Mercurial > hg > mpdl-xml-content
annotate texts/XML/echo/la/Archimedes_1565_4E7V2WGH.xml @ 29:90b1eda1b0a9
Some new special instructions
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
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| date | Fri, 02 Dec 2016 14:37:22 +0100 |
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1 <?xml version="1.0" encoding="utf-8"?><echo xmlns="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:de="http://www.mpiwg-berlin.mpg.de/ns/de/1.0/" xmlns:dcterms="http://purl.org/dc/terms" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:echo="http://www.mpiwg-berlin.mpg.de/ns/echo/1.0/" xmlns:xhtml="http://www.w3.org/1999/xhtml" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" version="1.0RC"> |
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2 <metadata> |
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3 <dcterms:identifier>ECHO:4E7V2WGH.xml</dcterms:identifier> |
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4 <dcterms:creator identifier="GND:118503863">Archimedes</dcterms:creator> |
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5 <dcterms:title xml:lang="la">Archimedis De iis quae ve huntur in aqua libri duo</dcterms:title> |
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6 <dcterms:date xsi:type="dcterms:W3CDTF">1565</dcterms:date> |
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7 <dcterms:language xsi:type="dcterms:ISO639-3">lat</dcterms:language> |
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8 <dcterms:rights>CC-BY-SA</dcterms:rights> |
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9 <dcterms:license xlink:href="http://creativecommons.org/licenses/by-sa/3.0/">CC-BY-SA</dcterms:license> |
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10 <dcterms:rightsHolder xlink:href="http://www.mpiwg-berlin.mpg.de">Max Planck Institute for the History of Science, Library</dcterms:rightsHolder> |
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11 </metadata> |
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12 <text xml:lang="la" type="free"> |
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13 <div xml:id="echoid-div1" type="section" level="1" n="1"><pb file="0001" n="1"/> |
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14 <pb file="0002" n="2"/> |
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15 <handwritten/> |
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16 <handwritten/> |
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17 <pb file="0003" n="3"/> |
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18 <pb file="0004" n="4"/> |
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19 <handwritten/> |
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20 <handwritten/> |
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21 <handwritten/> |
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22 <pb file="0005" n="5"/> |
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23 </div> |
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24 <div xml:id="echoid-div2" type="section" level="1" n="2"> |
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25 <head xml:id="echoid-head1" xml:space="preserve">ARCHIMEDIS</head> |
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26 <head xml:id="echoid-head2" xml:space="preserve">DE IIS QVAE VEHVNTVR <lb/>IN AQVA LIBRI DVO.</head> |
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27 <head xml:id="echoid-head3" xml:space="preserve">A FEDERICO COMMANDINO <lb/>VRBINATE IN PRISTINVM <lb/>NITOREM RESTITVTI, ET <lb/>COMMENTARIIS ILLVSTRATI.</head> |
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28 <figure> |
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29 <image file="0005-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0005-01"/> |
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30 </figure> |
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31 <figure> |
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32 <image file="0005-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0005-02"/> |
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33 </figure> |
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34 </div> |
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35 <div xml:id="echoid-div3" type="section" level="1" n="3"> |
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36 <head xml:id="echoid-head4" xml:space="preserve">CVM PRIVILEGIO IN ANNOS X. <lb/>BONONIAE,</head> |
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37 <p> |
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38 <s xml:id="echoid-s1" xml:space="preserve">Ex Officina Alexandri Benacii.</s> |
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39 <s xml:id="echoid-s2" xml:space="preserve"/> |
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40 </p> |
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41 <handwritten/> |
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42 </div> |
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43 <div xml:id="echoid-div4" type="section" level="1" n="4"> |
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44 <head xml:id="echoid-head5" xml:space="preserve">M D LXV.</head> |
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45 <figure> |
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46 <image file="0005-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0005-03"/> |
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47 </figure> |
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48 <pb file="0006" n="6"/> |
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49 <pb file="0007" n="7"/> |
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50 </div> |
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51 <div xml:id="echoid-div5" type="section" level="1" n="5"> |
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52 <head xml:id="echoid-head6" xml:space="preserve">RANVTIO FARNESIO <lb/>CARDINALI AMPLISSIMO <lb/>ET OPTIMO.</head> |
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53 <p> |
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54 <s xml:id="echoid-s3" xml:space="preserve"><emph style="sc">QVod</emph> tibi ſuperioribus diebus <lb/>pollicitus ſum, cum libellum <lb/>Ptolemæi de Analemmate in lu <lb/>cem proferrem, breui fore, vc<unsure/> <lb/>Archimedis etiam libri de ijs, <lb/>quæ in aqua vehuntur, & </s> |
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55 <s xml:id="echoid-s4" xml:space="preserve">emen <lb/>datiores, & </s> |
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56 <s xml:id="echoid-s5" xml:space="preserve">fortaſſe opera mea <lb/>illuſtriores ederentur: </s> |
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57 <s xml:id="echoid-s6" xml:space="preserve">mihi non committendum eſ-<lb/>ſe duxi, vt iure optimo malum nomen, præſertim à<unsure/> <lb/>te, cui tantopere debeo, exiſtimari poſſem. </s> |
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58 <s xml:id="echoid-s7" xml:space="preserve">quam-<lb/>uis cum mecum conſidero ſuſcepti negocij difficul <lb/>tates, quas multo plures, & </s> |
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59 <s xml:id="echoid-s8" xml:space="preserve">multo grauiores, quàm <lb/>in libello de Analemmate deprehendi; </s> |
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60 <s xml:id="echoid-s9" xml:space="preserve">vereor ne id <lb/>planè non aſſecutus ſim, quod ab initio ſpectaui, vt <lb/>mathematicarum diſciplinarum ſtudioſis hac in par-<lb/>te ſatisfacerem. </s> |
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61 <s xml:id="echoid-s10" xml:space="preserve">cum enim græcus Archimedis co-<lb/>dex nondum in lucem venerit, non ſolum is, qui eum <lb/>latinitate donauit, multis in locis fœ de lapſus eſt, ve-<lb/>rum etiam codex ipſe, vt etiam interpres fatetur, ve-<lb/>tuſtate corruptus, & </s> |
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62 <s xml:id="echoid-s11" xml:space="preserve">mancus eſt; </s> |
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63 <s xml:id="echoid-s12" xml:space="preserve">duæq́; </s> |
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64 <s xml:id="echoid-s13" xml:space="preserve">integræ <lb/>ἀποδείξεις, quas demonſtrationes dicimus, deperie-<lb/>runt. </s> |
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65 <s xml:id="echoid-s14" xml:space="preserve">quæ iactura quantam vim habeat ad pertur-<lb/>bandum admirabilem illum ordinem, quo inter ſe <lb/>mathematicæ diſciplinæ quodãmodo connexæ ſunt, |
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66 <pb file="0008" n="8"/> |
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67 tibi, qui iam in iis multam operam, multumq́; </s> |
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68 <s xml:id="echoid-s15" xml:space="preserve">ſtu-<lb/>dium poſuiſti, cogitandum relinquo. </s> |
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69 <s xml:id="echoid-s16" xml:space="preserve">nonnulla præ-<lb/>terea Archimedes vt perſpicua in his tractandis po-<lb/>nere non dubitauit, quæ veteres mathematici, qui <lb/>de conicis conſcripſerunt, plurimis, & </s> |
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70 <s xml:id="echoid-s17" xml:space="preserve">firmiſsimis <lb/>argumentis probauerũt. </s> |
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71 <s xml:id="echoid-s18" xml:space="preserve">Hæc autem idcirco à nobis <lb/>omnino ignorantur; </s> |
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72 <s xml:id="echoid-s19" xml:space="preserve">quòd poſtremi quatuor libri <lb/>conicorum Apollonii Pergæi adhuc in tenebris de-<lb/>liteſcunt. </s> |
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73 <s xml:id="echoid-s20" xml:space="preserve">Qua quidem in re (vt mea fert opinio) <lb/>ſingulari fato fuerunt mathematicæ diſciplinæ, cum <lb/>tot ſcriptorum præclara monumenta interierint, per <lb/>quæ non ſolum in ſtudioſos homines, uerum etiam <lb/>in humanũ genus mirabiles utilitates importatæ fuiſ-<lb/>ſent. </s> |
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74 <s xml:id="echoid-s21" xml:space="preserve">nam cum mecum conſidero quàm late pateant <lb/>hæ nobiliſsimæ ſcientiæ, quãtopere rebus publicis & </s> |
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75 <s xml:id="echoid-s22" xml:space="preserve"><lb/>priuatis admirabili quadã ratione, atque ordine gu-<lb/>bernandis neceſſariæ ſint, dubitãdum non exiſtimo, <lb/>quin magna ſit habenda gratia huius diuini boni au-<lb/>ctoribus, & </s> |
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76 <s xml:id="echoid-s23" xml:space="preserve">inuentoribus: </s> |
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77 <s xml:id="echoid-s24" xml:space="preserve">ueterumq́; </s> |
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78 <s xml:id="echoid-s25" xml:space="preserve">græcorum pru <lb/>dentiam ſatis admirari non poſſum, qui pueros cum <lb/>primum fari cœpiſſent, his diſciplinis imbuendos cu <lb/>rabant, ut à prima ætate multiplicis, ac ſubtilis ſcien-<lb/>tiæ contemplationi aſſueti nihil paruum, aut humile <lb/>cogitarent: </s> |
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79 <s xml:id="echoid-s26" xml:space="preserve">ſed uel ſe totos ijs artibus traderent, qua-<lb/>rum ope ciuitatibus ſuis & </s> |
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80 <s xml:id="echoid-s27" xml:space="preserve">præſidio, & </s> |
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81 <s xml:id="echoid-s28" xml:space="preserve">ornamento <lb/>eſſe poſſent: </s> |
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82 <s xml:id="echoid-s29" xml:space="preserve">uel humanis ſtudijs multam ſalutem di-<lb/>centes, diuinam philoſophiam toto animo amplexa-<lb/>rentur, cum ad eam per mathematicas diſciplinas fa- |
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83 <pb file="0009" n="9"/> |
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84 ciliorem ſibi aditum comparaſſent. </s> |
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85 <s xml:id="echoid-s30" xml:space="preserve">quamobrem gra <lb/>uisſimum damnum factum eſt in tot præſtãtisſimis <lb/>uiris: </s> |
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86 <s xml:id="echoid-s31" xml:space="preserve">quorũ ſcripta ſi in manus noſtras perueniſſent, <lb/>profecto multo præclarius cum rebus humanis age-<lb/>retur. </s> |
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87 <s xml:id="echoid-s32" xml:space="preserve">complures enim, qui nunctot difficultatibus <lb/>ab his ſtudijs deterrentur, hac ratione priuatis & </s> |
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88 <s xml:id="echoid-s33" xml:space="preserve">pu-<lb/>blicis rationibus optime conſuluiſſent. </s> |
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89 <s xml:id="echoid-s34" xml:space="preserve">Cum hæc ita <lb/>eſſent, tamen nullum mihi laborem ſubterfugiendũ <lb/>eſſe iudicaui, quo ſtudioſis hominibus, qui in mathe <lb/>maticis diſciplinis toto animo incumbũt, facilior pa <lb/>teret aditus ad abſtruſa, & </s> |
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90 <s xml:id="echoid-s35" xml:space="preserve">recondita ſenſa tanti ſcri-<lb/>ptoris intelligenda: </s> |
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91 <s xml:id="echoid-s36" xml:space="preserve">nec à uetere meo in ſtituto diſce-<lb/>dere uolui; </s> |
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92 <s xml:id="echoid-s37" xml:space="preserve">ſcis enim me multos abhinc annos hanc <lb/>eandem prouinciam, Archimedis quàm plurima ſcri <lb/>pta illuſtrandi ſuſcepiſſe. </s> |
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93 <s xml:id="echoid-s38" xml:space="preserve">quod neque arrogãtia, nec <lb/>inanis gloriæ ſpe adductus ſum, ut facerẽ, ſed me ue-<lb/>hementer in hanc mentem impulit honeſtisſima cu-<lb/>piditas de ſtudioſis hominibus benemerẽdi: </s> |
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94 <s xml:id="echoid-s39" xml:space="preserve">etenim <lb/>ſemper mea fuit ſentẽtia, mathematicum, qui libros <lb/>Archimedis accuratisſime non euoluerit, uix mathe-<lb/>maticum appellari debere: </s> |
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95 <s xml:id="echoid-s40" xml:space="preserve">cum eū neceſſe ſit in mul <lb/>tarum rerum ignoratione uerſari, ſine quibus mathe <lb/>maticæ diſciplinæ imperfectæ quodammodo, atque <lb/>in choatæ ſunt habendæ. </s> |
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96 <s xml:id="echoid-s41" xml:space="preserve">Dedi igitur operam, ut his <lb/>etiam Archimedis libris, quoad eius fieri poſſet, per <lb/>me aliqua lux afferretur. </s> |
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97 <s xml:id="echoid-s42" xml:space="preserve">quos ut Archimedis eſſe nõ <lb/>dubitarem, duæ non contemnendæ cauſſæ fuerunt. <lb/></s> |
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98 <s xml:id="echoid-s43" xml:space="preserve">una quòd in tanta obſcuritate ab interpretis inſcitia, |
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99 <pb file="0010" n="10"/> |
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100 & </s> |
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101 <s xml:id="echoid-s44" xml:space="preserve">à uetuſtate profecta, neſcio quod ueſtigium illius <lb/>acuti, & </s> |
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102 <s xml:id="echoid-s45" xml:space="preserve">perſpicacis ingenij, quo Archimedes excel-<lb/>luit, impreſſum apparet: </s> |
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103 <s xml:id="echoid-s46" xml:space="preserve">altera quòd tum græci, tum <lb/>latini ſcriptores grauisſimi hos ut Archimedis libros <lb/>recognoſcũt. </s> |
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104 <s xml:id="echoid-s47" xml:space="preserve">Strabo enim in primo libro hæc ad uer <lb/>bũ ſcribit. </s> |
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105 <s xml:id="echoid-s48" xml:space="preserve">ὁδὲ οὕτος ἡδὺς ἐ<unsure/>στὶν, ὥστε η{αὶ} μὴ μαθηματιηὸςὤν, οὐδὲ <lb/>τὴν Αρχιμήδουςβεβαιοῖ δόξαν, ὅτιφησὶνἐη{εῖ} νος ἐν τοῖς περἱ τῶνὀχον-<lb/>μένων, παντὸς ὑγροῦ καθεστηηότος, καἱ μένοντος τὴνἑ πιφάν{ει}αν σφαιρι-<lb/>κὴν {εῖ}ν{αι}, σφ{αὶ} ρας ταυτὸ ηέντρον ἐφούσης τῆ γῆ. </s> |
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106 <s xml:id="echoid-s49" xml:space="preserve">ταὺ την γάρ τὴν δοξαν <lb/>ἀποδέχονται πάντες οἱ μαθημάτων πῶς άψάμενοι. </s> |
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107 <s xml:id="echoid-s50" xml:space="preserve">& </s> |
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108 <s xml:id="echoid-s51" xml:space="preserve">Pappus Ale-<lb/>xandrinus in octauo mathematicarum collectionum <lb/>libro hæc ſcripta reliquit, ηαλοῦσι δὲ μηχανιηοὺςοἱ παλαιοὶ, <lb/>κ{αὶ} τοὺς θαυμασιουργοὺς, ὡνοἱ μὲν διὰ πνευμὰτων φιλοτεχνοῦσιν, ὡς <lb/>ἥρων πνευματιηοῖς, οἱ δὲ διὰ νευρίων καὶ σπάρτωνἐμψύχωνκινήσεις δο-<lb/>κοῦσι μιμ\~εισθαι, ὡςἥρων αὐτομάτοις, καὶ ζυγίοις: </s> |
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109 <s xml:id="echoid-s52" xml:space="preserve">ἄλλοι δὲ διὰ τῶν ἐφ<unsure/> <lb/>ὕδατος ὁχουμένων, ὡςὰρχιμήδης ὀχουμένοις. </s> |
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110 <s xml:id="echoid-s53" xml:space="preserve">Vitruuius etiam <lb/>in octauo libro de his eiſdem Archimedis libris me. <lb/></s> |
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111 <s xml:id="echoid-s54" xml:space="preserve">minit. </s> |
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112 <s xml:id="echoid-s55" xml:space="preserve">Fortaſſe, inquit, qui Archimedis libros legit, di <lb/>cet non poſſe fieri ueram ex aqua librationem: </s> |
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113 <s xml:id="echoid-s56" xml:space="preserve">ſed ei <lb/>placet aquam non eſſe libratam, ſed ſphæroides habe <lb/>re ſchema: </s> |
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114 <s xml:id="echoid-s57" xml:space="preserve">& </s> |
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115 <s xml:id="echoid-s58" xml:space="preserve">ibi habere centrum, quo loci habet or-<lb/>bis terrarum. </s> |
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116 <s xml:id="echoid-s59" xml:space="preserve">ut nemini dubium eſſe posſit, quin & </s> |
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117 <s xml:id="echoid-s60" xml:space="preserve"><lb/>genere ſcriptionis, & </s> |
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118 <s xml:id="echoid-s61" xml:space="preserve">tãtorum uirorum auctoritate, <lb/>ut germani Archimedis libri attente legendi, & </s> |
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119 <s xml:id="echoid-s62" xml:space="preserve">per-<lb/>pendendi ſint: </s> |
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120 <s xml:id="echoid-s63" xml:space="preserve">præſertim cum in ijs multa continean <lb/>tur cognitione dignisſirna, quæ nõ tam ad mathema <lb/>ticas diſciplinas, quàm ad naturæ obſcuritatem ſpe-<lb/>ctant. </s> |
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121 <s xml:id="echoid-s64" xml:space="preserve">Quamobrem ego ne tanto, & </s> |
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122 <s xml:id="echoid-s65" xml:space="preserve">tam fructuoſo <lb/>theſauro diutius ſtudioſi carerent, primum loca par- |
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123 <pb file="0011" n="11"/> |
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124 tim interpretis errore deprauata emendaui; </s> |
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125 <s xml:id="echoid-s66" xml:space="preserve">partim <lb/>uetuſtate corrupta & </s> |
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126 <s xml:id="echoid-s67" xml:space="preserve">conſumpta in priſtinam inte-<lb/>gritatem redegi, compluribus, quæ deſiderabantur, <lb/>meo, ut aiunt, marte ſuppletis. </s> |
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127 <s xml:id="echoid-s68" xml:space="preserve">Deinde quoniam Ar-<lb/>chimedes, quemadmodum ſupra dixi, non nulla po-<lb/>nit, ut perſpicua, & </s> |
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128 <s xml:id="echoid-s69" xml:space="preserve">quæ uel ipſe, uel ſuperiores ma-<lb/>thematici ἀποδείξεσι confirmauerunt, coactus ſum non <lb/>ſine maximo negotio ex ijs principijs conicæ diſcipli <lb/>næ Apollonij Pergæi, quæ in manus noſtras peruene-<lb/>rũt, nouas probationes adhibere, nequid eſſet, quod <lb/>diligentem lectorem in hac parte remorari poſſet. </s> |
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129 <s xml:id="echoid-s70" xml:space="preserve">re <lb/>ſtabat, ut theorema illud, quod ſine cognitione cen-<lb/>tri grauitatis corporum ſolidorũ percipinon poteſt, <lb/>uidelicet, Centrum grauitatis in portionibus conoi-<lb/>dis rectanguli axem ita diuidere, ut pars, quæ ad uer-<lb/>ticem terminatur, reliquæ partis, quæ ad baſim ſit du <lb/>pla, certisſimis rationibus comprobarem. </s> |
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130 <s xml:id="echoid-s71" xml:space="preserve">ſed huic <lb/>quoque rei prouiſum eſt à me: </s> |
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131 <s xml:id="echoid-s72" xml:space="preserve">ſeorſumq́ ab his li-<lb/>bris de cẽtro grauitatis ſolidorũ uberrime cõſcripſi. <lb/></s> |
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132 <s xml:id="echoid-s73" xml:space="preserve">denique nihil prætermiſi, quod ad Archimedem in <lb/>hac materia illuſtrandum attineret. </s> |
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133 <s xml:id="echoid-s74" xml:space="preserve">quod ſi, ut ſpero, <lb/>aſſecutus ſum, ſatis magnum fructum mihi cepiſſe ui <lb/>debor laborum, & </s> |
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134 <s xml:id="echoid-s75" xml:space="preserve">uigiliarum mearum: </s> |
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135 <s xml:id="echoid-s76" xml:space="preserve">ſin ſecus acci <lb/>derit, hoc me tamen conſolabor, quòd omnes intelli <lb/>gent, honeſtisſimo meo conſilio, non tã ingenij mei <lb/>imbecillitatem, quàm rei obſcuritatem, & </s> |
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136 <s xml:id="echoid-s77" xml:space="preserve">temporũ <lb/>iniurias obſtitiſſe. </s> |
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137 <s xml:id="echoid-s78" xml:space="preserve">Hoc loco ſuperuacaneum eſſe arbi <lb/>tror pluribus uerbis exponere, cur tibi amplisſime |
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138 <pb file="0012" n="12"/> |
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139 Cardinalis, has lucubrationes meas dicare conſtitue-<lb/>rim. </s> |
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140 <s xml:id="echoid-s79" xml:space="preserve">tantis enim beneficijs à te affectus, quanta fem-<lb/>per & </s> |
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141 <s xml:id="echoid-s80" xml:space="preserve">meminero, & </s> |
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142 <s xml:id="echoid-s81" xml:space="preserve">prædicabo; </s> |
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143 <s xml:id="echoid-s82" xml:space="preserve">tanta liberalitate cõ-<lb/>plexus, quantam ne optare quidem unquam auſus eſ <lb/>ſem. </s> |
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144 <s xml:id="echoid-s83" xml:space="preserve">cupio memorem, & </s> |
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145 <s xml:id="echoid-s84" xml:space="preserve">erga te gratum animũ qua <lb/>ratione poſſum, oſtendere. </s> |
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146 <s xml:id="echoid-s85" xml:space="preserve">quãuis ſi de te nihil aliud <lb/>præter auditum haberem, ſi amplitudini tuæ tanto-<lb/>pere deuinctus non eſſem; </s> |
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147 <s xml:id="echoid-s86" xml:space="preserve">tua in omni genere diſci-<lb/>plinarum excellentia, tua grauitas, atque innocentia <lb/>me magnopere hortata eſſet, ut te potisſimum deli-<lb/>gerem, ſub cuius clarisſimi nominis ſplendore hi Ar-<lb/>chimedis libri ab obliuione hominum, atque à ſilen-<lb/>tio uindicarentur. </s> |
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148 <s xml:id="echoid-s87" xml:space="preserve">uerecundius de te in præſentia di-<lb/>cerem, ne uiderer aſſentationi potius, quàm ueritati <lb/>ſeruire; </s> |
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149 <s xml:id="echoid-s88" xml:space="preserve">niſi omnibus perſuaſisſimum eſſet, diuinas & </s> |
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150 <s xml:id="echoid-s89" xml:space="preserve"><lb/>inauditas uirtutes tuas cum ſingulari eruditione con <lb/>iunctas in illo ſanctisſimo Reip. </s> |
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151 <s xml:id="echoid-s90" xml:space="preserve">chriſtianæ conſilio <lb/>tanquam lumen aliquod elucere. </s> |
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152 <s xml:id="echoid-s91" xml:space="preserve">quamobrem ea, <lb/>qua ſoles, benignitate, fidelisſimi clientis tui munus <lb/>accipies; </s> |
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153 <s xml:id="echoid-s92" xml:space="preserve">quod tibi, qui & </s> |
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154 <s xml:id="echoid-s93" xml:space="preserve">mathematicis diſciplinis, <lb/>& </s> |
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155 <s xml:id="echoid-s94" xml:space="preserve">phiſiologiæ ſtudijs tantopere delectaris, non iniu-<lb/>cundum fore confido.</s> |
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156 <s xml:id="echoid-s95" xml:space="preserve"/> |
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157 </p> |
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158 </div> |
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159 <div xml:id="echoid-div6" type="section" level="1" n="6"> |
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160 <head xml:id="echoid-head7" xml:space="preserve">Federicus Commandinus.</head> |
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161 <pb o="1" file="0013" n="13"/> |
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162 </div> |
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163 <div xml:id="echoid-div7" type="section" level="1" n="7"> |
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164 <head xml:id="echoid-head8" xml:space="preserve">ARCHIMEDIS DE IIS <lb/>QVAE VEHVNTVR IN AQVA <lb/>LIBER PRIMVS.</head> |
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165 <head xml:id="echoid-head9" xml:space="preserve">CVM COMMENTARIIS FEDERICI <lb/>COMMANDINI VRBINATIS.</head> |
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166 <head xml:id="echoid-head10" xml:space="preserve">POSITIO.</head> |
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167 <p> |
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168 <s xml:id="echoid-s96" xml:space="preserve">PONATVR humidi eam <lb/>eſſe naturam, vt partibus ip-<lb/>ſius æqualiter iacentibus, & </s> |
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169 <s xml:id="echoid-s97" xml:space="preserve"><lb/>continuatis inter ſe ſe, minus <lb/>preſſa à magis preſſa expella <lb/>tur. </s> |
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170 <s xml:id="echoid-s98" xml:space="preserve">Vnaquæque autem pars <lb/>eius premitur humido ſupra <lb/>ipſam exiſtente ad perpendiculum, ſi humidum <lb/>ſit deſcendens in aliquo, aut ab alio aliquo preſ-<lb/>ſum.</s> |
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171 <s xml:id="echoid-s99" xml:space="preserve"/> |
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172 </p> |
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173 </div> |
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174 <div xml:id="echoid-div8" type="section" level="1" n="8"> |
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175 <head xml:id="echoid-head11" xml:space="preserve">PROPOSITIO I.</head> |
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176 <p> |
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177 <s xml:id="echoid-s100" xml:space="preserve">SI ſuperficies aliqua plano ſecetur per idẽ ſem-<lb/>per punctum; </s> |
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178 <s xml:id="echoid-s101" xml:space="preserve">ſitq́; </s> |
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179 <s xml:id="echoid-s102" xml:space="preserve">ſectio circuli circunferen-<lb/>tia, centrum habens punctum illud, per quod pla <lb/>no ſecatur: </s> |
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180 <s xml:id="echoid-s103" xml:space="preserve">ſphæræ ſuperficies erit.</s> |
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181 <s xml:id="echoid-s104" xml:space="preserve"/> |
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182 </p> |
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183 <pb file="0014" n="14" rhead="ARCHIMEDIS"/> |
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184 <p> |
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185 <s xml:id="echoid-s105" xml:space="preserve">SECETVR ſuperficies aliqua plano per k punctum <lb/>ducto: </s> |
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186 <s xml:id="echoid-s106" xml:space="preserve">& </s> |
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187 <s xml:id="echoid-s107" xml:space="preserve">ſicſectio ſemper circuli circunferentia, centrum <lb/>habens punctum k. </s> |
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188 <s xml:id="echoid-s108" xml:space="preserve">Dico eam ſphæræ ſuperficiem eſſe. </s> |
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189 <s xml:id="echoid-s109" xml:space="preserve">Si <lb/>enim non eſt ſphæræ ſuperfi-<lb/> |
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190 <anchor type="figure" xlink:label="fig-0014-01a" xlink:href="fig-0014-01"/> |
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191 cies; </s> |
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192 <s xml:id="echoid-s110" xml:space="preserve">rectæ lineæ, quæ à pun-<lb/>cto k ad circunferentiam du-<lb/>cuntur non omnes æquales e-<lb/>runt. </s> |
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193 <s xml:id="echoid-s111" xml:space="preserve">Itaque ſint a b puncta <lb/>in ſuperficie; </s> |
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194 <s xml:id="echoid-s112" xml:space="preserve">& </s> |
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195 <s xml:id="echoid-s113" xml:space="preserve">inæquales li-<lb/>neæ a k k b: </s> |
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196 <s xml:id="echoid-s114" xml:space="preserve">per ipſas autem <lb/>a k k b planum ducatur, quod <lb/>ſectionem faciat in ſuperficie <lb/>lineam d a b c. </s> |
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197 <s xml:id="echoid-s115" xml:space="preserve">ergo d a b c cir <lb/>culi circunferentia eſt, cuius <lb/>centrum k; </s> |
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198 <s xml:id="echoid-s116" xml:space="preserve">quoniam ſuperficies eiuſmodi ponebatur: </s> |
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199 <s xml:id="echoid-s117" xml:space="preserve">& </s> |
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200 <s xml:id="echoid-s118" xml:space="preserve"><lb/>idcirco æquales inter ſe ſunt a k k b, ſed & </s> |
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201 <s xml:id="echoid-s119" xml:space="preserve">inæquales; </s> |
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202 <s xml:id="echoid-s120" xml:space="preserve">quod <lb/>fieri non poteſt. </s> |
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203 <s xml:id="echoid-s121" xml:space="preserve">conſtat igitur ſuperficiem eam eſſe ſphæ-<lb/>ræ ſuperficiem.</s> |
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204 <s xml:id="echoid-s122" xml:space="preserve"/> |
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205 </p> |
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206 <div xml:id="echoid-div8" type="float" level="2" n="1"> |
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207 <figure xlink:label="fig-0014-01" xlink:href="fig-0014-01a"> |
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208 <image file="0014-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0014-01"/> |
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209 </figure> |
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210 </div> |
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211 </div> |
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212 <div xml:id="echoid-div10" type="section" level="1" n="9"> |
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213 <head xml:id="echoid-head12" xml:space="preserve">PROPOSITIO II.</head> |
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214 <p> |
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215 <s xml:id="echoid-s123" xml:space="preserve"><emph style="sc">Omnis</emph> humidi conſiſtentis, atque manen-<lb/>tis ſuperficies ſphærica eſt; </s> |
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216 <s xml:id="echoid-s124" xml:space="preserve">cuius ſphæræ centrũ <lb/>eſtidem, quod centrum terræ.</s> |
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217 <s xml:id="echoid-s125" xml:space="preserve"/> |
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218 </p> |
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219 <p> |
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220 <s xml:id="echoid-s126" xml:space="preserve">INTELLIGATVR humidũ conſiſtens, manẽsq;</s> |
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221 <s xml:id="echoid-s127" xml:space="preserve">: <lb/>& </s> |
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222 <s xml:id="echoid-s128" xml:space="preserve">ſecetur ipſius ſuperficies plano per centrum terræ du-<lb/>cto. </s> |
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223 <s xml:id="echoid-s129" xml:space="preserve">ſit autem terræ centrum k: </s> |
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224 <s xml:id="echoid-s130" xml:space="preserve">& </s> |
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225 <s xml:id="echoid-s131" xml:space="preserve">ſuperficieiſectio, linea <lb/>a b c d. </s> |
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226 <s xml:id="echoid-s132" xml:space="preserve">Dico lineam a b c d circuli circunferentiam eſſe, cu <lb/>ius centrum k. </s> |
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227 <s xml:id="echoid-s133" xml:space="preserve">Si enim non eſt, rectæ lineæ à puncto k ad <lb/>lineam a b c d ductæ non erunt æquales. </s> |
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228 <s xml:id="echoid-s134" xml:space="preserve">Sumatur recta li <lb/>nea quibuſdam quidem à puncto k ad ipſam a b c d ductis <lb/>maior; </s> |
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229 <s xml:id="echoid-s135" xml:space="preserve">quibuſdam uero minor; </s> |
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230 <s xml:id="echoid-s136" xml:space="preserve">& </s> |
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231 <s xml:id="echoid-s137" xml:space="preserve">ex centro k, interual- |
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232 <pb o="2" file="0015" n="15" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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233 loq; </s> |
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234 <s xml:id="echoid-s138" xml:space="preserve">lineæ ſumptæ circulus deſcribatur. </s> |
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235 <s xml:id="echoid-s139" xml:space="preserve">cadet ergo ipſius <lb/>circunferentia partim <lb/> |
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236 <anchor type="figure" xlink:label="fig-0015-01a" xlink:href="fig-0015-01"/> |
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237 extra lineam a b c d, par <lb/>tim intra; </s> |
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238 <s xml:id="echoid-s140" xml:space="preserve">quoniam ea, <lb/>quæ ex centro quibuſ-<lb/>dam quidem à puncto <lb/>k ad ipſam ductis eſtma <lb/>ior; </s> |
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239 <s xml:id="echoid-s141" xml:space="preserve">& </s> |
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240 <s xml:id="echoid-s142" xml:space="preserve">quibuſdam mi-<lb/>nor. </s> |
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241 <s xml:id="echoid-s143" xml:space="preserve">Itaq; </s> |
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242 <s xml:id="echoid-s144" xml:space="preserve">ſit circuli de-<lb/>ſcripti circunferentia <lb/>fb h: </s> |
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243 <s xml:id="echoid-s145" xml:space="preserve">& </s> |
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244 <s xml:id="echoid-s146" xml:space="preserve">ex b ad k ducta <lb/>linea, iungãtur fk k h e, <lb/>quæ angulos æquales faciant. </s> |
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245 <s xml:id="echoid-s147" xml:space="preserve">deſcribatur autem & </s> |
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246 <s xml:id="echoid-s148" xml:space="preserve">ex cen <lb/>tro k circunferentia quædam x o p in plano, & </s> |
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247 <s xml:id="echoid-s149" xml:space="preserve">in humido. <lb/></s> |
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248 <s xml:id="echoid-s150" xml:space="preserve">ergo partes humidi, quæ ſunt ad circunferentiam x o p æ-<lb/>qualiter iacent, ac continuatæ inter ſe ſe: </s> |
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249 <s xml:id="echoid-s151" xml:space="preserve">& </s> |
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250 <s xml:id="echoid-s152" xml:space="preserve">premũtur qui <lb/>dem partes, quæ ad x o circunferentiam, humido, quod lo <lb/>co a b continetur: </s> |
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251 <s xml:id="echoid-s153" xml:space="preserve">quæ uero ad circunferentiam o p pre-<lb/>muntur humido, quod continetur b e. </s> |
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252 <s xml:id="echoid-s154" xml:space="preserve">inæqualiter igitur <lb/>premuntur partes humidi ad cir cunferentiã x o, & </s> |
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253 <s xml:id="echoid-s155" xml:space="preserve">ad o p. </s> |
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254 <s xml:id="echoid-s156" xml:space="preserve"><lb/>quare minus preſſæ à magis presſis expellentur. </s> |
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255 <s xml:id="echoid-s157" xml:space="preserve">non er-<lb/>go conſiſtet humidum. </s> |
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256 <s xml:id="echoid-s158" xml:space="preserve">Atqui ponebatur conſiſtens, & </s> |
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257 <s xml:id="echoid-s159" xml:space="preserve">ma <lb/>nens. </s> |
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258 <s xml:id="echoid-s160" xml:space="preserve">neceſſarium eſt igitur lineam a b c d eſſe circuli cir <lb/>cunferentiam, cuius centrum k. </s> |
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259 <s xml:id="echoid-s161" xml:space="preserve">Similiter autem demon-<lb/>ſtrabitur, & </s> |
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260 <s xml:id="echoid-s162" xml:space="preserve">ſi quomodocunque aliter ſuperficies humidi <lb/>plano ſecta fuerit per centrum terræ ſectionem circuli cir <lb/>cunferentiam eſſe: </s> |
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261 <s xml:id="echoid-s163" xml:space="preserve">& </s> |
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262 <s xml:id="echoid-s164" xml:space="preserve">centrum ipſius eſſe, quod & </s> |
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263 <s xml:id="echoid-s165" xml:space="preserve">terræ cen <lb/>trum. </s> |
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264 <s xml:id="echoid-s166" xml:space="preserve">Ex quibus conſtat ſuperficiem humidi conſiſtentis, <lb/> |
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265 <anchor type="note" xlink:label="note-0015-01a" xlink:href="note-0015-01"/> |
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266 atque manentis ſphæricam eſſe: </s> |
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267 <s xml:id="echoid-s167" xml:space="preserve">& </s> |
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268 <s xml:id="echoid-s168" xml:space="preserve">eius ſphæræ centrum <lb/>idem, quod centrum terræ: </s> |
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269 <s xml:id="echoid-s169" xml:space="preserve">quoniam eiuſmodi eſt, ut ſecta <lb/>per idem ſemper punctum ſectionem faciat circuli circun <lb/>ferentiam, centrum habentis punctum illud, per quod ipſa <lb/>plano ſecatur.</s> |
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270 <s xml:id="echoid-s170" xml:space="preserve"/> |
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271 </p> |
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272 <div xml:id="echoid-div10" type="float" level="2" n="1"> |
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273 <figure xlink:label="fig-0015-01" xlink:href="fig-0015-01a"> |
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274 <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0015-01"/> |
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275 </figure> |
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276 <note position="right" xlink:label="note-0015-01" xlink:href="note-0015-01a" xml:space="preserve">Prima hu <lb/>ius.</note> |
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277 </div> |
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278 <pb file="0016" n="16" rhead="ARCHIMEDIS"/> |
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279 </div> |
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280 <div xml:id="echoid-div12" type="section" level="1" n="10"> |
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281 <head xml:id="echoid-head13" xml:space="preserve">PROPOSITIO III.</head> |
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282 <p> |
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283 <s xml:id="echoid-s171" xml:space="preserve"><emph style="sc">Solidarvm</emph> magnitudinum, quæ æqualẽ <lb/>molem habentes æque graues ſunt, atque humi-<lb/>dum; </s> |
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284 <s xml:id="echoid-s172" xml:space="preserve">in humidum demiſſæ demergentur ita, vt <lb/>ex humidi ſuperficie nihil extet: </s> |
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285 <s xml:id="echoid-s173" xml:space="preserve">non tamen ad <lb/>huc deorſum ferentur.</s> |
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286 <s xml:id="echoid-s174" xml:space="preserve"/> |
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287 </p> |
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288 <p> |
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289 <s xml:id="echoid-s175" xml:space="preserve">SIT magnitudo aliqua æque grauis, atque humidum: <lb/></s> |
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290 <s xml:id="echoid-s176" xml:space="preserve">& </s> |
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291 <s xml:id="echoid-s177" xml:space="preserve">ſi fieri poteſt, in humidum demiſſa extet ex ſuperficie ip <lb/>ſius: </s> |
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292 <s xml:id="echoid-s178" xml:space="preserve">conſiſtat autem humidum, maneatq;</s> |
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293 <s xml:id="echoid-s179" xml:space="preserve">: & </s> |
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294 <s xml:id="echoid-s180" xml:space="preserve">intelligatur <lb/>aliquod planum ductũ <lb/> |
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295 <anchor type="figure" xlink:label="fig-0016-01a" xlink:href="fig-0016-01"/> |
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296 per cẽtrum terræ, & </s> |
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297 <s xml:id="echoid-s181" xml:space="preserve">hu <lb/>midi, ac per ſolidam ma <lb/>gnitudinem, ut ſit ſuper <lb/>ficiei quidem humidi ſe <lb/>ctio a b c d; </s> |
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298 <s xml:id="echoid-s182" xml:space="preserve">ſolidæ uero <lb/>magnitudinis inſiden-<lb/>tis e h t f; </s> |
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299 <s xml:id="echoid-s183" xml:space="preserve">& </s> |
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300 <s xml:id="echoid-s184" xml:space="preserve">terræ cen-<lb/>trum k: </s> |
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301 <s xml:id="echoid-s185" xml:space="preserve">ſitq; </s> |
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302 <s xml:id="echoid-s186" xml:space="preserve">ſolidæ ma-<lb/>gnitudinis pars, quæ in <lb/>humido eſt, b h t c; </s> |
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303 <s xml:id="echoid-s187" xml:space="preserve">& </s> |
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304 <s xml:id="echoid-s188" xml:space="preserve"><lb/>quæ extra humidum b e f c. </s> |
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305 <s xml:id="echoid-s189" xml:space="preserve">intelligatur etiam ſolida figu-<lb/>ra comprehenſa pyramide, baſim quidem habente paralle <lb/>logrammum, quod eſt in ſuperficie humidi; </s> |
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306 <s xml:id="echoid-s190" xml:space="preserve">uerticem au-<lb/>tem centrum terræ: </s> |
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307 <s xml:id="echoid-s191" xml:space="preserve">ſitq; </s> |
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308 <s xml:id="echoid-s192" xml:space="preserve">ſectio plani, in quo eſt a b c d cir-<lb/>cunferentia, & </s> |
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309 <s xml:id="echoid-s193" xml:space="preserve">planorum pyramidis k l, k m: </s> |
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310 <s xml:id="echoid-s194" xml:space="preserve">& </s> |
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311 <s xml:id="echoid-s195" xml:space="preserve">deſcriba-<lb/>tur quædam alterius ſphæræ ſuperficies x o p circa centrũ <lb/>k, in humido ſub e f h t, ut ſit ipſa x o p ſectio facta à ſuper fi <lb/>cie plani. </s> |
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312 <s xml:id="echoid-s196" xml:space="preserve">Sumatur præterea alia quædam pyramis æqua-<lb/>lis, & </s> |
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313 <s xml:id="echoid-s197" xml:space="preserve">ſimilis comprehendenti ſolidam figuram, ipſi con- |
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314 <pb o="3" file="0017" n="17" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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315 iuncta, & </s> |
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316 <s xml:id="echoid-s198" xml:space="preserve">continuata: </s> |
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317 <s xml:id="echoid-s199" xml:space="preserve">ſitq; </s> |
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318 <s xml:id="echoid-s200" xml:space="preserve">ſectio planorũ ipſius K m K n: <lb/></s> |
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319 <s xml:id="echoid-s201" xml:space="preserve">& </s> |
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320 <s xml:id="echoid-s202" xml:space="preserve">in humido intelligatur quædam magnitudo r s q y ex ip <lb/>ſo humido conſtans, æqualis, & </s> |
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321 <s xml:id="echoid-s203" xml:space="preserve">ſimilis ſolidæ b h t c, quæ <lb/>quidem pars eſt ſolidæ magnitudinis in humido demerſa. </s> |
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322 <s xml:id="echoid-s204" xml:space="preserve"><lb/>partes igitur humidi, quæ ſcilicet in prima pyramide ſuper <lb/>ficie x o continetur, & </s> |
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323 <s xml:id="echoid-s205" xml:space="preserve">quæ in altera continetur p o, æquali <lb/>ter ſunt poſitæ, & </s> |
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324 <s xml:id="echoid-s206" xml:space="preserve">continuatæ ſed non ſimiliter premun-<lb/>tur. </s> |
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325 <s xml:id="echoid-s207" xml:space="preserve">nam contenta quidem x o, premitur ſolido e h t f, & </s> |
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326 <s xml:id="echoid-s208" xml:space="preserve"><lb/>humido interiecto inter ſuperficies x o, l m, & </s> |
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327 <s xml:id="echoid-s209" xml:space="preserve">plana pyra-<lb/>midis; </s> |
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328 <s xml:id="echoid-s210" xml:space="preserve">contenta uero p o premitur ſolido r s q y, & </s> |
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329 <s xml:id="echoid-s211" xml:space="preserve">humi-<lb/>do inter ſuperficies o p, m n, & </s> |
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330 <s xml:id="echoid-s212" xml:space="preserve">pyramidis plana interiecto. </s> |
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331 <s xml:id="echoid-s213" xml:space="preserve"><lb/>minor autem eſt grauitas humidi, quod eſt inter m n, o p, <lb/>quàm eius, quod inter l m, x o. </s> |
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332 <s xml:id="echoid-s214" xml:space="preserve">ſolidum enim r s q y eſt mi <lb/>nus ſolido e h t f: </s> |
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333 <s xml:id="echoid-s215" xml:space="preserve">cum ſit æquale ipſi b h t c; </s> |
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334 <s xml:id="echoid-s216" xml:space="preserve">quia magnitu <lb/>dine æquale, & </s> |
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335 <s xml:id="echoid-s217" xml:space="preserve">æque graue ponitur ſolidum, atque humi-<lb/>dum: </s> |
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336 <s xml:id="echoid-s218" xml:space="preserve">reliquum autem reliquo inæquale eſt. </s> |
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337 <s xml:id="echoid-s219" xml:space="preserve">conſtatigitur <lb/>partem contentã ſuperficie o p, expelli ab ea, quæ ipſa x o <lb/>continetur: </s> |
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338 <s xml:id="echoid-s220" xml:space="preserve">& </s> |
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339 <s xml:id="echoid-s221" xml:space="preserve">non conſiſtere humidum. </s> |
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340 <s xml:id="echoid-s222" xml:space="preserve">ponebatur au-<lb/>tem conſiſtens, & </s> |
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341 <s xml:id="echoid-s223" xml:space="preserve">manens: </s> |
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342 <s xml:id="echoid-s224" xml:space="preserve">non ergo ex ſuperficie humidi <lb/>extat aliquid ſolidæ magnitudinis. </s> |
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343 <s xml:id="echoid-s225" xml:space="preserve">ſed neque demerſum <lb/>ſolidum ad inferiora feretur. </s> |
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344 <s xml:id="echoid-s226" xml:space="preserve">Similiter enim prementur <lb/>omnes partes humidi æqualiter poſitæ, cum ſolidum ſit æ-<lb/>que graue, atque humidum.</s> |
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345 <s xml:id="echoid-s227" xml:space="preserve"/> |
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346 </p> |
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347 <div xml:id="echoid-div12" type="float" level="2" n="1"> |
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348 <figure xlink:label="fig-0016-01" xlink:href="fig-0016-01a"> |
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349 <image file="0016-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0016-01"/> |
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350 </figure> |
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351 </div> |
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352 </div> |
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353 <div xml:id="echoid-div14" type="section" level="1" n="11"> |
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354 <head xml:id="echoid-head14" xml:space="preserve">PROPOSITIO IIII.</head> |
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355 <p> |
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356 <s xml:id="echoid-s228" xml:space="preserve"><emph style="sc">Solidarvm</emph> magnitudinum, quæcunque <lb/>leuior humido fuerit, demiſſa in humidum non <lb/>demergetur tota, ſed aliqua pars ipſius ex humi-<lb/>di ſuperficie extabit.</s> |
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357 <s xml:id="echoid-s229" xml:space="preserve"/> |
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358 </p> |
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359 <p> |
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360 <s xml:id="echoid-s230" xml:space="preserve">SIT magnitudo ſolida humido leuior; </s> |
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361 <s xml:id="echoid-s231" xml:space="preserve">& </s> |
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362 <s xml:id="echoid-s232" xml:space="preserve">demiſſa in hu <lb/>midum demergatur tota, ſi fieri poteſt, ut nulla pars ipſius |
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363 <pb file="0018" n="18" rhead="ARCHIMEDIS"/> |
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364 extet ex humidi ſuperficie. </s> |
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365 <s xml:id="echoid-s233" xml:space="preserve">conſiſtat autem humidum, ma <lb/>neatq;</s> |
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366 <s xml:id="echoid-s234" xml:space="preserve">: & </s> |
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367 <s xml:id="echoid-s235" xml:space="preserve">intelligatur aliquod planum ductum per centrũ <lb/>terræ, per humidum, & </s> |
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368 <s xml:id="echoid-s236" xml:space="preserve"><lb/> |
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369 <anchor type="figure" xlink:label="fig-0018-01a" xlink:href="fig-0018-01"/> |
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370 per magnitudinem ſoli-<lb/>dam: </s> |
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371 <s xml:id="echoid-s237" xml:space="preserve">à quo ſuperficies <lb/>quidem humidi ſecetur <lb/>ſecundum circunferen-<lb/>tiam a b c; </s> |
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372 <s xml:id="echoid-s238" xml:space="preserve">ſolida autem <lb/>magnitudo ſecundum fi <lb/>guram, in qua r: </s> |
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373 <s xml:id="echoid-s239" xml:space="preserve">& </s> |
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374 <s xml:id="echoid-s240" xml:space="preserve">cen-<lb/>trum terræ ſit K. </s> |
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375 <s xml:id="echoid-s241" xml:space="preserve">Intelli <lb/>gatur etiam quædam py <lb/>ramis comprehendens <lb/>figuram r, ſicuti prius, quæ pũctum K pro uertice habeat: <lb/></s> |
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376 <s xml:id="echoid-s242" xml:space="preserve">fecenturq; </s> |
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377 <s xml:id="echoid-s243" xml:space="preserve">ipſius plana à ſuperficie plani a b c ſecundum <lb/>a K K b: </s> |
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378 <s xml:id="echoid-s244" xml:space="preserve">& </s> |
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379 <s xml:id="echoid-s245" xml:space="preserve">ſumatur pyramis alia æ qualis, & </s> |
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380 <s xml:id="echoid-s246" xml:space="preserve">ſimilis ſuperio <lb/>ri, cuius plana ſecentur à plano a b c, ſecundum b K K c: </s> |
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381 <s xml:id="echoid-s247" xml:space="preserve"><lb/>deinde alterius ſphæræ ſuperficies quædam deſcribatur in <lb/>humido circa centrum K, ſub ſolida magnitudine: </s> |
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382 <s xml:id="echoid-s248" xml:space="preserve">& </s> |
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383 <s xml:id="echoid-s249" xml:space="preserve">ſece-<lb/>tur ab eodem plano ſecundum x o p: </s> |
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384 <s xml:id="echoid-s250" xml:space="preserve">poſtremo intelliga-<lb/>tur alia magnitudo h in poſteriori pyramide, quæ ex humi <lb/>do conſtet, & </s> |
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385 <s xml:id="echoid-s251" xml:space="preserve">ſolidæ magnitudini r ſit æ qualis. </s> |
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386 <s xml:id="echoid-s252" xml:space="preserve">partes igi-<lb/>tur humidi, & </s> |
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387 <s xml:id="echoid-s253" xml:space="preserve">quæ in prima pyramide continetur ſuperfi-<lb/>cie x o; </s> |
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388 <s xml:id="echoid-s254" xml:space="preserve">& </s> |
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389 <s xml:id="echoid-s255" xml:space="preserve">quæ in ſecunda ſuperficie o p continetur, æquali <lb/>ter iacent, & </s> |
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390 <s xml:id="echoid-s256" xml:space="preserve">continuatæ inter ſe ſe; </s> |
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391 <s xml:id="echoid-s257" xml:space="preserve">non tamen ſimiliter <lb/>premuntur: </s> |
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392 <s xml:id="echoid-s258" xml:space="preserve">nam quæ eſt in prima pyramide premitur ma <lb/>gnitudine ſolida r, & </s> |
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393 <s xml:id="echoid-s259" xml:space="preserve">humido cõtinente ipſam, quod eſt in <lb/>loco pyramidis a b o x: </s> |
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394 <s xml:id="echoid-s260" xml:space="preserve">quæ uero in altera pyramide pre-<lb/>mitur ſolida magnitudineh, & </s> |
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395 <s xml:id="echoid-s261" xml:space="preserve">humido ipſam continente <lb/>in loco pyramidis p o b c. </s> |
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396 <s xml:id="echoid-s262" xml:space="preserve">At grauitas ſolidæ magnitudi-<lb/>nis r, minor eſt grauitate humidi, in quo h: </s> |
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397 <s xml:id="echoid-s263" xml:space="preserve">quoniam ma-<lb/>gnitudo ſolida mole quidem æqualis, & </s> |
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398 <s xml:id="echoid-s264" xml:space="preserve">humido leuior po <lb/>nitur: </s> |
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399 <s xml:id="echoid-s265" xml:space="preserve">grauitas autem humidi continentis magnitudines <lb/>r h eſt æqualis; </s> |
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400 <s xml:id="echoid-s266" xml:space="preserve">cum pyramides æquales ſint. </s> |
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401 <s xml:id="echoid-s267" xml:space="preserve">magis ergo |
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402 <pb o="4" file="0019" n="19" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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403 premitur pars humidi, quæ eſt ſub ſuperficie o p. </s> |
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404 <s xml:id="echoid-s268" xml:space="preserve">quare ex-<lb/>pellet partem minus preſſam, & </s> |
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405 <s xml:id="echoid-s269" xml:space="preserve">non manebit humidum. <lb/></s> |
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406 <s xml:id="echoid-s270" xml:space="preserve">ponebatur autem manens. </s> |
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407 <s xml:id="echoid-s271" xml:space="preserve">non igitur demergetur tota, <lb/>ſed aliqua pars ipſius ex humidi ſuperficie extabit.</s> |
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408 <s xml:id="echoid-s272" xml:space="preserve"/> |
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409 </p> |
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410 <div xml:id="echoid-div14" type="float" level="2" n="1"> |
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411 <figure xlink:label="fig-0018-01" xlink:href="fig-0018-01a"> |
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412 <image file="0018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0018-01"/> |
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413 </figure> |
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414 </div> |
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415 </div> |
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416 <div xml:id="echoid-div16" type="section" level="1" n="12"> |
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417 <head xml:id="echoid-head15" xml:space="preserve">PROPOSITIO V.</head> |
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418 <p> |
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419 <s xml:id="echoid-s273" xml:space="preserve"><emph style="sc">Solidarvm</emph> magnitudinum quæcunque le <lb/>uior humido fuerit, demiſſa in humidum vſque <lb/>eô demergetur, vt tanta moles humidi, quanta <lb/>eſt partis demerſæ, eandem, quam tota magnitu-<lb/>do, grauitatem habeat.</s> |
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420 <s xml:id="echoid-s274" xml:space="preserve"/> |
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421 </p> |
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422 <p> |
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423 <s xml:id="echoid-s275" xml:space="preserve">DISPONANTVR eadem, quæſupra: </s> |
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424 <s xml:id="echoid-s276" xml:space="preserve">ſitq; </s> |
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425 <s xml:id="echoid-s277" xml:space="preserve">humi-<lb/>dum manens: </s> |
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426 <s xml:id="echoid-s278" xml:space="preserve">& </s> |
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427 <s xml:id="echoid-s279" xml:space="preserve">magnitudo e h t f humido leuior. </s> |
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428 <s xml:id="echoid-s280" xml:space="preserve">Si igitur <lb/>humidum manet, ſimiliter prementur eius partes, quæ æ-<lb/>qualiter iacent. </s> |
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429 <s xml:id="echoid-s281" xml:space="preserve">ſimiliter ergo premetur humidum ſub ſu-<lb/>perficiebus x o o p. <lb/></s> |
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430 <s xml:id="echoid-s282" xml:space="preserve"> |
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431 <anchor type="figure" xlink:label="fig-0019-01a" xlink:href="fig-0019-01"/> |
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432 quare æ qualis eſt graui-<lb/>tas, qua premuntur. </s> |
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433 <s xml:id="echoid-s283" xml:space="preserve">eſt <lb/>autem & </s> |
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434 <s xml:id="echoid-s284" xml:space="preserve">grauitas humi <lb/>di, quod in prima pyra-<lb/>mide abſque ſolido b h <lb/>t c, æqualis grauitati hu <lb/>midi, quod in altera py-<lb/>ramide abſq; </s> |
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435 <s xml:id="echoid-s285" xml:space="preserve">r s q y hu-<lb/>mido. </s> |
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436 <s xml:id="echoid-s286" xml:space="preserve">perſpicuum eſt <lb/>igitur grauitatem ma-<lb/>gnitudinis e h t f grauitati humidi r s q y æqualem eſſe. </s> |
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437 <s xml:id="echoid-s287" xml:space="preserve">ex <lb/>quibus conſtat, tantam humidi molem, quanta eſt pars de <lb/>merſa ſolidæ magnitudinis, eandem, quam tota magnitu-<lb/>do habere grauitatem.</s> |
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438 <s xml:id="echoid-s288" xml:space="preserve"/> |
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439 </p> |
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440 <div xml:id="echoid-div16" type="float" level="2" n="1"> |
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441 <figure xlink:label="fig-0019-01" xlink:href="fig-0019-01a"> |
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442 <image file="0019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0019-01"/> |
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443 </figure> |
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444 </div> |
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445 <pb file="0020" n="20" rhead="ARCHIMEDIS"/> |
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446 </div> |
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447 <div xml:id="echoid-div18" type="section" level="1" n="13"> |
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448 <head xml:id="echoid-head16" xml:space="preserve">PROPOSITIO VI.</head> |
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449 <p> |
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450 <s xml:id="echoid-s289" xml:space="preserve"><emph style="sc">Solidae</emph> magnitudines humido leuiores, in <lb/>humidum impulſæ ſurſum feruntur tanta ui, quã <lb/>to humidum molem habens magnitudini æqua-<lb/>lem, grauius eſt ipſa magnitudine.</s> |
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451 <s xml:id="echoid-s290" xml:space="preserve"/> |
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452 </p> |
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453 <p> |
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454 <s xml:id="echoid-s291" xml:space="preserve">SIT enim magnitudo aleuior humido: </s> |
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455 <s xml:id="echoid-s292" xml:space="preserve">& </s> |
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456 <s xml:id="echoid-s293" xml:space="preserve">ſit magnitu <lb/>dinis quidem a grauitas b: </s> |
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457 <s xml:id="echoid-s294" xml:space="preserve">humidi uero molem habentis <lb/>æqualem ipſi a, grauitas ſit b c. </s> |
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458 <s xml:id="echoid-s295" xml:space="preserve">demonſtrandum eſt magni <lb/>tudinem a in humidum impulſam tanta ui ſurſum ferri, <lb/>quanta eſt grauitas c. </s> |
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459 <s xml:id="echoid-s296" xml:space="preserve">accipiatur enim quædam magnitu-<lb/>do, in qua d habens grauitatem ipſi c æqualem. </s> |
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460 <s xml:id="echoid-s297" xml:space="preserve">Itaque <lb/>magnitudo ex utriſque magnitudinibus conſtans, in qui-<lb/>bus a d, leuior eſt humido: </s> |
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461 <s xml:id="echoid-s298" xml:space="preserve">nam magnitudinis quidem quæ <lb/>ex utriſque conſtat grauitas eſt b c; </s> |
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462 <s xml:id="echoid-s299" xml:space="preserve">humidi uero habentis <lb/>molem ipſis æ qualem grauitas maior eſt, quàm b c: </s> |
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463 <s xml:id="echoid-s300" xml:space="preserve">quo-<lb/>niam b c grauitas eſt humidi <lb/> |
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464 <anchor type="figure" xlink:label="fig-0020-01a" xlink:href="fig-0020-01"/> |
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465 molẽ habentis æqualem ipſia. <lb/></s> |
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466 <s xml:id="echoid-s301" xml:space="preserve">Si ergo demittatur in humidũ <lb/>magnitudo ex utriſque a d con <lb/>ſtans; </s> |
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467 <s xml:id="echoid-s302" xml:space="preserve">uſque eò demergetur, ut <lb/>tanta moles humidi, quanta eſt <lb/>pars magnitudinis demerſa eã <lb/>dem, quam tota magnitudo <lb/>grauitatem habeat. </s> |
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468 <s xml:id="echoid-s303" xml:space="preserve">hoc enim <lb/>iam demonſtratum eſt. </s> |
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469 <s xml:id="echoid-s304" xml:space="preserve">ſit autẽ <lb/>ſuperſicies humidi alicuius a b <lb/>c d circunferentia. </s> |
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470 <s xml:id="echoid-s305" xml:space="preserve">Quoniam igitur tanta moles humidi, <lb/>quanta eſt magnitudo a grauitatem habet eandem, quam <lb/>magnitudines a d: </s> |
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471 <s xml:id="echoid-s306" xml:space="preserve">perſpicuum eſt partem ipſius demer-<lb/>ſam eſſe magnitudinem a; </s> |
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472 <s xml:id="echoid-s307" xml:space="preserve">reliquam uero d totam ex hu- |
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473 <pb o="5" file="0021" n="21" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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474 midi ſuperficie extare. </s> |
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475 <s xml:id="echoid-s308" xml:space="preserve">Quare conſtat magnitudinem a <lb/>tanta ui ſurſum ferri, quãta deorſum premitur ab eo, quod <lb/>eſt ſupra; </s> |
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476 <s xml:id="echoid-s309" xml:space="preserve">uidelicet à d, cũ neutra ab altera expellatur, ſed <lb/>d fertur deorſum tanta grauitate, quanta eſt c: </s> |
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477 <s xml:id="echoid-s310" xml:space="preserve">ponebatur <lb/>enim grauitas eius, in quo d ipſi c æqualis. </s> |
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478 <s xml:id="echoid-s311" xml:space="preserve">patet igitur <lb/>illud quod demonſtrare oportebat.</s> |
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479 <s xml:id="echoid-s312" xml:space="preserve"/> |
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480 </p> |
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481 <div xml:id="echoid-div18" type="float" level="2" n="1"> |
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482 <figure xlink:label="fig-0020-01" xlink:href="fig-0020-01a"> |
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483 <image file="0020-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0020-01"/> |
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484 </figure> |
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485 </div> |
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486 </div> |
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487 <div xml:id="echoid-div20" type="section" level="1" n="14"> |
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488 <head xml:id="echoid-head17" xml:space="preserve">PROPOSITIO VII.</head> |
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489 <p> |
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490 <s xml:id="echoid-s313" xml:space="preserve"><emph style="sc">Solidae</emph> magnitudines humido grauiores <lb/>demiſſæ in humidum ferentur deorſum, donec <lb/>deſcendant: </s> |
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491 <s xml:id="echoid-s314" xml:space="preserve">& </s> |
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492 <s xml:id="echoid-s315" xml:space="preserve">erunt in humido tanto leuiores, <lb/>quanta eſt grauitas humidi molem habentis ſoli-<lb/>dæ magnitudini æqualem.</s> |
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493 <s xml:id="echoid-s316" xml:space="preserve"/> |
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494 </p> |
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495 <p> |
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496 <s xml:id="echoid-s317" xml:space="preserve">SOLIDAS magnitudines humido grauiores, in hu-<lb/>midum demiſſas deorſum quidam ferri, donec deſcẽdant, <lb/>manifeſtum eſt: </s> |
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497 <s xml:id="echoid-s318" xml:space="preserve">partes enim humidi, quæ ſub eis ſunt, pre-<lb/>muntur magis, quàm partes æqualiter ipſis adiacentes; <lb/></s> |
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498 <s xml:id="echoid-s319" xml:space="preserve">quoniam magnitudo ſolida humido grauior ponitur: </s> |
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499 <s xml:id="echoid-s320" xml:space="preserve">le-<lb/>uiores autem eſſe uti dictum eſt, demonſtrabitur hoc mo-<lb/>do. </s> |
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500 <s xml:id="echoid-s321" xml:space="preserve">Sit enim aliqua ma-<lb/> |
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501 <anchor type="figure" xlink:label="fig-0021-01a" xlink:href="fig-0021-01"/> |
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502 gnitudo a grauior hu-<lb/>mido: </s> |
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503 <s xml:id="echoid-s322" xml:space="preserve">& </s> |
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504 <s xml:id="echoid-s323" xml:space="preserve">fit magnitudi-<lb/>nis quidem a grauitas <lb/>b c: </s> |
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505 <s xml:id="echoid-s324" xml:space="preserve">humidi uero molẽ <lb/>habentis æqualem ipſi a <lb/>grauitas ſit b. </s> |
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506 <s xml:id="echoid-s325" xml:space="preserve">demon-<lb/>ſtrandum eſt magnitudi <lb/>nem a in humido exiſtē <lb/>tem habere grauitatem <lb/>æqualem ipſi c. </s> |
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507 <s xml:id="echoid-s326" xml:space="preserve">Accipia <lb/>tur enim alia aliqua magnitudo, in qua d, leuior humido;</s> |
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508 <s xml:id="echoid-s327" xml:space="preserve"> |
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509 <pb file="0022" n="22" rhead="ARCHIMEDIS"/> |
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510 cuius grauitas ſitipſi b æqualis: </s> |
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511 <s xml:id="echoid-s328" xml:space="preserve">humidi ucro molem ha-<lb/>bentis æqualem magnitudini d, ſit grauitas æqualis b c. <lb/></s> |
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512 <s xml:id="echoid-s329" xml:space="preserve">Itaque compoſitis magnitudinibus a d, magnitudo ex <lb/>utriſque conſtans æque grauis erit, atque ipſum humidũ: </s> |
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513 <s xml:id="echoid-s330" xml:space="preserve"><lb/>grauitas enim utrarũque magnitudinum eſt æqualis utriſ-<lb/>que grauitatibus, uidelicet b c, & </s> |
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514 <s xml:id="echoid-s331" xml:space="preserve">b: </s> |
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515 <s xml:id="echoid-s332" xml:space="preserve">grauitas autem humi <lb/>di habentis molem æqualem utriſque magnitudinibus, eſt <lb/>eiſdem grauitatibus æqualis. </s> |
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516 <s xml:id="echoid-s333" xml:space="preserve">Demisſis igitur magnitudini <lb/>bus, & </s> |
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517 <s xml:id="echoid-s334" xml:space="preserve">in humidum proiectis æque graues erunt, atque hu <lb/>midum: </s> |
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518 <s xml:id="echoid-s335" xml:space="preserve">neque ſurſum, neque deorſum ferentur: </s> |
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519 <s xml:id="echoid-s336" xml:space="preserve">quoniam <lb/>magnitudo quidem a <lb/> |
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520 <anchor type="figure" xlink:label="fig-0022-01a" xlink:href="fig-0022-01"/> |
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521 grauior humido feretur <lb/>deorſum; </s> |
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522 <s xml:id="echoid-s337" xml:space="preserve">& </s> |
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523 <s xml:id="echoid-s338" xml:space="preserve">eadem ui à <lb/>magnitudine d ſurſum <lb/>retrahetur: </s> |
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524 <s xml:id="echoid-s339" xml:space="preserve">magnitudo <lb/>autem d humido leuior <lb/>feretur ſurſum tanta ui, <lb/>quanta eſt grauitas c: <lb/></s> |
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525 <s xml:id="echoid-s340" xml:space="preserve">demõſtratũ enim eſt ma <lb/> |
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526 <anchor type="note" xlink:label="note-0022-01a" xlink:href="note-0022-01"/> |
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527 gnitudines ſolidas hu-<lb/>mido leuiores, impulſas <lb/>in humidum tanta uiretrahi ſurſum, quanto humidum ha <lb/>bens molem magnitudini æqualem grauius eſt ipſa magni <lb/>tudine. </s> |
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528 <s xml:id="echoid-s341" xml:space="preserve">Athumidum molem habens æqualem d, grauius <lb/>eſt, quam d, ipſa c grauitate. </s> |
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529 <s xml:id="echoid-s342" xml:space="preserve">Conſtatigitur magnitudinem <lb/>a deorſum ferri tanta grauitate, quanta eſt c. </s> |
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530 <s xml:id="echoid-s343" xml:space="preserve">quod de-<lb/>monſtrare oportebat.</s> |
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531 <s xml:id="echoid-s344" xml:space="preserve"/> |
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532 </p> |
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533 <div xml:id="echoid-div20" type="float" level="2" n="1"> |
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534 <figure xlink:label="fig-0021-01" xlink:href="fig-0021-01a"> |
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535 <image file="0021-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0021-01"/> |
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536 </figure> |
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537 <figure xlink:label="fig-0022-01" xlink:href="fig-0022-01a"> |
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538 <image file="0022-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0022-01"/> |
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539 </figure> |
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540 <note position="left" xlink:label="note-0022-01" xlink:href="note-0022-01a" xml:space="preserve">6. huius.</note> |
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541 </div> |
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542 </div> |
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543 <div xml:id="echoid-div22" type="section" level="1" n="15"> |
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544 <head xml:id="echoid-head18" xml:space="preserve">POSITIO II.</head> |
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545 <p> |
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546 <s xml:id="echoid-s345" xml:space="preserve"><emph style="sc">Ponatvr</emph> eorum, quæ in humido ſurſum <lb/>feruntur, vnumquodque ſurſum ferri ſecundum <lb/>perpendicularem, quæ per centrum grauitatis ip <lb/>ſorum ducitur.</s> |
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547 <s xml:id="echoid-s346" xml:space="preserve"/> |
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548 </p> |
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549 <pb o="6" file="0023" n="23" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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550 </div> |
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551 <div xml:id="echoid-div23" type="section" level="1" n="16"> |
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552 <head xml:id="echoid-head19" xml:space="preserve">COMMENTARIVS.</head> |
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553 <p style="it"> |
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554 <s xml:id="echoid-s347" xml:space="preserve">AT ucro ea, quæ feruntur deorſum, ſecundum perpendicula-<lb/>rem, quæ per centrum grauit atis ipſorum ducitur, ſimiliter ferri, <lb/>uel tanquam notum, uel ut ab alijs poſitum prætermiſit.</s> |
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555 <s xml:id="echoid-s348" xml:space="preserve"/> |
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556 </p> |
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557 </div> |
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558 <div xml:id="echoid-div24" type="section" level="1" n="17"> |
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559 <head xml:id="echoid-head20" xml:space="preserve">PROPOSITIO VIII.</head> |
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560 <p> |
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561 <s xml:id="echoid-s349" xml:space="preserve">SI aliqua magnitudo ſolida leuior humido, <lb/> |
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562 <anchor type="note" xlink:label="note-0023-01a" xlink:href="note-0023-01"/> |
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563 quæ figuram portionis ſphæræ habeat, in humi-<lb/> |
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564 <anchor type="note" xlink:label="note-0023-02a" xlink:href="note-0023-02"/> |
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565 dum demittatur, ita vt baſis portionis non tan-<lb/>gat humidum: </s> |
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566 <s xml:id="echoid-s350" xml:space="preserve">figura inſidebit recta, ita vt axis <lb/>portionis ſit ſecundum perpendicularem. </s> |
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567 <s xml:id="echoid-s351" xml:space="preserve">Et ſi <lb/>ab aliquo inclinetur figura, vt baſis portionis hu-<lb/>midum cõtingat; </s> |
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568 <s xml:id="echoid-s352" xml:space="preserve">non manebit inclinata ſi demit <lb/>tatur, ſed recta reſtituetur.</s> |
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569 <s xml:id="echoid-s353" xml:space="preserve"/> |
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570 </p> |
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571 <div xml:id="echoid-div24" type="float" level="2" n="1"> |
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572 <note position="right" xlink:label="note-0023-01" xlink:href="note-0023-01a" xml:space="preserve">A</note> |
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573 <note position="right" xlink:label="note-0023-02" xlink:href="note-0023-02a" xml:space="preserve">B</note> |
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574 </div> |
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575 <p> |
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576 <s xml:id="echoid-s354" xml:space="preserve">[INTELLIGATVR quædam magnitudo, qualis <lb/> |
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577 <anchor type="note" xlink:label="note-0023-03a" xlink:href="note-0023-03"/> |
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578 dicta eſt, in humidum demiſſa: </s> |
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579 <s xml:id="echoid-s355" xml:space="preserve">& </s> |
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580 <s xml:id="echoid-s356" xml:space="preserve">ducatur planum per axẽ <lb/>portionis, & </s> |
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581 <s xml:id="echoid-s357" xml:space="preserve">per terræ <lb/> |
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582 <anchor type="figure" xlink:label="fig-0023-01a" xlink:href="fig-0023-01"/> |
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583 centrum, ut ſit ſuperfi-<lb/>ciei humidi ſectio circũ <lb/>ferentia a b c d: </s> |
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584 <s xml:id="echoid-s358" xml:space="preserve">& </s> |
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585 <s xml:id="echoid-s359" xml:space="preserve">figu-<lb/>ræ ſectio e f h circunfe-<lb/>rentia: </s> |
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586 <s xml:id="echoid-s360" xml:space="preserve">ſit autem e h <lb/>recta linea; </s> |
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587 <s xml:id="echoid-s361" xml:space="preserve">& </s> |
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588 <s xml:id="echoid-s362" xml:space="preserve">f t axis <lb/>portionis. </s> |
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589 <s xml:id="echoid-s363" xml:space="preserve">Si igitur in-<lb/>clinetur figura, ita ut a-<lb/>xis portionis f t non ſit <lb/>ſecundum perpendicu-<lb/>larem. </s> |
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590 <s xml:id="echoid-s364" xml:space="preserve">demonſtrandum eſt, non manere ipſam figu-<lb/>ram; </s> |
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591 <s xml:id="echoid-s365" xml:space="preserve">ſed in rectum reſtitui. </s> |
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592 <s xml:id="echoid-s366" xml:space="preserve">Itaque centrum ſphæræ eſt |
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593 <pb file="0024" n="24" rhead="ARCHIMEDIS"/> |
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594 in linea ft. </s> |
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595 <s xml:id="echoid-s367" xml:space="preserve">nam ſit primum figura maior dimidia ſphære: <lb/></s> |
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596 <s xml:id="echoid-s368" xml:space="preserve">ſitq; </s> |
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597 <s xml:id="echoid-s369" xml:space="preserve">in dimidia ſphæra ſphæræ centrum t; </s> |
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598 <s xml:id="echoid-s370" xml:space="preserve">in minori por-<lb/>tioneſit centrum p; </s> |
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599 <s xml:id="echoid-s371" xml:space="preserve">& </s> |
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600 <s xml:id="echoid-s372" xml:space="preserve">in maiori _k_: </s> |
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601 <s xml:id="echoid-s373" xml:space="preserve">per _k_ uero, & </s> |
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602 <s xml:id="echoid-s374" xml:space="preserve">terræ cen <lb/>trum l ducatur _k_ l ſecans circunferentiam e f h in pun-<lb/>cto n. </s> |
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603 <s xml:id="echoid-s375" xml:space="preserve">Quoniam igitur unaquæque ſphæræportio axem <lb/> |
