </s>
          <s id="s.003362">Cui <lb></lb>
reſpondendum, quia ille à directis radijs, hic verò à reflexis progignitur: <lb></lb>
radios autem reflexos debiliores eſſe directis optici docent, <expan abbr="atq;">atque</expan> experien­<lb></lb>
tia confirmat.</s>
        </p>
        <p type="main">
          <s id="s.003363">Porrò quando luminaria ſunt oppoſita, vt ſi Luna eſſet in B, Sol verò in K, <lb></lb>
tunc maximus fit <expan abbr="vterq;">vterque</expan> fluxus, quia radij directi <expan abbr="vtriuſq;">vtriuſque</expan> vniuntur cum ra­<lb></lb>
dijs reflexis alterius; ita vt <expan abbr="vterq;">vterque</expan> æſtus fiat, &amp; per radium reflexum, &amp; per <lb></lb>
directum ſimul, v. g. æſtus, qui Lunæ ſubiacet fit per radium Lunæ directum, <lb></lb>
&amp; quia Sol eſt in oppoſitione cum Luna, ſit vt ipſius radij reflectantur, &amp; <lb></lb>
vniantur cum directis Lunæ ad eundem tumorem excitandum. </s>
          <s id="s.003364">ſimiliter in­<lb></lb>
fra Solem directè alius fit à directis ipſius radijs; &amp; quia Luna ei opponitur <lb></lb>
lumen eius ad <expan abbr="vſq;">vſque</expan> ſydera pertinens reuertitur, <expan abbr="vnaq́">vnaque</expan>; cum directa Solis lu­<lb></lb>
ce ad eundem efficiendum concurrit.</s>
        </p>
        <p type="main">
          <s id="s.003365">Exiſtentibus demum luminaribus circa quadratum aſpectum, vt ſi Luna <lb></lb>
eſſet in F, Sole exiſtente in K. exiguus, ac penè nullus fit fluxus, quia eorum <lb></lb>
vires non ſunt vnitæ, cùm radij nec incidentes, nec reflexi vniantur imò vi­<lb></lb>res eorum ſeparatæ maria in contrarias partes diſtrahunt, vnde fit, vt neu­<lb></lb>
tro alteri concedente, apud neutrum victoria conſtet.</s>
        </p>
        <p type="main">
          <s id="s.003366"><expan abbr="Atq;">Atque</expan> hæc eſt mea de æſtu maris per reflexionem ſententia. </s>
          <s id="s.003367">quam iamdiu <lb></lb>
inuentam, <expan abbr="atq;">atque</expan> auditoribus meis ſæpius explicatam, reperi tandem non ſine <pb pagenum="201" xlink:href="009/01/201.jpg"></pb>gaudio fuiſſe etiam ſubtiliſſimi Scoti opinionem, quam ipſe breuiter in pri­<lb></lb>
mum ſent. </s>
          <s id="s.003368">de creatione mundi tantummodo ſine vlla expoſitione, <expan abbr="atq;">atque</expan> con­<lb></lb>
firmatione proponit. </s>
          <s id="s.003369">in eadem prorſus ſententia eſt Rogerius Bachon inter <lb></lb>
Opticos probatiſſimus, cap. 5. de Speculis Mathematicis.</s>
        </p>
        <p type="main">
          <s id="s.003370">Aliorum demum opinationes, ſiue Angelo cuidam, ſiue virtuti totam <lb></lb>
terram peruadenti hunc æſtum aſcribentium, non eſt meum refellere, cum <lb></lb>
non phyſicum, ſed mathematicum agere inſtituerim.</s>
        </p>
        <p type="main">
          <s id="s.003371"><arrow.to.target n="marg263"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003372"><margin.target id="marg263"></margin.target>273</s>
        </p>
        <p type="main">
          <s id="s.003373">Cap. 7. <emph type="italics"></emph>(Quod Imagunculas animatas eſſe, &amp;c.)<emph.end type="italics"></emph.end> huiuſmodi imagines, &amp; <lb></lb>
ſtatuas, quæ ſpontè mouebantur Græci appellarunt Automata, ideſt ſpon­<lb></lb>
tanea, cuiuſmodi ſunt automata Heronis, Alexandrini, quæ adhuc extant.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003374"><emph type="italics"></emph>IN LIBELLVM <lb></lb>
De admirandis auditionibus.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003375"><arrow.to.target n="marg264"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003376"><margin.target id="marg264"></margin.target>274</s>
        </p>
        <p type="main">
          <s id="s.003377">Nvmero 82. Quæ de illa inſula extra Herculis columnas ſita narrat, <lb></lb>
eam putant recentiores Geographi, &amp; quidem meritò nouo orbi <lb></lb>
conuenire.</s>
        </p>
        <p type="main">
          <s id="s.003378"><arrow.to.target n="marg265"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003379"><margin.target id="marg265"></margin.target>275</s>
        </p>
        <p type="main">
          <s id="s.003380">Numero 100. Quæ de Iſtro, ſiue Dannubio tradit, eum ſcilicet <lb></lb>
eſſe bifidum, <expan abbr="alteroq́">alteroque</expan>; ramo in Pontum, altero verò in Mediterraneum ex­<lb></lb>
onerari: ſunt contra omnes recentiores Geographos; apparet tamen eam <lb></lb>
fuiſſe veterum nonnullorum opinionem, quos <expan abbr="ſeq́uutus">ſequutus</expan> Ariſt. deceptus eſt, <lb></lb>
à quibus etiam multò poſt falſi ſunt Diodorus, Pomponius, &amp; Solinus, qui <lb></lb>
Iſtrum Iſtriæ Prouincìæ fluuium faciunt, quem ex Iſtro Germaniæ veluti ra­<lb></lb>
mum contra omnem veritatem deriuant. </s>
          <s id="s.003381">Verùm hoc illis condonandum <lb></lb>
præſertim antiquioribus, cum tunc temporis Geographia parum eſſet <lb></lb>
exculta.</s>
        </p>
        <p type="main">
          <s id="s.003382">Primus Strabo hanc falſitatem libro 1. redarguit, &amp; poſt ipſum Plinius <lb></lb>
Iſtrum iſtum fabuloſum appellat.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003383"><emph type="italics"></emph>IN LIBELLVM <lb></lb>
De lineis inſecabilibus, ſiue indiuiduis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003384"><arrow.to.target n="marg266"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003385"><margin.target id="marg266"></margin.target>276</s>
        </p>
        <p type="main">
          <s id="s.003386">Diſputat libellus hic ſanè acutiſſimus, Vtrum quantitas conſtet ex <lb></lb>
indiuiſibilibus, quam quęſtionem recentiores agitant in Phyſicis <lb></lb>
tractatione de Quantitate; <expan abbr="atq;">atque</expan> hinc nonnulla ſumunt argumen­<lb></lb>
ta: plura ſumpturi niſi operis obſcuritas, &amp; mathematicarum, <lb></lb>
ignoratio hactenus obſtitiſſet.</s>
        </p>
        <p type="main">
          <s id="s.003387">Sciendum igitur primo loco, nos poſſe duo indiuiſibilium genera in quan­<lb></lb>
titate concipere. </s>
          <s id="s.003388">primum eorum, quæ verè indiuidua ſunt, <expan abbr="nullasq́">nullasque</expan>; habent <lb></lb>partes, ſiue nullo modo ſunt quanta; cuiuſmodi eſt <expan abbr="punctũ">punctum</expan> mathematicum.</s>
        </p>
        <p type="main">
          <s id="s.003389">Alterum quorumdam indiuiſibilium quidem, ſed tamen quantorum cu­<lb></lb>
iuſmodi eſſent, quædam adeò minimæ lineæ, quæ omnem effugiant diuiſio­<lb></lb>nem: ex quibus antiqui opinabantur lineas totales, ac diuiduas componi. <lb></lb>
</s>
          <s id="s.003390">atque de hoc ſecundo indiuiduorum, quantorum genere videtur opuſculum <pb pagenum="202" xlink:href="009/01/202.jpg"></pb>iſtud diſſerere. </s>
          <s id="s.003391">&amp; quia partim rationibus phyſicis, partim geometricis vti­<lb></lb>
tur, ideò nec omninò phyſicus nec omninò mathematicus eſt. </s>
          <s id="s.003392">Ego igitur, <lb></lb>quæ mathematica ſunt, ex inſtituto exponere aggrediar.</s>
        </p>
        <p type="main">
          <s id="s.003393">Ad intelligentiam igitur huius operis neceſſarium eſt nouiſſe, quæ nam <lb></lb>
ſint quantitates commenſurabiles, &amp; quæ in commenſurabiles. </s>
          <s id="s.003394">quæ prima, <lb></lb>
&amp; ſecunda definitione 10. Elem. explicantur; <expan abbr="egoq́">egoque</expan>; eas primo Priorum oc­<lb></lb>
caſione aſymetriæ diametri cum coſta ſatis expoſui: vtrumuis locum vide­<lb></lb>
ris præſenti neceſſitati conſultum erit.</s>
        </p>
        <p type="main">
          <s id="s.003395"><arrow.to.target n="marg267"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003396"><margin.target id="marg267"></margin.target>277</s>
        </p>
        <p type="main">
          <s id="s.003397">Primus locus Mathematicus eſt hic <emph type="italics"></emph>(Poſtremò ex ijs, quæ tradunt Mathe­<lb></lb>
maticis imbuti diſciplinis, quiuis lineam aliquam inſecabilem eſſe concedet. </s>
          <s id="s.003398">nam <lb></lb>
ſi, vt aiunt, illæ commenſurabiles ſunt lineæ, quæ eadem menſura dimetiri queunt, <lb></lb>&amp; nihil impedit, quin omnes commenſurabiles re ipſa dimetiantur, extabit profe­<lb></lb>
ctò longitudo aliqua, qua omnes commenſurabuntur; quæ neceſſario erit indiuidua, <lb></lb>
nam ſi dicatur eſſe diuidua, huius <expan abbr="quoq;">quoque</expan> menſuræ partes, menſuram aliquam com­<lb></lb>
munem habebunt, partes enim toti commenſurabiles ſunt ita, vt portio partis il­<lb></lb>
lius, quæ dimidium totius fuerat, efficiatur dupla alterius; quoniam autem hoc <lb></lb>
fieri nequit, atoma debet eſſe menſura hæc communis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003399"><emph type="italics"></emph>Eodem modo, &amp; quæ ſimul ab ipſa menſura commenſuratæ, tanquam omnes ex <lb></lb>
ea menſura compoſitæ ſunt lineæ, veluti ex atomis conflantur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003400">Affert rationem quandam ex Mathematicis, qua nonnulli probabant ex­<lb></lb>
tare lineas atomas, ex quibus cæteræ lineæ tanquam partibus conſtarent: <lb></lb>
ac proinde negabant lineas eſſe in infinitum diuiduas, ſeu quamlibet lineam <lb></lb>
ſecari poſſe, ſed aſſerebant <expan abbr="diuidẽdo">diuidendo</expan>, tandem ad indiuiduas <expan abbr="deueniendũ">deueniendum</expan> eſſe.</s>
        </p>
        <p type="main">
          <s id="s.003401">Præmiſſa igitur, vt monui commenſurabilium, &amp; incommenſurabilium <lb></lb>
linearum cognitione in hunc modum, &amp; textum Ariſtot. &amp; rationem ipſo­<lb></lb>
rum exponam.</s>
        </p>
        <p type="main">
          <s id="s.003402">Mathematici oſtendunt extare lineas commenſurabiles, quæ ſcilicet ea­<lb></lb>
dem communi menſura menſurantur: at nihil impedit quin omnes <expan abbr="cõmen-ſurabiles">commen­<lb></lb>
ſurabiles</expan> re ipſa menſurentur, debet ergò extare vna aliqua longitudo, qua <lb></lb>
omnes commenſurabiles dimetiamur. </s>
          <s id="s.003403">hanc autem neceſſe eſt eſſe atomam, <lb></lb>
nam ſi diuidua ſtatuatur, poterit ſemper ſecari, &amp; ſubſecari bifariam, qua­<lb></lb>
re cum partes huiuſmodi ſint toti commenſurabiles, ſequetur aliam exiſtere <lb></lb>
menſuram, qua omnes hæ partes, &amp; proinde tota linea commenſurentur. <lb></lb>
</s>
          <s id="s.003404">Verùm hoc fieri nequit, nam hoc pacto non eſſet vna tantum longitudo om­<lb></lb>
nium commenſurabilium linearum communis menſura, verùm plures, &amp; <lb></lb>
plures in infinitum, quod eſt contra Mathematicorum placita. </s>
          <s id="s.003405">dicendum, <lb></lb>
itaque, communem illam omnium menſuram eſſe omnis diuiſionis exper­<lb></lb>
tem; &amp; propterea etiam lineas omnes commenſurabiles ex atomis lineis <lb></lb>
componi, quæ nimirum prædictæ communi menſuræ æquales ſint. </s>
          <s id="s.003406"><expan abbr="atq;">atque</expan> hæc <lb></lb>
eſt illarum prima argumentatio.</s>
        </p>
        <p type="main">
          <s id="s.003407"><arrow.to.target n="marg268"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003408"><margin.target id="marg268"></margin.target>278</s>
        </p>
        <p type="main">
          <s id="s.003409">Secundus locus <emph type="italics"></emph>(Idem etiam contingit in figuris planis, quæ à lineis rationa­<lb></lb>
libus procreantur: nam omnes huiuſmodi figuræ erunt etiam inuicem commenſura­<lb></lb>
biles, quare <expan abbr="ēadem">eadem</expan> ratione, qua in lineis proximè vſi ſumus, ſequetur earum com­<lb></lb>
munem menſuram eſſe pariter indiuiduam.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003410">Sciendum eſt omnes lineas <expan abbr="cõmenſurabiles">commenſurabiles</expan> longitudine, eſſe etiam com­<lb></lb>
menſurabiles (vt aiunt Geometræ) potentia, ideſt ſecundum quadrata ea­


<pb pagenum="203" xlink:href="009/01/203.jpg"></pb>rum, ſiue dicas quadrata <expan abbr="quoq;">quoque</expan> earum eſſe commenſurabilia, v. g. linea dua­<lb></lb>
<figure id="id.009.01.203.1.jpg" place="text" xlink:href="009/01/203/1.jpg"></figure><lb></lb>
rum vnciarum, &amp; linea trium vnciarum ſunt <lb></lb>
commenſurabiles longitudine, &amp; potentia, <lb></lb>
quia potentia lineæ duarum vnciarum, ſiue <lb></lb>
<expan abbr="quadratũ">quadratum</expan>, eſt quatuor vnciarum ſuperficia­<lb></lb>
lium: &amp; quadratum lineæ trium vnciarum, <lb></lb>
eſt nouem vnciarum quadratarum, vt patet <lb></lb>
in figuris, quorum quadratorum communis <lb></lb>
menſura eſt vncia vna quadrata. </s>
          <s id="s.003411">atque hanc <lb></lb>
illi nullo modo diuidi poſſe contendebant.</s>
        </p>
        <p type="main">
          <s id="s.003412"><arrow.to.target n="marg269"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003413"><margin.target id="marg269"></margin.target>279</s>
        </p>
        <p type="main">
          <s id="s.003414">Tertius locus <emph type="italics"></emph>(Præterea ſi quis communem ſtatam, ac determinatam menſu­<lb></lb>ram faciat diuiduam, non erit amplius in rerum natura linea vlla rationalis, aut <lb></lb>
irrationalis, reſpectu expoſitæ, ac determinatæ lineæ; neque aliarum vlla erit, de <lb></lb>
quibus modo dictum eſt, veluti quam Apotomen vocant ex duobus nominibus. </s>
          <s id="s.003415">Ve­<lb></lb>
rùm neque ſecundum ſe aliquam definitam naturam habebunt, ſed collatæ ſibi ipſis <lb></lb>
tam rationales, quàm irrationales erunt omnes.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003416">Hæc eſt alia eorumdem ratio ad idem comprobandum: quam, vt benè <lb></lb>
percipiamus, nonnulla prius ex definitionibus 10. Elem. ſunt explicanda: <lb></lb>
vt quæ nam ſint lineæ rationales, quæ irrationales, quæ ex binis nomini­<lb></lb>
bus, quæ Apotomæ.</s>
        </p>
        <p type="main">
          <s id="s.003417">Propoſita igitur linea quapiam, v. g. trium palmorum qualis eſt linea A, <lb></lb>
poſſunt inueniri quamplurimæ lineæ, quarum aliæ ſint illi longitudine com­<lb></lb>
<figure id="id.009.01.203.2.jpg" place="text" xlink:href="009/01/203/2.jpg"></figure><lb></lb>
menſurabiles, ſiue quæ cum expoſita A, ha­<lb></lb>
beant communem menſuram. </s>
          <s id="s.003418">v. g. linea B, <lb></lb>
<expan abbr="quinq;">quinque</expan> palmorum eſt commenſurabilis lineæ <lb></lb>
A, quia vtramque communis menſura vnius <lb></lb>
palmi metitur: aliæ verò ſint eidem A, lon­<lb></lb>
gitudine incommenſurabiles, qualis eſſet diameter C D, quadrati lineæ A, <lb></lb>
quæ eſt cum latere A, incommenſurabilis ex vltima 10.</s>
        </p>
        <figure id="id.009.01.203.3.jpg" place="text" xlink:href="009/01/203/3.jpg"></figure>
        <p type="main">
          <s id="s.003419">Cæterum lineam primò expoſitam, vt eſt in præ­<lb></lb>
ſentia A, quod eſſet notæ quantitatis, Græci appella­<lb></lb>
runt <foreign lang="grc">Ρήτην,</foreign> ideſt rationalem, quemadmodum Latini <lb></lb>
eam appellant.</s>
        </p>
        <p type="main">
          <s id="s.003420">Linearum autem longitudine <expan abbr="incommẽſurabilium">incommenſurabilium</expan> <lb></lb>
cum expoſita rationali A, aliæ ſunt, quæ tamen ſunt <lb></lb>
commenſurabiles eidem potentia, ideſt conſtituunt <lb></lb>
quadrata, quæ ſunt commenſurabilia quadrato ra­<lb></lb>
tionali A, vt linea C D, cum ſit diameter quadrati li­<lb></lb>
neæ A, quadratum exhibet, quod eſt duplum quadrati lineæ A, ex 47. primi, <lb></lb>
quadratum autem lineæ A, eſt nouem, igitur quadratum eius duplum erit <lb></lb>
octodecim, quadratum ſcilicet lineæ C D. octodecim autem, &amp; nouem ſunt <lb></lb>
<expan abbr="cõmenſurabilia">commenſurabilia</expan> communi vnitatis menſura, huiuſmodi lineæ dicuntur com­<lb></lb>menſurabiles potentia tantum, potentia. </s>
          <s id="s.003421">n. </s>
          <s id="s.003422">lineæ dicuntur <expan abbr="quadratũ">quadratum</expan> illius.</s>
        </p>
        <p type="main">
          <s id="s.003423">Quæ igitur rationali propoſitæ ſunt commenſurabiles aliquo modo, ſiue <lb></lb>
longitudine, &amp; potentia (<expan abbr="quæcunq;">quæcunque</expan> enim commenſurabilis eſt longitudine, <lb></lb>
eſt etiam potentia) ſiue potentia ſolùm, rationales ipſæ quoque dicuntur.


<pb pagenum="204" xlink:href="009/01/204.jpg"></pb><figure id="id.009.01.204.1.jpg" place="text" xlink:href="009/01/204/1.jpg"></figure><lb></lb>
Aliæ verò (quarum permultæ in decimo reperiun­<lb></lb>
tur) quæ nec longitudine, nec potentia illi ſunt <lb></lb>
commenſurabiles, irrationales appellantur, qua­<lb></lb>
lis eſſet media proportionalis E F, inter duas A, <lb></lb>
&amp; C D, in præſenti figura ex 11. 10.</s>
        </p>
        <p type="main">
          <s id="s.003424">Sciendum præterea ex 37. 10. &amp; ſequentibus, <lb></lb>
quod ex duabus lineis rationalibus reſpectu rationalis expoſitæ. </s>
          <s id="s.003425">v. g. A, com­<lb></lb>
menſurabilibus inuicem tantum potentia, componitur linea, quæ cum ea­<lb></lb>
<figure id="id.009.01.204.2.jpg" place="text" xlink:href="009/01/204/2.jpg"></figure><lb></lb>
dem expoſita eſt irrationalis, <expan abbr="vocaturq́">vocaturque</expan>; ex <lb></lb>
duobus nominibus, ſiue Binomium, vt ſi ex <lb></lb>
latere A, &amp; diametro C D, componatur li­<lb></lb>
nea A C D, erit irrationalis cum rationali <lb></lb>
A, <expan abbr="diceturq́">diceturque</expan>; binomium. </s>
          <s id="s.003426">Amplius ex 74. 10. &amp; ſequentibus, ſi prædictum <lb></lb>
minus nomen, ſiue minor linea A, detrahatur ex maiori nomine C D, vt re­<lb></lb>
linquatur B D linea, erit ipſa reliqua B D, irrationalis, quam poſtea appel­<lb></lb>
lant Apotomen, ſiue latinè Reſiduum.</s>
        </p>
        <p type="main">
          <s id="s.003427">Poſtremò, &amp; hoc non ignorandum ex 43. 10. lineam, ſiue <expan abbr="binomiũ">binomium</expan> A C D, <lb></lb>
non poſſe diuidi in alio puncto, præter C, in duas lineas, quæ ſint rationales <lb></lb>
expoſitæ, &amp; potentia tantum inuicem commenſurabiles.</s>
        </p>
        <p type="main">
          <s id="s.003428">His præmiſſis textum, ac rationem illorum explicabo in hunc modum.</s>
        </p>
        <p type="main">
          <s id="s.003429">Si quis faciat diuiduam lineam illam, quæ eſt communis <expan abbr="mẽſura">menſura</expan> omnium <lb></lb>
commenſurabilium, ſequetur hoc abſurdum contra demonſtrationes 10. <lb></lb>
quod nulla erit amplius linea rationalis, nec irrationalis, quia ſi communis <lb></lb>
menſura diuidatur, tolletur ea de rerum natura; vnde non erit amplius in­<lb></lb>
ter lineas ſymetria vlla, quare neque vllæ erunt rationales, eſſe enim ratio­<lb></lb>
nale oritur ex commenſurabilitate. </s>
          <s id="s.003430">quare <expan abbr="neq;">neque</expan> extabit illa rationalis expo­<lb></lb>
ſita, ad quam cæteræ relatæ dicuntur rationales, vel irrationales: quapro­<lb></lb>
pter etiam irrationales nullæ erunt, <expan abbr="neq;">neque</expan> vlla alia erit ex prædictis, veluti <lb></lb>
nec irrationalis illa, quam vocant Apotomen ex Binomio, ſiue ex duobus <lb></lb>
nominibus, de qua Euclides propoſ. </s>
          <s id="s.003431">74. 10. &amp; ſequentibus pertractat.</s>
        </p>
        <p type="main">
          <s id="s.003432">Notandum in verſu illo <emph type="italics"></emph>(Apotomen ex duobus nominibus compoſitam)<emph.end type="italics"></emph.end> vni­<lb></lb>
ca voce illa <emph type="italics"></emph>(Compoſitam)<emph.end type="italics"></emph.end> addita ab Interprete Iatino, quæ non extat in tex. <lb></lb>
græco, magnum Ariſtoteli imponi erratum, cum hac ratione dicat apoto­<lb></lb>
men ex duobus nominibus eſſe compoſitam, quod falſiſſimum eſt. </s>
          <s id="s.003433">Apotome <lb></lb>
enim, vt ſupra dictum eſt, ne dum ex duobus nominibus conſtat, verum ip­<lb></lb>
ſa eſt reſiduum lineæ maioris, ſi minor ab ipſa detrahatur. </s>
          <s id="s.003434">Verumenimuero <lb></lb>
vox illa <emph type="italics"></emph>(Compoſitam)<emph.end type="italics"></emph.end> in nullo codice reperitur, quare pro arbitrio, atque <lb></lb>
ex Geometriæ inſcitia addita, tolli debet, ne tantæ inſcitiæ Ariſt. ipſe re­<lb></lb>
darguatur. </s>
          <s id="s.003435">hæc in hunc locum ſufficiant.</s>
        </p>
        <p type="main">
          <s id="s.003436"><arrow.to.target n="marg270"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003437"><margin.target id="marg270"></margin.target>280</s>
        </p>
        <p type="main">
          <s id="s.003438">Quartus locus <emph type="italics"></emph>(Quod verò de commenſurabilibus lineis poſtremò dicunt, om­<lb></lb>nes vna quadam, &amp; eadem menſura oportere menſurari, falſum eſt admodum, &amp; <lb></lb>
nequaquam Mathematicorum ſuppoſitionibus concordat. </s>
          <s id="s.003439">non enim ita ſupponunt <lb></lb>
Geometræ, <expan abbr="neq;">neque</expan> vtile ipſis iſtud foret, imò potius aduerſaretur, lineas omnes com­<lb></lb>menſurabiles eſſe, &amp; omnium commenſurabilium linearum communem menſuram <lb></lb>
exiſtimare. </s>
          <s id="s.003440">quamobrem ridiculum eſt eos, qui dicunt ſe demonstrare ex Geometra­<lb></lb>
rum decretis, &amp; ex quibus Mathematici docent in contentioſam pariter, ac falla­<emph.end type="italics"></emph.end>


<pb pagenum="205" xlink:href="009/01/205.jpg"></pb><emph type="italics"></emph>cem diuertere argumentationem, præſertim tam inualidam. </s>
          <s id="s.003441">nam multis modis im­<lb></lb>becillis eſt eiuſmodi ratio, &amp; quouis modo licet euitare, ne aut inuſitata dicere, aut <lb></lb>
argui videamur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003442">Refellit hoc loco ſuperiores rationes in tribus locis præmiſſis allatas, <lb></lb>
quibus nonnulli probabant quantitatem ex indiuiduis conſtare, &amp; proinde <lb></lb>
concedenda eſſe quædam Quanta, omninò atoma; ſic igitur inquit. </s>
          <s id="s.003443">Quod <lb></lb>verò de commenſurabilibus lineis dicunt, omnes videlicet vnica quadam, <lb></lb>
<expan abbr="eademq́">eademque</expan>; determinata menſura menſurari oportere, falſum omninò eſt, &amp; <lb></lb>
contra mathematicorum dogmata, non enim Geometræ hoc aſſerunt, cùm <lb></lb>ipſorum demonſtrationibus aduerſetur; ſed <expan abbr="tantũ">tantum</expan> dicunt omnes lineas, quæ <lb></lb>
ad inuicem ſunt commenſurabiles, commenſurari, vna <expan abbr="eademq́">eademque</expan>; menſura, <lb></lb>
<figure id="id.009.01.205.1.jpg" place="text" xlink:href="009/01/205/1.jpg"></figure><lb></lb>
ſed non tamen vnica, ideſt non vnica, ac determi­<lb></lb>
nata. </s>
          <s id="s.003444">poſſunt enim eſſe plures <expan abbr="eædemq́">eædemque</expan>; menſuræ <lb></lb>
communes plurium quantitatum commenſura­<lb></lb>
bilium, vt præſentium trium linearum 4. 6. 8. <lb></lb>
communis <expan abbr="mẽſura">menſura</expan> eſt linea 2. binarius enim tres <lb></lb>
numeros 4.6. &amp; 8. menſurat. </s>
          <s id="s.003445">&amp; ſi linea 2. bifariam <lb></lb>
ſecetur, erit dimidium eius linea 1. quæ pariter <lb></lb>
erit communis menſura trium prædictarum li­<lb></lb>
nearum, cûm vnitas ſit omnium numerorum communis menſura. benè ve­<lb></lb>
rum eſt, quod Geometræ, quando ſimpliciter loquuntur de huiuſmodi com­<lb></lb>
muni menſura, intelligunt de ea, quæ inter omnes eſt maxima: vt in prædi­<lb></lb>
ctis tribus lineis maxima earum communis menſura eſt linea 2. <expan abbr="Atq;">Atque</expan> hoc ſi­<lb></lb>
bi volunt Geometræ, ex quibus totus hic textus intelligi poteſt.</s>
        </p>
        <p type="main">
          <s id="s.003446"><arrow.to.target n="marg271"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003447"><margin.target id="marg271"></margin.target>281</s>
        </p>
        <p type="main">
          <s id="s.003448">Quintus locus <emph type="italics"></emph>(Ob rectæ verò lineæ motum in ſemicirculum, quam neceſſe eſt <lb></lb>
in rectum ita diuidere, vt infinitæ circunferentiæ, &amp; interualla totidem inuenian­<lb></lb>
tur)<emph.end type="italics"></emph.end> Interpres latinus ſic vertit <emph type="italics"></emph>(Ob rectæ verò lineæ motum in ſemicirculum <lb></lb>
diuiduas non credere, &amp;c.)<emph.end type="italics"></emph.end> vbi verba illa <emph type="italics"></emph>(Diuiduas non credere)<emph.end type="italics"></emph.end> pro arbitrio, <lb></lb>
ac ſine ratione, imò contra rationem addidit: tum quia in Græco textu non <lb></lb>
extant, tum quia ſenſus totius ſententiæ is eſt, vt potius debuiſſet affirmati­<lb></lb>
uè dicere <emph type="italics"></emph>(Diuiduas credere)<emph.end type="italics"></emph.end> nam Ariſtoteles videtur ſic <expan abbr="argumẽtari">argumentari</expan>, quan­<lb></lb>
<figure id="id.009.01.205.2.jpg" place="text" xlink:href="009/01/205/2.jpg"></figure><lb></lb>
do recta linea A B, vt in appoſita figura mo­<lb></lb>
uetur intrando in ſemicirculum C A D B, ita <lb></lb>
vt primò ſit in ſitu A B, ſecundò in E F, tertiò <lb></lb>
in G H, &amp; ſimiliter in alijs omnibus ſemicir­<lb></lb>
culi locis, neceſſariò accidit, vt infinitæ peri­<lb></lb>
phęriæ, quales <expan abbr="sũt">sunt</expan> A B, E A B F, G E A B F H, <lb></lb>
cadant inter infinitas partes lineæ ingredien­<lb></lb>
tis, vt ſunt A B, E F, G H, <expan abbr="atq;">atque</expan> tam tota recta <lb></lb>
ingrediens, quàm totus ſemicirculus, diuidatur in partes infinitas, ita vt <lb></lb>
nulla pars lineæ rectæ, <expan abbr="neq;">neque</expan> vlla ſemicirculi ſuperſit, quæ ſe ſe mutuò non <lb></lb>
diuidantur, ergò nihil tam in linea, quàm in ſemicirculo remanet, quod non <lb></lb>
ſecetur: tota igitur linea recta, &amp; periphæria illa diuidua eſt, quam ob rem <lb></lb>
nullo modo conſtare poteſt ex indiuiduis, ex quibus manifeſtum eſt perpe­<lb></lb>
ram additamentum illud factum eſſe, &amp; ſimul ratio, &amp; textus Ariſt. eadem <lb></lb>
opera patefacta ſunt.</s>
        </p>
        <pb pagenum="206" xlink:href="009/01/206.jpg"></pb>
        <p type="main">
          <s id="s.003449"><arrow.to.target n="marg272"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003450"><margin.target id="marg272"></margin.target>282</s>
        </p>
        <p type="main">
          <s id="s.003451">Sextus locus <emph type="italics"></emph>(Rurſus <expan abbr="quoq;">quoque</expan> facilè perſuaderi poteſt ex mota duorurm circulo­ <lb></lb>rum æqualium, nam quiſquis horum moueatur, oportet per maiorem ſemicirculum <lb></lb>moueri, &amp; quæcunque alia huiuſmodi constituta ſunt de lineis, fieri non poſſe, vt <lb></lb>
talis vllus motus peragatur, quin prius omnibus, &amp; ſingulis interiectis occurrat. <lb></lb>
</s>
          <s id="s.003452">Atque hæc Mathematicorum ſcita, multò magis ab omnibus conceſſa ſunt, quàm <lb></lb>
illorum dicta.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003453">Hæc eſt alia ratio, qua probat totam circuli periphæriam eſſe diuiduam. <lb></lb>
<figure id="id.009.01.206.1.jpg" place="text" xlink:href="009/01/206/1.jpg"></figure><lb></lb>
ſint enim duo circuli æquales primum in eo­<lb></lb>
dem loco, <expan abbr="vocenturq́">vocenturque</expan>; A, &amp; B, deinde circu­<lb></lb>
lus B, moueatur, &amp; diſcedat à circulo A, ma­<lb></lb>
nente; ſtatim <expan abbr="namq;">namque</expan> pars egreſſa E F G, erit <lb></lb>
maior ſemicirculo, &amp; ſemper fiet maior, ac <lb></lb>
maior. </s>
          <s id="s.003454"><expan abbr="atq;">atque</expan> in tali motu omnes partes egre­<lb></lb>
dientis circuli ſecantur ab omnibus partibus <lb></lb>
circuli manentis. </s>
          <s id="s.003455">vnde patet nihil eſſe in eo­<lb></lb>
rum periphærijs, quod non diuidatur. </s>
          <s id="s.003456">nul­<lb></lb>
lum igitur in eis eſt indiuiduum. </s>
          <s id="s.003457">falluntur igitur aduerſarij.</s>
        </p>
        <p type="main">
          <s id="s.003458"><arrow.to.target n="marg273"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003459"><margin.target id="marg273"></margin.target>283</s>
        </p>
        <p type="main">
          <s id="s.003460">Septimus locus <emph type="italics"></emph>(Quamuis autem ex confutatis nuper rationibus appareat, ne­<lb></lb>
que probabile, neque neceſſarium eſſe lineas vllas indiuiduas extare, tamen ex ijs <lb></lb>etiam, quæ deinceps ſubiungam, multò magis perſpicuum euadet. </s>
          <s id="s.003461">&amp; primò quidem <lb></lb>
per ea, quæ Mathematici demonſtrant, at que addiſcenda proponunt, quæ mutare <lb></lb>
non decet, niſt probabiliores rationes habeamus. </s>
          <s id="s.003462">Nam neque lineæ, neque rectæ li­<lb></lb>
neæ definitio cum inſecabili linea conſentit, vt quæ nec inter duo puncta extenſa <lb></lb>ſit, nec medium vllam habeat.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003463">Idem, ſed paulò mutatis verbis poſtea repetit, quæ fortè ab aliquo per <lb></lb>
errorem addita ſunt. </s>
          <s id="s.003464">Verumenimuerò maximè conſiderandum eſt, quan­<lb></lb>
tum hoc loco Ariſt. Mathematicis demonſtrationibus tribuat: quod dixe­<lb></lb>
rim propter recentiores quoſdam, qui eò audaciæ deuenerunt, vt Euclidis <lb></lb>
firmiſſimas, <expan abbr="atq;">atque</expan> Ariſtot. teſtimonio, <expan abbr="veterumq́">veterumque</expan>; Philoſophorum omnium <lb></lb>
comprobatas, negare non verentur Demonſtrationes.</s>
        </p>
        <p type="main">
          <s id="s.003465">Cæterùm Ariſt. iterum opinionem <expan abbr="aſſerẽtium">aſſerentium</expan> lineas inſecabiles hoc mo­<lb></lb>
do confutat: nam ſi inquit, lineam illam, quam vocant inſecabilem, eſt non <lb></lb>
ſolum linea, ſed etiam linea recta, illi conueniret rectæ lineæ definitio, ſed <lb></lb>
nullo modo poteſt ci conuenire, ergò tollendæ ſunt de rerum natura huiuſ­<lb></lb>
modi lineæ. </s>
          <s id="s.003466">Porrò definitio lineæ eſt, vt ſit longitudo latitudinis expers, &amp; <lb></lb>
ſi recta ſit ex æquo ſua interiacet puncta extrema, ergò ipſa linea media erit <lb></lb>
inter duo indiuidua extrema puncta; at verò linea, quam ipſi volunt eſſe <lb></lb>
indiuiduum quoddam, qua ratione medium erit inter alia duo indiuidua? <lb></lb>
</s>
          <s id="s.003467">ipſi enim <expan abbr="vidẽtur">videntur</expan> velle iſtam lineam non habere medium vllum, ſi enim con­<lb></lb>cederent habere medium, iam poſſet in medio ſecari, quod ipſi nequaquam <lb></lb>
concederent: patet igitur definitionem lineæ minimè illi conuenire, &amp; pro­<lb></lb>
pterea <expan abbr="neq;">neque</expan> eſſe inter lineas enumerandam.</s>
        </p>
        <p type="main">
          <s id="s.003468"><arrow.to.target n="marg274"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003469"><margin.target id="marg274"></margin.target>284</s>
        </p>
        <p type="main">
          <s id="s.003470">Octauus locus <emph type="italics"></emph>(Deinde omnes lineæ commenſurabiles erunt: nam omnes ab in­<lb></lb>
diuiduis lineis dimetientur, quæqué; longitudine, quæqué; potentia ſunt commenſurabi­<lb></lb>
les. </s>
          <s id="s.003471">indiuiduæ autem lineæ ſibi ipſis commenſurabiles ſunt longitudine, cum inter ſe <lb></lb>
fiat æquales; quare potentia quoque, quod ſi hoc eſt, diuiduum erit quadratum.<emph.end type="italics"></emph.end></s>
        </p>
        <pb pagenum="207" xlink:href="009/01/207.jpg"></pb>
        <p type="main">
          <s id="s.003472">Pergit adhuc nouis rationibus aduerſarios refellere, dicens, ſi extarent <lb></lb>
huiuſmodi indiuiduæ lineæ, ſequeretur omnes omninò lineas eſſe commen­<lb></lb>
ſurabiles, quod eſt contra demonſtrata in 10. Elem. quia cum omnes lineæ <lb></lb>
<expan abbr="conſtẽt">conſtent</expan> per ipſos ex lineis atomis, iſtæ atomæ eſſent omnium linearum com­<lb></lb>
munes menſuræ, vnde &amp; illæ, quæ dicuntur potentia tantum commenſura­<lb></lb>
biles, vt ſupra explicaui, erunt etiam commenſurabiles longitudine. </s>
          <s id="s.003473">indiui­<lb></lb>
duæ verò ipſæ, cum ſint inuicem æquales, erunt ipſæ <expan abbr="quoq;">quoque</expan> commenſurabi­<lb></lb>
les longitudine, quare &amp; potentia, omnes enim longitudine commenſura­<lb></lb>
biles, ſunt etiam potentia commenſurabiles, ex 9. 10. vnde ſequitur qua­<lb></lb>
drata earum omnia eſſe <expan abbr="quoq;">quoque</expan> commenſurabilia: <expan abbr="atq;">atque</expan> hinc conſequitur, in­<lb></lb>
quit, ea eſſe <expan abbr="quoq;">quoque</expan> diuidua (quam conſecutionem probat infra num. </s>
          <s id="s.003474">290.) <lb></lb>
vnde ſequeretur ipſam <expan abbr="quoq;">quoque</expan> lineam latus quadrati poſſe diuidi, non igitur <lb></lb>
ponenda erat indiuidua.</s>
        </p>
        <p type="main">
          <s id="s.003475"><arrow.to.target n="marg275"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003476"><margin.target id="marg275"></margin.target>285</s>
        </p>
        <p type="main">
          <s id="s.003477">Nonus Iocus, cuius latinam interpretationem, cum admodum eſſet de­<lb></lb>
prauata ex græco textu, in hunc modum correxi <emph type="italics"></emph>(Præterea cùm circa maio­<lb></lb>rem latitudinem facit applicata, æquale ei, quod ab indiuidua, &amp; pedali copulatis <lb></lb>
circa bipedalem, minorem faciet latitudinem, quàm ſit indiuidua: erit minus, quod <lb></lb>
circa indiuiduam)<emph.end type="italics"></emph.end> ideſt cùm minor linea applicata cum maiore, latitudinem <lb></lb>
<figure id="id.009.01.207.1.jpg" place="text" xlink:href="009/01/207/1.jpg"></figure><lb></lb>
faciat. </s>
          <s id="s.003478">v. g. linea minor A B, applicata cum ma­<lb></lb>
iori B C, vt in figura, ita vt contineant figuram <lb></lb>
A B C D. </s>
          <s id="s.003479">Minor A B, facit latitudinem figuræ, <lb></lb>
maior verò B C, facit longitudinem. </s>
          <s id="s.003480">Iam cum <lb></lb>
aduerſarij velint extare huiuſmodi lineas ato­<lb></lb>
mas, conſtituatur figura ſub vna ex illis, quæ ſit v. g. A B, &amp; altera maiori, <lb></lb>
quæ ſit pedalis, v. g. B C, vt in præcedenti figura, ſumatur deinde linea bi­<lb></lb>
<figure id="id.009.01.207.2.jpg" place="text" xlink:href="009/01/207/2.jpg"></figure><lb></lb>
pedalis E F, cui per 45. primi ap­<lb></lb>
plicetur ſpatium E F G H, æquale <lb></lb>
ſpatio ſuperiori A B C D, neceſſa­<lb></lb>
riò latitudo E H, huius ſecundæ fi­<lb></lb>
guræ minor erit quàm latitudo il­<lb></lb>lius, hoc eſt minor, quàm ſit indiuidua A B, quod eſt abſurdum. </s>
          <s id="s.003481">vel dicere <lb></lb>
oportet <expan abbr="ſpatiũ">ſpatium</expan> circa indiuiduam A B, eſſe minus quàm iſtud poſterius, quod <lb></lb>
eſt contra conſtructionem, &amp; propterea pariter inconueniens, non igitur <lb></lb>
huiuſmodi lineæ ſunt ponendæ.</s>
        </p>
        <p type="main">
          <s id="s.003482"><arrow.to.target n="marg276"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003483"><margin.target id="marg276"></margin.target>286</s>
        </p>
        <p type="main">
          <s id="s.003484">Decimus locus <emph type="italics"></emph>(Cum ex tribus datis lineis triangulus componatur, ex tribus <lb></lb>
<expan abbr="quoq;">quoque</expan> indiuiduis lineis componi poterit. </s>
          <s id="s.003485">in omni autem æquilatero perpendicularis <lb></lb>in mediam baſim incidit, quare, &amp; in medium indiuiduæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003486">Ex 22. primi Elem. ex tribus datis lineis, quarum quælibet duæ ſint, re­<lb></lb>
liqua maiores poteſt conſtitui triangulum: poterit igitur ex tribus indiui­<lb></lb>
<figure id="id.009.01.207.3.jpg" place="text" xlink:href="009/01/207/3.jpg"></figure><lb></lb>
duis conſtitui <expan abbr="triãgulum">triangulum</expan>, <expan abbr="illudq́">illudque</expan>; æquilaterum, cum omnes in­<lb></lb>
diuiduæ lineæ ſint æquales. </s>
          <s id="s.003487">ſit igitur ex eis triangulum A B C, <lb></lb>
ſi igitur ab angulo A, ducatur perpendicularis A D, ad baſim <lb></lb>
B C, eam bifariam ſecabit ex ſcholio 26. primi, erit igitur li­<lb></lb>
nea B C, ſecabilis, contra quam aduerſarij opinantur.</s>
        </p>
        <p type="main">
          <s id="s.003488"><arrow.to.target n="marg277"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003489"><margin.target id="marg277"></margin.target>287</s>
        </p>
        <p type="main">
          <s id="s.003490">Vndecimus locus <emph type="italics"></emph>(Si quadratum ex quatuor indiuiduis conſtituatur diametro <lb></lb>
protracta, &amp; perpendiculari ducta, quadrati coſta potentia <expan abbr="perpẽdicularem">perpendicularem</expan>, diame-<emph.end type="italics"></emph.end>


<pb pagenum="208" xlink:href="009/01/208.jpg"></pb><emph type="italics"></emph>trumqué mediam æquat: quare non erit minima. </s>
          <s id="s.003491">neque duplum erit ſpatium à diame­<lb></lb>
tro conſurgens illius, quod ab indiuidua procreatur: nans æquali ablato, reliquum <lb></lb>erit minus indiuidua, nam ſi æqualis, diameter quadruplum deſcriberet, &amp;c.) <emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.208.1.jpg" place="text" xlink:href="009/01/208/1.jpg"></figure><lb></lb>
ideſt ſi per 46 primi quadratum. </s>
          <s id="s.003492">v.g. A B C D, ex qua­<lb></lb>
tuor inſecabilibus componatur, cuius diametro B C, <lb></lb>
perpendicularis A E, inſiſtat, erit per 47. primi qua­<lb></lb>
dratum lineæ A B, æquale quadratis <expan abbr="linearũ">linearum</expan> A E, E B, <lb></lb>
quare tam E B, quàm A E, minores erunt ipſa A B; <lb></lb>
quare ipſa non erit minima cum ſit indiuidua, quod eſt <lb></lb>
abſurdum. </s>
          <s id="s.003493">Præterea ex ſcholio 47. primi, quadratum <lb></lb>
C B F G, diametri C B, duplum eſt quadrati A B C D, <lb></lb>
ergò diameter C B, maior quàm A B. </s>
          <s id="s.003494">Auferatur igitur ab ipſa, C B, æqua­<lb></lb>
lis ipſi A B, quæ igitur reliqua erit, vel erit æqualis ipſi A B, vel minor. </s>
          <s id="s.003495">non <lb></lb>
æqualis, quia tunc diameter dupla eſſet lateris A B, &amp; quadratum diametri <lb></lb>
quadruplum foret quadrati lateris A B. ex ſcholio 4. ſecundi, quod abſur­<lb></lb>
dum eſt, repugnat enim 47. primi. </s>
          <s id="s.003496">nec minor, quia hoc modo exiſteret linea <lb></lb>
quædam minor minima, ſcilicet atoma, quod pariter eſt inconueniens.</s>
        </p>
        <p type="main">
          <s id="s.003497"><arrow.to.target n="marg278"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003498"><margin.target id="marg278"></margin.target>288</s>
        </p>
        <p type="main">
          <s id="s.003499">Duodecimus locus <emph type="italics"></emph>(Amplius ſi quæuis linea præter inſectilem in partes diui­ <lb></lb>
di poteſt, tùm æquales, tùm inæquales, ſeindatur linea in tria fruſta, quæ non con­<lb></lb>
ſtet ex tribus atomis, ſed vniuerſaliter ex imparibus numero atomis, ſic diuiſa erit <lb></lb>
linea indiuidua. </s>
          <s id="s.003500">ſimiliter autem ſi in duo diuidatur linea, quæ ex imparibus <expan abbr="cõſtat">conſtat</expan>)<emph.end type="italics"></emph.end><lb></lb>
hoc eſt detur linea quæpiam ab aduerſario ex lineis indiuiduis numerò im­<lb></lb>
paribus, conſtans. </s>
          <s id="s.003501">v. g. ex quinque; hæc diuidi poteſt in tres æquas partes <lb></lb>
per 10.6. Si igitur diuidatur in tria æqualia, neceſſariò tres ex atomis illam <lb></lb>
integrantibus erunt diſſectæ, nam tertia quælibet pars continebit indiui­<lb></lb>
duam vnam cum duabus tertijs alterius partibus. </s>
          <s id="s.003502">idem accidet ſi bifariam <lb></lb>
per 10. primi, ſecetur quæuis ex imparibus numero atomis conflata.</s>
        </p>
        <p type="main">
          <s id="s.003503"><arrow.to.target n="marg279"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003504"><margin.target id="marg279"></margin.target>289</s>
        </p>
        <p type="main">
          <s id="s.003505">Decimustertius locus <emph type="italics"></emph>(Quod ſi bifariam quidem non omnis linea finditur, ſed <lb></lb>
quæ ſolum ex paribus conflata ſit. </s>
          <s id="s.003506">ſi iam in duas partes diuiſa, in <expan abbr="quæcunq;">quæcunque</expan> diuidi <lb></lb>
poteſt diuideretur, ſic <expan abbr="quoq;">quoque</expan> inſectilis linea diuideretur, quando ex paribus compo­<lb></lb>
ſita, per inæqualia ſcinderetur)<emph.end type="italics"></emph.end> ideſt, quod ſi dixerit aduerſarius, non omnem <lb></lb>
lineam bifariam diuidi poſſe, ſed eam ſolùm, quæ ex numero paribus atomis <lb></lb>
conſtiterit: ea igitur diuidatur primo bifariam. </s>
          <s id="s.003507">deinde iterum diuidatur <lb></lb>
quomodocunque, ideſt &amp; bifariam, &amp; non bifariam, nam hoc etiam pacto <lb></lb>
indiuidua diuidetur, quod eſt inconueniens.</s>
        </p>
        <p type="main">
          <s id="s.003508"><arrow.to.target n="marg280"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003509"><margin.target id="marg280"></margin.target>290</s>
        </p>
        <p type="main">
          <s id="s.003510">Decimusquartus locus <emph type="italics"></emph>(Amplius non eſſet cuiuſuis lineæ quadratum: habe­<lb></lb>
ret enim longitudinem, &amp; latitudinem; <expan abbr="atq;">atque</expan> idcircò diuiſibile erit, cum illa quidem <lb></lb>aliquid, hæc autem aliquid aliud; quod ſi quadratum diuiduum eſt, &amp; linea, vnde <lb></lb>
procreatur, diuidua erit)<emph.end type="italics"></emph.end> poſſe ſuper quamuis datam lineam quadratum de­<lb></lb>
ſcribi patet ex 46. primi, quadratum igitur deſcriptum ab indiuidua, cum <lb></lb>
ſit ſuperficies, latitudinem, ac longitudinem habebit, quæ diuerſæ ſunt di­<lb></lb>
menſiones. </s>
          <s id="s.003511">poterit ergò ſecundum <expan abbr="vtramq;">vtramque</expan> diuidi; ex qua diuiſione neceſ­<lb></lb>
ſariò latera ipſius, hoc eſt lineæ, quas indiuiduas illi ponunt diuidentur, <lb></lb>quod eſt inconueniens, non igitur indiuiduæ erunt.</s>
        </p>
        <p type="main">
          <s id="s.003512"><arrow.to.target n="marg281"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003513"><margin.target id="marg281"></margin.target>291</s>
        </p>
        <p type="main">
          <s id="s.003514">Decimusquintus locus <emph type="italics"></emph>(Adhuc etiam, vt linea ſic, &amp; ſuperficies, &amp; corpus <lb></lb>erit impartibile: vno quippe indiuiduo exiſtente, cætera <expan abbr="quoq;">quoque</expan> conſequentur, quia<emph.end type="italics"></emph.end> <pb pagenum="209" xlink:href="009/01/209.jpg"></pb><emph type="italics"></emph>vnum per aliud diuiditur, at corpus indiuiduum non eſt, cum in ſe latitudinem, &amp; <lb></lb>
profunditatem contineat: quare nec linea poteſt eſſe atoma. </s>
          <s id="s.003515">corpus ſiquidem in ſu­<lb></lb>perficies, ſuperficies verò in lineas ſoluitur)<emph.end type="italics"></emph.end> hoc eſt: præterea, quemadmodum <lb></lb>linea per aduerſarium extat indiuidua, ſic &amp; ſuperficies ab eadem linea de­<lb></lb>
ſcripta erit atoma, &amp; corpus ab hac ſuperficie deſcriptum erit impartibile. <lb></lb>
</s>
          <s id="s.003516">Sciendum enim, quod ex motu puncti deſcribitur linea: ex motu lineæ de­<lb></lb>
ſcribitur ſuperficies: ex motu tandem ſuperficiei corpus ortum habet, vt ſo­<lb></lb>
let in horum definitionibus explicari.</s>
        </p>
        <p type="main">
          <s id="s.003517">Si igitur horum vnum nempè linea ſit atoma, &amp; reliqua, quæ ab ipſa ma­<lb></lb>
nant erunt indiuiſa, quia corpus diuiditur per ſuperficiem, &amp; ſuperficies <lb></lb>
per lineam, ideſt ad diuiſionem corporis neceſſe eſt diuidi ſuperficiem, &amp; ad <lb></lb>
ſuperficiei diuiſionem diuidi lineam, quæ ipſam terminat. </s>
          <s id="s.003518">At cum omne <lb></lb>
corpus latitudinem, &amp; profunditatem habeat, nullum poterit extare cor­<lb></lb>
pus, quod diuidi nequeat; quare neque illud, quod ab atoma linea oriretur. <lb></lb>
</s>
          <s id="s.003519">Quare nec linea illa corporis procreatrix erit indiuidua; corpus ſiquidem <lb></lb>
in ſuperficies, &amp; ſuperficies in lineas quodammodo reſoluitur: &amp; ex diui­<lb></lb>
ſione ſolidi ſuperficies ſecari debet, &amp; demum ſuperficiei, ſectionem lineæ <lb></lb>
ſectio ſubſequitur. </s>
          <s id="s.003520">Tollendæ igitur ſunt de rerum natura lineæ atomæ.</s>
        </p>
        <p type="main">
          <s id="s.003521"><arrow.to.target n="marg282"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003522"><margin.target id="marg282"></margin.target>292</s>
        </p>
        <p type="main">
          <s id="s.003523">Decimusſextus locus <emph type="italics"></emph>(Quin etiam orbis circunferentia rectam lineam pluri­<lb></lb>
bus tanget punctis, punctus enim contactus, quiqué eſt in circulo, quiqué eſt in recta, <lb></lb>
ſe ſe mutuò tangunt. </s>
          <s id="s.003524">quod ſi hoc fieri nequit, <expan abbr="neq;">neque</expan> punctus punctum tangere valet: <lb></lb>
quod ſi ſe tangere nequeunt, <expan abbr="neq;">neque</expan> linea punctis conſtare poteſt, nam neque punctum <lb></lb>
tangere neceſſarium eſt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003525">In 2. 3. &amp; corollario eius demonſtratur circuli peripheriam tangere re­<lb></lb>
ctam lineam in vnico puncto. </s>
          <s id="s.003526">iam ſi linea conſtaret ex punctis indiuiduis <lb></lb>
tanquam partibus, poſſet circulus <expan abbr="tãgere">tangere</expan> rectam lineam in duobus punctis. <lb></lb>
<figure id="id.009.01.209.1.jpg" place="text" xlink:href="009/01/209/1.jpg"></figure><lb></lb>
Sit circulus, cuius centrum A, tangens lineam <lb></lb>
rectam B C, conſtantem ex punctis, quorum vnus <lb></lb>
ſit in extremo D, lineæ B D, alterum verò in E, <lb></lb>
principio lineæ E C, circulus A, tangere poterit <lb></lb>
in F, termino communi vtriuſque lineæ, hocque <lb></lb>
modo tanget <expan abbr="vtrunq;">vtrunque</expan> punctum D, &amp; E, quod eſt <lb></lb>
impoſſibile per 2. 3. ſequitur igitur <expan abbr="neq;">neque</expan> illa duo puncta D, E, ſe mutuò tan­<lb></lb>
gere, &amp; eadem ratione nulla alia <expan abbr="pũcta">puncta</expan> eiuſdem lineæ, ex quibus manifeſtum <lb></lb>
eſt, impoſſibile eſſe, lineam ex huiuſmodi punctis conſtare poſſe.</s>
        </p>
        <p type="main">
          <s id="s.003527">Reliqua huius opuſculi, quamuis Mathematica alicui videri poſſint, <lb></lb>
non tamen ſunt, non enim linearibus indigent demonſtrationi­<lb></lb>
bus, <expan abbr="neq;">neque</expan> ex Geometriæ principijs procedunt. </s>
          <s id="s.003528">ad Phyſi­<lb></lb>
cum igitur pertinebunt, cuius eſt diſputare, num <lb></lb>
indiuidua exiſtant, &amp; quomodo in quanti­<lb></lb>
tate, <expan abbr="idq́">idque</expan>; rationibus aliunde, quàm <lb></lb>
ex Geometria deductis.</s>
        </p>
        <pb pagenum="210" xlink:href="009/01/210.jpg"></pb>
        <p type="head">
          <s id="s.003529"><emph type="italics"></emph>In Librum de Propriet. Elementorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003530">Libellum de cauſis proprietatum Elementorum, quamuis nonnulla <lb></lb>
mathematica loca contineat, tamen, quia certò conſtat ex ijs, <lb></lb>
quæ in eo de Secta Arabum, de Sclauis, de Dalmatis, qui multis <lb></lb>
poſt Ariſtotelem ſæculis floruerunt, auctorem alium eſſe ab Ariſto­<lb></lb>
tele conſultò &amp; meritò omiſi.</s>
        </p>
        <p type="head">
          <s id="s.003531"><emph type="italics"></emph>In Librum de Cauſis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003532">Alterum de cauſis libellum pariter prætermiſi, cum is vocibus Arabi­<lb></lb>cam barbariem redolentibus ſcateat: phraſis præterea, &amp; quędam <lb></lb>de Deo dicta, planè indicant authorem non eſſe Ariſtotelem; ſed potius <lb></lb>
Arabem quempiam.</s>
        </p>
      </chap>
      <pb pagenum="211" xlink:href="009/01/211.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.003533">EX LIBRO NONO <lb></lb>
DE HIST. ANIMALIVM</s>
        </p>
        <p type="head">
          <s id="s.003534"><emph type="italics"></emph>Araneorum industriæ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003535"><arrow.to.target n="marg283"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003536"><margin.target id="marg283"></margin.target>293</s>
        </p>
        <p type="main">
          <s id="s.003537">Cap. 39. <emph type="italics"></emph>(Aranei ſtatim cum editi ſunt, fila mittunt, non ab intrinſeco <lb></lb>
tanquam excrementum, vt Democritus ait, ſed ab extrinſeco de ſuo cor­<lb></lb>
pore, veluti cortice; aut more eorum animalium, quæ ſuos villos iacu­<lb></lb>
lantur, vt hystricis)<emph.end type="italics"></emph.end> Cum olim in hunc locum incidiſſem, inceſſit <lb></lb>animum meum illa cupido, vt ſcilicet certò ſcirem, num iure, an iniuria <lb></lb>
Ariſt. Democritum hoc loco reijceret, Araneum fila ab intrinſeco emitte­<lb></lb>
re aſſerentem: quapropter ad magiſtram rerum experientiam confugi, ac­<lb></lb>
cepto manu bacillo Araneum quendam ex ijs, qui circulares telas, quas <lb></lb>
<expan abbr="nõnulli">nonnulli</expan>, &amp; quidem aptè labyrinthos appellant, ingenio <expan abbr="vtiq;">vtique</expan> mathematico <lb></lb>contexunt, ſic adij, vt Araneus pro arbitrio ſuper bacillum liberè inambu­<lb></lb>
laret, dum ipſe interim curioſius illum obſeruarem, quanam videlicet ex <lb></lb>
parte filum foras ederet; cum ecce tibi Araneus experienti mihi vltrò fa­<lb></lb>
uens ſe ſe ex baculo demiſit, ita tamen, vt ex filo ſuo in aere ſuſpenſus re­<lb></lb>
maneret. </s>
          <s id="s.003538">cum primum obſeruo ipſum inuerſum, hoc eſt capite deorſum, &amp; <lb></lb>
ventre ſurſum pendere. </s>
          <s id="s.003539">vt autem acutius cernerem, eum opacæ cuidam rei <lb></lb>
oppoſui, ne præ nimia luce tenuiſſimum aranei filum aciem oculorum effu­<lb></lb>
geret; quo facto in temperata luce illa, clariſſimè videbam filum ex ſeceſſu <lb></lb>
aranei prodire. </s>
          <s id="s.003540"><expan abbr="Araneumq́">Araneumque</expan>; vno pede filum illud retinere, ne amplius exi­<lb></lb>
ret, <expan abbr="longiusq́">longiusque</expan>; fieret, quàm ſuo conſilio par eſſet. </s>
          <s id="s.003541">coegi deinde ipſum aſcen­<lb></lb>
dere, &amp; deſcendere ſæpius, donec certò certius, mihi conſtitiſſet filum illud <lb></lb>
non ab extrinſeco, vt hoc loco Ariſt. affirmat, ſed ab intrinſeco quippe ex <lb></lb>
ſeceſſu prodire, ac proinde veriſſimam eſſe quamuis ab Ariſt. reiectam De­<lb></lb>
mocriti ſententiam. </s>
          <s id="s.003542">cum Ariſt. pariter errauit Vlyſſes Aldobrandus in ſuo <lb></lb>
de inſectis pulcherrimo, <expan abbr="atq;">atque</expan> doctiſſimo Opere.</s>
        </p>
        <p type="main">
          <s id="s.003543">Verumenimuerò opportunè accidit, vt huius dubitationis ſolutio, aliam <lb></lb>
mihi alterius quæſtionis, iam olim ſummis votis expetitam afferret expli­<lb></lb>
cationem. </s>
          <s id="s.003544">ea eſt huiuſmodi. </s>
          <s id="s.003545">ſæpius fueram expertus, Araneos quoſdam eſ­<lb></lb>
ſe, qui ex vno loco ad alium omninò ſibi inacceſſibilem, tranſeant, ſiue quod <lb></lb>
idem eſt, ex eo loco, ad illum fila deducant, vt ex vna arbore ad aliam; <lb></lb>
quamuis inter <expan abbr="vtramq;">vtramque</expan> aut aquæ, aut denſiſſima ſpineta, ac ſepes interpo­<lb></lb>
nantur. </s>
          <s id="s.003546">quod maximè mane æquitantes experimur, dum nobis fila per vias <lb></lb>
tranſuerſa, oculis, atque vultui obuiantia adhærent. </s>
          <s id="s.003547">Qua ratione id Ara­<lb></lb>neus perficeret, neminem, qui literis mandaſſet, reperi, ne ipſum quidem. <lb></lb>
</s>
          <s id="s.003548">Vlyſſem Aldobrandum, qui in hac eruditorum palæſtra, maiores noſtros <lb></lb>
omnes videtur ſuperaſſe. </s>
          <s id="s.003549">Phyſiologi à me hac de re interrogati, varij va­<lb></lb>ria, nec conſentientia reſpondebant. </s>
          <s id="s.003550">Alij aiebant Araneum ſe demittere, <lb></lb>
ac ſuſpendere ex vna arbore, &amp; deinde ad aliam à vento perferri, at ego his <lb></lb>
minimè aſſentiebar, quia m Araneo nullum eſſet naturale inſtrumentum,


<pb pagenum="212" xlink:href="009/01/212.jpg"></pb>veluti velum, in quod ventus poſſit impingere. </s>
          <s id="s.003551">Alij Araneum ex vna arbo­<lb></lb>
re deſcendere, &amp; poſtea alteram conſcendere, interim emiſſum retro filum <lb></lb>
raptando, ac deinde ſurſum attrahendo attollere, ac prætendere: ſed ho­<lb></lb>
rum reſponſionum ob plurima impedimenta, quæ tenuiſſimum filum ſæpius <lb></lb>
ſcidiſſent, ſubridens refellebam. </s>
          <s id="s.003552">Alij verò aiebant Araneum qualitate qua­<lb></lb>
dam præditum eſſe, qua ipſe per aera, non ſecus, ac per aquam piſces, &amp; <lb></lb>
per aerem volucres, ambulare poſſet. </s>
          <s id="s.003553">Verum opinatio iſta, ne riſu quidem <lb></lb>
digna videbatur. </s>
          <s id="s.003554">Huius igitur quæſiti ſolutionem, quam omnes ad hanc <lb></lb>
<expan abbr="vſq;">vſque</expan> diem latuiſſe putò, <expan abbr="quamq́">quamque</expan>; omnibus gratiſſimam fore cognoui tibi lo­<lb></lb>
co auctarij initio promiſſi, nunc perſoluam. </s>
          <s id="s.003555">accidit ergò, vt dicebam, vt <lb></lb>
dum Araneus fugæ cupidus ex bacillo in temperatæ lucis loco, nimirum è <lb></lb>
regione alicuius opaci penderet, vt cernerem ex filo illo, ex quo ſuſpende­<lb></lb>
batur plura alia fila hinc inde alternatim prodire, quemadmodum ex alter­<lb></lb>
nis arundinum nodis folia enaſci ſolent. </s>
          <s id="s.003556">quæ fila, innata læuitate, per ae­<lb></lb>
rem quoquo verſus ceu natantia diffundebantur. </s>
          <s id="s.003557">factum eſt autem, vt eo­<lb></lb>
rum vnum quendam arboris cuiuſdam ramum attingeret, <expan abbr="eiq́">eique</expan>; ſtatim adhæ­<lb></lb>
reret; quod illicò Araneus optimè perſenſit, quippe quod filum illud viſce­<lb></lb>
ribus eius ex altero capite affigeretur, atque per filum illud, alijs ommiſſis, <lb></lb>ſubitò, vti egregius funambulus accurrit, ſed tamen pedibus ſurſum, dorſo <lb></lb>
autem deorſum, non ſupra filum, ſed infra ad ramum illum ſe contulit, <expan abbr="ſicq́">ſicque</expan>; <lb></lb>
me hoſtem ſuum fuga ſæpius eluſit. </s>
          <s id="s.003558">Ex qua repetita ſæpius obſeruatione lu­<lb></lb>
ce clarius comperi Araneum non ſimplex filum, ſed ramoſum, ac multiplex <lb></lb>
emittere, <expan abbr="atq;">atque</expan> aliquando ex ſeceſſu etiam ipſo duo ſimul eijcere, <expan abbr="alterũ">alterum</expan> quo <lb></lb>
<figure id="id.009.01.212.1.jpg" place="text" xlink:href="009/01/212/1.jpg"></figure><lb></lb>
ſuſpendatur, alterum <lb></lb>
verò, quod ſorte hac, <lb></lb>
<expan abbr="atq;">atque</expan> illac volitans, ali­<lb></lb>
cui rei occurrat, <expan abbr="atq;">atque</expan> <lb></lb>
hæreat, per quod po­<lb></lb>
ſtea ipſe incedens, ad <lb></lb>
locum ſibi prius inac­<lb></lb>
ceſſum, aditum parat. <lb></lb>
</s>
          <s id="s.003559">qua inre fures eos per­<lb></lb>
bellè imitatur, qui <lb></lb>ſcholas ex funibus con­<lb></lb>
textas, ac hamis fer­<lb></lb>
reis munitas, ad fene­<lb></lb>
ſtras proijciunt, vt per <lb></lb>eas ibi affixas conſcen­<lb></lb>
dere queant. </s>
          <s id="s.003560">quæ om­<lb></lb>
nia ex appoſita figura <lb></lb>
melius percipies, vbi <lb></lb>
ex ſiniſtra arbore pen­<lb></lb>
det Araneus A, ex filo <lb></lb>
B A, ex quo tanquam <lb></lb>
rami alia fila C G, D H, <lb></lb>
E I, M O, F L, alter­


<pb pagenum="213" xlink:href="009/01/213.jpg"></pb>natim prodeunt, ac per aerem hinc inde volitant. </s>
          <s id="s.003561">Si ergò filum E I, dextræ <lb></lb>
arbori occurrerit, <expan abbr="eiq́">eique</expan>; hæſerit, vt in figura, illicò Araneus huius rei con­<lb></lb>
ſcius per filum A E I, aſcendit, <expan abbr="ſeq́">ſeque</expan>; ad prius inacceſſam ſibi dextram arbo­<lb></lb>
rem transfert; <expan abbr="atq;">atque</expan> deinde inter <expan abbr="vtramq;">vtramque</expan> ducto iam filo vno, poteſt vltrò, <lb></lb>
<expan abbr="citroq́">citroque</expan>; means, ſuam etiam circularem, ac labyrinthiacam telam in muſca­<lb></lb>
rum capturam contexere; quales aliquando inter duas arbores admira­<lb></lb>
ri ſolemus.</s>
        </p>
        <p type="main">
          <s id="s.003562">Quæres fortè, num Araneus filum intus tanquam in glomo, vel ſpira con­<lb></lb>
uolutum contineat? </s>
          <s id="s.003563">dicam, quod non ſine experientia conijcio, exiſtimo <lb></lb>
Araneum non continere intra ſe filum vllum, verum humorem quendam <lb></lb>
viſcoſum, qui in tenuiſſima fila ſit ductilis; quemadmodum videmus acci­<lb></lb>
dere gummi, quæ diſrupta exhibet lentorem quendam, qui ſolo attritu ita <lb></lb>
digitis hæret, vt amoto ſenſim digito, filum tenue, &amp; oblongum valdè de­<lb></lb>
ducatur, hoc inde conijcio, quia aliquando cum ventrem Araneorum ſecuiſ­<lb></lb>
ſem nullum intus filum, ſed ſolus humor quidam lentus apparuit.</s>
        </p>
        <p type="main">
          <s id="s.003564">Cùm ex paruulis hiſce meis obſeruationibus circa animalculum iſtud <lb></lb>
vnum tam præclara cognouiſſem, quæ nullus ad hanc <expan abbr="vſq;">vſque</expan> diem, quod ſciam <lb></lb>
obſeruaſſet; animaduerti latiſſimum patere campum ad animalium hiſto­<lb></lb>
riam ampliandam, ſi ij, qui huic pulcherrimæ cognitioni dant operam, non <lb></lb>
ijs ſolum, quæ ab alijs perſcripta ſunt contenti eſſent, verùm etiam certiſ­<lb></lb>
ſimis, <expan abbr="atq;">atque</expan> exploratiſſimis experientijs ea coniungerent.</s>
        </p>
        <p type="main">
          <s id="s.003565">Atque hæc de Araneo ſatis.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003566"><emph type="italics"></emph>De inceſſu animalium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003567"><arrow.to.target n="marg284"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003568"><margin.target id="marg284"></margin.target>294</s>
        </p>
        <p type="main">
          <s id="s.003569">Cap. 7. <emph type="italics"></emph>(Etenim habentia pedes, quoniam ſuper <expan abbr="vtrumq;">vtrumque</expan> oppoſitorum cru­<lb></lb>
rum viciſſim ſtant, pondusqué ſustinent, neceſſe habent altero progredien­<lb></lb>
te, inflectere alterum; æqualia namque longitudine nata ſunt habere op­<lb></lb>
poſita membra. </s>
          <s id="s.003570">&amp; quod ponderi ſubſtat rectum eſſe oportet, vt perpen­<lb></lb>
diculum ad terram. </s>
          <s id="s.003571">quando autem progreditur, fit hypotenuſa, valens manentem <lb></lb>
magnitudinem, &amp; eam, quæ interiacet. </s>
          <s id="s.003572">quoniam autem æqualia ſunt membra, ne­<lb></lb>
ceſſe eſt inflecti id, quod manet, aut in poplite, aut in conflexione)<emph.end type="italics"></emph.end> Vult probare <lb></lb>
in greſſu neceſſariam eſſe aliquam flexionem membrorum. </s>
          <s id="s.003573">verum prius <lb></lb>
ſciendum, quod lineam hypotenuſam, quemadmodum etiam Athenæus lib. <lb></lb>
10. teſtatur, eam appellant geometræ, quæ in triangulo rectangulo recto <lb></lb>
angulo ſubtenditur, vnde &amp; denominata eſt hypotenuſa, ideſt ſubtenſa, vt <lb></lb>
<figure id="id.009.01.213.1.jpg" place="text" xlink:href="009/01/213/1.jpg"></figure><lb></lb>
in triangulo A B C, cuius angulus B, rectus ſit, recta <lb></lb>
A C, angulo recto B, ſubtenſa, hypotenuſa dicitur. <lb></lb>
</s>
          <s id="s.003574">Ariſt. igitur ait, quod antequam animal ambulare in­<lb></lb>
cipiat, dum ſcilicet manet, habet crura, quæ manent <lb></lb>
recta, ſiue perpendicularia horizonti, cum autem in­<lb></lb>
cipit progredi neceſſe eſt <expan abbr="vtrũq;">vtrunque</expan> crus inclinari ad ho­<lb></lb>
rizontem. </s>
          <s id="s.003575">nam primum crus in ingreſſu prolatum fit <lb></lb>
hypotenuſa, quia ſcilicet ſubtendit angulum rectum, <lb></lb>
quem facit alterum crus adhuc quieſcens, cum hori­


<pb pagenum="214" xlink:href="009/01/214.jpg"></pb>zonte; vt in ſuperiori triangulo, ſi concipiamus crura fuiſſe duo latera A B, <lb></lb>
A D, quæ manente animali, fuiſſent ambo ſimul in ſitu A B, perpendicula­<lb></lb>
ria horizonti; incipiens autem animal ambulare, proferat primo crus A D, <lb></lb>
A D, fiet hypotenuſa trianguli A B C, &amp; quia crus hoc A D, factum hypo­<lb></lb>tenuſa æquale eſt alteri manenti A B, nequit totius veræ hypotenuſæ A C, <lb></lb>officio fungi, quæ æquiualet toti A D, &amp; præterea interiacenti D C, vt ea au­<lb></lb>
<figure id="id.009.01.214.1.jpg" place="text" xlink:href="009/01/214/1.jpg"></figure><lb></lb>tem hypotenuſa debet eſſe maior, quia opponitur <lb></lb>
maiori angulo nimirum recto B, quam latus A B, <lb></lb>
quod angulo acuto C, opponitur per 19. primi, &amp; <lb></lb>
propterea niſi alterum ſubſequens crus A B, incli­<lb></lb>
netur, vt in ſecunda figura, non poteſt hypotenuſa <lb></lb>
A D, terram attingere, <expan abbr="atq;">atque</expan> hac de cauſa neceſſe <lb></lb>
eſt, vt initio greſſus <expan abbr="vtrumq;">vtrumque</expan> crus, quod prius per­<lb></lb>
pendiculare erat, inclinetur; inclinato igitur crure <lb></lb>
A B, antrorſum tunc prolatum crus A C, terram <lb></lb>
contingit, <expan abbr="ſicq́">ſicque</expan>; factus eſt primus greſſus B C.</s>
        </p>
        <p type="main">
          <s id="s.003576"><arrow.to.target n="marg285"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003577"><margin.target id="marg285"></margin.target>295</s>
        </p>
        <p type="main">
          <s id="s.003578">Eodem loco <emph type="italics"></emph>(Signum autem, quod hoc ita ſe habet illud est. </s>
          <s id="s.003579">ſi quis enim iuxta <lb></lb>
parietem per terram ambulet, quæ deſignatur linea non eſt recta, ſed obtorta, quo­<lb></lb>
niam minorem quidem flectentis fieri deſcriptam neceſſe eſt; ſtantis autem, &amp; ere­<lb></lb>
cti maiorem)<emph.end type="italics"></emph.end> Vt probet, quod animal in gradiendo modo attollitur, modo <lb></lb>
deprimitur, ſignum hoc affert, quia ſi quis ſecus parietem per terram am­<lb></lb>
bulet, linea quam vertex capitis in pariete deſignat non eſt recta, ſeb obtor­<lb></lb>
ta: quæ linea optimè deſignatur, ſi ambulantis vmbra in pariete apparens <lb></lb>
ſimul, cum ipſo in pariete ambulet; videmus enim vmbram illam modo al­<lb></lb>
tiorem fieri, modo breuiorem; quod ſignum eſt ambulantem modo incli­<lb></lb>
nari, quando ſcilicet crus alterum profert, ſeu crura dilatat; modo erigi, <lb></lb>cum crus ſubſequens præcedenti coniungit, tune enim incedens fit horizon­<lb></lb>
ti perpendicularis.</s>
        </p>
        <p type="main">
          <s id="s.003580"><arrow.to.target n="marg286"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003581"><margin.target id="marg286"></margin.target>296</s>
        </p>
        <p type="main">
          <s id="s.003582">Eodem cap. <emph type="italics"></emph>(Quoniam autem fiat ad rectum, vel concidet recto minore effe­<lb></lb>
cto, vel non progredietur: ſi enim altero crure recto progreditur alterum, maius <lb></lb>
erit cum ſit æquale: hoc <expan abbr="nanq;">nanque</expan> poterit, &amp; id, quod quieſcit, &amp; ipſam hypotenu­<lb></lb>
ſam, neceſſe igitur eſt, &amp; inflectere id, quod procurrit, &amp; inflexum ſimul alterum <lb></lb>
extendere, membra enim triangulorum æquilaterorum efficiuntur, <expan abbr="caputq́">caputque</expan>, fit infe­<lb></lb>
rius, vbi perpendiculum fuerit, in quo firmatum eſt)<emph.end type="italics"></emph.end> Hæc ſunt ferè eadem cum <lb></lb>
ijs, quæ in primo huius capitis loco dicta ſunt. </s>
          <s id="s.003583">proinde ea cum duabus illis <lb></lb>
triangulorum figuris repetenda ſunt, vt breuius quæ nunc reſtant explicen­<lb></lb>
tur. </s>
          <s id="s.003584">quoniam igitur animal antequam gradiatur, maximè homo, ſtat hori­<lb></lb>
zonti perpendicularis, neceſſe eſt ad progrediendum, vt fiat aliqua mem­<lb></lb>
brorum inflexio, ſi enim homo ſine vlla ſui corporis flexura inclinet ſe ad <lb></lb>
horizontem, ita vt cum horizonte faciat ex anteriori parte. </s>
          <s id="s.003585">v. g. angulum <lb></lb>
recto minorem, ſiue acutum, vel concidet, vel non poterit progredi; ſi enim <lb></lb>
alterum crus præmitteretur, altero manente perpendiculari, <expan abbr="ſicq́">ſicque</expan>; progre­<lb></lb>
deretur quiſpiam, ſequeretur crus prolatum, quale eſt A D, iu priori trian­<lb></lb>
gulo, debere fieri maius altero crure A B, manente, quia fieret tota hypo­<lb></lb>
tenuſa A C, ſie enim terram attingeret; at non poteſt fieri illo maius, quia <lb></lb>
eſt illi æquale, ergò hac ratione inceſſus fieri nequit. </s>
          <s id="s.003586">neceſſe igitur refle­


<pb pagenum="215" xlink:href="009/01/215.jpg"></pb>ctere <expan abbr="vtrumq;">vtrumque</expan> crus non ſolum ad horizontem, ſed etiam circa aliquam cor­<lb></lb>
poris flexuram, vel nodum, vt circa genu, aut alia. </s>
          <s id="s.003587">crura enim in greſſu fiunt <lb></lb>
latera ſuperiora trianguli iſoſcelis, vt in ſecunda figura patuit, cuius baſis <lb></lb>
eſt paſſus. </s>
          <s id="s.003588">&amp; tunc caput ambulantis fit inferius, quàm antequam gradere­<lb></lb>
<figure id="id.009.01.215.1.jpg" place="text" xlink:href="009/01/215/1.jpg"></figure><lb></lb>
tur; quia tunc ambo crura erant horizonti perpen­<lb></lb>
dicularia. </s>
          <s id="s.003589">quando autem caput fuerit in linea <expan abbr="per-pẽdiculari">per­<lb></lb>
pendiculari</expan> trianguli iſoſcelis, tunc erit inferius quàm <lb></lb>
alibi, vt in pręſenti figura, linea <expan abbr="perpẽdicularis">perpendicularis</expan> trian­<lb></lb>
guli huius iſoſcelis eſt linea A E, quia baſi B C, per­<lb></lb>
pendicularis incidit; quando igitur caput ambulan­<lb></lb>
tis. </s>
          <s id="s.003590">v. g. D, fuerit in hac linea, <expan abbr="tũc">tunc</expan> erit inferius quàm <lb></lb>
in quauis alia greſſus parte: quia tunc crura A B, <lb></lb>
A C, ſunt maximè diuaricata, &amp; proinde angulus A, <lb></lb>
&amp; ſimul punctum D, maximè demiſſa.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003591"><emph type="italics"></emph>De motu animalium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003592"><arrow.to.target n="marg287"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003593"><margin.target id="marg287"></margin.target>297</s>
        </p>
        <p type="main">
          <s id="s.003594">Cap. 1. <emph type="italics"></emph>(Primum quidem in animalibus: oportet enim ſi moueatur aliqua <lb></lb>particularum quieſcere aliquam, &amp; propter hoc, &amp; flexus animalibus <lb></lb>inſunt: tanquam enim centro vtuntur flexibus &amp; fit tota pars, in qua <lb></lb>
eſt flexus &amp; vna, &amp; duæ; &amp; recta, &amp; flexa, quæ permutatur potentia, <lb></lb>
&amp; actu, propter flexum. </s>
          <s id="s.003595">cum autem flectitur, &amp; mouetur, hoc quidem ſignum mo-<emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.215.2.jpg" place="text" xlink:href="009/01/215/2.jpg"></figure><lb></lb>
<emph type="italics"></emph>vetur, illud autem manet in flexibus, quemadmodum <expan abbr="vtiq;">vtique</expan> ſi dia­<lb></lb>
metri, quæ quidem A D, maneat, quæ cutem B, moueatur, &amp; <lb></lb>
fiat A C, ſed hic quidem videtur, ſecundum omnem modum in­<lb></lb>
diuiſibile eſſe centrum. </s>
          <s id="s.003596">etenim moueri, vt aiunt, fingunt in ipſis, <lb></lb>
non enim mouetur mathematicorum aliquid.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003597">Intendit probare neceſſe eſſe ad motum animalium, vt <lb></lb>
vna pars quieſcat, dum altera mouetur. </s>
          <s id="s.003598">propter hoc enim inquit flexus ani­<lb></lb>
malibus inſunt, vbi in græco pro voce flexus legitur <foreign lang="grc">καμπτη,</foreign> quod ſignifi­<lb></lb>
cat nodum, articulum, &amp; <expan abbr="deniq;">denique</expan> locum ipſum, vbi fit membri flexura, tan­<lb></lb>
quam enim centro quodam vtuntur flexibus, ideſt nodis, ſeu iuncturæ ſunt <lb></lb>
in motu membrorum inſtar centri. </s>
          <s id="s.003599">v. g. nodus cubiti fit centrum, cum bra­<lb></lb>
chij parte, quæ eſt inter humerum, &amp; cubitum manente, reliquum brachij <lb></lb>
circumducimus; ſic manente genu tanquam centro, crus huc illud agita­<lb></lb>
mus, &amp; fit tota pars. </s>
          <s id="s.003600">v. g. totum brachium, in quo eſt cubiti iunctura, &amp; vna <lb></lb>
tota pars, quando manet rectum; &amp; duæ <expan abbr="quãdo">quando</expan> in flexura cubiti brachium <lb></lb>inflectitur; &amp; fit tota hæc longitudo recta prius, poſtea flexa: quæ propter <lb></lb>
flexuram modo vna eſt actu, ſed duæ potentia. </s>
          <s id="s.003601">modo duæ in actu, ſed vna in <lb></lb>
potentia. </s>
          <s id="s.003602">cum autem flectitur, &amp; mouetur brachium, vnum quidem ſignum, <lb></lb>
ſiue punctum, quod eſt extremum partis manentis, manet; alterum verò ſi­<lb></lb>
gnum, ſiue punctum, quod eſt extremum partis motæ <expan abbr="eſtq́">eſtque</expan>; alteri ſigno con­<lb></lb>
tiguum mouetur ſimul cum tota parte mota. </s>
          <s id="s.003603">quemadmodum, ſi diametri <lb></lb>
ſuperioris figuræ, pars D A, maneat, pars autem A B, moueatur ad A C, <lb></lb>
erit huius flexuræ centrum A, quod vt extremum lineæ D A, manentis, ma­


<pb pagenum="216" xlink:href="009/01/216.jpg"></pb>net: vt verò extremum motæ A B, mouetur. </s>
          <s id="s.003604">quamuis in mathematicis hæc <lb></lb>
quidem duorum centrorum diſtinctio nulla ſit, quia centrum mathemati­<lb></lb>
cum omninò indiuiduum eſt: neque in mathematicis eſt propriè motus, <lb></lb>
quamuis enim aliquando Mathematici dicant, ſi linea, vel ſi punctum mo­<lb></lb>
ueretur, vel moueatur, &amp; ſimilia, huiuſmodi tamen motus ſunt rebus ma­<lb></lb>
thematicis extrinſeci, nec quatenus hoc modo mouentur conſiderantur: <lb></lb>
patet igitur, qua ratione Ariſtot. partem manentem in motu neceſſariam <lb></lb>
eſſe velit.</s>
        </p>
        <p type="main">
          <s id="s.003605"><arrow.to.target n="marg288"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003606"><margin.target id="marg288"></margin.target>298 a</s>
        </p>
        <p type="main">
          <s id="s.003607">Cap. 5. <emph type="italics"></emph>(Quemadmodum autem ſpontanea mouentur paruo motu facto)<emph.end type="italics"></emph.end> Spon­<lb></lb>
tanea iſta erant machinæ, quæ à ſeipſis mouebantur, quas Græci automata <lb></lb>dixerunt, cuiuſmodi ſunt Automata Heronis Alexandrini, quæ adhuc <expan abbr="extãt">extant</expan>.</s>
        </p>
        <p type="main">
          <s id="s.003608">Cap. 8. Eſt ibi quoddam triangulum cum elementis more geometrarum <lb></lb>
depictum, vnde locus ille videri poſſit mathematicus, verumtamen nullo <lb></lb>
modo geometriæ auxilio indiget.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003609"><emph type="italics"></emph>De generatione animalium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003610"><arrow.to.target n="marg289"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003611"><margin.target id="marg289"></margin.target>298.b</s>
        </p>
        <p type="main">
          <s id="s.003612">Lib. 2. cap. 1. <emph type="italics"></emph>(Sitqué perinde ac admirabilia illa ſpontanea)<emph.end type="italics"></emph.end> Intelligit ma­<lb></lb>
chinas illas miro artificio confictas, quæ à ſe ipſis intrinſeco prin­<lb></lb>
cipio mouebantur, quas Græci veteres Automata, ideſt ſpontanea, <lb></lb>
vel ſpontina, vt vertit Interpres vocabant, cuiuſmodi ſunt Auto­<lb></lb>
mata Heronis Alexandrini, quæ adhuc extant græca, <expan abbr="quæq́">quæque</expan>; ab Abbate Gua­<lb></lb>
ſtallenſi in Italicum ſunt conuerſa. </s>
          <s id="s.003613">Automata hodie ſunt Horologia, quæ ex <lb></lb>
multis dentatis rotis Germani conſtruunt.</s>
        </p>
        <p type="main">
          <s id="s.003614"><arrow.to.target n="marg290"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003615"><margin.target id="marg290"></margin.target>299</s>
        </p>
        <p type="main">
          <s id="s.003616">Lib. 2. cap. 4. <emph type="italics"></emph>(Nam &amp; triangula figura duobus rectis æquale ſemper habet)<emph.end type="italics"></emph.end><lb></lb>
vide quæ de hac re ſcripſi lib. 1. Priorum, ſecto 3. cap. 1.</s>
        </p>
        <p type="main">
          <s id="s.003617"><arrow.to.target n="marg291"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003618"><margin.target id="marg291"></margin.target>300</s>
        </p>
        <p type="main">
          <s id="s.003619">Ibidem <emph type="italics"></emph>(Et diametrum incommenſurabilem eſſe cum coſta ſempiternum eſt: at­<lb></lb>
tamen cauſa eorum aliqua &amp; demonſtratio eſt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003620">Quæ libro 1. Priorum, ſecto 1. cap. 23. de hac re annotata ſunt, abundè <lb></lb>
huic etiam loco ſatisfaciunt.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003621"><emph type="italics"></emph>In Ethica, ſeu Moralia ad Nicomachum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003622"><arrow.to.target n="marg292"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003623"><margin.target id="marg292"></margin.target>301</s>
        </p>
        <p type="main">
          <s id="s.003624">Lib. 1. cap. 7. <emph type="italics"></emph>(Faber enim, &amp; Geometra diuerſo modo rectum angulum <lb></lb>
<expan abbr="vtriq;">vtrique</expan> conſiderant: ille quatenus <expan abbr="ſolũ">ſolum</expan> ad opus vtile eſt, hic verò cum ve­<lb></lb>
ritatis ſpeculator ſit, quid, &amp; qualis ſit, indagat)<emph.end type="italics"></emph.end> Id quod dicit Ariſt. <lb></lb>
confirmatur ex eo, quod Fabri omnes vtuntur amuſſi, ſeu norma, <lb></lb>
<figure id="id.009.01.216.1.jpg" place="text" xlink:href="009/01/216/1.jpg"></figure><lb></lb>
quæ nihil aliud eſt quàm angulus rectus, quæ vulgò <lb></lb>
ſquadra dicitur, vt eius auxilio angulum ipſum re­<lb></lb>
ctum in opus conferant, <expan abbr="ſicq́">ſicque</expan>; normæ, aut amuſſis du­<lb></lb>
ctu ſua ipſi opera ad angulos rectos, ideſt quadrata, <lb></lb>
conficiunt. </s>
          <s id="s.003625">Geometra verò conſiderat eundem an­<lb></lb>
gulum, quatenus fit à linea ſuper lineam aliam per­<lb></lb>
pendiculariter inſiſtente, vt eſt in definit. </s>
          <s id="s.003626">10. primi,


<pb pagenum="217" xlink:href="009/01/217.jpg"></pb>vt in figura, vbi linea A B, inſiſtens alteri D C, perpendiculariter, ideſt ita <lb></lb>
vt faciat angulos hinc inde æqualis A B D, A B C, prædictos inquam duos <lb></lb>
angulos conſiderat e. </s>
          <s id="s.003627">ſe rectos. </s>
          <s id="s.003628">contemplatur præterea Geometra omnes <lb></lb>
angulos rectos eſſe inter ſe æquales, vt in 12. axiomate primi Elem. ponitur, <lb></lb>
&amp; ſimilia plura alia, quorum conſiderationem Faber omninò negligit.</s>
        </p>
        <p type="main">
          <s id="s.003629"><arrow.to.target n="marg293"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003630"><margin.target id="marg293"></margin.target>302</s>
        </p>
        <p type="main">
          <s id="s.003631">Libro 2. capite 6. <emph type="italics"></emph>(Id quod ſecundum Arithmeticam rationem medium eſt)<emph.end type="italics"></emph.end><lb></lb>
Arithmetica ratio, fiue proportio ea eſt, cuius termini creſcunt per æqua­<lb></lb>
les exceſſus, vt 2. 6. 10. 14. horum enim terminorum exceſſus æquales ſunt, <lb></lb>
cum ſint omnes quaternarij. </s>
          <s id="s.003632">ſimiliter inter hos terminos 3. 6. 9. 12. eſt arith­<lb></lb>
metica analogia, cùm omnes ternario numero ſuperent præcedentes, &amp; à <lb></lb>
ſequentibus ſuperentur. </s>
          <s id="s.003633">Porrò apud Mathematicos tria ſunt genera pro­<lb></lb>
portionum, ſiue medietatum, Arithmetica quam modo ſuppoſui; Geome­<lb></lb>
trica, &amp; Harmonica, quas inferius oblata occaſione opportunius explicabo.</s>
        </p>
        <figure id="id.009.01.217.1.jpg" place="text" xlink:href="009/01/217/1.jpg"></figure>
        <p type="main">
          <s id="s.003634"><arrow.to.target n="marg294"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003635"><margin.target id="marg294"></margin.target>303</s>
        </p>
        <p type="main">
          <s id="s.003636">Lib. 2. cap. 9. <emph type="italics"></emph>(Vt circuli medium deprehendere non <lb></lb>
cuiuſlibet, ſed <expan abbr="ſciẽtis">ſcientis</expan> ſolummodo eſt)<emph.end type="italics"></emph.end> Reperire medium, <lb></lb>
ſiue centrum dati circuli docet Euclides propoſitio­<lb></lb>
ne prima 3. hoc modo. </s>
          <s id="s.003637">in dato circulo ducatur vt­<lb></lb>
cunque recta B C, quæ per 10. primi diuidatur bifa­<lb></lb>
riam in F, &amp; per F, ducatur <expan abbr="perpẽdicularis">perpendicularis</expan> A E F D, <lb></lb>
quæ ſecetur bifariam in E, <expan abbr="eritq́">eritque</expan>; punctum E, non ſo­<lb></lb>
lum ipſius lineæ medium; ſed etiam totius circuli <lb></lb>centrum, quemadmodum ibi demonſtrat Euclides.</s>
        </p>
        <p type="main">
          <s id="s.003638"><arrow.to.target n="marg295"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003639"><margin.target id="marg295"></margin.target>304</s>
        </p>
        <p type="main">
          <s id="s.003640">Lib. 3. cap. 3. <emph type="italics"></emph>(De æternis autem nemo conſultat, vt <lb></lb>
de mundo, aut diametro, &amp; latere, quod nulla inter ſe <lb></lb>
æquabilitate conueniant)<emph.end type="italics"></emph.end> Qua ratione diameter, &amp; latus eiuſdem quadrati <lb></lb>nulla æquabilitate, ideſt nulla communi menſura inter ſe conueniant, fusè <lb></lb>
explicatum eſt libro Priorum, ſecto 1. cap. 23.</s>
        </p>
        <p type="main">
          <s id="s.003641"><arrow.to.target n="marg296"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003642"><margin.target id="marg296"></margin.target>305</s>
        </p>
        <p type="main">
          <s id="s.003643">Eodem cap. <emph type="italics"></emph>(Qui enim conſultat quærere videtur, &amp; reſoluere prædicto modo, <lb></lb>
quemadmodum deſignationes)<emph.end type="italics"></emph.end> Per deſignationes Ariſt. intelligere geometri­<lb></lb>
cas demonſtrationes ſæpius dictum eſt in logicis textibus, quod pariter ex <lb></lb>
hoc loco confirmatur. </s>
          <s id="s.003644">quando autem ait <emph type="italics"></emph>(Reſoluere prædicto modo, quemad­<lb></lb>
modum deſignationes)<emph.end type="italics"></emph.end> innuit reſolutionem geometricam, de qua abundè di­<lb></lb>
ctum eſt in explicatione tituli librorum Reſolutoriorum; quam expoſui, ni­<lb></lb>
hil aliud eſſe, quam medij inquiſitionem ad id, quod propoſitum fuerit de­</s>
        </p>
        <p type="main">
          <s id="s.003645"><arrow.to.target n="marg297"></arrow.to.target><lb></lb>
monſtrandum. </s>
          <s id="s.003646">veram autem, <expan abbr="atq;">atque</expan> germanam fuiſſe huiuſmodi explicatio­<lb></lb>nem, hoc loco Ariſt. ipſe confirmat, cum hanc reſolutionem dicat eſſe ſimi­<lb></lb>
lem conſultationi, ſiue inquiſitioni mediorum ad finem in rebus practicis <lb></lb>conſequendum; ipſa verò eſt inquiſitio mediorum ad id, quod in rebus ſpe­<lb></lb>
culatiuis propoſitum eſt, demonſtrandum. </s>
          <s id="s.003647">conſultatio igitur eſt in rebus <lb></lb>
practicis, quod in ſpeculatiuis eſt reſolutio.<lb></lb>
<arrow.to.target n="marg298"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003648"><margin.target id="marg297"></margin.target>306</s>
        </p>
        <p type="margin">
          <s id="s.003649"><margin.target id="marg298"></margin.target>307</s>
        </p>
        <p type="main">
          <s id="s.003650">Lib. 5. cap. 3. <emph type="italics"></emph>(Quod enim proportione conſtat, id non tam vnitario numero, <lb></lb>
quàm numero in vniuerſum proprium eſt)<emph.end type="italics"></emph.end> Per vnitarium numerum intelligitur <lb></lb>numerus ex vnitatibus abſtractis conφlatus, ideſt, cuius vnitates non ſint res <lb></lb>
phyſicæ, ſed à naturalibus abſtractæ, qualis conſiderat Arithmeticus: omni <lb></lb>tamen numero ſiue abſtracto, ſiue non, conuenit proportiones ſuſcipere, <lb></lb>
id eſt &amp; numero, &amp; rebus numeratis.</s>
        </p>
        <pb pagenum="218" xlink:href="009/01/218.jpg"></pb>
        <p type="main">
          <s id="s.003651"><arrow.to.target n="marg299"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003652"><margin.target id="marg299"></margin.target>308</s>
        </p>
        <p type="main">
          <s id="s.003653">Ibidem <emph type="italics"></emph>(N im proportio æqualitas eſt rationum)<emph.end type="italics"></emph.end> Per proportionem hoc lo­<lb></lb>
co intelligenda eſt illa, quam nunc appellant proportionalitatem, quæ eſt <lb></lb>
duarum rationum, ſeu proportionum ſimilitudo, ſiue æqualitas, vt manife­<lb></lb>
ſtum eſt ex 4. definit. </s>
          <s id="s.003654">5. Elem. v. g. cum ſit eadem ratio 9. ad 6. quæ eſt 6. ad <lb></lb>
4. propterea hæc rationum ſimilitudo, vel æqualitas dicitur ipſa proportio, <lb></lb>
ſeu diſtinctionis gratia Proportionalitas.</s>
        </p>
        <p type="main">
          <s id="s.003655"><arrow.to.target n="marg300"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003656"><margin.target id="marg300"></margin.target>309</s>
        </p>
        <p type="main">
          <s id="s.003657">Ibidem <emph type="italics"></emph>(In quatuorqué minimis reperitur, diſiunctam ſanè in quatuor conſistere <lb></lb>perſpicuum eſt: ſed &amp; continentem nihilominus, vno enim hæc perinde, aψ duobus <lb></lb>vtitur, biſque id accipit in hunc modum, qualis primi reſpectus eſt ad ſecundum, <lb></lb>talis ſecundi ad tertium; bis enim hic, ſecundum dictum eſt, quare ſi ſecundum bis <lb></lb>
poſitum ſit, quatuor erunt ea, quæ conſtant proportione)<emph.end type="italics"></emph.end> Quæ hic ab Ariſtot. di­<lb></lb>
cuntur deſumpta ſunt, partim ex definit. </s>
          <s id="s.003658">6. 5. partim ex 9. definit. </s>
          <s id="s.003659">eiuſdem. <lb></lb>
</s>
          <s id="s.003660">breuiter autem ſic ſe habent. </s>
          <s id="s.003661">Ad conſtituendam proportionalitatem ne­<lb></lb>
ceſſarij ſunt omninò quatuor termini, quod quidem primum perſpicuum <lb></lb>eſt in ea proportionalitate, quam Diſiunctam vocant, quæ eſt huiuſmodi, <lb></lb>
vt 9. ad 6. ita 3. ad 2. deinde <expan abbr="verũ">verum</expan> eſt etiam in ea, quam continuam dicunt, <lb></lb>
quæ talis eſt, vt 9. ad 6. ita 6. ad 4. quæ in tribus quidem terminis 9. 6. 4. <lb></lb>conſiſtit, ſed tamen, quia medius 6. <expan abbr="vtrumq;">vtrumque</expan> reſpicit extremum, ideò vices <lb></lb>
duorum gerit, ac proinde eſt, ac ſi hoc modo termini diſponantur 9. 6. 6. 4. <lb></lb>
vbi 6. bis ponitur, <expan abbr="ſuntq́">ſuntque</expan>; quatuor huius etiam proportionalitatis termini. <lb></lb>
</s>
          <s id="s.003662">hinc Ariſt. textum ſatis intelligere poteris.</s>
        </p>
        <p type="main">
          <s id="s.003663"><arrow.to.target n="marg301"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003664"><margin.target id="marg301"></margin.target>310</s>
        </p>
        <p type="main">
          <s id="s.003665">Eodem cap. <emph type="italics"></emph>(Sicut igitur primus terminus ſe habebit ad ſecundum, ita tertius <lb></lb>
ad quartum; igitur etiam alterna vice, ſicut primus ad tertium, ita ſecundus ad <lb></lb>
quartum. </s>
          <s id="s.003666">quare etiam totum ad totum, quod diſtributio binatim copulat. </s>
          <s id="s.003667">quæ ſi <lb></lb>
etiam ita compoſita fuerint, iustè copulat)<emph.end type="italics"></emph.end> Accipit Ariſt. illum argumentandi <lb></lb>
modum, quem Geometræ alternam rationem vocant, <expan abbr="quàmq́">quàmque</expan>; definit. </s>
          <s id="s.003668">12. <lb></lb>
5. exponunt, vt eam rebus ipſis accommodet, <expan abbr="atq;">atque</expan> in praxim deducat; eſt <lb></lb>
autem huiuſmodi, ſint primum quatuor termini proportionales, ideſt, vt <lb></lb>
primus ad ſecundum, ita tertius ad quartum. </s>
          <s id="s.003669">v. g. vt 9. ad 6. ita 3. ad 2. <lb></lb>
valet conſequentia hæc, ergò etiam alternatim erit, vt primus ad tertium, <lb></lb>
ita ſecundus ad quartum, v. g. in allato exemplo, ita erit 9. ad 3. vt 6. ad 2. <lb></lb>
quam ſequelam eſſe validam probat deinde Euclides propoſit. </s>
          <s id="s.003670">16. 5. hinc <lb></lb>
aliam deducit conſequentiam, quam Euclides propoſit. </s>
          <s id="s.003671">12. 5. demonſtrat, <lb></lb>
dum ait, quare etiam totum ad totum erit. </s>
          <s id="s.003672">v. g. quia concluſum eſt ita eſſe <lb></lb>
9. ad 3. quemadmodum 6. ad 2. ita etiam erit totum ad totum, ideſt ita <lb></lb>
etiam erunt antecedentes termini ſimul ad conſequentes ſimul, v. g. ita erit <lb></lb>
etiam totum 15. quod eſt totum ex antecedentibus terminis 9. &amp; 6. ad to­<lb></lb>
tum 5. conflatum ex conſequentibus terminis 3. &amp; 2. In ſumma igitur ſi fue­<lb></lb>
rit vt 9. ad 3. ita 6. ad 2. ita etiam erit 15. ad 5. quod verum eſſe apparet in <lb></lb>
his numeris, cum tam 9. ad 3. quà 6. ad 2. &amp; 15. ad 5. habeant triplam <lb></lb>
proportionem.</s>
        </p>
        <p type="main">
          <s id="s.003673">Horum exemplum in rebus practicis ſit hoc: ſit vt Plato ad Proclum, ita <lb></lb>
mille aurei ad quingentos aureos, ergò alternatim ita erit Plato ad 1000. <lb></lb>
aureos, ſicuti Proclus ad 500. quare ita etiam totum erit ad totum, ſcilicet <lb></lb>
Plato, &amp; Proclus ſimul ad 1000. &amp; 500. ſimul, quæ duo tota, diſtributio mo­<lb></lb>ralis, ac practica diuidit, &amp; binatim copulat, hoc modo dicens, vt Plato ad <pb pagenum="219" xlink:href="009/01/219.jpg"></pb>Proclum, ita 1000. ad 500, &amp; poſtea, vt Plato ad 1000. ita Proclus ad 500. <lb></lb>
iuxta <expan abbr="vtriuſq;">vtriuſque</expan> merita, &amp; quidem iſta eſt huiuſmodi moralis diſtributio, cum <lb></lb>
modis argumentandi ab Euclide comprobatis, nitatur.</s>
        </p>
        <p type="main">
          <s id="s.003674"><arrow.to.target n="marg302"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003675"><margin.target id="marg302"></margin.target>311</s>
        </p>
        <p type="main">
          <s id="s.003676">Ibidem <emph type="italics"></emph>(Hanc verò proportionalitatem Mathematici Geometricam vocant: <lb></lb>
propterea quod in Geometrica euenit, vt eandem totum ad totum rationem habeat, <lb></lb>
quam habet alterutrum, ad alterutrum)<emph.end type="italics"></emph.end> ideſt, hanc duarum Geometricarum <lb></lb>
rationum ſimilitudinem Mathematici proportionalitatem Geometricam <lb></lb>
appellant, propterea quod in hac duarum rationum geometricarum ſimili­<lb></lb>
tudine accidit, vt ſit totum ad totum, quemadmodum etiam partes toto­<lb></lb>
rum, vt ſupra explicatum eſt; quod non accidit in duarum proportionum <lb></lb>
arithmeticarum ſimilitudine; ſi enim ponamus has duas rationes arithme­<lb></lb>
ticas ſimiles, vt 10. ad 8. ita 6. ad 4. quæ ſunt ſimiles, propter ſimiles exceſ­<lb></lb>
ſus primorum, &amp; ſecundorum terminorum, cum <expan abbr="vbiq;">vbique</expan> exceſſus ſit binarij. <lb></lb>
</s>
          <s id="s.003677">non erit tamen totum 16. ad totum 12. in eadem ratione cum diuiſis ter­<lb></lb>
minis, cum ibi ſit exceſſus binarij, hic verò quaternarij. </s>
          <s id="s.003678">hæc videtur eſſe <lb></lb>
Ariſt. ratio; quam adhuc melius declaraſſe libet. </s>
          <s id="s.003679">Geometrica igitur pro­<lb></lb>
portionalitas ita dicta eſt, quia quælibet proportio poteſt in materia Geo­<lb></lb>
metrica, lineis, ſuperficiebus, &amp; corporibus continuari in quatuor termi­<lb></lb>
nis, ita vt proportionalitas, ſeu ſimilitudo rationum exurgat, quod in nu­<lb></lb>meris fieri ſemper nequit, cum plures ſint proportiones, quæ numeris ex­<lb></lb>
primi nequeunt, vt ſunt eæ, quas irrationales appellant, cuiuſmodi eſt inter <lb></lb>
diametrum, &amp; coſtam eiuſdem quadrati, cuius nec proportio, nec propor­<lb></lb>
tionalitas in numeris reperiri poteſt, quæ tamen in lineis, ſuperficiebus, ac <lb></lb>
corporibus eſſe poſſunt: eſt enim vt diameter vnius quadrati ad latus eiuſ­<lb></lb>
dem, ita idem latus ad aliam lineam inuentam per 11. 6. vel vt diameter ad <lb></lb>
coſtam, ita quælibet alia linea ad aliam inuentam, per 12. 6. omnis igitur <lb></lb>
proportionalitas rebus Geometricis ineſſe poteſt; non autem numeris, in <lb></lb>
quibus ſolum poſſunt eſſe rationes rationales, ſeu <expan abbr="rerũ">rerum</expan> commenſurabilium; <lb></lb>
latius igitur patet Geometrica hæc ſimilitudo, quàm Arithmetica, cùm <lb></lb>
Geometrica complectatur tam rationales, quàm irrationales. </s>
          <s id="s.003680">meritò igi­<lb></lb>
tur talis proportionalitas appellari debuit à rebus Geometricis, in quibus <lb></lb>
ſemper reperitur, non autem ab Arithmeticis, cum quibus ſæpius reperiri <lb></lb>
nequit. </s>
          <s id="s.003681">Vide Campanum in explicatione definitionis 3. 5. Elemen.</s>
        </p>
        <p type="main">
          <s id="s.003682"><arrow.to.target n="marg303"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003683"><margin.target id="marg303"></margin.target>312</s>
        </p>
        <p type="main">
          <s id="s.003684">Ibidem <emph type="italics"></emph>(Non eſt autem continens hæc proportio: non enim vnus, &amp; idem ter­<lb></lb>
minus efficitur, &amp; cui, &amp; quod)<emph.end type="italics"></emph.end> ideſt, hæc proportionalitas contracta ad res <lb></lb>
practicas, non eſt continens, ideſt, quæ conſiſtat in tribus tantum terminis, <lb></lb>
quorum medius eſt, ad quem refertur primus, &amp; is qui refertur ad ter­<lb></lb>
tium; ſed eſt diſiuncta, quia conſtat ſemper quatuor terminis, quorum duo <lb></lb>
ſunt perſonæ aliquæ, reliqui verò duo ſunt res, quæ perſonis debentur, vt ſi <lb></lb>
ſint Plato, &amp; Proclus, quibus iuxta meritorum quantitatem debeant diuidi <lb></lb>
1500. aurei, debent diuidi aurei in duas partes, quæ habeant eam propor­<lb></lb>
tionem, quam habet Plato ad Proclum. </s>
          <s id="s.003685">quod ſi Plato duplum męruit quàm <lb></lb>
Proclus, erit vt Plato ad Proclum, ita 1000. ad 500.</s>
        </p>
        <p type="main">
          <s id="s.003686">Ex quibus patet hanc analogiam in rebus agendis non niſi in quatuor <lb></lb>
terminis conſiſtere poſſe, &amp; ideo non eſſe continuam, ſed diſiunctam, vt vo­<lb></lb>
lebat Ariſtot.</s>
        </p>
        <pb pagenum="220" xlink:href="009/01/220.jpg"></pb>
        <p type="main">
          <s id="s.003687"><arrow.to.target n="marg304"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003688"><margin.target id="marg304"></margin.target>313</s>
        </p>
        <p type="main">
          <s id="s.003689">Lib. 5. cap. 6. <emph type="italics"></emph>(Atque id vel proportione vel numero)<emph.end type="italics"></emph.end> ideſt, vel proportio­<lb></lb>
nalitate Geometrica, vel Arithmetica; quæ autem ſit proportionalitas <lb></lb>
Geometrica, dictum eſt paulò ante in prioribus locis Mathematicis huius <lb></lb>
quinti libri; quæ verò ſit proportionalitas Arithmetica dictum eſt ſuperius <lb></lb>
lib. 2. cap. 6. Verum hæc Arithmetica proportionalitas, meritò ab Ariſtot. <lb></lb>
hic contradiſtincta eſt à proportionalitate Geometrica: quia Arithmetica <lb></lb>
hæc analogia attenditur ſolum, iuxta eundem exceſſum numerorum, non, <lb></lb>
autem iuxta proportionem, ſeu habitudinem terminorum ad inuicem, quod <lb></lb>
maximè in Geometrica ſpectatur. </s>
          <s id="s.003690">propterea Mathematici <expan abbr="cẽſent">cenſent</expan> eam vo­<lb></lb>
candam eſſe potius medietatem Arithmeticam, quam proportionalita­<lb></lb>
tem, cum quibus nunc Ariſt. conſentit.</s>
        </p>
        <p type="main">
          <s id="s.003691"><arrow.to.target n="marg305"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003692"><margin.target id="marg305"></margin.target>314</s>
        </p>
        <p type="main">
          <s id="s.003693">Lib. 6. cap. 5. <emph type="italics"></emph>(Verbi cauſa triangulum tres angulos duobus rectis æquales ha­<lb></lb>bere, vel non habere)<emph.end type="italics"></emph.end> lib. 1. Priorum, ſecto 3. cap. 1. fusè hanc trianguli affe­<lb></lb>
ctionem expoſui.</s>
        </p>
        <p type="main">
          <s id="s.003694"><arrow.to.target n="marg306"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003695"><margin.target id="marg306"></margin.target>315</s>
        </p>
        <p type="main">
          <s id="s.003696">Lib. 6. cap. 8. <emph type="italics"></emph>(Nam illud etiam conſideratione dignum videtur. </s>
          <s id="s.003697">quid ſit, quod <lb></lb>
puer fieri Mathematicus poteſt, ſapiens autem naturalis non poteſt. </s>
          <s id="s.003698">An quia illa <lb></lb>
per abſtractionem ſunt, horum autem principia ab experientia ſumuntur)<emph.end type="italics"></emph.end> Ex hoc <lb></lb>
loco manifeſtè apparet Ariſt. exiſtimare principia Mathematica nullo mo­<lb></lb>
do nobis per experientiam innoteſcere, quod nonnulli negant.</s>
        </p>
        <p type="main">
          <s id="s.003699"><arrow.to.target n="marg307"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003700"><margin.target id="marg307"></margin.target>316</s>
        </p>
        <p type="main">
          <s id="s.003701">Lib. 7. cap. 8. <emph type="italics"></emph>(In actionibus autem principium illud eſt, cuius cauſa res fit, <lb></lb>
ſicut in Mathematicis ſuppoſitiones; nam neque illic ratio eſt, quæ doctrinam tra­<lb></lb>
dat principiorum, neque hic<emph.end type="italics"></emph.end>) Suppoſitionum, ſiue principiorum Mathemati­<lb></lb>
corum tria ſunt genera, definitiones, poſtulata, axiomata, quæ in ipſo primi <lb></lb>
Elementorum veſtibulo proponuntur: ſolaque terminorum explicatione <lb></lb>
<expan abbr="abſq;">abſque</expan> vllo diſcurſu, addiſcuntur.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003702"><emph type="italics"></emph>Ex primo Libro Magnorum Moralium.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003703"><arrow.to.target n="marg308"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003704"><margin.target id="marg308"></margin.target>317</s>
        </p>
        <p type="main">
          <s id="s.003705">Cap. 1. (<emph type="italics"></emph>Nec enim luſtitia eſt numerus pariter par<emph.end type="italics"></emph.end>) vt ſcilicet dicebat <lb></lb>
Pythagoras. </s>
          <s id="s.003706">Porrò definit. </s>
          <s id="s.003707">8. 7. ſic habetur: Pariter par nume­<lb></lb>
rus eſt, quem par numerus per numerum parem, ideſt paribus vi­<lb></lb>
cibus, metitur, cuiuſmodi eſt numerus 24. quem numerus 6. me­<lb></lb>
titur per numerum parem, nimirum per 4. quia ſcilicet numerus 6. paribus <lb></lb>
vicibus, quippe per 4. ſiue quater ipſum numerum 24. menſurat, quia to­<lb></lb>
ties in ipſo adæquatè continetur.</s>
        </p>
        <p type="main">
          <s id="s.003708"><arrow.to.target n="marg309"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003709"><margin.target id="marg309"></margin.target>318</s>
        </p>
        <p type="main">
          <s id="s.003710">Cap. 2. (<emph type="italics"></emph>Abſurdum enim ſit, volenti oſtendere triangulum duobus rectis æqua­<lb></lb>les habere angulos, ſumere principium huiuſmodi, anima immortalis est<emph.end type="italics"></emph.end>) Repete, <lb></lb>
quæ de hac trianguli proprietate fusè ſcripſi lib. 1. Priorum, ſect. </s>
          <s id="s.003711">3. cap. 1. <lb></lb>
quam affectionem debet Geometra demonſtrare ex Geometriæ principijs, <lb></lb>
quemadmodum facit Euclides in 32. primi, non autem ex principijs extrin­<lb></lb>
ſecis, vt quod anima ſit immortalis.</s>
        </p>
        <p type="main">
          <s id="s.003712"><arrow.to.target n="marg310"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003713"><margin.target id="marg310"></margin.target>319</s>
        </p>
        <p type="main">
          <s id="s.003714">Cap. 10. <emph type="italics"></emph>(Vt enim habuerint principia, ita, quæ de principijs ortum ducunt, <lb></lb>
Perſpicuè autem licet hoc in Geometria magis intueri, vbi cum aliqua ſumpſeris <lb></lb>
principia, vt ea habuerint, ita etiam, quæ ipſa conſequuntur: velut ſi triangulum <lb></lb>
duobus rectis æquales habet angulos, quadratum <expan abbr="quoq;">quoque</expan> quatuor angulis rectis ha-<emph.end type="italics"></emph.end>


<pb pagenum="221" xlink:href="009/01/221.jpg"></pb><emph type="italics"></emph>beat neceſſe eſt. </s>
          <s id="s.003715">&amp; ſi triangulum ſecus, ita etiam, &amp; quadratum commutabitur, <lb></lb>
ex altera parte enim ei reſpondet. </s>
          <s id="s.003716">&amp; ſi quadratum quatuor angulis rectis æquales, <lb></lb>
non habuerit angulos ne quidem triangulum duobus rectis habebit æquales)<emph.end type="italics"></emph.end> Hanc <lb></lb>
trianguli affectionem, habere ſcilicet, ſuos tres angulos æquales duobus re­<lb></lb>
ctis angulis abundè explicaui libro 1. Priorum, ſecto 3. cap. 1. quam Eucli­<lb></lb>
des propoſit. </s>
          <s id="s.003717">32. primi demonſtrauit, ex qua demonſtratione, tanquam ex <lb></lb>
Geometrico principio ſequitur omne <expan abbr="quoq;">quoque</expan> quadrangulum habere quatuor <lb></lb>
angulos æquales quatuor rectis angulis; omne <expan abbr="namq;">namque</expan> quadrangulum eſt po­<lb></lb>
tentia duo triangula, cum diuidatur ducta ipſius diametro in duo <expan abbr="trìãgula">trìangula</expan>. <lb></lb>
</s>
          <s id="s.003718">quod ſi triangulus proprietatem illam non haberet, <expan abbr="neq;">neque</expan> hæc quadrangulo <lb></lb>
conueniret. </s>
          <s id="s.003719">&amp; ſi quadrangulum non haberet quatuor angulos æquales qua­<lb></lb>
tuor rectis angulis, neque triangulum habere poſſet tres angulos æqua­<lb></lb>
les duobus rectis, cum nihil ſit aliud triangulum, quàm dimidiatum qua­<lb></lb>
drangulum.</s>
        </p>
        <p type="main">
          <s id="s.003720"><arrow.to.target n="marg311"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003721"><margin.target id="marg311"></margin.target>320</s>
        </p>
        <p type="main">
          <s id="s.003722">Cap. 16. <emph type="italics"></emph>(In Geometria ſi quidem cum quis dixerit quadrangulŭm quatuor rectis <lb></lb>æquales habere, &amp; percunctatur propter quid, occurrit, quia etiam triangulŭm duo­<lb></lb>
bus rectis æquales habet. </s>
          <s id="s.003723">in his igitur ex determinato ſibi principio propter quid <lb></lb>
aſſumpſerunt)<emph.end type="italics"></emph.end> Lege, quæ proximè in præcedenti loco expoſui, ea enim om­<lb></lb>
nia huc etiam pertinent. </s>
          <s id="s.003724">hoc ſolum addendum ad illorum verborum (<emph type="italics"></emph>Ex de­<lb></lb>
terminato ſibi principio propter quid aſſumpſerunt<emph.end type="italics"></emph.end>) intelligentiam, ideſt ex vna <lb></lb>
concluſione demonſtrata, tanquam principio alia demonſtrant; quod rectè <lb></lb>
fieri Ariſt. in primo Poſter. docet.</s>
        </p>
        <p type="main">
          <s id="s.003725"><arrow.to.target n="marg312"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003726"><margin.target id="marg312"></margin.target>321</s>
        </p>
        <p type="main">
          <s id="s.003727">Cap. 31. (<emph type="italics"></emph>A qui proportionale in quatuor nihilominus perficitur: nam quem­<lb></lb>
admodum A, ad B, ita C, ad D.<emph.end type="italics"></emph.end>) ideſt proportionalitas in quatuor terminis <lb></lb>
conſiſtit, quemadmodum pluribus ſupra lib. 5. cap. 3. Ethycorum explica­<lb></lb>
tum eſt: quò nunc Lectorem ablego.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003728"><emph type="italics"></emph>Ex primo Libro Moralium Eudemiorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003729"><arrow.to.target n="marg313"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003730"><margin.target id="marg313"></margin.target>322</s>
        </p>
        <p type="main">
          <s id="s.003731">Cap. 5 (<emph type="italics"></emph>Vt ſi duplum multiplicium primum est, non licet multiplex com­<lb></lb>
muniter prædicatum ſeparari, quippe, quod duplo prius eſt<emph.end type="italics"></emph.end>) Inter pro­<lb></lb>
portionum genera vnum eſt, quod dicitur multiplex, quod ſub ſe <lb></lb>
infinitas ſpecies continet, vt Duplum, Triplum, Quadruplum, &amp; c. <lb></lb>
in infinitum. </s>
          <s id="s.003732">vbi vides, cur Ariſt. dixerit duplum eſſe primum inter multi­<lb></lb>
plicia, cum verè naturali ordine numerorum ipſi primus debeatur locus. <lb></lb>
</s>
          <s id="s.003733">Vides etiam cur non liceat, Multiplex ipſum genus commune prædicatum <lb></lb>
omnibus ſpeciebus veluti Idæam ſeparari; tunc enim ait, ipſum mul­<lb></lb>
tiplex abſtractum eſſet prius ordine ipſo primo multiplici, ſci­<lb></lb>
licet duplo; &amp; Duplum non eſſet primum inter mul­<lb></lb>
tiplicia, quæ <expan abbr="vtraq;">vtraque</expan> ſunt abſurda; non igitur <lb></lb>
illud tanquam Idæam licet ſepa­<lb></lb>
ratum ponere.</s>
        </p>
      </chap>
      <pb pagenum="222" xlink:href="009/01/222.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.003734"><emph type="italics"></emph>Ex Secundo Moralium Eudem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003735"><arrow.to.target n="marg314"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003736"><margin.target id="marg314"></margin.target>323</s>
        </p>
        <p type="main">
          <s id="s.003737">Cap. 7. <emph type="italics"></emph>(Nam ſi habenti trigono duos rectos, neceſſe eſt tetragonum qua­<lb></lb>
tuor rectis conſtare, manifeſtum eſt, quod trigonus duos rectos habens <lb></lb>
cauſa eius exiſtat. </s>
          <s id="s.003738">Verùm ſi quid in trigono mutaris, neceſſarium eſt, &amp; <lb></lb>
in tetragono mutes, vt ſi tres habuerunt, ſex; &amp; ſi quatuor, octo; ſin <lb></lb>
verò non mutes, vt illud, ita hoc <expan abbr="quoq;">quoque</expan> habeat neceſſe eſt)<emph.end type="italics"></emph.end> Lege prius, quæ ſupra <lb></lb>
lib. 1. Magnor. moral. </s>
          <s id="s.003739">cap. 10. ſcripſi, ex quibus poſtea ſic locum hunc in­<lb></lb>
terpretaberis, ſi triangulum habet tres angulos æquales duobus rectis an­<lb></lb>
gulis, neceſſe eſt quodcunque quadrilaterum habere ſuos quatuor angulos <lb></lb>
æquales quatuor rectis, quia omne quadrangulum continet duo triangula; <lb></lb>
&amp; ſi natura trianguli fuerit immutata ita, vt habeat tres angulos æquales <lb></lb>
non duobus, ſed tribus rectis, tunc neceſſe erit tetragonum <expan abbr="quoq;">quoque</expan> mutatum <lb></lb>
eſſe, quia neceſſariò habebit ſuos angulos æquales non quatuor tantum re­<lb></lb>
ctis, ſed ſex: pariter ſi triangulum habeat tres angulos quatuor rectis pa­<lb></lb>
res, quadrangulum ſuos habebit angulos, octo rectis æquiualentes. </s>
          <s id="s.003740">His igi­<lb></lb>
tur ex Geometria ſatisfactum ſit.</s>
        </p>
        <p type="main">
          <s id="s.003741"><arrow.to.target n="marg315"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003742"><margin.target id="marg315"></margin.target>324</s>
        </p>
        <p type="main">
          <s id="s.003743">Cap. 10. <emph type="italics"></emph>(Multa verò opinione concipiunt, quæ penes nos non ſunt, vt diame­<lb></lb>
trum commenſurabilem eſſe)<emph.end type="italics"></emph.end> Quæ lib. 1. Priorum, ſecto 3. cap. 23. de aſyme­<lb></lb>
tria diametri, &amp; coſtæ eiuſdem quadrati allata ſunt, ſatis huic etiam loco <lb></lb>
facere poſſunt.</s>
        </p>
        <p type="main">
          <s id="s.003744"><arrow.to.target n="marg316"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003745"><margin.target id="marg316"></margin.target>325</s>
        </p>
        <p type="main">
          <s id="s.003746">Eodem cap. <emph type="italics"></emph>(Quapropter non deremotis apud Indos, nec de circuli quadratu­<lb></lb>
ra deliberamus: nam illa ad nos non ſpectant, hoc verò fieri nequit)<emph.end type="italics"></emph.end> Quid ſit cir­<lb></lb>
culi quadratio, &amp; qua ratione eam antiqui inueſtigauerint in Prædicamen­<lb></lb>
to Relationis, &amp; alibi in Logicis, pluribus explicatum eſt. </s>
          <s id="s.003747">An verò poſſi­<lb></lb>
bilis ſit circuli quadratura, reſpondendum eſt cum diſtinctione, nam theo­<lb></lb>
rematicè quidem facta eſt ab Archimede, cum ipſe probauerit circulum, <lb></lb>
quemuis æqualem eſſe triangulo, cuius vnum latus circa angulum rectum <lb></lb>ſit circuli ſemidiameter, alterum verò circunferentia. </s>
          <s id="s.003748">Problematicè verò, <lb></lb>
ideſt, vt opere ipſo efficiamus <expan abbr="triãgulum">triangulum</expan> illud, nondum à quoquam ritè per­<lb></lb>
actum eſt: &amp; propterea problema hoc difficile admodum cenſendum eſt, <lb></lb>
præſertim cum tota Geometrarum antiquitas, <expan abbr="atq;">atque</expan> poſteritas in ipſum fru­<lb></lb>
ſtra <expan abbr="hucuſq;">hucuſque</expan> inſudauerit, <expan abbr="atq;">atque</expan> adeò etiam moraliter impoſſibile exiſtiman­<lb></lb>
dum eſt. </s>
          <s id="s.003749">quo ſenſu locutum eſſe Ariſt. hoc loco crediderim, dum ait, illud <lb></lb>
fieri non poſſe. </s>
          <s id="s.003750">abſolutè tamen aſſerere non debemus eſſe impoſſibilem, cum <lb></lb>
nulla id demonſtratione certum ſit, imò ego ſimpliciter, vt aiunt, credo eſ­<lb></lb>
ſe poſſibilem, cum alia theoremata, <expan abbr="atq;">atque</expan> problemata (quale eſt pytagoreum <lb></lb>
illud celebre, quod 47. locum in primo Elemen. occupat, &amp; pro cuius adin­<lb></lb>uentione Pythagoras Muſis Hecatombas ſacrificauit) olim fuerint diù à <lb></lb>
multis incaſſum quæſita, <expan abbr="atq;">atque</expan> impoſſibilia habita, quæ poſtea tandem re­<lb></lb>
perta ſunt.</s>
        </p>
        <p type="main">
          <s id="s.003751"><arrow.to.target n="marg317"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003752"><margin.target id="marg317"></margin.target>326</s>
        </p>
        <p type="main">
          <s id="s.003753">Cap. 12. (<emph type="italics"></emph>Si triangulo duo recti, neceſſum eſt hoc conſequi<emph.end type="italics"></emph.end>) ideſt, ſi triangu­<lb></lb>
lum habet tres angulos æquales duobus rectis, neceſſe eſt conſequi, vt ſupe­<lb></lb>
rius ſepius dixit, quod quadrilaterum habeat quatuor angulos æquales qua­<lb></lb>
tuor rectis, lib. 1. Magn. moral. </s>
          <s id="s.003754">cap. 10. ſatis de hac re dictum eſt.</s>
        </p>
      </chap>
      <pb pagenum="223" xlink:href="009/01/223.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.003755"><emph type="italics"></emph>Ex Septimo Moralium Eudem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003756"><arrow.to.target n="marg318"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003757"><margin.target id="marg318"></margin.target>327</s>
        </p>
        <p type="main">
          <s id="s.003758">Cap. 12. (<emph type="italics"></emph>luxtaqué diametrum iungit<emph.end type="italics"></emph.end>) ideſt diametraliter opponit, quæ <lb></lb>
eſt omnium maxima oppoſitio, ita vt quæ diametraliter oppoſita <lb></lb>
ſunt, amplius diſtare nequeant, quia diameter eſt maxima om­<lb></lb>
nium diſtantia, ſiue fit diameter quadrilateræ figuræ, ſiue circuli.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003759"><emph type="italics"></emph>Ex Libro 3. Politicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003760"><arrow.to.target n="marg319"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003761"><margin.target id="marg319"></margin.target>328</s>
        </p>
        <p type="main">
          <s id="s.003762">Cap. 2. (<emph type="italics"></emph>Ceu harmoniam earumdem vocem aliam eſſe dicimus, &amp; modò <lb></lb>
Doricam, modò Phrygiam vocitamus<emph.end type="italics"></emph.end>) Harmonias Doricam, &amp; Phry­<lb></lb>
giam veteres Muſici, vt Ariſtoxenes, Euclides, Ptolæmeus vocant <lb></lb>
Tonos, &amp; Modos, Dorium ſcilicet, &amp; Phrygium. </s>
          <s id="s.003763">per muſicum au­<lb></lb>
tem modum intelligebant quandam vocum conſtitutionem, ſeu rithmum, <lb></lb>
quem nos hodie vulgò ariam vocamus, vt doctè explicat Ioſephus Zarlinus <lb></lb>
in 4. parte Inſtitut. Muſicalium, necnon in lib. 6. ſupplem. </s>
          <s id="s.003764">Denominati au­<lb></lb>
tem fuerunt prædicti, <expan abbr="alijq́">alijque</expan>; plures modi à nationibus illis, apud quas ma­<lb></lb>
ximè in vſu erant, vt Dorius à Dorienſibus; Phrygius à Phrygijs; Lydius à <lb></lb>
Lydijs. </s>
          <s id="s.003765">Porrò præter prædictos modos alij plures à veteribus Muſicis com­<lb></lb>
memorantur; variè tamen, alij enim tres, alij ſeptem, alij quindecim, vel <lb></lb>
ſeptemdecim etiam connumerarunt; Tres tamen præcipui, &amp; ad quos reli­<lb></lb>
qui reuocabantur, fuerunt Dorius, Phrygius, &amp; Lydius. </s>
          <s id="s.003766">quorum hæ fuerunt <lb></lb>
proprietates. </s>
          <s id="s.003767">Dorius erat grauis, ſeuerus, &amp; bellicoſus. </s>
          <s id="s.003768">vnde priſci exiſti­<lb></lb>
marunt ipſum in hominum animos prudentiam, caſtitatem, <expan abbr="atq;">atque</expan> virtutem <lb></lb>
inducere. </s>
          <s id="s.003769">Phrygius verò erat hilaris, lætus, placidus, ac propterea feſtis, <lb></lb>
&amp; choreis idoneus. </s>
          <s id="s.003770">vnde prouerbium illud vetus ortum habuit, à Dorio ad <lb></lb>
Phrygium, ideſt à rebus altiſſimis, &amp; ſerijs ad humiles, &amp; iucundas. </s>
          <s id="s.003771">Hos <lb></lb>
ambos ſolos Plato, &amp; Ariſt. in Rempublicam admiſerunt. </s>
          <s id="s.003772">Lydius demum <lb></lb>
modus erat horribilis, mœſtus, ac triſtis, <expan abbr="ideoq́">ideoque</expan>; lamentationibus, ac la­<lb></lb>
crymis aptus. </s>
          <s id="s.003773">Hoc in funeribus mortuos lamentantes vtebantur, ita vt pre­<lb></lb>
ſentibus lacrymas cierent, <expan abbr="vitaq́">vitaque</expan>; functos lacrymis proſequerentur.</s>
        </p>
        <p type="main">
          <s id="s.003774">Recentiores Muſici ſuos modos vocant Tonos, in quibus vtinam anti­<lb></lb>
quos imitarentur, illi enim ſuis rithmis, modiſuè auditorum animos varijs <lb></lb>
pro illorum varietate motibus mirè afficiebant: ſed noſtri, rithmos in ſuis <lb></lb>
cantilenis negligunt, nec illis curæ eſt, vt per rithmos hominum affectiones <lb></lb>
percellant, cum tamen Plato aſſerat Muſici officium eſſe rithmos adinueni­<lb></lb>
re; præterea quod animis ciendis valdè obſtat, cantilenæ verba, ac ſenſum <lb></lb>prorſus per ſuos, quos vocant, contrapunctos, omninò offuſcant, vt nihil <lb></lb>
præter magnum quendam vocum ſtrepitum concordem exaudiatur: <expan abbr="quiq́">quique</expan>; <lb></lb>
rithmis imitari hominum mores deberent, mimicis quibuſdam adinuentis <lb></lb>
id præſtare conantur.</s>
        </p>
        <p type="main">
          <s id="s.003775">Verùm hac de re legantur eruditiſſimi Dialogi de Muſica Vincentij Ga­<lb></lb>
lilæi, cuius præcipuas rationes in fine huius operis, &amp; chronologiæ videre <lb></lb>
poteris. </s>
          <s id="s.003776">Cæterum, qui plura de modis tam antiquis, quàm nouis deſiderat,


<pb pagenum="224" xlink:href="009/01/224.jpg"></pb>conſulat Ioſephum Zarlinum in 4. parte Inſtitutionum Muſicalium, necnon <lb></lb>
lib. 6. ſupplemen. virum vatia eruditione refertum, <expan abbr="deq́">deque</expan>; Muſica in primis <lb></lb>
optimè meritum.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003777"><emph type="italics"></emph>Ex Quarto Politicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003778"><arrow.to.target n="marg320"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003779"><margin.target id="marg320"></margin.target>329</s>
        </p>
        <p type="main">
          <s id="s.003780">Cap. 3. (<emph type="italics"></emph>Eodemqué modo in harmonijs, vt quidam tradunt: nam &amp; in illis <lb></lb>
poſuerunt duas ſpecies, vnam Doricam, alteram Phrygiam: cæteras <lb></lb>
verò omnes vel ad Doricam, vel ad Phrygiam referri.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003781">Vide proximè in præcedenti loco dicta, quæ omnia ita etiam <lb></lb>
huic loco quadrant, vt præterea nihil deſideretur.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003782"><emph type="italics"></emph>Ex Quinto Politicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003783"><arrow.to.target n="marg321"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003784"><margin.target id="marg321"></margin.target>330.a</s>
        </p>
        <p type="main">
          <s id="s.003785">Cap. 1. (<emph type="italics"></emph>Quare opus eſt partim arithmetica æquitate vti, partim ea, quæ <lb></lb>
eſt ſecundum dignitatem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003786">Arithmeticam medietatem ſupra explicaui lib. 2. cap. 6. Ethi­<lb></lb>
corum. </s>
          <s id="s.003787">per eam deinde, quæ eſt ſecundum dignitatem, intelligit <lb></lb>
Geometricam, quam ſupra lib. 5. cap. 3. Ethicorum expoſui. </s>
          <s id="s.003788">vtimur enim <lb></lb>
ea, quando opus eſt diſtribuere præmia non omnibus æqualiter, ſed habita <lb></lb>
ratione meritorum vniuſcuiuſque. </s>
          <s id="s.003789">quando autem non ſecundum perſona­<lb></lb>
rum dignitatem, ſed omnibus æqualiter fit diſtributio, illa dicitur Arithme­<lb></lb>
tica proportionalitas, quia ſeruat eandem <expan abbr="vbiq;">vbique</expan> differentiam terminorum.</s>
        </p>
        <p type="main">
          <s id="s.003790"><arrow.to.target n="marg322"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003791"><margin.target id="marg322"></margin.target>330.b</s>
        </p>
        <p type="main">
          <s id="s.003792">Cap. 12. &amp; vlt. (<emph type="italics"></emph>In Republica verò Platonis Socrates de mutationibus loqui­<lb></lb>
tur, nec tamen rectè. </s>
          <s id="s.003793">illius enim Reip. quæ eſt optima, <expan abbr="atq;">atque</expan> prima, mutatio nulla <lb></lb>
propria aſſignatur. </s>
          <s id="s.003794">inquit enim cauſam eſſe mutationis, quia ſic natura compara­<lb></lb>
tum ſit, vt nihil permaneat, ſed in ambitu quodam temporis, mutationem recipiat. <lb></lb>
</s>
          <s id="s.003795">eſſe verò principium horum, inquit, quorŭm ſeſquitertia radix quinario iuncta, duas <lb></lb>
exhibet harmonias. </s>
          <s id="s.003796">inquiens quando numerus huius diagrammatis efficiatur ſoli­<lb></lb>
dus<emph.end type="italics"></emph.end>) Quoad textus interpretationem, nonnulli pro (ſeſquitertia radix) ver­<lb></lb>
tunt (ſeſquitertius cubus) ſed qua id ratione ignoro. </s>
          <s id="s.003797">græcum verbum eſt <lb></lb>
<foreign lang="grc">πυθμὴν,</foreign> quod fundamentum, latus, &amp; radicem ſignificat, non autem cubum. <lb></lb>
</s>
          <s id="s.003798">præterea ſenſui radix, non autem cubus quadrare poteſt. </s>
          <s id="s.003799">Porrò ſciendum <lb></lb>
Ariſt. locum hunc ex Platonis lib. 8. de Rep. accepiſſe, loco Mathematico <lb></lb>
obſcuriſſimo, vbi ille de Reip. ſeu Gubernation is mutatione, ac duratione <lb></lb>
pertractat. </s>
          <s id="s.003800">quì locus adeò ſemper obſcurus habitus eſt, vt Cicero ipſe cum <lb></lb>
rem pœnitus obſcuram ſignificare vellet, dicere ſolitus eſſet, numero Plato­<lb></lb>
nis obſcurius. </s>
          <s id="s.003801">quam ob cauſam Theon Smyrnæus, qui Mathematica Plato­<lb></lb>
nis loca commentarijs illuſtrauit, hiſce tenebris lucem nullam afferre auſus <lb></lb>
eſt, verùm eas diſſimulans cautè declinauit. </s>
          <s id="s.003802">cùm igitur præſens Ariſt. locus <lb></lb>ſit illius Platonici particula quædam, quid mirum, ſi non minori obſcurita­<lb></lb>
te, ac difficultate impeditus ſit? </s>
          <s id="s.003803">vnde etiam ſequitor huius explicationem, <lb></lb>
ab illius explicatione petendam eſſe. </s>
          <s id="s.003804">Locum illum Platonis fuſi ſimè expli­<lb></lb>
cat Marſilius Ficinus to 2. operum ſuorum pag. </s>
          <s id="s.003805">1413. vbi pag. </s>
          <s id="s.003806">1421. cap. 12.


<pb pagenum="225" xlink:href="009/01/225.jpg"></pb>illius commentarij propè finem præſens Ariſt. locus ex præmiſſis ab eo bre­<lb></lb>
uiter, ac dilucidè declaratur. </s>
          <s id="s.003807">quæ explanatio, quoniam mihi præ cæteris ar­<lb></lb>
ridet, eam hoc loco, explicatiorem tamen, referam. </s>
          <s id="s.003808">Illud autem præſcien­<lb></lb>
dum eſt, hæc quæ a Socrate lib. 8. de Repub. recenſentur, confingi à Muſis, <lb></lb>
tanquam oraculum quoddam obſcuriſſimum effata; quo arcana quædam <lb></lb>
myſteria de Rerump. durationibus, ac mutationibus continerentur.</s>
        </p>
        <p type="main">
          <s id="s.003809">Aiebat igitur Socrates, Muſarum ſpiritu afflatus, optimam Politiam, op­<lb></lb>
timis ſcilicet legibus, ac moribus conſtitutam, ſua natura omninò immu­<lb></lb>
tabilem, <expan abbr="atq;">atque</expan> adeò diuturnam per ſe fore. </s>
          <s id="s.003810">Verumtamen mutationi obno­<lb></lb>xiam eſſe, quoniam ſie natura comparatum eſt, vt cuncta, quæ naturæ ſinu <lb></lb>
continentur, certa quadam annorum, vel ſæculorum periodo exacta, mu­<lb></lb>
tationem ſubire fatali lege, cogantur. </s>
          <s id="s.003811">tunc autem harum viceſſitudinum <lb></lb>
principium contingere, fatidicæ Muſæ ſignificare voluerunt, cùm is anno­<lb></lb>
rum, vel ſæculorum numerus ab illius Reip. exordio elapſus fuerit, qui ſit <lb></lb>
numerus ſolidus, &amp; cubus, eius numeri, in quo optima Reipub. conſtitutio <lb></lb>
conſiſtit. </s>
          <s id="s.003812">hic porrò numerus, in quo Reip. perfectio ſtatuitur, eſt Duodena­<lb></lb>
rius, quem multis in locis, varias ob rationes extulit Plato, præcipuè verò, <lb></lb>
quoniam in ſe ipſo duas continet harmonias, ſiue duas proportiones har­<lb></lb>
monicas, quæ ſimul iunctæ, perfectiſſimam omnium conflant harmoniam, <lb></lb>
quæ Diapaſon dicitur. </s>
          <s id="s.003813">duæ autem illæ rationes harmonicæ ſunt Seſquiter­<lb></lb>
tia, &amp; Seſquialtera. </s>
          <s id="s.003814">Seſquitertia reperitur primò inter hos numeros 4. 3. <lb></lb>
cùm enim ea inter duas voces, aut ſonos reperitur, ij edunt harmoniam, <lb></lb>
ſeu conſonantiam illam, quæ Diateſſaron appellatur. </s>
          <s id="s.003815">ſimul autem ijdem ad­<lb></lb>
diti efficiunt 7. qui numerus propterea in textu dicitur radix Epitrite, ſiue <lb></lb>
Seſquitertia, quoniam vt vidimus <expan abbr="cõponitur">componitur</expan> ex numeris 4. 3. Seſquitertiam <lb></lb>
rationem habentibus. </s>
          <s id="s.003816">Seſquialtera verò ratio reperitur primò inter hos <lb></lb>
numeros 3. 2. cùm enim duo ſoni in earum fuerint ratione ſuauem edent <lb></lb>
<expan abbr="conſonãtiam">conſonantiam</expan>, quæ Diapente nominatur; ſimul autem ijdem compoſiti Qui­<lb></lb>
narium efficiunt; cui quinario ſeſquitertia radix adiuncta, quæ eſt 7. Duo­<lb></lb>
denarium componunt: qui propterea duas exhibet harmonias. </s>
          <s id="s.003817">Præterea <lb></lb>
hæ duæ harmoniæ ſimul copulatæ conflant ſuauiſſimam Diapaſon conſonan­<lb></lb>
tiam, nam iunctæ ſimul prædictæ duæ rationes ſeſquialtera, &amp; ſeſquitertia, <lb></lb>
eo modo quo tradunt Muſici, hoc ſcilicet modo 4. 3. 2. oritur inter extre­<lb></lb>
mos numeros dupla ratio, quæ ipſius Diapaſon eſt forma. </s>
          <s id="s.003818">nam ratio 4.ad 3. <lb></lb>
eſt ſeſquitertia; ratio 3. ad 2. eſt ſeſquialtera; ratio verò 4. ad 2. quæ ex il­<lb></lb>
lis componitur, eſt dupla. </s>
          <s id="s.003819">quòd ſi duo ſoni duplam hanc rationem nacti fue­<lb></lb>rint, conſonantiam Diapaſon ſuauiſſimam reſonabunt. </s>
          <s id="s.003820">Cùm igitur nume­<lb></lb>
rus 12. harmonias haſce complectatur, per eum Muſæ optimum Reip. ini­<lb></lb>
tium, ac ſtatum ſignificare voluerunt. </s>
          <s id="s.003821">Verumenimuerò cum numerus hu­<lb></lb>
ius diagrammatis, ideſt huiuſcemodi conditionis, qui eſt 12. factus fuerit <lb></lb>
ſolidus, hoc eſt, quando Reſp. benè conſtituta ad eam annorum, vel ſæculo­<lb></lb>
rum periodum peruenerit, qui ſit numerus ſolidus numeri 12. tunc fatali <lb></lb>
ordine, mutationem pati incipiet, atque in peius, cùm optimi mutatio ſit <lb></lb>
peſsima, prolabi. </s>
          <s id="s.003822">porrò numerus ſolidus ipſius 12. eſt 1728. vti mox expli­<lb></lb>
cabo. </s>
          <s id="s.003823">vult igitur Socrates ibi myſticè ſignificare poſt tot annorum, aut ſæ­<lb></lb>
culorum numerum Remp. omnem quamuis optimam, in deterius prolapſu­<pb pagenum="226" xlink:href="009/01/226.jpg"></pb>ram, cùm enim ad ſummam perfectionem peruenerit, quæ in numero ſoli­<lb></lb>
do, &amp; cubico ſignificatur, ſi vlterius progreſsura ſit, neceſſariò ſummam <lb></lb>
perfectionem præteribit, ac derelinquet. </s>
          <s id="s.003824">Quòd autem numerus 1728. ſit <lb></lb>
numerus ſolidus, &amp; cubus ipſius Duodenarij ſic palàm fiet, ſi tamen prius, <lb></lb>
ea repetiueris, quæ ſupra in primo Poſter. num. </s>
          <s id="s.003825">33. marginali, de numero <lb></lb>
Quadrato, &amp; Cubo dicta ſunt: eſt autem cubus numerus is, qui ex gemina­<lb></lb>
to ductu alicuius numeri in ſe ipſum, producitur. </s>
          <s id="s.003826">multiplica igitur primò <lb></lb>
12. in 12. &amp; producetur numerus 144. qui quadratus, &amp; planus eſt. </s>
          <s id="s.003827">rurſus <lb></lb>
duc 12. in hunc 144. <expan abbr="produceturq́">produceturque</expan>; numerus 1728. quì cubus, ac proinde <lb></lb>
ſolidus eſt, vt loco citato explicauimus. </s>
          <s id="s.003828"><expan abbr="Atq;">Atque</expan> hæc Socratici huius myſterij <lb></lb>
explicatio ſufficiat.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.003829"><emph type="italics"></emph>Ex Octauo Politicorum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003830"><arrow.to.target n="marg323"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003831"><margin.target id="marg323"></margin.target>331</s>
        </p>
        <p type="main">
          <s id="s.003832">Cap. 5. (<emph type="italics"></emph>Muſicam verò omnes fatemur eſſe ex iucundiſſimis, ſiue nuda ſit, <lb></lb>
ſiue cum melodia<emph.end type="italics"></emph.end>) Quamuis latina interpretatio pro melodia, di­<lb></lb>
cat, modulatione, Gręcus tamen textus habet <foreign lang="grc">μετα μελωδιας,</foreign> ideſt <lb></lb>
<expan abbr="cũ">cum</expan> melodia. </s>
          <s id="s.003833">per Muſicam nudam, forte Ariſtoteles intelligit eam, <lb></lb>
quæ ſolis ſonis <expan abbr="abſq;">abſque</expan> oratione conſtat; per melodiam verò intelligit eam, <lb></lb>
quam Ioſephus Zarlinus in 2. parte ſuarum Inſtitutionum Muſicalium defi­<lb></lb>
nit, quæ eſt concentus plurium vocum harmonicus cum rithmo, &amp; oratio­<lb></lb>
ne, ideſt, qua canitur oratio aliqua ſub aliquo rithmo, aut modo, ſiue vt <lb></lb>nunc loquimur, conqualchearia.</s>
        </p>
        <p type="main">
          <s id="s.003834">Ex quibus liquet noſtros contrapuntiſtas toto cœlo aberrare, dum ſuas <lb></lb>
cantilenas, abſque vlla verborum intelligentia, atque abſque vllo rithmo <lb></lb>
diſperdunt.</s>
        </p>
        <p type="main">
          <s id="s.003835"><arrow.to.target n="marg324"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003836"><margin.target id="marg324"></margin.target>332</s>
        </p>
        <p type="main">
          <s id="s.003837">Eodem capite propè finem meminit harmoniæ Lydiæ, Mixtæ, Doricæ, <lb></lb>
Phrygiæ. </s>
          <s id="s.003838">de quibus ſupra 3. lib. Polit. cap. 2. tractaui, <expan abbr="earumq́">earumque</expan>; proprieta­<lb></lb>
tes, quas hic Ariſt. recenſet ibi connumeraui.</s>
        </p>
        <p type="main">
          <s id="s.003839"><arrow.to.target n="marg325"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003840"><margin.target id="marg325"></margin.target>333</s>
        </p>
        <p type="main">
          <s id="s.003841">Ibidem meminit etiam Rithmorum, &amp; Harmonie. </s>
          <s id="s.003842">Quid Rithmus dictum <lb></lb>
eſt ſuperius lib. 3. Politic. eſſe quem nunc vulgò ariam cantores, ac tibici­<lb></lb>
nes appellant.</s>
        </p>
        <p type="main">
          <s id="s.003843">Harmonia eſt plurium vocum ex acuto, &amp; graui concors modulatio.</s>
        </p>
        <p type="main">
          <s id="s.003844">Verùm de his fuſius in Problematibus Muſicis, ſect. </s>
          <s id="s.003845">19.</s>
        </p>
        <p type="main">
          <s id="s.003846"><arrow.to.target n="marg326"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003847"><margin.target id="marg326"></margin.target>334</s>
        </p>
        <p type="main">
          <s id="s.003848">Cap. 7 (<emph type="italics"></emph>Conſiderandum vtrum omnibus vtendum ſit harmonijs, &amp; rithmis<emph.end type="italics"></emph.end>) <lb></lb>
Vide quæ ſupra lib. 3. Politic. cap. 2. annotaui.</s>
        </p>
        <p type="head">
          <s id="s.003849"><emph type="italics"></emph>In Oeconomicis nihil Mathematicum reperi.<emph.end type="italics"></emph.end></s>
        </p>
      </chap>
      <pb pagenum="227" xlink:href="009/01/227.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.003850"><emph type="italics"></emph>EX PROBLEMATIBVS ARIST.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.003851"><emph type="italics"></emph>Ex Sectione Prima.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003852"><arrow.to.target n="marg327"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003853"><margin.target id="marg327"></margin.target>335</s>
        </p>
        <p type="main">
          <s id="s.003854">Sectione 1. num. </s>
          <s id="s.003855">3. <emph type="italics"></emph>(Quemadmodum tempora, ita ſyderum ortus, Orionis <lb></lb>
Arcturi, Virgiliarum, Caniculæ, qui flatus, imbresqué excitant, qui ſereni­<lb></lb>
tates, frigora, teporeſuè ſolent afferre)<emph.end type="italics"></emph.end> Intelligit de ortu coſmico, qui <lb></lb>
fit, quando aſtrum ſimul cum Sole oritur: quem ortum abundè in 2. <lb></lb>
Meter. ſumma 2. cap. 2. explicatum inuenies. </s>
          <s id="s.003856">Vt autem intelligas, quænam <lb></lb>
ſint Orionis, Arcturi, Virgiliarum, &amp; Caniculæ conſtellationes, &amp; in qua <lb></lb>
cœli parte ſint collocatæ, ſatius eſt globum aliquem aſtronomicum, in quo <lb></lb>
aſteriſmi omnes clarè depicti ſint, intueri, quàm hoc loco pluribus verbis <lb></lb>
rem per ſe claram, obſcurare. </s>
          <s id="s.003857">De Orione plura dicta ſunt 2. Meter. præ­<lb></lb>
ſertim quo tempore oriatur. </s>
          <s id="s.003858">Arcturus verò, ſiue Bootes, primæ magnitu­<lb></lb>
dinis ſtella, mane vnà cùm Sole in noſtro climate ex Magini Tabulis, circa <lb></lb>
28. diem Septembris oritur. </s>
          <s id="s.003859">De Virgilijs tamen illud exiſtimo <expan abbr="aduertẽdum">aduertendum</expan>, <lb></lb>quod in Tauri aſteriſmo, duæ aliæ partiales conſtellationes continentur; in <lb></lb>
capite enim ipſius, ſunt illæ, quæ &amp; Hiades, &amp; Atlantides, &amp; Succulæ nuncu­<lb></lb>
pantur. </s>
          <s id="s.003860">in dorſo autem ſunt illæ, quæ Pleiades, &amp; Virgiliæ ſunt appellatæ,</s>
        </p>
        <p type="main">
          <s id="s.003861"><emph type="italics"></emph>Quæ ſeptem dici, ſex tamen eſſe ſolent.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003862">Vt cecinit Ouidius, quandoquidem ſeptima ferè nunquam apparet. </s>
          <s id="s.003863">has vul­<lb></lb>
gus Gallinellam vocat. </s>
          <s id="s.003864">quod ſi eas per Teleſcopium inſpiciamus, mirum di­<lb></lb>
ctu, eaſdem plures eſſe, quàm quadraginta ſtellas minimas clarè videbimus, <lb></lb>
vt optimè primus omnium Galilæus obſeruauit. </s>
          <s id="s.003865">Porrò conſtellatio Tauri in <lb></lb>
noſtris regionibus oriri cum Sole, incipit vno circiter ſeſquimenſe poſt ver­<lb></lb>
num æquinoctium. </s>
          <s id="s.003866">De Canicula ſatis dixi in 2. Meteor. ſumma 2. cap. 2.</s>
        </p>
        <p type="main">
          <s id="s.003867"><arrow.to.target n="marg328"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003868"><margin.target id="marg328"></margin.target>336</s>
        </p>
        <p type="main">
          <s id="s.003869">Eadem ſect. </s>
          <s id="s.003870">num. </s>
          <s id="s.003871">17. <emph type="italics"></emph>(Cur à Virgiliarum occaſu ad Fauonij vſque flatus, bi <lb></lb>potiſſimum pereant, qui morbo longo laborant, &amp; ſenes, quam iuuenes potius?) <emph.end type="italics"></emph.end><lb></lb>
Lege, quæ ſcripſi de occaſu ſyderum lib. 2. Meteor. ſumma 2. cap. 2. dein­<lb></lb>
de, quæ in ſuperiore proximè loco de Virgilijs: quibus hæc pauca <expan abbr="addãtur">addantur</expan>. <lb></lb>
</s>
          <s id="s.003872">cum intelligat de coſmico Virgiliarum occaſu, qui noctu apparet, <expan abbr="incipitq́">incipitque</expan>; <lb></lb>
tunc primum, quando oriente Sole, ipſe occumbunt, neceſſe eſt occaſum <lb></lb>
hunc incipere poſt autumnale <expan abbr="æquinoctiũ">æquinoctium</expan> ferè ſeſquimenſe in noſtris regio­<lb></lb>
nibus; cum enim Virgiliæ ſint in Tauro, neceſſe eſt occidente Tauro initio <lb></lb>
dixi, vt Sol ſit in oppoſito ſigno, videlicet in Scorpione; in quo aſteriſmo <lb></lb>
Sol reperitur poſt prædictum æquinoctium vno ferè menſe cum dimidio. </s>
          <s id="s.003873">de <lb></lb>
hac re plura Plinius lib. 18. cap. 31.</s>
        </p>
        <p type="head">
          <s id="s.003874"><emph type="italics"></emph>Ex Sectione 15.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003875"><arrow.to.target n="marg329"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003876"><margin.target id="marg329"></margin.target>337</s>
        </p>
        <p type="main">
          <s id="s.003877">Nvm. 1. <emph type="italics"></emph>(Cur linea ab angulo ad angulum ducta, ſola ex omnibus, quæ fi­<lb></lb>guras rectilineas bifariam ſecant, diameter vocata eſt? </s>
          <s id="s.003878">An quod dia­<lb></lb>
meter, vt nomen ipſum deſignat, duas in partes figuram æquè dimetien­<lb></lb>do diuidit, nihil dimenſæ figuræ defiruens? </s>
          <s id="s.003879">igitur hæc, quæ per commiſ­<lb></lb>ſuras, hoc est, per angulos figuram diuidit, appellanda eſt diameter, quoniam hæc<emph.end type="italics"></emph.end> <pb pagenum="228" xlink:href="009/01/228.jpg"></pb><emph type="italics"></emph>figuram non deſtruit, quamuis diuidat. </s>
          <s id="s.003880">quemadmodum faciunt; qui vaſa militaria <lb></lb>
partiuntur. </s>
          <s id="s.003881">At cæteræ lineæ, quæ per lineas compoſitam figuram ſecant, eam cor­<lb></lb>rumpunt: committitur enim rectilinea figura in angulis, vel ſecundum angulos)<emph.end type="italics"></emph.end><lb></lb>
Vt rectè problema hoc percipiamus, proponenda eſt figura rectilinea, &amp; <lb></lb>vna ex ijs, quæ parallelogramma dicuntur, vt ſunt Quadratum, Quadrila­<lb></lb>
<figure id="id.009.01.228.1.jpg" place="text" xlink:href="009/01/228/1.jpg"></figure><lb></lb>
terum, Rhombus, Rhomboides, cuiuſmo­<lb></lb>
di eſt præſens, aliter verba Ariſt. illi non <lb></lb>
ſemper quadrarent, quia illarum diameter <lb></lb>
illas ſemper bifariam non ſe caret. </s>
          <s id="s.003882">quemad­<lb></lb>
modum videre eſt in trapezio. </s>
          <s id="s.003883">&amp; pentagono <lb></lb>
etiam æquilatero. </s>
          <s id="s.003884">Quærit igitur, cur ex om­<lb></lb>
nibus lineis, quæ quadrilaterum A B C D, <lb></lb>
bifariam diuidunt, quales ſunt E F, G H, &amp; <lb></lb>
D B. ſola D B, quæ ab angulo ad angulum <lb></lb>
ducta eſt, mœruit appellari diameter. </s>
          <s id="s.003885">Reſpondet autem, eam fortè appel­<lb></lb>
lationem hanc præ cæteris inde promeruiſſe, quòd, quamuis aliæ omnes <lb></lb>
æquè parallelogrammum dimetiantur, ſola tamen ipſa D B, ipſum non de­<lb></lb>
ſtruit, nec ſcindit, cùm ei nouam aliquam diuiſionem non inferat, ſed id per <lb></lb>
angulos ſecet, vbi prius laterum commiſſuræ <expan abbr="erãt">erant</expan>: reliquæ verò omnes no­<lb></lb>
uas figuræ ſectiones inferunt, cùm eius latera in punctis E, F, G, H, <expan abbr="diuidãt">diuidant</expan>, <lb></lb>
vbi nulla prius erat diuiſio; quapropter ipſam quodammodo deſtruunt, <lb></lb>
<expan abbr="atq;">atque</expan> corrumpunt. </s>
          <s id="s.003886">Aduertè vulgatam verſionem latinam hanc <emph type="italics"></emph>(Angulis enim <lb></lb>
constant, quæ rectis lineis continentur)<emph.end type="italics"></emph.end> malè græco textui <foreign lang="grc">συγκεινται γαρ τὸ <lb></lb>
ευθυγραμμον κατὰ τας γωνίας,</foreign> reſpondere, qui ſic latinè reddendus eſt: com­<lb></lb>
ponitur enim rectilineum iuxta angulos; quæ interpretatio vera eſt, quia <lb></lb>
anguli ſunt laterum commiſſuræ, vt dictum eſt.</s>
        </p>
        <p type="main">
          <s id="s.003887"><arrow.to.target n="marg330"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003888"><margin.target id="marg330"></margin.target>338</s>
        </p>
        <p type="main">
          <s id="s.003889">Eadem ſect. </s>
          <s id="s.003890">num. </s>
          <s id="s.003891">2. <emph type="italics"></emph>(cur diameter ita eſt appellata? </s>
          <s id="s.003892">Vtrum quoniam ſola bi­<lb></lb>
partitò figuram diuidat? </s>
          <s id="s.003893">An quod ſola figuram ſecat per partes, ſiue membra, qui­<lb></lb>
bus in flexa coarctatur, cùm cæteræ per latera diuidant?)<emph.end type="italics"></emph.end> præſentis problematis <lb></lb>expoſitio petatur ex præcedentis expoſitione.</s>
        </p>
        <p type="main">
          <s id="s.003894"><arrow.to.target n="marg331"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003895"><margin.target id="marg331"></margin.target>339</s>
        </p>
        <p type="main">
          <s id="s.003896">In problemate 3. <emph type="italics"></emph>(Cur homines omnes tam Græci, quàm Barbari ad decem <lb></lb>
<expan abbr="vſq;">vſque</expan> numerare conſueuere, &amp; c. </s>
          <s id="s.003897">Vtrum quod denarius numerus perfectus ſit: con­<lb></lb>tinet enim omnia numerorum genera. </s>
          <s id="s.003898">vt par, impar, quadratum, quadrantale, lon­<lb></lb>
gum, planum, primum, compoſitum)<emph.end type="italics"></emph.end> Cur omnes nationes miro quodam con­<lb></lb>
ſenſu ſuos numeros in denas, veluti in gradus quoſdam diuidant, Ariſtoteles <lb></lb>
cauſam indagaturus, reſpondet primò id fortè accidiſſe ob denarij numeri <lb></lb>perfectionem: cuius perfectionis hoc eſt indicium, quod denarius contineat <lb></lb>
omnes numerorum ſpecies. </s>
          <s id="s.003899">quæ quidem omnes numerorum ſpecies in defi­<lb></lb>
nitionibus 7. Elem. exponuntur, quas conſulere debes. </s>
          <s id="s.003900">in denario numero <lb></lb>
contineri numeros pares, ac impares, per ſe patet. </s>
          <s id="s.003901">continetur etiam in eo <lb></lb>
quadratus numerus, imò duo quadrati numeri, nam, &amp; quaternarius eſt <lb></lb>
numerus quadratus, quippe qui ex ductu binarij in binarium producatur: <lb></lb>
item nouenarius eſt quadratus, quippe qui ex multiplicatione ternarij in <lb></lb>
ternarium gignitur. </s>
          <s id="s.003902">Porrò pro quadrantali numero intelligendus eſt nume­<lb></lb>
rus cubus, erat. </s>
          <s id="s.003903">n. </s>
          <s id="s.003904">quadratal apud Romanos vas cubicæ figuræ: imò in græ­<lb></lb>
co textu voci huic quadrantali, reſpondet <foreign lang="grc">κυβος,</foreign> ideſt, cubus. </s>
          <s id="s.003905">vnde apud la­<pb pagenum="229" xlink:href="009/01/229.jpg"></pb>tinos quadrantal pro cubo ſolet vſurpari. </s>
          <s id="s.003906">in denario autem <expan abbr="cõtinetur">continetur</expan> etiam <lb></lb>
hic numerus, eſt enim octonarius numerus cubus, fit enim ex binario ter in <lb></lb>
ſe ipſum multiplicato, hoc modo; duo bis faciunt quatuor: rurſus duo qua­<lb></lb>
ter faciunt octo; quem ex definitione numeri cubi, conſtat eſſe cubum. </s>
          <s id="s.003907">qua <lb></lb>
ratione deinde reliqui numeri, longus, planus, primus, compoſitus, in de­<lb></lb>
nario exiſtant, facilè eſt cognoſcere, dummodo eorum definitiones tenean­<lb></lb>
tur, quæ initio 7. Elem. traduntur.</s>
        </p>
        <p type="main">
          <s id="s.003908"><arrow.to.target n="marg332"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003909"><margin.target id="marg332"></margin.target>340</s>
        </p>
        <p type="main">
          <s id="s.003910">Ibidem <emph type="italics"></emph>(An quod denarius fons, <expan abbr="atq;">atque</expan> principium eſt, quippe qui ex vno, duo­<lb></lb>
bus, tribus, &amp; quatuor conſtet)<emph.end type="italics"></emph.end> Aliam denarij dignitatem aſſignat, quam ex <lb></lb>Plutarcho lib. 1. cap. 3. de Placitis Philoſophorum, optimè poſſumus intel­<lb></lb>
ligere: vbi ſic ait: Pythagorei aiebant denarium eſſe Naturam, quoniam <lb></lb>
omnes gentes <expan abbr="vſq;">vſque</expan> ad decem natura duce numerabant. </s>
          <s id="s.003911">tum etiam, quia ex <lb></lb>
quaternario conſtabat, ideſt, ex his quatuor numeris 1. 2. 3. 4. qui ſimul ad­<lb></lb>
diti faciunt decem: quaternarium enim Pythagorei multis de cauſis, quas <lb></lb>
apud Petrum Bungum de myſterijs numerorum videre poteris, adeò extol­<lb></lb>
lebant, vt dicerent ex quaternario naturalia, omnia conſtare, vt quaterni­<lb></lb>
tati omnia accepta referrent. </s>
          <s id="s.003912">vnde etiam per ipſum conceptis his verſibus <lb></lb>
iurare ſolebant,</s>
        </p>
        <p type="main">
          <s id="s.003913"><emph type="italics"></emph>Iuro per omnipotentem animæ, qui Tetrada noſtræ <lb></lb>
Perpetuos fontes naturæ infudit habentem.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003914">Pręcipua verò cauſa, cur tantopere Pythagorei quaternitatem celebrarint, <lb></lb>
refertur à Marſilio Ficino cap. 24. compendij in Tymeum, his verbis: ex <lb></lb>
quatuor elementis, geometrica, &amp; harmonica ratione coniunctis inuicem, <lb></lb>
vniuerſum Mundum compoſitum Pythagorei omnes exiſtimant: conſonan­<lb></lb>tiam horum in cœlo ſemper eſſe perfectam, ſub cœlo autem aliquando diſ­<lb></lb>
ſonantem. </s>
          <s id="s.003915">volebant ergò totum mundum, tam æthereum, quàm elementa­<lb></lb>
rem, conſtare ex quatuor elementis; &amp; ideò ex quaternario omnia conſta­<lb></lb>
re dicebant. </s>
          <s id="s.003916">hac de cauſa Pythagorei in muſicis conſonantijs, vltra quater­<lb></lb>
narium progredi vetabant; hoc eſt nullam admittebant conſonantiam, quæ <lb></lb>
numeris quaternario <expan abbr="contẽtis">contentis</expan> non exprimeretur, idcircò ſupra quadruplam <lb></lb>
non aſcendebant. </s>
          <s id="s.003917">Verùm inter alias quaternitatis dignitates hanc maximi <lb></lb>
faciebant, quod denarius ex ipſa, vti modo dictum eſt, componeretur; cu­<lb></lb>
ius excellentiam in ipſum proinde denarium transfundebant, dicebantque <lb></lb>
denarium eſſe numerum perfectum, &amp; aliorum numerorum fontem, atque <lb></lb>
principium. </s>
          <s id="s.003918">quemadmodum natura ipſa quaternario <expan abbr="cõſtans">conſtans</expan>, erat omnium <lb></lb>
rerum origo. </s>
          <s id="s.003919">Ex quibus manifeſtum eſt Ariſt. Problema hoc, <expan abbr="atq;">atque</expan> eius ſo­<lb></lb>
lutionem ex Pythagoreorum philoſophia accepiſſe.</s>
        </p>
        <p type="main">
          <s id="s.003920"><arrow.to.target n="marg333"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003921"><margin.target id="marg333"></margin.target>341</s>
        </p>
        <p type="main">
          <s id="s.003922">Ibidem <emph type="italics"></emph>(An quia corpora, quæ feruntur numero nouenario continentur)<emph.end type="italics"></emph.end> Puto <lb></lb>
hæc nouem corpora, quæ mouentur, eſſe cœlos, primum ſcilicet Mobile, <lb></lb>
Firmamentum, &amp; ſeptem Planetarum orbes: quibus ſi addas ſphæram ele­<lb></lb>
mentarem, habebis denarium corporum perfectiſſimum, ex quo tota Mun­<lb></lb>
di machina componitur.</s>
        </p>
        <p type="main">
          <s id="s.003923"><arrow.to.target n="marg334"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003924"><margin.target id="marg334"></margin.target>342</s>
        </p>
        <p type="main">
          <s id="s.003925">Ibidem <emph type="italics"></emph>(An quod decem proportionibus, quatuor cubales numeri conſumun­<lb></lb>
tur, è quibus numeris vniuerſum conſtare Pythagoreis placet?)<emph.end type="italics"></emph.end> Aliam denarij <lb></lb>perfectionem affert, quam ex 8. 9. Elem. comprobare, <expan abbr="atq;">atque</expan> intelligere poſ­<lb></lb>
ſumus. </s>
          <s id="s.003926">eſt autem 8. 9. Elem. propoſitio hæc: ſi decem numeri in eadem pro­


<pb pagenum="230" xlink:href="009/01/230.jpg"></pb>portione progrediantur ab vnitate incipientes, erunt ex illis quatuor cubi, <lb></lb>
v.g. in ſerie duplæ proportionis progrediantur hi decem termini: 1. 2. 4. 8. <lb></lb>
16. 32. 64. 128. 256. 512. ex his decem numeris ſunt quatuor cubi, nimi­<lb></lb>
rum hi 1. 8. 64. 512. numerus cubus eſt, qui fit ex tribus æqualibus numeris <lb></lb>
in ſe multiplicatis. </s>
          <s id="s.003927">ſic vnitas eſt cubus, quia fit ex vnitatibus tribus in ſe du­<lb></lb>
ctis, nam 1. in 1. facit 1. &amp; rurſus iſtud 1. in 1. facit 1. ſic etiam 8. eſt cubus, <lb></lb>
quia fit ex tribus his numeris æqualibus 2. 2. 2. inuicem ductis hoc modo, <lb></lb>
2. in 2. facit 4. rurſus 4. in 2. facit 8. qui eſt cubus. </s>
          <s id="s.003928">ſic 64. fit ex tribus hiſce <lb></lb>
4. 4. 4. pariter. </s>
          <s id="s.003929">512. fit ex tribus his 8. 8. 8. <expan abbr="eſtq́">eſtque</expan>; propterea cubus. </s>
          <s id="s.003930">ſimiliter <lb></lb>
ſi alia progreſſio inſtituatur vſque ad decem terminos, erunt in ea quatuor <lb></lb>
cubi, eodem ordine, quo in ſuperiori progreſſione, ideſt primo loco, 4.7. &amp; <lb></lb>
10. v.g. ſit tripla progreſſio hæc 1. 3. 9. 27. 81. 243. 729. 2182. 6546. 19638. <lb></lb>
quatuor cubi erunt hi 1. 27. 729. 19638. quorum latera cubica, ſunt hi nu­<lb></lb>
meri 1. 3. 9. 27.</s>
        </p>
        <p type="main">
          <s id="s.003931">Poſtquam huius loci explicationem ex allegata Euclidiana demonſtratio­<lb></lb>
ne attuliſſem, incidi in Petri Apponenſis horum problematum commenta­<lb></lb>
ria, qui aliam à ſe confictam expoſitionem affert, <expan abbr="aitq́">aitque</expan>; ſe per quatuor inte­<lb></lb>
gros annos laboraſſe, antequam eam inuenire, <expan abbr="locumq́">locumque</expan>; hunc intelligere <lb></lb>
poſſet. </s>
          <s id="s.003932">eſt autem hæc; Denarius componitur ex quaternario, ideſt ex qua­<lb></lb>
tuor primis numeris cubis, ſcilicet 1. 9. 27. 64. qui ſunt cubi, &amp; ſimul additi <lb></lb>
conſtituunt decem denas, ideſt <expan abbr="cẽtum">centum</expan>. </s>
          <s id="s.003933">quæ cum nulli Mathematicæ demon­<lb></lb>ſtrationi innitatur, nec vniuerſalis ſit, ex ſe apparet, quàm ſit <expan abbr="commẽtitia">commentitia</expan>, <lb></lb>
&amp; ab Ariſt. mente aliena. </s>
          <s id="s.003934">Enim uerò me antea ipſius commentarijs caruiſ­<lb></lb>
ſe gauiſus ſum, quibus niſi caruiſſem, veritate ipſa caruiſſem, acquieuiſſem <lb></lb>
enim illius fictioni, quæ <expan abbr="vtcunq;">vtcunque</expan> videtur quadrare, <expan abbr="hocq́">hocque</expan>; pacto veritatis in­<lb></lb>quirendæ occaſio ſublata fuiſſet.</s>
        </p>
        <p type="main">
          <s id="s.003935"><arrow.to.target n="marg335"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003936"><margin.target id="marg335"></margin.target>343</s>
        </p>
        <p type="main">
          <s id="s.003937">Ibidem <emph type="italics"></emph>(An quod omnes homines digitis decem lege naturali creantur? </s>
          <s id="s.003938">it aque <lb></lb>ſui numeri quaſi calculos adipiſcentes hac eadem multitudine, cætera <expan abbr="quoq;">quoque</expan> nume­<lb></lb>
rant)<emph.end type="italics"></emph.end> hæc Ariſt. ratio confirmatur ex recentiorum relationibus de populis <lb></lb>
Braſiliæ, qui cum per ſummam barbariem, in omni rerum ignoratione ver­<lb></lb>
ſarentur, ex digitis tamen, <expan abbr="vtcunq;">vtcunque</expan> numerabant. </s>
          <s id="s.003939"><expan abbr="cumq́">cumque</expan>; vellent ſignificare <lb></lb>
quinque dicebant, manum vnam: cum verò decem dicebant, manus duas: <lb></lb>
cum viginti dicebant, manus, &amp; pedes. </s>
          <s id="s.003940">&amp; ſimiliter in alijs. </s>
          <s id="s.003941">non tamen hac <lb></lb>
ratione longè progrediebantur.</s>
        </p>
        <p type="main">
          <s id="s.003942"><arrow.to.target n="marg336"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003943"><margin.target id="marg336"></margin.target>344</s>
        </p>
        <p type="main">
          <s id="s.003944">In 4. problema. </s>
          <s id="s.003945">quoniam textus huius problematis, tam apud græcos, <lb></lb>
quàm latinos mendoſus apparet, eum propterea per ſequentem paraphra­<lb></lb>
ſim exponam, qua &amp; intelligi, &amp; reſtitui etiam poterit. </s>
          <s id="s.003946">Quæſtio autem eſt <lb></lb>
de inæquali incremento, ac decremento vmbrarum Solis, quæ propriè de <lb></lb>
vmbris in plano horizontali proiectis, quas rectas vmbras appellant intel­<lb></lb>ligenda eſt: hæ enim inæqualiter creſcunt, &amp; decreſcunt, ſi quidem mane <lb></lb>
plurimum, poſtea parum, tandem nihil ferè circa meridiem minuuntur, po­<lb></lb>
ſitis tamen æqualibus temporibus. </s>
          <s id="s.003947">Quærit igitur Ariſt. cur cum Sol eodem <lb></lb>
vigore feratur, non idem tamen incrementum, decrementumuè vmbrarum <lb></lb>
exultet? </s>
          <s id="s.003948">pro cuius intelligentia, ac ſolutione inſpicienda eſt figura ſequens: <lb></lb>
in qua ſemicirculus, ſiue arcus A B C, ſit is, per quem Sol incedit, dum ele­<lb></lb>
uatur ſupra horizontem H I; &amp; quia Sol vniformiter ſcandit hunc arcum,


<pb pagenum="231" xlink:href="009/01/231.jpg"></pb><figure id="id.009.01.231.1.jpg" place="text" xlink:href="009/01/231/1.jpg"></figure><lb></lb>
ſint duo arcus æquales A B, B C. <lb></lb>
in plano autem horizontis ere­<lb></lb>
ctum ſit corpus G D, quod à So­<lb></lb>
le conſpiciatur, ſiue illuminetur. <lb></lb>
</s>
          <s id="s.003949"><expan abbr="ſitq́">ſitque</expan>; primum Sol in A. radius ip­<lb></lb>
ſius per verticem D, tranſiliens <lb></lb>
erit A D I; vmbra verò erit G I. <lb></lb>
</s>
          <s id="s.003950">Sole deinde in B, exiſtente, erit <lb></lb>
radius B D E, &amp; vmbra G E. </s>
          <s id="s.003951">So­<lb></lb>
le tandem in C. <expan abbr="radioq́">radioque</expan>; C D F. <lb></lb>
vmbra erit G F. </s>
          <s id="s.003952">Dicit ergò Ariſt. quod cum anguli A D B, B D C, ſint ex <lb></lb>
æqualibus arcubus A B, B C. ad centrum D, conſtituti, erunt æquales, qua­<lb></lb>
re erunt etiam æquales alij duo anguli illis ad verticem oppoſiti, per 15. <lb></lb>
primi, qui ſunt contenti in triangulo G D I. quod <expan abbr="triã">tria</expan>ngulum fit à radio pri­<lb></lb>
mo D I, re conſpecta à Sole G D, &amp; vmbra G I. anguli inquam F D E, E D I, <lb></lb>
qui ſunt ad verticem prædictis angulis, erunt, &amp; ipſi æquales inuicem.</s>
        </p>
        <p type="main">
          <s id="s.003953">Supponit præterea pro certo radium D I, qui cæteris longius prolabitur, <lb></lb>
eſſe maiorem propinquiore D E. <expan abbr="ipſumq́">ipſumque</expan>; D E, maiorem eſſe reliquo radio <lb></lb>
D F. oportet autem <expan abbr="radiũ">radium</expan> B E, terminari in puncto E, quod ſit citra radium <lb></lb>
D I. &amp; radium D F, deſinere in F, citra radium D E, aliter ſequitur lineam <lb></lb>
rectam B E, vel C F, ſecare lineam recta A I, in pluribus punctis, quàm vno <lb></lb>
D. quod eſt impoſſibile. </s>
          <s id="s.003954">cùm ergò totus angulus F D I, diuidatur à linea <lb></lb>
D E, in duos angulos æquales F D E, E D I. <expan abbr="ſitq́">ſitque</expan>; latus D I, maius latere D F. <lb></lb>erit ex ſcholio 19. primi Elemen. linea E I, maior, quam F E, quæ ſunt duæ <lb></lb>
inæquales vmbræ, quæ tamen reſpondent æqualibus arcubus A B, B C. ſimi­<lb></lb>
liter vmbra F E, maior eſſe probaretur ſequenti qualibet vmbra, quæ tamen <lb></lb>
ex arcu æquali procederet. </s>
          <s id="s.003955">&amp; ſic deinceps vſquequò Sol eſſet in meridie, vbi <lb></lb>
vmbra eſſet omnium minima.</s>
        </p>
        <p type="main">
          <s id="s.003956">Atque ex his patet, cur quamuis Sol vniformiter in cœlo moueatur, vm­<lb></lb>
brarum tamen incrementa, ſint diſparia, nec vniformia. </s>
          <s id="s.003957">eadem intelligen­<lb></lb>
da ſunt de vmbrarum decrementis promeridianis Sole ad occaſum labente: <lb></lb>
tunc enim, vt ille cecinit.</s>
        </p>
        <p type="main">
          <s id="s.003958"><emph type="italics"></emph>Maioresqué cadunt altis de montibus vmbræ.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003959">Aduerte verba illa (<emph type="italics"></emph>Angulus D E, maior, quam E F, angulo D G, est<emph.end type="italics"></emph.end>) eſſe <lb></lb>
mendoſa, etiam in græco textu; vnde, &amp; malè in latinum tranſlata: <expan abbr="neq;">neque</expan> in <lb></lb>
græco eſt vox, angulus: aliter Ariſt. quæſtioni non ſatisfaceret. </s>
          <s id="s.003960">quare di­<lb></lb>
cendum, &amp; interpretandum videtur, quemadmodum à me factum eſt.</s>
        </p>
        <p type="main">
          <s id="s.003961"><arrow.to.target n="marg337"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.003962"><margin.target id="marg337"></margin.target>345</s>
        </p>
        <p type="main">
          <s id="s.003963">In 5. problema. </s>
          <s id="s.003964">Difficile admodum eſt problema iſtud, &amp; cuius ſolutio­<lb></lb>
nem nullus veterum, quod ſciam, perfectè attigit: quamuis Vitellio nu. </s>
          <s id="s.003965">39. <lb></lb>
lib. 2. hoc idem proponat, ac ſoluere contendat: Verùm nec Ariſt. nec Vi­<lb></lb>
tellio intellectui ſatisfaciunt mathematico: probabilia tamen afferunt. </s>
          <s id="s.003966">op­<lb></lb>timè hae de re Maurolicus, in ſuis Poſthumis, nuper editis Photiſmis, &amp; Ke­<lb></lb>
plerus, in paralip. </s>
          <s id="s.003967">ad Vitell. </s>
          <s id="s.003968">Senſum Ariſt. ac textum pariter per paraphra­<lb></lb>
ſim exponam, ita tamen, vt eius textus ex hac paraphraſi omninò clarus <lb></lb>
euadat. </s>
          <s id="s.003969">Quærit igitur; cur lumen Solis ingrediens per quadrangularia, ſeu <lb></lb>triangularia foramina, vel etiam per rimulas, cùm poſtea recipiatur in pla­<pb pagenum="232" xlink:href="009/01/232.jpg"></pb>no ſatis ab illo foramine remoto, vt in pariete, vel pauimento, non recipi­<lb></lb>
tur in eadem figura, per quam ingreſſum eſt; quamuis. </s>
          <s id="s.003970">n. </s>
          <s id="s.003971">foramen ſit angulo­<lb></lb>
ſum, illuminatio tamen in oppoſito plano facta eſt ſemper circularis, ſi pla­<lb></lb>
num ſit ſatis remotum, &amp; radio Solis directè, ſeu perpendiculariter obie­<lb></lb>
ctum: ſi enim non ſit perpendiculariter, ſed obliquè, tunc illuminationes <lb></lb>apparent non omnino circulares, ſed ouales; quemadmodum quotidie ac­<lb></lb>
cidere videmus in pauimentis, vbi omnes ferè huiuſmodi illuminationes <lb></lb>
ellipſes ſunt, quamuis Sol per anguloſa foramina ingrediatur. </s>
          <s id="s.003972">quæ ellipſes ſi <lb></lb>
in plano Solis radio <expan abbr="perpẽdiculariter">perpendiculariter</expan> obiecto recipiantur, perfecti euadunt <lb></lb>
orbes. </s>
          <s id="s.003973">hoc etiam, inquit Ariſt, in cratibus patet; crates enim illæ habebant <lb></lb>
foramina anguloſa, atque oblonga, fiunt enim crates ex virgis decuſſatis, <lb></lb>
quorum foramina ſunt quadrilatera, per quæ Sol ingrediens, non tamen <lb></lb>
recipit anguloſam figuram, ſed in debita remotione rotundatur. </s>
          <s id="s.003974">Reſpon­<lb></lb>
det propoſitæ quæſtioni dicens; radijs Solis fortè illud accidere, quod &amp; ra­<lb></lb>
dijs viſualibus, qui ab oculo ad rem conſpectam producuntur: ij enim in <lb></lb>
turbinis, ſeu coni figuram aguntur, cuius apex eſt in oculo, baſis autem eſt <lb></lb>
in re viſa: &amp; quamuis res viſa ſit anguloſa, vt triangula, vel quadrangula, <lb></lb>
tamen ſi à longè conſpiciatur, circularis apparet; vnde &amp; figura viſualium <lb></lb>
radiorum, quæ in proximum obiectum incidens baſim anguloſam, remoto, <lb></lb>
quantum ſatis eſt, obiecto, baſim habebit orbicularem. </s>
          <s id="s.003975">eodem igitur modo <lb></lb>
de Solis radijs exiſtimare debemus: qui quamuis per anguloſa foramina in­<lb></lb>
trent, tamen, ſi in remoto obiecto recipiantur, figuram circularem ſortien­<lb></lb>
tur: quod ſi non ſatis remoto plano occurrant, anguloſam etiam figuram <lb></lb>
pro foraminis qualitate efficient: &amp; quidem eo foramini ſimiliorem, quo ei <lb></lb>
propior erit coni luminoſi baſis. </s>
          <s id="s.003976">vel aliter etiam <expan abbr="reſpondẽdum">reſpondendum</expan> eſt, hoc mo­<lb></lb>
do; figura Solis, quæ orbicularis eſt, vndique lineis rectis, ſeu radijs, quos <lb></lb>
emittit, circundata eſt, qui radij cum intrent per foramina lineis <expan abbr="quoq;">quoque</expan> re­<lb></lb>
ctis contenta, accidunt lateribus figuræ foraminis, &amp; propterea cum rectæ <lb></lb>
lineæ lineis rectis applicentur, deberent hi radij in figuram rectilineam con­<lb></lb>
formari; quod &amp; faciunt, vt patet in cratium feneſtellis, vbi ſi radij poſt in­<lb></lb>
greſſum ſtatim in plano quopiam recipiantur, figuram efficiunt feneſtellæ ſi­<lb></lb>
milem. </s>
          <s id="s.003977">quod ſi in plano, ſatis remoto, deſinant, non amplius anguloſam, <lb></lb>
ſed circularem illuminationem efficient. </s>
          <s id="s.003978">Cuius cauſa eſt, quia, vt initio di­<lb></lb>
xi, eodem modo lumen, ſeu radij Solis producuntur, quo etiam radij vi­<lb></lb>
ſuales, quod inde patere poteſt, quia perſpectiui eadem de <expan abbr="vtriſq;">vtriſque</expan> &amp; ſuppo­<lb></lb>
nunt, &amp; oſtendunt. </s>
          <s id="s.003979">quemadmodum igitur quando oculi noſtri aſpectus ad <lb></lb>
figuram rectilineam, &amp; anguloſam, quæ propinqua ſit directus, eam angu­<lb></lb>loſam iudicat, vt re vera eſt, quam deinde longè ſemotam oualem, aut cir­<lb></lb>
cularem exiſtimat; propterea quod radij viſuales ad extremitates laterum <lb></lb>
figuræ, ſiue angulos ipſius protenſi euaneſcunt, quia imbecilli admodum <lb></lb>
ſunt; ſunt autem imbecilli, quia, <expan abbr="cũ">cum</expan> tendant ad angulos, quæ minimæ partes <lb></lb>
ſunt obiecti, &amp; quidem longè ſemoti, fit, vt ij anguli ſub angulo adeò paruo <lb></lb>ad oculum veniant, vt ſub eo viſio fieri nequeat; idcircò cùm anguli cerni <lb></lb>
nequeant, obiectum ſub circulari figura apparebit. </s>
          <s id="s.003980">ſed quantum lateris re­<lb></lb>
cti viſi in turbine viſuali continetur, ac ſub angulo viſioni idoneo ad ocu­<lb></lb>
lum defertur, id tantum in viſum agere poteſt. </s>
          <s id="s.003981">Reliquum, quod eſt in an­<pb pagenum="233" xlink:href="009/01/233.jpg"></pb>gulis ob dictam rationem non poteſt, quia radij viſuales, vt dictum eſt, in­<lb></lb>
diſcreti ſunt, &amp; confuſi. </s>
          <s id="s.003982"><expan abbr="neq;">neque</expan> hoc mirum, cùm multa videre nullo pacto poſ­<lb></lb>
ſimus, quamuis noſtro attingantur aſpectu, vt ea, quæ ſunt in tenebris, cui <lb></lb>
ſimile accidit, cum quadratum à longè viſum videtur habere plurimos an­<lb></lb>gulos; atq; adeò ad rotunditatem, ſi remoueatur adhuc, accedere, vt etiam <lb></lb>
circulus videatur. </s>
          <s id="s.003983">Cùm enim, vt ſupra dixi, aſpectus in turbinis modum <lb></lb>
procedat quoties figura conſpecta vlterius ſepoſita eſt, radij viſuales, qui ad <lb></lb>
angulos tendant, quoniam &amp; imbecilli, &amp; pauci ſunt, ob dictam cauſam rem <lb></lb>
aſſequi nequeunt: qui autem in mediam partem concurrunt, hi perſiſtere <lb></lb>poſſunt, vtpotè, conferti, &amp; validi: ergò cum figura propè eſt, anguli <expan abbr="quoq;">quoque</expan> <lb></lb>
aſpici poſſunt, aucta diſtantia non poſſunt. </s>
          <s id="s.003984">hac etiam de cauſa linea circu­<lb></lb>
laris valdè diſtans, &amp; in ſitu, quo conuexum ad oculum rectà vergat: &amp; in <lb></lb>
Luna die octauo, quando dimidia eſt linea illa, quæ illuminatam partem à <lb></lb>
non illuminata diuidit, recta videtur, quamuis circularis ſit, eſt enim in <lb></lb>
ſphærico corpore deſignata. </s>
          <s id="s.003985">quando enim circunferentia propè eſt, viſus <lb></lb>
diſcernere valet quanto pars altera, parte altera, ſit propior; vnde rotun­<lb></lb>
ditas apparet: at cùm procul abeſt rectè ſentire nequit, ſed æqualem par­<lb></lb>
tium ſitum cernere ſibi videtur, <expan abbr="eamq́">eamque</expan>; propterea rectam iudicat. </s>
          <s id="s.003986">hæc igi­<lb></lb>
tur, quæ accidere viſui certum eſt, eadem ſimiliter radijs Solis conuenire <lb></lb>
par eſt credere. </s>
          <s id="s.003987">ex quibus iam patere poteſt, cur lumen Solis per quadrila­<lb></lb>
teras figuras profluens illuminationem rotundam reddat.</s>
        </p>
        <p type="head">
          <s id="s.003988"><emph type="italics"></emph>De Lucis Figuratione.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.003989"><expan abbr="Atq;">Atque</expan> hæc eſt ſolutio admirandi huius effectus ab Ariſt. allata, quæ <lb></lb>
quoniam non paucas habet difficultates, aliam ex Maurolyco de­<lb></lb>
ſumptam, quæ ſatis demonſtratiua eſt, afferam.</s>
        </p>
        <p type="main">
          <s id="s.003990">Primò igitur illud Perſpectiuus principium ſtatuendum eſt, ex <lb></lb>
quolibet corporis lucidi puncto, ad quodlibet medij punctum, lumen rectis <lb></lb>
lineis quoquouerſus emicare, ita vt lumen à quouis puncto lucidi, tanquam <lb></lb>
à centro <expan abbr="circumquaq;">circumquaque</expan> effuſum in modum ſphæræ diffundatur.</s>
        </p>
        <p type="main">
          <s id="s.003991">Secundò, quò magis duorum vicinorum circulorum peripheriæ augen­<lb></lb>
tur, eò magis ad vnius circuli ſimilitudinem accedere; vt in hac figura cer­<lb></lb>
<figure id="id.009.01.233.1.jpg" place="text" xlink:href="009/01/233/1.jpg"></figure><lb></lb>
nere licet, vbi ſunt primò duo parui <lb></lb>
circuli circa centra A, &amp; B, deſcripti, <lb></lb>
quorum circunferentiæ creſcant <expan abbr="vſq;">vſque</expan> <lb></lb>
ad circunferentias C D, &amp; E F, quo in­<lb></lb>
cremento poſito, ſtatim vel ad ſenſum <lb></lb>
manifeſtum eſt, has duas maiores pe­<lb></lb>ripherias, magis referre vnius circuli <lb></lb>
ſimilitudinem, quàm referant duo pa­<lb></lb>
rui circelli. </s>
          <s id="s.003992">quod ſi eſſent tres circelli, <lb></lb>
qui augerentur, magis adhuc vnicum <lb></lb>
circulum imitarentur; <expan abbr="ſicq́">ſicque</expan>; deinceps, <lb></lb>quò plures eò perfectius: &amp; quò magis <pb pagenum="234" xlink:href="009/01/234.jpg"></pb>etiam augentur eò perfectius. </s>
          <s id="s.003993"><expan abbr="proptereaq́">proptereaque</expan>; poterunt aliquando exactè cir­<lb></lb>
culum quò ad ſenſum imitari. </s>
          <s id="s.003994">quod de circulis dictum eſt, intelligi etiam de­<lb></lb>
bet de omnibus alijs figuris eiuſdem ſpeciei, vt de duabus ellipſibus, aut <lb></lb>
de duobus triangulis, &amp;c.</s>
        </p>
        <p type="main">
          <s id="s.003995">Tertiò, lumen Solis per foramen tam <expan abbr="exiguũ">exiguum</expan>, quod ſit inſtar puncti tranſ­<lb></lb>
miſſum, <expan abbr="figurã">figuram</expan> Solis rotundam videlicet, quamuis conuerſam referre; quod <lb></lb>
<figure id="id.009.01.234.1.jpg" place="text" xlink:href="009/01/234/1.jpg"></figure><lb></lb>
hac deſcriptione patefiet. </s>
          <s id="s.003996">ſit Sol vbi A B, fora­<lb></lb>
men inſtar puncti vbi E. illuminatio, in planum <lb></lb>
radio Solis perpendiculariter obiectum, ſit C I D. <lb></lb>
aio eam eſſe inſtar Solis rotundam, inuerſam ta­<lb></lb>
men. </s>
          <s id="s.003997">nam ſi ab omnibus punctis ſolaris periphe­<lb></lb>
riæ radij per vnicum punctum E, rectà transfe­<lb></lb>
rantur ad planum rectà obiectum, vbi C D, con­<lb></lb>
flabunt duas conicas ſuperficies A E B, C E D, <lb></lb>
baſes habentes circulos A B, C D, verticem verò <lb></lb>
eundem E. </s>
          <s id="s.003998">Cùm igitur illuminatio C D, ſit ve­<lb></lb>
luti ſectio luminoſi coni C E D, quæ perpendicu­<lb></lb>
lariter eum ſecat, ex Apollonij conicis circulus <lb></lb>
erit, ac proinde Solis figuram imitabitur. </s>
          <s id="s.003999">erit <lb></lb>
tamen inuerſa, quia cum, vt dictum eſt in prima <lb></lb>
prænotatione, radij rectis lineis ferantur, pun­<lb></lb>
ctum A, ſiniſtrum, repreſentabitur in D, parte <lb></lb>
dextra. </s>
          <s id="s.004000">B, verò dexterum apparebit in C, parte <lb></lb>
ſiniſtra, &amp; H, in anteriore parte Solis, feretur in <lb></lb>
I, punctum illuminationis poſterius: <expan abbr="atq;">atque</expan> eodem <lb></lb>
modo reliqua puncta in contrarias partes trans­<lb></lb>
ferentur. </s>
          <s id="s.004001">Quod ſi planum terminans conum ra­<lb></lb>
dioſum non illi ſit perpendiculare, ſed obliquum, <lb></lb>
vti eſt G F, ſectionem faciet ellipticam ex eodem <lb></lb>
Apollonio, <expan abbr="ideoq́">ideoque</expan>; Solis il luminatio, quod pluri­<lb></lb>
mùm accidit, oualis apparet. </s>
          <s id="s.004002">Quod dictum eſt de Solis illuminatione, in­<lb></lb>telligi etiam debet de alijs quibuſuis lucidis, vel coloratis luce perfuſis, quæ <lb></lb>ſuas ſpecies emittunt, cuiuſuis ſint figuræ, eodem enim modo oſtendemus <lb></lb>
eorum illuminationes, ſeu ſpecies debere figuram ipſorum primitiuam re­<lb></lb>
ferre, quamuis inuerſam.</s>
        </p>
        <p type="main">
          <s id="s.004003">Quartò, dico, Cauſam huius apparentiæ primariam eſſe ipſam Solis ro­<lb></lb>tunditatem, quæ per ſingula foraminis cuiuſuis puncta in oppoſitum planum <lb></lb>
ſe ſe transfundit. </s>
          <s id="s.004004">quod enim nuper de vnico puncto oſtenſum eſt, idem intel­<lb></lb>
ligendum eſt de ſingulis foraminis punctis, per ſingula enim puncta ſingulæ <lb></lb>
illuminationes rotundæ in aduerſum planum tranſmittuntur, quæ quò lon­<lb></lb>
gius à foramine proceſſerint, cò perfectiorem rotunditatem aſſequentur, ob <lb></lb>
eam cauſam, quàm in ſecunda prænotatione innuimus. </s>
          <s id="s.004005">quæ vt explicatius <lb></lb>
tractentur, neuè in hac Solis luce cæcutiamus, linearem demonſtrationem <lb></lb>
afferemus. </s>
          <s id="s.004006">ſit ſolare corpus A B, foramen verò <expan abbr="qualiſcunq;">qualiſcunque</expan> figuræ, veluti ri­<lb></lb>
mula C D, per quam Solis ſplendor illapſus oppoſitum planum, in quo F E, <lb></lb>
colluſtret. </s>
          <s id="s.004007">iam ex infinitis punctis rimulæ C D, ſatis erit extrema duo C, D,


<pb pagenum="235" xlink:href="009/01/235.jpg"></pb>conſiderare. </s>
          <s id="s.004008">per punctum igitur D, ducantur radij A D E, B D K. per pun­<lb></lb>
ctum verò C, ducantur alij A C H, B C F, qui cùm ab extremitatibus Solis <lb></lb>
profluant, reliquos omnes radios intra ſe continebunt. </s>
          <s id="s.004009">ex tertia igitur prę­<lb></lb>
notatione per punctum C, procedit rotunda illuminatio, cuius diameter <lb></lb>
<figure id="id.009.01.235.1.jpg" place="text" xlink:href="009/01/235/1.jpg"></figure><lb></lb>
F H, ſimiliter per punctum D, illuminatio <lb></lb>
rotunda emanat, cuius diameter K E, &amp; pa­<lb></lb>
riter ex omnibus alijs rimulæ punctis ro­<lb></lb>
tundi ſplendores in ſuperficiem, vbi F E, <lb></lb>
tranſmittuntur. </s>
          <s id="s.004010">Iam dicimus has duas illu­<lb></lb>
minationes ex prænotatis ſecundo loco, <lb></lb>
quàm longius planum F E, à foramine de­<lb></lb>
ſtiterit, vt ſi eſſet in L M, ad vnius circuli <lb></lb>
rotunditatem magis accedere, vt apparet <lb></lb>
in plano L M, vbi maiores factæ ſunt illumi­<lb></lb>
nationes, &amp; ideò magis ad vnam circula­<lb></lb>
rem accedunt. </s>
          <s id="s.004011"><expan abbr="manifeſtũ">manifeſtum</expan> eft enim, quò lon­<lb></lb>
gius radij C F, C H, producti fuerint, eò <lb></lb>
maiorem fore <expan abbr="diametrũ">diametrum</expan> illuminationis F H. <lb></lb>
euadet enim L O, &amp; ſimiliter ex productio­<lb></lb>
ne radiorum D K, D E, diameter alterius <lb></lb>
illuminationis K E, augebitur, &amp; fiet N M. <lb></lb>
&amp; conſequenter duæ ipſarum peripheriæ ſi­<lb></lb>
mul maiores fient, ac proinde ad vnius cir­<lb></lb>
culi ſimilitudinem ex <expan abbr="ſecũda">ſecunda</expan> notatione per­<lb></lb>
uenient. </s>
          <s id="s.004012">&amp; quamuis ex radiorum produ­<lb></lb>
ctione augeantur non ſolum prædictæ dia­<lb></lb>
metri illuminationum, ſed etiam earum <lb></lb>
differentiæ F K, &amp; H E; hæ tamen differen­<lb></lb>
tiæ reſpectu illorum nihil, quod ſenſibile ſit <lb></lb>
augentur; quod inde oritur, quia angulus F C H, maior eſt angulo F B K, <lb></lb>
per 16. primi Elem. &amp; ideò crura F C, H C, magis dilatata ſunt quàm cru­<lb></lb>
ra F B, K B, &amp; ideò ſi producantur, multò magis creſcit F H, dum euadit <lb></lb>
M N, quàm F K, dum euadit M O. eodem modo magis creſcit K E, dum fit <lb></lb>
O L, quàm H E, dum fit K L. quare ex ſecunda notatione earum periphe­<lb></lb>
riæ ad vnius orbis figuram tandem concurrere videbuntur. </s>
          <s id="s.004013">multò autem <lb></lb>
euidentius ad rotunditatem accederent, ſi tertia illuminatio per tertium <lb></lb>
aliquod punctum tranſiens, ſic perueniret; &amp; quo plures, eò etiam perfectius, <lb></lb>
omnes enim rotundæ eſſent, &amp; ex radiorum proceſſu augerentur, atque ad <lb></lb>
vnius orbis formam ſe ſe reciperent.</s>
        </p>
        <p type="main">
          <s id="s.004014">Hæ porrò, quæ Geometricè comprobata ſunt, libet etiam iucunda qua­<lb></lb>
dam experientia confirmare; fiant igitur in feneſtra quapiam duo, vel tria <lb></lb>
parua admodum foramina, inuicem proxima, per quæ totidem illumina­<lb></lb>
tiones ad obiectam chartam transferantur, hæ admota foramini charta <lb></lb>
paruæ, ac ſibi mutuò parum incumbentes apparebunt, &amp; proinde vnicum <lb></lb>
circulum non præſeferent; quò autem longius charta remouebitur, eò ma­<lb></lb>
iores fient, ac ſibi mutuò magis incumbentes, ac idcircò in vnum ferè cir­