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604 <anchor type="note" xlink:label="note-0024-01a" xlink:href="note-0024-01"/> |
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605 habet in linea, quæ à cẽtro ſphæræ ad cius baſim perpen-<lb/>dicularis ducitur: </s> |
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606 <s xml:id="echoid-s376" xml:space="preserve">habetq; </s> |
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607 <s xml:id="echoid-s377" xml:space="preserve">in axe grauitatis centrum: <lb/></s> |
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608 <s xml:id="echoid-s378" xml:space="preserve">portionis in humido demerſæ, quæ ex duabus ſphæræ <lb/>portionibus conſtat, axis erit in perpendiculari per _k_ du-<lb/>cta. </s> |
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609 <s xml:id="echoid-s379" xml:space="preserve">& </s> |
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610 <s xml:id="echoid-s380" xml:space="preserve">idcirco centrum grauitatis ipſius erit in linea n _k_, <lb/>quod ſit r. </s> |
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611 <s xml:id="echoid-s381" xml:space="preserve">ſed totius portionis grauitatis centrum eſt in li <lb/> |
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612 <anchor type="note" xlink:label="note-0024-02a" xlink:href="note-0024-02"/> |
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613 nea f t inter _k_, & </s> |
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614 <s xml:id="echoid-s382" xml:space="preserve">f, quod ſit x. </s> |
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615 <s xml:id="echoid-s383" xml:space="preserve">reliquæ ergo figuræ, quæ eſt <lb/> |
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616 <anchor type="note" xlink:label="note-0024-03a" xlink:href="note-0024-03"/> |
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617 extra humidum, centrum erit in linea r x producta ad par <lb/>tes x; </s> |
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618 <s xml:id="echoid-s384" xml:space="preserve">& </s> |
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619 <s xml:id="echoid-s385" xml:space="preserve">aſſumpta ex ea, linea quadam, quæ ad r x eandem <lb/>proportionem habeat, quam grauitas portionis in humi-<lb/>do demerſæ habet ad grauitatem figuræ, quæ eſt extra hu-<lb/>midum. </s> |
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620 <s xml:id="echoid-s386" xml:space="preserve">Sit autem s centrum dictæ figuræ: </s> |
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621 <s xml:id="echoid-s387" xml:space="preserve">& </s> |
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622 <s xml:id="echoid-s388" xml:space="preserve">per s duca-<lb/>tur perpendicularis l s. </s> |
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623 <s xml:id="echoid-s389" xml:space="preserve">Feretur ergo grauitas figuræ qui-<lb/> |
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624 <anchor type="note" xlink:label="note-0024-04a" xlink:href="note-0024-04"/> |
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625 dem, quæ extra humidum per rectam s l deorſum; </s> |
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626 <s xml:id="echoid-s390" xml:space="preserve">portio <lb/>nis autem, quæ in humido, ſurſum per rectam r l. </s> |
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627 <s xml:id="echoid-s391" xml:space="preserve">quare <lb/>non manebit figura: </s> |
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628 <s xml:id="echoid-s392" xml:space="preserve">ſed partes eius, quæ ſunt ad e, deor-<lb/>ſum; </s> |
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629 <s xml:id="echoid-s393" xml:space="preserve">& </s> |
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630 <s xml:id="echoid-s394" xml:space="preserve">quæ ad h ſurſum ſerẽtur: </s> |
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631 <s xml:id="echoid-s395" xml:space="preserve">idq; </s> |
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632 <s xml:id="echoid-s396" xml:space="preserve">cõtinenter fiet, quoad <lb/>ſ t ſit ſecundum perpendicularem. </s> |
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633 <s xml:id="echoid-s397" xml:space="preserve">Eodem modo in aliis <lb/>portionibus idem demonſtrabitur.</s> |
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634 <s xml:id="echoid-s398" xml:space="preserve">]</s> |
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635 </p> |
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636 <div xml:id="echoid-div25" type="float" level="2" n="2"> |
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637 <note position="right" xlink:label="note-0023-03" xlink:href="note-0023-03a" xml:space="preserve">Suppleta <lb/>a Federi-<lb/>co Cõm.</note> |
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638 <figure xlink:label="fig-0023-01" xlink:href="fig-0023-01a"> |
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639 <image file="0023-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0023-01"/> |
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640 </figure> |
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641 <note position="left" xlink:label="note-0024-01" xlink:href="note-0024-01a" xml:space="preserve">C</note> |
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642 <note position="left" xlink:label="note-0024-02" xlink:href="note-0024-02a" xml:space="preserve">D</note> |
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643 <note position="left" xlink:label="note-0024-03" xlink:href="note-0024-03a" xml:space="preserve">E</note> |
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644 <note position="left" xlink:label="note-0024-04" xlink:href="note-0024-04a" xml:space="preserve">F</note> |
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645 </div> |
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646 <figure> |
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647 <image file="0024-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0024-01"/> |
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648 </figure> |
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649 <pb o="7" file="0025" n="25" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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650 </div> |
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651 <div xml:id="echoid-div27" type="section" level="1" n="18"> |
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652 <head xml:id="echoid-head21" xml:space="preserve">COMMENTARIVS.</head> |
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653 <p style="it"> |
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654 <s xml:id="echoid-s399" xml:space="preserve">H_<emph style="sc">VIVS</emph>_ propoſitionis demonſtratio iniuria temporum deſidera-<lb/>tur, quam nos ita reſtituimus, ut ex figuris, quæ remanſerunt Archi <lb/>medem ſcripſiſſe colligi potuit: </s> |
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655 <s xml:id="echoid-s400" xml:space="preserve">neque enim eas immutare uiſum est, <lb/>quæ uero ad declarationem, explicationémque addenda fuerant, in <lb/>commentarijs ſuppleuimus, id quod etiam præstitimus in ſecunda <lb/>propoſitione ſecundi libri.</s> |
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656 <s xml:id="echoid-s401" xml:space="preserve"/> |
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657 </p> |
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658 <p style="it"> |
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659 <s xml:id="echoid-s402" xml:space="preserve">_SI aliqua magnitudo ſolida leuior humido.</s> |
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660 <s xml:id="echoid-s403" xml:space="preserve">]_ Ea uerba, <lb/> |
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661 <anchor type="note" xlink:label="note-0025-01a" xlink:href="note-0025-01"/> |
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662 leuior bumido, nos addidimus, quæ in translatione non erant; </s> |
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663 <s xml:id="echoid-s404" xml:space="preserve">quo-<lb/>niam de eiuſmodi magnitudinibus in bac propoſitione agitur.</s> |
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664 <s xml:id="echoid-s405" xml:space="preserve"/> |
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665 </p> |
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666 <div xml:id="echoid-div27" type="float" level="2" n="1"> |
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667 <note position="right" xlink:label="note-0025-01" xlink:href="note-0025-01a" xml:space="preserve">A</note> |
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668 </div> |
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669 <p> |
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670 <s xml:id="echoid-s406" xml:space="preserve">In humidũ demittatur, ita ut baſis portionis nõ tangat hu <lb/> |
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671 <anchor type="note" xlink:label="note-0025-02a" xlink:href="note-0025-02"/> |
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672 midum.</s> |
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673 <s xml:id="echoid-s407" xml:space="preserve">] _Hoc est in humidum ita demitt atur, ut baſis ſurſum ſpe_ <lb/>_ctet; </s> |
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674 <s xml:id="echoid-s408" xml:space="preserve">uertex autem deorſum. </s> |
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675 <s xml:id="echoid-s409" xml:space="preserve">quod quidem opponitur ei, quod in ſe-_ <lb/>_quenti dixit._ </s> |
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676 <s xml:id="echoid-s410" xml:space="preserve">In humidum demittatur, ita ut baſis tota ſit in <lb/>humido. </s> |
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677 <s xml:id="echoid-s411" xml:space="preserve">_His enim uerbis ſignificat portionem oppoſito modo in_ <lb/>_humidum demitti, ut ſcilicet uertex ſurſum; </s> |
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678 <s xml:id="echoid-s412" xml:space="preserve">baſis autem deorſum_ <lb/>_uergat. </s> |
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679 <s xml:id="echoid-s413" xml:space="preserve">eodem dicendi modo frequenter uſus est in ſecundo libro; </s> |
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680 <s xml:id="echoid-s414" xml:space="preserve">in_ <lb/>_quo de portionibus conoidis rectangulitractatur._</s> |
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681 <s xml:id="echoid-s415" xml:space="preserve"/> |
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682 </p> |
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683 <div xml:id="echoid-div28" type="float" level="2" n="2"> |
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684 <note position="right" xlink:label="note-0025-02" xlink:href="note-0025-02a" xml:space="preserve">B</note> |
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685 </div> |
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686 <p style="it"> |
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687 <s xml:id="echoid-s416" xml:space="preserve">_Quoniã igitur unaquæq; </s> |
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688 <s xml:id="echoid-s417" xml:space="preserve">ſphæræ portio axẽ habet in linea,_ <lb/> |
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689 <anchor type="note" xlink:label="note-0025-03a" xlink:href="note-0025-03"/> |
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690 _quæ à cẽtro ſphæræ ad eius baſim perpẽdicularis ducitur.</s> |
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691 <s xml:id="echoid-s418" xml:space="preserve">]_ <lb/>Iungatur enim b c, & </s> |
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692 <s xml:id="echoid-s419" xml:space="preserve">k l ſecet circunferentiam a b c d in puncto g; <lb/></s> |
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693 <s xml:id="echoid-s420" xml:space="preserve">lineam uero rectam b c in m. </s> |
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694 <s xml:id="echoid-s421" xml:space="preserve">& </s> |
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695 <s xml:id="echoid-s422" xml:space="preserve">quoniam duo circuli a b c d, e f b <lb/>ſecant ſe ſe in punctis b c; </s> |
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696 <s xml:id="echoid-s423" xml:space="preserve">recta linea, quæ ipſorum centra coniun-<lb/>git, uidelicet k l lineam b c bifariam, & </s> |
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697 <s xml:id="echoid-s424" xml:space="preserve">ad angulos rectos ſecat: </s> |
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698 <s xml:id="echoid-s425" xml:space="preserve"><lb/>ut in commentarij s in Ptolemæi planiſpbærium oſtendimus. </s> |
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699 <s xml:id="echoid-s426" xml:space="preserve">quare <lb/>portionis circuli b n c diameter eſt m n; </s> |
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700 <s xml:id="echoid-s427" xml:space="preserve">& </s> |
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701 <s xml:id="echoid-s428" xml:space="preserve">portionis b g c diame-<lb/> |
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702 <anchor type="note" xlink:label="note-0025-04a" xlink:href="note-0025-04"/> |
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703 ter m g: </s> |
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704 <s xml:id="echoid-s429" xml:space="preserve">nam rectæ lineæ, quæ ipſi b c æquidistantes ex utraque <lb/>parte ducuntur, cum linea n g rectos angulos faciunt; </s> |
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705 <s xml:id="echoid-s430" xml:space="preserve">& </s> |
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706 <s xml:id="echoid-s431" xml:space="preserve">idcirco ab <lb/> |
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707 <anchor type="note" xlink:label="note-0025-05a" xlink:href="note-0025-05"/> |
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708 ipſa bifariam ſecantur. </s> |
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709 <s xml:id="echoid-s432" xml:space="preserve">portionis igitur ſpbæræ b n c axis eſt n m; <lb/></s> |
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710 <s xml:id="echoid-s433" xml:space="preserve">& </s> |
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711 <s xml:id="echoid-s434" xml:space="preserve">portionis b g c axis m g. </s> |
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712 <s xml:id="echoid-s435" xml:space="preserve">ex quo ſequitur, portionis in bumido <lb/>demerſæ axem eſſe in linea k l; </s> |
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713 <s xml:id="echoid-s436" xml:space="preserve">ipſam ſcilicet n g. </s> |
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714 <s xml:id="echoid-s437" xml:space="preserve">& </s> |
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715 <s xml:id="echoid-s438" xml:space="preserve">cum grauita-<lb/>tis centrum cuius libet ſpbæræ portionis ſit in axe; </s> |
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716 <s xml:id="echoid-s439" xml:space="preserve">quod nos in libro |
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717 <pb file="0026" n="26" rhead="ARCHIMEDIS"/> |
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718 @e centro grauitatis ſolidorum demonstrauimus: </s> |
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719 <s xml:id="echoid-s440" xml:space="preserve">erit magnitudi-<lb/>nis ex utriſque portionibus b n c, b g c conſtantis; </s> |
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720 <s xml:id="echoid-s441" xml:space="preserve">hoc eſt portionis <lb/>in humido demerſa grauitatis centrum in linea n g, quæ ipſarum <lb/>ſphæræ portionum centra graui-<lb/> |
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721 <anchor type="figure" xlink:label="fig-0026-01a" xlink:href="fig-0026-01"/> |
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722 tatis coniungit. </s> |
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723 <s xml:id="echoid-s442" xml:space="preserve">ſi enim fieri po-<lb/>teſt, ſit extra lineam n g, ut in <lb/>q: </s> |
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724 <s xml:id="echoid-s443" xml:space="preserve">sîtq; </s> |
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725 <s xml:id="echoid-s444" xml:space="preserve">portionis b n c centrum <lb/>grauitatis u; </s> |
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726 <s xml:id="echoid-s445" xml:space="preserve">& </s> |
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727 <s xml:id="echoid-s446" xml:space="preserve">ducatur u q.</s> |
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728 <s xml:id="echoid-s447" xml:space="preserve"/> |
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729 </p> |
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730 <div xml:id="echoid-div29" type="float" level="2" n="3"> |
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731 <note position="right" xlink:label="note-0025-03" xlink:href="note-0025-03a" xml:space="preserve">C</note> |
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732 <note position="right" xlink:label="note-0025-04" xlink:href="note-0025-04a" xml:space="preserve">29. primi</note> |
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733 <note position="right" xlink:label="note-0025-05" xlink:href="note-0025-05a" xml:space="preserve">3. tertii.</note> |
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734 <figure xlink:label="fig-0026-01" xlink:href="fig-0026-01a"> |
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735 <image file="0026-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0026-01"/> |
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736 </figure> |
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737 </div> |
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738 <p style="it"> |
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739 <s xml:id="echoid-s448" xml:space="preserve">Quoniam igitur à portione in bu-<lb/>mido demerſa aufertur ſphæræ <lb/>portio b n c, non habens idem cen <lb/>trum grauitatis: </s> |
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740 <s xml:id="echoid-s449" xml:space="preserve">erit ex octaua <lb/>primi libri Archimcdis de centro <lb/>grauitatis planorum, reliquæ por <lb/>tionis b g c centrum in linea u q <lb/>producta. </s> |
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741 <s xml:id="echoid-s450" xml:space="preserve">quod fieri non potest; </s> |
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742 <s xml:id="echoid-s451" xml:space="preserve">eſt enim in axe ipſius mg. </s> |
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743 <s xml:id="echoid-s452" xml:space="preserve">sequi-<lb/>tur ergo ut portionis in humido demerſæ centrum grauitatis ſit in li <lb/>nean k. </s> |
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744 <s xml:id="echoid-s453" xml:space="preserve">quod oſtendendum propoſuimus.</s> |
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745 <s xml:id="echoid-s454" xml:space="preserve"/> |
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746 </p> |
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747 <p style="it"> |
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748 <s xml:id="echoid-s455" xml:space="preserve">_Sed totius portionis grauitatis centrum eſt in linea ft, in-_ <lb/> |
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749 <anchor type="note" xlink:label="note-0026-01a" xlink:href="note-0026-01"/> |
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750 _ter_ k, _& </s> |
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751 <s xml:id="echoid-s456" xml:space="preserve">f, quod ſit x.</s> |
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752 <s xml:id="echoid-s457" xml:space="preserve">]_ Compleatur ſphæra, ut ſit portionis additæ <lb/>axis t y; </s> |
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753 <s xml:id="echoid-s458" xml:space="preserve">& </s> |
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754 <s xml:id="echoid-s459" xml:space="preserve">centrú grauitatis z. </s> |
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755 <s xml:id="echoid-s460" xml:space="preserve">Itaque quoniá à tota ſphæra, cuius <lb/>grauitatis cétrum eſt k, ut etiam in eodem libro demóſtrauimus, au <lb/> |
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756 <anchor type="note" xlink:label="note-0026-02a" xlink:href="note-0026-02"/> |
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757 fertur portio e y h centrú grauitatis habens z: </s> |
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758 <s xml:id="echoid-s461" xml:space="preserve">erit reliquæ portionis <lb/>e f h cétrú in linea z k producta. </s> |
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759 <s xml:id="echoid-s462" xml:space="preserve">quare inter k. </s> |
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760 <s xml:id="echoid-s463" xml:space="preserve">& </s> |
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761 <s xml:id="echoid-s464" xml:space="preserve">f neceſſario cadet.</s> |
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762 <s xml:id="echoid-s465" xml:space="preserve"/> |
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763 </p> |
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764 <div xml:id="echoid-div30" type="float" level="2" n="4"> |
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765 <note position="left" xlink:label="note-0026-01" xlink:href="note-0026-01a" xml:space="preserve">D</note> |
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766 <note position="left" xlink:label="note-0026-02" xlink:href="note-0026-02a" xml:space="preserve">8. primi <lb/>Archime <lb/>dis.</note> |
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767 </div> |
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768 <p> |
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769 <s xml:id="echoid-s466" xml:space="preserve">Reliquæ ergo figuræ, quæ eſt extra humidum, centrum erit <lb/> |
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770 <anchor type="note" xlink:label="note-0026-03a" xlink:href="note-0026-03"/> |
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771 in linea r x producta.</s> |
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772 <s xml:id="echoid-s467" xml:space="preserve">] _Ex eadem octaua primi libri Archime-_ <lb/>_dis de centro grauitatis planorum._</s> |
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773 <s xml:id="echoid-s468" xml:space="preserve"/> |
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774 </p> |
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775 <div xml:id="echoid-div31" type="float" level="2" n="5"> |
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776 <note position="left" xlink:label="note-0026-03" xlink:href="note-0026-03a" xml:space="preserve">E</note> |
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777 </div> |
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778 <p style="it"> |
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779 <s xml:id="echoid-s469" xml:space="preserve">_Feretur ergo grauitas, figuræ quidem quæ extra humi_-<lb/> |
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780 <anchor type="note" xlink:label="note-0026-04a" xlink:href="note-0026-04"/> |
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781 _dum per rectam s l deorſum; </s> |
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782 <s xml:id="echoid-s470" xml:space="preserve">portionis autem, quæ in_ <lb/>_humido ſurſum per rectam r l.</s> |
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783 <s xml:id="echoid-s471" xml:space="preserve">]_ Ex antecedenti poſitio-<lb/>ne. </s> |
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784 <s xml:id="echoid-s472" xml:space="preserve">magnitudo @enim, quæ in humido demerſa est, tanta ui per li-<lb/>neam r l ſurſum@fertur, quanta quæ extra humidum per li-<lb/>neam s l, deorſum: </s> |
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785 <s xml:id="echoid-s473" xml:space="preserve">id quod ex propoſitione ſexta huius li-<lb/>briconſtare poteſt. </s> |
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786 <s xml:id="echoid-s474" xml:space="preserve">& </s> |
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787 <s xml:id="echoid-s475" xml:space="preserve">quoniam feruntur per alias, atque alias li- |
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788 <pb o="8" file="0027" n="27" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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789 neas; </s> |
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790 <s xml:id="echoid-s476" xml:space="preserve">neutra alteri obſistit, quo minus moueatur; </s> |
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791 <s xml:id="echoid-s477" xml:space="preserve">ídq; </s> |
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792 <s xml:id="echoid-s478" xml:space="preserve">continenter <lb/>fiat, dum portio in rectum fuerit conſtituta: </s> |
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793 <s xml:id="echoid-s479" xml:space="preserve">tunc enim utrarumque <lb/>magnitudinum grauitatis centra in unam, eandémq; </s> |
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794 <s xml:id="echoid-s480" xml:space="preserve">perpendicula-<lb/>rum conueniunt, uidelicet in axem portionis: </s> |
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795 <s xml:id="echoid-s481" xml:space="preserve">& </s> |
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796 <s xml:id="echoid-s482" xml:space="preserve">quanto conatu, im <lb/>petùue ea, quæ in humido eſt ſurſum, tanto quæ extra humidum de-<lb/>orſum per eandem lineam contendit. </s> |
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797 <s xml:id="echoid-s483" xml:space="preserve">quare cum altera alteram non <lb/>ſuperet, non amplius mouebitur portio; </s> |
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798 <s xml:id="echoid-s484" xml:space="preserve">ſed conſiſtet, manebítq; </s> |
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799 <s xml:id="echoid-s485" xml:space="preserve">in <lb/>eodem ſemper ſitu; </s> |
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800 <s xml:id="echoid-s486" xml:space="preserve">niſi forte aliqua cauſſa extrinſecus acceſſerit.</s> |
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801 <s xml:id="echoid-s487" xml:space="preserve"/> |
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802 </p> |
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803 <div xml:id="echoid-div32" type="float" level="2" n="6"> |
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804 <note position="left" xlink:label="note-0026-04" xlink:href="note-0026-04a" xml:space="preserve">F</note> |
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805 </div> |
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806 </div> |
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807 <div xml:id="echoid-div34" type="section" level="1" n="19"> |
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808 <head xml:id="echoid-head22" xml:space="preserve">PROPOSITIO IX.</head> |
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809 <p> |
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810 <s xml:id="echoid-s488" xml:space="preserve"><emph style="sc">Qvòd</emph> ſi figura humido leuior in humidum <lb/>demittatur, ita ut baſis tota ſit in humido; </s> |
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811 <s xml:id="echoid-s489" xml:space="preserve">inſide <lb/>bit recta, ita ut axis ipſius ſecundum perpendicu <lb/>larem conſtituatur.</s> |
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812 <s xml:id="echoid-s490" xml:space="preserve"/> |
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813 </p> |
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814 <p> |
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815 <s xml:id="echoid-s491" xml:space="preserve">INTELLIGATVR enim magnitudo aliqua, qua-<lb/>lis dicta eſt, in humidum demiſſa: </s> |
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816 <s xml:id="echoid-s492" xml:space="preserve">& </s> |
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817 <s xml:id="echoid-s493" xml:space="preserve">intelligatur planum <lb/>per axem portionis, & </s> |
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818 <s xml:id="echoid-s494" xml:space="preserve">per centrum terræ ductum: </s> |
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819 <s xml:id="echoid-s495" xml:space="preserve">ſitq; </s> |
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820 <s xml:id="echoid-s496" xml:space="preserve">ſu <lb/>perficiei quidem humidi ſectio a b c d circunferentia; </s> |
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821 <s xml:id="echoid-s497" xml:space="preserve">figu <lb/>ræ autem ſectio circun ferentia e f h: </s> |
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822 <s xml:id="echoid-s498" xml:space="preserve">& </s> |
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823 <s xml:id="echoid-s499" xml:space="preserve">ſit e h recta linea: <lb/></s> |
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824 <s xml:id="echoid-s500" xml:space="preserve">& </s> |
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825 <s xml:id="echoid-s501" xml:space="preserve">axis portionis f t. </s> |
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826 <s xml:id="echoid-s502" xml:space="preserve">Si igitur fieri poteſt, non ſit f t ſecun <lb/>dum perpendicularem.</s> |
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827 <s xml:id="echoid-s503" xml:space="preserve"/> |
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828 </p> |
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|
829 <figure> |
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830 <image file="0027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0027-01"/> |
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831 </figure> |
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832 <p> |
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833 <s xml:id="echoid-s504" xml:space="preserve">Demonſtrandum eſt non <lb/>manerefiguram; </s> |
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834 <s xml:id="echoid-s505" xml:space="preserve">ſed in re <lb/>ctum reſtitui. </s> |
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835 <s xml:id="echoid-s506" xml:space="preserve">eſt autem <lb/>centrum ſphæræ in linea <lb/>f t: </s> |
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836 <s xml:id="echoid-s507" xml:space="preserve">rurſus enim ſit figu-<lb/>ra primo maior dimidia <lb/>ſphæra: </s> |
|
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837 <s xml:id="echoid-s508" xml:space="preserve">& </s> |
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838 <s xml:id="echoid-s509" xml:space="preserve">ſphæræ centrũ <lb/>in dimidia ſphæra ſit pun-<lb/>ctum t; </s> |
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839 <s xml:id="echoid-s510" xml:space="preserve">in minore portione p; </s> |
|
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840 <s xml:id="echoid-s511" xml:space="preserve">in maiori uero ſit _k_: </s> |
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841 <s xml:id="echoid-s512" xml:space="preserve">& </s> |
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842 <s xml:id="echoid-s513" xml:space="preserve">per <lb/>_k_, & </s> |
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843 <s xml:id="echoid-s514" xml:space="preserve">terræ centrum l ducatur _k_ l. </s> |
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844 <s xml:id="echoid-s515" xml:space="preserve">Itaque figura quæ eſt <lb/> |
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|
845 <anchor type="note" xlink:label="note-0027-01a" xlink:href="note-0027-01"/> |
|
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|
846 <pb file="0028" n="28" rhead="ARCHIMEDIS"/> |
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847 extra humidi ſuperficiem, axem habetin perpendiculari <lb/>per _k_: </s> |
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848 <s xml:id="echoid-s516" xml:space="preserve">& </s> |
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849 <s xml:id="echoid-s517" xml:space="preserve">propter ea, quæ ſuperius dicta ſunt, centrum gra-<lb/>uitatis ipſius eſt in linea n _k_, quod ſitr; </s> |
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850 <s xml:id="echoid-s518" xml:space="preserve">totius autem por-<lb/>tionis centrum grauitatis eſt in linea f t, inter _k_ & </s> |
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851 <s xml:id="echoid-s519" xml:space="preserve">f, quod <lb/>ſit x. </s> |
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852 <s xml:id="echoid-s520" xml:space="preserve">reliquæ ergo figuræ, eius ſcilicet, quæ eſt in humido, <lb/>centrum erit in rectalinea r x producta ad partes x; </s> |
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853 <s xml:id="echoid-s521" xml:space="preserve">& </s> |
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854 <s xml:id="echoid-s522" xml:space="preserve">aſ-<lb/> |
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855 <anchor type="figure" xlink:label="fig-0028-01a" xlink:href="fig-0028-01"/> |
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856 ſumpta ex ea linea quadam, quæ ad x r eandem habeat pro <lb/>portionem, quam grauitas portionis, quæ eſt extra humi-<lb/>dum, ad grauitatem figuræ, quæ in humido. </s> |
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857 <s xml:id="echoid-s523" xml:space="preserve">Sit autem o <lb/>centrum dictæ figuræ: </s> |
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858 <s xml:id="echoid-s524" xml:space="preserve">& </s> |
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859 <s xml:id="echoid-s525" xml:space="preserve">per o perpendicularis ducatur <lb/>l o. </s> |
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860 <s xml:id="echoid-s526" xml:space="preserve">Feretur ergo grauitas portionis quidem, quæ eſt ex-<lb/>tra humidum, per rectam r l deorſum; </s> |
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861 <s xml:id="echoid-s527" xml:space="preserve">figuræ autem, quæ <lb/>in humido, per rectam o l ſurſum. </s> |
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862 <s xml:id="echoid-s528" xml:space="preserve">non manet igitur flgu-<lb/>ra; </s> |
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863 <s xml:id="echoid-s529" xml:space="preserve">ſed partes eius, quæ ſuntad h, deorſum ferẽtur; </s> |
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864 <s xml:id="echoid-s530" xml:space="preserve">& </s> |
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865 <s xml:id="echoid-s531" xml:space="preserve">quæ <lb/>ad e ſurſum. </s> |
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866 <s xml:id="echoid-s532" xml:space="preserve">atque hoc ſemper erit, donec f t ſecundum <lb/>perpendicularem fiat.</s> |
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867 <s xml:id="echoid-s533" xml:space="preserve"/> |
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868 </p> |
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869 <div xml:id="echoid-div34" type="float" level="2" n="1"> |
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870 <note position="right" xlink:label="note-0027-01" xlink:href="note-0027-01a" xml:space="preserve">A</note> |
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871 <figure xlink:label="fig-0028-01" xlink:href="fig-0028-01a"> |
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872 <image file="0028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0028-01"/> |
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873 </figure> |
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874 </div> |
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875 </div> |
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876 <div xml:id="echoid-div36" type="section" level="1" n="20"> |
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877 <head xml:id="echoid-head23" xml:space="preserve">COMMENTARIVS.</head> |
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878 <p> |
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879 <s xml:id="echoid-s534" xml:space="preserve">ITAQVE figura, quæ extra humidi ſuperficiem, <lb/> |
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880 <anchor type="note" xlink:label="note-0028-01a" xlink:href="note-0028-01"/> |
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881 axem habet in perpendiculari per _k_.</s> |
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882 <s xml:id="echoid-s535" xml:space="preserve">]</s> |
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883 </p> |
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884 <div xml:id="echoid-div36" type="float" level="2" n="1"> |
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885 <note position="left" xlink:label="note-0028-01" xlink:href="note-0028-01a" xml:space="preserve">A</note> |
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886 </div> |
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887 <p style="it"> |
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888 <s xml:id="echoid-s536" xml:space="preserve"><emph style="sc">D_vcatvr_</emph> enim b c, quæ ſecet lineam n k in m: </s> |
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889 <s xml:id="echoid-s537" xml:space="preserve">ipſa uero <lb/>n k circunferentiam a b c d ſecet in g. </s> |
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890 <s xml:id="echoid-s538" xml:space="preserve">eodem modo, quo ſupra, de- |
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891 <pb o="9" file="0029" n="29" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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892 monſtrabimus portionis ſphæræ b n c axem eſſe ipſam n m: </s> |
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893 <s xml:id="echoid-s539" xml:space="preserve">& </s> |
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894 <s xml:id="echoid-s540" xml:space="preserve"><lb/>portionis b g c axem g m. <lb/></s> |
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895 <s xml:id="echoid-s541" xml:space="preserve"> |
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896 <anchor type="figure" xlink:label="fig-0029-01a" xlink:href="fig-0029-01"/> |
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897 quare centrum grauitatis utri <lb/>uſque, erit in linea n m. </s> |
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898 <s xml:id="echoid-s542" xml:space="preserve">& </s> |
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899 <s xml:id="echoid-s543" xml:space="preserve"><lb/>quoniam à portione b n c au-<lb/>fertur portio b g c, non ha-<lb/>bens@idem grauitatis centrú: <lb/></s> |
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900 <s xml:id="echoid-s544" xml:space="preserve">reliquæ magnitudinis, quæ est <lb/>extra humidi ſuperficiem, cen-<lb/>trum grauitatis erit in linea <lb/>n k; </s> |
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901 <s xml:id="echoid-s545" xml:space="preserve">quæ ſcilicet earum portionum centra grauitatis coniungit: </s> |
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902 <s xml:id="echoid-s546" xml:space="preserve">ex <lb/>eadem octaua Archimedis.</s> |
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903 <s xml:id="echoid-s547" xml:space="preserve"/> |
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904 </p> |
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905 <div xml:id="echoid-div37" type="float" level="2" n="2"> |
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906 <figure xlink:label="fig-0029-01" xlink:href="fig-0029-01a"> |
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907 <image file="0029-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0029-01"/> |
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908 </figure> |
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909 </div> |
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910 <pb file="0030" n="30"/> |
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911 </div> |
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912 <div xml:id="echoid-div39" type="section" level="1" n="21"> |
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913 <head xml:id="echoid-head24" xml:space="preserve">ARCHIMEDIS DE IIS <lb/>QVAE VEHVNTVR IN AQVA <lb/>LIBER SECVNDVS.</head> |
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914 <head xml:id="echoid-head25" xml:space="preserve">CVM COMMENTARIIS FEDERICI <lb/>COMMANDINI VRBINATIS.</head> |
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915 <head xml:id="echoid-head26" xml:space="preserve">PROPOSITIO I.</head> |
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916 <p> |
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917 <s xml:id="echoid-s548" xml:space="preserve">SI magnitudo aliqua humido <lb/>leuior demittatur in humi-<lb/>dum, eam in grauitate pro-<lb/>portionem habebit ad humi-<lb/>dum æqualis molis, quã pars <lb/>magnitudinis demerſa habet <lb/>ad totam magnitudinem.</s> |
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918 <s xml:id="echoid-s549" xml:space="preserve"/> |
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919 </p> |
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920 <p> |
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921 <s xml:id="echoid-s550" xml:space="preserve">DEMITTATVR enim in humidum aliqua magni-<lb/>tudo ſolida, quæ ſit fa, leuior humido: </s> |
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922 <s xml:id="echoid-s551" xml:space="preserve">& </s> |
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923 <s xml:id="echoid-s552" xml:space="preserve">pars quidem ip-<lb/>ſius demerſa ſit a; </s> |
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924 <s xml:id="echoid-s553" xml:space="preserve">quæ autem extra humidum f. </s> |
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925 <s xml:id="echoid-s554" xml:space="preserve">demon-<lb/>ſtrandum eſt, ma <lb/> |
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926 <anchor type="figure" xlink:label="fig-0030-01a" xlink:href="fig-0030-01"/> |
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927 gnitudinem f a <lb/>ad humidum æ-<lb/>qualis molis eam <lb/>in grauitate pro-<lb/>portionem habe <lb/>re, quam habet <lb/>a ad fa. </s> |
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928 <s xml:id="echoid-s555" xml:space="preserve">accipiatur enim aliqua humidi magnitudo n i |
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929 <pb o="10" file="0031" n="31" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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930 æqualis magnitudini f a; </s> |
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931 <s xml:id="echoid-s556" xml:space="preserve">ſitq, ipſi f æqualis n: </s> |
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932 <s xml:id="echoid-s557" xml:space="preserve">& </s> |
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933 <s xml:id="echoid-s558" xml:space="preserve">ipſi a æ-<lb/>qualis i. </s> |
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934 <s xml:id="echoid-s559" xml:space="preserve">magnitudinis autem f a grauitas ſit b: </s> |
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935 <s xml:id="echoid-s560" xml:space="preserve">& </s> |
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936 <s xml:id="echoid-s561" xml:space="preserve">magni-<lb/>tudinis n i grauitas o r; </s> |
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937 <s xml:id="echoid-s562" xml:space="preserve">& </s> |
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938 <s xml:id="echoid-s563" xml:space="preserve">ipſius i ſit r. </s> |
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939 <s xml:id="echoid-s564" xml:space="preserve">magnitudo igi-<lb/>tur f a ad n i eam proportionem habet, quam grauitas b <lb/>ad grauitatem or. </s> |
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940 <s xml:id="echoid-s565" xml:space="preserve">Sed quoniam magnitudo f a in humi-<lb/>dum demiſſa leuior eſt humido; </s> |
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941 <s xml:id="echoid-s566" xml:space="preserve">patet tantam humidi mo-<lb/>lem, quanta eſt pars magnitudin_i_s demerſa, eandem quam <lb/>magnitudo f a habere grauitatem. </s> |
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942 <s xml:id="echoid-s567" xml:space="preserve">hoc enim ſuperius de-<lb/> |
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943 <anchor type="note" xlink:label="note-0031-01a" xlink:href="note-0031-01"/> |
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944 monſtratum eſt. </s> |
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945 <s xml:id="echoid-s568" xml:space="preserve">Atipſi a reſpondet humidum i, cuius qui <lb/>dem grauitas eſt r; </s> |
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946 <s xml:id="echoid-s569" xml:space="preserve">& </s> |
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947 <s xml:id="echoid-s570" xml:space="preserve">ipſius f a grauitas b. </s> |
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948 <s xml:id="echoid-s571" xml:space="preserve">ergo b graui-<lb/>tas eius, quod habet molem æqualem toti magnitudini <lb/>f a, æqualis erit grauitati humidi i, uidelicetipſi r. </s> |
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949 <s xml:id="echoid-s572" xml:space="preserve">Et quo <lb/>niam ut magnitudo f a ad humidum n i ſibi reſpondens, <lb/>ita eſt b ad o r: </s> |
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950 <s xml:id="echoid-s573" xml:space="preserve">eſt autem b æqualis ipſi r: </s> |
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951 <s xml:id="echoid-s574" xml:space="preserve">& </s> |
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952 <s xml:id="echoid-s575" xml:space="preserve">utr ad o r, ita <lb/>i ad n i; </s> |
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953 <s xml:id="echoid-s576" xml:space="preserve">& </s> |
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954 <s xml:id="echoid-s577" xml:space="preserve">a ad f a. </s> |
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955 <s xml:id="echoid-s578" xml:space="preserve">Sequitur ut f a ad humidum æqualis <lb/> |
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956 <anchor type="note" xlink:label="note-0031-02a" xlink:href="note-0031-02"/> |
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957 molis eam in grauitate proportionem habeat, quam ma-<lb/>gnitudo a habet ad f a. </s> |
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958 <s xml:id="echoid-s579" xml:space="preserve">quod demonſtrare oportebat.</s> |
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959 <s xml:id="echoid-s580" xml:space="preserve"/> |
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960 </p> |
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961 <div xml:id="echoid-div39" type="float" level="2" n="1"> |
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962 <figure xlink:label="fig-0030-01" xlink:href="fig-0030-01a"> |
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963 <image file="0030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0030-01"/> |
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964 </figure> |
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965 <note position="right" xlink:label="note-0031-01" xlink:href="note-0031-01a" xml:space="preserve">5. priml <lb/>huius.</note> |
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966 <note position="right" xlink:label="note-0031-02" xlink:href="note-0031-02a" xml:space="preserve">11. quinta<unsure/></note> |
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967 </div> |
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968 </div> |
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969 <div xml:id="echoid-div41" type="section" level="1" n="22"> |
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970 <head xml:id="echoid-head27" xml:space="preserve">PROPOSITIO II.</head> |
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971 <p> |
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972 <s xml:id="echoid-s581" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/> |
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973 <anchor type="note" xlink:label="note-0031-03a" xlink:href="note-0031-03"/> |
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974 axem habuerit minorem, quam ſeſquialterum <lb/>eius, quæ uſque ad axem, quamcunque propor-<lb/>tionem habens ad humidum in grauitate; </s> |
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975 <s xml:id="echoid-s582" xml:space="preserve">demiſ <lb/>ſa in humidum, ita ut baſis ipſius humidum non <lb/>contingat; </s> |
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976 <s xml:id="echoid-s583" xml:space="preserve">& </s> |
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977 <s xml:id="echoid-s584" xml:space="preserve">poſita inelinata, non manebit incli <lb/>nata; </s> |
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978 <s xml:id="echoid-s585" xml:space="preserve">ſed recta reſtituetur. </s> |
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979 <s xml:id="echoid-s586" xml:space="preserve">Rectam dico conſi-<lb/>ſtere talem portionem, quando planum quod ip <lb/>ſam ſecuit, ſuperficiei humidi fuerit æquidiſtans.</s> |
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980 <s xml:id="echoid-s587" xml:space="preserve"/> |
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981 </p> |
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982 <div xml:id="echoid-div41" type="float" level="2" n="1"> |
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983 <note position="right" xlink:label="note-0031-03" xlink:href="note-0031-03a" xml:space="preserve">A</note> |
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984 </div> |
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985 <p> |
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986 <s xml:id="echoid-s588" xml:space="preserve">SIT portio rectanguli conoidis, qualis dicta eſt; </s> |
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987 <s xml:id="echoid-s589" xml:space="preserve">& </s> |
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988 <s xml:id="echoid-s590" xml:space="preserve">ia- |
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989 <pb file="0032" n="32" rhead="ARCHIMEDIS"/> |
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990 ceatinclinata. </s> |
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991 <s xml:id="echoid-s591" xml:space="preserve">Demonſtrandum eſt non manere ipſam; </s> |
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992 <s xml:id="echoid-s592" xml:space="preserve">ſed <lb/>rectam reſtitui. </s> |
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993 <s xml:id="echoid-s593" xml:space="preserve">Itaque ſecta ipſa plano per axem, recto ad <lb/>planum, quod eſt in ſuperficie humidi, portionis ſectio ſit <lb/>a p o l rectanguli coni ſectio: </s> |
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994 <s xml:id="echoid-s594" xml:space="preserve">axis portionis, & </s> |
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995 <s xml:id="echoid-s595" xml:space="preserve">ſectionis <lb/>diameter n o: </s> |
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996 <s xml:id="echoid-s596" xml:space="preserve">ſuperficiei autem humidi ſectio ſit i s. </s> |
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997 <s xml:id="echoid-s597" xml:space="preserve">Si <lb/>igitur portio non eſt recta; </s> |
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998 <s xml:id="echoid-s598" xml:space="preserve">non utique erit a l ipſi i s æ-<lb/>quidiſtans. </s> |
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999 <s xml:id="echoid-s599" xml:space="preserve">quare n o cum i s non faciet angulos rectos. <lb/></s> |
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1000 <s xml:id="echoid-s600" xml:space="preserve">ducatur crgo k ω contingens ſectionem coni in p [quæ <lb/> |
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1001 <anchor type="note" xlink:label="note-0032-01a" xlink:href="note-0032-01"/> |
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1002 ipſi i s æquidiſtet: </s> |
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1003 <s xml:id="echoid-s601" xml:space="preserve">& </s> |
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1004 <s xml:id="echoid-s602" xml:space="preserve">à puncto p ad i s ducatur p f æquidi <lb/>ſtans ipſi o n, quæ erit ſectionis i p o s diameter, & </s> |
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1005 <s xml:id="echoid-s603" xml:space="preserve">axis por <lb/> |
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1006 <anchor type="note" xlink:label="note-0032-02a" xlink:href="note-0032-02"/> |
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1007 tionis in humido demerſæ. </s> |
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1008 <s xml:id="echoid-s604" xml:space="preserve">ſumantur deinde centra graui <lb/>tatum: </s> |
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1009 <s xml:id="echoid-s605" xml:space="preserve">ſitq; </s> |
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1010 <s xml:id="echoid-s606" xml:space="preserve">ſolidæ magnitudinis a p o l grauitatis centrũ <lb/> |
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1011 <anchor type="note" xlink:label="note-0032-03a" xlink:href="note-0032-03"/> |
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1012 r; </s> |
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1013 <s xml:id="echoid-s607" xml:space="preserve">ipſius uero i p o s centrum ſit b: </s> |
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1014 <s xml:id="echoid-s608" xml:space="preserve">& </s> |
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1015 <s xml:id="echoid-s609" xml:space="preserve">iuncta b r produca-<lb/> |
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1016 <anchor type="note" xlink:label="note-0032-04a" xlink:href="note-0032-04"/> |
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1017 tur ad g, quod ſit centrum grauitatis reliquæ figuræ i s l a. <lb/></s> |
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1018 <s xml:id="echoid-s610" xml:space="preserve">Quoniam igitur n o ipſius quidem r o ſeſquialtera eſt; </s> |
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1019 <s xml:id="echoid-s611" xml:space="preserve"><lb/>eius autẽ, quæ uſque ad axẽ minor, quam ſeſquialtera; </s> |
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1020 <s xml:id="echoid-s612" xml:space="preserve">erit <lb/> |
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1021 <anchor type="note" xlink:label="note-0032-05a" xlink:href="note-0032-05"/> |
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1022 r o minor, quàm quæ uſque ad axem. </s> |
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1023 <s xml:id="echoid-s613" xml:space="preserve">Quare angulus r p ω <lb/> |
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1024 <anchor type="note" xlink:label="note-0032-06a" xlink:href="note-0032-06"/> |
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1025 acutus erit: </s> |
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1026 <s xml:id="echoid-s614" xml:space="preserve">cum enim linea, quæ uſque ad axem maior ſit <lb/>ipſa r o; </s> |
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1027 <s xml:id="echoid-s615" xml:space="preserve">quæ à puncto r ad k ω perpendicularis ducitur, <lb/>uidelicet r t, cũ <lb/> |
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1028 <anchor type="figure" xlink:label="fig-0032-01a" xlink:href="fig-0032-01"/> |
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1029 linea f p extra <lb/>ſectionem con <lb/>ueniet: </s> |
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1030 <s xml:id="echoid-s616" xml:space="preserve">& </s> |
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1031 <s xml:id="echoid-s617" xml:space="preserve">pro-<lb/>pterea inter p <lb/>& </s> |
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1032 <s xml:id="echoid-s618" xml:space="preserve">ω puncta ca-<lb/>datneceſſe eſt. <lb/></s> |
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1033 <s xml:id="echoid-s619" xml:space="preserve">Itaq; </s> |
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1034 <s xml:id="echoid-s620" xml:space="preserve">ſi per b g <lb/>ducantur lineæ <lb/>ipſi r t æquidi-<lb/>ſtantes; </s> |
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1035 <s xml:id="echoid-s621" xml:space="preserve">angu-<lb/>los rectos cum <lb/>ſuperficie humidi continebunt: </s> |
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1036 <s xml:id="echoid-s622" xml:space="preserve">& </s> |
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1037 <s xml:id="echoid-s623" xml:space="preserve">quod in humido eſt ſur-<lb/> |
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1038 <anchor type="note" xlink:label="note-0032-07a" xlink:href="note-0032-07"/> |
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1039 ſum feretur ſecundum perpendicularem, quæ per b ducta <lb/>eſt, ipſi r t æquidiſtans: </s> |
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1040 <s xml:id="echoid-s624" xml:space="preserve">quod uero eſt extra humi dum ſe- |
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1041 <pb o="11" file="0033" n="33" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1042 cundum eam, quæ per g, deorſum ferctur; </s> |
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1043 <s xml:id="echoid-s625" xml:space="preserve">& </s> |
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1044 <s xml:id="echoid-s626" xml:space="preserve">non ita mane <lb/>bit ſolidum a p o l: </s> |
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1045 <s xml:id="echoid-s627" xml:space="preserve">nam quod eſt ad a feretur ſurſum; </s> |
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1046 <s xml:id="echoid-s628" xml:space="preserve">& </s> |
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1047 <s xml:id="echoid-s629" xml:space="preserve"><lb/>quod ad b deorſum, donec n o ſecundum perpendicu-<lb/>larem conſtituatur.</s> |
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1048 <s xml:id="echoid-s630" xml:space="preserve">]</s> |
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1049 </p> |
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1050 <div xml:id="echoid-div42" type="float" level="2" n="2"> |
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1051 <note position="left" xlink:label="note-0032-01" xlink:href="note-0032-01a" xml:space="preserve">Suppleta <lb/>a. Federi-<lb/>co Cõm.</note> |
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1052 <note position="left" xlink:label="note-0032-02" xlink:href="note-0032-02a" xml:space="preserve">B</note> |
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1053 <note position="left" xlink:label="note-0032-03" xlink:href="note-0032-03a" xml:space="preserve">C</note> |
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1054 <note position="left" xlink:label="note-0032-04" xlink:href="note-0032-04a" xml:space="preserve">D</note> |
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1055 <note position="left" xlink:label="note-0032-05" xlink:href="note-0032-05a" xml:space="preserve">E</note> |
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1056 <note position="left" xlink:label="note-0032-06" xlink:href="note-0032-06a" xml:space="preserve">F</note> |
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1057 <figure xlink:label="fig-0032-01" xlink:href="fig-0032-01a"> |
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1058 <image file="0032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0032-01"/> |
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1059 </figure> |
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1060 <note position="left" xlink:label="note-0032-07" xlink:href="note-0032-07a" xml:space="preserve">G</note> |
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1061 </div> |
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1062 </div> |
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1063 <div xml:id="echoid-div44" type="section" level="1" n="23"> |
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1064 <head xml:id="echoid-head28" xml:space="preserve">COMMENTARIVS.</head> |
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1065 <p style="it"> |
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1066 <s xml:id="echoid-s631" xml:space="preserve"><emph style="sc">D_esideratvr_</emph> propoſitionis huius demonstratio, quam nos <lb/>etiam ad Archimedis figuram appoſite restituimus, commentarijs-<lb/>que illustrauimus.</s> |
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1067 <s xml:id="echoid-s632" xml:space="preserve"/> |
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1068 </p> |
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1069 <p style="it"> |
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1070 <s xml:id="echoid-s633" xml:space="preserve">_Recta portio conoidis rectanguli, quando axem habue_ <lb/> |
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1071 <anchor type="note" xlink:label="note-0033-01a" xlink:href="note-0033-01"/> |
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1072 _rit minorem, quàm ſeſquialterum eius, quæ uſque ad axẽ]_ <lb/>In tranſlatione mendoſe legebatur. </s> |
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1073 <s xml:id="echoid-s634" xml:space="preserve">maiorem quàm ſeſquialterum: <lb/></s> |
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1074 <s xml:id="echoid-s635" xml:space="preserve">& </s> |
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1075 <s xml:id="echoid-s636" xml:space="preserve">ita legebatur in ſequenti propoſitione. </s> |
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1076 <s xml:id="echoid-s637" xml:space="preserve">est autem recta portio co <lb/>noidis, quæ plano ad axem recto abſcinditur: </s> |
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1077 <s xml:id="echoid-s638" xml:space="preserve">eâmque rectam tunc <lb/>conſiſtere dicimus, quando planum abſcindens, uidelicet baſis pla-<lb/>num, ſuperficiei humidi æquidiſtans fuerit.</s> |
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1078 <s xml:id="echoid-s639" xml:space="preserve"/> |
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1079 </p> |
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1080 <div xml:id="echoid-div44" type="float" level="2" n="1"> |
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1081 <note position="right" xlink:label="note-0033-01" xlink:href="note-0033-01a" xml:space="preserve">A</note> |
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1082 </div> |
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1083 <p> |
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1084 <s xml:id="echoid-s640" xml:space="preserve">Quæ erit ſectionis i p o s diameter, & </s> |
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1085 <s xml:id="echoid-s641" xml:space="preserve">axis portionis in <lb/> |
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1086 <anchor type="note" xlink:label="note-0033-02a" xlink:href="note-0033-02"/> |
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1087 humido demerſæ] _ex_ 46 _primi conicorum Apollonij: </s> |
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1088 <s xml:id="echoid-s642" xml:space="preserve">uel ex co-_ <lb/>_rollario_ 51 _eiuſdem_.</s> |
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1089 <s xml:id="echoid-s643" xml:space="preserve"/> |
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1090 </p> |
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1091 <div xml:id="echoid-div45" type="float" level="2" n="2"> |
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1092 <note position="right" xlink:label="note-0033-02" xlink:href="note-0033-02a" xml:space="preserve">B</note> |
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1093 </div> |
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1094 <p style="it"> |
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1095 <s xml:id="echoid-s644" xml:space="preserve">_Sitque ſolidæ magnitudinis a p o l grauitatis centrum r,_ <lb/> |
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1096 <anchor type="note" xlink:label="note-0033-03a" xlink:href="note-0033-03"/> |
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1097 _ipſius uero i p o s centrum ſit b.</s> |
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1098 <s xml:id="echoid-s645" xml:space="preserve">]_ Portionis enim conoidis <lb/>rectanguli centrum grauitatis eſt in axe, quem ita diuidit, ut pars <lb/>eius, quæ ad uerticem terminatur, reliquæ partis, quæ ad baſim, ſit <lb/>dupla: </s> |
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1099 <s xml:id="echoid-s646" xml:space="preserve">quod nos in libro de centro grauitatis ſolidorum propoſitio-<lb/>ne 29 demonstrauimus. </s> |
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1100 <s xml:id="echoid-s647" xml:space="preserve">Cum igitur portionis a p o l centrum gra-<lb/>uitatis ſit r, erit o r dupla r n: </s> |
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1101 <s xml:id="echoid-s648" xml:space="preserve">& </s> |
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1102 <s xml:id="echoid-s649" xml:space="preserve">propterea n o ipſius o r ſeſqui-<lb/>altera. </s> |
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1103 <s xml:id="echoid-s650" xml:space="preserve">Eadem ratione b centrum grauitatis portionis i p o s est in <lb/>axe p f, ita ut p b dupla ſit b f.</s> |
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1104 <s xml:id="echoid-s651" xml:space="preserve"/> |
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1105 </p> |
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1106 <div xml:id="echoid-div46" type="float" level="2" n="3"> |
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1107 <note position="right" xlink:label="note-0033-03" xlink:href="note-0033-03a" xml:space="preserve">C</note> |
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1108 </div> |
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1109 <p style="it"> |
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1110 <s xml:id="echoid-s652" xml:space="preserve">_Etiuncta b r producatur ad g, quod ſit centrum graui_ <lb/> |
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1111 <anchor type="note" xlink:label="note-0033-04a" xlink:href="note-0033-04"/> |
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1112 _tatis reliquæ figuræ i s l a]_ Si enim linea b r in g producta, ha <lb/>beat g r ad r b proportionem eam, quam conoidis portio i p o s ad <lb/>reliquam figuram, quæ ex humidi ſuperficie extat: </s> |
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1113 <s xml:id="echoid-s653" xml:space="preserve">erit punctum g <lb/>ipſius grauitatis centrum, ex octaua Archimedis.</s> |
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1114 <s xml:id="echoid-s654" xml:space="preserve"/> |
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1115 </p> |
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1116 <div xml:id="echoid-div47" type="float" level="2" n="4"> |
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1117 <note position="right" xlink:label="note-0033-04" xlink:href="note-0033-04a" xml:space="preserve">D</note> |
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1118 </div> |
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1119 <pb file="0034" n="34" rhead="ARCHIMEDIS"/> |
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1120 <p style="it"> |
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1121 <s xml:id="echoid-s655" xml:space="preserve">_Erit r o minor, quàm, quæ uſque ad axem]_ Ex decima <lb/> |
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1122 <anchor type="note" xlink:label="note-0034-01a" xlink:href="note-0034-01"/> |
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1123 propoſitione quinti libri elementorum. </s> |
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1124 <s xml:id="echoid-s656" xml:space="preserve">Linea, quæ uſque ad axem <lb/>apud Archimedem, eſt dimidia eius, iuxta quam poſſunt, quæ à ſe-<lb/>ctione ducuntur; </s> |
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1125 <s xml:id="echoid-s657" xml:space="preserve">ut ex quarta propoſitione libri de conoidibus, & </s> |
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1126 <s xml:id="echoid-s658" xml:space="preserve"><lb/>ſphæroidibus apparet. </s> |
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1127 <s xml:id="echoid-s659" xml:space="preserve">cur uero ita appellata ſit, nos in commentarijs <lb/>in eam editis tradidimus.</s> |
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1128 <s xml:id="echoid-s660" xml:space="preserve"/> |
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1129 </p> |
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1130 <div xml:id="echoid-div48" type="float" level="2" n="5"> |
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1131 <note position="left" xlink:label="note-0034-01" xlink:href="note-0034-01a" xml:space="preserve">E</note> |
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1132 </div> |
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1133 <p style="it"> |
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1134 <s xml:id="echoid-s661" xml:space="preserve">_Quare angulus r p ω acutus erit]_ producatur linea n o ad <lb/> |
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1135 <anchor type="note" xlink:label="note-0034-02a" xlink:href="note-0034-02"/> |
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1136 h, ut ſit r h æqualis ei, quæ uſque ad axem. </s> |
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1137 <s xml:id="echoid-s662" xml:space="preserve">ſi igitur à puncto h du-<lb/>catur linea ad rectos angulos ipſi n h, conueniet cum f p extra ſe-<lb/>ctionem: </s> |
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1138 <s xml:id="echoid-s663" xml:space="preserve">ducta enim per o ipſi a l æquidiſtans, extra ſectionem ca <lb/>dit ex decima ſepti-<lb/> |
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1139 <anchor type="figure" xlink:label="fig-0034-01a" xlink:href="fig-0034-01"/> |
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1140 ma primi libri coni-<lb/>corum. </s> |
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1141 <s xml:id="echoid-s664" xml:space="preserve">Itaque con-<lb/>ueniat in u. </s> |
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1142 <s xml:id="echoid-s665" xml:space="preserve">& </s> |
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1143 <s xml:id="echoid-s666" xml:space="preserve">quo <lb/>niam f p est æqui-<lb/>distans diametro; <lb/></s> |
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1144 <s xml:id="echoid-s667" xml:space="preserve">h u uero ad diame-<lb/>trum perpendicula-<lb/>ris; </s> |
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1145 <s xml:id="echoid-s668" xml:space="preserve">& </s> |
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1146 <s xml:id="echoid-s669" xml:space="preserve">r h æqualis <lb/>ei, quæ uſq; </s> |
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1147 <s xml:id="echoid-s670" xml:space="preserve">ad axẽ, <lb/>linea à puncto r ad <lb/>u ducta angulos re-<lb/>ctos faciet cum ea, quæ ſectionem in puncto p contingit, hoc eſt cum <lb/>k ω, ut mox demonstrabitur. </s> |
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1148 <s xml:id="echoid-s671" xml:space="preserve">quare perpendicularis r t inter p & </s> |
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1149 <s xml:id="echoid-s672" xml:space="preserve"><lb/>ω cadet; </s> |
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1150 <s xml:id="echoid-s673" xml:space="preserve">erítque r p ω angulus acutus.</s> |
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1151 <s xml:id="echoid-s674" xml:space="preserve"/> |
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|
1152 </p> |
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1153 <div xml:id="echoid-div49" type="float" level="2" n="6"> |
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1154 <note position="left" xlink:label="note-0034-02" xlink:href="note-0034-02a" xml:space="preserve">F</note> |
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1155 <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a"> |
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1156 <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0034-01"/> |
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1157 </figure> |
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1158 </div> |
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1159 <p style="it"> |
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1160 <s xml:id="echoid-s675" xml:space="preserve">Sit rectanguli coni ſectio, ſeu parabole a b c, cuius <lb/>diameter b d: </s> |
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1161 <s xml:id="echoid-s676" xml:space="preserve">atque ipſam contingat linea e f in pun-<lb/>cto g: </s> |
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1162 <s xml:id="echoid-s677" xml:space="preserve">ſumatur autem in diametro b d linea h k æqua-<lb/>lis ei, quæ uſque ad axem: </s> |
|
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1163 <s xml:id="echoid-s678" xml:space="preserve">& </s> |
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1164 <s xml:id="echoid-s679" xml:space="preserve">per g ducta g l, diame-<lb/>tro æquidistante, à puncto _k_ ad rectos angulos ipſi b d <lb/>ducatur _k_ m, ſecans g l in m. </s> |
|
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1165 <s xml:id="echoid-s680" xml:space="preserve">Dico lineam ab h ad |
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1166 <pb o="12" file="0035" n="35" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1167 m productam per pendicularem eſſe ad ipſam e f, quam <lb/>quidem ſecet in n.</s> |
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1168 <s xml:id="echoid-s681" xml:space="preserve"/> |
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1169 </p> |
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1170 <p style="it"> |
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1171 <s xml:id="echoid-s682" xml:space="preserve"><emph style="sc">D_vcatvr_</emph> enim à puncto g linea g o ad rectos angulos ipſi <lb/>e f, diametrum in o ſecans: </s> |
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1172 <s xml:id="echoid-s683" xml:space="preserve">& </s> |
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1173 <s xml:id="echoid-s684" xml:space="preserve">rurſus ab eodem puncto ducatur g p <lb/>ad diametrum perpendicularis: </s> |
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1174 <s xml:id="echoid-s685" xml:space="preserve">ſecet autem ipſa diameter producta <lb/>lineã e f in q. </s> |
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1175 <s xml:id="echoid-s686" xml:space="preserve">erit p b ipſi b q æqualis, ex trigeſimaquinta primi co <lb/>nicorum: </s> |
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1176 <s xml:id="echoid-s687" xml:space="preserve">& </s> |
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1177 <s xml:id="echoid-s688" xml:space="preserve">g p pro-<lb/> |
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1178 <anchor type="note" xlink:label="note-0035-01a" xlink:href="note-0035-01"/> |
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1179 <anchor type="figure" xlink:label="fig-0035-01a" xlink:href="fig-0035-01"/> |
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|
1180 portionalis ĩter q p, p o <lb/>quare quadratũ g p re-<lb/> |
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1181 <anchor type="note" xlink:label="note-0035-02a" xlink:href="note-0035-02"/> |
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1182 ctangulo o p q æquale <lb/>erit: </s> |
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1183 <s xml:id="echoid-s689" xml:space="preserve">ſed etiã æquale est <lb/>rectangulo cõtento ipſa <lb/>p b, & </s> |
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1184 <s xml:id="echoid-s690" xml:space="preserve">linea, iuxta quã <lb/>poſſunt, quæ à ſectione <lb/>ad diametrũ ordinatim <lb/>ducuntur, ex undecima <lb/>primi conicorum. </s> |
|
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1185 <s xml:id="echoid-s691" xml:space="preserve">ergo <lb/> |
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|
1186 <anchor type="note" xlink:label="note-0035-03a" xlink:href="note-0035-03"/> |
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|
1187 quæ est proportio q p <lb/>ad p b eadem est lineæ, <lb/>iuxta quã poſſunt, quæ <lb/>à ſectione ducũtur ad ip <lb/>ſam p o: </s> |
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1188 <s xml:id="echoid-s692" xml:space="preserve">est autem q p <lb/>dupla p b: </s> |
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1189 <s xml:id="echoid-s693" xml:space="preserve">cũ ſint p b, <lb/>b q æquales, ut dictum <lb/>est. </s> |
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1190 <s xml:id="echoid-s694" xml:space="preserve">Linea igitur iuxta <lb/>quam poſſunt, quæ à ſe-<lb/>ctione ducuntur ipſi-<lb/>us p o dupla erit: </s> |
|
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1191 <s xml:id="echoid-s695" xml:space="preserve">& </s> |
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1192 <s xml:id="echoid-s696" xml:space="preserve"><lb/>propterea p o æqualis <lb/>ei, quæ uſque ad axem, <lb/>uidelicet ipſi k h: </s> |
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1193 <s xml:id="echoid-s697" xml:space="preserve">ſed eſt p g æqualis k m; </s> |
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1194 <s xml:id="echoid-s698" xml:space="preserve">& </s> |
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1195 <s xml:id="echoid-s699" xml:space="preserve">angulus o p g angu-<lb/> |
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1196 <anchor type="note" xlink:label="note-0035-04a" xlink:href="note-0035-04"/> |
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1197 lo h k m; </s> |
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1198 <s xml:id="echoid-s700" xml:space="preserve">quòd uterque rectus. </s> |
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1199 <s xml:id="echoid-s701" xml:space="preserve">quare & </s> |
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1200 <s xml:id="echoid-s702" xml:space="preserve">o g ipſi h m est œqualis: <lb/></s> |
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1201 <s xml:id="echoid-s703" xml:space="preserve"> |
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1202 <anchor type="note" xlink:label="note-0035-05a" xlink:href="note-0035-05"/> |
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1203 & </s> |
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1204 <s xml:id="echoid-s704" xml:space="preserve">angulus p o g angulo _k_ h m. </s> |
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1205 <s xml:id="echoid-s705" xml:space="preserve">æquidistantes igitur ſunt o g, h n: <lb/></s> |
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1206 <s xml:id="echoid-s706" xml:space="preserve"> |
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1207 <anchor type="note" xlink:label="note-0035-06a" xlink:href="note-0035-06"/> |
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1208 <pb file="0036" n="36" rhead="ARCHIMEDIS"/> |
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1209 angulus b n f œqualis angulo o g f: </s> |
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1210 <s xml:id="echoid-s707" xml:space="preserve">quòd cum ſit g o perpendi-<lb/> |
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1211 <anchor type="note" xlink:label="note-0036-01a" xlink:href="note-0036-01"/> |
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1212 cularis ad e f, & </s> |
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1213 <s xml:id="echoid-s708" xml:space="preserve">h n ad eandem perpendicularis erit. </s> |
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1214 <s xml:id="echoid-s709" xml:space="preserve">quod de-<lb/>monstrare oportebat.</s> |
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1215 <s xml:id="echoid-s710" xml:space="preserve"/> |
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1216 </p> |
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1217 <div xml:id="echoid-div50" type="float" level="2" n="7"> |
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1218 <note position="right" xlink:label="note-0035-01" xlink:href="note-0035-01a" xml:space="preserve">cor. 8. ſe-<lb/>xti.</note> |
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1219 <figure xlink:label="fig-0035-01" xlink:href="fig-0035-01a"> |
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1220 <image file="0035-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0035-01"/> |
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1221 </figure> |
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1222 <note position="right" xlink:label="note-0035-02" xlink:href="note-0035-02a" xml:space="preserve">17. ſextĩ.</note> |
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1223 <note position="right" xlink:label="note-0035-03" xlink:href="note-0035-03a" xml:space="preserve">14. ſexti.</note> |
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1224 <note position="right" xlink:label="note-0035-04" xlink:href="note-0035-04a" xml:space="preserve">32. primi</note> |
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1225 <note position="right" xlink:label="note-0035-05" xlink:href="note-0035-05a" xml:space="preserve">4. primi.</note> |
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1226 <note position="right" xlink:label="note-0035-06" xlink:href="note-0035-06a" xml:space="preserve">28</note> |
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1227 <note position="left" xlink:label="note-0036-01" xlink:href="note-0036-01a" xml:space="preserve">29, primi</note> |
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1228 </div> |
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1229 <p> |
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1230 <s xml:id="echoid-s711" xml:space="preserve">Et quod in humido eſt ſurſum ſeretur ſecundum per-<lb/> |
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1231 <anchor type="note" xlink:label="note-0036-02a" xlink:href="note-0036-02"/> |
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1232 pendicularem, quæ per b ducta eſtipſi rt æquidiſtans.</s> |
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1233 <s xml:id="echoid-s712" xml:space="preserve">] <lb/>_Cur hoc quidem ſurſum, illud uero deorſum per lineam perpen-_ <lb/>_dicularem feratur, diximus ſupra in octauam primi libri buius. </s> |
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1234 <s xml:id="echoid-s713" xml:space="preserve">qua_ <lb/>_re neque in hac, neque in alijs, quæ ſequuntur, eadem iterare neceſſa_ <lb/>_rium exiſtimauimus._</s> |
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1235 <s xml:id="echoid-s714" xml:space="preserve"/> |
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1236 </p> |
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1237 <div xml:id="echoid-div51" type="float" level="2" n="8"> |
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1238 <note position="left" xlink:label="note-0036-02" xlink:href="note-0036-02a" xml:space="preserve">G</note> |
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1239 </div> |
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1240 </div> |
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1241 <div xml:id="echoid-div53" type="section" level="1" n="24"> |
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1242 <head xml:id="echoid-head29" xml:space="preserve">PROPOSITIO III.</head> |
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1243 <p> |
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1244 <s xml:id="echoid-s715" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli quando <lb/>axem habuerit minorem, quam ſeſquialterum <lb/>eius, quæ uſque ad axem, quamcunque propor-<lb/>tionem habens ad humidum in grauitate; </s> |
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1245 <s xml:id="echoid-s716" xml:space="preserve">demiſ-<lb/>ſa in humidum, ita ut baſis ipſius tota ſit in humi <lb/>do; </s> |
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1246 <s xml:id="echoid-s717" xml:space="preserve">& </s> |
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1247 <s xml:id="echoid-s718" xml:space="preserve">poſita inclinata, non manebit inclinata, ſed <lb/>ita reſtituetur, ut axis ipſius ſecundum perpendi <lb/>cularem fiat.</s> |
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1248 <s xml:id="echoid-s719" xml:space="preserve"/> |
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1249 </p> |
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1250 <p> |
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1251 <s xml:id="echoid-s720" xml:space="preserve">DEMITTATVR enim aliqua portio in humidum, <lb/>qualis dicca eſt: </s> |
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1252 <s xml:id="echoid-s721" xml:space="preserve">ſitq; </s> |
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1253 <s xml:id="echoid-s722" xml:space="preserve">ipſius baſis in humido: </s> |
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1254 <s xml:id="echoid-s723" xml:space="preserve">& </s> |
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1255 <s xml:id="echoid-s724" xml:space="preserve">ſecta ipſa <lb/>plano per axẽ, recto ad ſuperficiẽ humidi, ſit ſectio a p ol <lb/>rectanguli coniſectio: </s> |
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1256 <s xml:id="echoid-s725" xml:space="preserve">axis portionis, & </s> |