<pb pagenum="236" xlink:href="009/01/236.jpg"></pb>culum coaleſcent. </s>
          <s id="s.004015">nunquam tamen ad geometricam rotunditatem perue­<lb></lb>
nient, quamuis illam ſenſui obijciant.</s>
        </p>
        <p type="main">
          <s id="s.004016">Aliter Ioannes Keplerus totam hanc demonſtrationem inſtituit, quem tu <lb></lb>
in ſuis ad Vitellionem Paralipom. conſule. </s>
          <s id="s.004017">eius tantum experientiam non <lb></lb>
iniucundam, qua iſtud probat, non grauabor referre. </s>
          <s id="s.004018">cap. igitur ſecundo <lb></lb>
de Figuratione lucis hæc habet: librum in ſublimi locaui, qui eſſet loco lu­<lb></lb>
centis corporis, hunc inter &amp; pauimentum figebatur tabella foramine mul­<lb></lb>tangulo. </s>
          <s id="s.004019">filum deinde ex vno libri angulo per foramen in pauimentum de­<lb></lb>
miſſum, ita incidebat in pauimento, vt terminos foraminis raderet, cuius <lb></lb>
veſtigia creta imitabar; qua ratione creabatur figura in pauimento ſimilis <lb></lb>
foramini. </s>
          <s id="s.004020">Idem accidebat, annexo filo ex altero, tertio, quarto libri angu­<lb></lb>
lo, <expan abbr="adeoq́">adeoque</expan>; ex infinitis marginum punctis. </s>
          <s id="s.004021"><expan abbr="Itaq;">Itaque</expan> infinitarum in pauimento <lb></lb>
figurarum foraminis exilium ſeries adumbrabat magnam, &amp; quadrangulam <lb></lb>
libri figuram. </s>
          <s id="s.004022">hic primus eſt in hoc labore ſucceſſus. </s>
          <s id="s.004023">hæc ille; ex quibus po­<lb></lb>
ſtea ſuam demonſtrationem adornauit. </s>
          <s id="s.004024">His igitur perſpicuè demonſtratis <lb></lb>
facilè erit nonnulla corollaria inde contexere.</s>
        </p>
        <p type="main">
          <s id="s.004025">Primum, ſi ad planum F E, radius perpendiculariter incidat, illuminatio <lb></lb>
erit circulus, ſi verò obliquè ellipſis, vt in tertio loco vidimus. </s>
          <s id="s.004026">cùm igitur <lb></lb>
pauimentis, ac parietibus hæ illuminationes, vt plurimum obliquè acci­<lb></lb>
dant, ideò ferè ſemper ouales apparent.</s>
        </p>
        <p type="main">
          <s id="s.004027">Secundum, quod quidem magni momenti eſt, eſt enim, vti ſcientiam de­<lb></lb>
cet, vniuerſale, quod enim oſtenſum eſt de Sole, eodem modo oſtendi poteſt <lb></lb>
de quouis lucido, &amp; de quouis corpore illuminato ſuam ſpeciem diffunden­<lb></lb>
te. </s>
          <s id="s.004028">ſimili enim modo demonſtrare poſſumus cur Sol eclypſim patiens, illu­<lb></lb>
minationem pariter eclypſatam efficiat, &amp; inuerſam. </s>
          <s id="s.004029">eadem eſt ratio de <lb></lb>
Lunæ illuminationibus.</s>
        </p>
        <p type="main">
          <s id="s.004030">Tertium, &amp; quidem ſcitu digniſſimum <expan abbr="quodq;">quodque</expan> hactenus doctorum viro­<lb></lb>
rum ingenia latuit, rationem reddere hinc poſſumus, cur ſi feneſtris omni­<lb></lb>
bus obſeratis, conclaue obſcurum reddatur, tenui tantùm relicto forami­<lb></lb>
ne, per quod externo lumini aditus patent, formæ externarum rerum pro­<lb></lb>
priæ, quamuis inuerſæ, in oppoſito plano, appareant. </s>
          <s id="s.004031">eadem ſcilicet de cau­<lb></lb>
ſa, qua &amp; Solis imago propria, <expan abbr="quoniã">quoniam</expan> videlicet per ſingula foraminis pun­<lb></lb>
cta, vt tertio loco patuit, <expan abbr="vnaquæq;">vnaquæque</expan> res, ſeu lucida, ſeu illuminata tantùm <lb></lb>
ſit, per ſingula foraminis puncta, ſingulas proprias emittit imagines, quæ <lb></lb>
omnes poſtea in vnam ex iuſta diſtantia coaleſcunt. </s>
          <s id="s.004032"><expan abbr="atq;">atque</expan> eadem ratione in­<lb></lb>uertuntur. </s>
          <s id="s.004033">ob quam etiam rationem ſolares maculæ in Solis ſplendoribus, <lb></lb>
non eodem ſitu quem in Solis diſco obtinent, ſed inuerſo ſpectantur. </s>
          <s id="s.004034">atque <lb></lb>
hæc pro inſtituto dicta ſufficiant.</s>
        </p>
        <p type="main">
          <s id="s.004035"><arrow.to.target n="marg338"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004036"><margin.target id="marg338"></margin.target>346</s>
        </p>
        <p type="main">
          <s id="s.004037">In 6. Problema. </s>
          <s id="s.004038">quoniam vulgata interpretatio videtur mendoſa, cum in <lb></lb>
multis textui græco non conſentiat, eam ſic emendatam accipe <emph type="italics"></emph>(Cur Lunæ <lb></lb>
ſphærica exiſtente, rectam cum ſemiplena eſt, cernimus? </s>
          <s id="s.004039">An quoniam eodem in <lb></lb>
plano aſpectus noster verſatur, vt circuli ambitus, quem Lunæ Solingruens facit, <lb></lb>quod cùm accidit, Sol recta linea videtur; cum enim quid ſuum aſpectum ſphæræ <lb></lb>
admouerit, orbem videre neceſſe ſit; Luna autem ſphærica ſit, eamqué Sol aſpiciat; <lb></lb>orbis profectò id eſſe debet, quod à Sole efficitur. </s>
          <s id="s.004040">Hic ergò cum è regione ſe nobis <lb></lb>
præbet, totus videtur, &amp; ſic plenilunium apparet, cùm autem mutatur propter<emph.end type="italics"></emph.end> <pb pagenum="237" xlink:href="009/01/237.jpg"></pb><emph type="italics"></emph>Solis diſceſſum peripheria eius aſpici poteſt, ita vt recta appareat. </s>
          <s id="s.004041">altera verò pars <lb></lb>
circularis, quoniam ex aduersò noſtri aſpectus hemiſphærium eſt; talis verò appa­<lb></lb>ret ſemicirculus. </s>
          <s id="s.004042">ſemper enim Luna aſpectui nostro oppoſita eſt, ſed quando Sol in­<lb></lb>
cubuerit, non videtur, &amp; repletur post diem octauum ſecundum dimidium; quo­<lb></lb>niam paulatim Sol euadens, orbem nobis facit inclinatiorem; ita verò circulus ad <lb></lb>
oculum noſtrum diſpoſitus, ſimilis videtur ſectioni conicæ. </s>
          <s id="s.004043">lunaris verò apparet <lb></lb>
iam Sole amoto; cùm enim ad extrema puncta peruenerit, iuxta quæ dimidiata <lb></lb>
apparet, circulus fit Solis, &amp; Solis circunferentia videtur; non enim amplius in <lb></lb>
directum viſui iacet, ſed præterit. </s>
          <s id="s.004044">quo facto, &amp; per eadem puncta ducto circulo, ne­<lb></lb>
ceſſe eſt lunularem apparere: pars enim aliqua circuli ſtatim aſpectui patet, priori è <lb></lb>
contra exiſtente, ita vt de ſplendido reſecetur. </s>
          <s id="s.004045">tum etiam extrema <expan abbr="manẽt">manent</expan> in eodem, <lb></lb>
vt oporteat lunularem apparere magis, &amp; minus, ſecundum Solis motum. </s>
          <s id="s.004046">per­<lb></lb>
moto enim Sole, &amp; circulus, ſecundum quem conſpicitur, reuertitur ad eadem <lb></lb>
puncta. </s>
          <s id="s.004047">ſecundum enim infinitas inclinationes accidit inclinari: ſi quidem maxi­<lb></lb>
mi circuli per eadem puncta duci poſſunt infiniti)<emph.end type="italics"></emph.end> Vt rectè textum hunc intel­<lb></lb>
ligas, lege prius, quæ de Lunæ illuminatione lib. 1. Poſt. tex. 30. dicta ſunt. <lb></lb>
</s>
          <s id="s.004048">&amp; ante omnia experire in pila aliqua lumini lucernæ, aut candelæ obiecta, <lb></lb>
&amp; circumlata, omnes illius ſpheræ illuminationes, vt ibi docui. </s>
          <s id="s.004049">videbis enim <lb></lb>
qua ratione linea illa, quæ confinium eſt partis illuminatæ, &amp; partis obſcu­<lb></lb>
ræ, aliquando videatur lunularis, aliquando oualis, aliquando recta linea, <lb></lb>
quorum rationem Ariſt. in præſenti problemate inquirit. </s>
          <s id="s.004050">lege præterea, ſi <lb></lb>
plenam huius rei cognitionem deſideras, propoſit. </s>
          <s id="s.004051">74. 75. 76. 77. libri 4. <lb></lb>Vitellionis, vbi hæc omnia exactè, &amp; non leui brachio, vt hic fit ab Ariſtot. <lb></lb>
demonſtrantur. </s>
          <s id="s.004052">Interim tamen huius loci explicationem hanc accipe. </s>
          <s id="s.004053">Cur <lb></lb>
cùm Luna ſemiplena eſt, linea illa, quæ terminus eſt partis illuminatæ, &amp; <lb></lb>
partis obſcuræ, <expan abbr="quæq́">quæque</expan>; Lunam bifariam diuidit, videtur linea recta, cùm ta­<lb></lb>
men non ſit; cùm enim fit in globoſa ſuperficie Lunæ, neceſſariò circularis <lb></lb>
<figure id="id.009.01.237.1.jpg" place="text" xlink:href="009/01/237/1.jpg"></figure><lb></lb>
eſt? </s>
          <s id="s.004054">vt autem rem hanc melius intelli­<lb></lb>
gamus, præſens figura illuminationis <lb></lb>
Lunæ inſpicienda eſt: vbi oculus noſter <lb></lb>
eſt in centro mundi A; vnde varias Lu­<lb></lb>
næ illuminationes aſpicit: è quibus <lb></lb>
octo tantum, in figura ſunt depictæ: in <lb></lb>
quibus videre eſt Lunæ ſemper dimi­<lb></lb>
dium illud, ſiue hemiſphærium, quod <lb></lb>
Solem aſpicit, eſſe illuminatum, cuius <lb></lb>
terminus, ſiue baſis eſt linea K L, <expan abbr="eſtq́">eſtque</expan>; <lb></lb>confinium illuſtratæ partis, &amp; opacæ. <lb></lb>
</s>
          <s id="s.004055">quæ linea ſemper in Luna eſt circula­<lb></lb>
ris, cum ſit in ſphęrico corpore: quan­<lb></lb>
do tamen Luna videtur ſemiplena, vt <lb></lb>
<expan abbr="quãdo">quando</expan> eſt in D, vel in K. hæc linea K L, <lb></lb>
videtur recta. </s>
          <s id="s.004056">ratio huius eſt, quia exi­<lb></lb>
ſtente Luna ſemiplena, circulus K D L, <lb></lb>
qui eſt baſis illuminationis ſolaris, eſt in eodem plano cum oculo A, ſiue in <lb></lb>eadem rectitudine, vt apparet in figura, vbi, fi linea K D L, ſumatur loco <pb pagenum="238" xlink:href="009/01/238.jpg"></pb>diametri prædicti circuli, &amp; produci intelligatur verſus oculum A, per ip­<lb></lb>ſum tranſilit; quo in ſitu, ſi circulus oculo ſubijciatur, non planam ipſius <lb></lb>
ſuperficiem, ſed circunferentiam <expan abbr="tantũ">tantum</expan> aſpicit, <expan abbr="fitq́">fitque</expan>; vt non lineam curuam, <lb></lb>
ſed rectam videre videatur, vt in præcedenti problemate diximus, &amp; Per­<lb></lb>
ſpectiui demonſtrant, &amp; Vitellio lib. 4. propoſit. </s>
          <s id="s.004057">5. &amp; propoſit. </s>
          <s id="s.004058">50.</s>
        </p>
        <p type="main">
          <s id="s.004059">Quidquid porrò ſphæram aſpexerit, neceſſariò ita illam aſpicit, vt quod <lb></lb>
de ipſa videt, ſit orbiculare, cum ergò Sol Lunam aſpiciat, ſiue illuminet, <lb></lb>
debet illuminatio illa eſſe orbicularis, hoc eſt habere orbicularem baſim, vt <lb></lb>
in figura patet, in qua Sol aſpiciens Lunam, quamuis in diuerſis locis poſi­<lb></lb>
tam, eius tamen ſemper dimidium illuſtrat, cuius dimidij baſis eſt circula­<lb></lb>
ris, <expan abbr="repreſentaturq́">repreſentaturque</expan>; in lineis K B L, K C L, K D L, &amp; cęteris ſimilibus, quan­<lb></lb>
do igitur Luna eſt in tali poſitione, vt totus ille orbis illuminationis oculis <lb></lb>
noſtris in A, poſitis obijciatur, totus vna cum tota illuminatione conſpici­<lb></lb>
tur, vt accidit, quando Luna eſt in F. <expan abbr="tuncq́">tuncque</expan>; eſt oppoſita diametraliter So­<lb></lb>
li, <expan abbr="eſtq́">eſtque</expan>; Plenilunium. </s>
          <s id="s.004060">Cùm autem Luna vetus mutatur in nouam, receden­<lb></lb>
do à Sole, vt quando tranſit à B, in C, tunc <expan abbr="circũferentia">circunferentia</expan> K B L, prædicti or­<lb></lb>
bis, quæ Luna in B, exiſtente, videri non poterat, incipit videri quando fue­<lb></lb>
rit in C. <expan abbr="cerniturq́">cerniturque</expan>; pars illius illuminationis circa punctum L, quæ videtur <lb></lb>
falcata; quæ pars recedente adhuc magis Luna à Sole, ſemper augetur, ideſt <lb></lb>
ſemper maior illuminationis pars cernitur: ita vt cùm fuerit in D, ſemiple­<lb></lb>
na appareat, &amp; linea K D L, quæ ibi orbicularis eſt, oculo in A, videtur re­<lb></lb>
ctà, ob cauſam ſuperius dictam; tunc igitur lumen Lunæ ex vna parte vide­<lb></lb>
tur terminari linea recta, ex altera circulari, ita vt figura luminis ſit ſemi­<lb></lb>
circulus. </s>
          <s id="s.004061">Porrò Luna ſemper ex ſe oculis noſtris opponitur, quamuis non <lb></lb>
ſemper cernatur, vt accidit in Nouilunio, quando ſcilicet Luna eſt infra <lb></lb>Solem in B, quia cum Sol ſit ſupra ipſam, illuminat hemiſphærium eius ſu­<lb></lb>
perius, quod oculo in A, eſt auerſum; &amp; ideò videri nequit; poſtea paula­<lb></lb>
tim recidendo à Sole, incipit hemiſphærium illuſtratum ad oculum A, ver­<lb></lb>
gere, &amp; ideo conſpici, <expan abbr="ſicq́">ſicque</expan>; primo apparet lunularis, ſeu falcata, deinde mi­<lb></lb>
nus, ac minus falcata, quia linea interior falcis minus curuatur, &amp; ſectio­<lb></lb>
nem conicam, quam oualem dicunt, refert: deinde magis ad rectitudinem <lb></lb>
accedit, ita vt circa octauum diem, ſeu circa primum Lunæ quadrantem, <lb></lb>linea illa videatur recta, Luna autem dixotomos, ſeu dimidiata; vbi enim <lb></lb>
circunferentia illuminationis Solis, ad puncta deuenit vltima, per quæ Lu­<lb></lb>
na bifariam diuiditur, apparet tantum oculo circunferentia illius, &amp; nullo <lb></lb>
modo ipſum circuli planum, qui baſis eſt: ſed, vt ſupra etiam dictum eſt, <lb></lb>
planum eius productum ſecaret oculum in A, exiſtentem, &amp; ſtatim ab hoc <lb></lb>
ſitu mutatur, &amp; præterit, quod cùm fit, neceſſe eſt, vt prædictus circulus per <lb></lb>ſumma puncta K L, deſignatus, non amplius recta linea, ſed curua, &amp; lunu­<lb></lb>
laris appareat, quia aliquo modo planum prædicti circuli ad oculos incli­<lb></lb>
natur, priori tamen circunferentia ex aduersò oculorum, vt dictum eſt, exi­<lb></lb>
ſtente, <expan abbr="atq;">atque</expan> hoc modo ex inclinatione baſis ad oculum aliquid lucis amplius <lb></lb>
reſecatur, ideſt ab oculo cernitur. </s>
          <s id="s.004062">tum etiam extrema huius circunferentiæ <lb></lb>in codem perſiſtunt, ideſt in eiſdem punctis, &amp; propterea linea illa magis, <lb></lb>&amp; minus incuruatur pro Solis remotione; ita vt tandem reuertatur ad ea­<lb></lb>
dem puncta. </s>
          <s id="s.004063">fieri enim poteſt, vt infinitas inclinationes ſuſcipiat, ſi quidem <pb pagenum="239" xlink:href="009/01/239.jpg"></pb>per eadem duo extrema puncta K, L, duci poſſunt infiniti circuli maximi. <lb></lb>
</s>
          <s id="s.004064"><expan abbr="Atq;">Atque</expan> hæc eſt Ariſtotelis ſententia, non ſine ingrata tautologia, tandem <expan abbr="vt-cunq;">vt­<lb></lb>
cunque</expan> expreſſa.</s>
        </p>
        <p type="main">
          <s id="s.004065"><arrow.to.target n="marg339"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004066"><margin.target id="marg339"></margin.target>347</s>
        </p>
        <p type="main">
          <s id="s.004067">In 7. problema <emph type="italics"></emph>(Cur Sol, &amp; Luna plana eſſe videntur, cùm tamen ſphærica <lb></lb>
ſint? </s>
          <s id="s.004068">An, vt ea omnia, quorum quodnam plus, minuſuè, diſtet, incertum ſit, æquè <lb></lb>
poſita eſſe videntur? </s>
          <s id="s.004069">ſic etiam res, quamuis vna, cùm plures tamen habeat partes, <lb></lb>
niſi varius color adſit, partes illæ omnes, ex æquo collocatas videri neceſſe eſt: quod <lb></lb>
autem ex æquo videtur, neceſſarium etiam eſt æquabile, ac planum apparere)<emph.end type="italics"></emph.end><lb></lb>
Quæſtionem hanc demonſtratiuè pertractat Vitellio lib. 4. propoſit. </s>
          <s id="s.004070">65. Eu­<lb></lb>
clides etiam theor. </s>
          <s id="s.004071">25. optices. </s>
          <s id="s.004072">cæterùm textus ſatis clarus videtur: vbi au­<lb></lb>
tem ait <emph type="italics"></emph>(niſi varius color adſit)<emph.end type="italics"></emph.end> hoc ait, quia nonnulli colores ſunt, qui fa­<lb></lb>
ciunt, vt obiecta appareant prominentiora, &amp; proinde propinquiora; ta­<lb></lb>
les ſunt colores, qui præ cæteris Iucidiores ſunt: alij verò ſunt, qui obiecta <lb></lb>
deprimunt, &amp; proinde remouent; cuiuſmodi ſunt colores omnes tenebri­<lb></lb>
coſi. </s>
          <s id="s.004073">poſito igitur in re viſa eodem colore, partes illius ob magnam diſtan­<lb></lb>
tiam <expan abbr="vidẽtur">videntur</expan> æqualiter à viſu diſtare, &amp; ideo res plana apparet. </s>
          <s id="s.004074">quia, quam­<lb></lb>
uis diſtantiæ illæ partium ab oculo ab inuicem differant, tamen parum dif­<lb></lb>
ferunt, idcircò eas ſenſus iudicat æquales, <expan abbr="ſicq́">ſicque</expan>; æqualiter iudicamus diſtare <lb></lb>
partes remotiſſimæ ſphæræ, quamuis pars illa, cui linea viſualis perpendi­<lb></lb>
culariter accidit, ſit propinquior; ſiue illa, quæ eſt in medio hemiphærij viſi: <lb></lb>
partes autem, quæ ſunt circa baſim dicti hemiſphærij ſint remotiores. </s>
          <s id="s.004075">reli­<lb></lb>
qua ex ſe manifeſta ſunt.</s>
        </p>
        <p type="main">
          <s id="s.004076"><arrow.to.target n="marg340"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004077"><margin.target id="marg340"></margin.target>348</s>
        </p>
        <p type="main">
          <s id="s.004078">In 8. problema <emph type="italics"></emph>(Cur Sol oriens, <expan abbr="etq;">etque</expan> occidens vmbras efficit longas; efferens <lb></lb>
ſe, minores: obtinens cœli medium minimas? </s>
          <s id="s.004079">An quod oriens primo vmbram ter­<lb></lb>
ræ æquidiſtantem reddit, ac infinitam pęnè protrahit, deinde longam, &amp; poſtea mi­<lb></lb>
norem ſubinde? </s>
          <s id="s.004080">quia linea recta, quæ de ſuperiori puncto elicitur, interius cadit.<emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.239.1.jpg" place="text" xlink:href="009/01/239/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>ſit Gnomon A B. Sol, vbi C, &amp; vbi D. <lb></lb>
radius igitur ex C, preficiſcens, eſt C F, <lb></lb>&amp; exterius procedit, quàm radius D E. <lb></lb>
eſt autem vmbra B E, Sole ſublimiori <lb></lb>
exiſtente: vmbra verò B F, Sole humi­<lb></lb>
liori. </s>
          <s id="s.004081">ergò quò Sol altior fuerit, eò mi­<lb></lb>
nor vmbra erit, minimaqué tunc erit, cum <lb></lb>
Sol ſuper caput noſtrum verſabitur)<emph.end type="italics"></emph.end><lb></lb>
Problema præſens eſt idem <expan abbr="cũ">cum</expan> quar­<lb></lb>
to huius ſectionis: eadem igitur ex­<lb></lb>
poſitio <expan abbr="vtriq;">vtrique</expan> inſeruiat. </s>
          <s id="s.004082">hoc ſolum addendum eſt, Gnomonem apud græcos <lb></lb>
inter cætera ſignificare ſtylum ſolaris horologij: in quo ſenſu hoc loco po­<lb></lb>
nitur. </s>
          <s id="s.004083">ſignificat præterea amuſſim, ſeu normam, quæ nihil aliud eſt, quam <lb></lb>
quidam angulus rectus materialis: &amp; quoniam ſtylus horologij figitur ad <lb></lb>
angulos rectos in plano horizontali, propterea ipſe <expan abbr="quoq;">quoque</expan> Gnomon appel­<lb></lb>
latus eſt: imò <expan abbr="pleriq;">plerique</expan> amuſſim quandam horologijs præſertim viatorijs, lo­<lb></lb>
co ſtyli accommodant.</s>
        </p>
        <p type="main">
          <s id="s.004084"><arrow.to.target n="marg341"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004085"><margin.target id="marg341"></margin.target>349</s>
        </p>
        <p type="main">
          <s id="s.004086">In 9. <emph type="italics"></emph>(Cur vmbræ Lunæ maiores, quam Solis ſunt cùm eodem proueniant per­<lb></lb>
pendiculo? </s>
          <s id="s.004087">An quod Sol ſuperior, quam Luna eſt? </s>
          <s id="s.004088"><expan abbr="itaq;">itaque</expan> neceſſe eſt radium à ſupe­<lb></lb>
riore procedentem intus cadere. </s>
          <s id="s.004089">ſit Gnomon A D, Luna B Sol C, Lunæ radius B F.<emph.end type="italics"></emph.end>


<pb pagenum="240" xlink:href="009/01/240.jpg"></pb><figure id="id.009.01.240.1.jpg" place="text" xlink:href="009/01/240/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>ergò vmbra Lunæ D F, Solis radius <lb></lb>
C E; vmbra igitur neceſſariò minor <lb></lb>
eſt, eſt enim D E.)<emph.end type="italics"></emph.end> de hac re vide <lb></lb>
Spheram P. Clauij, cap. de Ordine <lb></lb>
cœlorum, vnde huius textus expo­<lb></lb>
ſitionem in hunc modum licebit af­<lb></lb>
ferre. </s>
          <s id="s.004090">Quando igitur Ariſt. quærit, <lb></lb>
cur Luna maiores proijciat vm­<lb></lb>
bras, qua Sol, debet ſupponere So­<lb></lb>
lem, &amp; Lunam eſſe in eadem altitu­<lb></lb>
dine ſupra horizontem, v. g. eſſe <expan abbr="vtrunq;">vtrunque</expan> in eadem linea recta C D, ducta à <lb></lb>
centro mundi ad Solem; ſic enim habebunt eandem ambo eleuationem ſu­<lb></lb>
pra horizontem G H, vt factum eſt in figura: aliter Sol etiam faciet vmbras <lb></lb>
Lunæ vmbris modo longiores, modo breuiores. </s>
          <s id="s.004091">cur igitur, inquit, exiſtente <lb></lb>
<expan abbr="vtroq;">vtroque</expan> in eadem altitudine, ſiue in eadem recta C D. vmbræ lunares factæ à <lb></lb>
Gnomone D A, ſunt vmbris ſolaribus longiores?</s>
        </p>
        <p type="main">
          <s id="s.004092">Reſpondet id fortè accidere, quia Sol multò remotior ſit à centro mun­<lb></lb>
di D, quàm Luna. </s>
          <s id="s.004093">vnde quamuis æquè ſupra horizontem ſint eleuata, cum <lb></lb>
ambo ſint in linea C D. tamen propter Lunæ propinquitatem ad centrum <lb></lb>
mundi, fit vt magnitudo ſtyli D A, reſpectu Lunæ ſit valdè ſenſibilis, quæ ta­<lb></lb>
men collata <expan abbr="cũ">cum</expan> Sole à terra remotiſſimo nullius euadit ſenſibilitatis; <expan abbr="idemq́">idemque</expan>; <lb></lb>
eſt punctum D, ac punctum A. ex quo fit, vt radius Lunæ B F, cadat extra <lb></lb>
radium Solis C E: <expan abbr="hincq́">hincque</expan>; rurſus neceſſe eſt vmbram Solis D E, minorem eſ­<lb></lb>
ſe vmbra Lunæ D F. </s>
          <s id="s.004094">Quod ſi concipiamus Lunam magis à terris diſtantem, <lb></lb>
<expan abbr="Soliq́">Solique</expan>; propinquiorem, vt in puncto R, tunc amborum radij ſimul ferè co­<lb></lb>
incident: <expan abbr="ſicq́">ſicque</expan>; æquales ferè vmbræ <expan abbr="vtriuſq;">vtriuſque</expan> erunt.</s>
        </p>
        <p type="main">
          <s id="s.004095">Huius rei facilè experientiam facere poteris, ſi per quadrantem Solis al­<lb></lb>
titudine notata, ſimul etiam ipſius vmbram ex quopiam magno Gnomone <lb></lb>
proiectam obſeruaueris: deinde eadem Lunæ ſplendentis altitudine per cal­<lb></lb>
culum obſeruata vmbram lunarem eiuſdem Gnomonis cum vmbra ſolari <lb></lb>
contuleris: inuenies enim lunarem ſolari longiorem.</s>
        </p>
        <p type="main">
          <s id="s.004096"><arrow.to.target n="marg342"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004097"><margin.target id="marg342"></margin.target>350</s>
        </p>
        <p type="main">
          <s id="s.004098">In 10. problem. <emph type="italics"></emph>(Propter quid in Solis eclypſibus, ſi quis ſpectet per cribrum, <lb></lb>aut per folium, veluti platani, vel alterius latifolij, vel per digitos altera manu <lb></lb>
ſuper alteram coniungens ſplendores, qui in terra fiunt ſunt lunulæ? </s>
          <s id="s.004099">An quod ſicu­<lb></lb>ti lux per foramen anguloſum ſplendens, turbo, &amp; conus fit: cauſa verò, quia duo <lb></lb>
efficiuntur coni, vnus à Sole ad foramen, &amp; alter hinc ad terram, qui ſimul <expan abbr="habẽt">habent</expan> <lb></lb>
vertices. </s>
          <s id="s.004100">quando igitur ſic ſe habet; &amp; ſuperiori parte circulari detrahitur, erunt <lb></lb>
è contrariò lucis lunulæ in terra; ex peripheria enim lunulari procedunt radij. </s>
          <s id="s.004101">quæ <lb></lb>
autem in digitis, aut cribris, veluti foramina fiunt, manifeſtius id faciunt, quàm <lb></lb>
magna foramina. </s>
          <s id="s.004102">A Luna autem hoc non fit, <expan abbr="neq;">neque</expan> ipſa deficiente, <expan abbr="neq;">neque</expan> creſcente, <lb></lb>
<expan abbr="neq;">neque</expan> decreſcente, quia ſplendores extremitatum eius non ſunt manifeſti, &amp; certi. <lb></lb>
</s>
          <s id="s.004103">ſed in medio potiſſimum <expan abbr="ſplẽdet">ſplendet</expan>. </s>
          <s id="s.004104">lunula autem falcata exiguam habet latitudinem)<emph.end type="italics"></emph.end><lb></lb>
Vt rectè, atque non ſine delectatione problema præſens intelligas, lege ea, <lb></lb>
quæ in additione ad problema 5. huius ſectionis ſcripſimus de Figuratione <lb></lb>
lucis: deinde operæpretium erit audire, quid dicat Gemma Friſius in tra­<lb></lb>ctatu de Radij aſtronomici ſtructura, cap. 18. vbi loquitur de Solis deliquij <pb pagenum="241" xlink:href="009/01/241.jpg"></pb>dimenſione, his verbis, extat, inquit, alius modus dimetiéndæ ſolaris eclyp­<lb></lb>
ſis, omnium facillimus, ac certiſſimus, cuius nos admonuit Eraſmus Rei­<lb></lb>
noldus in comm. in Theoricas Peurbarhij. </s>
          <s id="s.004105">tempore igitur ſolaris defectus, <lb></lb>
intra parietes vſpiam, clauſis omnibus feneſtris, admittatur Solis radius, <lb></lb>
per anguſtum foramen rotundum; <expan abbr="excipiaturq́">excipiaturque</expan>; radius hic in plana tabella, <lb></lb>
vbi certo quantum Sol defecerit ad vnguem, licet videre, <expan abbr="abſq;">abſque</expan> vlla intui­<lb></lb>tus moleſtia, ac tam perfectè, atque fi in cœlo coram adeſſes) hæc ille, licet <lb></lb>
autem videre, quia illuminatio in tabella excepta, quæ alias ſolet eſſe cir­<lb></lb>
cularis, erit tempore eclypſis ipſa pariter cum Sole defectiua, <expan abbr="atq;">atque</expan> inſtar fal­<lb></lb>
catæ lunulæ. </s>
          <s id="s.004106">deinde ſubdit; verum hoc omninò ſcire neceſſarium eſt, con­<lb></lb>
trario modo apparere defectum illum in tabula per radios Solis, quàm in <lb></lb>
cœlo contingit: hoc eſt, ſi in cœlo ſuperior pars deliquium patiatur, in ra­<lb></lb>
dijs apparebit inferior deficere, vt ratio exigit optica) <expan abbr="hucuſq;">hucuſque</expan> Gemma Fri­<lb></lb>
ſius; ex quo etiam placuit accipere totius huius <expan abbr="experiẽtiæ">experientiæ</expan> figuram, quam <lb></lb>
<figure id="id.009.01.241.1.jpg" place="text" xlink:href="009/01/241/1.jpg"></figure><lb></lb>
ipſe <expan abbr="cuiuſdã">cuiuſdam</expan> eclypſis an­<lb></lb>
ni 1544. apponit. </s>
          <s id="s.004107">eſt au­<lb></lb>
tem hæc. </s>
          <s id="s.004108">in qua Sol <expan abbr="de-ficiẽs">de­<lb></lb>
ficiens</expan> eſt A B C. pars in­<lb></lb>
ferior B D P C, ipſa eſt <lb></lb>
lumine priuata; ſuperior <lb></lb>
B A P D, ſplendens, quæ <lb></lb>
ſimilis eſt falcatæ lunu­<lb></lb>
læ. </s>
          <s id="s.004109">radij Solis <expan abbr="ingrediũ-tur">ingrediun­<lb></lb>
tur</expan> in cubiculum per fo­<lb></lb>
ramen E. <expan abbr="excipiunturq́">excipiunturque</expan>; <lb></lb>
in tabella K M N O, fo­<lb></lb>
ramini, ſeu radio Solis <lb></lb>
<expan abbr="perpẽdiculariter">perpendiculariter</expan> oppo­<lb></lb>
ſita: in qua propterea <lb></lb>
apparet Solis illuminatio, non vt alias circularis, ſed manca, ac defectiua, <lb></lb>
lunulæ inſtar: <expan abbr="eſtq́">eſtque</expan>; G H I L, quæ inuerſo modo ſe habet, ac in cœlo, quem­<lb></lb>
admodum figura oſtendit: cuius cauſa eſt, quia radij Solis A E, D E, C E, <lb></lb>
poſt foraminis ingreſſum commutantur, quia ſe mutuò ſecant, vnde qui ſu­<lb></lb>
periores erant, fiunt inferiores intra foramen, &amp; in tabella; ſic radius A E, <lb></lb>
omnibus ſuperior, poſt ingreſſum fit omnibus inferior; eſt enim E H, <expan abbr="deſi-nitq́">deſi­<lb></lb>
nitque</expan>; in puncto H, omnium illuminationis infimo. </s>
          <s id="s.004110">reliqua autem pars cir­<lb></lb>
culi illuminationis G L I F, deficit, quia pars Solis B D P C, quæ ipſam illu­<lb></lb>ſtrare ſolet, propter eclypſim nullos per foramen E, immittit radios. </s>
          <s id="s.004111">Ve­<lb></lb>
rum eclypſis tempore, etiam ſi huiuſmodi illuminationes intra cubiculum <lb></lb>
non obſeruentur, ſed foris, manifeſtè omnes apparent, non ſecus, ac ipſe ſol <lb></lb>
defectiuæ: tales ſùnt omnes, quæ per quælibet foramina, in quolibet paui­<lb></lb>mento, aut oppoſito pariete apparent: de quibus etiam Ariſtot. in præſenti <lb></lb>
problemate loquitur. </s>
          <s id="s.004112">ex his facilè eſt verborum Ariſt. ſenſum aſſequi. </s>
          <s id="s.004113">Quæ­<lb></lb>
rit igitur, cur tempore ſolaris deliquij, ſi Solis illuminationes per cribri fo­<lb></lb>
ramina, aut inter alicuius arboris folia, ex ijs, quæ lata habent folia, aut <lb></lb>
inter manuum decuſſatos digitos, ingredientes, atque in terra apparentes,


<pb pagenum="242" xlink:href="009/01/242.jpg"></pb>ſpectemus, eas falcatas, ac lunulatas, videamus; non autem, vt ſolemus, <lb></lb>
rotundas. </s>
          <s id="s.004114">Reſpondet, id fortè accidere, quia lux per foramen intrans, fit <lb></lb>
conus natura ſua, vt in 5. problemate pręcedenti, explicatum eſt. </s>
          <s id="s.004115">&amp; in præ­<lb></lb>
ſenti figura conus lucis intrantis per foramen E, figuratur à lineis E F, E H, <lb></lb>
quibus ſimiles alias plurimas debemus concipere ab E, ad circularem ba­<lb></lb>
ſim F G L I, quæ turbinem perfectum efficiunt. </s>
          <s id="s.004116">alius præterea conus eſt à fo­<lb></lb>
ramine ad Solem, cuius baſis eſt A B C P, circulus Solis: &amp; continetur ſub <lb></lb>
infinitis radijs, quorum duo ſunt A E, C E; <expan abbr="vterq́">vterque</expan>; autem habet verticem ad <lb></lb>
E, quia igitur plures radij ſuperioris coni deficiunt, ideò etiam in inferiori <lb></lb>
deficient: <expan abbr="eritq́">eritque</expan>; ſitus eorum inuerſus ob radiorum interſectionem ad pun­<lb></lb>
ctum E, <expan abbr="eritq́">eritque</expan>; ſplendor in tabella apparens lunulatus, quia ex parte Solis <lb></lb>
pariter lunulata producitur. </s>
          <s id="s.004117">cætera ſatis ſunt per ſe clara.</s>
        </p>
        <p type="head">
          <s id="s.004118"><emph type="italics"></emph>Ex Sectione 16.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.004119"><arrow.to.target n="marg343"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004120"><margin.target id="marg343"></margin.target>351</s>
        </p>
        <p type="main">
          <s id="s.004121">In 1. problema <emph type="italics"></emph>(Cur baſes bullarum in aquis ſunt albæ; &amp; ſi in Sole ponan­<lb></lb>
tur, non faciunt vmbram; ſed bullæ reliquum vmbram facit, baſis verò non <lb></lb>
facit, ſed circulariter à Sole illuminatur. </s>
          <s id="s.004122">quod verò mirabilius eſt, quod <expan abbr="neq;">neque</expan> <lb></lb>
ſi quodpiam lignum in aquam inferatur in Sole, hæc ſub aqua diuiduntur. <lb></lb>
</s>
          <s id="s.004123">An non fit vmbra, ſed à Sole diſſipatur vmbra? </s>
          <s id="s.004124">fi igitur vmbra est non inſpectum, <lb></lb>&amp; à Sole circulariter inſpicitur moles: hoc verò impoſſibile eſſe oſtenditur in Op­<lb></lb>
ticis. </s>
          <s id="s.004125"><expan abbr="neq;">neque</expan> enim minimum, à maximo totum conſpici poteſt)<emph.end type="italics"></emph.end> Cùm ex ipſius textus <lb></lb>verbis ſatis perſpicuè appareat, quid proponatur, reliqua ſic breuiter ex­<lb></lb>
ponam. </s>
          <s id="s.004126">quod igitur de ligno ait, exiſtimo hoc modo <expan abbr="accipiendũ">accipiendum</expan>, vt lignum <lb></lb>
illud in aqua ponatur ſub bulla, ita vt vmbra bullæ cadat ſuper ipſum, <expan abbr="tũcq́">tuncque</expan>; <lb></lb>
vmbra illius ſimiliter apparebit defectiua, quia baſis illuminatio ipſam ex <lb></lb>
parte deſtruet. </s>
          <s id="s.004127">Reſpondet, An non fit vmbra, ſed à Sole vmbra fugatur? <lb></lb>
</s>
          <s id="s.004128">quæ verba ſubobſcura ſunt; <expan abbr="neq;">neque</expan> reſponſio videtur allata ad ſoluendum pro­<lb></lb>
blema, ſed ad eum magis confirmandum. </s>
          <s id="s.004129">deinde ait: ſi igitur nihil aliud eſt <lb></lb>
vmbra, quam id, quod non aſpicitur à Sole, &amp; à Sole tamen videamus illu­<lb></lb>
minari totam bullæ baſim circulariter, neceſſe eſt totam etiam bullam <expan abbr="vn-diq;">vn­<lb></lb>
dique</expan> à Sole illuminari, &amp; conſpici, quod tamen impoſſibile eſſe demonſtra­<lb></lb>
tur ab opticis: ipſi enim demonſtrant, nullum corpus, quantumuis mini­<lb></lb>
mum, totum poſſe circumſpici à quamuis maximo illuminante. </s>
          <s id="s.004130">quod qui­<lb></lb>
dem antiquitus demonſtrauit Ariſtarchus Samius in libello de diſtantijs So­<lb></lb>
<figure id="id.009.01.242.1.jpg" place="text" xlink:href="009/01/242/1.jpg"></figure><lb></lb>
lis, &amp; Lunæ: &amp; <lb></lb>
poſtea Vitellio <lb></lb>
lib. 2. propoſ. </s>
          <s id="s.004131">27. <lb></lb>
&amp; ex figura præ­<lb></lb>
ſenti facilè eſt id <lb></lb>
intelligere: <expan abbr="ĩ">i</expan>n qua <lb></lb>
ſit Sol ſphæra A, <lb></lb>
illuminans ſphæ­<lb></lb>
rulam B, extre­<lb></lb>
mi radij DF, EF.


<pb pagenum="243" xlink:href="009/01/243.jpg"></pb>vmbra erit igitur G F H, ad partes C, Soli auerſas. </s>
          <s id="s.004132">quas nunquam Sol, etiam <lb></lb>
ſi ſphæra B, arenulæ vnius grano minor fuerit poterit illuſtrare. </s>
          <s id="s.004133">quæ quidem <lb></lb>
non ſoluunt quæſtionem, ſed eam difficiliorem reddunt. </s>
          <s id="s.004134">Quapropter non <lb></lb>
videtur Ariſt. voluiſſe hoc diſcutere, ſed ſolum tanquam mirum quodam <lb></lb>
proponere. </s>
          <s id="s.004135">quod ſi quid mutire liceat, vbi tantus philoſophus admirabun­<lb></lb>dus obmuteſcit, dixerim propterea baſim bullæ non adumbrari ab vmbra <lb></lb>
ipſius bullæ, quia cum bulla ſit ſphærica, &amp; tranſparens, Solis lumen eam <lb></lb>
peruadit, <expan abbr="atq;">atque</expan> ex ſuperficie concaua ad illius baſim partim reflectitur, <expan abbr="ſicq́">ſicque</expan>; <lb></lb>
eam illuminat. </s>
          <s id="s.004136">quamuis enim ſit diaphana, non tamen omninò tranſparens <lb></lb>
eſt, cum aqua ſit aere craſſior: bulla autem eſt ex aqua. </s>
          <s id="s.004137">ſuperficiem autem <lb></lb>
bullæ concauam id efficere debere, patet ex concaua figura, quæ maximè <lb></lb>
reflexioni eſt apta.</s>
        </p>
        <p type="main">
          <s id="s.004138"><arrow.to.target n="marg344"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004139"><margin.target id="marg344"></margin.target>352</s>
        </p>
        <p type="main">
          <s id="s.004140">In 3. problem. <emph type="italics"></emph>(Cur in magnitudinibus, quæ pondere ſunt inæquali, accidit, vt <lb></lb>
ſi partem moueas læuiorem, circunferatur, quod iacitur; vt in talis fieri opplum­<lb></lb>
batis videmus)<emph.end type="italics"></emph.end> Ariſtotelis tempore tales tali opplumbati erant in vſu, qui <lb></lb>
exemplo præſenti queſtioni eſſe poſſent: Aptius nunc exemplum deſumi po­<lb></lb>
teſt ex bacillo aliquo, cuius altera extremitas ſit cæteris partibus multò <lb></lb>
grauior, qui ſi per aerem manibus eiaculatur, ſolet, dum per aerem fertur, <lb></lb>
circumuerti.</s>
        </p>
        <p type="main">
          <s id="s.004141"><arrow.to.target n="marg345"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004142"><margin.target id="marg345"></margin.target>353</s>
        </p>
        <p type="main">
          <s id="s.004143">Ibidem <emph type="italics"></emph>(Sin autem alterum altero fertur cælerius, circulo ferri neceſſe eſt, cùm <lb></lb>
in hoc ſolo figuræ genere efficiatur, vt puncta eadem ſubalterna, lineas inæquales <lb></lb>poſſint eodem tempore permeare)<emph.end type="italics"></emph.end> Quando, inquit, duo puncta in eadem magni­<lb></lb>
tudine poſita mouentur ad motum illius, &amp; tamen non æqualiter progre­<lb></lb>
diuntur, ſignum eſt, illam magnitudinem moueri circulariter, &amp; proinde <lb></lb>
vel eſſe circulum, vel ſaltem circuli in modum conuerti; cum in ſolo orbi­<lb></lb>
culari motu contingat, vt duo puncta inæqualiter a centro remota, poſſint <lb></lb>
inæquales lineas eodem tempore permeare, punctum enim, quod <expan abbr="cẽtro">centro</expan> pro­<lb></lb>
pinquius eſt, breuiorem deſcribit lineam, quod autem remotius, maiorem.</s>
        </p>
        <p type="main">
          <s id="s.004144"><arrow.to.target n="marg346"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004145"><margin.target id="marg346"></margin.target>354</s>
        </p>
        <p type="main">
          <s id="s.004146">In 4. problem. </s>
          <s id="s.004147">ſatis eſſe exiſtimo per paraphraſim præſens problema ex­<lb></lb>
ponere, ex qua tamen, vbi opus fuerit, textus corrigatur. </s>
          <s id="s.004148">Cur ea, quæ in <lb></lb>
terram cadunt, <expan abbr="atq;">atque</expan> reſiliunt angulos ad planitiem, faciunt ſimiles vtraque <lb></lb>
<figure id="id.009.01.243.1.jpg" place="text" xlink:href="009/01/243/1.jpg"></figure><lb></lb>
ex parte, qua planum tetigerint? </s>
          <s id="s.004149">v. g. ſi <lb></lb>
corpus quodpiam cadat ex puncto D, per <lb></lb>
lineam D C, ſuper planum A B, ex puncto <lb></lb>
C, vbi cæciderat, reſilit per lineam C E, <lb></lb>
ita vt faciat duos angulos æquales vtrin­<lb></lb>
que ad punctum. </s>
          <s id="s.004150">C, angulum ſcilicet in­<lb></lb>
cidentiæ D C B, &amp; angulum reflexionis <lb></lb>
E C A? </s>
          <s id="s.004151">An quod omnia iſta, natura qui­<lb></lb>
dem ſua feruntur per rectam lineam, vi­<lb></lb>
demus enim grauia omnia deorſum re­<lb></lb>
ctà tendere; ſi autem aliquod impedi­<lb></lb>
mentum occurrat, vt fit, quando plano <lb></lb>
terræ occurrunt, tunc lineam illam, quam infra terram facerent <expan abbr="eundemq́">eundemque</expan>; <lb></lb>
angulum, quem infra <expan abbr="planũ">planum</expan> facerent, ſupra faciunt, v. g. mobile per lineam <lb></lb>D C, cadens, niſi obſtitiſſet planum A B, tetendiſſet per lineam rectam <pb pagenum="244" xlink:href="009/01/244.jpg"></pb>D C G, <expan abbr="feciſſetq́">feciſſetque</expan>; propterea angulum A C G, infra planum, æqualem angulo <lb></lb>
D C B, quo ceciderat. </s>
          <s id="s.004152">cum igitur nequeat prædictum angulum infra pla­<lb></lb>
num conficere, par eft, vt eum reſiliendo ſupra planum efficiat; propterea <lb></lb>
reſilit per lineam C E; quæ <expan abbr="angulũ">angulum</expan> A C E, ſupra cum plano conſtituit æqua­<lb></lb>
lem angulo A C G, infra, &amp; proinde æqualem <expan abbr="etiã">etiam</expan> angulo incidentiæ D C B. <lb></lb>
</s>
          <s id="s.004153">Duobus porrò modis grauia ſuper terræ planitiem cadunt; aut enim per­<lb></lb>
pendiculariter, &amp; fecundum mundi diametrum decidunt; aut obliquè, ſeu <lb></lb>
in latera. </s>
          <s id="s.004154">quæ igitur primo modo deſcendunt, ideſt perpendiculariter, ſeu <lb></lb>
quæ angulos rectos cum plano faciunt, ea etiam reſiliunt perpendiculariter, <lb></lb>ſeu ad angulos rectos, &amp; ideo neceſſariò per eandem lineam, qua decide­<lb></lb>
rant, repercutiuntur; cuius cauſa eſſe poteſt, quia diameter ſcilicet mundi, <lb></lb>
per quam delapſa ſunt, ea bifariam diuidit, vt in figura, graue E, per D C, <lb></lb>
<figure id="id.009.01.244.1.jpg" place="text" xlink:href="009/01/244/1.jpg"></figure><lb></lb>
plano A B, perpendicularem deſcendat; <lb></lb>
quæ <expan abbr="perpẽdicularis">perpendicularis</expan> coincidit cum mun­<lb></lb>
di diametro, perpendicula enim omnia <lb></lb>
ad mundi centrum tendunt; graue igi­<lb></lb>
tur E, dum puncto C, alliditur, diuidi­<lb></lb>
tur bifariam, à diametro mundi D C: <lb></lb>
vnde &amp; in æquilibrio conſtituitur, ita vt <lb></lb>
nulla ſit maior ratio, cur ad partem <lb></lb>
vnam, quàm ad alteram reſultet, &amp; ideo <lb></lb>
<expan abbr="conueniẽs">conueniens</expan> eſt, ipſum per eandem lineam <lb></lb>
D C, reuerti; ſic enim faciet etiam an­<lb></lb>
gulos incidentiæ, &amp; reflexionis æquales. <lb></lb>
</s>
          <s id="s.004155">quæ verò ſecundo modo decidunt, ideſt <lb></lb>
obliquè, &amp; ſecundum latera: quoniam non ſecundum perpendiculum, ſed <lb></lb>
ex puncto extra perpendiculum poſito, planum feriunt, accidit vt à puncto <lb></lb>
incidentiæ C, vt in priori figura, in contrariam partem repulſa reſulcent; <lb></lb>deſcenderant enim ex D, &amp; in contrariam partem, ſcilicet ad E, prioris fi­<lb></lb>
guræ reflectuntur. </s>
          <s id="s.004156">quòd ſi huiuſmodi grauia ſint rotunda, facilius in contra­<lb></lb>
rias partes exurgunt, propter ipſorum figuram motui, ac reſultationi ap­<lb></lb>tiſſimam; ſiue moueantur, ita vt centrum eorum etiam locum permutet, <lb></lb>
ſiue ita vt quieſcat. </s>
          <s id="s.004157">quæ verò non ſunt rotunda, ſed rectilinea, idem faciunt, <lb></lb>
quoniam perpendiculum ipſorum, ideſt linea, per quam deberent perpen­<lb></lb>
diculariter reſultare ob impulſum eliditur, &amp; flectitur ad altiorem partem, <lb></lb>
vbi nimirum eſt linea C E, in priori figura, ita vt à perpendiculo quodam­<lb></lb>
modo deflectantur. </s>
          <s id="s.004158">quemadmodum ij, quorum pars altera inferior, v. g. al­<lb></lb>
terum crus abſcinditur, qui coguntur à rectitudine priori in alteram par­<lb></lb>
tem, vel etiam retrorſum cadere: quando, vt dixi, eorum perpendiculum, <lb></lb>
quod perpendiculariter eleuatum eſt, &amp; ſecundum quod corpus ipſum de­<lb></lb>
beret æquari, ſeu in æquilubrio conſtitui, antrorſum pellitur. </s>
          <s id="s.004159">Porrò in his, <lb></lb>
quæ ob grauitatem deſcendunt; deorſum, &amp; retrorſum, opponuntur; deor­<lb></lb>
ſum enim eſt pars, quæ ad terram, anterior verò ea pars eſt, quæ ſurſum, ſeu <lb></lb>
retrorſum, vergit. </s>
          <s id="s.004160">quod igitur in his grauibus, ſiue rotundis, ſiue rectilineis <lb></lb>
eſt caſus ex vna parte, idem præſtat ex oppoſita parte latio, qua reſurgunt, <lb></lb>ideſt ad angulos pares fit caſus, &amp; latio: propterea neutra eorum ad rectum <pb pagenum="245" xlink:href="009/01/245.jpg"></pb>angulum repercutiuntur, <expan abbr="neq;">neque</expan> ſecundum perpendicularem, quia perpendi­<lb></lb>
culum eductum ex puncto caſus diuidit ea bifariam. </s>
          <s id="s.004161">fieri autem nequit, vt <lb></lb>
ad idem <expan abbr="punctũ">punctum</expan> C, plani A B. plura perpendicula erigantur ex 13. 11. Elem. <lb></lb>
(vt in ſecunda figura, ſola linea C D, perpendicularis eſſe poteſt ad idem, <lb></lb>
punctum C,) quibus perpendiculis grauia <expan abbr="diuidãtur">diuidantur</expan> bifariam, <expan abbr="atq;">atque</expan> in æqui­<lb></lb>
librio conſtituantur: quod tamen ſequeretur, ſi linea reflexionis eorum, quæ <lb></lb>
obliquè cadunt, eſſet perpendicularis, ab hac enim diuiderentur bifariam, <lb></lb>
&amp; præterea etiam ab alia, quæ perpendicularis verè eſt: &amp; præterea etiam <lb></lb>
à priori linea incidentiæ, quæ pariter eſſet perpendicularis, cum ſimiliter <lb></lb>
cadat, <expan abbr="atq;">atque</expan> reflexa, <expan abbr="diuiderẽtur">diuiderentur</expan> bipartitò. </s>
          <s id="s.004162">Quod abſurdum eſt. </s>
          <s id="s.004163">ſed cùm ad­<lb></lb>
uerſam quidem in partem <expan abbr="ferãtur">ferantur</expan>, &amp; non ad angulum rectum, reliquum eſt, <lb></lb>
vt ad acutum angulum reſiliant, ex altera puncti incidentiæ parte, quia an­<lb></lb>
gulus rectus eſt terminus, intra quem omnes anguli aduerſi <expan abbr="continẽtur">continentur</expan>, qua­<lb></lb>
les ſunt in prima figura ij, quorum vnum angulum incidentiæ, alterum re­<lb></lb>
flexionis appellant.</s>
        </p>
        <p type="main">
          <s id="s.004164">Notandum quoad verſionem latinam Theodori Gazæ, quod vbi ſunt ver­<lb></lb>
ba illa <emph type="italics"></emph>(Aut colei violantur)<emph.end type="italics"></emph.end> in græco eſt, <foreign lang="grc">κολυτρους ὐφαρπαζουσι.</foreign> quorum ver­<lb></lb>borum paraphraſim omiſi, quia theſauri, aut lexica græca nullam huius ver­<lb></lb>
bi notionem afferunt, quæ huic loco quadret, dicunt enim ex Athenæo <foreign lang="grc">τους <lb></lb>κολυθρους</foreign>, ſignificare ficus maturas: Suidas verò ait eſſe quoddam plantæ ge­<lb></lb>
nus: quorum neutrum ad rem facit. </s>
          <s id="s.004165">propterea vel textus corruptus eſt, vel <lb></lb>
metaphoricè vſus eſt Ariſt. hoc verbo, cuius metaphoræ modò intelligen­<lb></lb>
tiam non habemus.</s>
        </p>
        <p type="main">
          <s id="s.004166"><arrow.to.target n="marg347"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004167"><margin.target id="marg347"></margin.target>355</s>
        </p>
        <p type="main">
          <s id="s.004168">In 5. problem. <emph type="italics"></emph>(Cur Cylinder propulſus fertur in directum, ſuisqué terminanti­<lb></lb>
bus orbibus lineas rectas deſcribit, turbo verò ſuo manente murone circunfertur, <lb></lb>
at que in ſuo terminante orbe orbem deſcribit? </s>
          <s id="s.004169">&amp;c.)<emph.end type="italics"></emph.end> Ad huius textus intelligen­<lb></lb>
tiam fatis eſt noſſe, quid Cylindrus, &amp; quid Conus, ſiue turbo ſit. </s>
          <s id="s.004170">conum igi­<lb></lb>
tur ex definitione 8. 11. ſic poſſumus deſcribere, eſſe corpus ex vna parte <lb></lb>acuminatum, ex altera verò planum, quod planum dicitur baſis coni, <expan abbr="eſtq́">eſtque</expan>; <lb></lb>
circulus: vulgò appellatur Pyramis rotunda. </s>
          <s id="s.004171">Cylindrum verò ex definit. </s>
          <s id="s.004172">21. <lb></lb>
eiuſdem 11. ſic poſſumus explicare: eſſe corpus rotundum <expan abbr="oblongũ">oblongum</expan>, æqua­<lb></lb>
lis vtrinque craſſitiei, cuius duæ baſes ſunt circuii: <expan abbr="eſtq́">eſtque</expan>; veluti fruſtum co­<lb></lb>
lumnæ. </s>
          <s id="s.004173">his cognitis poteris nullo negotio totius problematis ſolutionem <lb></lb>
ex textu Ariſt. percipere.</s>
        </p>
        <p type="main">
          <s id="s.004174"><arrow.to.target n="marg348"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004175"><margin.target id="marg348"></margin.target>356</s>
        </p>
        <p type="main">
          <s id="s.004176">In 6. probl. <emph type="italics"></emph>(Cur voluminum ſectio plana, &amp; recta, ſi quidem fuerit baſi volu­<lb></lb>minis æquidiſtans, explicata lineam rectam oſtendit: ſi verò fuerit baſi inclinata, <lb></lb>tortuoſam? </s>
          <s id="s.004177">An quia accidit, vt circulis illius ſectionis in eodem plano existenti­<lb></lb>
bus, hanc quidem ſectionem non adiacentem eſſe. </s>
          <s id="s.004178">ſed hic quidem pius, illic verò mi­<lb></lb>
nus ab eadem distare. </s>
          <s id="s.004179">Ita vt explicato volumine circuli quidem ij, qui in eodem, <lb></lb>ſunt plano, &amp; principium habent in eodem plano, ex ſe ipſis euolutis faciět rectam <lb></lb>
lineam: eſt enim facta recta ex circulis, qui ſunt in eodem plano; ita vt etiam re­<lb></lb>
cta exiſtat in plano. </s>
          <s id="s.004180">At verò obliquæ illius ſectionis linea explicata, non exiſtens <lb></lb>primæ æquidiſtans, ſed hic quidem plus, illic verò minus ab ea recedens, propterea <lb></lb>quod etiam ipſa ſectio ita ſe habeat ad eandem non εrit in eodem plano: <expan abbr="itaq;">itaque</expan> <expan abbr="neq;">neque</expan> <lb></lb>
recta, <expan abbr="neq;">neque</expan> enim eiuſdem rectæ pars in vno plano, pars verò alia in alio plano eſſe <lb></lb>
poteſt)<emph.end type="italics"></emph.end> ſi Theodorus Gaza loco horum verborum <emph type="italics"></emph>(Cur ſectio chartarum, ſiue <emph.end type="italics"></emph.end>e <pb pagenum="246" xlink:href="009/01/246.jpg"></pb><emph type="italics"></emph>papyri)<emph.end type="italics"></emph.end> verſiſſet; cur voluminum ſectio, quemadmodum ego feci, quod, &amp; <lb></lb>
facere debebat, iuxta <expan abbr="græcorũ">græcorum</expan> <expan abbr="verborũ">verborum</expan> notionem, <foreign lang="grc">Δια τί τῶν βιβλίων ὴ τομη</foreign>, <lb></lb>
locum hunc non ſolum non obſcuraſſet, verum etiam clarum omninò red­<lb></lb>
didiſſet, eſt enim Problema de ſectione voluminis papyracei, quibus vete­<lb></lb>
res illi ſcribebant. </s>
          <s id="s.004181">quapropter optimè intelliges textum hunc, ſi huiuſmodi <lb></lb>
volumen bis ſecueris, primo quidem ſectione baſi voluminis parallela; ſe­<lb></lb>
cundo verò ſectione tranſuerſali, ſeu obliqua ad baſim: nam explicata pri­<lb></lb>
ma ſectione apparebit eam eſſe lineam rectam: euoluta verò <expan abbr="ſecũda">ſecunda</expan> ſectio­<lb></lb>ne apparebit eam eſſe tortuoſam, &amp; flexuoſam; Ariſt. reddens rationem, <lb></lb>
cur hæc ſit tortuoſa, ait id eſſe, quia ſectione obliqua exiſtente, ideſt ex vna <lb></lb>
parte depreſſiori, &amp; ex altera altiori, ſequitur, quod circuli, qui ex tali ſe­<lb></lb>
ctione oriuntur non remanent in eodem plano, dum euoluuntur; quare <expan abbr="neq;">neque</expan> <lb></lb>
linea, ex qua illi circuli conſtant, poterit eſſe in eodem plano, &amp; ideo <expan abbr="neq;">neque</expan> <lb></lb>
recta eſſe poterit, quia fieri nequit, vt eiuſdem lineæ pars ſit in plano vno, <lb></lb>
pars verò in altero; quod oſtenditur in prima 11. Elem. quæ eſt hæc; rectæ <lb></lb>
lineæ pars quædam non eſt in ſubiecto plano, pars verò in ſublimi.</s>
        </p>
        <p type="main">
          <s id="s.004182"><arrow.to.target n="marg349"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004183"><margin.target id="marg349"></margin.target>357</s>
        </p>
        <p type="main">
          <s id="s.004184">In 12. problem. </s>
          <s id="s.004185">quod eſt idem cum tertio ſuperiori, videnda ſunt, quæ ibi <lb></lb>
annotaui, hic tamen aliter ſoluitur, ſed tanta facilitate, vt nihil præte­<lb></lb>
rea opus ſit.</s>
        </p>
        <p type="main">
          <s id="s.004186"><arrow.to.target n="marg350"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004187"><margin.target id="marg350"></margin.target>358</s>
        </p>
        <p type="main">
          <s id="s.004188">In 13. probl. </s>
          <s id="s.004189">quod eſt idem cum quarto præcedenti, repetenda eſt illius. <lb></lb>
</s>
          <s id="s.004190">explicatio, vt huic inſeruiat. </s>
          <s id="s.004191">Ariſt. autem pulchrè, &amp; aptè aſſimilat refle­<lb></lb>
xionem corporum reflexioni radiorum viſualium ex ſpeculis; vbi, vt docent <lb></lb>
Perſpectiui, radius viſualis ſpeculo incidens, facit ſemper angulum æqua­<lb></lb>
lem ei, quem facit radius reflexus; eſt enim apud eos axioma, angulus in­<lb></lb>
cidentiæ æqualis eſt angulo reflexionis.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.004192"><emph type="italics"></emph>SECTIO XIX.</s>
          <lb></lb>
          <s id="s.004193">De Muſica.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.004194"><arrow.to.target n="marg351"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004195"><margin.target id="marg351"></margin.target>359</s>
        </p>
        <p type="main">
          <s id="s.004196">Problema primum ex ſe clarum eſt.</s>
        </p>
        <p type="main">
          <s id="s.004197">In 2. problema. </s>
          <s id="s.004198">In verba illa <emph type="italics"></emph>(Sed quemadmodum linea bipedalis non <emph.end type="italics"></emph.end><lb></lb>
<figure id="id.009.01.246.1.jpg" place="text" xlink:href="009/01/246/1.jpg"></figure><lb></lb>
<emph type="italics"></emph>duplum, ſed quadruplum quoddam deſcribit, <lb></lb>
ſic, &amp;c.)<emph.end type="italics"></emph.end> ideſt, quemadmodum linea bi­<lb></lb>
pedalis, quæ quamuis ſit dupla lineæ pedalis non <lb></lb>
tamen deſcribit quadratum duplum quadrati il­<lb></lb>
lius, ſed quadruplum: vt probatur in ſcholio 4. <lb></lb>
2. Elem. &amp; videre eſt in hac figura, vbi linea A B, <lb></lb>
eſt dupla lineæ A C. quadratum verò lineæ A B, <lb></lb>
ſcilicet quadratum A B D E, eſt <expan abbr="quadruplũ">quadruplum</expan> qua­<lb></lb>
drati lineæ A C, quadrati nimirum A C F G. re­<lb></lb>
liqua huius textus manifeſta ſunt</s>
        </p>
        <p type="main">
          <s id="s.004199">Scias Lector, me nullum, horum de Muſica Problematum (quemadmo­<lb></lb>
dum &amp; in pluribus alijs mathematicis locis accidit) vidiſſe expoſitorem, <lb></lb>
præter vnum Petrum Aponentem, quem tamen tanquam omninò his rebus <lb></lb>
elucidandis ineptum, reieci.</s>
        </p>
        <pb pagenum="247" xlink:href="009/01/247.jpg"></pb>
        <p type="main">
          <s id="s.004200">Vt autem cætera problemata rectè, ac facilè ſoluantur, operæpretium <lb></lb>
eſt, ortum, ac generationem totius Muſicæ breuiter præmittere; natura <lb></lb>
enim ipſius ritè perſpecta, ea deinde, quæ ipſam conſequuntur nullo nego­<lb></lb>
tio percipi poſſunt.</s>
        </p>
        <p type="main">
          <s id="s.004201">Primò igitur ſciendum eſt, duplicem nos poſſe conſiderare in voce, ſono­<lb></lb>
uè varietatem. </s>
          <s id="s.004202">prima eſt, qua eadem vox, aut ſonus modo maior, modo mi­<lb></lb>
nor efficitur, vt quando eadem chorda lentè pulſata ſonum edit exiguum; <lb></lb>
vehementer verò percuſſa, maiorem emittit ſonum. </s>
          <s id="s.004203">huiuſmodi vocem ap­<lb></lb>
pellant Muſici continuam. </s>
          <s id="s.004204">Philoſophi fortè eam vocarent vocis extenſio­<lb></lb>
nem. </s>
          <s id="s.004205">ſecunda vocis differentia, aut varietas eſt, cum ab vna voce ad aliam <lb></lb>
tranſimus, vt cum à graui ad acutiorem aſcendimus, vel contra, ab acuta <lb></lb>
ad grauem deſcendimus, ita vt hæc ſit mutatio ab vna voce ad aliam, hanc <lb></lb>
harmonici diſcretam vocem dicunt. </s>
          <s id="s.004206">quam fortè Philoſophi ex varia vocis <lb></lb>
intenſione prouenire iure dixerint. </s>
          <s id="s.004207">Hanc porrò diſcretam vocem, altera <lb></lb>
omiſſa, conſiderant Muſici. </s>
          <s id="s.004208">qua ratione autem, &amp; in quot, quaſuè voces eam <lb></lb>
diuiſerint antiqui, paucis accipe.</s>
        </p>
        <p type="main">
          <s id="s.004209">Cape duas chordas æreas, ex ijs ſcilicet, quas in cytharis adhibere <expan abbr="ſolẽt">ſolent</expan>; <lb></lb>nam quæ ex inteſtinis ouium fiunt, vt plurimum aut falſæ ſunt, aut aeris mu­<lb></lb>
tationi obnoxiæ. </s>
          <s id="s.004210">ſint hæ duæ chordæ æquales omninò, atque æquè intenſæ, <lb></lb>
<figure id="id.009.01.247.1.jpg" place="text" xlink:href="009/01/247/1.jpg"></figure><lb></lb>
ita vt ſint vniſonæ, hoc eſt, vna <expan abbr="tan-tũ">tan­<lb></lb>
tum</expan> vox ſit. </s>
          <s id="s.004211">quamuis duæ fides. </s>
          <s id="s.004212">opor­<lb></lb>
tet autem, vt ſint ſuper regula ali­<lb></lb>
qua lignea oblonga, &amp; perpolita, <lb></lb>
collocatæ, <expan abbr="quemadmodũ">quemadmodum</expan> ſuper ma­<lb></lb>
nubrio alicuius muſici inſtrumenti. <lb></lb>
</s>
          <s id="s.004213">hanc regulam veteres appellant regulam harmonicam, vel etiam mono­<lb></lb>
chordium; quo inſtrumento omnes conſonantias, ac <expan abbr="diſſonãtias">diſſonantias</expan>, <expan abbr="atq;">atque</expan> etiam <lb></lb>
interualla muſica experiebantur. </s>
          <s id="s.004214">altera iam ex illis diuidatur bifariam in <lb></lb>
E. deinde ſub puncto E, pone quem vulgò Tactum dicunt, veteres autem, <lb></lb>
hemiſphærium à figura denominabant, erat autem inſtar vnius Tacti mobi­<lb></lb>
lis: ſuppoſito igitur in E, Tacto, preme ibi chordam, ita vt altera tantum <lb></lb>
ipſius medietas, v. g. E D, tota pulſari, atque reſonare poſſit; pulſa igitur <lb></lb>
chordam vtranque ſimul, ſcilicet totam A B, &amp; dimidiam E D, ita vt ſimul <lb></lb>
reſonent. </s>
          <s id="s.004215">&amp; audies ſuauiſſimam omnium conſonèntiam, ex ſono totius A B, <lb></lb>
&amp; ſono dimidiæ E D, conflatam. </s>
          <s id="s.004216">hunc veteres Diapaſon, ideſt per omnes <lb></lb>
ſubaudi, chordas appellabant, quia in <expan abbr="antiquorũ">antiquorum</expan> muſicis inſtrumentis chor­<lb></lb>
dæ duæ omnium extremæ, ideſt grauiſſima, &amp; acutiſſima omnium fidium, <lb></lb>
conſonantiam hanc continebant: ita vt à grauiſſima omnium facto tranſitu <lb></lb>
per omnes chordas ad omnium ſupremam, &amp; acutiſſiman, conſonantiam <lb></lb>
hanc ſuauiſſimam exaudirent. </s>
          <s id="s.004217">appellatur etiam Dupla ratione proportio­<lb></lb>nis vnius vocis ad alteram, vox enim chordæ A B, eſt duplo maior, aut gra­<lb></lb>
uior voce dimidiæ E D. quemadmodum enim corpora ſononantia ſe habent <lb></lb>
ad inuicem, ita ſe ſoni eorum. </s>
          <s id="s.004218">chorda autem A B, dupla eſt chordæ E D. <lb></lb>nunc eam vulgus Octauam appellat, eo quod à prima voce, <expan abbr="eaq́">eaque</expan>; grauiſſi­<lb></lb>
ma, quæ Vt dicitur, <expan abbr="vſq;">vſque</expan> ad eam vocem, quæ ei in conſonantia diapaſon re­<lb></lb>
ſpondent, ſunt hæ octo voces, Vt, Re, Mi, Fa, Sol, Re, Mi, Fa. </s>
          <s id="s.004219">ex quibus <pb pagenum="248" xlink:href="009/01/248.jpg"></pb>prima, Vt, &amp; vltima, Fa, quæ octaua eſt, conſonantiam diapaſon, aut du­<lb></lb>
plam, aut octauam reddunt.</s>
        </p>
        <p type="main">
          <s id="s.004220">Rurſus eadem chorda C D, diuidatur in tres partes æquales in punctis <lb></lb>
<figure id="id.009.01.248.1.jpg" place="text" xlink:href="009/01/248/1.jpg"></figure><lb></lb>
F, G. </s>
          <s id="s.004221">F D igitur erit duæ ter­<lb></lb>
tiæ tam totius C D, quàm to­<lb></lb>
tius A B. ponatur iam tactus <lb></lb>
in F, <expan abbr="percutiãturq́">percutianturque</expan>; ſimul A B, <lb></lb>
&amp; F D; audietur conſonantia <lb></lb>
ſuauis admodum, &amp; perfecta <lb></lb>
quidem, ſed <expan abbr="nõ">non</expan> tamen, vt Dia­<lb></lb>
paſon. </s>
          <s id="s.004222">hanc priſci Diapente dixerunt, ideſt per <expan abbr="quinq;">quinque</expan> ſubaudi chordas, eò <lb></lb>
quod prima, &amp; quinta chorda, hanc conſonarent. </s>
          <s id="s.004223">ſecundum proportionem <lb></lb>
verò dicitur ſeſquialtera, quoniam chorda A B, ad chordam F D, eſt ſeſqui­<lb></lb>
altera, &amp; conſequenter etiam earum ſoni erunt in eadem ratione. </s>
          <s id="s.004224">ſeſquial­<lb></lb>
tera autem proportio eſt, quando maior quantitas A B, continet minorem <lb></lb>
F D, ſemel,&amp; adhuc dimidium ipſius. </s>
          <s id="s.004225">vulgò quinta, quia ex prima voce, Vt, <lb></lb>
&amp; quinta, Sol, conſtat.</s>
        </p>
        <p type="main">
          <s id="s.004226">Eadem iterum chorda in quatuor æquas partes ſecetur in punctis H, E, I, <lb></lb>
<figure id="id.009.01.248.2.jpg" place="text" xlink:href="009/01/248/2.jpg"></figure><lb></lb>
ita vt chorda H D, ſit tres quartæ <lb></lb>
totius A D. facto deinde tactu in <lb></lb>
H, pulſentur ſimul A B, H D, &amp; au­<lb></lb>
dietur conſonantia quidem, ſed <lb></lb>
duabus præcedentibus imperfe­<lb></lb>
ctior. </s>
          <s id="s.004227">hæc antiquitus Diateſſaron, <lb></lb>
ideſt per quatuor, ſubaudi chor­<lb></lb>
das, aut voces, ſimili ratione, qua ſuperiores dicta fuit. </s>
          <s id="s.004228">reſpectu autem pro­<lb></lb>
portionis chordarum, ac ſonorum dicitur ſeſquitertia, quia maior A B, mi­<lb></lb>
norem H D, ſemel, &amp; adhuc tertiam ipſius partem continet. </s>
          <s id="s.004229">vulgò nunc di­<lb></lb>
citur, quarta, quid inter primam vocem, Vt, &amp; quartam, Fa, reperiatur. <lb></lb>
</s>
          <s id="s.004230">Iam verò ſi in eadem chorda C D, ponantur puncta H, &amp; F, vt in præcedenti <lb></lb>
figura, &amp; ſimul duæ chordæ H D, &amp; F D, hoc eſt tres quartæ, &amp; duæ tertiæ, <lb></lb>
arithmeticis rationibus comparentur, reperiemus maiorem H D, ad mi­<lb></lb>
norem F D, proportionem habere ſeſquioctauam, &amp; ſonum maioris H D, <lb></lb>
ad minorem F D, eandem habebit rationem, hoc eſt, vt nouis vocabulis <lb></lb>
vtamur inter Fa, &amp; Sol, eſſe ſeſquioctauam proportionem; ſi autem ſimul <lb></lb>
hi duo ſoni exaudiantur diſſonantiam auribus facient. </s>
          <s id="s.004231">diſtantiam porrò <lb></lb>hanc inter voces Fa, Sol, ſiue inter chordas H D, F D, ſiue inter duo inter­<lb></lb>
ualla H D, F D, harmonici, quorum ratio eſſet ſeſquioctaua Tonum appel­<lb></lb>
larunt. </s>
          <s id="s.004232">Diuiſerunt poſtea totam C D, in nouem partem æquales, quarum <lb></lb>
prima ſit in puncto K, diuiſa, ita vt tota C D, ad reliquam K D, quæ conti­<lb></lb>
net octo partes ex illis, habeat rationem ſeſquioctauam, hoc pariter inter­<lb></lb>uallum Toni erit, cuius primum ſonum, ideſt totius C D, nunc dicunt, Vt, <lb></lb>
ſecundum verò ſonum reliquæ chordæ K D, dicunt, Re Reliquam poſtea <lb></lb>
K D, ſimiliter in nouem partes diuiſerunt, cuius prima pars ſit in puncto L, <lb></lb>
notata. </s>
          <s id="s.004233">&amp; eadem ratione inter chordam K D, &amp; chordam L D, <expan abbr="earumq́">earumque</expan>; ſo­<lb></lb>
nos erit ſeſquioctaua proportio. </s>
          <s id="s.004234">ſonum chordæ L D, nunc appellant, Mi.


<pb pagenum="249" xlink:href="009/01/249.jpg"></pb>Interuallum verò, quod inter chordam L D, &amp; chordam H D, remanet, non <lb></lb>
habet proportionem ſeſquioctauam, ſed dimidio ferè minorem, &amp; propte­<lb></lb>
rea huiuſmodi interuallum ſemitonium, &amp; etiam dieſis, ſiue diuiſio, appel­<lb></lb>
latur. </s>
          <s id="s.004235">Interuallum verò illud, quod inter puncta F, &amp; E, remanet, diuiſe­<lb></lb>
runt eodem modo, quo diuiſum fuit ſpatium inter C, &amp; H, &amp; repercerunt <lb></lb>
eaſdem iterum voces; ſint diuiſiones illæ punctis M, &amp; N, notatæ; &amp; pari­<lb></lb>
ter hic etiam inter N, &amp; E, ſiue inter Mi, &amp; Fa, eſt alterum ſemitonium. <lb></lb>
</s>
          <s id="s.004236">ſunt igitur hæ octo voces, Vt, Re, Mi, Fa, Sol, Re, Mi, Fa, quæ totam Dia­<lb></lb>
paſon componunt, vt enim ſupra dictum eſt inter Vt, &amp; Fa, <expan abbr="vltimũ">vltimum</expan>, eſt con­<lb></lb>
ſonantia diapaſon, ſiue dicamus inter chordam C D, vel A B, &amp; chordam <lb></lb>
E D. ex interuallis autem,quæ ſunt inter voces, duo ſunt ſemitonia, ſcilicet <lb></lb>
vnum inter Mi, &amp; Fa, notatum literis L, H. &amp; alterum inter vltima Mi, &amp; <lb></lb>
Fa, ſignatum notis N, E. reliqua <expan abbr="quinq;">quinque</expan> interualla ſunt integri toni. </s>
          <s id="s.004237">Aduer­<lb></lb>
tendum præterea eſt, ab Vt, <expan abbr="vſq;">vſque</expan> ad primum Sol, eſſe conſonantiam Dia­<lb></lb>
pente, quæ continet tria interualla toniaca, &amp; vnum ſemitonium; in vni­<lb></lb>
uerſum tamen ſunt <expan abbr="quinq;">quinque</expan> voces, Vt, Re, Mi, Fa, Sol.</s>
        </p>
        <p type="main">
          <s id="s.004238">Notandum etiam, quod à Sol, <expan abbr="vſq;">vſque</expan> ad vltimum Fa, ſunt quatuor voces, <lb></lb>
Sol, Re, Mi, Fa,quæ omninò ſimiles ſunt primis quatuor, Vt, Re, Mi Fa. </s>
          <s id="s.004239">hæ <lb></lb>
tamen ſunt grauiores, illæ verò acutiores, &amp; quemadmodum ab Vt, ad pri­<lb></lb>
mum Fa, eſt Diateſſaron, ita etiam à Sol, <expan abbr="vſq;">vſque</expan> ad vltimum Fa, eſt altera Dia­<lb></lb>
teſſaron. </s>
          <s id="s.004240">Ex quibus vltimò notandum ſequitur, duas conſonantias Diateſ­<lb></lb>
ſaron, &amp; Diapente totam conſtituere Diapaſon: ſiue Diapaſon diuidi in <lb></lb>
vnam Diateſſaron, &amp; vnam Diapente; nam ab Vt, ad Sol, eſt Diapente; à <lb></lb>
Sol, verò in Fa, poſtremum, eſt Diateſſaron. </s>
          <s id="s.004241">quod etiam aliter conſtabit, ſi <lb></lb>
dicamus ab Vt, ad primum Fa, eſſe Diateſſaron, vt patet ex chordæ diuiſio­<lb></lb>
ne: ex Fa, autem primò ad vltimum Fa, eſſe Diapente: vt manifeſtum eſt <lb></lb>
ex quatuor ipſius interuallis, quorum tria ſunt Toni, reliquum verò ſemito­<lb></lb>
nium, quæ etiam erant in altera Diapente inter Vt, &amp; Sol, contenta.</s>
        </p>
        <p type="main">
          <s id="s.004242">Nunc rurſus fiat tactus in I, eſt autem I D, quarta pars totius C D. per­<lb></lb>
cutiantur ſimul A B, &amp; I D; <expan abbr="edeturq́">edeturque</expan>; ſuauiſſima conſonantia Diſdiapaſon <lb></lb>
appellata, propterea quod ex duabus Diapaſon conſtet; quarum prima eſt <lb></lb>
inter A B, ſiue C D, &amp; E D: ſecunda verò inter ipſam E D, &amp; I D. harum <lb></lb>
enim proportio dupla eſt, ſicuti illarum. </s>
          <s id="s.004243">proportio huius eſt quadrupla, vt <lb></lb>
ex diuiſione conſtat; vulgò dicitur decimaquinta, quia à primo Vt, <expan abbr="vſq;">vſque</expan> ad <lb></lb>
hanc vocem,quæ etiam Fa, nominatur, eſſent quindecim voces, ſi interual­<lb></lb>
lum E I, eo modo diuideretur, quo diuiſum eſt primum C E.</s>
        </p>
        <p type="main">
          <s id="s.004244">Poſtremò ſit G D, tertia pars totius C D, <expan abbr="fiatq́">fiatque</expan>; in G, tactus, pulſentur ſi­<lb></lb>mul A B, G D. audietur ſuauis conſonantia, quæ Diapaſondiapente nomi­<lb></lb>
natur, quod conſtet ex vna Diapaſon contenta interuallo C E, ſiue duabus <lb></lb>
chordis C D, E D, &amp; vna Diapente contenta interuallo E G, ſiue chordis <lb></lb>
E D, G D; nam chorda E D, ad chordam G D, ſeſquialtera eſt; quæ propor­<lb></lb>
tio naturam ipſius diapente conſtituit huius conſonantiæ proportio eſt tri­<lb></lb>
pla, eſt enim chorda A B, vel C D, tripla ipſius G D. vulgò dicitur duode­<lb></lb>cima, cò quod inter Vt, &amp; Sol, notatum litera G, ſint duodecim voces, ſi in­<lb></lb>
teruallum E O, ſuas recipiat diuiſiones. </s>
          <s id="s.004245">ex quibus omnibus manifeſtum eſt <lb></lb>
auris experimento, eſſe omninò quinque conſonantias, tres ſimplices Dia­


<pb pagenum="250" xlink:href="009/01/250.jpg"></pb>paſon, Diapente, Diateſſaron; duas verò compoſitas, Diſdiapaſon, &amp; Dia­<lb></lb>
paſondiapente.</s>
        </p>
        <p type="main">
          <s id="s.004246">Illud poſtremò loco non ignorandum, aliter has voces Vt, Re, &amp;c. </s>
          <s id="s.004247">vete­<lb></lb>
res illos Græcos denominaſſe, nam primam, ideſt grauiſſimam vocem, ſiue <lb></lb>
chordam, quam modò Vt, dicunt, eam ipſi Hypaten vocarunt, &amp; reliquas <lb></lb>
ordine ſequenti.</s>
        </p>
        <figure id="id.009.01.250.1.jpg" place="text" xlink:href="009/01/250/1.jpg"></figure>
        <p type="main">
          <s id="s.004248">Vt, Hypate — ideſt Principalis.</s>
        </p>
        <p type="main">
          <s id="s.004249">Re, Parhypate —— Poſtprincipalis.</s>
        </p>
        <p type="main">
          <s id="s.004250">Mi, Lychanos —— Index.</s>
        </p>
        <p type="main">
          <s id="s.004251">Fa, Meſe ———— Media.</s>
        </p>
        <p type="main">
          <s id="s.004252">Sol, Parameſe —— Poſtmedia.</s>
        </p>
        <p type="main">
          <s id="s.004253">Re, Trite———— Tertia.</s>
        </p>
        <p type="main">
          <s id="s.004254">Mi, Paranete —— Antepenultima.</s>
        </p>
        <p type="main">
          <s id="s.004255">Fa, Nete ———— Vltima, vel ſuprema.</s>
        </p>
        <p type="main">
          <s id="s.004256">His paucis ex magno Muſicæ Campo decerptis problematum declara­<lb></lb>
tionem ſatis inſtructi aggrediamur.</s>
        </p>
        <p type="main">
          <s id="s.004257"><arrow.to.target n="marg352"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004258"><margin.target id="marg352"></margin.target>360</s>
        </p>
        <p type="main">
          <s id="s.004259">Problema 3. <emph type="italics"></emph>(Cur maximè in cantando Parhypatem, vox rumpi non minus ſo­<lb></lb>
leat, quam in Nete, ſupremisqué quamquam cùm interuallo ampliori? </s>
          <s id="s.004260">An quod eius <lb></lb>
cantus &amp; perdifficilis est, &amp; cantandi primordium obtinet? </s>
          <s id="s.004261">difficilis autem pro­<lb></lb>
pter intenſionem, &amp; compreſſionem vocis eſt, quibus in rebus eſt labor, &amp; difficul­<lb></lb>
tas: corrumpi autem quæqué maximè ſolent, quoties labore acrius opprimuntur)<emph.end type="italics"></emph.end><lb></lb>
Quærit Ariſt. cur qui eam vocem cantant, quæ Parhypate dicitur, non mi­<lb></lb>
nus defatigetur, <expan abbr="vocemq́">vocemque</expan>; perdat, quam qui Neten, aut aliam ex acutiori­<lb></lb>
bus vocibus, ipſi Nete proximam, vt Paranetem. </s>
          <s id="s.004262">ratio dubitandi eſt, quia <lb></lb>
Parhypate eſt vox grauis, &amp; quæ paruo interuallo diſtat à grauiſſima om­<lb></lb>
nium Hypate: at verò Nete, <expan abbr="aliæq́">aliæque</expan>; illi viciniores ſunt acutiſſimæ, <expan abbr="magnisq́">magnisque</expan>; <lb></lb>
ab Hypate diſtant interuallis. </s>
          <s id="s.004263">Reſpondet, id accidere ob difficultatem, quæ <lb></lb>
in eius cantu reperitur; quæ difficultas laborem infert, labor autem vocem <lb></lb>
<expan abbr="corrũpit">corrumpit</expan>. </s>
          <s id="s.004264">ſed vnde hæc difficultas? </s>
          <s id="s.004265">reſpondet inde prouenire, quia hæc vox <lb></lb>
cantandi principium eſt. </s>
          <s id="s.004266">vbi per cantandi principium puto ipſum intellige­<lb></lb>re ſemitonium, ſiue dieſim, nam, vt ipſe ait primo <lb></lb>
<figure id="id.009.01.250.2.jpg" place="text" xlink:href="009/01/250/2.jpg"></figure><lb></lb>
re ſemitonium, ſiue dieſim, nam, vt ipſe ait primo <lb></lb>
Poſter. cap. 38. dieſis hęc, ſiue ſemitonium dicitur <lb></lb>
principium cantus, quia minimum eſt omnium in­<lb></lb>
teruallorum, quæ voce poſſint exprimi: <expan abbr="atq;">atque</expan> ex eo <lb></lb>
alia interualla conſtant, <expan abbr="eſtq́">eſtque</expan>; veluti illorum ele­<lb></lb>
mentum. </s>
          <s id="s.004267">vide illius loci explanationem. </s>
          <s id="s.004268">Iam ve­<lb></lb>
rò difficile admodum eſſe cantare per ſemitonia, <lb></lb>
perſpicuum eſt cantoribus, quod oporteat, vt ait <lb></lb>
Ariſt. vocem, quantum opus eſt, intendere, ſimul <lb></lb>
ac comprimere, ne ſcilicet in maius iuſto inter­<lb></lb>
uallum erumpat.</s>
        </p>
        <p type="main">
          <s id="s.004269">Verum dubitabis, cur Ariſt. ponat ſemitonium <lb></lb>
ab Hypate ad Parhypatem, cùm ſuperius <expan abbr="dictũ">dictum</expan> ſit, <lb></lb>ſemitonium eſſe tantummodo inter Mi, &amp; Fa, ideſt <lb></lb>
inter Lychanon, &amp; Meſen. </s>
          <s id="s.004270">Scias igitur alios aliter <lb></lb>
<expan abbr="interuallorũ">interuallorum</expan> ordinem feciſſe: inter quos Lychaon <pb pagenum="251" xlink:href="009/01/251.jpg"></pb>antiquiſſimus Muſicus ſic ea diſpoſuit; vt in præſenti ordine, vbi, vt vides <lb></lb>
inter hypatem, &amp; perhypatem, eſt ſemitonij interuallum. </s>
          <s id="s.004271">ad hunc Lichaonis <lb></lb>igitur ordinem videtur Ariſt. reſpexiſſe. </s>
          <s id="s.004272">ego verò ſuperius communiorem <lb></lb>
viam, nec adeò antiquam ſequutus ſum. </s>
          <s id="s.004273">ex Boethio, &amp; Zarlino.</s>
        </p>
        <p type="main">
          <s id="s.004274"><arrow.to.target n="marg353"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004275"><margin.target id="marg353"></margin.target>361</s>
        </p>
        <p type="main">
          <s id="s.004276">Probl. 4. <emph type="italics"></emph>(Séd cur hæc difficile, hypate facilè cantatur, cum non niſi dieſi di­<lb></lb>
ſcrepent? </s>
          <s id="s.004277">An quod hypate remiſſior eſt, <expan abbr="atq;">atque</expan> etiam læuius à conſtitutione aſcendi­<lb></lb>
tur? </s>
          <s id="s.004278">hæc eadem cauſa est, cur ad vnam cantari videantur, quæ ad hanc parane­<lb></lb>
temqué cantantur. </s>
          <s id="s.004279">agendum enim eſt, cum intentione, conditionequé moribus idonea <lb></lb>pro voluntate. </s>
          <s id="s.004280">quæ verò cauſa eſt, vt cum conſonantia ſit?)<emph.end type="italics"></emph.end> Ideſt, cur parhypa­<lb></lb>
te, de qua in præcedenti problemate dictum eſt, difficilius canitur, quam <lb></lb>
hypate, cùm tamen ab inuicem diſtent non niſi ſemitonij interuallo? </s>
          <s id="s.004281">forſi­<lb></lb>
tan id accidit, quia hypate eſt remiſſior, cùm ſit omnium grauiſſima, hoc <lb></lb>
eſt, non eſt opus in ea decantanda, ita vocem intendere, quemadmodum in <lb></lb>
parhypate, quæ acutior eſt. </s>
          <s id="s.004282">reliqua huius loci verba exiſtimo eſſe admodum <lb></lb>
mendoſa tam græcè, quàm latinè, cùm nonnulla in eis ſint, quæ nullo pacto <lb></lb>
ad rem faciunt, præſertim extrema ſententia. </s>
          <s id="s.004283">&amp; conſultius eſſe exiſtimo fa­<lb></lb>
teri me ea non intelligere, quam ea violenter huc, <expan abbr="atq;">atque</expan> illuc diducere.</s>
        </p>
        <p type="main">
          <s id="s.004284"><arrow.to.target n="marg354"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004285"><margin.target id="marg354"></margin.target>362</s>
        </p>
        <p type="main">
          <s id="s.004286">Probl. 5. <emph type="italics"></emph>(Cur ſuauius cantilenam, quam nouimus, audire ſolemus, quam eam, <lb></lb>
quam ignoramus? </s>
          <s id="s.004287">Vtrum, quia cùm quod cantatur, agnoſcimus, tunc magis ma­<lb></lb>
nifeſtus eſt, qui veluti ſcopum aſſequitur. </s>
          <s id="s.004288">id autem contemplatu ſuaue eſt. </s>
          <s id="s.004289">An quia <lb></lb>
diſcere ſeu intelligere ſuaue eſt, cuius ratio eſt, quia hoc quidem verſatur in acci­<lb></lb>
pienda ſcientia, illud verò in vtenda. </s>
          <s id="s.004290">præterea ſolitum inſolito ſuauius eſt?)<emph.end type="italics"></emph.end> Vbi <lb></lb>
Theodorus Gaza poſuerat, calcem reſtitui ex græco textu, ſcopum, vt res <lb></lb>
ipſa etiam poſtulabat. </s>
          <s id="s.004291">Porrò tres affert rationes, cur ſuauius ſit notam <lb></lb>
iam cantilenam auſcultare, quàm ignotam. </s>
          <s id="s.004292">prima eſt, quia, cùm cognoſci­<lb></lb>
mus, quod cantatur, tunc ſcopus, ac finis, in quem cantor, ac tota tendit <lb></lb>
cantilena manifeſtus eſt, <expan abbr="ſicq́">ſicque</expan>; eam melius percipimus; quia dum ipſam au­<lb></lb>
dimus, ſcopum etiam ipſius, cuius contemplatio iucunda eſt, contempla­<lb></lb>
mur. </s>
          <s id="s.004293">quemadmodum iucundius eſt ſpectare currentem canem, &amp; iam cap­<lb></lb>tantem feram, ſi ſimul feram etiam ipſam, quæ ſcopus ipſius eſt, quam ſi fe­<lb></lb>
ram minimè videamus. </s>
          <s id="s.004294">ſecunda eſt, quia ipſum diſcere, ac intelligere dele­<lb></lb>
ctabile eſt, &amp; huius quidem ratio manifeſta eſt tam in accipienda, quàm in <lb></lb>
vtenda ſcientia: dum igitur cantilenam <expan abbr="primũ">primum</expan> audimus, quam prius igno­<lb></lb>
rabamus, ſcientiam illius tantum accipimus; dum autem notam auſculta­<lb></lb>
mus, non ſolum ipſam, ſed ipſius etiam ſcopum contemplantes, ea perfectè <lb></lb>
vtimur. </s>
          <s id="s.004295">tertia ratio eſt, quia res ſolitæ plerumque, quam inſolitæ iucundio­<lb></lb>
res exiſtunt.</s>
        </p>
        <p type="main">
          <s id="s.004296">Probl. 6. &amp; per ſe ſatis clarum eſt; &amp; ad harmonicam non ſpectat.</s>
        </p>
        <p type="main">
          <s id="s.004297"><arrow.to.target n="marg355"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004298"><margin.target id="marg355"></margin.target>363</s>
        </p>
        <p type="main">
          <s id="s.004299">Probl. 7. <emph type="italics"></emph>(Cur veteres cum ſeptem fidibus concentus diſponerent, hypaten, non <lb></lb>
neten relinquebant? </s>
          <s id="s.004300">An falsò id dicitur (earum enim <expan abbr="vtramq;">vtramque</expan> ſeruarunt; ſed tri­<lb></lb>
ten adimere ſolebant) An non? </s>
          <s id="s.004301">ſed quia grauior ſonum poteſt acutioris. </s>
          <s id="s.004302">ergò hypa­<lb></lb>
te magis antiphonum, quam nete reddebat; nam vt acutum vim deſiderat plenio­<lb></lb>
rem, ſic graue exprimi facilius poteſt)<emph.end type="italics"></emph.end> Propter quid, inquit, antiquiſſimi Mu­<lb></lb>ſicorum cùm ex ſeptem tantum chordis muſica inſtrumenta <expan abbr="componerẽt">componerent</expan>, <lb></lb>
neten non hypatèn omittebant? </s>
          <s id="s.004303">ſupra octo voces, ſeu chordas recenſui, <expan abbr="ra-tionemq́">ra­<lb></lb>
tionemque</expan>; ipſarum, vnà cum antiquis appellationibus explicaui, quarum <pb pagenum="252" xlink:href="009/01/252.jpg"></pb><figure id="id.009.01.252.1.jpg" place="text" xlink:href="009/01/252/1.jpg"></figure><lb></lb>
prima eſt hypate, vltima verò note, quibus re­<lb></lb>
petitis facilè eſt intelligere, quod reſpondet <lb></lb>
Ariſt. </s>
          <s id="s.004304">Reſpondet enim id non omninò verum <lb></lb>
cenſeri debere, nam vtramque quidem hypa­<lb></lb>
tem, ſcilicet &amp; neten aſſumebant; triten verò <lb></lb>
<expan abbr="omittebãt">omittebant</expan>. </s>
          <s id="s.004305">quibus verbis ordinem, quem Ter­<lb></lb>
pander inuexit, inſinuare videtur, nam vt ait <lb></lb>
Pauſanias in Lachonicis, Timothæus quatuor <lb></lb>
chordas, antiquis ſeptem chordis à Terpan­<lb></lb>
dro ordinatis addidit, quarum ſeptem chor­<lb></lb>
darum hic erat ordo, &amp; nomenclatura, &amp; in­<lb></lb>
terualla; è quibus triten ademptam videre eſt, <lb></lb>
vt Ariſt. innuit.</s>
        </p>
        <p type="main">
          <s id="s.004306">Subdit poſtea aliam rationem dicens; fortè ſatius eſſe dicere neten qui­<lb></lb>
dem antiquitus fuiſſe prætermiſſam, relicta hypate, co quod hypate, cum <lb></lb>
diſtet per octauam, ſeu per Diapaſon à Nete, erat illius Antiphonum, ideſt, <lb></lb>
erat vox eiuſdem naturæ, &amp; ferè eadem cum ea. </s>
          <s id="s.004307">ſciendum. </s>
          <s id="s.004308">n. </s>
          <s id="s.004309">Muſicos docere <lb></lb>
voces omnes <expan abbr="vſq;">vſque</expan> ad ſeptem eſſe ab inuicem differentes, &amp; diuerſæ naturæ <lb></lb>cùm autem ad octauam ventum eſt, tunc redire voces iterum eiuſdem na­<lb></lb>
turæ, &amp; ferè eædem cum præcedentibus: ita vt octaua ſit eadem cum pri­<lb></lb>
ma, &amp; nona cum ſecunda, &amp; decima cum tertia, &amp; ſic de reliquis, quæ om­<lb></lb>
nes diſtant per octonarium, ſine ſunt octauæ. </s>
          <s id="s.004310"><expan abbr="dicebanturq́">dicebanturque</expan>; huiuſmodi voces <lb></lb>
Antiphonæ, quaſi contraſonantes, vel viciſſim ſonantes (vide infra annota­<lb></lb>
ta in 14. Probl.) quarum vox grauior, cùm dupla ſit, acutioris edit ſonum, <lb></lb>
qui duplus eſt ſoni acutioris, ſiue qui bis in ſe continet <expan abbr="ſonũ">ſonum</expan> acutioris. </s>
          <s id="s.004311">Qua­<lb></lb>
re relicta hypate, &amp; dempta nete, quarum illa eſt huius dupla, nihil ferè ad­<lb></lb>
emptum fuiſſe videbatur, cùm ſonus nete contineretur in ſono hypates. </s>
          <s id="s.004312">hac <lb></lb>
igitur de cauſa veteres illi netem potius, quam hypatem omiſerunt. </s>
          <s id="s.004313">præte­<lb></lb>
rea dici poteſt, eos hypatem potius retinuiſſe, quia cùm remiſſior ſit, fa­<lb></lb>
cilius cantatur; Nete autem cùm acutiſſima ſit maiore vi, vt cantetur, <lb></lb>
opus habet.</s>
        </p>
        <p type="main">
          <s id="s.004314"><arrow.to.target n="marg356"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004315"><margin.target id="marg356"></margin.target>364</s>
        </p>
        <p type="main">
          <s id="s.004316">Probl. 8. <emph type="italics"></emph>(Cur grauis ſonum poteſt acutæ? </s>
          <s id="s.004317">An quia maius eſt; etenim quemad­<lb></lb>
modum graue obtuſo, ſic acutum acuto angulo ſimile eſt)<emph.end type="italics"></emph.end> Ex intelligentia præ­<lb></lb>cedentis problematis, præſens fatis ferè clarum eſt: imò ex illo ortum iſtud <lb></lb>
eſſe videtur. </s>
          <s id="s.004318">quærit, cur vox grauior poſſit vocem acutiorem, ſiuè illi æqui­<lb></lb>
ualeat, vt dictum eſt, in præcedenti de Antiphonis. </s>
          <s id="s.004319">cauſa eſt, inquit, quia <lb></lb>grauis maior eſt, quàm acuta; grauis enim oritur à maiori corpore, vt à <lb></lb>
chorda maiori, vt ſuperius apparuit; vel à maiori canna, vt patet in Orga­<lb></lb>
nis. </s>
          <s id="s.004320">voces autem, &amp; ſoni eandem habent cum corporibus ſonantibus pro­<lb></lb>
portionem. </s>
          <s id="s.004321">quare grauis ſonus maior eſt acuto; <expan abbr="cũ">cum</expan> igitur maior ſit, eum in <lb></lb>
ſe continebit, <expan abbr="eumq́">eumque</expan>; poterit. </s>
          <s id="s.004322">eſt enim grauis ſonus ſimilis angulo obtuſo, &amp; <lb></lb>
acutus ſonus ſimilis acuto angulo: obtuſus autem angulus maior eſt acuto, <lb></lb>
<expan abbr="eumq́">eumque</expan>; in ſe continet. </s>
          <s id="s.004323"><expan abbr="eumq́">eumque</expan>; propterea poteſt.</s>
        </p>
        <p type="main">
          <s id="s.004324"><arrow.to.target n="marg357"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004325"><margin.target id="marg357"></margin.target>365</s>
        </p>
        <p type="main">
          <s id="s.004326">Probl. 9. <emph type="italics"></emph>(Cur ſolitarias cantilenas ſuauius audire ſolemus, ſi ad tybiam, aut <lb></lb>ad lyram vnam cantatur, cùm tamen ad fides, canticumqué idem, modo <expan abbr="vtroq;">vtroque</expan> per­<lb></lb>
agatur? </s>
          <s id="s.004327">nam ſi idem ita amplius fit, plus ad plures tibias, <expan abbr="atq;">atque</expan> etiam ſuauius eſſe<emph.end type="italics"></emph.end> <pb pagenum="253" xlink:href="009/01/253.jpg"></pb><emph type="italics"></emph>oportet? </s>
          <s id="s.004328">An quoniam manifeſtior eſt ſcopus eius, cùm ad vnam lyram, vel tibiam <lb></lb>
cantatur? </s>
          <s id="s.004329">ad plures verò ſuauitas ſeruari non poteſt, cùm cantilena offuſcetur, te­<lb></lb>
taqué penè deleatur)<emph.end type="italics"></emph.end> Cur ſolitariæ cantilenæ (quas Græci Monodias appella­<lb></lb>
bant, &amp; ab vna tantum perſona cantabantur) ſuauiores ſunt, ſi ad lyram <lb></lb>
vnam, vel ad tibiam vnam, quam ſi ad plures lyras, aut tibias accinantur; <lb></lb>cùm tamen vtroque modo, ideſt tam ad lyram, quàm ad tibiam, &amp; tam ad <lb></lb>
vnam, quàm ad plures idem canticum perſonetur. </s>
          <s id="s.004330">&amp; cùm idem canticum <lb></lb>
ad plures lyras, aut tibias decantatum in maius exereſcat, deberet etiam <lb></lb>
ſuauius auribus accidere. </s>
          <s id="s.004331">Reſpondet fortè monodiam iucundiorem eſſe ad <lb></lb>vnum inſtrumentum, quia ipſius ſcopus tunc manifeſtior eſt: pluribus au­<lb></lb>
tem adhibitis inſtrumentis ſuauitas ſeruari nequit, cùm cantilena tot ſonis <lb></lb>
offuſcetur, ac tota penè obruatur. </s>
          <s id="s.004332">Verumenimuerò vtinam recentiores Mu­<lb></lb>
ſicæ contrapuntiſtæ, iſta, quæ hoc loco ab Ariſt tradita ſunt ritè animad­<lb></lb>
uertent. </s>
          <s id="s.004333">non <expan abbr="vtiq;">vtique</expan> tanta verborum, <expan abbr="atq;">atque</expan> rithmorum confuſione, <expan abbr="atq;">atque</expan> pluri­<lb></lb>
morum inſtrumentorum ſtrepitu gauderent: ex quibus eorum cantilena ita <lb></lb>
offuſcatur, vt nulla omninò reddatur; <expan abbr="ſolusq́">ſolusque</expan>; ſtrepitus quidam ingens aures <lb></lb>
obtundat; quem modum non ſine huius ætatis dedecore, futura ſecula non <lb></lb>
ſine irriſione mirabuntur: non aliter, ac nos ſemipriſcæ ætatis architectu­<lb></lb>
ram, &amp; ſculpturam irridere ſolemus.</s>
        </p>
        <p type="main">
          <s id="s.004334"><arrow.to.target n="marg358"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004335"><margin.target id="marg358"></margin.target>366</s>
        </p>
        <p type="main">
          <s id="s.004336">Probl. 10. ſatis clarum ex ſe. </s>
          <s id="s.004337">illud ſolum notatione dignum eſt, Teretiza­<lb></lb>
re, quod eſt canere, vt modo aiunt, non verba, ſed notas, fuiſſe idem, quod <lb></lb>nunc ſolmifationem, aut lalagen decantare.</s>
        </p>
        <p type="main">
          <s id="s.004338"><arrow.to.target n="marg359"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004339"><margin.target id="marg359"></margin.target>367</s>
        </p>
        <p type="main">
          <s id="s.004340">Probl. 11. <emph type="italics"></emph>(Cur vox, aut ſonus deſinens acutoir fit? </s>
          <s id="s.004341">An quia minor, vt quæ fa­<lb></lb>
cta ſit imbecillior?)<emph.end type="italics"></emph.end> lege quæ in probl. </s>
          <s id="s.004342">8. annotata ſunt, &amp; huic quoque ſatis­<lb></lb>
factum erit.</s>
        </p>
        <p type="main">
          <s id="s.004343"><arrow.to.target n="marg360"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004344"><margin.target id="marg360"></margin.target>368</s>
        </p>
        <p type="main">
          <s id="s.004345">Probl. 12. <emph type="italics"></emph>(Quamobrem quæ grauior è fidibus eft ſemper modulationem, aut <lb></lb>
cantilenam ſuſcipit: nam ſi oporteat canere parameſen cùm ſola meſe, nihilominus <lb></lb>
medium gignetur: ſi verò meſen, neceſſarium ambo, ſola non gignitur? </s>
          <s id="s.004346">An quia <lb></lb>
graue magnum eſt, itaque validius: &amp; in magno paruum ineſi. </s>
          <s id="s.004347">&amp; per interceptio­<lb></lb>
nem duæ netæ ex hypate fiunt)<emph.end type="italics"></emph.end> Vt facilè præſens problema intelligatur, ha­<lb></lb>
bendus eſt ob oculos ordo antiquarum chordarum, quem ſupra ante pro­<lb></lb>
blema tertium expoſui. </s>
          <s id="s.004348">Quærit cur fidicines ſolerent modulationem tam <lb></lb>
cum cautu, quàm ſine cantu à grauiſſima omnium chordarum exordiri:<lb></lb>
ideſt primam omnium grauiſſimam chordam pulſare; vt ipſa reliquis acu­<lb></lb>
tioribus, quaſi dux præiret, quam reliquæ ſequerentur: &amp; ſi oporteat cane­<lb></lb>
re parameſen vnà cùm meſe, quæ grauior eſt, gignitur, non ſonus parame­<lb></lb>
ſes, ſed ipſius meſes redditur. </s>
          <s id="s.004349">ſi verò oporteat canere meſen, id non poteſt <lb></lb>
fieri per ſolam parameſen, ſed <expan abbr="vtraq;">vtraque</expan> neceſſaria eſt, vel ſaltem ipſa meſe. <lb></lb>
</s>
          <s id="s.004350">Ratio huius, inquit, eſt, quia quod graue eſt, magnum eſt, &amp; proinde acuto <lb></lb>
etiam validius. </s>
          <s id="s.004351">præterea in magno etiam paruum ineſt: grauior igitur vox <lb></lb>
maior eſt, ac validior, quàm acuta, vt ſuperius explicatum eſt; meritò igi­<lb></lb>
tur ſonus grauior pręire debet, <expan abbr="atq;">atque</expan> ad modulationem alios prouocare, cùm <lb></lb>
reliquas ſecum tanquam partes proprias naturaliter trahat. </s>
          <s id="s.004352">quando autem <lb></lb>
parameſe, ac meſe ſimul canuntur, tunc meſe ſola videtur exaudiri; quia <lb></lb>
cùm ipſa grauior ſit, quàm parameſe, erit etiam ipſa maior, ac validior, &amp; <lb></lb>propterea ſonus parameſes in ſono meſes euaneſcit, ſiue ſuperuacaneus eſt,<pb pagenum="254" xlink:href="009/01/254.jpg"></pb>At verò ſola parameſe nequit præter proprium ſonum, etiam ſonum meſes <lb></lb>
efficere; quia cùm parameſe ſit acutior, quàm meſe, vt patet ex præceden­<lb></lb>
tibus, erit etiam ipſa minor, ac imbecillior: idcircò ad ſonum meſes effi­<lb></lb>
ciendum, aut meſe cum parameſe, aut ſaltem ſola meſe neceſſaria eſt. </s>
          <s id="s.004353">quòd <lb></lb>
autem grauior ſonus ſit acuto maior, hinc patet, quia duæ netæ in hypate <lb></lb>
continentur; ſi enim ſonus hypates bifariam diuidatur, v. g. ſi flatus ex ali­<lb></lb>
qua grauiſſima canna exiens ita intercipiatur, vt medius tantum per can­<lb></lb>
nam effletur, fit ſonus ex hypate nete, ex dimidio nimirum flatu hypates fit <lb></lb>
nete; idem patet in chordis, quia dimidium alicuius chordæ, vt ſupra pa­<lb></lb>
tuit, ad totam, eſt nete ad hypatem. </s>
          <s id="s.004354">duæ igitur nete in hypate continentur <lb></lb>
<emph type="italics"></emph>(Cantilenam ſuſcipit)<emph.end type="italics"></emph.end> ideſt ſoliti erant ad grauiorem vocem canere. </s>
          <s id="s.004355">hic eſt <lb></lb>
fortè ſenſus Ariſtotelis.</s>
        </p>
        <p type="main">
          <s id="s.004356"><arrow.to.target n="marg361"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004357"><margin.target id="marg361"></margin.target>369</s>
        </p>
        <p type="main">
          <s id="s.004358">Problema 13. <emph type="italics"></emph>(Cur in conſonantia Diapaſon graue quidem acuti antiphonum <lb></lb>
accipi potest; grauis verò acutum non poteſt? </s>
          <s id="s.004359">An maximè, quia in vtroque mo­<lb></lb>
dus <expan abbr="vtriuſq;">vtriuſque</expan> contentus est? </s>
          <s id="s.004360">Sed ſi minus, certè in graui acutum eſt, maius enim <lb></lb>
graue eſt)<emph.end type="italics"></emph.end> Cur inquit, ex duobus ſonis, qui Diapaſon efficiant, grauis qui­<lb></lb>
dem habet in ſe Antiphonum acuti, ideſt, in ſe continet etiam acutum: at <lb></lb>
verò acutus non habet antiphonum grauis, ideſt non continet in ſe grauem. <lb></lb>
</s>
          <s id="s.004361">Ratio eſt, inquit, quia in <expan abbr="vtraq;">vtraque</expan> continetur ſonus, ſeu modus <expan abbr="vtriuſq;">vtriuſque</expan> qua­<lb></lb>
tenus voces huius conſonantiæ ſunt eiuſdem naturæ, vt in 7. Probl. dictum <lb></lb>
eſt. </s>
          <s id="s.004362">Sed melius eſt Dicere, quia in graui tanquam in magno acutum veluti <lb></lb>
paruum includitur, vt paulò ante fuſius explicatum eſt.</s>
        </p>
        <p type="main">
          <s id="s.004363"><arrow.to.target n="marg362"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004364"><margin.target id="marg362"></margin.target>370</s>
        </p>
        <p type="main">
          <s id="s.004365">Problema 14. <emph type="italics"></emph>(Cur Antiphonŭm conſonantiæ Diapaſon ita latitat, vt vniſonum <lb></lb>
eſſe videatur, veluti in Punico, aut homine? </s>
          <s id="s.004366">Quæ <expan abbr="namq;">namque</expan> poſita in acutis ſunt, non <lb></lb>
vniſona, ſed ex proportione ſibi Diapaſon concinentia reſpondent? </s>
          <s id="s.004367">An modus pro­<lb></lb>
portionis facit, vt ſonus quaſi <expan abbr="idẽ">idem</expan> eſſe appareat? </s>
          <s id="s.004368">Proportio. </s>
          <s id="s.004369">n. </s>
          <s id="s.004370">in ſonis æqualitas eſt; <lb></lb>
a quale autem omne ad vnitatem <expan abbr="referẽdum">referendum</expan> eſt. </s>
          <s id="s.004371">hoc idem in fiſtolis etiam euenit, vt <lb></lb>
falli aures poſſint)<emph.end type="italics"></emph.end> Quænam ſint voces Antiphonæ in 7. Probl. dictum eſt. </s>
          <s id="s.004372">quod <lb></lb>
ad textum attinet pro verbo, atropo, quod in Gazæ translatione legitur, <lb></lb>
repoſui, Homine, Græcè enim eſt, <foreign lang="grc">ανθρωπῳ,</foreign> ex quo fortè mendosè factum <lb></lb>
eſt illud, atropo. </s>
          <s id="s.004373">quod quid ſibi velit, nuſquam reperitur: Verbum præte­<lb></lb>
rea Punicum, puto ſignificare inſtrumentum aliquod muſicum Phęnicibus <lb></lb>
vſitatum, vel ab eis repertum, Græcè enim legitur in <foreign lang="grc">φοίνικῳ</foreign>. His præmiſ­<lb></lb>
ſis, quæritur, cur Antiphonum, ideſt vocum corriſpondentia in Diapaſon <lb></lb>
ita latitat, vt non duæ voces differentes, ſed duæ vniſonæ, fiue vniſonum <lb></lb>
videatur? </s>
          <s id="s.004374">vt manifeſtè audire eſt in inſtrumento Punico, &amp; in humana voce? <lb></lb>
</s>
          <s id="s.004375">Dubitationis cauſa eſt, quia voces acutæ nullo modo cum grauioribus ſibi <lb></lb>
Antiphonis vniſonæ ſunt, ſed per octo voces ab illis in acutum diſtant.</s>
        </p>
        <p type="main">
          <s id="s.004376">Reſpondet modum proportionis, ideſt duplam proportionem, quæ inter <lb></lb>
huiuſmodi voces reperitur in cauſa eſſe, vt voces illæ videantur vniſonæ. </s>
          <s id="s.004377">eſt <lb></lb>
enim proportio dupla (quæ forma ipſius Diapaſon eſt) ſimpliciſsima, &amp; pri­<lb></lb>
ma inter omnes muſicales proportiones. </s>
          <s id="s.004378">dupla enim proportio eſt omnium <lb></lb>
prima, ac ſimpliciſſima, reliquæ enim, vt ſunt tripla, ſelquialtera, ſeſqui­<lb></lb>
tertia, &amp; huiuſmodi aliæ, ſunt ipſa compoſitiores. </s>
          <s id="s.004379">In dupla enim propor­<lb></lb>
tione altera quantitas diuiditur tantum bifariam, vt ſuperius patuit: diuiſio <lb></lb>porrò bifariam, ſiue in partes æquales eſt prima <expan abbr="omniũ">omnium</expan>, quia magis ad vni­


<pb pagenum="255" xlink:href="009/01/255.jpg"></pb><expan abbr="tateẽ">tatem</expan>, siue ad indiuiſum, &amp; ad æquale accedit, cùm in partes æquas diuidat: <lb></lb>
&amp; cùm ad eam opus ſit unica tantum diuisione. </s>
          <s id="s.004380">In alijs proportionalibus, vt <lb></lb>
in tripla, opus eſt duabus diuiſionibus, vt supra patuit: ſimiliter in alijs, ſeſ­<lb></lb>
quialtera, ſeſquitertia, opus eſt pluribus diuiſionibus: cùm igitur ipſa præ­<lb></lb>cæteris magis ad æqualitatem, &amp; vnitatem accedat, facit,vt voces ipſius <lb></lb>videantur ferè æquales, hoc eſt ferè eædem, &amp; vniſonæ, &amp; eiuſdem naturæ: <lb></lb>id, quod etiam in  ſiſstolis adeò verum eſt, vt aliquando aures decipiant, cùm <lb></lb>nimirum aures iudicent duas fiſstulas eſſe aut vnam tantum, aut duas vni­<lb></lb>
ſonas, quæ tamen re vera ſunt in conſonantia Diapaſon.<arrow.to.target n="marg363a"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004381"><margin.target id="marg363a"></margin.target>371</s>
        </p>
        <p type="main">
          <s id="s.004382">Probl. 15 (Cur genus cantilenæ, quod lex apellatum eſt non per antiſtrophos <lb></lb>
olim agebatur, cùm tamen cæteris chorici canticis antiſtrophi vſus non deeſſet? <lb></lb>
</s>
          <s id="s.004383">An quod olim leges à certatoribus, &amp; pugilibus agebantur, qui cùm iam egregiè <lb></lb>
imitari, <expan abbr="valenterqq́">valenterque</expan>; pertendere poſſent, cantum prolixum, ac varium efficiebant <lb></lb>
<expan abbr="itaqq́">itaque</expan> vt verba, ita etiam moduli, <expan abbr="numeriqq́">numerique</expan> variè ſubinde imitationem inſequeban­<lb></lb>
tur; imitari <expan abbr="namqq́">namque</expan> modulamine potius, quam vocabulis neceſſe eſt. </s>
          <s id="s.004384">Quamobrem <lb></lb>
dithryambi etiam poſteaquam imitari coeperunt, antiſtrophis amplius non vtuntur <lb></lb>
quamquam plurimum ante vterentur. </s>
          <s id="s.004385">Cuius rei cauſa eſt, quod olim homines li­<lb></lb>
beri, atque ingenui ſolebant ipſi tripudiare, atque choreas ducere: ïtaque multos <lb></lb>
eſſe, qui certatorio cantu fungi poſſent, erat difficile: quapropter illis in more fue­<lb></lb>
rat, vt modulos enharmonios cantarent. </s>
          <s id="s.004386">Vnus enim crebrò cantilenam mutare, <lb></lb>
<expan abbr="variamqq́">variamque</expan>, contexere facilius poteſt, quàm multi; &amp; certator, quàm qui mores con­<lb></lb>
ſeruat: quocirca ſimplicius illi modulari debuerant. </s>
          <s id="s.004387">Antiſtrophus autem simplex <lb></lb>
eſt, eſt enim numerus, &amp; ab vno menſuratur: hæc eadem causa eſt, cur in ſcena nul­<lb></lb>
li ſint antiſtrophi: in choro verò maximè. </s>
          <s id="s.004388">Hiſtrio namque ſimul &amp; certator, &amp; <lb></lb>
imitator eſt: chorus verò minus imitatur) </s>
          <s id="s.004389">Cur cantilenæ quædam antiquitus <lb></lb>
leges apellarentur, infra Probl. 28. explicabitur. </s>
          <s id="s.004390">Antiſtrophon hoc loco <lb></lb>
ſumitur pro ſtropha, ſtrophæ autem nihil aliud ſunt, quàm Odarum partes <lb></lb>
illæ ſibi numerò, &amp; genere carminum conſimiles, ex quibus tota ode <expan abbr="coñ">conſtat</expan>. <lb></lb>
 </s>
          <s id="s.004391">Dithryambi erant hymni in honorem Bacchi decantari ſoliti. </s>
          <s id="s.004392">Tandem, vt <lb></lb>
intelligamus quidnam eſſent modi enharmonij, sciendum eſt veteres illos <lb></lb>
Muſicos tria totius muſicæ genera feciſſe, <expan abbr="Diatonicuũ">Diatonicum</expan>, Chromaticum, Enhar­<lb></lb>
monicum. </s>
          <s id="s.004393">quæ genera ab invicem diſtinguebantur, ſecundum variam Te­<lb></lb>
trachordorum <expan abbr="coñ">conſtitutionem</expan>; ex tetrachordis enim totam ſeriem, ſeu Mo­<lb></lb>
nochordium, ſeu regulam harmonicam componebant. </s>
          <s id="s.004394">Erat autem tetra­<lb></lb>
chordum intervallum Diateſſaron, conſtans ex quatuor chordis, ſeu voci­<lb></lb>
bus, vt, re, mi, fa; <expan abbr="quaruũ">quarum</expan> vocum interualla vnius tetrachordi generis, vnius, <lb></lb>
differebant ab interuallis alterius tetrachordi alterius generis, v.g. in ge­<lb></lb>nere Diatonico erat huiuſmodi tetra chordium, cuius primum interualllum <lb></lb>
<figure id="id.009.01.255.1.jpg" place="text" xlink:href="009/01/255/1.jpg"></figure>erat ſemitonium, reliqua verò duo erant to­<lb></lb>
ni.  </s>
          <s id="s.004395">&amp; à prima voce Hypate, ad vltimam Me­<lb></lb>
ſen, erat conſonantia Diateſſeron huic tetra­<lb></lb>
chordo addebant aliud ſimile, mediante tono <lb></lb>
vno inter vtrunque: ita vt ex duobus confla­<lb></lb>
retur tota diapaſon à gravi hypate, ad ſupre­<lb></lb>
mam Neten. </s>
          <s id="s.004396">his interuallis, ac tetrachordis <lb></lb>in genere Diatonico cantabatur. Genus verò <pb pagenum="256" xlink:href="009/01/256.jpg"></pb><figure id="id.009.01.256.1.jpg" place="text" xlink:href="009/01/256/1.jpg"></figure><lb></lb>
<expan abbr="chromaticũ">chromaticum</expan> inter chordas ſui tetrachor­<lb></lb>
di <expan abbr="ſequẽtia">ſequentia</expan> interualla ſeruabat. </s>
          <s id="s.004397"><expan abbr="trihemi-toniũ">trihemi­<lb></lb>
tonium</expan> autem interuallum ex tribus ſemi­<lb></lb>
tonijs conſtabat, ſeu ex vno toto, &amp; vno <lb></lb>
ſemitonio ex duobus huiuſmodi tetra­<lb></lb>
chordis ſuum monochordium, ſeu ſuas <lb></lb>
octo voces, ſeu ſuam Diapaſon genus <lb></lb>
chromaticum componebat. </s>
          <s id="s.004398">Enharmo­<lb></lb>
nicum tandem genus tetrachordo vtebatur, cuius interualla erant ea, quæ <lb></lb>
ſequuntur.</s>
        </p>
        <figure id="id.009.01.256.2.jpg" place="text" xlink:href="009/01/256/2.jpg"></figure>
        <p type="main">
          <s id="s.004399">Erat hic etiam inter hypate, &amp; Meſe Diateſſaron; huic aliud tetrachor­<lb></lb>
dum pariter addebatur, vt in alijs generibus, ex quibus tota Diapaſon <expan abbr="cõ-flabatur">con­<lb></lb>
flabatur</expan>. </s>
          <s id="s.004400">Huiuſcemodi igitur tetrachordis <expan abbr="vnumquodq;">vnumquodque</expan> genus ſuum ſyſte­<lb></lb>
ma, ſiue <expan abbr="conſtitutionẽ">conſtitutionem</expan> Diapaſon componebat, <expan abbr="addẽdo">addendo</expan> priori tetrachordo <lb></lb>
aliud <expan abbr="tetrachordũ">tetrachordum</expan>, ita vt Meſe vltima chorda primi tetrachordi, cùm Nete <lb></lb>
vltima ſecundi tetrachordi Diapente reſonaret; prima verò hypate, cùm <lb></lb>
vltima Nete Diapaſon efficerent, vt ſuperius in ſerie Lychaonis videre eſt.</s>
        </p>
        <p type="main">
          <s id="s.004401">Ex quibus patet quinam eſſent enharmonij moduli, ſiue interualla, qui­<lb></lb>
bus enharmonium genus decantaretur. </s>
          <s id="s.004402">Sciendum præterea ex lib. 3. Muſi­<lb></lb>
corum Ptol. modulos enharmonios fuiſſe graues, &amp; ſeueros, vt idcirco Do­<lb></lb>
rienſes, quorum modi grauitate, ac ſeueritate præditi erant, ipſis maximè <lb></lb>
delectarentur. </s>
          <s id="s.004403">Vnde etiam patere poteſt enharmonios modos minimè cer­<lb></lb>tatorijs canticis idoneos fuiſſe. </s>
          <s id="s.004404">His præmiſſis, ſic textum facilè exponere <lb></lb>
eſt: cur cantilenæ genus illud, quod lex appellatur, non per antiſtrophos, <lb></lb>
ſeu ſtrophas olim agebatur, cùm tamen cæteris choręarum, ac chori can­<lb></lb>
ticis antiſtrophi, ſeu ſtrophæ non deeſſent. </s>
          <s id="s.004405">Ratio huius forſitan hæc eſt; <lb></lb>
quia vſus antiſtrophorum, ſeu ſtropharum eundem ſemper modum per to­<lb></lb>
tam cantilenam ſeruat, cùm cantilena conſtet ex pluribus ſtrophis ſibi ſi­<lb></lb>
milibus: quapropter ſtropharum vſus maximè ei conuenit, qui <expan abbr="eundẽ">eundem</expan> ſem­<lb></lb>
per morem in cantando retinet, è contra verò ei, qui varios mores, <expan abbr="variũq́">variunque</expan>; <lb></lb>
<expan abbr="cantũ">cantum</expan> ſtudet efficere minimè quadrat: talis enim non indiget ſtatutis ſtro­<lb></lb>
phis, nec rithmis, vt ſunt odæ, ſed potius carmine libero, vt ſunt heroica <lb></lb>
poemata hexametris verſibus contexta. </s>
          <s id="s.004406">quia igitur olim certatores, ac pu­<lb></lb>
giles, qui viribus pollebant, <expan abbr="quiq́">quique</expan>; egregiè varios mores imitabantur; cùm <lb></lb>
cantum varium, ac prolixum, intentum, ac remiſſum efficere valerent, hu­<lb></lb>
iuſmodi leges decantabant, propterea nullis ſtrophis vtebantur, vt ſcilicet <lb></lb>
facilius in omnes partes poſſet vox, &amp; cantus excurrere. </s>
          <s id="s.004407"><expan abbr="Itaq;">Itaque</expan> vt verba, ita <lb></lb>etiam modulos, ac numeros, prout imitatio requirebat, ſubinde varios red­<lb></lb>
debant; modulatione enim melius, quam verbis ipſis imitatio perficitur.


<pb pagenum="257" xlink:href="009/01/257.jpg"></pb>hac eadem de cauſa hymni Dithyrambici, poſtquam ad imitationem adhi­<lb></lb>
beri cœperunt, vt liberius imitationi inſeruirent, ſtrophis, quibus antea, <lb></lb>plurimum abundabant, priuati ſunt. </s>
          <s id="s.004408">cur autem olim ſtrophas habuerint, <lb></lb>
quibus modo carent, cauſa eſt, quia olim nobiles viri ſolebant ipſi choros, <lb></lb>
<expan abbr="choreasq́">choreasque</expan>; adire, <expan abbr="atq;">atque</expan> in ipſis tripudiare, ac canere; chori autem, &amp; choreæ <lb></lb>
ſtrophas ſemper habuerunt, in choris enim eundem ſemper morem, ac mo­<lb></lb>
dum, rithmumuè conſeruant; quapropter difficile erat inuenire multos, qui <lb></lb>
certatorio, ac vario ſemper cantu, <expan abbr="variaq́">variaque</expan>; imitatione decantarent: talis <lb></lb>
enim cantus ſtrophas reijcit: eadem de cauſa modulos enharmonios vte­<lb></lb>
bantur in ſuis canticis, quippe qui graues, ac ſeueri erant, <expan abbr="neq;">neque</expan> idonei va­<lb></lb>
rijs rationibus, ac moribus. </s>
          <s id="s.004409">vnus enim, vt accidit in cantu certatorio can­<lb></lb>
tilenam facilius pro libito in omnes partes immutare poteſt, quàm multi, <lb></lb>
vt ſolent eſſe in choro. </s>
          <s id="s.004410">&amp; certator etiam facilius id præſtat, quam qui eun­<lb></lb>
dem ſemper morem, ac modum retinet, quocirca ſimplicius, quod fit per <lb></lb>
ſtrophas illi, qui in choris <expan abbr="canebãt">canebant</expan>, modulari debuerant; ſtropha enim ſim­<lb></lb>
plex eſt, <expan abbr="vnoq́">vnoque</expan>; tempore, ac menſura ſemper eadem menſuratur. </s>
          <s id="s.004411">hæc eadem <lb></lb>cauſa eſt, cur in ſcena nullus, vbi variæ imitationes aguntur, in choro verò, <lb></lb>
vbi ſemper eodem tenore proceditur, plurimus ſtropharum vſus ſit: Hiſtrio <lb></lb>
namque, quì in ſcena agit, &amp; certator, &amp; imitator ſimul eſt; chorus autem <lb></lb>
minus imitatur, hoc eſt ſimplici, <expan abbr="atq;">atque</expan> vniformi ſemper imitatione procedit.</s>
        </p>
        <p type="main">
          <s id="s.004412"><arrow.to.target n="marg363"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004413"><margin.target id="marg363"></margin.target>372</s>
        </p>
        <p type="main">
          <s id="s.004414">Probl. 16. <emph type="italics"></emph>(Qua de cauſa Antiphonum ſuauius est ſymphono? </s>
          <s id="s.004415">An quia in an­<lb></lb>tiphono manifeſtior est ipſa conſonantia, quàm eùm ad ſymphoniam cantatur: ne­<lb></lb>ceſſe enim eſt in ſymphonia alteram vocem alteri vniſonam eſſe; ita vt duæ in ean­<lb></lb>
dem coaleſcentes altera alteram offuſcare poſſit)<emph.end type="italics"></emph.end> Per antiphonum intelligit nunc <lb></lb>
Ariſt. conſonantiam ex vocibus <expan abbr="differẽtibus">differentibus</expan> conflatam, cuiuſmodi eſt Dia­<lb></lb>
paſon, Diapente, &amp; Diateſſaron: per ſymphonum intelligit conſonantiam <lb></lb>
ex vocibus eiuſdem intenſionis, ſiue ex vniſonis. </s>
          <s id="s.004416">non me latet aliter Muſi­<lb></lb>
cos antiphonas, ſymphonas, ac homophonas accipere. </s>
          <s id="s.004417">vide Prolæm. lib. 1. <lb></lb>
cap. 7. harm. </s>
          <s id="s.004418">ſed hoc loco ita accipiendas eſſe, vti dixi, manifeſtum eſt ex <lb></lb>
Ariſt. contextu. </s>
          <s id="s.004419">Ait igitur ſuauiorem eſſe antiphonarum conſonantiam, <lb></lb>
quam vniſonarum; quia ibi conſonantia melius percipitur; nam in vocibus <lb></lb>vniſonis, vox alteri voci conſonans, eundem cum illa edit ſonum, ita vt duæ <lb></lb>
in vnam, <expan abbr="eandemq́">eandemque</expan>; prorſus coaleſcant, <expan abbr="ſicq́">ſicque</expan>; altera alteram offuſcet, vnde <lb></lb>
conſonantia, quæ ex pluribus conſtare debet, non percipitur.</s>
        </p>
        <p type="main">
          <s id="s.004420"><arrow.to.target n="marg364"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004421"><margin.target id="marg364"></margin.target>373</s>
        </p>
        <p type="main">
          <s id="s.004422">Probl. 17. <emph type="italics"></emph>(Cur ſola Diapaſon conſonantia cantatur? </s>
          <s id="s.004423">ſecundum hanc enim, &amp; <lb></lb>
nullam aliam magadare ſolent. </s>
          <s id="s.004424">An quod hæc ſola ex fidibus inuicem antiphonis <lb></lb>
conſtat? </s>
          <s id="s.004425">in antiphonis autem etiamſi alteram tantum canis, idem efficies, voces <lb></lb>enim vtriuſque chordæ vna ſola continet. </s>
          <s id="s.004426">ita vt in hac conſonantia, quamuis vox <lb></lb>
vna tantum cantetur, tota tamen conſonantia quodammodo canitur. </s>
          <s id="s.004427">ita vt in hac <lb></lb>
ſymphonia, &amp; vnica voce canente, &amp; duabus, exurgat quodammodo harmonia. <lb></lb>
</s>
          <s id="s.004428">vel vna decantante, altera verò per tibiam ſonante, veluti vnam ambæ, conſtituunt. <lb></lb>
</s>
          <s id="s.004429">propterea in ſola Diapaſon canere ſolemus, quoniam, inquam, voces antiphonæ <lb></lb>
vnius, eiuſdemqué chordæ vocem obtinent)<emph.end type="italics"></emph.end> Sciendum primò apud veteres vſui <lb></lb>
fuiſſe inſtrumentum quoddam muſicum, quod Magadis, &amp; Magas appella­<lb></lb>
batur, ad quod ſuas cantilenas canere ſolebant, <expan abbr="atq;">atque</expan> hoc erat <foreign lang="grc">μαγαδίζειν</foreign><lb></lb>
magadiſſare. </s>
          <s id="s.004430">Erat autem vnius chordæ tantum, vel vnius vocis, ſi fortè fue­


<pb pagenum="258" xlink:href="009/01/258.jpg"></pb>rit tibia, ideſt, quod vnicam vocem, &amp; non plures ſimul ederet, quemad­<lb></lb>
modum refert Zarlinus; quamuis varias voces ſucceſſiuè poſſet edere. </s>
          <s id="s.004431">hoc <lb></lb>
enim pacto ad ipſum canentes, Diapaſon cum ipſo facilè effeciſſent. </s>
          <s id="s.004432">Notan­<lb></lb>
dum præterea Ariſt. ſumere in textu Antiphonum pro ſola Diapaſon. </s>
          <s id="s.004433">Quæ­<lb></lb>
rit igitur, cur canentes ſoliti ſint per ſolam Diapaſon conſonantiam cane­<lb></lb>
re, quod probat ex vſu Magadis, quod vulgò ad cantum adhibere ſolebant, <lb></lb>
cùm eo enim omnes in Diapaſon conueniebant. </s>
          <s id="s.004434">Cauſam huius in identita­<lb></lb>
tem, vt aiunt, <expan abbr="vocũ">vocum</expan>, ex quibus Diapaſon conſtat, refert. </s>
          <s id="s.004435">quamuis enim non <lb></lb>
ſint vniſonæ duæ voces octauam conſtituentes, ſunt tamen eiuſdem naturæ, <lb></lb>
&amp; acutior, vt ſupra dictum eſt, reſpectu grauioris eſt eadem cum graui, in <lb></lb>
acutiori vocum ordine, quaſi renata. </s>
          <s id="s.004436">ob quam adeò perfectam duorum vo­<lb></lb>
cum ſimilitudinem fit, vt illarum altera cantata, aut ſonata, altera natura­<lb></lb>
liter ad illius præſentiam excitetur, &amp; decantetur: vnde huiuſmodi voces <lb></lb>
mutuam obtinent altera alterius vim. </s>
          <s id="s.004437">Hinc fit, vt paſſim in agris, ac pra­<lb></lb>
tis ipſi meſſores, <expan abbr="atq;">atque</expan> paſtores naturalia quadam harum vocum ſimilitudi­<lb></lb>
ne prouocati, ſolam Diapaſon conſonantiam ſuauiter ſimul canentes, ſuos <lb></lb>
labores fœliciter fallant.</s>
        </p>
        <p type="main">
          <s id="s.004438"><arrow.to.target n="marg365"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004439"><margin.target id="marg365"></margin.target>374</s>
        </p>
        <p type="main">
          <s id="s.004440">Probl. 18. <emph type="italics"></emph>(Sed cur Solis Antiphonis vocibus hoc ineſt? </s>
          <s id="s.004441">An quod ſole pari in­<lb></lb>
teruallo diſtant à Meſe. </s>
          <s id="s.004442">Medietas igitur ſimilitudinem quandam tonorum efficit, <lb></lb>
vt ſenſus aurium dicat, quod eadem, &amp; quod ambæ extremæ)<emph.end type="italics"></emph.end> Quærit cauſam <lb></lb>
tantæ ſimilitudinis inter voces Diapaſon conſtituentes, de qua ſimilitudi­<lb></lb>
ne <expan abbr="dictũ">dictum</expan> eſt in præcedenti problemate: ait igitur fortè hanc ſimilitudinem <lb></lb>
inde prouenire, quod <expan abbr="vtraq́">vtraque</expan>; illarum <expan abbr="duarũ">duarum</expan> vocum, quæ Diapaſon efficiunt, <lb></lb>
æquidiſtat à Meſe, ſeu Media: grauis deorſum, acuta verò ſurſum: quare <lb></lb>
tot gradus grauitatis grauis obtinebit, quot acuta acuminis, igitur ſimilli­<lb></lb>
mæ erunt, &amp; propterea auditus iudicat vnam eſſe, quæ quidem ratio iuxta <lb></lb>
ordinem <expan abbr="Terpãdri">Terpandri</expan>, &amp; antiquorum illius æui nullam habet <expan abbr="difficultatẽ">difficultatem</expan>, cùm <lb></lb>
ſeptem tantum fidibus, quarum media Meſe erat totum <expan abbr="Monochordiũ">Monochordium</expan> con­<lb></lb>
ſtituerent. </s>
          <s id="s.004443">At verò in ordine Lychaonis, &amp; poſteriorum, qui octo chordas <lb></lb>
aſſumebant, aliter re ſe haberet.</s>
        </p>
        <p type="main">
          <s id="s.004444"><arrow.to.target n="marg366"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004445"><margin.target id="marg366"></margin.target>375</s>
        </p>
        <p type="main">
          <s id="s.004446">Probl. 19. <emph type="italics"></emph>(Cur non canunt Diapente, &amp; Diateſſaron in Antiphonis? </s>
          <s id="s.004447">An quod <lb></lb>
non eadem conſonandi ratio ijs ineſt, quæ in Diapaſon: in qua vox grauis eundem <lb></lb>
habet in grauitate modum, quem acuta in acumine: ita vt, &amp; eadem vox quidem, <lb></lb>
&amp; ſimul diuerſa oriatur. </s>
          <s id="s.004448">At però in Diapente, &amp; Diateſſaron non ita est, quam­<lb></lb>
obrem ſonus vocis oppoſitæ non apparet; non enim eſt idem)<emph.end type="italics"></emph.end> Cur in quotidia­<lb></lb>
nis cantilenis, in quibus voces non vniſonæ, ſed diuerſæ, ſeu antiphonæ ad­<lb></lb>
hibentur, non vtuntur vocibus Diapente, aut Diateſſaron reſonantibus, ſed <lb></lb>
tantum, vt antea dictum eſt, Diapaſon. </s>
          <s id="s.004449">Ratio, inquit, eſt, quia inter voces <lb></lb>
illarum non eſt tanta ſimilitudo, quanta in vocibus Diapaſon conſonantiæ, <lb></lb>
in qua vox grauis tanta eſt in grauitate, quanta acuta in acumine; &amp; proin­<lb></lb>
de non ita naturaliter, ac facilè ſe produnt, &amp; canuntur, quemadmodum <lb></lb>
Diapaſon, vbi vox altera alteram ob naturalem ſimilitudinem prouocat.</s>
        </p>
        <p type="main">
          <s id="s.004450"><arrow.to.target n="marg367"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004451"><margin.target id="marg367"></margin.target>376</s>
        </p>
        <p type="main">
          <s id="s.004452">Probl. 20. <emph type="italics"></emph>(Cur ſi quis, mota Meſe, alijs quamuis omnibus chordis benè con­<lb></lb>
ſonantibus, inſtrumento vtatur, non ſolŭ cum ad Meſes ſonŭ peruenerit, ſed etiam <lb></lb>
in reliqua melodia, aures anget, modumque <expan abbr="incõinnum">incom<gap></gap>innum</expan> efficient: ſi verò Lycha­<lb></lb>
nos, aut alia quæpiam mota fuerit, tunc diſcrimen, aut inc<expan abbr="õ">on</expan>cinnitas ſolum appare-<emph.end type="italics"></emph.end>


<pb pagenum="259" xlink:href="009/01/259.jpg"></pb><emph type="italics"></emph>bit, cùm ipſam, quis pulſauerit? </s>
          <s id="s.004453">An non ratione id optima accidit? </s>
          <s id="s.004454">quandoqui­<lb></lb>
dem optima <expan abbr="quæq;">quæque</expan> melodiæ gratia ſæpè Meſe vtuntur: omneſque probi Poetæ cre­<lb></lb>
brò ad meſen veniunt: &amp; ſi ab ea diſceſſerint, ad eam ſtatim reuertuntur: nec vllam <lb></lb>
aliam toties repetunt. </s>
          <s id="s.004455">quemadmodum igitur demptis ex oratione quibuſdam con­<lb></lb>
iunctionibus (veluti <foreign lang="grc">τὲ, &amp; ἤ</foreign>) non eſt amplius ſermo græcus; alijs verò detractis <lb></lb>nihil ſermoni detrahitur; eò, quod illis vti ſæpè neceſſe eſt, his verò perrarò. </s>
          <s id="s.004456">ſic <lb></lb>etiam ſonus meſes eſt veluti aliorum ſonorum coniunctio, maximeque pulchriorŭ; <lb></lb>
propterea eius ſonus ſæpiſſimè aſſumi ſolet)<emph.end type="italics"></emph.end> Si quis ea, quæ initio dicta ſunt, pro­<lb></lb>
bè tenuerit, facilè ad huius problematis intelligentiam perueniet; per me­<lb></lb>
ſen motam intellige de ſuo ſibi conuente ſtatu <expan abbr="dimotã">dimotam</expan>, &amp; ideò ab alijs chor<lb></lb>
dis diſſonantem. </s>
          <s id="s.004457">Idem quærit Probl. 38.</s>
        </p>
        <p type="main">
          <s id="s.004458"><arrow.to.target n="marg368"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004459"><margin.target id="marg368"></margin.target>377</s>
        </p>
        <p type="main">
          <s id="s.004460">Probl. 21. <emph type="italics"></emph>(Cur qui grauius căntant, ſi abſonant deprehendi facilius poſſunt, quă, <lb></lb>
qui cantant acutius: nec verò ſecus in rithmis accidit. </s>
          <s id="s.004461">euidentiores enim, qui pec­<lb></lb>
cant in grauiori. </s>
          <s id="s.004462">Vtrum, quòd plus temporis graue obtinet? </s>
          <s id="s.004463">plus autem plenius à <lb></lb>ſenſu auriŭm percipi poteſt. </s>
          <s id="s.004464">Vel quia illud in ampliori tempore agitur, &amp; ideò am­<lb></lb>
pliorem etiam ſui ſenſationem exhibet. </s>
          <s id="s.004465">Velox autem, &amp; acutŭ facilè ob ſuam ve­<lb></lb>
locitatem latitat)<emph.end type="italics"></emph.end> Quid eſſet rithmus explicabitur in problemate 27. <expan abbr="ſequẽ-ti">ſequen­<lb></lb>
ti</expan> ait: <emph type="italics"></emph>(Velox autem, &amp; acutum)<emph.end type="italics"></emph.end> Cur vox acuta ſit velox, <expan abbr="dictũ">dictum</expan> eſt in 1. Top. <lb></lb>
cap. 13. reliqua ſunt ſatis clara.</s>
        </p>
        <p type="main">
          <s id="s.004466">Probl. 22. ex ſe manifeſtum eſt: atque idem cum ſequenti num. </s>
          <s id="s.004467">46.</s>
        </p>
        <p type="main">
          <s id="s.004468"><arrow.to.target n="marg369"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004469"><margin.target id="marg369"></margin.target>378</s>
        </p>
        <p type="main">
          <s id="s.004470">Probl. 23. <emph type="italics"></emph>(Cur Nete duplo acutior eſt hypate? </s>
          <s id="s.004471">An primum, quod cum ner­<lb></lb>
uus parte ſui dimidia, &amp; totus ſimul pulſatur, Diapaſon concinentia exultat: quod <lb></lb>
pariter in fiſtulis apparet, ſonus enim, qui per medium foramen emergit Diapaſon <lb></lb>
cum eo reſonat, qui per totam fiſtulam exit. </s>
          <s id="s.004472">In cæteris etiam duplo interuallo Dia­<lb></lb>
paſon continetur, nam, &amp; qui tibias perforant, it a eas oſſumunt. </s>
          <s id="s.004473">&amp; qui fiſtulas aptè <lb></lb>elaborant, ſumitatem extremam tăntum hypates circumlinunt: netem verò ad <expan abbr="vſq;">vſque</expan> <lb></lb>
dimidium obturant. </s>
          <s id="s.004474">&amp; in Triquetris Pſalterijs, nerui, quorum alter ſit alterius <lb></lb>
longitudine duplus, æquè intenti Diapaſon reddunt. </s>
          <s id="s.004475">Diapente verò ſeſquialtera <lb></lb>
proportione; Diateſſaron autem ſeſquitertio interuallo continetur)<emph.end type="italics"></emph.end> Ex ijs, quæ <lb></lb>
initio huius tractationis de Monochordij diuiſione, <expan abbr="deq́">deque</expan>; Diapaſon, <expan abbr="Diapẽ-te">Diapen­<lb></lb>
te</expan>, Diateſſaron conſonantiarum ordine, ac proportione dicta ſunt, perſpi­<lb></lb>
cua omnino redduntur omnia, quæ hic quæruntur, &amp; redduntur. </s>
          <s id="s.004476">Illud no­<lb></lb>
tandum Triquetrum Pſalterium inſtrumentum muſicum fuiſſe, à triangula­<lb></lb>
ri figura denominatum, noſtræ forſan Harpæ, perſimile: in quo fides eſſent <lb></lb>
eo modo diſpoſitæ, ac intentæ, vt in Harpa.</s>
        </p>
        <p type="main">
          <s id="s.004477"><arrow.to.target n="marg370"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004478"><margin.target id="marg370"></margin.target>379</s>
        </p>
        <p type="main">
          <s id="s.004479">Probl. 24. <emph type="italics"></emph>(Cur ſi quis pſallens netem pulſatam apprehenderit, ſolam Hypa­<lb></lb>
tem reſonare videbitur? </s>
          <s id="s.004480">An quod tinnitus huius maximè connaturalis eſt ſono il­<lb></lb>
lius, illique conſonus. </s>
          <s id="s.004481">quia igitur cum ſuo conſimili augetur, hoc ceſſante, ille ſo­<lb></lb>
lus apparet ſoni verò alij propter paruitatem euaneſcunt)<emph.end type="italics"></emph.end> Cur ſi quis dum pſal­<lb></lb>
terium pulſatur, neten pulſatam ſonantem manu apprehenderit, ita vt ſo­<lb></lb>num ipſius interpellet, ſonus ille intermortuus, ac dimidiatus, videbitur ſo­<lb></lb>nus hypates, &amp; non alterius chordæ, quia, vt dictum eſt, hypates, &amp; nete, <lb></lb>
Diapaſon reſonant; cuius conſonantiæ voces ſunt eiuſdem naturæ, aut val­<lb></lb>
dè <expan abbr="connaturãles">connaturales</expan>; imò ſonus hypates duplus eſt ſoni netes. </s>
          <s id="s.004482">Interpellato igi­<lb></lb>
tur acutioris ſono, reliquus qui ipſi adeò ſimilis eſt meritò videbitur hypa­<lb></lb>tes: ſoni verò aliarum chordarum ob ipſorum paruitatem, quia nimirum <pb pagenum="260" xlink:href="009/01/260.jpg"></pb>minores, quam ſub dupli illius ſunt, omninò euaneſcunt. </s>
          <s id="s.004483">Hic eſt ſenſus he­<lb></lb>
rum verborum; vtrum autem allata ratio ſit bona, aliorum eſto iudicium. <lb></lb>
</s>
          <s id="s.004484">Idem quærit num. </s>
          <s id="s.004485">43.</s>
        </p>
        <p type="main">
          <s id="s.004486"><arrow.to.target n="marg371"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004487"><margin.target id="marg371"></margin.target>380</s>
        </p>
        <p type="main">
          <s id="s.004488">Probl. 25. <emph type="italics"></emph>(Cur in harmonijs chorda illa, quæ dicitur Meſe, ſeu media, ſic ap­<lb></lb>pellata eſt? </s>
          <s id="s.004489">cum inter octo nullum medium ſit? </s>
          <s id="s.004490">An quoniam olim harmoniæ ſep­<lb></lb>
tem fidibus conſtabant, qui numerus medium habet)<emph.end type="italics"></emph.end> Ex ordine chordarum Li­<lb></lb>
chaonis, &amp; <expan abbr="Terpãdri">Terpandri</expan>, quorum alter ſeptem, alter verò octo chordis mono­<lb></lb>
chordium conflabat, vt ſupra recenſui, huic loco abundè ſatisfieri poteſt.</s>
        </p>
        <p type="main">
          <s id="s.004491">Probl. 26. ſatis ex ſe clarum eſt, <expan abbr="atq;">atque</expan> idem cum num. </s>
          <s id="s.004492">47.</s>
        </p>
        <p type="main">
          <s id="s.004493"><arrow.to.target n="marg372"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004494"><margin.target id="marg372"></margin.target>381</s>
        </p>
        <p type="main">
          <s id="s.004495">Probl. 27. <emph type="italics"></emph>(Cur inter omnia, quæ ſub ſenſus cadunt, ſola audibilia mores obti­<lb></lb>
nent? </s>
          <s id="s.004496">quamuis ſine ſermone aliquid modulemur, mores tamen præ ſe ipſa modula­<lb></lb>
tio fert, ſed nec color, nec odor, nec ſapor id habet. </s>
          <s id="s.004497">An quia motum non ſolŭ eum <lb></lb>obtinet, quo ipſe ſtrepitus aures mouet (talis enim motio reliquis etiam ſenſibus <lb></lb>
ineſt, nam, &amp; color mouet viſum) ſed illum etiam quem poſt prædictum, <expan abbr="ſubſequẽ-tem">ſubſequen­<lb></lb>
tem</expan> percipimus: hic. </s>
          <s id="s.004498">n. </s>
          <s id="s.004499">ſimilitudinem habet, &amp; in rithmis, &amp; in ſonorum grauium, <lb></lb>
&amp; acutorum ordine: non autem in eorum mixtione; quod in alijs ſenſibilibus non <lb></lb>
eſt. </s>
          <s id="s.004500">porrò motus ipſi practici ſunt, praxis autem morum index est)<emph.end type="italics"></emph.end> Mores obti­<lb></lb>
nere, aut præſeferre nihil aliud eſt, quàm mores illius referre, &amp; in mentem <lb></lb>
reuocare, à quo talis motus, aut ſonus prouenire ſolet, qui ſonus mores il­<lb></lb>
los refert. </s>
          <s id="s.004501">propterea videmus cantilenas nonnullas turpes mores reddere, <lb></lb>
vt laſciuiam, procacitatem, leuitatem, quia à natura <expan abbr="hominũ">hominum</expan> turpium, vt <lb></lb>
laſciuorum proficiſci <expan abbr="ſolẽt">ſolent</expan>, <expan abbr="eosq́">eosque</expan>; decent. </s>
          <s id="s.004502">alij <expan abbr="cãtus">cantus</expan> ex oppoſito bonos mo­<lb></lb>res referunt, vt grauitatem, temperantiam, æſtitatem; qui quidem ex pro­<lb></lb>
borum hominum natura prodire ſolent, <expan abbr="eosq́">eosque</expan>; decent. </s>
          <s id="s.004503">Illud in prophanis <lb></lb>
canticis, iſtud verò in Eccleſiaſticis quotidie experimur; cur autem Audi­<lb></lb>
bilia præ cætéris mores referant, cauſam Ariſt. refert in motum illum, qui <lb></lb>
in ſonis, &amp; vocibus percipitur. </s>
          <s id="s.004504"><expan abbr="neq;">neque</expan> hic motus eſt is, quo ſonus aures immu­<lb></lb>
tat, hoc enim commune eſt omnibus ſenſorijs, vt à ſuis obiectis immuten­<lb></lb>tur, &amp; afficiantur: ſed is eſt, qui prædictam aurium immutationem ſubſe­<lb></lb>
quitur, <expan abbr="intellectuq́">intellectuque</expan>; percipitur, v.g. quando audimus cantilenam, ſonus ip­<lb></lb>
ſe primò aures ferit, <expan abbr="easq́">easque</expan>; afficit; deinde percipimus vocis ipſius motum, <lb></lb>&amp; quaſi curſum, quo à graui in acutum, &amp; è cóntra, aliquando celeriter, ali­<lb></lb>
quando tardè vario modulamine mouetur. </s>
          <s id="s.004505">huiuſmodi motus habet in ſe <lb></lb>morum ſimilitudinem; hac igitur de cauſa audibilia mores referunt. </s>
          <s id="s.004506">Vide <lb></lb>
infra probl. </s>
          <s id="s.004507">39.</s>
        </p>
        <p type="main">
          <s id="s.004508">Iam explicandum eſt breuiter, quid ſit rithmus, quem Latini numerum <lb></lb>
dicunt partim ex Platone, partim ex Ariſt. </s>
          <s id="s.004509">Plato lib. 2. de leg. ſic. </s>
          <s id="s.004510">alia qui­<lb></lb>
dem animalia non habent ſenſationem ordinationis, &amp; inordinationis mo­<lb></lb>
tuum, quibus rithmus, &amp; harmonia nomen eſt. </s>
          <s id="s.004511">Ariſt. infra probl. </s>
          <s id="s.004512">38. ſic. <lb></lb>
</s>
          <s id="s.004513">rithmo verò gaudemus, quia habet numerum manifeſtum, ordinatum, ra­<lb></lb>
tumque: vnde &amp; nos ordinatè mouet. </s>
          <s id="s.004514">Ex quibus patet, rithmum eſſe cer­<lb></lb>tam, ac ſtatam periodum aliquot interuallerum ſibi ſuccedéntium in quouis <lb></lb>
motu in determinata menſura temporis. </s>
          <s id="s.004515">quæ periodus ſolet ſæpius recur­<lb></lb>
rere, aut repeti. </s>
          <s id="s.004516">dictum eſt in quouis motu, quia in choreis pedum pulſa­<lb></lb>
tione, ac motu, rithmi complures efficiuntur, quos choreæ magiſtri <expan abbr="docẽt">docent</expan>, <lb></lb>
qualis eſt is, quem vulgò <expan abbr="dicũt">dicunt</expan> Gagliarda. </s>
          <s id="s.004517">Digitorum etiam motu, &amp; mal­


<pb pagenum="261" xlink:href="009/01/261.jpg"></pb>leorum ictibus poteſt rithmus fieri: <expan abbr="atq;">atque</expan> adeò cæteris omnibus, quæ in ſuo <lb></lb>
motu certis interuallis mouentur; ita vt etiam piſtores ipſi machina ſua il­<lb></lb>
la, qua maſſam ſubigunt, rithmum quendam efficere ſoleant. </s>
          <s id="s.004518">His porrò mo­<lb></lb>
tus ſi in vocibus, ac ſonis muſicis, ſeu in cantilenis exiſtat, præcipuè rithmus <lb></lb>
dicitur, quod ſi concinnus ſit, &amp; elegans aures ſuauiter mulcet, <expan abbr="animumq́">animumque</expan>; <lb></lb>
in varias paſſiones inducit: rithmum hunc, qui in cantilenis eſt, vulgò can­<lb></lb>
tores appellant Ariam. </s>
          <s id="s.004519">Vnde qui intelligit, quid ſint Ariæ, quæ paſſim can­<lb></lb>tantur, ac ſonantur, facilè etiam quid ſit rithmus, intelliget. </s>
          <s id="s.004520">Hic igitur <lb></lb>
rithmus miram habet in ſe morum ſimilitudinem, quæ conſiſtit in motu <lb></lb>
rithmi, ſeu in ordine interuallorum aptiſſimo, per quæ vox aſcendit, &amp; de­<lb></lb>
ſcendit: nullo autem modo conſiſtit in mixtione ſonorum grauium, &amp; acu­<lb></lb>
torum; ex hac enim mixtione non rithmus, ſed conſonantia exurgit. </s>
          <s id="s.004521">mo­<lb></lb>
tus autem omnis fit per aliquam actionem, actio verò omnis eſt morum il­<lb></lb>
lius, cuius eſt actio manifeſtatrix. </s>
          <s id="s.004522">ex quibus patet, cur in cantilenis rithmi­<lb></lb>
cis mores appareant, non autem in cæteris ſenſuum obiectis.</s>
        </p>
        <p type="main">
          <s id="s.004523"><arrow.to.target n="marg373"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004524"><margin.target id="marg373"></margin.target>382</s>
        </p>
        <p type="main">
          <s id="s.004525">Problema 28. <emph type="italics"></emph>(Cur pleræque cantilenæ leges appellantur? </s>
          <s id="s.004526">An quod homines <lb></lb>
priusqué, literas ſcirent, leges cantabant, ne eas obliuione traderent, quod etiam <lb></lb>noſtra ætate Agathyrſis in more eſt. </s>
          <s id="s.004527">ergò primas poſteriorum cantilenarum, eodem <lb></lb>
appellauerunt nomine, quo omnes ſuperiores vocantur)<emph.end type="italics"></emph.end> Agathyrſi populi à Pli­<lb></lb>
nio, &amp; Pomponio Mela ſupra paludem Meotidem inter Scythicas nationes <lb></lb>
numerantur. </s>
          <s id="s.004528">cur autem cantilenæ nonnullæ leges dicerentur, præter ratio­<lb></lb>nem hic ab Ariſtot. allatam, aliam Plutarchus de Muſica affert, vbi ſic ait: <lb></lb>
Non enim antiquitus pro libidine cuiuſque, vti nunc, licebat fidibus canere, <lb></lb>
nec rithmos, <expan abbr="concentusq́">concentusque</expan>; transferre; in ipſis <expan abbr="namq;">namque</expan> legibus accommoda­<lb></lb>
tam cuique tentionem tuebantur, cuius rei cauſa id nominis inditum erat; <lb></lb>
leges enim ſunt vocatæ quoniam præſcriptum, quaſi lege, <expan abbr="cautumq́">cautumque</expan>; erat, ne <lb></lb>
quis pro qualibet, vnam ſpeciem, <expan abbr="formamq́">formamque</expan>; tentionis lege ſancitam, tranſ­<lb></lb>
grederetur. </s>
          <s id="s.004529">hæc ille. </s>
          <s id="s.004530">fubdit poſtea alias fuiſſe ſimiles illis harmonijs, quas <lb></lb>
nunc ſonatas dicimus, fuiſſe tamen ſtatas, ac determinatas numero, quibus <lb></lb>
ſolis vti liceret.</s>
        </p>
        <p type="main">
          <s id="s.004531">Probl. 29. Idem eſt cum præcedenti 27. eadem igitur <expan abbr="quoq;">quoque</expan> ſit explicatio.</s>
        </p>
        <p type="main">
          <s id="s.004532"><arrow.to.target n="marg374"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004533"><margin.target id="marg374"></margin.target>383</s>
        </p>
        <p type="main">
          <s id="s.004534">Probl. 30. <emph type="italics"></emph>(Cur <expan abbr="neq;">neque</expan> hypodorium, <expan abbr="neq;">neque</expan> hypophrygium est in tragœdiarum obo­<lb></lb>
ro? </s>
          <s id="s.004535">An quia non habet antiſtrophon, vtpotè quæ ſcenica ſunt, imitationiqué, accom­<lb></lb>
modata)<emph.end type="italics"></emph.end> huc pertinent ea, quæ ad cap. 2. lib. 3. Polit. ſcripſi, de tonis, Do­<lb></lb>
rio, Phrygio, Lydio. </s>
          <s id="s.004536">quibus nunc hæc addo, ex Boetio lib. 4. tonus, ſen mo­<lb></lb>
dus erat quædam cantus conſtitutio, ab hypate <expan abbr="vſq;">vſque</expan> ad netem, proprio rith­<lb></lb>
mo modificata: ita vt modos Dorius alium rithmum, à Phrygio, &amp; reliquis <lb></lb>
diſcrepantem haberet. </s>
          <s id="s.004537">quilibet preterea modus ſuam certam ſedem in Mo­<lb></lb>
nochordio obtinebat, vnde ſequebatur vnum eſſe reliquis omnibus <lb></lb>
grauiorem, alium eſſe omnium acutiſſimum, reliquos verò in­<lb></lb>
termedios, alijs grauiores fuiſſe, vt in ſequenti figura, <lb></lb>
in qua, tanquam in tabella, omnia, quæ de hiſce <lb></lb>
modis dici ſolent, perſpicuè licet <lb></lb>intueri.</s>
        </p>
        <pb pagenum="262" xlink:href="009/01/262.jpg"></pb>
        <p type="head">
          <figure id="id.009.01.262.1.jpg" place="text" xlink:href="009/01/262/1.jpg"></figure>
          <s id="s.004538"><emph type="italics"></emph>ORDO ANTIQVORVM MODORVM.<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table6"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table6"></table.target>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>Nete.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>T.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>Nete.</cell>
            <cell>Paranete.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>S.</cell>
            <cell>S.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>Nete.</cell>
            <cell>Paranete.</cell>
            <cell>Trite.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>Nete.</cell>
            <cell>Paranete.</cell>
            <cell>Trite.</cell>
            <cell>Meſe.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>Nete.</cell>
            <cell>Paranete.</cell>
            <cell>Trite.</cell>
            <cell>Meſe.</cell>
            <cell>Lychanos.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>S.</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell>Nete.</cell>
            <cell>Paranete.</cell>
            <cell>Trite.</cell>
            <cell>Meſe.</cell>
            <cell>Lychanos</cell>
            <cell>Parhypate</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>Nete.</cell>
            <cell>Paranete.</cell>
            <cell>Trite.</cell>
            <cell>Meſe.</cell>
            <cell>Lychanos.</cell>
            <cell>Parhypat.</cell>
            <cell>Hypate.</cell>
          </row>
          <row>
            <cell></cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
          </row>
          <row>
            <cell>Nete.</cell>
            <cell>Paranete.</cell>
            <cell>Trite.</cell>
            <cell>Meſe.</cell>
            <cell>Lychanos.</cell>
            <cell>Parhypat.</cell>
            <cell>Hypatc.</cell>
            <cell>Proſlamb.</cell>
          </row>
          <row>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>— —</cell>
          </row>
          <row>
            <cell>Paranete.</cell>
            <cell>Trite.</cell>
            <cell>Meſe.</cell>
            <cell>Lychanos</cell>
            <cell>Parhypat.</cell>
            <cell>Hypate.</cell>
            <cell>Proſlamb.</cell>
            <cell>Hypermix</cell>
          </row>
          <row>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>— —</cell>
            <cell>tolydius.</cell>
          </row>
          <row>
            <cell>Trite.</cell>
            <cell>Meſe.</cell>
            <cell>Lychanos.</cell>
            <cell>Parhypat.</cell>
            <cell>Hypate.</cell>
            <cell>Proſlamb.</cell>
            <cell>Mixtolyd.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>— —</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>Meſe.</cell>
            <cell>Lychanos.</cell>
            <cell>Parhypat.</cell>
            <cell>Hypate.</cell>
            <cell>Proſlamb.</cell>
            <cell>Lydius.</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>— —</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>Lychanos</cell>
            <cell>Parhypat.</cell>
            <cell>Hypate.</cell>
            <cell>Proſlamb.</cell>
            <cell>Phrygius.</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>S.</cell>
            <cell>— —</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>Parhypate</cell>
            <cell>Hypate.</cell>
            <cell>Proſlamb.</cell>
            <cell>Dorius.</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>T.</cell>
            <cell>T.</cell>
            <cell>— —</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>Hypate.</cell>
            <cell>Proſlamb.</cell>
            <cell>Hypolyd.</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>T.</cell>
            <cell>— —</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>Proſlamb.</cell>
            <cell>Hypophry</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>— —</cell>
            <cell>gius.</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>Hypodor.</cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
        </table>
        <pb pagenum="263" xlink:href="009/01/263.jpg"></pb>
        <p type="main">
          <s id="s.004539">In qua apparet Hypodorium fuiſſe omnium <expan abbr="grauiſſimũ">grauiſſimum</expan>, quo acutior erat <lb></lb>tono vno Hypophrygius; ſic reliqui præcedentibus erant vel tono, vel ſe­<lb></lb>
mitonio acutiores. </s>
          <s id="s.004540">literæ T, &amp; S, ſignificant Tonos, &amp; Semitonia, quibus <lb></lb>
voces ſingulorum modorum diſtabant. </s>
          <s id="s.004541">ex quibus etiam apparet vario or­<lb></lb>
dine interualla vnius Modi ſe habuiſſe, <expan abbr="atq;">atque</expan> in alio. </s>
          <s id="s.004542">præterea <expan abbr="vnumquemq;">vnumquemque</expan> <lb></lb>
modum vnius Diapaſon conſtitutionem habuiſſe. </s>
          <s id="s.004543">tres illi Hypodorius, Hy­<lb></lb>
pophrygius, Hypolydius, ita ſunt appellati, quod collocati eſſent infra Do­<lb></lb>
rium, Phrygium, Lydium per vnum Tetrachordum, vt patet in figura. </s>
          <s id="s.004544">ſed <lb></lb>
vt adhuc melius hanc rem intelligamus, dicendum eſt cum Zarlino lib. 4. <lb></lb>
Inſtit. </s>
          <s id="s.004545">Modos fuiſſe varias ſpecies cantilenarum proprios rithmos haben­<lb></lb>
tes, certo ordine, ac certo carmine, <expan abbr="certoq́">certoque</expan>; etiam inſtrumento decantari <lb></lb>
ſolitas: denominabantur autem Doriæ, Phrygiæ, &amp;c. </s>
          <s id="s.004546">ab illis ſcilicet natio­<lb></lb>nibus, apud quas magis eſſent in vſu. </s>
          <s id="s.004547">huiuſmodi modos nos hodie Arias ap­<lb></lb>
pellamus, <expan abbr="easq́">easque</expan>; pariter à varijs nationibus denominamus, vt quas dicimus <lb></lb>
Spagnolettam, Franceſcam, Græcam, Neapolitanam, Siculam, &amp;c.</s>
        </p>
        <p type="main">
          <s id="s.004548">De qualitatibus horum modorum plura veteres, ac Zarlinus ipſe citato <lb></lb>
loco: <expan abbr="nosq́">nosque</expan>; nonnulla ſupra in Politicis diximus. </s>
          <s id="s.004549">Quod ad hunc locum ſpe­<lb></lb>
ctat, videndum quales eſſent Hypodorius, &amp; Hypophrygius: quod Ariſt. ip­<lb></lb>
ſe infra Problem. 49. docet, ait enim, modum hypophrygium animos Lym­<lb></lb>
phatis, ſimiles reddere, <expan abbr="cogereq́">cogereque</expan>; debacchari: Hypodorium verò eſſe mo­<lb></lb>
dum magnificum, conſtantem, grauemque. </s>
          <s id="s.004550"><expan abbr="vtrumq;">vtrumque</expan> autem fuiſſe variæ imi­<lb></lb>
tationi aptum, <expan abbr="ideoq́">ideoque</expan>; caruiſſe ſtrophis: quæ ad eandem ſemper imitatio­<lb></lb>
nem, <expan abbr="eundemq́">eundemque</expan>; morem tendunt: vt ſupra Probl. 15. explicaui. </s>
          <s id="s.004551">ex quibus <lb></lb>
intelligere poſſumus Problema præſens; modus ſcilicet Hypodorius, &amp; Hy­<lb></lb>
pophrygius à choro Tragœdiarum arcebantur, quia carebant antiſtrophis, <lb></lb>
quibus chorus gaudebat; chorus enim non imitabatur varios mores, <expan abbr="va-riosq́">va­<lb></lb>
riosque</expan>; hominum affectus, ſed eodem ſeruato affectu per eaſdem ſtrophas ad <lb></lb>
finem <expan abbr="vſq;">vſque</expan> perueniebat. </s>
          <s id="s.004552">erant autem prædicti duo modi ſcenis idonei, quia <lb></lb>
in ſcena varios mores, affectus, &amp; animi paſſiones imitabantur, <expan abbr="atq;">atque</expan> ad eoſ­<lb></lb>
dem variè auditorum animos impellebant; ad quod peragendum ipſi erant <lb></lb>
idonei teſte Ariſt. citato loco. </s>
          <s id="s.004553">cùm præſertim antiſtrophis carerent, quæ ob­<lb></lb>
ſiſtere variæ, ac multiplici imitationi poterant. </s>
          <s id="s.004554">ſi plura de modis, aut tonis <lb></lb>
deſideras, conſule Ptolæm. lib. 2. harm. </s>
          <s id="s.004555">Boetium lib. 4. Ioſephum Zarli­<lb></lb>
num lib. 4. Inſtit. &amp; lib. 6. Supplem. muſicorum.</s>
        </p>
        <p type="main">
          <s id="s.004556">Illud nunc occurrit maximè notandam. </s>
          <s id="s.004557">Veteres non ſolum in choris, ſed <lb></lb>in ipſa ſcena etiam cantare, aut ſonorare ſolitos fuiſſe, quod manifeſtè ap­<lb></lb>
paret ex problemate 75. ex 30. præſenti, necnon ex 31. &amp; 49. ſequentibus.</s>
        </p>
        <p type="main">
          <s id="s.004558"><arrow.to.target n="marg375"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004559"><margin.target id="marg375"></margin.target>384</s>
        </p>
        <p type="main">
          <s id="s.004560">Probl. 31. <emph type="italics"></emph>(Cur Phrynicus, cæteriqué, illius ætatis Muſici magis Melopæi erăt? <lb></lb>
</s>
          <s id="s.004561">An quoniam tunc temporis in tragædijs carmine contextis maior erat <expan abbr="cātilenarum">cantilenarum</expan> <lb></lb>
vſus)<emph.end type="italics"></emph.end> Apud Suidam inter plures Phrynicos, vnus recenſetur patria Athe­<lb></lb>nienſis, &amp; Poeta Tragicus, qui circa Olympiadem 67. floruit: quem puto <lb></lb>hunc eſſe, de quo in hoc problemate agitur. </s>
          <s id="s.004562">hic enim Poeta Tragicus quo­<lb></lb>que erat, vt apparet ex illis verbis (in tragœdijs carmine contextis) quod <lb></lb>
autem ſimul Muſicus eſſet, non videtur dubium; antiquitus enim, vt rectè <lb></lb>
etiam Zarlinus obſeruabit, ijdem erant Poetæ, &amp; Muſici, quod optimè ex <lb></lb>
Plutarco dé muſica confirmatur, vbi plures connumerat antiquos Muſicos,


<pb pagenum="264" xlink:href="009/01/264.jpg"></pb>quì ſimul Poetę extiterunt, ſic ait; Steſichorus, &amp; veteres alij <expan abbr="Poetarũ">Poetarum</expan>, qui <lb></lb>
carmina adhibitis modulis condidere. </s>
          <s id="s.004563">ſed quid erat Melopæia? </s>
          <s id="s.004564">ex Ariſto­<lb></lb>
xeno, <expan abbr="atq;">atque</expan> Euclide; Melopæia eſt vſus harmonicæ tractationi ſubiectorum, <lb></lb>
ad decorum propoſiti argumenti. </s>
          <s id="s.004565">ex qua definitione patet Melopæum eum <lb></lb>
fuiſſe, quem modo vocant Compoſitorem. </s>
          <s id="s.004566">dicitur Melopæia, quaſi cantus <lb></lb>
effectrix. </s>
          <s id="s.004567">is igitur erat Melopæus, qui res ſubiectas harmonicæ ſcientiæ, vt <lb></lb>
ſunt ſonus, interualla, genera, modi, conſonantiæ, diſſonantiæ ritè in vſum <lb></lb>
vocabat: vnde cantilenas humana oratione conſtructas ad decorum, ideſt <lb></lb>
pro rei argumento conuenientibus rythmis modulabatur. </s>
          <s id="s.004568">Antiquitus igi­<lb></lb>
tur Melopæiæ magis ſtudebant, quàm Ariſt. tempeſtate, quia tunc tempo­<lb></lb>
ris magis erant in tragædijs cantilenæ in vſu, quam poſtea.</s>
        </p>
        <p type="main">
          <s id="s.004569"><arrow.to.target n="marg376"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004570"><margin.target id="marg376"></margin.target>385</s>
        </p>
        <p type="main">
          <s id="s.004571">Probl. 32. <emph type="italics"></emph>(Cur Diapaſon conſonantiam dicimus, non ratione numeri Diao­<lb></lb>cto, vti Diateſſaron, &amp; Diapente? </s>
          <s id="s.004572">An quod antiquitus non pluribus, quàm ſep­<lb></lb>
tem vterentur numeris? </s>
          <s id="s.004573">Deinde Terpander tritè exempta, Neten adiunxit, eiusqué <lb></lb>temporibus conſonantia hæc dicta eſt Diapaſon, non Diaocto: quippe quæ ſeptem <lb></lb>non octo conſtaret)<emph.end type="italics"></emph.end> Lege, quæ ſupra ad 7. problem. </s>
          <s id="s.004574">ſunt annotata de ordine <lb></lb>
chordarum, quem Terpander induxit. </s>
          <s id="s.004575">Septem nimirum chordas <expan abbr="cõſtituit">conſtituit</expan>, <lb></lb>
inter quas Trite deſiderabatur, vt ibi explicaui. </s>
          <s id="s.004576">quare Terpander non im­<lb></lb>mutauit numerum chordarum antiquum, ſed tantummodo Neten cum Tri<lb></lb>
te commutauit. </s>
          <s id="s.004577">Tempore igitur Terpandri cum ſeptem eſſent <expan abbr="tantũ">tantum</expan> chor­<lb></lb>
dæ in pſalterijs, etiamſi prima cum vltima conſonantiam Diapaſon reſona­<lb></lb>
ret, non tamen potuit hæc conſonantia appellari Diaocto. </s>
          <s id="s.004578">Boetius lib. 1. <lb></lb>
cap. 20. Muſicæ, prædicta aſſerit de Terpandro. </s>
          <s id="s.004579">Suidas ait <expan abbr="Terpandrũ">Terpandrum</expan> fuiſſe <lb></lb>
<expan abbr="Lesbiũ">Lesbium</expan>, &amp; Poetam Lyricum, qui primus lyram ex ſeptem chordis fecit, cùm <lb></lb>
prius à Mercurio ex quatuor tantum conſtructa fuiſſet. </s>
          <s id="s.004580">Cæterum ipſa Dia­<lb></lb>
paſon ſic dicta eſt, quaſi per omnes, quia à prima chorda per omnes aſcen­<lb></lb>
dendo ad vltimam perueniebatur, cum qua prima Diapaſon reſonabat. </s>
          <s id="s.004581">vel <lb></lb>
quia intra Diapaſon reliquæ omnes conſonantiæ concinentur, quæ dicuntur <lb></lb>
primæ: quæ enim ſupra Diapaſon ſunt, eædem ſunt cum prædictis, ſiue eiuſ­<lb></lb>
dem naturæ; ſed quæ repetuntur, vt ſupra ſæpe dictum eſt.</s>
        </p>
        <p type="main">
          <s id="s.004582"><arrow.to.target n="marg377"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004583"><margin.target id="marg377"></margin.target>386</s>
        </p>
        <p type="main">
          <s id="s.004584">Probl. 33. <emph type="italics"></emph>(Cur aptius de acuto in graue canitur, quam de graue in acutum? <lb></lb>
</s>
          <s id="s.004585">Vtrum, quod ita fit, vt à ſuo inchoetur principio? </s>
          <s id="s.004586">neruus enim, qui medius, &amp; dux <lb></lb>
eſt ſecundi tetrachordi, acutiſſimus est. </s>
          <s id="s.004587">illo autem modo non à principio, ſed à fine <lb></lb>
exordiretur. </s>
          <s id="s.004588">An quod graue generoſius, &amp; ſonantius ab acuto oriri poteſt)<emph.end type="italics"></emph.end> Na­<lb></lb>
turale eſt omnibus, cùm canere incipiunt, ab acuto incipere; cum autem <lb></lb>
deſinunt, in graui deſinere: quod ſi quis <expan abbr="contrariũ">contrarium</expan> faciat, ineptè agere æſti­<lb></lb>
mabitur? </s>
          <s id="s.004589">Huius quæritur cauſa. </s>
          <s id="s.004590">Vbi explicandum quid ſit tetrachordum. <lb></lb>
</s>
          <s id="s.004591">Tetrachordum igitur erat ſyſtema, vel <expan abbr="cõſtitutio">conſtitutio</expan> quatuor chordarum, qui­<lb></lb>
bus Diateſſaron conſtabat. </s>
          <s id="s.004592">in maximo autem ſyſtemate, quod erat duarum <lb></lb>
Diapaſon, ſiue Diſdiapaſon, erant plura tetrachorda. </s>
          <s id="s.004593">horum primum illud <lb></lb>
erat, quod in parte grauiſſima <expan abbr="collocatũ">collocatum</expan> erat, cuius hæ erant chordæ, Hy­<lb></lb>
pate, Parhypate, Lychanos, Meſe. </s>
          <s id="s.004594">ſi igitur inſtrumentum habuerit tantum <lb></lb>duo tetrachorda, neruus medius crit ipſa Meſe, quæ eſt acutiſſima primi te­<lb></lb>
trachordi, eſt præterea hæc Meſe veluti dux reliquarum chordarum, nam, <lb></lb>
vt dictum eſt in Probl. 20. eſt in medio carum vti dux; ſæpiſſimè omnium <lb></lb>pulſatur: ca ſola ab alijs diſſonante, reliquæ omnes videntur diſſonare. </s>
          <s id="s.004595">cùm <pb pagenum="265" xlink:href="009/01/265.jpg"></pb>igitur alijs præſtet <expan abbr="ſitq́">ſitque</expan>; ſui tetrachordi acutiſſima, <expan abbr="cõuenienter">conuenienter</expan> natura du­<lb></lb>
ce fit, vt ab acuta voce cantum exordiamur. </s>
          <s id="s.004596">ideſt ſicut in tetrachoreo prin­<lb></lb>
cipalis eſt acuta, ſiue principium tetrachordi eſt acutum, ita etiam princi­<lb></lb>
pium cantus debet eſſe acutum. </s>
          <s id="s.004597">Quod ſi à graui cantandi principium fa­<lb></lb>
ceremus, à fine potius, quàm à principio contra naturæ ordinem <expan abbr="principiũ">principium</expan> <lb></lb>
faceremus. </s>
          <s id="s.004598">Theodorus Gaza vertit, primi tetrachordi, verum in vulgatis, <lb></lb>
atque correctis codicibus græcis legitur, <foreign lang="grc">παράτετραχορδου,</foreign> quod non pri­<lb></lb>
mum, ſed potius ſubſequens tetrachordum, ſignificare videtur. </s>
          <s id="s.004599">vtrumuis le­<lb></lb>
gamus, explicatio allata ſufficere poteſt. </s>
          <s id="s.004600">Subdit poſtea aliam reſponſio­<lb></lb>
nem, quod nimirum hoc modo grauis vox cantilenam claudens, quando ex <lb></lb>
acuto quodammodo orta eſt, generoſior, <expan abbr="atq;">atque</expan> ſonantior euadit.</s>
        </p>
        <p type="main">
          <s id="s.004601"><arrow.to.target n="marg378"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004602"><margin.target id="marg378"></margin.target>387</s>
        </p>
        <p type="main">
          <s id="s.004603">Probl. 34. <emph type="italics"></emph>(Cur bis Diapente, aut bis Diateſſaron <expan abbr="cõſonantia">conſonantia</expan> <expan abbr="cõponi">componi</expan> non poteſt, <lb></lb>
bis <expan abbr="aũt">autem</expan> Diapaſon poteſt? </s>
          <s id="s.004604">An, quòd bis Diapente, non bis Diateſſaron eſt: ſed Dia­<lb></lb>
teſſaron, &amp; Diapente in vnă Diapaſon concurrunt)<emph.end type="italics"></emph.end> Quamuis textus <expan abbr="aliquantulũ">aliquantulum</expan> <lb></lb>
<expan abbr="ẽt">et</expan> græcus corruptus ſit, verumtamen ſenſum Ariſt. ex ſequentibus percipie­<lb></lb>
mus. </s>
          <s id="s.004605">Pro intelligentia igitur huius problematis placet hic deſcribere <expan abbr="demõ-ſtrationem">demon­<lb></lb>
ſtrationem</expan> 16. lib. 3. doctiſſimi Fabri ſtapulentis, qua ipſe veſtigijs <expan abbr="antiquo-rũ">antiquo­<lb></lb>
rum</expan> inhærens optimè præſenti quæſtioni ſatisfacit. </s>
          <s id="s.004606">eſt <expan abbr="aũt">aut</expan> huiuſmodi: Biſdia­<lb></lb>
teſſaron, aut bis <expan abbr="Diapẽte">Diapente</expan> <expan abbr="nullã">nullam</expan> conſonantiam <expan abbr="cõponere">componere</expan> poteſt, omnis <expan abbr="namq;">namque</expan> <lb></lb>
conſonantia, aut in proportione multiplici, aut in ſuperparticulari collo­<lb></lb>
canda eſt, ex Pythagoreorum, <expan abbr="aliorumq́">aliorumque</expan>; Muſicorum traditione; ſed ſi duæ <lb></lb>
Diateſſaron, aut duæ Diapente componantur, <expan abbr="neq;">neque</expan> multiplicem, neque ſu­<lb></lb>
perarticularem creant rationem, ergò additæ nullam efficere valent <expan abbr="cõſo-nantiam">conſo­<lb></lb>
nantiam</expan>. </s>
          <s id="s.004607">duas Diapentes nullam facere rationem multiplicem, aut ſuper­<lb></lb>
particularem patet ex numeris earum rationem continentibus ſimul addi­<lb></lb>
tis, eo modo, quo Muſici ſolent addere. </s>
          <s id="s.004608">ratio Diapentes eſt ſeſquialtera, ſi <lb></lb>
ergo duæ ſeſquialteræ ſimul continuentur, vt in his numeris. </s>
          <s id="s.004609">9. 6. 4. ratio <lb></lb>
primi 9 ad vltimum 4. erit compoſita ex duabus ſeſquialteris; ratio autem <lb></lb>
9. &amp; 4. <expan abbr="neq;">neque</expan> eſt multiplex, <expan abbr="neq;">neque</expan> ſuperarticularis, vt oporteret, ſed eſt multi­<lb></lb>
plex ſuperparticularis, quæ ad conſonantiam inepta eſt. </s>
          <s id="s.004610">propterea igitur <lb></lb>
duæ Diapentæ additæ nullam faciunt conſonantiam. </s>
          <s id="s.004611">quod præterea expe­<lb></lb>
rientia ipſa manifeſtat. </s>
          <s id="s.004612">ſed cur proprio multiplex, &amp; ſuperarticularis ſunt <lb></lb>
harmonicæ; multiplex verò ſuperarticularis, aut quælibet alia non? </s>
          <s id="s.004613">fortè <lb></lb>
quia in illis maior ſeruatur integritas, quæ perfectio eſt: in cæteris verò mi­<lb></lb>
nor integritas, quæ imperfectio eſt. </s>
          <s id="s.004614">quod melius in ſequenti problem. </s>
          <s id="s.004615">ex­<lb></lb>
plicabitur. </s>
          <s id="s.004616">ſimiliter duas Diateſſaron nullam facere <expan abbr="rationẽ">rationem</expan> conſonantem, <lb></lb>
patet ex numeris illarum additis: eorum proportio eſt ſeſquitertia, <expan abbr="addã-tur">addan­<lb></lb>
tur</expan>; ergò duæ ſeſquitertiæ, vt in his numeris 16. 12. 9. ratio primi 16. ad <lb></lb>
extremum 9. nec multiplex, nec ſuperparticularis eſt, vt oporteret: ergò <lb></lb>
nullam conſonantiam efficient. </s>
          <s id="s.004617">At verò, ſi vna Diateſſaron, &amp; vna Dia­<lb></lb>pente, componantur, efficiunt Diapaſon; quia ipſarum rationes additæ du­<lb></lb>
plam, quæ eſt ratio Diapaſon, efficiunt: dupla autem eſt multiplex. </s>
          <s id="s.004618"><expan abbr="ponã-tur">ponan­<lb></lb>
tur</expan> hi tres numeri 6. 4. 3. proportio primi 6. &amp; ſecundi 4. eſt ſeſquialtera, <lb></lb>
pro Diapente. </s>
          <s id="s.004619">proportio ſecundi 4. &amp; 3. eſt ſeſquitertia pro Diateſſaron. <lb></lb>
</s>
          <s id="s.004620">Iam proportio inter primum 6. &amp; vltimum 3. eſt dupla: quæ eſt ratio ipſius <lb></lb>
perfectiſſimæ conſonantiæ Diapaſon. </s>
          <s id="s.004621">Ex quibus Ariſt. ſententia manifeſta <pb pagenum="266" xlink:href="009/01/266.jpg"></pb>eſt. </s>
          <s id="s.004622">idem quærit etiam problemate 42. Hæc de ratione multiplici, &amp; ſuper­<lb></lb>
particulari dicta ſunt ex veterum ſententia: recentiores enim muſicæ de­<lb></lb>
prauatores plures alias rationes perperam inter harmonicas intruſerunt.</s>
        </p>
        <p type="main">
          <s id="s.004623"><arrow.to.target n="marg379"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004624"><margin.target id="marg379"></margin.target>388</s>
        </p>
        <p type="main">
          <s id="s.004625">Probl. 35. <emph type="italics"></emph>(Cur Diapaſon conſonantia omnium pulcherrima eſt? </s>
          <s id="s.004626">An quod <lb></lb>
integris terminis huius proportiones continentur: cæterarum autem non integris? <lb></lb>
</s>
          <s id="s.004627">cùm enim Nete dupla ad hypaten ſit, quocunque in genere Nete duo tenuerit, hy­<lb></lb>
pate vnum habebit; &amp; vbi hypate duo, Nete quatuor reſonabit, &amp; ita deinceps. <lb></lb>
</s>
          <s id="s.004628">At verò eadem Nete meſes ſeſquialtera eſt: proportio <expan abbr="namq;">namque</expan> ſeſquialtera, qua <expan abbr="cõ-ſonantia">con­<lb></lb>
ſonantia</expan> diapente concluditur, non integris numeris poſita eſt: maior enim mino­<lb></lb>
rem intra ſe continet totum, &amp; partem eius dimidiam. </s>
          <s id="s.004629">quamobrem non integri <lb></lb>
cùm integris comparantur, ſed partes ſuperſunt. </s>
          <s id="s.004630">Conſonantia quoque Diateſſa­<lb></lb>ron proportione ſeſquitertia continetur, quæ terminis conſtat, quorum maior mi­<lb></lb>
norem totum continet, &amp; inſuper tertiam eius partem. </s>
          <s id="s.004631">An quod ex amba­<lb></lb>
bus conſiſtit, perfectiſſima eſt? </s>
          <s id="s.004632">&amp; quoniam modulandi menſuram hæc tenet, meri­<lb></lb>
tò omnium elegantiſſima)<emph.end type="italics"></emph.end> Proportio conſonantiæ Diapaſon eſt ſicuti 2. ad 1. <lb></lb>
vbi vides vtrunque terminum eſſe integrum, quia maior minorem bis inte­<lb></lb>
grè continet. </s>
          <s id="s.004633">proportio verò conſonantiæ Diapente, eſt ſicuti 3. ad 2. vbi <lb></lb>
maior terminus minorem non integrè continet, ſed ſemel, &amp; adhuc <expan abbr="dimidiũ">dimidium</expan> <lb></lb>
illius. </s>
          <s id="s.004634">proportio <expan abbr="deniq́">denique</expan>; Diateſſaron eſt ſicuti 4. ad 3. vbi maior <expan abbr="minorẽ">minorem</expan> non <lb></lb>
integrè continet, ſed ſemel, &amp; adhuc tertiam ipſius partem: breuiter deno­<lb></lb>
minationes <expan abbr="harũ">harum</expan> <expan abbr="proportionũ">proportionum</expan> ſunt hi, 2/1. 1 1/2. 1 1/3. vbi vides, <expan abbr="primũ">primum</expan>, qui eſt <lb></lb>
Diapaſon conſtare ex integris numeris. </s>
          <s id="s.004635">ſecundum verò, &amp; tertium, qui ſunt <lb></lb>
Diapente, &amp; Diateſſaron exintegro cum fractione. </s>
          <s id="s.004636">maior autem perfectio <lb></lb>
eſt integritas, quam fractio, aut diuiſio. </s>
          <s id="s.004637">propterea perfectior reliquis eſt <lb></lb>
conſonantia Diapaſon: &amp; Diapente adhuc perfectior, quam Diateſſaron, <lb></lb>
quia illius numeri minorem habent fractionem, quam huius. </s>
          <s id="s.004638">Aliter reſpon­<lb></lb>
det poſtea dicens, Diapaſon perfectam eſſe adeò conſonantiam; quoniam <lb></lb>ex duabus Diapente, &amp; Diateſſaron conſtat, vt ſupra ex diuiſione mono­<lb></lb>
chordij, &amp; in præcedenti etiam problemate patuit. </s>
          <s id="s.004639">quæ ratio, quantum va­<lb></lb>
leat, alij viderint. </s>
          <s id="s.004640">Reſpondet tandem Diapaſon ideò perfectiſſimam eſſe, <lb></lb>
quia ipſa ſit modulandi menſura, ideſt, quia intra terminos huius <expan abbr="conſonã-tiæ">conſonan­<lb></lb>
tiæ</expan> omnes aliæ ſimplices conſonantiæ continentur, vt ſupra initio explica­<lb></lb>
ui. </s>
          <s id="s.004641">meritò igitur omnium elegantiſſima eſt. </s>
          <s id="s.004642">In græco textu ſuperſunt <expan abbr="nō-nulla">non­<lb></lb>
nulla</expan>, quæ meritò Gaza omiſit, cum nullo pacto cùm præcedentibus cohæ­<lb></lb>
reant. </s>
          <s id="s.004643">Verba illa <emph type="italics"></emph>(Cum enim nete ad hypatem dupla ſit, quocunque in genere <lb></lb>duo tenuerit, hypate vnŭμ habebit &amp;c.)<emph.end type="italics"></emph.end> Videntur <foreign lang="grc">νστερογ πρωτερον·</foreign> cum debuiſ­<lb></lb>
ſet dicere, hypatem duplam eſſe ipſius netes, vt ſupra patuit ex diuiſione <lb></lb>
regulæ harmonicæ. </s>
          <s id="s.004644">Fortè vult dicere neten eſſe duplò acutiorem, <expan abbr="quã">quam</expan> hy­<lb></lb>
pate: vel fuit memoriæ lapſus. </s>
          <s id="s.004645">quod ait <emph type="italics"></emph>(At verò eadem nete Meſes ſeſqui­<lb></lb>
altera eſt)<emph.end type="italics"></emph.end> vult dicere Meſen ad neten habere ſeſquialteram proportionem, <lb></lb>
quamuis inuersè loquatur: qua ratione verò Meſe ad netem ſeſquialtera <lb></lb>
ſit, ex diuiſione monochordij initio tradita ſatis patere poteſt.</s>
        </p>
        <p type="main">
          <s id="s.004646"><arrow.to.target n="marg380"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004647"><margin.target id="marg380"></margin.target>389</s>
        </p>
        <p type="main">
          <s id="s.004648">Probl. 36. <emph type="italics"></emph>(Cur ſi neruus medius ex ſuo intentionis modo dimotus fuerit, cæte­<lb></lb>
ris <expan abbr="quoq;">quoque</expan> omnes nerui, ſonos diſſonos reddent: ſed ſi, immoto illo manente, ali­<lb></lb>
quis ex cæteris dimotus fuerit, ſolus hic, qui modo ſuo caruerit, aberrabit? </s>
          <s id="s.004649">An, <lb></lb>quod ratio concinendi, aptaneruorum omnium intentione continetur, quæ non niſ<emph.end type="italics"></emph.end>i <pb pagenum="267" xlink:href="009/01/267.jpg"></pb><emph type="italics"></emph>per habitudinem quandam ad Meſen, ſeu ad Medium, <expan abbr="accommodāda">accommodanda</expan> omnibus eſt, <lb></lb>
ordoqué, ratione illius diſponi ſingulis debet? </s>
          <s id="s.004650">ergo ſublata concinendi cauſa, <expan abbr="concē-tus">concen­<lb></lb>
tus</expan> æquè cuſtodiri præterεa nequit. </s>
          <s id="s.004651">Veruntamen Meſe ſibi conſtante, ſi quis alius <lb></lb>
diſcreparit, meritò illius ſola pars deeſt: cæteri <expan abbr="uamq;">νamque</expan> omnes modum ſuæ <expan abbr="concinẽ-tiæ">concinen­<lb></lb>
tiæ</expan> ſeruant integrum) (Neruus medius)<emph.end type="italics"></emph.end> ideſt, Meſe, ſic appellata, quod me­<lb></lb>
dia eſſet. <emph type="italics"></emph>(Quæ non niſi per habitudinem quandam ad Meſen)<emph.end type="italics"></emph.end> hypate cum Me­<lb></lb>
ſe conſonabat Diateſſaron: nete cum eadem Meſe conſonabat Diapente, <lb></lb>
quæ ſunt duæ præcipuæ conſonantiæ, Diapaſon integrantes; ergo ſublata <lb></lb>
Meſe de ſuo ſtatu, illas pariter tolli neceſſe eſt. </s>
          <s id="s.004652">eandem quæſtionem ſupra <lb></lb>
Probl. 20. pertractauit, quàm nunc reuiſere conſultum erit.</s>
        </p>
        <p type="main">
          <s id="s.004653"><arrow.to.target n="marg381"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004654"><margin.target id="marg381"></margin.target>390</s>
        </p>
        <p type="main">
          <s id="s.004655">Probl. 37. <emph type="italics"></emph>(Cur existente vocum acumine, ſecundum paruum: grauitate au­<lb></lb>
tem ſecundum multum<emph.end type="italics"></emph.end> (<emph type="italics"></emph>quod enim graue eſt, ob vaſtitatem graue eſt: quod verò <lb></lb>
acutum ob paruitatem<emph.end type="italics"></emph.end>) <emph type="italics"></emph>difficilius eſt acutas voces canere, quàm graues; &amp; pauci <lb></lb>
ſunt, qui ſuperna cantare valeant; &amp; leges orthiæ, &amp; acutæ cantu difficiles ſunt, <lb></lb>
quod ſint valdè intenſæ. </s>
          <s id="s.004656">Quamquam facilius ſit mouere exiguum, quam <expan abbr="magnũ">magnum</expan>: <lb></lb>
idem <expan abbr="itaq;">itaque</expan> in aere deberet accidere. </s>
          <s id="s.004657">An quia non idem eſt eſſe acutæ vocis à natu­<lb></lb>
ræ, <expan abbr="atq;">atque</expan> <expan abbr="acutũ">acutum</expan> canere: verùm naturaliter imbecilla omnia acutæ ſunt vocis; prop­<lb></lb>
terea ectici ſunt acutæ vocis, quia parum aeris non multum ciere poſſunt: paucus <lb></lb>verò velociter fertur; in cantu verò acutum canere ſignum eſt roboris, quod enim <lb></lb>
valdè <expan abbr="fērtur">fertur</expan>, velociter fertur: &amp; difficilè eſt alta canere, at grauia ſunt humilia)<emph.end type="italics"></emph.end><lb></lb>
Vt intelligas pręſens Problema, lege, quæ lib. 1. Top. c. 3. ſcripſi. </s>
          <s id="s.004658">Leges Or­<lb></lb>
thiæ, erant cantilenæ (vt ſupra probl. </s>
          <s id="s.004659">28. patuit) intenſa admodum, <expan abbr="altaq́">altaque</expan>; <lb></lb>
voce decantari ſolitæ, vnde, &amp; Orthiæ ſunt dictæ; de quibus vide Herodo­<lb></lb>
tum lib. I. &amp; Agell. lib. 16. Plutarchus <expan abbr="quoq;">quoque</expan> de muſica ſæpè meminit Or­<lb></lb>
thiæ legis.</s>
        </p>
        <p type="main">
          <s id="s.004660">Difficilius deberet eſſe canere graue, quàm <expan abbr="acutũ">acutum</expan>, quia graue eſt in mul­<lb></lb>
to, &amp; acutum in paruo, vt patet in cannis. </s>
          <s id="s.004661">canna enim grauis eſt maior, &amp; <lb></lb>
ideo plus aeris mouet. </s>
          <s id="s.004662">chorda etiam grauior, eſt maior, ergò etiam plus <lb></lb>
aeris impellit; idem in cæteris. </s>
          <s id="s.004663">facilius tamen eſt graue, quam acutum: <lb></lb>
præterea imbecilla, vt Ectici, mulieres, pueri, vocem habent naturaliter <lb></lb>
acutam, ergò facilius deberet eſſe acutum canere, cùm exigua vis id præ­<lb></lb>
ſtare videatur? </s>
          <s id="s.004664">Reſpondet aliud eſſe canere acutum, &amp; aliud à natura ha­<lb></lb>
bere vocem acutam. </s>
          <s id="s.004665">qui enim cantat acutum, oportet, vt validè vocem in­<lb></lb>
tendat extenſiuè, <expan abbr="atq;">atque</expan> intenſiuè, ideſt opus eſt acumine, &amp; vociferatione, <lb></lb>
quam debiles edere nequeunt; quia quamuis vocem <expan abbr="habeãt">habeant</expan> acutam, tamen <lb></lb>
paruam habent. </s>
          <s id="s.004666"><expan abbr="Neq;">Neque</expan> difficile eſt canere graue, quia a natura eſt habere ar­<lb></lb>
teriam magnam, &amp; ideo multum aeris ciere, &amp; proinde canere, quæ enim <lb></lb>
naturaliter fiunt, facilè fiunt.</s>
        </p>
        <p type="main">
          <s id="s.004667">Obijces, Ariſt. in Probl. 26. &amp; 47. dixiſſe contrarium, ſcilicet facilius <lb></lb>
eſſe canere acutum, quam graue, ibi enim reſpondet: vtrum, quod facilius <lb></lb>
acutum, quam graue cantatur? </s>
          <s id="s.004668">Reſpondeo primùm, Ariſt. ibi non aſſere­<lb></lb>
re, ſed dubitanter loqui. </s>
          <s id="s.004669">ſecundò, hæc ab eo dicta eſſe problematicè, ideſt <lb></lb>
non conſequenter, ſed quæ poſſint in <expan abbr="vtramq;">vtramque</expan> partem diſputari.</s>
        </p>
        <p type="main">
          <s id="s.004670"><arrow.to.target n="marg382"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004671"><margin.target id="marg382"></margin.target>391</s>
        </p>
        <p type="main">
          <s id="s.004672">Probl. 38. <emph type="italics"></emph>(Cur rithmo, modulo, cantico, &amp; omninò ſymphonijs gaudent om­<lb></lb>
nes? </s>
          <s id="s.004673">An quia motibus naturalibus naturaliter gaudemus. </s>
          <s id="s.004674">iudicium, quod infantes <lb></lb>nuper editi, ipſis delectantur. </s>
          <s id="s.004675">ob conſuetudinem verò canticorum modis gaudemus.<emph.end type="italics"></emph.end>


<pb pagenum="268" xlink:href="009/01/268.jpg"></pb><emph type="italics"></emph>rithmo autem gaudemus, quod habeat numerum ratum, &amp; ordinatum, &amp; quod <lb></lb>
nos ordinatè moueat. </s>
          <s id="s.004676">magis enim proprium naturæ eſt ordinatus motus, quam in­<lb></lb>
ordinatus: &amp; ideò magis etiam ſecundum naturam eſt. </s>
          <s id="s.004677">argumentum, quod cùm la­<lb></lb>
boramus, &amp; bibimus, &amp; comedimus ordinatè, naturam, viresqué noſtras, &amp; ſerua­<lb></lb>
mus, &amp; augemus: cùm verò inordinatè eam corrumpimus, &amp; dimouemus. </s>
          <s id="s.004678">morbi <lb></lb>
enim dimotiones ſunt naturalis conſtitutionis corporis. </s>
          <s id="s.004679">conſonantia verò lætamur, <lb></lb>
quod ſit mixtio quędam contrariorum, proportionem habentium ad inuicem. </s>
          <s id="s.004680">ſi qui­<lb></lb>
dem proportio ordo eſt, qui naturà quidem ſuauis est. </s>
          <s id="s.004681">mixtum verò omne ſuauius <lb></lb>
eſt immixto. </s>
          <s id="s.004682">præſertim ſi cùm ſenſibile ſit, æquè <expan abbr="vtriuſq;">vtriuſque</expan> extremi vim retineat, in <lb></lb>
conſonantia porrò proportio eſt)<emph.end type="italics"></emph.end> Quid rithmus ſit, ſupra num. </s>
          <s id="s.004683">27. explicaui. <lb></lb>
</s>
          <s id="s.004684">ſed optimè ex hoc loco elicitur rithmum eſſe certam quandam in aliquo <lb></lb>
motu periodum, ſcilicet determinatorum ictuum, &amp; temporum. </s>
          <s id="s.004685">Sympho­<lb></lb>
niam, Muſici dicunt eſſe plurium ſonorum conuenientium mixtionem, ſe­<lb></lb>
cundum aliquem canendi modum. </s>
          <s id="s.004686">quod ait, pueri nuper editi ipſis delecta­<lb></lb>
ri ſolent. </s>
          <s id="s.004687">patet hoc modo; ſolo rithmo lætantur, quando incunabulum or­<lb></lb>
dinatè agitatur: modis muſicis, cùm illis cantilena quæpiam etiam ſolita­<lb></lb>
ria, vti eſt Nenia accinitur: ſymphonia tandem, quando muſico aliquo in­<lb></lb>ſtrumento addita etiam humana voce concinnitur. </s>
          <s id="s.004688">reliqua per ſe patent.</s>
        </p>
        <p type="main">
          <s id="s.004689"><arrow.to.target n="marg383"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004690"><margin.target id="marg383"></margin.target>392</s>
        </p>
        <p type="main">
          <s id="s.004691">Probl. 39. <emph type="italics"></emph>(Cur ſuauius eſt ſymphonum, quàm vniſonum? </s>
          <s id="s.004692">An quod antipho­<lb></lb>
num ipſum quoque <expan abbr="conſonã">conſonam</expan> eſt per Diapaſon, quippe cùm ex pueris, virisqué fiat an­<lb></lb>
tiphonum, qui ita inter ſe vocibus distant, vt Nete, &amp; Hypate. </s>
          <s id="s.004693">omnis autem ſym­<lb></lb>
phonia ſeno ſimplici ſuauior eſt, cur autem ita dictum eſt: quarum ſuauiſſima est <lb></lb>
Diapaſon: Vniſonum autem ſimplicem ſonum habet)<emph.end type="italics"></emph.end> Cur ſuauior eſt conſonan­<lb></lb>tia, quæ oritur ex vocibus ſymphonis, ideſt, diuerſis, quam quæ ijſdem ſiue <lb></lb>
vniſonis? </s>
          <s id="s.004694">An quia talis conſonantia magis ad naturam Diapaſon accedit; <lb></lb>
imò Diapaſon ipſa vna eſt ex ſymphonis; ipſa autem fit ex puerorum, ac vi­<lb></lb>
rorum vocibus, quæ inuicem diſtant, vt Nete, &amp; Hypate, ideſt in dupla ra­<lb></lb>
tione; omnis autem conſonantia ſuauior eſt ſono ſimplici: at verò ſympho­<lb></lb>
num continet diuerſos ſonos: vniſonum autem quamuis plures contineat, <lb></lb>
tamen propter earum nimiam ſimilitudinem, perinde ac vna ſimplex vox, <lb></lb>
reſpectu illius reputatur. </s>
          <s id="s.004695">non me latet aliter exponi voces ſymphonon, &amp; <lb></lb>
omophon à Ptolæm. </s>
          <s id="s.004696">primo harm. </s>
          <s id="s.004697">cap. 7. &amp; alijs: ſed illa Ariſtot. ſententiæ <lb></lb>
minimè quadrant. </s>
          <s id="s.004698">Probl. 16. ſuperius eſt ferè idem cum hoc.</s>
        </p>
        <p type="main">
          <s id="s.004699"><arrow.to.target n="marg384"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004700"><margin.target id="marg384"></margin.target>393</s>
        </p>
        <p type="main">
          <s id="s.004701">Probl. 40. <emph type="italics"></emph>(Cur in ſola Diapaſon conſonantia magadari ſolitum eſt? </s>
          <s id="s.004702">An quia, <lb></lb>
vt pedes carminum proportionem, aut æqualis ad æqualem, aut duo ad vnum, aut <lb></lb>aliam aliquam obtinent; ita ſoni, quibus conſonantia confiat, motus rationem in­<lb></lb>
ter ſe aliquam ſeruant. </s>
          <s id="s.004703">cæterarum igitur <expan abbr="conſonātiarum">conſonantiarum</expan> alterius quidem fines ſunt <lb></lb>
imperfecti, cùm finiant ad dimidium. </s>
          <s id="s.004704">propterea nequeunt eſſe eiuſdem facultatis. <lb></lb>
</s>
          <s id="s.004705">eumqué ſint diſpares, diſcrepantia illa ſenſui occurrit; quemadmodum in choris in <lb></lb>
ipſo fine alium maiori voce abundare accidit. </s>
          <s id="s.004706">Præterea ipſi hypate accidit, vt eun­<lb></lb>dem finem habeat periodorum in ſonis cùm nete: vltimus enim à nete ictus ceris <lb></lb>
factus hypate eſt. </s>
          <s id="s.004707">quod cùm finiant in idem, quamuis non idem fecerint, euenit, vt <lb></lb>
opus abſolui vnum, communequé poſſit, vt eis accidit, qui ſub extremam cantilenam <lb></lb>
pulſant; nam etiamſi prius non ſonuerint, tamen quòd in idem deſierint hoc extre­<lb></lb>
mo magis delectent, quam contriſtauerint ante finem diſcrepantijs. </s>
          <s id="s.004708">quoniam igitur <lb></lb>in Diapaſon, quod commune exultat cum differentijs ſuauiſſimum eſt; magadari <emph.end type="italics"></emph.end> <pb pagenum="269" xlink:href="009/01/269.jpg"></pb><emph type="italics"></emph>autem ex contrarijs vocibus conſiſtat, propterea in Diapaſon magadari ſolě)<emph.end type="italics"></emph.end> Hic <lb></lb>
repetenda ſunt, quæ probl. </s>
          <s id="s.004709">17. annotaui, vbi quid Magadis, &amp; Magadari <lb></lb>
explicatum eſt. </s>
          <s id="s.004710">repetenda ſunt pariter, quæ in 35. probl. </s>
          <s id="s.004711">de præſtantia con­<lb></lb>
ſonantiæ Diapaſon ſunt dicta. </s>
          <s id="s.004712">verba illa <emph type="italics"></emph>(Cum finiat ad dimidiam)<emph.end type="italics"></emph.end> <expan abbr="intelligē-da">intelligen­<lb></lb>
da</expan> ſunt de Diapente, cuius rationis termini ſunt 1 1/2. qui ad dimidium fi­<lb></lb>niunt; poſtea videtur aliquid addendum pro Diateſſaron, cuius <expan abbr="nimirū">nimirum</expan> ter­<lb></lb>
mini ſunt 1 1/3. qui ad vnam tertiam finiunt; nequeunt igitur horum termi­<lb></lb>
norum conſonantiæ Diapente, &amp; Diateſſaron, eſſe eiuſdem facultatis cum <lb></lb>
Diapaſon, cuius rationis termini ſunt integri, vt 2. ad 1. verba illa <emph type="italics"></emph>(Præte­<lb></lb>rea ipſi hypati accidit, &amp; c.)<emph.end type="italics"></emph.end> vt intelligantur, vide problem. </s>
          <s id="s.004713">24. cum ſua ex­<lb></lb>
plicatione.</s>
        </p>
        <p type="main">
          <s id="s.004714"><arrow.to.target n="marg385"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004715"><margin.target id="marg385"></margin.target>394</s>
        </p>
        <p type="main">
          <s id="s.004716">Probl. 41. <emph type="italics"></emph>(Cur ſuauius cantum audimus, quem ſcimus, quam quem ignora­<lb></lb>
mus? </s>
          <s id="s.004717">Vtrum, quoniam cum <expan abbr="cantilenã">cantilenam</expan> agnoſcimus, manifeſtior eſt, qui veluti ſco­<lb></lb>
pum, aſſequatur. </s>
          <s id="s.004718">cognoſcentium autem ſpeculatio ſuauis eſt. </s>
          <s id="s.004719">An quia accidit, vt <lb></lb>
auditor vnà cum cantore afficiatur, qui notam cantat cantilenam, nam tunc audi­<lb></lb>
tor illi quaſi ſuccinit. </s>
          <s id="s.004720">Solet autem quiſque alacriter canere, niſi ob aliquam neceſ­<lb></lb>
ſitatem id faciat)<emph.end type="italics"></emph.end> Lege problem. </s>
          <s id="s.004721">5. eiuſque explicationem, <expan abbr="eritq́">eritque</expan> huic etiam <lb></lb>
ſatisfactum.</s>
        </p>
        <p type="main">
          <s id="s.004722"><arrow.to.target n="marg386"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004723"><margin.target id="marg386"></margin.target>395</s>
        </p>
        <p type="main">
          <s id="s.004724">Probl. 42. <emph type="italics"></emph>(Cur nec bis Diapente, nec bis Diateſſaron conſonant, ſed bis Dia­<lb></lb>
paſon. </s>
          <s id="s.004725">An quod Diapente conſonantia est in proportione ſeſquialtera? </s>
          <s id="s.004726">quod ſi tres <lb></lb>
ſeſquialteri, aut ſeſquitertij numeri ordine diſponantur, extremi <expan abbr="nullã">nullam</expan> inuicem pro­<lb></lb>
portionem habebunt, <expan abbr="neq;">neque</expan> enim multiplices, <expan abbr="neq;">neque</expan> ſuperparticulares erunt: At con­<lb></lb>ſonantia Diapaſon, quoniam in dupla ratione conſiſtit, qua duplicata, quadruplam <lb></lb>
extremi rationem obtinebunt. </s>
          <s id="s.004727">Itaque cum conſonantia ex ſonis conſiet proportio­<lb></lb>
nem, habentibus, <expan abbr="proportionemq́">proportionemque</expan>; habeant ij, qui interuallo bis Diapaſon compo­<lb></lb>
nuntur; minimè autem, ij, qui bis Diateſſaron, aut bis Diapente interuallis conti­<lb></lb>
nentur; idcirco ſoni bis Diapaſon conſoni ſunt, cæteri verò <expan abbr="nequaquã">nequaquam</expan> ob prædicta)<emph.end type="italics"></emph.end><lb></lb>
Quæ dicta ſunt ad probl. </s>
          <s id="s.004728">34. &amp; 35. totum ferè hunc locum illuſtrant. </s>
          <s id="s.004729">Cæ­<lb></lb>
terum <expan abbr="quãdo">quando</expan> Muſici volunt duas ſeſquialteras ratione ſimul addere, diſpo­<lb></lb>
nunt ordine tres numeros habentes inuicem ſeſquialteram rationem, vt ſe­<lb></lb>
quentes 9. 6. 4. quam deinde habent rationem extremi 9. &amp; 4. eam dicunt <lb></lb>
<expan abbr="compoſitã">compoſitam</expan> ex duabus ſeſquialteris: quæ quidem eſt dupla ſeſquiquarta, quæ <lb></lb>
conſonantiæ faciendæ inepta eſt; ſiue quæ non eſt conſonantia harmonica; <lb></lb>
vnde Muſici dicunt eoſdem numero nullam habere rationem, ideſt harmo­<lb></lb>
nicam, cum omnis harmonica ſit, aut multiplex, aut ſuperparticularis, vn­<lb></lb>de patet cur bis Diapente nullam pariat conſonantiam. </s>
          <s id="s.004730">Similiter, duæ ra­<lb></lb>tiones ſeſquitertiæ, 16. 12. 9. additæ efficiunt rationem, quæ eſt inter 16. &amp; <lb></lb>
9. quæ non eſt harmonica, quia neque multiplex, neque ſuperpendicularis <lb></lb>
eſt; ideo apud Muſicos nulla eſt; quamuis re vera ab Arithmeticis dicatur, <lb></lb>
&amp; ſit ſuperpartiens, &amp; in ſpecie ſuperquintupartiens nonas. </s>
          <s id="s.004731">Duæ verò du­<lb></lb>
plæ, vti 4. 2. 1. eo modo conflant rationem inter 4. &amp; 1. quæ multiplex eſt, <lb></lb>&amp; quadrupla dicitur, ideò harmonica eſt portio: vnde patet ratio, cur duæ <lb></lb>
Diapaſon conſonant. </s>
          <s id="s.004732">talis autem conſonantia appellatur Diſdiapaſon, cu­<lb></lb>
ius forma eſt in ratione quadrupla.</s>
        </p>
        <p type="main">
          <s id="s.004733"><arrow.to.target n="marg387"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004734"><margin.target id="marg387"></margin.target>396</s>
        </p>
        <p type="main">
          <s id="s.004735">Probl. 43. <emph type="italics"></emph>(Cur ſi quis <expan abbr="pſallẽs">pſallens</expan> neten apprehědat, ſola hypate ſubſonare videtur? <lb></lb>
</s>
          <s id="s.004736">An <expan abbr="quoniã">quoniam</expan> nete deſinens &amp; elangueſcens euadit hypate; <expan abbr="indiciũ">indicium</expan>, quòd poſt hypaten <emph.end type="italics"></emph.end>-<pb pagenum="270" xlink:href="009/01/270.jpg"></pb><emph type="italics"></emph>neten canore aptiſſimè licet. </s>
          <s id="s.004737">quaſi. </s>
          <s id="s.004738">n. </s>
          <s id="s.004739">cantus illius, ſit etiam huius ſimilitudinem ex <lb></lb>
illa capiunt. </s>
          <s id="s.004740">cùm <expan abbr="aũt">aut</expan> Echo eius cantus quidam ſit (eſt. </s>
          <s id="s.004741">n. </s>
          <s id="s.004742">tactus vocis netes deſinen­<lb></lb>
tis) ſonus idem exiſtens ſono hypates, meritò ob ſimilitudinem nete videtur moue­<lb></lb>
re hypaten. </s>
          <s id="s.004743">ſcimus enim neten appræhenſam non moueri; videntes verò hypatem <lb></lb>
non appræhenſam, &amp; ſonitum ipſius audientes, ipſam ſonare credimus. </s>
          <s id="s.004744">quod qui­<lb></lb>
dem in multis nobis accidit, in quibus <expan abbr="neq;">neque</expan> ratione, <expan abbr="neq;">neque</expan> ſenſu poſſumus certi ali­<lb></lb>
quid videre. </s>
          <s id="s.004745">Præterea ſi nete maximè intenta percutiatur, accidit iugum tremere, <lb></lb>
nihil igitur mirum. </s>
          <s id="s.004746">ipſo commoto, omnes chordas ſimul commoueri, nec abſurdum <lb></lb>
eas ſonum facere. </s>
          <s id="s.004747">ſonus quidem netes, &amp; deſinens, &amp; incipiens alienus eſt à cæte­<lb></lb>ris: deſinens tamen idem cùm hypate: quo addito propriæ ipſius motioni, illius to­<lb></lb>
tum videri, nihil abſurdi. </s>
          <s id="s.004748">eſt verò maior, quam communis reliquarum chordarum <lb></lb>ſonus, quod illæ quidem, quaſi à nete, propulſæ molliter ſonant: nete verò totis vi­<lb></lb>
ribus, omnium quippe vehementiſſima. </s>
          <s id="s.004749">it aque ſecundarius eius ſonus ſuperior reli­<lb></lb>
quis erit; præſertim cùm læuiſſimo motu moueantur)<emph.end type="italics"></emph.end> Idem quæſiuit num. </s>
          <s id="s.004750">24. quæ <lb></lb>
ibi dicta ſunt, huc etiam pertinent. </s>
          <s id="s.004751">quibus repetitis melius ſequentem pa­<lb></lb>
raphraſim percipies. </s>
          <s id="s.004752">Verum ante omnia antiquæ lyræ ex antiquis monu­<lb></lb>
mentis figuram oculis ſubijciam.</s>
        </p>
        <figure id="id.009.01.270.1.jpg" place="text" xlink:href="009/01/270/1.jpg"></figure>
        <p type="main">
          <s id="s.004753">Porrò, vt tradit Vincentius Galilæus in ſuis Dialogis, erat eius figura, <lb></lb>
quaſi ex caprino capite conſtructa, cuius duo brachia erant capræ cornua; <lb></lb>
inferior pars cranium, quæ tota baſi complanatæ ita ſuperponebatur, vt in <lb></lb>
quouis poſita plano recta conſiſteret, <expan abbr="neq;">neque</expan> vt geſtaretur, opus erat. </s>
          <s id="s.004754">chordæ <lb></lb>
ipſius, quæ eſſent, &amp; qua ratione eſſent collocatæ, in figura apparet; quot <lb></lb>
autem fuerint, pro temporum varietate determinandum eſt, nam primo 4. <lb></lb>
deinde 7. demum 8. fuerunt, &amp; plures etiam. </s>
          <s id="s.004755">Iugum autem, cuius cauſa fi­<lb></lb>
guram appoſui, erat ſupernum illud tranſuerſarium, cui fides annecteban­<lb></lb>
tur, vt idem Vincentius aſſerit. </s>
          <s id="s.004756">nunc ad textum.</s>
        </p>
        <pb pagenum="271" xlink:href="009/01/271.jpg"></pb>
        <p type="main">
          <s id="s.004757">Cur ſi quis neten cæteris intactis, percuſſam, ac ſonantem manu compre­<lb></lb>
hendat, ac ſiſtat, videbitur audire hypaten? </s>
          <s id="s.004758">primo reſpondet, id accidere, <lb></lb>
quia ſonus ille extremus, quo nete ceſſat, euadit ſonus ipſius hypates. </s>
          <s id="s.004759">pro­<lb></lb>
pterea igitur <expan abbr="tũc">tunc</expan> exiſtimamus audire hypatem. </s>
          <s id="s.004760">cuius rei indicium eſt, quod <lb></lb>
qui cantant hypaten, facilè ad neten cantandam tranſeunt; cùm enim can­<lb></lb>
tus hypates, ſit etiam cantus netes, &amp; veluti illius echo, facilè eſt ex hypa­<lb></lb>
te ſimilitudinem netes accipere. </s>
          <s id="s.004761">præterea in hoc decipimur, quia cum au­<lb></lb>
diamus ſonum hypates, <expan abbr="eamq́">eamque</expan>; minimè tentam videamus, quemadmodum <lb></lb>
neten videmus, eam ſonare meritò credimus; quod quidem in multis acci­<lb></lb>
dit, vbi nec ratio, nec ſenſus attingit, ſic in ſcena aliquando putamus quem­<lb></lb>
piam tuba ſonare, quod eam ori ipſius admotam videamus, cùm tamen <lb></lb>
alius poſt ſcenam lateat, qui tuba ſonet. </s>
          <s id="s.004762">ſimile accidit in nete, &amp; hypate. <lb></lb>
</s>
          <s id="s.004763">tertiò reſpondet, quod quando quis neten percutit, quæ omnium intenſiſſi­<lb></lb>
ma eſt, accidit, vt iugum, cui illa nectitur, moueatur, tremetque, ex quo <lb></lb>
tremore fit, vt reliquæ omnes chordæ moueantur, ac tremant, &amp; proinde <lb></lb>
ſonum edant. </s>
          <s id="s.004764">cùm autem ſonus netis, &amp; incipiens, &amp; deſinens ſit ferè idem <lb></lb>
cum ſono hypates, accidit in hoc caſu, vt ſonus deſinentis netis, vniatur <lb></lb>
cum ſono hypates, <expan abbr="ſicq́">ſicque</expan>; manifeſtè totus ille ſonus hypates eſſe videatur. <lb></lb>
</s>
          <s id="s.004765">cæteræ verò chordæ non audiuntur ob eorum ſonorum paruitatem, qui ideò <lb></lb>
exigui ſunt, quia excitati ſunt ab impulſu, &amp; motu iugi, qui exiguus erat. <lb></lb>
</s>
          <s id="s.004766">ſonus autem netes illis omnibus ſuperior eſt, &amp; quia ipſa primò percuſſa eſt, <lb></lb>
&amp; quia intenſiſſima eſt, &amp; celerrimè mouetur.</s>
        </p>
        <p type="main">
          <s id="s.004767"><arrow.to.target n="marg388"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004768"><margin.target id="marg388"></margin.target>397</s>
        </p>
        <p type="main">
          <s id="s.004769">Probl. 44. <emph type="italics"></emph>(Cur ſuauius ſolitariam cantilenam audimus, cùm ad tibiam, quam <lb></lb>
cùm ad lyram cantatur? </s>
          <s id="s.004770">An quod omne ſuaue, quod <expan abbr="mixiũ">mixtum</expan> est cum ſuauiori, ſua­<lb></lb>
uius redditur? </s>
          <s id="s.004771">Atqui tibia, quam lyra ſuauior eſt: ergò cantilena tibiæ admixta <lb></lb>
ſuauior erit, quam lyræ. </s>
          <s id="s.004772">quoniam omne mixtum, immixto ſuauius, modo quis ſen­<lb></lb>
ſum amborum percipiat. </s>
          <s id="s.004773">Vinum enim oximele ſuauius eſt, quoniam quæ natura <lb></lb>permiſcet, longè melius temperantur, quam quæ à nobis miſcentur. </s>
          <s id="s.004774">Vinum enim <lb></lb>
ex acuto, &amp; dulci ſapore mixtum eſt. </s>
          <s id="s.004775">Idem manifestant mala punica, quæ vinoſa <lb></lb>
appellantur. </s>
          <s id="s.004776">Enimuerò cantilena, &amp; tibia inuicem miſcentur, ob ſimilitudinem, <lb></lb>
ſpiritu enim <expan abbr="vtraq;">vtraque</expan> perficitur. </s>
          <s id="s.004777">ſonus autem lyræ, quoniam non ſpiritu ſit, <expan abbr="eſtq́">eſtque</expan> mi­<lb></lb>
nus ſenſibilis, quam tibiæ, minus voci immiſcetur: quare ſenſui diſcrimen inferens, <lb></lb>
minus ſuauis eſt; quemadmodum de ſaporibus dictum eſt. </s>
          <s id="s.004778">Adde, quod tibia ſoni­<lb></lb>
tu ſuo, &amp; humanæ vocis ſimilitudine, <expan abbr="plerosq́">plerosque</expan> cantilenæ errores occultare poteſt. <lb></lb>
</s>
          <s id="s.004779">Sonus autem lyræ cùm exiguus, <expan abbr="atq;">atque</expan> voci immixtus, <expan abbr="ideoq;">ideoque</expan> manifeſtus per ſe, ma­<lb></lb>
nifeſtum cantilenæ errorem, quaſt appoſita regula facit. </s>
          <s id="s.004780">cùm verò multa in can­<lb></lb>
tando peccentur, quod ex <expan abbr="vtriſq;">vtriſque</expan> compoſitum eſt, neceſſariò peius eſt)<emph.end type="italics"></emph.end> ſatis ex ſe <lb></lb>
clarum eſt.</s>
        </p>
        <p type="main">
          <s id="s.004781"><arrow.to.target n="marg389"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004782"><margin.target id="marg389"></margin.target>398</s>
        </p>
        <p type="main">
          <s id="s.004783">Probl. 45. <emph type="italics"></emph>(Cur neruum illum, quem Meſen, ſeu medium dicimus, ſic appella­<lb></lb>
mus; cùm inter ſeptem, non autem inter octo ſit <expan abbr="mediũ">medium</expan>? </s>
          <s id="s.004784">An quod olim harmoniæ <lb></lb>
ſeptem neruis conſtabant, quorum medium eſt. </s>
          <s id="s.004785">Præterea eorum, quæ inter quæuis <lb></lb>
extrema continentur illud ſolum, quod medium eſt, principium ctiam quoddam eſt: <lb></lb>
quod enim in medio eorum eſt, quæ in aliquo interuallo ad vtrumuis extremorum <lb></lb>
vergunt; illud ſolum, &amp; <expan abbr="mediũ">medium</expan>, &amp; principium est. </s>
          <s id="s.004786">cùm igitur in harmoniæ inter­<lb></lb>
uillo extrema ſint hypate, &amp; nete, <expan abbr="hisq́">hisque</expan> interiaceant reliqui ſoni, quorum is, qui <lb></lb>
Meſe dicitur, eſt etiam principium, quippe <expan abbr="principiũ">principium</expan> alterius tetrachordi, idcircò<emph.end type="italics"></emph.end> <pb pagenum="272" xlink:href="009/01/272.jpg"></pb><emph type="italics"></emph>meritò Meſe, ſeu medius dictus eſt. </s>
          <s id="s.004787">principium enim, &amp; medium vnum ſolum eſſe <lb></lb>
potuit eorum, quæ inter extrema aliqua continentur)<emph.end type="italics"></emph.end> Idem quæſiuit ſupra num, <lb></lb>
25. vide igitur, quæ ibi annotaui. </s>
          <s id="s.004788">vide præterea, quæ nu. 35. de Tetrachor­<lb></lb>
dis dicta ſunt, vbi cur Meſe ſit dux, &amp; principium primi tetrachordi appa­<lb></lb>
rebit. </s>
          <s id="s.004789">Quæſtioni autem reſpondet duplici modo. </s>
          <s id="s.004790">primò, quemadmodum <lb></lb>
etiam in 25. ſecundò, reſpondet idcircò Meſen ita eſſe appellatam, quod in­<lb></lb>
ter eos ſonos inter extrema contentos rationem principij haberet, ſolent <lb></lb>
enim ea, quæ inter extrema aliqua ſunt <expan abbr="cæterorũ">cæterorum</expan> principia, eſſe <expan abbr="etiã">etiam</expan> media.</s>
        </p>
        <p type="main">
          <s id="s.004791">Probl. 46. eſt idem cum ſuperiori 22. <expan abbr="vtrumq;">vtrumque</expan> autem ex ſe ita manifeſtum <lb></lb>
eſt, vt <expan abbr="abſq;">abſque</expan> harmonica facultate probè intelligatur.</s>
        </p>
        <p type="main">
          <s id="s.004792">Probl. 47. idem cum 26. in quibus ait, facilius eſſe canere acutum, quàm <lb></lb>
graue: in 37. verò contrarium: difficilius eſſe cantare acutum, quàm gra­<lb></lb>
ue. </s>
          <s id="s.004793">ſibi conciliabitur, ſi dixeris, hæc dicta eſſe problematicè.</s>
        </p>
        <p type="main">
          <s id="s.004794"><arrow.to.target n="marg390"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004795"><margin.target id="marg390"></margin.target>399</s>
        </p>
        <p type="main">
          <s id="s.004796">Probl. 48. <emph type="italics"></emph>(Cur veteres cùm ſeptem neruis concentus facerent, hypaten, non <lb></lb>
neten reliquerunt? </s>
          <s id="s.004797">An non hypate, ſed nunc <expan abbr="vocatã">vocatam</expan> paraneten, toniqué interuallum <lb></lb>
abſtulerunt, vltima verò Acutidenſi pro Meſe vtebantur propterea ipſam Meſen <lb></lb>
appellarunt. </s>
          <s id="s.004798">An quod ſuperioris Tetrachordi finis <expan abbr="principiũ">principium</expan> erat inferioris, &amp; me­<lb></lb>
dium extremorum habebat ſecundum ſoni proportionem)<emph.end type="italics"></emph.end> Idem quæſiuit ſupra <lb></lb>
num. </s>
          <s id="s.004799">7. vide igitur, quæ ibi expoſui; hoc loco quærendum reſtat, quid ſit <lb></lb>
illud Acutidenſum. </s>
          <s id="s.004800">pro qua re vide Ariſtoxenum lib. 3. &amp; Euclidem in Iſa­<lb></lb>
goge ad Muſicam: Zarlinum tandem lib. 2. ſupplem. </s>
          <s id="s.004801">&amp; 5. quæ res, quamuis <lb></lb>
plura dicant, adhuc ob antiquitatem non ſatis intelliguntur. </s>
          <s id="s.004802">Quid eſſet Den­<lb></lb>
ſum, exponit ſic Euclides: Denſum eſt certa trium ſonorum, vel duorum <lb></lb>
interuallorum ex ijs, qui Diateſſaron componunt diſpoſitio talis, vt inter­<lb></lb>
uallum, quod conſtituunt hæ tres voces, vel hæc duo interualla, ſit maius <lb></lb>
reliquo interuallo ipſius Diateſſaron. </s>
          <s id="s.004803">ponit præterea ſonorum alios eſſe <lb></lb>
Grauidenſos, alios Mediodenſos, alios Acutidenſos. </s>
          <s id="s.004804">quibus conſonant, quæ <lb></lb>
Ariſtoxenus ait, dum ait denſum fuiſſe illam partem Diateſſaron, in qua <lb></lb>
erant duo toni; ſic enim reliquum, quod erat <expan abbr="ſemitoniũ">ſemitonium</expan> multò minus erat. <lb></lb>
</s>
          <s id="s.004805">erant autem variæ Diateſſaron diuiſiones pro Generum varietate. </s>
          <s id="s.004806">Antiqui <lb></lb>
igitur ſecundum aliquam eorum diuiſionem, quæ denſum in parte acuta po­<lb></lb>
nebat vltimam chordam illius denſi, quæ pariter vltima erat illius Tetra­<lb></lb>
chordi, ſiue Diateſſaron pro media vtebantur, <expan abbr="eamq́">eamque</expan>; idcircò Meſen appel­<lb></lb>
larunt. </s>
          <s id="s.004807">quæ de Tetrachordis ſubdit clara ſunt ex dictis num. </s>
          <s id="s.004808">33.</s>
        </p>
        <p type="main">
          <s id="s.004809">Per ſuperius Tetrachordum intelligere debemus Acutius, ſic enim finis <lb></lb>
illius erit principium inferioris, ideſt grauioris Tetrachordi; vult enim <lb></lb>
Ariſt. vt ſupra non ſemel viſum eſt, acutiorem ſonum Tetrachordi eſſe illius <lb></lb>
principium. </s>
          <s id="s.004810">vide præſertim num. </s>
          <s id="s.004811">45. Antiqui igitur Paraneten omittentes, <lb></lb>
aliam, quæ vltima erat in parte denſa Tetrachordi, <expan abbr="quæq́">quæque</expan>; principium pri­<lb></lb>
mi, &amp; finis ſecundi erat, pro Meſe vtebantur, ex quibus quæſtioni vtcunque <lb></lb>
inuolutè ſatis reſpondet.</s>
        </p>
        <p type="main">
          <s id="s.004812"><arrow.to.target n="marg391"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004813"><margin.target id="marg391"></margin.target>400</s>
        </p>
        <p type="main">
          <s id="s.004814">Probl. 49. <emph type="italics"></emph>(Cur Tragœdiarum choris, <expan abbr="neq;">neque</expan> ſubdorio, <expan abbr="neq;">neque</expan> ſubphrygio cantandi <lb></lb>
genere, vti mos eſt? </s>
          <s id="s.004815">An quod modulŭm præſtare hæ harmoniæ nequeunt, quo choris <lb></lb>
valdè opus eſt; mores habet hypophrygius practicos (<expan abbr="quamobrẽ">quamobrem</expan> in Gerione excur­<lb></lb>ſus, &amp; armatio ipſo perficiunter) Subdorius verò magnificus, constans, <expan abbr="gramſq;">grauiſque</expan> <lb></lb>eſt, quocirca omnium harmoniarum maximè cytharæ conuenit. </s>
          <s id="s.004816">ſed hæc ambo, vt<emph.end type="italics"></emph.end> <pb pagenum="273" xlink:href="009/01/273.jpg"></pb><emph type="italics"></emph>choris minimè congruunt, ſic ſcenis ſunt magis domeſtica; etenim ſcena Heroum <lb></lb>
facta, dictaqué ſimulat. </s>
          <s id="s.004817">Veterum autem duces ſoli Heroes fuerunt. </s>
          <s id="s.004818">populi verò erant <lb></lb>
Homines, ex quibus chorus conſtat. </s>
          <s id="s.004819">quapropter choro competunt mores flebiles, &amp; <lb></lb>æquales, &amp; moduli; hæc enim humana ſunt. </s>
          <s id="s.004820">quæqué harmoniæ cæteræ aliæ non ha­<lb></lb>
bent. </s>
          <s id="s.004821">minimè verò hypophrygius, qui lymphaticus, <expan abbr="atq;">atque</expan> bacchicus eſt. </s>
          <s id="s.004822">At mixto­ <lb></lb>
lydius illa præſtare poteſt; propterea eo aliquo modo afficimur. </s>
          <s id="s.004823">magis autem debi­<lb></lb>
les afficiuntur, quàm fortes, quapropter ille choris conuenit. </s>
          <s id="s.004824">Hypodorio verò, &amp; <lb></lb>
hypophrygio agimus, qui choro non conueniunt; est enim chorus, veluti curator <lb></lb>
quidam otioſus, ijs ſolum beneuolentiam præbens, quibus adeſt)<emph.end type="italics"></emph.end> lege Probl. 30. <lb></lb>
vbi idem quæſiuit. </s>
          <s id="s.004825">lege præterea, quæ ad num. </s>
          <s id="s.004826">15. annotaui: ex quibus lo­<lb></lb>
cum hunc intelliges. </s>
          <s id="s.004827">quod ait <emph type="italics"></emph>(In Gerione excurſus, &amp; armatio)<emph.end type="italics"></emph.end> exiſtimo, <lb></lb>
Gerionem hunc Tragœdiam fuiſſe illam, quam Suidas in Nicomacho inter <lb></lb>
Nicomachi Alexandrini Tragici Tragœdias recenſet. </s>
          <s id="s.004828">excurſus verò, &amp; ar­<lb></lb>
matio erant partes, quibus conſtabat Tragœdia, quemadmodum noſtræ in <lb></lb>
<expan abbr="quinq;">quinque</expan> actus diuiduntur. </s>
          <s id="s.004829">vide Zarlinum cap. 5. primæ partis Inſtit. vbi tra­<lb></lb>
dit fabulam quandam Delonam dictam, quæ in modum Tragœdiæ habeba­<lb></lb>
tur, fuiſſe diuiſam in 5. partes, Explorationem, Prouocationem, Iambicum, <lb></lb>
Spondeum, &amp; Ouationem, aut Saltationem.</s>
        </p>
        <p type="main">
          <s id="s.004830"><arrow.to.target n="marg392"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004831"><margin.target id="marg392"></margin.target>401</s>
        </p>
        <p type="main">
          <s id="s.004832">Probl. 50. <emph type="italics"></emph>(Cur in ſonis grauioribus ſymphonia mollior euadit? </s>
          <s id="s.004833">An quod mo­<lb></lb>
dulatus cantus ſua quidem natura mollis eſt, &amp; quietus: ſed admixti numeri, ſeu <lb></lb>
rithmi ratione aſperior redditur, &amp; mouentior. </s>
          <s id="s.004834">cùm igitur ſonus granis, mollis, &amp; <lb></lb>quietus ſit; ſonus autem acutus mouens, &amp; irritans; omninò ſequitur admixito­<lb></lb>
ne eiuſdem numeri grauiorem cantum debere eſſe <expan abbr="quoq;">quoque</expan> molliorem; eſt enim modu­<lb></lb>
latus cantus ex ijs, quæ mollia ſunt)<emph.end type="italics"></emph.end> ſoni grauiores natura molliores ſunt, ideſt, <lb></lb>
molles mores reddunt, ſeu molles, <expan abbr="effœminatosq́">effœminatosque</expan>; decent magis, quàm ſo­<lb></lb>
ni acuti: moduli præterea, ſiue cantilenæ modulatæ, aut rithmi, molles na­<lb></lb>
tura ſunt, vt ſuperius num. </s>
          <s id="s.004835">38. explicatum eſt: ſi igitur vtrique &amp; graui, &amp; <lb></lb>
acuto addatur numerus, neceſſariò graue mollius euadet.</s>
        </p>
        <p type="main">
          <s id="s.004836"><arrow.to.target n="marg393"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004837"><margin.target id="marg393"></margin.target>402</s>
        </p>
        <p type="main">
          <s id="s.004838">Probl. 51. <emph type="italics"></emph>(Cur æqualium, &amp; ſimilium doliorum, ſi vnum ſit inane; alterum <lb></lb>
verò dimidiatum tinnitus eorum Diapaſon reſonabunt? </s>
          <s id="s.004839">An quoniam ſonus dimi­<lb></lb>
diati cum ſono vacui duplam habent inuicem proportionem: quid enim in iſtis po­<lb></lb>
tius, quàm in fiſtulis res euariat? </s>
          <s id="s.004840">motum <expan abbr="namq;">namque</expan> eŭndem acutiorem putamus, quem <lb></lb>
velociorem. </s>
          <s id="s.004841">in maioribus verò aer tardius occurrit, vt in duplis duplò, &amp; cæteris <lb></lb>
ſecundum proportionem. </s>
          <s id="s.004842">in vtris etiam, duplus cùm dimidio Diapaſon conſonat)<emph.end type="italics"></emph.end><lb></lb>
quæ initio dixi in diuiſione monochordij, &amp; alibi, ſed præſertim in Probl. 23. <lb></lb>
locum hunc abundè declarant. </s>
          <s id="s.004843">Vbicunque enim corpus ſonans duplum eſt <lb></lb>
alterius corporis ſonantis, ſiue ſint chordæ, ſiue fiſtulæ, ſiue dolia, ſiue vtres, <lb></lb>
reſonant Diapaſon, cuius forma conſiſtit in proportione dupla, quæ in hu­<lb></lb>
iuſmodi corporum ſonis reperitur.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.004844"><emph type="italics"></emph>Ex Sectione 23.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.004845"><arrow.to.target n="marg394"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004846"><margin.target id="marg394"></margin.target>403</s>
        </p>
        <p type="main">
          <s id="s.004847">Problema 2. <emph type="italics"></emph>(Cur nauigia onuſtiora in portu, quàm in alto eſſe videntur? </s>
          <s id="s.004848">cæ­<lb></lb>
lerius enim de alto in terram veniunt, quàm de terra in altum prouehantur? <lb></lb>
</s>
          <s id="s.004849">An quod plus aquæ, quàm minus reniti validius poteſt? </s>
          <s id="s.004850">parua enim oppreſſa onere<emph.end type="italics"></emph.end> <pb pagenum="274" xlink:href="009/01/274.jpg"></pb><emph type="italics"></emph>cedit, vt demergi neceſſe ſit: multa è contrariò repellit, ac ſustinet. </s>
          <s id="s.004851">vis enim ea <lb></lb>
eſt aquæ. </s>
          <s id="s.004852">vt ſurſum verſus compellat inferius; ergò, vt in portu maris parùm, ſic <lb></lb>
multùm in alto eſt: <expan abbr="itaq;">itaque</expan> plus oneris conuehi in portu videbitur, etiam mouebitur <lb></lb>ægrius, quia magis immergitur, &amp; aqua minus reniti poteſt: at verò in alto res <lb></lb>contra vſu venit)<emph.end type="italics"></emph.end> ſenſus verborum Ariſt. ſatis perſpicuus eſt, res tamen ſunt <lb></lb>
magis expendendæ. </s>
          <s id="s.004853">primò <expan abbr="namq;">namque</expan> maximè ambigo de experientia ipſa, quæ <lb></lb>
huic quæſtioni ſubijcitur, ſi enim vera ſunt ea, quæ ab Aſtronomis afferun­<lb></lb>tur, vt maris ſphæricitatem aſſerant, falſa neceſſariò erit experientia hæc: <lb></lb>
aiunt autem ipſi pari velocitate nauigia è portu in altum euehi, <expan abbr="atq;">atque</expan> ex al­<lb></lb>to in portum appellant; quod ſignum manifeſtum eſt, ſuperficiem maris <expan abbr="ex-timã">ex­<lb></lb>
timam</expan> æquè <expan abbr="vndiq;">vndique</expan> à centro mundi diſtare, ac proinde omninò <expan abbr="ſphæricã">ſphæricam</expan> eſſe.</s>
        </p>
        <p type="main">
          <s id="s.004854">Illud poſtea, quod pro ſolutione Problematis affert, dum ait, nauim ma­<lb></lb>
gis in portu, quàm in alto demergi (quoniam plus aquæ, valeat magis, quàm <lb></lb>
minus, nauigij onus ſuſtinere, parua enim aqua oppreſſa onere cædit faci­<lb></lb>lius, quàm multa) non paruam habet difficultatem. </s>
          <s id="s.004855">refragantur enim ma­<lb></lb>
ximorum <expan abbr="Matnematicorũ">Mathematicorum</expan> demonſtrationes. </s>
          <s id="s.004856">Archimedes enim demonſt. </s>
          <s id="s.004857">5. <lb></lb>
lib. 1. de ijs, quæ vehuntur in aqua acutiſſimè demonſtrat; ſolidarum ma­<lb></lb>
gnitudinum humido læuiorum, in humidum eò <expan abbr="vſq;">vſque</expan> demergi, vt tanta moles <lb></lb>
humidi, quanta eſt partis demerſæ, eandem quam tota magnitudo, graui­<lb></lb>
tatem habeat. </s>
          <s id="s.004858">quod idem Galilæus Galilæus, in Italico Diſcurſu de rebus, <lb></lb>
quæ aquæ innatunt, ſubtiliter <expan abbr="cõprobauit">comprobauit</expan>, vt videre eſt apud ipſum pag. </s>
          <s id="s.004859">14. <lb></lb>
quæ cum certa ſint, ſequitur neceſſariò falſum eſſe, maiorem aquæ copiam <lb></lb>
altiùs nauim quàm minorem, extollere. </s>
          <s id="s.004860">dummodo tamen aqua <expan abbr="vtrobiq;">vtrobique</expan> ſit <lb></lb>
eiuſdem grauitatis. </s>
          <s id="s.004861">quare Galilæus pag. </s>
          <s id="s.004862">17. ſic orationem claudit: valeant <lb></lb>
inquit, eorum falsè opiniones, qui exiſtimant nauigium facilius à magna <lb></lb>
aquæ copia ſuſtineri, quàm à parua: quod Ariſt. ſect. </s>
          <s id="s.004863">23. probl. </s>
          <s id="s.004864">2. credidit: <lb></lb>cum contrà verum ſit, nauim æquè facilè in oceano, <expan abbr="atq;">atque</expan> in decem doliorum <lb></lb>
aqua innatare, ac ſuſtineri hæc ille.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.004865"><emph type="italics"></emph>Ex Sectione 30.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.004866"><arrow.to.target n="marg395"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004867"><margin.target id="marg395"></margin.target>404</s>
        </p>
        <p type="main">
          <s id="s.004868">Ad 6 Probl. <emph type="italics"></emph>(Cur nihil in eo delectamur, quod triangulum duobus rectis pa­<lb></lb>
res angulos internos habere ſpectamus)<emph.end type="italics"></emph.end> vide quæ lib. 1. Priorum, ſecto <lb></lb>
3. cap. 3. de hac trianguli proprietate annotaui, cuius etiam ſæpius Ariſt. <lb></lb>
meminit, nunquam tamen verbum illud, internos, præterquam hic, addi­<lb></lb>
dit; vt autem benè intelligas quinam ſint hi anguli interni, &amp; qui externi, <lb></lb>
&amp; quod etiam rectis externi æquiualeat, lege quæ ad tex. 39. primi Poſter, <lb></lb>
ſunt annotata.</s>
        </p>
      </chap>
      <chap>
        <p type="head">
          <s id="s.004869"><emph type="italics"></emph>Ex Sectione 31.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.004870">Eorum, quæ ad oculos pertinent,</s>
        </p>
        <p type="main">
          <s id="s.004871"><arrow.to.target n="marg396"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004872"><margin.target id="marg396"></margin.target>405</s>
        </p>
        <p type="main">
          <s id="s.004873">Probl. 7. <emph type="italics"></emph>(Quam ob cauſam <expan abbr="vtrunq;">vtrunque</expan> aſpectum ſimul diuertere deſtrorſum, &amp; <lb></lb>
ſiniſtrorſum, &amp; ad nares demittere valemus, &amp; alterum ad dextram, &amp; ad <lb></lb>
ſiniſtram, ſimul verò vnum dextrorſum, alterum ſiniſtrorſum nequimus; ſimiliter <lb></lb>
<expan abbr="neq;">neque</expan> deorſum, <expan abbr="neq;">neque</expan> ſurſum. </s>
          <s id="s.004874">ſimul verò ad idem poſſumus, ſeparatim verò nequa-<emph.end type="italics"></emph.end>