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1257 <s xml:id="echoid-s726" xml:space="preserve">ſectionis diame-<lb/>ter p f: </s> |
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1258 <s xml:id="echoid-s727" xml:space="preserve">ſuperficiei autem humidi ſectio ſit is. </s> |
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1259 <s xml:id="echoid-s728" xml:space="preserve">Quòd ſi incli <lb/>nata iaceat portio, non erit axis ſecundum perpendicula-<lb/>rem. </s> |
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1260 <s xml:id="echoid-s729" xml:space="preserve">ergo p f cum is angulos rectos non faciet. </s> |
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1261 <s xml:id="echoid-s730" xml:space="preserve">Itaque <lb/>ducatur linea quædã k ω æquidiſtans ipſi is; </s> |
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1262 <s xml:id="echoid-s731" xml:space="preserve">contingensq; <lb/></s> |
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1263 <s xml:id="echoid-s732" xml:space="preserve">ſectionẽ ap ol in o: </s> |
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1264 <s xml:id="echoid-s733" xml:space="preserve">& </s> |
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1265 <s xml:id="echoid-s734" xml:space="preserve">ſolidæ quidẽ magnitudinis a p o l <lb/>ſit r grauitatis centrum: </s> |
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1266 <s xml:id="echoid-s735" xml:space="preserve">ipſius autem i p o s centrum ſit |
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1267 <pb o="13" file="0037" n="37" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1268 b: </s> |
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1269 <s xml:id="echoid-s736" xml:space="preserve">iunctaq; </s> |
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1270 <s xml:id="echoid-s737" xml:space="preserve">br producatur: </s> |
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1271 <s xml:id="echoid-s738" xml:space="preserve">& </s> |
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1272 <s xml:id="echoid-s739" xml:space="preserve">ſit g centrum grauitatis <lb/>reliquæ figuræ isla. </s> |
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1273 <s xml:id="echoid-s740" xml:space="preserve">ſimiliter demonſtrabitur angulum <lb/>rok acutu eſ-<lb/> |
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1274 <anchor type="figure" xlink:label="fig-0037-01a" xlink:href="fig-0037-01"/> |
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1275 ſe: </s> |
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1276 <s xml:id="echoid-s741" xml:space="preserve">& </s> |
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1277 <s xml:id="echoid-s742" xml:space="preserve">perpendi <lb/>culare ab r ad <lb/>k ω ductam ca <lb/>dereinter k & </s> |
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1278 <s xml:id="echoid-s743" xml:space="preserve"><lb/>o, quæ ſit rt. <lb/></s> |
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1279 <s xml:id="echoid-s744" xml:space="preserve">ſi autem à pun <lb/>ctis g b ducan <lb/>tur ipſi r t æqui <lb/>diſtantes; </s> |
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1280 <s xml:id="echoid-s745" xml:space="preserve">pars <lb/>quidem ſolidæ <lb/>magnitudinis, <lb/>quæ in humido eſt, ſurſum feretur ſecundum perpendicu-<lb/>larem per g ductam: </s> |
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1281 <s xml:id="echoid-s746" xml:space="preserve">quæ autem extra humidum ſecundũ <lb/>perpendicularem per b deorſum feretur: </s> |
|
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1282 <s xml:id="echoid-s747" xml:space="preserve">& </s> |
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1283 <s xml:id="echoid-s748" xml:space="preserve">non manebit <lb/>ſolidum a p o l ſic habens in humido: </s> |
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1284 <s xml:id="echoid-s749" xml:space="preserve">ſed quod quidem <lb/>eſt ad a feretur ſurſum: </s> |
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1285 <s xml:id="echoid-s750" xml:space="preserve">quod autem ad l deorſum, donec <lb/>p f fiat ſecundum perpendicularem.</s> |
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1286 <s xml:id="echoid-s751" xml:space="preserve"/> |
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1287 </p> |
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1288 <div xml:id="echoid-div53" type="float" level="2" n="1"> |
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1289 <figure xlink:label="fig-0037-01" xlink:href="fig-0037-01a"> |
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1290 <image file="0037-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0037-01"/> |
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1291 </figure> |
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1292 </div> |
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1293 </div> |
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1294 <div xml:id="echoid-div55" type="section" level="1" n="25"> |
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1295 <head xml:id="echoid-head30" xml:space="preserve">PROPOSITIO IIII.</head> |
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1296 <p> |
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1297 <s xml:id="echoid-s752" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>fuerit humido leuior, & </s> |
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1298 <s xml:id="echoid-s753" xml:space="preserve">axem habuerit maiorẽ, <lb/>qnàm ſeſquialterum eius, quæ uſque ad axem: </s> |
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1299 <s xml:id="echoid-s754" xml:space="preserve">ſi <lb/>in grauitate ad humidum æqualis molis non mi-<lb/>norem proportionem habeat ea, quàm quadra-<lb/>tũ, quod fit ab exceſſu, quo axis maior eſt, quàm <lb/>ſeſquialter eius, quæ uſque ad axẽ, habet ad qua-<lb/>dratum, quod ab axe; </s> |
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1300 <s xml:id="echoid-s755" xml:space="preserve">demiſſa in humidum, ita |
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1301 <pb file="0038" n="38" rhead="ARCHIMEDIS"/> |
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1302 ut baſis ipſius humidum non contingat; </s> |
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1303 <s xml:id="echoid-s756" xml:space="preserve">& </s> |
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1304 <s xml:id="echoid-s757" xml:space="preserve">poſi-<lb/>ta inclinata, non manebit inclinata, ſed recta re-<lb/>ſtituetur.</s> |
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1305 <s xml:id="echoid-s758" xml:space="preserve"/> |
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1306 </p> |
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1307 <p> |
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1308 <s xml:id="echoid-s759" xml:space="preserve">SIT portio conoidis rectanguli, qualis dicta eſt: </s> |
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1309 <s xml:id="echoid-s760" xml:space="preserve">& </s> |
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1310 <s xml:id="echoid-s761" xml:space="preserve">de-<lb/>miſſa in humidum, ſi fieri poteſt, non ſitrecta; </s> |
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1311 <s xml:id="echoid-s762" xml:space="preserve">ſed inclina-<lb/>ta: </s> |
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1312 <s xml:id="echoid-s763" xml:space="preserve">ſecta autem ipſa plano per axem, recto ad ſuperficiem <lb/>humidi, portionis quidem ſectio ſit rectanguli coni ſectio <lb/>a p o l, axis portionis, & </s> |
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1313 <s xml:id="echoid-s764" xml:space="preserve">ſectionis diameter n o; </s> |
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1314 <s xml:id="echoid-s765" xml:space="preserve">& </s> |
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1315 <s xml:id="echoid-s766" xml:space="preserve">ſuper-<lb/>ficiei humidi ſectio ſit is. </s> |
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1316 <s xml:id="echoid-s767" xml:space="preserve">ſi igitur portio non eſt recta, nõ <lb/>faciet n o cum is angulos æquales. </s> |
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1317 <s xml:id="echoid-s768" xml:space="preserve">Ducatur k ω contin-<lb/>gens rectanguli coni ſectionem in p; </s> |
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1318 <s xml:id="echoid-s769" xml:space="preserve">æquidiſtanſq; </s> |
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1319 <s xml:id="echoid-s770" xml:space="preserve">ipſi <lb/>is: </s> |
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1320 <s xml:id="echoid-s771" xml:space="preserve">& </s> |
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1321 <s xml:id="echoid-s772" xml:space="preserve">à puncto p ipſi o n æquidiſtans ducatur p f. </s> |
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1322 <s xml:id="echoid-s773" xml:space="preserve">Itaque <lb/>ſumantur centra grauitatum: </s> |
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1323 <s xml:id="echoid-s774" xml:space="preserve">& </s> |
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1324 <s xml:id="echoid-s775" xml:space="preserve">ſolidi quidem a p o l cen <lb/>trum ſit r; </s> |
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1325 <s xml:id="echoid-s776" xml:space="preserve">eius autem, quod intra humidum, centrum b: <lb/></s> |
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1326 <s xml:id="echoid-s777" xml:space="preserve">iunctaq; </s> |
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1327 <s xml:id="echoid-s778" xml:space="preserve">b r pro-<lb/> |
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1328 <anchor type="figure" xlink:label="fig-0038-01a" xlink:href="fig-0038-01"/> |
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1329 ducatur ad g, ut <lb/>g ſit centrũ graui <lb/>tatis ſolidi, quod <lb/>extra humidum. <lb/></s> |
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1330 <s xml:id="echoid-s779" xml:space="preserve">Quoniam igitur <lb/>n o ipſius quidem <lb/>r o ſeſquialtera ẽ; </s> |
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1331 <s xml:id="echoid-s780" xml:space="preserve"><lb/>eius autẽ, quæ uſ-<lb/>que ad axẽ maior, <lb/>quàm ſeſquialte-<lb/>ra: </s> |
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1332 <s xml:id="echoid-s781" xml:space="preserve">patet r o maio <lb/> |
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1333 <anchor type="note" xlink:label="note-0038-01a" xlink:href="note-0038-01"/> |
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1334 rẽ eſſe, quàm quæ <lb/>uſq; </s> |
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1335 <s xml:id="echoid-s782" xml:space="preserve">ad axẽ. </s> |
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1336 <s xml:id="echoid-s783" xml:space="preserve">Sit ei, <lb/> |
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1337 <anchor type="note" xlink:label="note-0038-02a" xlink:href="note-0038-02"/> |
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1338 quæ uſque ad axẽ <lb/>æqualis r h: </s> |
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1339 <s xml:id="echoid-s784" xml:space="preserve">& </s> |
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1340 <s xml:id="echoid-s785" xml:space="preserve">o h dupla ipſius h m. </s> |
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1341 <s xml:id="echoid-s786" xml:space="preserve">quòd cū n o ipſius r o <lb/> |
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1342 <anchor type="note" xlink:label="note-0038-03a" xlink:href="note-0038-03"/> |
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1343 ſeſquialtera ſit; </s> |
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1344 <s xml:id="echoid-s787" xml:space="preserve">item q; </s> |
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1345 <s xml:id="echoid-s788" xml:space="preserve">m o ipſius o h: </s> |
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1346 <s xml:id="echoid-s789" xml:space="preserve">& </s> |
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1347 <s xml:id="echoid-s790" xml:space="preserve">reliqua n m reli <lb/> |
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1348 <anchor type="note" xlink:label="note-0038-04a" xlink:href="note-0038-04"/> |
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1349 quæ r h ſeſquialtera erit. </s> |
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1350 <s xml:id="echoid-s791" xml:space="preserve">ergo axis tanto maior eſt, quàm |
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1351 <pb o="14" file="0039" n="39" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1352 ſeſquialter eius, quæ uſque ad axem, quanta eſt linea m o. <lb/></s> |
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1353 <s xml:id="echoid-s792" xml:space="preserve">Ponebatur autem portio ad humidum æqualis molis non <lb/>minorem in grauitate proportionem habere, quam qua-<lb/>dratum, quod fit ab exceſſu, quo axis eſt maior, quam ſeſ-<lb/>quialter eius, quæ uſque ad axem, ad quadratum, quod ab <lb/>axe. </s> |
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1354 <s xml:id="echoid-s793" xml:space="preserve">quare conſtat portionem ad humidum in grauitate <lb/>non minorem proportionem habere, quàm quadratum li <lb/>neæ m o ad quadratum ipſius n o. </s> |
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1355 <s xml:id="echoid-s794" xml:space="preserve">Sed quam proportio-<lb/>nem habet portio ad humidum in grauitate, eandem por-<lb/>tio ipſius demerla habet ad totam portionem: </s> |
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1356 <s xml:id="echoid-s795" xml:space="preserve">hoc enim <lb/> |
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1357 <anchor type="note" xlink:label="note-0039-01a" xlink:href="note-0039-01"/> |
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1358 ſupra demonſtratum eſt: </s> |
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1359 <s xml:id="echoid-s796" xml:space="preserve">& </s> |
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1360 <s xml:id="echoid-s797" xml:space="preserve">quam proportionem habet de <lb/> |
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1361 <anchor type="note" xlink:label="note-0039-02a" xlink:href="note-0039-02"/> |
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1362 merſa portio ad totam, eam quadratum p f habet ad n o <lb/>quadratum: </s> |
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1363 <s xml:id="echoid-s798" xml:space="preserve">cum demonſtratum ſit in iis, quæ de conoidi <lb/>bus, & </s> |
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1364 <s xml:id="echoid-s799" xml:space="preserve">ſphæroidibus, ſi à rectangulo conoide duæ portio-<lb/>nes planis quomodocunque ductis abſcindantur, portio-<lb/>nes inter ſe eandem habere proportionem, qnàm quadra-<lb/>ta, quæ ab ipſorum axibus conſtituuntur. </s> |
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1365 <s xml:id="echoid-s800" xml:space="preserve">non minorem <lb/>ergo proportionẽ habet quadratum pf ad quadratũ n o, <lb/>quàm quadratum m o ad idem n o quadratum. </s> |
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1366 <s xml:id="echoid-s801" xml:space="preserve">quare <lb/> |
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1367 <anchor type="note" xlink:label="note-0039-03a" xlink:href="note-0039-03"/> |
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1368 p f non eſt minor ipſa m o; </s> |
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1369 <s xml:id="echoid-s802" xml:space="preserve">nec b p item minor h o. </s> |
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1370 <s xml:id="echoid-s803" xml:space="preserve">Si <lb/> |
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1371 <anchor type="note" xlink:label="note-0039-04a" xlink:href="note-0039-04"/> |
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1372 igitur ab h ducatur linea ad rectos angulos ipſi n o, coi-<lb/> |
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1373 <anchor type="note" xlink:label="note-0039-05a" xlink:href="note-0039-05"/> |
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1374 bit cum b p, atque inter b, & </s> |
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1375 <s xml:id="echoid-s804" xml:space="preserve">p cadet. </s> |
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1376 <s xml:id="echoid-s805" xml:space="preserve">coeat in t. </s> |
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1377 <s xml:id="echoid-s806" xml:space="preserve">& </s> |
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1378 <s xml:id="echoid-s807" xml:space="preserve">quo <lb/> |
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1379 <anchor type="note" xlink:label="note-0039-06a" xlink:href="note-0039-06"/> |
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1380 niam p f quidem æquidiſtans eſt diametro, h t autem ad <lb/>diametrum perpendicularis; </s> |
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1381 <s xml:id="echoid-s808" xml:space="preserve">& </s> |
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1382 <s xml:id="echoid-s809" xml:space="preserve">r h æqualis ei, quæ uſque <lb/>ad axem: </s> |
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1383 <s xml:id="echoid-s810" xml:space="preserve">ducta linea ab r ad t & </s> |
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1384 <s xml:id="echoid-s811" xml:space="preserve">producta angulos rectos <lb/>faciet cum linea ſectionem in puncto p contingente. </s> |
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1385 <s xml:id="echoid-s812" xml:space="preserve">qua-<lb/>re & </s> |
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1386 <s xml:id="echoid-s813" xml:space="preserve">cum is, & </s> |
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1387 <s xml:id="echoid-s814" xml:space="preserve">cum humidi ſuperficie, quæ per is tran-<lb/>ſit. </s> |
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1388 <s xml:id="echoid-s815" xml:space="preserve">Itaque ſi per b g puncta lineæ ipſi r t æquidiſtantes du <lb/>cantur, angulos rectos facient cum ſuperficie humidi: </s> |
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1389 <s xml:id="echoid-s816" xml:space="preserve">& </s> |
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1390 <s xml:id="echoid-s817" xml:space="preserve"><lb/>quod quidem in humido eſt ſolidum conoidis feretur ſur-<lb/>ſum ſecundum eam, quæ per b ducta fuerit ipſi r t æquidi <lb/>ſtans: </s> |
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1391 <s xml:id="echoid-s818" xml:space="preserve">quod autem extra humidum, ſecundum eam, quæ <lb/>per g deorſum feretur. </s> |
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1392 <s xml:id="echoid-s819" xml:space="preserve">atque hoc tandiu fiet, quoad co-<lb/>noides rectum conſtituatur.</s> |
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1393 <s xml:id="echoid-s820" xml:space="preserve"/> |
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1394 </p> |
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1395 <div xml:id="echoid-div55" type="float" level="2" n="1"> |
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1396 <figure xlink:label="fig-0038-01" xlink:href="fig-0038-01a"> |
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1397 <image file="0038-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0038-01"/> |
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1398 </figure> |
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1399 <note position="left" xlink:label="note-0038-01" xlink:href="note-0038-01a" xml:space="preserve">10. quinti</note> |
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1400 <note position="left" xlink:label="note-0038-02" xlink:href="note-0038-02a" xml:space="preserve">A</note> |
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1401 <note position="left" xlink:label="note-0038-03" xlink:href="note-0038-03a" xml:space="preserve">B</note> |
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1402 <note position="right" xlink:label="note-0038-04" xlink:href="note-0038-04a" xml:space="preserve">19. quinti</note> |
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1403 <note position="right" xlink:label="note-0039-01" xlink:href="note-0039-01a" xml:space="preserve">C</note> |
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1404 <note position="right" xlink:label="note-0039-02" xlink:href="note-0039-02a" xml:space="preserve">D</note> |
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1405 <note position="right" xlink:label="note-0039-03" xlink:href="note-0039-03a" xml:space="preserve">E</note> |
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1406 <note position="right" xlink:label="note-0039-04" xlink:href="note-0039-04a" xml:space="preserve">F</note> |
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1407 <note position="right" xlink:label="note-0039-05" xlink:href="note-0039-05a" xml:space="preserve">G</note> |
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1408 <note position="right" xlink:label="note-0039-06" xlink:href="note-0039-06a" xml:space="preserve">H</note> |
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1409 </div> |
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1410 <pb file="0040" n="40" rhead="ARCHIMEDIS"/> |
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1411 </div> |
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1412 <div xml:id="echoid-div57" type="section" level="1" n="26"> |
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1413 <head xml:id="echoid-head31" xml:space="preserve">COMMENTARIVS.</head> |
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1414 <p style="it"> |
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1415 <s xml:id="echoid-s821" xml:space="preserve">_Sit ei, quæ uſque ad axem æqualis r h.</s> |
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1416 <s xml:id="echoid-s822" xml:space="preserve">]_ Ita legendum eſt, <lb/> |
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1417 <anchor type="note" xlink:label="note-0040-01a" xlink:href="note-0040-01"/> |
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1418 non r m, ut tranſlatio habet, quod ex ijs, quæ ſequuntur, manifeſte <lb/>conſtare poteſt.</s> |
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1419 <s xml:id="echoid-s823" xml:space="preserve"/> |
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1420 </p> |
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1421 <div xml:id="echoid-div57" type="float" level="2" n="1"> |
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1422 <note position="left" xlink:label="note-0040-01" xlink:href="note-0040-01a" xml:space="preserve">A</note> |
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1423 </div> |
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1424 <p style="it"> |
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1425 <s xml:id="echoid-s824" xml:space="preserve">_Et oh dupla ipſius h m.</s> |
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1426 <s xml:id="echoid-s825" xml:space="preserve">]_ In tranſlatione mendoſe legeba-<lb/> |
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1427 <anchor type="note" xlink:label="note-0040-02a" xlink:href="note-0040-02"/> |
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1428 tur, on dupla ipſius rm.</s> |
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1429 <s xml:id="echoid-s826" xml:space="preserve"/> |
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|
1430 </p> |
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1431 <div xml:id="echoid-div58" type="float" level="2" n="2"> |
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1432 <note position="left" xlink:label="note-0040-02" xlink:href="note-0040-02a" xml:space="preserve">B</note> |
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1433 </div> |
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1434 <p> |
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1435 <s xml:id="echoid-s827" xml:space="preserve">Hoc enim ſupra demonſtratum eſt.</s> |
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1436 <s xml:id="echoid-s828" xml:space="preserve">] _In prima huius_.</s> |
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1437 <s xml:id="echoid-s829" xml:space="preserve"/> |
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|
1438 </p> |
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1439 <note position="left" xml:space="preserve">C</note> |
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|
1440 <p> |
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1441 <s xml:id="echoid-s830" xml:space="preserve">Et quam proportionem habet demerſa portio ad totã, <lb/> |
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1442 <anchor type="note" xlink:label="note-0040-04a" xlink:href="note-0040-04"/> |
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1443 eam quadratum p f habet ad n o quadratum.</s> |
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1444 <s xml:id="echoid-s831" xml:space="preserve">] _Hoc loco in_ <lb/>_tranſlatione non nulli deſider abantur, quænos reſtituimus. </s> |
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1445 <s xml:id="echoid-s832" xml:space="preserve">Illud au_ <lb/>_tem ab Archimede demonſtratum eſt in libro de conoidibus & </s> |
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1446 <s xml:id="echoid-s833" xml:space="preserve">ſphæ_ <lb/>_roidibus propoſitione_ 26.</s> |
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1447 <s xml:id="echoid-s834" xml:space="preserve"/> |
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|
1448 </p> |
|
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1449 <div xml:id="echoid-div59" type="float" level="2" n="3"> |
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1450 <note position="left" xlink:label="note-0040-04" xlink:href="note-0040-04a" xml:space="preserve">D</note> |
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1451 </div> |
|
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1452 <p style="it"> |
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1453 <s xml:id="echoid-s835" xml:space="preserve">_Quare p f non eſt minor ipſa m o.</s> |
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1454 <s xml:id="echoid-s836" xml:space="preserve">]_ Nam ex decima quinti <lb/> |
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1455 <anchor type="note" xlink:label="note-0040-05a" xlink:href="note-0040-05"/> |
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|
1456 ſequitur, quadratum p f non eſſe minus quadrato m o. </s> |
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1457 <s xml:id="echoid-s837" xml:space="preserve">quare neque <lb/>linea p f minor erit linea m o ex 22 ſexti.</s> |
|
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1458 <s xml:id="echoid-s838" xml:space="preserve"/> |
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|
1459 </p> |
|
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|
1460 <div xml:id="echoid-div60" type="float" level="2" n="4"> |
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|
1461 <note position="left" xlink:label="note-0040-05" xlink:href="note-0040-05a" xml:space="preserve">E</note> |
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|
1462 </div> |
|
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|
1463 <p style="it"> |
|
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1464 <s xml:id="echoid-s839" xml:space="preserve">_Nec b p item minor h o.</s> |
|
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1465 <s xml:id="echoid-s840" xml:space="preserve">]_ Eſt enim ut p f ad p b, ita m o, <lb/> |
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|
1466 <anchor type="note" xlink:label="note-0040-06a" xlink:href="note-0040-06"/> |
|
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|
1467 ad h o & </s> |
|
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|
1468 <s xml:id="echoid-s841" xml:space="preserve">permutando, ut p f ad mo, ita b p, ad b o. </s> |
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|
1469 <s xml:id="echoid-s842" xml:space="preserve">ſed p f non <lb/>est minor m o, ut oſtenſiim cst. </s> |
|
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1470 <s xml:id="echoid-s843" xml:space="preserve">ergo neque b p ipſa h o minor erit.</s> |
|
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|
1471 <s xml:id="echoid-s844" xml:space="preserve"/> |
|
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|
1472 </p> |
|
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|
1473 <div xml:id="echoid-div61" type="float" level="2" n="5"> |
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|
1474 <note position="left" xlink:label="note-0040-06" xlink:href="note-0040-06a" xml:space="preserve">F</note> |
|
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|
1475 </div> |
|
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|
1476 <note position="left" xml:space="preserve">14. quinti</note> |
|
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|
1477 <p> |
|
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|
1478 <s xml:id="echoid-s845" xml:space="preserve">Si igitur ab h <lb/> |
|
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|
1479 <anchor type="note" xlink:label="note-0040-08a" xlink:href="note-0040-08"/> |
|
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|
1480 <anchor type="figure" xlink:label="fig-0040-01a" xlink:href="fig-0040-01"/> |
|
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|
1481 ducatur linea ad <lb/>rectos angulos ip <lb/>ſi n o, coibit cum <lb/>b p, atque inter <lb/>b & </s> |
|
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|
1482 <s xml:id="echoid-s846" xml:space="preserve">p cadet.</s> |
|
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|
1483 <s xml:id="echoid-s847" xml:space="preserve">] <lb/>_Corruptus erat hic_ <lb/>_locus in tranſlatio-_ <lb/>_ne. </s> |
|
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|
1484 <s xml:id="echoid-s848" xml:space="preserve">Illud uero ita de-_ <lb/>_monſtr abitur. </s> |
|
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|
1485 <s xml:id="echoid-s849" xml:space="preserve">Quo-_ <lb/>_niam p f non eſt mi-_ <lb/>_nor o m, nec p b ip-_ <lb/>_ſa h o; </s> |
|
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|
1486 <s xml:id="echoid-s850" xml:space="preserve">ſi ponatur p f_ <lb/>_æqualis o m; </s> |
|
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|
1487 <s xml:id="echoid-s851" xml:space="preserve">& </s> |
|
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|
1488 <s xml:id="echoid-s852" xml:space="preserve">p b,_ <lb/>_ipſi h o æqualis erit._</s> |
|
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|
1489 <s xml:id="echoid-s853" xml:space="preserve"> |
|
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|
1490 <pb o="15" file="0041" n="41" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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|
1491 _quare per o ductaipſi al æquidiſtans cadet extra ſectionem ex 17._ <lb/></s> |
|
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|
1492 <s xml:id="echoid-s854" xml:space="preserve">_primi conicorum: </s> |
|
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|
1493 <s xml:id="echoid-s855" xml:space="preserve">& </s> |
|
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|
1494 <s xml:id="echoid-s856" xml:space="preserve">cum b p producta coibit inſra p. </s> |
|
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|
1495 <s xml:id="echoid-s857" xml:space="preserve">ergò & </s> |
|
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|
1496 <s xml:id="echoid-s858" xml:space="preserve">per-_ <lb/>_pendicularis ducta per b cum eadem infra b coibit, at que inter b &_</s> |
|
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|
1497 <s xml:id="echoid-s859" xml:space="preserve"> <lb/>_p neceſſario cadet. </s> |
|
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|
1498 <s xml:id="echoid-s860" xml:space="preserve">multo autem magis illud idem ſequetur, ſi pona-_ <lb/>_mus pf ipſa om maiorem eſſe._</s> |
|
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|
1499 <s xml:id="echoid-s861" xml:space="preserve"/> |
|
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|
1500 </p> |
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1501 <div xml:id="echoid-div62" type="float" level="2" n="6"> |
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1502 <note position="left" xlink:label="note-0040-08" xlink:href="note-0040-08a" xml:space="preserve">G</note> |
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1503 <figure xlink:label="fig-0040-01" xlink:href="fig-0040-01a"> |
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1504 <image file="0040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0040-01"/> |
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1505 </figure> |
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1506 </div> |
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1507 <p> |
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1508 <s xml:id="echoid-s862" xml:space="preserve">Et quoniam p f quidem æquidiſtans eſt diametro, htau <lb/> |
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1509 <anchor type="note" xlink:label="note-0041-01a" xlink:href="note-0041-01"/> |
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1510 tem ad diametrum perpendicularis; </s> |
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1511 <s xml:id="echoid-s863" xml:space="preserve">& </s> |
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1512 <s xml:id="echoid-s864" xml:space="preserve">rh æqualis ei, quæ <lb/>uſque ad axem, ducta linea ab r ad e, & </s> |
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1513 <s xml:id="echoid-s865" xml:space="preserve">producta angulos <lb/>rectos facere cum linea ſectionem in p contingente.</s> |
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1514 <s xml:id="echoid-s866" xml:space="preserve">] <lb/>_Hoc ſuperius à nobis demonſtratum eſt in ſecundam buin<unsure/>s._</s> |
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1515 <s xml:id="echoid-s867" xml:space="preserve"/> |
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1516 </p> |
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1517 <div xml:id="echoid-div63" type="float" level="2" n="7"> |
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1518 <note position="right" xlink:label="note-0041-01" xlink:href="note-0041-01a" xml:space="preserve">H</note> |
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1519 </div> |
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1520 </div> |
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1521 <div xml:id="echoid-div65" type="section" level="1" n="27"> |
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1522 <head xml:id="echoid-head32" xml:space="preserve">PROPOSITIO V.</head> |
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1523 <p> |
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1524 <s xml:id="echoid-s868" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>leuior humido axem habuerit maiorem, quàm <lb/>ſeſquialterum eius, quæ uſque ad axem; </s> |
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1525 <s xml:id="echoid-s869" xml:space="preserve">ſi ad hu-<lb/>midum in grauitate non maiorem proportionẽ <lb/>habeat, quàm exceſſus, quo quadratum quod ſit <lb/>ab axe maius eſt quadrato, quod ab exceſſu, quo <lb/>axis maior eſt, quàm ſeſquialter eius, quæ uſque <lb/>ad axem, ad quadratum, quod ab axe: </s> |
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1526 <s xml:id="echoid-s870" xml:space="preserve">demiſſa in <lb/>humidum, ita ut baſis ipſius tota ſit in humido; <lb/></s> |
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1527 <s xml:id="echoid-s871" xml:space="preserve">& </s> |
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1528 <s xml:id="echoid-s872" xml:space="preserve">poſita inclinata non manebit inclinata, ſed re-<lb/>ſtituetur ita, ut axis ipſius ſecundum perpendi-<lb/>cularem fiat.</s> |
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1529 <s xml:id="echoid-s873" xml:space="preserve"/> |
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1530 </p> |
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1531 <p> |
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1532 <s xml:id="echoid-s874" xml:space="preserve">DEMITTATVR enim in humidum portio aliqna, <lb/>qualis dicta eſt: </s> |
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1533 <s xml:id="echoid-s875" xml:space="preserve">& </s> |
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1534 <s xml:id="echoid-s876" xml:space="preserve">ſit baſis ipſius tota in humido. </s> |
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1535 <s xml:id="echoid-s877" xml:space="preserve">Secta au-<lb/>tem ipſa plano per axem, recto ad ſuperſiciem humidi, erit <lb/>ſectio rectanguli coniſectio, quæ ſit apol: </s> |
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1536 <s xml:id="echoid-s878" xml:space="preserve">axis portionis, |
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1537 <pb file="0042" n="42" rhead="ARCHIMEDIS"/> |
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1538 & </s> |
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1539 <s xml:id="echoid-s879" xml:space="preserve">ſectionis diameter no: </s> |
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1540 <s xml:id="echoid-s880" xml:space="preserve">ſuperſiciei autem humidi ſectio <lb/>ſit is. </s> |
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1541 <s xml:id="echoid-s881" xml:space="preserve">Quoniam igitur axis non eſt ſecundum perpendicu <lb/>larem; </s> |
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1542 <s xml:id="echoid-s882" xml:space="preserve">ipſa no cum is non faciet angulos æquales. </s> |
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1543 <s xml:id="echoid-s883" xml:space="preserve">Du-<lb/>catur k ω contingens ſectionem apol in p; </s> |
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1544 <s xml:id="echoid-s884" xml:space="preserve">atque ipſi is <lb/>æquidiſtans: </s> |
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1545 <s xml:id="echoid-s885" xml:space="preserve">per p autem ducatur p f æquidiſtās ipſi n o: <lb/></s> |
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1546 <s xml:id="echoid-s886" xml:space="preserve">& </s> |
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1547 <s xml:id="echoid-s887" xml:space="preserve">ſumantur grauitatum centra: </s> |
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1548 <s xml:id="echoid-s888" xml:space="preserve">ſitq; </s> |
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1549 <s xml:id="echoid-s889" xml:space="preserve">ipſius a p o l ſolidi <lb/>centrum r; </s> |
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1550 <s xml:id="echoid-s890" xml:space="preserve">eius quod extra humidum ſit b: </s> |
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1551 <s xml:id="echoid-s891" xml:space="preserve">& </s> |
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1552 <s xml:id="echoid-s892" xml:space="preserve">iuncta br <lb/>producatur adg, <lb/> |
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1553 <anchor type="figure" xlink:label="fig-0042-01a" xlink:href="fig-0042-01"/> |
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1554 quodſit centrum <lb/>grauitatis ſolidi ĩ <lb/>humido demerſi: <lb/></s> |
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1555 <s xml:id="echoid-s893" xml:space="preserve">ſumatur præterea <lb/>r h æ qualis ei, quæ <lb/>uſque ad axẽ: </s> |
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1556 <s xml:id="echoid-s894" xml:space="preserve">o h <lb/>autem dupla ipſi-<lb/>us h m; </s> |
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1557 <s xml:id="echoid-s895" xml:space="preserve">& </s> |
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1558 <s xml:id="echoid-s896" xml:space="preserve">alia fiãt, <lb/>ſicuti ſuperius di-<lb/>ctum eſt. </s> |
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1559 <s xml:id="echoid-s897" xml:space="preserve">Itaque <lb/>cum portio ad hu <lb/>midum in grauita <lb/>te non maiorem <lb/>proportionem ha <lb/>bere ponatur, quã <lb/>exceſſus, quo quadratum n o excedit quadratum m o, ad <lb/>ipſum n o quadratum: </s> |
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1560 <s xml:id="echoid-s898" xml:space="preserve">& </s> |
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1561 <s xml:id="echoid-s899" xml:space="preserve">quam proportionem in grauita <lb/>te portio habet ad humidum æqualis molis, eandem ha-<lb/>beat magnitudo portionis demerſa ad totam portio-<lb/>nem, quod demonſtratum eſt in prima propoſitione: </s> |
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1562 <s xml:id="echoid-s900" xml:space="preserve"><lb/>magnitudo demerſa non maiorem proportionem ha-<lb/> |
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1563 <anchor type="note" xlink:label="note-0042-01a" xlink:href="note-0042-01"/> |
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1564 bebit ad totam portionem, quàm ſit dicta illa propor-<lb/>portio. </s> |
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1565 <s xml:id="echoid-s901" xml:space="preserve">quare non maiorem proportionem habet tota <lb/> |
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1566 <anchor type="note" xlink:label="note-0042-02a" xlink:href="note-0042-02"/> |
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1567 portio ad eam quæ eſt extra humidum, quàm quadratum <lb/>no ad quadratum m o. </s> |
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1568 <s xml:id="echoid-s902" xml:space="preserve">habet autem tota portio ad eam, <lb/> |
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1569 <anchor type="note" xlink:label="note-0042-03a" xlink:href="note-0042-03"/> |
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1570 quæ extra humidum proportionem eandem, quam qua- |
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1571 <pb o="16" file="0043" n="43" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1572 dratum n o ad quadratum p f. </s> |
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1573 <s xml:id="echoid-s903" xml:space="preserve">quadratum igitur n o ad <lb/>quadratum p f non maiorem proportionem habet, quàm <lb/>ad quadratum m o. </s> |
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1574 <s xml:id="echoid-s904" xml:space="preserve">ex quo eſſicitur, ut p f non ſit minor <lb/> |
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1575 <anchor type="note" xlink:label="note-0043-01a" xlink:href="note-0043-01"/> |
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1576 ipſa o m; </s> |
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1577 <s xml:id="echoid-s905" xml:space="preserve">neque p b ipſa o h. </s> |
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1578 <s xml:id="echoid-s906" xml:space="preserve">quæ ergo ab h ducitur ad <lb/> |
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1579 <anchor type="note" xlink:label="note-0043-02a" xlink:href="note-0043-02"/> |
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1580 rectos angulos ipſi n o, coibit cum b p inter p & </s> |
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1581 <s xml:id="echoid-s907" xml:space="preserve">b. </s> |
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1582 <s xml:id="echoid-s908" xml:space="preserve">co-<lb/>eatin t. </s> |
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1583 <s xml:id="echoid-s909" xml:space="preserve">& </s> |
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1584 <s xml:id="echoid-s910" xml:space="preserve">quoniam in rectanguli coniſectione p f eſt æqui <lb/>diſtans diametro n o; </s> |
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1585 <s xml:id="echoid-s911" xml:space="preserve">h t autem ad diametrum perpẽ-<lb/>dicularis: </s> |
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1586 <s xml:id="echoid-s912" xml:space="preserve">& </s> |
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1587 <s xml:id="echoid-s913" xml:space="preserve">r h æqualis ei, quæ uſque ad axem: </s> |
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1588 <s xml:id="echoid-s914" xml:space="preserve">conſtat r t <lb/>productam ſacere angulos rectos cum ipſa k p ω. </s> |
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1589 <s xml:id="echoid-s915" xml:space="preserve">quare <lb/>& </s> |
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1590 <s xml:id="echoid-s916" xml:space="preserve">cum is. </s> |
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1591 <s xml:id="echoid-s917" xml:space="preserve">ergo rt perpendicularis eſt ad ſuperſiciem hu <lb/>midi. </s> |
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1592 <s xml:id="echoid-s918" xml:space="preserve">et ſi per b g puncta ducantur æquidiſtantes ipſirt, <lb/>ad ſuperſiciem humidi perpendicular es erunt. </s> |
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1593 <s xml:id="echoid-s919" xml:space="preserve">portio igi <lb/>tur, qnæ eſt extra humidum, deorſum in humidum feretur <lb/>ſecundum perpendicularem per b ductam; </s> |
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1594 <s xml:id="echoid-s920" xml:space="preserve">quæ uero in-<lb/>tra humidum ſecundum perpendicularem per g ſurſum <lb/>feretur: </s> |
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1595 <s xml:id="echoid-s921" xml:space="preserve">& </s> |
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1596 <s xml:id="echoid-s922" xml:space="preserve">non manebit ſolida portio a p o l, ſedintra hu <lb/>midum mouebitur, donecutique ipſa n o ſecundum per-<lb/>pendicularem ſiat.</s> |
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1597 <s xml:id="echoid-s923" xml:space="preserve"/> |
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1598 </p> |
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1599 <div xml:id="echoid-div65" type="float" level="2" n="1"> |
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1600 <figure xlink:label="fig-0042-01" xlink:href="fig-0042-01a"> |
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1601 <image file="0042-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0042-01"/> |
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1602 </figure> |
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1603 <note position="left" xlink:label="note-0042-01" xlink:href="note-0042-01a" xml:space="preserve">11. quin-<lb/>ti.</note> |
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1604 <note position="left" xlink:label="note-0042-02" xlink:href="note-0042-02a" xml:space="preserve">A</note> |
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1605 <note position="left" xlink:label="note-0042-03" xlink:href="note-0042-03a" xml:space="preserve">B</note> |
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1606 <note position="right" xlink:label="note-0043-01" xlink:href="note-0043-01a" xml:space="preserve">C</note> |
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1607 <note position="right" xlink:label="note-0043-02" xlink:href="note-0043-02a" xml:space="preserve">D</note> |
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1608 </div> |
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1609 </div> |
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1610 <div xml:id="echoid-div67" type="section" level="1" n="28"> |
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1611 <head xml:id="echoid-head33" xml:space="preserve">COMMENTARIVS.</head> |
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1612 <p style="it"> |
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1613 <s xml:id="echoid-s924" xml:space="preserve">_Quare non maiorem proportionem habet tota portio_ <lb/> |
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1614 <anchor type="note" xlink:label="note-0043-03a" xlink:href="note-0043-03"/> |
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1615 _ad eam, quæ eſt extra humidum, quam quadratum n o ad_ <lb/>_quadratum m o]_ cum enim magnitudo portionis in bumidum <lb/>demerſa ad totam portionem non maiorem proportionem babeat, <lb/>quàm exceſſus, quo quadratum n o excedit quadratum m o, ad ip-<lb/>ſum no quadratum: </s> |
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1616 <s xml:id="echoid-s925" xml:space="preserve">conuertendo per uigeſimáſextam quinti ele-<lb/>mentorum ex traditione Campani, tota portio ad magnitudinem de <lb/>merſam non minorem proportionem babebit, quàm quadratum n o <lb/>ad exceſſum, quo ipſum quadratum no excedit quadratum m o. </s> |
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1617 <s xml:id="echoid-s926" xml:space="preserve">In <lb/>telligatur portio, quæ extra bumidum, magnitudo prima: </s> |
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1618 <s xml:id="echoid-s927" xml:space="preserve">quæ in bu <lb/>mido demerſa est, ſecunda: </s> |
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1619 <s xml:id="echoid-s928" xml:space="preserve">tertia autem magnitudo ſit quadratum <lb/>mo: </s> |
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1620 <s xml:id="echoid-s929" xml:space="preserve">& </s> |
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1621 <s xml:id="echoid-s930" xml:space="preserve">exceſſus, quo quadratum n o excedit quadratum m o ſit <lb/>quarta. </s> |
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1622 <s xml:id="echoid-s931" xml:space="preserve">ex his igitur magnitudinibus, primæ & </s> |
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1623 <s xml:id="echoid-s932" xml:space="preserve">ſecundæ ad ſecun- |
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1624 <pb file="0044" n="44" rhead="ARCHIME DIS"/> |
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1625 dam non minor est proportio, quàm tertiæ & </s> |
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1626 <s xml:id="echoid-s933" xml:space="preserve">quartæ ad quartam; <lb/></s> |
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1627 <s xml:id="echoid-s934" xml:space="preserve">est enim quadratum m o unà cum exceſſu, quo quadratum n o exce <lb/>dit quadratum m o æquale ipſi n o quadrato. </s> |
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1628 <s xml:id="echoid-s935" xml:space="preserve">quare per conuerſio <lb/>nem rationis ex 30 eiuſdem, primæ & </s> |
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1629 <s xml:id="echoid-s936" xml:space="preserve">ſecundæ ad primam non ma-<lb/>ior proportio erit, quàm tertiæ & </s> |
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1630 <s xml:id="echoid-s937" xml:space="preserve">quartæ ad tertiam: </s> |
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1631 <s xml:id="echoid-s938" xml:space="preserve">& </s> |
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1632 <s xml:id="echoid-s939" xml:space="preserve">idcirco to-<lb/>ta portio ad portionem eam, quæ est extra bumidum non maiorem <lb/>proportionem babebit, quàm quadratum n o ad quadratum mo. </s> |
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1633 <s xml:id="echoid-s940" xml:space="preserve"><lb/>quod demonstrandum proponebatur.</s> |
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1634 <s xml:id="echoid-s941" xml:space="preserve"/> |
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1635 </p> |
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1636 <div xml:id="echoid-div67" type="float" level="2" n="1"> |
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1637 <note position="right" xlink:label="note-0043-03" xlink:href="note-0043-03a" xml:space="preserve">A</note> |
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1638 </div> |
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1639 <p> |
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1640 <s xml:id="echoid-s942" xml:space="preserve">Habet autem tota portio ad eam, quæ extra humidum <lb/> |
|
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1641 <anchor type="note" xlink:label="note-0044-01a" xlink:href="note-0044-01"/> |
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1642 proportionem eandem, quam quadratum n o ad quadra <lb/>tum p f.</s> |
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1643 <s xml:id="echoid-s943" xml:space="preserve">] _Ex uigeſimaſexta libri de conoidibus, & </s> |
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1644 <s xml:id="echoid-s944" xml:space="preserve">ſpbæroi-_ <lb/>_dibus._</s> |
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1645 <s xml:id="echoid-s945" xml:space="preserve"/> |
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1646 </p> |
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1647 <div xml:id="echoid-div68" type="float" level="2" n="2"> |
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1648 <note position="left" xlink:label="note-0044-01" xlink:href="note-0044-01a" xml:space="preserve">B</note> |
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1649 </div> |
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1650 <p> |
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1651 <s xml:id="echoid-s946" xml:space="preserve">Ex quo eſſicitur, ut p ſ non ſit minor ipſa o m; </s> |
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1652 <s xml:id="echoid-s947" xml:space="preserve">neque <lb/> |
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1653 <anchor type="note" xlink:label="note-0044-02a" xlink:href="note-0044-02"/> |
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1654 pb ipſa o h.</s> |
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1655 <s xml:id="echoid-s948" xml:space="preserve">] _Sequitur illud ex decima & </s> |
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1656 <s xml:id="echoid-s949" xml:space="preserve">decimaquarta quinti,_ <lb/>_& </s> |
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1657 <s xml:id="echoid-s950" xml:space="preserve">ex uigeſimaſecunda ſexti elementorum, ut ſuperius dictum eſt._</s> |
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1658 <s xml:id="echoid-s951" xml:space="preserve"/> |
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1659 </p> |
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1660 <div xml:id="echoid-div69" type="float" level="2" n="3"> |
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1661 <note position="left" xlink:label="note-0044-02" xlink:href="note-0044-02a" xml:space="preserve">C</note> |
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1662 </div> |
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1663 <p> |
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1664 <s xml:id="echoid-s952" xml:space="preserve">Quæ ergo ab h ducitur ad rectos angulos ipſi n o coi-<lb/> |
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1665 <anchor type="note" xlink:label="note-0044-03a" xlink:href="note-0044-03"/> |
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1666 bit cum p b inter p & </s> |
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1667 <s xml:id="echoid-s953" xml:space="preserve">b.</s> |
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1668 <s xml:id="echoid-s954" xml:space="preserve">] _Cur boc ita contingat, nos proxi-_ <lb/>_me explicauimus._</s> |
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1669 <s xml:id="echoid-s955" xml:space="preserve"/> |
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1670 </p> |
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1671 <div xml:id="echoid-div70" type="float" level="2" n="4"> |
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1672 <note position="left" xlink:label="note-0044-03" xlink:href="note-0044-03a" xml:space="preserve">D</note> |
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1673 </div> |
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1674 </div> |
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1675 <div xml:id="echoid-div72" type="section" level="1" n="29"> |
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1676 <head xml:id="echoid-head34" xml:space="preserve">PROPOSITIO VI.</head> |
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1677 <p> |
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1678 <s xml:id="echoid-s956" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>leuior humido axem habuerit maiorem quidem <lb/>quàm ſeſquialterum eius, quæ uſque ad axem, <lb/>minorem nero, quàm ut ad eam, quæ uſque ad <lb/>axem proportionem habeat, quam quindecim <lb/>ad quatuor; </s> |
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1679 <s xml:id="echoid-s957" xml:space="preserve">in humidum demiſſa adeo, ut baſis <lb/>ipſius contingat humidum, nunquam conſiſtet <lb/>inclinata ita, ut baſis in uno puncto humidum <lb/>contingat.</s> |
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1680 <s xml:id="echoid-s958" xml:space="preserve"/> |
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1681 </p> |
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1682 <pb o="17" file="0045" n="45" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1683 <p> |
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1684 <s xml:id="echoid-s959" xml:space="preserve">SIT portio, qualis dicta eſt, & </s> |
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1685 <s xml:id="echoid-s960" xml:space="preserve">in humidum demittatur, <lb/>ſicuti diximus, adeo ut baſis eius in uno puncto contingat <lb/>humidum. </s> |
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1686 <s xml:id="echoid-s961" xml:space="preserve">demonſtrandum eſtnon manere ipſam portio-<lb/>nem, ſed reuoluiita, ut baſis nullo modo humidi ſuperſicie <lb/> |
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1687 <anchor type="note" xlink:label="note-0045-01a" xlink:href="note-0045-01"/> |
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|
1688 contingat. </s> |
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1689 <s xml:id="echoid-s962" xml:space="preserve">Secta enim ipſa per axem, plano ad ſuper ſiciem <lb/>humidi recto, ſit ſectio ſuperſiciei portionis a p o l re-<lb/>ctãguli coni ſe <lb/> |
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1690 <anchor type="figure" xlink:label="fig-0045-01a" xlink:href="fig-0045-01"/> |
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|
1691 ctio: </s> |
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1692 <s xml:id="echoid-s963" xml:space="preserve">ſuperſi-<lb/>ciei humidi ſe-<lb/>ctio ſit a s: </s> |
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1693 <s xml:id="echoid-s964" xml:space="preserve">axis <lb/>autem portio-<lb/>nis, ac ſectio-<lb/>nis diameter n <lb/>o: </s> |
|
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1694 <s xml:id="echoid-s965" xml:space="preserve">& </s> |
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1695 <s xml:id="echoid-s966" xml:space="preserve">ſccetur in <lb/>f quidẽ ita, ut <lb/>o f ſit dupla ip <lb/>ſius ſn; </s> |
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1696 <s xml:id="echoid-s967" xml:space="preserve">in ω ue <lb/>ro, ut n o ad <lb/>f ω eandem ha <lb/>beat proportionem, quam quindecim ad quatuor: </s> |
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1697 <s xml:id="echoid-s968" xml:space="preserve">& </s> |
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1698 <s xml:id="echoid-s969" xml:space="preserve">ipſi <lb/>n o ad rectos angulos ducatur ω k. </s> |
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1699 <s xml:id="echoid-s970" xml:space="preserve">Itaque quoniam n o <lb/> |
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1700 <anchor type="note" xlink:label="note-0045-02a" xlink:href="note-0045-02"/> |
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1701 ad f ω maiorem habet proportionem, quàm ad eam, quæ <lb/>uſque ad axem; </s> |
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1702 <s xml:id="echoid-s971" xml:space="preserve">ſit ei, quæ uſque ad axem æqualis f b: </s> |
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1703 <s xml:id="echoid-s972" xml:space="preserve">& </s> |
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1704 <s xml:id="echoid-s973" xml:space="preserve">du <lb/>catur p c quidem ipſi a s æquidiſtans, cõtingensq; </s> |
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1705 <s xml:id="echoid-s974" xml:space="preserve">ſectio-<lb/>nem a p o l in p; </s> |
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1706 <s xml:id="echoid-s975" xml:space="preserve">pi uero æquidiſtans ipſi n o: </s> |
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1707 <s xml:id="echoid-s976" xml:space="preserve">& </s> |
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1708 <s xml:id="echoid-s977" xml:space="preserve">primum <lb/>ſecet pi ipſam κ ω in h. </s> |
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1709 <s xml:id="echoid-s978" xml:space="preserve">Quoniã ergo in portione a p o l, <lb/> |
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1710 <anchor type="note" xlink:label="note-0045-03a" xlink:href="note-0045-03"/> |
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1711 quæ continetur recta linea, & </s> |
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1712 <s xml:id="echoid-s979" xml:space="preserve">rectanguli coni ſectione, κ ω <lb/>quidem æ quidiſtans eſtipſi a l; </s> |
|
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1713 <s xml:id="echoid-s980" xml:space="preserve">p i uero diametro æquidi-<lb/>ſtat: </s> |
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1714 <s xml:id="echoid-s981" xml:space="preserve">ſecaturq; </s> |
|
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1715 <s xml:id="echoid-s982" xml:space="preserve">ab ipſa κ ω in h: </s> |
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1716 <s xml:id="echoid-s983" xml:space="preserve">& </s> |
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1717 <s xml:id="echoid-s984" xml:space="preserve">a s æquidiſtat contingen-<lb/>ti in p: </s> |
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1718 <s xml:id="echoid-s985" xml:space="preserve">neceſſarium eſtipſam p i ad p h uel ean dem pro-<lb/>portionem habere, quam habet n ω ad ω o, uel maiorem: <lb/></s> |
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1719 <s xml:id="echoid-s986" xml:space="preserve">hocenim iam demonſtratum eſt. </s> |
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1720 <s xml:id="echoid-s987" xml:space="preserve">At uero n ω ſeſquialtera <lb/>eſt ipſius ω o. </s> |
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1721 <s xml:id="echoid-s988" xml:space="preserve">& </s> |
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1722 <s xml:id="echoid-s989" xml:space="preserve">pi igitur uel ſeſquialtera eſt ipſius h p; </s> |
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1723 <s xml:id="echoid-s990" xml:space="preserve"><lb/>uel maior, quàm ſeſquialtera. </s> |
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1724 <s xml:id="echoid-s991" xml:space="preserve">Quare ph ipſius h i aut du <lb/> |
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1725 <anchor type="note" xlink:label="note-0045-04a" xlink:href="note-0045-04"/> |
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1726 <pb file="0046" n="46" rhead="ARCHIMEDIS"/> |
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1727 pla eſt, aut minor, quàm dupla. </s> |
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1728 <s xml:id="echoid-s992" xml:space="preserve">Sit autem p t dupla t i. </s> |
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1729 <s xml:id="echoid-s993" xml:space="preserve">erit <lb/>centrum grauitatis eius, quod eſt in humido, punctum t. <lb/></s> |
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1730 <s xml:id="echoid-s994" xml:space="preserve">Itaque iuncta t f producatur; </s> |
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1731 <s xml:id="echoid-s995" xml:space="preserve">ſitq; </s> |
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1732 <s xml:id="echoid-s996" xml:space="preserve">eius, quod extra humi <lb/>dum grauitatis centrum g: </s> |
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1733 <s xml:id="echoid-s997" xml:space="preserve">& </s> |
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1734 <s xml:id="echoid-s998" xml:space="preserve">à puncto b ad rectos angu-<lb/>los ipſi n o ducatur b r. </s> |
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1735 <s xml:id="echoid-s999" xml:space="preserve">Quòd cum p i quidem ſit æqui-<lb/>diſtans diametro n o: </s> |
|
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1736 <s xml:id="echoid-s1000" xml:space="preserve">br autem ad diametrum perpendi <lb/>cularis. </s> |
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1737 <s xml:id="echoid-s1001" xml:space="preserve">& </s> |
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1738 <s xml:id="echoid-s1002" xml:space="preserve">f b æqualis ei, quæ uſque ad axem: </s> |
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1739 <s xml:id="echoid-s1003" xml:space="preserve">perſpicuum <lb/>eſt f r productam æquales facere angulos cum ea, quæ ſe-<lb/>ctionem a p o l in puncto p contingit. </s> |
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1740 <s xml:id="echoid-s1004" xml:space="preserve">quare & </s> |
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1741 <s xml:id="echoid-s1005" xml:space="preserve">cum a s: </s> |
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1742 <s xml:id="echoid-s1006" xml:space="preserve"><lb/>& </s> |
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1743 <s xml:id="echoid-s1007" xml:space="preserve">cum ſuperficie humidi. </s> |
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1744 <s xml:id="echoid-s1008" xml:space="preserve">lineæ autem ductæ per tg æqui-<lb/>diſtantes ipſi f r, erunt & </s> |
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1745 <s xml:id="echoid-s1009" xml:space="preserve"><lb/> |
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1746 <anchor type="figure" xlink:label="fig-0046-01a" xlink:href="fig-0046-01"/> |
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1747 ad humidi ſuperficiẽ per-<lb/>pendiculares: </s> |
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1748 <s xml:id="echoid-s1010" xml:space="preserve">& </s> |
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1749 <s xml:id="echoid-s1011" xml:space="preserve">ſolidi <lb/>a p o l magnitudo, quæ ẽ <lb/>intra humidum ſurſum fe <lb/>retur ſecundum perpen-<lb/>dicularem per t ductam; <lb/></s> |
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1750 <s xml:id="echoid-s1012" xml:space="preserve">quæ uero extra humidum <lb/>ſecundum eam, quæ per g <lb/>deorſum feretur. </s> |
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1751 <s xml:id="echoid-s1013" xml:space="preserve">reuolue <lb/> |
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1752 <anchor type="note" xlink:label="note-0046-01a" xlink:href="note-0046-01"/> |
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1753 tur ergo ſolidum a p o l: <lb/></s> |
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1754 <s xml:id="echoid-s1014" xml:space="preserve">& </s> |
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1755 <s xml:id="echoid-s1015" xml:space="preserve">baſis ipſius nullo modo <lb/>humidi ſuperficiem con-<lb/>tinget. </s> |
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1756 <s xml:id="echoid-s1016" xml:space="preserve">At ſi pi lineam k ω <lb/>non ſecet, ut in ſecunda <lb/>figura; </s> |
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1757 <s xml:id="echoid-s1017" xml:space="preserve">manifeſtum eſt punctum t, quod eſt centrum gra-<lb/>uitatis demerſæ portionis, cadere inter p & </s> |
|
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1758 <s xml:id="echoid-s1018" xml:space="preserve">i: </s> |
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1759 <s xml:id="echoid-s1019" xml:space="preserve">& </s> |
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1760 <s xml:id="echoid-s1020" xml:space="preserve">reliqua <lb/>ſimiliter demonſtrabuntur.</s> |
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1761 <s xml:id="echoid-s1021" xml:space="preserve"/> |
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|
1762 </p> |
|
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1763 <div xml:id="echoid-div72" type="float" level="2" n="1"> |
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1764 <note position="right" xlink:label="note-0045-01" xlink:href="note-0045-01a" xml:space="preserve">A</note> |
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1765 <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a"> |
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|
1766 <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0045-01"/> |
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|
1767 </figure> |
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1768 <note position="right" xlink:label="note-0045-02" xlink:href="note-0045-02a" xml:space="preserve">B</note> |
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1769 <note position="right" xlink:label="note-0045-03" xlink:href="note-0045-03a" xml:space="preserve">C</note> |
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1770 <note position="right" xlink:label="note-0045-04" xlink:href="note-0045-04a" xml:space="preserve">D</note> |
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1771 <figure xlink:label="fig-0046-01" xlink:href="fig-0046-01a"> |
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|
1772 <image file="0046-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0046-01"/> |
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|
1773 </figure> |
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1774 <note position="left" xlink:label="note-0046-01" xlink:href="note-0046-01a" xml:space="preserve">E</note> |
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|
1775 </div> |
|
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|
1776 </div> |
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|
1777 <div xml:id="echoid-div74" type="section" level="1" n="30"> |
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|
1778 <head xml:id="echoid-head35" xml:space="preserve">COMMENTARIVS.</head> |
|
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|
1779 <p> |
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1780 <s xml:id="echoid-s1022" xml:space="preserve">Demonſtrandum eſt non manere ipſam portionem, ſed <lb/> |
|
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1781 <anchor type="note" xlink:label="note-0046-02a" xlink:href="note-0046-02"/> |
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|
1782 reuolui ita, ut baſis nullo modo ſuperficiem humidi con-<lb/>tingat.</s> |
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|
1783 <s xml:id="echoid-s1023" xml:space="preserve">] _Hæcnos addidimus tanquam ab interprete omiſſa_.</s> |
|
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|
1784 <s xml:id="echoid-s1024" xml:space="preserve"/> |
|
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|
1785 </p> |
|
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1786 <div xml:id="echoid-div74" type="float" level="2" n="1"> |
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1787 <note position="left" xlink:label="note-0046-02" xlink:href="note-0046-02a" xml:space="preserve">A</note> |
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1788 </div> |
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1789 <pb o="18" file="0047" n="47" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1790 <p> |
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1791 <s xml:id="echoid-s1025" xml:space="preserve">Itaque quoniam no ad f ω maiorem habetproportio-<lb/> |
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1792 <anchor type="note" xlink:label="note-0047-01a" xlink:href="note-0047-01"/> |
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1793 nem, quam ad eam, quæ uſque ad axem.</s> |
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1794 <s xml:id="echoid-s1026" xml:space="preserve">] _Habet enim diame-_ <lb/>_ter portioms n o ad f ω proportionem eandem, quam quindeeim ad_ <lb/>_quatuor; </s> |
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1795 <s xml:id="echoid-s1027" xml:space="preserve">ad eam uero, quæ uſque ad axem minorem proportionem_ <lb/>_habere ponitur, quàm quindecim ad quatuor. </s> |
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1796 <s xml:id="echoid-s1028" xml:space="preserve">quare n o ad f ω ma_ <lb/>_iorem habebit proportionem, quàm ad eam, quæ uſque ad axem: </s> |
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1797 <s xml:id="echoid-s1029" xml:space="preserve">&_</s> |
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1798 <s xml:id="echoid-s1030" xml:space="preserve"> <lb/>_propterea quæ uſque ad axem ipſa f ω maior erit_. <lb/></s> |
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1799 <s xml:id="echoid-s1031" xml:space="preserve"/> |
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1800 </p> |
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1801 <div xml:id="echoid-div75" type="float" level="2" n="2"> |
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1802 <note position="right" xlink:label="note-0047-01" xlink:href="note-0047-01a" xml:space="preserve">B</note> |
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1803 </div> |
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1804 <note position="right" xml:space="preserve">10. quinti</note> |
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1805 <p> |
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1806 <s xml:id="echoid-s1032" xml:space="preserve">Quoniam ergo in portione a p o l, quæ continetur re-<lb/>cta linea, & </s> |
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1807 <s xml:id="echoid-s1033" xml:space="preserve">rectanguli coni ſectione, _k_ ω quidem æ quidi-<lb/>ſtans eſt ipſi a l; </s> |
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1808 <s xml:id="echoid-s1034" xml:space="preserve">p i uero diametro æquidiſtat; </s> |
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1809 <s xml:id="echoid-s1035" xml:space="preserve">ſecaturq; <lb/></s> |
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1810 <s xml:id="echoid-s1036" xml:space="preserve">ab ipſa k ω in h: </s> |
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1811 <s xml:id="echoid-s1037" xml:space="preserve">& </s> |
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1812 <s xml:id="echoid-s1038" xml:space="preserve">a c æquidiſtat contingenti in p neceſ-<lb/>ſarium eſt ipſam p i ad p h uel eandem proportionem ha <lb/>bere, quam habet n ω ad ω o, uel maiorem. </s> |
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1813 <s xml:id="echoid-s1039" xml:space="preserve">hoc enim iam <lb/>demonſtratum eſt] _Vbi hoc demonſtratum ſit uel ab ipſo Ar-_ <lb/>_chimede, uel ab alio, numdum apparet, quocircanos demonstra-_ <lb/>_tionem afferemus, poſteaquam non nulla, quæ ad eam pertinent ex-_ <lb/>_plicauerimus_.</s> |
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1814 <s xml:id="echoid-s1040" xml:space="preserve"/> |
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1815 </p> |
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1816 </div> |
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1817 <div xml:id="echoid-div77" type="section" level="1" n="31"> |
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1818 <head xml:id="echoid-head36" xml:space="preserve">LEMMAI.</head> |
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1819 <p style="it"> |
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1820 <s xml:id="echoid-s1041" xml:space="preserve">Sint lineæ a b, a c angulum b a c continentes: </s> |
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1821 <s xml:id="echoid-s1042" xml:space="preserve">& </s> |
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1822 <s xml:id="echoid-s1043" xml:space="preserve">à <lb/>puncto d, quod in linea a c ſumptum ſit, ducantur d e, <lb/>d f utcunque ad ipſam a b. </s> |
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1823 <s xml:id="echoid-s1044" xml:space="preserve">Sumptis uero in eadem li. <lb/></s> |
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1824 <s xml:id="echoid-s1045" xml:space="preserve">nea quotlibet punctis g l, ducantur g h, l m ipſi d e <lb/>æquidistantes; </s> |
|
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1825 <s xml:id="echoid-s1046" xml:space="preserve">& </s> |
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1826 <s xml:id="echoid-s1047" xml:space="preserve">g k, l n æquidiſtantes f d. </s> |
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1827 <s xml:id="echoid-s1048" xml:space="preserve">deinde à <lb/>punctis d, g uſque ad lineam m l ducantur, d o p qui <lb/>dem ſecans g h in o; </s> |
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1828 <s xml:id="echoid-s1049" xml:space="preserve">& </s> |
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1829 <s xml:id="echoid-s1050" xml:space="preserve">g q, quæ æquidistent ipſi b a. </s> |
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1830 <s xml:id="echoid-s1051" xml:space="preserve"><lb/>Dico lineas, quæ inter æquidiſtantes ipſi f d ad eas, quæ <lb/>inter æquidiſtantes d e interiiciuntur, uidelicet k n ad g q, <lb/>uel ad o p; </s> |
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1831 <s xml:id="echoid-s1052" xml:space="preserve">f k ad d o; </s> |
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1832 <s xml:id="echoid-s1053" xml:space="preserve">& </s> |
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1833 <s xml:id="echoid-s1054" xml:space="preserve">f n ad d p eandem inter ſe ſe <lb/>proportionem habere: </s> |
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1834 <s xml:id="echoid-s1055" xml:space="preserve">nempe eam, quã habet a f ad a e.</s> |
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1835 <s xml:id="echoid-s1056" xml:space="preserve"> |
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1836 <pb file="0048" n="48" rhead="ARCHIMEDIS"/> |
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1837 Quoniam enim triangula afd, akg, anl ſi-<lb/> |
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1838 <anchor type="figure" xlink:label="fig-0048-01a" xlink:href="fig-0048-01"/> |
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1839 milia ſunt; </s> |
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1840 <s xml:id="echoid-s1057" xml:space="preserve">itémq; </s> |
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1841 <s xml:id="echoid-s1058" xml:space="preserve">ſimilia efd, h k g, mnl: <lb/></s> |
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1842 <s xml:id="echoid-s1059" xml:space="preserve">erit ut af ad fd, ita ak ad kg; </s> |
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1843 <s xml:id="echoid-s1060" xml:space="preserve">ut autem fd <lb/> |
|
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1844 <anchor type="note" xlink:label="note-0048-01a" xlink:href="note-0048-01"/> |
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1845 ad fe, ita kg ad kh. </s> |
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1846 <s xml:id="echoid-s1061" xml:space="preserve">quare ex æquali ut af <lb/>ad fe, ita ak ad kh: </s> |
|
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1847 <s xml:id="echoid-s1062" xml:space="preserve">& </s> |
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1848 <s xml:id="echoid-s1063" xml:space="preserve">per conuerſionem ra-<lb/>tionis ut af ad ae, ita ak ad ah. </s> |
|
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1849 <s xml:id="echoid-s1064" xml:space="preserve">eodem <lb/>modo oſtendetur, ut af ad a e, ita an ad am. <lb/></s> |
|
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1850 <s xml:id="echoid-s1065" xml:space="preserve">cum igitur an ad am ſit, ut a k ad a h; </s> |
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1851 <s xml:id="echoid-s1066" xml:space="preserve">erit <lb/> |
|
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1852 <anchor type="note" xlink:label="note-0048-02a" xlink:href="note-0048-02"/> |
|
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1853 reliqua kn ad reliquam h m, hoc eſt ad g q, <lb/>uel o p, ut a n ad a m; </s> |
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1854 <s xml:id="echoid-s1067" xml:space="preserve">hoc estut a f ad a e. <lb/></s> |
|
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1855 <s xml:id="echoid-s1068" xml:space="preserve">rurſus a k ad a h est, ut a f ad a e. </s> |
|
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1856 <s xml:id="echoid-s1069" xml:space="preserve">er-<lb/>go reliqua f k ad e h reliquam, uidelicet <lb/>ad do, ut a f ad a e. </s> |
|
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1857 <s xml:id="echoid-s1070" xml:space="preserve">Similiter demonſtrabi-<lb/>mus ita eſſe fn ad d p. </s> |
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1858 <s xml:id="echoid-s1071" xml:space="preserve">quod quidem demonſtra <lb/>re oportebat.</s> |
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1859 <s xml:id="echoid-s1072" xml:space="preserve"/> |
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1860 </p> |
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1861 <div xml:id="echoid-div77" type="float" level="2" n="1"> |
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1862 <figure xlink:label="fig-0048-01" xlink:href="fig-0048-01a"> |
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1863 <image file="0048-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0048-01"/> |
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1864 </figure> |
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1865 <note position="left" xlink:label="note-0048-01" xlink:href="note-0048-01a" xml:space="preserve">4. ſexti.</note> |
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1866 <note position="left" xlink:label="note-0048-02" xlink:href="note-0048-02a" xml:space="preserve">19. quinti</note> |
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1867 </div> |
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1868 </div> |
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1869 <div xml:id="echoid-div79" type="section" level="1" n="32"> |
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1870 <head xml:id="echoid-head37" xml:space="preserve">LEMMA II.</head> |
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1871 <p style="it"> |
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1872 <s xml:id="echoid-s1073" xml:space="preserve">Sint in eadem linea a b puncta <lb/> |
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1873 <anchor type="figure" xlink:label="fig-0048-02a" xlink:href="fig-0048-02"/> |
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1874 duo r s ita diſpoſita, ut a s ad a r <lb/>eandem proportionem habeat, quam <lb/>a f ad ae: </s> |
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1875 <s xml:id="echoid-s1074" xml:space="preserve">& </s> |
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1876 <s xml:id="echoid-s1075" xml:space="preserve">per r ducatur rtipſi <lb/>e d æquidiſtans; </s> |
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1877 <s xml:id="echoid-s1076" xml:space="preserve">per s uero ducatur <lb/>s t æquidiſtans fd, ita ut cum r t in <lb/>t puncto conueniat. </s> |
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1878 <s xml:id="echoid-s1077" xml:space="preserve">Dico punctum t <lb/>cadere in lineam a c.</s> |
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1879 <s xml:id="echoid-s1078" xml:space="preserve"/> |
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1880 </p> |
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1881 <div xml:id="echoid-div79" type="float" level="2" n="1"> |
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1882 <figure xlink:label="fig-0048-02" xlink:href="fig-0048-02a"> |
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1883 <image file="0048-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0048-02"/> |
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1884 </figure> |
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1885 </div> |
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1886 <p style="it"> |
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1887 <s xml:id="echoid-s1079" xml:space="preserve">Si enim fieri potest, cadat citra: </s> |
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1888 <s xml:id="echoid-s1080" xml:space="preserve">& </s> |
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1889 <s xml:id="echoid-s1081" xml:space="preserve">produca <lb/>tur rt uſque ad ipſam a c in u. </s> |
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1890 <s xml:id="echoid-s1082" xml:space="preserve">deinde per u <lb/>ducatur u x ipſi f d æquidiſtans. </s> |
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1891 <s xml:id="echoid-s1083" xml:space="preserve">Itaque ex <lb/>ijs, quæ proxime demonstrauimus a x ad ar |
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1892 <pb o="19" file="0049" n="49" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1893 eam proportionem babebit, quam a f ad a e. </s> |
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1894 <s xml:id="echoid-s1084" xml:space="preserve">Sed & </s> |
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1895 <s xml:id="echoid-s1085" xml:space="preserve">eandem habet <lb/>a s ad a r. </s> |
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1896 <s xml:id="echoid-s1086" xml:space="preserve">quare a s ipſi a x eſt æqualis, pars toti, quod fieri non <lb/> |
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1897 <anchor type="note" xlink:label="note-0049-01a" xlink:href="note-0049-01"/> |
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1898 poteſt. </s> |
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1899 <s xml:id="echoid-s1087" xml:space="preserve">Idem abſurdum ſequetur, ſi ponamus punctum t cadere ul-<lb/>tra lineam a c. </s> |
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1900 <s xml:id="echoid-s1088" xml:space="preserve">neceſſarium igitur est, ut in ipſam a c cadat. </s> |