<pb pagenum="275" xlink:href="009/01/275.jpg"></pb><emph type="italics"></emph>quam? </s>
          <s id="s.004875">An quia quamuis ſint duo aſpectus, ex vnico tamen principio eodem modo <lb></lb>
dependent? </s>
          <s id="s.004876"><expan abbr="quæcunq;">quæcunque</expan> autem ita ſe habent, quoties alterum extremŭμ mouetur, ne­<lb></lb>ceſſe est alterum conſequi ad idem; alterius enim extremum eſt alterius extremi <lb></lb>
principium. </s>
          <s id="s.004877">ſi igitur res vna nequit, ſimul in contraria moueri; nec aſpectus pote­<lb></lb>
runt: cùm ita accidat, vt extrema in partes aduerſas moueantur, ſi quidem alter <lb></lb>
ſurſum, alter deorſum moueatur, initiumqué ſequi alterum aſpectum; quod impoſſi­<lb></lb>
bile. </s>
          <s id="s.004878">Oculorum verò limitas inde oritur, quia oculorum globi principio continen­<lb></lb>
tur, quo &amp; ſurſum, &amp; deorſum, &amp; ad latera conuerti poſſint. </s>
          <s id="s.004879">cùm igitur ita ſint <lb></lb>
collocati, vt ſitu inuicem ſimili reſpondeant, <expan abbr="atq;">atque</expan> ſint in medio ſe ſe mouendi ſur­<lb></lb>
ſum, deorſum, &amp; ad latera, eodemqué in puncto viſum habeant, tali ſitu præcipuè <lb></lb>
ab inuicem ſunt inuariabiles. </s>
          <s id="s.004880">qui verò in eodem puncto pupillas habent, limi non <lb></lb>ſunt, ſed tamen ab inuicem differŭνt: nam alijs nigri aliquid occultatur, &amp; ſurſum <lb></lb>proijciunt alba, veluti ſternutaturi. </s>
          <s id="s.004881">alijs in angulŭμ oculi exteriorem, nigrŭμ vergit, <lb></lb>
ſicuti furioſis: alijs in interiorem ad nares, vt perſonis tragicis, &amp; ſeueris, qui ſunt <lb></lb>
contuitu graui. </s>
          <s id="s.004882">Quibus verò ſitu diſſimili globi ſunt poſiti, ſed eodem puncto ſtant <lb></lb>
pupillæ: aut quibus ſitus ſimilis eſt, ſed non idem punctum pupillarum hi neceſſariò <lb></lb>
limi ſunt. </s>
          <s id="s.004883">propterea toruè aſpiciunt, &amp; oculos contrahunt; conantur enim in eun­<lb></lb>
dem habitum collocare globum, alterum firmum continentes, alterŭ verò agitan­<lb></lb>
tes. </s>
          <s id="s.004884">neceſſariò enim limus eſt, cui non eodem de puncto viſus prodeunt, quippe qui <lb></lb>
dimotum contuendi principium, perinde ac ille, cui ſuppreſſo oculo res vna gemi­<lb></lb>
nata videtur. </s>
          <s id="s.004885">ergò ſi oculus ſurſum dimotus eſt, terminus inſpiciendi deorſum eſt: <lb></lb>
ſed ſi oculus deorſum lapſus eſt, terminus ſurſum. </s>
          <s id="s.004886">Vno verò oculo à ſitu ſuo di­<lb></lb>
moto, moueri quidem res viſa ſurſum, deorſumuè ob id videtur, quia &amp; pupilla: <lb></lb>ſed geminata nunquam apparebit, niſi duo ſint viſus, qui contorqueantur. </s>
          <s id="s.004887">talis ap­<lb></lb>
paret limo <foreign lang="grc">ετεροφθαλμῳ,</foreign> ſeu ſtraboni, vt duplicata illi videatur. </s>
          <s id="s.004888">propter poſitio­<lb></lb>nem verò id fit, quia ſcilicet oculus ſuo medio non ſit constitutus)<emph.end type="italics"></emph.end> Quæcunque ab <lb></lb>
Ariſt. hoc loco læuiter attinguntur, exactè ab opticis Alhazeno, &amp; Vitell. <lb></lb>
pertractantur. </s>
          <s id="s.004889">vide propoſ. </s>
          <s id="s.004890">26. lib. 3. Vitell. quæ eſt hæc. </s>
          <s id="s.004891">Vno oculo moto <lb></lb>
neceſſe eſt alium eidem conformiter moueri.</s>
        </p>
        <p type="main">
          <s id="s.004892">Quando ait <emph type="italics"></emph>(Et alterum ad dextram, &amp; ad ſiniſtram)<emph.end type="italics"></emph.end> ſignificat nos poſſe <lb></lb>
mouere alterum oculum, altero manente, quoquouerſus: quod non video <lb></lb>
quomodo verum ſit, alij fortè videbunt.</s>
        </p>
        <p type="main">
          <s id="s.004893">Quando ait <emph type="italics"></emph>(<expan abbr="Atq;">Atque</expan> ſint in medio mouendi ſe ſe)<emph.end type="italics"></emph.end> per medium mouendi intel­<lb></lb>
ligit Ariſt. punctum, quod concipitur eſſe in medio inter ſurſum, &amp; deor­<lb></lb>
ſum; necnon inter dextrum, &amp; ſiniſtrum oculorum in naturali poſitione <lb></lb>
manentium.</s>
        </p>
        <p type="main">
          <s id="s.004894">Quando ait <emph type="italics"></emph>(Eodemque in puncto viſum habent)<emph.end type="italics"></emph.end> &amp; <emph type="italics"></emph>(Quiverò in eodem pun­<lb></lb>
cto pupillas habent)<emph.end type="italics"></emph.end> per idem <expan abbr="pũctum">punctum</expan> intelligo illud, quod in vno oculorum <lb></lb>
habet eandem poſitionem cum altero puncto alterius oculi, ſic duo oculi <lb></lb>
habebunt pupillas in codem puncto, quando eas habebunt conſimiliter lo­<lb></lb>
catas, &amp; habebunt eandem in <expan abbr="vtroq;">vtroque</expan> oculo poſitionem.</s>
        </p>
        <p type="main">
          <s id="s.004895">Quando ait <emph type="italics"></emph>(Dimotum contuendi principiŭm habet<emph.end type="italics"></emph.end>) ideſt, locum pupillæ non <lb></lb>
habet in eodem ſitu, in quo oporteret, ideſt non habet conſimilem ſitum re­<lb></lb>
ſpectu ſurſum, &amp; deorſum, dextrorſum, &amp; ſiniſtrorſum, alteri pupillæ.</s>
        </p>
        <p type="main">
          <s id="s.004896">Quando ait <emph type="italics"></emph>(Perinde vt ille, cui res vna geminari oculo ſuppreſſo videtur)<emph.end type="italics"></emph.end> vt <lb></lb>rectius ea, quæ hoc loco ab Ariſt. dicuntur, percipi poſſint, explicandum <pb pagenum="276" xlink:href="009/01/276.jpg"></pb>prius exiſtimo, cur quamuis geminatos oculos habeamus, res tamen vnicæ <lb></lb>
non ſolent geminatæ videri, dummodo oculi à naturali ſuo ſitu non luxen­<lb></lb>
tur, quod etiam à Vitell. propoſit. </s>
          <s id="s.004897">28. &amp; 46. lib. 3. pertractatur: quamuis <lb></lb>
commentum illud Vitell. &amp; Alaz. non placeat de neruorum opticorum <lb></lb>
vnione, eò quod Anatomici refragentur.</s>
        </p>
        <p type="main">
          <s id="s.004898">Dicendum igitur, quod cùm anima vna ſit, &amp; obiectum etiam ſit vnum, <lb></lb>
&amp; cùm <expan abbr="vterq́">vterque</expan>; oculus habeat conſimilem omninò ſitum, ſit etiam, vt ſpecies <lb></lb>obiecti repreſentatiua eodem modo in <expan abbr="vtroq;">vtroque</expan> oculo ſituetur, ob quem con­<lb></lb>
ſimilem ſitum, tum oculi, tum ſpeciei ſit, vt anima vtatur duobus oculis, <lb></lb>tanquam vno oculo, &amp; duabus ſpeciebus tanquam vna ſpecie: ſi enim alter <lb></lb>
oculus alteri oculo imponeretur, eſſent omninò partes vnius congruentes <lb></lb>
partibus alterius, &amp; ſpecies vnius oculi congrueret, &amp; vniretur pœnitus <lb></lb>
cum altera alterius, ſecundum ſingulas earum partes conſimiles. </s>
          <s id="s.004899">vt autem <lb></lb>
ſpecies ſituentur conſimiliter in vtroque oculo neceſſe eſt, vt <expan abbr="vterq́">vterque</expan>; oculus <lb></lb>
eodem modo aſpiciat <expan abbr="obiectũ">obiectum</expan>; quod tunc ſit, quando axes viſuales vtriuſq; <lb></lb>
oculi vniuntur in obiecto. </s>
          <s id="s.004900">axis porrò viſualis eſt linea ab obiecto tendens ad <lb></lb>
centrum oculi, quæ tamen tranſeat per centra corneæ, vueæ, &amp; pupillæ. </s>
          <s id="s.004901">tunc <lb></lb>
partes ſpecierum erunt omninò conſimiliter collocatæ in <expan abbr="vtroq;">vtroque</expan> viſu: ita <lb></lb>
vt pars ſpeciei, quæ dextra eſt in vno, ſit etiam dextra in altero. </s>
          <s id="s.004902">Intelligo <lb></lb>
eandem partem reſpectu obiecti, quæ refert eandem obiecti partem. </s>
          <s id="s.004903">quem­<lb></lb>
admodum igitur nec duabus auribus audimus duas voces, nec duabus nari­<lb></lb>
bus geminos odores, nec duplici manu duplicatas res tactas: ita anima, <lb></lb>
ſeruatis, quæ nuper dixi, duobus viſibus res vnam vnicè videre debet.</s>
        </p>
        <p type="main">
          <s id="s.004904">Hinc facilius cognoſcemus, qua de cauſa res viſa aliquando geminetur. <lb></lb>
</s>
          <s id="s.004905">quoties enim ſpecies eiuſdem obiecti in altero oculorum habet alium ſitum, <lb></lb>
quàm in altero, geminatio accidit, quia non habet conſimilem ſitum, &amp; ſi <lb></lb>
vna alteri ſupponeretur, non reſponderent partes vnius dexteræ, v. g. par­<lb></lb>
tibus dextris alterius; vnde non identificarentur; nec quæ eſſent in eadem <lb></lb>
parti oculi repreſentarent eandem obiecti partem; &amp; propterea oculi non <lb></lb>
eſſent quodammodo vnus oculus, cùm alter ab altero diuerſimodè à ſpecie <lb></lb>
informaretur. </s>
          <s id="s.004906">vt autem ſpecies <expan abbr="vtrumq;">vtrumque</expan> oculum conſimiliter <expan abbr="informẽt">informent</expan>, ne­<lb></lb>
<figure id="id.009.01.276.1.jpg" place="text" xlink:href="009/01/276/1.jpg"></figure><lb></lb>
ceſſe eſt, vt axes viſuales, quales <lb></lb>
ſunt in appoſita figura, C B, D B, <lb></lb>
ab oculis C, D, ad obiectum B, <lb></lb>ducti, in ipſo obiecto B. imò in <lb></lb>
eodem ipſius puncto vniantur: <lb></lb>
quoties. </s>
          <s id="s.004907">n. </s>
          <s id="s.004908">res viſa <expan abbr="nõ">non</expan> eſt in con­<lb></lb>
curſu axium, vt eſt res A. tunc <lb></lb>
diſſimiliter ſpeciem ad oculos <lb></lb>
mittit, nam ſpecies puncti A, in oculo D, erit ad ſiniſtram centri pupillæ; <lb></lb>
in oculo verò C, erit ad dextram.</s>
        </p>
        <p type="main">
          <s id="s.004909">Quando pariter alterum oculorum digito ſurſum, aut deorſum compri­<lb></lb>
mimus, fit, vt ille aliquantulum à loco ſuo naturali, &amp; conſimili ſitui alte­<lb></lb>
rius dimoueatur; quare neceſſe, vt axis ipſius ſimiliter ad motum oculi di­<lb></lb>
moueatur, nec amplius concurrat cum altero axe, in eodem obiecti puncto, <lb></lb>
in quod alter <expan abbr="tẽdet">tendet</expan>, vel <expan abbr="ẽt">et</expan> in <expan abbr="alterũ">alterum</expan> <expan abbr="obiectũ">obiectum</expan>. </s>
          <s id="s.004910">vide Vitell. prop. 103. &amp; 104. li. </s>
          <s id="s.004911">4.</s>
        </p>
        <pb pagenum="277" xlink:href="009/01/277.jpg"></pb>
        <p type="main">
          <s id="s.004912">Quando ait <emph type="italics"></emph>(Si oculus ſurſum dimotus eſt, terminus inſpiciendi deorſum est)<emph.end type="italics"></emph.end><lb></lb>Per terminum inſpiciendi intelligo rem illam, quæ prius videbatur, &amp; poſt <lb></lb>
oculi dimotionem infra axem viſualem remanet.</s>
        </p>
        <p type="main">
          <s id="s.004913">Quando ait <emph type="italics"></emph>(Sed geminata nunquam apparebit, niſi duo ſint viſus, qui contor­<lb></lb>
queantur)<emph.end type="italics"></emph.end> niſi duplex ſit conſpectus, ideſt, niſi oculus vnus ab altero diffe­<lb></lb>renter ſituetur, ſic enim ſpeciem diuersè reſpiciunt, non videbitur res du­<lb></lb>
plicata.</s>
        </p>
        <p type="main">
          <s id="s.004914">Quando ait <emph type="italics"></emph>(Tali apparet limo, ſeu ſtraboni)<emph.end type="italics"></emph.end> græcè ait <foreign lang="grc">ετεροφθσλμω</foreign>, quod <lb></lb>
propriè ſignificat eum, quem Latini Luſcum dicunt, qui vnius <expan abbr="tantũ">tantum</expan> eſt ocu­<lb></lb>
li. </s>
          <s id="s.004915">videtur tamen vſurpaſſe illud pro limo, ſeu ſtrabone, vt Gaza etiam ac­<lb></lb>
cipit, ſecus enim non poſſet illi res geminari, cùm ad id neceſſarij ſint duo <lb></lb>
oculi, vt modo dixerat.</s>
        </p>
        <p type="main">
          <s id="s.004916">Quando ait <emph type="italics"></emph>(Propter poſitionem verò id fit)<emph.end type="italics"></emph.end> ex paulò ante dictis poſſunt <lb></lb>
intelligi. </s>
          <s id="s.004917">verumtamen, &amp; illud addam; Duplicis conſpectus, vel gemina­<lb></lb>tionis cauſa eſſe poteſt, vel diuerſus oculorum ſitus, vel etiam ſitus ſpecie­<lb></lb>
rum diuerſus, vt quando obiectum eſt intra concurſionem axium, vt in <expan abbr="præ-cedẽti">præ­<lb></lb>
cedenti</expan> figura, vbi etiamſi oculi naturalem <expan abbr="ſitũ">ſitum</expan> conſeruent, res geminabitur.</s>
        </p>
        <p type="main">
          <s id="s.004918"><arrow.to.target n="marg397"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004919"><margin.target id="marg397"></margin.target>406</s>
        </p>
        <p type="main">
          <s id="s.004920">Probl. 11. <emph type="italics"></emph>(Cur diſtractis oculis res vna duæ apparent? </s>
          <s id="s.004921">An quod radij <expan abbr="vtriuſq;">vtriuſque</expan> <lb></lb>
oculi ad idem <expan abbr="pũctum">punctum</expan> non concurrunt? </s>
          <s id="s.004922">quaſi ergò duo videat, bis idem videre ani­<lb></lb>
ma exiſtimat ſimile eſt in permutatis digitis, vnum&#039; enim duo apparet, tanquam bis <lb></lb>
tactum<emph.end type="italics"></emph.end>) Præſens Problema ex dictis in præcedenti problemate ſatis clarum <lb></lb>
euadit: imò illa ex his viciſſim confirmantur.</s>
        </p>
        <p type="main">
          <s id="s.004923">Quando ait (<emph type="italics"></emph>Radij <expan abbr="vtriuſq;">vtriuſque</expan> oculi ad idem punctum non concurrunt<emph.end type="italics"></emph.end>) intelligit <lb></lb>
de axibus viſualibus, quos in ſuperiori declaratos habes.</s>
        </p>
        <p type="main">
          <s id="s.004924">Quando ait (<emph type="italics"></emph>Simile eſt in permutatis digitis<emph.end type="italics"></emph.end>) vt pulcherrimum iſtud experi­<lb></lb>mentum, quo res vna tacta, duæ videntur, experiaris oportet, vt globulum <lb></lb>
quempiam duobus proximis digitis eiuſdem manus tangas, ita vt vnus al­<lb></lb>
terum decuſſet, ſiue tranſcendat, vel ei conuoluatur, ita vt extremitates di­<lb></lb>
gitorum permutent loca, vel vt extremum vnius ſit, vbi deberet eſſe extre­<lb></lb>
mum alterius; deinde globulum inter <expan abbr="vtriuſq;">vtriuſque</expan> digiti extrema locatum, ſimul <lb></lb>
tangant, tunc enim exiſtimabis te duos globulos tangere.</s>
        </p>
        <p type="main">
          <s id="s.004925"><arrow.to.target n="marg398"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004926"><margin.target id="marg398"></margin.target>407</s>
        </p>
        <p type="main">
          <s id="s.004927">Probl. 17. <emph type="italics"></emph>(Cur res vna non videtur geminari, ſi oculum in latera contorquent? <lb></lb>
An quia conſpiciendi <expan abbr="principiũ">principium</expan> ab eadem linea ſumendum eſt. </s>
          <s id="s.004928">duo autem videntur, <lb></lb>quando illud ſurſum, aut deorſum mutatur; in latus verò nihil refert, niſi ſimul <lb></lb>
ſurſum, aut deorſum)<emph.end type="italics"></emph.end> quod præſenti problemate proponitur, non videtur <expan abbr="vſ-quequaq;">vſ­<lb></lb>
quequaque</expan> verum, expertus enim ſum, moto etiam in latus oculo, res viſas, <lb></lb>
quamuis magna cum difficultate, geminari. </s>
          <s id="s.004929">per lineam illam, à qua princi­<lb></lb>
pium ſumitur conſpiciendi, intelligit lineam rectam tranſeuntem per cen­<lb></lb>
tra <expan abbr="vcriuſq;">vtriuſque</expan> pupillæ. </s>
          <s id="s.004930">quod autem ait nihil referri, ſi oculus in latus, ſiue ad <lb></lb>
prædictam lineam luxetur, falſum omninò puto ex dictis ſupra ad Probl. 7. <lb></lb>hoc enim modo alter oculus diſſimiliter ab altero collocatur, vnde neceſſe <lb></lb>
eſt <expan abbr="cõſequi">conſequi</expan> geminationem ſecus ac ſi ſurſum, aut deorſum <expan abbr="alterũ">alterum</expan> luxaueris.</s>
        </p>
        <p type="main">
          <s id="s.004931"><arrow.to.target n="marg399"></arrow.to.target></s>
        </p>
        <p type="margin">
          <s id="s.004932"><margin.target id="marg399"></margin.target>408</s>
        </p>
        <p type="main">
          <s id="s.004933">Probl 21. <emph type="italics"></emph>(Cur alia quidem ambobus oculis potius inſpicimus; rectitudinem <lb></lb>
verò, quæ eſt in verſibus, vnum oculum literis admouentes potius conſpicimus? <lb></lb>
</s>
          <s id="s.004934">An quia verſus quidem coincidentes quemadmodum tradunt Optici, perturbatio­<lb></lb>nem quandam afferunt; quando verò vnico viſu inſpicimus, ſecundum vnicam re-<emph.end type="italics"></emph.end>


<pb pagenum="278" xlink:href="009/01/278.jpg"></pb><emph type="italics"></emph>ctam viſualem lineam inſpicimus, qua tanquam recta regula melius verſuum re­<lb></lb>ctitudinem dignoſcimus; rectum enim recto dijudicatur.)<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.004935">Quando ait <emph type="italics"></emph>(V num oculum literis admouentes)<emph.end type="italics"></emph.end> quando volumus inſpicere <lb></lb>
num rectus ſit ſcripturæ alicuius verſus, oculum alterum altero clauſo, prin­<lb></lb>
cipio, aut extremo illius verſus admouemus, vt hoc modo ſecundum longi­<lb></lb>
tudinem, non autem è regione illum intueamur, ſic enim linea viſualis re­<lb></lb>
cta, quaſi linea <expan abbr="quædã">quædam</expan> materialis rectitudini verſus coaptata,illã examinat.</s>
        </p>
        <p type="main">
          <s id="s.004936">Libet his opticis Problematibus, auctarij loco, tractationem quandam <lb></lb>
de Oculi pupilla, cùm ſit eiuſdem argumenti, apponere, in qua nonnulla <lb></lb>
ſcitu digna, <expan abbr="atq;">atque</expan> iucunda, ac nuper obſeruata traduntur.</s>
        </p>
        <p type="head">
          <s id="s.004937"><emph type="italics"></emph>De humani Oculi Pupilla.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.004938">Vt ea, quæ dicenda ſunt, facilius percipi poſſint, neceſſe eſt breuiter <lb></lb>
oculi fabricam, in qua mirum totius naturæ Opificis artificium elu­<lb></lb>
cet, auxilio ſequentium figurarum præmittere.</s>
        </p>
        <figure id="id.009.01.278.1.jpg" place="text" xlink:href="009/01/278/1.jpg"></figure>
        <p type="main">
          <s id="s.004939">A B C, cornea. </s>
          <s id="s.004940">ſpatium A B, eſt propriè corneæ pars tranſlucida.</s>
        </p>
        <p type="main">
          <s id="s.004941">E G F, vuea. </s>
          <s id="s.004942">H I, pupilla, ſeu foramen vueæ.</s>
        </p>
        <p type="main">
          <s id="s.004943">L M N O P, aranea, quæ medium orbiculum M N Q, ambit.</s>
        </p>
        <p type="main">
          <s id="s.004944">Referat ergò prima hæc figura humanum oculum, ſeu potius oculi ſe­<lb></lb>
ctionem, qui nimirum diſſectus ſit ab anteriori, <expan abbr="vſq;">vſque</expan> ad poſteriorem partem <lb></lb>
in duas æquas partes. </s>
          <s id="s.004945">qua ſectione appareant omnes tunicæ, &amp; humores,ex <lb></lb>
quibus ille conflatur. </s>
          <s id="s.004946">conſtat autem ſecundum Anatomicos ex tribus tuni­<lb></lb>
cis, <expan abbr="totidemq́">totidemque</expan>; humoribus. </s>
          <s id="s.004947">verum figuram explicemus. </s>
          <s id="s.004948">Pedunculus ille C, <lb></lb>
neruus eſt opticus è cerebro manans, ex quo tanquam ex radice totus ena­<lb></lb>
ſcitur oculus.</s>
        </p>
        <p type="main">
          <s id="s.004949">Exteriori illa circunferentia A B C, ſignificatur membrana totum ocu­<lb></lb>
lum complectens, quæ cornea propter duritiem appellatur. </s>
          <s id="s.004950">cuius pars pun­<lb></lb>
ctis A, B, terminata, inſtar Laternæ cornu, pellucida eſt. </s>
          <s id="s.004951">hanc vulgò lucem <lb></lb>
oculi, Medici Iridem maiorem appellant.</s>
        </p>
        <p type="main">
          <s id="s.004952">Media illa, &amp; imperfecta peripheria E H I F G, vuea ab acini vuæ nigræ <lb></lb>
ſimilitudine nuncupatur; eſt enim plurimum nigra. </s>
          <s id="s.004953">hæc vbi ad partem cor­<lb></lb>
neæ tranſlucidam A B, peruenit, eam quaſi fugiens intra oculum ſubſidit,


<pb pagenum="279" xlink:href="009/01/279.jpg"></pb>&amp; tendens per loca E H I F, ipſi corneæ, veluti <expan abbr="infundibulũ">infundibulum</expan> quoddam ſuppo­<lb></lb>
nitur. </s>
          <s id="s.004954">hinc <expan abbr="aliorũ">aliorum</expan> anatomicorum figuras corrigere licebit, in quibus mem­<lb></lb>
brana E H I E, <expan abbr="tanquã">tanquam</expan> plana ſuperficies ipſi corneæ ſupponitur.nõ eſt tamen <lb></lb>
hac parte tota integra, nam, vt ait Plinius, medium eius feneſtrauit Pupilla. <lb></lb>
</s>
          <s id="s.004955">ea eſt paruum foramen rotundum inter puncta H, I.porrò ſi liceat <expan abbr="hãc">hanc</expan> vueę <lb></lb>partem E, H, I, F, è directò integram aſpicere, ſimilis videbitur huic cir­<lb></lb>
culo ſecundæ figuræ A B C; in cuius medio circulus G H I, foramen eſt, cui <lb></lb>tum Iridi minori, tum Pupillæ nomen eſt; quæ noſtri eſt materia ſermonis: <lb></lb>
res quidem exigua, ſed planè admirabilis: tantillo enim foramine, maria, <lb></lb>
montes, innumera animalia, ac plantas graphicè, <expan abbr="atq;">atque</expan> adeò locis diſtincta <lb></lb>
animus noſter inſpicit; imò, vt cecinit Manilius.</s>
        </p>
        <p type="main">
          <s id="s.004956"><emph type="italics"></emph>Paruula ſic magnum peruiſit pupula Cœlum.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.004957">Sic olim admirationi fuit Homeri Ilias, exiguis adeò litoris conſcripta, vt <lb></lb>
vnius nucis cortice clauderetur.</s>
        </p>
        <p type="main">
          <s id="s.004958">Superficies huius membranæ E H I F, exterior, quæ ſcilicet corneam re­<lb></lb>
ſpicit, in homine varia eſt, cæſia, glauca, ſubalbida, nigra. </s>
          <s id="s.004959">Excipiendi ſunt <lb></lb>
à cæteris Sinarum gentes, quæ, vt poſtremò perlatum eſt à noſtris PP. ſunt <lb></lb>
omnes ſpectandi nigris oculis. </s>
          <s id="s.004960">Tartari etiam omnes virides habent oculos. <lb></lb>
</s>
          <s id="s.004961"><expan abbr="vtriq́">vtrique</expan>; ſcilicet tales habent oculos, quòd tales habeant vueas. </s>
          <s id="s.004962">ex hac enim <lb></lb>
varius oculorum color: quippe qui non in exteriori ſuperficie corneæ, quæ <lb></lb>
omninò diaphana eſt, &amp; propterea excolor, ſed vueæ inſidet. </s>
          <s id="s.004963">In nocturnis <lb></lb>
tamen animalibus lucida eſt; <expan abbr="atq;">atque</expan> hinc lux illa, cuius ope, circumfuſus aer <lb></lb>
adeò illuſtratur, vt noctu videre queant. </s>
          <s id="s.004964">ſi qui etiam <expan abbr="hominũ">hominum</expan> noctu videant, <lb></lb>
ij flaua, ac lucida vuea, vt obſeruaui, præditi ſunt: &amp; ideò interdiù maiorem <lb></lb>
Iridem flauum oſtendunt. </s>
          <s id="s.004965">ſuperficies tandem illius, quæ oculi interiora con­<lb></lb>
ſpicit, nigerrimo colore, quì vel <expan abbr="Anatomicorũ">Anatomicorum</expan> digitos inficiat, intincta eſt.</s>
        </p>
        <p type="main">
          <s id="s.004966">Tertia tandem, quæ per P L M N O, incedit Aranea dicitur, eſt enim in­<lb></lb>
ſtar araneæ tenuiſſima, præſertim, quà vuæ E H I F, ſupponitur. </s>
          <s id="s.004967">hæc præte­<lb></lb>rea globuli M K I N Q, anteriorem partem M H I N, circumueſtit, qui in <lb></lb>
ea affixus, non ſecus ac Araneus in ſuæ araneæ centro, immobilis hæret. <lb></lb>
</s>
          <s id="s.004968">hæc de tunicis.</s>
        </p>
        <p type="main">
          <s id="s.004969">Reliqui ſunt humores tres, quibus oculus repleatur; poſterius iſtud ſpa­<lb></lb>tium P L Q O, humori vitreo ob vitri ſimilitudinem dicto, natura attri­<lb></lb>
buit: anteriorem oculi ſedem inter corneam, &amp; vueam, humor aqueus oc­<lb></lb>
cupauit, ſic dictus, quod ſit natura limpidiſſimus; quippe qui primus ingre­<lb></lb>
dientia rerum ſimulacra excipiat. </s>
          <s id="s.004970">medium locum, prædictum ſcilicet glo­<lb></lb>
bulum, quem aranea complectitur, humor chriſtallinus ſibi vindicauit. </s>
          <s id="s.004971">hic <lb></lb>
quà corneam ſpectat ſphæricæ eſt figuræ, atque ex hac parte foramen H I, <lb></lb>
vueæ, ſeu pupillam obſidet, vt aduentantibus rerum ſimulacris ſit obuius, <lb></lb>
<expan abbr="eaq́">eaque</expan>; ſiſtat, vnde factum eſt illi <expan abbr="quoq;">quoque</expan> pupillæ nomen. </s>
          <s id="s.004972">Huic <expan abbr="quoq;">quoque</expan> Aranea ſi­<lb></lb>
mul, &amp; vuea tenuiſſimis fibris in orbem connectuntur; quæ connexio non in <lb></lb>
ora pupillæ extrema, ſed circa ipſam, vt in circulo D E F, ſecundæ figuræ: <lb></lb>
qui fibrarum circulus apparet etiam in vuea, eſt enim veluti ſutura quædam <lb></lb>
circularis circa pupillam, non longè tamen ab ipſa. </s>
          <s id="s.004973">porrò iunctura hæc <lb></lb>
adeò fortis eſt, vt non ſine aliqua vi vuea, &amp; aranea inde à chriſtallino di­<lb></lb>
uellantur. </s>
          <s id="s.004974">Hæc de oculi fabrica nunc ſufficiant.</s>
        </p>
      </chap>
      <pb pagenum="280" xlink:href="009/01/280.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.004975"><emph type="italics"></emph>PROBLEMATA NONNVLLA <lb></lb>
De Oculi Pupilla.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.004976">1. Cvr ei Pupillæ nomen inditum eſt? </s>
          <s id="s.004977">Admiratione ſanè non caret <lb></lb>apud præcipuas linguas, miro quodam conſenſu, idem etymon <lb></lb>
obtinere: ſcilicet denominatam eſſe ab imaguncula illa, quæ <lb></lb>
veluti parua puppa, ſeu pupula, ſeu pupilla, qualis in figura <lb></lb>
ſpectatur, perpetuò in paruo hoc vuæ circello ſpectatur. </s>
          <s id="s.004978">propter hanc igitur <lb></lb>
puppam Hebræi circellum G B I, Bath, ideſt filiolam, Græci <foreign lang="grc">κορην</foreign>, ideſt puel­<lb></lb>
lam, Latini demum pupillam cognominarunt.</s>
        </p>
        <p type="main">
          <s id="s.004979">2. Atqui vnde imaguncula hæc, quæ in oculis noſtris perpetuò ſpectatur? <lb></lb>
</s>
          <s id="s.004980">ſcilicet ob terſitiem, &amp; ſphæricitatem cornea eſt inſtar <expan abbr="cõuexi">conuexi</expan> ſpeculi, quod <lb></lb>
ſpectanti imaginem reddat, quam ergò videmus in aliorum oculis puppam, <lb></lb>
noſtra eſt imago, quæ propterea tamen parua eſt, quoniam oculus ſpeculum <lb></lb>
paruum, ac conuexum ſimul eſt, cuius eſt imagines rebus ipſis multò mino­<lb></lb>
res reflectere. </s>
          <s id="s.004981">vt in tractatu de Speculis optici demonſtrant.</s>
        </p>
        <p type="main">
          <s id="s.004982">3. Cùm tota cornea, quæ Iris maior eſt, ſit æquè torſa, ac perpolita, cur <lb></lb>
non æquè tota hanc pupulam oſtendit? </s>
          <s id="s.004983">ſed è regione minoris Iridis ferè tan­<lb></lb>
tum? </s>
          <s id="s.004984">cauſa eſt in promptu, quia nimirum ſpeculum debet eſſe omnis colo­<lb></lb>
ris expers, ne colores ſpeculi coloribus imaginum miſceantur; taliſque eſt <lb></lb>
Iris minor,quæ etiamſi videatur nigra, non tamen verè colorata eſt, vt mox <lb></lb>
oſtendam: at verò maiori Iridi colores vueæ ſubſunt, qui ne ſpeculi officio <lb></lb>
fungatur, ſunt impedimento. </s>
          <s id="s.004985">eſt præterea Iris minor admodum opacata, <lb></lb>
quæ altera conditio, maximè ſpeculo neceſſaria eſt: illa enim nigredo Iri­<lb></lb>
dis minoris, ſeu pupillæ, non nigredo, ſed opacitas eſt, vt dicetur poſtea.</s>
        </p>
        <p type="main">
          <s id="s.004986">4. Cùm iam <expan abbr="cõſtet">conſtet</expan> foramini vuæ G H I, à pupilla in ipſo verſante nomen <lb></lb>
inditum eſſe; nec non vnde ſit ea pupilla, &amp; cur tam parua; quæritur iam, <lb></lb>
quid ſit pupilla, ſeu Iris minor, an ſcilicet ſit foramen illud vueæ, an potius <lb></lb>
chriſtallinus humor, qui in illud intruditur, <expan abbr="vacuumq́">vacuumque</expan>; illius replet? </s>
          <s id="s.004987">Re­<lb></lb>
ſpondeo, Ariſt. humorem ipſum <expan abbr="chriſtallinũ">chriſtallinum</expan> appellaſſe pupillam: Galenum<lb></lb>
tum chriſtallinum, tum foramen ipſum: aptè tamen <expan abbr="vtrumq;">vtrumque</expan> dici exiſtimo. <lb></lb>
</s>
          <s id="s.004988">Chriſtallinum quidem &amp; quia replet vacuum illud, <expan abbr="atq;">atque</expan> è regione illius pu­<lb></lb>
pillæ imaguncula ſpectatur. </s>
          <s id="s.004989">foramen verò, quia terminos illius rotunditatis <lb></lb>
circumſcribat. </s>
          <s id="s.004990">vnde aptius fortè dixeris, vtrumque, chriſtallinum ſcilicet, <lb></lb>
&amp; foramen veluti partes ad totam pupillam conſtituendam eſſe neceſſaria: <lb></lb>
ita vt nihil aliud ipſa ſit, quàm ſuperficies illa chriſtallini; quæ vueæ fora­<lb></lb>
mine continetur.</s>
        </p>
        <p type="main">
          <s id="s.004991">5. Cur in omnibus hominibus nigra videtur? </s>
          <s id="s.004992">cum tamen nigri nihil ibi <lb></lb>
eſſe ex anotomia conſtet: imò ibi chriſtallinus eſt omninò pelluidus; &amp; vl­<lb></lb>
tra, <expan abbr="citraq́">citraque</expan>; alij duo humores, vitreus, &amp; aqueus, æquè tranſparentes, <expan abbr="atq;">atque</expan> <lb></lb>
omnis nigredinis expertes. </s>
          <s id="s.004993">vnde igitur nigredo illa? </s>
          <s id="s.004994">Dicendum eſt nigre­<lb></lb>
dinem hanc non eſſe veram, ſed apparentem, <expan abbr="eamq́">eamque</expan>; ex interna oculi opa­<lb></lb>
citate; opacitatem verò ex foraminis paruitate, quæ lumen non admittat, <lb></lb>prouenire: quotidiana enim nos docet experientia feneſtellas, &amp; huiuſmodi <pb pagenum="281" xlink:href="009/01/281.jpg"></pb>alia foramina, quæ intus non ſint illuminata, ſed tenebroſa, nigra quamuis <lb></lb>
minimè ſint, apparere. </s>
          <s id="s.004995">Idem præterea mihi ex anatome manifeſtè patuit, <lb></lb>
cùm enim per ſectionem caſu quodam pupilla oculi, quem ſecabam, facta <lb></lb>
fuiſſet aliquanto mior, illicò nigredine omni exuta, alba viſa eſt; quia ſci­<lb></lb>
licet patuit lumini aditus, quod internam oculi opacitatem fugauit. </s>
          <s id="s.004996">pro­<lb></lb>
pterea in bobus, &amp; capris, quia magna, &amp; oblonga eſt, quæ multum lucis <lb></lb>
admittat, aloa ſimiliter, non vt in nobis nigra conſpicitur.</s>
        </p>
        <p type="main">
          <s id="s.004997">6. Cur in clariſſima luce Solis pupilla omninò euaneſcit? </s>
          <s id="s.004998">ex dictis in præ­<lb></lb>
cedenti problemate, huic <expan abbr="quoq;">quoque</expan> ſatisfieri poteſt. </s>
          <s id="s.004999">cùm enim clariſſimo lumi­<lb></lb>ni obijcitur, fit, vt oculi interiora illuſtrentur; vnde fugatis tenebris, &amp; om­<lb></lb>
ni opacitate, etiam nigredo illa nulla ſit.</s>
        </p>
        <p type="main">
          <s id="s.005000">7. Poſſumus ne dum in oculos alterius intuemur, quanta ſit re vera, co­<lb></lb>
gnoſcere? </s>
          <s id="s.005001">exiſtimo certam ipſius quantitatem oculos noſtros omninò late­<lb></lb>
re: videtur enim per refractionem, cùm ſit infra humorem aqueum aere <lb></lb>denſiorem, at quæ refractè videntur ex medio denſiore, ea maiora, quàm <lb></lb>
ſint, apparere, demonſtrant Perſpectiui. </s>
          <s id="s.005002">attamen cùm in luce ſatis tempe­<lb></lb>
rata verſamur, ſi oculum directè, vt minor fiat refractio, ſpectemus, ipſam <lb></lb>
non multò verò maiorem ſpectabimus.</s>
        </p>
        <p type="main">
          <s id="s.005003">8. Sed vnde nam illud mirum illi accidit, vt modo maior, modo minor <lb></lb>
nullo fermè tempore interpoſito, euadat; &amp; aliquando ad tantam magni­<lb></lb>
tudinem exereſcat, vt totam forè maiorem Iridem occupet; vt ſi in ſecun­<lb></lb>
da figura foramen B G I, <expan abbr="vſq;">vſque</expan> ad circulum L M N O, dilataretur? </s>
          <s id="s.005004">Antiquio­<lb></lb>
res, vt Galenus, cùm obſeruaſſent eam, multò maiorem eſſe noctu, quam <lb></lb>
interdiù, id ſolummodo ratione noctis, &amp; diei contingere, <expan abbr="atq;">atque</expan> foramen il­<lb></lb>
lud verè augeri, &amp; minui exiſtimarunt. </s>
          <s id="s.005005">ſuſpenſum tamen Galenum reddit, <lb></lb>
nulla huius motus organa reperiri, qua propter ad ſpiritus animales confu­<lb></lb>
git, <expan abbr="eisq́">eisque</expan>; huius augmenti, &amp; decrementi cauſam attribuit: eiuſdem ſenten­<lb></lb>
tiæ eſt 10. Baptiſta Porta, inter huius ætatis Perſpectiuos celebris. </s>
          <s id="s.005006">Hiero­<lb></lb>
nymus ab Aquapendente ab antiquioribus in eo diſſentit, quod cauſam hu­<lb></lb>
ius referat non in ſpiritus, ſed in proprietatem quandam ipſius vueæ natu­<lb></lb>
ralem. </s>
          <s id="s.005007">Porrò cum ego ſententias horum mente verſarem opportunè acci­<lb></lb>
dit, vt vel ipſo meridie cùm quodam in loco ſatis opaco colloquerer, <expan abbr="atq;">atque</expan> <lb></lb>
eo illius capitis ſitu, vt oculi illius in multa eſſent opacitate, cùm ecce tibi <lb></lb>
pupillas illius, magna cum admiratione, adeò magnas conſpexi, vt totam <lb></lb>
ferè maiorem Iridem adæquarent; illicò hominem in claram lucem, atque <lb></lb>
Solem deduxi; <expan abbr="atq;">atque</expan> ecce tibi repentè eædem pupillæ minimæ factæ ſunt. <lb></lb>
</s>
          <s id="s.005008">eandem ſubinde experientiam ſexcenties obſeruaui, vnde duo notatu di­<lb></lb>
gna innotuerunt.</s>
        </p>
        <p type="main">
          <s id="s.005009">Primum eſt: Maiores noſtros hallucinatos eſſe cùm nocte tantummodo <lb></lb>
magnas, per diem verò paruas fieri pupillas opinati ſunt: Verum id ratio­<lb></lb>
ne lucis, &amp; tenebrarum quouis tempore accidere patuit. </s>
          <s id="s.005010">vnde etiam in ob­<lb></lb>
ſcuriſſima nocte admota oculis accenſa candella, minuantur; amota ſtatim <lb></lb>
augeantur. </s>
          <s id="s.005011">hinc accidit, pupillam hanc, Medicum quendam ætate noſtra <lb></lb>
celebrem fefelliſſe, quì dum ægrotum quendam in cubiculo ſatis tenebroſo, <lb></lb>
<expan abbr="oculisq́">oculisque</expan>; laborantem curaret, animaduertit illius pupillas eſſe iuſto maio­<lb></lb>
res, quapropter plurima illi medicamenta pro pupillarum reſtrictione ad­


<pb pagenum="282" xlink:href="009/01/282.jpg"></pb>hibuit; ſed omnia tamen irrita; cui ad mota luce, ſtatim ſunt imminutæ: <lb></lb>
nec tamen æger conualuit. </s>
          <s id="s.005012">erat ſcilicet apparentia, non veritas.</s>
        </p>
        <p type="main">
          <s id="s.005013">Secundum hoc modo patuit. </s>
          <s id="s.005014">Cęperam <expan abbr="namq;">namque</expan> pariter de principali quæ­<lb></lb>
ſtione ambigere circa ſententiam eorundem, num ſcilicet verè pupillæ mo­<lb></lb>
dò dilatarentur, modò conſtringerentur; an potius aliqua ſit, quæ eos fe­<lb></lb>
fellerit apparentia. </s>
          <s id="s.005015"><expan abbr="atq;">atque</expan> tandem poſt diuturnam obſeruationem, poſt <expan abbr="plu-rimorũ">plu­<lb></lb>
rimorum</expan> oculorum diſſectionem, tutò auſus ſum primò aſſerere, pupillas ve­<lb></lb>
rè nec augeri, nec minui, ſed meram illam eſſe apparentiam, quod ſequen­<lb></lb>
tibus rationibus comprobabam.</s>
        </p>
        <p type="main">
          <s id="s.005016">Primò, ſi verè tunc cum fermè maiorem totam Iridem occupant, relicta <lb></lb>
tantummodo gracili in orbem armilla, dilatarentur, tunc neceſſariò con­<lb></lb>
nexio vueæ cum aranea ſcinderetur, cùm tunc pupillæ gyrum D E F, illius <lb></lb>
connexionis veſtigium tranſcendant; ac præterea tenuiſſimæ illæ fibræ, qui­<lb></lb>
bus conſuitur, frangerentur, <expan abbr="ſtatimq́">ſtatimque</expan>; iterum nemine auctore reſarcirentur. <lb></lb>
</s>
          <s id="s.005017">quod nec ſine oculi detrimento, nec ſine ſenſu doloris aliquo fieri poſſe, quis <lb></lb>
dicat? </s>
          <s id="s.005018">quæ omnia nullo modo conſequuntur.</s>
        </p>
        <p type="main">
          <s id="s.005019">Secundò, nulla extant huius organa motus, quod plures Anatomicos, <lb></lb>
<expan abbr="atq;">atque</expan> Galenum ipſum dubios reddit.</s>
        </p>
        <p type="main">
          <s id="s.005020">Tertiò, Medici omnes volunt eos, qui anguſtiori ſunt pupilla, acie ocu­<lb></lb>
lorum plus valere, quàm qui ſunt latiori; ſi ergò iſta eſſet vera conſtrictio, <lb></lb>
&amp; dilatatio, accideret nos eodem pęnè temporis momento, modò acutius, <lb></lb>modò hebetius videre: imò in temperata luce, vbi maior apparet, minus, <lb></lb>
quam in clariſſima luce, &amp; Sole, vbi minima apparet, videremus. </s>
          <s id="s.005021">quibus <lb></lb>
quotidiana refragatur experientia.</s>
        </p>
        <p type="main">
          <s id="s.005022">Quartò, plura obiecta compræhenderet oculus in opacitate, quàm in cla­<lb></lb>
ritate, &amp; Sole: omnia enim illa videmus, quæ intra pyramidem viſualem <lb></lb>
continentur, quæ eò capacior, &amp; latior eſt, quò pupilla maior eſt: habet <lb></lb>
<expan abbr="namq;">namque</expan> hæc pyramis verticem in centro oculi, &amp; poſtea dilatatur ad dilata­<lb></lb>
tionem foraminis vueæ, quod eſt pupilla. </s>
          <s id="s.005023">verùm nos nunquam experimur <lb></lb>
plura obiecta compræhendere in vmbra, quam in Sole.</s>
        </p>
        <p type="main">
          <s id="s.005024">Quintò, ſi illa dilatatio vera eſſet, pupilla non ſemper in homine videre­<lb></lb>
tur nigra; magnitudo enim foraminis multum luminis intra oculum admit­<lb></lb>
teret, quod opacitatem, &amp; nigredinem illam omnem fugaret; hac enim de <lb></lb>
cauſa in bobus, &amp; capris alba cernitur. </s>
          <s id="s.005025">quapropter certò certius, &amp; luce <lb></lb>
clarius, motum hunc non verum, ſed apparentem eſſe mihi, <expan abbr="atq;">atque</expan> alijs per­<lb></lb>
ſuadebam.</s>
        </p>
        <p type="main">
          <s id="s.005026">Verumenimuerò Græcorum illud adagium, ſecundæ cogitationes ſunt <lb></lb>
ſapientiores, veriſſimum eſt, nam quinquennio poſtquam hæc de pupillæ di­<lb></lb>
latatione conſcripſeram, cùm iterum opticam publicè aggrederer, oculi <lb></lb>
fabricam, <expan abbr="atq;">atque</expan> pupillæ motum iſtum attentius conſiderans, conatus ſum ob <lb></lb>
ſequentes rationes prædictis euidentiores mutare ſententiam, <expan abbr="atq;">atque</expan> aſſerere <lb></lb>
verè pupillam augeri, ac minui, eò quod vuea ipſa in luce verè conſtringa­<lb></lb>
tur, in opacitate verò, ac tenebris magno naturæ miraculo dilatetur.</s>
        </p>
        <p type="main">
          <s id="s.005027">Prima ratio, cùm pupilla minuitur, eam perpetuò vueę colores <expan abbr="cõſequun-tur">conſequun­<lb></lb>
tur</expan>, <expan abbr="eamq́">eamque</expan>; circundant, ſiue cum ipſa conſtringuntur; quod minimè fieret, <lb></lb>
niſi vuea ipſa conſtringeretur, <expan abbr="ſicq́">ſicque</expan>, foramen illud imminueret.</s>
        </p>
        <pb pagenum="283" xlink:href="009/01/283.jpg"></pb>
        <p type="main">
          <s id="s.005028">Secunda, ex hac experientia; fac vt aliquis alterum ſibi oculum manum <lb></lb>
illi applicans tegat: &amp; illicò ſine vlla lucis mutatione videbis alterius ocu­<lb></lb>
li pupillam modicùm, ſed tamen ſenſibiliter ſatis dilatari.</s>
        </p>
        <p type="main">
          <s id="s.005029">Tertia, animalia quædam, vt Catus, quando pupillam ſuam hanc dila­<lb></lb>
tant, eam in orbem dilatant: quando verò conſtringunt, eam in oualem, ac <lb></lb>
tandem rimulam quandam conſtringunt: <expan abbr="idq́">idque</expan>; in eadem opacitate, quæ ar­<lb></lb>
gumento ſunt vueam ipſam moueri.</s>
        </p>
        <p type="main">
          <s id="s.005030">Quarta, eſt quidam oculorum morbus, quo æger <expan abbr="abſq;">abſque</expan> magno dolore lu­<lb></lb>
cem nequit aſpicere: ergò ſignum lucem aliquid intra oculum mouere poſ­<lb></lb>
ſe, ex quo motu æger doleat; huiuſmodi ægrotum quendam ego aliquando <lb></lb>
magna admiratione inuiſi. </s>
          <s id="s.005031">hæc adeò miranda in nobis ſummus naturæ opi­<lb></lb>
fex perpetuò operatur.</s>
        </p>
        <p type="main">
          <s id="s.005032">Reliquum eſt, vt ſuperioribus rationibus, quibus me diù deceptum fuiſſe <lb></lb>
exiſtimo, ſatisfaciam.</s>
        </p>
        <p type="main">
          <s id="s.005033">Ad primam igitur reſpondeo, etiamſi vuea verè dilatetur, non indè ne­<lb></lb>
ceſſariò ſequi connexionem illam ſcindi debere: poſſunt enim tenuiſſimæ il­<lb></lb>
læ fibræ intendi, ac remitti: <expan abbr="ſicq́">ſicque</expan>; motui vueæ obtemperare.</s>
        </p>
        <p type="main">
          <s id="s.005034">Ad ſecundam, cauſam efficientem huius dilatationis eſſe ſpiritus; non <lb></lb>
partem quampiam, aut organum materiale.</s>
        </p>
        <p type="main">
          <s id="s.005035">Ad tertiam negandum eſt illud Medicorum placitum.</s>
        </p>
        <p type="main">
          <s id="s.005036">Ad quartam concedenda eſt tota, ſed tamen addendum eſt, nos non ad­<lb></lb>
uertere modò plura obiecta, modò pauciora videre, quia valde difficile eſt <lb></lb>
id obſeruare, porrò fateor difficile eſſe huic rationi ſatisfacere, quia non, <lb></lb>
omninò conſtat, qua ratione, &amp; quo loco oculi fiat viſio.</s>
        </p>
        <p type="main">
          <s id="s.005037">Ad quintam dicendum eſt, nunquam foramen illud pupillæ ad <expan abbr="tãtam">tantam</expan> ma­<lb></lb>
gnitudinem euadere, vt ſatis lucis admittat ad oculi internam opacitatem <lb></lb>
fugandam. </s>
          <s id="s.005038">Præterea, quando dilatatur eſt in loco ſatis opaco, vnde fit, vt <lb></lb>
opacitas ambientis aeris minimè expellat oculi opacitatem; quin potius <lb></lb>
eam iuuet neceſſe eſt.</s>
        </p>
        <p type="main">
          <s id="s.005039">9. Cur altero oculorum tecto, alterius pupilla aliquantulum dilatatur? <lb></lb>
</s>
          <s id="s.005040">R. ſciendum miram eſſe oculorum ſocietatem, quò vnus intuetur, alter eò <lb></lb>
etiam conſpirat: quorſum alter conuertitur, eorſum &amp; alter: anima enim <lb></lb>
vtitur duobus oculis tanquam vno. </s>
          <s id="s.005041">quia igitur dum alter tegitur, ei ſimul <lb></lb>
tenebræ offunduntur, quibus præſentibus pupilla illius ampliatur, vt ſupra <lb></lb>
vidimus, neceſſe eſt, ob oculorum fidam ſocietatem, vt alterius etiam pu­<lb></lb>
pilla augeatur. </s>
          <s id="s.005042">huic <expan abbr="reſpõſioni">reſponſioni</expan> quis in hunc modum obijciet: cùm alter ocu­<lb></lb>
lus ſit in lumine, ac eapropter pupillam conſtringere debeat; cur non ei po­<lb></lb>
tius alter morem gerit, <expan abbr="ſicq́">ſicque</expan>; pupillam, præſentibus etiam tenebris in ſuo <lb></lb>
ſtatu continet. </s>
          <s id="s.005043">huic iterum reſpondeo, quoniam alter, qui obtenebratur pu­<lb></lb>
pillam ampliare, alter verò, qui illuminatur coarctare ſtudet, fit vt <expan abbr="vterq;">vterque</expan> <lb></lb>
pupillam, modicum tamen, vt experientia docet, in orbem diducat.</s>
        </p>
        <p type="main">
          <s id="s.005044"><expan abbr="Atq;">Atque</expan> hæc ſunt, quæ nuper circa oculi pupillam obſeruata, huic loco ad­<lb></lb>
denda exiſtimaui.</s>
        </p>
        <p type="head">
          <s id="s.005045">LAVS DEO.</s>
        </p>
      </chap>
      <pb xlink:href="009/01/284.jpg"></pb>
      <pb xlink:href="009/01/285.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.005046">DE <lb></lb>
MATHEMATICARVM <lb></lb>
NATVRA DISSERTATIO.</s>
        </p>
        <p type="head">
          <s id="s.005047">VNA CVM CLARORVM <lb></lb>
<emph type="italics"></emph>MATHEMATICORVM <lb></lb>
CHRONOLOGIA.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.005048">AD ILLVSTRISSIMVM AC NOBILISSIMVM <lb></lb>
PETRVMFRANCISCVM MALASPINAM <lb></lb>
ÆDIFICIORVM MARCHIONEM.</s>
        </p>
        <p type="head">
          <s id="s.005049"><emph type="italics"></emph>Authore eodem Ioſepho Blancano è Societate IESV, <lb></lb>
Mathematicarum in Parmenſi Academia profeſſore.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.005050">BONONIÆ M. DC. XV.</s>
        </p>
        <p type="head">
          <s id="s.005051">Apud Bartholomæum Cochium. </s>
          <s id="s.005052">Superiorum permiſſu.</s>
        </p>
        <p type="head">
          <s id="s.005053">Sumptibus Hieronymi Tamburini.</s>
        </p>
        <pb xlink:href="009/01/286.jpg"></pb>
        <pb pagenum="3" xlink:href="009/01/287.jpg"></pb>
        <p type="head">
          <s id="s.005054">ILLVSTRISSIMO <lb></lb>
AC NOBILISSIMO <lb></lb>
PETROFRANCISCO <lb></lb>
MALASPINAE<emph type="italics"></emph><lb></lb>ÆDIFICIORVM MARCHIONI.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.005055">Ioſeph Blancanus è Societate Ieſu. S.P.D.</s>
        </p>
        <p type="main">
          <s id="s.005056"><emph type="italics"></emph>Meam hanc de Mathematicarum natu­<lb></lb>
ra Diſſertationem, vnà cum illustrium, <lb></lb>
Mathematicorum Chronologia, tibi Illu­<lb></lb>
striß. Marchio iure meritò dicare, ac ſub <lb></lb>
clarißimi tui nominis patrocinio in lucem <lb></lb>dare constitui. </s>
          <s id="s.005057">primum quidem, vt mei perpetui erga te <lb></lb>
amoris, &amp; obſeruantiæ hoc vnum ſaltem specimen exta­<lb></lb>
ret; tum vt idoneum, æquumqué propoſitæ Quæstionis iu­<lb></lb>
dicem nanciſcerer. </s>
          <s id="s.005058">Cùm enim ad iuſtum arbitrum duo <lb></lb>
potißimum requirantur, rerum ſcilicet cognitio, atque pru­<lb></lb>
dentia, quem te rei, de qua agitur peritiorem, quemque pru­<lb></lb>
dentiorem; inuenire potuerim? </s>
          <s id="s.005059">tu enim cùm Phyſiologiæ, <lb></lb>
ac Mathematicarum omnium Encyclopædiam mirum, <lb></lb>
in modum excolueris, ad intima Mathematicarum pene-<emph.end type="italics"></emph.end>


<pb pagenum="4" xlink:href="009/01/288.jpg"></pb><emph type="italics"></emph>tralia ita peruaſisti, vt Archimedis, &amp; Apollonÿ ad­<lb></lb>mirandis, ac ſubtilißimis demonſtrationibus detinearis. <lb></lb>
</s>
          <s id="s.005060">Quanta porrò in rebus agendis prudentia valeas toti penè <lb></lb>Europæ innotuit, cùm pro noſtris Sereniß. Ducibus non <lb></lb>
ſolùm ad omnes ferè Italiæ, atque Germaniæ Principes, <lb></lb>
verùm etiam ad Cæſaream Maieſtatem, rebus fœliciter <lb></lb>geſtis, Legatus decimùm extiteris: ac demum à Sereniß. <lb></lb>
Duce Ranutio inter primarios de Rep. </s>
          <s id="s.005061">Conſiliorum Au­<lb></lb>
thores, adſcitus fueris. </s>
          <s id="s.005062">Cæterùm in Clarorum Mathe­<lb></lb>
maticorum Chronologia perlegenda, ſæpißimè tibi nobiliſ­<lb></lb>ſimi æquè, ac doctißimi Viri, tui omninò perſimiles, oc­<lb></lb>
current, quod tibi nonniſi gratißimum accidere poſſe arbi­<lb></lb>
tror. </s>
          <s id="s.005063">Complectere igitur,quà ſoles benignitate, atque clemen­<lb></lb>tia noſtra hæc quantulacunque munuſcula, quæ ſi tibi acce­<lb></lb>
pta eſſe intellexero, tùm demum maximorum mu­<lb></lb>
nerum loco habenda eſſe cenſebo. </s>
          <s id="s.005064">incolu­<lb></lb>
mem tibi, ac fœlicem D. O. M. <lb></lb>
long æuitatem tueatur. <lb></lb>
</s>
          <s id="s.005065">Vale.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.005066"><emph type="italics"></emph>Parmæ Idibus Nouembris M. DC. XIIII.<emph.end type="italics"></emph.end><lb></lb>
</s>
        </p>
        <pb pagenum="5" xlink:href="009/01/289.jpg"></pb>
        <p type="head">
          <s id="s.005067"><emph type="bold"></emph>ADDITAMENTVM.<emph.end type="bold"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.005068"><emph type="italics"></emph>DE NATVRA SCIENTIARVM <lb></lb>
MATHEMATICARVM.<emph.end type="italics"></emph.end></s>
        </p>
      </chap>
      <chap>
        <p type="main">
          <s id="s.005069">Qvoniam in hoc Opere multa ad Mathematicarum natu­<lb></lb>ram ſpectantia ſparſim dicta ſunt, non ab re, <expan abbr="neq;">neque</expan> ingratum <lb></lb>
Lectori fore duxi, ea quodammodo huc in vnum congerere, <lb></lb>
quæ ad earum naturam ritè percipiendam neceſſaria eſſe vi­<lb></lb>
derentur. </s>
          <s id="s.005070">præſertim cùm recentiorum quamplurimi, qui eas <lb></lb>
læuiter nimis attigerunt, hac de re, veluti cæci de colore, plu­<lb></lb>
ribus ad internam tamen earum naturam minimè ſpectantibus, garrire ge­<lb></lb>
ſtiant. </s>
          <s id="s.005071">Vt autem tractatio euadat planior, eam ſic commodè partiemur, vt</s>
        </p>
        <p type="main">
          <s id="s.005072">Primo, De materia, ſeu ſubiecto harum diſciplinarum agamus.</s>
        </p>
        <p type="main">
          <s id="s.005073">2. De medio Demonſtrationum Geometricarum, ſeu, vtrum ſint De­<lb></lb>
monſtrationes potiſſimæ.</s>
        </p>
        <p type="main">
          <s id="s.005074">3. De præſtantia ſcientiæ, quam nobis pariunt.</s>
        </p>
        <p type="main">
          <s id="s.005075">4. Aliquot calumniarum dilutio.</s>
        </p>
        <p type="main">
          <s id="s.005076">5. De Mathematicis medijs.</s>
        </p>
        <p type="head">
          <s id="s.005077"><emph type="italics"></emph>Deſubiecto Geometræ, &amp; Arithmeticæ, quod ſolet dici <lb></lb>
Materia intelligibilis. </s>
          <s id="s.005078">Cap. 1.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005079">Primò, agemus de puris Mathematicis Geometria, &amp; Arithmetica, <lb></lb>
quarum eſt diuerſa ratio à medijs, Aſtronomia, ſcilicet Perſpecti­<lb></lb>
ua, Mechanica, &amp; Muſica. </s>
          <s id="s.005080">Quantitas igitur abſtracta à materia <lb></lb>
ſenſibili dupliciter conſiderari ſolet. </s>
          <s id="s.005081">conſideratur enim à Phyſico, <lb></lb>
&amp; Mathematico ſecundum ſe, ideſt, abſolutè, quatenus Quantitas eſt; ſiue <lb></lb>terminata ſit, ſiue non; qua ratione affectiones ipſius ſunt, diuiſibilitas, lo­<lb></lb>
cabilitas, figurabilitas, &amp;c. </s>
          <s id="s.005082">à Geometra verò, &amp; Arithmetico conſidera­<lb></lb>tur non abſolutè, ſed quatenus eſt terminata, vt ſunt in quantitate continua <lb></lb>
lineæ finitæ rectæ, aut curuæ, vt ſunt ſuperficies terminatæ, ex quibus variæ <lb></lb>
fiunt figuræ, vt circulus, triangulum, &amp;c. </s>
          <s id="s.005083">vt tandem ſunt ſolida item termi­<lb></lb>
nata, ex quibus variæ exiſtunt ſpecies ſolidarum figurarum, veluti, pyramis, <lb></lb>
cubus, conus, cylindrus, &amp;c. </s>
          <s id="s.005084">quæ ad Geometram pertinent. </s>
          <s id="s.005085">Quæ omnia in <lb></lb>
quantitate etiam diſcreta, ſeu in numeris proportionaliter reperiuntur,


<pb pagenum="6" xlink:href="009/01/290.jpg"></pb>quos ſolùm terminatos Arithmeticus accipit. </s>
          <s id="s.005086">eſſe autem genera hæc termi­<lb></lb>
natæ Quantitatis Geometriæ, aut Arithmeticæ ſubiectum, ex eo patet, quod <lb></lb>
eas ſolas quantitates ipſi definiunt, <expan abbr="deq́">deque</expan>; ipſis varias paſſiones <expan abbr="demonſtrãt">demonſtrant</expan>, <lb></lb>
<expan abbr="easq́">easque</expan>; omninò ab eis diuerſas, quas Phyſicus, &amp; Metaphyſicus in ea abſolu­<lb></lb>
tè ſpectata conſiderant. </s>
          <s id="s.005087">Vnde manifeſtum eſt, has affectiones, quas Ma­<lb></lb>
thematicus contemplatur ab ipſa Quantitate, quatenus terminata eſt ema­<lb></lb>
nare; ſunt autem æqualitas, inæqualitas, talis diuiſio, transfiguratio, pro­<lb></lb>
portiones variæ, commenſuratio, incommenſuratio, figurarum <expan abbr="cõſtructio-nes">conſtructio­<lb></lb>
nes</expan>, &amp;c. </s>
          <s id="s.005088">Quæ ſanè affectiones ab intrinſeca Quantitatis natura minimè <lb></lb>
fluunt, poſita enim ea interminata, prædictæ paſſiones non conſequuntur, <lb></lb>
nihil enim, ea ſic poſita, eſt æquale, aut inæquale, &amp;c. </s>
          <s id="s.005089">ſed addita Quantita­<lb></lb>
ti terminatione, eæ ab ea per emanationem profluunt. </s>
          <s id="s.005090">Quapropter inrè di­<lb></lb>
xeris formalem rationem Mathematicæ conſiderationis eſſe Terminatio­<lb></lb>
nem; &amp; obiectum totale adæquatum eſſe Quantitatem terminatam, qua­<lb></lb>
tenus terminata eſt. </s>
          <s id="s.005091">Ex hac enim terminatione variæ oriuntur figuræ, &amp; <lb></lb>
numeri, quas Mathematicus definit, <expan abbr="deq́">deque</expan>; ipſis varia demonſtrat. </s>
          <s id="s.005092"><expan abbr="Atq;">Atque</expan> hæc <lb></lb>eſt illa Quantitas, quæ dici ſolet materia intelligibilis, ad differentiam ma­<lb></lb>
teriæ ſenſibilis, quæ ad Phyſicum ſpectat; illa enim ab hac per intellectum <lb></lb>
ſeparatur, ac ſolo intellectu percipitur. </s>
          <s id="s.005093">Continuum igitur, &amp; diſcretum, <lb></lb>
<expan abbr="vtrumq;">vtrumque</expan> <expan abbr="terminatũ">terminatum</expan>, eſt materia intelligibilis, illud Geometriæ, iſtud Arith­<lb></lb>
meticæ. </s>
          <s id="s.005094">Hinc etiam patet, cur dicatur Mathematicus conſiderare Quan­<lb></lb>
titatem finitam, quia accipit terminatam, quæ finita eſt: quod enim habet <lb></lb>
terminus, ſeu fines, finitum eſt. </s>
          <s id="s.005095">quod ſi dari poſſet quantitas aliqua termi­<lb></lb>
nata, &amp; ſimul infinita, de ea etiam Demonſtrationes Euclidis fieri poſſent; <lb></lb>
ſi enim daretur triangulum infinitum, eodem modo de eo oſtendi poſſet ha­<lb></lb>
bere tres angulos æquales duobus rectis. </s>
          <s id="s.005096">Porrò hanc terminatam Quanti­<lb></lb>tatem eſſe Geometriæ, &amp; Arithmeticæ ſubiectum minimè cognouerunt ij, <lb></lb>
qui Geometricas demonſtrationes impugnarunt, vt in eorum ſcriptis vide­<lb></lb>
re eſt, quæ prima eis fuit errandi occaſio.</s>
        </p>
        <p type="main">
          <s id="s.005097">Porrò ex hac mathematica abſtractione à materia ſenſibili, fit vt materia <lb></lb>
hæc abſtracta perfectionem quandam acquirat, quam perfectionem mathe­<lb></lb>
maticam appellant. </s>
          <s id="s.005098">v. g. triangulum abſtractum eſt omninò planum ex tri­<lb></lb>
bus lineis omninò rectis, <expan abbr="tribusq́">tribusque</expan>; angulis punctis omninò indiuiduis con­<lb></lb>
ſtitutum, quale in rerum natura (exceptis fortè cœleſtibus) vix puto repe­<lb></lb>
riri poſſe. </s>
          <s id="s.005099">vnde nonnulli ſolent Mathematicis illud obijcere; entia ſcilicet <lb></lb>
mathematica non extare, niſi per ſolum intellectum. </s>
          <s id="s.005100">Verumenimuerò ſcien­<lb></lb>
dum eſt entia hæc mathematica, quamuis in ea perfectione non extent, id <lb></lb>
tamen eſſe per accidens, conſtat enim naturam, &amp; artem figuras mathema­<lb></lb>
ticas præcipuè intendere, quamuis propter materiæ ſenſibilis ruditatem, &amp; <lb></lb>
imperfectionem, quæ perfectas omninò figuras ſuſcipere nequit, ſuo ſine <lb></lb>
fruſtrentur; natura enim in truncis arborum cylindri figuram affectat, in <lb></lb>pomis, &amp; vuarum acinis aut ſphæricam, aut ſphæroidem, in cornea oculi <lb></lb>
circulum; imò oculus ipſæ maximè ſphæricus eſt. </s>
          <s id="s.005101">Sol, <expan abbr="reliquaq́">reliquaque</expan>; aſtra com­<lb></lb>
muni omnium conſenſu omninò ſphærica ſunt. </s>
          <s id="s.005102">ipſa aquæ ſuperficies globo­<lb></lb>
ſa eſt. </s>
          <s id="s.005103"><expan abbr="terraq́">terraque</expan>; ipſa niſi obſtaret materiæ craſſities, &amp; diuerſitas, rotunda pla­<lb></lb>
nè euaderet. </s>
          <s id="s.005104">lineæ ſpirales conicæ nonne manifeſtè in marinis cochlæis de­<pb pagenum="7" xlink:href="009/01/291.jpg"></pb>ſignantur? </s>
          <s id="s.005105">Cylindricæ, &amp; planæ in nonnullis herbis? </s>
          <s id="s.005106">Ars præterea palàm <lb></lb>
magis eaſdem figuras proſequitur; artifices enim omnia ferè opificia qua­<lb></lb>dratis figuris, aut rotundis, aut circulis, aut ellipſibus induunt. </s>
          <s id="s.005107">Verum ipſa <lb></lb>
<expan abbr="quoq;">quoque</expan> ars, non ſecus ac natura, quam imitatur ſuo fine ob materiæ rudita­<lb></lb>
tem defraudatur. </s>
          <s id="s.005108">Quamuis igitur re ipſa non exiſtant, quia tamen tamin <lb></lb>
mente Auctoris naturæ, quàm in humana, eorum Ideæ tamquam exactiſſi­<lb></lb>
mi rerum typi, necnon tamquam exacta entia Mathematica exiſtunt; Ideo <lb></lb>
de ipſis eorum idæis, quæ per ſe primò intenduntur, &amp; quæ vera ſunt entia, <lb></lb>
agit Mathematicus. </s>
          <s id="s.005109">Quapropter dicendum eſt, entia hæc Geometrica om­<lb></lb>
nibus numeris abſoluta eſſe entia per ſe, &amp; vera; figuræ verò tum natura­<lb></lb>
les, tum artificiales, quæ in rebus exiſtunt, cùm à nullo efficiente intendan­<lb></lb>
tur, eſſe entia per accidens, imperfecta, &amp; falſa. </s>
          <s id="s.005110">v. g. triangulum in aliqua <lb></lb>
charta depictum, non eſt verum triangulum, ſed verum <expan abbr="triangulũ">triangulum</expan> illud eſt, <lb></lb>
quod in idæa diuina eſt. </s>
          <s id="s.005111">ex quibus obiter illud intelligas, cur ſcilicet ali­<lb></lb>
quando Plato dixerit Deum geometrizare, ideſt tanquam verum Geome­<lb></lb>
tram non niſi perfectiſſimas idæas contemplari. </s>
          <s id="s.005112">Eodem etiam modo, Poe­<lb></lb>
tæ, quì res perfectas imitari debent, eas ſaltem vt plurimum, non vt exti­<lb></lb>
terunt verſibus decantant; ſed quales eſſe debuerunt, confingunt, <expan abbr="atq;">atque</expan> lecto­<lb></lb>
ribus, aut ſpectatoribus repræſentant. </s>
          <s id="s.005113">Vltimò dici poteſt; hæc entia eſſe <lb></lb>
poſſibilia, quis enim neget Angelum, aut Deum ea poſſe efficere? </s>
          <s id="s.005114">ad obie­<lb></lb>
ctum autem ſcientiæ ſatis eſt eſſe poſſibile; ſcientia enim abſtrahit ab exi­<lb></lb>
ſtentia ſubiecti.</s>
        </p>
        <p type="main">
          <s id="s.005115">In hac præterea intelligibili materia, alio modo materia alia accipitur, <lb></lb>
cùm partes ſcilicet dicuntur materia totius, vt quando duo triangula com­<lb></lb>
ponunt quodpiam quadrilaterum, ſunt illa duo trigona materia totius il­<lb></lb>
lius quadrilateri.</s>
        </p>
        <p type="main">
          <s id="s.005116">Similiter aliquando plures anguli partiales componunt tanquam mate­<lb></lb>
riam totalem angulum. </s>
          <s id="s.005117">Idem dicendum de alijs ſimilibus, pariter eſſe ali­<lb></lb>
cuius dimidium, aut tertiam partem, aut duplum, aut reliquum cuiuſpiam <lb></lb>
totius, referuntur ad veram cauſam materialem, <expan abbr="idq́">idque</expan>; teſte Ariſt. tex. 31.2. <lb></lb>
&amp; tex. 3. 5. Metaph. &amp; omnibus Philoſophis. </s>
          <s id="s.005118">quæ quidem materiæ accep­<lb></lb>
tio, ſimilis eſt acceptioni materiæ phyſicæ, ex qua tanquam ex parte <expan abbr="cõpo-ſitum">compo­<lb></lb>
ſitum</expan> conſtatur: conflatur enim ex materia, &amp; forma, tanquam partibus. <lb></lb>
</s>
          <s id="s.005119">diuerſa verò eſt ab ea, quàm phyſici paſſim vſurpant, dum conſiderant ma­<lb></lb>teriam, in qua, aut circa quam, vt aiunt. </s>
          <s id="s.005120">hoc tamen non obſtat, quominus <lb></lb>
illa veram cauſæ materialis rationem non obtineat. </s>
          <s id="s.005121">Quod etiam Geome­<lb></lb>
tricarum demonſtrationum impugnatores videntur minimè aduertiſſe. </s>
          <s id="s.005122">quæ <lb></lb>
illis ſecunda errandi cauſa extitit.</s>
        </p>
        <p type="main">
          <s id="s.005123">Poſtremò aduertendum, quòd magni <expan abbr="momẽti">momenti</expan> eſt, definitiones tam Geo­<lb></lb>
metricæ, quàm Arithmeticæ eſſe omnino eſſentiales, quæ ſcilicet totam rei <lb></lb>quid ditatem explicent; minimè verò eſſe tantummodo nominis explicatio­<lb></lb>
nes, aut definitiones, vt ijdem perperam exiſtimarunt, qui eorum tertius <lb></lb>
eſt error. </s>
          <s id="s.005124">quod quidem Ariſt. ſenſiſſe manifeſtum eſt, <expan abbr="quotieſcunq;">quotieſcunque</expan> enim in <lb></lb>Analyticis de ſcientiarum principijs loquitur, inter ea definitiones Geome<lb></lb>
triæ, &amp; Arithmeticæ ſemper <expan abbr="cõnumerat">connumerat</expan>, quod minimè feciſſet, ſi ſolùm no­<lb></lb>
minis eſſent explicationes. </s>
          <s id="s.005125">Verum quidem eſt eas, vt plurimum eſſe ſimul,


<pb pagenum="8" xlink:href="009/01/292.jpg"></pb>&amp; rei, &amp; nominis expoſitiones. </s>
          <s id="s.005126">quod ſæpè accidit, cùm ſcilicet nomina val­<lb></lb>
dè perfecta, ac rei omnino conuenientia ſunt; nam</s>
        </p>
        <p type="main">
          <s id="s.005127"><emph type="italics"></emph>Conueniunt rebus nomina ſæpè ſuis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005128">Huiuſmodi ſæpè ſunt, quæ perfectam continent etymologiam, vbi ipſa <lb></lb>
nominis expoſitio, ſimul etiam eſt rei eſſentialis definitio. </s>
          <s id="s.005129">tales ſunt ſæpè <lb></lb>
nomina, &amp; definitiones Geometricæ. </s>
          <s id="s.005130">Exempli cauſa, talis eſt definitio qua­<lb></lb>
drati, nam quando dico, quadratum eſt figura plana quatuor rectis lineis, <lb></lb>
&amp; quatuor angulis rectis conſtans, explico ſimul rationem nominis, &amp; ra­<lb></lb>
tionem rei: dicitur enim quadratum á quatuor illis lineis. </s>
          <s id="s.005131">Explico deinde <lb></lb>
totam eius eſſentiam, quando dico ipſum conſtare ex quatuor lineis rectis, <lb></lb>
&amp; quatuor angulis rectis, quæ duo ſimul iuncta conſtituunt totam quadrati <lb></lb>
eſſentiam, ſunt enim ipſius differentia conſtitutiua; loco autem generis eſt <lb></lb>
figura plana quadrilatera: quapropter erit hæc perfectiſſima definitio, cùm <lb></lb>non ſolum nominis, ſed etiam rei eſſentiam rotam patefaciat; ſtatim enim, <lb></lb>
ac cognoſcimus quadratum ex prædictis <expan abbr="cõſtare">conſtare</expan> nihil amplius de ipſius eſ­<lb></lb>
ſentia animus ſcire deſiderat, ſed acquieſcit, vnde eam eſſe optimam defi­<lb></lb>
nitionem manifeſtum eſt. </s>
          <s id="s.005132">Huiuſmodi <expan abbr="quoq;">quoque</expan> eſt definitio figuræ altera par­<lb></lb>
te longioris, nam cùm dicitur, ea eſt figura plana quadrilatera, quæ <expan abbr="rectã-gula">rectan­<lb></lb>
gula</expan> quidem, &amp; æquilatera non eſt, patet inde, cur dicatur altera parte <expan abbr="lõ-gior">lon­<lb></lb>
gior</expan>, quòd eſt ipſius etymon: deinde ipſius eſſentia, ita innoteſcit, vt nihil <lb></lb>
amplius de ea quærendum ſuperſit. </s>
          <s id="s.005133">Similiter cum dicitur, <expan abbr="æquilaterũ">æquilaterum</expan> trian­<lb></lb>
gulum eſt, quod tria latera habet æqualia, ecce tibi, &amp; nominis, &amp; rei cau­<lb></lb>
ſa. </s>
          <s id="s.005134">talis eſt etiam prima 6. definitio, ſimiles figuræ rectilineæ ſunt, quæ &amp; an­<lb></lb>
gulos ſingulos ſingulis æquales habent, atque etiam latera, quæ <expan abbr="circũ">circum</expan> æqua­<lb></lb>
les proportionalia; hic enim etymologia, &amp; rei natura manifeſtatur. </s>
          <s id="s.005135">talis <lb></lb>
adhuc eſt prima definitio 10. commenſurabiles magnitudines <expan abbr="dicũtur">dicuntur</expan>, quas <lb></lb>
eadem menſura metitur. </s>
          <s id="s.005136">innumeras huiuſmodi alias, quæ apud alios Geo­<lb></lb>
metras reperiuntur, miſſas facio, ne in re tam clara longior ſim. </s>
          <s id="s.005137">ſed alias <lb></lb>
contemplemus, quæ nullo modo ſunt nominis definitionis, ſed rei tantum, <lb></lb>prima Euclidis definitio, quæ eſt Puncti, iuxta puncti naturam bipartita eſt, <lb></lb>
habet enim eſſe partim abſolutum, partim relatiuum; cùm in prima defi­<lb></lb>
nitione dicitur. </s>
          <s id="s.005138">Punctum eſt, cuius nulla pars eſt, definitur quatenus abſo­<lb></lb>
lutum, cùm poſtea in tertia definitione dicitur, termini lineæ ſunt puncta, <lb></lb>
definitur quatenus eſt quid alterius: ex quibus tota puncti natura fit mani­<lb></lb>
feſta; etymologia verò ne <expan abbr="quaquã">quaquam</expan>; nam dicitur punctum à pungendo, quaſi <lb></lb>
ſit punctura quædam, quæ notio in Euclidis definitione minimè attingitur. <lb></lb>
</s>
          <s id="s.005139">Similiter cum dicitur, line a eſt longitudo latitudinis expers, vbinam nomi­<lb></lb>
nis ratio? </s>
          <s id="s.005140">nam linea dicitur à lino, quaſi lineum filum; antiquitus enim ex <lb></lb>lino fila fiebant, quibus fabri ad deſignationes vtebantur, quemadmodum <lb></lb>
nunc ex cannabe: at in Euclidis definitione ridiculum eſt hanc <expan abbr="rationẽ">rationem</expan> que­<lb></lb>
rere; in qua tamen lineæ eſſentia perfectè apparet.</s>
        </p>
        <p type="main">
          <s id="s.005141">Pariter quando definit ſuperficiem eſſe eam, quæ longitudinem, <expan abbr="latitudi-nemq́">latitudi­<lb></lb>
nemque</expan>; <expan abbr="tãtum">tantum</expan> habet, apparet quidem rei natura, at verò vbi nominis defi­<lb></lb>
nitio, quæ eſt, dici ſuperficiem, quaſi ſupremam faciem? </s>
          <s id="s.005142">Cùm dicitur An­<lb></lb>
gulus eſt duarum linearum, ſe mutuò tangentium inclinatio, vbinam vocis <lb></lb>
notio? </s>
          <s id="s.005143">aperitur tamen rei natura, &amp; quidditas. </s>
          <s id="s.005144">Sed magis manifeſtum eſt<pb pagenum="9" xlink:href="009/01/293.jpg"></pb>in linea perpendiculari, quæ proculdubio denominata eſt à perpendiculo, <lb></lb>
in definitione tamen nullum huius veſtigium: at verò quid ipſa ſit, optimè <lb></lb>
explicatur. </s>
          <s id="s.005145">Definitio porrò circuli videtur aſſignari non per intrinſeca, <lb></lb>
ſed tamen æquiualet intrinſecæ definitioni; quando enim dicitur circulus <lb></lb>
eſt figura plana, vncia linea contenta, ad quam ab vno puncto eorum, quæ <lb></lb>
intra figuram ſunt, ductæ omnes lineæ ſunt æquales, perinde eſt, ac ſi dice­<lb></lb>
ret, circulus eſt figura plana, cuius medium æquidiſtat ab extremis, quę eſt <lb></lb>
eſſentialis; poſita enim hac æquidiſtantia, ponitur neceſſariò circulus. </s>
          <s id="s.005146">Ve­<lb></lb>
rùm centri definitionem eſſe tantum nominis explicationem; abſurdum eſt: <lb></lb>centrum enim vox gręca eſt, quæ primo ſignificat ſtimulum, vel aculeum il­<lb></lb>
lum. </s>
          <s id="s.005147">quo Boues agit bubulcus.</s>
        </p>
        <p type="main">
          <s id="s.005148">At Rhombi definitionem, quàm ridiculum eſt, eam nominis ſolum expli­<lb></lb>
cationem continere, cùm nihil minus. </s>
          <s id="s.005149">Dicitur enim Rhombus à cuiuſdam <lb></lb>
piſcis, vel cuiuſdam textorij inſtrumenti <expan abbr="ſimilitudinẽ">ſimilitudinem</expan>, cuius figuram refert. <lb></lb>
</s>
          <s id="s.005150">naturam tamen ipſius definitio aperit, ideſt Rhombus eſt figura plana qua­<lb></lb>
drilatera, æquilatera, ſed non rectangula. </s>
          <s id="s.005151">Idem perſpicere licet in defini­<lb></lb>tionibus corporum, quarum prima eſt, ſolidum eſt, quòd longitudinem, la­<lb></lb>
titudinem, &amp; craſſitudinem habet; ex qua clarè tota rei natura perſpicitur. <lb></lb>
</s>
          <s id="s.005152">Sed ne longior ſim, innumeras pene alias apud omnes Geometras reperies <lb></lb>
omnino eſſentiales, quas prætero: eadem prorſus de Arithmeticę defini­<lb></lb>tionibus ſunt intelligenda, vt eas conſideranti ſtatim patebit. </s>
          <s id="s.005153">Quod ſi quis <lb></lb>iam fateatur haſce definitiones eſſentiales eſſe, ſed tamen adhuc Mathema­<lb></lb>
ticis illas definitiones cauſales, quas demonſtratio requirit, deneget; is ſi­<lb></lb>
bi refragrantem audiat Ariſt. qui tex. 12. 2. de Anima; ait Tetragoniſmi <lb></lb>extare duas definitiones, vnam formalem, ſeu eſſentialem, quę eſt, Tetrago­<lb></lb>
niſmus eſt effectio quadrati æqualis dato æquilatero; altera verò cauſalis, <lb></lb>ſcilicet Tetragoniſmus eſt inuentio medię proportionalis, quia linea illa me <lb></lb>dia proportionali, eſt cauſa quadrati æqualis datæ figuræ: vide noſtram hu­<lb></lb>
ius loci explicationem. </s>
          <s id="s.005154">Concedat is igitur oportet, Geometricas de fini­<lb></lb>
tiones non ſolum nominales, ſed etiam formales, &amp; cauſales eſſe. </s>
          <s id="s.005155">quo no­<lb></lb>
mine Mathematicas definitiones reliquarum ſcientiarum definitiones an­<lb></lb>
tecellere iam certum eſſe poteſt; cùm apud omnes Philoſophos in confeſſo <lb></lb>
ſit, vltimas rerum <expan abbr="differẽtias">differentias</expan> nos latere, ſine quibus vera definitio nulla eſt; <lb></lb>adeò, vt etiam apud eoſdem ambigatur, vtrum illa definitio hominis, ani­<lb></lb>
mal rationale, vera ſit definitio nec ne.</s>
        </p>
        <p type="main">
          <s id="s.005156">Obijces fortè iterum, definitiones haſce Mathematicarum eſſe vt pluri­<lb></lb>mum definitiones ſubiecti: at in potiſſima demonſtratione, ad quam tendi­<lb></lb>
mus, requiri cauſales definitiones paſſionis primò, &amp; per ſe; definitionem <lb></lb>verò ſubiecti per accidens, vt quando aliquid immediatè ab ea procedens <lb></lb>
de ſubiecto ipſo <expan abbr="demonſrandũ">demonſtrandum</expan> eſt. </s>
          <s id="s.005157">reſpondendum cenſeo, primò, quod cùm <lb></lb>definitio cauſalis paſſionis non ſit aliud, quam cauſa ipſius, ſi in definitione <lb></lb>
ſubiecti continetur causa paſſionis, aſſumendo definitionem ſubiecti, aſſu­<lb></lb>metur etiam definitio cauſalis paſſionis. </s>
          <s id="s.005158">ſecundò, quod in Mathematicis <lb></lb>definitiones ipſius ſubiecti ſæpe euadunt definitiones paſſionis, vt infra cla­<lb></lb>rè patebit, quando nimirum ipſum ſubiectum. </s>
          <s id="s.005159">v. g. quadratum veluti paſſio <lb></lb>de figuratione quapiam demonſtratur; ſiue quando oſtenditur ex quapiam <pb pagenum="10" xlink:href="009/01/294.jpg"></pb>conſtructione rectè fieri quadratum, triangulum, lineam perpendicularem, <lb></lb>
&amp; ſimilia. </s>
          <s id="s.005160">tertiò, in præcedenti dubitatione dictum eſſe ex mente Ariſtot. <lb></lb>
etiam in Mathematicis eſſe definitiones cauſales, <expan abbr="idq́">idque</expan>; <expan abbr="exẽplo">exemplo</expan> tetragoniſmi <lb></lb>
confirmatum. </s>
          <s id="s.005161">Ex his, quæ de ſcientiarum definitionibus dicta ſunt, notan­<lb></lb>
da eſt quædam diſparitas inter alias ſcientias, &amp; Mathematicas in modo <lb></lb>
procedendi ad ſubiecti proprij cognitionem. </s>
          <s id="s.005162">nam in demonſtrationibus à <lb></lb>
ſigno, à quibus incipiunt vt plurimum aliæ ſcientiæ, ſola cognitio nominis <lb></lb>
ſubiecti requiritur, non autem eſſentialis definitio; eius enim eſſentia, quæ <lb></lb>
occulta eſt, per accidentia, &amp; proprietates à poſteriori indagatur; qua de­<lb></lb>tecta ab ea iterum ad demonſtrandas paſſiones diſtinctè, &amp; ſcientificè re­<lb></lb>
gredimur. </s>
          <s id="s.005163">quod ſi primò perfecta obiecti cognitio obijceretur, vt ſit in Ma­<lb></lb>
thematicis ob perfectus earum definitiones pulcherrimo naturæ ordine ab <lb></lb>
eſſentia ipſius ad paſſiones demonſtrandas procederemus, vt fit in demon­<lb></lb>
ſtrationibus à cauſa, quales ferè ſemper ſunt in Geometria, &amp; Arithmetica <lb></lb>
exceptis demonſtrationibus ab impoſsibili, vbi nobis primò tota ſubiecti <lb></lb>
natura ex præmiſsis definitionibus obijcitur, ex qua deinde ſemper à priori <lb></lb>
ad inueſtigandas illius paſsiones procedimus; in quo proceſſu definitio ſu­<lb></lb>
biecti pręmitti, <expan abbr="eiusq́">eiusque</expan>; quidditas ſupponi debet. </s>
          <s id="s.005164">vnde etiam ſequitur Mathe­<lb></lb>
maticas haſce à notioribus nobis, &amp; natura, vt vult Auerroes, &amp; cæteri ferè <lb></lb>
omnes, &amp; ex noſtris maximè Toletus quęſt. </s>
          <s id="s.005165">4. ſecundi Phyſ. procedere. </s>
          <s id="s.005166">no­<lb></lb>tioribus nobis, quia primum omnium manifeſta eſt tota figuræ eſſentia ex <lb></lb>
definitione ipſius allata, ignotis adhuc ipſius affectionibus, notioribus na­<lb></lb>
tura, quia prius natura eſt ſubiecti eſſentia, quàm paſsiones, quæ ab ea ma­<lb></lb>
nant, <expan abbr="deq́">deque</expan>; ea demonſtrantur: <expan abbr="atq;">atque</expan> hæc cauſæ eſt, cur ſemper tanti factæ ſine <lb></lb>
Geometricæ demonſtrationes, <expan abbr="primumq́">primumque</expan>; certitudinis gradum obtineant.</s>
        </p>
        <p type="head">
          <s id="s.005167"><emph type="italics"></emph>De medio Demonstrationum Geometriæ, &amp; Arith­<lb></lb>
meticæ, ſeu, An ſint potißimæ Demonſtrationes. <lb></lb>
</s>
          <s id="s.005168">Cap. 2.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005169">Coguntur huius tempeſtatis Mathematici ea, quæ antiquiſsimo poſ­<lb></lb>
ſeſsionis iure tutò hactenus poſsiderunt, à nonnullis recentioribus <lb></lb>
ea diripere volentibus, omni conatu tutari. </s>
          <s id="s.005170">quis enim vnquam <lb></lb>alicuius nominis philoſophus ante Alexandrum Piccolomineum <lb></lb>
extitit, qui Geometris potiſsimas Demonſtrationes eripere tentauerit? <lb></lb>
</s>
          <s id="s.005171">profectò nullus fatetur ipſe ſe primum inter recentiores hanc veritatem <lb></lb>
olfeciſſe, ſed verè omnium etiam antiquorum primus ipſe fuit, nam duos, <lb></lb>
vel tres, quos ex antiquis in ſuam ſententiam pertrahere conatur, re vera, <lb></lb>
vt infra patebit, minimè pertrahit.</s>
        </p>
        <p type="main">
          <s id="s.005172">Primo igitur antiquorum auctoritates, præcipuè verò Ariſt. pro parte af­<lb></lb>
firmatiua afferemus. </s>
          <s id="s.005173">&amp; verò indignum, <expan abbr="atq;">atque</expan> ſuperuacaneum exiſtimo, cùm <lb></lb>eo, qui Ariſt. Analyticos poſteriores legerit, de ipſius ſententia diſputare, <lb></lb>
<expan abbr="ciusq́">eiusque</expan>; mentem, quaſi in fruſta locis aliquot citandis ſecare, cùm totis duo­<lb></lb>
bus libris nihil aliud agere videatur, quàm <expan abbr="perfectã">perfectam</expan> Demonſtrationis idęam <pb pagenum="11" xlink:href="009/01/295.jpg"></pb>ex Geometricis delineare; quippe qui omnes conditiones, <expan abbr="omniaq́">omniaque</expan>; ad per­<lb></lb>
fectam demonſtrationem neceſſaria, <expan abbr="vbiq;">vbique</expan> Geometricis demonſtrationibus <lb></lb>
attribuat, <expan abbr="idq́">idque</expan>; non ſolùm præceptis, ſed etiam exemplis perpetuò confir­<lb></lb>
met: ego quidem nihil vnquam Ariſt. clarius expreſsiſſe, nihil fuſius com­<lb></lb>
probaſſe exiſtimo, quàm Geometriæ demonſtrationes omnibus numeris ab­<lb></lb>
ſolutas eſſe, ita vt Philoſopho indignum omninò videatur ſententiam ipſius <lb></lb>
adeò manifeſtam aliò detorquere; ſatius eſſet, meo iudicio, palàm peri­<lb></lb>
patetici nomen ex hac parte deponere, quàm hoc modo Peripateticorum <lb></lb>
doctrinam, vt nonnulli faciunt, vel diſsimulare, vel tam perperam interpre­<lb></lb>
tari. </s>
          <s id="s.005174">quamuis igitur ſatis eſſet lectorem ad libros analyticos, <expan abbr="corumq́">eorumque</expan>; in­<lb></lb>
terpretes amandare, non grauabor tamen loca aliquot ſelecta, atque adeò <lb></lb>
manifeſta in medium afferre, vt magnopere mirandum ſit <expan abbr="cõtrariæ">contrariæ</expan> opinio­<lb></lb>nis authores ea pro libito interpretari. </s>
          <s id="s.005175">quorum primus ſit tex. 23. primi Po­<lb></lb>
ſter. <emph type="italics"></emph>(<expan abbr="Vnumquodq;">Vnumquodque</expan> autem ſcimus non ſecundum accidens, quando ſecundum illud <lb></lb>cognoſcimus, ſecundum quod ineſt, ex principijs illius, ſecundum quod illud, vt <lb></lb>
duobus rectis æquales habere, cui inest per ſe ex principijs huius)<emph.end type="italics"></emph.end> vbi manifeſtè <lb></lb>
vides Ariſt. aſſerere demonſtrationem illam, qua Geometra oſtendit, omne <lb></lb>
<expan abbr="triãgulum">triangulum</expan> habere tres angulos æquales duobus rectis procedere ex primis, <lb></lb>
immediatis, ex ijs, quæ ſunt per ſe, &amp; ſecundum quod ipſum; nullo autem <lb></lb>
modo ex ijs, quæ ſunt per accidens. </s>
          <s id="s.005176">perinde ac ſi diceret eam eſſe potiſsi­<lb></lb>
mam, <expan abbr="atq;">atque</expan> omnibus numeris abſolutam. </s>
          <s id="s.005177">verùm de hac demonſtratione in­<lb></lb>
ferius pluribus dicetur, interim vide citati loci noſtram explicationem ſu­<lb></lb>
pra in locis Mathem.</s>
        </p>
        <p type="main">
          <s id="s.005178">Textu deinde 29. primi Poſter. <emph type="italics"></emph>(Conuertuntur autem magis quæ ſunt in Ma­<lb></lb>thematicis, quoniam nullum accidens (ſed &amp; hoc differunt ab ijs, quæ ſunt in di­<lb></lb>
ſputationibus) ſed definitiones)<emph.end type="italics"></emph.end> vbi vides Mathematicos nullum <expan abbr="accidẽs">accidens</expan>, ſen <lb></lb>
contingens accipere, ſed definitiones, ideſt, non per aliquod contingens, <lb></lb>
ſed per cauſam formalem. </s>
          <s id="s.005179">tex. verò 31. <emph type="italics"></emph>(Figurarum autem maximè ſcientialis <lb></lb>
est prima, mathematicæ <expan abbr="namq;">namque</expan> <expan abbr="ſciẽtiæ">ſcientiæ</expan> per hanc demonſtrationes ferunt, vt Arith­<lb></lb>
metica, &amp; Geometria, &amp; Perſpectiua, &amp; ferè dixerim quæcunque ipſius Propter <lb></lb>
quid conſiderationem faciunt)<emph.end type="italics"></emph.end> poſtea tex. 11. 2. Poſter. aſſerit demonſtratio­<lb></lb>nem illam, qua Geometra probat, angulum in ſemicirculo eſſe rectum, eſſe <lb></lb>
à cauſa materiali, imò eam tanquam optimum huiuſmodi demonſtrationis <lb></lb>
exemplum affert. </s>
          <s id="s.005180">ſed de hac eadem demonſtratione infra iterum dicendum <lb></lb>
erit; vide interim prædicti loci explicationem ſupra in locis Mathemat. </s>
          <s id="s.005181">al­<lb></lb>
latam. </s>
          <s id="s.005182">hæc ex Logica ſufficiant, ne huc toti poſteriores inſerantur. </s>
          <s id="s.005183">tex. 68. <lb></lb>
2. Phyſ. <emph type="italics"></emph>(Aut enim ad ipſum Quid est, reducitur ipſum Propter quid vltimum in <lb></lb>immobilibus, vt in mathematicis, ad definitionem enim recti, aut commenſurabi­<lb></lb>lis, aut alterius cuiuſpiam reducitur vltimum)<emph.end type="italics"></emph.end> ecce iterum cauſa formalis in <lb></lb>
Mathematicis demonſtrationibus. </s>
          <s id="s.005184">vide huius loci explicationem ſupra à <lb></lb>
nobis allatam. </s>
          <s id="s.005185">6. Metaph. tex. 1. <emph type="italics"></emph>(Mathematicorum <expan abbr="quoq;">quoque</expan> principia, elementa, <lb></lb>
&amp; cauſæ ſunt)<emph.end type="italics"></emph.end> 11. Metaph. cap. 1. ſummæ 3. <emph type="italics"></emph>(Patet igitur tria eſſe genera ſpe­<lb></lb>
culatiuarum ſcientiarum, Naturalem, Mathematicam, Theologiam)<emph.end type="italics"></emph.end> ecce tibi, <lb></lb>
quam clara ſit Ariſt. ſententia.</s>
        </p>
        <p type="main">
          <s id="s.005186">Quod ad Platonis auctoritatem attinet, certum eſt, eum in Mathematicis <lb></lb>
cauſam materialem, &amp; formalem agnouiſſe, nam teſte Ariſt. primo Metaph.


<pb pagenum="12" xlink:href="009/01/296.jpg"></pb>cap. 7. ipſe non credebat aliarum cauſarum, quàm formalis, &amp; materialis, <lb></lb>
quas Mathematici tractant, ſpeculationem <expan abbr="philoſophicã">philoſophicam</expan> eſſe magnifacien­<lb></lb>
dam; efficientem enim, &amp; finalem <expan abbr="nũquam">nunquam</expan> explicuit, proptereaquod à Ma­<lb></lb>
thematicis, nunquam tractarentur. </s>
          <s id="s.005187">Proclus præterea cap. 10. lib. 7. in Eu­<lb></lb>
clidem, ait; Mathematicam verò omninò rerum ſempiternarum vim ha­<lb></lb>
bentem, ſcientiam appellat Plato. </s>
          <s id="s.005188">&amp; paulo poſt, ne dicamus igitur, quod <lb></lb>
Mathematicam à ſcientiarum numero Plato expellit. </s>
          <s id="s.005189">&amp; in fine cap. ait. <lb></lb>
<emph type="italics"></emph>(Mathematica tamen eſt ſcientia, non vt à ſuppoſitione immunis, ſed vt propria­<lb></lb>rum in anima rationum cognitrix, &amp; vt cauſas concluſionum afferens)<emph.end type="italics"></emph.end> nota illud, <lb></lb>
cauſas concluſionum afferens. </s>
          <s id="s.005190">concludit poſtea ſic, hæc omnia de Platonis <lb></lb>
ſententia pro Mathematicis dicta ſint.</s>
        </p>
        <p type="main">
          <s id="s.005191">Ipſum præterea exiſtimaſſe eas eſſe abſolutiſsimas ſcientias ex multis ip­<lb></lb>
ſius dictis par eſt credere; cur enim dixiſſet, Deum Geometrizare, niſi ob <lb></lb>
ſummam Geometriæ excellentiam? </s>
          <s id="s.005192">cur omnes ageometretos è gymnaſio <lb></lb>
ſuo arcebat? </s>
          <s id="s.005193">cur eas in aſcenſu ad ſummi Boni cognitionem naturali Phi­<lb></lb>
loſophiæ prætulit? </s>
          <s id="s.005194">quàm autem immeritò in <expan abbr="contrariã">contrariam</expan> ſententiam alij eum <lb></lb>
ad ſe pertrahant, infra apparebit, cùm calumnias diluemus.</s>
        </p>
        <p type="main">
          <s id="s.005195">Sequatur tertio loco Procli ipſius authoritas, qui in primo, &amp; ſecundo <lb></lb>
libro <expan abbr="cõm">comm</expan>. in Euclidem, totus eſt in Mathematicis, præcipuè verò in Geo­<lb></lb>
metria ſummis laudibus cumulandis, <expan abbr="easq́">easque</expan>; eſſe perfectiſſimas ſcientias ſæ­<lb></lb>
pius non ſolum aſſerit, ſed etiam demonſtrat. </s>
          <s id="s.005196">Id igitur primum cap. 10. lib. <lb></lb>
primi aggreditur, vbi fusè oſtendit ex Platone eſſe ſcientias, quæ <expan abbr="cõcluſio-num">concluſio­<lb></lb>
num</expan> cauſas afferant. </s>
          <s id="s.005197">i. </s>
          <s id="s.005198">perfectiſſimas habere <expan abbr="demõſtrationes">demonſtrationes</expan>. </s>
          <s id="s.005199">&amp; cap. 5. lib. 2. <lb></lb>
de Euclide loquens, ait <emph type="italics"></emph>(Præcipuè verò circa Geometricam elementorum inſti­<lb></lb>
tutionem eum quiſpiam admirabitur propter ordinem, &amp; electionem eorum, quæ <lb></lb>
per elementa diſtribuit, etenim non ca aſſumpſit omnia, quæ poterat dicere, ſed ea <lb></lb>
duntaxat, quæ elementari tradere potuit ordine. </s>
          <s id="s.005200">Adhuc autem omnis generis ſyl­<lb></lb>
logiſmorum modus, alios quidem à cauſis fidem ſuſcipientes, alios verò à certis no­<lb></lb>
tis perfectos, omnes autem inuincibiles, &amp; certos, ad ſcientiamqué, accommodatos)<emph.end type="italics"></emph.end><lb></lb>notanda ſunt illa; à cauſis fidem ſuſcipientes, quibus præcipuè indicat, ſe in <lb></lb>
Demonſtrationibus Euclidianis cauſas agnoſcere.</s>
        </p>
        <p type="main">
          <s id="s.005201">Lib. deinde 3. in comm. ad primam Euclidis propoſit. </s>
          <s id="s.005202">hæc habet; <expan abbr="Quã-do">Quan­<lb></lb>
do</expan> igitur ſyllogiſmus Geometris per impoſſibile fuerit, ſymptoma tantùm <lb></lb>
inuenire cupiunt, quando autem per præcipuam Demonſtrationem, tunc <lb></lb>
rurſus ſiquidem in particulari demonſtrationes fiant, cauſa nondum mani­<lb></lb>
feſta eſt, ſi verò in vniuerſali, in omnibus ſimilibus continuò &amp; ipſum pro­<lb></lb>
pter quid manifeſtam fit. </s>
          <s id="s.005203">Ecce tibi iterum ipſum Propter quid in Geome­<lb></lb>
tricis. </s>
          <s id="s.005204">&amp; in eod. </s>
          <s id="s.005205">com. </s>
          <s id="s.005206">poſt multa; illam autem, quæ demonſtratio dicitur, <lb></lb>
quandoquidem propria <expan abbr="Demõſtrationi">Demonſtrationi</expan> habentem inueniemus, &amp; definitio­<lb></lb>nibus Medijs quæſitum oſtendentem; hæc enim Demonſtrationis perfectio <lb></lb>
eſt. </s>
          <s id="s.005207">Vbi obſeruandum eſt apud Proclum Geometram vti definitionibus pro <lb></lb>
Medio; quòd requiritur ad exactiſſimam <expan abbr="Demonſtrationẽ">Demonſtrationem</expan>, vt ipſe ait: quod <lb></lb>
declarat exemplo primæ <expan abbr="Demõſtrationis">Demonſtrationis</expan> Euclidis, cùm ait, quando autem <lb></lb>
per deſcriptionem circulorum, quod conſtitutum eſt Triangulum æquilate­<lb></lb>
rum eſſe oſtenditur, à cauſa apprehenſio fit, æqualitatem enim circulorum <lb></lb>cauſam æqualitatis laterum illius eſſe dicemus. </s>
          <s id="s.005208">Quid igitur apud Proclum <pb pagenum="13" xlink:href="009/01/297.jpg"></pb>clarius dici poterat? </s>
          <s id="s.005209">quæ tamen omnia aduerſarij videntur clauſis de indu­<lb></lb>
ſtria oculis præterijſſe; conſtat enim ex opuſculo Piccolominei ipſum dili­<lb></lb>
genter hoc conſilio Proclum perlegiſſe; quì igitur fieri potuit, quin ea vi­<lb></lb>
derit. </s>
          <s id="s.005210">Sed hodie plurimi non ad verum, ſed ad libitum philoſophamur.</s>
        </p>
        <p type="main">
          <s id="s.005211">Placuit hos tres ſolos Platonem, Ariſtotilem, &amp; Proclum ex veteribus pro <lb></lb>
noſtra ſententia in <expan abbr="mediũ">medium</expan> adducere, propterea quod eos ſibi adiungere <expan abbr="cõ-tra">con­<lb></lb>
tra</expan> omnem rationem nitantur aduerſarij, vt ex prædictis iam ſatis liquidè <lb></lb>
conſtat. </s>
          <s id="s.005212">Reliquorum verò Philoſophorum, tàm Græcorum, quàm <expan abbr="Arabũ">Arabum</expan>, <lb></lb>
aut Latinorum placita citare ſuperſedeo, etiam ſi omnes vno ore Geome­<lb></lb>tricas demonſtrationes tanquam omnium exactiſſimas celebrauerint, vel <lb></lb>teſte ipſo Piccolomineo, qui initio libelli de certitudine, Mathematica, ſic <lb></lb>
ait; omnes ferè Latini, vt D. Albertus, D. Tho nas, Marſilius, Egydius, Zi­<lb></lb>
mara, &amp; <expan abbr="pleriq́">plerique</expan>, alij vno ore Auerroem interpretati ſunt, dicere Mathema­<lb></lb>
ticas demonſtrationes eſſe in primo gradu certitudinis, quod Mathemati­<lb></lb>
cus ex notioribus nobis, &amp; natura demonſtret, quippequi vel ſolus, vel ma­<lb></lb>
ximè demonſtratione illa, quam potiſſimam appellant, vtatur, qua. </s>
          <s id="s.005213">ſ. </s>
          <s id="s.005214">ſimul <lb></lb>
&amp; quod effectus ſit, &amp; cur ſit liquidò innoteſcit. </s>
          <s id="s.005215">Verum ipſe omnium pri­<lb></lb>mus abſolutè dici poteſt, cum nullus ante ipſum, cuius opera extent, id di­<lb></lb>
cat; quamuis ipſe fallo Proclum, Ariſt. &amp; Platonem ſibi conetur adiungere. <lb></lb>
</s>
          <s id="s.005216">Poſt ipſum verò ſoli duo ferè Pererius, &amp; Conimbric. </s>
          <s id="s.005217">eum ſequuti ſunt. </s>
          <s id="s.005218">At <lb></lb>
verò contrariam <expan abbr="ſentẽtiam">ſententiam</expan> reliqui omnes poſt ipſum amplexi ſunt; ex qui­<lb></lb>
bus ſolos duos, <expan abbr="eosq́">eosque</expan>; præſtantiſſios huiuſce tempeſtatis philoſophos alle­<lb></lb>
gaſſe ſit ſatis. </s>
          <s id="s.005219">Toletum. </s>
          <s id="s.005220">ſ. </s>
          <s id="s.005221">&amp; Zabarellam. </s>
          <s id="s.005222">Toletus enim quæſt. </s>
          <s id="s.005223">4. 2. Phyſ. <lb></lb>
in 3. concluſione habet iſta; Phyſicus, &amp; Mathematicus differunt in modo <lb></lb>demonſtrandi, Phyſicus enim frequenter vtitur demonſtratione ſigni, &amp; ef­<lb></lb>fectus, quia ipſius cauſæ frequentius ſunt occultæ, nec per ſe ſenſibiles, effe­<lb></lb>
ctus verò ſunt ſenſibiles, vt mors, motus, &amp;c. </s>
          <s id="s.005224">quæ ad ſenſum patent, <expan abbr="quorũ">quorum</expan> <lb></lb>
cauſæ à ſenſibus ſunt remotæ. </s>
          <s id="s.005225">At Mathematicus frequentius à prioribus <lb></lb>
procedit cum eius cauſæ notiores ſint effectibus, à ſenſu enim abſtrahit, &amp; <lb></lb>
in intellectu notius eſt, quod prius eſt. </s>
          <s id="s.005226">videas Lector, quàm ſyncerè natu­<lb></lb>
ralis philoſophiæ profeſſor vera de Mathematicis loquatur, ita vt etiam eas <lb></lb>
illi præferat. </s>
          <s id="s.005227">Iacobus autem Zabarella in toto ſuo opere logico, perpetuò <lb></lb>Mathematicas demonſtrationes, vt potiſſimas agnoſcit, <expan abbr="exemplaq́">exemplaque</expan>; Ariſt. <lb></lb>
geometrica exponit tanquam vera, &amp; omnino rebus ipſis accommodata; <lb></lb>
quare non eſt, cur vnum, aut alterum ipſius locum hic deſcribamus. </s>
          <s id="s.005228">illud <lb></lb>
non prætermittam, ipſum fateri ſe bis, teruè totum Euclidem ſedulò perle­<lb></lb>
giſſe, vt probè poſſet Ariſt. mentem circa <expan abbr="demõſtrationis">demonſtrationis</expan> naturam aſſequi, <lb></lb>
cùm videret Ariſt. <expan abbr="quæcunq;">quæcunque</expan> de demonſtratione præciperet, omnia ad Geo­<lb></lb>
metricam normam, tanquam ad lydium lapidem examinare. </s>
          <s id="s.005229">Locus, vbi <lb></lb>
hæc ait, mihi è memoria excidit, certus tamen ſum apud ipſum ea me le­<lb></lb>
giſſe. </s>
          <s id="s.005230">quarto tandem loco, communi authoritate omnium antiquorum idem <lb></lb>
comprobatur, apud quos ſemper demonſtrationes Geometricę appellatæ <lb></lb>
ſunt per antonomaſiam demonſtrationes, non rationes, non opiniones, non <lb></lb>
ſententiæ, quemadmodum in reliquis philoſophiæ partibus fieri ſolet. </s>
          <s id="s.005231">Sed <lb></lb>
iam ab authoritatibus ad rationes.</s>
        </p>
        <p type="main">
          <s id="s.005232">Prima, Vera, &amp; perfecta demonſtratio ex Auerrois ſententia debet à no­<pb pagenum="14" xlink:href="009/01/298.jpg"></pb>tioribus nobis, &amp; natura procedere, tales ſunt Geometricæ vt paulo ſupra <lb></lb>
patuit, ergò ipſæ potiſſimæ erunt demonſtrationes.</s>
        </p>
        <p type="main">
          <s id="s.005233">2. Ex Themiſtio cap, 2. ſuæ paraphr. </s>
          <s id="s.005234">2. Poſter. Demonſtratio potiſſima <lb></lb>
debet oſtendere, &amp; quod, &amp; Propter quid. </s>
          <s id="s.005235">quod profectò cæteris demon­<lb></lb>
ſtrationibus melius præſtant Geometricæ, &amp; Arithmeticæ. </s>
          <s id="s.005236">v. g. demonſtra­<lb></lb>
tio 32. 3. oſtendit angulum in ſemicirculo eſſe rectum, quod omninò igno­<lb></lb>
tum erat; &amp; affert cauſam, quæ pariter ignorabatur. </s>
          <s id="s.005237">Idem ferè faciunt aliæ <lb></lb>
omnes. </s>
          <s id="s.005238">in Mathematicis verò medijs, in Phyſica, &amp; Metaphyſica effectus <lb></lb>
<expan abbr="plerumq;">plerumque</expan> noti ſunt, ſed cauſæ latent; dicendum igitur Geometricas demon­<lb></lb>ſtrationes ex Auerroe, &amp; Themiſtio præſtantiſſimas eſſe.</s>
        </p>
        <p type="main">
          <s id="s.005239">Tertia, quæ eſt euidentiſſima, ſumatur ex reſolutione aliquot demonſtra­<lb></lb>
tionum. </s>
          <s id="s.005240">Quid enim opus eſt diſputationem hanc per extraneas ambages <lb></lb>
agitare? </s>
          <s id="s.005241">cum licear quaſi in rem præſentem ire, &amp; veluti demonſtrationum <lb></lb>
anatome facta oculis ipſis earum media contueri. </s>
          <s id="s.005242">ſed prius in memoriam <lb></lb>
redigendum eſt, illam eſſe perfectiſſimam demonſtrationem, quæ non ſolùm <lb></lb>
rei demonſtrandæ cauſam propriam, &amp; adæquatam affert, verùm etiam <lb></lb>
euidentiſſimè oſtendit talem paſſionem ab illa cauſa procedere, ita vt non <lb></lb>
poſsit, vt ait Ariſt. aliter res ſe habere, in quo profectò Mathematicæ ex­<lb></lb>
cellunt. </s>
          <s id="s.005243">Cauſa verò hæc in Geometria, &amp; Arithmetica aliquando eſt mate­<lb></lb>
rialis, quando ſcilicet vtuntur pro Medio partibus, reſpectu totius; vel eſt <lb></lb>
formalis, quando nimirum Medium eſt definitio ſubiecti, aut paſsionis. </s>
          <s id="s.005244">non <lb></lb>
me tamen latet omnem perfectam demonſtrationem alio ſenſu dici à qui­<lb></lb>
buſdam procedere per cauſam formalem, quia in ea continetur cauſalis de­<lb></lb>
finitio paſsionis, quæ definitio cauſam ipſius exhibet, &amp; proinde tanquam <lb></lb>
forma ipſius eſt, quæ rem in eſſe conſtituat.</s>
        </p>
        <p type="main">
          <s id="s.005245">Secundò notandum eſt: Omnem demonſtrationis diſcurſum reſolui tan­<lb></lb>
dem in aliquid, aut per ſe notum, aut à poſteriori comprobatum. </s>
          <s id="s.005246">Satis. </s>
          <s id="s.005247">n. <lb></lb>
</s>
          <s id="s.005248">eſt, vt cauſa euidentur appareat, <expan abbr="quocunq;">quocunque</expan> id modo fiat. </s>
          <s id="s.005249">hoc dixi propter <lb></lb>nonnullos, qui cùm in Geometricę demonſtrationibus lineam, aut diuiſionem <lb></lb>
aliquam, rei, quæ oſtenditur, non intrinſecam animaduertunt, ſtatim exi­<lb></lb>
ſtimant eas per extrinſeca <expan abbr="demõſtrare">demonſtrare</expan>: ſed decipiuntur; quia non animad­<lb></lb>uertunt lineas illas, aut partitiones, non eſſe medium demonſtrationis, ſed <lb></lb>
adhiberi ad medij inuentionem, &amp; connexionem cum paſſione. </s>
          <s id="s.005250">Quod au­<lb></lb>
tem eorum dubitatio omninò vana ſit ex eo patet, quod plurimæ ſunt de­<lb></lb>
monſtrationes, quæ ſine vlla linearum conſtructione, aut diuiſione compro­<lb></lb>
bentur, vti ſunt 15.33 34.42. 36. in ſolo primo elementorum. </s>
          <s id="s.005251">atque hæc <lb></lb>
erroris eorum præcipua cauſa eſt.</s>
        </p>
        <p type="main">
          <s id="s.005252">His præmiſſis, primò oſtendemus cauſam formalem in Geometricæ de­<lb></lb>
monſtrationibus reperiri deinde materialem. </s>
          <s id="s.005253"><expan abbr="idq́">idque</expan>; primò per reſolutionem <lb></lb>
primę Euclidis, quæ à cauſa formali procedit. </s>
          <s id="s.005254">&amp; quia hæc demonſtratio <lb></lb>
non theorema, ſed problema eſt, ideò ſciendum, quòd minimè aduerſarij <lb></lb>
animaduerterunt, in omni problemate per <expan abbr="quandã">quandam</expan> <expan abbr="linearũ">linearum</expan> conſtructionem <lb></lb>
doceri aliquid effici. </s>
          <s id="s.005255">v. g. in præſenti docet Euclides, qua ratione deſcrip­<lb></lb>
tis <expan abbr="quibuſdã">quibuſdam</expan> circulis circa datam lineam, <expan abbr="ductisq́">ductisque</expan>; aliquot lineis modo prę­<lb></lb>
ſcripto, gignatur <expan abbr="triangulũ">triangulum</expan> æquilaterum, vt rem conſideranti manifeſtum <lb></lb>
eſt. </s>
          <s id="s.005256">quare nullo modo lineamenta illa, vt illę circulorum ſemidiametri ſunt <pb pagenum="15" xlink:href="009/01/299.jpg"></pb>extrinſeca rei, de qua demonſtratur; quinimò ſubiectum ipſius ſunt. </s>
          <s id="s.005257">Quia <lb></lb>
verò facta conſtructione, ſtatim perſpicuè apparet ortum eſſe triangulum, <lb></lb>
æquilaterum, non eſt illi curę probare illud eſſe triangulum, ſed quia an ſit <lb></lb>
æquilaterum ignoratur, idcircò totus demonſtrationis diſcurſus verſatur in <lb></lb>
demonſtranda trium illarum linearum æqualitate.</s>
        </p>
        <p type="main">
          <s id="s.005258">Quem <expan abbr="quidẽ">quidem</expan> diſcurſum continere cauſam, quamuis per ſe pateat, vt mox <lb></lb>
apparebit, non deeſt <expan abbr="tamẽ">tamen</expan> Procli authoritas adeò clara, vt magnopere mi­<lb></lb>
rer Piccolomineum Procli ſtudioſum, eam non vidiſſe: Proclus enim in <expan abbr="cõ-men">conm<lb></lb>
men</expan>. huius demonſtrationis hæc habet; quando autem per deſcriptionem <lb></lb>circulorum, quod conſtructum eſt triangulum æquilaterum eſſe oſtenditur, <lb></lb>
à cauſa apprehenſio fit; ſimilitudinem. </s>
          <s id="s.005259">n. </s>
          <s id="s.005260">&amp; æqualitatem circulorum cauſam <lb></lb>
dicimus eſſe æqualitatis laterum illius trianguli. </s>
          <s id="s.005261">Quibus verbis non ſolum <lb></lb>
authoritas, ſed ratio etiam optima, cur hæc ſit demonſtratio à cauſa, con­<lb></lb>tinetur, quia nimirum oſtendit cauſam æqualitatis laterum eſſe, quia ſint <lb></lb>
ſemidiametri æqualium circulorum. </s>
          <s id="s.005262">Quæ argumentatio procedit à defini­<lb></lb>
tione ſubiecti, quod eſt circulus: quamuis non tota, ſed tantum quatenus <lb></lb>neceſſaria eſt, afferatur, ideſt definitio ſemidiametrorum, quod ad demon­<lb></lb>
ſtrandum ſufficit, vt benè notat Zabarella, loquens de hac ipſa demonſtra­<lb></lb>
tione; cùm igitur medium ſit definitio ſubiecti, patet eam eſſe perfectam <lb></lb>
demonſtrationem, in qua paſſionis oſtenſæ allata eſt propria, &amp; adæquata <lb></lb>
cauſa, quæ eſt natura circuli. </s>
          <s id="s.005263"><expan abbr="ſicq́">ſicque</expan>; Euclides optimè demonſtraujt ex con­<lb></lb>
ſtructione, quàm præceperat, gigni triangulum æquilaterum. </s>
          <s id="s.005264">Subiectum <lb></lb>
igitur eſt il a circulorum, ac linearum configuratio, medium definitio cir­<lb></lb>
culi, paſſio triangulum <expan abbr="æquilaterũ">æquilaterum</expan>. </s>
          <s id="s.005265">ex qua <expan abbr="demõſtratione">demonſtratione</expan> erui poteſt etiam <lb></lb>
definitio paſſionis cauſalis, ideſt, eſſe triangulum æquilaterum ex tali <expan abbr="cõſtru-ctione">conſtru­<lb></lb>
ctione</expan> ortum. </s>
          <s id="s.005266">Quare huic nihil deeſt ad perfectam <expan abbr="demonſtrationẽ">demonſtrationem</expan>. </s>
          <s id="s.005267">ex qui­<lb></lb>
bus videas, quàm immeritò nonnulli eam impugnent, putantes eam eſſe per <lb></lb>
extranea; cauſa erroris fuit, quia exiſtimarunt abſolutè demonſtrari <expan abbr="triã-gulum">trian­<lb></lb>
gulum</expan> illud eſſe æquilaterum. </s>
          <s id="s.005268">verùm decepti ſunt, quia in hoc, &amp; in omni­<lb></lb>
bus alijs problematis, demonſtratur talem conſtructionem parere triangu­<lb></lb>
lum, vel <expan abbr="quadratũ">quadratum</expan>, vel quid aliud, vt patet Euclidem, vel obiter <expan abbr="inſpiciẽti">inſpicienti</expan>.</s>
        </p>
        <p type="main">
          <s id="s.005269">Placet adhuc alteram à formali cauſa procedentem expendere. </s>
          <s id="s.005270">ea eſt 46. <lb></lb>
primi elem. </s>
          <s id="s.005271">quæ ſimiliter problema eſt, quo docet Euclides, qua ratione ſu­<lb></lb>
pra data recta linea quadratum deſcribatur. </s>
          <s id="s.005272">tradit igitur quandam linea­<lb></lb>
rum conſtructionem, ex qua poſtea demonſtrat ortum eſſe quadratum, ita <lb></lb>
vt conſtructio illa ſit loco ſubiecti, de qua demonſtratvr eſſe quadratum. </s>
          <s id="s.005273">non <lb></lb>
igitur intendit, vt nonnulli falsò putant, <expan abbr="demõſtrare">demonſtrare</expan> abſolutè illud eſſe qua­<lb></lb>
dratum, ſed ex tali conſtructione ortum eſſe quadratum duo autem ſunt de <lb></lb>eſſentia quadrati, primum habere quatuor latera æqualia, ſecundum habe­<lb></lb>
re quatuor angulos rectos, vt ex definitione conſtat. </s>
          <s id="s.005274">Neutrum autem ſine <lb></lb>
altero ſufficit, <expan abbr="nã">nam</expan> &amp; Rhombus quatuor latera æqualia habet, &amp; Altera par­<lb></lb>
te longius habet quatuor angulos rectos, neutrum tamen quadratum eſt. </s>
          <s id="s.005275">ſi <lb></lb>
verò <expan abbr="vtrunq;">vtrunque</expan> ſimul cuipiam figuræ competat, illam neceſſariò quadratum <lb></lb>
eſſe efficient. </s>
          <s id="s.005276">Probat igitur Euclid. vtraq, euidenter ineſſe illi figuræ ex vi <lb></lb>
illius conſtructionis, &amp; ideò illi quadrati definitionem competere. </s>
          <s id="s.005277">Quare <lb></lb>
hęc erit potiſſima demonſtratio, cùm cauſam afferat <expan abbr="intrinſecã">intrinſecam</expan>, propriam,


<pb pagenum="16" xlink:href="009/01/300.jpg"></pb>&amp; adæquatam, propter quam res eſt. </s>
          <s id="s.005278">Vbi notandum effectum re vera diſtin­<lb></lb>
gui à ſua cauſa, eſſe enim quadratum (qui effectus eſt) non eſt habere, qua­<lb></lb>
tuor angulos rectos ſolum: <expan abbr="neq;">neque</expan> habere quatuor latera æqualia ſolum, ſed <lb></lb>
<expan abbr="vtrunq;">vtrunque</expan> ſimul in eodem; vnde reſultat totum, ſeu <expan abbr="compoſitũ">compoſitum</expan>, quod eſt quid <lb></lb>
diuerſum à partibus ſeorſum ſumptis. </s>
          <s id="s.005279">in demonſtratione autem hac, cauſa <lb></lb>
ſunt partes ſeorſim ſumptæ; effectus verò eſt compoſitum, ex earum vnione <lb></lb>
reſultans. </s>
          <s id="s.005280">Notandum præterea eandem demonſtrationem procedere à de­<lb></lb>
finitione ſubiecti, nam illa duo quadrati eſſentialia, ex definitione eorum, <lb></lb>quæ ſunt in conſtitutione petuntur, quæ conſtitutio eſt inſtar ſubiecti, vt ſu­<lb></lb>
pra monui: ex hac autem definitione partium ſubiecti in demonſtratione <lb></lb>
contenta, eruitur definitio cauſalis ipſius paſsionis, quæ eſt, quadratum eſt <lb></lb>
figura habens quatuor angulos rectos, &amp; quatuor latera æqualia, ex tali <expan abbr="cõ-ſtructione">con­<lb></lb>
ſtructione</expan> producta. </s>
          <s id="s.005281">Notandum tandem quouis modo ſiue à cauſa, ſiue ab <lb></lb>effectu oſtendantur illa duo eſſentialia quadrati, ineſſe ipſi, nihil referre ad <lb></lb>
demonſtrationis perfectionem. </s>
          <s id="s.005282">Satis. </s>
          <s id="s.005283">n. </s>
          <s id="s.005284">eſt, ſi habeamus rei cauſam <expan abbr="propriã">propriam</expan>, <lb></lb>
ita vt aliter ſe habere nequeat. </s>
          <s id="s.005285">ſexcentæ huiuſmodi per formalem cauſam, <lb></lb>
apud Euclid. Archim Appoll. </s>
          <s id="s.005286">&amp; alios Geometras reperies. </s>
          <s id="s.005287">vide Appendi­<lb></lb>
cem, ad finem operis, in qua omnes primi elem. </s>
          <s id="s.005288">demonſtrationes reſolutas <lb></lb>
inuenies, <expan abbr="plurimasq́">plurimasque</expan>; à cuſa formali.</s>
        </p>
        <p type="main">
          <s id="s.005289">Sed iam materialem cauſam indagemus, <expan abbr="idq́">idque</expan>; duce Ariſt. accipiamus igi<lb></lb>tur celeberrimam illam 32. primi elem. </s>
          <s id="s.005290">quam Mathematicis <expan abbr="ſoiẽt">ſolent</expan> aduerſa­<lb></lb>
rij opponere. </s>
          <s id="s.005291">&amp; quoniam ſupra tex. 23. 1. Poſter. nos eam per cauſam ma­<lb></lb>terialem procedere oſtendimus, ideò ne actum agamus, <expan abbr="explicationẽ">explicationem</expan> illam <lb></lb>
nunc opus eſt relegere. </s>
          <s id="s.005292">Hoc tamen loco partem ipſius primam, angulum, <lb></lb>
videlicet externum cuiuſuis trianguli, æqualem eſſe duobus internis, &amp; op­<lb></lb>
poſitis, examinabo; cuius medium, ſi ad rigorem demonſtrationis rediga­<lb></lb>
tur, eſt hoc; externus angulus eſt diuiſibilis in duos angulos, quorum ſingu­<lb></lb>
li ſingulis internis ſunt æ quales, ergo <expan abbr="etiã">etiam</expan> totalis anguius erit æqualis am­<lb></lb>
bobus internis ſimul ſumptis. </s>
          <s id="s.005293">Quod autem externus angulus ſit diuiſibilis <lb></lb>
in duas partes æquales internis angulis probat diuidendo illum per lineam <lb></lb>
illam oppoſito trianguli lateri parallelam, vnde ſtatim ex parallelarum na <lb></lb>tura apparet partiales angulos anguli externi æquales eſſe internis triangu<lb></lb>
li; ex quo ſequitur totum externum angulum eſſe æqualem duobus internis <lb></lb>
ſimul ſumptis. </s>
          <s id="s.005294">Hic autem modus argumentandus, à partibus poſsibilibus ad <lb></lb>
totum, eſſe à cauſa materiali, apud omnes Philoſophos in <expan abbr="cõfeſſo">confeſſo</expan> eſt, &amp; Ari­<lb></lb>
ſtot. ipſe tex. 3. 5. Metaph. id aſſerit. </s>
          <s id="s.005295">&amp; tex. 11. 2. Poſter. vtitur ſimili <expan abbr="exẽ-plo">exem­<lb></lb>
plo</expan> ad <expan abbr="materialẽ">materialem</expan> cauſam explicandam. </s>
          <s id="s.005296">quamuis autem Geometræ non di­<lb></lb>
cant talem angulum, vel talem figuram eſſe <expan abbr="diuiſibilẽ">diuiſibilem</expan> in partes æquales alijs <lb></lb>
quibuſdam, ſed ſtatim diuidant, id faciunt breuitatis cauſa; vtuntur enim <lb></lb>
actu pro potentia, quia actus potentiam ſupponit, quòd optimè Ariſtot. 9. <lb></lb>
Metaphyſ. tex. 20. annotauit, ſic; Deſcriptiones quoque actu inueniuntur, <lb></lb>diuidentes namque inueniunt, quòd ſi diuiſæ eſſent, manifeſtæ eſſent, nunc <lb></lb>
autem inſunt potentia, &amp;c. </s>
          <s id="s.005297">Cuius loci noſtram ſuperius allatam explicatio­<lb></lb>
nem habes. </s>
          <s id="s.005298">per deſcriptiones autem intelligit Geometricas demonſtratio­<lb></lb>
nes, vt ſæpius ſupra in opere oſtenſum eſt. </s>
          <s id="s.005299">Innumeræ ſunt apud Geometras, <lb></lb>
quę per hanc poſsibilem diuiſionem procedunt, <expan abbr="quęq;">quęque</expan> ideò ſunt à cauſa ma­


<pb pagenum="17" xlink:href="009/01/301.jpg"></pb>teriali; plures autem eſſe in primo elem. </s>
          <s id="s.005300">conſtat ex appendice in fine ope­<lb></lb>
ris addita. </s>
          <s id="s.005301">Notandum hic <expan abbr="quoq;">quoque</expan> cauſam eſſe natura ſua diſtinctam ab effe­<lb></lb>
ctu, non ſecus ac potentia ab actu; nam ex eo, quòd poſſit aliquid diuidi in <lb></lb>
partes æquales aliquibus, ſequitur illud totum eſſe actu æquale alteri, &amp; eſt <lb></lb>à priori, quia partes natura prius ſunt toto, cùm ſint ipſius cauſa. </s>
          <s id="s.005302">Notan­<lb></lb>
dum hic etiam parallelam illam, qua angulus diuiditur, duci ad <expan abbr="mediũ">medium</expan> de­<lb></lb>
monſtrationis indagandum, nequaquam verò ipſam eſſe medium, &amp; idcir­<lb></lb>
cò demonſtrationem hanc non eſſe per extrinſeca, niſi velis minorem <expan abbr="pro-poſitionẽ">pro­<lb></lb>
poſitionem</expan> per extrinſeca oſtendi, quòd libenter concedimus, cùm iſtud de­<lb></lb>
monſtrationi nihil deroget. </s>
          <s id="s.005303">Eſt autem per intrinſecam, propriam, &amp; adæ­<lb></lb>
quatam cauſam illius æqualitatis, partes enim reſpectu totius ſunt tales. <lb></lb>
</s>
          <s id="s.005304">eſt igitur potiſſima demonſtratio, quòd erat demonſtrandum.</s>
        </p>
        <p type="main">
          <s id="s.005305">Poſtquam Euclides hanc primam propoſitionis partem demonſtrauit, <lb></lb>
oſtendit alteram. </s>
          <s id="s.005306">ſ. </s>
          <s id="s.005307">omne triangulum habere tres, &amp;c. </s>
          <s id="s.005308">quoniam partes duo­<lb></lb>
rum rectorum ſunt æquales tribus angulis illis. </s>
          <s id="s.005309">quod medium pariter eſt à <lb></lb>
cauſa materiali, à partibus ad totum. </s>
          <s id="s.005310">Vide huius explicationem tex. 23. <lb></lb>
1. Poſter. vbi etiam videbis eam poſſe demonſtrari modo Pythagoreorum, <lb></lb>
abſque vlla diuiſione, ſed per partes actu exiſtentes. </s>
          <s id="s.005311">hoc dico propter eos, <lb></lb>
qui per haſce diuiſiones timent, ne non inueniatur medium á priori. </s>
          <s id="s.005312">ſed vt <lb></lb>deponant penitus hunc ſcrupulum, ſciant in huiuſmodi demonſtrationibus, <lb></lb>
quibus aliquid ęquale alteri adhibita diuiſione demonſtratur, ſępè accide­<lb></lb>
re, vt non diuidatur, niſi vnus terminorum ęqualitatis, quare ex parte in­<lb></lb>
diuiſi ęqualitas cauſabitur à partibus actu pręcedentibus, &amp; <expan abbr="conſtituẽtibus">conſtituentibus</expan> <lb></lb>
totum; quod videre eſt in <expan abbr="vtraq;">vtraque</expan> parte huius. </s>
          <s id="s.005313">32. ſecundum Euclidem, &amp; in <lb></lb>
47. primi elem. </s>
          <s id="s.005314">&amp; alijs plurimis.</s>
        </p>
        <p type="main">
          <s id="s.005315">Sed primò Piccolom. ex Proclo obijcit hęc <emph type="italics"></emph>(Quando enim eo, quòd extrin­<lb></lb>ſecus angulus duobus internis, &amp; oppoſitis æqualis est, oſtenditur triangulum ha­<lb></lb>
bere tres angulos æquales duobus rectis, quomodo à cauſa eſt <expan abbr="demõstratio">demonstratio</expan> hæc? </s>
          <s id="s.005316">non <lb></lb>
ne medium certum ſignum est? </s>
          <s id="s.005317">etenim <expan abbr="neq;">neque</expan> externo exiſtente angulo cùm interni <lb></lb>
exiſtant, duobus rectis æquales ſunt; eſt. </s>
          <s id="s.005318">n. </s>
          <s id="s.005319">triangulum latere etiam non producto)<emph.end type="italics"></emph.end><lb></lb>
Pergit deinde Proclus demonſtrare primam Euclidìs demonſtrationem eſſe <lb></lb>
per cauſam, &amp; proinde veram demonſtrationem, quòd Piccolomin. in ſua <lb></lb>citatione callidè videtur reticuiſſe. </s>
          <s id="s.005320">Ad <expan abbr="obiectionẽ">obiectionem</expan> reſpondeo primò. </s>
          <s id="s.005321">angu­<lb></lb>
lum externum in Euclidiana demonſtratione minimè extraneum eſſe, quia <lb></lb>
in hac ſecunda parte aſſumitur pro ſubiecto demonſtrationis, ideſt pro par­<lb></lb>te duorum rectorum, ipſe enim cum angulo ſibi deinceps facit duos angulos <lb></lb>
rectos, quibus tres anguli trianguli probantur ęquales: quod Proclus <expan abbr="nõ">non</expan> vi­<lb></lb>
detur vidiſſe. </s>
          <s id="s.005322">Secundò, ſi hęc Euclidiana illi <expan abbr="nõ">non</expan> probatur, accipiat de eadem <lb></lb>
re Pythagoricam, quę abſque angulo externo, &amp; ab <expan abbr="q;">que</expan> vlla diuiſione probat <lb></lb>intentum; &amp; erit omnis ſublata dubitatio. </s>
          <s id="s.005323">Tertiò, ſi conuincerent aduer­<lb></lb>
farij, quòd nequaquam faciunt, hanc non eſſe à priori, ſequitur ne propte­<lb></lb>
rea reliquas omnes eſſe ei ſimiles, vt ipſi inferre conantur? </s>
          <s id="s.005324">minimè <expan abbr="gentiũ">gentium</expan>. <lb></lb>
</s>
          <s id="s.005325">quo logico iure ab vno particulari inferre volunt vniuerſale?</s>
        </p>
        <p type="main">
          <s id="s.005326">Secundò, obijcies, paſſionem hanc, habere tres angulos, &amp;c. </s>
          <s id="s.005327">non recipro­<lb></lb>
cari cum triangulo, ſeu non eſſe ſecundum quod ipſum, vt aiunt Logici: re­<lb></lb>peritur enim figura quędam pręter triangulum, vt patet apud Proclum, quę <pb pagenum="18" xlink:href="009/01/302.jpg"></pb>eandem habet proprietatem. </s>
          <s id="s.005328">Reſpondeo habere tres angulos rectilineos <lb></lb>
(de his. </s>
          <s id="s.005329">n. </s>
          <s id="s.005330">Euclides agit) æquales, &amp;c. </s>
          <s id="s.005331">cùm triangulo conuerti, nam Proclus <lb></lb>
eam conuertit. </s>
          <s id="s.005332">figura autem illa alia, quę habet tres angulos ęquales duo­<lb></lb>
bus rectis, non habet angulos rectilineos, <expan abbr="neq;">neque</expan>. </s>
          <s id="s.005333">n. </s>
          <s id="s.005334">rectilinea eſt, vt apud Pro­<lb></lb>
clum videre eſt: &amp; ideò non eſt ad mentem Euclidis, aut Pythagorę. </s>
          <s id="s.005335">ſed iam <lb></lb>
cum Proclo concludamus, ſic; quia etiam illud <expan abbr="quoq;">quoque</expan> dicendum eſt, quòd <lb></lb>
internos angulos duobus rectis æquales habere, per ſe, &amp; <expan abbr="ſecundũ">ſecundum</expan> quod ip­<lb></lb>
ſum triangulo ineſt: idcircò &amp; Ariſtot. in tractatu de demonſtratione hoc <lb></lb>
exemplum habet in promptu, ſecundum quod ipſum conſiderans, hęc ille.</s>
        </p>
        <p type="main">
          <s id="s.005336">Aliam per cauſam <expan abbr="materialẽ">materialem</expan> ex mente Ariſt. expendimus tex. 11. 2. Po­<lb></lb>
ſterior. vbi ait, angulum in ſemicirculo eſſe rectum, quoniam eſt dimidium <lb></lb>
duorum rectorum, quod medium eſt in cauſa materiali, eſſe enim dimidium <lb></lb>
eſt eſſe partem. </s>
          <s id="s.005337">Cauſa igitur, quæ facit angulum illum eſſe rectum, eſt di­<lb></lb>
midia quantitas duorum rectorum, quę ipſum conſtituit; ſed fortè melius <lb></lb>
dicemus, ſi dixerimus, ideò eſſe rectum, quia eſt diuiſibilis in duas partes, <lb></lb>
quę ſimul ſumptę, ſunt æquales dimidio duorum rectorum, ſiue vni recto. <lb></lb>
</s>
          <s id="s.005338">Linea verò per quam diuiditur, non eſt medium, ſed medij manifeſtiua. <lb></lb>
</s>
          <s id="s.005339">In ſequenti appendice ad finem Operis plures alias videbis in ſolo primo <lb></lb>
elem. </s>
          <s id="s.005340">à cauſa materiali.</s>
        </p>
        <p type="main">
          <s id="s.005341"><expan abbr="Neq;">Neque</expan> verò neceſſe eſſe exiſtimo <expan abbr="demonſtrationẽ">demonſtrationem</expan> quampiam ex Arithme­<lb></lb>
tica examinare, cùm <expan abbr="cõſtet">conſtet</expan> eam eodem prorſus modo cum Geometria de­<lb></lb>
monſtrare, vt planè in 78. &amp; 9. elem. </s>
          <s id="s.005342">videre licet: imò <expan abbr="quęcunq;">quęcunque</expan> hactenus <lb></lb>
de altera ſunt dicta, de vtraque intelligenda eſſe volumus, nam vt eſt apud <lb></lb>
Eutocium in comm. Apollonij: <foreign lang="grc">ταυτα γαρ μαθηματα, δοκουντι ειμεν αδελφα</foreign>.</s>
        </p>
        <p type="head">
          <s id="s.005343"><emph type="italics"></emph>Contra prædicta generatim obijciuntur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005344">Primò, cauſæ iſtæ Geometricę non videntur veræ cauſę, <expan abbr="non.n.">nonnisi</expan></s>
          <s id="s.005345">ſatis <lb></lb>
videntur ab effectibus ſuis diſtingui: nam in cauſa formali partes <lb></lb>
definitionis ſunt <expan abbr="idẽ">idem</expan> cum definito; &amp; in cauſa materiali partes ſunt <lb></lb>
idem cum toto, ergo non ſunt verę cauſæ, &amp; proinde <expan abbr="neq;">neque</expan> veræ de­<lb></lb>
monſtrationes. </s>
          <s id="s.005346">Reſpondeo primò, ſupra dictum eſt, quando partes ſeor­<lb></lb>ſim, &amp; non vt totum componentes ſumuntur, diſtingui à toto, quod partes <lb></lb>
vnitas ſignificat, &amp; præterea formam compoſiti; quæ diſtinctio non eſt ſo­<lb></lb>
lius rationis.</s>
        </p>
        <p type="main">
          <s id="s.005347">Secundò, licet non appareat tanta diſtinctio hic, quanta in Mathemati­<lb></lb>
cis medijs, &amp; Phyſica, eſt tamen tanta, quę ſufficiat ad perfectiſſimam de­<lb></lb>
monſtrationem, quod patet authoritate Ariſt. Platonis, Procli, &amp; omnium <lb></lb>Gręcorum, Arabum, &amp; Latinorum (præter duos, vel tres recentiores) qui <lb></lb>
<expan abbr="oẽs">omnes</expan> haſce demonſtrationes perſectiſſimas, eſſe conſentiunt, vt ſup. </s>
          <s id="s.005348">diximus.</s>
        </p>
        <p type="main">
          <s id="s.005349">Tertiò, <expan abbr="quantacunq;">quantacunque</expan> ſit hæc diſtinctio, certum eſt eam non eſſe ſolius ra­<lb></lb>
tionis, quod clarum eſt, primo apud eos, qui putant relationem diſtingui à <lb></lb>
fundamento, vt aiunt, realiter, modaliter, vel formaliter. </s>
          <s id="s.005350">Secundò, <expan abbr="cõſtat">conſtat</expan> <lb></lb>
etiam apud reliquos omnes, præſertim apud recentiores, qui variæ eam de­<lb></lb>
nominant, alij. </s>
          <s id="s.005351">n. </s>
          <s id="s.005352">eam formalem, alij realem, alij modalem, alij ex natura <pb pagenum="19" xlink:href="009/01/303.jpg"></pb>rei, alij realem modalem, &amp; alij alijs formalitatibus eam appellant; qui­<lb></lb>
bus ſingulis aliquam realitatem illi ineſſe ſignificant, quæ ſufficit ad perſe­<lb></lb>
ctam demonſtrationem. </s>
          <s id="s.005353">ſatis enim eſt ad perfectam demonſtrationem, vt <lb></lb>
per eam cauſa propria, &amp; adæquata effectus, iuxta rei naturam, detegatur, <lb></lb>
ſic enim intellectui noſtro fit ſatis, vt acquieſcat, &amp; verum intueatur, quod <lb></lb>
eſt finis perfectæ demonſtrationis. </s>
          <s id="s.005354"><expan abbr="neq;">neque</expan> verò maior, aut minor diſtinctio fa­<lb></lb>
cit, vt cauſa ſit magis, aut minus vera, ſed vera illa eſt cauſa, quæ verè cau­<lb></lb>
ſat effectum à ſe non ratione tantum diſtinctum: &amp; proinde vera illa demon­<lb></lb>
ſtratio eſt, quæ pér eam demonſtrat. </s>
          <s id="s.005355">quartò hæc, quamuis parua diſtinctio, <lb></lb>multum tamen ex alia parte conducit ad demonſtrationis perfectionem, ex <lb></lb>
ea enim fit, vt in demonſtratione liquidò appareat, cauſam illam eſſe ve­<lb></lb>
ram, &amp; propriam affectionis demonſtratæ, ita vt non poſſit à propinquiori <lb></lb>
procedere; quod in nulla alia ſcientia tam euidenter apparet.</s>
        </p>
        <p type="main">
          <s id="s.005356">Obiectio 2. Geometra oſtendit eandem concluſionem per plures demon­<lb></lb>
ſtrationes, ergò per diuerſa media, atqui vnius effectus eſt vna tantum cau­<lb></lb>
ſa propria, &amp; adæquata.</s>
        </p>
        <p type="main">
          <s id="s.005357">Reſpondeo primò, eandem rem oſtendi quidem per plures demonſtratio­<lb></lb>
nes, quarum vna eſt à priori, altera verò à poſteriori. </s>
          <s id="s.005358">ſecundò, ſi omnes ſint <lb></lb>
à priori, tunc eſſentialiter eſſe vnam tantum, plures verò accidentaliter, <lb></lb>
quia in omnibus erit idem medium præcipuum, ſed conſtructio, qua illud <lb></lb>
detegitur diuerſa, vt patet in 32. primi, quam aliter Pythagorici, aliter Eu­<lb></lb>
clides, aliter Proclus demonſtrarunt, ſed tamen in omnibus eſt idem Me­<lb></lb>
dium, cauſa ſcilicet materialis, quamuis diuerſa ſit conſtructio.</s>
        </p>
        <p type="main">
          <s id="s.005359">Obiectio 3. Demonſtrationes Geometricæ non conſtant ex proprijs, &amp; <lb></lb>
per ſe, non enim Geometra conſiderat eſſentiam Quantitatis, neque eius <lb></lb>
paſsiones, quatenus ab illius eſſentia manant, quare ex communibus qui­<lb></lb>
buſdam, &amp; merè extrinſecis neceſſe eſt procedere. </s>
          <s id="s.005360">Reſpondeo ex dictis cap. <lb></lb>
1. de materia intelligibili, &amp; definitionibus Geometricis huic obiectioni <lb></lb>
abundè fieri ſatis. </s>
          <s id="s.005361">materia enim Geometriæ non eſt quantitas ſecundum ſe, <lb></lb>ſed quatenus terminata, cuius totam eſſentiam ex definitionibus eſſentiali­<lb></lb>
bus Geometra cognoſcit: quorum <expan abbr="vtrumq;">vtrumque</expan> aduerſarios latuit.</s>
        </p>
        <p type="main">
          <s id="s.005362">Præterea falſum eſt, Geometram ex communibus pluribus ſcientijs pro­<lb></lb>
cedere, quod vetat Ariſt. 1. Poſter. procedit enim ex principijs communi­<lb></lb>
bus quantitatibus terminatis, ideſt figuris, &amp; numeris; quod non ſolum li­<lb></lb>
cet, ſed etiam debet fieri ex 1. Poſter. tex. 20. &amp; 25. <expan abbr="neq;">neque</expan> vnquam idem prin­<lb></lb>
cipium repetit, niſi vbi eſt effectus formalis ipſius, &amp; non niſi contrahendo <lb></lb>
ad illud particulare.</s>
        </p>
        <p type="head">
          <s id="s.005363"><emph type="italics"></emph>Recentiorum calumniæ aduerſus Mathematicas <lb></lb>
diluuntur. </s>
          <s id="s.005364">Cap. 3.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005365">Prima eſt, qua Alexander Piccolomineus, &amp; eius ſectatores malè con­<lb></lb>
tra Mathematicos Proclum adducunt, quod, vt manifeſtè videas, <lb></lb>
hic tibi deſcribam integram Procli ſententiam, quam ille mutila­<lb></lb>
tam citat, &amp; quidem græcè, vt melius lateat. </s>
          <s id="s.005366">libro igitur 3. in Eu­


<pb pagenum="20" xlink:href="009/01/304.jpg"></pb>clid. ſic habet: At cauſam, &amp; ipſum Propter quid Geometriam minimè <lb></lb>
contemplari pluribus viſum eſt, huiuſce enim ſententiæ eſt Amphinomus <lb></lb>
Ariſt. duce. </s>
          <s id="s.005367">hæc ſunt, quæ Piccolomineus græcè citat, quibus deinde ſubdit, <lb></lb>
quid amplius quærimus pro hac ſententia? </s>
          <s id="s.005368">ſed non ne aduertis Lector, præ­<lb></lb>
dictum ſententiam non eſſe Procli, ſed Amphinomi cuiuſdam nullius nomi­<lb></lb>nis philoſophi, qui ſub falsò Ariſt. patrocinio Geometricæ cauſas auferre <lb></lb>
conabatur, quæ tamen ac ſi Procli ipſius eſſet à Piccolomineo in medium, <lb></lb>
affertur. </s>
          <s id="s.005369">ſed videamus, quæ Piccolomineus prætermiſit, pergit poſtea <lb></lb>
Procius, inueniet autem aliquis (inquit Geminus) huius etiam inquiſitio­<lb></lb>
nem in Geometria, quo modo Geometræ non eſt quærere qua de cauſa, in <lb></lb>
circulis quidem infinita multiangula inſcribantur: in ſphæris verò multian­<lb></lb>gula ſolida æquilatera, &amp; æquiangula ex ſimilibus planis conſtructa infinita <lb></lb>
inſcribere eſt impoſsibile. </s>
          <s id="s.005370">ad quem enim ſpectaret hoc inueſtigare præter <lb></lb>
Geometram? </s>
          <s id="s.005371">Hæc eſt ſententia Gemini à Proclo allata ad Amphinomi opi­<lb></lb>
nionem confutandam. </s>
          <s id="s.005372">Pergit poſtea Proclus ex propria ſententia ſic; Quan­<lb></lb>
do igitur Geometris ſyllogiſmus per impoſsibile fuerit, ſymptoma tàntum <lb></lb>
inuenire cupiunt, <expan abbr="quãdo">quando</expan> autem per præcipuam demonſtrationem, tunc rur­<lb></lb>
ſus, ſi quidem in particulari demonſtrationes fiunt, cauſa nondum manife­<lb></lb>
ſta eſt, ſi verò in vniuerſali, in <expan abbr="omnibusq́">omnibusque</expan>; ſimilibus <expan abbr="cõtinuò">continuò</expan> &amp; ipſum Propter <lb></lb>
quid manifeſtum eſt. </s>
          <s id="s.005373">hæc eſt tota illius loci integra ſeries, quàm aduerſarius <lb></lb>
mutilatam obtrudebat. </s>
          <s id="s.005374">vbi apertè vides, Procli de Geometria ſententiam.</s>
        </p>
        <p type="main">
          <s id="s.005375">Secunda calumnia eſt, qua contra Mathematicos Ariſt. quamuis inuitum <lb></lb>
interpretantur. </s>
          <s id="s.005376"><expan abbr="neq;">neque</expan> ſolum loca ipſius clariſsima ex analyticis, phyſicis, &amp; <lb></lb>
metaphyſicis, quæ ſuperius recitauimus in alienum ſenſum detorquent; ſed <lb></lb>
præter ea vnicum tantum locum, qui expreſsè Mathematicis refragari vi­<lb></lb>
deatur, afferunt, <expan abbr="cumq́">cumque</expan>; ſibi falsò fauere <expan abbr="cõfingunt">confingunt</expan>. </s>
          <s id="s.005377">is eſt in angulo quodam <lb></lb>
operum ipſius, in ſecundò videlicet Moral. Eudem. cap. 7. ſic, in immobi­<lb></lb>
libus autem, vt in Mathematicis non per ſe, ſed ſimilitudine quadam prin­<lb></lb>
cipia appellantur. </s>
          <s id="s.005378">cuius germana interpretatio non contra, ſed pro Mathe­<lb></lb>
maticis eſt: loquitur enim ibi Ariſt. de principio <expan abbr="efficiẽte">efficiente</expan>, à quo fiunt actio­<lb></lb>
nes humanæ, &amp; per motum, vt doceret in homine eſſe principia quarundam <lb></lb>
actionum, quæ propriæ ſunt hominis, &amp; liberæ. </s>
          <s id="s.005379">talia autem principia ne­<lb></lb>
gant eſſe in immobilibus, quoniam ſunt entia neceſſaria, &amp; libertatis exper­<lb></lb>
tia; eſſe tamen ſecundum ſimilitudinem, ideſt, vt ſe habent principia libera <lb></lb>
ad actiones liberas, ita principia neceſſaria ad actiones neceſſarias: vnde <lb></lb>
ſequitur, quod <expan abbr="quemadmodũ">quemadmodum</expan> principia libera ſunt adæquata cauſa illarum <lb></lb>
actionum; ſic etiam principia Mathematica erunt cauſæ adæquatæ <expan abbr="paſsio-nũ">paſsio­<lb></lb>
num</expan> Mathematicarum; quod ipſe Ariſt. teſtatur paulo poſt, his verbis: nam <lb></lb>
ſi habente trigono duos rectos, neceſſe eſt, tetragonum quatuor rectis con­<lb></lb>
ſtare, clarum eſt, quod trigonus duos rectos habens cauſa eius exiſtit. </s>
          <s id="s.005380">&amp; vt <lb></lb>
intelligamus eum loqui de propria cauſa, quæ ſit medium demonſtrationis, <lb></lb>
ſubdit; id autem neceſſariò euenire patet ex analyticis. </s>
          <s id="s.005381">quare Mathemati­<lb></lb>
cis maximè fauet hic locus, contra quàm aduerſarij autumabant. </s>
          <s id="s.005382">nullus igi­<lb></lb>
tur locus apud Ariſt. reliquus eſt, qui noſtræ ſententiæ apertè non faueat.</s>
        </p>
        <p type="main">
          <s id="s.005383">Tertia, qua contra Mathematicos Platonem ipſum eximium Mathema­<lb></lb>ticarum fautorem, ac ſtudioſum excitare conantur; aiunt enim ipſum 7. de <pb pagenum="21" xlink:href="009/01/305.jpg"></pb>Rep. dicere Mathematicos circa quantitatem ſomniare. </s>
          <s id="s.005384">Verùm, ne falla­<lb></lb>
cia ſubſit, ecce tibi Platonis verba è græco ſyncerè tranſlata. </s>
          <s id="s.005385">Reliquæ ve­<lb></lb>
rò, quas diximus verarum <expan abbr="rerũ">rerum</expan> quoquo modo eſſe participes, Geometriam <lb></lb>
ſcilicet, <expan abbr="eiusq́">eiusque</expan>; comites circa ipſam eſſentiam quodammodo ſomniant: ſyn­<lb></lb>
cerè autem quicquam ab illis cernere impoſſibile eſt, tantiſper dum ſuppo­<lb></lb>
ſitionibus hærent, <expan abbr="easq́">easque</expan>; ratas, &amp; immobiles adeo ſeruant, vt illarum ratio­<lb></lb>
nem reddere nequeant. </s>
          <s id="s.005386">Aduerte primò Platonem non dicere Geometram <lb></lb>
abſolutè ſomniare, ſed quodammodo ſomniare: deinde non dicere circa <lb></lb>
quantitatem, ſed circa eſſentiam: vnde ſi liberet funem cum eis trahere <lb></lb>
contentioſum, <expan abbr="verbisq́">verbisque</expan>; hærere, vt ipſi faciunt, nihil contra Mathematicos <lb></lb>
facerent. </s>
          <s id="s.005387">ſed eſto intelligat ipſam quantitatem, videamus quid ſibi velit. </s>
          <s id="s.005388">vt <lb></lb>
autem iſtud videamus, ſciendum eſt, totius 7. de Rep. diſcurſum eſſe in con­<lb></lb>
ſtituendo Reip. Cuſtode, &amp; Gubernatore, quem in primis vult eſſe optimè <lb></lb>
natura, tam ad agendum, quàm ad ſpeculandum idoneum: vult præterea <lb></lb>
ipſum ſapientiſſimum eſſe, ideſt Theologiam, quàm etiam Dialecticam no­<lb></lb>
minat, optimè callere, per quam <expan abbr="abſq;">abſque</expan> vllo diſcurſu <expan abbr="rerũ">rerum</expan>, ac præcipuè ſum­<lb></lb>
mi Boni eſſentias contempletur. </s>
          <s id="s.005389">ad hanc diuinam contemplationem, vt <lb></lb>
peruenire poſſit, opus ait eſſe eum humanis aliquot diſciplinis imbutum eſ­<lb></lb>
ſe. </s>
          <s id="s.005390">totam autem Philoſophiam inibi partitur in tres partes, in Dialecticam, <lb></lb>
ſeu Theologiam, quam intellectui attribuit abſque vlla ſuppoſitione, &amp; di­<lb></lb>
ſcurſu: <expan abbr="hancq́">hancque</expan>; ſolam ſcientiæ nomine dignam eſſe exiſtimat. </s>
          <s id="s.005391">ſecundò, in <lb></lb>
Mathematicam, quam in cognitione, ſeu ratiocinatione collocat, &amp; pro­<lb></lb>
pterea principia ſupponit. </s>
          <s id="s.005392">tertiò tandem in opinionem, quæ verſatur circa <lb></lb>
res naturales, quæ in imaginatione ab eo collocatur. </s>
          <s id="s.005393">Iuxta hanc trimem­<lb></lb>
brem diuiſionem ex Platonis ſententia ſic habet Marſilius Ficinus in argu­<lb></lb>
mento huius 7. de Rep. </s>
          <s id="s.005394">Diuina ſiquidem in tribus aquis repręſentari viden­<lb></lb>
tur, primò quidem licet <expan abbr="cõfuſius">confuſius</expan> in rationibus phyſicis. </s>
          <s id="s.005395">ſecundò, diſtinctius <lb></lb>
in Mathematicis. </s>
          <s id="s.005396">tertiò, clariſſimè in Metaphyſicis. </s>
          <s id="s.005397">&amp; paulo poſt: Diuinas <lb></lb>
<expan abbr="deniq;">denique</expan> formas omninò immortales, <expan abbr="resq́">resque</expan>; veras Plato exiſtimat, quarum, <lb></lb>
imagines quidem ſunt mathematicæ formæ, vmbræ verò res naturales. </s>
          <s id="s.005398">&amp; <lb></lb>
poſtea: Cùm autem animaduertat Thaletem, Democritum, Anaxagoram, <lb></lb>
tam negligentes in rebus diuinis fuiſſe, quàm <expan abbr="diligẽtes">diligentes</expan> in naturalibus; Phæ­<lb></lb>
recidem contra, &amp; Pythagoram, <expan abbr="Pythagoreosq́">Pythagoreosque</expan>; omnes Mathematicorum <lb></lb>
Principes pariter ſummos extitiſſe Theologos, <expan abbr="moribusq́">moribusque</expan>; diuinos, meritò <lb></lb>
partim ratione, partim experientia concluſit curioſum naturalium ſtudium <lb></lb>
animum à diuinis ſæpè auertere. </s>
          <s id="s.005399">&amp; poſt nonnulla: Quare ſeptimus hic liber <lb></lb>de Rep. vbi animum ad ſummum Bonum, Solemque, ideſt Deum, <expan abbr="idæasq́">idæasque</expan>; <lb></lb>
diuinas, quaſi ſtellas, conuenientibus producit gradibus, nullam in ipſo <lb></lb>
aſcenſu naturalis peritiæ facit mentionem, ſed mathematicos quoſdam gra­<lb></lb>
dus ad Diuina commodius perducentes adducit in medium, inter quos duo <lb></lb>
ſunt puri, Arithmetica, &amp; Geometria, &amp;c. </s>
          <s id="s.005400">reliquas deinde Mathematicas <lb></lb>
Plato enumerat, ac ſingillatim commendat, <expan abbr="atq;">atque</expan> in Rempub. admittit. </s>
          <s id="s.005401">poſt <lb></lb>
multa iterum Ficinus ſic, cùm ergò dicit, animam à lucerna ad Lunam, à <lb></lb>
Luna ad Solem attolli, ſignificat Plato à formis naturalibus ad Mathemati­<lb></lb>
cas, ab his ad diuinas <expan abbr="deniq;">denique</expan> eleuari. </s>
          <s id="s.005402">&amp; nonnullis intermiſſis, <expan abbr="deniq;">denique</expan> Theo­<lb></lb>
logiam, quam etiam Metaphyſicam, &amp; Dialecticam dicit, omnibus facul­


<pb pagenum="22" xlink:href="009/01/306.jpg"></pb>tatibus anteponit, vt ducentem omnes, &amp; ad ſuum officium vtentem mini­<lb></lb>
ſterio ſingularum. </s>
          <s id="s.005403">officium autem eius eſſe per totam entis progredi latitu­<lb></lb>
dinem, <expan abbr="atq;">atque</expan> ad ipſum Bonum totius entis cauſam, ſe <expan abbr="cõferre">conferre</expan>, &amp; quid <expan abbr="vnum-quodq;">vnum­<lb></lb>
quodque</expan> ſit definire, rationem <expan abbr="cuiusq́">cuiusque</expan>; eſſentiæ aſſignando, quiduè quamlibet <lb></lb>
ſequatur eſſentiam demonſtrare. </s>
          <s id="s.005404">Cæteras autem facultates præ huius no­<lb></lb>
bilitate iudicat eſſe ſeruiles: aut enim ad opiniones hominum declinant, aut <lb></lb>
ſaltem etſi ad incorporea ſe ſe pro viribus erigunt, nihilominus circa illa <lb></lb>
quodammodo ſomniant, quales eſſe inquit Mathematicas. </s>
          <s id="s.005405">ex quibus iam <lb></lb>
clarè vides Platonem dixiſſe Mathematicas quodam modo ſomniare circa <lb></lb>
eſſentiam rerum, non abſolutè, ſed comparatione ad Theologiam: intelli­<lb></lb>
gis etiam cur Mathematicas nolit ſcientias appellari, quia nimirum ſolam <lb></lb>
Theologiam hoc nomine dignam cenſebat: qua de cauſa minus Phyſicam, <lb></lb>
eodem nomine dignam putauit, cùm eam opinionem, Mathematicam verò <lb></lb>
cogitationem, vel ratiocinationem dicat. </s>
          <s id="s.005406">ob eandem comparationem aſſe­<lb></lb>
rit etiam Mathematicas minus eſſe certas, quam Theologiam, quoniam, <lb></lb>
ſcilicet hæc nihil ſupponit, <expan abbr="nihilq́">nihilque</expan>; diſcurrit, ſed intuetur: illæ verò iactis <lb></lb>
quibuſdam principijs, quæ probari nequeunt, diſcurrunt, <expan abbr="proptereaq́">proptereaque</expan>; in <lb></lb>
ipſo diſcurſu poteſt error aliquis contingere. </s>
          <s id="s.005407">eandem huius loci explica­<lb></lb>
tionem habes apud Proclum cap. 10. lib. 1. in Euclid. ſic; Verum quid ſibi <lb></lb>
velit Plato, quando in libris de Rep. à Mathematica ſcientiæ nomen abſtu­<lb></lb>
lit breuiter dicam. </s>
          <s id="s.005408">&amp; paulo poſt; hanc <expan abbr="deniq;">denique</expan> ſcientiam, quam ab artibus <lb></lb>
diſtinguimus diuidens, vnam quidem ſuppoſitionis expertem eſſe vult, alte­<lb></lb>
ram verò ex ſuppoſitione ſcaturire. </s>
          <s id="s.005409">&amp; poſt pauca: &amp; ſic ait Mathematicam <lb></lb>tanquam ſuppoſitionibus vtentem ab ea, quæ ſuppoſitionibus caret, <expan abbr="perfe-ctaq́">perfe­<lb></lb>
ctaque</expan>; eſt ſcientia deficere: vna enim verè eſt ſcientia, per quam omnia, quæ <lb></lb>
ſunt cognoſcere apti ſumus. </s>
          <s id="s.005410">perſpicuè igitur vides auctoritate horum phi­<lb></lb>
loſophorum hæc omnia Platonem non abſolutè, ſed comparatè dixiſſe.</s>
        </p>
        <p type="main">
          <s id="s.005411">Poſt hæc Ficinus iterum ex ſententia Platonis ſic pergit: quoniam verò <lb></lb>
diſſerendi facultas, ſi adoleſcentibus tradatur opinionem honeſti debilitat, <lb></lb>
vnde euadunt <expan abbr="intemperãtes">intemperantes</expan>, imò &amp; ſuperbi, &amp; impij, vt in Philebo quoque, <lb></lb>
&amp; legibus dicitur, idcircò ante trigeſimum ætatis annum in Mathematicis <lb></lb>erudiendi ſunt, <expan abbr="atq;">atque</expan> in publicis negotijs per interualla pariter exercendi. <lb></lb>
</s>
          <s id="s.005412">concludamus tandem hæc pulcherrimo eiuſdem 7. de Rep. loco, quem etiam <lb></lb>
Proclus lib. 1. cap. 8. ad verbum ferè ſic recitat: Ideo &amp; in Repub. Socrates <lb></lb>
rectè dixit, oculus, nimirum animæ, qui ab alijs ſtudijs excœcatur, <expan abbr="defodi-turq́">defodi­<lb></lb>
turque</expan>; à Mathematicis tantum diſciplinis recreari, <expan abbr="excitariq́">excitarique</expan>; rurſus ad eius, <lb></lb>
qui eſt, contemplationem, &amp; à ſimulacris ad ea, quæ vera ſunt: nam pulchri­<lb></lb>
tudo, &amp; ordo Mathematicarum rationum, firmitudoque, ac ſtabilitas con­<lb></lb>
templationis, nos ipſis coniungit intellectibus, <expan abbr="perfectèq́">perfectèque</expan>; in ipſis obfirmat, <lb></lb>
perpetuò quidem manentibus, &amp; diuina pulchritudine collucentibus, ac mu­<lb></lb>
tuum ordinem ſeruantibus. </s>
          <s id="s.005413">Animaduertiſti Lector ex his paucis, quot lau­<lb></lb>
dibus in hoc 7. de Repub. Mathematicæ à Platone cumulentur, vt totus ferè <lb></lb>
liber quoddam ipſarum encomium videatur: vnde mirum ſit, iſtos ex eo­<lb></lb>
dem ſeptimum locum illum contra Mathematicas, inter tot ipſarum præ­<lb></lb>
conia ſedulò emendicaſſe, ac perperam interpretatos eſſe: ſicque Araneos <lb></lb>imitatos eſſe, qui ex mellifluis floribus, ex quibus Apes mella, venenum col­<pb pagenum="23" xlink:href="009/01/307.jpg"></pb>ligunt. </s>
          <s id="s.005414">Verumenimuerò quis vnquam de Platonis mente erga Mathemati­<lb></lb>
cas dubitare poterit, cùm ipſe omnes ageometretos è ſuo gymnaſio reijce­<lb></lb>
ret; cùm quotidie, vt ex Philopono refert ipſe Piccolomineus auditoribus <lb></lb>
ſuis aliquod Problema Mathematicum proponeret. </s>
          <s id="s.005415">qui de legibus 6. &amp; 7. <lb></lb>
de ſingulis Mathematicis addiſcendis leges ſancit, vbi <expan abbr="Geometriã">Geometriam</expan> adeò ex­<lb></lb>
tollit, vt aſſerat, aſymetriam quantitatum ignorare, non <expan abbr="hominũ">hominum</expan>, ſed por­<lb></lb>
corum, ac pecorum ignorantiam eſſe. </s>
          <s id="s.005416">In Epinomide tandem quàm digna, <lb></lb>
<expan abbr="quamq́">quamque</expan>; præclarè de Aſtronomia, <expan abbr="deq́">deque</expan>; Geometria, &amp; Arithmetica, quæ ad <lb></lb>
eam conferunt, prædicat. </s>
          <s id="s.005417">præcipua autem Aſtronomiæ laus ibi tradita eſt, <lb></lb>
quod ea, inter omnes ſcientias, animum ad cœleſtia, <expan abbr="atq;">atque</expan> diuina attollit, <expan abbr="in-deq́">in­<lb></lb>
deque</expan>; ad ſummi Boni cognitionem, <expan abbr="atq;">atque</expan> amorem allicit, quam veram eſſe <expan abbr="ſa-piẽtiam">ſa­<lb></lb>
pientiam</expan> diuinitus inibi Plato pluribus fatetur. </s>
          <s id="s.005418">plura alia Platonis loca bre­<lb></lb>
uitatis cauſa, prætereo; quis enim, vel leuiter eum attingit, quì non per­<lb></lb>
ſpicuè videat, eum ſuper omnes Philoſophos eſſe Mathematicarum com­<lb></lb>
mendatorem eximium.</s>
        </p>
        <p type="main">
          <s id="s.005419">Quarta eſt Mathematicas, Geometriam preſertim conſiſtere in imagi­<lb></lb>
natione potius, quàm in diſcurſu, &amp; proinde ſcientias eſſe puerilis ingenij, <lb></lb>
cùm pueri valeant imaginatione. </s>
          <s id="s.005420">accedit authoritas Ariſt. qui 6. Eth. cap. 8, <lb></lb>
ait, quid eſt, quòd puer fieri Mathem. poteſt, ſapiens, aut naturalis non po­<lb></lb>
teſt? </s>
          <s id="s.005421">præterea, quia antiquitus pueris ante alias tradebantur. </s>
          <s id="s.005422">Reſpondeo, <lb></lb>
quòd, vt in præcedenti reſponſione dictum eſt ex Plat. ſententia, non ima­<lb></lb>ginatio, ſed ratiocinatio, ſeu cogitatio verſatur circa Mathematicas, ima­<lb></lb>
ginatio autem circa naturalem Philoſophiam; ſed audi Platonem in eodem <lb></lb>
ſeptimo. </s>
          <s id="s.005423">Placet igitur, ait, primam partem vocare ſcientiam, ſecundam <lb></lb>
cogitationem, tertiam fidem, poſtremam imaginationem. </s>
          <s id="s.005424">Conſtat autem <lb></lb>
vltimas duas ab ipſo collocari in naturali peritia. </s>
          <s id="s.005425">adeſt etiam Procli autho<lb></lb>
ritas, qui cap. 5. lib. 1. ſic habet; Inſtrumentum <expan abbr="itaq;">itaque</expan> aptum ad iudicandum <lb></lb>
cunctas res Mathematicas cogitationem ex Platonis ſententia ſtatuimus, <lb></lb>
quippequæ opinione quidem ſuperior eſt, ab intelligentia verò ſuperatur. <lb></lb>
</s>
          <s id="s.005426">Per cogitationem verò intelligendum eſſe quendam mentis motum, ideſt <lb></lb>
diſcurſum, tum ex vi græcè vocis <foreign lang="grc">Nοηματος,</foreign> tum ex vi Latinæ vocis manife­<lb></lb>
ſtum eſt; cogitatio enim dicitur quaſi coagitatio. </s>
          <s id="s.005427">ſ <expan abbr="mẽtis">mentis</expan>, quæ idem eſt cùm <lb></lb>
diſcurſu, aut ratiocinatione: quare manifeſtum eſt horum Philoſophorum <lb></lb>
authoritate ratiocinationem verſari circa Mathematicas, imaginationem <lb></lb>
verò circa res phyſicas, contra quam ipſi contendebat. </s>
          <s id="s.005428">Verum quid opus <lb></lb>
eſt authoritate, vbi res ipſa videri poteſt, conſideret quilibet Geometricas <lb></lb>
demonſtrationes, clarè videbit opus quidem eſſe non mediocri imagina­<lb></lb>
tione, ſed multo maiori diſcurſu, ſunt enim in nonnullis, 50. &amp; 60. conſe­<lb></lb>
quentiæ, vna poſt alteram inuicem connexæ. </s>
          <s id="s.005429">Sed quid dico in nonnullis cùm <lb></lb>
totum Euclidis opus ſit perpetua quædam illationum catena mirabilis, ita <lb></lb>vt vltimæ ipſius demonſtrationes contineant, ſi reſoluantur conſecutionum <lb></lb>
miriadas; at verò omnis illationis expers prorſus eſt imaginatio. </s>
          <s id="s.005430">Si verò <lb></lb>
ipſarum inuentionem conſideremus, admirabiles omninò videbuntur, tum <lb></lb>
quia res omnino abſtruſas, &amp; abditas <expan abbr="demõſtrant">demonſtrant</expan>, tum quia media, quibus <lb></lb>
eas <expan abbr="cõprobant">comprobant</expan>, diuino ingenij acumine indigent, vt inueſtigentur; vt prop­<lb></lb>
terea earum authores nomina ſua immortalitati conſecrarint; ſic Thale­


<pb pagenum="24" xlink:href="009/01/308.jpg"></pb>tis Mileſij 5. primi; Oenipodis 11. Pythagoreorum 32. Pythagoræ ipſius <lb></lb>
47. <expan abbr="inuentorũ">inuentorum</expan> adhuc nomina celebrantur: Hippocrati quadraſſe lunulam, <lb></lb>
Archimedi Parabolam, quantam gloriam peperit? </s>
          <s id="s.005431">Apollonij Pergęi Coni­<lb></lb>
ca magni Geometræ nomen ei compararunt. </s>
          <s id="s.005432">hæc &amp; plura alia non ſolum <lb></lb>
puerilis ingenij acumen, verum etiam virilis captum magnis ſpatijs ſupe­<lb></lb>
rare videntur. </s>
          <s id="s.005433">Cur autem Ariſt. dixerit puerum fieri poſſe Mathematicum <lb></lb>non autem ſapientem, aut naturalem, ipſe declarat, quia <expan abbr="nimirũ">nimirum</expan> in <expan abbr="vtraq;">vtraque</expan> <lb></lb>
eorum opus eſt experientia, quæ in puero non eſt, experientiam enim affert <lb></lb>
temporis longitudo: facilius tamen puer moralia intelligit, quam Geome­<lb></lb>
trica, facilius enim eſt intelligere quid virtus, quid vitium, &amp;c. </s>
          <s id="s.005434">quam quin­<lb></lb>
tam, aut ſeptimam primi; non ideo tamen puer erit prudens, quia pruden­<lb></lb>
tia non ſpeculatiua, ſed practica eſt, <expan abbr="Itaq;">Itaque</expan> quod puer ſapiens, aut natura­<lb></lb>
lis eſſe nequeat, defectus non eſt ex parte intellectus, ſed ex parte experien­<lb></lb>
tię. </s>
          <s id="s.005435">Neque præterea dicendæ pueriles ſunt, quòd antiquitus, vt vult etiam <lb></lb>
Plato, pueris primò <expan abbr="traderẽtur">traderentur</expan>, quandoquidem in illis totius ætatis robur, <lb></lb>&amp; florem inſumebant, cùm ad 30. vſq; ætatis annum in eis, occuparentur. <lb></lb>
</s>
          <s id="s.005436">pueriles meritò dicerentur, ſi in pueritia tantùm eis operam dediſſent. </s>
          <s id="s.005437">Di­<lb></lb>
cam ſyncerè, quod ipſe, dum eas per plures annos docerem expertus ſum; <lb></lb>
<expan abbr="quoſcunq;">quoſcunque</expan> reperi ingenio in Mathematicis pollere, hi pariter in alijs omni­<lb></lb>
bus excellebant. </s>
          <s id="s.005438">Requirit enim ſtudium iſtud omnes ingenij partes, imagi­<lb></lb>
nationem, diſcurſum, &amp; memoriam. </s>
          <s id="s.005439">Idcircò veteres puerorum ingenium <lb></lb>
ad Mathematicas quaſi ad Lydium lapidem experiebantur; <expan abbr="ijsq́">ijsque</expan> inepti à <lb></lb>
reliquis ſtudijs arcebantur. </s>
          <s id="s.005440">audi Platonem 7. de Repub. </s>
          <s id="s.005441">An &amp; hoc aduer­<lb></lb>
tiſti, quod homines natura Arithmetici ad omnes doctrinas, vt ita dixerim <lb></lb>acuti videantur.) &amp; poſtea concludens ait, propter omnes, quas adduxi­<lb></lb>
mus rationes, haud quaquam negligenda hæc ſunt, ſed in his præcipuè eru­<lb></lb>
diendi, qui optimis ſunt ingenijs.</s>
        </p>
        <p type="main">
          <s id="s.005442">Quinta, Geometria carpitur, quòd plures habeat demonſtrationes per <lb></lb>
ſuperpoſitionem factas, qui modus demonſtrandi videtur aduerſarijs valdè <lb></lb>
imperfectus, ac penè ridiculus. </s>
          <s id="s.005443">Sed <expan abbr="ſciẽdum">ſciendum</expan> primò in toto Euclide eſſe <expan abbr="tan-tũmodo">tan­<lb></lb>
tummodo</expan> tres per ſuperpoſitionem. </s>
          <s id="s.005444">Secundò, eas eſſe tam perfectas, ac eui­<lb></lb>dentes, quàm reliquæ; falluntur, qui putant illam ſuperpoſitionem eſſe de­<lb></lb>
monſtrationis medium, eſt enim loco conſtructionis: neque, quæ proban­<lb></lb>
da ſunt æqualia, ea ſuperponuntur, vt ipſi putant, hæc enim ratio nullius eſ­<lb></lb>
ſet momenti, nec Geometrica, ſed Phyſica potius, niteretur enim ſenſibus: <lb></lb>
ſed ſuperponuntur quædam, quæ æqualia ſunt, vt ex eorum ſuperpoſitione <lb></lb>
appareat æqualitas eorum, quæ non ſuperponuntur. </s>
          <s id="s.005445">Conſidera quartam <lb></lb>
primi videbis ibi ſuperponi quædam latera æqualia duorum triangulorum, <lb></lb>
vt deinde baſes, quæ non ſuperponuntur, inferantur eſſe æquales: &amp; ratio, <lb></lb>
qua probantur æquales eſt, quia congruunt, non quia ſuperponuntur, vt ipſi <lb></lb>
putant, nec intelligunt, quodnam ſit illius demonſtrationis medium.</s>
        </p>
        <p type="main">
          <s id="s.005446">Sexta calumnia ridicula, eſt cuiuſdam, qui Geometras reprehendit, quòd <lb></lb>
ſæpè vtantur circulo, vt patet, inquit, in prima, 6. 4. &amp; 8. primi elem. </s>
          <s id="s.005447">ſi. </s>
          <s id="s.005448">n. </s>
          <s id="s.005449">lo­<lb></lb>
quatur de circulo, qui figura eſt, in ſola <expan abbr="carũ">carum</expan> prima is adhibetur, vt patet, vel <lb></lb>
figuras ipſas more puerorum ſpectanti: ſi loquatur de circulo, quòd vitium <lb></lb>
eſt in <expan abbr="demõſtrando">demonſtrando</expan>, id multo magis falſum eſt, cùm in nulla <expan abbr="earũ">earum</expan> reperiatur.</s>
        </p>
        <pb pagenum="25" xlink:href="009/01/309.jpg"></pb>
        <p type="main">
          <s id="s.005450">Septima eſt, qua dicunt Geometras non habere materiam veram, &amp; pro­<lb></lb>
priam, ea enim Phyſica eſt, &amp; proinde <expan abbr="neq;">neque</expan> cauſam materialem. </s>
          <s id="s.005451">Sed dicé­n<lb></lb>
dum eſt Geometras <expan abbr="quidẽ">quidem</expan> carere propria materia phyſica, non carere ta­<lb></lb>
men propria materia Mathematica, quæ eſt illa intelligibilis, de qua cap. 1. <lb></lb>
dictum eſt.</s>
        </p>
        <p type="main">
          <s id="s.005452">Octaua, eſt cuiuſdam dicentis, opinionem <expan abbr="communẽ">communem</expan> eſſe, Mathematicam <lb></lb>non eſſe propriè ſcientiam; ſed hoc manifeſtè falſum eſt, cum inter tot Phi­<lb></lb>loſophos Græcos, Arabes, Latinos, ſolùm ipſe duos, vel tres huius ſenten­<lb></lb>
tiæ in medium poſſit afferre, Piccolom. .ſ. </s>
          <s id="s.005453">quem ſequitur Pererius.</s>
        </p>
        <p type="main">
          <s id="s.005454">Nona, eſt alterius, qui Geometras damnat, quòd plura reuocant ad illud <lb></lb>
vniuerſale axioma, quæ ſunt eadem vni tertio, ſunt eadem inter ſe. </s>
          <s id="s.005455">Verùm <lb></lb>
iſte malè Geometrarum principia nouit, axioma enim illud, nuſquam apud <lb></lb>
Mathematicos reperitur, <expan abbr="neq;">neque</expan> reperiri <expan abbr="põt">pot.</expan>, cum <expan abbr="quantitatẽ">quantitatem</expan> non inuoluat.</s>
        </p>
        <p type="main">
          <s id="s.005456">Decima, qua dicunt entia Mathematica non extare: ſed ex initio dictis <lb></lb>
de materia intelligibili hęc nota ſatis detergitur.</s>
        </p>
        <p type="main">
          <s id="s.005457">Vndecima, abſtractionem à materia <expan abbr="multũ">multum</expan> derogare perfectioni Mathe­<lb></lb>
maticarum demonſtrationum; cui reſpondeat eruditiſſimus Toletus, qui <lb></lb>
in 2. Phyſ. quæſt. </s>
          <s id="s.005458">4. ſic ait: Phyſicus frequentet vtitur demonſtratione effe­<lb></lb>
ctus, &amp; ſigni, quia ipſius cauſæ frequentius ſunt occultæ nec per ſe ſenſibiles, <lb></lb>
at Mathematicus frequentius à prioribus procedit, cùm eius cauſæ notio­<lb></lb>
res ſint effectibus, à ſenſu .n. </s>
          <s id="s.005459">abſtrahit, &amp; in intellectu notius eſt, quòd prius <lb></lb>
eſt. </s>
          <s id="s.005460">Poſtea in 4. concluſ. </s>
          <s id="s.005461">ſic ait: omnem phyſicæ imperfectionem à materia <lb></lb>
pendere, vnde Ariſt. 2. Metaphy. tex. 16. tradens huius non exactæ certitu­<lb></lb>
dinis rationem ait: natura materiam habet: &amp; poſt pauca: At res Mathe­<lb></lb>
maticæ, cùm ab hac materia ſeparent ſimpliciter neceſſariæ ſunt, ſemper <lb></lb>
enim omnis triangulus habet tres angulos æquales duobus rectis. </s>
          <s id="s.005462">ex quibus <lb></lb>
apparet omnem Mathematicarum perfectionem oriri ex abſtractione, con­<lb></lb>
trà quàm putabat aduerſarius.</s>
        </p>
        <p type="main">
          <s id="s.005463">Duodecima eſt, Mathematicas abſtrahere à Bono: verùm eas ab ea libe­<lb></lb>
rat Ariſt. dum lib. 13. Metaphyſ. ait: qui dicunt Mathematicas ſcientias ni­<lb></lb>
hil de bono, vel pulchro dicere, falſum dicunt: dicunt enim, &amp; maximè <expan abbr="oſtẽ-dunt">oſten­<lb></lb>
dunt</expan>, nam etiamſi non nominant, quia tamen opera, &amp; rationes oſtendunt, <lb></lb>
non ne dicunt de eis? </s>
          <s id="s.005464">pulchri <expan abbr="namq́">namque</expan>; maximè ſpecies ſunt, ordo, commen­<lb></lb>
ſuratio, &amp; definitum, quæ maximè à Mathematicis ſcientijs oſtenduntur.</s>
        </p>
        <p type="main">
          <s id="s.005465">Decimatertia eſt, <expan abbr="Geometriã">Geometriam</expan>, &amp; <expan abbr="Arithmeticã">Arithmeticam</expan>, vt ſunt ſpeculatiuæ ſcien­<lb></lb>
tię eſſe inutiles, <expan abbr="atq;">atque</expan> iniucundas. </s>
          <s id="s.005466">Sed hæc oppoſitio in omnes quadrat ſpe­<lb></lb>
culatiuas, quæ non vtiles, ſed gratia ſui ſunt. </s>
          <s id="s.005467">quod maximè ij ſolent oppo­<lb></lb>
nere, qui ſcientias, vt ille cecinit cauponantur, ſeu qui eas quæſtuoſas fa­<lb></lb>
ciunt. </s>
          <s id="s.005468">Verùm hos animo mercatores potius, quàm Philoſophos amande­<lb></lb>
mus ad cap. 8. 9. &amp; 10. libri primi Procli, vbi fusè de vtilitate <expan abbr="earũ">earum</expan> omnium <lb></lb>
diſſerit. </s>
          <s id="s.005469">Quod ſi Philoſophus ſit, qui hæc opponat; huic illa ſufficiat vtili­<lb></lb>
tas, qua loca omnia Ariſt. Mathematica, quę ferè <expan abbr="quadringẽta">quadringenta</expan> ſunt, facilè <lb></lb>
Mathematicarum auxilio intelliguntur, <expan abbr="ſicq́">ſicque</expan>; ad plenam totius Ariſt. intel­<lb></lb>
ligentiam, tandem perueniri poteſt.</s>
        </p>
        <p type="main">
          <s id="s.005470">Quod attinet ad delectationem inuenient, inibi apud Proclum hæc, cui <lb></lb>Mathematicarum diſciplinarum cognitionem ſpernunt, voluptates, quæ in <pb pagenum="26" xlink:href="009/01/310.jpg"></pb>ipſis ſunt, minimè deguſtarunt. </s>
          <s id="s.005471">&amp; quì Mathematicas ignorant iecunas ca­<lb></lb>
piunt voluptates. </s>
          <s id="s.005472">Ex quibus fit, vt viri nobiles, ac Principes, quì non lu­<lb></lb>
crandi, ſed philoſophandi cauſa literis dant operam, Mathematicis maxi­<lb></lb>
mè ſtudijs delectentur. </s>
          <s id="s.005473">inter quos celeberrimi extiterunt, ex antiquis qui­<lb></lb>
dem Archimedes Regum Siciliæ conſanguineus; Boetius vir conſularis; <lb></lb>
Alphonſus Rex <expan abbr="Hiſpaniarũ">Hiſpaniarum</expan>: noſtra verò ætate Marchio Guidobaldus, Prin­<lb></lb>
ceps Ticho Brahe; Franciſcus Candalla, &amp; alij complures, quorum monu­<lb></lb>
menta in omne æuum permanſura mundus admirabitur.</s>
        </p>
        <p type="main">
          <s id="s.005474">Decimaquarta, ſubiecti ignobilitatem Mathematicis exprobrant, quod <lb></lb>
videlicet ſit accidens. </s>
          <s id="s.005475">ſed reſpondetur primò, quod quamuis ſit accidens, <lb></lb>
eſt tamen immateriale, &amp; abſtractum, qua ratione inter ſubiectum Phyſicæ, <lb></lb>
&amp; Mathematicæ collocatur. </s>
          <s id="s.005476">ſecundò, melius eſſe de aliquo accidente veri­<lb></lb>tates innumeras cognoſcere, <expan abbr="easq́">easque</expan>; admirabiles, quàm circa <expan abbr="ſubſtantiã">ſubſtantiam</expan> ma­<lb></lb>
terialem præſertim, mille opinionum turbis, ac diſſenſionibus perpetuò huc <lb></lb>
illuc agitari, <expan abbr="neq;">neque</expan> vnquam ad vllius ſubſtantiæ cognitionem peruenire. </s>
          <s id="s.005477">ter­<lb></lb>
tiò, in Mathematicis medijs aliter ſe habere, vbi non nudam quantitatem, <lb></lb>
ſed vel cœleſtia corpora omnium nobiliſſima, vel ſonos muſicos, vel viſionis <lb></lb>
modos, ac deceptiones, vel cauſas virium machinarum eodem fine, ac ſco­<lb></lb>
po, quo cæteri Philoſophi cætera contemplantur.</s>
        </p>
        <p type="main">
          <s id="s.005478">Decimaquinta eſt, quod ſiue ioco, ſiue ex eruditionis ignoratione addunt <lb></lb>
Mathematicos legibus tum prophanis, tum ſacris ſæpius proſcriptos, ac <lb></lb>
damnatos fuiſſe; <expan abbr="atq;">atque</expan> olim non rarò Imperatorum editis Romano imperio <lb></lb>
pulſos. </s>
          <s id="s.005479">Verùm iſtis nequaquam opus eſſet reſpondere, cùm vix nullus adeò <lb></lb>
eruditionis expers reperiatur, quì ignoret illos pſeudomathematicos fuiſſe <lb></lb>
eos, quì, &amp; quidem aptius, &amp; Genethliaci, &amp; Chaldæi, &amp; Iudiciarij dice­<lb></lb>
bantur. </s>
          <s id="s.005480">quorum doctrina nullo mihi pacto probari poteſt, cùm nullis nec <lb></lb>
experientijs, nec rationibus fulciatur, ſed mera vanitas, <expan abbr="atq;">atque</expan> impoſtura, ſæ­<lb></lb>
pè etiam ſuperſtitio ſit. </s>
          <s id="s.005481">vt propterea mirandum ſit, cur non pęnitus huiuſ­<lb></lb>
modi artes de medio tollantur, ſed quod ait Cor. Tacitus lib. 1. hiſtor. </s>
          <s id="s.005482">hoc <lb></lb>
genus hominum potentibus infidum, ſperantibus fallax, in Ciuitate no­<lb></lb>
ſtra, &amp; vetabitur ſemper, &amp; retinebitur. </s>
          <s id="s.005483">lege libros 12. Pici Mirandulani <lb></lb>
contra Aſtrologos. </s>
          <s id="s.005484">accedunt præterea Tycho, &amp; Keplerus, quì quamuis <lb></lb>
Aſtronomi, Aſtronomiam tamen iſtam pluribus improbarunt. </s>
          <s id="s.005485">Calumniosè <lb></lb>
tamen ij faciunt, quì illorum nebulonum culpam in omnes Mathematicos <lb></lb>
transferre geſtiunt. </s>
          <s id="s.005486"><expan abbr="Atq;">Atque</expan> hæc tantummodo dicta velim, ne ſimpliciores ab <lb></lb>
iſtis calumniatoribus decipiantur.</s>
        </p>
        <p type="main">
          <s id="s.005487">Decimaſexta, qua in vniuerſum proponunt hoc modo, vtrum Mathema­<lb></lb>
ticæ h. </s>
          <s id="s.005488">beant perfectas demonſtrationes, poſtea in diſcurſu multa contra <lb></lb>eas adducunt, quæ tandem in fine tractationis contra ſolas Geometriam, &amp; <lb></lb>
Arithmeticam valere fatentur. </s>
          <s id="s.005489">Quare niſi lector ad finem <expan abbr="vſq;">vſque</expan> omnia per­<lb></lb>
legerit, quod rarò accidit, decipitur, putat enim in omnes Mathematicas <lb></lb>
illa quadrare, cùm tamen ipſi fateantur, ſe nunquam loquutos eſſe de me­<lb></lb>
dijs Aſtronomia, Muſica, Optica, Mechanica, quibus ineſſe veram ſcientiæ <lb></lb>
demonſtratiuæ rationem libenter concedunt.</s>
        </p>
        <p type="main">
          <s id="s.005490">Decimaſeptima, eſt cuiuſdam recentiſſimi Philoſophi, quì vbi pluribus <lb></lb>
contra Mathematicas diſſeruit, nihil, vt fieri ſolet, ſibi obijcit, <expan abbr="verũ">verum</expan> com­