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1901 <s xml:id="echoid-s1089" xml:space="preserve">quod <lb/>demonſtrandum propoſuimus.</s> |
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1902 <s xml:id="echoid-s1090" xml:space="preserve"/> |
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1903 </p> |
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1904 <div xml:id="echoid-div80" type="float" level="2" n="2"> |
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1905 <note position="right" xlink:label="note-0049-01" xlink:href="note-0049-01a" xml:space="preserve">9. quinti</note> |
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1906 </div> |
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1907 </div> |
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1908 <div xml:id="echoid-div82" type="section" level="1" n="33"> |
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1909 <head xml:id="echoid-head38" xml:space="preserve">LEMMA III.</head> |
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1910 <p style="it"> |
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1911 <s xml:id="echoid-s1091" xml:space="preserve">Sit parabole, cuius diameter a b: </s> |
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1912 <s xml:id="echoid-s1092" xml:space="preserve">atque eam cŏtingen <lb/>tes rectæ lineæ a c, b d; </s> |
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1913 <s xml:id="echoid-s1093" xml:space="preserve">a c quidem in puncto c, b d ue <lb/>ro in b: </s> |
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1914 <s xml:id="echoid-s1094" xml:space="preserve">& </s> |
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1915 <s xml:id="echoid-s1095" xml:space="preserve">per c ductis duabus lineis; </s> |
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1916 <s xml:id="echoid-s1096" xml:space="preserve">quarum alter a c e <lb/>diametro æquidiſtet, alter a c f æquidiſtet ipſi b d: </s> |
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1917 <s xml:id="echoid-s1097" xml:space="preserve">ſuma <lb/>tur quod uis punctum g in diametro: </s> |
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1918 <s xml:id="echoid-s1098" xml:space="preserve">fiatque ut f b, ad <lb/>b g, ita b g ad b h: </s> |
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1919 <s xml:id="echoid-s1099" xml:space="preserve">& </s> |
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1920 <s xml:id="echoid-s1100" xml:space="preserve">per g h ducantur g k l, h e m, <lb/>æquidiſtantes b d: </s> |
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1921 <s xml:id="echoid-s1101" xml:space="preserve">per m uero ducatur m n o ipſi a c <lb/>æquidistans, quæ diametrum ſecet in o: </s> |
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1922 <s xml:id="echoid-s1102" xml:space="preserve">& </s> |
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1923 <s xml:id="echoid-s1103" xml:space="preserve">per n ducta <lb/>n p uſque ad diametrum, ipſi b d æquidistet. </s> |
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1924 <s xml:id="echoid-s1104" xml:space="preserve">Dico h o <lb/>ipſius g b duplam eſſe.</s> |
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1925 <s xml:id="echoid-s1105" xml:space="preserve"/> |
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1926 </p> |
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1927 <p style="it"> |
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1928 <s xml:id="echoid-s1106" xml:space="preserve">V_EL_ igitur linea m n o ſccat diametrum in g, uel in alijs pun-<lb/>ctis: </s> |
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1929 <s xml:id="echoid-s1107" xml:space="preserve">& </s> |
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1930 <s xml:id="echoid-s1108" xml:space="preserve">ſi quidem ſecat in g, unum at que idem punctum duabus li-<lb/>teris go notabitur. </s> |
|
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1931 <s xml:id="echoid-s1109" xml:space="preserve">Itaque quoniam f c, p n, h e m ſibiipſis æqui <lb/>distant: </s> |
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1932 <s xml:id="echoid-s1110" xml:space="preserve">& </s> |
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1933 <s xml:id="echoid-s1111" xml:space="preserve">ipſi a c æquidiſtat m n o: </s> |
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1934 <s xml:id="echoid-s1112" xml:space="preserve">fient triangula a f c, o p n, <lb/>o h m inter ſe ſimilia. </s> |
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1935 <s xml:id="echoid-s1113" xml:space="preserve">quare erit o h ad h m, ut a f ad fc: </s> |
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1936 <s xml:id="echoid-s1114" xml:space="preserve">& </s> |
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1937 <s xml:id="echoid-s1115" xml:space="preserve">per-<lb/> |
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1938 <anchor type="note" xlink:label="note-0049-02a" xlink:href="note-0049-02"/> |
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1939 mut ando o h ad a f, ut h m ad fc. </s> |
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1940 <s xml:id="echoid-s1116" xml:space="preserve">est autem quadratum h m ad <lb/>quadratum g l, ut linea h b ad lineam b g, ex uigeſima primi libri <lb/>conicorum: </s> |
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1941 <s xml:id="echoid-s1117" xml:space="preserve">& </s> |
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1942 <s xml:id="echoid-s1118" xml:space="preserve">quadratum g l ad quadratum fc, ut linea g b ad <lb/>ipſam b f: </s> |
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1943 <s xml:id="echoid-s1119" xml:space="preserve">ſuntq; </s> |
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1944 <s xml:id="echoid-s1120" xml:space="preserve">h b, b g, b f lineæ deinceps proportionales. </s> |
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1945 <s xml:id="echoid-s1121" xml:space="preserve">er-<lb/> |
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1946 <anchor type="note" xlink:label="note-0049-03a" xlink:href="note-0049-03"/> |
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1947 go & </s> |
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1948 <s xml:id="echoid-s1122" xml:space="preserve">quadrata h m, g l, f c, & </s> |
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1949 <s xml:id="echoid-s1123" xml:space="preserve">ipſorum latera proportionalia <lb/>erunt. </s> |
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1950 <s xml:id="echoid-s1124" xml:space="preserve">atque idcirco ut quadratum h m ad quadratum g l, ita li- |
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1951 <pb file="0050" n="50" rhead="ARCHIMEDIS"/> |
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1952 nea h m ad li-<lb/> |
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1953 <anchor type="figure" xlink:label="fig-0050-01a" xlink:href="fig-0050-01"/> |
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1954 neam fc. </s> |
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1955 <s xml:id="echoid-s1125" xml:space="preserve">at uero <lb/>ut h m ad f c, ita <lb/>o h ad a f: </s> |
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1956 <s xml:id="echoid-s1126" xml:space="preserve">& </s> |
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1957 <s xml:id="echoid-s1127" xml:space="preserve">ut <lb/>quadratum h m <lb/>ad quadratú g l, <lb/>ita linea h b ad <lb/>b g; </s> |
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1958 <s xml:id="echoid-s1128" xml:space="preserve">hoc est b g <lb/>ad b f. </s> |
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1959 <s xml:id="echoid-s1129" xml:space="preserve">ex quibus <lb/>ſequitur o h ad <lb/>a f ita eſſe, ut b g <lb/>ad b f: </s> |
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1960 <s xml:id="echoid-s1130" xml:space="preserve">& </s> |
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1961 <s xml:id="echoid-s1131" xml:space="preserve">permu <lb/>tando oh ad b g, <lb/>ut a f ad f b. </s> |
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1962 <s xml:id="echoid-s1132" xml:space="preserve">ſed <lb/>eſt a f dupla ip-<lb/>ſius fb: </s> |
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1963 <s xml:id="echoid-s1133" xml:space="preserve">ſunt eni <lb/>a b, b f æquales <lb/>ex 35 primi libri <lb/>conicorum. </s> |
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1964 <s xml:id="echoid-s1134" xml:space="preserve">ergo <lb/>& </s> |
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1965 <s xml:id="echoid-s1135" xml:space="preserve">h o ipſius g b <lb/>eſt dupla. </s> |
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1966 <s xml:id="echoid-s1136" xml:space="preserve">quod demonſtrare oportebat.</s> |
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1967 <s xml:id="echoid-s1137" xml:space="preserve"/> |
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1968 </p> |
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1969 <div xml:id="echoid-div82" type="float" level="2" n="1"> |
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1970 <note position="right" xlink:label="note-0049-02" xlink:href="note-0049-02a" xml:space="preserve">4. ſexti.</note> |
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1971 <note position="right" xlink:label="note-0049-03" xlink:href="note-0049-03a" xml:space="preserve">22. ſexti. <lb/>cor. 20. ſe <lb/>xti.</note> |
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1972 <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a"> |
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1973 <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0050-01"/> |
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1974 </figure> |
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1975 </div> |
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1976 </div> |
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1977 <div xml:id="echoid-div84" type="section" level="1" n="34"> |
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1978 <head xml:id="echoid-head39" xml:space="preserve">LEMMA IIII.</head> |
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1979 <p style="it"> |
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1980 <s xml:id="echoid-s1138" xml:space="preserve">Iiſdem manentibus, & </s> |
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1981 <s xml:id="echoid-s1139" xml:space="preserve">à puncto m ducta m q uſque <lb/>ad diametrum, quæ ſectionem in puncto m conting at; <lb/></s> |
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1982 <s xml:id="echoid-s1140" xml:space="preserve">Dico h q ad q o eandem proportionem habere, quam <lb/>habet g h ad c n.</s> |
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1983 <s xml:id="echoid-s1141" xml:space="preserve"/> |
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1984 </p> |
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1985 <p style="it"> |
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1986 <s xml:id="echoid-s1142" xml:space="preserve">F_IAT_ enim h r æqualis g f. </s> |
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1987 <s xml:id="echoid-s1143" xml:space="preserve">& </s> |
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1988 <s xml:id="echoid-s1144" xml:space="preserve">cumtriangula a f c, o p n ſimi <lb/>lia ſint, & </s> |
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1989 <s xml:id="echoid-s1145" xml:space="preserve">p n ſit æqualis f c; </s> |
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1990 <s xml:id="echoid-s1146" xml:space="preserve">eodem modo demonſtrabimus p o, f a <lb/>inter ſe æquales eſſe. </s> |
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1991 <s xml:id="echoid-s1147" xml:space="preserve">quare p o ipſius f b dupla erit. </s> |
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1992 <s xml:id="echoid-s1148" xml:space="preserve">Sed eſt h o du <lb/>pla g b. </s> |
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1993 <s xml:id="echoid-s1149" xml:space="preserve">ergo & </s> |
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1994 <s xml:id="echoid-s1150" xml:space="preserve">reliqua p h reliquæ f g; </s> |
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1995 <s xml:id="echoid-s1151" xml:space="preserve">uidelicet ipſius r h eſt du- |
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1996 <pb o="20" file="0051" n="51" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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1997 pla. </s> |
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1998 <s xml:id="echoid-s1152" xml:space="preserve">ex quo fit ut pr, rh, fg inter ſe ſint æquales; </s> |
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1999 <s xml:id="echoid-s1153" xml:space="preserve">itémq; </s> |
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2000 <s xml:id="echoid-s1154" xml:space="preserve">æquales <lb/>rg, pf. </s> |
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2001 <s xml:id="echoid-s1155" xml:space="preserve">eſt enim pg utrique r p, gf communis. </s> |
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2002 <s xml:id="echoid-s1156" xml:space="preserve">Quoniam igitur <lb/>hb ad bg est, ut <lb/> |
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2003 <anchor type="figure" xlink:label="fig-0051-01a" xlink:href="fig-0051-01"/> |
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2004 gb ad bf; </s> |
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2005 <s xml:id="echoid-s1157" xml:space="preserve">per c<gap/> <lb/>uerſionem ratio-<lb/>mis erit b h ad <lb/>h g, ut b g ad gf. <lb/></s> |
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2006 <s xml:id="echoid-s1158" xml:space="preserve">eſt autem q h ad <lb/>h b, ut h o ad gb. </s> |
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2007 <s xml:id="echoid-s1159" xml:space="preserve"><lb/>nam ex 35 primi <lb/>libri conicorum, <lb/>cum linea qm có <lb/>tingat ſectionem <lb/>in punctom; </s> |
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2008 <s xml:id="echoid-s1160" xml:space="preserve">erút <lb/>h b, bq æquales; </s> |
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|
2009 <s xml:id="echoid-s1161" xml:space="preserve"><lb/>& </s> |
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2010 <s xml:id="echoid-s1162" xml:space="preserve">gh ipſius h b <lb/>dupla. </s> |
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2011 <s xml:id="echoid-s1163" xml:space="preserve">ergo ex æ-<lb/>quali q h ad hg, <lb/>ut ho ad g f; </s> |
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2012 <s xml:id="echoid-s1164" xml:space="preserve">hoc <lb/>eſt ad hr: </s> |
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2013 <s xml:id="echoid-s1165" xml:space="preserve">& </s> |
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2014 <s xml:id="echoid-s1166" xml:space="preserve">per <lb/>mutando q h ad <lb/>h o, ut g h ad h r. </s> |
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2015 <s xml:id="echoid-s1167" xml:space="preserve"><lb/>rurſus per conuerſionem rationis h q ad qo, ut h g ad g r; </s> |
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2016 <s xml:id="echoid-s1168" xml:space="preserve">hoc eſt <lb/>p f: </s> |
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2017 <s xml:id="echoid-s1169" xml:space="preserve">& </s> |
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2018 <s xml:id="echoid-s1170" xml:space="preserve">propterea ad ipſam cn, quod demonstrandum fuerat.</s> |
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2019 <s xml:id="echoid-s1171" xml:space="preserve"/> |
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|
2020 </p> |
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2021 <div xml:id="echoid-div84" type="float" level="2" n="1"> |
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2022 <figure xlink:label="fig-0051-01" xlink:href="fig-0051-01a"> |
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2023 <image file="0051-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0051-01"/> |
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|
2024 </figure> |
|
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2025 </div> |
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|
2026 <p style="it"> |
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2027 <s xml:id="echoid-s1172" xml:space="preserve">His igitur explicatis, iam adid, quod propoſitum fue <lb/>rat, accedamus. </s> |
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2028 <s xml:id="echoid-s1173" xml:space="preserve">Itaque dico primum nc ad c k eandem <lb/>proportionem babere, quam h g ad g b.</s> |
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2029 <s xml:id="echoid-s1174" xml:space="preserve"/> |
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|
2030 </p> |
|
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|
2031 <p style="it"> |
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2032 <s xml:id="echoid-s1175" xml:space="preserve">Quoniam enim h q ad qo eſt, ut h g ad c n, hoc eſt ad a o ipſi <lb/>cn æqualem; </s> |
|
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2033 <s xml:id="echoid-s1176" xml:space="preserve">erit reliqua gq ad reliquam q a, ut h q ad q o: </s> |
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2034 <s xml:id="echoid-s1177" xml:space="preserve">& </s> |
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2035 <s xml:id="echoid-s1178" xml:space="preserve"><lb/>ob eam cauſſam lineæ a c g l productæ ex ijs, quæ ſupra demonſtra <lb/>uimus in linea q m conueniunt. </s> |
|
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2036 <s xml:id="echoid-s1179" xml:space="preserve">Rurſus gq ad qa eſt, ut h q ad |
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|
2037 <pb file="0052" n="52" rhead="ARCHIMEDIS"/> |
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2038 q o; </s> |
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2039 <s xml:id="echoid-s1180" xml:space="preserve">uidelicet ut h g ad f p: </s> |
|
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2040 <s xml:id="echoid-s1181" xml:space="preserve">quod proxime demonſtr atum eſt. </s> |
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2041 <s xml:id="echoid-s1182" xml:space="preserve">At <lb/> |
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2042 <anchor type="note" xlink:label="note-0052-01a" xlink:href="note-0052-01"/> |
|
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2043 ueroipſi g q æquales ſunt duæ lineæ ſimul ſumptæ qb, hoc eſt h b, <lb/> |
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2044 <anchor type="note" xlink:label="note-0052-02a" xlink:href="note-0052-02"/> |
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2045 & </s> |
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2046 <s xml:id="echoid-s1183" xml:space="preserve">b g: </s> |
|
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2047 <s xml:id="echoid-s1184" xml:space="preserve">atque ipſi q a æqualis eſt h f. </s> |
|
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2048 <s xml:id="echoid-s1185" xml:space="preserve">Sienim ab æqualibus h b, <lb/>bq, æqualia fb, <lb/> |
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|
2049 <anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/> |
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|
2050 ba demantur, re <lb/>manentia æqua-<lb/>lia erunt. </s> |
|
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2051 <s xml:id="echoid-s1186" xml:space="preserve">ergo <lb/>dempta h g ex <lb/>duabus lineis h <lb/>b, h g, relinqui-<lb/>tur dupla ipſius <lb/>b g; </s> |
|
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2052 <s xml:id="echoid-s1187" xml:space="preserve">hoc eſt o h: <lb/></s> |
|
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|
2053 <s xml:id="echoid-s1188" xml:space="preserve">& </s> |
|
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2054 <s xml:id="echoid-s1189" xml:space="preserve">dempta p f ex <lb/>f h, reliqua est <lb/>b p. </s> |
|
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2055 <s xml:id="echoid-s1190" xml:space="preserve">quare o h <lb/> |
|
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|
2056 <anchor type="note" xlink:label="note-0052-03a" xlink:href="note-0052-03"/> |
|
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2057 ad h p, eſt ut g q <lb/>ad q a. </s> |
|
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2058 <s xml:id="echoid-s1191" xml:space="preserve">Sed ut <lb/>g q ad q a, ita <lb/>h q ad q o; </s> |
|
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2059 <s xml:id="echoid-s1192" xml:space="preserve">hoc <lb/>eſt h g ad n c: <lb/></s> |
|
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|
2060 <s xml:id="echoid-s1193" xml:space="preserve">& </s> |
|
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|
2061 <s xml:id="echoid-s1194" xml:space="preserve">ut o h ad h p, <lb/> |
|
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|
2062 <anchor type="note" xlink:label="note-0052-04a" xlink:href="note-0052-04"/> |
|
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|
2063 ita g b ad c k. </s> |
|
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2064 <s xml:id="echoid-s1195" xml:space="preserve">eſt <lb/>cnim o h dupla <lb/>g b, & </s> |
|
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2065 <s xml:id="echoid-s1196" xml:space="preserve">h p item <lb/>dupla gf; </s> |
|
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|
2066 <s xml:id="echoid-s1197" xml:space="preserve">hoc eſt <lb/>c k. </s> |
|
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|
2067 <s xml:id="echoid-s1198" xml:space="preserve">eandem igitur proportionem habet h g ad n c, qnam g b ad <lb/>c k: </s> |
|
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|
2068 <s xml:id="echoid-s1199" xml:space="preserve">& </s> |
|
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|
2069 <s xml:id="echoid-s1200" xml:space="preserve">permutando n c ad c k eandem habet, quam b g ad g b.</s> |
|
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|
2070 <s xml:id="echoid-s1201" xml:space="preserve"/> |
|
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|
2071 </p> |
|
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|
2072 <div xml:id="echoid-div85" type="float" level="2" n="2"> |
|
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|
2073 <note position="left" xlink:label="note-0052-01" xlink:href="note-0052-01a" xml:space="preserve">2. lem:</note> |
|
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|
2074 <note position="left" xlink:label="note-0052-02" xlink:href="note-0052-02a" xml:space="preserve">4. lem.</note> |
|
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|
2075 <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a"> |
|
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|
2076 <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0052-01"/> |
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2077 </figure> |
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2078 <note position="left" xlink:label="note-0052-03" xlink:href="note-0052-03a" xml:space="preserve">19. quinti</note> |
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2079 <note position="left" xlink:label="note-0052-04" xlink:href="note-0052-04a" xml:space="preserve">15. quin-<lb/>ti.</note> |
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2080 </div> |
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2081 <p style="it"> |
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2082 <s xml:id="echoid-s1202" xml:space="preserve">Sumatur deinde aliud quod uis punctum in ſectum in ſectione, <lb/>quod ſit s: </s> |
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2083 <s xml:id="echoid-s1203" xml:space="preserve">& </s> |
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2084 <s xml:id="echoid-s1204" xml:space="preserve">per s duæ lineæ ducantur: </s> |
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2085 <s xml:id="echoid-s1205" xml:space="preserve">st quidem <lb/>æquidistans ipſi db, diametrumque in puncto t ſecans; <lb/></s> |
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2086 <s xml:id="echoid-s1206" xml:space="preserve">s u uero æquidistans ac, & </s> |
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2087 <s xml:id="echoid-s1207" xml:space="preserve">ſecans c e in u. </s> |
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2088 <s xml:id="echoid-s1208" xml:space="preserve">Dico u c <lb/>ad ck maiorem proportionem habere, quamtg ad gb.</s> |
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2089 <s xml:id="echoid-s1209" xml:space="preserve"/> |
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2090 </p> |
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2091 <pb o="21" file="0053" n="53" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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2092 <p style="it"> |
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2093 <s xml:id="echoid-s1210" xml:space="preserve">Producatur enim u s ad lineam qm in x: </s> |
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2094 <s xml:id="echoid-s1211" xml:space="preserve">& </s> |
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2095 <s xml:id="echoid-s1212" xml:space="preserve">à puncto x duca <lb/>tur ad diametrum x y ipſi bd æquidistans. </s> |
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2096 <s xml:id="echoid-s1213" xml:space="preserve">erit gt minor quàm <lb/>gy, quoniam u s minor eſt quàm ux: </s> |
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2097 <s xml:id="echoid-s1214" xml:space="preserve">& </s> |
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2098 <s xml:id="echoid-s1215" xml:space="preserve">ex primo lemmate yg <lb/>ad uc erit, ut h g ad n c; </s> |
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2099 <s xml:id="echoid-s1216" xml:space="preserve">uidelicet ut g b ad c k, quod proxime de <lb/>monstrauimus: </s> |
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2100 <s xml:id="echoid-s1217" xml:space="preserve">& </s> |
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2101 <s xml:id="echoid-s1218" xml:space="preserve">permutando yg ad gb, ut uc ad c k. </s> |
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2102 <s xml:id="echoid-s1219" xml:space="preserve">Sed t g <lb/>cum ſit ipſa y g minor, habet ad g b proportionem minorem, quàm <lb/>y g ad eandem. </s> |
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2103 <s xml:id="echoid-s1220" xml:space="preserve">ergo u c ad c K maiorem proportioné habet, quàm <lb/>t g ad g b. </s> |
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2104 <s xml:id="echoid-s1221" xml:space="preserve">quod demonstraſſe oportuit. </s> |
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2105 <s xml:id="echoid-s1222" xml:space="preserve">Itaque poſitione data g K <lb/>unum duntaxat erit in ſectione punctum, uidelicet m, à quo ductis <lb/>duabus lineis m e h, mno, habeat n c ad c K proportionem ean-<lb/>dem, quam h g ad g b. </s> |
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2106 <s xml:id="echoid-s1223" xml:space="preserve">nam ſi ab alijs omnibus ducantur, ſemper <lb/>ea, quæ inter a c, & </s> |
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2107 <s xml:id="echoid-s1224" xml:space="preserve">lineam ipſi æquidistantem interijcitur, ad c K <lb/>proportionem maiorem habebit, quàm quæ inter g K atque ei æqui <lb/>diſtantem, ad ipſam g b. </s> |
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2108 <s xml:id="echoid-s1225" xml:space="preserve">Conſtat igitur id, quod ab Archimede di-<lb/>ctum est; </s> |
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2109 <s xml:id="echoid-s1226" xml:space="preserve">nempe lineam pi ad p h uel eandem, quam n ω ad ω o, <lb/>uel maiorem habere proportionem.</s> |
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2110 <s xml:id="echoid-s1227" xml:space="preserve"/> |
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|
2111 </p> |
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2112 <p> |
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2113 <s xml:id="echoid-s1228" xml:space="preserve">Quare p h ipſius h i aut dupla eſt, aut minor quàm du <lb/> |
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2114 <anchor type="note" xlink:label="note-0053-01a" xlink:href="note-0053-01"/> |
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2115 pla.</s> |
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2116 <s xml:id="echoid-s1229" xml:space="preserve">] _Si quidé_ <lb/> |
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2117 <anchor type="figure" xlink:label="fig-0053-01a" xlink:href="fig-0053-01"/> |
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2118 _minor, quàm du-_ <lb/>_pla, ſit pt dupl. </s> |
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2119 <s xml:id="echoid-s1230" xml:space="preserve">2_ <lb/>_ti. </s> |
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2120 <s xml:id="echoid-s1231" xml:space="preserve">erit centrum_ <lb/>_grauitatis eius,_ <lb/>_quod in humido_ <lb/>_est, punctumt. </s> |
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2121 <s xml:id="echoid-s1232" xml:space="preserve">ſi_ <lb/>_uero p h ſit ip-_ <lb/>_ſius h i dupla,_ <lb/>_erit h grauitatis_ <lb/>_centrum: </s> |
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2122 <s xml:id="echoid-s1233" xml:space="preserve">ductâq;_ <lb/></s> |
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2123 <s xml:id="echoid-s1234" xml:space="preserve">_h f, & </s> |
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2124 <s xml:id="echoid-s1235" xml:space="preserve">producta_ <lb/>_ad centrum eius,_ <lb/>_quod est extra humidum, uidelicct ad g, alia ſimiliter demonstra-_ <lb/>_buntur. </s> |
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2125 <s xml:id="echoid-s1236" xml:space="preserve">atque idem intelligendum est in propoſitione, quæ ſe_-<lb/>_quitur._</s> |
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2126 <s xml:id="echoid-s1237" xml:space="preserve"/> |
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|
2127 </p> |
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2128 <div xml:id="echoid-div86" type="float" level="2" n="3"> |
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2129 <note position="right" xlink:label="note-0053-01" xlink:href="note-0053-01a" xml:space="preserve">D</note> |
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2130 <figure xlink:label="fig-0053-01" xlink:href="fig-0053-01a"> |
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2131 <image file="0053-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0053-01"/> |
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2132 </figure> |
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2133 </div> |
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2134 <p> |
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2135 <s xml:id="echoid-s1238" xml:space="preserve">Reuoluetur ergo ſolidum a p o l, & </s> |
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2136 <s xml:id="echoid-s1239" xml:space="preserve">baſis ipſius nullo <lb/> |
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2137 <anchor type="note" xlink:label="note-0053-02a" xlink:href="note-0053-02"/> |
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|
2138 <pb file="0054" n="54" rhead="ARCHIMEDIS"/> |
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|
2139 modo hum idi ſuperficiem continget.</s> |
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2140 <s xml:id="echoid-s1240" xml:space="preserve">] _In translatione le-_ <lb/>_gebatur ut baſis ipſius non tangat ſuperficiem humidi ſecundum_ <lb/>_unum ſignum. </s> |
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2141 <s xml:id="echoid-s1241" xml:space="preserve">nos autem ita uertere maluimus, & </s> |
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2142 <s xml:id="echoid-s1242" xml:space="preserve">hic & </s> |
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2143 <s xml:id="echoid-s1243" xml:space="preserve">in ijs,_ <lb/>_quæ ſequuntur, quoniam græci οὐδὲ \~εις, οὐδὲ ἓν, pro οὐ δεὶς, &_</s> |
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2144 <s xml:id="echoid-s1244" xml:space="preserve"> <lb/>_οὐδε ν frequenter utútur. </s> |
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2145 <s xml:id="echoid-s1245" xml:space="preserve">ut οὐδ ε ςιν οὐ δεὶς, nullus eſt: </s> |
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2146 <s xml:id="echoid-s1246" xml:space="preserve">οὐ \’δ ὑφ<unsure/> ἑ νὸς,_ <lb/>_ànullo & </s> |
|
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2147 <s xml:id="echoid-s1247" xml:space="preserve">alia eiuſmodi._</s> |
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2148 <s xml:id="echoid-s1248" xml:space="preserve"/> |
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2149 </p> |
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2150 <div xml:id="echoid-div87" type="float" level="2" n="4"> |
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2151 <note position="right" xlink:label="note-0053-02" xlink:href="note-0053-02a" xml:space="preserve">E</note> |
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2152 </div> |
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2153 </div> |
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2154 <div xml:id="echoid-div89" type="section" level="1" n="35"> |
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2155 <head xml:id="echoid-head40" xml:space="preserve">PROPOSITIO VII.</head> |
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2156 <p> |
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2157 <s xml:id="echoid-s1249" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>leuior humido axem habuerit maiorem quidem <lb/>quàm ſeſquialtérum eius, quæ uſque ad axem; <lb/></s> |
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2158 <s xml:id="echoid-s1250" xml:space="preserve">minorem uero, quàm ut ad eam, quæ uſque ad <lb/>axem proportionem habeat, quam quindecim <lb/>ad quatuor: </s> |
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2159 <s xml:id="echoid-s1251" xml:space="preserve">in humidum demiſſa, adeo ut baſis <lb/>ipſius tota ſitin humido; </s> |
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2160 <s xml:id="echoid-s1252" xml:space="preserve">nunquam conſiſtet ita, <lb/>ut baſis contingat humidi ſuperficiem: </s> |
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2161 <s xml:id="echoid-s1253" xml:space="preserve">ſed ut to-<lb/>ta in humido ſit, & </s> |
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2162 <s xml:id="echoid-s1254" xml:space="preserve">nullo modo eius ſuperficiem <lb/>contingat.</s> |
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2163 <s xml:id="echoid-s1255" xml:space="preserve"/> |
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2164 </p> |
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2165 <p> |
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2166 <s xml:id="echoid-s1256" xml:space="preserve">SIT portio qualis dicta eſt: </s> |
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2167 <s xml:id="echoid-s1257" xml:space="preserve">& </s> |
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2168 <s xml:id="echoid-s1258" xml:space="preserve">demittatur in humidũ, <lb/>ut diximus, adeo ut baſis ipſius in uno puncto contingat <lb/>humidi ſuperficiem. </s> |
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2169 <s xml:id="echoid-s1259" xml:space="preserve">Demonſtrandum eſt non manere ip-<lb/>ſam: </s> |
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2170 <s xml:id="echoid-s1260" xml:space="preserve">ſed reuolui ita ut baſis fuperficiem humidi nullo mo-<lb/>do contingat. </s> |
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2171 <s xml:id="echoid-s1261" xml:space="preserve">Secta enim ipſa plano per axem, recto ad ſu <lb/>perficiem humidi, ſectio ſit a p o l rectanguli coni ſectio: <lb/></s> |
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2172 <s xml:id="echoid-s1262" xml:space="preserve">ſuperficiei humidi ſectio ſit s 1: </s> |
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2173 <s xml:id="echoid-s1263" xml:space="preserve">axis portionis, & </s> |
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2174 <s xml:id="echoid-s1264" xml:space="preserve">ſectio-<lb/>nis diameter p f: </s> |
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2175 <s xml:id="echoid-s1265" xml:space="preserve">ſeceturq; </s> |
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2176 <s xml:id="echoid-s1266" xml:space="preserve">p f in r quidem ita ut r p ſit <lb/>dupla ipſius r f; </s> |
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2177 <s xml:id="echoid-s1267" xml:space="preserve">in ω autem ut p f ad r ω proportionem <lb/>habeat, quam quindecim ad quatuor: </s> |
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2178 <s xml:id="echoid-s1268" xml:space="preserve">& </s> |
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2179 <s xml:id="echoid-s1269" xml:space="preserve">ω k ipſi p f ad re-<lb/>ctos angulos ducatur erit r ω minor, quàm quæ uſque ad <lb/>axem. </s> |
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2180 <s xml:id="echoid-s1270" xml:space="preserve">Itaque accipiatur ei, quæ uſque ad axem æqualis rh:</s> |
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2181 <s xml:id="echoid-s1271" xml:space="preserve"> |
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2182 <pb o="22" file="0055" n="55" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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2183 & </s> |
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2184 <s xml:id="echoid-s1272" xml:space="preserve">c o quidẽ <lb/> |
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2185 <anchor type="figure" xlink:label="fig-0055-01a" xlink:href="fig-0055-01"/> |
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2186 ducatur con <lb/>tingẽs ſectio <lb/>nẽin o, quæ <lb/>ipſi s l æqui-<lb/>diſtet; </s> |
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2187 <s xml:id="echoid-s1273" xml:space="preserve">n o au <lb/>tem æquidi-<lb/> |
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2188 <anchor type="note" xlink:label="note-0055-01a" xlink:href="note-0055-01"/> |
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2189 ſtet p f: </s> |
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2190 <s xml:id="echoid-s1274" xml:space="preserve">& </s> |
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2191 <s xml:id="echoid-s1275" xml:space="preserve">pri <lb/>mum ipſam <lb/>k ω ſecet, at-<lb/>quein pũcto <lb/>i ſimiliter ut <lb/>in ſuperiori-<lb/>bus demonſtrabitur no, uel ſeſquialtera ipſius oi, uel <lb/>maior, quàm ſeſquialtera. </s> |
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2192 <s xml:id="echoid-s1276" xml:space="preserve">Sit autem o i minor, quam du-<lb/>pla ipſius in: </s> |
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2193 <s xml:id="echoid-s1277" xml:space="preserve">ſitq; </s> |
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2194 <s xml:id="echoid-s1278" xml:space="preserve">o b dupla b n: </s> |
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2195 <s xml:id="echoid-s1279" xml:space="preserve">& </s> |
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2196 <s xml:id="echoid-s1280" xml:space="preserve">diſponantur eadem, <lb/>quæſupra. </s> |
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2197 <s xml:id="echoid-s1281" xml:space="preserve">Similiter demonſtrabimus, ſi ducatur linea r t, <lb/>facere eam angulos rectos cum linea c o, & </s> |
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2198 <s xml:id="echoid-s1282" xml:space="preserve">cum ſuperficie <lb/>humidi. </s> |
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2199 <s xml:id="echoid-s1283" xml:space="preserve">quare à punctis b g lineæ ductæipſi r t æquidiſtã <lb/>tes, etiã ad humidi ſuperfi-<lb/> |
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2200 <anchor type="figure" xlink:label="fig-0055-02a" xlink:href="fig-0055-02"/> |
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2201 ciẽ perpẽdiculares erunt. <lb/></s> |
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2202 <s xml:id="echoid-s1284" xml:space="preserve">portio igitur quæ eſt extra <lb/>humidũ deorſum feretur <lb/>ſecundum eam perpendi-<lb/>cularem, quæ per b tran-<lb/>ſit; </s> |
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2203 <s xml:id="echoid-s1285" xml:space="preserve">quæ uero intra humi-<lb/>dum ſecundum eam, quæ <lb/>per g ſurſum ſeretur. </s> |
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2204 <s xml:id="echoid-s1286" xml:space="preserve">ex <lb/>quibus conſtat reuolui ſo-<lb/>lidum, ita ut baſis ipſius <lb/>nullo modo humidi ſuper <lb/>ficiem contingat: </s> |
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2205 <s xml:id="echoid-s1287" xml:space="preserve">quo-<lb/>niam nuncin uno puncto <lb/>contingens deorſum fer- |
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2206 <pb file="0056" n="56" rhead="ARCHIMEDIS"/> |
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2207 tur ex parte 1. </s> |
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2208 <s xml:id="echoid-s1288" xml:space="preserve">Quod ſi n o non ſecuerit ipſam ω k, <lb/>eadem nihilominus demonſtrabuntur.</s> |
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2209 <s xml:id="echoid-s1289" xml:space="preserve"/> |
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2210 </p> |
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2211 <div xml:id="echoid-div89" type="float" level="2" n="1"> |
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2212 <figure xlink:label="fig-0055-01" xlink:href="fig-0055-01a"> |
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2213 <image file="0055-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0055-01"/> |
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2214 </figure> |
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2215 <note position="right" xlink:label="note-0055-01" xlink:href="note-0055-01a" xml:space="preserve">10. quinti</note> |
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2216 <figure xlink:label="fig-0055-02" xlink:href="fig-0055-02a"> |
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2217 <image file="0055-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0055-02"/> |
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2218 </figure> |
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2219 </div> |
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2220 </div> |
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2221 <div xml:id="echoid-div91" type="section" level="1" n="36"> |
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2222 <head xml:id="echoid-head41" xml:space="preserve">PROPOSITIO VIII.</head> |
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2223 <p> |
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2224 <s xml:id="echoid-s1290" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>axem habuerit maiorem quidem, quàm ſeſqui-<lb/>alterum eius, quæ uſque ad axem; </s> |
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2225 <s xml:id="echoid-s1291" xml:space="preserve">minorem ue-<lb/>ro, quàm ut ad eam, quæ uſque ad axem propor-<lb/>tionem habeat, quam quindecim ad quatuor: </s> |
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2226 <s xml:id="echoid-s1292" xml:space="preserve">ſi <lb/>in grauitate ad humidum habeat proportionem <lb/>minorem ea, quam quadratum, quod fit ab exceſ <lb/>ſu, quo axis maior eſt, quàm ſeſquialter eius, quæ <lb/>uſque ad axem, habet ad quadratum, quod ab <lb/>axe: </s> |
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2227 <s xml:id="echoid-s1293" xml:space="preserve">demiſſa in humidum, ita ut baſis ipſius humi <lb/>dum non contingat; </s> |
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2228 <s xml:id="echoid-s1294" xml:space="preserve">neque in rectum reſtitue-<lb/>tur, neque manebit inclinata, niſi quando axis <lb/>cum ſuperficie humidi angulum fecerit æqualẽ <lb/>ei, de quo infra dicetur.</s> |
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2229 <s xml:id="echoid-s1295" xml:space="preserve"/> |
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2230 </p> |
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2231 <p> |
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2232 <s xml:id="echoid-s1296" xml:space="preserve">SIT portio qualis dicta eſt; </s> |
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2233 <s xml:id="echoid-s1297" xml:space="preserve">ſitque b d æqualis axi: </s> |
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2234 <s xml:id="echoid-s1298" xml:space="preserve">& </s> |
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2235 <s xml:id="echoid-s1299" xml:space="preserve"><lb/>b k quidem dupla ipſius _K_ d: </s> |
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2236 <s xml:id="echoid-s1300" xml:space="preserve">r _K_ uero æqualis ei, quæ uſ-<lb/>que ad axem: </s> |
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2237 <s xml:id="echoid-s1301" xml:space="preserve">& </s> |
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2238 <s xml:id="echoid-s1302" xml:space="preserve">ſit c b ſeſquialtera b r. </s> |
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2239 <s xml:id="echoid-s1303" xml:space="preserve">erit & </s> |
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2240 <s xml:id="echoid-s1304" xml:space="preserve">c d ipſius <lb/>_k_ r ſeſquialtera. </s> |
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2241 <s xml:id="echoid-s1305" xml:space="preserve">Quam uero portionem habet portio ad <lb/> |
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2242 <anchor type="note" xlink:label="note-0056-01a" xlink:href="note-0056-01"/> |
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2243 humidum in grauitate, habeat quadratum f q ad quadra-<lb/>tum d b: </s> |
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2244 <s xml:id="echoid-s1306" xml:space="preserve">& </s> |
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2245 <s xml:id="echoid-s1307" xml:space="preserve">ſit f dupla ipſius q. </s> |
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2246 <s xml:id="echoid-s1308" xml:space="preserve">perſpicuum igitur eſt f q <lb/>ad d b proportionem minorem habere ea, quam habet <lb/>c b ad b d. </s> |
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2247 <s xml:id="echoid-s1309" xml:space="preserve">eſt enim c b exceſſus, quo axis maior eſt, quàm <lb/>ſeſquialter eins, quæ uſque ad axem: </s> |
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2248 <s xml:id="echoid-s1310" xml:space="preserve">quare f q minor eſt <lb/> |
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2249 <anchor type="note" xlink:label="note-0056-02a" xlink:href="note-0056-02"/> |
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2250 <pb o="23" file="0057" n="57" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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2251 ipſa b c: </s> |
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2252 <s xml:id="echoid-s1311" xml:space="preserve">& </s> |
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2253 <s xml:id="echoid-s1312" xml:space="preserve">idcirco f minor ipſa b r. </s> |
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2254 <s xml:id="echoid-s1313" xml:space="preserve">ſit ipſi f æqualis r ψ: <lb/></s> |
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2255 <s xml:id="echoid-s1314" xml:space="preserve"> |
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2256 <anchor type="note" xlink:label="note-0057-01a" xlink:href="note-0057-01"/> |
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2257 ducaturq; </s> |
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2258 <s xml:id="echoid-s1315" xml:space="preserve">ad b d perpendicularis ψ e, quæ posſit dimidiũ <lb/>eius, quod lineis _k_ r, ψ b continetur: </s> |
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2259 <s xml:id="echoid-s1316" xml:space="preserve">& </s> |
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2260 <s xml:id="echoid-s1317" xml:space="preserve">iungatur b c. </s> |
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2261 <s xml:id="echoid-s1318" xml:space="preserve">De-<lb/>monſtrandum eſt portionem in humidum demiſſam, ſicu-<lb/>ti dictum eſt, conſiſtere inclinatam ita, ut axis cum ſuperſi-<lb/>cie humidi angulum faciat angulo c b ψ æqualem. </s> |
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2262 <s xml:id="echoid-s1319" xml:space="preserve">demit-<lb/>tatur enim aliqua portio in humidum, ut baſis ipſius hu-<lb/>midi ſuperficiem non contingat: </s> |
|
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2263 <s xml:id="echoid-s1320" xml:space="preserve">& </s> |
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2264 <s xml:id="echoid-s1321" xml:space="preserve">ſi fieri poteſt, axis cum <lb/>ſuperficie humidi non faciat angulum æqualem angulo <lb/>e b ψ; </s> |
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2265 <s xml:id="echoid-s1322" xml:space="preserve">ſed primo maiorem. </s> |
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2266 <s xml:id="echoid-s1323" xml:space="preserve">ſecta autẽ portione plano per <lb/>axem, recto ad ſu-<lb/> |
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2267 <anchor type="figure" xlink:label="fig-0057-01a" xlink:href="fig-0057-01"/> |
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2268 perficiem humi-<lb/>di, ſit ſectio a p o l <lb/>rectanguli coni ſe <lb/>ctio: </s> |
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2269 <s xml:id="echoid-s1324" xml:space="preserve">ſuperficiei <lb/>humidi ſectio x s: <lb/></s> |
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2270 <s xml:id="echoid-s1325" xml:space="preserve">ſitq; </s> |
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2271 <s xml:id="echoid-s1326" xml:space="preserve">axis portio-<lb/>nis, & </s> |
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2272 <s xml:id="echoid-s1327" xml:space="preserve">ſectiõis dia <lb/>meter n o: </s> |
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2273 <s xml:id="echoid-s1328" xml:space="preserve">& </s> |
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2274 <s xml:id="echoid-s1329" xml:space="preserve">du-<lb/>catur p y quidem <lb/>ipſi x s æquidi-<lb/>ſtans, quæ ſectio-<lb/>nem a p o l contin <lb/>gat in p: </s> |
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2275 <s xml:id="echoid-s1330" xml:space="preserve">p m ue-<lb/>ro æquidiſtans ip-<lb/>ſi n o: </s> |
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2276 <s xml:id="echoid-s1331" xml:space="preserve">& </s> |
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2277 <s xml:id="echoid-s1332" xml:space="preserve">p i ad <lb/>n o perpendicularis. </s> |
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2278 <s xml:id="echoid-s1333" xml:space="preserve">ſit præterea b r æqualis o ω. </s> |
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2279 <s xml:id="echoid-s1334" xml:space="preserve">itemq; </s> |
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2280 <s xml:id="echoid-s1335" xml:space="preserve"><lb/> |
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2281 <anchor type="note" xlink:label="note-0057-02a" xlink:href="note-0057-02"/> |
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2282 r k ipſi t _a_ & </s> |
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2283 <s xml:id="echoid-s1336" xml:space="preserve">ω h perpendicularis ad axem. </s> |
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2284 <s xml:id="echoid-s1337" xml:space="preserve">Itaque quo-<lb/>niam ponitut axis portionis cum ſuperficie humidi facere <lb/>angulum maiorem angulo b: </s> |
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2285 <s xml:id="echoid-s1338" xml:space="preserve">erit angulus p y i angulo b <lb/> |
|
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|
2286 <anchor type="note" xlink:label="note-0057-03a" xlink:href="note-0057-03"/> |
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2287 maior. </s> |
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2288 <s xml:id="echoid-s1339" xml:space="preserve">maiorem ergo proportionem habet quadratum <lb/>p i ad quadratum y i, quam quadratum e ψ ad ψ b qua-<lb/> |
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2289 <anchor type="note" xlink:label="note-0057-04a" xlink:href="note-0057-04"/> |
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2290 dratum. </s> |
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2291 <s xml:id="echoid-s1340" xml:space="preserve">Sed quam proportionem habet quadratum p i <lb/>ad quadratum i y, eandem linea k r habet ad lineam i y:</s> |
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2292 <s xml:id="echoid-s1341" xml:space="preserve"> |
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2293 <pb file="0058" n="58" rhead="ARCHIMEDIS"/> |
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2294 & </s> |
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2295 <s xml:id="echoid-s1342" xml:space="preserve">quam proportionem habet quadratum e ψ ad quadra-<lb/> |
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2296 <anchor type="note" xlink:label="note-0058-01a" xlink:href="note-0058-01"/> |
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2297 tum ψ b, eandem habet dimidium lineæ _k_ r ad lineã ψ b. <lb/></s> |
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2298 <s xml:id="echoid-s1343" xml:space="preserve">quare maiorem babet proportionem _k_ r ad i y, quàm di-<lb/> |
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2299 <anchor type="note" xlink:label="note-0058-02a" xlink:href="note-0058-02"/> |
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2300 midium k r ad ψ b: </s> |
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2301 <s xml:id="echoid-s1344" xml:space="preserve">& </s> |
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2302 <s xml:id="echoid-s1345" xml:space="preserve">idcirco i y minor eſt, quàm dupla <lb/> |
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2303 <anchor type="note" xlink:label="note-0058-03a" xlink:href="note-0058-03"/> |
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2304 ψ b. </s> |
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2305 <s xml:id="echoid-s1346" xml:space="preserve">eſt autem ipſius o i dupla. </s> |
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2306 <s xml:id="echoid-s1347" xml:space="preserve">ergo o i minor eſt, quàm <lb/>ψ b: </s> |
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2307 <s xml:id="echoid-s1348" xml:space="preserve">& </s> |
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2308 <s xml:id="echoid-s1349" xml:space="preserve">i ω maior, quàm ψ r. </s> |
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2309 <s xml:id="echoid-s1350" xml:space="preserve">ſed ψ r eſt æqualis ipſi f. </s> |
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2310 <s xml:id="echoid-s1351" xml:space="preserve">maior <lb/> |
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2311 <anchor type="note" xlink:label="note-0058-04a" xlink:href="note-0058-04"/> |
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2312 igitur eſt i ω, quàm f. </s> |
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2313 <s xml:id="echoid-s1352" xml:space="preserve">& </s> |
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2314 <s xml:id="echoid-s1353" xml:space="preserve">quoniam portio ad humidum in <lb/>grauitate eam ponitur habere proportionem, quam qua-<lb/>dratum f q ad quadratum b d: </s> |
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2315 <s xml:id="echoid-s1354" xml:space="preserve">quam uero proportionem <lb/>habet portio ad humidum in grauitate, eam habet pars ip <lb/>ſius demerſa ad totam portionem: </s> |
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2316 <s xml:id="echoid-s1355" xml:space="preserve">& </s> |
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2317 <s xml:id="echoid-s1356" xml:space="preserve">quam pars ipſius de-<lb/>merſa habet ad totam, eandem habet quadratum p m ad <lb/>quadratnm o n: </s> |
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2318 <s xml:id="echoid-s1357" xml:space="preserve">ſequitur quadratum p m ad quadratum <lb/>o n eam proportionem habere, quam quadratum f q ad <lb/>b d quadratum. <lb/></s> |
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2319 <s xml:id="echoid-s1358" xml:space="preserve"> |
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2320 <anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/> |
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2321 atque ideo ſ q æ-<lb/> |
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2322 <anchor type="note" xlink:label="note-0058-05a" xlink:href="note-0058-05"/> |
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2323 qualis eſt ipſi p m. <lb/></s> |
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2324 <s xml:id="echoid-s1359" xml:space="preserve">demõſtrata eſt au <lb/> |
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2325 <anchor type="note" xlink:label="note-0058-06a" xlink:href="note-0058-06"/> |
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2326 tem p h maior, <lb/>quàm f. </s> |
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2327 <s xml:id="echoid-s1360" xml:space="preserve">cõſtat igi <lb/>tur p m minorem <lb/>eſſe, quàm ſeſqui-<lb/>alterã ipſius p h: <lb/></s> |
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2328 <s xml:id="echoid-s1361" xml:space="preserve">& </s> |
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2329 <s xml:id="echoid-s1362" xml:space="preserve">idcirco p h ma <lb/>iorem, quàm du-<lb/>plam h m. </s> |
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2330 <s xml:id="echoid-s1363" xml:space="preserve">Sit p z <lb/>ipſius z m dupla. </s> |
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2331 <s xml:id="echoid-s1364" xml:space="preserve"><lb/>erit t quidem cẽ-<lb/>trũ grauitatis to-<lb/>tius ſolidi: </s> |
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2332 <s xml:id="echoid-s1365" xml:space="preserve">centrũ <lb/>eius partis, quæ intra humidum, punctumz: </s> |
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2333 <s xml:id="echoid-s1366" xml:space="preserve">reliquæ uero <lb/>partis centrum erit in linea z t producta uſque ad g. </s> |
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2334 <s xml:id="echoid-s1367" xml:space="preserve">Eodẽ <lb/> |
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2335 <anchor type="note" xlink:label="note-0058-07a" xlink:href="note-0058-07"/> |
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2336 modo demonſtrabitur linea th perpendicularis ad ſuper-<lb/>ficiem humidi. </s> |
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2337 <s xml:id="echoid-s1368" xml:space="preserve">& </s> |
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2338 <s xml:id="echoid-s1369" xml:space="preserve">portio demerſa in humido ſeretur extra |
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|
2339 <pb o="24" file="0059" n="59" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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2340 humidum ſecundum perpendicularem, quæ per z ad hu-<lb/>midi ſuperficiem ducta fuerit: </s> |
|
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2341 <s xml:id="echoid-s1370" xml:space="preserve">quæ autem eſt extra humi-<lb/>dum ſecundum eam, quæ per gintra humidum feretur. </s> |
|
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|
2342 <s xml:id="echoid-s1371" xml:space="preserve">nõ <lb/>ergo manebit portio ſic inclinata, ut ponitur: </s> |
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2343 <s xml:id="echoid-s1372" xml:space="preserve">ſed neque re <lb/>ſtituecur recta: </s> |
|
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2344 <s xml:id="echoid-s1373" xml:space="preserve">quoniam perpendicularium per z g ducta <lb/>rum, quæ quidem per z ducitur ad eas partes cadit, in qui <lb/>bus eſt l; </s> |
|
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|
2345 <s xml:id="echoid-s1374" xml:space="preserve">& </s> |
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2346 <s xml:id="echoid-s1375" xml:space="preserve">quæ per g ad eas, in quibus eſt a. </s> |
|
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2347 <s xml:id="echoid-s1376" xml:space="preserve">quare ſequi-<lb/>tur centrum z ſurſum ferri: </s> |
|
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|
2348 <s xml:id="echoid-s1377" xml:space="preserve">& </s> |
|
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|
2349 <s xml:id="echoid-s1378" xml:space="preserve">g deorſum. </s> |
|
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|
2350 <s xml:id="echoid-s1379" xml:space="preserve">ergo partes to <lb/>tius ſolidi, quæ ſunt ad a deorſum, quæ uero ad l ſurſum <lb/>ferentur. </s> |
|
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|
2351 <s xml:id="echoid-s1380" xml:space="preserve">Rurſus alia eadem ponantur: </s> |
|
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|
2352 <s xml:id="echoid-s1381" xml:space="preserve">axis autem <lb/>portionis cum ſuperficie humidi angulum faciat minorẽ <lb/>eo, qui eſt ad b. </s> |
|
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|
2353 <s xml:id="echoid-s1382" xml:space="preserve">minorem igitur proportionem habet qua <lb/> |
|
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|
2354 <anchor type="note" xlink:label="note-0059-01a" xlink:href="note-0059-01"/> |
|
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|
2355 dratum p i ad quadratum i y, quàm quadratum e ψ ad <lb/>ψ b quadratum: </s> |
|
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|
2356 <s xml:id="echoid-s1383" xml:space="preserve">quare k r ad i y minorem proportionẽ <lb/>habet, quàm dimidium k r ad ψ b: </s> |
|
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2357 <s xml:id="echoid-s1384" xml:space="preserve">& </s> |
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2358 <s xml:id="echoid-s1385" xml:space="preserve">propterea i y maior <lb/>eſt, quam dupla ψ b. </s> |
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2359 <s xml:id="echoid-s1386" xml:space="preserve">eſt autem ipſius o i dupla. </s> |
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2360 <s xml:id="echoid-s1387" xml:space="preserve">ergo o i <lb/>ipſa ψ b maior e-<lb/> |
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2361 <anchor type="figure" xlink:label="fig-0059-01a" xlink:href="fig-0059-01"/> |
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2362 rit. </s> |
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2363 <s xml:id="echoid-s1388" xml:space="preserve">ſed tota o ω eſt <lb/>æqualis ipſi r b: <lb/></s> |
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2364 <s xml:id="echoid-s1389" xml:space="preserve">& </s> |
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2365 <s xml:id="echoid-s1390" xml:space="preserve">reliqua ω i mi-<lb/>nor quàm ψ r. </s> |
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2366 <s xml:id="echoid-s1391" xml:space="preserve">qua <lb/>re & </s> |
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2367 <s xml:id="echoid-s1392" xml:space="preserve">p h minor e-<lb/>rit, quàm f. </s> |
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2368 <s xml:id="echoid-s1393" xml:space="preserve">Quòd <lb/>cum m p ipſi f q <lb/>ſit æqualis, cõſtat <lb/>p m maiorẽ eſſe, <lb/>quàm ſeſquialterã <lb/>ipſius p h: </s> |
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2369 <s xml:id="echoid-s1394" xml:space="preserve">& </s> |
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2370 <s xml:id="echoid-s1395" xml:space="preserve">p h <lb/>minorem, quam <lb/>duplam h m. </s> |
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2371 <s xml:id="echoid-s1396" xml:space="preserve">Sit <lb/>p z ipſius z m du <lb/>pla. </s> |
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2372 <s xml:id="echoid-s1397" xml:space="preserve">Rurſus to-<lb/>tius quidem ſolidi centrum grauitatis erit pũctum t; </s> |
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2373 <s xml:id="echoid-s1398" xml:space="preserve">eius <lb/>uero partis, quæ intra humidum z: </s> |
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2374 <s xml:id="echoid-s1399" xml:space="preserve">& </s> |
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2375 <s xml:id="echoid-s1400" xml:space="preserve">iuncta z t inuenia- |
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2376 <pb file="0060" n="60" rhead="ARCHIMEDIS"/> |
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2377 tur centrum grauitatis eius, quæ extra humidum in pro-<lb/>tracta, quod ſit g. </s> |
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2378 <s xml:id="echoid-s1401" xml:space="preserve">Itaque per z g ductis perpendiculari-<lb/> |
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2379 <anchor type="note" xlink:label="note-0060-01a" xlink:href="note-0060-01"/> |
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2380 bus ad humidi ſuperficiem, quæ ipſi t h æquidiſtent; </s> |
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2381 <s xml:id="echoid-s1402" xml:space="preserve">ſequi-<lb/>tur portionem ipſam non manere, fed reuolui adeo, ut a-<lb/>xis cum ſuperficie <lb/> |
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2382 <anchor type="figure" xlink:label="fig-0060-01a" xlink:href="fig-0060-01"/> |
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2383 humidi angulum <lb/>faciat maiorẽ eo, <lb/>quem nunc facit.</s> |
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2384 <s xml:id="echoid-s1403" xml:space="preserve"/> |
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2385 </p> |
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2386 <div xml:id="echoid-div91" type="float" level="2" n="1"> |
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2387 <note position="left" xlink:label="note-0056-01" xlink:href="note-0056-01a" xml:space="preserve">A</note> |
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2388 <note position="left" xlink:label="note-0056-02" xlink:href="note-0056-02a" xml:space="preserve">B</note> |
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2389 <note position="right" xlink:label="note-0057-01" xlink:href="note-0057-01a" xml:space="preserve">C</note> |
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2390 <figure xlink:label="fig-0057-01" xlink:href="fig-0057-01a"> |
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2391 <image file="0057-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0057-01"/> |
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2392 </figure> |
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2393 <note position="right" xlink:label="note-0057-02" xlink:href="note-0057-02a" xml:space="preserve">D</note> |
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2394 <note position="right" xlink:label="note-0057-03" xlink:href="note-0057-03a" xml:space="preserve">E</note> |
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2395 <note position="right" xlink:label="note-0057-04" xlink:href="note-0057-04a" xml:space="preserve">F</note> |
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2396 <note position="left" xlink:label="note-0058-01" xlink:href="note-0058-01a" xml:space="preserve">G</note> |
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2397 <note position="left" xlink:label="note-0058-02" xlink:href="note-0058-02a" xml:space="preserve">13. quin-<lb/>ti.</note> |
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2398 <note position="left" xlink:label="note-0058-03" xlink:href="note-0058-03a" xml:space="preserve">H</note> |
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2399 <note position="left" xlink:label="note-0058-04" xlink:href="note-0058-04a" xml:space="preserve">K</note> |
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2400 <figure xlink:label="fig-0058-01" xlink:href="fig-0058-01a"> |
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2401 <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0058-01"/> |
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2402 </figure> |
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2403 <note position="left" xlink:label="note-0058-05" xlink:href="note-0058-05a" xml:space="preserve">L</note> |
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2404 <note position="left" xlink:label="note-0058-06" xlink:href="note-0058-06a" xml:space="preserve">M</note> |
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2405 <note position="left" xlink:label="note-0058-07" xlink:href="note-0058-07a" xml:space="preserve">N</note> |
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2406 <note position="right" xlink:label="note-0059-01" xlink:href="note-0059-01a" xml:space="preserve">O</note> |
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2407 <figure xlink:label="fig-0059-01" xlink:href="fig-0059-01a"> |
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2408 <image file="0059-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0059-01"/> |
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2409 </figure> |
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2410 <note position="left" xlink:label="note-0060-01" xlink:href="note-0060-01a" xml:space="preserve">P</note> |
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2411 <figure xlink:label="fig-0060-01" xlink:href="fig-0060-01a"> |
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2412 <image file="0060-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0060-01"/> |
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2413 </figure> |
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|
2414 </div> |
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2415 <p> |
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2416 <s xml:id="echoid-s1404" xml:space="preserve">Et quoniam cũ <lb/>antea poſuiſſem´<unsure/> <lb/>facere angulũ ma <lb/>iorem angulo b, <lb/>portio neque tũc <lb/>cõſiſtebat; </s> |
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2417 <s xml:id="echoid-s1405" xml:space="preserve">perſpi <lb/>cuũ eſt ipſam con <lb/>ſiſtere, ſi angulum <lb/>fecerit angulo b <lb/>æqualem. </s> |
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2418 <s xml:id="echoid-s1406" xml:space="preserve">Sic e-<lb/> |
|
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|
2419 <anchor type="note" xlink:label="note-0060-02a" xlink:href="note-0060-02"/> |
|
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2420 nim eriti o æqua-<lb/>lis ψ b: </s> |
|
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2421 <s xml:id="echoid-s1407" xml:space="preserve">itemq; </s> |
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2422 <s xml:id="echoid-s1408" xml:space="preserve">ω i <lb/>æqualis ψ r: </s> |
|
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2423 <s xml:id="echoid-s1409" xml:space="preserve">& </s> |
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2424 <s xml:id="echoid-s1410" xml:space="preserve">p h ipſi f. </s> |
|
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|
2425 <s xml:id="echoid-s1411" xml:space="preserve">erit igitur m p ſeſquialtera p h; <lb/></s> |
|
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|
2426 <s xml:id="echoid-s1412" xml:space="preserve">& </s> |
|
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|
2427 <s xml:id="echoid-s1413" xml:space="preserve">p h dupla h m. </s> |
|
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2428 <s xml:id="echoid-s1414" xml:space="preserve">quare cum h ſit centrum grauitatis eius <lb/>partis, quæ eſt in humido, per eandem perpendicularem, <lb/>& </s> |
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2429 <s xml:id="echoid-s1415" xml:space="preserve">ipſa ſurſum, & </s> |
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2430 <s xml:id="echoid-s1416" xml:space="preserve">quæ extra eſt feretur deorſum. </s> |
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2431 <s xml:id="echoid-s1417" xml:space="preserve">mane-<lb/>bitigitur portio; </s> |
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2432 <s xml:id="echoid-s1418" xml:space="preserve">quoniam altera pars ab altera non re-<lb/>pelletur.</s> |
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2433 <s xml:id="echoid-s1419" xml:space="preserve"/> |
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2434 </p> |
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2435 <div xml:id="echoid-div92" type="float" level="2" n="2"> |
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2436 <note position="left" xlink:label="note-0060-02" xlink:href="note-0060-02a" xml:space="preserve">Q</note> |
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2437 </div> |
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2438 </div> |
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2439 <div xml:id="echoid-div94" type="section" level="1" n="37"> |
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2440 <head xml:id="echoid-head42" xml:space="preserve">COMMENTARIVS.</head> |
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2441 <p style="it"> |
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2442 <s xml:id="echoid-s1420" xml:space="preserve">_E T ſit c b ſeſquialtera b r. </s> |
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2443 <s xml:id="echoid-s1421" xml:space="preserve">erit & </s> |
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2444 <s xml:id="echoid-s1422" xml:space="preserve">c d ipſius k r ſeſqui-_ <lb/> |
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2445 <anchor type="note" xlink:label="note-0060-03a" xlink:href="note-0060-03"/> |
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2446 _altera_.</s> |
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2447 <s xml:id="echoid-s1423" xml:space="preserve">] In translatione ita legebatur. </s> |
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2448 <s xml:id="echoid-s1424" xml:space="preserve">ſit autem & </s> |
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2449 <s xml:id="echoid-s1425" xml:space="preserve">c b quidem <lb/>hemiolia ipſius b r: </s> |
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2450 <s xml:id="echoid-s1426" xml:space="preserve">c d autem ipſius K r. </s> |
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2451 <s xml:id="echoid-s1427" xml:space="preserve">Sed nos quod postremo <lb/>loco legitur, idcirco corrigendum duximus, quoniam illud non po-<lb/>nitur ita eſſe, ſed ex ijs, quæ poſita ſunt, neceſſario colligitur. </s> |
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2452 <s xml:id="echoid-s1428" xml:space="preserve">ſi enim |
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2453 <pb o="25" file="0061" n="61" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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2454 b ψ dupla ſit ψ d, erit d b ipſius b ψ ſeſquialtera. </s> |
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2455 <s xml:id="echoid-s1429" xml:space="preserve">& </s> |
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2456 <s xml:id="echoid-s1430" xml:space="preserve">quoniam e b ſeſ <lb/>quialtera est b r, ſequitur reliquam c d ipſius ψ r, boc est eius, quæ <lb/> |
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2457 <anchor type="note" xlink:label="note-0061-01a" xlink:href="note-0061-01"/> |
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2458 uſque ad axem ſeſquialteram eſſe. </s> |
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2459 <s xml:id="echoid-s1431" xml:space="preserve">quare b c erit exceſſus, quo axis <lb/>maior est, quàm ſeſquialter eius, quæ uſque ad axem.</s> |
|
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2460 <s xml:id="echoid-s1432" xml:space="preserve"/> |
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2461 </p> |
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2462 <div xml:id="echoid-div94" type="float" level="2" n="1"> |
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2463 <note position="left" xlink:label="note-0060-03" xlink:href="note-0060-03a" xml:space="preserve">A</note> |
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2464 <note position="right" xlink:label="note-0061-01" xlink:href="note-0061-01a" xml:space="preserve">12. quinti</note> |
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2465 </div> |
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2466 <p style="it"> |
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2467 <s xml:id="echoid-s1433" xml:space="preserve">_Quare f q minor éſtipſa b c.</s> |
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2468 <s xml:id="echoid-s1434" xml:space="preserve">]_ Nam cum portio ad bumi-<lb/> |
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2469 <anchor type="note" xlink:label="note-0061-02a" xlink:href="note-0061-02"/> |
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2470 dum in grauitate proportionem habeat eandem, quàm quadratum <lb/>f q ad quadratum d b: </s> |
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2471 <s xml:id="echoid-s1435" xml:space="preserve">habeatq, minorem proportionem, quàm qua <lb/>dratum factum ab exceſſu, quo axis maior eſt, quàm ſeſquialter eius, <lb/>quæ uſque ad axem, ad quadratum ab axe; </s> |
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2472 <s xml:id="echoid-s1436" xml:space="preserve">boc eſt minorem, quàm <lb/>quadratum c b ad quadratum b d: </s> |
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2473 <s xml:id="echoid-s1437" xml:space="preserve">ponitur enim linea b d æqualis <lb/>axi: </s> |
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2474 <s xml:id="echoid-s1438" xml:space="preserve">quadratum f q ad quadratum d b proportionem minorem ha-<lb/>bebit, quàm quadratum c b ad idem b d quadratum. </s> |
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2475 <s xml:id="echoid-s1439" xml:space="preserve">ergo quadra-<lb/> |
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2476 <anchor type="note" xlink:label="note-0061-03a" xlink:href="note-0061-03"/> |
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2477 tum f q minus erit quadrato c b: </s> |
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2478 <s xml:id="echoid-s1440" xml:space="preserve">& </s> |
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2479 <s xml:id="echoid-s1441" xml:space="preserve">propterea linea f q ipſa b c <lb/>minor.</s> |
|
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2480 <s xml:id="echoid-s1442" xml:space="preserve"/> |
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|
2481 </p> |
|
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2482 <div xml:id="echoid-div95" type="float" level="2" n="2"> |
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2483 <note position="right" xlink:label="note-0061-02" xlink:href="note-0061-02a" xml:space="preserve">B</note> |
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2484 <note position="right" xlink:label="note-0061-03" xlink:href="note-0061-03a" xml:space="preserve">8. quinti.</note> |
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|
2485 </div> |
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|
2486 <p style="it"> |
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2487 <s xml:id="echoid-s1443" xml:space="preserve">_Etidcirco f minor ipſa b r.</s> |
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2488 <s xml:id="echoid-s1444" xml:space="preserve">]_ Quoniam enim c b ſeſquial-<lb/> |
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2489 <anchor type="note" xlink:label="note-0061-04a" xlink:href="note-0061-04"/> |
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|
2490 tera eſt b r, & </s> |
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2491 <s xml:id="echoid-s1445" xml:space="preserve">f q ipſius f ſeſquialtera: </s> |
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2492 <s xml:id="echoid-s1446" xml:space="preserve">estq; </s> |
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2493 <s xml:id="echoid-s1447" xml:space="preserve">f q minor b c; </s> |
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2494 <s xml:id="echoid-s1448" xml:space="preserve">& </s> |
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2495 <s xml:id="echoid-s1449" xml:space="preserve">f <lb/> |
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|
2496 <anchor type="note" xlink:label="note-0061-05a" xlink:href="note-0061-05"/> |
|
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|
2497 ipſa b r minor erit.</s> |
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|
2498 <s xml:id="echoid-s1450" xml:space="preserve"/> |
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|
2499 </p> |
|
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|
2500 <div xml:id="echoid-div96" type="float" level="2" n="3"> |
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|
2501 <note position="right" xlink:label="note-0061-04" xlink:href="note-0061-04a" xml:space="preserve">C</note> |
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2502 <note position="right" xlink:label="note-0061-05" xlink:href="note-0061-05a" xml:space="preserve">14. quin-<lb/>ti.</note> |
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2503 </div> |
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2504 <p style="it"> |
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2505 <s xml:id="echoid-s1451" xml:space="preserve">_Itaque quoniam ponitur axis portionis cum ſuperficie_ <lb/> |
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2506 <anchor type="note" xlink:label="note-0061-06a" xlink:href="note-0061-06"/> |
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2507 _humidi facere angulum maiorem angulo b: </s> |
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2508 <s xml:id="echoid-s1452" xml:space="preserve">erit angulus_ <lb/>_p y i angulo b maior.</s> |
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2509 <s xml:id="echoid-s1453" xml:space="preserve">]_ Nam cum linea p y ſuperficiei bumidi <lb/>æ quidistet; </s> |
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2510 <s xml:id="echoid-s1454" xml:space="preserve">uidelicet ipſi x s: </s> |
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2511 <s xml:id="echoid-s1455" xml:space="preserve">angulus p y i æqualis erit angulo, qui <lb/> |
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2512 <anchor type="note" xlink:label="note-0061-07a" xlink:href="note-0061-07"/> |
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2513 diametro portionis n o, & </s> |
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2514 <s xml:id="echoid-s1456" xml:space="preserve">linea x s continetur. </s> |
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2515 <s xml:id="echoid-s1457" xml:space="preserve">quare & </s> |
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2516 <s xml:id="echoid-s1458" xml:space="preserve">angulo <lb/>b maior erit.</s> |
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2517 <s xml:id="echoid-s1459" xml:space="preserve"/> |
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2518 </p> |
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2519 <div xml:id="echoid-div97" type="float" level="2" n="4"> |
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2520 <note position="right" xlink:label="note-0061-06" xlink:href="note-0061-06a" xml:space="preserve">D</note> |
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2521 <note position="right" xlink:label="note-0061-07" xlink:href="note-0061-07a" xml:space="preserve">29. primi</note> |
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2522 </div> |
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2523 <p style="it"> |
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2524 <s xml:id="echoid-s1460" xml:space="preserve">_Maiorem igitur proportionem habet quadratum p i ad_ <lb/> |
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2525 <anchor type="note" xlink:label="note-0061-08a" xlink:href="note-0061-08"/> |
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2526 _quadratum i y, quàm quadratum e ψ ad ψ b quadratu.</s> |
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2527 <s xml:id="echoid-s1461" xml:space="preserve">]_ <lb/>Deſcribantur ſeorſum triangula p i y, e ψ b. </s> |
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2528 <s xml:id="echoid-s1462" xml:space="preserve">& </s> |
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2529 <s xml:id="echoid-s1463" xml:space="preserve">cum angulus p y i <lb/>maior ſit angulo e b ψ, ad lineam i y, atque ad punctum y in ea da-<lb/>tum fiat angulus u y i æqualis angulo e b ψ. </s> |
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2530 <s xml:id="echoid-s1464" xml:space="preserve">est autem angulus ad <lb/>i rectus æqualis recto ad ψ. </s> |
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2531 <s xml:id="echoid-s1465" xml:space="preserve">reliquus igitur y u i reliquo b c ψ est <lb/>æqualis. </s> |
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2532 <s xml:id="echoid-s1466" xml:space="preserve">quare linea u i ad lineam i y eandem proportionem ha-<lb/> |
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2533 <anchor type="note" xlink:label="note-0061-09a" xlink:href="note-0061-09"/> |
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2534 bet, quam linea e ψ ad ψ b. </s> |
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2535 <s xml:id="echoid-s1467" xml:space="preserve">Sed linea p i, quæ maior est ipſa u i ad <lb/> |
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2536 <anchor type="note" xlink:label="note-0061-10a" xlink:href="note-0061-10"/> |
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2537 lineam in maiorem habet proportionem quam u i ad eandem. </s> |
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2538 <s xml:id="echoid-s1468" xml:space="preserve">ergo <lb/> |
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2539 <anchor type="note" xlink:label="note-0061-11a" xlink:href="note-0061-11"/> |