<pb pagenum="27" xlink:href="009/01/311.jpg"></pb>plura Ariſt. &amp; veterum philoſophorum loca ſibi aduerſantia diſſimulat: tan­<lb></lb>
dem, quod nullus adhuc auſus eſt, concludit Mathematicam nullam eſſe <lb></lb>
Philoſophiæ partem, cùm tamen apud Ariſt. &amp; omnes peripateticos nihil <lb></lb>
frequentius occurrat, quam tres eſſe philoſophiæ partes, Phyſicam, Mathe­<lb></lb>
maticam, &amp; Metaphyſicam. </s>
          <s id="s.005491">ait præterea ſæpius certitudinem Mathemati­<lb></lb>
cam ex eo prouenire, quod ad ſenſum oſtendant, ſeu quod ſenſu ipſarum ve­<lb></lb>
ritates percipiantur; quod omninò falſum eſſe ſæpius ſupra probatum eſt, <lb></lb>
cùm eorum materia ſit omninò intelligibilis, non autem ſenſibilis, &amp; nullus <lb></lb>
eſt, qui eas, vel leuiter attigerit, quì iſtud palàm non fateatur.</s>
        </p>
        <p type="main">
          <s id="s.005492">Scias <expan abbr="deniq;">denique</expan> candide Lector, me ſyncerè omnia, &amp; ſolius veritatis amo­<lb></lb>
re hucuſque dixiſſe, vt experiri poteris, ſi authores, quos citaui, adieris, <lb></lb>quod vt facias, obſecro, plura enim, quam dixi, reperies. </s>
          <s id="s.005493">è contrariò vi­<lb></lb>
diſti, quàm ſolicitè, alij, quorum munus eſſet eas fouere, ac tueri, <expan abbr="cõtra">contra</expan> pul­<lb></lb>cherrimas haſce facultates, neſcio quo conſilio tam ſolicitè egerint.</s>
        </p>
        <p type="head">
          <s id="s.005494"><emph type="italics"></emph>De præstantia ſcientiæ, quam nobis pariunt Geometria, <lb></lb>
&amp; Arithmetica. </s>
          <s id="s.005495">Cap. 4.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005496">Scientiarum nobilitas, <expan abbr="atq;">atque</expan> præſtantia non ex ſubiecto, &amp; medio ſo­<lb></lb>
lum, verùm etiam, &amp; quidem multò magis ex notitia, quæ per illas <lb></lb>
acquiritur, <expan abbr="quippeq́">quippeque</expan>; illarum finis eſt, penſanda eſt: quanto enim hæc <lb></lb>
præſtantior, ac nobilior eſt, tanto etiam illæ excellentiores ſunt ha­<lb></lb>
bendæ. </s>
          <s id="s.005497">Verumenimuerò, vt ait Ariſt. initio 2. Metaphyl. </s>
          <s id="s.005498">Finis ſcientiæ eſt <lb></lb>
veritas. </s>
          <s id="s.005499">quod &amp; Plato confirmat, dum in Sympoſio aſſerit, animi cibum eſ­<lb></lb>
ſe veritatem, atqui Geometria, &amp; Arithmetica, cùm ſemper euidentiam <lb></lb>
pariant, ſemper etiam veritatem conſequentur, vnde Ariſt. 1. Ethic. cap. 7. <lb></lb>
Geometram appellat veritatis ſpeculatorem. </s>
          <s id="s.005500">cùm præterea reliquæ ſcien­<lb></lb>
tiæ (Mathematicis medijs exceptis) rarò euidentiam pariant, rarò etiam <lb></lb>
veritatem aſſequentur, ſed ferè ſemper opiniones gignent, vnde eas meritò <lb></lb>
non ſcientias, ſed opiniones Plato voluit appellari: ſequitur abſolutè di­<lb></lb>
cendum eſſe, Mathematicas reliquarum ſcientiarum præſtantiſſimas eſſe; <lb></lb>
quemadmodum inter opiniones præſtantiſſimum quid eſt veritas.</s>
        </p>
        <p type="main">
          <s id="s.005501"><expan abbr="Neq;">Neque</expan> verò ſolum huius notitiæ excellentia à veritate deſumitur, verum <lb></lb>
etiam ab admirabilitate, ſunt enim concluſiones Mathematicæ (paruis ex­<lb></lb>
ceptis) admirabiles veritates, licet enim cùm Cicer. primo de orat ſic ad­<lb></lb>
mirati; quis ignorat, ij, qui Mathematici vocantur quanta in obſcuritate <lb></lb>
rerum, &amp; quàm in recondita arte, &amp; multiplici, <expan abbr="ſubtiliq́">ſubtilique</expan>; verſantur? </s>
          <s id="s.005502">Et ve­<lb></lb>
rò quis non admiratur, cùm intelligit, omne triangulum habere tres angu­<lb></lb>
los æquales duobus rectis? </s>
          <s id="s.005503">&amp; omnem figuram rectilineam habere angulos <lb></lb>
externos, etiamſi mille ſint, æquales tantummodo quatuor rectis? </s>
          <s id="s.005504">Item <lb></lb>
duo parallelogramma ſuper eadem baſi, &amp; in ijſdem parallelis conſtituta <lb></lb>
æqualia eſſe, etiamſi alterum quantumuis longum efficiatur? </s>
          <s id="s.005505">quàm admira­<lb></lb>
bilis eſt 47. primi, pro cuius inuentione Pythagoras Muſis Hecatombas <lb></lb>
immolauit? </s>
          <s id="s.005506">In ſecundò deinde Elem. lib. quàm ſubtilis eſt 14. quæ rectili­<lb></lb>
neo cuiuis quadratum exhibet æquale. </s>
          <s id="s.005507">In tertiò poſtea, quot <expan abbr="quantaq́">quantaque</expan>; an­


<pb pagenum="28" xlink:href="009/01/312.jpg"></pb>gulus ille contingentiæ continet miracula, qui quamuis quantus ſit, neque <lb></lb>
tamen à linea recta partiri. </s>
          <s id="s.005508">ſed ne longior ſim, quis in 10. libro conſiderans <lb></lb>
illam tot magnitudinum aſymetriam, &amp; diametrum eſſe coſtæ incommen­<lb></lb>
ſurabilem, non magnopere obſtupeſcit, <expan abbr="atq;">atque</expan> cum Platone non aſſerit illius <lb></lb>
ignorantiam, non hominum, ſed ſuum, ac pecorum eſſe? </s>
          <s id="s.005509">In 13. quanto <lb></lb>
ſtupore afficimur in illa <expan abbr="quinq;">quinque</expan> regularium corporum in eadem ſphæra in­<lb></lb>
ſcriptione? </s>
          <s id="s.005510">&amp; cum intelligimus <expan abbr="quinq;">quinque</expan> tantum in tota rerum natura repe­<lb></lb>
riri poſſe ſolida regularia? </s>
          <s id="s.005511">meritò igitur Cardanus lib. 16. de ſubtilit. </s>
          <s id="s.005512">ait; <lb></lb>
Euclidis ſunt duæ præcipuæ laudes, inconcuſſa dogmatum firmitas libri <lb></lb>
elementorum, <expan abbr="perfectioq́">perfectioque</expan>; adeò abſoluta, vt nullum opus iurè huic compa­<lb></lb>
rare audeas, quibus fit, vt adeò veritatis lux in eo refulgeat, vt ij ſoli in ar­<lb></lb>
duis quæſtionibus videantur poſſe verum à falsò diſcernere, quì Euclidem <lb></lb>
habeant familiarem.</s>
        </p>
        <p type="main">
          <s id="s.005513">Quod ſi ad Archimedis opera oculum conuertamus, quam ſæpè nos ea <lb></lb>
reddunt ſtupefactos? </s>
          <s id="s.005514">vt dum oſtendit triangulum quoddam eſſe dato cir­<lb></lb>
culo æquale. </s>
          <s id="s.005515">dum Parabolam ad quadratum redigit: dum planorum centra <lb></lb>
grauitatis rimatur: dum totius arenæ mundum vniuerſum complentis nu­<lb></lb>
merum ſubducit: dum quodlibet pondus, <expan abbr="atq;">atque</expan> adeò mundi machinam loco <lb></lb>
dimoueri poſſe, vel ab vnica formica demonſtrat. </s>
          <s id="s.005516">quamquam hæc duo ad <lb></lb>
medias pertinent. </s>
          <s id="s.005517">At libellus de lineis ſpiralibus, &amp; alter de ijs, quæ in aqua <lb></lb>
vehuntur, quam admirandi ſunt? </s>
          <s id="s.005518">De ſphæra poſtea, &amp; cylindro varia de­<lb></lb>monſtrans, quanto &amp; alios, ac ſe ipſum ſpatio ſuperat, vt dum inter cætera <lb></lb>
diuino planè acumine oſtendit cuiuſlibet ſphæræ ſuperficiem quadruplam <lb></lb>eſſe circuli eiuſdem maximi. </s>
          <s id="s.005519">Item ſi tria hæc, cylindrus, ſphæra, &amp; conus, <lb></lb>
ſint in eadem altitudine, <expan abbr="eorumq́">eorumque</expan>; baſes ſint circuli maximi illius ſphæræ, <lb></lb>
habere inuicem proportionem, quam <expan abbr="habẽt">habent</expan> hi numeri 3. 2. 1. quare ob tam <lb></lb>
præclarum ingenij monumentum, ſepulcro ipſius marmoreo ſphæra, &amp; cy­<lb></lb>lindrus marmorei ſunt impoſiti, quemadmodum Cicero lib. 5. Tuſcinarrat, <lb></lb>
vbi etiam magnopere gloriatur, ſe cum in Sicilia Quæſtor eſſet, illud igno­<lb></lb>ratum ab Syracuſanis ſeptum vndique, ac veſtitum vepribus, &amp; dumetis in­<lb></lb>
dagaſſe ſepulcrum. </s>
          <s id="s.005520">meritò igitur Cardanus lib. 16. de ſubtil. </s>
          <s id="s.005521">eum tanquam <lb></lb>
ingeniorum Phænicem ſupra omnes ſubtilitate præſtantes viros, <expan abbr="atq;">atque</expan> adeo <lb></lb>
ſupra Ariſtot. ipſum duplici ordine euexit; Archimedes, inquit, primus ſit <lb></lb>
non ſolum ob monumenta illius nunc vulgata, ſed ob mechanica, quibus <lb></lb>
vires Romanorum ſæpius fregit. </s>
          <s id="s.005522">Apollonius deinde pergæus cognomento <lb></lb>
Magnus Geometra, nulla ratione Archimede inferior, quam mira, quam <lb></lb>
abſtruſa in ſuis conicis in lucem profert? </s>
          <s id="s.005523">ſed inter cætera illud <expan abbr="admirãdum">admirandum</expan>; <lb></lb>
inueniri duas lineas, quas vocat aſymptotos, quæ ſi in infinitum producan­<lb></lb>
tur, ſemper magis inuicem accedunt, nunquam tamen concurrunt. </s>
          <s id="s.005524">miſſos <lb></lb>
facio Hipſiclem, Theodoſium tripolitam, Menelaum, Serenum, Pappum, &amp; <lb></lb>
alios, quorum opera omnem ſuperant admirationem, <expan abbr="eaq́">eaque</expan>; mirabili adeo <lb></lb>
connexione, ac certitudine tradita, vt nullus ſit, qui priſcis illis Geometris <lb></lb>
ingenio cedere libenter nolit. </s>
          <s id="s.005525">Quapropter cum Cardano lib. 16. de ſubtil. <lb></lb>
</s>
          <s id="s.005526">hanc partem concludamus; nihil mirum igitur, inquit, Geometriam eſſe <lb></lb>omnium ſcientiarum ſubtiliſſimam, quæ cùm tamen à manifeſtiſſimis ini­<lb></lb>
tium ducat, meritò anſam præbuit, vt prima omnium etiam pueros doce­


<pb pagenum="29" xlink:href="009/01/313.jpg"></pb>retur. </s>
          <s id="s.005527">mirum eſt, quam breui ex apertiſſimis ad obſcuriſſima trahat, &amp; ex <lb></lb>
humillimis in altiſſima ſtatim aſſurgat.</s>
        </p>
        <p type="main">
          <s id="s.005528">Sed iam Arithmeticæ etiam fructus inſpiciamus: in qua pręter ea, quæ <lb></lb>
Euclides, Iordanus, &amp; Maurolicus egregia ſanè adinuenerunt, quàm mira­<lb></lb>
bile eſt illud veluti ſcientiarum monſtrum, ac portentum, quod Algebram <lb></lb>
vocant? </s>
          <s id="s.005529">nihil fortaſſe in tota peritiæ Encyclopedia ſubtilius, profundius ni­<lb></lb>
hil, non humano ingenio par eſt, ſed quid cœlitus reuelatum dixeris: nume­<lb></lb>
ros illos, quos ſurdos vocant, &amp; qui nullo modo exprimi poſſunt, addit, ſub­<lb></lb>
trahit, multiplicat, diuidit, perinde ac ſi numeri communes eſſent: illis ve­<lb></lb>
rò, quos minores, quàm nihil <expan abbr="cõfingit">confingit</expan>, quid abſtruſius? </s>
          <s id="s.005530">quibus tamen <expan abbr="vtriſq́">vtriſque</expan>; <lb></lb>admirandas adeò diſſoluit quæſtiones, &amp; enigmata, vt ij, qui hanc callent <lb></lb>
eruditionem, nihil in numerorum infinita ditione <expan abbr="obſcurũ">obſcurum</expan>, nihil arduum ti­<lb></lb>
meant; vt propterea eos non homines, ſed vel intelligentias quaſdam ſepa­<lb></lb>
ratas, aut præſtigiatores quoſdam eſſe exiſtimes.</s>
        </p>
        <p type="main">
          <s id="s.005531">Hanc tandem Geometriæ, &amp; Arithmeticæ tractationem abſoluentes, ex <lb></lb>
prędictis breuiter earum prærogatiuas ſic perſtringamus: quarum.</s>
        </p>
        <p type="main">
          <s id="s.005532">Prima ſit, quòd omnes ſcientiæ partes ab inuicem diſtinctas obtinent, vi­<lb></lb>
delicet primo loco definitiones, <expan abbr="easq́">easque</expan>; eſſentiales. </s>
          <s id="s.005533">Secundò, poſtulata. </s>
          <s id="s.005534">Ter­<lb></lb>tiò, axiomata, quę ſunt tria principiorum genera; ex quibus ſcientia dedu<lb></lb>
citur. </s>
          <s id="s.005535">quapropter quarto loco ſuccedunt propoſitiones cum ſuis demon­<lb></lb>ſtrationibus: quæ partim problemata, partim Theoremata ſunt. </s>
          <s id="s.005536">hunc por­<lb></lb>
rò doctrinæ ordinem pulcherrimum ab Ariſt. etiam traditum, præterquam <lb></lb>
in Mathematicis, &amp; maximè hiſce puris nuſquam eſt reperire.</s>
        </p>
        <p type="main">
          <s id="s.005537">Secunda, ex principiorum autem præmiſſorum certitudine fit, vt proce­<lb></lb>
dant à notioribus nobis, &amp; natura.</s>
        </p>
        <p type="main">
          <s id="s.005538">Tertia, quod omnes earum comprobationes ſunt <expan abbr="demõſtrationes">demonſtrationes</expan>, partim <lb></lb>
à priori, partim à poſteriori, vbi nihil probabile, nulla opinionum diſcordia, <lb></lb>
ſed totum euidens, concors, &amp; verum cernitur.</s>
        </p>
        <p type="main">
          <s id="s.005539">Quarta, demonſtrationes à priori ſunt tantum à cauſis intrinſecis mate­<lb></lb>
ria, &amp; forma.</s>
        </p>
        <p type="main">
          <s id="s.005540">Quinta, demonſtrationes earum, vt plurimum, &amp; quod, &amp; propter quid <lb></lb>
ſimul manifeſtant.</s>
        </p>
        <p type="main">
          <s id="s.005541">Sexta, eſt mirabilis, &amp; perpetua demonſtrationum connexio, &amp; depen­den<lb></lb>
tia ab inuicem.</s>
        </p>
        <p type="head">
          <s id="s.005542"><emph type="italics"></emph>De Mathematicis medijs, Aſtronomia, Perſpectiua, <lb></lb>
Mathematica, Muſica. </s>
          <s id="s.005543">Cap. 5.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005544">De materia harum, cùm apud Philoſophos conueniat, nihil eſt, <lb></lb>
cur de ea dicamus. </s>
          <s id="s.005545">Eas verò habere perfectiſſimas demonſtra­<lb></lb>
tiones, quibus etiam mirabiles veritates, <expan abbr="easq́">easque</expan>; ſcitu iucundiſſi­<lb></lb>
mas nobis patefaciunt, vna, <expan abbr="eademq́">eademque</expan>, opera breuiter demonſtra­<lb></lb>
bimus. </s>
          <s id="s.005546">non deeſt tamen Ariſt. authoritas 1. Poſter. tex. 30. aſſerentis eas <lb></lb>
habere cauſas demonſtrationum, ſic. </s>
          <s id="s.005547">Hic enim ipſum quidem quòd ſenſi­<lb></lb>
tiuorum eſt ſcire, ipſum verò propter quid Mathematicorum, hi <expan abbr="namq́">namque</expan>; ha­


<pb pagenum="30" xlink:href="009/01/314.jpg"></pb>bent cauſarum demonſtrationes. </s>
          <s id="s.005548">loquitur ibi de medijs hiſce facultatibus. <lb></lb>
</s>
          <s id="s.005549">vide noſtram illius loci explicationem ſuperius allatam; aut ſi mais <expan abbr="aliorũ">aliorum</expan>. <lb></lb>
</s>
          <s id="s.005550">ſed iam rem ipſam oculis inſpiciamus.</s>
        </p>
        <p type="main">
          <s id="s.005551">Et, vt ab Aſtronomia <expan abbr="initiũ">initium</expan> faciamus, <expan abbr="demõſtratio">demonſtratio</expan> eclypſis Lunæ (etiam <lb></lb>
Ariſt. <expan abbr="eiusq́">eiusque</expan>; interpretibus præcipuè Zabarella teſtibus) non ne potiſſima? <lb></lb>
</s>
          <s id="s.005552">nam affectionis illius, ſeu defectus propriam, &amp; adæquatam cauſam euiden<lb></lb>
tem facit, interpoſitionem, videlicet terrę. </s>
          <s id="s.005553">Idem de ſolari defectu dicen­<lb></lb>
dum, cuius cauſam oſtendunt eſſe Lunæ obiectionem. </s>
          <s id="s.005554">quas demonſtratio­<lb></lb>
nes ab Aſtronomis inuentas eſſe ex <expan abbr="ipſorũ">ipſorum</expan> libris conſtat. </s>
          <s id="s.005555">&amp; quòd medio <expan abbr="vtã-tur">vtan­<lb></lb>
tur</expan> Geometrico, nimirum circulo, &amp; diametris, &amp; diametrali oppoſitione. <lb></lb>
</s>
          <s id="s.005556">quàm deinde certæ ſint, patet ex eclypſium infallibili prædictione.</s>
        </p>
        <p type="main">
          <s id="s.005557">Secundò, cur Sol plures dies in parte Zodiaci æſtiua; quàm in hyberna <lb></lb>
moratur? </s>
          <s id="s.005558">cauſam afferunt Apogæum.</s>
        </p>
        <p type="main">
          <s id="s.005559">Tertiò, cur Luna ſucceſſiuè illuminatur? </s>
          <s id="s.005560">quia ſphærica eſt.</s>
        </p>
        <p type="main">
          <s id="s.005561">Quartò, cur in horologijs ſolaribus tropici ſunt lineæ curuę? </s>
          <s id="s.005562">æquator <lb></lb>
verò linea recta? </s>
          <s id="s.005563">quia illi ſunt ſectiones conicę, æquator verò eſt ſectio duo­<lb></lb>
rum planorum.</s>
        </p>
        <p type="main">
          <s id="s.005564">Quintò, cur Sol non totam ſimul terram illuminat, ſed ſucceſſiuè? </s>
          <s id="s.005565">quia <lb></lb>
terra ſphærica eſt.</s>
        </p>
        <p type="main">
          <s id="s.005566">Quam deinde mirabiles, ac iucundæ ſunt cognitiones illæ, de quibus ipſæ <lb></lb>
ſacræ literæ mirabundæ loquuntur? </s>
          <s id="s.005567">altitudinem videlicet Cœli, <expan abbr="atq;">atque</expan> pro­<lb></lb>
funditatem abyſſi improbo ſanè auſu perſcrutari? </s>
          <s id="s.005568">terræ, Lunæ, Solis magni­<lb></lb>
tudines, ac diſtantias acumine planè diuino nobis euidenter tradidiſſe? </s>
          <s id="s.005569">to­<lb></lb>
tius <expan abbr="deniq;">denique</expan> mundi fabricam, ac ſymmetriam, qua cognitione nihil præſtan­<lb></lb>
tius, potiſsima hæc nobis philoſophia manifeſtum facit; vt jure liceat illud <lb></lb>
accinere:</s>
        </p>
        <p type="main">
          <s id="s.005570"><emph type="italics"></emph>Felices animæ, quibus hæc cognoſcere primum<emph.end type="italics"></emph.end>, <lb></lb>
<emph type="italics"></emph>Inque domos ſuper as ſcandere cura fuit. <lb></lb>
</s>
          <s id="s.005571">Admouere oculis diſtantia ſydera noſtris<emph.end type="italics"></emph.end>, <lb></lb>
<emph type="italics"></emph>Aetheraqué, ingenio ſuppoſuere ſuo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005572">Aſtronomiæ pars iucunda æquè, ac mirabilis eſt Geographia: qua, vel in <lb></lb>
globo, vel in tabula, terras omnes, maria omnia, quaſi præſentes licèt <expan abbr="cõ-templari">con­<lb></lb>
templari</expan>. </s>
          <s id="s.005573">quoque loco vnumquodque ſit, qua zona, quo climate, magna iu­<lb></lb>
cunditate percipitur.</s>
        </p>
        <p type="main">
          <s id="s.005574">In perſpectiua etiam <expan abbr="neq;">neque</expan> deſunt perfectæ demonſtrationes, v. g. cur ocu­<lb></lb>
lus ſphæricus? </s>
          <s id="s.005575">vt illi <expan abbr="vndiq;">vndique</expan> lineæ perpendiculares poſſint accidere. </s>
          <s id="s.005576">Sed cur <lb></lb>
lineæ perpendiculares? </s>
          <s id="s.005577">vt diſtincta fieret viſio; ecce cauſæ finales. </s>
          <s id="s.005578">Cur <expan abbr="cõ-cauũ">con­<lb></lb>
cauum</expan> <expan abbr="ſpeculũ">ſpeculum</expan> ibi vrit? </s>
          <s id="s.005579">quia ſolares radij reflexi illuc congregantur. </s>
          <s id="s.005580">Cur bacu<lb></lb>
lus in aqua fractus apparet? </s>
          <s id="s.005581">quia per lineas refractas videtur. </s>
          <s id="s.005582">Cur Iris <expan abbr="rotũ-da">rotun­<lb></lb>
da</expan>? </s>
          <s id="s.005583">quia non videtur, niſi ſub ſtatuto angulo, qui non niſi in orbem collo­<lb></lb>
cari poteſt. </s>
          <s id="s.005584">vbi ſimul vides, quàm dignæ <expan abbr="quoq;">quoque</expan> ſint hæ cognitiones.</s>
        </p>
        <p type="main">
          <s id="s.005585">In mechanicis poſtea: cur cuneus tantas obtinet vires? </s>
          <s id="s.005586">quia eſt vectis <lb></lb>
geminatus. </s>
          <s id="s.005587">Vnde cochleæ tanta vis? </s>
          <s id="s.005588">quia conſtat cuneo, &amp; vecte. </s>
          <s id="s.005589">Verum <lb></lb>
quid admirabilius, quàm quodlibet pondus, vel ipſum vniuerſum, vnius for­<lb></lb>
micæ vi poſſe commoueri? </s>
          <s id="s.005590"><expan abbr="ipſamq́">ipſamque</expan>; naturam, vt ait Ariſt. vel inuitam ſupe­<lb></lb>
rare. </s>
          <s id="s.005591">quàm ſubtìlia ſunt ea, quæ de centro grauitatis Archimedes olim, nu­<pb pagenum="31" xlink:href="009/01/315.jpg"></pb>per verò Commandinus, &amp; Lucas Valerius demonſtrarunt.</s>
        </p>
        <p type="main">
          <s id="s.005592">Muſica tandem ſuas habet <expan abbr="demõſtrationes">demonſtrationes</expan>, v. g. Tonum conſtare ex duo­<lb></lb>
bus ſemitonijs minoribus, &amp; commate, quia ratio ſeſquioctaua duobus ſe­<lb></lb>
mitonijs minoribus, &amp; vno commate conſtat. </s>
          <s id="s.005593">Tonus autem in ſeſquiocta­<lb></lb>
ua ratione conſiſtit. </s>
          <s id="s.005594">Item Diapentem conſtare ex tribus tonis, &amp; ſemitonio <lb></lb>
minori; quia ſi ex ſeſquialtero interuallo, quòd eſt diapentes demas ſeſqui­<lb></lb>
tertium, reſtat ſeſquioctauum. </s>
          <s id="s.005595"><expan abbr="Seſquitertiũ">Seſquitertium</expan> verò continet duos tonos cum <lb></lb>
ſemitonio minori; ecce cauſæ materiales. </s>
          <s id="s.005596">Cur bis diapente, aut bis dia­<lb></lb>
teſſaron conſonantia <expan abbr="cõponi">componi</expan> non poteſt? </s>
          <s id="s.005597">cauſam huius habes ſupra ſectio­<lb></lb>
ne 19. problem. </s>
          <s id="s.005598">nu. </s>
          <s id="s.005599">34. ſed quàm mirum eſt Pythagoram ſonos in propor­<lb></lb>
tiones diuiſiſſe, non ſecus, ac ſi quantitates quædam permanentes eſſent?</s>
        </p>
        <p type="main">
          <s id="s.005600">Reliquum eſſet de Mathematicis etiam practicis <expan abbr="nõnulla">nonnulla</expan> dicere, in qui­<lb></lb>
bus omnes <expan abbr="quoq;">quoque</expan> cauſæ manifeſtæ reperiuntur; ex eò enim, quòd practicæ <lb></lb>
ſunt, neceſſariò finem inuoluunt. </s>
          <s id="s.005601">efficientem verò <expan abbr="materiã">materiam</expan>, &amp; formam ſæ­<lb></lb>
pè adhibent ad præmiſſas probandas, quas aſſumunt ad concludendum id, <lb></lb>
quod principaliter intendunt. </s>
          <s id="s.005602">Porrò inter practicas omnium præſtantiſſi­<lb></lb>
ma eſt Geometria practica; quis enim non admiratur, cùm audit Geome­<lb></lb>
tram ſolo viſu inacceſſas etiam magnitudines quaſcunque, vt turres, vel <lb></lb>
montes menſurare?</s>
        </p>
        <p type="main">
          <s id="s.005603">Ex quibus liquidò conſtant Mathematicas habere perfectiſſimas <expan abbr="domõ-ſtrationes">demon­<lb></lb>
ſtrationes</expan>, quarum cauſę ita ab effectu diſtinguntur, vt nullis calumnijs ſint <lb></lb>
obnoxiæ: quare etiam ſi aduerſarij conuincant, quòd neutiquam faciunt, <lb></lb>
Geometriam, &amp; Arithmeticam illis carere; reliquis tamen prædictis con­<lb></lb>
cedere coguntur: <expan abbr="easq́">easque</expan>; per omne <expan abbr="cauſarũ">cauſarum</expan> genus excurrere, quòd tan­<lb></lb>ta præterea euidentia præſtant, vt nihil ambiguum, nihil contro­<lb></lb>
uerſum relinquatur: Mathematicę <expan abbr="namq;">namque</expan> teſte etiam Ariſt. <lb></lb>
1. Elenchorum non ſunt contentioſæ. </s>
          <s id="s.005604">Vnde ſit, vt to­<lb></lb>
ta hæc adeò digna, <expan abbr="atq;">atque</expan> admiranda cognitio ſit <lb></lb>
mera veritas, quæ omnium ſcientiarum finis <lb></lb>
atque animæ noſtræ cibus eſt.</s>
        </p>
        <p type="head">
          <s id="s.005605">LAVS DEO.</s>
        </p>
      </chap>
      <pb pagenum="32" xlink:href="009/01/316.jpg"></pb>
      <chap>
        <p type="head">
          <s id="s.005606">APPENDIX.</s>
        </p>
        <p type="main">
          <s id="s.005607"><emph type="italics"></emph>Placet nunc demum, vt melius àdhuc Mathematica­<lb></lb>
rum natur a pateat, locaqué Arist. Mathematica ma­<lb></lb>
gis illustrentur, Demonſtrationes primi Elemento­<lb></lb>rum Euclidis breuiter expendere, atque vnamquamque <lb></lb>
ad ſuum demonſtrationis genus referre.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005608">Prima igitur Demonſtratione Euclides oſtendit Triangulum il­<lb></lb>
lud eo modo <expan abbr="cõſtrnctum">conſtructum</expan> eſſe æquilaterum, hoc proximo medio, <lb></lb>
quia ſcilicet habet tria latera æqualia, quod medium eſt ipſius <lb></lb>
ſubiecti demonſtrationis, ſiue trianguli æquilateri definitio: <lb></lb>
quare hæc demonſtratio erit per cauſam formalem.</s>
        </p>
        <p type="main">
          <s id="s.005609">Secunda Demonſtratione oſtendit duas lineas eſſe æquales, quoniam am­<lb></lb>
bæ ſunt vni tertiæ æquales, quæ ratio nititur illi axiomati, quæ ſunt æqualia <lb></lb>
vni tertio, ſunt etiam inter ſe. </s>
          <s id="s.005610">eſt quidem demonſtratio oſtenſiua, ſed non <lb></lb>
per cauſam, verum à ſigno: eſſe enim æquales vni tertiæ, eſt ſignum æquali­<lb></lb>
tatis earum.</s>
        </p>
        <p type="main">
          <s id="s.005611">Tertia Demonſtratio eodem medio vtitur, quo ſecunda.</s>
        </p>
        <p type="main">
          <s id="s.005612">Quarta Demonſtratio oſtendit, primò de illis duobus triangulis, quod <lb></lb>
habent baſes æquales, quia baſes congruunt ſibi mutuo. </s>
          <s id="s.005613">ſecundò, oſtendit <lb></lb>
alios duos angulos eſſe æquales alijs duobus <expan abbr="vtrumq́">vtrumque</expan>; <expan abbr="vtriq́">vtrique</expan>; eadem ratione, <lb></lb>
quia nimirum ſibi mutuò congruunt. </s>
          <s id="s.005614">ſi dixeris igitur, quod ſibi mutuò con­<lb></lb>
gruere ſit definitio æqualis, erit demonſtratio per cauſam formalem; ſi au­<lb></lb>
tem dixeris eſſe ſignum æqualitatis, erit à ſigno, &amp; à poſteriori.</s>
        </p>
        <p type="main">
          <s id="s.005615">5. Oſtendit de Triangulo Iſoſcele, primò, quod Anguli, qui ſunt ad ba­<lb></lb>
ſim, ſunt æquales, ratio eſt, quia ablatis æqualibus ab æqualibus ipſi ſunt <lb></lb>
reliqui. </s>
          <s id="s.005616">Quæ quidem ratio etiam Ariſt. teſte, eſt per cauſam materialem; <lb></lb>
nam eſſe dimidium, tertiam partem, duplum, reliquum, alicuius totius, &amp; <lb></lb>
ſimilia, nihil aliud eſt, quàm eſſe partes reſpectu totius; partes autem ſunt <lb></lb>
materia, vt apertè docet Ariſt. tex. 3. lib. 5. Metaph. quem ſupra <expan abbr="cũ">cum</expan> alijs <expan abbr="ex-plicatũ">ex­<lb></lb>
plicatum</expan> habes. </s>
          <s id="s.005617">ſecundò, demonſtrat de eodem Iſoſcele, angulos infra baſim <lb></lb>
eſſe æquales, ratio, quia opponuntur ęqualibus lateribus <expan abbr="triangulorũ">triangulorum</expan> quar­<lb></lb>
tæ præcedentis, quæ ratio videtur ſignum quoddam æqualitatis eorum eſſe.</s>
        </p>
        <p type="main">
          <s id="s.005618">6. Probat duo illa latera illius trianguli eſſe æqualia, ab impoſſibili, quia <lb></lb>
ſequeretur partem eſſe æqualem toti.</s>
        </p>
        <p type="main">
          <s id="s.005619">7. Duas poſteriores lineas cum duabus prioribus neceſſariò coincidere <lb></lb>
demonſtrat, quia aliter ſequeretur, vel partem eſſe æqualem toti: vel angu­<lb></lb>
los lſolcelis ſub baſi eſſe inæquales, vel etiam eos, qui ſupra baſim, contra <lb></lb>
quàm oſtenſum eſt in quinta.</s>
        </p>
        <p type="main">
          <s id="s.005620">8. Probat angulos illos fore æquales, quia congruunt: per 8. ſcilicet <lb></lb>
axioma: videtur à ſigno.</s>
        </p>
        <pb pagenum="33" xlink:href="009/01/317.jpg"></pb>
        <p type="main">
          <s id="s.005621">9 Probat angulum illum diuiſum eſſe bifariam, per <expan abbr="octauã">octauam</expan> pręcedentem: <lb></lb>
eſt ergo eiuſdem naturæ.</s>
        </p>
        <p type="main">
          <s id="s.005622">10 Probat lineam <expan abbr="illã">illam</expan> eſſe diuiſam in duas lineas æquales, quia illæ duæ <lb></lb>
ſunt baſes triangulorum quartę; hoc autem, eſſe baſes talium <expan abbr="triangulorũ">triangulorum</expan>, <lb></lb>
videtur eſſe definitio; quare hæc demonſtratio eſſet à definitione ſubiecti, <lb></lb>
&amp; per cauſam formalem.</s>
        </p>
        <p type="main">
          <s id="s.005623">11 Probat illam lineam facere angulos rectos, quia facit angulos cum <lb></lb>
ſubiecta linea aquales; nam ex decima definitione ſi illi anguli ſint æquales, <lb></lb>
qui fiunt à tali linea, erunt ipſi quoque recti. </s>
          <s id="s.005624">demonſtratio igitur eſt à de­<lb></lb>
finitione.</s>
        </p>
        <p type="main">
          <s id="s.005625">12 Probat lineam illam eſſe perpendicularem ex definitione lineæ per­<lb></lb>pendicularis, quia nimirum facit angulos, cum ſubiecta linea æquales, re­<lb></lb>
ctoſuè; eſt igitur demonſtratio à definitione, à priori, per cauſam formalem.</s>
        </p>
        <p type="main">
          <s id="s.005626">13 Probat duos angulos eſſe æquales duobus angulis rectis, <expan abbr="quoniã">quoniam</expan> <expan abbr="vtri-q́ue">vtri­<lb></lb>
que</expan> ſunt æquales vni tertiæ rei. </s>
          <s id="s.005627">quare eſt à ſigno.</s>
        </p>
        <p type="main">
          <s id="s.005628">14 Probat intentum, quia aliter ſequeretur, partem toti æqualem eſſe.</s>
        </p>
        <p type="main">
          <s id="s.005629">15 Probat angulos ad verticem æquales eſſe, quia ſi ab æqualibus, æqua­<lb></lb>lia demas ipſi remaneat: vel ſunt reliqui. </s>
          <s id="s.005630">Eſt igitur demonſtratio per cau­<lb></lb>
ſam materialem, vt dictum eſt in quinta.</s>
        </p>
        <p type="main">
          <s id="s.005631">16 Probat angulum externum maiorem eſſe interno, quia eſt maior alio <lb></lb>
angulo æquali ipſi interno. </s>
          <s id="s.005632">eſt à ſigno.</s>
        </p>
        <p type="main">
          <s id="s.005633">17 Probat duos angulos eſſe minores alijs duobus angulis, ex 4. axiom. <lb></lb>
</s>
          <s id="s.005634">quia. </s>
          <s id="s.005635">ſ. </s>
          <s id="s.005636">ſi inæqualibus adiecta ſint æqualia, tota erunt inæqualia: vbi cauſa <lb></lb>
inæqualitatis totorum, eſt adiectum illud, quo adiecto conflatur duo tota: <lb></lb>
quare adiectum illud eſt; pars autem eſt materia totius. </s>
          <s id="s.005637">demonſtrat igitur <lb></lb>
per cauſam materialem.</s>
        </p>
        <p type="main">
          <s id="s.005638">18 Probat angulum vnum eſſe altero <expan abbr="maiorẽ">maiorem</expan>, quia ille ſit veluti totum, <lb></lb>
iſte verò illius pars. </s>
          <s id="s.005639">reducitur ad cauſam materialem.</s>
        </p>
        <p type="main">
          <s id="s.005640">19 Probat propoſitum ab impoſſibili.</s>
        </p>
        <p type="main">
          <s id="s.005641">20 Probat duo illa latera eſſe reliquo maiora, quia ſunt æqualia vni li­<lb></lb>
neæ, quæ ipſa reliquo latere maior eſt. </s>
          <s id="s.005642">eſt à ſigno.</s>
        </p>
        <p type="main">
          <s id="s.005643">21 Probat illas duas rectas eſſe minores alijs duabus, ex eo, quòd ſint <lb></lb>
minores quadam quantitate, quæ quantitas minor eſt illis duabus. </s>
          <s id="s.005644">à ſigno.</s>
        </p>
        <p type="main">
          <s id="s.005645">Secundò, probat <expan abbr="angulũ">angulum</expan> illum eſſe maiorem altero, quia. </s>
          <s id="s.005646">f. </s>
          <s id="s.005647">eſt maior quo­<lb></lb>
dam angulo, qui maior eſt illo altero. </s>
          <s id="s.005648">pariter à ſigno.</s>
        </p>
        <p type="main">
          <s id="s.005649">22 Probat per illud axioma, quæ ſunt æqualia vni tertio, &amp;c.</s>
        </p>
        <p type="main">
          <s id="s.005650">23 Probat duos angulos eſſe æquales, quòd ſint anguli oppoſiti baſibus <lb></lb>
triangulorum octauæ. </s>
          <s id="s.005651">videtur à definitione horum angulorum.</s>
        </p>
        <p type="main">
          <s id="s.005652">24 Probat latus illud eſſe maius altero latere, ex eo, quòd ſit æquale cui­<lb></lb>
dam lateri, quod etiam eſt maius illo latere.</s>
        </p>
        <p type="main">
          <s id="s.005653">25 Probat propoſitionem, deducens ad abſurdum.</s>
        </p>
        <p type="main">
          <s id="s.005654">26 Demonſtrat deducendo ad inconueniens.</s>
        </p>
        <p type="main">
          <s id="s.005655">27 Probat illas eſſe parallelas, quia nunquam concurrere poſſunt; eſt à <lb></lb>
definitione parallelarum.</s>
        </p>
        <p type="main">
          <s id="s.005656">28 Puto à cauſa demonſtrare, oſtendit enim duas rectas eſſe æquidiſtan­<lb></lb>
tes, quia earum anguli alterni ſint æquales, illi enim anguli ſunt cauſa æqui­


<pb pagenum="34" xlink:href="009/01/318.jpg"></pb>diſtantiæ linearum. </s>
          <s id="s.005657">ſimile dicendum eſt de ſecunda parte demonſtrationis.</s>
        </p>
        <p type="main">
          <s id="s.005658">29 Prima pars probatur ab impoſſibili. </s>
          <s id="s.005659">ſecunda à ſigno, quæ ſuat æqua­<lb></lb>
lia vni tertio &amp;c. </s>
          <s id="s.005660">Idem dicendum de tertia parte.</s>
        </p>
        <p type="main">
          <s id="s.005661">30 Probat eſſe parallelas ex 27. primi, quare est <expan abbr="eiuſdẽ">eiuſdem</expan> naturæ cum illa.</s>
        </p>
        <p type="main">
          <s id="s.005662">31 <expan abbr="Eandẽ">Eandem</expan> habet <expan abbr="rationẽ">rationem</expan>, quam 27. primi. </s>
          <s id="s.005663">per cauſam igitur formalem.</s>
        </p>
        <p type="main">
          <s id="s.005664">32 Primò, probat <expan abbr="anguiũ">anguium</expan> externum eſſe æqualem duabus internis, &amp; ap­<lb></lb>
poſitis ex eo, quòd partes anguli externi, ſint æquales partibus illorum: ex <lb></lb>
æqualitate.ſ partium infert <expan abbr="æqualitatẽ">æqualitatem</expan> totorum: quæ demonſtratio eſt per <lb></lb>
cauſam materialem. </s>
          <s id="s.005665">Secundò, probat illam adeò celeberrimam, omnis <lb></lb>
triangulus habet tres, &amp;c. </s>
          <s id="s.005666">quàm fuſiſſimè explicaui ſupra ad tex. 23. primi <lb></lb>
Poſter. vbi Ariſt. eam in exemplum perfectiſſimæ demonſtrationis adducit.</s>
        </p>
        <p type="main">
          <s id="s.005667">33 Partim per 4. primi, partim per 27. primi <expan abbr="demõſtrat">demonſtrat</expan>: quapropter ad <lb></lb>
earum naturam ſunt referendæ.</s>
        </p>
        <p type="main">
          <s id="s.005668">34 Tria probat. </s>
          <s id="s.005669"><expan abbr="primũ">primum</expan>, per 26. primi, <expan abbr="ſecundũ">ſecundum</expan> per illud axioma, ſi æqua­<lb></lb>libus æqualia adijcias, tota ſunt æqualia, quod duobus angulis applicat. <lb></lb>
</s>
          <s id="s.005670">quæ demonſtratio eſt à partibus ad tota: à cauſa nimirum materiali. </s>
          <s id="s.005671">ter­<lb></lb>
tium per 4. primi concludit.</s>
        </p>
        <p type="main">
          <s id="s.005672">35 Procedit per cauſam materialem: in omni enim caſu probat illa duo <lb></lb>
parallelogramma eſſe æqualia, quia ſi æqualibus æqualia adijciantur, tota <lb></lb>
erunt æqualia: vt in præcedenti dictum eſt.</s>
        </p>
        <p type="main">
          <s id="s.005673">36 Probat duo eſſe æqualia, quia ſunt vni tertio æqualia: videlicet à ſi­<lb></lb>
gno, à poſteriori.</s>
        </p>
        <p type="main">
          <s id="s.005674">37 Probat duo triangula eſſe æqualia, quòd ſint dimidia duorum paral­<lb></lb>
lelogrammorum æqualium: eſt <expan abbr="itaq;">itaque</expan> à cauſa materiali.</s>
        </p>
        <p type="main">
          <s id="s.005675">38 Eadem ratione demonſtrat in hac, <expan abbr="atq;">atque</expan> in præcedenti.</s>
        </p>
        <p type="main">
          <s id="s.005676">39 Propoſitum probat, ad abſurdum deducendo aduerſarium.</s>
        </p>
        <p type="main">
          <s id="s.005677">40 Similiter demonſtrat ac in præcedenti 39.</s>
        </p>
        <p type="main">
          <s id="s.005678">41 Probat vnum eſſe duplum alterius, quòd ſit duplum alterius, quod il­<lb></lb>
li æquale eſt. </s>
          <s id="s.005679">videtur à ſigno.</s>
        </p>
        <p type="main">
          <s id="s.005680">42 Probat parallelogrammum, &amp; <expan abbr="triangulũ">triangulum</expan> eſſe æqualia, quoniam <expan abbr="vtrũ-que">vtrun­<lb></lb>
que</expan> duplum ſit eiuſdem trianguli: videlicet per cauſam materialem.</s>
        </p>
        <p type="main">
          <s id="s.005681">43 Probat duo parallelogramma eſſe ęqualia, quoniam ablatis æquali­<lb></lb>
bus ab æqualibus ſint reſidua. </s>
          <s id="s.005682">cauſa eſt materialis.</s>
        </p>
        <p type="main">
          <s id="s.005683">44 Probat parallelogrammum æquari triangulo, quia <expan abbr="vtrunq;">vtrunque</expan> cuidam <lb></lb>
tertio æquatur. </s>
          <s id="s.005684">à ſigno videlicet.</s>
        </p>
        <p type="main">
          <s id="s.005685">45 Probat totum parallelogrammum æquari toti rectilineo; eo, quòd <lb></lb>
partes amborum totorum ſint æquales: eſt perſpicua cauſa materialis.</s>
        </p>
        <p type="main">
          <s id="s.005686">46 Probat quadrilaterum quoddam eſſe quadratum ex definitione qua­<lb></lb>
drati, quia ſ habet quatuor angulos rectos, &amp; quatuor latera æqualia. </s>
          <s id="s.005687">eſt <lb></lb>
igitur à cauſa formali.</s>
        </p>
        <p type="main">
          <s id="s.005688">47 Probat quadratum lateris angulo recto ſubſenſi, eſſe æquale duobus <lb></lb>
quadratis reliquorum <expan abbr="laterũ">laterum</expan> trianguli illius: &amp; ratio deſumpta eſt à parti­<lb></lb>
bus, quia. </s>
          <s id="s.005689">ſ. </s>
          <s id="s.005690">partes prædicti quadrati æquales ſunt ſingillatim prędictis qua­<lb></lb>
dratis; ergo totum quadratum totis illis quadratis æquale eſt. </s>
          <s id="s.005691">manifeſta eſt <lb></lb>
cauſa materialis.</s>
        </p>
        <p type="main">
          <s id="s.005692">48 Probat angulum quendam eſſe rectum, eo, quòd æqualis ſit cuidam <lb></lb>
angulo recto. </s>
          <s id="s.005693">eſt à ſigno.</s>
        </p>
        <pb pagenum="35" xlink:href="009/01/319.jpg"></pb>
        <p type="main">
          <s id="s.005694">Hæc pauca ſufficiant, vt Philoſophi habeant, vnde poſſint de Geometri­<lb></lb>
cis demonſtrationibus dijudicare. </s>
          <s id="s.005695">non tamen quiſpiam exiſtimet idem iu­<lb></lb>
dicium de reliquis Mathematicis eſſe faciendum, Aſtronomia enim, Opti­<lb></lb>
ca, &amp; alię vtuntur etiam alijs cauſarum generibus in demonſtrando, vt ſup. <lb></lb>
</s>
          <s id="s.005696">cap. 5. de natura Mathem. patuit. </s>
          <s id="s.005697">Et quamuis ſępè demonſtrent ab effe­<lb></lb>ctu, perpetuò tamen euidentiam efficiunt eam, vt nullam, vt ait Themiſtius, <lb></lb>
patiantur inſtantiam.</s>
        </p>
        <p type="main">
          <s id="s.005698">Hortabantur me nonnulli, vt eandem operam locis Mathematicis, quæ <lb></lb>
apud Platonem ſunt, impenderem: quibus dum obtemperare vellem repe­<lb></lb>
ri Theonem quendam Smirnæum Scriptorem Græcum, iam pridem idem <lb></lb>
præſtitiſſe, cuius opus adhuc Græcum aſſeruatur in Vaticana Bibliotheca, <lb></lb>
vt ait Ioſephus Auria in præf. </s>
          <s id="s.005699">ad ſuum Theodoſium Tripolitam; vbi <lb></lb>ſpondet ſe breui eum è Gręco à ſe conuerſum, in lucem editu­<lb></lb>
rum, quod an pręſtiterit ignoro. </s>
          <s id="s.005700">Curandum igitur eſt, <lb></lb>
à recentioribus Platonicis, vt tandem aliquan­<lb></lb>
do, ne à quopiam actum agatur latina <lb></lb>
voce, ac luce pariter donetur.</s>
        </p>
        <p type="head">
          <s id="s.005701"><emph type="italics"></emph>LAVS DEO.<emph.end type="italics"></emph.end></s>
        </p>
        <pb xlink:href="009/01/320.jpg"></pb>
        <pb xlink:href="009/01/321.jpg"></pb>
        <p type="head">
          <s id="s.005702">CLARORVM <lb></lb>
MATHEMATICORVM <lb></lb>
CHRONOLOGIA</s>
        </p>
        <p type="head">
          <s id="s.005703">Eorum videlicet, qui rebus, aut ſcriptis cla­<lb></lb>
ruerunt, ex certis hiſtorijs deprompta.</s>
        </p>
        <p type="head">
          <s id="s.005704"><emph type="italics"></emph>Omißis tum fabuloſis, tum ob nimiam antiquitatem incertis, <lb></lb>veluti ſunt ea, quæ de Athlante, Zoroaſtro, Endimione, <lb></lb>
Orpheo, Lino, alijsqué traduntur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.005705">Iubal verò pater canentium cithara, &amp; organo, hoc eſt Muſicæ <lb></lb>
auctor, omiſſus eſt eò, quòd nimio interuallo <lb></lb>
cæteros antecedat.</s>
        </p>
        <pb xlink:href="009/01/322.jpg"></pb>
        <pb pagenum="39" xlink:href="009/01/323.jpg"></pb>
        <p type="head">
          <s id="s.005706"><emph type="italics"></emph>PRIMVM SECVLVM INCIPIENS<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table7"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table7"></table.target>
          <row>
            <cell>Ab Orbe cond. anno</cell>
            <cell>3237</cell>
          </row>
          <row>
            <cell>Ante primam Olymp. ann.</cell>
            <cell>76</cell>
          </row>
          <row>
            <cell>Ante Vrbem cond. ann.</cell>
            <cell>100</cell>
          </row>
          <row>
            <cell>Ante Chriſti natiuitatem ann.</cell>
            <cell>852</cell>
          </row>
          <row>
            <cell>Ioſaphat Iudæorum Rege.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Aremulo Latinorum Rege.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.005707"><emph type="italics"></emph>Singula porrò ſecula ex 100. annis constant. </s>
          <s id="s.005708">Anno huius ſeculi 76. <lb></lb>
vel ante Vrb. cond. </s>
          <s id="s.005709">24. Olympiades initium ſumpſerunt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005710">TERPANDER THEBANVS muſicus celeberrimus, Ho­<lb></lb>
meri pronepos, de quo Ariſt. in probl. </s>
          <s id="s.005711">&amp; alij omnes veteres me­<lb></lb>
minerunt: claruit paulo ante primam Olympiad. hoc eſt ante <lb></lb>
Chriſti natiuitatem 800. circiter annis. </s>
          <s id="s.005712">Lyram, quam olim <lb></lb>
Orpheus tetrachordum fecerat, ipſe heptachordum reddidit. <lb></lb>
</s>
          <s id="s.005713">modulos lyricos, &amp; leges fidium inuenit. </s>
          <s id="s.005714">adiecit ſuis, &amp; Homeri carminibus <lb></lb>
modos, quibus canerentur. </s>
          <s id="s.005715">Primus de Muſica ſcripſit. </s>
          <s id="s.005716">Lacædemonios in­<lb></lb>
ter ſe diſſidentes cantus ſuauitate ſedauit. </s>
          <s id="s.005717">quaternas in ludis Pythijs victo­<lb></lb>
rias retuliſſe publicis monumentis traditum eſt. </s>
          <s id="s.005718">ex Plutar. de Muſica.</s>
        </p>
        <p type="main">
          <s id="s.005719">Vt autem intelligas quidnam eſſent leges, de quibus hic, &amp; infra paſſim <lb></lb>
fit mentio, audi Plutarchum de Muſica. </s>
          <s id="s.005720">Omninò inquit cytharæ <expan abbr="cãtus">cantus</expan> Ter­<lb></lb>
pandri ratio omni ex parte ſimplex perexit eſſe <expan abbr="vſq;">vſque</expan> ad ætatem Phrynidis, <lb></lb>
non enim antiquitus pro libidine cuiuſque (vti nunc) licebat fidibus cane­<lb></lb>
re, nec rithmus, concentuſuè transferre: in ipſis <expan abbr="namq;">namque</expan> legibus accommo­<lb></lb>
datam <expan abbr="cuiq;">cuique</expan> tentionem tuebatur, cuius rei cauſa id nominis inditum erat: <lb></lb>
leges enim ſunt vocatæ, quoniam præſcriptum quaſi lege, <expan abbr="cautumq́">cautumque</expan>; erat, <lb></lb>
ne quis per quamlibet vnam ſpeciem, <expan abbr="formamq́">formamque</expan>; tentionis tranſgrederetur.</s>
        </p>
        <p type="main">
          <s id="s.005721">XENOCRATES Italus Locrenſis, poſt Terpandrum, &amp; ante Saca­<lb></lb>
dam vixit, fuit Pæanum conditor. </s>
          <s id="s.005722">Patritius in Poetica.</s>
        </p>
        <p type="main">
          <s id="s.005723">ARDVLVS TRÆZENIVS prior Clona, Tibianam muſicam <lb></lb>
conſtituit. </s>
          <s id="s.005724">Plutarchus de Muſica.</s>
        </p>
        <p type="main">
          <s id="s.005725">CLONAS ad imitationem Terpandri, primus leges Tibiarum fecit. <lb></lb>
</s>
          <s id="s.005726">circa primam Olymp. Plut. de Muſica.</s>
        </p>
        <pb pagenum="40" xlink:href="009/01/324.jpg"></pb>
        <p type="head">
          <s id="s.005727"><emph type="italics"></emph>SECVNDVM SECVLVM INCIPIENS<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table8"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table8"></table.target>
          <row>
            <cell>Ab Vrbecondita.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Ab Orbe cond. ann.</cell>
            <cell>3337</cell>
          </row>
          <row>
            <cell>Olympiadis 6. ann.</cell>
            <cell>4</cell>
          </row>
          <row>
            <cell>Ante Chriſti natiuitatem ann.</cell>
            <cell>752</cell>
          </row>
          <row>
            <cell>Ozia ludæorum Rege.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Romulo Latinorum Rege.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.005728">SACADAS ARGIVVS conditor Modorum: &amp; inuentor legis <lb></lb>
Tripartilis. </s>
          <s id="s.005729">circa Ro. cond. </s>
          <s id="s.005730">Pindaro antiquior. </s>
          <s id="s.005731">ter vicit in ludis Py­<lb></lb>
thijs. </s>
          <s id="s.005732">Plaut. &amp; Patric.</s>
        </p>
        <p type="main">
          <s id="s.005733">EVPHORBVS PHRYX ante Thaletem contemplationem <lb></lb>
de lineis fecit, &amp; triangulum Scalenum inuenit, ideſt (ve opinor) modum <lb></lb>
ipſum conſtruendi excogitauit, hic igitur primus geometrizare cœpit. <lb></lb>
</s>
          <s id="s.005734">Laertius in Thalete.</s>
        </p>
        <p type="head">
          <s id="s.005735"><emph type="italics"></emph>TERTIVM SECVLVM INCIPIENS<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table9"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table9"></table.target>
          <row>
            <cell>Ab Vrbe cond. ann.</cell>
            <cell>101</cell>
          </row>
          <row>
            <cell>Ante Chriſti natiuitatem ann.</cell>
            <cell>652</cell>
          </row>
          <row>
            <cell>Manaſſe Iudæorum.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Tullo Hoſtilio Romanorum Rege.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.005736">THALES MILESIVS ab Vrbe cond. </s>
          <s id="s.005737">ann. </s>
          <s id="s.005738">120. natus. </s>
          <s id="s.005739">hic pri­<lb></lb>
mus omnium ſeptem Sapientum appellatus eſt Sapiens. </s>
          <s id="s.005740">ex Aegy­<lb></lb>
pto primus in Græciam Geometriam tranſtulit. </s>
          <s id="s.005741">inuenit triangu­<lb></lb>
lum in circulo orthogonicum, hoc eſt, ni fallor, 31. tertij Elem. <lb></lb>
</s>
          <s id="s.005742">Quintam, &amp; 15. &amp; 26. primi Elemen. adinuenit. </s>
          <s id="s.005743">illud etiam demonſtrauit, <lb></lb>
Diametrum circulum bifariam ſecare. </s>
          <s id="s.005744">tropicos, &amp; æquinoctialem deſigna­<lb></lb>uit. </s>
          <s id="s.005745">primus eclypſes Solis prædixit: quarum prima teſte Piin. lib 2. cap. 12. <lb></lb>
contigit ann. </s>
          <s id="s.005746">V. C. 170. menſus eſt Aegypti Pyramides ex vmbra. </s>
          <s id="s.005747">inuentor <lb></lb>
fuit Vrſæ minoris, ideſt, eam primus obſeruauit, &amp; alios docuit. </s>
          <s id="s.005748">ex præno­<lb></lb>
tione, atque oliuarum emptione diuitias ſibi comparauit. </s>
          <s id="s.005749">ante Methonem <lb></lb>
ann. </s>
          <s id="s.005750">132.</s>
        </p>
        <p type="main">
          <s id="s.005751">SIMONIDES LYRICVS octauam lyræ chordam addidit: hic <lb></lb>
fuit inuentor Artis memoriæ. </s>
          <s id="s.005752">Plin.</s>
        </p>
        <p type="main">
          <s id="s.005753">LYCAON Muſicus, nouum chordarum ordinem inuexit, quem habes <lb></lb>
loco. </s>
          <s id="s.005754">344. locorum Mathematicorum, &amp; octauam lyræ chordam addidit. <lb></lb>
</s>
          <s id="s.005755">Boetius. </s>
          <s id="s.005756">Zarlinus.</s>
        </p>
        <p type="main">
          <s id="s.005757">MAMERTINVS inſignis Geometra, <expan abbr="quiq́">quique</expan>; multa geometrica ad­<lb></lb>
inuenit. </s>
          <s id="s.005758">Thaleti ſucceſſit.</s>
        </p>
        <p type="main">
          <s id="s.005759">ANAXIMANDER MILESIVS Thaleti ſucceſſor. </s>
          <s id="s.005760">Horologium <pb pagenum="41" xlink:href="009/01/325.jpg"></pb>Solare, Sphæram, <expan abbr="Gnomonemq́">Gnomonemque</expan>, reperit, obliquitatem Zodiaci obſeruauit. <lb></lb>
</s>
          <s id="s.005761">Terræ circuitum reperit. </s>
          <s id="s.005762">primus tabulam Geographicam expoſuit. </s>
          <s id="s.005763">primus <lb></lb>
Lunam aliena luce lucere demonſtrauit. </s>
          <s id="s.005764">ſphæram conſtruxit. </s>
          <s id="s.005765">Plin. Laert.</s>
        </p>
        <p type="main">
          <s id="s.005766">CLEOSTRATVS Zodiacum in 12. ſigna diuiſit. </s>
          <s id="s.005767">paulo poſt Ana­<lb></lb>
ximandrum. </s>
          <s id="s.005768">Plin.</s>
        </p>
        <p type="main">
          <s id="s.005769">HECATEVS MILESIVS primus codicem de ſitu orbis reliquit. <lb></lb>
</s>
          <s id="s.005770">paulo poſt Thaletem. </s>
          <s id="s.005771">Proclus in comm. Euclidis.</s>
        </p>
        <p type="main">
          <s id="s.005772">AMETISTVS ſummus Geometra, atque rerum geometricarum in­<lb></lb>
uentor, frater Steſichori poetæ. </s>
          <s id="s.005773">inter Thaletem, &amp; Pyth. </s>
          <s id="s.005774">Proclus.</s>
        </p>
        <p type="main">
          <s id="s.005775">POLEMON auditor Panetij Rhodij, tempore Ariſtophanis gram­<lb></lb>
matiei, orbis deſcriptionem fecit. </s>
          <s id="s.005776">Suidas.</s>
        </p>
        <p type="main">
          <s id="s.005777">SAPPHO Poetria, &amp; Muſica. </s>
          <s id="s.005778">prima Plectri vſum inuexit, cùm prius <lb></lb>
digitis ſolum pulſarent. </s>
          <s id="s.005779">inuentrix etiam Mixtolydij concentus. </s>
          <s id="s.005780">Plut.</s>
        </p>
        <p type="main">
          <s id="s.005781">PYTHAGORAS SAMIVS, Aegypto, ac Perſide perluſtrata, in <lb></lb>
Mathematicis excelluit. </s>
          <s id="s.005782">primus Numerorum ſcientiam apud græcos illu­<lb></lb>
ſtrauit. </s>
          <s id="s.005783">Muſicæ theoricam ex Fabri malleis adinuenit. </s>
          <s id="s.005784">Luciferum, <expan abbr="atq;">atque</expan> He­<lb></lb>
ſperum, quæ duo ſydera putabantur, eſſe vnum, <expan abbr="atq;">atque</expan> idem Veneris aſtrum <lb></lb>
docuit. </s>
          <s id="s.005785">omne triangulum habere tres, &amp;c. </s>
          <s id="s.005786">quæ eſt 32. primi Elem. primi <lb></lb>
Pythag. demonſtrarunt. </s>
          <s id="s.005787">47. primi Elem. reperit, pro qua Muſis Hecatombas <lb></lb>
immolauit. </s>
          <s id="s.005788">primus Mathematicæ ludum aperuit. </s>
          <s id="s.005789">Proclus. </s>
          <s id="s.005790">Plin.</s>
        </p>
        <p type="main">
          <s id="s.005791">TELAVGES filius Pythagoræ, &amp; magiſter Empedoclis, ſcripſit de <lb></lb>
Quaternario libros 4.</s>
        </p>
        <p type="main">
          <s id="s.005792">ANAXIMENAS MILESIVS tempore Pythagoræ, dixit ſydera <lb></lb>
non ſolum ſupra terram, ſed circa terram moueri. </s>
          <s id="s.005793">Laert.</s>
        </p>
        <p type="main">
          <s id="s.005794">DAMON Muſicus, Pythag. <expan abbr="adoleſcẽtes">adoleſcentes</expan>, aliquot luxuriæ deditos, har­<lb></lb>
monicis canticis ad bonam frugem reuocauit. </s>
          <s id="s.005795">Zarlinus.</s>
        </p>
        <p type="head">
          <s id="s.005796"><emph type="italics"></emph>QVARTVM SECVLVM INCIPIENS<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table10"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table10"></table.target>
          <row>
            <cell>Ab Vrbe cond. ann.</cell>
            <cell>201</cell>
          </row>
          <row>
            <cell>Ante Chriſtum ann.</cell>
            <cell>552</cell>
          </row>
          <row>
            <cell>Sub Babylonica captiuitate Iudæorum.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Seruio Tullio Romanorum Rege.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.005797"><emph type="italics"></emph>Anno 44. huius ſeculi, Romæ exactis Regibus Conſſ. ſufficiuntur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005798">ANAXAGORAS CLAZOMENIVS primus lunarem <lb></lb>
eclypſim prædixit, <expan abbr="eiusq́">eiusque</expan>; cauſam patefecit. </s>
          <s id="s.005799">primus librum in lu­<lb></lb>cem edidit. </s>
          <s id="s.005800">dixit Solem eſſe maiorem Pelopomneſo. </s>
          <s id="s.005801">ſcripſit de <lb></lb>
radijs viſiuis, vel de ratione ſcenographices. </s>
          <s id="s.005802">Vitr. lib. 7. Val. <lb></lb>
Max. Diog. Laert.</s>
        </p>
        <p type="main">
          <s id="s.005803">OENIPEDES CHIVS Democriti ſyncronus, inuenit 12. &amp; 23. <lb></lb>
primi Elem. huius diſcipulus fuit Zenodorus.</s>
        </p>
        <p type="main">
          <s id="s.005804">ZENODORVS auctor tractatus de figuris Iſoperimetris, qui eſt <lb></lb>
apud Clauium in ſphæra, &amp; in Geometric. pract. </s>
          <s id="s.005805">Theon enim, ex quo de­<pb pagenum="42" xlink:href="009/01/326.jpg"></pb>ſumpſit Clauius, eum Zenodoro attribuit.</s>
        </p>
        <p type="main">
          <s id="s.005806">PERICLES diſcipulus Anaxagoræ, &amp; Athenienſium Princeps; Athe­<lb></lb>
nienſes ob tetram Solis eclypſim trepidantes, &amp; palantes, eclypſis natura <lb></lb>
expoſita, ſedauit. </s>
          <s id="s.005807">Val. Max.</s>
        </p>
        <p type="main">
          <s id="s.005808">HIPPOCRATES CHIVS, qui dum circulum quadrare conare­<lb></lb>
tur, lunulam quadrauit. </s>
          <s id="s.005809">eius circuli quadraturam Ariſt. ſæpè in paralogiſmi <lb></lb>
exemplum adducit, <expan abbr="eamq́">eamque</expan>; breuiter expoſuit. </s>
          <s id="s.005810">primus Elementa Geometri­<lb></lb>
ca conſcripſit. </s>
          <s id="s.005811">primus inſpexit duabus medijs proportionalibus <expan abbr="inuẽtis">inuentis</expan> cu­<lb></lb>bum duplari poſſe. </s>
          <s id="s.005812">Eratoſthenes apud Eutocium in commen. Archimedis. <lb></lb>
</s>
          <s id="s.005813">Proclus etiam.</s>
        </p>
        <p type="main">
          <s id="s.005814">THEODORVS CYRENEVS ſodalis Hippocratis Chij, multis <lb></lb>
Geometriam auxit. </s>
          <s id="s.005815">Proclus.</s>
        </p>
        <p type="main">
          <s id="s.005816">PHRINIS inſignis Cytharedus. </s>
          <s id="s.005817">primus apud Athenienſes cithara ce­<lb></lb>
cinit. </s>
          <s id="s.005818"><expan abbr="primasq́">primasque</expan>; in Panathenaicis tulit. </s>
          <s id="s.005819">fuit diſcipulus Ariſtoclis, qui ex Ter­<lb></lb>
pandri familia ortum trahebat.</s>
        </p>
        <p type="main">
          <s id="s.005820">PHRINICVS, cuius Ariſt. in Problem. muſicis cecinit: inſignis mu­<lb></lb>
ſicus, ac tetrametri carminis inuentor.</s>
        </p>
        <p type="main">
          <s id="s.005821">LASVS HERMINÆVS primus omnium de Muſica ſcripſit. </s>
          <s id="s.005822">Darij <lb></lb>
Hidaſpis tempore. </s>
          <s id="s.005823">Suida.</s>
        </p>
        <p type="main">
          <s id="s.005824">DIOCLES de Muſica ſcripſit. </s>
          <s id="s.005825">Suida.</s>
        </p>
        <p type="main">
          <s id="s.005826">ISMENIVS CHORAVLES, teſte Boetio, multis ægritudine la­<lb></lb>
borantibus, ſono, &amp; cantu omnes animi moleſtias deterſit. </s>
          <s id="s.005827">eius æqualis fuit <lb></lb>
Dionyſiodorus, &amp; Nicomachus. </s>
          <s id="s.005828">Plin.</s>
        </p>
        <p type="main">
          <s id="s.005829">NICOMACHVS Arithmeticus, quem Boetius ſequitur, &amp; cuius ex­<lb></lb>
tat græca arithmetica, vbi &amp; de muſica tractat ex Zarlino 8. ſupplem. </s>
          <s id="s.005830">mu­<lb></lb>
ſicalium. </s>
          <s id="s.005831">Pappus lib. 3. eum Pythagoricum appellat; Iſidorus lib. 3. ethym. <lb></lb>
</s>
          <s id="s.005832">videtur eum paulo poſt Pythagoram collocare. </s>
          <s id="s.005833">eiuſdem meminit Eutocius.</s>
        </p>
        <p type="main">
          <s id="s.005834">EMPEDOCLES AGRIGENTINVS Pythagoricus, cantu furibundum <lb></lb>
adoleſcentem, ac nudo ferro hoſtem impetentem compreſſit, ac ſedauit.</s>
        </p>
        <p type="main">
          <s id="s.005835">TIMÆVS LOCRVS Pythagoricus, Mathemata ſcripſit, teſte Sui­<lb></lb>
da. </s>
          <s id="s.005836">paulo maior Platone, à quo Plato ſuum Timæum inſcripſit, ac partim <lb></lb>
deſcripſit. </s>
          <s id="s.005837">extat adhuc ipſius monumentum de natura mundi.</s>
        </p>
        <p type="main">
          <s id="s.005838">SIMON Philoſophus Socratis amicus; ſcripſit Dialogum de muſica. <lb></lb>
</s>
          <s id="s.005839">Diog. Laert.</s>
        </p>
        <p type="main">
          <s id="s.005840">CRATISTVS, qui ſolo naturæ acumine, quoduis geometricum Pro­<lb></lb>
blema, quamuis difficile reſoluebat. </s>
          <s id="s.005841">Proclus.</s>
        </p>
        <pb pagenum="43" xlink:href="009/01/327.jpg"></pb>
        <p type="head">
          <s id="s.005842"><emph type="italics"></emph>QVINTVM SECVLVM INCIPIENS<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table11"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table11"></table.target>
          <row>
            <cell>Ab Vrbe cond. ann.</cell>
            <cell>301</cell>
          </row>
          <row>
            <cell>Ante Chriſtum ann.</cell>
            <cell>452</cell>
          </row>
          <row>
            <cell>P. Sextio, T. Memnio Conſſ.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.005843"><emph type="italics"></emph>In quo Socrates ſeptuagenarius anno ab Vrbe cond. </s>
          <s id="s.005844">353. <lb></lb>
Olymp. 95. ann. </s>
          <s id="s.005845">1. moruur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005846">DEMOCRITVS MILESIVS anno vno maior natu quàm <lb></lb>
Socrates. </s>
          <s id="s.005847">ſcripſit de contactu circuli, &amp; ſphæræ. </s>
          <s id="s.005848">de Geometria. <lb></lb>
</s>
          <s id="s.005849">de lineis irrationalibus. </s>
          <s id="s.005850">de ſolidis. </s>
          <s id="s.005851">de numeris geometricis. </s>
          <s id="s.005852">Har­<lb></lb>
monica, ſiue de Muſica. </s>
          <s id="s.005853">de concentu, &amp; harmonia. </s>
          <s id="s.005854">Actinogra­<lb></lb>
phiam, ſiue de radijs, ſiue de Perſpectiua, ſiue de Scenographice. </s>
          <s id="s.005855">certamen <lb></lb>
Clepſydræ. </s>
          <s id="s.005856">de Planetis, de anno magno. </s>
          <s id="s.005857">Cœli, <expan abbr="tęrræq́">tęrræque</expan>; <expan abbr="deſcriptionẽ">deſcriptionem</expan>. </s>
          <s id="s.005858">Laert.</s>
        </p>
        <p type="main">
          <s id="s.005859">ANTISTHENES Socratis auditor de Muſica <expan abbr="cõmentatus">commentatus</expan> eſt. </s>
          <s id="s.005860">Laer.</s>
        </p>
        <p type="main">
          <s id="s.005861">SIMMIAS Thebanus diſcipulus Socratis, de Muſica. </s>
          <s id="s.005862">Laert. </s>
          <s id="s.005863">Suid.</s>
        </p>
        <p type="main">
          <s id="s.005864">PARMENIDES ELEATES primus dixit terram eſſe <expan abbr="rotũdam">rotundam</expan>, <lb></lb>
&amp; in medio mundi ſitam. </s>
          <s id="s.005865">Laert.</s>
        </p>
        <p type="main">
          <s id="s.005866">PROTAGORAS de Mathematicis ſcripſit. </s>
          <s id="s.005867">Laert.</s>
        </p>
        <p type="main">
          <s id="s.005868">EPICVRVS ille Epicureorum Sectarius, de Muſica ſcripſit.</s>
        </p>
        <p type="main">
          <s id="s.005869">METON, &amp; EVCTEMON ante Alex. Mag. obitum ann. </s>
          <s id="s.005870">108. Athenis ſolſti­<lb></lb>
tium obſeruarunt. </s>
          <s id="s.005871">Methon primus tempeſtatum <lb></lb>
prognoſtica ſingulis annis edidit. </s>
          <s id="s.005872">primus cyclum Enneadecaterida, vel Au­<lb></lb>
reum numerum in Græcia inſtituit; qui annus Methonis dicitur.</s>
        </p>
        <p type="main">
          <s id="s.005873">PLATO, Socratis auditor, fuit maximè Mathematum ſtudioſus; nam <lb></lb>
ſingulis diebus auditorib. </s>
          <s id="s.005874">ſuis geometricum problema proponebat. </s>
          <s id="s.005875">Ageo­<lb></lb>
metretos è ſchola arcebat. </s>
          <s id="s.005876">primus ſectiones conicas, &amp; cylindericas inchoa­<lb></lb>
uit. </s>
          <s id="s.005877">Modum demonſtrandi per Analyſim inuenit, item modum agri dime­<lb></lb>
tiendi pulcherrimum, vt apud Vitr. conſtat. </s>
          <s id="s.005878">Delij eum tanquam oraculum <lb></lb>
conſuluerunt de modo aræ, ſiue cubi duplicandi, quos tamen ipſe ad Eucli­<lb></lb>
dem Geometram modeſtè abire iuſſit: non <expan abbr="tamẽ">tamen</expan> omnino à problemate ab­<lb></lb>
ſtinuit, extat enim apud <expan abbr="Eutociũ">Eutocium</expan> Platonis modus inuendi duas medias pro­<lb></lb>
portionales, quibus inuentis, cubi duplicatio peracta eſſet. </s>
          <s id="s.005879">in ſuis pręterea <lb></lb>
Dialogis <expan abbr="cõplura">complura</expan> habet Mathematica, quæ olim Theon Smyrneus, ac Phi­<lb></lb>
lippus Mendæus commentarijs illuſtrarunt. </s>
          <s id="s.005880">Proclus Lucr.</s>
        </p>
        <p type="main">
          <s id="s.005881">AMICLAS HERACLEOTES Platonis familiaris geometricas <lb></lb>
inuentiones amplificauit.</s>
        </p>
        <p type="main">
          <s id="s.005882">LEODAMAS THASIVS à Platone Analyſim didicit, cuius ope <lb></lb>
multa geometrica excogitauit.</s>
        </p>
        <p type="main">
          <s id="s.005883">NEOCLIDES LEODAMANTE iunior, inter rerum geome­<lb></lb>
tricarum repertores connumeratur. </s>
          <s id="s.005884">Proc.</s>
        </p>
        <p type="main">
          <s id="s.005885">LEON diſcipulus Neoclidis: Determinationem geometricam inuenit, <lb></lb>
quæ diſtinguit problema poſſibile ab impoſſibili. </s>
          <s id="s.005886">Geometrica elementa, <lb></lb>ſecundus ab Hippocrate, ſed accuratius conſtruxit. </s>
          <s id="s.005887">Procl.</s>
        </p>
        <pb pagenum="44" xlink:href="009/01/328.jpg"></pb>
        <p type="main">
          <s id="s.005888">EVDOXIVS GNIDVIS Aſtronomus, Leonte iunior, &amp; Platonis <lb></lb>
Comes in Aegyptum. </s>
          <s id="s.005889">Quintum elem. </s>
          <s id="s.005890">Euclidis de proportionibus inuenit. <lb></lb>
</s>
          <s id="s.005891">Theoremata multa vniuerſalia reddidit, inuenit etiam Arachnen, horolo­<lb></lb>
gium, videlicet ſolare, in quo lineæ horariæ, &amp; arcus ſignorum in modum <lb></lb>araneæ ſe ſecant. </s>
          <s id="s.005892">Vitr. </s>
          <s id="s.005893">Octaetidem, ideſt, Solis, ac Lunæ per octonos an­<lb></lb>
nos recurſus docuit. </s>
          <s id="s.005894">ſcripſit de Geometria, &amp; Aſtronomia. </s>
          <s id="s.005895">Mathematicas <lb></lb>
ad vſum mechanicum vnà cum Archita traducere conatus eſt: quos am­<lb></lb>
bos Plato redarguit, quòd Philoſophiam proſtituiſſent.</s>
        </p>
        <p type="main">
          <s id="s.005896">ARCHITA TERENTIVS Mechanice inuentor: reprehenſus à <lb></lb>Platone, vt modo dictum eſt. </s>
          <s id="s.005897">Cubum reperit. </s>
          <s id="s.005898">ligneam <expan abbr="columbã">columbam</expan> volantem <lb></lb>
exhibuit. </s>
          <s id="s.005899">qua præterea ratione duas medias reperiret, extat apud <expan abbr="Eutociũ">Eutocium</expan>.</s>
        </p>
        <p type="main">
          <s id="s.005900">THEÆTETVS ATHENIENSIS Architæ Tarentini ſodalis, <lb></lb>
cum quo Geometrica auxit. </s>
          <s id="s.005901">Procl. primus de <expan abbr="quinq;">quinque</expan> ſolidis tractauit. </s>
          <s id="s.005902">Laer. <lb></lb>
inuenit 10. decimi.</s>
        </p>
        <p type="main">
          <s id="s.005903">BRYSO, &amp; ANTIPHON Circuli quadrationem inuenire conantur. </s>
          <s id="s.005904">extant <lb></lb>
ipſorum conatus apud Ariſt. quos explicauimus in <lb></lb>
locis Math.</s>
        </p>
        <p type="main">
          <s id="s.005905">PHILIPPVS MENDÆVS diſcipulus Platonis. </s>
          <s id="s.005906">loca Mathema­<lb></lb>
tica operum Platonis explicauit. </s>
          <s id="s.005907">Comperit Iridem inſequentes ſe fugere, <lb></lb>
fugientes verò inſequi.</s>
        </p>
        <p type="main">
          <s id="s.005908">HELICO CYGICENVS Platonis familiaris, cùm Dyoniſio Re­<lb></lb>
gi ſolis defectum, qui tunc accidit, antea multò prænunciaſſet, Rex ſumma <lb></lb>
admiratione affectus, argenti talentum ei donauit.</s>
        </p>
        <p type="main">
          <s id="s.005909">PHILOSOPHVS, Platonis auditor. </s>
          <s id="s.005910">De interuallo Solis, &amp; Lunæ. </s>
          <s id="s.005911">de <lb></lb>
eclypſi. </s>
          <s id="s.005912">de magnitudine Solis, Lunæ, &amp; terræ. </s>
          <s id="s.005913">de Planetis. </s>
          <s id="s.005914">de Arithmetica. <lb></lb>
</s>
          <s id="s.005915">de numeris fecundis, de opticis, de circularibus, &amp; medietatibus egit. </s>
          <s id="s.005916">Suida.</s>
        </p>
        <p type="head">
          <s id="s.005917"><emph type="italics"></emph>SEXTVM SECVLVM INCIPIENS<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table12"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table12"></table.target>
          <row>
            <cell>Ab Vrbe cond. ann.</cell>
            <cell>401</cell>
          </row>
          <row>
            <cell>Ante Chriſti Natiuitatem ann.</cell>
            <cell>352</cell>
          </row>
          <row>
            <cell>Tito Manlio Dictatore.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.005918"><emph type="italics"></emph>In quo Alex. Mag. imperauit: <expan abbr="obijtq́">obijtque</expan>; ann. </s>
          <s id="s.005919">ab Orbe cond. </s>
          <s id="s.005920">3791. <lb></lb>
ab Vrbe cond. </s>
          <s id="s.005921">425.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005922">THEVDIVS MAGNES, elementa geometrica tertius con­<lb></lb>
ſtruxit. </s>
          <s id="s.005923">Procl.</s>
        </p>
        <p type="main">
          <s id="s.005924">CYGICINVS ATHENIENSIS geometrica amplia­<lb></lb>
uit. </s>
          <s id="s.005925">Procl.</s>
        </p>
        <p type="main">
          <s id="s.005926">HERMOTIMVS COLOPHONIVS, quartus elementa geo­<lb></lb>
metrica vberiora reddidit.</s>
        </p>
        <p type="main">
          <s id="s.005927">ARISTÆVS ſenior, ante Euclidem demonſtrauit de conicis: quem <lb></lb>Euclides in ijſdem ſequutus eſt. </s>
          <s id="s.005928">Item de reſolutione. </s>
          <s id="s.005929">Item de locis ſolidis, <lb></lb>
lib. 5. Pappus.</s>
        </p>
        <pb pagenum="45" xlink:href="009/01/329.jpg"></pb>
        <p type="main">
          <s id="s.005930">GEMINVS demonſtrauit linearum tres tantùm eſſe ſimilares, <expan abbr="rectã">rectam</expan>, <lb></lb>
circularem, &amp; ſpiralem cylindricam. </s>
          <s id="s.005931">Ortus <expan abbr="quoq;">quoque</expan> linearum ſpiricarum, &amp; <lb></lb>
conchoidum, &amp; ciſſoidum, tradidit. </s>
          <s id="s.005932">Propoſitionem quintam primi ele­<lb></lb>
mentorum vniuerſalius, quam Thales, demonſtrauit; oſtendit enim æqua­<lb></lb>les lineas rectas ab vno puncto ad vnam ſimilium partium lineam, ideſt, vel <lb></lb>
ad rectam, vel ad circularem, vel ad cylindricam, <expan abbr="incidẽtes">incidentes</expan>, facere angulos <lb></lb>
ad baſim æquales. </s>
          <s id="s.005933">Scripſit præterea lib. 6. <expan abbr="geometricarũ">geometricarum</expan> <expan abbr="enarrationũ">enarrationum</expan>. </s>
          <s id="s.005934">Procl.</s>
        </p>
        <p type="main">
          <s id="s.005935">PERSEVS CITTICVS Poſt Geminum: inuenit lineas ſpiricas. <lb></lb>
</s>
          <s id="s.005936">Proclus.</s>
        </p>
        <p type="main">
          <s id="s.005937">MENECHMVS EVDOXI diſcipulus, ſectiones conicas reperit. <lb></lb>
</s>
          <s id="s.005938">Tribus proportionibus tres alias adiecit. </s>
          <s id="s.005939">modus ipſius <expan abbr="inueniẽdi">inueniendi</expan> duas me­<lb></lb>
dias, extat apud Eutocium.</s>
        </p>
        <p type="main">
          <s id="s.005940">DINOSTRATVS Menechmi frater, geometrica <expan abbr="cõplura">complura</expan> reperit.</s>
        </p>
        <p type="main">
          <s id="s.005941">XENOCRATES CHALCEDONIVS Platonis audit. </s>
          <s id="s.005942">de geo­<lb></lb>
metria primum duos, deinde <expan abbr="quinq;">quinque</expan> lib. compoſuit. </s>
          <s id="s.005943">Item de numeris lib. 1. <lb></lb>
De aſtrologia lib. 6. Diog. Laert.</s>
        </p>
        <p type="main">
          <s id="s.005944">EVCLIDES Megarenſis geometra, Platonis tempore, ſiquidem te­<lb></lb>
ſte Valer. Maxim. ad eum Plato, Delios aræ ſacrę conductores amandauit. <lb></lb>
</s>
          <s id="s.005945">Alexandriæ longo tempore dedit operatu diſcipulis, vnde excellentem in <lb></lb>
Mathematicis habitum conſequutus eſt, <expan abbr="neq;">neque</expan> <expan abbr="vſquã">vſquam</expan> deceptus eſt. </s>
          <s id="s.005946">Papp. lib. 7. <lb></lb>
vixit autem, &amp; claruít, vſque ad Ptolæmeum primum Aegypti Regem, vt <lb></lb>vult Procl ſit ne idem cum Euclide Megarenſi auctore ſectæ Megaricæ, du­<lb></lb>
bitatur. </s>
          <s id="s.005947">Quintus geometrica elementa mira methodo contexuit. </s>
          <s id="s.005948">ipſius ex­<lb></lb>
tant etiam Phœnomena, optica; catoptrica. </s>
          <s id="s.005949">muſica. </s>
          <s id="s.005950">data. </s>
          <s id="s.005951">Item de reſolu­<lb></lb>
tione, de fallacijs, de locis ad ſuperficiem lib. 2. Conicorum lib. 4. Item <lb></lb>
Poriſmatum lib. 3. quæ perierunt. </s>
          <s id="s.005952">Pappus, &amp; Procl.</s>
        </p>
        <p type="main">
          <s id="s.005953">L. PAPYRIVS curſor Romę primum ſolare horologium publico lo­<lb></lb>
co conſtruxit. </s>
          <s id="s.005954">Plin.</s>
        </p>
        <p type="main">
          <s id="s.005955">HERMOPHILVS cœcus Theopompum Geometriam ſine abaco, <lb></lb>
ac radio docuit.</s>
        </p>
        <p type="main">
          <s id="s.005956">ARATVS Poeta Græcus, ante Hipparchum centum ferè annis, de <lb></lb>
Cœlo, <expan abbr="ſtellisq́">ſtellisque</expan>; eleganter cecinit.</s>
        </p>
        <p type="main">
          <s id="s.005957">CALIPPVS Cygicenus aſtronomus inſignis, cuius Ariſtot. in Meta­<lb></lb>
phyſic. meminit. </s>
          <s id="s.005958">periodum 76. annorum ex quatuor Methonis cyclis con­<lb></lb>
flauit, qua Sol, &amp; Luna iterum ad priſtina reſtituantur: initium prime pe­<lb></lb>
riodi ſtatuit <expan abbr="obitũ">obitum</expan> Darij Regis, ſeu initium Monarchiæ Gręcorum. </s>
          <s id="s.005959">ex Al­<lb></lb>
mag. Ptol.</s>
        </p>
        <p type="main">
          <s id="s.005960">ARISTOT. </s>
          <s id="s.005961">Platonis auditor, &amp; Alex. Magn. præceptor, ſcripſit Me­<lb></lb>
chanicas quæſtiones. </s>
          <s id="s.005962">librum vnum, quem appellauit <expan abbr="Mathematicũ">Mathematicum</expan>. </s>
          <s id="s.005963">ſect. </s>
          <s id="s.005964">19. <lb></lb>
problematum de muſica Item alium librum de muſica. </s>
          <s id="s.005965">Halonis, &amp; Iridis <lb></lb>
demonſtrationes apud ipſum primum reperiuntur. </s>
          <s id="s.005966">paſſim in ſuis operibus <lb></lb>omnis generis Mathemata ingerit.</s>
        </p>
        <p type="main">
          <s id="s.005967">AVTOLYCVS præceptor Arceſilai, floruit circa Olymp. 120. extat <lb></lb>eius ſubtilis admodum liber de ſphæra, quæ mouetur, &amp; alter de vario ortu <lb></lb>
&amp; occaſu ſyderum. </s>
          <s id="s.005968">Diog Laer. in Arcelilao.</s>
        </p>
        <p type="main">
          <s id="s.005969">THEOPHRASTVS Eriſſius Ariſt. diſcipulus, &amp; succeſſor, reliquit <pb pagenum="46" xlink:href="009/01/330.jpg"></pb>tres libros de muſica, vnum de muſicis. </s>
          <s id="s.005970">Harmonicorum vnum. </s>
          <s id="s.005971">de menſu­<lb></lb>
ris vnum. </s>
          <s id="s.005972">de numeris vnum. </s>
          <s id="s.005973">Hiſtoriarum geometricarum quatuor. </s>
          <s id="s.005974">Aſtro­<lb></lb>
logicæ hiſtoriæ 6. Arithmeticarum hiſtoriarum vnum. </s>
          <s id="s.005975">de lineis indiuiduis. <lb></lb>
</s>
          <s id="s.005976">Diog. Laert.</s>
        </p>
        <p type="main">
          <s id="s.005977">HERACLIDES Ponticus, Speuſippi, &amp; Ariſt. auditor. </s>
          <s id="s.005978">de Muſica <lb></lb>
lib. duos. </s>
          <s id="s.005979">de Geometria etiam ſcripſit. </s>
          <s id="s.005980">Diog. Laert.</s>
        </p>
        <p type="main">
          <s id="s.005981">DICEARCHVS Siculus, Ariſt, auditor, primus montium altitudi­<lb></lb>
nem perpendicularem dimenſus eſt; altiſſimum prodidit Pelion, nimirum <lb></lb>
1250. paſſuum. </s>
          <s id="s.005982">Plinius lib. 2. c. 67.</s>
        </p>
        <p type="main">
          <s id="s.005983">ARISTOXENVS Tarentinus Muſicus, auditor Ariſt. eius extant <lb></lb>
harmonicorum lib. 3. Suid.</s>
        </p>
        <p type="main">
          <s id="s.005984">CONON Geometra, &amp; Aſtronomus inſignis. </s>
          <s id="s.005985">Ptolæmeo Philadelpho <lb></lb>
gratificaturus, Berenices Comam in Cœlum tranſulit. </s>
          <s id="s.005986">libros 6. de Aſtrolo­<lb></lb>
gia compoſuit. </s>
          <s id="s.005987">de eo Virgil.</s>
        </p>
        <p type="main">
          <s id="s.005988"><emph type="italics"></emph>In medio duo ſigna, Conon. &amp; quis fuit alter.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.005989">ille verò alter Pontano noſtro in Virgilium eſt Archimedes Cononi ſynero­<lb></lb>
nus, &amp; familiaris. </s>
          <s id="s.005990">Hunc plurimi faciebat Archimedes, <expan abbr="eiusq́">eiusque</expan>; propterea <lb></lb>
mortem in lib. de Quadrat. </s>
          <s id="s.005991">Parab. deflet.</s>
        </p>
        <p type="main">
          <s id="s.005992">TIMOTHEVS Muſicus, cùm ad Alexandri magni menſam orthium <lb></lb>
modum caneret, regem velut inſanum coegit ad arma: rurſus remittente <lb></lb>
cantu, regis etiam furorem remiſit. </s>
          <s id="s.005993">Chromaticum genus inuexit. </s>
          <s id="s.005994">ſeptem <lb></lb>
Terpandri chordis quatuor addidit. </s>
          <s id="s.005995">Plut. de muſic.</s>
        </p>
        <p type="main">
          <s id="s.005996">ARCHELAVS Chorographus, omnem terram ab Alex. Mag. pera­<lb></lb>
gratam deſcripſit. </s>
          <s id="s.005997">Diog. Laert.</s>
        </p>
        <p type="main">
          <s id="s.005998">XENOPHANTVS Muſicus, certis modis Alex. Magn. ad arma pro <lb></lb>
libito concitabat.</s>
        </p>
        <p type="main">
          <s id="s.005999">ARISTARCHVS Samius, hoc tempore ante <expan abbr="Hipparchũ">Hipparchum</expan> 200. ann. <lb></lb>
</s>
          <s id="s.006000">Scaphen, ſeu Hemiſphærium, hoc eſt horologium ſciothericum in concauo <lb></lb>
hemiſphærico deſcripſit Vitr. extat ipſius egregium monumentum ingenij, <lb></lb>
libellus de magnitudine, &amp; diſtantijs Solis, &amp; Lunæ.</s>
        </p>
        <p type="main">
          <s id="s.006001">BEROSVS Chaldæus, tempore Antiochi Sotheris hemicyclium ex­<lb></lb>
cauatum, genus horologij ſolaris reperit. </s>
          <s id="s.006002">dicebat Lunam eſſe pilam ex di­<lb></lb>
midia parte candentem: reliqua habere ceruleo colore. </s>
          <s id="s.006003">cætera apud Vitr. <lb></lb>
lib. 9. ei Athenienſes ob diuinas prædictiones publicè in gymnaſio ſtatuam <lb></lb>
inaurata lingua ſtatuere. </s>
          <s id="s.006004">Plin.</s>
        </p>
        <p type="main">
          <s id="s.006005">ARISTILLVS Aſtronomus, cuius obſeruationes circa inerrantes <lb></lb>
ſtellas, ſæpè Ptol. 7. Magnæ conſtr. </s>
          <s id="s.006006">recenſet. </s>
          <s id="s.006007"><expan abbr="videturq́">videturque</expan>; eum Timocharide <lb></lb>
paulò antiquiorem facere.</s>
        </p>
        <p type="main">
          <s id="s.006008">TIMOCHARIS ante Hipparchum ann. </s>
          <s id="s.006009">156. ab obitu verò Alexan. <lb></lb>
Mag. ann. </s>
          <s id="s.006010">41. ſuas obſeruationes habuit, quas Ptolæmeus in Almageſto re­<lb></lb>
cenſet. </s>
          <s id="s.006011">obſeruauit primam ſtellam Arietis poſt ſectionem vernalem gr. 2.</s>
        </p>
        <pb pagenum="47" xlink:href="009/01/331.jpg"></pb>
        <p type="head">
          <s id="s.006012"><emph type="italics"></emph>SEPTIMVM SECVLVM INCIPIENS<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table13"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table13"></table.target>
          <row>
            <cell>Ab Vrbe cond. anno</cell>
            <cell>501</cell>
          </row>
          <row>
            <cell>Ante Chriſti natiuitatem ann.</cell>
            <cell>252</cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006013">Conſſ. D. Lunio.</s>
        </p>
        <p type="main">
          <s id="s.006014">L. Poſthumio.</s>
        </p>
        <p type="main">
          <s id="s.006015">ERATOSTHENES Cyreneus ſub Ptolymæo Euergete primo, &amp; <lb></lb>
duobus ſequentibus regibus, ab obitu Alex. ann. </s>
          <s id="s.006016">90. &amp; totidem an­<lb></lb>
te Hipparchum, aſtronomicis in Aegypto vacabat, <expan abbr="reperitq́">reperitque</expan>; Solis <lb></lb>
declinationem gr. 23. 51. primus terræ ambitum ratione vmbrarum <lb></lb>
Solis inueſtigauit, vt ex Cleomede refert Clauius in ſphæra, vbi eam fusè <lb></lb>
explicat. </s>
          <s id="s.006017">Duplicandi cubi ſummus fuit artifex, vt patet ex eius Meſolabio <lb></lb>
apud Pappum, &amp; Eutocium, <expan abbr="atq;">atque</expan> ob id votiuam tabellam in templo conſe­<lb></lb>
crauit. </s>
          <s id="s.006018">extat epiſtola ipſius ad Regem Ptolæmeum apud Eutocium, de ra­<lb></lb>
tione cubi duplicandi.</s>
        </p>
        <p type="main">
          <s id="s.006019">ARCHIMEDES Syracuſanus ingeniorum Phœnix: quadraginta <lb></lb>
ipſius mira adinuenta Mechanica fuiſſe, tradit Pappus lib. 8. quorum vnum <lb></lb>
fuit; datum pondus data potentia mouere, in quo fertur dixiſſe.</s>
        </p>
        <p type="main">
          <s id="s.006020"><emph type="italics"></emph>Dic vbi contestam, &amp; cœlum terramqué mouebo.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006021">Alterum, quo portionem argenti auro mixtam in Corona illa deprehendit; <lb></lb>
vnde præ lætitia è balneo nudus exiliuit; <expan abbr="atq;">atque</expan> per Vrbem domum <expan abbr="properãs">properans</expan> <lb></lb>
clamabat <foreign lang="grc">ἑυρηκα, ἑυρηκα.</foreign></s>
        </p>
        <p type="main">
          <s id="s.006022">Tertium ſit ſphæra illa vitrea, Automa celebre, quæ ſyderum <expan abbr="omniũ">omnium</expan> mo­<lb></lb>
tus mirè imitabatur. </s>
          <s id="s.006023">de qua Claudianus pulcherrimum illud texit epigram­<lb></lb>
ma: Iupiter in paruo, &amp;c.</s>
        </p>
        <p type="main">
          <s id="s.006024">Quartum, ſpecula parabolica conſtruxit, quibus hoſtium naues procul <lb></lb>
comburerentur.</s>
        </p>
        <p type="main">
          <s id="s.006025">Quintum, chocleam aquaticam (ex Vitruuio, &amp; Diodoro) qua in altum <lb></lb>
aqua effertur excogitauit. </s>
          <s id="s.006026">quam Ioſephus Cedretus noſtra tempeſtate re­<lb></lb>
ſtaurauit.</s>
        </p>
        <p type="main">
          <s id="s.006027">Sextum, nauim graui pondere oneratam, machina quadam facillimè in <lb></lb>
litus attraxit.</s>
        </p>
        <p type="main">
          <s id="s.006028">Septimum, complures bellicas machinas fabricatus eſt, quibus per trien­<lb></lb>
nium contra hoſtes Romanos patriam ſolus tutatus eſt. </s>
          <s id="s.006029">reliqua ipſius adin­<lb></lb>
uenta perierunt. </s>
          <s id="s.006030">Verumenimuerò admiranda mihi magis ipſius ſcripta mo­<lb></lb>
numenta videntur, in quibus quidquid eſt, totum Archimedis eſt. </s>
          <s id="s.006031">quorum <lb></lb>
memoria extat, hæc ſunt. </s>
          <s id="s.006032">De æquaponderantibus lib. 2. Circuli dimenſio. <lb></lb>
</s>
          <s id="s.006033">de lineis ſpiralibus. </s>
          <s id="s.006034">quadratura paraboles. </s>
          <s id="s.006035">de conoidibus, &amp; ſphæroidibus. <lb></lb>
</s>
          <s id="s.006036">de arenæ numero. </s>
          <s id="s.006037">de ijs, quæ vehuntur in aqua. </s>
          <s id="s.006038">de ſphæra, &amp; cylindro. </s>
          <s id="s.006039">de li­<lb></lb>
bra. </s>
          <s id="s.006040">viaticum. </s>
          <s id="s.006041">de ſphæræ <expan abbr="cõſtructione">conſtructione</expan>. </s>
          <s id="s.006042">de 13. ſolidis à ſe inuentis. </s>
          <s id="s.006043">lemmata. <lb></lb>
</s>
          <s id="s.006044">de ſectione circuli. </s>
          <s id="s.006045">de ſpeculis comburentibus. </s>
          <s id="s.006046">M. Marcellus interdixerat <lb></lb>
ne ille vnus captis Syracuſis occideretur: tantus virtuti honos, vel ab hoſti­<lb></lb>
bus haberi par eſt. </s>
          <s id="s.006047">occiſus eſt autem ab ignaro milite, dum in patriæ dire­<lb></lb>
ptione totus cuidam demonſtrationi vacaret.</s>
        </p>
        <pb pagenum="48" xlink:href="009/01/332.jpg"></pb>
        <p type="main">
          <s id="s.006048">CTESIBVS Machinator ſubtiliſſimus: Pneumatica inuenit, ideſt, <lb></lb>
quæ ſpiritu, ac vento motus efficerent, quaſi ſpiritalia. </s>
          <s id="s.006049">hydraulicas machi­<lb></lb>
nas primus conſtruxit. </s>
          <s id="s.006050">adhuc viget machina illa Cteſibij, de qua Vitr. Hy­<lb></lb>
draulica etiam horologia primus exhibuit.</s>
        </p>
        <p type="main">
          <s id="s.006051">SVLPITIVS GALLVS <expan abbr="Cõſul">Conſul</expan>, primus Romani generis rationem <lb></lb>
eclypſium in vulgus edidit, &amp; pridie quam P. Aemilius Perſen Regem ſupe­<lb></lb>
raret, animos militum ob futuram ſequenti die eclypſim trepidaturos, bre­<lb></lb>
ui futuri euentus, admonitione habita, confirmauit. </s>
          <s id="s.006052">Plin. Val. </s>
          <s id="s.006053">Max.</s>
        </p>
        <p type="head">
          <s id="s.006054"><emph type="italics"></emph>OCTAVVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table14"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table14"></table.target>
          <row>
            <cell>Ab Vrbe cond. ann.</cell>
            <cell>601</cell>
          </row>
          <row>
            <cell>Ante Chriſti natiuitatem</cell>
            <cell>152</cell>
          </row>
          <row>
            <cell>Conſ. L. Mummio.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006055">APOLLONIVS PERGÆVS ſub Ptolæmeo euergete ſecun­<lb></lb>
do cognomento magnus Geometra, quod vniuerſaliter de omni <lb></lb>
Cono <expan abbr="elemẽta">elementa</expan> conica octo libris ſubtiliſſimis demonſtraſſet. </s>
          <s id="s.006056">ſcri­<lb></lb>
pſit præterea de ſectione proportionis, &amp; ſpatij. </s>
          <s id="s.006057">de locis planis <lb></lb>
lib. 2. de perturbatis rationibus. </s>
          <s id="s.006058">de tactionibus. </s>
          <s id="s.006059">de inclinationibus. </s>
          <s id="s.006060">de cho­<lb></lb>
clea. </s>
          <s id="s.006061">Pappus. </s>
          <s id="s.006062">modus ipſius inueniendi duas medias, extat apud Eutocium in <lb></lb>
comm. Archimedis. </s>
          <s id="s.006063">pharetram. </s>
          <s id="s.006064">ſolaris horologij genus reperit. </s>
          <s id="s.006065">Vitr.</s>
        </p>
        <p type="main">
          <s id="s.006066">ISIDORVS Philoſophus, Hypſiclis Alexandrini præceptor; nam <lb></lb>
Hypſicles in 15. Elem. vbi ponit inclinationes <expan abbr="quinq;">quinque</expan> <expan abbr="corporũ">corporum</expan> regularium, <lb></lb>
ait ſe eas ab Iſidoro Magno præceptore ſuo accepiſſe. </s>
          <s id="s.006067">Plinius eum citat in <lb></lb>
Geographicis. </s>
          <s id="s.006068">Suidas verò ſic, Iſidorus Philoſophus, vt ſi quis alius philo­<lb></lb>
ſophatus eſt in Mathematis.</s>
        </p>
        <p type="main">
          <s id="s.006069">YPSICLES Alexandrinus, Iſidori diſcipulus, qui libros duos Elemen­<lb></lb>
torum 14. &amp; 15. Euclidi addidit. </s>
          <s id="s.006070">nominat Apollonium. </s>
          <s id="s.006071">videtur his tempo­<lb></lb>
ribus extitiſſe.</s>
        </p>
        <p type="main">
          <s id="s.006072">PHILO BIZANTIVS mechanicus, mechanica fecit ante Hero­<lb></lb>nem, à quo memoratur; hunc exiſtimo eum eſſe, cuius Proclus ad octauam <lb></lb>
primi meminit, referens ipſius demonſtrationem. </s>
          <s id="s.006073">modus ipſius inueniendi <lb></lb>
duas medias legitur apud Eutocium in Archim.</s>
        </p>
        <p type="main">
          <s id="s.006074">POSSIDONIVS Philoſophus Panetij diſcipulus, qui Rhodi tempo­<lb></lb>
re Ciceronis docebat, à Plinio Mathematicus appellatur: à Strabone ve­<lb></lb>
rò citatur in Geographicis. </s>
          <s id="s.006075">Geographica igitur ſcripſit. </s>
          <s id="s.006076">huius ianuæ cùm <lb></lb>
ad eum audiendum Pomponius Magnus adiret, Imperij faſces ſubmiſit.</s>
        </p>
        <p type="main">
          <s id="s.006077">SERENVS Antinſenſis: cuius ſunt Cylindricorum lib. 2. ſubtiliſſimi. <lb></lb>
</s>
          <s id="s.006078">videtur in hæc tempora poſt Apollonium incidiſſe.</s>
        </p>
        <p type="main">
          <s id="s.006079">HERO <expan abbr="Alexãdrinus">Alexandrinus</expan> Cteſibij diſcipulus, eius ſunt Automata; Spirita­<lb></lb>
lia: de Balliſtis. </s>
          <s id="s.006080">Mechanica. </s>
          <s id="s.006081">Barulcos. </s>
          <s id="s.006082">de Rotulis. </s>
          <s id="s.006083">de Horologijs aquaticis. <lb></lb>
</s>
          <s id="s.006084">Camaricha. </s>
          <s id="s.006085">Cambeſtria. </s>
          <s id="s.006086">modus ipſius <expan abbr="inueniẽdi">inueniendi</expan> duas medias legitur apud <lb></lb>
Eutocium. </s>
          <s id="s.006087">Geometrumenon, ideſt Geometria practica. </s>
          <s id="s.006088">de eo fit mentio ad <lb></lb>
ſecundam, &amp; 25. primi elem. </s>
          <s id="s.006089">Proclus. </s>
          <s id="s.006090">Pappus. </s>
          <s id="s.006091">Vitr.</s>
        </p>
        <pb pagenum="49" xlink:href="009/01/333.jpg"></pb>
        <p type="main">
          <s id="s.006092">HIPPARCVS, qui &amp; Abrachis dicitur ab obitu Alexandri ann. </s>
          <s id="s.006093">100. <lb></lb>
ante Ptolemæum 280. obſeruauit maximam Solis declinationem gr. 23. 51. <lb></lb>
Inuenit primam Arietis poſt æquinotij Venrni punctum, gr. 4. <expan abbr="nouã">nouam</expan> ſtellam <lb></lb>
ſuo æuo genitam deprehendit, cuius occaſione in ſyderalem ſcientiam ſeriò <lb></lb>
incubuit. </s>
          <s id="s.006094">primus igitur ſtellas numerauit, <expan abbr="ſuisq́">ſuisque</expan>; locis aſſignauit, organis <lb></lb>
ad id excogitatis. </s>
          <s id="s.006095">Plin. Scripſit de motu Lunæ in latitudinem, &amp; de Arati <lb></lb>
phænomenis. </s>
          <s id="s.006096">ex Suida. </s>
          <s id="s.006097">Tabulas etiam aſtronomicas, teſte Ptol. condidit. <lb></lb>
</s>
          <s id="s.006098">Adhuc extant eius lib. 3. in Arati Phænomena: &amp; vnus Aſteriſmorum, <expan abbr="ſuntq́">ſuntque</expan>; <lb></lb>
Græcè nuper editi.</s>
        </p>
        <p type="main">
          <s id="s.006099">CLEOMEDES his ſeculis gręcè ſcribit Meteora, quibus tractat ea, <lb></lb>
quæ in ſphęra ſolent doceri. </s>
          <s id="s.006100">extat græcolatinus interprete, &amp; ſcholiaſte Ro­<lb></lb>
berto Balforeo, qui eum inter Poſſidonium, &amp; Ptolemæum collocat. </s>
          <s id="s.006101">Item <lb></lb>
Arithmeticam, &amp; Harmonicam, quæ aſſeruantur in Bibliotecha Vaticana, <lb></lb>
&amp; S. Floræ. </s>
          <s id="s.006102">ex eodem Roberto.</s>
        </p>
        <p type="head">
          <s id="s.006103"><emph type="italics"></emph>NONVM SEMISECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table15"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table15"></table.target>
          <row>
            <cell>Continens ann.</cell>
            <cell>52</cell>
          </row>
          <row>
            <cell>Ab Vrbe cond. ann.</cell>
            <cell>701</cell>
          </row>
          <row>
            <cell>Ante Chriſti ortum ann.</cell>
            <cell>52</cell>
          </row>
          <row>
            <cell>Conſ. C. Pomp. Magno. <expan abbr="q.">que</expan> Cæcilio.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006104">THEODOSIVS Tripolita de habitationibus. </s>
          <s id="s.006105">de diebus, &amp; no­<lb></lb>
ctibus. </s>
          <s id="s.006106">ſphæricorum lib. 3. de lineatione ædium. </s>
          <s id="s.006107">Commentaria <lb></lb>
in Theudę capita. </s>
          <s id="s.006108">in viaticum Archimedis. </s>
          <s id="s.006109">Sceptica capita aſtro <lb></lb>
logica. </s>
          <s id="s.006110">de vere. </s>
          <s id="s.006111">Horologium ad omne clima, ideſt vniuerſale ex­<lb></lb>
cogitauit. </s>
          <s id="s.006112">Vitr.</s>
        </p>
        <p type="main">
          <s id="s.006113">SOSIGENES Aſtronomus, cuius opera Iulius Cæſ. Calend. correxit.</s>
        </p>
        <p type="main">
          <s id="s.006114">Sequentes quinque extiterunt ante Vitr. ex quo eos deſumpſimus: ſed <lb></lb>
quanto ignoratur.</s>
        </p>
        <p type="main">
          <s id="s.006115">ATHENÆVS de machinis, cuius extant duo fragmenta græca apud <lb></lb>
Vitr in fine. </s>
          <s id="s.006116">non eſt ille Dipnoſophiſtarum, ille enim vixit in ſecundo Chri­<lb></lb>
ſti ſeculo. </s>
          <s id="s.006117">Eiuſdem mechanica.</s>
        </p>
        <p type="main">
          <s id="s.006118">DIONYSIODORVS, cuius fragmentum extat apud Eut. in <expan abbr="cõ-men">com­<lb></lb>
men</expan>. Archimedis, quo ſubtiliſſima demonſtratio continetur ſecandi ſphæ­<lb></lb>
ram in datam rationem. </s>
          <s id="s.006119">Inuenit conum, ideſt horologij ſolaris genus, fi­<lb></lb>
guram conicam referens, vel in cono deſcriptam.</s>
        </p>
        <p type="main">
          <s id="s.006120">SCOPAS Siracuſanus Plinthum reperit genus horologij in Plintho de­<lb></lb>
ſcripti: inſtar quadratæ trabis erectæ, in cuius ſummo erat horizontale, in <lb></lb>
quatuor verò lateribus erant duo verticalia, auſtrale, &amp; boreale. </s>
          <s id="s.006121">necnon <lb></lb>
duo meridiana, orientale, &amp; occidentale.</s>
        </p>
        <p type="main">
          <s id="s.006122">PATROCLES fuit inuentor <foreign lang="grc">πελεκίνου</foreign>, ideſt <expan abbr="bipẽnis">bipennis</expan>, quòd genus ho­<lb></lb>
rologij ſolaris erat, figuram bipennis referens.</s>
        </p>
        <p type="main">
          <s id="s.006123">PARMENION <foreign lang="grc">προς τα ισορουμενα</foreign> excogitauit, horologia videlicet, <lb></lb>
quæ cœli hiſtoriam narrarent, horas, dies, menſes, ſigna Zodiaci, &amp; c.</s>
        </p>
        <pb pagenum="50" xlink:href="009/01/334.jpg"></pb>
        <p type="main">
          <s id="s.006124">ANDRONICVS CYRESTES Athenis in turri octogona <expan abbr="Ane-moſcopiũ">Ane­<lb></lb>
moſcopium</expan> primus collocauit. </s>
          <s id="s.006125">ex Vitruuio. </s>
          <s id="s.006126">ponendus igitur ante <expan abbr="Vitruuiũ">Vitruuium</expan>, <lb></lb>
quanto tamen ignoratur. </s>
          <s id="s.006127"><expan abbr="Anemoſcopiũ">Anemoſcopium</expan> eſt machina, continens ventorum <lb></lb>
figuras, ac ſitus. </s>
          <s id="s.006128">cum indice mobili, qui ventum perflantem commonſtrat: <lb></lb>
quale Bononiæ eſt in Epiſcopio.</s>
        </p>
        <p type="main">
          <s id="s.006129">M. AGRIPPA, Auguſti gener, &amp; Conſ. terrarum orbem proprijs <expan abbr="cõ-mentarijs">com<lb></lb>
mentarijs</expan> deſcriptum, poſtea in porticu depictum Pop. Rom. ſpectandum <lb></lb>
propoſuit. </s>
          <s id="s.006130">Plin. lib. 3. cap. 2.</s>
        </p>
        <p type="main">
          <s id="s.006131">C. IVLIVS CÆSAR Monarcha, primus ſcripſit Metaphraſim in <lb></lb>
Arati Phœmena. </s>
          <s id="s.006132">Suid.</s>
        </p>
        <p type="main">
          <s id="s.006133">VITRVVIVS, qui in ſuo de architectura opere <expan abbr="cõplura">complura</expan> miſcet ma­<lb></lb>
thematica. </s>
          <s id="s.006134">præcipuè illud, quòd de horologijs ſolaribus ex Analemmate <lb></lb>
primus ex latinis, literis conſignauit. </s>
          <s id="s.006135">ait Venerem, &amp; Mercurium circa So­<lb></lb>
lem, tanquam centrum circumferri. </s>
          <s id="s.006136">ſuum opus Auguſto dicauit.</s>
        </p>
        <p type="main">
          <s id="s.006137">C. MANILIVS Antiochenus Aſtrologus, &amp; Poeta, primus latinis car­<lb></lb>
minibus, quamuis Græcus, aſtronomica cecinit. </s>
          <s id="s.006138">extat ipſius aſtronomicon. <lb></lb>
</s>
          <s id="s.006139">floruit ſub Auguſto.</s>
        </p>
        <p type="head">
          <s id="s.006140"><emph type="italics"></emph>DECIMVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table16"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table16"></table.target>
          <row>
            <cell>Sed primum à natiuitate Chriſti.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Ab Orbe cond. ann.</cell>
            <cell>4089</cell>
          </row>
          <row>
            <cell>Ab Vrbe cond. ann.</cell>
            <cell>752</cell>
          </row>
          <row>
            <cell>Olympiade exacta</cell>
            <cell>194</cell>
          </row>
          <row>
            <cell>Octauiani Auguſti Imper. ann.</cell>
            <cell>42</cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006141">DIONYSIVS AFER, qui Græco poemate orbis ſitum de­<lb></lb>
cantauit.</s>
        </p>
        <p type="main">
          <s id="s.006142">MARINVS TYRIVS ſcripſit de Geographia. </s>
          <s id="s.006143">eum Pto­<lb></lb>
lemæus reprehendit.</s>
        </p>
        <p type="main">
          <s id="s.006144">STRABO eruditiſſimè, ac fusè orbis ſitum, cuius <expan abbr="magnã">magnam</expan> partem pe­<lb></lb>
ragrauerat, deſcripſit.</s>
        </p>
        <p type="main">
          <s id="s.006145">SOLINVS, &amp; P. MELA De ſitu orbis pariter conſcripſerat.</s>
        </p>
        <p type="main">
          <s id="s.006146">STRATON AMASENVS Philoſophus, lib. 7. Geographicos <lb></lb>
edidit Suid.</s>
        </p>
        <p type="main">
          <s id="s.006147">PLINIVS, omnis generis Mathemata ſuo operi <expan abbr="cõmiſcuit">commiſcuit</expan>. </s>
          <s id="s.006148">ſed præ­<lb></lb>
cipuè Geographica à 2. lib. <expan abbr="vſq;">vſque</expan> ad 6.</s>
        </p>
        <p type="main">
          <s id="s.006149">ARTEMIDORVS tempore Strabonis, ſcripſit Geographica, vt <lb></lb>
patet ex Plinio, &amp; Strabone, eum ſæpè citante.</s>
        </p>
        <p type="main">
          <s id="s.006150">IV. HIGINIVS de ſignis cœleſtibus. </s>
          <s id="s.006151">de mundo, &amp; ſphæra ad Quin­<lb></lb>
tilianum.</s>
        </p>
        <p type="main">
          <s id="s.006152">ANDROMACHVS Creteuſis quem Theoricarum inuentorem fa­<lb></lb>
cit Clauius.</s>
        </p>
        <p type="main">
          <s id="s.006153">MENELAVS, qui &amp; Mileſius, poſt Hipparchum a. </s>
          <s id="s.006154">224. ante Ptole­