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2540 p i ad i y maiorem proportionem habebit, quàm e ψ ad ψ b: </s> |
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2541 <s xml:id="echoid-s1469" xml:space="preserve">& </s> |
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2542 <s xml:id="echoid-s1470" xml:space="preserve"><lb/>propterea quadratum p i ad quadratum i y maiorem habebit, quàm |
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2543 <pb file="0062" n="62" rhead="ARCHIMEDIS"/> |
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2544 quadratum e ψ ad quadr. </s> |
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2545 <s xml:id="echoid-s1471" xml:space="preserve">itum ψ b.</s> |
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2546 <s xml:id="echoid-s1472" xml:space="preserve"/> |
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2547 </p> |
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2548 <div xml:id="echoid-div98" type="float" level="2" n="5"> |
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2549 <note position="right" xlink:label="note-0061-08" xlink:href="note-0061-08a" xml:space="preserve">E</note> |
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2550 <note position="right" xlink:label="note-0061-09" xlink:href="note-0061-09a" xml:space="preserve">4. ſexti.</note> |
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2551 <note position="right" xlink:label="note-0061-10" xlink:href="note-0061-10a" xml:space="preserve">8. quinti.</note> |
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2552 <note position="right" xlink:label="note-0061-11" xlink:href="note-0061-11a" xml:space="preserve">13. quin-<lb/>ti.</note> |
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2553 </div> |
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2554 <p style="it"> |
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2555 <s xml:id="echoid-s1473" xml:space="preserve">_Sed quam proportionem habet qua-_ <lb/> |
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2556 <anchor type="figure" xlink:label="fig-0062-01a" xlink:href="fig-0062-01"/> |
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2557 <anchor type="note" xlink:label="note-0062-01a" xlink:href="note-0062-01"/> |
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2558 _dratum p i ad quadratum i y, eandem li_ <lb/>_nea k r habet ad lineam i y.</s> |
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2559 <s xml:id="echoid-s1474" xml:space="preserve">]_ Est enim ex <lb/>undecima primi conicorum quadratum p i æqua <lb/>le rectangulo contento linea i o, & </s> |
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2560 <s xml:id="echoid-s1475" xml:space="preserve">ea, iuxta quam poſſunt quæ à <lb/>ſectione ad diametrum ducuntur, uidelicet duplaipſius k r. </s> |
|
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|
2561 <s xml:id="echoid-s1476" xml:space="preserve">atque <lb/>est i y dupla i o, extrigeſimatertia eiuſdem: </s> |
|
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2562 <s xml:id="echoid-s1477" xml:space="preserve">quare ex decimaſext a <lb/>ſexti elementorum, rectangulum, quod fit ex k r, & </s> |
|
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2563 <s xml:id="echoid-s1478" xml:space="preserve">i y æ quale eſt <lb/>rectangulo contento linea i o & </s> |
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2564 <s xml:id="echoid-s1479" xml:space="preserve">ea, iuxta quam poſſunt: </s> |
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2565 <s xml:id="echoid-s1480" xml:space="preserve">hoc eſt qua <lb/>drato p i. </s> |
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2566 <s xml:id="echoid-s1481" xml:space="preserve">Sed ut rectangulnm ex k r, & </s> |
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2567 <s xml:id="echoid-s1482" xml:space="preserve">i y ad quadratum i y, ita <lb/> |
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|
2568 <anchor type="note" xlink:label="note-0062-02a" xlink:href="note-0062-02"/> |
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|
2569 linea κ r ad ipſam i y. </s> |
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|
2570 <s xml:id="echoid-s1483" xml:space="preserve">ergo linea κ r ad i y eandem proportionem <lb/>habebit, quam rectangulum ex κ r & </s> |
|
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|
2571 <s xml:id="echoid-s1484" xml:space="preserve">i y, hoc eſt quadratum p i ad <lb/>quadratum i y.</s> |
|
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|
2572 <s xml:id="echoid-s1485" xml:space="preserve"/> |
|
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|
2573 </p> |
|
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|
2574 <div xml:id="echoid-div99" type="float" level="2" n="6"> |
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2575 <figure xlink:label="fig-0062-01" xlink:href="fig-0062-01a"> |
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2576 <image file="0062-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0062-01"/> |
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2577 </figure> |
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2578 <note position="left" xlink:label="note-0062-01" xlink:href="note-0062-01a" xml:space="preserve">F</note> |
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2579 <note position="left" xlink:label="note-0062-02" xlink:href="note-0062-02a" xml:space="preserve">lem. 22. <lb/>decimi.</note> |
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2580 </div> |
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2581 <p> |
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2582 <s xml:id="echoid-s1486" xml:space="preserve">Et quam proportionem habet quadratũ e ψ ad quadra <lb/> |
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2583 <anchor type="note" xlink:label="note-0062-03a" xlink:href="note-0062-03"/> |
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2584 tum ψ b, eandem habet dimidium lineæ K r ad lineã ψ b.</s> |
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2585 <s xml:id="echoid-s1487" xml:space="preserve">]</s> |
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2586 </p> |
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2587 <div xml:id="echoid-div100" type="float" level="2" n="7"> |
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2588 <note position="left" xlink:label="note-0062-03" xlink:href="note-0062-03a" xml:space="preserve">G</note> |
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2589 </div> |
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2590 <p style="it"> |
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2591 <s xml:id="echoid-s1488" xml:space="preserve">Nam cum quadratum e ψ poſitum ſit æquale dimidio rectanguli <lb/>contenti linea κ r, & </s> |
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2592 <s xml:id="echoid-s1489" xml:space="preserve">ψ b; </s> |
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2593 <s xml:id="echoid-s1490" xml:space="preserve">hoc est ei, quod dimidia ipſius κ r <lb/>& </s> |
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2594 <s xml:id="echoid-s1491" xml:space="preserve">linea ψ b continetur: </s> |
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2595 <s xml:id="echoid-s1492" xml:space="preserve">& </s> |
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2596 <s xml:id="echoid-s1493" xml:space="preserve">ut rectangulum ex dimidia κ r, & </s> |
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2597 <s xml:id="echoid-s1494" xml:space="preserve">ψ b <lb/> |
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2598 <anchor type="note" xlink:label="note-0062-04a" xlink:href="note-0062-04"/> |
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2599 ad quadratum ψ b, ita ſit dimidia κ r ad line am ψ b: </s> |
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2600 <s xml:id="echoid-s1495" xml:space="preserve">habebit dimi-<lb/>dia κ r ad ψ b proportionem eandem, quam quadratum e ψ ad qua-<lb/>dratum ψ b.</s> |
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2601 <s xml:id="echoid-s1496" xml:space="preserve"/> |
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2602 </p> |
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2603 <div xml:id="echoid-div101" type="float" level="2" n="8"> |
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2604 <note position="left" xlink:label="note-0062-04" xlink:href="note-0062-04a" xml:space="preserve">lem. 22. <lb/>decimi</note> |
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2605 </div> |
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2606 <p style="it"> |
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2607 <s xml:id="echoid-s1497" xml:space="preserve">_Etidcirco i y minor eſt, quàm dupla ψ b.</s> |
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2608 <s xml:id="echoid-s1498" xml:space="preserve">]_ Quam enim pro <lb/> |
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2609 <anchor type="note" xlink:label="note-0062-05a" xlink:href="note-0062-05"/> |
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2610 portionem habet dimidium κ r ad ψ b, habeat κ r ad aliam lineam. <lb/></s> |
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2611 <s xml:id="echoid-s1499" xml:space="preserve">erit ea maior, quàm i y; </s> |
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2612 <s xml:id="echoid-s1500" xml:space="preserve">nempe ad quam κ r minorem proportioné <lb/> |
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2613 <anchor type="note" xlink:label="note-0062-06a" xlink:href="note-0062-06"/> |
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2614 habet: </s> |
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2615 <s xml:id="echoid-s1501" xml:space="preserve">at que erit dupla ψ b. </s> |
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2616 <s xml:id="echoid-s1502" xml:space="preserve">ergo i y minor eſt, quam dupla ψ b.</s> |
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2617 <s xml:id="echoid-s1503" xml:space="preserve"/> |
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2618 </p> |
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2619 <div xml:id="echoid-div102" type="float" level="2" n="9"> |
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2620 <note position="left" xlink:label="note-0062-05" xlink:href="note-0062-05a" xml:space="preserve">H</note> |
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2621 <note position="left" xlink:label="note-0062-06" xlink:href="note-0062-06a" xml:space="preserve">10. quinti.</note> |
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2622 </div> |
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2623 <p style="it"> |
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2624 <s xml:id="echoid-s1504" xml:space="preserve">_Et i ω maior, quam ψ r.</s> |
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2625 <s xml:id="echoid-s1505" xml:space="preserve">]_ Cum enim o ω poſita ſit æ qualis b r <lb/> |
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2626 <anchor type="note" xlink:label="note-0062-07a" xlink:href="note-0062-07"/> |
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2627 ſi ex b r dematur ψ b, & </s> |
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2628 <s xml:id="echoid-s1506" xml:space="preserve">ex o ω dematur o i, quæ minor eſt ψ b: </s> |
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2629 <s xml:id="echoid-s1507" xml:space="preserve">erit <lb/>reliqua i ω maior reliqua ψ r.</s> |
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2630 <s xml:id="echoid-s1508" xml:space="preserve"/> |
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2631 </p> |
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2632 <div xml:id="echoid-div103" type="float" level="2" n="10"> |
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2633 <note position="left" xlink:label="note-0062-07" xlink:href="note-0062-07a" xml:space="preserve">K</note> |
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2634 </div> |
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2635 <p style="it"> |
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2636 <s xml:id="echoid-s1509" xml:space="preserve">_Atqueideo f q æqualis eſt ipſi p m.</s> |
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2637 <s xml:id="echoid-s1510" xml:space="preserve">]_ Ex decimaquarta <lb/> |
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2638 <anchor type="note" xlink:label="note-0062-08a" xlink:href="note-0062-08"/> |
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2639 quinti elementorum, nam linea o n ipſi b d eſt æ qualis.</s> |
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2640 <s xml:id="echoid-s1511" xml:space="preserve"/> |
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|
2641 </p> |
|
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|
2642 <div xml:id="echoid-div104" type="float" level="2" n="11"> |
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2643 <note position="left" xlink:label="note-0062-08" xlink:href="note-0062-08a" xml:space="preserve">L</note> |
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|
2644 </div> |
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2645 <p style="it"> |
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2646 <s xml:id="echoid-s1512" xml:space="preserve">_Demonſtrata eſt autem p h maior, quàm f.</s> |
|
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2647 <s xml:id="echoid-s1513" xml:space="preserve">]_ Etenim de-<lb/> |
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2648 <anchor type="note" xlink:label="note-0062-09a" xlink:href="note-0062-09"/> |
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|
2649 monstrata est i ω maior, quàm f; </s> |
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2650 <s xml:id="echoid-s1514" xml:space="preserve">atque est p h æqualis ipſi i ω.</s> |
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2651 <s xml:id="echoid-s1515" xml:space="preserve"/> |
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2652 </p> |
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2653 <div xml:id="echoid-div105" type="float" level="2" n="12"> |
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2654 <note position="left" xlink:label="note-0062-09" xlink:href="note-0062-09a" xml:space="preserve">M</note> |
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2655 </div> |
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2656 <p style="it"> |
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2657 <s xml:id="echoid-s1516" xml:space="preserve">_Eodem modo demonſtrabitur t h perpendicularis ad_ <lb/> |
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2658 <anchor type="note" xlink:label="note-0062-10a" xlink:href="note-0062-10"/> |
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2659 <pb o="26" file="0063" n="63" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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2660 _humidi ſuperficiem.</s> |
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2661 <s xml:id="echoid-s1517" xml:space="preserve">]_ Est enim t ω æqualis κ r, hoc eſt ei, quæ <lb/>uſque ad axem. </s> |
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2662 <s xml:id="echoid-s1518" xml:space="preserve">quare ex ijs, quæ ſuperius demonſtrata ſunt, linea <lb/>t h ducta erit ad humidi ſuperficiem perpendicularis.</s> |
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2663 <s xml:id="echoid-s1519" xml:space="preserve"/> |
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2664 </p> |
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2665 <div xml:id="echoid-div106" type="float" level="2" n="13"> |
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2666 <note position="left" xlink:label="note-0062-10" xlink:href="note-0062-10a" xml:space="preserve">N</note> |
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2667 </div> |
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2668 <p style="it"> |
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2669 <s xml:id="echoid-s1520" xml:space="preserve">_Minorem igitur proportionem habet quadratum p i_ <lb/> |
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2670 <anchor type="note" xlink:label="note-0063-01a" xlink:href="note-0063-01"/> |
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2671 _ad quadratum i y, quàm quadratum e ψ ad ψ b quadratũ]_ <lb/>Hæc & </s> |
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2672 <s xml:id="echoid-s1521" xml:space="preserve">alia, quæ ſequuntur, tum in hac, tum in ſequenti propoſitio-<lb/>ne non alio, quàm quo ſupra modo demonstrabimus.</s> |
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2673 <s xml:id="echoid-s1522" xml:space="preserve"/> |
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2674 </p> |
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2675 <div xml:id="echoid-div107" type="float" level="2" n="14"> |
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2676 <note position="right" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve">O</note> |
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2677 </div> |
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2678 <p style="it"> |
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2679 <s xml:id="echoid-s1523" xml:space="preserve">_Itaque per z g ductis perpendicularibus ad humidi ſu-_ <lb/> |
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2680 <anchor type="note" xlink:label="note-0063-02a" xlink:href="note-0063-02"/> |
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2681 _perficiem, quæ i pſi t h æ quidiftent; </s> |
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2682 <s xml:id="echoid-s1524" xml:space="preserve">ſequitur portionem ip_ <lb/>_ſam non manere, ſed reuolui adeo, ut axis cum ſuperſicie_ <lb/>_humidi angulum faciat maiorem eo, quem nunc facit.</s> |
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2683 <s xml:id="echoid-s1525" xml:space="preserve">]_ <lb/>Nam cum perpendicularis, quæ per g, ducitur ad eas partes cadat, <lb/>in quibus eſt l; </s> |
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2684 <s xml:id="echoid-s1526" xml:space="preserve">quæ autem per Z ad eis in quibus a: </s> |
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2685 <s xml:id="echoid-s1527" xml:space="preserve">neceſſarium eſt <lb/>centrum g deorſum ferri, & </s> |
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2686 <s xml:id="echoid-s1528" xml:space="preserve">Z ſurſum. </s> |
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2687 <s xml:id="echoid-s1529" xml:space="preserve">quare partes ſolidi, quæ <lb/>ſunt ad l deorſum; </s> |
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2688 <s xml:id="echoid-s1530" xml:space="preserve">quæ uero ad a ſurſum ferentur, ut axis cum ſu-<lb/>perficie humidi maiorem angulum contineat.</s> |
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2689 <s xml:id="echoid-s1531" xml:space="preserve"/> |
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2690 </p> |
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2691 <div xml:id="echoid-div108" type="float" level="2" n="15"> |
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2692 <note position="right" xlink:label="note-0063-02" xlink:href="note-0063-02a" xml:space="preserve">P</note> |
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2693 </div> |
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2694 <p> |
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2695 <s xml:id="echoid-s1532" xml:space="preserve">Sic enim erit i o æ qualis ψ b, itẽq; </s> |
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2696 <s xml:id="echoid-s1533" xml:space="preserve">ω i æ qualis ψ r, & </s> |
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2697 <s xml:id="echoid-s1534" xml:space="preserve">p h <lb/> |
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2698 <anchor type="note" xlink:label="note-0063-03a" xlink:href="note-0063-03"/> |
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2699 ipſi f.</s> |
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2700 <s xml:id="echoid-s1535" xml:space="preserve">] _Hoc in tertia figura, quam nos addidimus, perſpicue apparet_.</s> |
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2701 <s xml:id="echoid-s1536" xml:space="preserve"/> |
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2702 </p> |
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2703 <div xml:id="echoid-div109" type="float" level="2" n="16"> |
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2704 <note position="right" xlink:label="note-0063-03" xlink:href="note-0063-03a" xml:space="preserve">Q</note> |
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2705 </div> |
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2706 </div> |
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2707 <div xml:id="echoid-div111" type="section" level="1" n="38"> |
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2708 <head xml:id="echoid-head43" xml:space="preserve">PROPOSITIO IX.</head> |
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2709 <p> |
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2710 <s xml:id="echoid-s1537" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>axem habuerit maiorem quidem, quàm ſeſquial-<lb/>terum eius, quæ uſque ad axem; </s> |
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2711 <s xml:id="echoid-s1538" xml:space="preserve">minorem uero, <lb/>quàm ut ad eam, quæ uſque ad axem proportio-<lb/>nem habeat, quam quindecim ad quatuor; </s> |
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2712 <s xml:id="echoid-s1539" xml:space="preserve">& </s> |
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2713 <s xml:id="echoid-s1540" xml:space="preserve">in <lb/>grauitate ad humidum proportionem habeat ma <lb/>iorem, quàm exceſſus, quo quadratum, quod fit <lb/>ab axe maius eſt quadrato, quod ab exceſſu, quo <lb/>axis eſt maior, quàm ſeſquialter eius, quæ uſq; </s> |
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2714 <s xml:id="echoid-s1541" xml:space="preserve">ad <lb/>axem, habet ad quadratum, quod ab axe: </s> |
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2715 <s xml:id="echoid-s1542" xml:space="preserve">in hu- |
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|
2716 <pb file="0064" n="64" rhead="ARCHIMEDIS"/> |
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2717 midum demiſſa adeo, ut baſis ipſius tota ſit in hu <lb/>mido, & </s> |
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2718 <s xml:id="echoid-s1543" xml:space="preserve">poſita inclinata, nec conuertetur ita, ut <lb/>axis ipſius ſecũdum perpendicularem ſit, nec ma <lb/>nebit inclinata, niſi quãdo axis cum ſuperficie hu <lb/>midi angulum fecerit æqualem angulo ſimiliter <lb/>ut prius, aſſumpto.</s> |
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2719 <s xml:id="echoid-s1544" xml:space="preserve"/> |
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|
2720 </p> |
|
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|
2721 <p> |
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|
2722 <s xml:id="echoid-s1545" xml:space="preserve">SIT portio, qualis dicta eſt: </s> |
|
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2723 <s xml:id="echoid-s1546" xml:space="preserve">ponaturq; </s> |
|
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|
2724 <s xml:id="echoid-s1547" xml:space="preserve">d b æ qualis axi <lb/>portionis: </s> |
|
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|
2725 <s xml:id="echoid-s1548" xml:space="preserve">& </s> |
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|
2726 <s xml:id="echoid-s1549" xml:space="preserve">b _k_ quidem ſit dupla ipſius _k_ d; </s> |
|
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|
2727 <s xml:id="echoid-s1550" xml:space="preserve">k r autem <lb/>æqualis ei, quæ uſque ad axem: </s> |
|
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|
2728 <s xml:id="echoid-s1551" xml:space="preserve">& </s> |
|
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parents:
diff
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|
2729 <s xml:id="echoid-s1552" xml:space="preserve">c b ſeſquialtera b r. <lb/></s> |
|
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|
2730 <s xml:id="echoid-s1553" xml:space="preserve">Quam uero proportionem habet portio ad humidum in <lb/>grauitate, eam habeat exceſſus, quo quadratum b d exce-<lb/>dit quadratum f q, ad ipſum b d quadratum: </s> |
|
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|
2731 <s xml:id="echoid-s1554" xml:space="preserve">& </s> |
|
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diff
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|
2732 <s xml:id="echoid-s1555" xml:space="preserve">ſit f ipſius <lb/>q dupla. </s> |
|
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diff
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|
2733 <s xml:id="echoid-s1556" xml:space="preserve">conſtat igitur exceſſum, quo quadratum b d ex-<lb/>cedit quadratum <lb/> |
|
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|
2734 <anchor type="figure" xlink:label="fig-0064-01a" xlink:href="fig-0064-01"/> |
|
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|
2735 b c ad quadratum <lb/>b d, minorem ha-<lb/>bere proportio-<lb/>nem, quàm exceſ-<lb/>ſus, quo quadratũ <lb/>b d excedit qua-<lb/>dratum f q ad b d <lb/>quadratum. </s> |
|
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|
2736 <s xml:id="echoid-s1557" xml:space="preserve">eſt e-<lb/>nim b c exceſſus <lb/>quo axis portiõis <lb/>maior eſt, quã ſeſ-<lb/>quialter eius, quæ <lb/>uſque ad axem. <lb/></s> |
|
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|
2737 <s xml:id="echoid-s1558" xml:space="preserve">quare quadr atum <lb/> |
|
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|
2738 <anchor type="note" xlink:label="note-0064-01a" xlink:href="note-0064-01"/> |
|
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|
2739 b d magis excedit <lb/>quadratum f q, quàm b c quadratum: </s> |
|
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|
2740 <s xml:id="echoid-s1559" xml:space="preserve">& </s> |
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diff
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|
2741 <s xml:id="echoid-s1560" xml:space="preserve">idcirco linea f q <lb/>minor eſt, quàm b c itemq; </s> |
|
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|
2742 <s xml:id="echoid-s1561" xml:space="preserve">f minor, quàm br. </s> |
|
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|
2743 <s xml:id="echoid-s1562" xml:space="preserve">Sit ipſi ſ |
|
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|
2744 <pb o="27" file="0065" n="65" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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|
2745 æqualis r ψ: </s> |
|
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|
2746 <s xml:id="echoid-s1563" xml:space="preserve">& </s> |
|
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diff
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|
2747 <s xml:id="echoid-s1564" xml:space="preserve">ducatur ψ r perpendicularis ad b d, quæ <lb/>posſit dimidium eius, quod ipſis k r, ψ b, continetur. </s> |
|
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|
2748 <s xml:id="echoid-s1565" xml:space="preserve">Dico <lb/>portionem in humidum demiſſam adeo, ut baſis ipſius to-<lb/>ta ſit in humido, ita conſiſtere, ut axis cum ſuperficie humi <lb/>di faciat angulum angulo b æqualem. </s> |
|
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|
2749 <s xml:id="echoid-s1566" xml:space="preserve">Demittatur enim <lb/>portio in humidum, ſicuti dictum eſt; </s> |
|
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|
2750 <s xml:id="echoid-s1567" xml:space="preserve">& </s> |
|
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|
2751 <s xml:id="echoid-s1568" xml:space="preserve">axis cum humidi <lb/>ſuperficie non faciat angulum æqualẽ ipſi b, ſed primo ma <lb/>iorem: </s> |
|
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|
2752 <s xml:id="echoid-s1569" xml:space="preserve">ſecta autem ipſa plano per axem, recto ad ſuperfi-<lb/>ciem humidi, ſectio portionis ſit a p o l rectanguli coni ſe-<lb/>ctio; </s> |
|
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|
2753 <s xml:id="echoid-s1570" xml:space="preserve">ſuperficiei humidi ſectio c i; </s> |
|
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diff
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|
2754 <s xml:id="echoid-s1571" xml:space="preserve">ſitq, axis portionis, & </s> |
|
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|
2755 <s xml:id="echoid-s1572" xml:space="preserve">ſe <lb/>ctionis diameter n o, quæ fecetur in punctis ω t, ut prius. </s> |
|
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2756 <s xml:id="echoid-s1573" xml:space="preserve">& </s> |
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|
2757 <s xml:id="echoid-s1574" xml:space="preserve"><lb/>ducantur y p quidem ipſi ci æquidiſtans, contingensq; </s> |
|
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2758 <s xml:id="echoid-s1575" xml:space="preserve">ſe <lb/>ctionem in p; </s> |
|
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|
2759 <s xml:id="echoid-s1576" xml:space="preserve">m p uero æquidiſtans n o: </s> |
|
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|
2760 <s xml:id="echoid-s1577" xml:space="preserve">& </s> |
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|
2761 <s xml:id="echoid-s1578" xml:space="preserve">p s ad axem <lb/>perpendicularis. </s> |
|
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|
2762 <s xml:id="echoid-s1579" xml:space="preserve">Quoniam igitur axis portionis cum ſu-<lb/>perficie humidi facit angulum maiorem angulo b; </s> |
|
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|
2763 <s xml:id="echoid-s1580" xml:space="preserve">erit & </s> |
|
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|
2764 <s xml:id="echoid-s1581" xml:space="preserve"><lb/>angulus s y p angulo b maior. </s> |
|
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|
2765 <s xml:id="echoid-s1582" xml:space="preserve">quare quadratum p s ad <lb/>quadratum s y maiorem habet proportionem, quàm qua <lb/>dratum ψ e ad quadratum ψ b: </s> |
|
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|
2766 <s xml:id="echoid-s1583" xml:space="preserve">& </s> |
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|
2767 <s xml:id="echoid-s1584" xml:space="preserve">propterea _K_ r ad s y ma <lb/> |
|
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|
2768 <anchor type="note" xlink:label="note-0065-01a" xlink:href="note-0065-01"/> |
|
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|
2769 iorem habet, quàm dimidium ipſius κ r ad ψ b. </s> |
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|
2770 <s xml:id="echoid-s1585" xml:space="preserve">ergo s y <lb/>minor eſt, quam dupla ψ b; </s> |
|
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|
2771 <s xml:id="echoid-s1586" xml:space="preserve">& </s> |
|
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|
2772 <s xml:id="echoid-s1587" xml:space="preserve">s o minor, quam ψ b. </s> |
|
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|
2773 <s xml:id="echoid-s1588" xml:space="preserve">quare <lb/> |
|
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|
2774 <anchor type="note" xlink:label="note-0065-02a" xlink:href="note-0065-02"/> |
|
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|
2775 s ω maior, quàm r ψ; </s> |
|
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|
2776 <s xml:id="echoid-s1589" xml:space="preserve">& </s> |
|
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|
2777 <s xml:id="echoid-s1590" xml:space="preserve">p h maior, quàm f. </s> |
|
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|
2778 <s xml:id="echoid-s1591" xml:space="preserve">Itaque quoniã <lb/>portio ad humidum in grauitate eam habet proportionẽ, <lb/> |
|
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|
2779 <anchor type="note" xlink:label="note-0065-03a" xlink:href="note-0065-03"/> |
|
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|
2780 quam exceſſus, quo quadratum b d excedit quadratum f q <lb/>ad quadratum b d: </s> |
|
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|
2781 <s xml:id="echoid-s1592" xml:space="preserve">quam uero proportionem habet por-<lb/>tio ad humidum in grauitate, eandem pars ipſius demerſa <lb/>habet ad totam portionẽ: </s> |
|
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|
2782 <s xml:id="echoid-s1593" xml:space="preserve">ſequitur partẽ demerſam ad to <lb/>tam portionem, eam proportionem habere, quã exceſſus, <lb/>quo quadratum b d excedit quadratũ f q, ad quadratū b d. <lb/></s> |
|
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|
2783 <s xml:id="echoid-s1594" xml:space="preserve">habebit ergo tota portio ad eam, quæ eſt extra humidum <lb/> |
|
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|
2784 <anchor type="note" xlink:label="note-0065-04a" xlink:href="note-0065-04"/> |
|
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|
2785 proportionem eandem, quam quadratum b d ad quadra-<lb/>tum f q. </s> |
|
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|
2786 <s xml:id="echoid-s1595" xml:space="preserve">Sed quam proportionem habet tota portio ad eã, <lb/>quæ eſt extra humidum, eandem habet quadratum n o ad <lb/>quadratum p m. </s> |
|
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|
2787 <s xml:id="echoid-s1596" xml:space="preserve">ergo p m ipſi f q æ qualis etit. </s> |
|
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|
2788 <s xml:id="echoid-s1597" xml:space="preserve">demonſtra <lb/>ta eſt autem p h maior, quàm f: </s> |
|
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|
2789 <s xml:id="echoid-s1598" xml:space="preserve">quare m h minor erit, |
|
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|
2790 <pb file="0066" n="66" rhead="ARCHIMEDIS"/> |
|
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|
2791 quàm q; </s> |
|
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|
2792 <s xml:id="echoid-s1599" xml:space="preserve">& </s> |
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2793 <s xml:id="echoid-s1600" xml:space="preserve">p h maior, quàm dupla h m. </s> |
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2794 <s xml:id="echoid-s1601" xml:space="preserve">Sit igitur <lb/>p z dupla ip-<lb/> |
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2795 <anchor type="figure" xlink:label="fig-0066-01a" xlink:href="fig-0066-01"/> |
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2796 ſius z m: </s> |
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2797 <s xml:id="echoid-s1602" xml:space="preserve">& </s> |
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2798 <s xml:id="echoid-s1603" xml:space="preserve">iun <lb/>cta z t produca <lb/>tur ad g. </s> |
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2799 <s xml:id="echoid-s1604" xml:space="preserve">erit <lb/>totius quidem <lb/>portionis gra-<lb/>uitatis centrũ <lb/>t: </s> |
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2800 <s xml:id="echoid-s1605" xml:space="preserve">eius, quæ eſt <lb/>extra humidũ <lb/>z: </s> |
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2801 <s xml:id="echoid-s1606" xml:space="preserve">reliquæ uero <lb/>partis, quæ in <lb/>humido, cen-<lb/>trum erit in li-<lb/>nea z t produ-<lb/>cta; </s> |
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2802 <s xml:id="echoid-s1607" xml:space="preserve">quod ſit g. <lb/></s> |
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2803 <s xml:id="echoid-s1608" xml:space="preserve">demõſtrabitur <lb/>ſimiliter, ut <lb/>prius, th per-<lb/>pẽdicularis ad <lb/>ſuperficiem hu <lb/>midi: </s> |
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2804 <s xml:id="echoid-s1609" xml:space="preserve">& </s> |
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2805 <s xml:id="echoid-s1610" xml:space="preserve">quæ <lb/>per z, g ducun-<lb/>tur æquidiſtan-<lb/>tes ipſi th, ad <lb/>eandem perpẽ <lb/>diculares. </s> |
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2806 <s xml:id="echoid-s1611" xml:space="preserve">ergo <lb/>portio, quæ eſt <lb/>extra humidũ <lb/>deorſum fere-<lb/>tur ſecundum <lb/>eam quæ per z <lb/>tranſit; </s> |
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2807 <s xml:id="echoid-s1612" xml:space="preserve">quæ ue <lb/>to intra ſecun- |
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2808 <pb o="22" file="0067" n="67" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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2809 dum eam, quæ per g ſurſum eleuabitur. </s> |
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2810 <s xml:id="echoid-s1613" xml:space="preserve">non igitur manebit <lb/>portio ſic inclinata, nec conuertetur ita, ut axis ad ſuperfi-<lb/>ciem humidi ſit perpendicularis: </s> |
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2811 <s xml:id="echoid-s1614" xml:space="preserve">quoniam quæ ex parte 1 <lb/> |
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2812 <anchor type="note" xlink:label="note-0067-01a" xlink:href="note-0067-01"/> |
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2813 deorſum; </s> |
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2814 <s xml:id="echoid-s1615" xml:space="preserve">quæ uero ex parte a ſurſum ferentur, ut ex iam de <lb/>monſtratis apparere poteſt. </s> |
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2815 <s xml:id="echoid-s1616" xml:space="preserve">Quòd ſi axis cum ſuperficie <lb/>humidi fecerit angulum minorem angulo b, ſimiliter de-<lb/> |
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2816 <anchor type="note" xlink:label="note-0067-02a" xlink:href="note-0067-02"/> |
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2817 monſtrabitur, nõ manere portionem, ſed inclinari, donec <lb/>utique axis cum ſuperficie humidi faciat angulum angulo <lb/>b æqualem.</s> |
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2818 <s xml:id="echoid-s1617" xml:space="preserve"/> |
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2819 </p> |
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2820 <div xml:id="echoid-div111" type="float" level="2" n="1"> |
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2821 <figure xlink:label="fig-0064-01" xlink:href="fig-0064-01a"> |
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2822 <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0064-01"/> |
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2823 </figure> |
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2824 <note position="left" xlink:label="note-0064-01" xlink:href="note-0064-01a" xml:space="preserve">A</note> |
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2825 <note position="right" xlink:label="note-0065-01" xlink:href="note-0065-01a" xml:space="preserve">B</note> |
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2826 <note position="right" xlink:label="note-0065-02" xlink:href="note-0065-02a" xml:space="preserve">C</note> |
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2827 <note position="right" xlink:label="note-0065-03" xlink:href="note-0065-03a" xml:space="preserve">D</note> |
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2828 <note position="right" xlink:label="note-0065-04" xlink:href="note-0065-04a" xml:space="preserve">E</note> |
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2829 <figure xlink:label="fig-0066-01" xlink:href="fig-0066-01a"> |
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2830 <image file="0066-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0066-01"/> |
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2831 </figure> |
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2832 <note position="right" xlink:label="note-0067-01" xlink:href="note-0067-01a" xml:space="preserve">F</note> |
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2833 <note position="right" xlink:label="note-0067-02" xlink:href="note-0067-02a" xml:space="preserve">G</note> |
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2834 </div> |
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2835 </div> |
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2836 <div xml:id="echoid-div113" type="section" level="1" n="39"> |
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2837 <head xml:id="echoid-head44" xml:space="preserve">COMMENTARIVS.</head> |
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2838 <p> |
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2839 <s xml:id="echoid-s1618" xml:space="preserve">QVARE quadratum b d magis excedit quadratum <lb/> |
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2840 <anchor type="note" xlink:label="note-0067-03a" xlink:href="note-0067-03"/> |
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2841 f q, quàm b c quadratum: </s> |
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2842 <s xml:id="echoid-s1619" xml:space="preserve">& </s> |
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2843 <s xml:id="echoid-s1620" xml:space="preserve">idcirco linea f q minor eſt, <lb/>quàm b c: </s> |
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2844 <s xml:id="echoid-s1621" xml:space="preserve">itemq; </s> |
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2845 <s xml:id="echoid-s1622" xml:space="preserve">f minor quam b r.</s> |
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2846 <s xml:id="echoid-s1623" xml:space="preserve">] _Quoniani exceſſus, quo_ <lb/>_quadratum b d excedit quadratum b c ad quadratum b d minorem_ <lb/>_proportionem habet, quàm exceſſus, quo quadratum b d excedit qua_ <lb/>_dratum f q, ad idem quadratum: </s> |
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2847 <s xml:id="echoid-s1624" xml:space="preserve">erit ex octaua quinti exceſſus, quo_ <lb/>_quadratum b d excedit quadratum b c, minor quàm exceſſus, quo ex_ <lb/>_cedit quadratum f q. </s> |
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2848 <s xml:id="echoid-s1625" xml:space="preserve">ergo quadratum f q minus est quadrato b c: </s> |
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2849 <s xml:id="echoid-s1626" xml:space="preserve">&_</s> |
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2850 <s xml:id="echoid-s1627" xml:space="preserve"> <lb/>_propterea linea f q minor linea b c. </s> |
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2851 <s xml:id="echoid-s1628" xml:space="preserve">Sed f q ad f eandem proportionẽ_ <lb/>_habet, quam b c ad b r; </s> |
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2852 <s xml:id="echoid-s1629" xml:space="preserve">utraque enim utriuſque ſeſquialtera est. </s> |
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2853 <s xml:id="echoid-s1630" xml:space="preserve">cum_ <lb/> |
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2854 <anchor type="note" xlink:label="note-0067-04a" xlink:href="note-0067-04"/> |
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2855 _igitur f q ſit minor b c, & </s> |
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2856 <s xml:id="echoid-s1631" xml:space="preserve">f ipſa b r minor erit_.</s> |
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2857 <s xml:id="echoid-s1632" xml:space="preserve"/> |
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2858 </p> |
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2859 <div xml:id="echoid-div113" type="float" level="2" n="1"> |
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2860 <note position="right" xlink:label="note-0067-03" xlink:href="note-0067-03a" xml:space="preserve">A</note> |
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2861 <note position="right" xlink:label="note-0067-04" xlink:href="note-0067-04a" xml:space="preserve">14. quinti</note> |
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2862 </div> |
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2863 <p> |
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2864 <s xml:id="echoid-s1633" xml:space="preserve">Et propterea k r ad ſ y maiorem habet, quàm dimidium <lb/> |
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2865 <anchor type="note" xlink:label="note-0067-05a" xlink:href="note-0067-05"/> |
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2866 ipſius k r ad ψ b.</s> |
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2867 <s xml:id="echoid-s1634" xml:space="preserve">] _Est enim k r ad ſ y, ut quadratum p s ad qua_ <lb/>_dratum ſ y<unsure/>: </s> |
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2868 <s xml:id="echoid-s1635" xml:space="preserve">& </s> |
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2869 <s xml:id="echoid-s1636" xml:space="preserve">dimidium lineæ K r ad lineam ψ b, ut quadratum e ψ_ <lb/>_ad quadratum ψ b_.</s> |
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2870 <s xml:id="echoid-s1637" xml:space="preserve"/> |
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2871 </p> |
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2872 <div xml:id="echoid-div114" type="float" level="2" n="2"> |
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2873 <note position="right" xlink:label="note-0067-05" xlink:href="note-0067-05a" xml:space="preserve">B</note> |
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2874 </div> |
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2875 <p> |
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2876 <s xml:id="echoid-s1638" xml:space="preserve">Et s o minor quàm ψ b] _Est enim ſ y dupla ipſius ſ o._</s> |
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2877 <s xml:id="echoid-s1639" xml:space="preserve"/> |
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2878 </p> |
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2879 <note position="right" xml:space="preserve">C</note> |
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2880 <p> |
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2881 <s xml:id="echoid-s1640" xml:space="preserve">Et p h maior, quàm f.</s> |
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2882 <s xml:id="echoid-s1641" xml:space="preserve">] _Nam p h eſt æqualis ſ ω, & </s> |
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2883 <s xml:id="echoid-s1642" xml:space="preserve">r ψ_ <lb/> |
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2884 <anchor type="note" xlink:label="note-0067-07a" xlink:href="note-0067-07"/> |
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2885 _ipſi f_.</s> |
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2886 <s xml:id="echoid-s1643" xml:space="preserve"/> |
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|
2887 </p> |
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2888 <div xml:id="echoid-div115" type="float" level="2" n="3"> |
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2889 <note position="right" xlink:label="note-0067-07" xlink:href="note-0067-07a" xml:space="preserve">D</note> |
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|
2890 </div> |
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2891 <p> |
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2892 <s xml:id="echoid-s1644" xml:space="preserve">Habebit ergo tota portio ad eam, quæ eſt extra humi-<lb/> |
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2893 <anchor type="note" xlink:label="note-0067-08a" xlink:href="note-0067-08"/> |
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2894 dum proportionem eandem, quam quadratum b d ad qua <lb/>dratum f q.</s> |
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2895 <s xml:id="echoid-s1645" xml:space="preserve">] _Cum pars demerſa ad totam portionem ita ſit, ut_ <lb/>_exceſſus, quo quadratum b d excedit quadratum f q ad b d quadratu<unsure/>:_</s> |
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2896 <s xml:id="echoid-s1646" xml:space="preserve"> |
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2897 <pb file="0068" n="68" rhead="ARCHIMEDIS"/> |
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2898 _erit conuertendo tota portio ad partem ipſius demerſam, ut quadrd-_ <lb/>_tum b d ad exceſſum, quo quadratum f q excedit. </s> |
|
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2899 <s xml:id="echoid-s1647" xml:space="preserve">quare per conuer-_ <lb/>_ſionem rationis tota portio ad eam, quæ extra humidum est ut_ <lb/>_qu.</s> |
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2900 <s xml:id="echoid-s1648" xml:space="preserve">idratum b d ad quadratum f q: </s> |
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2901 <s xml:id="echoid-s1649" xml:space="preserve">nam quadratum b d tanto maius_ <lb/>_est exceſſu, quo excedit quadratum f q, quantum est ipſum f q qua-_ <lb/>_dratum_.</s> |
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2902 <s xml:id="echoid-s1650" xml:space="preserve"/> |
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|
2903 </p> |
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2904 <div xml:id="echoid-div116" type="float" level="2" n="4"> |
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2905 <note position="right" xlink:label="note-0067-08" xlink:href="note-0067-08a" xml:space="preserve">E</note> |
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2906 </div> |
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|
2907 <p style="it"> |
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2908 <s xml:id="echoid-s1651" xml:space="preserve">_Quoniam quæ ex parte 1 deorſum, quæ uero ex parte a_ <lb/> |
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2909 <anchor type="note" xlink:label="note-0068-01a" xlink:href="note-0068-01"/> |
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2910 _ſurſum ferentur.</s> |
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2911 <s xml:id="echoid-s1652" xml:space="preserve">]_ Hæc nos ita correximus, nam in translatione <lb/>mendoſe, ut opinor, legebatur, quoniam quæ ex parte l ad ſuperiora <lb/>ferentur, perpendicularis enim quæ tranſit per z ad partes l, & </s> |
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2912 <s xml:id="echoid-s1653" xml:space="preserve">quæ <lb/>per g ad partes a cadit. </s> |
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2913 <s xml:id="echoid-s1654" xml:space="preserve">quare centrum z unà cum p. </s> |
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2914 <s xml:id="echoid-s1655" xml:space="preserve">trtibus ijs, quæ <lb/>ſunt ad l deorſum feretur, centrum uero g unà cum partibus quæ ad <lb/>a ſurſum.</s> |
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2915 <s xml:id="echoid-s1656" xml:space="preserve"/> |
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|
2916 </p> |
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2917 <div xml:id="echoid-div117" type="float" level="2" n="5"> |
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2918 <note position="left" xlink:label="note-0068-01" xlink:href="note-0068-01a" xml:space="preserve">F</note> |
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2919 </div> |
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2920 <p> |
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2921 <s xml:id="echoid-s1657" xml:space="preserve">Similiter demonſtrabitur non manere portionem, ſed <lb/> |
|
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2922 <anchor type="note" xlink:label="note-0068-02a" xlink:href="note-0068-02"/> |
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2923 inclinari, donec utique axis cum ſuperficie humidi faciat <lb/>angulum angulo b æqualem.</s> |
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2924 <s xml:id="echoid-s1658" xml:space="preserve">] _Illud uero tum ex ijs, quæ in an_ <lb/>_tecedenti dicta ſunt, tum ex figuris, quas appoſuimus, facile demon-_ <lb/>_strari potest._</s> |
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2925 <s xml:id="echoid-s1659" xml:space="preserve"/> |
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|
2926 </p> |
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|
2927 <div xml:id="echoid-div118" type="float" level="2" n="6"> |
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2928 <note position="left" xlink:label="note-0068-02" xlink:href="note-0068-02a" xml:space="preserve">G</note> |
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|
2929 </div> |
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|
2930 </div> |
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|
2931 <div xml:id="echoid-div120" type="section" level="1" n="40"> |
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|
2932 <head xml:id="echoid-head45" xml:space="preserve">PROPOSITIO X.</head> |
|
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|
2933 <p> |
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|
2934 <s xml:id="echoid-s1660" xml:space="preserve"><emph style="sc">Recta</emph> portio conoidis rectanguli, quando <lb/>leuior humido axem habuerit maiorem, quàm <lb/>ut ad eam, quæ uſque ad axem proportionem ha-<lb/>beat, quam quindecim ad quatuor: </s> |
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|
2935 <s xml:id="echoid-s1661" xml:space="preserve">in humidum <lb/>demiſſa, ita ut baſis ipſius non contingat humi-<lb/>dum: </s> |
|
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|
2936 <s xml:id="echoid-s1662" xml:space="preserve">non nunquam quidem recta conſiſtet; </s> |
|
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|
2937 <s xml:id="echoid-s1663" xml:space="preserve">non <lb/> |
|
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|
2938 <anchor type="note" xlink:label="note-0068-03a" xlink:href="note-0068-03"/> |
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|
2939 nunquam inclinata: </s> |
|
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|
2940 <s xml:id="echoid-s1664" xml:space="preserve">& </s> |
|
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|
2941 <s xml:id="echoid-s1665" xml:space="preserve">interdum adeo inclinata, <lb/> |
|
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|
2942 <anchor type="note" xlink:label="note-0068-04a" xlink:href="note-0068-04"/> |
|
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|
2943 ut baſis ipſius in uno puncto contingat ſuperfi-<lb/>ciem humidi: </s> |
|
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|
2944 <s xml:id="echoid-s1666" xml:space="preserve">idq; </s> |
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|
2945 <s xml:id="echoid-s1667" xml:space="preserve">in duabus diſpoſitionibus:</s> |
|
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2946 <s xml:id="echoid-s1668" xml:space="preserve"> |
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|
2947 <pb o="29" file="0069" n="69" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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|
2948 interdũ quidem ita, ut baſis in humidum magis <lb/> |
|
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|
2949 <anchor type="note" xlink:label="note-0069-01a" xlink:href="note-0069-01"/> |
|
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|
2950 demergatur: </s> |
|
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|
2951 <s xml:id="echoid-s1669" xml:space="preserve">interdum uero ita, ut ſuperficiem <lb/> |
|
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|
2952 <anchor type="note" xlink:label="note-0069-02a" xlink:href="note-0069-02"/> |
|
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|
2953 humidi nullo modo contingat; </s> |
|
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2954 <s xml:id="echoid-s1670" xml:space="preserve">ſecundum pro-<lb/> |
|
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|
2955 <anchor type="note" xlink:label="note-0069-03a" xlink:href="note-0069-03"/> |
|
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|
2956 portionem, quam habet ad humidum in grauita-<lb/>te. </s> |
|
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|
2957 <s xml:id="echoid-s1671" xml:space="preserve">Eorum quæ dicta ſunt, ſingula inferius de-<lb/>monſtrabuntur.</s> |
|
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|
2958 <s xml:id="echoid-s1672" xml:space="preserve"/> |
|
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|
2959 </p> |
|
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|
2960 <div xml:id="echoid-div120" type="float" level="2" n="1"> |
|
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|
2961 <note position="left" xlink:label="note-0068-03" xlink:href="note-0068-03a" xml:space="preserve">A</note> |
|
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|
2962 <note position="left" xlink:label="note-0068-04" xlink:href="note-0068-04a" xml:space="preserve">B</note> |
|
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|
2963 <note position="right" xlink:label="note-0069-01" xlink:href="note-0069-01a" xml:space="preserve">C</note> |
|
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|
2964 <note position="right" xlink:label="note-0069-02" xlink:href="note-0069-02a" xml:space="preserve">D</note> |
|
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|
2965 <note position="right" xlink:label="note-0069-03" xlink:href="note-0069-03a" xml:space="preserve">E</note> |
|
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|
2966 </div> |
|
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|
2967 <p> |
|
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|
2968 <s xml:id="echoid-s1673" xml:space="preserve">SIT portio qualis dicta eſt: </s> |
|
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|
2969 <s xml:id="echoid-s1674" xml:space="preserve">& </s> |
|
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|
2970 <s xml:id="echoid-s1675" xml:space="preserve">ſecta ipſa plano per axẽ. <lb/></s> |
|
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|
2971 <s xml:id="echoid-s1676" xml:space="preserve">recto ad ſuperficiem humidi, ſectio ſit a p o l rectanguli co <lb/>ni ſeccio: </s> |
|
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|
2972 <s xml:id="echoid-s1677" xml:space="preserve">axis portionis, & </s> |
|
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|
2973 <s xml:id="echoid-s1678" xml:space="preserve">ſectionis diameter b d: </s> |
|
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|
2974 <s xml:id="echoid-s1679" xml:space="preserve">ſece-<lb/>turq; </s> |
|
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|
2975 <s xml:id="echoid-s1680" xml:space="preserve">b d in puncto quidem _k_ ita, ut b k dupla ſitipſius <lb/>_k_ d: </s> |
|
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|
2976 <s xml:id="echoid-s1681" xml:space="preserve">in c uero ita, ut b d ad <emph style="sc">K C</emph> proportionẽ habeat ean-<lb/>dem, quam quindecim ad quatuor. </s> |
|
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|
2977 <s xml:id="echoid-s1682" xml:space="preserve">conſtat igitur k c ma-<lb/> |
|
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|
2978 <anchor type="note" xlink:label="note-0069-04a" xlink:href="note-0069-04"/> |
|
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|
2979 iorem eſſe, quàm quæ uſque ad axem. </s> |
|
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|
2980 <s xml:id="echoid-s1683" xml:space="preserve">Sit ei quæ uſque ad <lb/> |
|
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|
2981 <anchor type="note" xlink:label="note-0069-05a" xlink:href="note-0069-05"/> |
|
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|
2982 axem æqualis k r: </s> |
|
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|
2983 <s xml:id="echoid-s1684" xml:space="preserve">& </s> |
|
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|
2984 <s xml:id="echoid-s1685" xml:space="preserve">ipſius k r ſeſquialtera d s. </s> |
|
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|
2985 <s xml:id="echoid-s1686" xml:space="preserve">Eſt autem <lb/> |
|
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|
2986 <anchor type="note" xlink:label="note-0069-06a" xlink:href="note-0069-06"/> |
|
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|
2987 & </s> |
|
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|
2988 <s xml:id="echoid-s1687" xml:space="preserve">s b ſeſquial-<lb/> |
|
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|
2989 <anchor type="figure" xlink:label="fig-0069-01a" xlink:href="fig-0069-01"/> |
|
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|
2990 tera ipſius b r. <lb/></s> |
|
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|
2991 <s xml:id="echoid-s1688" xml:space="preserve">Itaque iũgatur <lb/>a b, & </s> |
|
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|
2992 <s xml:id="echoid-s1689" xml:space="preserve">per c du <lb/>catur c e per-<lb/>pẽdicularis ad <lb/>b d, quæ lineã <lb/>a b in puncto <lb/>e ſecet: </s> |
|
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|
2993 <s xml:id="echoid-s1690" xml:space="preserve">& </s> |
|
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|
2994 <s xml:id="echoid-s1691" xml:space="preserve">per <lb/>e ducatur e z <lb/>æquidiſtãs b d. </s> |
|
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|
2995 <s xml:id="echoid-s1692" xml:space="preserve"><lb/>Rurſus ipſa a b <lb/>bifariã in t di-<lb/>uiſa, ducatur t <lb/>h eidem b d æ-<lb/>quidiſtans: </s> |
|
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|
2996 <s xml:id="echoid-s1693" xml:space="preserve">& </s> |
|
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|
2997 <s xml:id="echoid-s1694" xml:space="preserve"><lb/>intelligantur rectanguli coni ſectiones deſcriptæ a e i qui- |
|
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|
2998 <pb file="0070" n="70" rhead="ARCHIMEDIS"/> |
|
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|
2999 dem circa e z diametrum; </s> |
|
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|
3000 <s xml:id="echoid-s1695" xml:space="preserve">a t d uero circa diametrum t h; <lb/></s> |
|
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|
3001 <s xml:id="echoid-s1696" xml:space="preserve"> |
|
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|
3002 <anchor type="note" xlink:label="note-0070-01a" xlink:href="note-0070-01"/> |
|
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|
3003 quæ ſimiles ſint portioni a b l. </s> |
|
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|
3004 <s xml:id="echoid-s1697" xml:space="preserve">tranſibit igitur a e i coni <lb/> |
|
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|
3005 <anchor type="note" xlink:label="note-0070-02a" xlink:href="note-0070-02"/> |
|
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|
3006 ſectio per _K_: </s> |
|
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|
3007 <s xml:id="echoid-s1698" xml:space="preserve">& </s> |
|
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|
3008 <s xml:id="echoid-s1699" xml:space="preserve">quæ ab r ducta eſt perpendicularis ad b d, <lb/>ipſam a e i ſecabit. </s> |
|
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|
3009 <s xml:id="echoid-s1700" xml:space="preserve">ſecet in punctis y g: </s> |
|
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|
3010 <s xml:id="echoid-s1701" xml:space="preserve">& </s> |
|
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changeset
|
3011 <s xml:id="echoid-s1702" xml:space="preserve">per y g ducan <lb/>tur ipſi b d æquidiſtantes p y q, o g n, quæ ſecent a t d in <lb/>f x. </s> |
|
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3012 <s xml:id="echoid-s1703" xml:space="preserve">ducantur poſtremo, & </s> |
|
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|
3013 <s xml:id="echoid-s1704" xml:space="preserve">p χ, o φ contingentes ſectionẽ <lb/>a p o l in punctis p o. </s> |
|
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3014 <s xml:id="echoid-s1705" xml:space="preserve">cũ<unsure/> ergo tres portiones ſint a p o l, <lb/> |
|
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|
3015 <anchor type="note" xlink:label="note-0070-03a" xlink:href="note-0070-03"/> |
|
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|
3016 a e i, a t d, contentæ rectis lineis, & </s> |
|
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|
3017 <s xml:id="echoid-s1706" xml:space="preserve">rectangulorum cono-<lb/>rum ſectionibus; </s> |
|
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|
3018 <s xml:id="echoid-s1707" xml:space="preserve">rectæq, ſimiles, & </s> |
|
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|
3019 <s xml:id="echoid-s1708" xml:space="preserve">inæquales, quæ contin <lb/>gunt ſe ſe ſuper unamquanque baſim: </s> |
|
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3020 <s xml:id="echoid-s1709" xml:space="preserve">à puncto autem n <lb/>ſurſum ducta ſit n x g o; </s> |
|
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|
3021 <s xml:id="echoid-s1710" xml:space="preserve">& </s> |
|
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3022 <s xml:id="echoid-s1711" xml:space="preserve">à q ipſa q fy p: </s> |
|
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3023 <s xml:id="echoid-s1712" xml:space="preserve">habebit o g ad <lb/>g x proportionem compoſitam ex proportione, quam ha <lb/>bet i l ad l a; </s> |
|
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3024 <s xml:id="echoid-s1713" xml:space="preserve">& </s> |
|
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3025 <s xml:id="echoid-s1714" xml:space="preserve">ex proportione, quam a d habet ad d i. <lb/></s> |
|
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3026 <s xml:id="echoid-s1715" xml:space="preserve">Sed i l ad l a <lb/> |
|
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|
3027 <anchor type="figure" xlink:label="fig-0070-01a" xlink:href="fig-0070-01"/> |
|
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3028 habet eandem, <lb/>quam duo ad <lb/>quinque. </s> |
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3029 <s xml:id="echoid-s1716" xml:space="preserve">ete-<lb/>nim c b ad b d <lb/> |
|
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|
3030 <anchor type="note" xlink:label="note-0070-04a" xlink:href="note-0070-04"/> |
|
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3031 eſt, ut ſex ad <lb/>quĩdecim; </s> |
|
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3032 <s xml:id="echoid-s1717" xml:space="preserve">hoc <lb/>eſt ut duo ad <lb/>quinque: </s> |
|
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|
3033 <s xml:id="echoid-s1718" xml:space="preserve">& </s> |
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|
3034 <s xml:id="echoid-s1719" xml:space="preserve">ut <lb/> |
|
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|
3035 <anchor type="note" xlink:label="note-0070-05a" xlink:href="note-0070-05"/> |
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3036 c b ad b d, ita <lb/>e b ad b a: </s> |
|
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|
3037 <s xml:id="echoid-s1720" xml:space="preserve">& </s> |
|
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3038 <s xml:id="echoid-s1721" xml:space="preserve"><lb/>d z ad d a. </s> |
|
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|
3039 <s xml:id="echoid-s1722" xml:space="preserve">ha-<lb/>rum autẽ d z, <lb/> |
|
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|
3040 <anchor type="note" xlink:label="note-0070-06a" xlink:href="note-0070-06"/> |
|
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|
3041 d a duplæ ſunt <lb/>ipſæ l i, l a: </s> |
|
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3042 <s xml:id="echoid-s1723" xml:space="preserve">& </s> |
|
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|
3043 <s xml:id="echoid-s1724" xml:space="preserve"><lb/> |
|
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|
3044 <anchor type="note" xlink:label="note-0070-07a" xlink:href="note-0070-07"/> |
|
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|
3045 a d ad d i eã pro <lb/>portionem habet, quam quinque ad unum. </s> |
|
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3046 <s xml:id="echoid-s1725" xml:space="preserve">ſed proportio <lb/>compoſita ex proportione, quam habet duo ad quinque; <lb/></s> |
|
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|
3047 <s xml:id="echoid-s1726" xml:space="preserve">& </s> |
|
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3048 <s xml:id="echoid-s1727" xml:space="preserve">ex proportione, quam quinque ad unum; </s> |
|
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3049 <s xml:id="echoid-s1728" xml:space="preserve">eſt eadem, <lb/>quam habent duo ad unum: </s> |
|
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3050 <s xml:id="echoid-s1729" xml:space="preserve">duo autem ad unum duplam <lb/>proportionem habent. </s> |
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3051 <s xml:id="echoid-s1730" xml:space="preserve">dupla eſt igitur g b ipſius g x: </s> |
|
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|
3052 <s xml:id="echoid-s1731" xml:space="preserve">&</s> |
|
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3053 <s xml:id="echoid-s1732" xml:space="preserve"> |
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|
3054 <pb o="30" file="0071" n="71" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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|
3055 eadem ratione oſtẽdetur p y ipſius y f dupla. </s> |
|
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|
3056 <s xml:id="echoid-s1733" xml:space="preserve">Itaque quo <lb/>niam d s ſeſquialtera eſt ipſius _k_ r; </s> |
|
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|
3057 <s xml:id="echoid-s1734" xml:space="preserve">erit b s exceſſus, quo <lb/>axis eſt maior, quàm ſeſquialter eius, quæ uſque ad axem. <lb/></s> |
|
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|
3058 <s xml:id="echoid-s1735" xml:space="preserve">Si igitur portio ad humidũ in grauitate eã habet propor-<lb/>tionem, quam quadratum, quod fit à linea b ſ ad quadra-<lb/>tum, quod à b d, aut maiorem; </s> |
|
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|
3059 <s xml:id="echoid-s1736" xml:space="preserve">in hnmidum demiſſa, ita <lb/>ut baſis ipſius non contingat humidum, recta conſiſtet. </s> |
|
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|
3060 <s xml:id="echoid-s1737" xml:space="preserve">de <lb/>monſtratum eſt enim ſuperius, portionem, cuius axis eſt <lb/> |
|
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|
3061 <anchor type="note" xlink:label="note-0071-01a" xlink:href="note-0071-01"/> |
|
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|
3062 maior, quàm ſeſquiaiter eius, quæ uſque ad axem, ſi ad hu-<lb/>midum in grauitate non minorem proportionem habeat, <lb/>quàm quadratum, quod fit ab exceſſu, quo axis maior eſt, <lb/>quam ſeſquialter eius, quæ uſque ad axem, ad quadratum, <lb/>quod ab axe; </s> |
|
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|
3063 <s xml:id="echoid-s1738" xml:space="preserve">demiſſam in humidum, ita ut dictum eſt, re-<lb/>ctam conſiſtere.</s> |
|
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|
3064 <s xml:id="echoid-s1739" xml:space="preserve"/> |
|
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changeset
|
3065 </p> |
|
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|
3066 <div xml:id="echoid-div121" type="float" level="2" n="2"> |
|
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|
3067 <note position="right" xlink:label="note-0069-04" xlink:href="note-0069-04a" xml:space="preserve">F</note> |
|
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|
3068 <note position="right" xlink:label="note-0069-05" xlink:href="note-0069-05a" xml:space="preserve">G</note> |
|
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|
3069 <note position="right" xlink:label="note-0069-06" xlink:href="note-0069-06a" xml:space="preserve">H</note> |
|
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|
3070 <figure xlink:label="fig-0069-01" xlink:href="fig-0069-01a"> |
|
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|
3071 <image file="0069-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0069-01"/> |
|
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|
3072 </figure> |
|
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|
3073 <note position="left" xlink:label="note-0070-01" xlink:href="note-0070-01a" xml:space="preserve">K</note> |
|
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|
3074 <note position="left" xlink:label="note-0070-02" xlink:href="note-0070-02a" xml:space="preserve">L</note> |
|
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|
3075 <note position="left" xlink:label="note-0070-03" xlink:href="note-0070-03a" xml:space="preserve">M</note> |
|
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|
3076 <figure xlink:label="fig-0070-01" xlink:href="fig-0070-01a"> |
|
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|
3077 <image file="0070-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0070-01"/> |
|
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changeset
|
3078 </figure> |
|
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|
3079 <note position="left" xlink:label="note-0070-04" xlink:href="note-0070-04a" xml:space="preserve">N</note> |
|
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|
3080 <note position="left" xlink:label="note-0070-05" xlink:href="note-0070-05a" xml:space="preserve">O</note> |
|
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|
3081 <note position="left" xlink:label="note-0070-06" xlink:href="note-0070-06a" xml:space="preserve">P</note> |
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|
3082 <note position="left" xlink:label="note-0070-07" xlink:href="note-0070-07a" xml:space="preserve">Q</note> |
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|
3083 <note position="right" xlink:label="note-0071-01" xlink:href="note-0071-01a" xml:space="preserve">R</note> |
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|
3084 </div> |
|
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|
3085 </div> |
|
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|
3086 <div xml:id="echoid-div123" type="section" level="1" n="41"> |
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|
3087 <head xml:id="echoid-head46" xml:space="preserve">COMMENTARIVS.</head> |
|
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|
3088 <p style="it"> |
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|
3089 <s xml:id="echoid-s1740" xml:space="preserve">_<emph style="sc">Qvae</emph>_ hac decima propoſitione continentur, Archimedes in <lb/>quinque partes diſſecuit, & </s> |
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|
3090 <s xml:id="echoid-s1741" xml:space="preserve">ſingulas ſeorſum demonſtrauit.</s> |
|
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|
3091 <s xml:id="echoid-s1742" xml:space="preserve"/> |
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|
3092 </p> |
|
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|
3093 <p> |
|
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|
3094 <s xml:id="echoid-s1743" xml:space="preserve">Nonnunquam quidem recta conſiſtat.</s> |
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3095 <s xml:id="echoid-s1744" xml:space="preserve">] _Hæc eſt prima_ <lb/> |
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|
3096 <anchor type="note" xlink:label="note-0071-02a" xlink:href="note-0071-02"/> |
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|
3097 _pars, cuius demonstr ationem ſtatim ſubiungit._</s> |
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|
3098 <s xml:id="echoid-s1745" xml:space="preserve"/> |
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|
3099 </p> |
|
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|
3100 <div xml:id="echoid-div123" type="float" level="2" n="1"> |
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|
3101 <note position="right" xlink:label="note-0071-02" xlink:href="note-0071-02a" xml:space="preserve">A</note> |
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|
3102 </div> |
|
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|
3103 <p> |
|
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|
3104 <s xml:id="echoid-s1746" xml:space="preserve">Etinterdum adeo inclinata, ut baſis ipſius in uno pun-<lb/> |
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|
3105 <anchor type="note" xlink:label="note-0071-03a" xlink:href="note-0071-03"/> |
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|
3106 cto contingat ſuperficiem humidi; </s> |
|
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|
3107 <s xml:id="echoid-s1747" xml:space="preserve">idq; </s> |
|
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|
3108 <s xml:id="echoid-s1748" xml:space="preserve">in duabus diſpoſi-<lb/>tionibus.</s> |
|
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|
3109 <s xml:id="echoid-s1749" xml:space="preserve">] _Demonſtratum eſt illud in tertia parte._</s> |
|
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|
3110 <s xml:id="echoid-s1750" xml:space="preserve"/> |
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|
3111 </p> |
|
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|
3112 <div xml:id="echoid-div124" type="float" level="2" n="2"> |
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|
3113 <note position="right" xlink:label="note-0071-03" xlink:href="note-0071-03a" xml:space="preserve">B</note> |
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|
3114 </div> |
|
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|
3115 <p> |
|
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3116 <s xml:id="echoid-s1751" xml:space="preserve">Interdum ita, ut baſis in humidum magis demergatur.</s> |
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|
3117 <s xml:id="echoid-s1752" xml:space="preserve">] <lb/> |
|
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|
3118 <anchor type="note" xlink:label="note-0071-04a" xlink:href="note-0071-04"/> |
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|
3119 _Pertinet id ad quartam partem._</s> |
|
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|
3120 <s xml:id="echoid-s1753" xml:space="preserve"/> |
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|
3121 </p> |
|
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|
3122 <div xml:id="echoid-div125" type="float" level="2" n="3"> |
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|
3123 <note position="right" xlink:label="note-0071-04" xlink:href="note-0071-04a" xml:space="preserve">C</note> |
|
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|
3124 </div> |
|
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|
3125 <p> |
|
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3126 <s xml:id="echoid-s1754" xml:space="preserve">Interdum uero ita, ut ſuperficiem humidi nullo modo <lb/> |
|
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|
3127 <anchor type="note" xlink:label="note-0071-05a" xlink:href="note-0071-05"/> |
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|
3128 contingat.</s> |
|
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|
3129 <s xml:id="echoid-s1755" xml:space="preserve">] _Hoc duobus item modis fit, quorum unus in ſecunda,_ <lb/>_alter in quarta parte explicatur._</s> |
|
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|
3130 <s xml:id="echoid-s1756" xml:space="preserve"/> |
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|
3131 </p> |
|
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|
3132 <div xml:id="echoid-div126" type="float" level="2" n="4"> |
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|
3133 <note position="right" xlink:label="note-0071-05" xlink:href="note-0071-05a" xml:space="preserve">D</note> |
|
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|
3134 </div> |
|
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|
3135 <p> |
|
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|
3136 <s xml:id="echoid-s1757" xml:space="preserve">Secundum proportionem, quam habet ad humidum in <lb/> |
|
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|
3137 <anchor type="note" xlink:label="note-0071-06a" xlink:href="note-0071-06"/> |
|
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|
3138 grauitate.</s> |
|
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|
3139 <s xml:id="echoid-s1758" xml:space="preserve">] _In translatione ita legebatur, quam autem proportio_ <lb/>_nem habet ad humidum in grauitate._</s> |
|
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|
3140 <s xml:id="echoid-s1759" xml:space="preserve"/> |
|
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|
3141 </p> |
|
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|
3142 <div xml:id="echoid-div127" type="float" level="2" n="5"> |
|
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|
3143 <note position="right" xlink:label="note-0071-06" xlink:href="note-0071-06a" xml:space="preserve">E</note> |
|
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|
3144 </div> |
|
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|
3145 <p style="it"> |
|
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|
3146 <s xml:id="echoid-s1760" xml:space="preserve">_Conſtat igitur k c maiorem eſſe, quàm quæ uſque ad_ <lb/> |
|
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|
3147 <anchor type="note" xlink:label="note-0071-07a" xlink:href="note-0071-07"/> |
|
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|
3148 _axem.</s> |
|
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|
3149 <s xml:id="echoid-s1761" xml:space="preserve">]_ Nam cum b d ad k c eandem habeat proportionem, quam |
|
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|
3150 <pb file="0072" n="72" rhead="ARCHIMEDIS"/> |
|
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|
3151 quindecim ad quatuor; </s> |
|
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|
3152 <s xml:id="echoid-s1762" xml:space="preserve">& </s> |
|
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|
3153 <s xml:id="echoid-s1763" xml:space="preserve">ad eam, quæ uſque ad axem maiorem pro <lb/>portionem habeat: </s> |
|
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|
3154 <s xml:id="echoid-s1764" xml:space="preserve">erit quæ uſ que ad axem minor ipſa k c.</s> |
|
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|
3155 <s xml:id="echoid-s1765" xml:space="preserve"/> |
|
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|
3156 </p> |
|
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|
3157 <div xml:id="echoid-div128" type="float" level="2" n="6"> |
|
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|
3158 <note position="right" xlink:label="note-0071-07" xlink:href="note-0071-07a" xml:space="preserve">F</note> |
|
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|
3159 </div> |
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3160 <note position="left" xml:space="preserve">10. quinti</note> |
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3161 <p> |
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3162 <s xml:id="echoid-s1766" xml:space="preserve">Sit ei, quæ uſque ad axem æ qualis k r.</s> |
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3163 <s xml:id="echoid-s1767" xml:space="preserve">] _Hac nos addidimus,_ <lb/> |
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3164 <anchor type="note" xlink:label="note-0072-02a" xlink:href="note-0072-02"/> |
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3165 _quæ in translatione non erant._</s> |
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3166 <s xml:id="echoid-s1768" xml:space="preserve"/> |
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3167 </p> |
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3168 <div xml:id="echoid-div129" type="float" level="2" n="7"> |
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3169 <note position="left" xlink:label="note-0072-02" xlink:href="note-0072-02a" xml:space="preserve">G</note> |
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3170 </div> |
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3171 <p style="it"> |
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3172 <s xml:id="echoid-s1769" xml:space="preserve">_Eſt autem & </s> |
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3173 <s xml:id="echoid-s1770" xml:space="preserve">s b ſeſquialtera ipſius b r.</s> |
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3174 <s xml:id="echoid-s1771" xml:space="preserve">]_ Ponitur enim <lb/> |
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3175 <anchor type="note" xlink:label="note-0072-03a" xlink:href="note-0072-03"/> |
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3176 d b ſeſquialtera ipſius b k; </s> |
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3177 <s xml:id="echoid-s1772" xml:space="preserve">itémq; </s> |
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3178 <s xml:id="echoid-s1773" xml:space="preserve">d ſ ſeſquialtera k r. </s> |
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3179 <s xml:id="echoid-s1774" xml:space="preserve">quare ut to <lb/>ta d b ad totam b K, ita pars d s ad partem K r. </s> |
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3180 <s xml:id="echoid-s1775" xml:space="preserve">ergo & </s> |
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3181 <s xml:id="echoid-s1776" xml:space="preserve">reliqua <lb/> |
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3182 <anchor type="note" xlink:label="note-0072-04a" xlink:href="note-0072-04"/> |
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3183 s b ad reliquim b r, ut d b ad b k.</s> |
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3184 <s xml:id="echoid-s1777" xml:space="preserve"/> |
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|
3185 </p> |
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3186 <div xml:id="echoid-div130" type="float" level="2" n="8"> |
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3187 <note position="left" xlink:label="note-0072-03" xlink:href="note-0072-03a" xml:space="preserve">H</note> |
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3188 <note position="left" xlink:label="note-0072-04" xlink:href="note-0072-04a" xml:space="preserve">19. quinti</note> |
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|
3189 </div> |
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|
3190 <p style="it"> |
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3191 <s xml:id="echoid-s1778" xml:space="preserve">_Quæ ſimiles ſint portioni a b l.</s> |
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3192 <s xml:id="echoid-s1779" xml:space="preserve">]_ Similes portiones coni ſe-<lb/> |
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3193 <anchor type="note" xlink:label="note-0072-05a" xlink:href="note-0072-05"/> |
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|
3194 ctionum Apollonius it. </s> |
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3195 <s xml:id="echoid-s1780" xml:space="preserve">i diffiniuit in ſexto libro conicorum, ut ſcri-<lb/>bit Eutocius, εν οἱς α χ θεισω<unsure/>νἐν ἑηάστω παραλλήλων τῆ βάσει, ἵσωι <lb/>τὸ πλῆθος, ὰι παρὰλληλοι, καὶ αἱ βάσ{ει}ς πρ<unsure/>ὸς τὰςἀποτεμνομένας <lb/>ἀπὸ τῶν διαμέ τρων ταῖς νορυφαῖς ἐν τοῖς αὐτοῖ ς λὄγοιςεἰσἰ, καὶἁι <lb/>ἀποτεμνόμεναι πρ<unsure/>ὸς τάς ἀποτεμνομένας; </s> |
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3196 <s xml:id="echoid-s1781" xml:space="preserve">hoc est. </s> |
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3197 <s xml:id="echoid-s1782" xml:space="preserve">in quibus ſi du-<lb/>cantnr lineæ æquidistantes baſi numero æquales: </s> |
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3198 <s xml:id="echoid-s1783" xml:space="preserve">æquidiſtantes atq; <lb/></s> |
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3199 <s xml:id="echoid-s1784" xml:space="preserve">baſes ad partes diametrorum, quæ ab ipſis ad uerticem abſcindũtur, <lb/>eandem proportionem babent: </s> |
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3200 <s xml:id="echoid-s1785" xml:space="preserve">it émq; </s> |
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3201 <s xml:id="echoid-s1786" xml:space="preserve">partes abſciſſæ ad abſciſſas. </s> |
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3202 <s xml:id="echoid-s1787" xml:space="preserve"><lb/>ducuntur autem lineæ baſi æquidistantes: </s> |
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3203 <s xml:id="echoid-s1788" xml:space="preserve">ut opinor, deſcripta in ſin <lb/>gulis plane rectilinea figura, quæ lateribus numero æqualibus conti <lb/> |
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3204 <anchor type="note" xlink:label="note-0072-06a" xlink:href="note-0072-06"/> |
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|
3205 neatur. </s> |
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3206 <s xml:id="echoid-s1789" xml:space="preserve">Itaq; </s> |
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3207 <s xml:id="echoid-s1790" xml:space="preserve">portiones ſimiles à ſimilibus coni ſectionibus abſcindũ <lb/>tur: </s> |
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3208 <s xml:id="echoid-s1791" xml:space="preserve">& </s> |
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3209 <s xml:id="echoid-s1792" xml:space="preserve">earum diametri ſiue ad baſes rectæ, ſiue cum baſibus æ qua-<lb/>les angulos facientes, ad ipſas baſes eandem habent proportionem.</s> |
|
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3210 <s xml:id="echoid-s1793" xml:space="preserve"/> |
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|
3211 </p> |
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|
3212 <div xml:id="echoid-div131" type="float" level="2" n="9"> |
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3213 <note position="left" xlink:label="note-0072-05" xlink:href="note-0072-05a" xml:space="preserve">K</note> |
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3214 <note position="left" xlink:label="note-0072-06" xlink:href="note-0072-06a" xml:space="preserve">γνωρίμως</note> |
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|
3215 </div> |
|
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|
3216 <p style="it"> |
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3217 <s xml:id="echoid-s1794" xml:space="preserve">_Tranſibit igitur a e i coni ſectio per k.</s> |
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3218 <s xml:id="echoid-s1795" xml:space="preserve">]_ Sienim fieri po <lb/> |
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3219 <anchor type="note" xlink:label="note-0072-07a" xlink:href="note-0072-07"/> |
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3220 teſt non tranſeat per k, ſed per aliud punctum lineæ d b, ut per u. <lb/></s> |
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3221 <s xml:id="echoid-s1796" xml:space="preserve">Quoniam igitur in rectáguli coni ſectione a e i, cuius diameter e z, <lb/>ducta eſt a e, & </s> |
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3222 <s xml:id="echoid-s1797" xml:space="preserve">producta: </s> |
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3223 <s xml:id="echoid-s1798" xml:space="preserve">& </s> |
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3224 <s xml:id="echoid-s1799" xml:space="preserve">d b diametro æquidistans utraſque <lb/>a e, a i ſecat; </s> |
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3225 <s xml:id="echoid-s1800" xml:space="preserve">a e quidem in b, ai uero in d: </s> |
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3226 <s xml:id="echoid-s1801" xml:space="preserve">habebit d b ad b u <lb/>proportionem eandem, quam a z, ad z d, ex quarta propoſitione li <lb/>bri. </s> |
|
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3227 <s xml:id="echoid-s1802" xml:space="preserve">Archimedis de quadratura parabol<unsure/>æ. </s> |
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3228 <s xml:id="echoid-s1803" xml:space="preserve">Sed a z ſeſquialtera eſt <lb/>ipſius z d: </s> |
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|
3229 <s xml:id="echoid-s1804" xml:space="preserve">eſt enim ut tria ad duo, quod mox demonſtrabimus. </s> |
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|
3230 <s xml:id="echoid-s1805" xml:space="preserve">ergo <lb/>d b ſeſquialtera eſt ipſius b u. </s> |
|
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3231 <s xml:id="echoid-s1806" xml:space="preserve">eſt auté d b & </s> |
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3232 <s xml:id="echoid-s1807" xml:space="preserve">ipſius b k ſeſquialte <lb/>ra. </s> |
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3233 <s xml:id="echoid-s1808" xml:space="preserve">quare lineæ b u, b k inter ſe æ quales ſunt; </s> |
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3234 <s xml:id="echoid-s1809" xml:space="preserve">quod fieri non po-<lb/> |
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3235 <anchor type="note" xlink:label="note-0072-08a" xlink:href="note-0072-08"/> |
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3236 teſt. </s> |
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3237 <s xml:id="echoid-s1810" xml:space="preserve">restanguli igitur com ſectio a e i per punctum k tranſibit. <lb/></s> |
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3238 <s xml:id="echoid-s1811" xml:space="preserve">quod demonstrare uolebamus.</s> |
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3239 <s xml:id="echoid-s1812" xml:space="preserve"/> |
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3240 </p> |
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3241 <div xml:id="echoid-div132" type="float" level="2" n="10"> |
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3242 <note position="left" xlink:label="note-0072-07" xlink:href="note-0072-07a" xml:space="preserve">L</note> |
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3243 <note position="left" xlink:label="note-0072-08" xlink:href="note-0072-08a" xml:space="preserve">2. quinti.</note> |
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3244 </div> |
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3245 <pb o="37" file="0073" n="73" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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3246 <p> |
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3247 <s xml:id="echoid-s1813" xml:space="preserve">Cum ergo tres portiones ſint a p o i, a ei, atd, con-<lb/> |
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3248 <anchor type="note" xlink:label="note-0073-01a" xlink:href="note-0073-01"/> |
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3249 tentæ rectis lineis, & </s> |
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3250 <s xml:id="echoid-s1814" xml:space="preserve">rectãgulorum conorum ſectionibus; <lb/></s> |
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3251 <s xml:id="echoid-s1815" xml:space="preserve">rectæq;</s> |
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3252 <s xml:id="echoid-s1816" xml:space="preserve">, ſimiles, & </s> |
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3253 <s xml:id="echoid-s1817" xml:space="preserve">inæquales, quæ contingunt ſe ſe ſuper <lb/>unam quamque baſim.</s> |
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3254 <s xml:id="echoid-s1818" xml:space="preserve">] _Poſt ea uerba, ſuper unamquanque_ <lb/>_baſim, in trans latione aliqua deſiderari uidentur. </s> |
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3255 <s xml:id="echoid-s1819" xml:space="preserve">Ad borum autem_ <lb/>_demonſtrationem non nulla præmittere oportet, quæ etiam ad alia,_ <lb/>_quæ ſequuntur, neceſſaria erunt._</s> |
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3256 <s xml:id="echoid-s1820" xml:space="preserve"/> |
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|
3257 </p> |
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3258 <div xml:id="echoid-div133" type="float" level="2" n="11"> |
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3259 <note position="right" xlink:label="note-0073-01" xlink:href="note-0073-01a" xml:space="preserve">M</note> |
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|
3260 </div> |
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|
3261 </div> |
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3262 <div xml:id="echoid-div135" type="section" level="1" n="42"> |
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3263 <head xml:id="echoid-head47" xml:space="preserve">LEMMA I.</head> |
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3264 <p style="it"> |
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3265 <s xml:id="echoid-s1821" xml:space="preserve">Sit recta linea a b, quam ſecent duæ lineæ inter ſeſe <lb/>æquidiſtantes a c, d e, ita ut quam proportionem ba-<lb/>bet a b ad b d, eandern haheat a c ad de. </s> |
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3266 <s xml:id="echoid-s1822" xml:space="preserve">Dico li-<lb/>neam, quæ c b puncta coniungit, etiam per ipſum e <lb/>tr anſire.</s> |
|
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3267 <s xml:id="echoid-s1823" xml:space="preserve"/> |
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|
3268 </p> |
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3269 <p style="it"> |
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3270 <s xml:id="echoid-s1824" xml:space="preserve">SI enim fieri poteſt, non tranſeat pere, ſed nel ſupra, uel infra. <lb/></s> |
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3271 <s xml:id="echoid-s1825" xml:space="preserve">tranſeat primum infra, ut per f. </s> |
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3272 <s xml:id="echoid-s1826" xml:space="preserve">erunt triangula a b c, d b f inter ſe <lb/>ſimilia. </s> |
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3273 <s xml:id="echoid-s1827" xml:space="preserve">quare ut a b ad b d, ita a c ad d f. </s> |
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3274 <s xml:id="echoid-s1828" xml:space="preserve">ſed ut a b ad bd, ita <lb/> |
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3275 <anchor type="note" xlink:label="note-0073-02a" xlink:href="note-0073-02"/> |
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|
3276 erat a c ad d e. </s> |
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3277 <s xml:id="echoid-s1829" xml:space="preserve">ergo d f ipſi d e æqualis erit, uidelicet pars to-<lb/> |
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3278 <anchor type="note" xlink:label="note-0073-03a" xlink:href="note-0073-03"/> |
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|
3279 ti, quod eſt <lb/> |
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|
3280 <anchor type="figure" xlink:label="fig-0073-01a" xlink:href="fig-0073-01"/> |
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|
3281 cbſurdum. <lb/></s> |
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3282 <s xml:id="echoid-s1830" xml:space="preserve">Idem ab-<lb/>ſurdum ſe <lb/>quetur, ſi <lb/>linea c b <lb/>ſupra e pú <lb/>ctum tran <lb/>ſire pona-<lb/>tur. </s> |
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3283 <s xml:id="echoid-s1831" xml:space="preserve">quare <lb/>c b etiam <lb/>per e ne-<lb/>ceſſario tranſibit. </s> |
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3284 <s xml:id="echoid-s1832" xml:space="preserve">quod oportebat demonſtrare.</s> |
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3285 <s xml:id="echoid-s1833" xml:space="preserve"/> |
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|
3286 </p> |
|
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|
3287 <div xml:id="echoid-div135" type="float" level="2" n="1"> |
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|
3288 <note position="right" xlink:label="note-0073-02" xlink:href="note-0073-02a" xml:space="preserve">4. ſexti.</note> |
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3289 <note position="right" xlink:label="note-0073-03" xlink:href="note-0073-03a" xml:space="preserve">9. quinti.</note> |
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|
3290 <figure xlink:label="fig-0073-01" xlink:href="fig-0073-01a"> |
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|
3291 <image file="0073-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0073-01"/> |
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|
3292 </figure> |
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|
3293 </div> |
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|
3294 <pb file="0074" n="74" rhead="ARCHIMEDIS"/> |
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|
3295 </div> |
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|
3296 <div xml:id="echoid-div137" type="section" level="1" n="43"> |
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|
3297 <head xml:id="echoid-head48" xml:space="preserve">LEMMA II.</head> |
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|
3298 <p style="it"> |
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|
3299 <s xml:id="echoid-s1834" xml:space="preserve">Sint duæ portionis ſimiles, contentæ rectis lineis, & </s> |
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|
3300 <s xml:id="echoid-s1835" xml:space="preserve"><lb/>rectangulorum conorum ſectionibus; </s> |
|
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|
3301 <s xml:id="echoid-s1836" xml:space="preserve">a b c quidem ma-<lb/>ior, cuius diameter b d; </s> |
|
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|
3302 <s xml:id="echoid-s1837" xml:space="preserve">e f c uero minor, cuius diameter <lb/>fg: </s> |
|
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|
3303 <s xml:id="echoid-s1838" xml:space="preserve">aptenturq; </s> |
|
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|
3304 <s xml:id="echoid-s1839" xml:space="preserve">inter ſeſe, ita ut maior minorem includat <lb/>& </s> |
|
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3305 <s xml:id="echoid-s1840" xml:space="preserve">ſint earum baſes a c, e c in eadem recta linea, ut idẽ <lb/>punctum c ſit utriuſque terminus: </s> |
|
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|
3306 <s xml:id="echoid-s1841" xml:space="preserve">ſumatur deinde in ſe <lb/>ctione a b c quodlibet punctum b: </s> |
|
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|
3307 <s xml:id="echoid-s1842" xml:space="preserve">& </s> |
|
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|
3308 <s xml:id="echoid-s1843" xml:space="preserve">iungatur h c. </s> |
|
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|
3309 <s xml:id="echoid-s1844" xml:space="preserve">Di <lb/>co lineam h c ad partem ſui ipſius, quæ inter c, & </s> |
|
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|
3310 <s xml:id="echoid-s1845" xml:space="preserve">ſe-<lb/>ctionem e f c interiicitur, eam proportionẽ habere, quam <lb/>habet a c ad c e.</s> |
|
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|
3311 <s xml:id="echoid-s1846" xml:space="preserve"/> |
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|
3312 </p> |
|
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|
3313 <p style="it"> |
|
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|
3314 <s xml:id="echoid-s1847" xml:space="preserve">_<emph style="sc">Dvcatvr</emph>_ b c, quæ tranſibit per f. </s> |
|
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|
3315 <s xml:id="echoid-s1848" xml:space="preserve">quoniam enim portiones <lb/>ſimiles ſunt, diametri cú baſibus æquales continent angulos. </s> |
|
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|
3316 <s xml:id="echoid-s1849" xml:space="preserve">quare <lb/>æquidiſtant inter ſe ſe b d, f g: </s> |
|
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|
3317 <s xml:id="echoid-s1850" xml:space="preserve">éſtq; </s> |
|
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|
3318 <s xml:id="echoid-s1851" xml:space="preserve">b d ad a c, ut f g ad e c: <lb/></s> |
|
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|
3319 <s xml:id="echoid-s1852" xml:space="preserve">& </s> |
|
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|
3320 <s xml:id="echoid-s1853" xml:space="preserve">permu-<lb/> |
|
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|
3321 <anchor type="figure" xlink:label="fig-0074-01a" xlink:href="fig-0074-01"/> |
|
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|
3322 tando b d ad <lb/>f g, ut a c ad <lb/>c e: </s> |
|
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|
3323 <s xml:id="echoid-s1854" xml:space="preserve">hoc eſt <lb/> |
|
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|
3324 <anchor type="note" xlink:label="note-0074-01a" xlink:href="note-0074-01"/> |
|
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changeset
|
3325 ut earum di-<lb/>midiæ d c ad <lb/>c g. </s> |
|
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|
3326 <s xml:id="echoid-s1855" xml:space="preserve">ergo ex <lb/>antecedēti lé <lb/>mate ſequi-<lb/>tur lineá b c <lb/>per punctum <lb/>f tranſire. <lb/></s> |
|
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3327 <s xml:id="echoid-s1856" xml:space="preserve">Ducatur præ <lb/>terea à puncto h ad diametrum b d linea h K, æquidiſtans baſi <lb/>a c: </s> |
|
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|
3328 <s xml:id="echoid-s1857" xml:space="preserve">& </s> |
|
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|
3329 <s xml:id="echoid-s1858" xml:space="preserve">iuncta k c, quæ diametrum f g ſecet in l; </s> |
|
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|
3330 <s xml:id="echoid-s1859" xml:space="preserve">per l ducatur |
|
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|
3331 <pb o="32" file="0075" n="75" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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|
3332 ad ſectionem e f g ex parte e linea l m, eidem a c baſi æquidi-<lb/>stans. </s> |
|
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|
3333 <s xml:id="echoid-s1860" xml:space="preserve">Sit autem ſectionis a b c, linea b n iuxta quam poſſunt, quæ <lb/>à ſectione ducuntur: </s> |
|
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|
3334 <s xml:id="echoid-s1861" xml:space="preserve">& </s> |
|
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|
3335 <s xml:id="echoid-s1862" xml:space="preserve">ſectionis e f c ſit ipſa f o. </s> |
|
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|
3336 <s xml:id="echoid-s1863" xml:space="preserve">quoniam igi-<lb/>tur triangula c d b, c f g ſimilia ſunt, erit ut b c ad c f, ita d c <lb/> |
|
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|
3337 <anchor type="note" xlink:label="note-0075-01a" xlink:href="note-0075-01"/> |
|
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|
3338 ad c g; </s> |
|
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|
3339 <s xml:id="echoid-s1864" xml:space="preserve">& </s> |
|
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|
3340 <s xml:id="echoid-s1865" xml:space="preserve">b d ad f g. </s> |
|
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|
3341 <s xml:id="echoid-s1866" xml:space="preserve">rurſus quoniam triangula c k b, c l f etiã <lb/>inter ſe ſunt ſimilia, ut b c ad c f, boc eſt ut b d ad f g, ita erit k c <lb/>ad c l; </s> |
|
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|
3342 <s xml:id="echoid-s1867" xml:space="preserve">& </s> |
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|
3343 <s xml:id="echoid-s1868" xml:space="preserve">b K ad f l. </s> |
|
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|
3344 <s xml:id="echoid-s1869" xml:space="preserve">quare K c ad c l, & </s> |
|
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|
3345 <s xml:id="echoid-s1870" xml:space="preserve">b k ad f l ſunt ut d c <lb/>ad c g: </s> |
|
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|
3346 <s xml:id="echoid-s1871" xml:space="preserve">hoc eſt ut earum duplæ a c ad c e. </s> |
|
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|
3347 <s xml:id="echoid-s1872" xml:space="preserve">ſed ut b d ad f g, ita d c <lb/> |
|
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|
3348 <anchor type="note" xlink:label="note-0075-02a" xlink:href="note-0075-02"/> |
|
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|
3349 ad c g; </s> |
|
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|
3350 <s xml:id="echoid-s1873" xml:space="preserve">hoc ẽ a d ad e g: </s> |
|
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|
3351 <s xml:id="echoid-s1874" xml:space="preserve">& </s> |
|
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|
3352 <s xml:id="echoid-s1875" xml:space="preserve">permutãdo ut b d ad a d, ita f g ad e g. <lb/></s> |
|
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|
3353 <s xml:id="echoid-s1876" xml:space="preserve">quadratum autem a d æquale eſt rectangulo d b n ex undecima pri <lb/>mi conicorum. </s> |
|
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|
3354 <s xml:id="echoid-s1877" xml:space="preserve">ergo tres lineæ b d, a d, b n inter ſe ſunt proportio <lb/> |
|
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|
3355 <anchor type="note" xlink:label="note-0075-03a" xlink:href="note-0075-03"/> |
|
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|
3356 nales. </s> |
|
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|
3357 <s xml:id="echoid-s1878" xml:space="preserve">eadem quoque ratione cum quadratum e g æquale ſit rectan <lb/>gulo g f o, tres aliæ lineæ f g, e g, f o, deinceps proportionales <lb/>erũt. </s> |
|
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|
3358 <s xml:id="echoid-s1879" xml:space="preserve">& </s> |
|
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|
3359 <s xml:id="echoid-s1880" xml:space="preserve">ut b d ad, a d, ita f g ad e g. </s> |
|
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|
3360 <s xml:id="echoid-s1881" xml:space="preserve">quare ut a d ad b n, ita e g <lb/>ad f o. </s> |
|
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|
3361 <s xml:id="echoid-s1882" xml:space="preserve">ex æquali igitur, ut d b ad b n, ita g f ad f o: </s> |
|
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|
3362 <s xml:id="echoid-s1883" xml:space="preserve">& </s> |
|
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|
3363 <s xml:id="echoid-s1884" xml:space="preserve">permu-<lb/>tando ut d b ad g f, ita b n ad f o. </s> |
|
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|
3364 <s xml:id="echoid-s1885" xml:space="preserve">ut autem d b ad g f, ita b k <lb/>ad f l. </s> |
|
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|
3365 <s xml:id="echoid-s1886" xml:space="preserve">ergo b k ad f l, ut b n ad f o: </s> |
|
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|
3366 <s xml:id="echoid-s1887" xml:space="preserve">& </s> |
|
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|
3367 <s xml:id="echoid-s1888" xml:space="preserve">permutando, ut b k ad <lb/>bn, ita f l ad f o. </s> |
|
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|
3368 <s xml:id="echoid-s1889" xml:space="preserve">Rurſus quoniá quadratú h K æquale eſt rectan <lb/> |
|
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|
3369 <anchor type="note" xlink:label="note-0075-04a" xlink:href="note-0075-04"/> |
|
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changeset
|
3370 gulo k b n: </s> |
|
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|
3371 <s xml:id="echoid-s1890" xml:space="preserve">& </s> |
|
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changeset
|
3372 <s xml:id="echoid-s1891" xml:space="preserve">quadratum m l rectangulo l f o æquale: </s> |
|
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|
3373 <s xml:id="echoid-s1892" xml:space="preserve">erunt tres <lb/>lineæ b k, k h, b n proportionales: </s> |
|
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changeset
|
3374 <s xml:id="echoid-s1893" xml:space="preserve">itémq; </s> |
|
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3375 <s xml:id="echoid-s1894" xml:space="preserve">proportionales inter ſe <lb/>f l, l m, f o. </s> |
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3376 <s xml:id="echoid-s1895" xml:space="preserve">quare ut linea b K ad lineam b n, ita quadratum b K <lb/> |
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3377 <anchor type="note" xlink:label="note-0075-05a" xlink:href="note-0075-05"/> |
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3378 ad quadratum h k: </s> |
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3379 <s xml:id="echoid-s1896" xml:space="preserve">& </s> |
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3380 <s xml:id="echoid-s1897" xml:space="preserve">ut linea f l ad ipſam f o, ita quadratú f l <lb/>ad quadratum l m. </s> |
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3381 <s xml:id="echoid-s1898" xml:space="preserve">Itaque quoniam, ut b K ad b n, ita eſt f l ad <lb/>f o; </s> |
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3382 <s xml:id="echoid-s1899" xml:space="preserve">erit ut quadratum b K ad quadratum k h, ita quadratum f l <lb/>ad l m quadratum. </s> |
|
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3383 <s xml:id="echoid-s1900" xml:space="preserve">ergo ut linea b k, ad lineam K h, ita linea f l <lb/> |
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3384 <anchor type="note" xlink:label="note-0075-06a" xlink:href="note-0075-06"/> |
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3385 ad ipsã lm: </s> |
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3386 <s xml:id="echoid-s1901" xml:space="preserve">& </s> |
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3387 <s xml:id="echoid-s1902" xml:space="preserve">permutãdo ut b k ad f l, ita k h ad lm. </s> |
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3388 <s xml:id="echoid-s1903" xml:space="preserve">ſed b k ad <lb/>f l erat ut k c ad c l. </s> |
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3389 <s xml:id="echoid-s1904" xml:space="preserve">ergo k h ad lm, ut K c ad c l. </s> |
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3390 <s xml:id="echoid-s1905" xml:space="preserve">quare ex eo <lb/>dem lemmate patet lineam h c, & </s> |
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3391 <s xml:id="echoid-s1906" xml:space="preserve">per m punctum tranſire. </s> |
|
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3392 <s xml:id="echoid-s1907" xml:space="preserve">ut igi-<lb/>tur K c ad c l: </s> |
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3393 <s xml:id="echoid-s1908" xml:space="preserve">hoc eſt ut a c ad c e, ita h c ad c m; </s> |
|
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3394 <s xml:id="echoid-s1909" xml:space="preserve">hoc eſt ad eam <lb/>ipſius partem, quæ inter c, & </s> |
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3395 <s xml:id="echoid-s1910" xml:space="preserve">e g c ſectionem interyeitur. </s> |
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3396 <s xml:id="echoid-s1911" xml:space="preserve">ſimiliter <lb/>demonſtrabimus idem contingere in alijs lineis, quæ à puncto c ad <lb/>a b c ſectionem perducuntur. </s> |
|
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|
3397 <s xml:id="echoid-s1912" xml:space="preserve">At uero b c ad e f eandern propor-<lb/>tionem habere, liquido apparet; </s> |
|
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|
3398 <s xml:id="echoid-s1913" xml:space="preserve">nam b c ad c f, eſt ut d c ad c g; <lb/></s> |
|
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3399 <s xml:id="echoid-s1914" xml:space="preserve">uidelicet ut earum duplæ, a c ad c e.</s> |
|
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|
3400 <s xml:id="echoid-s1915" xml:space="preserve"/> |
|
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|
3401 </p> |
|
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|
3402 <div xml:id="echoid-div137" type="float" level="2" n="1"> |
|
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|
3403 <figure xlink:label="fig-0074-01" xlink:href="fig-0074-01a"> |
|
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|
3404 <image file="0074-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0074-01"/> |
|
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|
3405 </figure> |
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3406 <note position="left" xlink:label="note-0074-01" xlink:href="note-0074-01a" xml:space="preserve">15. quin-<lb/>ti.</note> |
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3407 <note position="right" xlink:label="note-0075-01" xlink:href="note-0075-01a" xml:space="preserve">4. ſexti.</note> |
|
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3408 <note position="right" xlink:label="note-0075-02" xlink:href="note-0075-02a" xml:space="preserve">15. quin-<lb/>ti.</note> |
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3409 <note position="right" xlink:label="note-0075-03" xlink:href="note-0075-03a" xml:space="preserve">17. ſexti.</note> |
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3410 <note position="right" xlink:label="note-0075-04" xlink:href="note-0075-04a" xml:space="preserve">11. primi <lb/>conicorũ</note> |
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3411 <note position="right" xlink:label="note-0075-05" xlink:href="note-0075-05a" xml:space="preserve">cor. 20. ſe <lb/>xti.</note> |
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3412 <note position="right" xlink:label="note-0075-06" xlink:href="note-0075-06a" xml:space="preserve">22. ſexti</note> |
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|
3413 </div> |
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3414 <pb file="0076" n="76" rhead="ARCHIMEDIS"/> |
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3415 <p style="it"> |
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3416 <s xml:id="echoid-s1916" xml:space="preserve">Ex quibus perſpicuum eſt lineas omnes ſic ductas ab <lb/>ipſis ſectionibus in eandem proportionem ſecari. </s> |
|
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3417 <s xml:id="echoid-s1917" xml:space="preserve">eſt enim <lb/>diuidendo, conuertendoque cm ad mb, & </s> |
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3418 <s xml:id="echoid-s1918" xml:space="preserve">cf ad fb, ut <lb/>ce ad ea.</s> |
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3419 <s xml:id="echoid-s1919" xml:space="preserve"/> |
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3420 </p> |
|
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3421 </div> |
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3422 <div xml:id="echoid-div139" type="section" level="1" n="44"> |
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3423 <head xml:id="echoid-head49" xml:space="preserve">LEMMA III.</head> |
|
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3424 <p style="it"> |
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3425 <s xml:id="echoid-s1920" xml:space="preserve">Sed & </s> |
|
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3426 <s xml:id="echoid-s1921" xml:space="preserve">illud constare potest; </s> |
|
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3427 <s xml:id="echoid-s1922" xml:space="preserve">lineas, quæ in portioni-<lb/>bus eiuſmodi ſimilibus ita ducuntur, ut cú baſibus æqua-<lb/>les angulos contineant, ab ipſis ſimiles quoque portiones <lb/>abſcindere: </s> |
|
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3428 <s xml:id="echoid-s1923" xml:space="preserve">hoc eſt, ut in propoſita figura, portiones h b c, <lb/>m f c, quas lineæ c h, c m abſcindunt, etiam inter ſe <lb/>ſimiles eſſe.</s> |
|
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|
3429 <s xml:id="echoid-s1924" xml:space="preserve"/> |
|
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|
3430 </p> |
|
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|
3431 <p style="it"> |
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|
3432 <s xml:id="echoid-s1925" xml:space="preserve"><emph style="sc">D_ividantvr_</emph> enim ch, cm bifariam in punctis p q: </s> |
|
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|
3433 <s xml:id="echoid-s1926" xml:space="preserve">& </s> |
|
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|
3434 <s xml:id="echoid-s1927" xml:space="preserve">per <lb/>ipſa ducantur lineæ r p s, t q u diametris æquidiſtantes. </s> |
|
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|
3435 <s xml:id="echoid-s1928" xml:space="preserve">erit portio-<lb/>nis b s c diameter p s, & </s> |
|
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|
3436 <s xml:id="echoid-s1929" xml:space="preserve">portionis m u c diameter q u. </s> |
|
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|
3437 <s xml:id="echoid-s1930" xml:space="preserve">Itaque fiat <lb/>ut quadratum c r ad quadratum c p, ita linea b n ad aliam lineam, <lb/>quæ ſit s x: </s> |
|
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|
3438 <s xml:id="echoid-s1931" xml:space="preserve">& </s> |
|
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|
3439 <s xml:id="echoid-s1932" xml:space="preserve">ut quadratum c t ad quadratum c q, ita fiat f o ad <lb/>u y. </s> |
|
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|
3440 <s xml:id="echoid-s1933" xml:space="preserve">iam exijs <lb/> |
|
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|
3441 <anchor type="figure" xlink:label="fig-0076-01a" xlink:href="fig-0076-01"/> |
|
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|
3442 quæ demóſtra <lb/>uimus in com-<lb/>mentarijs in <lb/>quartam pro-<lb/>poſitioné. </s> |
|
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|
3443 <s xml:id="echoid-s1934" xml:space="preserve">Ar-<lb/>chrmedis de co <lb/>noidibus, & </s> |
|
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|
3444 <s xml:id="echoid-s1935" xml:space="preserve"><lb/>ſphæroidibus, <lb/>patet quadra-<lb/>tum c p æqua-<lb/>le eſſe rectan-<lb/>gulo p s x:</s> |
|
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|
3445 <s xml:id="echoid-s1936" xml:space="preserve"> |
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3446 <pb o="25" file="0077" n="77" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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3447 itêmq; </s> |
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3448 <s xml:id="echoid-s1937" xml:space="preserve">quadratum c q æquale rectangulo q u y, hoc eſt ſectionum <lb/>h s c, m u c lineas s x, u y, eas eſſe, iuxta quas poſſunt, quæ à ſectio-<lb/>ne ad diametrum ducuntur. </s> |
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3449 <s xml:id="echoid-s1938" xml:space="preserve">ſed cú triangula c p r, c q t ſimilia ſint, <lb/>habebit c r ad c p eandem proportionem, quam c t ad c q: </s> |
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3450 <s xml:id="echoid-s1939" xml:space="preserve">& </s> |
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3451 <s xml:id="echoid-s1940" xml:space="preserve">id-<lb/> |
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3452 <anchor type="note" xlink:label="note-0077-01a" xlink:href="note-0077-01"/> |
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3453 circo quadratum c r ad quadratum c p eandem habebit, quam <lb/>quadratum c t ad quadratum c q. </s> |
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3454 <s xml:id="echoid-s1941" xml:space="preserve">ergo & </s> |
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3455 <s xml:id="echoid-s1942" xml:space="preserve">linea b n, ad lineam <lb/>ſ x ita erit, ut linea fo ad ipſam u y. </s> |
|
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3456 <s xml:id="echoid-s1943" xml:space="preserve">erat autem b c ad c m, ut a c <lb/>ad c e. </s> |
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3457 <s xml:id="echoid-s1944" xml:space="preserve">quare & </s> |
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3458 <s xml:id="echoid-s1945" xml:space="preserve">earum dimidiæ c p ad c q, ut a d ad e g: </s> |
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3459 <s xml:id="echoid-s1946" xml:space="preserve">& </s> |
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3460 <s xml:id="echoid-s1947" xml:space="preserve"><lb/>permutando c p ad a d, ut c q ad e g. </s> |
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3461 <s xml:id="echoid-s1948" xml:space="preserve">Sed oſtenſum est a d ad b n <lb/>ita eſſe, ut e g ad f o: </s> |
|
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3462 <s xml:id="echoid-s1949" xml:space="preserve">& </s> |
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3463 <s xml:id="echoid-s1950" xml:space="preserve">b n ad s x, ut f o ad u y. </s> |
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3464 <s xml:id="echoid-s1951" xml:space="preserve">ergo ex <lb/>æquali c p ad ſ x erit, ut c q ad u y. </s> |
|
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3465 <s xml:id="echoid-s1952" xml:space="preserve">Quòd cum quadratú c p æqua <lb/>le ſit rectangulo p s x & </s> |
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3466 <s xml:id="echoid-s1953" xml:space="preserve">quadratum c q rectangulo q u y, erunt <lb/>tres lineæ ſ p, p c, ſ x proportionales; </s> |
|
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3467 <s xml:id="echoid-s1954" xml:space="preserve">itemq; </s> |
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3468 <s xml:id="echoid-s1955" xml:space="preserve">proportionales ip-<lb/>ſæ u q, q c, u y. </s> |
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3469 <s xml:id="echoid-s1956" xml:space="preserve">quare & </s> |
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3470 <s xml:id="echoid-s1957" xml:space="preserve">ſ p ad p c, ut u q ad q c: </s> |
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3471 <s xml:id="echoid-s1958" xml:space="preserve">& </s> |
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3472 <s xml:id="echoid-s1959" xml:space="preserve">ut p c ad <lb/>c h, ita q c ad c m. </s> |
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3473 <s xml:id="echoid-s1960" xml:space="preserve">ex æquali igitur ut portionis h ſ c diameter ſ p <lb/>ad eius baſim c h, ita portionis m u s diameter u q ad baſim c m. <lb/></s> |
|
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3474 <s xml:id="echoid-s1961" xml:space="preserve">& </s> |
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3475 <s xml:id="echoid-s1962" xml:space="preserve">anguli, quos diametri cum baſibus continent, ſunt æquales, quòd <lb/>lineæ ſ p, u q ſibi ipſis æquidiſtent, ergo & </s> |
|
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3476 <s xml:id="echoid-s1963" xml:space="preserve">portiones h ſ c, m u c <lb/>inter ſe ſimiles erunt. </s> |
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3477 <s xml:id="echoid-s1964" xml:space="preserve">id quod demonstrandum proponebatur.</s> |
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3478 <s xml:id="echoid-s1965" xml:space="preserve"/> |
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|
3479 </p> |
|
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3480 <div xml:id="echoid-div139" type="float" level="2" n="1"> |
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3481 <figure xlink:label="fig-0076-01" xlink:href="fig-0076-01a"> |
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3482 <image file="0076-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0076-01"/> |
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3483 </figure> |
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3484 <note position="right" xlink:label="note-0077-01" xlink:href="note-0077-01a" xml:space="preserve">22. fexti</note> |
|
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3485 </div> |
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|
3486 </div> |
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3487 <div xml:id="echoid-div141" type="section" level="1" n="45"> |
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3488 <head xml:id="echoid-head50" xml:space="preserve">LEMMA IIII.</head> |
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3489 <p style="it"> |
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3490 <s xml:id="echoid-s1966" xml:space="preserve">Sint duæ lineæ a b, c d, quæ ſecentur in punctis e f, <lb/>ita ut quam proportionem habet a e ad e b, habeat c f <lb/>ad f d: </s> |
|
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3491 <s xml:id="echoid-s1967" xml:space="preserve">rurſus ſecentur in aliis duobus punctis g h; </s> |
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3492 <s xml:id="echoid-s1968" xml:space="preserve">& </s> |
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3493 <s xml:id="echoid-s1969" xml:space="preserve"><lb/>habeat c h ad h d eandem proportionem, quam a g ad <lb/>g b. </s> |
|
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3494 <s xml:id="echoid-s1970" xml:space="preserve">Dico c f ad f h ita eſſe, ut a e ad e g.</s> |
|
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3495 <s xml:id="echoid-s1971" xml:space="preserve"/> |
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|
3496 </p> |
|
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|
3497 <p style="it"> |
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3498 <s xml:id="echoid-s1972" xml:space="preserve"><emph style="sc">Q_voniam_</emph> enim ut a e ad e b, ita c f ad f d, erit componen <lb/>do ut a b ad e b, ita c d ad f d. </s> |
|
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3499 <s xml:id="echoid-s1973" xml:space="preserve">Rurſus cum ſit ut a g ad g b, ita <lb/>c h ad h d; </s> |
|
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3500 <s xml:id="echoid-s1974" xml:space="preserve">componendo, conuertendoq; </s> |
|
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3501 <s xml:id="echoid-s1975" xml:space="preserve">ut g b ad a b, ita erit h d <lb/>ad c d. </s> |
|
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3502 <s xml:id="echoid-s1976" xml:space="preserve">ergo ex æquali, conuertendoq; </s> |
|
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|
3503 <s xml:id="echoid-s1977" xml:space="preserve">ut e b ad g b, ita f d ad h d:</s> |
|
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3504 <s xml:id="echoid-s1978" xml:space="preserve"> |
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|
3505 <pb file="0078" n="78" rhead="ARCHIMEDIS"/> |
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|
3506 & </s> |
|
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|
3507 <s xml:id="echoid-s1979" xml:space="preserve">per conuer-<lb/> |
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|
3508 <anchor type="figure" xlink:label="fig-0078-01a" xlink:href="fig-0078-01"/> |
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|
3509 ſionem rationis <lb/>ut e b ad e g, <lb/>ita f d ad f h. <lb/></s> |
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3510 <s xml:id="echoid-s1980" xml:space="preserve">eſt autem ut a e <lb/>ad e b, ita c f <lb/>ad f d. </s> |
|
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3511 <s xml:id="echoid-s1981" xml:space="preserve">ex æqua <lb/>li igitur ut a e <lb/>ad e g, ita c f <lb/>ad f h.</s> |
|
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3512 <s xml:id="echoid-s1982" xml:space="preserve"/> |
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|
3513 </p> |
|
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|
3514 <div xml:id="echoid-div141" type="float" level="2" n="1"> |
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|
3515 <figure xlink:label="fig-0078-01" xlink:href="fig-0078-01a"> |
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|
3516 <image file="0078-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0078-01"/> |
|
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3517 </figure> |
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3518 </div> |
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3519 <p style="it"> |
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3520 <s xml:id="echoid-s1983" xml:space="preserve"><emph style="sc">A_liter_</emph>. </s> |
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3521 <s xml:id="echoid-s1984" xml:space="preserve">Aptentur lineæ a b, c d inter ſe ſe, ita ut ad partes <lb/>a c angulum faciant; </s> |
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3522 <s xml:id="echoid-s1985" xml:space="preserve">& </s> |
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3523 <s xml:id="echoid-s1986" xml:space="preserve">ſint a c in uno atque eodem puncto: </s> |
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3524 <s xml:id="echoid-s1987" xml:space="preserve">deinde <lb/>iungantur d b, h g, fe. </s> |
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3525 <s xml:id="echoid-s1988" xml:space="preserve">cum igitur ſit ut a e ad e b, ita c f, hoc eſt <lb/>a f ad f d; </s> |
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3526 <s xml:id="echoid-s1989" xml:space="preserve">æquidiſtabit fe ipſi d b: </s> |
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3527 <s xml:id="echoid-s1990" xml:space="preserve">& </s> |
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3528 <s xml:id="echoid-s1991" xml:space="preserve">ſimiliter h g eidem d b <lb/> |
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3529 <anchor type="note" xlink:label="note-0078-01a" xlink:href="note-0078-01"/> |
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3530 æquidiſtabit: </s> |
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3531 <s xml:id="echoid-s1992" xml:space="preserve">quoniam a h ad h d eſt, ut a g ad g b. </s> |
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3532 <s xml:id="echoid-s1993" xml:space="preserve">ergo f c, h g <lb/> |
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3533 <anchor type="note" xlink:label="note-0078-02a" xlink:href="note-0078-02"/> |
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3534 inter ſe ſe æquidiſtant: </s> |
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3535 <s xml:id="echoid-s1994" xml:space="preserve">& </s> |
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3536 <s xml:id="echoid-s1995" xml:space="preserve">idcirco ut a e ad e g, ita a f; </s> |
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3537 <s xml:id="echoid-s1996" xml:space="preserve">hoc eſt c f ad <lb/>fh. </s> |
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3538 <s xml:id="echoid-s1997" xml:space="preserve">quod demonſtrare oportebat.</s> |
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3539 <s xml:id="echoid-s1998" xml:space="preserve"/> |
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|
3540 </p> |
|
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3541 <div xml:id="echoid-div142" type="float" level="2" n="2"> |
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3542 <note position="left" xlink:label="note-0078-01" xlink:href="note-0078-01a" xml:space="preserve">2. ſexti:</note> |
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3543 <note position="left" xlink:label="note-0078-02" xlink:href="note-0078-02a" xml:space="preserve">30. primi</note> |
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3544 </div> |
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3545 </div> |
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3546 <div xml:id="echoid-div144" type="section" level="1" n="46"> |
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3547 <head xml:id="echoid-head51" xml:space="preserve">LEMMA V.</head> |
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3548 <p style="it"> |
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3549 <s xml:id="echoid-s1999" xml:space="preserve">Sint rurſus duæ portiones ſimiles, contentæ rectis li-<lb/>neis, & </s> |
|
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3550 <s xml:id="echoid-s2000" xml:space="preserve">rectangulorum conorum ſectionibus, ut in ſupe-<lb/>riori figura a b c, cuius diameter b d: </s> |
|
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3551 <s xml:id="echoid-s2001" xml:space="preserve">& </s> |
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3552 <s xml:id="echoid-s2002" xml:space="preserve">e f c, cuius <lb/>diameter f g: </s> |
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3553 <s xml:id="echoid-s2003" xml:space="preserve">ducaturque à puncto e linea e h, diame-<lb/>tris b d, f g æquidiſtans, quæ ſectionem a b c in _k_ ſe-<lb/>cet: </s> |
|
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3554 <s xml:id="echoid-s2004" xml:space="preserve">& </s> |
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3555 <s xml:id="echoid-s2005" xml:space="preserve">à puncto c ducatur c h contingens ſectionem <lb/>a b c in c conueniensque cumlinea e h in h, quæ ſectio <lb/>nem quoque e f c in eodem c puncto continget, ut demon <lb/>strabitur. </s> |
|
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3556 <s xml:id="echoid-s2006" xml:space="preserve">Dico lineam ductam ab ipſa c h uſque ad ſe-<lb/>ctionem e f c, ita ut lineæ e h æquidistet, in eandem pro <lb/>portionem diuidi à ſectione a b c; </s> |
|
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3557 <s xml:id="echoid-s2007" xml:space="preserve">in quam linea c a à |
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|
3558 <pb o="34" file="0079" n="79" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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3559 ſectione e f c diuiditur: </s> |
|
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|
3560 <s xml:id="echoid-s2008" xml:space="preserve">pars uero lineæ c a, quæ eſt in-<lb/>ter duas ſectiones proportione reſpondebit parti lineæ <lb/>ductæ, quæ itidem inter eaſdem ſectiones interiicitur; </s> |
|
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3561 <s xml:id="echoid-s2009" xml:space="preserve">hoc <lb/>est ut in propoſita figura, ſi producatur d b uſque ad c h <lb/>in l, ut ſectioni e f c in puncto m occurrat; </s> |
|
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|
3562 <s xml:id="echoid-s2010" xml:space="preserve">lineam l <lb/>b ad b m eãdem proportionem habere, quàm c e ad e a.</s> |
|
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3563 <s xml:id="echoid-s2011" xml:space="preserve"/> |
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|
3564 </p> |
|
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|
3565 <p style="it"> |
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3566 <s xml:id="echoid-s2012" xml:space="preserve">Produc. </s> |
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3567 <s xml:id="echoid-s2013" xml:space="preserve">itur enim <lb/> |
|
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3568 <anchor type="figure" xlink:label="fig-0079-01a" xlink:href="fig-0079-01"/> |
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3569 q f ad eandem lineá <lb/>c h in n, ſecás a b c <lb/>ſectionem in o: </s> |
|
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3570 <s xml:id="echoid-s2014" xml:space="preserve">& </s> |
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3571 <s xml:id="echoid-s2015" xml:space="preserve"><lb/>iuncta b c, quæ tran <lb/>ſibit per f, ut oſten-<lb/>ſum eſt, erunt trian-<lb/>gula c g f, c d b ſi-<lb/>milia: </s> |
|
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3572 <s xml:id="echoid-s2016" xml:space="preserve">itémq; </s> |
|
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|
3573 <s xml:id="echoid-s2017" xml:space="preserve">ſimi-<lb/>lia íter ſe, c f n, c b l. <lb/></s> |
|
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3574 <s xml:id="echoid-s2018" xml:space="preserve">quare ut g f ad d b, <lb/> |
|
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|
3575 <anchor type="note" xlink:label="note-0079-01a" xlink:href="note-0079-01"/> |
|
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|
3576 ita erit c f ad c b: <lb/></s> |
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|
3577 <s xml:id="echoid-s2019" xml:space="preserve">& </s> |
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3578 <s xml:id="echoid-s2020" xml:space="preserve">ut c f ad c b, ita <lb/>f n ad b l. </s> |
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3579 <s xml:id="echoid-s2021" xml:space="preserve">ergo g f <lb/> |
|
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|
3580 <anchor type="note" xlink:label="note-0079-02a" xlink:href="note-0079-02"/> |
|
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3581 ad d b, ut f n ad b l: <lb/></s> |
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3582 <s xml:id="echoid-s2022" xml:space="preserve">& </s> |
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3583 <s xml:id="echoid-s2023" xml:space="preserve">permutando g f <lb/>ad f n, ut d b ad b l. </s> |
|
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|
3584 <s xml:id="echoid-s2024" xml:space="preserve"><lb/>eſt autem d b æqua <lb/>lis ipſi b l ex trigeſi <lb/>maquinta primi li-<lb/> |
|
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|
3585 <anchor type="note" xlink:label="note-0079-03a" xlink:href="note-0079-03"/> |
|
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|
3586 bri conicorum. </s> |
|
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|
3587 <s xml:id="echoid-s2025" xml:space="preserve">ergo <lb/>& </s> |
|
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|
3588 <s xml:id="echoid-s2026" xml:space="preserve">g f ipſi p i æqua <lb/>lis erit: </s> |
|
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|
3589 <s xml:id="echoid-s2027" xml:space="preserve">& </s> |
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3590 <s xml:id="echoid-s2028" xml:space="preserve">ex trige <lb/>ſimatertia eiuſdem <lb/>linea c h ſectionem <lb/>e f c in eodem pun- |
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3591 <pb file="0080" n="80" rhead="ARCHIMEDIS"/> |
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3592 cto continget. </s> |
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3593 <s xml:id="echoid-s2029" xml:space="preserve">Itaque iuncta cm producatur ad ſectionem a b c in p: <lb/></s> |
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3594 <s xml:id="echoid-s2030" xml:space="preserve">& </s> |
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3595 <s xml:id="echoid-s2031" xml:space="preserve">à p ad a c ducatur p q, quæ ipſi b d æquidiſtet. </s> |
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3596 <s xml:id="echoid-s2032" xml:space="preserve">quoniam igi-<lb/>tur linea c h contingit ſectionem e f c in c puncto; </s> |
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3597 <s xml:id="echoid-s2033" xml:space="preserve">habebit l m <lb/>ad m d proportionem eandem, quam c d ad d e, ex quinta propoſi-<lb/>tione Archimedis in libro de quadratura parabolæ. </s> |
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3598 <s xml:id="echoid-s2034" xml:space="preserve">& </s> |
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3599 <s xml:id="echoid-s2035" xml:space="preserve">propter triá <lb/>gulorum c m d, c p q <lb/> |
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3600 <anchor type="figure" xlink:label="fig-0080-01a" xlink:href="fig-0080-01"/> |
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3601 ſimilitudinem, ut c m <lb/>ad c d, ita erit c p ad <lb/>c q: </s> |
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3602 <s xml:id="echoid-s2036" xml:space="preserve">permutandôq; <lb/></s> |
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3603 <s xml:id="echoid-s2037" xml:space="preserve">ut c m ad c p, ita c d <lb/>ad c q. </s> |
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3604 <s xml:id="echoid-s2038" xml:space="preserve">ut autem c m <lb/>ad c p, ſic c e ad c a: </s> |
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3605 <s xml:id="echoid-s2039" xml:space="preserve"><lb/>quod proxime demó-<lb/>ſtrauimus. </s> |
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3606 <s xml:id="echoid-s2040" xml:space="preserve">quare ut <lb/>c e ad c a, ſit c d ad <lb/>c q: </s> |
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3607 <s xml:id="echoid-s2041" xml:space="preserve">hoc eſt ut totum <lb/>ad totum, ſic pars ad <lb/>partem, reliquum igi <lb/>tur d e ad reliquum <lb/>q a eſt ut c e ad c a; </s> |
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3608 <s xml:id="echoid-s2042" xml:space="preserve"><lb/>uidelicet ut c d ad <lb/>c q: </s> |
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3609 <s xml:id="echoid-s2043" xml:space="preserve">& </s> |
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3610 <s xml:id="echoid-s2044" xml:space="preserve">permutando <lb/>c d ad d e, ut c q ad <lb/>q a. </s> |
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3611 <s xml:id="echoid-s2045" xml:space="preserve">êſtq; </s> |
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3612 <s xml:id="echoid-s2046" xml:space="preserve">l m ad m <lb/>d, ut c d ad d e. </s> |
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3613 <s xml:id="echoid-s2047" xml:space="preserve">ergo <lb/>l m ad m d, ut c q ad <lb/>q a. </s> |
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3614 <s xml:id="echoid-s2048" xml:space="preserve">ſed l b ad b d <lb/>ex quinta Archime-<lb/>dis, quam diximus; </s> |
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3615 <s xml:id="echoid-s2049" xml:space="preserve"><lb/>eſt ut c d ad d a. </s> |
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3616 <s xml:id="echoid-s2050" xml:space="preserve">con <lb/>ſtat igitur ex antece-<lb/>denti lemmate c d ad d q ita eſſe, ut l b ad b m. </s> |
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3617 <s xml:id="echoid-s2051" xml:space="preserve">ut autem c d ad d q, <lb/> |
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3618 <anchor type="note" xlink:label="note-0080-01a" xlink:href="note-0080-01"/> |
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3619 ita c m ad m p. </s> |
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3620 <s xml:id="echoid-s2052" xml:space="preserve">ergo l b ad b m, ut c m ad m p. </s> |
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3621 <s xml:id="echoid-s2053" xml:space="preserve">Quòd cum demon <lb/>ſtratum fuerit, c m ad m p, ut c e ad e a: </s> |
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3622 <s xml:id="echoid-s2054" xml:space="preserve">habebit l b ad b m eandé |
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3623 <pb o="35" file="0081" n="81" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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3624 proportionem, quam c e ad e a. </s> |
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3625 <s xml:id="echoid-s2055" xml:space="preserve">ſimiliter demonſtrabitur eandem <lb/>babere n o ad o f: </s> |
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3626 <s xml:id="echoid-s2056" xml:space="preserve">& </s> |
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3627 <s xml:id="echoid-s2057" xml:space="preserve">reliquas eiuſmodi, at uero b K ad K e eam <lb/>habere proportionem, quam habet c e ad e a, ex eadem quinta. </s> |
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3628 <s xml:id="echoid-s2058" xml:space="preserve">Ar-<lb/>chimedis perſpicue apparet. </s> |
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3629 <s xml:id="echoid-s2059" xml:space="preserve">at que illud eſt, quod demonſtr andum <lb/>propoſuimus.</s> |
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3630 <s xml:id="echoid-s2060" xml:space="preserve"/> |
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3631 </p> |
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3632 <div xml:id="echoid-div144" type="float" level="2" n="1"> |
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3633 <figure xlink:label="fig-0079-01" xlink:href="fig-0079-01a"> |
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3634 <image file="0079-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0079-01"/> |
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3635 </figure> |
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3636 <note position="right" xlink:label="note-0079-01" xlink:href="note-0079-01a" xml:space="preserve">4. ſexti:</note> |
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3637 <note position="right" xlink:label="note-0079-02" xlink:href="note-0079-02a" xml:space="preserve">11. quinti</note> |
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3638 <note position="right" xlink:label="note-0079-03" xlink:href="note-0079-03a" xml:space="preserve">14. quinti</note> |
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3639 <figure xlink:label="fig-0080-01" xlink:href="fig-0080-01a"> |
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3640 <image file="0080-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0080-01"/> |
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3641 </figure> |
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3642 <note position="left" xlink:label="note-0080-01" xlink:href="note-0080-01a" xml:space="preserve">2. ſexti.</note> |
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3643 </div> |
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3644 </div> |
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3645 <div xml:id="echoid-div146" type="section" level="1" n="47"> |
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3646 <head xml:id="echoid-head52" xml:space="preserve">LEMMA VI.</head> |
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3647 <p style="it"> |
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3648 <s xml:id="echoid-s2061" xml:space="preserve">Itaque maneant eadem, quæ ſupra: </s> |
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3649 <s xml:id="echoid-s2062" xml:space="preserve">& </s> |
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3650 <s xml:id="echoid-s2063" xml:space="preserve">itidem deſcri-<lb/>batur alia portio ſimilis contenta recta linea & </s> |
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3651 <s xml:id="echoid-s2064" xml:space="preserve">rectan-<lb/>guli coni ſectione d r c; </s> |
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3652 <s xml:id="echoid-s2065" xml:space="preserve">cuius diameter r s, ut ſecet li-<lb/>neam f g in t: </s> |
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3653 <s xml:id="echoid-s2066" xml:space="preserve">producaturque s r ad lineam c h in u; <lb/></s> |
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3654 <s xml:id="echoid-s2067" xml:space="preserve">cuiſectio a b c occurrat in x, & </s> |
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3655 <s xml:id="echoid-s2068" xml:space="preserve">e f c in y. </s> |
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3656 <s xml:id="echoid-s2069" xml:space="preserve">Dico b m <lb/>ad m d proportionem habere compoſitam ex propor-<lb/>tione, quam babet e a ad a c; </s> |
|
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3657 <s xml:id="echoid-s2070" xml:space="preserve">& </s> |
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3658 <s xml:id="echoid-s2071" xml:space="preserve">ex ea, quam c d ba-<lb/>bet ad de.</s> |
|
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3659 <s xml:id="echoid-s2072" xml:space="preserve"/> |
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|
3660 </p> |
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3661 <p style="it"> |
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3662 <s xml:id="echoid-s2073" xml:space="preserve"><emph style="sc">S_imiliter_</emph> enim ut ſupra, demonſtrabimus lineam c h con-<lb/>tingere ſectioné d r c in c puncto: </s> |
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3663 <s xml:id="echoid-s2074" xml:space="preserve">& </s> |
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3664 <s xml:id="echoid-s2075" xml:space="preserve">l m ad m d, itêmq; </s> |
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3665 <s xml:id="echoid-s2076" xml:space="preserve">n f ad f t; <lb/></s> |
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3666 <s xml:id="echoid-s2077" xml:space="preserve">& </s> |
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3667 <s xml:id="echoid-s2078" xml:space="preserve">u y ad y r ita eſſe, ut c d ad d e. </s> |
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3668 <s xml:id="echoid-s2079" xml:space="preserve">Quoniam igitur lb ad b m eſt, <lb/>ut c e ad e a; </s> |
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3669 <s xml:id="echoid-s2080" xml:space="preserve">erit componendo, conuertendôq; </s> |
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3670 <s xml:id="echoid-s2081" xml:space="preserve">bm ad lm, ut e a ad <lb/>a c: </s> |
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3671 <s xml:id="echoid-s2082" xml:space="preserve">& </s> |
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3672 <s xml:id="echoid-s2083" xml:space="preserve">ut lm ad m d, ita c d ad d e. </s> |
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3673 <s xml:id="echoid-s2084" xml:space="preserve">proportio autem b m ad m d <lb/>compoſita eſt ex proportione, quam habet b m ad l m, & </s> |
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3674 <s xml:id="echoid-s2085" xml:space="preserve">ex propor <lb/>tione, quam l m habet ad m d. </s> |
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3675 <s xml:id="echoid-s2086" xml:space="preserve">ergo proportio b m ad m d etiam com <lb/>poſita erit ex proportione, quam habet e a, ad a c; </s> |
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3676 <s xml:id="echoid-s2087" xml:space="preserve">& </s> |
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3677 <s xml:id="echoid-s2088" xml:space="preserve">ex ea, quam <lb/>c d habet ad d e. </s> |
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3678 <s xml:id="echoid-s2089" xml:space="preserve">Eadem ratione demonſtrabitur o f ad f t; </s> |
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3679 <s xml:id="echoid-s2090" xml:space="preserve">itêmq; </s> |
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3680 <s xml:id="echoid-s2091" xml:space="preserve"><lb/>x y ad y r proportionem habere ex eiſdem proportionibus compo-<lb/>ſitam: </s> |
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3681 <s xml:id="echoid-s2092" xml:space="preserve">& </s> |
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|
3682 <s xml:id="echoid-s2093" xml:space="preserve">ita in aijs. </s> |
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3683 <s xml:id="echoid-s2094" xml:space="preserve">quod demonſtrare oportebat.</s> |
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|
3684 <s xml:id="echoid-s2095" xml:space="preserve"/> |
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|
3685 </p> |
|
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|
3686 <p style="it"> |
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3687 <s xml:id="echoid-s2096" xml:space="preserve">Ex quibus apparet lineas ſic ductas, quæ inter ſectio <lb/>nes a b c, d r c interiiciuntur à ſectione e f c in eandem <lb/>proportionem diuidi.</s> |
|
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|
3688 <s xml:id="echoid-s2097" xml:space="preserve"/> |
|
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|
3689 </p> |
|
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|
3690 <pb file="0082" n="82" rhead="ARCHIMEDIS"/> |
|
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|
3691 <p style="it"> |
|
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3692 <s xml:id="echoid-s2098" xml:space="preserve">_Etenim c b ad b d eſt ut ſex ad quindecim.</s> |
|
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3693 <s xml:id="echoid-s2099" xml:space="preserve">]_ Poſuimus <lb/> |
|
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|
3694 <anchor type="note" xlink:label="note-0082-01a" xlink:href="note-0082-01"/> |
|
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|
3695 enim b K duplam eſſe ipſius K d. </s> |
|
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3696 <s xml:id="echoid-s2100" xml:space="preserve">quare componendo b d ad k d erit, <lb/>ut tria ad unum; </s> |
|
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3697 <s xml:id="echoid-s2101" xml:space="preserve">hoc eſt ut quindecim ad quinque. </s> |
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3698 <s xml:id="echoid-s2102" xml:space="preserve">ſed b d ad K c <lb/>erat ut quídecim <lb/> |
|
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|
3699 <anchor type="figure" xlink:label="fig-0082-01a" xlink:href="fig-0082-01"/> |
|
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|
3700 ad quatuor. </s> |
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3701 <s xml:id="echoid-s2103" xml:space="preserve">ergo <lb/>b d ad d c, ut quin <lb/>decim ad nouem: <lb/></s> |
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3702 <s xml:id="echoid-s2104" xml:space="preserve">& </s> |
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|
3703 <s xml:id="echoid-s2105" xml:space="preserve">per conuerſio <lb/>nem rationis, con <lb/>uertendôq; </s> |
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3704 <s xml:id="echoid-s2106" xml:space="preserve">c b ad <lb/>b d, ut ſex ad quí <lb/>decim.</s> |
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|
3705 <s xml:id="echoid-s2107" xml:space="preserve"/> |
|
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|
3706 </p> |
|
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|
3707 <div xml:id="echoid-div146" type="float" level="2" n="1"> |
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3708 <note position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve">N</note> |
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3709 <figure xlink:label="fig-0082-01" xlink:href="fig-0082-01a"> |
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|
3710 <image file="0082-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0082-01"/> |
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|
3711 </figure> |
|
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|
3712 </div> |
|
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|
3713 <p> |
|
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3714 <s xml:id="echoid-s2108" xml:space="preserve">Etut c b ad <lb/> |
|
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|
3715 <anchor type="note" xlink:label="note-0082-02a" xlink:href="note-0082-02"/> |
|
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|
3716 b d, ita e b ad <lb/>b a, & </s> |
|
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3717 <s xml:id="echoid-s2109" xml:space="preserve">d z ad <lb/>d a.</s> |
|
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|
3718 <s xml:id="echoid-s2110" xml:space="preserve">] _Nam cum_ <lb/>_triangula c b e,_ <lb/>_d b a ſint ſimilia,_ <lb/>_erit ut c b ad b e,_ <lb/>_ita d b, ad b a & </s> |
|
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|
3719 <s xml:id="echoid-s2111" xml:space="preserve">permutando, ut c b ad b d; </s> |
|
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|
3720 <s xml:id="echoid-s2112" xml:space="preserve">ita e b ad b a. </s> |
|
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|
3721 <s xml:id="echoid-s2113" xml:space="preserve">Rurſus_ <lb/>_ut b c ad c e, ita b d ad d a: </s> |
|
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|
3722 <s xml:id="echoid-s2114" xml:space="preserve">permutandôq; </s> |
|
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|
3723 <s xml:id="echoid-s2115" xml:space="preserve">ut c b ad b d, ita c e, hoc_ <lb/>_eſt d z ei æqualis ad d a._</s> |
|
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|
3724 <s xml:id="echoid-s2116" xml:space="preserve"/> |
|
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|
3725 </p> |
|
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|
3726 <div xml:id="echoid-div147" type="float" level="2" n="2"> |
|
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|
3727 <note position="left" xlink:label="note-0082-02" xlink:href="note-0082-02a" xml:space="preserve">O</note> |
|
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|
3728 </div> |
|
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|
3729 <p> |
|
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|
3730 <s xml:id="echoid-s2117" xml:space="preserve">Harum autem d z d a duplæ ſuntipſæ l i, la.</s> |
|
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|
3731 <s xml:id="echoid-s2118" xml:space="preserve">] _Lineam_ <lb/> |
|
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|
3732 <anchor type="note" xlink:label="note-0082-03a" xlink:href="note-0082-03"/> |
|
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|
3733 _quidem l a duplam eſſe ipſius d a, cum b d ſit portionis diameter,_ <lb/>_manifeſte conſtat. </s> |
|
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|
3734 <s xml:id="echoid-s2119" xml:space="preserve">At uero l i ipſius d z dupla hoc pacto demon-_ <lb/>_ſtrabitur. </s> |
|
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|
3735 <s xml:id="echoid-s2120" xml:space="preserve">Quoniam enim z d ad d a eſt, ut duo ad quinque; </s> |
|
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|
3736 <s xml:id="echoid-s2121" xml:space="preserve">erit có_ <lb/>_uertendo, diuidendôq; </s> |
|
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|
3737 <s xml:id="echoid-s2122" xml:space="preserve">a z, hoc eſt i z ad z d, ut tria ad duo: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
3738 <s xml:id="echoid-s2123" xml:space="preserve">&_</s> |
|
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parents:
diff
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|
3739 <s xml:id="echoid-s2124" xml:space="preserve"> <lb/>_rurſus diuidendo i d ad d z, ut <gap/>m ad duo. </s> |
|
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parents:
diff
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|
3740 <s xml:id="echoid-s2125" xml:space="preserve">erat autem z d ad_ <lb/>_d a, hoc eſt ad d l, ut duo ad quinque. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
3741 <s xml:id="echoid-s2126" xml:space="preserve">ergo ex æquali, conuertendóq;_ <lb/></s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
3742 <s xml:id="echoid-s2127" xml:space="preserve">_l d ad d i, ut quinque ad unum: </s> |
|
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parents:
diff
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|
3743 <s xml:id="echoid-s2128" xml:space="preserve">& </s> |
|
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parents:
diff
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|
3744 <s xml:id="echoid-s2129" xml:space="preserve">per conuerſionem rationis d l ad_ <lb/>_li, ut quinque ad quatuor. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
3745 <s xml:id="echoid-s2130" xml:space="preserve">ſed d z ad d l erat, ut duo ad quinque._ </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
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|
3746 <s xml:id="echoid-s2131" xml:space="preserve"><lb/>_ergo rurſus ex æquali d z ad l i, ut duo ad quatuor. </s> |
|
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parents:
diff
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|
3747 <s xml:id="echoid-s2132" xml:space="preserve">dupla eſt igitur_ <lb/>_l i ipſius d z. </s> |
|
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diff
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|
3748 <s xml:id="echoid-s2133" xml:space="preserve">quod demonſtrandum fuerat._</s> |
|
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|
3749 <s xml:id="echoid-s2134" xml:space="preserve"/> |
|
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diff
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|
3750 </p> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3751 <div xml:id="echoid-div148" type="float" level="2" n="3"> |
|
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diff
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|
3752 <note position="left" xlink:label="note-0082-03" xlink:href="note-0082-03a" xml:space="preserve">P</note> |
|
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|
3753 </div> |
|
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diff
changeset
|
3754 <p> |
|
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diff
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|
3755 <s xml:id="echoid-s2135" xml:space="preserve">Et a d ad d i eam proportionem habet, quã quinque <lb/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3756 <anchor type="note" xlink:label="note-0082-04a" xlink:href="note-0082-04"/> |
|
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|
3757 <pb o="36" file="0083" n="83" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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changeset
|
3758 ad unum.</s> |
|
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diff
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|
3759 <s xml:id="echoid-s2136" xml:space="preserve">] _Hoc nos proxime demonſtrauimus._</s> |
|
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|
3760 <s xml:id="echoid-s2137" xml:space="preserve"/> |
|
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diff
changeset
|
3761 </p> |
|
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|
3762 <div xml:id="echoid-div149" type="float" level="2" n="4"> |
|
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|
3763 <note position="left" xlink:label="note-0082-04" xlink:href="note-0082-04a" xml:space="preserve">Q</note> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3764 </div> |
|
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diff
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|
3765 <p> |
|
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|
3766 <s xml:id="echoid-s2138" xml:space="preserve">Demonſtratum eſt enim ſuperius portionem cuius axis <lb/> |
|
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|
3767 <anchor type="note" xlink:label="note-0083-01a" xlink:href="note-0083-01"/> |
|
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|
3768 eſt maior, quàm ſeſquialter eius, quæ uſque ad axem, ſiad <lb/>humidum in grauitate non minorem proportionem ha-<lb/>beat &</s> |
|
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|
3769 <s xml:id="echoid-s2139" xml:space="preserve">c.</s> |
|
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|
3770 <s xml:id="echoid-s2140" xml:space="preserve">] _Illud uero demonſtrauit in quarta propoſitione buius_ <lb/>_libri_.</s> |
|
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|
3771 <s xml:id="echoid-s2141" xml:space="preserve"/> |
|
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|
3772 </p> |
|
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|
3773 <div xml:id="echoid-div150" type="float" level="2" n="5"> |
|
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|
3774 <note position="right" xlink:label="note-0083-01" xlink:href="note-0083-01a" xml:space="preserve">R</note> |
|
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|
3775 </div> |
|
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|
3776 </div> |
|
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|
3777 <div xml:id="echoid-div152" type="section" level="1" n="48"> |
|
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|
3778 <head xml:id="echoid-head53" xml:space="preserve">II.</head> |
|
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|
3779 <p> |
|
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|
3780 <s xml:id="echoid-s2142" xml:space="preserve">Si portio ad humidum in grauitate minorem <lb/> |
|
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|
3781 <anchor type="note" xlink:label="note-0083-02a" xlink:href="note-0083-02"/> |
|
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|
3782 quidem proportionem habeat, quàm quadra-<lb/>tum ſ b ad quadratum b d; </s> |
|
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|
3783 <s xml:id="echoid-s2143" xml:space="preserve">maiorem uero, <lb/>quàm quadratum x o ad quadratum b d; </s> |
|
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|
3784 <s xml:id="echoid-s2144" xml:space="preserve">de-<lb/>miſſa in humidum, adeo inclinata, ut baſis ip-<lb/>ſius non contingat humidum, inclinata conſi-<lb/>ſtet; </s> |
|
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diff
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|
3785 <s xml:id="echoid-s2145" xml:space="preserve">ita ut baſis ſuperficiem humidi nullo modo <lb/>contingat; </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3786 <s xml:id="echoid-s2146" xml:space="preserve">& </s> |
|
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|
3787 <s xml:id="echoid-s2147" xml:space="preserve">axis cum humidi ſuperficie angu-<lb/>lum faciat maiorem angulo χ.</s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3788 <s xml:id="echoid-s2148" xml:space="preserve"/> |
|
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changeset
|
3789 </p> |
|
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|
3790 <div xml:id="echoid-div152" type="float" level="2" n="1"> |
|
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|
3791 <note position="right" xlink:label="note-0083-02" xlink:href="note-0083-02a" xml:space="preserve">A</note> |
|
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|
3792 </div> |
|
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|
3793 </div> |
|
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|
3794 <div xml:id="echoid-div154" type="section" level="1" n="49"> |
|
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|
3795 <head xml:id="echoid-head54" xml:space="preserve">III.</head> |
|
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changeset
|
3796 <p> |
|
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|
3797 <s xml:id="echoid-s2149" xml:space="preserve">Si portio ad humidum in grauitate, eam ha-<lb/>beat proportionem, quam quadratum x o ad <lb/>quadratum b d; </s> |
|
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|
3798 <s xml:id="echoid-s2150" xml:space="preserve">demiſſa in humidum inclinata <lb/>adeo, ut baſis ipſius non contingat humidum; <lb/></s> |
|
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|
3799 <s xml:id="echoid-s2151" xml:space="preserve">conſiſtet, & </s> |
|
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changeset
|
3800 <s xml:id="echoid-s2152" xml:space="preserve">manebit ita, ut baſis in uno pun-<lb/>cto humidi ſuperficiem contingat: </s> |
|
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|
3801 <s xml:id="echoid-s2153" xml:space="preserve">& </s> |
|
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|
3802 <s xml:id="echoid-s2154" xml:space="preserve">axis cum <lb/>ſuperficie humidi angulũ faciat angulo χ æqualẽ. </s> |
|
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|
3803 <s xml:id="echoid-s2155" xml:space="preserve"><lb/>Quòd ſi portio ad humidum in grauitate cam <lb/>proportionem habeat, quam quadratum p f ad |
|
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|
3804 <pb file="0084" n="84" rhead="ARCHIMEDIS"/> |
|
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|
3805 quadratum b d; </s> |
|
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|
3806 <s xml:id="echoid-s2156" xml:space="preserve">in humidum demiſſa, & </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
3807 <s xml:id="echoid-s2157" xml:space="preserve">poſi-<lb/>ta inclinata adeo, ut baſis ipſius non contingat <lb/>humidum; </s> |
|
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3808 <s xml:id="echoid-s2158" xml:space="preserve">conſiſtet inclinata, ita ut baſis in uno <lb/>puncto humidi ſuperficiem contingat: </s> |
|
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|
3809 <s xml:id="echoid-s2159" xml:space="preserve">& </s> |
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|
3810 <s xml:id="echoid-s2160" xml:space="preserve">axis cũ <lb/>ea faciat angulum angulo φ æqualem.</s> |
|
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3811 <s xml:id="echoid-s2161" xml:space="preserve"/> |
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diff
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|
3812 </p> |
|
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|
3813 </div> |
|
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|
3814 <div xml:id="echoid-div155" type="section" level="1" n="50"> |
|
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|
3815 <head xml:id="echoid-head55" xml:space="preserve">IIII.</head> |
|
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|
3816 <p> |
|
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3817 <s xml:id="echoid-s2162" xml:space="preserve">Si portio ad humidum in grauitate maiorem <lb/> |
|
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|