<pb pagenum="51" xlink:href="009/01/335.jpg"></pb>męum 41. aſtronomicis obſeruationibus dedit operam. </s>
          <s id="s.006155"><expan abbr="primã">primam</expan> Arietis poſt <lb></lb>
æquinoctium gr. 6. 12. deprehendit. </s>
          <s id="s.006156">lib. 6. de ſubtenſis, ſeu chordis. </s>
          <s id="s.006157">Item <lb></lb>
lib. 3. de ſphęricis triangulis, qui extant.</s>
        </p>
        <p type="main">
          <s id="s.006158">PLVTARCHVS libellum de muſica optima eruditione, ac doctrina <lb></lb>
refertum reliquit; quem ſuperius ſæpè citauimus.</s>
        </p>
        <p type="head">
          <s id="s.006159"><emph type="italics"></emph>VNDECIMVM SECVLVM<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006160">Secundum verò à Chriſto incipiens ab ann. <lb></lb>
</s>
          <s id="s.006161">Chriſti 101.</s>
        </p>
        <p type="main">
          <s id="s.006162">Clemente ſummo Chriſtianorum Pont.Imperante Traiano.</s>
        </p>
        <p type="main">
          <s id="s.006163">DIOPHANTES Alexandrinus Algebræ, quam hodie vocant <lb></lb>
mirabilis artifex: extant eius 13. libri Græci Arithmeticorum. <lb></lb>
</s>
          <s id="s.006164">ponitur ſub Antonino à Raphaele Bombello in Algebræ præfa­<lb></lb>
tione.</s>
        </p>
        <p type="main">
          <s id="s.006165">PTOLEMÆVS Alexandr. Aſtronomorum princeps. </s>
          <s id="s.006166">obſeruabat à <lb></lb>
Natiuit. </s>
          <s id="s.006167">Chriſti ann. </s>
          <s id="s.006168">130. maximam Solis declinationem gr. 23. 50. primam <lb></lb>
Arietis poſt æquinot. </s>
          <s id="s.006169">gr. 6. 40. ſcripſit magnam conſtructionem, quam Al­<lb></lb>
mageſtum vocant. </s>
          <s id="s.006170">de Annalemmate. </s>
          <s id="s.006171">de momentis. </s>
          <s id="s.006172">Geographiam. </s>
          <s id="s.006173">Planiſ­<lb></lb>
phærium. </s>
          <s id="s.006174">de ſpeculis. </s>
          <s id="s.006175">libros mechanicos 3. canonem expeditum. </s>
          <s id="s.006176">de iudi­<lb></lb>
cijs 4. centiloquium. </s>
          <s id="s.006177">ſtellarum inenarrantium ſignificationes.</s>
        </p>
        <p type="main">
          <s id="s.006178">SEXTVS EMPIRICVS, qui dum de Mathematicis in vtranque <lb></lb>
partem ſubtiliter diſputat, de eis plura doctè in medium profert. </s>
          <s id="s.006179">ex Gen­<lb></lb>
tiano Herueto eius interprete.</s>
        </p>
        <p type="head">
          <s id="s.006180"><emph type="italics"></emph>DVODECIMVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table17"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table17"></table.target>
          <row>
            <cell>Tertium autem à Chriſto incipiens</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Ab ann. Chriſti</cell>
            <cell>201</cell>
          </row>
          <row>
            <cell>Victore ſummo Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Imperante Septimio Seuero.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006181">PORPHIRIVS Philoſophus Platonicus ſcripſit Iſagogem aſtro­<lb></lb>
nomicarum rerum lib. 3. Suid. </s>
          <s id="s.006182">is eſt, cuius eſt Iſagoge, de quinque <lb></lb>
vniuerſalibus. </s>
          <s id="s.006183">eius Proclus meminit ad 14. 18. &amp; 20. propoſitionem <lb></lb>
primi elem. </s>
          <s id="s.006184">vbi illius demonſtrationes affert. </s>
          <s id="s.006185">Item Hypothipoſes <lb></lb>
aſtronomicarum poſitionum, ideſt expoſitio in Almageſtum.</s>
        </p>
        <p type="main">
          <s id="s.006186">CENSORINVS in eruditiſſimo libello de die Natali, plura habet <lb></lb>
ad Mathematicas, præſertim verò ad Aſtronomum ſpectantia.</s>
        </p>
        <p type="main">
          <s id="s.006187">HIPPOLYTVS Epiſcopus, ob diſcordias inter Latinos, &amp; Græcos <lb></lb>de celebrando Paſchate paulo ante excitatas, primus ſcribit de cyclo Paſ­<lb></lb>
cali, <expan abbr="eiusq́">eiusque</expan>; inuentor celebratur. </s>
          <s id="s.006188">Iſidorus.</s>
        </p>
        <pb pagenum="52" xlink:href="009/01/336.jpg"></pb>
        <p type="head">
          <s id="s.006189"><emph type="italics"></emph>DECIMVMTERTIVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table18"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table18"></table.target>
          <row>
            <cell>Quartum verò Chr. ab ann. Chriſti</cell>
            <cell>301</cell>
          </row>
          <row>
            <cell>Marcellino ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Diocletiano, &amp; Maximiano Impp.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006190"><emph type="italics"></emph>Quo Scholæ florentiſſimæ Romæ, Athenis, Cæſareæ, <lb></lb>
Constantinopoli, frequentabantur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006191">SEXTVS AVIENVS RVFFVS Arati phęnomena, &amp; Dyoni­<lb></lb>
ſij Afri de ſitu orbis poema latine interpretatus eſt.</s>
        </p>
        <p type="main">
          <s id="s.006192">IVL. MATERNVS Siculus latinè ſcripſit: ſed de iudicijs.</s>
        </p>
        <p type="main">
          <s id="s.006193">THEOPHILVS Epiſcopus <expan abbr="Alexãdrinus">Alexandrinus</expan> inter Aegyptios Ma­<lb></lb>
thematicos celebris iuſſu Theodoſij ſenioris Imper. cyclum Paſchalem or­<lb></lb>
dinauit: cui alium paulo poſt contrarium Romanis Dyoniſius Abbas pro­<lb></lb>
poſuit.</s>
        </p>
        <p type="main">
          <s id="s.006194">ABIFELDEA Princeps Syriæ, Aſsyriæ, &amp; Perſidis, Geographus in­<lb></lb>
ſignis. </s>
          <s id="s.006195">eius Geographia aſſeruatur in Bibliotheca Palatina Arabicè ſcripta. <lb></lb>
</s>
          <s id="s.006196">Corradus Geſnerus in Alfraganum.</s>
        </p>
        <p type="main">
          <s id="s.006197">VALENS huius ætatis inſignis Mathematicus, qui iuſſu Conſtantini <lb></lb>
Magni, Vrbis Conſtantinopolitanæ, quam tunc ipſe ædificabat, genituram <lb></lb>
ex cœleſti Themate inani labore dijudicauit. </s>
          <s id="s.006198">Zonaras.</s>
        </p>
        <p type="main">
          <s id="s.006199">EVSEBIVS Cæſarienſis Epiſcopus ſcribit de cyclo Paſchali. </s>
          <s id="s.006200">Iſid.</s>
        </p>
        <p type="main">
          <s id="s.006201">MAXIMVS Epirota ſub Iuliano Apoſtata ſcripſit de numeris. </s>
          <s id="s.006202">Suid.</s>
        </p>
        <p type="main">
          <s id="s.006203">Quinque <expan abbr="ſequẽtes">ſequentes</expan> ponendi ſunt inter Archimedem, &amp; Proclum; quo au­<lb></lb>
tem loco ignoratur; Proclus enim recenſens Mathematicos vſque ad Ar­<lb></lb>
chimedem, de eis ſilet.</s>
        </p>
        <p type="main">
          <s id="s.006204">NICOMEDES, qui de line is conchoidibus ſcripſit, per quas duas <lb></lb>
medias proportionales exhibeat, <expan abbr="atq;">atque</expan> hinc cubum duplicabat. </s>
          <s id="s.006205">ijſdem con­<lb></lb>choidibus angulum datum rectilineum trifariam ſecabat. </s>
          <s id="s.006206">extant ipſius ſub­<lb></lb>
tiliſſimi conatus apud Eutocium, &amp; Pappum, &amp; P. Clauium in Geometria <lb></lb>
practica.</s>
        </p>
        <p type="main">
          <s id="s.006207">EVDEMVS, qui Geometricas enarrationes conſcripſit. </s>
          <s id="s.006208">Item libel­<lb></lb>
lum de angulo. </s>
          <s id="s.006209">Proclus.</s>
        </p>
        <p type="main">
          <s id="s.006210">MENELAVS Alexandrinus, cuius demonſtrationes affert Proclus <lb></lb>
ad vigeſimamquintam primi elementi.</s>
        </p>
        <p type="main">
          <s id="s.006211">GEMINVS RHODIVS, Procli Diadochi præceptor, græcè ſcri­<lb></lb>
pſit phænomena: quæ Mediolani in Bibliotheca Ambroſiana aſſeruantur:<lb></lb>
&amp; quidem græcolatina, Edone Stildario interprete. </s>
          <s id="s.006212">Præterea de ortu li­<lb></lb>
nearum ſpiralium, conchoidarum, ciſſoidarum, <expan abbr="earumq́">earumque</expan>; paſſionibus. </s>
          <s id="s.006213">Item <lb></lb>
de Mathematicarum ordine.</s>
        </p>
        <p type="main">
          <s id="s.006214">Circa finem huius quarti ſeculi fiunt <expan abbr="vndiq;">vndique</expan> Barbarorum irruptiones in <lb></lb>
Rom. </s>
          <s id="s.006215">Imperium: Gothi ſub Alarico Græcias inuadunt, <expan abbr="Athenasq́">Athenasque</expan>; <expan abbr="capiũt">capiunt</expan>, <lb></lb>
ac diripiunt.</s>
        </p>
        <pb pagenum="53" xlink:href="009/01/337.jpg"></pb>
        <p type="head">
          <s id="s.006216"><emph type="italics"></emph>DECIMVMQVARTVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table19"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table19"></table.target>
          <row>
            <cell>Quintum verò Chriſti.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Ab ann. Chriſti</cell>
            <cell>401</cell>
          </row>
          <row>
            <cell>Anaſtaſio ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Impp. Arcadio orienti, &amp; Honorio occidenti.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006217"><emph type="italics"></emph>Quo Roma ter capta, &amp; imperio occidentis ab Odoacre exciſo literæ, <lb></lb>
&amp; artes paſſim peſſundari incipiunt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006218">Sequentes duo ponendi ſunt ante Eutocium, ſed quanto non comperio.</s>
        </p>
        <p type="main">
          <s id="s.006219">DIOCLES, cuius modum inueniendi duas medias proportio­<lb></lb>
nales, &amp; modum ſecandi ſphæram in datam rationem, refert Euto­<lb></lb>
cius, deſumptum ex libro de Pyrijs, ſeu igniarijs.</s>
        </p>
        <p type="main">
          <s id="s.006220">SPORVS Nicenus, cuius etiam duarum mediarum inuentio eſt apud <lb></lb>
Eutocium in comm. Archim.</s>
        </p>
        <p type="main">
          <s id="s.006221">PROCLVS DIADOCHVS, qui Athenis Platonicæ ſcholæ præ­<lb></lb>
fuit. </s>
          <s id="s.006222">ſcripſit comm. in Euclidem eruditiſs. </s>
          <s id="s.006223">Suid. </s>
          <s id="s.006224">Georgius Heniſchius. </s>
          <s id="s.006225">Hy­<lb></lb>
potypoſes aſtronomicas. </s>
          <s id="s.006226">ſphæram. </s>
          <s id="s.006227">Archimedem imitatus vſtorijs ſpeculis, <lb></lb>
Valentis naues Conſtantinopolim obſidentes combuſſit. </s>
          <s id="s.006228">Zenoras, qui eum <lb></lb>
mirè commendat.</s>
        </p>
        <p type="main">
          <s id="s.006229">CYRILLVS Epiſcopus Alexandrinus ſcribit de Cyclo paſchali. </s>
          <s id="s.006230">Iſid.</s>
        </p>
        <p type="main">
          <s id="s.006231">S. AVGVSTINVS Epiſcopus lib. 6. de Muſica, Item de principijs <lb></lb>
Geometriæ, &amp; Arithmeticæ ſcribit.</s>
        </p>
        <p type="main">
          <s id="s.006232">MARINVS Philoſophus Neapolitanus Procli diſcipulus. </s>
          <s id="s.006233">eius eſt Pro­<lb></lb>
theoria in data Euclidis.</s>
        </p>
        <p type="main">
          <s id="s.006234">DEMETRIVS <expan abbr="Alexãdrinus">Alexandrinus</expan>, lineares aggreſſiones fecit. </s>
          <s id="s.006235">Papp. p.61.</s>
        </p>
        <p type="main">
          <s id="s.006236">PHILO TYANÆVS de lineis genitis ex implicatione <foreign lang="grc">πληκτοειδων</foreign>,<lb></lb>
&amp; aliarum varij generis ſuperficierum. </s>
          <s id="s.006237">Pappus p. </s>
          <s id="s.006238">61.</s>
        </p>
        <p type="main">
          <s id="s.006239">S. PROSPER Aquitanus de Cyclo paſchali. </s>
          <s id="s.006240">compoſuit Cyclum ma­<lb></lb>
gnum annorum 532. 10. Lucidus. </s>
          <s id="s.006241">Iſidorus.</s>
        </p>
        <p type="main">
          <s id="s.006242">PAPPVS Alexandrinus: cuius Mathematicæ collectiones extant. </s>
          <s id="s.006243">&amp; <lb></lb>
comm. in 5. Ptolæmei magnæ ſyntaxis. </s>
          <s id="s.006244">fecit etiam vniuerſalem orbis deſcri­<lb></lb>
ptionem. </s>
          <s id="s.006245">de fluuijs Lybiæ. </s>
          <s id="s.006246">extant eius lemmata in Apollonium Pergæum.</s>
        </p>
        <p type="main">
          <s id="s.006247">THEON Alexandrinus, cuius ſunt græca comm. in magnam ſyntaxim <lb></lb>
Ptolæmei, præterea de Arithmetica. </s>
          <s id="s.006248">de ortu Caniculæ. </s>
          <s id="s.006249">de Nili aſcenſu. <lb></lb>
</s>
          <s id="s.006250">comm. in paruum Aſtrolabium. </s>
          <s id="s.006251">ex Suid. fuit Pappi ſyncronus.</s>
        </p>
        <p type="main">
          <s id="s.006252">HYPATIA Theonis Geometræ filia Alex. Diophantis Arithmeticam <lb></lb>
comment. </s>
          <s id="s.006253">illuſtrauit. </s>
          <s id="s.006254">pręterea in Conica Apollonij ſcripſit. </s>
          <s id="s.006255">aſtronomicum <lb></lb>
canonem conſtruxit. </s>
          <s id="s.006256">claruit ſub Arcadio, &amp; Honorio.</s>
        </p>
        <p type="main">
          <s id="s.006257">VICTORINVS Aquitanus Aſtronomus, ab Hilario Papa Romam <lb></lb>
inuitatur ad Calendarij correctionem.</s>
        </p>
        <p type="main">
          <s id="s.006258">EVTOCIVS Aſcalonita poſt Theonem, &amp; Pappum ſcripſit, eos enim <lb></lb>
nominat. </s>
          <s id="s.006259">ſcripſit commentaria in conica Apollonij, &amp; in Archimedem de <lb></lb>
ſphæra, &amp; cylindro, de circuli dimenſione; &amp; de æqueponderantibus.</s>
        </p>
        <pb pagenum="54" xlink:href="009/01/338.jpg"></pb>
        <p type="head">
          <s id="s.006260"><emph type="italics"></emph>DECIMUMQVINTVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table20"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table20"></table.target>
          <row>
            <cell>Sextum verò Chriſti.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Ab ann. Chriſti</cell>
            <cell>501</cell>
          </row>
          <row>
            <cell>Symmacho ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Imperante Anaſtaſio orienti.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Theodorico Rege Goth. Italiæ.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006261"><emph type="italics"></emph>Hoc ſæculo Roma quartò capitur à Totila: &amp; Longobardi <lb></lb>
Italiam inuadunt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006262">BOETIVS vir clariſſimus, &amp; Conſularis, latinè de Arithmetica, <lb></lb>
de Muſica, de Geometria practica: inuenit Chiterinum muſicum <lb></lb>
inſtrumentum. </s>
          <s id="s.006263">Supplem. chron.</s>
        </p>
        <p type="main">
          <s id="s.006264">CASSIODORVS vir clariſſimus, &amp; Conſularis, ſcribit de <lb></lb>
Arithmetica, de Geometria, de Muſica, de Aſtronomia, de Paſchali com­<lb></lb>
puto. </s>
          <s id="s.006265">Baronius.</s>
        </p>
        <p type="main">
          <s id="s.006266">IOANNES <expan abbr="Grãmaticus">Grammaticus</expan> cognominato Philoponus, ſcripſit de Arith­<lb></lb>
metics. </s>
          <s id="s.006267">modus etiam quidam inueniendi duas medias ei tribuitur, vt apud <lb></lb>
Clau. in Geometria practica. </s>
          <s id="s.006268">comm. in Nicomachi arithmeticam ſcripſit.</s>
        </p>
        <p type="main">
          <s id="s.006269">HERO Mechanicus, qui eſt alius ab Herone Philoſopho mechanico, de <lb></lb>
quo ſuperius. </s>
          <s id="s.006270">eius extat liber de Geodæſia, &amp; alter de machinis bellicis. </s>
          <s id="s.006271">ait <lb></lb>ipſe in Geodæſia ſtellas fixas poſt Ptolemæum, vſque ad ſuam ætatem pro­<lb></lb>
greſſos eſſe grad. 7. qui progreſſus, ſi Albatignio credimus annos ſaltem 460. <lb></lb>
importat, qui Ptolemæi ætati aditi Heronem in hoc ſeculum transferunt.</s>
        </p>
        <p type="main">
          <s id="s.006272">HELIODORVS Lariſſeus, cuius extant optica græca. </s>
          <s id="s.006273">citat Hero­<lb></lb>
nem mechanicum.</s>
        </p>
        <p type="main">
          <s id="s.006274">DIONYSIVS exiguus Abbas Romanus, computum, &amp; cyclum pa­<lb></lb>
ſchalem aliter ac Græci ordinauit a. </s>
          <s id="s.006275">D. 532. quem latina Eccleſia poſtea <lb></lb>
<expan abbr="vſq;">vſque</expan> ad Calendarij correctionem ſub Greg. 13. factam, ſequuta eſt: primus <lb></lb>
annos à Chriſto Domino numerare cœpit, qui prius à Diocletiano, ſiue à <lb></lb>
perſecutione Diocletiani numerabantur. </s>
          <s id="s.006276">Chriſtmanus in Alfrag.</s>
        </p>
        <p type="main">
          <s id="s.006277">S. GREGORIVS Magnus Papa in muſicis excelluit, <expan abbr="eiq́">eique</expan>, adeo fuit <lb></lb>
addictus, vt Clericos ipſe muſicam doceret. </s>
          <s id="s.006278">canticum Eccleſiaſticum ordi­<lb></lb>
nauit, qui ab eo denominatur. </s>
          <s id="s.006279">choro etiam modum conſtituit.</s>
        </p>
        <pb pagenum="55" xlink:href="009/01/339.jpg"></pb>
        <p type="head">
          <s id="s.006280"><emph type="italics"></emph>DECIMUMSEXTVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table21"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table21"></table.target>
          <row>
            <cell>Septimum verò Chri. ab ann. Chriſti</cell>
            <cell>601</cell>
          </row>
          <row>
            <cell>Gregorio magno ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Mauritio Imperatore orientis.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Longobardis in Italia regnantibus.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006281"><emph type="italics"></emph>Arabes in Aſia, &amp; Africa, &amp; Europa complura regna occupant.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006282">ISIDORVS Hiſpalentis Epiſcopus, in ſuis de originibus libris om­<lb></lb>
nium Mathematicarum compendia inſerit: &amp; de cyclo paſchali plu­<lb></lb>
ribus agit: &amp; in libro de Mundo breuiter tractatum de ſphæra per­<lb></lb>
ſtringit.</s>
        </p>
        <p type="main">
          <s id="s.006283">MARTIANVS CAPELLA, qui etiam Fœlix Mineus dicitur, ad <lb></lb>
hæc tempora à Patricio in Poetica refertur, ſcilicet paulo ante Eraclium <lb></lb>
Imper. ſcripſit in ſuis nuptijs Mercurij cum Philologia, de 4. Mathematicis <lb></lb>
Geometria, Arithmetica, Muſica, Aſtronomia.</s>
        </p>
        <p type="head">
          <s id="s.006284"><emph type="italics"></emph>DECIMVMSEPTIMVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table22"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table22"></table.target>
          <row>
            <cell>Octauum autem Chr. ab ann. Chriſti</cell>
            <cell>701</cell>
          </row>
          <row>
            <cell>Sergio ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Imper. Tiberio Abſimero orientis.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Longobardis in Italia.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006285">VENERABILIS BEDA de Arithmetica. </s>
          <s id="s.006286">de Muſica. </s>
          <s id="s.006287">de <lb></lb>
Aſtrolabio. </s>
          <s id="s.006288">de Horologio ſolari. </s>
          <s id="s.006289">de computo Eccleſiaſtico. <lb></lb>
</s>
          <s id="s.006290">Ecce tibi quanta literatorum paucitas Barbaris Imperium de­<lb></lb>
uaſtantibus.</s>
        </p>
        <p type="head">
          <s id="s.006291"><emph type="italics"></emph>DECIMUMOCTAVVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table23"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table23"></table.target>
          <row>
            <cell>Nonum verò Chr. ab ann. Chriſti</cell>
            <cell>801</cell>
          </row>
          <row>
            <cell>Leone ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Imper. occidenti Carolo Magno.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Irene verò orienti.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006292"><emph type="italics"></emph>Literæ apud Arabes florere incipiunt.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006293">ALMEON, ſiue Almamom Rex Arabum, ante Albategnium <lb></lb>
ann. </s>
          <s id="s.006294">50. obſeruauit Solis maximam declinationem 23. 51. repe­<lb></lb>
rit vni gradui terræ deberi mill. 56. in campis Singar propè Ba­<lb></lb>
byloniam.</s>
        </p>
        <pb pagenum="56" xlink:href="009/01/340.jpg"></pb>
        <p type="main">
          <s id="s.006295">ALBATEGNIVS ARACENSIS Arabs, poſt Ptolemæum ann. <lb></lb>
</s>
          <s id="s.006296">750. à nat. </s>
          <s id="s.006297">Chriſti 880. ante Alfarabium 381. obſeruat Solis maximam de­<lb></lb>
clinationem 23. 35. &amp; primam Arietis poſt æquinoctium grad. 18. 2. Ara­<lb></lb>
cta eſt vrbs Syriæ, &amp; patria ipſius, à qua dicitur Aracenſis. </s>
          <s id="s.006298">extat liber eius <lb></lb>
de ſcientia ſtellarum.</s>
        </p>
        <p type="main">
          <s id="s.006299">MICHAEL PSELLVS Græcus Quadriuium, hoc eſt de 4. Mathe­<lb></lb>
maticis compendiosè ſcripſit, &amp; extat. </s>
          <s id="s.006300">Docuit filios Imperatoris. </s>
          <s id="s.006301">hic po­<lb></lb>
nitur à Baronio.</s>
        </p>
        <p type="main">
          <s id="s.006302">Sequentes 5. ponendi ſunt ante ſeculum 10. quo Suida ſcribens, eos me­<lb></lb>
morat: quanto autem, neſcire fateor.</s>
        </p>
        <p type="main">
          <s id="s.006303">PAVLVS Philoſophus, introductionem Aſtrologiæ compoſuit.</s>
        </p>
        <p type="main">
          <s id="s.006304">PETOSCIRIS Aegyptius, Aſtrologica è ſacris libris pertractauit.</s>
        </p>
        <p type="main">
          <s id="s.006305">ACHILLES STATIVS Alexandrinus Epiſcopus. </s>
          <s id="s.006306">de ſphæra.</s>
        </p>
        <p type="main">
          <s id="s.006307">ZOROMASDVS Chaldæus. </s>
          <s id="s.006308">Mathematica ſcripſit.</s>
        </p>
        <p type="main">
          <s id="s.006309">PELLES Aegienſis. </s>
          <s id="s.006310">Arithmeticorum lib. 2.</s>
        </p>
        <p type="main">
          <s id="s.006311">GEBER Arabs, cuius extat Opus aſtronomicum 9. libris diſtinctum, <lb></lb>quo Ptolemæi Almageſtum exponit, ac corrigit. </s>
          <s id="s.006312">initio agit de Triangulis <lb></lb>
ſphæricis, quantum aſtronomicis calculis opus eſt.</s>
        </p>
        <p type="head">
          <s id="s.006313"><emph type="italics"></emph>DECIMUMNONUM SECULUM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table24"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table24"></table.target>
          <row>
            <cell>Decimum verò Chr. ab ann. Chriſti</cell>
            <cell>901</cell>
          </row>
          <row>
            <cell>Ioanne ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Impp. Ludouico 4. occid.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Leone 6. orien.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006314"><emph type="italics"></emph>Lingua vulgaris Italica incipit emergere. </s>
          <s id="s.006315">Baron.<emph.end type="italics"></emph.end>GVIDO ARETINVS Monachus S. Benedicti, Romæ ſcripſit <lb></lb>
de Muſica. </s>
        </p>
        <p type="main">
          <s id="s.006316">nouam rationem cantus excogitauit. </s>
          <s id="s.006317">eius opera ſunt <lb></lb>
Introductorium muſicæ, in quo ipſe primus vocibus nomina in­<lb></lb>
didit, Vt, Re, Mi, Fa, Sol, La: Item Micrologus de Muſica.</s>
        </p>
        <p type="main">
          <s id="s.006318">ALFARABIVS Arabs, Aſtronomus celebris.</s>
        </p>
        <p type="main">
          <s id="s.006319">ALBVMASAR Arabs, Aſtronomus celebris. </s>
          <s id="s.006320">de magnis coniunctio­<lb></lb>
nibus, &amp; alia iudiciaria.</s>
        </p>
        <p type="main">
          <s id="s.006321">ALFRAGANVS Arabs, Aſtronomica elementa edidit.</s>
        </p>
        <p type="main">
          <s id="s.006322">BAGDADINVS Arabs. </s>
          <s id="s.006323">de diuiſione figurarum, extat.</s>
        </p>
        <p type="main">
          <s id="s.006324">BEN MVSA Arabs. </s>
          <s id="s.006325">de figuris planis, &amp; ſphæricis.</s>
        </p>
        <pb pagenum="57" xlink:href="009/01/341.jpg"></pb>
        <p type="head">
          <s id="s.006326"><emph type="italics"></emph>VIGESIMVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table25"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table25"></table.target>
          <row>
            <cell>Vndecimum <expan abbr="verõ">verom</expan> Chr. ab ann. Chriſti</cell>
            <cell>1001</cell>
          </row>
          <row>
            <cell>Silueſtro ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Impp. Ottone 3. occid.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Baſilio, &amp; Conſt. orient.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006327">ALHAZENVS Arabs: eius extant optica doctè, ac ſubtiliter <lb></lb>
pertractata. </s>
          <s id="s.006328">Item opuſculum de crepuſculis, vbi aeris ſuprema <lb></lb>
altitudinem acutiſſimè rimatur.</s>
        </p>
        <p type="main">
          <s id="s.006329">CAMPANVS Italus, ac Nouarenſis, primus Euclidem ex <lb></lb>
Arabico in latinum tranſtulit, ac ſcholijs illuſtrauit. </s>
          <s id="s.006330">Fuit optimus Aſtro­<lb></lb>
nomus: ſcripſit computum minorem, &amp; maiorem, anno 1200. vt ipſe ait. <lb></lb>
</s>
          <s id="s.006331">Item ſphæram, &amp; theoricas planetarum.</s>
        </p>
        <p type="main">
          <s id="s.006332">ARZAEL Arabs Poſt Albategnium ann. </s>
          <s id="s.006333">190. reperit Solis maximam <lb></lb>
declinationem gr. 23. 34.</s>
        </p>
        <p type="main">
          <s id="s.006334">IS A CIVS ARGYRVS Græcus, de Paſchatis correctione. </s>
          <s id="s.006335">Cla­<lb></lb>
uius in Calendario.</s>
        </p>
        <p type="head">
          <s id="s.006336"><emph type="italics"></emph>VIGESIMVMPRIMVM SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table26"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table26"></table.target>
          <row>
            <cell>Duodec. verò à Chr. ab ann. Chriſti</cell>
            <cell>1001.</cell>
          </row>
          <row>
            <cell>Paſchali ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Impp. Henrico 3. occid.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Alexio Commeno orient.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006337">RABBI Abraham de Sphæra. </s>
          <s id="s.006338">Chriſtm. in Alfrag.</s>
        </p>
        <p type="main">
          <s id="s.006339">IORDANVS Nemorarius, qui ſcribit de ponderibus, citat <lb></lb>
Campanum, &amp; Campanus in def. </s>
          <s id="s.006340">5. <expan abbr="elemẽ">elemen</expan>. </s>
          <s id="s.006341">citat Iordanum, qui <lb></lb>
ſcripſit de Arithmetica lib. 12. &amp; data Arithmetica. </s>
          <s id="s.006342">&amp; de Aſtro­<lb></lb>
labio. </s>
          <s id="s.006343">qui ſcribit de Arithmetica appellatur etiam Nemorarius, vnde vnus, <lb></lb>
&amp; idem Iordanus videtur eſſe.</s>
        </p>
        <p type="main">
          <s id="s.006344">AVERROES Arabs magnus commentator, fecit Epitomen Alma­<lb></lb>
giſti. </s>
          <s id="s.006345">Picus Mir. contra Aſtrologos.</s>
        </p>
        <p type="main">
          <s id="s.006346">ALMEON Almanſorius Arabs, poſt Arzaelem ann. </s>
          <s id="s.006347">70. anno Domi­<lb></lb>
ni 1140. declinationem Solis maximam reperit 23. 33.</s>
        </p>
        <p type="main">
          <s id="s.006348">ALPETRAGIVS Arabs Almconis coætaneus ann. </s>
          <s id="s.006349">Domini 1145. <lb></lb>
eandem cum eo inſpexit declinationem.</s>
        </p>
        <p type="main">
          <s id="s.006350">HVMENVS Aegyptius, cuius tabulæ aſtronomiæ arabicè ſcriptæ aſ­<lb></lb>
ſeruàntur in Biblioth. Palatina. </s>
          <s id="s.006351">Chriſtmanus in Alfr.</s>
        </p>
        <p type="main">
          <s id="s.006352">IOANNES Hiſpalenſis circa 1142. conuertit Alfragnum in latinum. <lb></lb>
</s>
          <s id="s.006353">ex Chriſtmanno.</s>
        </p>
        <p type="main">
          <s id="s.006354">THEON Smyrneus, circa hæc ſecula, Græcè loco Mathematica apud <lb></lb>
Platonem interpretatur. </s>
          <s id="s.006355">opus eius Græcum extat in Vatic. ex Ioſ. Auria.</s>
        </p>
        <pb pagenum="58" xlink:href="009/01/342.jpg"></pb>
        <p type="head">
          <s id="s.006356"><emph type="italics"></emph>XXII. SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table27"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table27"></table.target>
          <row>
            <cell>Decimum autem tertium Chr. ab ann. Domini</cell>
            <cell>1201</cell>
          </row>
          <row>
            <cell>Innocentio ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Impp. Ottone 4. occid.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Iſacio orient.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006357">VITELLIO, qui maiorum Optica in vnum congeſſit, ac digeſ­<lb></lb>
ſit. </s>
          <s id="s.006358">Riſnerus.</s>
        </p>
        <p type="main">
          <s id="s.006359">NICOLAVS CABASILLA Græcus, Ptolemæi ſynta­<lb></lb>
xis commentator.</s>
        </p>
        <p type="main">
          <s id="s.006360">FEDERICVS Secundus Imperat. primus Almageſtum ex Arabo in <lb></lb>
latinum conuerti curauit, <expan abbr="adeoq́">adeoque</expan>; aſtronomiam omnem excoluit. </s>
          <s id="s.006361">Chriſtm.</s>
        </p>
        <p type="main">
          <s id="s.006362">ALPHONSVS Rex Hiſpaniarum, cuius ſunt tabulæ Alphonſinæ. </s>
          <s id="s.006363">ob. <lb></lb>
</s>
          <s id="s.006364">ſeruauit ann. </s>
          <s id="s.006365">Domini 1250. primam Arietis poſt æquinoctium gra. </s>
          <s id="s.006366">23. 40. <lb></lb>
hic quadraginta <expan abbr="aureorũ">aureorum</expan> millia ad <expan abbr="aſtronomiã">aſtronomiam</expan> in lucem reuocandam cum <lb></lb>
ſempiterna ſui nominis gloria contulit.</s>
        </p>
        <p type="main">
          <s id="s.006367">IOAN. de Sacroboſco angulus ſcripſit de ſphæra; &amp; de <expan abbr="cõputo">computo</expan> Eccleſ.</s>
        </p>
        <p type="main">
          <s id="s.006368">IOANNES autor ſummæ Angel.</s>
        </p>
        <p type="main">
          <s id="s.006369">THEBIT Arabs, poſt Almeonem an. 50. an. 1270. primus motum tre­<lb></lb>
pidationis octauæ ſphærę rimatus eſt.</s>
        </p>
        <p type="main">
          <s id="s.006370">PROFATIVS Iudæus ann. </s>
          <s id="s.006371">Dom. 1300. poſt Almeon ann. </s>
          <s id="s.006372">160. Solis <lb></lb>
declinationem maximam annotauit. </s>
          <s id="s.006373">gr. 23. 32.</s>
        </p>
        <p type="main">
          <s id="s.006374">IOANNES GIRA Amalphenſis inuen it <expan abbr="mirã">miram</expan> illam magnetis pro­<lb></lb>
prietatem, qua ad polum ſemper conuertitur: vnde maxima rei Nauticæ <lb></lb>
vtilitas, &amp; acceſſio facta eſt:</s>
        </p>
        <p type="main">
          <s id="s.006375"><emph type="italics"></emph>Prima dedit nautis vſum magnetis Amalphis.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006376">Panormitanus. </s>
          <s id="s.006377">Ortelius tab. </s>
          <s id="s.006378">6.</s>
        </p>
        <p type="head">
          <s id="s.006379"><emph type="italics"></emph>XXIII. SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table28"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table28"></table.target>
          <row>
            <cell>Decimum verò quartum Chr. ab ann. Domini</cell>
            <cell>1301</cell>
          </row>
          <row>
            <cell>Bonifacio ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Impp. Alberto primo Auſtriaco occid.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Andronico Paleologo orient.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="main">
          <s id="s.006380">BARLAAM Monacus. </s>
          <s id="s.006381">Græcè de Arithmetica, nondum editus.</s>
        </p>
        <p type="main">
          <s id="s.006382">ROGERIVS BACCON ſub Clem. 5. perſpectiuam laudatiſſi­<lb></lb>
mam ſcribit. </s>
          <s id="s.006383">Item de loco ſtellarum. </s>
          <s id="s.006384">Item ſpecula Mathemati­<lb></lb>
ca. </s>
          <s id="s.006385">10. Lucidus. </s>
          <s id="s.006386">Collim.MARCVS POLVS Venetus, per totum Orientem peruagatus, plurima <lb></lb>
ſcitu digniſſima, de regnis Aſiæ Orientalibus breui comm. materna lingua <lb></lb>
complexus eſt.</s>
        </p>
        <p type="main">
          <s id="s.006387">IOANNES Archiep. Cantuar. </s>
          <s id="s.006388">auctor perſpectiuæ communis. </s>
          <s id="s.006389">poſt Vitell. <lb></lb>
&amp; multò ante 1500. ann. </s>
          <s id="s.006390">ex Gaurico.</s>
        </p>
        <pb pagenum="59" xlink:href="009/01/343.jpg"></pb>
        <p type="head">
          <s id="s.006391"><emph type="italics"></emph>XXIIII. SECVLVM<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006392">Decimum verò quintum Chr. ab an. Dom. 1401. incipiens.</s>
        </p>
        <p type="main">
          <s id="s.006393">Bonifacio ſum. Pont. <lb></lb>Impp. Ruberto Auſtriaco occid. &lt;/lb&gt;Emanuele Paleologo orien. <lb></lb><emph type="italics"></emph>Anno 1453. Imperium orientis capitur à Mahumeto 2. <lb></lb>
Turcarum Imperatore.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="head">
          <s id="s.006394"><emph type="italics"></emph>Nouus orbis circa finem huius ſeculi detegitur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006395">LEONARDVS Piſanus, primus ex recentioribus de Algebra la­<lb></lb>
tinè ſcripſit. </s>
          <s id="s.006396">nondum editus.</s>
        </p>
        <p type="main">
          <s id="s.006397">GEORGIVS PVRBACHIVS, Theoricas planetarum, edidit. <lb></lb>
</s>
          <s id="s.006398">epitomen almageſti inchoauit, quam poſtea Ioannes de Montere­<lb></lb>
gio abſoluit. </s>
          <s id="s.006399">Item tabul. </s>
          <s id="s.006400">eclypſium. </s>
          <s id="s.006401">declinationem Solis maximam, 23.28. <lb></lb>
prodidit. </s>
          <s id="s.006402">ſcripſit de horologio ſolari, &amp; aſſeruatur in Bibliotheca <expan abbr="Viennẽſi">Viennenſi</expan>. <lb></lb>
</s>
          <s id="s.006403">publicè Mathematicas, Viennæ, &amp; Ferrarię docuit.</s>
        </p>
        <p type="main">
          <s id="s.006404">IACOBVS FABER Stapulenſis edit <expan abbr="cõmentaria">commentaria</expan> in arthmeticam <lb></lb>
Iordani. </s>
          <s id="s.006405">Item elementa muſicæ libris quatuor.</s>
        </p>
        <p type="main">
          <s id="s.006406">FRANCHINVS GAFFVRIVS Laudenſis, latinè <expan abbr="Muſicã">Muſicam</expan>, Theoricam, <lb></lb>
&amp; Practicam ſcribit 1496.</s>
        </p>
        <p type="main">
          <s id="s.006407">IOANNES de Monteregio, Purbacchij diſcipulus. </s>
          <s id="s.006408">Epitomen alme­<lb></lb>
giſti abſoluit. </s>
          <s id="s.006409">opus de triangulis planis, &amp; ſphæricis. </s>
          <s id="s.006410">Tabulas directionum <lb></lb>
fecit. </s>
          <s id="s.006411">primus ephemerides aſtronomicas ad plures annos edidit. </s>
          <s id="s.006412">tangentes <lb></lb>
lineas inuenit. </s>
          <s id="s.006413">item libellum de Cometa. </s>
          <s id="s.006414">Mathias Rex Vngariæ multis eum <lb></lb>
auxit honoribus, &amp; diuitijs. </s>
          <s id="s.006415">tandem Romam à Summo Pontifice ad Calen­<lb></lb>
darij correctionem euocatus, ibi obijt, <expan abbr="ſepultusq́">ſepultusque</expan>; eſt in Pantheone. </s>
          <s id="s.006416">declina­<lb></lb>
tionem Solis maximam. </s>
          <s id="s.006417">23.30. edixit. </s>
          <s id="s.006418">Monteregio plurimum debet omnis <lb></lb>literatorum poſteritas, quòd veterum Græcorum ferè omnium, Archime­<lb></lb>
dis, Apollonij, Sereni, Ptolemæi, &amp; aliorum opera numero ferè triginta, in <lb></lb>
latinum conuerſa, Typis mandari curauerit. </s>
          <s id="s.006419">Ex Collimitij indice ante ta­<lb></lb>
bulam primi mobilis Monteregij.</s>
        </p>
        <p type="main">
          <s id="s.006420">PETRVS de ALIACO Card. Cameracenſis 1414. ſuaſit Concilio <expan abbr="Cõ-ſtantienſi">Con­<lb></lb>
ſtantienſi</expan> correctionem Calendarij Romani. </s>
          <s id="s.006421">ſcripſit de Calendarij corre­<lb></lb>
ctione. </s>
          <s id="s.006422">de parallelis, &amp;c. </s>
          <s id="s.006423">Chriſtmanus in Alfraganum.</s>
        </p>
        <p type="main">
          <s id="s.006424">F. LVCAS de Burgo edidit magnum volumen Italica lingua de arith­<lb></lb>
metica, in quo algebram ex Leonardo Piſano partim acceptam vulgauit: <lb></lb>
ibi etiam de geometria practica. </s>
          <s id="s.006425">Item librum de diuina proportione.</s>
        </p>
        <p type="main">
          <s id="s.006426">CHRISTOPHORVS, COLVMBVS Ligur, <expan abbr="Argonautũ">Argonautum</expan> princeps Aſtro­<lb></lb>nomiæ, &amp; Geographiæ ſcientia fretus nouum Orbem, magno, ac fęlici au­<lb></lb>
ſu detexit.</s>
        </p>
        <p type="main">
          <s id="s.006427">NICOLAVS CVSANVS Cardinalis, de Transformatione figurarum.</s>
        </p>
        <pb pagenum="60" xlink:href="009/01/344.jpg"></pb>
        <p type="head">
          <s id="s.006428"><emph type="italics"></emph>XXV. SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table29"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table29"></table.target>
          <row>
            <cell>Decimum verò ſextum Chr. ab ann. Chriſti</cell>
            <cell>1501</cell>
          </row>
          <row>
            <cell>Alexandro 6. ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Imper. Maximiliano occid.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006429"><emph type="italics"></emph>Totius orbis circumnauigatio: &amp; nouarum ſtellarum <lb></lb>
fixarum apparitio.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006430">IOANNES VERNERVS Germanus, aſtronomicas Tabulas, quibus <lb></lb>loca ſtellarum exponit, declinationi Solis max. tribuit grad. 23.28. <lb></lb>
<expan abbr="primã">primam</expan> Arietis poſt æquinoct. </s>
          <s id="s.006431">gr. 26. an. 15 4. de motu octauæ ſphæræ.</s>
        </p>
        <p type="main">
          <s id="s.006432">FERDINANDVS MEGALANES rei nauticæ, ac proinde aſtro­<lb></lb>
nomiæ peritiſſimus, fretum ſibi cognomen inueſtigauit, vnde poſtea nauis <lb></lb>
ipſius verè Victoria totius terræ <expan abbr="globũ">globum</expan> prima <expan abbr="omniũ">omnium</expan> <expan abbr="circũnauigauit">circunnauigauit</expan>. </s>
          <s id="s.006433">1519.</s>
        </p>
        <p type="main">
          <s id="s.006434">IOAN. BLANCHINVS Ferrarienſis, Tabulas aſtronomicas compoſuit.</s>
        </p>
        <p type="main">
          <s id="s.006435">LVDOVICVS FOLIANVS Mutinen. latinè de muſica Theorica. </s>
          <s id="s.006436">1529.</s>
        </p>
        <p type="main">
          <s id="s.006437">NICOLAVS COPERNICVS, nouis obſeruationibus cœleſtes motus <lb></lb>
corrigit. </s>
          <s id="s.006438">1515. antiquam Cleantis opinione de motu terræ ſuſcitauit. </s>
          <s id="s.006439">ait <lb></lb>
præterea, Solem in centro mundi quieſcere. </s>
          <s id="s.006440">1530.</s>
        </p>
        <p type="main">
          <s id="s.006441">ORONTIVS FINÆVS, Pariſijs Mathematicas docuit. </s>
          <s id="s.006442">varia compo­<lb></lb>
ſuit, quæ paſſim reperiuntur; legenda tamen cum antidoto Petri Nonij de <lb></lb>
erratis Orenſij. </s>
          <s id="s.006443">1530.</s>
        </p>
        <p type="main">
          <s id="s.006444">ERASMVS REINOLDVS eruditiſſimas Tabulas prutenicas. </s>
          <s id="s.006445">Item <expan abbr="cõ-mentaria">com­<lb></lb>
mentaria</expan> in Theoricas Purbachij edidit.</s>
        </p>
        <p type="main">
          <s id="s.006446">BARTHOLOMÆVS ZAMBERTVS, qui Euclidis Elementa, Optica, <lb></lb>
Catoptrica, Phænomena, &amp; Data ex Græcis Latina fecit.</s>
        </p>
        <p type="main">
          <s id="s.006447">PAVLVS Epiſcopus Foroſempronij. </s>
          <s id="s.006448">opus de Calendarij correctione, <lb></lb>
quod Paulina dicitur. </s>
          <s id="s.006449">ſub Leone X. conſcripſit.</s>
        </p>
        <p type="main">
          <s id="s.006450">ANDREAS SCHONERVS. de Gnomonica acutiſſimè ſcribit.</s>
        </p>
        <p type="main">
          <s id="s.006451">PETRVS APPIANVS de Geographia.</s>
        </p>
        <p type="main">
          <s id="s.006452">GEMMA Friſius arithmeticam practicam, aſtrolabium, &amp;c. </s>
          <s id="s.006453">ſcribit.</s>
        </p>
        <p type="main">
          <s id="s.006454">MICHAEL STIFELIVS arithmeticam integram, in qua Algebram op­<lb></lb>
tima methodo tradit.</s>
        </p>
        <p type="main">
          <s id="s.006455">ALOYSIVS LILIVS, alter noſtri æui Soſigenes, Calendarij <expan abbr="correctionẽ">correctionem</expan> <lb></lb>
excogitauit, qua cyclum Lunæ perpetuum, necnon ſtabilem æquinoctij ſe­<lb></lb>
dem faſtis Eccleſiaſticis indidit, quem ſequutus eſt Greg. XIII. Papa, dum <lb></lb>
anno Chriſti 1572. exemptis decem diebus, vniuerſo Chriſtiano orbi <expan abbr="Calẽ-darium">Calen­<lb></lb>
darium</expan> in perpetuum emendatum exhibuit. </s>
          <s id="s.006456">Eius frater Antonius Lilius <lb></lb>
vixit ſub Gregorio XIII.</s>
        </p>
        <p type="main">
          <s id="s.006457">RAPHAEL BOMBELLVS Bononienſis. </s>
          <s id="s.006458">Italicè de Algebra.</s>
        </p>
        <p type="main">
          <s id="s.006459">PETRVS NONIVS Salacienſis, vnico volumine varia pertractat. </s>
          <s id="s.006460">De <lb></lb>
nautica. </s>
          <s id="s.006461">in Theoricas Parbachij. </s>
          <s id="s.006462">de erratis Orontij. </s>
          <s id="s.006463">de crepuſculis.</s>
        </p>
        <p type="main">
          <s id="s.006464">LVCAS GAVRICVS Epiſcopus Ciuitatenſis, de Calendarij correctio­<lb></lb>
ne. </s>
          <s id="s.006465">Schol. in almag.</s>
        </p>
        <pb pagenum="61" xlink:href="009/01/345.jpg"></pb>
        <p type="main">
          <s id="s.006466">IOANNES BVTEO Logiſtices, lib. 5. de arca Noe. </s>
          <s id="s.006467">de quadraturis cir­<lb></lb>
culorum. </s>
          <s id="s.006468">tam antiquis, quam nouis.</s>
        </p>
        <p type="main">
          <s id="s.006469">FRANCISCVS MAVROLY VS Abbas Siculus, <expan abbr="Coſmographiã">Coſmographiam</expan>. </s>
          <s id="s.006470">Ari­<lb></lb>
thmeticorum lib. 3. de lineis horarijs Photiſmos. </s>
          <s id="s.006471">&amp; alia nonnulla, partim <lb></lb>
nondum edita, quorum index habetur in ſua Coſmographia. </s>
          <s id="s.006472">primus de li­<lb></lb>
neis ſecantibus ſcripſit.</s>
        </p>
        <p type="main">
          <s id="s.006473">HIERONYMVS CARDANVS, artem magnam ſcribit, in qua de al­<lb></lb>
gebra. </s>
          <s id="s.006474">obſeruauit Cometas eſſe in cœlo. </s>
          <s id="s.006475">in libris de ſubtilitate, &amp; varieta­<lb></lb>
te plurima miſcet ex omnibus Mathematicis.</s>
        </p>
        <p type="main">
          <s id="s.006476">IOANNES Padouanus de horolgijs.</s>
        </p>
        <p type="main">
          <s id="s.006477">FRANCISCVS FLVSSATES CANDALLA Gallus, nobiliſſimo gene­<lb></lb>
re ortus, commentaria in Euclidem: cui propria Minerua adiecit. </s>
          <s id="s.006478">16. li­<lb></lb>
brum. </s>
          <s id="s.006479">hic in Academia Burdigalenſi Mathematicarum profeſſori, annuum <lb></lb>
ſtipendium in perpetuum reliquit. </s>
          <s id="s.006480">ſi<expan abbr="q́">que</expan>; eam Cathedram fundauit.</s>
        </p>
        <p type="main">
          <s id="s.006481">FEDERICVS COMMANDINVS optimè meritus, ſi quiſquam alius <lb></lb>
de Mathematicis. </s>
          <s id="s.006482">Græcorum enim egregia monumenta nobis mira fęlici­<lb></lb>
tate traduxit, &amp; expoſuit. </s>
          <s id="s.006483">elementa Euclidis. </s>
          <s id="s.006484">conica Apollonij. </s>
          <s id="s.006485">opera Ar­<lb></lb>
chimedis. </s>
          <s id="s.006486">Ariſtarchum Samium. </s>
          <s id="s.006487">Bagdadinum de diuiſione figurarum. </s>
          <s id="s.006488">Ne­<lb></lb>
ronis ſpiritalia. </s>
          <s id="s.006489">Pappum Alexandrinum. </s>
          <s id="s.006490">Analemma Ptolemæi. </s>
          <s id="s.006491">ex proprijs <lb></lb>
verò. </s>
          <s id="s.006492">de centro grauit: ſolidorum. </s>
          <s id="s.006493">de lineis horarijs.</s>
        </p>
        <p type="main">
          <s id="s.006494">IOANNES de ROIAS aſtrolabium.</s>
        </p>
        <p type="main">
          <s id="s.006495">IOANNES STOFLERVS de fabrica, &amp; vſu aſtrolabij. </s>
          <s id="s.006496">commentaria <lb></lb>
in ſphæram Procli. </s>
          <s id="s.006497">de calendario.</s>
        </p>
        <p type="main">
          <s id="s.006498">ABRAHAMVS ORTELIVS Geographus. </s>
          <s id="s.006499">Theatrum mundi, &amp; theſau­<lb></lb>
rum geographicum.</s>
        </p>
        <p type="main">
          <s id="s.006500">GERARDVS MERCATOR Geographus, Ptolemæi geographiam re­<lb></lb>
ſtituit. </s>
          <s id="s.006501">Atlas, opus geographicum eius eſt.</s>
        </p>
        <p type="main">
          <s id="s.006502">ALEXANDER PICCOLOMIN. ſcripfit Italicè ſphæram. </s>
          <s id="s.006503">Theoricas <lb></lb>
planetarum. </s>
          <s id="s.006504">de ſtellis fixis. </s>
          <s id="s.006505">de magnitudine terræ, &amp; aquæ.</s>
        </p>
        <p type="main">
          <s id="s.006506">IOSEPHVS ZARLINVS de muſica duos tomos Italicè.</s>
        </p>
        <p type="main">
          <s id="s.006507">VINCENTIVS GALILEVS Florentinus. </s>
          <s id="s.006508">Italicè ſcribit quinque Dia­<lb></lb>
logos de muſica veteri, &amp; noua: vbi optimè <expan abbr="recentiorũ">recentiorum</expan> Contrapuntiſtarum <lb></lb>
(vt vocant) errata abſurdiſſima manifeſtat.</s>
        </p>
        <p type="main">
          <s id="s.006509">IO. BAPTISTA BENEDICTVS Gnomonica, &amp; ſpeculationes varias.</s>
        </p>
        <p type="main">
          <s id="s.006510">M. IACOBVS CHRISTMANVS comm. in Alfraganum: cui addidit <expan abbr="cō-mentum">com<lb></lb>
mentum</expan> eruditiſſimum de Calendarijs, &amp; temporum connexione.</s>
        </p>
        <p type="main">
          <s id="s.006511">IOSEPHVS AVRIA Neapolitanus optimè de Mathematicis meritus, <lb></lb>
ſiquidem quaſi alter <expan abbr="Cõmandmus">Commandinus</expan> priſcorum monumenta Græca nobis ex­<lb></lb>
ponere laborauit. </s>
          <s id="s.006512">eius ſunt: Autolycus de ſphera, quæ mouetur. </s>
          <s id="s.006513">Euclidis <lb></lb>
phænomena. </s>
          <s id="s.006514">Theodoſius Tripolita de habitationibus: &amp; de dicbus, &amp; no­<lb></lb>
ctibus. </s>
          <s id="s.006515">Item data Euclidis, nondum edita, quæ vt edantur, ſatago. </s>
          <s id="s.006516">plura <lb></lb>
alia dediſſet, ni mors interceſſiſſet.</s>
        </p>
        <p type="main">
          <s id="s.006517">NICOLAVS RAIMARVS, libellum edit, quo acutè per ſolam proſtha­<lb></lb>
phereſim, totum ſphæricorum triangulorum calculum abſoluit. </s>
          <s id="s.006518">P. Clauius <lb></lb>
in aſtrolabio.</s>
        </p>
        <p type="main">
          <s id="s.006519">IOANNES BAPT. Vicomercatus, de horologio ſolari inuenit modum <pb pagenum="62" xlink:href="009/01/346.jpg"></pb>deſcribendi plura horologia, &amp; varia vna, &amp; eadem opera ad Solem.</s>
        </p>
        <p type="main">
          <s id="s.006520">FRANCISCVS BAROCIVS patricius Venetus, cui plutimum debe­<lb></lb>
mus, tum ob Procli in Euclidem commentaria in latinum diligenter tran­<lb></lb>
ſlata, tum propter Heronis Mechanici de machinis bellicis, necnon de Geo­<lb></lb>
dæſia tranſlationem, <expan abbr="atq;">atque</expan> illuſtrationem. </s>
          <s id="s.006521">edidit præterea Coſmographiam.</s>
        </p>
        <p type="main">
          <s id="s.006522">P. ALEXANDER FLORAVANTVS Capucinus, ingenioſiſſimum, ac <lb></lb>
commodiſſimum inſtrumentum ad horologia in muri deſcribenda excogi­<lb></lb>
tauit; quod Retehorarium appellauit. </s>
          <s id="s.006523">F. </s>
          <s id="s.006524">Cherubinus in ſuo de horologijs <lb></lb>
Thaumalemmate.</s>
        </p>
        <p type="main">
          <s id="s.006525">GVIDVS VBALDVS Marchio, ex nobiliſſima familia de Monte. </s>
          <s id="s.006526">edidit <lb></lb>
Mechanica, Paraphraſim in æquepond. </s>
          <s id="s.006527">Archimedis. </s>
          <s id="s.006528">Aſtrolabium, Perſpe­<lb></lb>
ctiuam, omnia probatiſſima, &amp; proprio marte adinuenta. </s>
          <s id="s.006529">Poſthuma ſunt <lb></lb>
Problem. aſtron. </s>
          <s id="s.006530">&amp; opus de Cochlea.</s>
        </p>
        <p type="main">
          <s id="s.006531">TICHO BRAHE Baro Danus, verus Aſtronomiæ inſtaurator. </s>
          <s id="s.006532">in id ad <lb></lb>200. aureorum millia inſumpſit; nam, &amp; Palatium, &amp; inſtrumenta ſumptuo­<lb></lb>
ſa conſtruxit, &amp; operas plurimas aluit. </s>
          <s id="s.006533">opera eius edita ſunt, tomus primus <lb></lb>
de ſtella noua. </s>
          <s id="s.006534">alter de cometis, quas in cœlo reperit. </s>
          <s id="s.006535">epiſtolæ. </s>
          <s id="s.006536">mechanica. <lb></lb>
</s>
          <s id="s.006537">alia expectantur. </s>
          <s id="s.006538">aſſerit cœlum eſſe liquidum, &amp; quartum ignis elementum <lb></lb>
irridet. </s>
          <s id="s.006539">Venerem, &amp; Martem modò ſupra Solem, modò infra ferri obſerua­<lb></lb>
uit. </s>
          <s id="s.006540">obijt 1601.</s>
        </p>
        <p type="main">
          <s id="s.006541">IO. BAPT. VILLALPVNDVS Soc. Ieſu. in tertio tomo commentario­<lb></lb>
rum in Ezechielem, librum vnum iuſtæ magnitudinis habet, nouis demon­<lb></lb>
ſtrationibus Geometricis, &amp; alijs pluribus, tum ad mechanicam, tum ad <lb></lb>
menſuras geometricas pertinentibus refertum.</s>
        </p>
        <p type="main">
          <s id="s.006542">FRANCISCVS VIETA Gallus edidit, Canonem mathematicum, opus <lb></lb>
reſtitutæ Mathematicæ analyſeos, munimen aduerſus nouam Cyclometri­<lb></lb>
cam, Pſeudomeſolabum. </s>
          <s id="s.006543">Apollonius Gallus. </s>
          <s id="s.006544">Zetetica: &amp; alia nonnulla.</s>
        </p>
        <p type="main">
          <s id="s.006545">SIMON STEVINIVS Brugenſis, edidit Problem. Geometric. lib. 5.</s>
        </p>
        <p type="head">
          <s id="s.006546"><emph type="italics"></emph>XXVI. SECVLVM<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table30"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table30"></table.target>
          <row>
            <cell>Decimum verò ſeptimum Chr. ab ann. Domini</cell>
            <cell>1601</cell>
          </row>
          <row>
            <cell>Clemente 8. ſum. Pont.</cell>
            <cell></cell>
          </row>
          <row>
            <cell>Imp. Rodulpho 2. occid.</cell>
            <cell></cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006547"><emph type="italics"></emph>Reperitur Teloſcopium, quo in cœlo admiranda, ac noua <lb></lb>
primum ſpectantur.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006548">CHRISTOPHORVS CLAVIVS Bambergienſis è Soc. Ieſu, <lb></lb>
præceptor meus. </s>
          <s id="s.006549">ipſius opera ſunt: ſphæra. </s>
          <s id="s.006550">gnomonica. </s>
          <s id="s.006551">commen­<lb></lb>
taria in Euclidem. </s>
          <s id="s.006552">in Theodoſij ſphærica. </s>
          <s id="s.006553">de Triangulis planis, &amp; <lb></lb>
ſphæricis. </s>
          <s id="s.006554">Aſtrolabium. </s>
          <s id="s.006555">inſtrumentum ad horologia deſcriben­<lb></lb>
da. </s>
          <s id="s.006556">nona horologij deſcriptio per <expan abbr="Tangẽtes">Tangentes</expan>. </s>
          <s id="s.006557">Arithmetica practica. </s>
          <s id="s.006558">Geome­<lb></lb>
tria practica. </s>
          <s id="s.006559">Calendarij Romani à Greg. 13. reſtituti explicatio. </s>
          <s id="s.006560">Apologia <lb></lb>eiuſdem Calendarij contra Mæſtlinum, &amp; contra Ioſephum Scaligerum. <pb pagenum="63" xlink:href="009/01/347.jpg"></pb>Algebra. </s>
          <s id="s.006561">in quibus multa partim à ſe inuenta optimè demonſtrat. </s>
          <s id="s.006562">obijt ann. <lb></lb>
</s>
          <s id="s.006563">Domini 1612. 5. Februarij paulo poſt mediam noctem annorum 75. fere.</s>
        </p>
        <p type="main">
          <s id="s.006564">IO. ANTONIVS MAGINVS Bononiæ publicus <expan abbr="Mathematicarũ">Mathematicarum</expan> pro­<lb></lb>
feſſor. </s>
          <s id="s.006565">Geometriam practicam. </s>
          <s id="s.006566">Theoricas planetarum nouas, iuxta obſer­<lb></lb>
uationes Copernici. </s>
          <s id="s.006567">Tabulas ſecundorum mobilium. </s>
          <s id="s.006568">Primum mobile. </s>
          <s id="s.006569">Ta­<lb></lb>
bulas directionum. </s>
          <s id="s.006570">commentaria in Ptolemæi Geographiam. </s>
          <s id="s.006571">Ephemerides <lb></lb>
ad annos 50. &amp; Italicè de admirandis effectibus ſpeculi ſphærici ſcripſit. <lb></lb>
</s>
          <s id="s.006572">nunc Italiam magnum opus adornat.</s>
        </p>
        <p type="main">
          <s id="s.006573">MARINVS GHETALDVS patricius Raguſinus. </s>
          <s id="s.006574">Promotus Archime­<lb></lb>
des. </s>
          <s id="s.006575">de parabola, &amp; ſpeculo vſtorio. </s>
          <s id="s.006576">item Apollonius rediuiuus. </s>
          <s id="s.006577">&amp; ſupplem. <lb></lb>
</s>
          <s id="s.006578">Apoll. Galli. </s>
          <s id="s.006579">adhuc viuit.</s>
        </p>
        <p type="main">
          <s id="s.006580">LVCAS VALERIVS Romæ publicus Mathematicarum profeſſor. </s>
          <s id="s.006581">de <lb></lb>
centro grauit. </s>
          <s id="s.006582">ſolidorum. </s>
          <s id="s.006583">opus magno acumine conſcriptum. </s>
          <s id="s.006584">Item Qua­<lb></lb>
dratura Paraboles aliter, quàm Archimedes adhuc viuit.</s>
        </p>
        <p type="main">
          <s id="s.006585">ADRIANVS ROMANVS Belga, eius ſunt, Idæa Mathematica. </s>
          <s id="s.006586">Vra­<lb></lb>
nographia. </s>
          <s id="s.006587">expoſitio Archimedis de circuli dimenſione. </s>
          <s id="s.006588">exercitationes cy­<lb></lb>
clicæ. </s>
          <s id="s.006589">de Triangulis ſphæricis.</s>
        </p>
        <p type="main">
          <s id="s.006590">Nobiliſſimus CAROLVS GESVALDVS Princeps Venuſinus, noſtræ <lb></lb>
tempeſtatis Muſicorum, ac Melopæorum princeps, ac veteris Muſicæ re­<lb></lb>
ſtaurator. </s>
          <s id="s.006591">hic enim rithmis in Muſicam reuocatis, eos tum ad cantum, tum <lb></lb>
ad ſonum Modulos adhibuit, vt cæteri omnes Muſici, ei primas libenter de­<lb></lb>
tulerint, <expan abbr="eiusq́">eiusque</expan>; Modos Cantores, ac Fidicines omnes, reliquis poſthabitis, <lb></lb>
<expan abbr="vbiq;">vbique</expan> auidè complectantur. </s>
          <s id="s.006592">obijt 1614.</s>
        </p>
        <p type="main">
          <s id="s.006593">IO. BAPT. PORTA, eruditiſſimus æquè, ac nobiliſſimus. </s>
          <s id="s.006594">editi ſunt eius <lb></lb>
lib. 9. de Refractione optices. </s>
          <s id="s.006595">elementorum curuilineorum lib. 3. Interpre­<lb></lb>
tatio primi Almageſti, cum comm. Theonis. </s>
          <s id="s.006596">de Munitione lib. 3. Pneuma­<lb></lb>
ticorum lib. 3. Catoptrica nondum edita.</s>
        </p>
        <p type="main">
          <s id="s.006597">P. BERNARDINVS SALINVS de Soc. Ieſu. libri 11. in quibus ſuppo­<lb></lb>
ſita recta æquali circumferentiæ plurima veluti corollaria demonſtrantur. <lb></lb>
</s>
          <s id="s.006598">de horologijs lib. 2. varia problemata aſtronomica lib. 1. de menſuris geo­<lb></lb>
metricis lib. 1. quæ nondum edita aſſeruantur Genuæ in Colleg. Soc. noſtræ. <lb></lb>
</s>
          <s id="s.006599">obijt ann. </s>
          <s id="s.006600">Domini circiter 1608.</s>
        </p>
        <p type="main">
          <s id="s.006601">PETRVS ANTONIVS CATTALDVS Bononienſis, <expan abbr="publicusq́">publicusque</expan>; Bo­<lb></lb>
noniæ Mathematicarum profeſſor. </s>
          <s id="s.006602">cuius opera iam edita ſunt, Elementa <lb></lb>
numerorum arithmeticorum. </s>
          <s id="s.006603">Elementa Geometricorum. </s>
          <s id="s.006604">Algebra pro­<lb></lb>
portionalis. </s>
          <s id="s.006605">de lineis rectis æquidiſtantibus, &amp; non æquidiſtantibus; vbi <lb></lb>
Poſtulatum quintum, &amp; ſeptimum primi Euclid. oſtenſiuè, ac breuiter de­<lb></lb>
monſtrat. </s>
          <s id="s.006606">De numeris perfectis. </s>
          <s id="s.006607">Transformatio Geometrica, qua oſten­<lb></lb>
dit datum rectilineum, illud ipſum reducere ad formam propoſiti rectilinei. <lb></lb>
</s>
          <s id="s.006608">De radice quadrata breuiſſimè inuenienda. </s>
          <s id="s.006609">De quadratura circuli. </s>
          <s id="s.006610">Plures <lb></lb>
lectiones mathematicæ. </s>
          <s id="s.006611">Apud ipſum verò abſoluta, <expan abbr="atq;">atque</expan> ad Typum para­<lb></lb>
ta hec ſunt: Archimedis defenſio. </s>
          <s id="s.006612">Euclidis defenſio. </s>
          <s id="s.006613">Algebra numeralis, <lb></lb>
linealis, &amp; applicata. </s>
          <s id="s.006614">Elementa numerorum denominatorum. </s>
          <s id="s.006615">De regula <lb></lb>
aurea ſumma breuitate. </s>
          <s id="s.006616">Transformatio geometrica figurę, in aliam cuius <lb></lb>
ambitus, ac laterum numerus ſit propoſitus. </s>
          <s id="s.006617">Algebra triangularis. </s>
          <s id="s.006618">Hor­<lb></lb>
tus mathematicus. </s>
          <s id="s.006619">Continuatio algebræ proportionalis, vbi acutiſſimum <pb pagenum="64" xlink:href="009/01/348.jpg"></pb>opus zeteticorum doctiſſimi Franciſci Vietæ exponit. </s>
          <s id="s.006620">Examen geometriæ <lb></lb>
Caroli Bouilij.</s>
        </p>
        <p type="main">
          <s id="s.006621">IOANNES KEPLERVS Mathematicus Cæſareus, à quo edita ſunt; <lb></lb>
Myſterium coſmographicum. </s>
          <s id="s.006622">De ſtellis nouis. </s>
          <s id="s.006623">Paralipomena ad Vitell. <lb></lb>
vnà cum Optica aſtronomica. </s>
          <s id="s.006624">Opus de ſtella Martis. </s>
          <s id="s.006625">Dioprice.</s>
        </p>
        <p type="main">
          <s id="s.006626">GALILÆVS GALILÆVS Florentinus, cui plurimùm debet tota poſte­<lb></lb>
ritas, nam ope Teleſcopij nuper à Belgis inuenti, reperit quatuor planetas <lb></lb>
circa Iouem errantes; &amp; innumeras alias fixas; in Luna montes, ac valles; <lb></lb>nebuloſas eſſe ſtellularum greges, Gallaxiam eſſe exiguorum aſteriſcorum <lb></lb>
agmen; Venerem inſtar Lunæ augeri, &amp; minui; Saturnum duobus ſtipari <lb></lb>
ſatellitibus; hæc partim in ſuo Sydereo Nuncio exponit; partim in libro <lb></lb>
Italicè ſcripto de Maculis ſolaribus, vbi ſe primum earum repertorem eſſe <lb></lb>
contendit. </s>
          <s id="s.006627">Item Italicè de ijs, quæ natant, aut mouentur in aqua; opus <lb></lb>
acutiſſimum; vbi aliquot Ariſt. loca Mathematica expendit. </s>
          <s id="s.006628">adhuc viuit, <lb></lb>
&amp; nouum mundi Syſtema adornat.</s>
        </p>
        <p type="main">
          <s id="s.006629">APELLES poſt tabulam latens (ſic ficto nomine appellari voluit P. Chri­<lb></lb>ſtophorus Scheiner Germanus è Societate noſtra) maculas ſolares proprio <lb></lb>
Marte animaduertit, quid circa eas eodem ferè tempore alij agerent, om­<lb></lb>
ninò neſcius. </s>
          <s id="s.006630">eas tamen primus, libello ficti nominis, publici iuris fecit. <lb></lb>
</s>
          <s id="s.006631">item libellum de Sole elliptico. </s>
          <s id="s.006632">1612.</s>
        </p>
        <p type="main">
          <s id="s.006633">MARCVSANTONIVS de DOMINIS Archiepiſcopus Spalatri. </s>
          <s id="s.006634">de ra­<lb></lb>
dijs viſus, &amp; lucis: vbi inquirit Teleſcopij demonſtrationem.</s>
        </p>
        <p type="main">
          <s id="s.006635">P. CHRISTOPHORVS GREIMBERGERVS è Societ. noſtra, qui ad <lb></lb>
Aſtrolabium, &amp; Horologia attulit non pauca ipſo Clauio teſte. </s>
          <s id="s.006636">nuper edi­<lb></lb>
dit Catalogum veteres affixarum longitudines, &amp; latitudines <expan abbr="conferẽs">conferens</expan> cum <lb></lb>
nouis. </s>
          <s id="s.006637">Item libellum de ſpeculo vſtorio; &amp; Appendicem ad practicam Co­<lb></lb>
ni ſectionem, cui annexa ſunt conſectaria, quæ circulorum contactui, <expan abbr="ſectio-nemq́">ſectio­<lb></lb>
nemque</expan>; angulorum curuilineorum concernunt.</s>
        </p>
        <p type="main">
          <s id="s.006638">P. Fr. AGVILLONIVS BELGA è noſtra Societ. edidit elegantiſſimum <lb></lb>
Opticæ volumen; &amp; alterum adornat.</s>
        </p>
        <p type="main">
          <s id="s.006639">Huius ſeculi præcedens pars deſinit in anno Dom. 1614. quo ipſa Chro­<lb></lb>
nologia pariter abſoluta eſt.</s>
        </p>
        <p type="main">
          <s id="s.006640"><expan abbr="Atq;">Atque</expan> hic finis eſto breuis huius Chronologiæ, quæ continet auctores ferè <lb></lb>
257. annos verò 2464. in 26. ſecula diſtributa; in quam ex antiquis omnes <lb></lb>
quot quot reperire potuimus, ex recentioribus ſelectiores, qui aut ſcrip­<lb></lb>tis, aut rebus claruerint, cooptauimus; alioquin recenſendi fuiſſent omnes <lb></lb>
Pythagorici, <expan abbr="atq;">atque</expan> omnes Platonici, qui omnes Mathematicis eximié naua­<lb></lb>
bant operam. </s>
          <s id="s.006641">omnes præterea Poetæ ab Homero <expan abbr="vſq;">vſque</expan> ad Chriſt<emph type="italics"></emph>i<emph.end type="italics"></emph.end> ferè ſecu­<lb></lb>lum annumerandi fuiſſent, erant enim antiquitus poetæ omnes ſimul etiam <lb></lb>
muſici, vt perſpicuum eſt ex ijs, quæ in libro de muſica Plutarchus in hunc <lb></lb>
modum ait; non equidem fuiſſe immunem metri, numeriuè rati dictionem <lb></lb>
poematum muſicorum, ſed qualis Stetichori fuit, &amp; veterum aliorum poe­<lb></lb>
tarum, qui carmina adhibitis modulis condidere, Terpandrum <expan abbr="nanq;">nanque</expan> tra­<lb></lb>
dunt adiectis ad ſua, <expan abbr="atq;">atque</expan> Homeri carmina, per ſingulas leges modis, ſoli­<lb></lb>
tum in ludis cum concertatione editis canere.</s>
        </p>
        <p type="main">
          <s id="s.006642"><expan abbr="Neq;">Neque</expan> mireris, quòd <expan abbr="recẽtcs">recentes</expan> muſicos omnes, quos Contrapuntiſtas appel­


<pb pagenum="65" xlink:href="009/01/349.jpg"></pb>lant, omiſerim, id enim conſultò, ac meritò feci, cùm mihi obſequentes ra­<lb></lb>
tiones, quas breuiter ex Dialogis de muſica Vincentij Galilæi decerpſi, <lb></lb>
nomine muſici, indigni videantur.</s>
        </p>
        <p type="main">
          <s id="s.006643">Primò, quia officio muſici minimè funguntur: eſt autem ex Platonis, <expan abbr="atq;">atque</expan> <lb></lb>
Ariſtot. ſententia, officium muſici, rithmis, ſiue numeris vti ad auditorum <lb></lb>
affectus excitandos: Contrapuntiſtæ verò iſti rithmum omnem, aut nume­<lb></lb>
rum a ſpernantur.</s>
        </p>
        <p type="main">
          <s id="s.006644">Secundò, quia ſuas illas quatuor, aut quinque partes ſic ſimul <expan abbr="confundũt">confundunt</expan>, <lb></lb>
vt nullum verbum, <expan abbr="nullusq́">nullusque</expan>; rithmus percipiatur: ſed mera tantummodo <lb></lb>
muſica quædam confuſio: quam, qui audit neſcit quid audiat.</s>
        </p>
        <p type="main">
          <s id="s.006645">Tertiò, quia eodem modo carmina, ac ſolutam orationem canunt, vt in­<lb></lb>
telligere nequeas carmina ne, an proſam decantent. </s>
          <s id="s.006646">quod quidem maximè <lb></lb>
eſt inconueniens, &amp; veteri muſicæ contrarium. </s>
          <s id="s.006647"><expan abbr="neq;">neque</expan> enim muſica carminis <lb></lb>
numerum offuſcare; ſed eum magis exornare, <expan abbr="atq;">atque</expan> viuidum reddere debet.</s>
        </p>
        <p type="main">
          <s id="s.006648">Quartò, quia cantilenæ ſenſum omnem, dum diuerſa verba ſimul plures, <lb></lb>
in cantu pronunciant, ita tollunt, vt nihil omnino intelligatur: cùm tamen <lb></lb>
muſici officium ſit, cantilenæ ſententiam cantu, &amp; rithmo auditorum ani­<lb></lb>
mis ita in genere, vt eos iuxta ſententiæ illius <expan abbr="affectũ">affectum</expan>, afficiat, ac <expan abbr="cõmoueat">commoueat</expan>.</s>
        </p>
        <p type="main">
          <s id="s.006649">Quintò, quia de induſtria <expan abbr="cõtra">contra</expan> leges antiquas repetitiones eiuſdem, vel <lb></lb>
<expan abbr="conſonãtiæ">conſonantiæ</expan>, vel cadentiæ, aut aiunt, <expan abbr="atq;">atque</expan> <expan abbr="etiã">etiam</expan> rithmi maximè vitant: quod <lb></lb>
tamen ad animorum motus ciendos plurimùm valet.</s>
        </p>
        <p type="main">
          <s id="s.006650">Sextò, quia non muſicè, ſed mimicè, ideſt non rithmis, ſed modis à natu­<lb></lb>
ra muſicæ alienis, ac ridiculis frequenter imitari geſtiunt.</s>
        </p>
        <p type="main">
          <s id="s.006651">Hac igitur noſtra <expan abbr="qualicunq;">qualicunque</expan> fruere lucubratiuncula: <expan abbr="atq;">atque</expan> in ea contem­<lb></lb>
plare, quo <expan abbr="tẽpore">tempore</expan>, &amp; à quibus non ſolùm Mathematica, ſed aliæ etiam <expan abbr="ſciẽ-tiæ">ſcien­<lb></lb>
tiæ</expan> ortum habuerunt: quando &amp; apud quos floruerint, aut deſierint; ac <expan abbr="tã-dem">tan­<lb></lb>
dem</expan> iterum reuiuiſcere cęperint.</s>
        </p>
        <p type="main">
          <s id="s.006652">Quòd ſi quæras, <expan abbr="quibuſnã">quibuſnam</expan> ſtudijs, tribus illis annorum millibus, quæ chro­<lb></lb>
nologiam hanc noſtram præceſſerunt, homines vacauerint, ac proinde, cur <lb></lb>
tam ſerò literis operam nauare cæperint: reſpondendum eſſe arbitror, <lb></lb>
toto illo tempore homines fuiſſe totos, tum in artibus inue­<lb></lb>
niendis, atque excolendis, tum in vrbibus, atque <lb></lb>
rebus publicis conſtituendis: quippe­<lb></lb>
quæ magis humanæ vitæ ne­<lb></lb>
ceſſaria erant. <lb></lb>
</s>
          <s id="s.006653">Vale.</s>
        </p>
        <p type="head">
          <s id="s.006654">DEO OPT. MAX. LAVS.</s>
        </p>
        <pb xlink:href="009/01/350.jpg"></pb>
        <p type="head">
          <s id="s.006655">INDEX<lb></lb>
In præcedentem Chronologiam.<lb></lb>
<arrow.to.target n="table31"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table31"></table.target>
          <row>
            <cell><emph type="italics"></emph>A<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ardalus muſicus.<emph.end type="italics"></emph.end></cell>
            <cell><emph type="italics"></emph>ſec.<emph.end type="italics"></emph.end> 1</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Anaximander aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ametiſtus geomctra.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Anaximenes aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Anaxagoras astron.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Antiſthenes muſicus.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Amiclas geom.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Antiphon geom.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ariſtæus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Aratus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Aristoteles mathem.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Autolycus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ariſtoxenes muſ.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Archelaws geographus.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ariſtarchus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Arçhimedes mathem.<emph.end type="italics"></emph.end></cell>
            <cell>7</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Apollónius magnus geometra.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Athenæus mechan.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Andronicus mechan.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Artemidorus geogr.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Andromachus astron.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Abifeldea geogr.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>S. Auguſtmus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Almaon Rex aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>18</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Albaregnius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>18</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Achilles aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>18</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Alfarabius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Alhumaſar aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>19</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Alfraganus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>19</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Alhazenus opticus.<emph.end type="italics"></emph.end></cell>
            <cell>20</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Arzael aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>20</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Auerroes aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>21</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Almeon Almanſorius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>21</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Alpetragius aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>21</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>R. Abraham aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>21</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Alfonſus Rex aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Andreas Schonerus gnonom.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Aloyſius Lihus aſir.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Abrahamus Ortelius geogr.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Alexand. Florauantes gnom.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Adrianus R. B. mathem.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Apelles Latens aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>B<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Bryſo geom.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Beroſus aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Boetius mathem.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>V. Beda mathem.<emph.end type="italics"></emph.end></cell>
            <cell>17</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Bagdadinus geom.<emph.end type="italics"></emph.end></cell>
            <cell>19</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ben Muſa geom.<emph.end type="italics"></emph.end></cell>
            <cell>19</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Barlaam arithm.<emph.end type="italics"></emph.end></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Bart. Zambertus geom.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Bernardinus Salinus geom.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>C<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Carolus Geſualdus muſic.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Clonas muſic. ſec.<emph.end type="italics"></emph.end></cell>
            <cell>1</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cleoſtratus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cratistus geom.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cygicinus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Calippus astron.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Conon mathem.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cteſibius mechan.<emph.end type="italics"></emph.end></cell>
            <cell>7</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cleomedes aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>C. Manlius aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cenſorinu aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>12</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cyrillus Epiſc. astr.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Caſſiodorus math.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Campanus geom.<emph.end type="italics"></emph.end></cell>
            <cell>20</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Chriſtoph. Columb. nautic.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cbr. Clauius mathem.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Chr. Gruenbergeius mathem.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cardanus aſtr.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Cattaldus arithm.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>D<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Damon muſic.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diocles muſic.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Democritus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Dinoſtratus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Dicearcbus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Dionyſiodorus geom.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <pb xlink:href="009/01/351.jpg"></pb>
          <row>
            <cell><emph type="italics"></emph>Dionyſius Afer geogr.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diophantes arithm.<emph.end type="italics"></emph.end></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Diocles geom.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Demetrius geom.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Dionyſius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>E<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Evshorbus geometra.<emph.end type="italics"></emph.end></cell>
            <cell>2</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Empedocles muſ.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Epicurus muſ.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Euctemon aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Eudoxius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Euclides geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Eratoſthenes aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>7</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Euſebius Epiſc. aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Eudemus geom.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Eutocius geom.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Eraſmus Reinoldus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>F<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Franchinus muſ.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Fr. Lucas arithm.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ferd. Megalanes naut.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Fr. Maurolycus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Fr. Fluſſas geom.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Feder. Command. mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Fr. Barocius mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Fr. Vieta mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Fr. Aguillonius optic.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>G<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Geminus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Geminus Rhodius aſtr. &amp; geom.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>S. Greg. mag. muſ.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Geber aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>18</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Guido Aret. muſ.<emph.end type="italics"></emph.end></cell>
            <cell>19</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Geor. Purbachius astron.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Gemma Friſius mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Gerardus Mercator geogr.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Guiduſubaldus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Galilæus Galilæus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>H<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hecateus geogr.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hippocr. Chius geom.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Helicon aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hermotimus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hermophilus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Heraclides muſ.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hero mechan.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hipparchus astron.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hippolytus Epiſc. aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>12</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hypatia aſtron. &amp; arithm.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Hero mechan. alter.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Heliodorus optic.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>I<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ismenius muſ.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iſidorus geom.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iul. Cæſar astron.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iul. Higinius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iul. Maternus aſtrol.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Grammat. arithm.<emph.end type="italics"></emph.end></cell>
            <cell>15</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iſidorus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>16</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iſacius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>20</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iordanes arithm.<emph.end type="italics"></emph.end></cell>
            <cell>21</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. de Sacroboſco aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Gira nautic.<emph.end type="italics"></emph.end></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Epiſc. optic.<emph.end type="italics"></emph.end></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iacob. Faber muſ.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Monteregius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Vernerus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Blanchinus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Buteo mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Paduanus gnomon:<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Roias aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Stoſterus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ioſephus Zarlinus muſ.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Bapt. Benedictus gnomon.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ioſephus Auria mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Iacob. Chriſtm. aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Bapt. Vicomerc. gnomon.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Villalpandus geom.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Ant. Maginus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Io. Keplerus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>L<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Licaon muſ.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Laſus muſ.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Leodamas geom.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Leon geom.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>L. Papirius gnomon.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Leonardus algebrat.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ludouicus Folianus muſ.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Lucas Gaur. aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Lucas Valerius mechan.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <pb xlink:href="009/01/352.jpg"></pb>
          <row>
            <cell><emph type="italics"></emph>M<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mamertimis geom.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Methon aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Menechmus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>M. Agrippa geogr.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Marinus geogr.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Menelaus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Maximus arithm.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Menelaus geom.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Marinus Neapolit. geom.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Martianus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>16</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Michael Pſeltus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>18</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Mar. Polus geogr.<emph.end type="italics"></emph.end></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Michael Stif. arithra.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Marinus Ghetaldus mechan.<emph.end type="italics"></emph.end></cell>
            <cell>26</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>N<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Nicomachus arithm.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Neoclides geom.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Nicomedes geom.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Nicolaus Cabaſilla aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Nicolaus Cuſanus geom.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Nic. Copern. astron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Nic. Raimarus geom.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>O<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Oenipedes geom.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Orontius matbem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>P<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Polemon geogr.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Pythagoras mathem. ſummus.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Pericles aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Phrinis muſ.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Phrinicus muſ.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Parmenides aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Protagoras mathem.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Plato mathem.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Philippus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Philoſophus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Perſeus geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Philo mechan.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Poſſidonius mathem.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Patroclus gnomon.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Parmenion gnomon.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>P. Mela geogr.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Plinius geogr.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Plutarchus muſ.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ptolemæus astron.<emph.end type="italics"></emph.end></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Porphirius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>12</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Philo geom.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Proclus geom.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>S. Proſper aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Pappus geom.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Paulus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>18</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Petoſiris astron.<emph.end type="italics"></emph.end></cell>
            <cell>18</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Pelles arithm.<emph.end type="italics"></emph.end></cell>
            <cell>18</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Petrus de Aliaco aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>24</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Paulus Epiſc. aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Petrus Appianus geogr.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Petrus Nonius mathem.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>R<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Rogerius Baccon optic.<emph.end type="italics"></emph.end></cell>
            <cell>23</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Raphael arithm.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>S<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Sacadas muſ.<emph.end type="italics"></emph.end></cell>
            <cell>2</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Simonides lyric.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Sappho muſic.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Simon muſic.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Simmias muſic.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Sulpitius aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>7</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Serenus geom.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Soſigenes aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Scopas gnomon.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Strabo geogr.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Solinus geogr.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Straton geogr.<emph.end type="italics"></emph.end></cell>
            <cell>10</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Sextus Empir. mathem.<emph.end type="italics"></emph.end></cell>
            <cell>11</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>S. Auienus geogr.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Sporus geom.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>T<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Terpander muſic.<emph.end type="italics"></emph.end></cell>
            <cell>1</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Thales Mileſius.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Telauges arithm.<emph.end type="italics"></emph.end></cell>
            <cell>3</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Theodorus geom.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Timeus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Theætetus geom.<emph.end type="italics"></emph.end></cell>
            <cell>5</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Theudius geom.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Theophraſtus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Timotheus muſic.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Timocharis aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Theodoricus aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Theophilus Epiſc. astron.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Theon aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Theon Smyrnæus mathem.<emph.end type="italics"></emph.end></cell>
            <cell>21</cell>
          </row>
          <pb xlink:href="009/01/353.jpg"></pb>
          <row>
            <cell><emph type="italics"></emph>Thebit aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ticho aſtron.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>V<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vincentius Galilæus muſic.<emph.end type="italics"></emph.end></cell>
            <cell>25</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vitellio optic.<emph.end type="italics"></emph.end></cell>
            <cell>22</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Victorinus astron.<emph.end type="italics"></emph.end></cell>
            <cell>14</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Vitruuius gnomon.<emph.end type="italics"></emph.end></cell>
            <cell>9</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Valens aſtrol.<emph.end type="italics"></emph.end></cell>
            <cell>13</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>X<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Xenocrates muſic.<emph.end type="italics"></emph.end></cell>
            <cell>1</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Xenocrates mathem.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Xenophantus muſic.<emph.end type="italics"></emph.end></cell>
            <cell>6</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Y<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Ypſicles geometra.<emph.end type="italics"></emph.end></cell>
            <cell>8</cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Z<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell><emph type="italics"></emph>Zenodorus geometra.<emph.end type="italics"></emph.end></cell>
            <cell>4</cell>
          </row>
        </table>
        <p type="head">
          <s id="s.006656"><emph type="italics"></emph>FINIS.<emph.end type="italics"></emph.end></s>
        </p>
        <p type="main">
          <s id="s.006657">Ego Fr. Hieronymus Onuphrius Romanus, ex Conuentu S. Mariæ Gra­<lb></lb>
tiarum, Doctor Colleg. &amp; Lector publicus, ac Sanctiſs. Inquiſitionis <lb></lb>
Conſultor, vel libentiſſimè vidi, ac perlegi Opus hoc aureum inſcriptum, <lb></lb>
LOCA ARISTOTELIS MATHEMATICA, &amp; conſcriptum <lb></lb>
ab Excellentiſs. P. Ioſepho Blancano de Societ. IESV. &amp; cùm in eo nihil re­<lb></lb>
periatur, quod aut ſit contra Eccleſiaſticas diſciplinas, aut quod pias aures <lb></lb>
offendat, quinimo maxima emergat vtilitas ijs, qui Ariſtotelicum textum <lb></lb>
conſultò amplectuntur, ideò poſſe typis dari cenſui, &amp;c.</s>
        </p>
        <p type="head">
          <s id="s.006658">Imprimatur</s>
        </p>
        <p type="head">
          <s id="s.006659"><emph type="italics"></emph>Idem qui &amp; ſupra nomine Reuerendiſſ. P. Inquiſit. Bonon.<emph.end type="italics"></emph.end></s>
        </p>
        <pb xlink:href="009/01/354.jpg"></pb>
        <p type="head">
          <s id="s.006660"><emph type="italics"></emph>Errata, quæ Lectorem ſisterent, ſic corrigantur.<emph.end type="italics"></emph.end><lb></lb>
<arrow.to.target n="table32"></arrow.to.target></s>
        </p>
        <table>
          <table.target id="table32"></table.target>
          <row>
            <cell><emph type="italics"></emph>pag. linea. erratum. correctum.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>4</cell>
            <cell>2</cell>
            <cell>perſuaſiſti</cell>
            <cell>peruaſiſti</cell>
          </row>
          <row>
            <cell>18</cell>
            <cell>22</cell>
            <cell>quadratum</cell>
            <cell>quadrangulum</cell>
          </row>
          <row>
            <cell>36</cell>
            <cell>34</cell>
            <cell>inueſtare</cell>
            <cell>inueftigare</cell>
          </row>
          <row>
            <cell>38</cell>
            <cell>15</cell>
            <cell>A G,</cell>
            <cell>A C G,</cell>
          </row>
          <row>
            <cell>39</cell>
            <cell>22</cell>
            <cell>æquatiorem</cell>
            <cell>æquatorem</cell>
          </row>
          <row>
            <cell>51</cell>
            <cell>46</cell>
            <cell><expan abbr="vnoq;">vnoque</expan></cell>
            <cell><expan abbr="vnoquoq;">vnoquoque</expan></cell>
          </row>
          <row>
            <cell>70</cell>
            <cell>in figura defiderantur duæ lineolæ rectæ A I, &amp; I E,</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>80</cell>
            <cell>13</cell>
            <cell>Tex.</cell>
            <cell>Textus 59.</cell>
          </row>
          <row>
            <cell>ibid.</cell>
            <cell>in figura pro O, ponatur C,</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>82</cell>
            <cell>13</cell>
            <cell>Tex.</cell>
            <cell>Tex. 110.</cell>
          </row>
          <row>
            <cell>ibid.</cell>
            <cell>45</cell>
            <cell>Vrſam</cell>
            <cell>ad Vrſam</cell>
          </row>
          <row>
            <cell>83</cell>
            <cell>42</cell>
            <cell>Tex.</cell>
            <cell>Tex. 65.</cell>
          </row>
          <row>
            <cell>107</cell>
            <cell>10</cell>
            <cell>fides</cell>
            <cell>fidem</cell>
          </row>
          <row>
            <cell>ibid.</cell>
            <cell>11</cell>
            <cell>ſaxiatritu</cell>
            <cell>ſciſſa, &amp; attrita</cell>
          </row>
          <row>
            <cell>166</cell>
            <cell>in 2 figura deſideratur linea P R.</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>127</cell>
            <cell>28</cell>
            <cell>L,</cell>
            <cell>C,</cell>
          </row>
          <row>
            <cell>155</cell>
            <cell>in 2. figura deeſt linea K P:</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell></cell>
            <cell>altera verò, quæ dextrorſum vergit, ſuperflua eſt.</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>220</cell>
            <cell>45</cell>
            <cell>quadratum</cell>
            <cell>quadrangulum</cell>
          </row>
          <row>
            <cell>221</cell>
            <cell>1</cell>
            <cell>quadratum</cell>
            <cell>quadrangulum</cell>
          </row>
          <row>
            <cell>231</cell>
            <cell>30</cell>
            <cell>incrementa</cell>
            <cell>decrementa</cell>
          </row>
          <row>
            <cell>ibid.</cell>
            <cell>31</cell>
            <cell>decrementis</cell>
            <cell>incrementis</cell>
          </row>
          <row>
            <cell>235</cell>
            <cell>19</cell>
            <cell>L O,</cell>
            <cell>M N,</cell>
          </row>
          <row>
            <cell>ibid.</cell>
            <cell>21</cell>
            <cell>N M,</cell>
            <cell>O L,</cell>
          </row>
          <row>
            <cell>ibid.</cell>
            <cell>34</cell>
            <cell>H L,</cell>
            <cell>N L,</cell>
          </row>
          <row>
            <cell>237</cell>
            <cell>in figura linea K D L, deberet directè ad punctum A, tendere</cell>
            <cell></cell>
            <cell></cell>
          </row>
          <row>
            <cell>256</cell>
            <cell>4</cell>
            <cell>toto</cell>
            <cell>tono</cell>
          </row>
          <row>
            <cell>264</cell>
            <cell>14</cell>
            <cell>numeris</cell>
            <cell>neruis</cell>
          </row>
          <row>
            <cell>266</cell>
            <cell>44</cell>
            <cell>cæteris</cell>
            <cell>cæteri</cell>
          </row>
          <row>
            <cell>267</cell>
            <cell>45</cell>
            <cell>indicium</cell>
            <cell>indicium</cell>
          </row>
          <row>
            <cell>274</cell>
            <cell>27</cell>
            <cell>contrà verum</cell>
            <cell>contrà, verum</cell>
          </row>
          <row>
            <cell>279</cell>
            <cell>30</cell>
            <cell>MKINQ,</cell>
            <cell>MHINQ,</cell>
          </row>
          <row>
            <cell></cell>
            <cell></cell>
            <cell><emph type="italics"></emph>In Additamento.<emph.end type="italics"></emph.end></cell>
            <cell></cell>
          </row>
          <row>
            <cell>9</cell>
            <cell>5</cell>
            <cell>vncìa</cell>
            <cell>vnica</cell>
          </row>
          <row>
            <cell>9</cell>
            <cell>28</cell>
            <cell>proportionali</cell>
            <cell>proportionalis</cell>
          </row>
          <row>
            <cell>14</cell>
            <cell>28</cell>
            <cell>diuionem</cell>
            <cell>diuiſionem</cell>
          </row>
          <row>
            <cell>18</cell>
            <cell>22</cell>
            <cell>78.</cell>
            <cell>7. 8.</cell>
          </row>
        </table>
        <pb xlink:href="009/01/355.jpg"></pb>
        <p type="head">
          <s id="s.006661">BONONIÆ</s>
        </p>
        <p type="head">
          <s id="s.006662">Apud Bartholomæum Cochìum. </s>
          <s id="s.006663">1615.</s>
        </p>
        <p type="head">
          <s id="s.006664">Superiorum permiſſu.<lb></lb>
</s>
        </p>
      </chap>
    </body>
  </text>
</archimedes>