3818 <anchor type="note" xlink:label="note-0084-01a" xlink:href="note-0084-01"/> |
|
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|
3819 quidem proportionem habeat, quàm quadra-<lb/>tum f p ad quadratum b d; </s> |
|
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|
3820 <s xml:id="echoid-s2163" xml:space="preserve">minorem uero, <lb/>quàm quadratum x o ad b d quadratum; </s> |
|
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|
3821 <s xml:id="echoid-s2164" xml:space="preserve">in hu-<lb/>midum demiſſa, & </s> |
|
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|
3822 <s xml:id="echoid-s2165" xml:space="preserve">inclinata adeo, ut baſis ipſius <lb/>non contingat humidum conſiſtet, & </s> |
|
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|
3823 <s xml:id="echoid-s2166" xml:space="preserve">manebit <lb/>ita, ut baſis in humidum magis demergatur.</s> |
|
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|
3824 <s xml:id="echoid-s2167" xml:space="preserve"/> |
|
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|
3825 </p> |
|
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|
3826 <div xml:id="echoid-div155" type="float" level="2" n="1"> |
|
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|
3827 <note position="left" xlink:label="note-0084-01" xlink:href="note-0084-01a" xml:space="preserve">B</note> |
|
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|
3828 </div> |
|
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|
3829 </div> |
|
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|
3830 <div xml:id="echoid-div157" type="section" level="1" n="51"> |
|
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|
3831 <head xml:id="echoid-head56" xml:space="preserve">V.</head> |
|
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diff
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|
3832 <p> |
|
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|
3833 <s xml:id="echoid-s2168" xml:space="preserve">Si portio ad humidum in grauitate proportio <lb/>nem habeat minorem, quàm quadratum f p ad <lb/>quadratum b d: </s> |
|
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diff
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|
3834 <s xml:id="echoid-s2169" xml:space="preserve">demiſſa in humidum, & </s> |
|
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diff
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|
3835 <s xml:id="echoid-s2170" xml:space="preserve">poſita <lb/>inclinata adeo ut baſis ipſius non contingat humi <lb/>dum: </s> |
|
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diff
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|
3836 <s xml:id="echoid-s2171" xml:space="preserve">conſiſtet inclinata, ita ut axis ipſius cum <lb/>humidi ſuperficie angulum faciat minorem an-<lb/>gulo φ: </s> |
|
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|
3837 <s xml:id="echoid-s2172" xml:space="preserve">& </s> |
|
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diff
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|
3838 <s xml:id="echoid-s2173" xml:space="preserve">baſis nullo modo ſuperficiem humi-<lb/>di contingat. </s> |
|
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|
3839 <s xml:id="echoid-s2174" xml:space="preserve">Hæc autem omnia deinceps de-<lb/>monſtrabuntur.</s> |
|
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|
3840 <s xml:id="echoid-s2175" xml:space="preserve"/> |
|
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diff
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|
3841 </p> |
|
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|
3842 <pb o="37" file="0085" n="85" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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|
3843 </div> |
|
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|
3844 <div xml:id="echoid-div158" type="section" level="1" n="52"> |
|
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|
3845 <head xml:id="echoid-head57" xml:space="preserve">DEMONSTRATIO SECVNDAE PARTIS.</head> |
|
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|
3846 <p> |
|
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|
3847 <s xml:id="echoid-s2176" xml:space="preserve">ITAQVE primum habeat portio ad humidum in <lb/>grauitate proportionem quidem maiorem, quàm qua dra <lb/>tum x o ad quadratum b d; </s> |
|
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|
3848 <s xml:id="echoid-s2177" xml:space="preserve">minorem uero, quàm quadra <lb/>tum, quod fit ab exceſſu, quo axis eſt maior, quàm ſeſquial-<lb/>ter eius, quæ uſque ad axem, ad quadratum b d: </s> |
|
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|
3849 <s xml:id="echoid-s2178" xml:space="preserve">& </s> |
|
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|
3850 <s xml:id="echoid-s2179" xml:space="preserve">quam <lb/>proportionem habet portio ad humidum in grauitate, eã <lb/>habeat quadratum, quod fit à linea ψ ad quadratum b d: <lb/></s> |
|
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|
3851 <s xml:id="echoid-s2180" xml:space="preserve">erit ψ maior quidem, quàm x o, minor uero, quàm exceſ-<lb/> |
|
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|
3852 <anchor type="note" xlink:label="note-0085-01a" xlink:href="note-0085-01"/> |
|
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changeset
|
3853 ſus, quo axis eſt maior, quàm ſeſquialter eius, quæ uſque ad <lb/>axem. </s> |
|
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|
3854 <s xml:id="echoid-s2181" xml:space="preserve">aptetur quædam recta linea m n conicis ſectioni-<lb/>bus a m q l, <lb/> |
|
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|
3855 <anchor type="figure" xlink:label="fig-0085-01a" xlink:href="fig-0085-01"/> |
|
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changeset
|
3856 a x d interiecta, <lb/>ac media, quæ li <lb/>neæ ψ ſit æqua-<lb/>lis; </s> |
|
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|
3857 <s xml:id="echoid-s2182" xml:space="preserve">ſecetq; </s> |
|
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|
3858 <s xml:id="echoid-s2183" xml:space="preserve">reli-<lb/>quã coni ſectio <lb/>nem in pun cto <lb/>h; </s> |
|
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|
3859 <s xml:id="echoid-s2184" xml:space="preserve">& </s> |
|
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diff
changeset
|
3860 <s xml:id="echoid-s2185" xml:space="preserve">rectam li-<lb/>neam r g in u. <lb/></s> |
|
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changeset
|
3861 <s xml:id="echoid-s2186" xml:space="preserve">demõſtrabitur <lb/> |
|
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|
3862 <anchor type="note" xlink:label="note-0085-02a" xlink:href="note-0085-02"/> |
|
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changeset
|
3863 m h dupla ip-<lb/>ſius h n, ſicuti <lb/>demonſtratum <lb/>eſt o g ipſius g x <lb/>duplam eſſe. </s> |
|
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diff
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|
3864 <s xml:id="echoid-s2187" xml:space="preserve">à <lb/>puncto autẽ m <lb/>ducatur m y contingens ſectionem a m q l in m: </s> |
|
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|
3865 <s xml:id="echoid-s2188" xml:space="preserve">& </s> |
|
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changeset
|
3866 <s xml:id="echoid-s2189" xml:space="preserve">m c a d <lb/>b d perpendicularis. </s> |
|
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changeset
|
3867 <s xml:id="echoid-s2190" xml:space="preserve">poſtea ducta a n, & </s> |
|
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diff
changeset
|
3868 <s xml:id="echoid-s2191" xml:space="preserve">producta ad q li <lb/>neæ a n, n q inter ſe æquales erunt. </s> |
|
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|
3869 <s xml:id="echoid-s2192" xml:space="preserve">quoniã enim in ſimi-<lb/> |
|
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changeset
|
3870 <anchor type="note" xlink:label="note-0085-03a" xlink:href="note-0085-03"/> |
|
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changeset
|
3871 libus portionibus a m q l, a x d ductæ ſunt à baſibus ad <lb/>portiones lineæ a q, a n, quæ æquales angulos continent <lb/>cum ipſis baſibus, eandem proportionem habebit q a ad <lb/>an, quam la ad a d. </s> |
|
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changeset
|
3872 <s xml:id="echoid-s2193" xml:space="preserve">æqualis eſt ergo a n ipſi n q; </s> |
|
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|
3873 <s xml:id="echoid-s2194" xml:space="preserve">& </s> |
|
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changeset
|
3874 <s xml:id="echoid-s2195" xml:space="preserve">a q <lb/> |
|
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changeset
|
3875 <anchor type="note" xlink:label="note-0085-04a" xlink:href="note-0085-04"/> |
|
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diff
changeset
|
3876 <pb file="0086" n="86" rhead="ARCHIMEDIS"/> |
|
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changeset
|
3877 ipſi my æquidiſtans. </s> |
|
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diff
changeset
|
3878 <s xml:id="echoid-s2196" xml:space="preserve">Demonſtrandum eſt portionem in <lb/> |
|
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diff
changeset
|
3879 <anchor type="note" xlink:label="note-0086-01a" xlink:href="note-0086-01"/> |
|
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parents:
diff
changeset
|
3880 humidum demiſſam, inclinatamq; </s> |
|
22d6a63640c6
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parents:
diff
changeset
|
3881 <s xml:id="echoid-s2197" xml:space="preserve">adeo, ut baſis ipſius nõ <lb/>contingat humidum, inclinatam conſiſtere ita, ut baſis ſu-<lb/>perficiem humidi nullo modo contingat: </s> |
|
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parents:
diff
changeset
|
3882 <s xml:id="echoid-s2198" xml:space="preserve">& </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3883 <s xml:id="echoid-s2199" xml:space="preserve">axis cum ea fa <lb/>ciat angulum angulo χ maiorem. </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3884 <s xml:id="echoid-s2200" xml:space="preserve">Demittatur enim in hu-<lb/>midum, conſiſtatq; </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3885 <s xml:id="echoid-s2201" xml:space="preserve">ita, ut baſis ipſius in uno puncto cõtin <lb/>gat humidi ſuperficiem: </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
3886 <s xml:id="echoid-s2202" xml:space="preserve">& </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3887 <s xml:id="echoid-s2203" xml:space="preserve">ſecta ipſa portione per axem, <lb/>plano ad humidi ſuperficiem recto; </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3888 <s xml:id="echoid-s2204" xml:space="preserve">ſuperficiei quidẽ por-<lb/>tionis ſectio ſit a p o l rectanguli coni ſectio: </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3889 <s xml:id="echoid-s2205" xml:space="preserve">ſuperficiei <lb/>humidi ſectio ſit a o: </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3890 <s xml:id="echoid-s2206" xml:space="preserve">axis autem portionis, & </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
3891 <s xml:id="echoid-s2207" xml:space="preserve">ſectionis dia <lb/>meter b d: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
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|
3892 <s xml:id="echoid-s2208" xml:space="preserve">& </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
3893 <s xml:id="echoid-s2209" xml:space="preserve">ſecetur b d in punctis k r, ut dictum eſt: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3894 <s xml:id="echoid-s2210" xml:space="preserve">du-<lb/> |
|
22d6a63640c6
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parents:
diff
changeset
|
3895 <anchor type="note" xlink:label="note-0086-02a" xlink:href="note-0086-02"/> |
|
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parents:
diff
changeset
|
3896 catur etiam p g æquidiſtans ipſi a o, quæ ſectionem a p o l <lb/>contingat in p: </s> |
|
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parents:
diff
changeset
|
3897 <s xml:id="echoid-s2211" xml:space="preserve">atque ab eo puncto ducatur p t æquidiſtãs <lb/>ipſi b d; </s> |
|
22d6a63640c6
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parents:
diff
changeset
|
3898 <s xml:id="echoid-s2212" xml:space="preserve">& </s> |
|
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parents:
diff
changeset
|
3899 <s xml:id="echoid-s2213" xml:space="preserve">p s ad b d perpendicularis. </s> |
|
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parents:
diff
changeset
|
3900 <s xml:id="echoid-s2214" xml:space="preserve">Itaque quoniam <lb/>portio ad humidum in grauitate eam proportionem ha-<lb/>bet, quam qua-<lb/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
3901 <anchor type="figure" xlink:label="fig-0086-01a" xlink:href="fig-0086-01"/> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
3902 dratũ, quod fit <lb/>à linea χ ad qua <lb/>dratum b d: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3903 <s xml:id="echoid-s2215" xml:space="preserve">quã <lb/>uero proportio <lb/>nem habet por-<lb/>tio ad humidũ, <lb/>eandem pars ip <lb/>ſius demerſa ha <lb/>bet ad totã por <lb/>tionẽ: </s> |
|
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diff
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|
3904 <s xml:id="echoid-s2216" xml:space="preserve">& </s> |
|
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diff
changeset
|
3905 <s xml:id="echoid-s2217" xml:space="preserve">quam <lb/>pars demerſa ad <lb/>totam, eandem <lb/>habet quadra-<lb/>tum t p ad b d <lb/>quadratum: </s> |
|
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diff
changeset
|
3906 <s xml:id="echoid-s2218" xml:space="preserve">erit <lb/>linea ψ æqualis <lb/>ipſi t p. </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3907 <s xml:id="echoid-s2219" xml:space="preserve">quare & </s> |
|
22d6a63640c6
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diff
changeset
|
3908 <s xml:id="echoid-s2220" xml:space="preserve">lineæ m n, p t; </s> |
|
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diff
changeset
|
3909 <s xml:id="echoid-s2221" xml:space="preserve">itemq, portiones a m q, <lb/>a p o inter ſe ſunt æquales. </s> |
|
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|
3910 <s xml:id="echoid-s2222" xml:space="preserve">Quòd cumin portionibus <lb/> |
|
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diff
changeset
|
3911 <anchor type="note" xlink:label="note-0086-03a" xlink:href="note-0086-03"/> |
|
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changeset
|
3912 <pb o="38" file="0087" n="87" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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changeset
|
3913 æqualibus, & </s> |
|
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|
3914 <s xml:id="echoid-s2223" xml:space="preserve">ſimilibus, a p o l, a m q l ab extremitati-<lb/>bus baſium ductæ ſint a o, a q ita, ut portiones ablatæ <lb/>faciant cum diametris angulos æquales; </s> |
|
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|
3915 <s xml:id="echoid-s2224" xml:space="preserve">& </s> |
|
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diff
changeset
|
3916 <s xml:id="echoid-s2225" xml:space="preserve">anguli, qui <lb/>ad y g: </s> |
|
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|
3917 <s xml:id="echoid-s2226" xml:space="preserve">& </s> |
|
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diff
changeset
|
3918 <s xml:id="echoid-s2227" xml:space="preserve">lineæ y b, g b, & </s> |
|
22d6a63640c6
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changeset
|
3919 <s xml:id="echoid-s2228" xml:space="preserve">b c, b s inter ſe æquales erunt. <lb/></s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
3920 <s xml:id="echoid-s2229" xml:space="preserve">quare & </s> |
|
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diff
changeset
|
3921 <s xml:id="echoid-s2230" xml:space="preserve">ipſæ c r, s r: </s> |
|
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changeset
|
3922 <s xml:id="echoid-s2231" xml:space="preserve">& </s> |
|
22d6a63640c6
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diff
changeset
|
3923 <s xml:id="echoid-s2232" xml:space="preserve">m u, p z: </s> |
|
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changeset
|
3924 <s xml:id="echoid-s2233" xml:space="preserve">& </s> |
|
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diff
changeset
|
3925 <s xml:id="echoid-s2234" xml:space="preserve">u n, z t. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3926 <s xml:id="echoid-s2235" xml:space="preserve">Quo-<lb/> |
|
22d6a63640c6
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diff
changeset
|
3927 <anchor type="note" xlink:label="note-0087-01a" xlink:href="note-0087-01"/> |
|
22d6a63640c6
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diff
changeset
|
3928 niam igitur m u minor eſt, quàm dupla u n; </s> |
|
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diff
changeset
|
3929 <s xml:id="echoid-s2236" xml:space="preserve">conſtat p z ip-<lb/>ſius z t minorem eſſe, quàm duplam. </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3930 <s xml:id="echoid-s2237" xml:space="preserve">Sit p α dupla ipſius <lb/>ω t: </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
3931 <s xml:id="echoid-s2238" xml:space="preserve">& </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3932 <s xml:id="echoid-s2239" xml:space="preserve">iuncta α k ad e producatur. </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
changeset
|
3933 <s xml:id="echoid-s2240" xml:space="preserve">ergo totius quidem por <lb/>tionis centrum grauitatis erit puntum κ; </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3934 <s xml:id="echoid-s2241" xml:space="preserve">partis eius, quæ <lb/>in humido eſt, centrum ω; </s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3935 <s xml:id="echoid-s2242" xml:space="preserve">eius uero, quæ extra humidum <lb/>in linea k e, quod ſit e. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3936 <s xml:id="echoid-s2243" xml:space="preserve">Sed linea k z perpendicularis erit <lb/>ad ſuperficiem humidi. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
3937 <s xml:id="echoid-s2244" xml:space="preserve">quare & </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3938 <s xml:id="echoid-s2245" xml:space="preserve">lineæ quæ per puncta e, <lb/>ω, æ quidiſtantes ipſi κ z ducuntur. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
3939 <s xml:id="echoid-s2246" xml:space="preserve">non ergo manebit por-<lb/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3940 <anchor type="note" xlink:label="note-0087-02a" xlink:href="note-0087-02"/> |
|
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diff
changeset
|
3941 tio, ſed reuoluetur ita, ut baſis ipſius ſuperficiem humidi <lb/>nullo modo contingat: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3942 <s xml:id="echoid-s2247" xml:space="preserve">quoniã nuncin uno puncto contin <lb/>gens, ſurſum fertur ex parte a. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
3943 <s xml:id="echoid-s2248" xml:space="preserve">perſpicuum eſt igitur por-<lb/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
3944 <anchor type="note" xlink:label="note-0087-03a" xlink:href="note-0087-03"/> |
|
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changeset
|
3945 tionem conſiſt ere ita, ut axis cum ſuperſicie humidi faciat <lb/>angulum maiorem angulo χ.</s> |
|
22d6a63640c6
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changeset
|
3946 <s xml:id="echoid-s2249" xml:space="preserve"/> |
|
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parents:
diff
changeset
|
3947 </p> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff
changeset
|
3948 <div xml:id="echoid-div158" type="float" level="2" n="1"> |
|
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3949 <note position="right" xlink:label="note-0085-01" xlink:href="note-0085-01a" xml:space="preserve">C</note> |
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3950 <figure xlink:label="fig-0085-01" xlink:href="fig-0085-01a"> |
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3951 <image file="0085-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0085-01"/> |
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3952 </figure> |
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3953 <note position="left" xlink:label="note-0085-02" xlink:href="note-0085-02a" xml:space="preserve">D</note> |
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3954 <note position="right" xlink:label="note-0085-03" xlink:href="note-0085-03a" xml:space="preserve">E</note> |
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3955 <note position="right" xlink:label="note-0085-04" xlink:href="note-0085-04a" xml:space="preserve">F</note> |
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3956 <note position="left" xlink:label="note-0086-01" xlink:href="note-0086-01a" xml:space="preserve">G</note> |
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3957 <note position="left" xlink:label="note-0086-02" xlink:href="note-0086-02a" xml:space="preserve">H</note> |
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3958 <figure xlink:label="fig-0086-01" xlink:href="fig-0086-01a"> |
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3959 <image file="0086-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0086-01"/> |
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3960 </figure> |
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3961 <note position="left" xlink:label="note-0086-03" xlink:href="note-0086-03a" xml:space="preserve">K</note> |
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3962 <note position="right" xlink:label="note-0087-01" xlink:href="note-0087-01a" xml:space="preserve">L</note> |
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3963 <note position="right" xlink:label="note-0087-02" xlink:href="note-0087-02a" xml:space="preserve">M</note> |
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3964 <note position="right" xlink:label="note-0087-03" xlink:href="note-0087-03a" xml:space="preserve">N</note> |
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3965 </div> |
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3966 </div> |
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3967 <div xml:id="echoid-div160" type="section" level="1" n="53"> |
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3968 <head xml:id="echoid-head58" xml:space="preserve">COMMENTARIVS.</head> |
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3969 <p> |
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3970 <s xml:id="echoid-s2250" xml:space="preserve">Siportio ad humidum in grauitate minorẽ proportio-<lb/> |
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3971 <anchor type="note" xlink:label="note-0087-04a" xlink:href="note-0087-04"/> |
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3972 nem habeat; </s> |
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3973 <s xml:id="echoid-s2251" xml:space="preserve">quàm quadratum s b ad quadratum b d; </s> |
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3974 <s xml:id="echoid-s2252" xml:space="preserve">ma-<lb/>iorem uero, quàm quadratum x o ad b d quadratum.</s> |
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3975 <s xml:id="echoid-s2253" xml:space="preserve">] _Hæc_ <lb/>_eſt ſecunda pars propoſitionis, quam aliæ deinceps, postea ipſarum_ <lb/>_demonstrationes eodem ordine ſequuntur_.</s> |
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3976 <s xml:id="echoid-s2254" xml:space="preserve"/> |
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3977 </p> |
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3978 <div xml:id="echoid-div160" type="float" level="2" n="1"> |
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3979 <note position="right" xlink:label="note-0087-04" xlink:href="note-0087-04a" xml:space="preserve">A</note> |
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3980 </div> |
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3981 <p> |
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3982 <s xml:id="echoid-s2255" xml:space="preserve">SI portio ad humidum in grauitate maiorem quidem <lb/> |
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3983 <anchor type="note" xlink:label="note-0087-05a" xlink:href="note-0087-05"/> |
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3984 proportionẽ habeat, quàm quadratũ f p ad quadratũb d.</s> |
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3985 <s xml:id="echoid-s2256" xml:space="preserve">] <lb/>_Hãc quartá parté nos reſtituimus, quæ ĩ trãſlatione deſiderabatur_.</s> |
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3986 <s xml:id="echoid-s2257" xml:space="preserve"/> |
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3987 </p> |
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3988 <div xml:id="echoid-div161" type="float" level="2" n="2"> |
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3989 <note position="right" xlink:label="note-0087-05" xlink:href="note-0087-05a" xml:space="preserve">B</note> |
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|
3990 </div> |
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3991 <p> |
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3992 <s xml:id="echoid-s2258" xml:space="preserve">Erit ψ maior quidem, quàm x o, minor uero, quàm ex-<lb/> |
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3993 <anchor type="note" xlink:label="note-0087-06a" xlink:href="note-0087-06"/> |
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3994 ceſſus, quo axis eſt maior, quam ſeſquialter eius, quæ uſque <lb/>ad axem,] _Sequitur illud ex decima quinti libri elementornm_.</s> |
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3995 <s xml:id="echoid-s2259" xml:space="preserve"/> |
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3996 </p> |
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3997 <div xml:id="echoid-div162" type="float" level="2" n="3"> |
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3998 <note position="right" xlink:label="note-0087-06" xlink:href="note-0087-06a" xml:space="preserve">C</note> |
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3999 </div> |
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4000 <p> |
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4001 <s xml:id="echoid-s2260" xml:space="preserve">Demonſtrabitur m h duplaipſius h n, ſicuti demonſtra <lb/> |
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4002 <anchor type="note" xlink:label="note-0087-07a" xlink:href="note-0087-07"/> |
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4003 tũ eſt o gipſius g x duplam eſſe.</s> |
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4004 <s xml:id="echoid-s2261" xml:space="preserve">] _Vt in prima parte huius, &_</s> |
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4005 <s xml:id="echoid-s2262" xml:space="preserve"> <lb/>_exijs, quæ nos proxime in ipſam conſcripſimus_.</s> |
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4006 <s xml:id="echoid-s2263" xml:space="preserve"/> |
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4007 </p> |
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4008 <div xml:id="echoid-div163" type="float" level="2" n="4"> |
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4009 <note position="right" xlink:label="note-0087-07" xlink:href="note-0087-07a" xml:space="preserve">D</note> |
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4010 </div> |
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4011 <p> |
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4012 <s xml:id="echoid-s2264" xml:space="preserve">Quoniam enim in ſimilibus portionibus a p o l, a x d, <lb/> |
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4013 <anchor type="note" xlink:label="note-0087-08a" xlink:href="note-0087-08"/> |
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|
4014 <pb file="0088" n="88" rhead="ARCHIMEDIS"/> |
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4015 ductæ ſunt à baſibus ad portiones lineæ a n, a q, quæ angu <lb/>los æquales continent cum ipſis baſibus, eandem propor-<lb/>tionem habebit q a ad a n, quam l a ad a d.</s> |
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4016 <s xml:id="echoid-s2265" xml:space="preserve">] _Hoc nos ſu_-<lb/>_pra demonstrauimus_.</s> |
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4017 <s xml:id="echoid-s2266" xml:space="preserve"/> |
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|
4018 </p> |
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|
4019 <div xml:id="echoid-div164" type="float" level="2" n="5"> |
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4020 <note position="right" xlink:label="note-0087-08" xlink:href="note-0087-08a" xml:space="preserve">E</note> |
|
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4021 </div> |
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|
4022 <p> |
|
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|
4023 <s xml:id="echoid-s2267" xml:space="preserve">Aequalis eſt ergo a n ipſi n q.</s> |
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|
4024 <s xml:id="echoid-s2268" xml:space="preserve">] _Cum enim q a ad a n ſit_, <lb/> |
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diff
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|
4025 <anchor type="note" xlink:label="note-0088-01a" xlink:href="note-0088-01"/> |
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diff
changeset
|
4026 _ut l a ad a d; </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
4027 <s xml:id="echoid-s2269" xml:space="preserve">diuidendo, conuertendoq; </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
4028 <s xml:id="echoid-s2270" xml:space="preserve">erit an ad n q, ut a d ad_ <lb/>_d l. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
4029 <s xml:id="echoid-s2271" xml:space="preserve">eſt autem a d æqualis ipſi d l, quoniam d b ponitur diameter_ <lb/>_portionis. </s> |
|
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diff
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|
4030 <s xml:id="echoid-s2272" xml:space="preserve">ergo & </s> |
|
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diff
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|
4031 <s xml:id="echoid-s2273" xml:space="preserve">a n ipſi n q eſt æqualis_. <lb/></s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4032 <s xml:id="echoid-s2274" xml:space="preserve"/> |
|
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diff
changeset
|
4033 </p> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
4034 <div xml:id="echoid-div165" type="float" level="2" n="6"> |
|
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diff
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|
4035 <note position="left" xlink:label="note-0088-01" xlink:href="note-0088-01a" xml:space="preserve">F</note> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
4036 </div> |
|
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diff
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|
4037 <note position="left" xml:space="preserve">14. quinti</note> |
|
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diff
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|
4038 <p> |
|
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diff
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|
4039 <s xml:id="echoid-s2275" xml:space="preserve">Et a q ipſi m y æquidiſtans.</s> |
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|
4040 <s xml:id="echoid-s2276" xml:space="preserve">] _Ex quinta ſecundi libri coni_-<lb/> |
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|
4041 <anchor type="note" xlink:label="note-0088-03a" xlink:href="note-0088-03"/> |
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diff
changeset
|
4042 _corum. </s> |
|
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|
4043 <s xml:id="echoid-s2277" xml:space="preserve">Apollonij_.</s> |
|
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|
4044 <s xml:id="echoid-s2278" xml:space="preserve"/> |
|
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diff
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|
4045 </p> |
|
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diff
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|
4046 <div xml:id="echoid-div166" type="float" level="2" n="7"> |
|
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diff
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|
4047 <note position="left" xlink:label="note-0088-03" xlink:href="note-0088-03a" xml:space="preserve">G</note> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
4048 </div> |
|
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diff
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|
4049 <p> |
|
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|
4050 <s xml:id="echoid-s2279" xml:space="preserve">Etſecetur b d in punctis k r, ut dictum eſt.</s> |
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diff
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|
4051 <s xml:id="echoid-s2280" xml:space="preserve">] _In prima_ <lb/> |
|
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diff
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|
4052 <anchor type="note" xlink:label="note-0088-04a" xlink:href="note-0088-04"/> |
|
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diff
changeset
|
4053 _parte huius propoſitionis. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
4054 <s xml:id="echoid-s2281" xml:space="preserve">ſecetur autem in K ita, ut b k ſit dupla ip_-<lb/>_ſius k d; </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4055 <s xml:id="echoid-s2282" xml:space="preserve">& </s> |
|
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diff
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|
4056 <s xml:id="echoid-s2283" xml:space="preserve">in r, ut K r ſit æqualis ei, quæ uſque ad axcm_.</s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4057 <s xml:id="echoid-s2284" xml:space="preserve"/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
4058 </p> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
4059 <div xml:id="echoid-div167" type="float" level="2" n="8"> |
|
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diff
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|
4060 <note position="left" xlink:label="note-0088-04" xlink:href="note-0088-04a" xml:space="preserve">H</note> |
|
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diff
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|
4061 </div> |
|
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diff
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|
4062 <p> |
|
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diff
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|
4063 <s xml:id="echoid-s2285" xml:space="preserve">Quòd cũ in portionibus æqualibus, & </s> |
|
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|
4064 <s xml:id="echoid-s2286" xml:space="preserve">ſimilibus, a p o l, <lb/> |
|
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|
4065 <anchor type="note" xlink:label="note-0088-05a" xlink:href="note-0088-05"/> |
|
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diff
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|
4066 a m q l ab extremitatibus baſium ductæ ſint a o, a q, ita ut <lb/>portiones ablatæ faciant cum diametris angulos æquales: <lb/></s> |
|
22d6a63640c6
moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4067 <s xml:id="echoid-s2287" xml:space="preserve">& </s> |
|
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|
4068 <s xml:id="echoid-s2288" xml:space="preserve">anguli, qui ad y g: </s> |
|
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|
4069 <s xml:id="echoid-s2289" xml:space="preserve">& </s> |
|
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diff
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|
4070 <s xml:id="echoid-s2290" xml:space="preserve">lineæ y b, g b inter ſe æquales erũt.</s> |
|
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|
4071 <s xml:id="echoid-s2291" xml:space="preserve">] <lb/>_Secet linea a q diametrum d b in θ, & </s> |
|
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|
4072 <s xml:id="echoid-s2292" xml:space="preserve">a o ſecet in η. </s> |
|
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|
4073 <s xml:id="echoid-s2293" xml:space="preserve">Itaque quo_-<lb/>_niam in portionibus æqualibus, & </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4074 <s xml:id="echoid-s2294" xml:space="preserve">ſimilibus a p o l, a m q l ab ex_-<lb/>_tremitatibus baſium_ <lb/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
4075 <anchor type="figure" xlink:label="fig-0088-01a" xlink:href="fig-0088-01"/> |
|
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diff
changeset
|
4076 _ducũtur a o, a q, quæ_ <lb/>_æquales angulos con_ <lb/>_tinent cum ipſis baſi_ <lb/>_bus: </s> |
|
22d6a63640c6
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|
4077 <s xml:id="echoid-s2295" xml:space="preserve">& </s> |
|
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diff
changeset
|
4078 <s xml:id="echoid-s2296" xml:space="preserve">anguli ad d_ <lb/>_utrique ſunt recti_: <lb/></s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4079 <s xml:id="echoid-s2297" xml:space="preserve">_erũt & </s> |
|
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diff
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|
4080 <s xml:id="echoid-s2298" xml:space="preserve">reliqui a η d_, <lb/>_a θ d inter ſe æqua_-<lb/>_les. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4081 <s xml:id="echoid-s2299" xml:space="preserve">linea autem p g_ <lb/>_æquidiſtat lineæ a o_: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4082 <s xml:id="echoid-s2300" xml:space="preserve"><lb/>_itémq; </s> |
|
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changeset
|
4083 <s xml:id="echoid-s2301" xml:space="preserve">m y ipſi a q_: </s> |
|
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|
4084 <s xml:id="echoid-s2302" xml:space="preserve"><lb/>_& </s> |
|
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changeset
|
4085 <s xml:id="echoid-s2303" xml:space="preserve">p s, m c ipſis a d_. </s> |
|
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|
4086 <s xml:id="echoid-s2304" xml:space="preserve"><lb/>_triágula igitur p g s_, <lb/>_m y c triãgulis a η d_ <lb/>_a θ d, atque inter ſe_ <lb/>_ſunt ſimilia: </s> |
|
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changeset
|
4087 <s xml:id="echoid-s2305" xml:space="preserve">& </s> |
|
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diff
changeset
|
4088 <s xml:id="echoid-s2306" xml:space="preserve">ut a d ad a η, ita a d ad a θ: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4089 <s xml:id="echoid-s2307" xml:space="preserve">& </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4090 <s xml:id="echoid-s2308" xml:space="preserve">permutando. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4091 <s xml:id="echoid-s2309" xml:space="preserve">li_-<lb/> |
|
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parents:
diff
changeset
|
4092 <anchor type="note" xlink:label="note-0088-06a" xlink:href="note-0088-06"/> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
4093 <pb o="39" file="0089" n="89" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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changeset
|
4094 _neæ autem a d inter ſe æquales ſunt. </s> |
|
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changeset
|
4095 <s xml:id="echoid-s2310" xml:space="preserve">ergo & </s> |
|
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parents:
diff
changeset
|
4096 <s xml:id="echoid-s2311" xml:space="preserve">ipſæ a η, a θ. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
changeset
|
4097 <s xml:id="echoid-s2312" xml:space="preserve">Sed ſunt_ <lb/>_æquales a o, a q: </s> |
|
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4098 <s xml:id="echoid-s2313" xml:space="preserve">& </s> |
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4099 <s xml:id="echoid-s2314" xml:space="preserve">earum dimidiæ a t a n. </s> |
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4100 <s xml:id="echoid-s2315" xml:space="preserve">ergo & </s> |
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4101 <s xml:id="echoid-s2316" xml:space="preserve">reliquæ t η, n θ_ <lb/>_boc eſt p g, m y. </s> |
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4102 <s xml:id="echoid-s2317" xml:space="preserve">ut autem p g ad g h, ita m y ad y c; </s> |
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4103 <s xml:id="echoid-s2318" xml:space="preserve">& </s> |
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4104 <s xml:id="echoid-s2319" xml:space="preserve">permutan_ <lb/> |
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4105 <anchor type="note" xlink:label="note-0089-01a" xlink:href="note-0089-01"/> |
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4106 _do, ut p g ad m y, ita g s ad y c. </s> |
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4107 <s xml:id="echoid-s2320" xml:space="preserve">quare g s, y c æquales ſunt: </s> |
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4108 <s xml:id="echoid-s2321" xml:space="preserve">&_</s> |
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4109 <s xml:id="echoid-s2322" xml:space="preserve"> <lb/>_ipſarum dimidiæ b s, b c: </s> |
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4110 <s xml:id="echoid-s2323" xml:space="preserve">ex quibus ſequitur ut & </s> |
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4111 <s xml:id="echoid-s2324" xml:space="preserve">reliquæ s r, c r_: <lb/></s> |
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4112 <s xml:id="echoid-s2325" xml:space="preserve">_& </s> |
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4113 <s xml:id="echoid-s2326" xml:space="preserve">idcirco p z, m u & </s> |
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4114 <s xml:id="echoid-s2327" xml:space="preserve">u n, z t inter ſe ſunt æquales_.</s> |
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4115 <s xml:id="echoid-s2328" xml:space="preserve"/> |
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|
4116 </p> |
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4117 <div xml:id="echoid-div168" type="float" level="2" n="9"> |
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4118 <note position="left" xlink:label="note-0088-05" xlink:href="note-0088-05a" xml:space="preserve">K</note> |
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4119 <figure xlink:label="fig-0088-01" xlink:href="fig-0088-01a"> |
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4120 <image file="0088-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0088-01"/> |
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|
4121 </figure> |
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4122 <note position="left" xlink:label="note-0088-06" xlink:href="note-0088-06a" xml:space="preserve">4. ſexti.</note> |
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4123 <note position="right" xlink:label="note-0089-01" xlink:href="note-0089-01a" xml:space="preserve">34. primi</note> |
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4124 </div> |
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4125 <p> |
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4126 <s xml:id="echoid-s2329" xml:space="preserve">Quoniam igitur m u minor eſt, quàm dupla u n.</s> |
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4127 <s xml:id="echoid-s2330" xml:space="preserve">] _Eſt_ <lb/> |
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4128 <anchor type="note" xlink:label="note-0089-02a" xlink:href="note-0089-02"/> |
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4129 _enim m h ipſius h n dupla, & </s> |
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4130 <s xml:id="echoid-s2331" xml:space="preserve">m u minor ipſa m h. </s> |
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4131 <s xml:id="echoid-s2332" xml:space="preserve">ergo m u minor_ <lb/>_eſt, quàm dupla h n; </s> |
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4132 <s xml:id="echoid-s2333" xml:space="preserve">& </s> |
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4133 <s xml:id="echoid-s2334" xml:space="preserve">multo minor, quàm dupla ipſius u n_.</s> |
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4134 <s xml:id="echoid-s2335" xml:space="preserve"/> |
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|
4135 </p> |
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4136 <div xml:id="echoid-div169" type="float" level="2" n="10"> |
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4137 <note position="right" xlink:label="note-0089-02" xlink:href="note-0089-02a" xml:space="preserve">L</note> |
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|
4138 </div> |
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|
4139 <p> |
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4140 <s xml:id="echoid-s2336" xml:space="preserve">Non ergo manebit portio, ſed reuoluetur, ita ut baſis ip <lb/> |
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|
4141 <anchor type="note" xlink:label="note-0089-03a" xlink:href="note-0089-03"/> |
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|
4142 ſius humidi ſuperſiciem nullo modo contingat. </s> |
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4143 <s xml:id="echoid-s2337" xml:space="preserve">quoniam <lb/>nunc in uno puncto contingens ſurſum fertur ex parte a.</s> |
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4144 <s xml:id="echoid-s2338" xml:space="preserve">] <lb/>_Tranſlatio ſic habet. </s> |
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4145 <s xml:id="echoid-s2339" xml:space="preserve">non ergo manet portio ſed inclinabitur, ut ba_-<lb/>_ſis ipſius nec ſecundum unum tang at ſuperficiem humidi, quoniam_ <lb/>_nunc ſecundum unum tacta ipſa reclinatur. </s> |
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4146 <s xml:id="echoid-s2340" xml:space="preserve">Quæ nos ex alijs Ar_-<lb/>_chimedis locis, & </s> |
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|
4147 <s xml:id="echoid-s2341" xml:space="preserve">perſpicuitatis cauſſa in eum modum corrigenda_ <lb/>_duximus. </s> |
|
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4148 <s xml:id="echoid-s2342" xml:space="preserve">In ſexta enim propoſitione huius ita ſcribit, ut habetur in_ <lb/>_tranſlatione. </s> |
|
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|
4149 <s xml:id="echoid-s2343" xml:space="preserve">reuoluetur ergo ſolidum a p o l, & </s> |
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4150 <s xml:id="echoid-s2344" xml:space="preserve">baſis ipſius nó tan_ <lb/>_get ſuperficiem humidi ſecundum unum ſignum. </s> |
|
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|
4151 <s xml:id="echoid-s2345" xml:space="preserve">Rurſus in ſeptima_ <lb/>_propoſitione. </s> |
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4152 <s xml:id="echoid-s2346" xml:space="preserve">manifeſtum igitur, quòd reuoluetur ſolidum ita ut ba_-<lb/>_ſis ipſius nec ſecundum unum ſignum contingat ſuperficiem humidi_, <lb/>_quoniam nunc ſecundum unum tangens deorſum fertur ex parte l_. <lb/></s> |
|
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|
4153 <s xml:id="echoid-s2347" xml:space="preserve">_At uero portionem ſurſum ferri ex parte a manifeſte constat. </s> |
|
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|
4154 <s xml:id="echoid-s2348" xml:space="preserve">nam_ <lb/>_cumperpendicularis ad ſuperficiem humidi, quæ tranſit per ω ad_ <lb/>_partes a cadat, & </s> |
|
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4155 <s xml:id="echoid-s2349" xml:space="preserve">quæ per e ad partes l, neceſſe eſt ut centrum ω_ <lb/>_ſurſum, e uero deorſum feratur_.</s> |
|
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|
4156 <s xml:id="echoid-s2350" xml:space="preserve"/> |
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|
4157 </p> |
|
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|
4158 <div xml:id="echoid-div170" type="float" level="2" n="11"> |
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|
4159 <note position="right" xlink:label="note-0089-03" xlink:href="note-0089-03a" xml:space="preserve">M</note> |
|
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|
4160 </div> |
|
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|
4161 <p> |
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|
4162 <s xml:id="echoid-s2351" xml:space="preserve">Perſpicuum eſtigitur portionem conſiſtere ita, ut axis <lb/> |
|
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|
4163 <anchor type="note" xlink:label="note-0089-04a" xlink:href="note-0089-04"/> |
|
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|
4164 cum ſuperficie humidi faciat angulum maiorem angu-<lb/>10 χ.</s> |
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|
4165 <s xml:id="echoid-s2352" xml:space="preserve">] _Iuncta enim a x producatur, ut diametrum b d ſe_-<lb/>_cet in λ, & </s> |
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4166 <s xml:id="echoid-s2353" xml:space="preserve">ab o puncto ipſi æquidistans ducatur o χ. </s> |
|
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|
4167 <s xml:id="echoid-s2354" xml:space="preserve">con_-<lb/>_tinget eaſectionem in o, ut in prima figura: </s> |
|
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|
4168 <s xml:id="echoid-s2355" xml:space="preserve">atque erit angu_-<lb/> |
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4169 <anchor type="note" xlink:label="note-0089-05a" xlink:href="note-0089-05"/> |
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4170 _lus ad χ angulo ad λ æqualis. </s> |
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4171 <s xml:id="echoid-s2356" xml:space="preserve">Sed angulus ad y æqualis est_ <lb/>_angulo ad θ: </s> |
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4172 <s xml:id="echoid-s2357" xml:space="preserve">& </s> |
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4173 <s xml:id="echoid-s2358" xml:space="preserve">angulus a θ d maior angulo a λ d; </s> |
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4174 <s xml:id="echoid-s2359" xml:space="preserve">quod ex_-<lb/> |
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4175 <anchor type="note" xlink:label="note-0089-06a" xlink:href="note-0089-06"/> |
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4176 _traipſum cadat. </s> |
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4177 <s xml:id="echoid-s2360" xml:space="preserve">ergo angulus ad y eo, qui ad χ maior erit_.</s> |
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4178 <s xml:id="echoid-s2361" xml:space="preserve"> |
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4179 <pb file="0090" n="90" rhead="ARCHIMEDIS"/> |
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4180 _Quoniam igi_-<lb/> |
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4181 <anchor type="figure" xlink:label="fig-0090-01a" xlink:href="fig-0090-01"/> |
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4182 _tur portio con_-<lb/>_uertitur, ita ut_ <lb/>_baſis humidum_ <lb/>_non contingat_, <lb/>_axis cum ſuper_ <lb/>_ficie eius faciet_ <lb/>_angulum maio_-<lb/>_rem angulo g;_ <lb/></s> |
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4183 <s xml:id="echoid-s2362" xml:space="preserve">_hoc est angulo_ <lb/>_y: </s> |
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4184 <s xml:id="echoid-s2363" xml:space="preserve">& </s> |
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4185 <s xml:id="echoid-s2364" xml:space="preserve">propter_-<lb/>_ea multo maio_-<lb/>_rem angulo χ_.</s> |
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4186 <s xml:id="echoid-s2365" xml:space="preserve"/> |
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4187 </p> |
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4188 <div xml:id="echoid-div171" type="float" level="2" n="12"> |
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4189 <note position="right" xlink:label="note-0089-04" xlink:href="note-0089-04a" xml:space="preserve">N</note> |
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4190 <note position="right" xlink:label="note-0089-05" xlink:href="note-0089-05a" xml:space="preserve">29. primi</note> |
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4191 <note position="right" xlink:label="note-0089-06" xlink:href="note-0089-06a" xml:space="preserve">16. primi</note> |
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4192 <figure xlink:label="fig-0090-01" xlink:href="fig-0090-01a"> |
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4193 <image file="0090-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0090-01"/> |
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4194 </figure> |
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4195 </div> |
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4196 </div> |
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4197 <div xml:id="echoid-div173" type="section" level="1" n="54"> |
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4198 <head xml:id="echoid-head59" xml:space="preserve">DEMONSTRATIO TERTIAE PARTIS.</head> |
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4199 <p> |
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4200 <s xml:id="echoid-s2366" xml:space="preserve">HABEAT deinde portio ad humidum eam in graui-<lb/>tate proportionem, quam quadratũ x o habet ad quadra-<lb/>tum b d: </s> |
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4201 <s xml:id="echoid-s2367" xml:space="preserve">& </s> |
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4202 <s xml:id="echoid-s2368" xml:space="preserve">in humidum demittatur adeo inclinata, ut ba-<lb/>ſis ipſius non con <lb/> |
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4203 <anchor type="figure" xlink:label="fig-0090-02a" xlink:href="fig-0090-02"/> |
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4204 tingat humidum. <lb/></s> |
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4205 <s xml:id="echoid-s2369" xml:space="preserve">Secta aũt ipſa per <lb/>axem plano ad hu <lb/>midi ſuperficiem <lb/>recto, ſolidi ſectio <lb/>ſit rectanguli co-<lb/>ni ſectio a p m l: </s> |
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4206 <s xml:id="echoid-s2370" xml:space="preserve">ſu <lb/>perficiei humidi <lb/>ſectio ſit i m: </s> |
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4207 <s xml:id="echoid-s2371" xml:space="preserve">axis <lb/>portionis, & </s> |
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4208 <s xml:id="echoid-s2372" xml:space="preserve">ſe-<lb/>ctionis diameter <lb/>b d: </s> |
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4209 <s xml:id="echoid-s2373" xml:space="preserve">ſeceturq; </s> |
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4210 <s xml:id="echoid-s2374" xml:space="preserve">b d <lb/>ſicuti prius: </s> |
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4211 <s xml:id="echoid-s2375" xml:space="preserve">& </s> |
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4212 <s xml:id="echoid-s2376" xml:space="preserve">du-<lb/>catur p n quidem |
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4213 <pb o="40" file="0091" n="91" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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4214 ipſi i m æquidiſtãs, & </s> |
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4215 <s xml:id="echoid-s2377" xml:space="preserve">contingens ſectionem in p; </s> |
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4216 <s xml:id="echoid-s2378" xml:space="preserve">pt uero <lb/>æquidiſtans b d, & </s> |
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4217 <s xml:id="echoid-s2379" xml:space="preserve">p s ad ipſam b d perpendicularis. </s> |
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4218 <s xml:id="echoid-s2380" xml:space="preserve">Demõ <lb/>ſtrandũ eſt, portionẽ non cõſiſtereita, ſed inclinari, donec <lb/>baſis in uno puncto ſuperficiem humidi cõtingat. </s> |
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4219 <s xml:id="echoid-s2381" xml:space="preserve">Maneãt <lb/>enim eadem, quæ in ſuperiori figura: </s> |
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4220 <s xml:id="echoid-s2382" xml:space="preserve">ducaturq; </s> |
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4221 <s xml:id="echoid-s2383" xml:space="preserve">o c ad b d <lb/>perpendicularis: </s> |
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4222 <s xml:id="echoid-s2384" xml:space="preserve">& </s> |
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4223 <s xml:id="echoid-s2385" xml:space="preserve">iuncta a x ad q producatur. </s> |
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4224 <s xml:id="echoid-s2386" xml:space="preserve">erit a x <lb/>æqualis ipſi x q. </s> |
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4225 <s xml:id="echoid-s2387" xml:space="preserve">deĩde ducatur o χ ipſi a q æquidiſtãs. </s> |
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4226 <s xml:id="echoid-s2388" xml:space="preserve">Quo <lb/>niã igitur portio ad humidũ eã in grauitate proportione <lb/>habere ponitur, quam quadratum x o ad quadratum b d: <lb/></s> |
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4227 <s xml:id="echoid-s2389" xml:space="preserve">& </s> |
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4228 <s xml:id="echoid-s2390" xml:space="preserve">eandem proportionem habet pars ipſius demerſa ad to <lb/>tam; </s> |
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4229 <s xml:id="echoid-s2391" xml:space="preserve">hoc eſt quadratum t p ad quadratum b d: </s> |
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4230 <s xml:id="echoid-s2392" xml:space="preserve">æqualis uti <lb/>que erit t p ipſi x o: </s> |
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4231 <s xml:id="echoid-s2393" xml:space="preserve">cumq; </s> |
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4232 <s xml:id="echoid-s2394" xml:space="preserve">portionum i p m, a o q diame-<lb/>tri ſint æquales, & </s> |
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4233 <s xml:id="echoid-s2395" xml:space="preserve">portiones ipſæ æquales erunt. </s> |
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4234 <s xml:id="echoid-s2396" xml:space="preserve">Rurſus <lb/> |
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|
4235 <anchor type="note" xlink:label="note-0091-01a" xlink:href="note-0091-01"/> |
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|
4236 quoniam in por <lb/> |
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|
4237 <anchor type="note" xlink:label="note-0091-02a" xlink:href="note-0091-02"/> |
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4238 <anchor type="figure" xlink:label="fig-0091-01a" xlink:href="fig-0091-01"/> |
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|
4239 tionibus æquali <lb/>bus, & </s> |
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4240 <s xml:id="echoid-s2397" xml:space="preserve">ſimilibus <lb/>a o q l, a p m l, <lb/>ductæ ſunt lineæ <lb/>a q, i m, quæ æ-<lb/>quales portio-<lb/>nes auferunt; </s> |
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4241 <s xml:id="echoid-s2398" xml:space="preserve">il-<lb/>la quidem ab ex <lb/>tremitate baſis, <lb/>hæc autem non <lb/>ab extremitate: <lb/></s> |
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4242 <s xml:id="echoid-s2399" xml:space="preserve">cõſtat eam, quæ <lb/>ab extremitate <lb/>baſis ducta eſt, <lb/>minorem facere <lb/>angulum acutũ <lb/>cum diametro totius portionis. </s> |
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4243 <s xml:id="echoid-s2400" xml:space="preserve">& </s> |
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4244 <s xml:id="echoid-s2401" xml:space="preserve">quoniam angulus, qui <lb/> |
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4245 <anchor type="note" xlink:label="note-0091-03a" xlink:href="note-0091-03"/> |
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4246 ad χ minor eſt angulo, qui ad n; </s> |
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4247 <s xml:id="echoid-s2402" xml:space="preserve">maior erit b c, quàm b s: <lb/></s> |
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4248 <s xml:id="echoid-s2403" xml:space="preserve">cr autem, quàm ſr minor. </s> |
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4249 <s xml:id="echoid-s2404" xml:space="preserve">quare & </s> |
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4250 <s xml:id="echoid-s2405" xml:space="preserve">o g minor, quàm p z: </s> |
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4251 <s xml:id="echoid-s2406" xml:space="preserve"><lb/>& </s> |
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4252 <s xml:id="echoid-s2407" xml:space="preserve">g x maior, quàm z t. </s> |
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4253 <s xml:id="echoid-s2408" xml:space="preserve">ergo p z maior eſt, quàm dupla z t;</s> |
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4254 <s xml:id="echoid-s2409" xml:space="preserve"> |
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4255 <pb file="0092" n="92" rhead="ARCHIMEDIS"/> |
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4256 quia o g ipſius g x eſt dupla. </s> |
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4257 <s xml:id="echoid-s2410" xml:space="preserve">Sit p h dupla h t: </s> |
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4258 <s xml:id="echoid-s2411" xml:space="preserve">& </s> |
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4259 <s xml:id="echoid-s2412" xml:space="preserve">iun-<lb/>cta h κ ad ω producatur. </s> |
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4260 <s xml:id="echoid-s2413" xml:space="preserve">erit totius quidem portionis cen <lb/>trum grauitatis k; </s> |
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4261 <s xml:id="echoid-s2414" xml:space="preserve">partis eius, quæ intra humidum h; </s> |
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4262 <s xml:id="echoid-s2415" xml:space="preserve">eius <lb/>uero, quæ extra humidum in linea κ ω, quod ſit ω. </s> |
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4263 <s xml:id="echoid-s2416" xml:space="preserve">Itaque <lb/>demonſtrabitur <lb/> |
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4264 <anchor type="figure" xlink:label="fig-0092-01a" xlink:href="fig-0092-01"/> |
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4265 ſimiliter & </s> |
|
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4266 <s xml:id="echoid-s2417" xml:space="preserve">k z ad <lb/>humidi ſuperſi-<lb/>ciem perpẽdicu-<lb/>laris, & </s> |
|
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4267 <s xml:id="echoid-s2418" xml:space="preserve">quæ per <lb/>puncta h ω æqui-<lb/>diſtantes ipſi κ z <lb/>ducuntur. </s> |
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4268 <s xml:id="echoid-s2419" xml:space="preserve">quare <lb/>nõ manebit por <lb/>tio, ſed inclinabi <lb/>tur, donec baſis <lb/>ipſius in uno pũ <lb/>cto contingat ſu <lb/>perficiem humi-<lb/>di: </s> |
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4269 <s xml:id="echoid-s2420" xml:space="preserve">atque ita con <lb/>ſiſtet. </s> |
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4270 <s xml:id="echoid-s2421" xml:space="preserve">nam in por <lb/>tionibus æquali-<lb/>bus a o q l, a p m l, ductæ erunt ab extremitatibus baſium <lb/>a q, a m, quæ æquales portiones abſcindunt: </s> |
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4271 <s xml:id="echoid-s2422" xml:space="preserve">etenim a o q <lb/>ipſi a p m, utin ſuperioribus æqualis demonſtrabitur. </s> |
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4272 <s xml:id="echoid-s2423" xml:space="preserve">ergo <lb/> |
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4273 <anchor type="note" xlink:label="note-0092-01a" xlink:href="note-0092-01"/> |
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4274 æquales faciunt acutos angulos a q, a m cum diametris ba <lb/>ſium: </s> |
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4275 <s xml:id="echoid-s2424" xml:space="preserve">quòd anguli ad χ & </s> |
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4276 <s xml:id="echoid-s2425" xml:space="preserve">n æquales ſint. </s> |
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4277 <s xml:id="echoid-s2426" xml:space="preserve">quare ſi ducta <lb/>h k ad ω producatur, erit totius portionis grauitatis cen-<lb/>trum k; </s> |
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4278 <s xml:id="echoid-s2427" xml:space="preserve">partis eius, quæ in humido h; </s> |
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4279 <s xml:id="echoid-s2428" xml:space="preserve">at eius, quæ extra <lb/>humidum in linea h κ; </s> |
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4280 <s xml:id="echoid-s2429" xml:space="preserve">quod ſit ω: </s> |
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4281 <s xml:id="echoid-s2430" xml:space="preserve">& </s> |
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4282 <s xml:id="echoid-s2431" xml:space="preserve">h k ad humidi ſuper-<lb/>ficiem perpendicularis. </s> |
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4283 <s xml:id="echoid-s2432" xml:space="preserve">per eaſdem igitur rectas lineas, <lb/>quod quidem in humido eſt, ſurſum, & </s> |
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4284 <s xml:id="echoid-s2433" xml:space="preserve">quod extra humi-<lb/>dum deorſum feretur. </s> |
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4285 <s xml:id="echoid-s2434" xml:space="preserve">quare manebit portio, cuius baſis <lb/>humidi ſuperficiem in uno puncto continget: </s> |
|
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4286 <s xml:id="echoid-s2435" xml:space="preserve">& </s> |
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4287 <s xml:id="echoid-s2436" xml:space="preserve">axis cum <lb/>ipſa angulum faciet æqualem angulo χ. </s> |
|
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4288 <s xml:id="echoid-s2437" xml:space="preserve">Similiter demon-<lb/> |
|
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4289 <anchor type="note" xlink:label="note-0092-02a" xlink:href="note-0092-02"/> |
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4290 <pb o="41" file="0093" n="93" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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4291 ſtrabitur portionem, quæ ad humidum in grauitate eandẽ <lb/>proportionem habeat, quàm quadratum p f ad quadratũ <lb/>b d in humidum demiſſam, ita ut baſis ipſius nõ cõtingat <lb/>humidum, inclinatam conſiſtere adeo, ut baſis in uno pun <lb/>cto humidi ſuperficiem contingat. </s> |
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4292 <s xml:id="echoid-s2438" xml:space="preserve">& </s> |
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4293 <s xml:id="echoid-s2439" xml:space="preserve">axis cum ipſa faciat <lb/>angulum angulo φ æqualem.</s> |
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4294 <s xml:id="echoid-s2440" xml:space="preserve"/> |
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|
4295 </p> |
|
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4296 <div xml:id="echoid-div173" type="float" level="2" n="1"> |
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|
4297 <figure xlink:label="fig-0090-02" xlink:href="fig-0090-02a"> |
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|
4298 <image file="0090-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0090-02"/> |
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|
4299 </figure> |
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4300 <note position="right" xlink:label="note-0091-01" xlink:href="note-0091-01a" xml:space="preserve">B</note> |
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4301 <note position="right" xlink:label="note-0091-02" xlink:href="note-0091-02a" xml:space="preserve">C</note> |
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4302 <figure xlink:label="fig-0091-01" xlink:href="fig-0091-01a"> |
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4303 <image file="0091-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0091-01"/> |
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|
4304 </figure> |
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4305 <note position="right" xlink:label="note-0091-03" xlink:href="note-0091-03a" xml:space="preserve">D</note> |
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4306 <figure xlink:label="fig-0092-01" xlink:href="fig-0092-01a"> |
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4307 <image file="0092-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0092-01"/> |
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|
4308 </figure> |
|
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4309 <note position="left" xlink:label="note-0092-01" xlink:href="note-0092-01a" xml:space="preserve">E</note> |
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4310 <note position="left" xlink:label="note-0092-02" xlink:href="note-0092-02a" xml:space="preserve">F</note> |
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4311 </div> |
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4312 </div> |
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4313 <div xml:id="echoid-div175" type="section" level="1" n="55"> |
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4314 <head xml:id="echoid-head60" xml:space="preserve">COMMENTARIVS.</head> |
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4315 <p style="it"> |
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4316 <s xml:id="echoid-s2441" xml:space="preserve">_Hoc eſt quadratum t p ad quadratum b d.</s> |
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4317 <s xml:id="echoid-s2442" xml:space="preserve">]_ Ex uigeſima <lb/> |
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4318 <anchor type="note" xlink:label="note-0093-01a" xlink:href="note-0093-01"/> |
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4319 ſexta libri Archimedis de conoidibus, & </s> |
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4320 <s xml:id="echoid-s2443" xml:space="preserve">ſphæroidibus. </s> |
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4321 <s xml:id="echoid-s2444" xml:space="preserve">ergo ex no <lb/>na quinti erit quadratum t p æquale quadrato x o: </s> |
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4322 <s xml:id="echoid-s2445" xml:space="preserve">& </s> |
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4323 <s xml:id="echoid-s2446" xml:space="preserve">propterea li <lb/>nea t p lineæ x o æqualis.</s> |
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4324 <s xml:id="echoid-s2447" xml:space="preserve"/> |
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4325 </p> |
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4326 <div xml:id="echoid-div175" type="float" level="2" n="1"> |
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4327 <note position="right" xlink:label="note-0093-01" xlink:href="note-0093-01a" xml:space="preserve">A</note> |
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4328 </div> |
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4329 <p style="it"> |
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4330 <s xml:id="echoid-s2448" xml:space="preserve">_Et portiones ipſæ æquales erunt.</s> |
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4331 <s xml:id="echoid-s2449" xml:space="preserve">]_ Ex uigeſimaquinta eiuſ-<lb/> |
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4332 <anchor type="note" xlink:label="note-0093-02a" xlink:href="note-0093-02"/> |
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|
4333 dem libri.</s> |
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4334 <s xml:id="echoid-s2450" xml:space="preserve"/> |
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4335 </p> |
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4336 <div xml:id="echoid-div176" type="float" level="2" n="2"> |
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4337 <note position="right" xlink:label="note-0093-02" xlink:href="note-0093-02a" xml:space="preserve">B</note> |
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4338 </div> |
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4339 <p> |
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4340 <s xml:id="echoid-s2451" xml:space="preserve">Rurſus <lb/> |
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4341 <anchor type="note" xlink:label="note-0093-03a" xlink:href="note-0093-03"/> |
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4342 <anchor type="figure" xlink:label="fig-0093-01a" xlink:href="fig-0093-01"/> |
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4343 quoniam <lb/>in portio <lb/>nibus æ-<lb/>qualibus, <lb/>& </s> |
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4344 <s xml:id="echoid-s2452" xml:space="preserve">ſimili-<lb/>bus a o q <lb/>l, a p m l.</s> |
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4345 <s xml:id="echoid-s2453" xml:space="preserve">] <lb/>_In portio-_ <lb/>_ne enim a p_ <lb/>_m l deſcri-_ <lb/>_batur por-_ <lb/>_tio a o q æ-_ <lb/>_qualis por_ <lb/>_tioni i p m_, <lb/>_cadet pun-_ <lb/>_ctum q in-_ <lb/>_fram, alio-_ <lb/>_qui totum parti eſſet æquale. </s> |
|
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4346 <s xml:id="echoid-s2454" xml:space="preserve">Ducatur deinde i u æquidiſtans a q_, |
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|
4347 <pb file="0094" n="94" rhead="ARCHIMEDIS"/> |
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4348 _quæ diametrum ſecet in ψ; </s> |
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4349 <s xml:id="echoid-s2455" xml:space="preserve">ſecet autem i m eandem in σ: </s> |
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4350 <s xml:id="echoid-s2456" xml:space="preserve">& </s> |
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4351 <s xml:id="echoid-s2457" xml:space="preserve">a q in_ <lb/>_v. </s> |
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4352 <s xml:id="echoid-s2458" xml:space="preserve">Dico angulum a ν d angulo i σ d minoré eſſe. </s> |
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4353 <s xml:id="echoid-s2459" xml:space="preserve">angulus enimi ψ d_ <lb/>_æqualis est angulo a ν d. </s> |
|
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4354 <s xml:id="echoid-s2460" xml:space="preserve">ſed angulus interior i ψ d minor eſt exte-_ <lb/> |
|
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|
4355 <anchor type="note" xlink:label="note-0094-01a" xlink:href="note-0094-01"/> |
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|
4356 _riore i σ d. </s> |
|
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|
4357 <s xml:id="echoid-s2461" xml:space="preserve">ergo & </s> |
|
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4358 <s xml:id="echoid-s2462" xml:space="preserve">a ν d ipſo i σ d minor erit_.</s> |
|
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|
4359 <s xml:id="echoid-s2463" xml:space="preserve"/> |
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|
4360 </p> |
|
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|
4361 <div xml:id="echoid-div177" type="float" level="2" n="3"> |
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|
4362 <note position="right" xlink:label="note-0093-03" xlink:href="note-0093-03a" xml:space="preserve">C</note> |
|
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|
4363 <figure xlink:label="fig-0093-01" xlink:href="fig-0093-01a"> |
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|
4364 <image file="0093-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0093-01"/> |
|
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|
4365 </figure> |
|
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|
4366 <note position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">29. primi</note> |
|
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|
4367 </div> |
|
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|
4368 <note position="left" xml:space="preserve">16. primi</note> |
|
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|
4369 <p style="it"> |
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|
4370 <s xml:id="echoid-s2464" xml:space="preserve">_Et quoniam angulus, qui ad χ minor eſt angulo, qui ad_ <lb/> |
|
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|
4371 <anchor type="note" xlink:label="note-0094-03a" xlink:href="note-0094-03"/> |
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|
4372 _n.</s> |
|
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4373 <s xml:id="echoid-s2465" xml:space="preserve">]_ Ducantur per o duæ lineæ, o c quidem ad diametrum b d per-<lb/>pendicularis: </s> |
|
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|
4374 <s xml:id="echoid-s2466" xml:space="preserve">& </s> |
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|
4375 <s xml:id="echoid-s2467" xml:space="preserve">o χ in puncto o ſectionem contingens, quæ diame <lb/>trum ſecet in χ. </s> |
|
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|
4376 <s xml:id="echoid-s2468" xml:space="preserve">æquidiſtabit o χ ipſi a q: </s> |
|
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|
4377 <s xml:id="echoid-s2469" xml:space="preserve">atque erit angulus ad <lb/> |
|
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|
4378 <anchor type="note" xlink:label="note-0094-04a" xlink:href="note-0094-04"/> |
|
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|
4379 χ æqualis ei, qui ad ν. </s> |
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|
4380 <s xml:id="echoid-s2470" xml:space="preserve">ergo angulus ad χ angulo ad σ, uidelicet eo, <lb/> |
|
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|
4381 <anchor type="note" xlink:label="note-0094-05a" xlink:href="note-0094-05"/> |
|
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|
4382 qui ad n minor erit: </s> |
|
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|
4383 <s xml:id="echoid-s2471" xml:space="preserve">& </s> |
|
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|
4384 <s xml:id="echoid-s2472" xml:space="preserve">propterea χ infra n cadet. </s> |
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4385 <s xml:id="echoid-s2473" xml:space="preserve">linea igitur χ b <lb/> |
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4386 <anchor type="note" xlink:label="note-0094-06a" xlink:href="note-0094-06"/> |
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4387 maior eſt, quàm n b. </s> |
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4388 <s xml:id="echoid-s2474" xml:space="preserve">Sed cum b c ſit æqualis χ b, & </s> |
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4389 <s xml:id="echoid-s2475" xml:space="preserve">b s ipſi n b: <lb/></s> |
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4390 <s xml:id="echoid-s2476" xml:space="preserve">erit b c ipſa b s maior.</s> |
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4391 <s xml:id="echoid-s2477" xml:space="preserve"/> |
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4392 </p> |
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4393 <div xml:id="echoid-div178" type="float" level="2" n="4"> |
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4394 <note position="left" xlink:label="note-0094-03" xlink:href="note-0094-03a" xml:space="preserve">D</note> |
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4395 <note position="left" xlink:label="note-0094-04" xlink:href="note-0094-04a" xml:space="preserve">5. ſecũdi <lb/>conicoiũ</note> |
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4396 <note position="left" xlink:label="note-0094-05" xlink:href="note-0094-05a" xml:space="preserve">29. primi.</note> |
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4397 <note position="left" xlink:label="note-0094-06" xlink:href="note-0094-06a" xml:space="preserve">35. primi <lb/>conicorũ</note> |
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4398 </div> |
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4399 <p> |
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4400 <s xml:id="echoid-s2478" xml:space="preserve">Ergo æquales faciunt angulos a q, a m cum diametris <lb/> |
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4401 <anchor type="note" xlink:label="note-0094-07a" xlink:href="note-0094-07"/> |
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4402 portionum.</s> |
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4403 <s xml:id="echoid-s2479" xml:space="preserve">] _Hoc demonstrabimus ut in commentarijs in ſecun-_ <lb/>_dam partem_.</s> |
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4404 <s xml:id="echoid-s2480" xml:space="preserve"/> |
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|
4405 </p> |
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4406 <div xml:id="echoid-div179" type="float" level="2" n="5"> |
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4407 <note position="left" xlink:label="note-0094-07" xlink:href="note-0094-07a" xml:space="preserve">E</note> |
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4408 </div> |
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4409 <p style="it"> |
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4410 <s xml:id="echoid-s2481" xml:space="preserve">_Similiter demonſtrabitur, portionem, quæ ad humidũ_ <lb/> |
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4411 <anchor type="note" xlink:label="note-0094-08a" xlink:href="note-0094-08"/> |
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4412 _in grauitate ean-_ <lb/> |
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4413 <anchor type="figure" xlink:label="fig-0094-01a" xlink:href="fig-0094-01"/> |
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4414 _dem proportio-_ <lb/>_nem habeat, quã_ <lb/>_quadratum p fad_ <lb/>_quadratũ b d; </s> |
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4415 <s xml:id="echoid-s2482" xml:space="preserve">in_ <lb/>_humidum demiſ-_ <lb/>_ſam, ita ut baſis ip_ <lb/>_ſius non cõtingat_ <lb/>_humidum, incli-_ <lb/>_natam conſiſtere_ <lb/>_adeo, ut baſis in_ <lb/>_uno pũcto humi-_ <lb/>_di ſuperficiem cõ_ <lb/>_tingat: </s> |
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4416 <s xml:id="echoid-s2483" xml:space="preserve">& </s> |
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4417 <s xml:id="echoid-s2484" xml:space="preserve">axis cũ_ <lb/>_ipſa faciat angulũ_ <lb/>_angulo φ æqualẽ]_ <lb/>Habeat portio ad humidum in grauitate proportionem eam, quam <lb/>p f quadratum ad quadratum b d: </s> |
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4418 <s xml:id="echoid-s2485" xml:space="preserve">& </s> |
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4419 <s xml:id="echoid-s2486" xml:space="preserve">demiſſa in humidum adeo in- |
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4420 <pb o="42" file="0095" n="95" rhead="DE IIS QVAE VEH. IN AQVA."/> |
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4421 clinata, ut baſis humidum non contingat, ſectur plano per axem, <lb/>recto ad ſuperficiem humidi, ut ſectio ſit a m o l rectanguli coni ſe-<lb/>ctio: </s> |
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4422 <s xml:id="echoid-s2487" xml:space="preserve">ſuperficiei humidi ſectio ſit i o: </s> |
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4423 <s xml:id="echoid-s2488" xml:space="preserve">axis portionis, & </s> |
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4424 <s xml:id="echoid-s2489" xml:space="preserve">ſectionis <lb/>diameter b d; </s> |
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4425 <s xml:id="echoid-s2490" xml:space="preserve">quæ in eaſdem, quas diximus, partes ſecetur: </s> |
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4426 <s xml:id="echoid-s2491" xml:space="preserve">duca-<lb/>turq; </s> |
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4427 <s xml:id="echoid-s2492" xml:space="preserve">m n quidem ipſi i o æquidiſtans, ut in puncto m ſectionem <lb/>cótingat: </s> |
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4428 <s xml:id="echoid-s2493" xml:space="preserve">mt uero æquidiſtans ipſi b d: </s> |
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4429 <s xml:id="echoid-s2494" xml:space="preserve">& </s> |
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4430 <s xml:id="echoid-s2495" xml:space="preserve">m s ad eandem perpen <lb/>dicularis. </s> |
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4431 <s xml:id="echoid-s2496" xml:space="preserve">Demonſtrandum eſt non manere portionem, ſed inclinari <lb/>ita, ut in uno puncto contingat ſuperficiem humidi. </s> |
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4432 <s xml:id="echoid-s2497" xml:space="preserve">ducatur enim p c <lb/>ad ipſam b d perpendicularis: </s> |
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4433 <s xml:id="echoid-s2498" xml:space="preserve">& </s> |
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4434 <s xml:id="echoid-s2499" xml:space="preserve">iuncta a f uſque ad ſectionem <lb/>producatur in q: </s> |
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4435 <s xml:id="echoid-s2500" xml:space="preserve">& </s> |
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4436 <s xml:id="echoid-s2501" xml:space="preserve">per p ducatur p φ ipſi a q æquidiſtans. </s> |
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4437 <s xml:id="echoid-s2502" xml:space="preserve">erunt <lb/>iam ex ijs, quæ demonſtrauimus a f, f q inter ſe ſe æquales. </s> |
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4438 <s xml:id="echoid-s2503" xml:space="preserve">& </s> |
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4439 <s xml:id="echoid-s2504" xml:space="preserve">cum <lb/>portio ad humi-<lb/> |
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|
4440 <anchor type="figure" xlink:label="fig-0095-01a" xlink:href="fig-0095-01"/> |
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|
4441 dum eam in gra-<lb/>uitate proportio <lb/>nem habeat, quá <lb/>quadratú p f ad <lb/>b d quadratum: <lb/></s> |
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4442 <s xml:id="echoid-s2505" xml:space="preserve">atque eandem ha <lb/>beat portio ipſi-<lb/>us demerſa ad to <lb/>tam portionem; </s> |
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4443 <s xml:id="echoid-s2506" xml:space="preserve"><lb/>hoc eſt quadratú <lb/>m t ad quadratú <lb/> |
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|
4444 <anchor type="note" xlink:label="note-0095-01a" xlink:href="note-0095-01"/> |
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|
4445 b d: </s> |
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|
4446 <s xml:id="echoid-s2507" xml:space="preserve">erit quadra <lb/>tum m t quadra-<lb/>to p f æquale: </s> |
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4447 <s xml:id="echoid-s2508" xml:space="preserve">& </s> |
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|
4448 <s xml:id="echoid-s2509" xml:space="preserve"><lb/>idcirco linea m t <lb/>æqualis lmeæ p <lb/>f. </s> |
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4449 <s xml:id="echoid-s2510" xml:space="preserve">Itaque quoniam in portionibus æqualibus, & </s> |
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|
4450 <s xml:id="echoid-s2511" xml:space="preserve">ſimilibus a p q l, a <lb/>m o l ductæ ſunt lineæ a q, i o, quæ æquales portiones abſcindunt; <lb/></s> |
|
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|
4451 <s xml:id="echoid-s2512" xml:space="preserve">illa quidem ab extremitate baſis; </s> |
|
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|
4452 <s xml:id="echoid-s2513" xml:space="preserve">hæc uero non ab extremitate: </s> |
|
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|
4453 <s xml:id="echoid-s2514" xml:space="preserve">ſe-<lb/>quitur ut a q, quæ ab extremitate ducitur, minorem acutum angulú <lb/>contineat cum diametro portionis, quàm ipſa i o. </s> |
|
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|
4454 <s xml:id="echoid-s2515" xml:space="preserve">Sed linea p φ li-<lb/>neæ a q æquidiſtat, & </s> |
|
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4455 <s xml:id="echoid-s2516" xml:space="preserve">m n ipſi i o. </s> |
|
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|
4456 <s xml:id="echoid-s2517" xml:space="preserve">angulus igitur ad φ angulo ad n |
|
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|
4457 <pb file="0096" n="96" rhead="ARCHIMEDIS"/> |
|
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|
4458 minor erit: </s> |
|
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|
4459 <s xml:id="echoid-s2518" xml:space="preserve">linea uero b c maior, quàm b s: </s> |
|
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|
4460 <s xml:id="echoid-s2519" xml:space="preserve">& </s> |
|
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|
4461 <s xml:id="echoid-s2520" xml:space="preserve">s r; </s> |
|
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|
4462 <s xml:id="echoid-s2521" xml:space="preserve">hoc eſt m χ ma-<lb/>ior, quàm c r, hoc eſt, quàm p y: </s> |
|
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4463 <s xml:id="echoid-s2522" xml:space="preserve">& </s> |
|
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4464 <s xml:id="echoid-s2523" xml:space="preserve">propterea χ t minor, quàm y f. <lb/></s> |
|
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|
4465 <s xml:id="echoid-s2524" xml:space="preserve">quòd cum p y ſit dupla y f, erit m χ maior, quàm dupla y f; </s> |
|
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4466 <s xml:id="echoid-s2525" xml:space="preserve">& </s> |
|
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|
4467 <s xml:id="echoid-s2526" xml:space="preserve"><lb/>multo maior, quàm dupla χ t. </s> |
|
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4468 <s xml:id="echoid-s2527" xml:space="preserve">fiat m h dupla ipſius h t: </s> |
|
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4469 <s xml:id="echoid-s2528" xml:space="preserve">& </s> |
|
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|
4470 <s xml:id="echoid-s2529" xml:space="preserve">copu-<lb/>lata h k producatur. </s> |
|
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4471 <s xml:id="echoid-s2530" xml:space="preserve">I am grauitatis centrum totius portionis erit <lb/>punctum k: </s> |
|
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4472 <s xml:id="echoid-s2531" xml:space="preserve">eius, quæ in humido est, h: </s> |
|
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|
4473 <s xml:id="echoid-s2532" xml:space="preserve">at rel iquæ partis, quæ ex-<lb/>tra humidum in linea h k producta; </s> |
|
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4474 <s xml:id="echoid-s2533" xml:space="preserve">quod ſit ω. </s> |
|
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4475 <s xml:id="echoid-s2534" xml:space="preserve">eodem modo demon <lb/>strabitur, & </s> |
|
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|
4476 <s xml:id="echoid-s2535" xml:space="preserve">lineam k h, & </s> |
|
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|
4477 <s xml:id="echoid-s2536" xml:space="preserve">quæ per h ω puncta ipſi k h æquidi-<lb/>ſtantes ducuntur, ad humidi ſuperficiem perpendiculares eſſe. </s> |
|
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|
4478 <s xml:id="echoid-s2537" xml:space="preserve">non <lb/>igitur maneb it <lb/> |
|
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|
4479 <anchor type="figure" xlink:label="fig-0096-01a" xlink:href="fig-0096-01"/> |
|
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|
4480 portio, ſed cum <lb/>uſque eò inclina-<lb/>ta fuerit, ut in <lb/>uno puncto con-<lb/>tingat ſuperfi-<lb/>cié humidi, tunc <lb/>conſiſtet. </s> |
|
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|
4481 <s xml:id="echoid-s2538" xml:space="preserve">an-<lb/>gulus enim ad n <lb/>angulo ad φ æ-<lb/>qualis erit; </s> |
|
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4482 <s xml:id="echoid-s2539" xml:space="preserve">li-<lb/>neáq; </s> |
|
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4483 <s xml:id="echoid-s2540" xml:space="preserve">b s lineæ <lb/>b c; </s> |
|
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|
4484 <s xml:id="echoid-s2541" xml:space="preserve">& </s> |
|
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|
4485 <s xml:id="echoid-s2542" xml:space="preserve">s r ipſi <lb/>c r. </s> |
|
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|
4486 <s xml:id="echoid-s2543" xml:space="preserve">quare & </s> |
|
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|
4487 <s xml:id="echoid-s2544" xml:space="preserve">m h <lb/>ipſi p y eſt æqua <lb/>lis. </s> |
|
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|
4488 <s xml:id="echoid-s2545" xml:space="preserve">Itaque ducta <lb/>h k producatur. <lb/></s> |
|
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|
4489 <s xml:id="echoid-s2546" xml:space="preserve">erit totius portionis grauitatis centrum K; </s> |
|
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|
4490 <s xml:id="echoid-s2547" xml:space="preserve">eius, quæ in humido eſt <lb/>h; </s> |
|
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|
4491 <s xml:id="echoid-s2548" xml:space="preserve">& </s> |
|
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|
4492 <s xml:id="echoid-s2549" xml:space="preserve">reliquæ partis centrum in linea producta; </s> |
|
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|
4493 <s xml:id="echoid-s2550" xml:space="preserve">ſit autem ω. </s> |
|
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|
4494 <s xml:id="echoid-s2551" xml:space="preserve">per ean <lb/>dem igitur rectam lineam k h, quæ eſt ad humidi ſuperficiem perpen <lb/>dicularis, id quod in humido eſt ſurſum; </s> |
|
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|
4495 <s xml:id="echoid-s2552" xml:space="preserve">& </s> |
|
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|
4496 <s xml:id="echoid-s2553" xml:space="preserve">quod extra humidum de <lb/>orſum feretur. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
4497 <s xml:id="echoid-s2554" xml:space="preserve">atque ob hác cauſſam portio non amplius mouebitur; </s> |
|
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|
4498 <s xml:id="echoid-s2555" xml:space="preserve"><lb/>ſed conſiſtet, manebítq, ita, ut eius baſis ſuperficiem humidi in uno <lb/>punsto contingat; </s> |
|
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|
4499 <s xml:id="echoid-s2556" xml:space="preserve">& </s> |
|
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|
4500 <s xml:id="echoid-s2557" xml:space="preserve">axis, cum ipſa angulum faciat æqualem angulo <lb/>φ. </s> |
|
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|
4501 <s xml:id="echoid-s2558" xml:space="preserve">at que illud eſt, quod demonſtrare oportebat.</s> |
|
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|
4502 <s xml:id="echoid-s2559" xml:space="preserve"/> |
|
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|
4503 </p> |
|
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|
4504 <div xml:id="echoid-div180" type="float" level="2" n="6"> |
|
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|
4505 <note position="left" xlink:label="note-0094-08" xlink:href="note-0094-08a" xml:space="preserve">F</note> |
|
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|
4506 <figure xlink:label="fig-0094-01" xlink:href="fig-0094-01a"> |
|
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|
4507 <image file="0094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0094-01"/> |
|
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|
4508 </figure> |
|
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|
4509 <figure xlink:label="fig-0095-01" xlink:href="fig-0095-01a"> |
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|
4510 <image file="0095-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0095-01"/> |
|
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|
4511 </figure> |
|
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|
4512 <note position="right" xlink:label="note-0095-01" xlink:href="note-0095-01a" xml:space="preserve">8. quinti.</note> |
|
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|
4513 <figure xlink:label="fig-0096-01" xlink:href="fig-0096-01a"> |
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|
4514 <image file="0096-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0096-01"/> |
|
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|
4515 </figure> |
|
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|
4516 </div> |
|
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|
4517 <pb o="43" file="0097" n="97" rhead="DE I _IS_ QVAE VEH. IN AQVA."/> |
|
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|
4518 </div> |
|
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|
4519 <div xml:id="echoid-div182" type="section" level="1" n="56"> |
|
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|
4520 <head xml:id="echoid-head61" xml:space="preserve">DEMONSTRATIO QVARTAE PARTIS.</head> |
|
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|
4521 <p> |
|
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|
4522 <s xml:id="echoid-s2560" xml:space="preserve">HABEAT rurſum portio ad humidum in grauitate <lb/>proportionem quidem maiorem, quàm quadratum f p ad <lb/>quadratum b d; </s> |
|
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|
4523 <s xml:id="echoid-s2561" xml:space="preserve">minorem uero, quàm quadratum x o ad <lb/>b d quadratum: </s> |
|
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|
4524 <s xml:id="echoid-s2562" xml:space="preserve">& </s> |
|
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changeset
|
4525 <s xml:id="echoid-s2563" xml:space="preserve">quam proportionem habet portio ad <lb/>humidum in grauitate, eandem habeat quadratum, quod <lb/>fit à linea ψ ad quadratum b d. </s> |
|
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|
4526 <s xml:id="echoid-s2564" xml:space="preserve">erit ψ maior, quàm f p, & </s> |
|
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diff
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|
4527 <s xml:id="echoid-s2565" xml:space="preserve">mi <lb/>nor, quàm x o. </s> |
|
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diff
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|
4528 <s xml:id="echoid-s2566" xml:space="preserve">aptetur ergo quæ dam rectalinea i u inter <lb/>portiones a u q l, a x d interiecta, quæ ſit æqualis ψ, & </s> |
|
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diff
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|
4529 <s xml:id="echoid-s2567" xml:space="preserve">ipſi <lb/>b d æquidiſtans: </s> |
|
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diff
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|
4530 <s xml:id="echoid-s2568" xml:space="preserve">occurratq; </s> |
|
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diff
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|
4531 <s xml:id="echoid-s2569" xml:space="preserve">reliquæ ſectioni in y. </s> |
|
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diff
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|
4532 <s xml:id="echoid-s2570" xml:space="preserve">rurſus <lb/>u y dupla ipſius y i demonſtrabitur, ſicuti demonſtrata eſt <lb/>o g ipſius g x dupla. </s> |
|
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parents:
diff
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|
4533 <s xml:id="echoid-s2571" xml:space="preserve">ducatur autem ab u linea u ο, quæ ſe <lb/>ctionem a u q l in u contingat: </s> |
|
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diff
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|
4534 <s xml:id="echoid-s2572" xml:space="preserve">& </s> |
|
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diff
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|
4535 <s xml:id="echoid-s2573" xml:space="preserve">iuncta a i ad q produca <lb/>tur. </s> |
|
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diff
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|
4536 <s xml:id="echoid-s2574" xml:space="preserve">eodem modo oſtendemus lineam a i ipſi i q æqualem <lb/>eſſe: </s> |
|
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|
4537 <s xml:id="echoid-s2575" xml:space="preserve">& </s> |
|
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diff
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|
4538 <s xml:id="echoid-s2576" xml:space="preserve">a q ipſi <lb/> |
|
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diff
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|
4539 <anchor type="figure" xlink:label="fig-0097-01a" xlink:href="fig-0097-01"/> |
|
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|
4540 u ω æquidiſtan-<lb/>tem. </s> |
|
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diff
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|
4541 <s xml:id="echoid-s2577" xml:space="preserve">Demon-<lb/>ſtrãdum eſt por <lb/>tionem in humi <lb/>dum demiſſam, <lb/>ĩclinatãq; </s> |
|
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diff
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|
4542 <s xml:id="echoid-s2578" xml:space="preserve">adeo, <lb/>ut baſis ipſius <lb/>non contingat <lb/>humidũ, ita con <lb/>ſiſtere, ut baſis <lb/>in humidũ ma-<lb/>gis demergatur <lb/>quam ut in uno <lb/>puncto eius ſu-<lb/>perficiem cõtin <lb/>gat. </s> |
|
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diff
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|
4543 <s xml:id="echoid-s2579" xml:space="preserve">Demittatur enim in humidum, ut dictum eſt; </s> |
|
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|
4544 <s xml:id="echoid-s2580" xml:space="preserve">& </s> |
|
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diff
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|
4545 <s xml:id="echoid-s2581" xml:space="preserve">iaceat <lb/>primo ſic inclinata, ut baſis nullo modo contingat ſuperfi-<lb/>ciem humidi. </s> |
|
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diff
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|
4546 <s xml:id="echoid-s2582" xml:space="preserve">ſecta autem ipſa plano per axem ad humidi |
|
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diff
changeset
|
4547 <pb file="0098" n="98" rhead="ARCHIMEDIS"/> |
|
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|
4548 ſuperficiem recto, ſit portionis ſectio anzg; </s> |
|
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|
4549 <s xml:id="echoid-s2583" xml:space="preserve">ſuperficiei <lb/>humidi ez: </s> |
|
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|
4550 <s xml:id="echoid-s2584" xml:space="preserve">a-<lb/> |
|
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|
4551 <anchor type="figure" xlink:label="fig-0098-01a" xlink:href="fig-0098-01"/> |
|
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diff
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|
4552 xis portionis, <lb/>& </s> |
|
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diff
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|
4553 <s xml:id="echoid-s2585" xml:space="preserve">ſectionis dia-<lb/>meter b d: </s> |
|
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|
4554 <s xml:id="echoid-s2586" xml:space="preserve">ſece-<lb/>turq, b d in pũ-<lb/>ctis _K_r, ſicuti <lb/>prius; </s> |
|
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|
4555 <s xml:id="echoid-s2587" xml:space="preserve">& </s> |
|
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|
4556 <s xml:id="echoid-s2588" xml:space="preserve">duca-<lb/>tur n l quidem <lb/>ipſi e z æquidi-<lb/>ſtans, quæ con-<lb/>tingat ſectionẽ <lb/>a n z g in n; </s> |
|
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|
4557 <s xml:id="echoid-s2589" xml:space="preserve">& </s> |
|
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|
4558 <s xml:id="echoid-s2590" xml:space="preserve"><lb/>n t æquidiſtans <lb/>ipſi b d; </s> |
|
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diff
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|
4559 <s xml:id="echoid-s2591" xml:space="preserve">n s ue-<lb/>ro ad b d perpẽ <lb/>dicularis. </s> |
|
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|
4560 <s xml:id="echoid-s2592" xml:space="preserve">Itaq; <lb/></s> |
|
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|
4561 <s xml:id="echoid-s2593" xml:space="preserve">quoniam portio ad humidum in grauitate eam proportio <lb/>nem habet, quam quadratum, quod fit à linea ψ ad quadra <lb/>tum b d: </s> |
|
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|
4562 <s xml:id="echoid-s2594" xml:space="preserve">erit ψ ipſi n t æqualis: </s> |
|
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diff
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|
4563 <s xml:id="echoid-s2595" xml:space="preserve">quod ſimiliter demonſtrabi <lb/>tur, ut ſuperius. </s> |
|
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|
4564 <s xml:id="echoid-s2596" xml:space="preserve">quare & </s> |
|
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diff
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|
4565 <s xml:id="echoid-s2597" xml:space="preserve">n t eſt æqualis ipſi u i. </s> |
|
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diff
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|
4566 <s xml:id="echoid-s2598" xml:space="preserve">portiones <lb/>igitur a u q, e n z inter ſe ſunt æquales. </s> |
|
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|
4567 <s xml:id="echoid-s2599" xml:space="preserve">Et cum in æquali-<lb/>bus, & </s> |
|
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|
4568 <s xml:id="echoid-s2600" xml:space="preserve">ſimilibus portionibus a u q l, a n z g ductæ ſint a q <lb/>e z, quæ æquales portiones auferunt; </s> |
|
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|
4569 <s xml:id="echoid-s2601" xml:space="preserve">illa quidem ab extre <lb/>mitate baſis; </s> |
|
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|
4570 <s xml:id="echoid-s2602" xml:space="preserve">hæc autem non ab extremitate: </s> |
|
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|
4571 <s xml:id="echoid-s2603" xml:space="preserve">minorem fa-<lb/>ciet acutum angulum cum portionis diametro, quæ ab ex-<lb/>tremitate baſis ducitur. </s> |
|
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|
4572 <s xml:id="echoid-s2604" xml:space="preserve">At triangulorum n l s, u ω c angu <lb/>lus ad l angulo ad ω maior eſt. </s> |
|
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|
4573 <s xml:id="echoid-s2605" xml:space="preserve">ergo b s minor erit, quam <lb/>b c: </s> |
|
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|
4574 <s xml:id="echoid-s2606" xml:space="preserve">& </s> |
|
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|
4575 <s xml:id="echoid-s2607" xml:space="preserve">ſ r maior, quàm c r: </s> |
|
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|
4576 <s xml:id="echoid-s2608" xml:space="preserve">ideoq; </s> |
|
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|
4577 <s xml:id="echoid-s2609" xml:space="preserve">n χ maior, quam u h; </s> |
|
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|
4578 <s xml:id="echoid-s2610" xml:space="preserve">& </s> |
|
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|
4579 <s xml:id="echoid-s2611" xml:space="preserve"><lb/>χ t minor, quàm h i. </s> |
|
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diff
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|
4580 <s xml:id="echoid-s2612" xml:space="preserve">Quoniam igitur u y dupla eſt ipſius <lb/>y i; </s> |
|
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|
4581 <s xml:id="echoid-s2613" xml:space="preserve">conſtat n χ maiorem eſſe, quàm duplã χ t. </s> |
|
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|
4582 <s xml:id="echoid-s2614" xml:space="preserve">Sit n m dupla <lb/>ipſius m t. </s> |
|
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|
4583 <s xml:id="echoid-s2615" xml:space="preserve">perſpicuũ eſt ex iis, quæ dicta ſunt, non manere <lb/>portionẽ; </s> |
|
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|
4584 <s xml:id="echoid-s2616" xml:space="preserve">ſed in clinari, donec eius baſis contingat ſuperfi-<lb/>ciem humidi: </s> |
|
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|
4585 <s xml:id="echoid-s2617" xml:space="preserve">contingat autem in puncto uno, ut patet in fi |
|
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|
4586 <pb o="44" file="0099" n="99" rhead="DE IIS QVAE VEH. IN AQVA."/> |
|
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changeset
|
4587 gura: </s> |
|
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|
4588 <s xml:id="echoid-s2618" xml:space="preserve">& </s> |
|
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|
4589 <s xml:id="echoid-s2619" xml:space="preserve">alia eadem diſponantur demonſtrabimus rurſum <lb/>n t æqualem eſſe ipſi u i: </s> |
|
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|
4590 <s xml:id="echoid-s2620" xml:space="preserve">& </s> |
|
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|
4591 <s xml:id="echoid-s2621" xml:space="preserve">portiones a u q, a n z inter <lb/>ſe ſe æquales. <lb/></s> |
|
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|
4592 <s xml:id="echoid-s2622" xml:space="preserve"> |
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4593 <anchor type="figure" xlink:label="fig-0099-01a" xlink:href="fig-0099-01"/> |
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4594 Itaque quoniã <lb/>ĩ portionibus <lb/>æqualibus, & </s> |
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4595 <s xml:id="echoid-s2623" xml:space="preserve">ſi <lb/>milibus a u q l, <lb/>a n z g ductæ <lb/>sũt a q, a z, por <lb/>tiones æqua-<lb/>les auferentes; <lb/></s> |
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4596 <s xml:id="echoid-s2624" xml:space="preserve">cum diametris <lb/>portionum æ-<lb/>quales angu-<lb/>los cõtinebũt. </s> |
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4597 <s xml:id="echoid-s2625" xml:space="preserve"><lb/>ergo triangulo <lb/>rum n l s, u ω c <lb/>anguli, qui cõ-<lb/>ſiſtũt ad l ω pũ-<lb/>cta, æquales ſunt: </s> |
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4598 <s xml:id="echoid-s2626" xml:space="preserve">& </s> |
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4599 <s xml:id="echoid-s2627" xml:space="preserve">b s recta linea æqualis ipſi b c: </s> |
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4600 <s xml:id="echoid-s2628" xml:space="preserve">ſ r ipſi cr, <lb/>n χ ipſi u h: </s> |
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4601 <s xml:id="echoid-s2629" xml:space="preserve">& </s> |
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4602 <s xml:id="echoid-s2630" xml:space="preserve">χ tipſi h i. </s> |
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4603 <s xml:id="echoid-s2631" xml:space="preserve">quòd cum u y dupla ſit ipſius y i, <lb/>erit n χ maior, quàm dupla χ t. </s> |
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4604 <s xml:id="echoid-s2632" xml:space="preserve">Sit igitur n m ipſius m t du <lb/>pla. </s> |
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4605 <s xml:id="echoid-s2633" xml:space="preserve">Rurſus ex his manifeſtum eſt, non manere ipſam por-<lb/>tionem; </s> |
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4606 <s xml:id="echoid-s2634" xml:space="preserve">ſed inclinari ex parte a: </s> |
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4607 <s xml:id="echoid-s2635" xml:space="preserve">ponebatur autem portio <lb/>humidi ſuperficiem in uno puncto contingere. </s> |
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4608 <s xml:id="echoid-s2636" xml:space="preserve">ergo ne-<lb/>ceſſe eſt, ut eius baſis in humidum magis demergatur.</s> |
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4609 <s xml:id="echoid-s2637" xml:space="preserve"/> |
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4610 </p> |
|
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4611 <div xml:id="echoid-div182" type="float" level="2" n="1"> |
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4612 <figure xlink:label="fig-0097-01" xlink:href="fig-0097-01a"> |
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4613 <image file="0097-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0097-01"/> |
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4614 </figure> |
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4615 <figure xlink:label="fig-0098-01" xlink:href="fig-0098-01a"> |
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4616 <image file="0098-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0098-01"/> |
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4617 </figure> |
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4618 <figure xlink:label="fig-0099-01" xlink:href="fig-0099-01a"> |
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4619 <image file="0099-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0099-01"/> |
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4620 </figure> |
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4621 </div> |
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4622 </div> |
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4623 <div xml:id="echoid-div184" type="section" level="1" n="57"> |
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4624 <head xml:id="echoid-head62" xml:space="preserve">DEMONSTRATIO QVINT AE PARTIS.</head> |
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4625 <p> |
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4626 <s xml:id="echoid-s2638" xml:space="preserve">HABEAT denique portio ad humidum in grauitate <lb/>minorem proportionem, quàm quadratum f p ad quadra-<lb/>tum b d: </s> |
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4627 <s xml:id="echoid-s2639" xml:space="preserve">& </s> |
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4628 <s xml:id="echoid-s2640" xml:space="preserve">quam proportionem habet portio ad humidũ <lb/>in grauitate, eandem quadratum, quod fit à linea ψ habeat <lb/>ad quadratum b d. </s> |
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4629 <s xml:id="echoid-s2641" xml:space="preserve">erit χ minor ipſa p f. </s> |
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4630 <s xml:id="echoid-s2642" xml:space="preserve">Rurſus aptetur |
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4631 <pb file="0100" n="100" rhead="ARCHIMEDIS"/> |
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4632 quædam recta linea g i, ſectionibus a g q l, a x d interiecta, <lb/>& </s> |
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4633 <s xml:id="echoid-s2643" xml:space="preserve">ipſi b d æquidiſtans; </s> |
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4634 <s xml:id="echoid-s2644" xml:space="preserve">quæ mediam coni ſectionem in pun <lb/>cto h, & </s> |
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4635 <s xml:id="echoid-s2645" xml:space="preserve">rectam <lb/> |
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4636 <anchor type="figure" xlink:label="fig-0100-01a" xlink:href="fig-0100-01"/> |
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4637 lineam r y in y <lb/>ſecet. </s> |
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4638 <s xml:id="echoid-s2646" xml:space="preserve">demonſtra <lb/>bitur g h dupla <lb/>h i, quemadmo-<lb/>dum demonſtra <lb/>ta eſt o g ipſius <lb/>g x dupla. </s> |
|
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4639 <s xml:id="echoid-s2647" xml:space="preserve">duca-<lb/>tur poſtea g ω cõ <lb/>tingens a g q l ſe <lb/>ctioneming: </s> |
|
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4640 <s xml:id="echoid-s2648" xml:space="preserve">& </s> |
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|
4641 <s xml:id="echoid-s2649" xml:space="preserve"><lb/>g c ad b d perpé <lb/>dicularis: </s> |
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4642 <s xml:id="echoid-s2650" xml:space="preserve">iun-<lb/>ctaq; </s> |
|
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4643 <s xml:id="echoid-s2651" xml:space="preserve">ai produ-<lb/>catur ad q. </s> |
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4644 <s xml:id="echoid-s2652" xml:space="preserve">erit <lb/>ergo a i æqualis <lb/>i q: </s> |
|
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4645 <s xml:id="echoid-s2653" xml:space="preserve">& </s> |
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4646 <s xml:id="echoid-s2654" xml:space="preserve">a q ipſi g ω <lb/>æquidiſtans. </s> |
|
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4647 <s xml:id="echoid-s2655" xml:space="preserve">Demonſtrandũ eſt portionẽ in humidũ demiſ <lb/>fam, inclinatamq; </s> |
|
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4648 <s xml:id="echoid-s2656" xml:space="preserve">adeo, ut baſis ipſius non cõtingat humi-<lb/>dũ, conſiſtere inclinatã ita, ut axis cum ſuperficie humidi <lb/>angulum faciat minorem angulo φ: </s> |
|
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|
4649 <s xml:id="echoid-s2657" xml:space="preserve">& </s> |
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|
4650 <s xml:id="echoid-s2658" xml:space="preserve">baſis humidi ſuper-<lb/>ficiem nullo modo contingat. </s> |
|
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4651 <s xml:id="echoid-s2659" xml:space="preserve">Demittatur enim in humi-<lb/>dum; </s> |
|
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4652 <s xml:id="echoid-s2660" xml:space="preserve">& </s> |
|
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|
4653 <s xml:id="echoid-s2661" xml:space="preserve">conſiſtat ita, ut baſis ipſius in uno puncto contin-<lb/>gat ſuperficiem humidi. </s> |
|
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|
4654 <s xml:id="echoid-s2662" xml:space="preserve">ſecta autem portione per axem, <lb/>plano ad humidi ſuperficiem recto, ſit portionis ſectio a n <lb/>z l rectanguli coni ſectio: </s> |
|
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|
4655 <s xml:id="echoid-s2663" xml:space="preserve">ſuperficiei humidi a z: </s> |
|
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|
4656 <s xml:id="echoid-s2664" xml:space="preserve">axis autẽ <lb/>portionis, & </s> |
|
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|
4657 <s xml:id="echoid-s2665" xml:space="preserve">ſectionis diameter b d: </s> |
|
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|
4658 <s xml:id="echoid-s2666" xml:space="preserve">ſeceturq; </s> |
|
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|
4659 <s xml:id="echoid-s2667" xml:space="preserve">b d in pun-<lb/>ctis _K_ r, ut ſuperius dictum eſt: </s> |
|
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|
4660 <s xml:id="echoid-s2668" xml:space="preserve">& </s> |
|
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|
4661 <s xml:id="echoid-s2669" xml:space="preserve">ducatur n f quidem ipſi <lb/>a z æquidiſtans, & </s> |
|
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|
4662 <s xml:id="echoid-s2670" xml:space="preserve">contingens coni ſectionem in pũcto n; <lb/></s> |
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4663 <s xml:id="echoid-s2671" xml:space="preserve">n t uero æquidiſtans ipſi b d: </s> |
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4664 <s xml:id="echoid-s2672" xml:space="preserve">& </s> |
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4665 <s xml:id="echoid-s2673" xml:space="preserve">n s ad eandem perpendi-<lb/>cularis. </s> |
|
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4666 <s xml:id="echoid-s2674" xml:space="preserve">Quoniam igitur portio ad humidum in grauitate, <lb/>cam habet proportionem, quam quadratum, quod fit à χ |
|
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4667 <pb o="43" file="0101" n="101" rhead="DEIIS QVAE VEH. IN AQVA."/> |
|
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|
4668 ad quadratum bd: </s> |
|
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4669 <s xml:id="echoid-s2675" xml:space="preserve">& </s> |
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4670 <s xml:id="echoid-s2676" xml:space="preserve">quam habet portio ad humidum in <lb/>grauitate, eandem quadratum nt habet ad bd quadratũ, <lb/>ex iis, quæ dicta ſunt: </s> |
|
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4671 <s xml:id="echoid-s2677" xml:space="preserve">conſtat n t lineæ ψ æqualem eſſe, <lb/>quare & </s> |
|
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4672 <s xml:id="echoid-s2678" xml:space="preserve">portio-<lb/> |
|
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4673 <anchor type="figure" xlink:label="fig-0101-01a" xlink:href="fig-0101-01"/> |
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4674 nes a n z, a g q <lb/>ſunt æquales. </s> |
|
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4675 <s xml:id="echoid-s2679" xml:space="preserve">Et <lb/>quoniam in por <lb/>tionibus æquali <lb/>bus, & </s> |
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4676 <s xml:id="echoid-s2680" xml:space="preserve">ſimilibus <lb/>a g q l, a n z l, ab <lb/>extremitatibus <lb/>baſiũ ductæ ſunt <lb/>a q, a z, quæ æ-<lb/>quales portiões <lb/>abſcindunt: </s> |
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4677 <s xml:id="echoid-s2681" xml:space="preserve">per <lb/>ſpicuum eſt an-<lb/>gulos facere æ-<lb/>quales cum por <lb/>tionum diame-<lb/>tris: </s> |
|
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4678 <s xml:id="echoid-s2682" xml:space="preserve">& </s> |
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4679 <s xml:id="echoid-s2683" xml:space="preserve">triangu-<lb/>lorum n fs, g ω c, angulos, qui ad f ω æquales eſſe: </s> |
|
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4680 <s xml:id="echoid-s2684" xml:space="preserve">itemque <lb/>æquales inter ſe, s b, c b; </s> |
|
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4681 <s xml:id="echoid-s2685" xml:space="preserve">& </s> |
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4682 <s xml:id="echoid-s2686" xml:space="preserve">s r, c r, quare & </s> |
|
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4683 <s xml:id="echoid-s2687" xml:space="preserve">n χ, g y æquales: <lb/></s> |
|
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|
4684 <s xml:id="echoid-s2688" xml:space="preserve">& </s> |
|
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|
4685 <s xml:id="echoid-s2689" xml:space="preserve">χ t y i. </s> |
|
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4686 <s xml:id="echoid-s2690" xml:space="preserve">cũq; </s> |
|
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|
4687 <s xml:id="echoid-s2691" xml:space="preserve">g h dupla ſit ipſius h i, erit n χ minor, quàm <lb/>duplaipſius χ t. </s> |
|
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4688 <s xml:id="echoid-s2692" xml:space="preserve">Sit igitur n m ipſius m t dupla: </s> |
|
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4689 <s xml:id="echoid-s2693" xml:space="preserve">& </s> |
|
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|
4690 <s xml:id="echoid-s2694" xml:space="preserve">iuncta <lb/>m K protrahatur ad e. </s> |
|
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|
4691 <s xml:id="echoid-s2695" xml:space="preserve">Itaque centrum grauitatis totius <lb/>erit punctum K: </s> |
|
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|
4692 <s xml:id="echoid-s2696" xml:space="preserve">partis eius, quæ eſt in humido, punctũ m: </s> |
|
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|
4693 <s xml:id="echoid-s2697" xml:space="preserve"><lb/>eius autem, quæ extra humidum in linea protracta, quod <lb/>ſit e. </s> |
|
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|
4694 <s xml:id="echoid-s2698" xml:space="preserve">ergo ex proxime demonſtratis patet, nõ manere por <lb/>tionem, ſed inclinari adeo, ut baſis nullo modo ſuperficiẽ <lb/>humidi contingat. </s> |
|
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4695 <s xml:id="echoid-s2699" xml:space="preserve">At uero portionem conſiſtere ita, uta-<lb/>xis cum ſuperficie humidi faciat angulum angulo φ mino-<lb/>rem, ſic demonſtrabitur. </s> |
|
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|
4696 <s xml:id="echoid-s2700" xml:space="preserve">conſiſtat enim, ſi fieri poteſt, ut <lb/>non faciat angulum minorem angulo φ: </s> |
|
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|
4697 <s xml:id="echoid-s2701" xml:space="preserve">& </s> |
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|
4698 <s xml:id="echoid-s2702" xml:space="preserve">alia eadem diſ-<lb/>ponantur; </s> |
|
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|
4699 <s xml:id="echoid-s2703" xml:space="preserve">ut in ſubiecta figura. </s> |
|
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|
4700 <s xml:id="echoid-s2704" xml:space="preserve">eodem modo demonſtra |
|
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|
4701 <pb file="0102" n="102" rhead="ARCHIMEDIS"/> |
|
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|
4702 bimus n t æqualem eſſe ψ, & </s> |
|
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|
4703 <s xml:id="echoid-s2705" xml:space="preserve">propterea ipſi gi. </s> |
|
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|
4704 <s xml:id="echoid-s2706" xml:space="preserve">& </s> |
|
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|
4705 <s xml:id="echoid-s2707" xml:space="preserve">quo-<lb/>niam triangulornm p φ c, n f s angulus f non eſt minor an <lb/>gulo φ, non erit b f maior, quam b c. </s> |
|
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4706 <s xml:id="echoid-s2708" xml:space="preserve">ergo neque s r mi-<lb/>nor, quàm c r: </s> |
|
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|
4707 <s xml:id="echoid-s2709" xml:space="preserve">neque n χ minor, quàm p y. </s> |
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|
4708 <s xml:id="echoid-s2710" xml:space="preserve">Sed cum p f ſit <lb/>maior, quàm n t: <lb/></s> |
|
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|
4709 <s xml:id="echoid-s2711" xml:space="preserve"> |
|
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|
4710 <anchor type="figure" xlink:label="fig-0102-01a" xlink:href="fig-0102-01"/> |
|
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|
4711 ſitq; </s> |
|
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|
4712 <s xml:id="echoid-s2712" xml:space="preserve">p f ſeſquialte <lb/>ra p y: </s> |
|
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|
4713 <s xml:id="echoid-s2713" xml:space="preserve">erit n t mi-<lb/>nor, quàm ſeſquial <lb/>tera n χ: </s> |
|
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|
4714 <s xml:id="echoid-s2714" xml:space="preserve">& </s> |
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|
4715 <s xml:id="echoid-s2715" xml:space="preserve">idcir-<lb/>co n χ maior, quã <lb/>dupla χ t. </s> |
|
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|
4716 <s xml:id="echoid-s2716" xml:space="preserve">ſit autẽ <lb/>n m dupla m t: </s> |
|
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|
4717 <s xml:id="echoid-s2717" xml:space="preserve">& </s> |
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|
4718 <s xml:id="echoid-s2718" xml:space="preserve"><lb/>iuncta m K produ <lb/>catur. </s> |
|
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|
4719 <s xml:id="echoid-s2719" xml:space="preserve">conſtat igi-<lb/>tur ex iam dictis <lb/>non manere por-<lb/>tionem; </s> |
|
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|
4720 <s xml:id="echoid-s2720" xml:space="preserve">ſed reuol <lb/>ui ita, ut axis cum <lb/>ſuperficie humidi <lb/>faciat angulum an <lb/>gulo φ minorem.</s> |
|
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|
4721 <s xml:id="echoid-s2721" xml:space="preserve"/> |
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|
4722 </p> |
|
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|
4723 <div xml:id="echoid-div184" type="float" level="2" n="1"> |
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|
4724 <figure xlink:label="fig-0100-01" xlink:href="fig-0100-01a"> |
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|
4725 <image file="0100-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0100-01"/> |
|
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|
4726 </figure> |
|
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|
4727 <figure xlink:label="fig-0101-01" xlink:href="fig-0101-01a"> |
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|
4728 <image file="0101-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0101-01"/> |
|
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|
4729 </figure> |
|
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|
4730 <figure xlink:label="fig-0102-01" xlink:href="fig-0102-01a"> |
|
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|
4731 <image file="0102-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0102-01"/> |
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4732 </figure> |
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4733 </div> |
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4734 </div> |
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4735 <div xml:id="echoid-div186" type="section" level="1" n="58"> |
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4736 <head xml:id="echoid-head63" xml:space="preserve">FINIS LIBRORVM ARCHIMEDIS DE <lb/>IIS, QVAE IN AQVA VEHVNTVR.</head> |
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4737 <pb file="0103" n="103"/> |
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4738 <pb file="0104" n="104"/> |
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4739 <pb file="0105" n="105"/> |
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4740 </div> |
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4741 <div xml:id="echoid-div187" type="section" level="1" n="59"> |
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4742 <head xml:id="echoid-head64" xml:space="preserve">FEDERICI <lb/>COMMANDINI <lb/>VRBINATIS</head> |
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4743 <head xml:id="echoid-head65" xml:space="preserve">LIBER DE CENTRO <lb/>GRAVITATIS <lb/>SOLIDORV M.</head> |
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4744 <figure> |
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4745 <image file="0105-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0105-01"/> |
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4746 </figure> |
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4747 </div> |
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4748 <div xml:id="echoid-div188" type="section" level="1" n="60"> |
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4749 <head xml:id="echoid-head66" xml:space="preserve">CVM PRIVILEGIO IN ANNOS X. <lb/>BONONIAE, <lb/>Ex Officina Alexandri Benacii.</head> |
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4750 <head xml:id="echoid-head67" xml:space="preserve">M D LXV.</head> |
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4751 <pb file="0106" n="106"/> |
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4752 <pb file="0107" n="107"/> |
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4753 </div> |
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4754 <div xml:id="echoid-div189" type="section" level="1" n="61"> |
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4755 <head xml:id="echoid-head68" xml:space="preserve">ALEXANDRO FARNESIO <lb/>CARDINALI AMPLISSIMO <lb/>ET OPTIMO.</head> |
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4756 <p> |
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4757 <s xml:id="echoid-s2722" xml:space="preserve">CVM multæ res in mathematicis <lb/>diſciplinis nequaquam ſatis ad-<lb/>huc explicatæ ſint, tum perdif-<lb/>ficilis, & </s> |
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4758 <s xml:id="echoid-s2723" xml:space="preserve">perobſcura quæſtio <lb/>eſt de centro grauitatis corpo-<lb/>rum ſolidorum; </s> |
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4759 <s xml:id="echoid-s2724" xml:space="preserve">quæ, & </s> |
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4760 <s xml:id="echoid-s2725" xml:space="preserve">ad co-<lb/>gnoſcendum pulcherrima eſt, <lb/>& </s> |
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4761 <s xml:id="echoid-s2726" xml:space="preserve">ad multa, quæ à mathematicis proponuntur, præ-<lb/>clare intelligenda maximum affert adiumentum. </s> |
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4762 <s xml:id="echoid-s2727" xml:space="preserve">de <lb/>qua neminem ex mathematicis, neque noſtra, neque <lb/>patrum noſtrorum memoria ſcriptum reliquiſſe ſci-<lb/>mus. </s> |
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4763 <s xml:id="echoid-s2728" xml:space="preserve">& </s> |
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4764 <s xml:id="echoid-s2729" xml:space="preserve">quamuis in earum monumentis literarum nõ <lb/>nulla reperiantur, ex quibus in hanc ſententiam addu <lb/>ci poſſumus, vt exiſtimemus hanc rem ab ijſdẽ vber-<lb/>rime tractatam eſſe; </s> |
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4765 <s xml:id="echoid-s2730" xml:space="preserve">tamen neſcio quo fato adhuc <lb/>in eiuſmodi librorum ignoratione verſamur. </s> |
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4766 <s xml:id="echoid-s2731" xml:space="preserve">Archi-<lb/>medes quidem mathematicorũ princeps in libello, <lb/>cuius inſcriptio eſt, κέντρα βάρων ἐπιπέδων, de centro pla-<lb/>norum copioſiſsime, atque acutiſsime conſcripſit: </s> |
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4767 <s xml:id="echoid-s2732" xml:space="preserve">& </s> |
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4768 <s xml:id="echoid-s2733" xml:space="preserve"><lb/>in eo explicando ſummã ingenii, & </s> |
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4769 <s xml:id="echoid-s2734" xml:space="preserve">ſcientiæ gloriã eſt <lb/>cõſecutus. </s> |
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4770 <s xml:id="echoid-s2735" xml:space="preserve">Sed de cognitione cẽtri grauitatis corporũ <lb/>ſolidorũ nulla in eius libris litera inuenitur. </s> |
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4771 <s xml:id="echoid-s2736" xml:space="preserve">non mul <lb/>tos abhinc annos <emph style="sc">Marcellvs i</emph>I. </s> |
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4772 <s xml:id="echoid-s2737" xml:space="preserve"><emph style="sc">Pont</emph>. </s> |
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4773 <s xml:id="echoid-s2738" xml:space="preserve"><emph style="sc">Max</emph>.</s> |
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4774 <s xml:id="echoid-s2739" xml:space="preserve"> |
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4775 <pb file="0108" n="108"/> |
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4776 cum adhuc Cardinalis eſſet, mihi, quæ ſua erat hu-<lb/>manitas, libros eiuſdem Archimedis de ijs, quæ ve-<lb/>huntur in aqua, latine redditos dono dedit. </s> |
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4777 <s xml:id="echoid-s2740" xml:space="preserve">hos cum <lb/>ego, ut aliorum ſtudia incitarem, emendãdos, & </s> |
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4778 <s xml:id="echoid-s2741" xml:space="preserve">cõ-<lb/>mentariis illuſtrandos ſuſcepiſſem, animaduerti dubi <lb/>tari non poſſe, quin Archimedes vel de hac materia <lb/>ſcripſiſſet, vel aliorum mathematicorum ſcripta per-<lb/>legiſſet. </s> |
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4779 <s xml:id="echoid-s2742" xml:space="preserve">nam in iis tum alia nonnulla, tum maxime <lb/>illam propoſitionem, ut euidentem, & </s> |
|
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4780 <s xml:id="echoid-s2743" xml:space="preserve">aliàs proba-<lb/>tam aſſumit, Centrũ grauitatis in portionibus conoi <lb/>dis rectan guli axem ita diuidere, vt pars, quæ ad verti <lb/>cem terminatur, alterius partis, quæ ad baſim dupla <lb/>ſit. </s> |
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4781 <s xml:id="echoid-s2744" xml:space="preserve">Verum hæc ad eam partem mathematicarum <lb/>diſciplinarum præcipue refertur, in qua de centro <lb/>grauitatis corporum ſolidorum tractatur. </s> |
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4782 <s xml:id="echoid-s2745" xml:space="preserve">non eſt au <lb/>tem conſentaneum Archimedem illum admirabilem <lb/>virum hanc propoſitionem ſibi argumentis con-<lb/>firmandam exiſtimaturum non fuiſſe, niſi eam vel <lb/>aliis in locis probauiſſet, vel ab aliis probatam eſſe <lb/>comperiſſet. </s> |
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4783 <s xml:id="echoid-s2746" xml:space="preserve">quamobrem nequid in iis libris intel-<lb/>ligendis deſiderari poſſet, ſtatui hanc etiam partem <lb/>vel à veteribus prætermiſſam, vel tractatam quidem, <lb/>ſed in tenebris iacentem, non intactam relinquere; <lb/></s> |
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4784 <s xml:id="echoid-s2747" xml:space="preserve">atque ex aſsidua mathematicorum, præſertim Archi-<lb/>medis lectione, quæ mihi in mentem venerunt, ea in <lb/>medium afferre; </s> |
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4785 <s xml:id="echoid-s2748" xml:space="preserve">ut centri grauitatis corporum ſoli-<lb/>dorum, ſi non perfectam, at certe aliquam noti- |
|
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4786 <pb file="0109" n="109"/> |
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4787 tiam haberemus. </s> |
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4788 <s xml:id="echoid-s2749" xml:space="preserve">Q uem meum laborem nō mathe-<lb/>maticis ſolum, verum iis etiam, qui naturæ obſcuri-<lb/>tate delectantur, nó iniucundam fore ſperaui: </s> |
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4789 <s xml:id="echoid-s2750" xml:space="preserve">multa <lb/>enim προβλήματα cognitiòne digniſsima, quæ ad vtrã-<lb/>que ſcientiam attinent, ſeſe legentibus obtuliſſent. <lb/></s> |
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4790 <s xml:id="echoid-s2751" xml:space="preserve">neque id vlli mirandum videri debet. </s> |
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4791 <s xml:id="echoid-s2752" xml:space="preserve">vt enim in cor-<lb/>poribus noſtris omnia membra, ex quibus certa quæ <lb/>dam officia naſcuntur, diuino quodam ordine inter <lb/>ſe implicata, & </s> |
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4792 <s xml:id="echoid-s2753" xml:space="preserve">colligata ſunt: </s> |
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4793 <s xml:id="echoid-s2754" xml:space="preserve">in iisq́; </s> |
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4794 <s xml:id="echoid-s2755" xml:space="preserve">admirabilis il-<lb/>la conſpiratio, quam σύμπνοιαν græci vocant, eluceſcit, <lb/>ita tres illæ Philoſophiæ (ut Ariſtotelis verbo vtar) <lb/>quæ veritatem ſ<unsure/>olam propoſitam habent, licet qui-<lb/>buſdam quaſi finibus ſuis regantur: </s> |
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4795 <s xml:id="echoid-s2756" xml:space="preserve">tamen earũ vna-<lb/>quæque per ſe ipſam quodammodo imperfecta eſt: </s> |
|
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4796 <s xml:id="echoid-s2757" xml:space="preserve"><lb/>neque altera ſine alterius auxilio plene comprehen-<lb/>di poteſt. </s> |
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4797 <s xml:id="echoid-s2758" xml:space="preserve">complures præterea mathematicorum no-<lb/>di ante hac explicatu difficillimi nullo negotio expe <lb/>diti eſſent: </s> |
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4798 <s xml:id="echoid-s2759" xml:space="preserve">atque (ut vno, verbo complectar) niſi <lb/>mea valde amo, tractationem hanc meam ſtudioſis <lb/>non mediocrem vtilitatem, & </s> |
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4799 <s xml:id="echoid-s2760" xml:space="preserve">magnam volupta-<lb/>tem allaturam eſſe mihi perſuaſi. </s> |
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4800 <s xml:id="echoid-s2761" xml:space="preserve">cum autem ad hoc <lb/>ſcribendum aggreſſus eſsem, allatus eſt ad me liber <lb/>Franciſci Maurolici Meſſanenſis, in quo vir ille do-<lb/>ctiſsimus, & </s> |
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4801 <s xml:id="echoid-s2762" xml:space="preserve">in iis diſciplinis exercitatiſsimus af-<lb/>firmabat ſe de centro grauitatis corporum ſolido-<lb/>rum conſcripſiſſe. </s> |
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4802 <s xml:id="echoid-s2763" xml:space="preserve">cum hoc intellexiſſem, ſuſtinui <lb/>me pauliſper: </s> |
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4803 <s xml:id="echoid-s2764" xml:space="preserve">tacitusque expectaui, dum opus cla- |
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4804 <pb file="0110" n="110"/> |
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4805 risſimi uiri, quem ſemper honoris cauſſa nomino, <lb/>in lucem proferretur: </s> |
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4806 <s xml:id="echoid-s2765" xml:space="preserve">mihi enim exploratisſimum <lb/>erat: </s> |
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4807 <s xml:id="echoid-s2766" xml:space="preserve">Franciſcum Maurolicum multo doctius, & </s> |
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4808 <s xml:id="echoid-s2767" xml:space="preserve"><lb/>exquiſitius hoc diſciplinarum genus ſcriptis ſuis tra <lb/>diturum. </s> |
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4809 <s xml:id="echoid-s2768" xml:space="preserve">ſed cum id tardius fieret, hoc eſt, ut ego <lb/>interpretor, diligentius, mihi diutius hac ſcriptione <lb/>non ſuperſedendum eſſe duxi, præſertim cum iam li-<lb/>bri Archimedis de iis, quæ uehuntur in aqua, opera <lb/>mea illuſtrati typis excudẽdi eſſent. </s> |
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4810 <s xml:id="echoid-s2769" xml:space="preserve">nec me alia cauſ <lb/>ſa impuliſſet, ut de centro grauitatis corporum ſoli-<lb/>dorum ſcriberem, niſi ut hac etiam ratione lux eis <lb/>quâm maxime fieri poſſet afferretur. </s> |
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4811 <s xml:id="echoid-s2770" xml:space="preserve">atq; </s> |
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4812 <s xml:id="echoid-s2771" xml:space="preserve">id eò mihi <lb/>faciendum exiſtimaui, quòd in ſpem ueniebam fore, <lb/>ut cum ego ex omnibus mathematicis primus, hanc <lb/>materiam explicandam ſuſcepiſſem; </s> |
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4813 <s xml:id="echoid-s2772" xml:space="preserve">ſi quid errati for <lb/>te à me commiſſum eſſet, boni uiri potius id meæ de <lb/>ſtudioſis hominibus bene merẽdi cupiditati, quàm <lb/>arrogantiæ aſcriberent. </s> |
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4814 <s xml:id="echoid-s2773" xml:space="preserve">reſtabat ut conſiderarem, cui <lb/>potisſimum ex principibus uiris contemplationem <lb/>hanc, nunc primum memoriæ, ac literis proditam de <lb/>dicarem. </s> |
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4815 <s xml:id="echoid-s2774" xml:space="preserve">harum mearum cogitationum ſumma fa-<lb/>cta, exiſtimaui nemini conuenientius de centro graui <lb/>tatis corporum opus dicari oportere, quàm <emph style="sc">Ale-<lb/>xandro</emph> <emph style="sc">Farnesio</emph> grauisſimo, ac prudentisſi-<lb/>mo Cardinali, quo in uiro ſumma fortuna ſemper cũ <lb/>ſumma uirtute certauit. </s> |
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4816 <s xml:id="echoid-s2775" xml:space="preserve">quid enim maxime in te ad-<lb/>mirari debeant homines, obſcurum eſt; </s> |
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4817 <s xml:id="echoid-s2776" xml:space="preserve">uſuḿne re- |
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4818 <pb file="0111" n="111"/> |
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4819 rum, qui pueritiæ tempus extremum principium ha <lb/>buiſti, & </s> |
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4820 <s xml:id="echoid-s2777" xml:space="preserve">imperiorũ, & </s> |
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4821 <s xml:id="echoid-s2778" xml:space="preserve">ad Reges, & </s> |
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4822 <s xml:id="echoid-s2779" xml:space="preserve">Imperatores ho-<lb/>norificentiſsimarum legationum; </s> |
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4823 <s xml:id="echoid-s2780" xml:space="preserve">an excellentiam <lb/>in omni genere literarum, qui vix adoleſcẽtulus, quæ <lb/>homines iam confirmata ætate ſummo ſtudio, diu-<lb/>turnisq́; </s> |
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4824 <s xml:id="echoid-s2781" xml:space="preserve">laboribus didicerunt, ſcientia, & </s> |
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4825 <s xml:id="echoid-s2782" xml:space="preserve">cogaitione <lb/>comprehendiſti: </s> |
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4826 <s xml:id="echoid-s2783" xml:space="preserve">an conſilium, & </s> |
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4827 <s xml:id="echoid-s2784" xml:space="preserve">ſapientiam in re-<lb/>gendis, & </s> |
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4828 <s xml:id="echoid-s2785" xml:space="preserve">gubernãdis Ciuitatibus, cuius grauiísimæ <lb/>ſententiæ in ſanctiſsimo Reip. </s> |
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4829 <s xml:id="echoid-s2786" xml:space="preserve">Chrſtianæ conſilio di-<lb/>ctæ, potius diuina oracula, quàm ſententiæ habitæ <lb/>ſunt, & </s> |
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4830 <s xml:id="echoid-s2787" xml:space="preserve">habentur. </s> |
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4831 <s xml:id="echoid-s2788" xml:space="preserve">prætermitto liberalitatem, & </s> |
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4832 <s xml:id="echoid-s2789" xml:space="preserve">mu-<lb/>nificentiam tuam, quam in ſtudio ſiſsimo quoque ho <lb/>neſtando quotidie magis oſtendis, ne videar auribus <lb/>tuis potius, quàm veritati ſeruire. </s> |
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4833 <s xml:id="echoid-s2790" xml:space="preserve">quamuis à te in tot <lb/>præclaros viros tanta beneficia collata ſunt, & </s> |
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4834 <s xml:id="echoid-s2791" xml:space="preserve">confe-<lb/>rũtur, vt omnibus teſtatum ſit, nihil tibi eſſe charius, <lb/>nihil iucundius, quàm eximia tua liberalitate homi-<lb/>nes ad amplexandam virtutem, licet currentes incita-<lb/>re. </s> |
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4835 <s xml:id="echoid-s2792" xml:space="preserve">nihil dico de ceteris virtutibus tuis, quæ tantæ <lb/>ſunt, quantæ ne cogitatione quidem comprehendi <lb/>poſſunt. </s> |
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4836 <s xml:id="echoid-s2793" xml:space="preserve">Quamobrem hac præcipue de cauſſa te hu-<lb/>ius meæ lucubrationis patronum eſſe volui, quam ea, <lb/>qua ſoles, humanitate accipies. </s> |
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4837 <s xml:id="echoid-s2794" xml:space="preserve">te enim ſemper ob <lb/>diuinas virtutes tuas colui, & </s> |
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4838 <s xml:id="echoid-s2795" xml:space="preserve">obſeruaui: </s> |
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4839 <s xml:id="echoid-s2796" xml:space="preserve">nihilq́; </s> |
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4840 <s xml:id="echoid-s2797" xml:space="preserve">mi-<lb/>hi fuit optatius; </s> |
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4841 <s xml:id="echoid-s2798" xml:space="preserve">quàm tibi perſpectum eſſe meum <lb/>erga te animum; </s> |
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4842 <s xml:id="echoid-s2799" xml:space="preserve">ſingularemq́; </s> |
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4843 <s xml:id="echoid-s2800" xml:space="preserve">obſeruantiam. </s> |
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4844 <s xml:id="echoid-s2801" xml:space="preserve">cœ-<lb/>lum igitur digito attingam, ſi poſt grauiſsimas oc- |
|
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4845 <pb file="0112" n="112"/> |
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4846 cupationes tuas legendo Federici tui libro aliquid <lb/>impertiri temporis non grauaberis: </s> |
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4847 <s xml:id="echoid-s2802" xml:space="preserve">eumq́; </s> |
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4848 <s xml:id="echoid-s2803" xml:space="preserve">in iis, qui <lb/>tibi ſemper addicti erunt, numerare. </s> |
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4849 <s xml:id="echoid-s2804" xml:space="preserve">Vale.</s> |
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4850 <s xml:id="echoid-s2805" xml:space="preserve"/> |
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4851 </p> |
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4852 <p> |
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4853 <s xml:id="echoid-s2806" xml:space="preserve">Federicus Commandinus.</s> |
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4854 <s xml:id="echoid-s2807" xml:space="preserve"/> |
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|
4855 </p> |
|
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4856 <pb o="1" file="0113" n="113"/> |
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4857 </div> |
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4858 <div xml:id="echoid-div190" type="section" level="1" n="62"> |
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4859 <head xml:id="echoid-head69" xml:space="preserve">FEDERICI COMMANDINI <lb/>VRBINATIS LIBER DE CENTRO <lb/>GRAVITATIS SOLIDORVM. <lb/>DIFFINITIONES.</head> |
|
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4860 <p> |
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4861 <s xml:id="echoid-s2808" xml:space="preserve"><emph style="sc">CEntrvm</emph> grauitatis, Pappus <lb/>Alexandrinus in octauo ma-<lb/>thematicarum collectionum <lb/>libro ita diffiniuit.</s> |
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4862 <s xml:id="echoid-s2809" xml:space="preserve"/> |
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4863 </p> |
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4864 <p style="it"> |
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4865 <s xml:id="echoid-s2810" xml:space="preserve">λέγομεν δἐ κἐντρον βάρ<unsure/>ους ἑκά στου σἀ<unsure/> <lb/>ματος {εἶ}ναι σημ\~ειον τικείμενον ἑντὸς, ἀφ<unsure/> <lb/>οὖ κα\’τ ἐποίνιαν ἀρτηθέν τό βάρ<unsure/>ος ἡμερ{εἶ} <lb/>φερ<unsure/>όμενον, καὶ φυλὰσσει τήν ἐξἀρχ\~νςθἐ-<lb/>σιν, οὐ μὴ περιτρ<unsure/>επ ὸμενον ἐντῆ φορ<unsure/>ᾶ. </s> |
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4866 <s xml:id="echoid-s2811" xml:space="preserve">hoc eſt,</s> |
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4867 </p> |
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4868 <p> |
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4869 <s xml:id="echoid-s2812" xml:space="preserve">Dicimus autem centrum grauitatis uniuſcu-<lb/>inſque corporis punctum quoddam intra poſi-<lb/>cum, à quo ſi graue appenſum mente concipia-<lb/>tur, dum fertur quieſcit; </s> |
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4870 <s xml:id="echoid-s2813" xml:space="preserve">& </s> |
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4871 <s xml:id="echoid-s2814" xml:space="preserve">ſeruat eam, quam in <lb/>principio habebat poſitionem: </s> |
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4872 <s xml:id="echoid-s2815" xml:space="preserve">neque in ipſa la-<lb/>tione circumuertitur.</s> |
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4873 <s xml:id="echoid-s2816" xml:space="preserve"/> |
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4874 </p> |
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4875 <p> |
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4876 <s xml:id="echoid-s2817" xml:space="preserve">Poſſumus etiam hoc modo diffinire.</s> |
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4877 <s xml:id="echoid-s2818" xml:space="preserve"/> |
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4878 </p> |
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4879 <p> |
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4880 <s xml:id="echoid-s2819" xml:space="preserve">Centrum grauitatis uniuſcuiuſque ſolidæ figu <lb/>ræ eſt punctum illud intra poſitum, circa quod <lb/>undique partes æqualium momentorum conſi-<lb/>ſtunt. </s> |
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4881 <s xml:id="echoid-s2820" xml:space="preserve">ſi enim per tale centrum ducatur planum <lb/>figuram quomodocunque ſecans ſemper in par- |
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4882 <pb file="0114" n="114" rhead="FED. COMMANDINI"/> |
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4883 tes æqueponderantes ipſam diuidet.</s> |
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4884 <s xml:id="echoid-s2821" xml:space="preserve"/> |
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|
4885 </p> |
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4886 <p> |
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4887 <s xml:id="echoid-s2822" xml:space="preserve">2 Priſmatis, cylindri, & </s> |
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4888 <s xml:id="echoid-s2823" xml:space="preserve">portionis cylindri axem <lb/>appello rectam lineam, quæ oppoſitorum plano-<lb/>rum centra grauitatis coniungit.</s> |
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4889 <s xml:id="echoid-s2824" xml:space="preserve"/> |
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|
4890 </p> |
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4891 <p> |
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4892 <s xml:id="echoid-s2825" xml:space="preserve">3 Pyramidis, coni, & </s> |
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4893 <s xml:id="echoid-s2826" xml:space="preserve">portionis coni axem dico li <lb/>neam, quæ à uertice ad centrum grauitatis baſis <lb/>perducitur.</s> |
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4894 <s xml:id="echoid-s2827" xml:space="preserve"/> |
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|
4895 </p> |
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4896 <p> |
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4897 <s xml:id="echoid-s2828" xml:space="preserve">4 Si pyramis, conus, portio coni, uel conoidis ſe-<lb/>cetur plano baſi æquidiſtante, pars, quæ eſt ad ba-<lb/>ſim, fruſtum pyramidis, coni, portionis coni, uel <lb/>conoidis dicetur; </s> |
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4898 <s xml:id="echoid-s2829" xml:space="preserve">quorum plana æquidiſtantia, <lb/>quæ opponuntur ſimilia ſunt, & </s> |
|
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4899 <s xml:id="echoid-s2830" xml:space="preserve">inæqualia: </s> |
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4900 <s xml:id="echoid-s2831" xml:space="preserve">axes <lb/>uero ſunt axium figurarum partes, quæ in ipſis <lb/>comprehenduntur.</s> |
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4901 <s xml:id="echoid-s2832" xml:space="preserve"/> |
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|
4902 </p> |
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|
4903 </div> |
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|
4904 <div xml:id="echoid-div191" type="section" level="1" n="63"> |
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4905 <head xml:id="echoid-head70" xml:space="preserve">PETITIONES.</head> |
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4906 <p> |
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4907 <s xml:id="echoid-s2833" xml:space="preserve">1 Solidarum figurarum ſimilium centra grauita-<lb/>tis ſimiliter ſunt poſita.</s> |
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4908 <s xml:id="echoid-s2834" xml:space="preserve"/> |
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|
4909 </p> |
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4910 <p> |
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4911 <s xml:id="echoid-s2835" xml:space="preserve">2 Solidis figuris ſimilibus, & </s> |
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4912 <s xml:id="echoid-s2836" xml:space="preserve">æqualibus inter ſe <lb/>aptatis, centra quoque grauitatis ipſarum inter ſe <lb/>aptata erunt.</s> |
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4913 <s xml:id="echoid-s2837" xml:space="preserve"/> |
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|
4914 </p> |
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|
4915 </div> |
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|
4916 <div xml:id="echoid-div192" type="section" level="1" n="64"> |
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4917 <head xml:id="echoid-head71" xml:space="preserve">THEOREMA I. PROPOSITIO I.</head> |
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4918 <p> |
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4919 <s xml:id="echoid-s2838" xml:space="preserve">Omnis figuræ rectilineæ in circulo deſcriptæ, <lb/>quæ æqualibus lateribus, & </s> |
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4920 <s xml:id="echoid-s2839" xml:space="preserve">angulis contine- |
|
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4921 <pb o="2" file="0115" n="115" rhead="DE CENTRO GRAVIT. SOLID."/> |
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4922 tur, centrum grauitatis eſt idem, quod circuli cen <lb/>trum.</s> |
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4923 <s xml:id="echoid-s2840" xml:space="preserve"/> |
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4924 </p> |
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|
4925 <p> |
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|
4926 <s xml:id="echoid-s2841" xml:space="preserve">Sit primo triangulum æquilaterum a b c in circulo de-<lb/>ſcriptum: </s> |
|
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|
4927 <s xml:id="echoid-s2842" xml:space="preserve">& </s> |
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|
4928 <s xml:id="echoid-s2843" xml:space="preserve">diuiſa a c bifariam in d, ducatur b d. </s> |
|
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|
4929 <s xml:id="echoid-s2844" xml:space="preserve">erit in li-<lb/>nea b d centrum grauitatis triãguli a b c, ex tertia decima <lb/>primi libri Archimedis de centro grauitatis planorum. </s> |
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4930 <s xml:id="echoid-s2845" xml:space="preserve">Et <lb/>quoniam linea a b eſt æqualis <lb/> |
|
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|
4931 <anchor type="figure" xlink:label="fig-0115-01a" xlink:href="fig-0115-01"/> |
|
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|
4932 lineæ b c; </s> |
|
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|
4933 <s xml:id="echoid-s2846" xml:space="preserve">& </s> |
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|
4934 <s xml:id="echoid-s2847" xml:space="preserve">a d ipſi d c; </s> |
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4935 <s xml:id="echoid-s2848" xml:space="preserve">eſtq́; <lb/></s> |
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4936 <s xml:id="echoid-s2849" xml:space="preserve">b d utrique communis: </s> |
|
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4937 <s xml:id="echoid-s2850" xml:space="preserve">trian-<lb/>gulum a b d æquale erit trian <lb/> |
|
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|
4938 <anchor type="note" xlink:label="note-0115-01a" xlink:href="note-0115-01"/> |
|
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|
4939 gulo c b d: </s> |
|
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4940 <s xml:id="echoid-s2851" xml:space="preserve">& </s> |
|
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4941 <s xml:id="echoid-s2852" xml:space="preserve">anguli angulis æ-<lb/>quales, qui æqualibus lateri-<lb/>bus ſubtenduntur. </s> |
|
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4942 <s xml:id="echoid-s2853" xml:space="preserve">ergo angu <lb/> |
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4943 <anchor type="note" xlink:label="note-0115-02a" xlink:href="note-0115-02"/> |
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4944 li ad d utriq; </s> |
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4945 <s xml:id="echoid-s2854" xml:space="preserve">recti ſunt. </s> |
|
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4946 <s xml:id="echoid-s2855" xml:space="preserve">quòd <lb/>cum linea b d ſecet a c biſa-<lb/>riam, & </s> |
|
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4947 <s xml:id="echoid-s2856" xml:space="preserve">ad angulos rectos; </s> |
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4948 <s xml:id="echoid-s2857" xml:space="preserve">in <lb/> |
|
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4949 <anchor type="note" xlink:label="note-0115-03a" xlink:href="note-0115-03"/> |
|
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|
4950 ipſa b d eſt centrum circuli. <lb/></s> |
|
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4951 <s xml:id="echoid-s2858" xml:space="preserve">quare in eadem b d linea erit <lb/>centrum grauitatis trianguli, & </s> |
|
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4952 <s xml:id="echoid-s2859" xml:space="preserve">circuli centrum. </s> |
|
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4953 <s xml:id="echoid-s2860" xml:space="preserve">Similiter <lb/>diuiſa a b bifariam in e, & </s> |
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|
4954 <s xml:id="echoid-s2861" xml:space="preserve">ducta c e, oſtendetur in ipſa utrũ <lb/>que centrum contineri. </s> |
|
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4955 <s xml:id="echoid-s2862" xml:space="preserve">ergo ea erunt in puncto, in quo li-<lb/>neæ b d, c e conueniunt. </s> |
|
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|
4956 <s xml:id="echoid-s2863" xml:space="preserve">trianguli igitur a b c centrum gra <lb/>uitatis eſt idem, quod circuli centrum.</s> |
|
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|
4957 <s xml:id="echoid-s2864" xml:space="preserve"/> |
|
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|
4958 </p> |
|
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|
4959 <div xml:id="echoid-div192" type="float" level="2" n="1"> |
|
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|
4960 <figure xlink:label="fig-0115-01" xlink:href="fig-0115-01a"> |
|
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|
4961 <image file="0115-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0115-01"/> |
|
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|
4962 </figure> |
|
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|
4963 <note position="right" xlink:label="note-0115-01" xlink:href="note-0115-01a" xml:space="preserve">8. primi.</note> |
|
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4964 <note position="right" xlink:label="note-0115-02" xlink:href="note-0115-02a" xml:space="preserve">13. primi.</note> |
|
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4965 <note position="right" xlink:label="note-0115-03" xlink:href="note-0115-03a" xml:space="preserve">corol. p@@ <lb/>mæ tertii</note> |
|
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|
4966 </div> |
|
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|
4967 <p> |
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4968 <s xml:id="echoid-s2865" xml:space="preserve">Sit quadratum a b c d in cir-<lb/> |
|
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|
4969 <anchor type="figure" xlink:label="fig-0115-02a" xlink:href="fig-0115-02"/> |
|
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|
4970 culo deſcriptum: </s> |
|
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|
4971 <s xml:id="echoid-s2866" xml:space="preserve">& </s> |
|
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|
4972 <s xml:id="echoid-s2867" xml:space="preserve">ducantur <lb/>a c, b d, quæ conueniant in e. </s> |
|
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|
4973 <s xml:id="echoid-s2868" xml:space="preserve">er-<lb/>go punctum e eſt centrum gra <lb/>uitatis quadrati, ex decima eiuſ <lb/>dem libri Archimedis. </s> |
|
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|
4974 <s xml:id="echoid-s2869" xml:space="preserve">Sed cum <lb/>omnes anguli ad a b c d recti <lb/>ſint; </s> |
|
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|
4975 <s xml:id="echoid-s2870" xml:space="preserve">erit a b c femicirculus: <lb/></s> |
|
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|
4976 <s xml:id="echoid-s2871" xml:space="preserve"> |
|
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|
4977 <anchor type="note" xlink:label="note-0115-04a" xlink:href="note-0115-04"/> |
|
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|
4978 itemq́; </s> |
|
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|
4979 <s xml:id="echoid-s2872" xml:space="preserve">b c d: </s> |
|
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|
4980 <s xml:id="echoid-s2873" xml:space="preserve">& </s> |
|
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|
4981 <s xml:id="echoid-s2874" xml:space="preserve">propterea li-<lb/>neæ a c, b d diametri circuli:</s> |
|
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|
4982 <s xml:id="echoid-s2875" xml:space="preserve"> |
|
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|
4983 <pb file="0116" n="116" rhead="FED. COMMANDINI"/> |
|
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|
4984 quæ quidem in centro conueniunt. </s> |
|
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|
4985 <s xml:id="echoid-s2876" xml:space="preserve">idem igitur eſt centrum <lb/>grauitatis quadrati, & </s> |
|
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|
4986 <s xml:id="echoid-s2877" xml:space="preserve">circuli centrum.</s> |
|
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|
4987 <s xml:id="echoid-s2878" xml:space="preserve"/> |
|
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|
4988 </p> |
|
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|
4989 <div xml:id="echoid-div193" type="float" level="2" n="2"> |
|
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|
4990 <figure xlink:label="fig-0115-02" xlink:href="fig-0115-02a"> |
|
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|
4991 <image file="0115-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0115-02"/> |
|
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|
4992 </figure> |
|
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|
4993 <note position="right" xlink:label="note-0115-04" xlink:href="note-0115-04a" xml:space="preserve">51. tortil.</note> |
|
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|
4994 </div> |
|
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|
4995 <p> |
|
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|
4996 <s xml:id="echoid-s2879" xml:space="preserve">Sit pentagonum æquilaterum, & </s> |
|
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|
4997 <s xml:id="echoid-s2880" xml:space="preserve">æquiangulum in circu-<lb/>lo deſcriptum a b c d e: </s> |
|
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|
4998 <s xml:id="echoid-s2881" xml:space="preserve">& </s> |
|
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|
4999 <s xml:id="echoid-s2882" xml:space="preserve">iun-<lb/> |
|
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|
5000 <anchor type="figure" xlink:label="fig-0116-01a" xlink:href="fig-0116-01"/> |
|
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|
5001 cta b d, bifariamq́; </s> |
|
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|
5002 <s xml:id="echoid-s2883" xml:space="preserve">in ſ diuiſa, <lb/>ducatur c f, & </s> |
|
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|
5003 <s xml:id="echoid-s2884" xml:space="preserve">producatur ad <lb/>circuli circumferentiam in g; <lb/></s> |
|
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|
5004 <s xml:id="echoid-s2885" xml:space="preserve">quæ lineam a e in h ſecet: </s> |
|
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|
5005 <s xml:id="echoid-s2886" xml:space="preserve">de-<lb/>inde iungantur a c, c e. </s> |
|
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|
5006 <s xml:id="echoid-s2887" xml:space="preserve">Eodem <lb/>modo, quo ſupra demonſtra-<lb/>bimus angulum b c f æqualem <lb/>eſſe angulo d c f; </s> |
|
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|
5007 <s xml:id="echoid-s2888" xml:space="preserve">& </s> |
|
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|
5008 <s xml:id="echoid-s2889" xml:space="preserve">angulos <lb/>ad f utroſque rectos: </s> |
|
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|
5009 <s xml:id="echoid-s2890" xml:space="preserve">& </s> |
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|
5010 <s xml:id="echoid-s2891" xml:space="preserve">idcir-<lb/>colineam c f g per circuli cen <lb/>trum tranſire. </s> |
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5011 <s xml:id="echoid-s2892" xml:space="preserve">Quoniam igi-<lb/>tur latera c b, b a, & </s> |
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5012 <s xml:id="echoid-s2893" xml:space="preserve">c d, d e æqualia ſunt; </s> |
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|
5013 <s xml:id="echoid-s2894" xml:space="preserve">& </s> |
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|
5014 <s xml:id="echoid-s2895" xml:space="preserve">æquales anguli <lb/>c b a, c d e: </s> |
|
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|
5015 <s xml:id="echoid-s2896" xml:space="preserve">erit baſis c a baſi c e, & </s> |
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5016 <s xml:id="echoid-s2897" xml:space="preserve">angulus b c a angulo <lb/> |
|
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|
5017 <anchor type="note" xlink:label="note-0116-01a" xlink:href="note-0116-01"/> |
|
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|
5018 d c e æqualis. </s> |
|
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|
5019 <s xml:id="echoid-s2898" xml:space="preserve">ergo & </s> |
|
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|
5020 <s xml:id="echoid-s2899" xml:space="preserve">reliquus a c h, reliquo e c h. </s> |
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5021 <s xml:id="echoid-s2900" xml:space="preserve">eſt au-<lb/>tem c h utrique triangulo a c h, e c h communis. </s> |
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5022 <s xml:id="echoid-s2901" xml:space="preserve">quare <lb/>baſis a h æqualis eſt baſi h e: </s> |
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5023 <s xml:id="echoid-s2902" xml:space="preserve">& </s> |
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5024 <s xml:id="echoid-s2903" xml:space="preserve">anguli, quiad h recti: </s> |
|
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5025 <s xml:id="echoid-s2904" xml:space="preserve">ſuntq́; <lb/></s> |
|
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5026 <s xml:id="echoid-s2905" xml:space="preserve">recti, qui ad f. </s> |
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|
5027 <s xml:id="echoid-s2906" xml:space="preserve">ergo lineæ a e, b d inter ſe ſe æquidiſtant. </s> |
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|
5028 <s xml:id="echoid-s2907" xml:space="preserve"><lb/> |
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|
5029 <anchor type="note" xlink:label="note-0116-02a" xlink:href="note-0116-02"/> |
|
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|
5030 Itaque cum trapezij a b d e latera b d, a e æquidiſtantia à li <lb/>nea fh bifariam diuidantur; </s> |
|
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|
5031 <s xml:id="echoid-s2908" xml:space="preserve">centrum grauitatis ipſius erit <lb/>in linea f h, ex ultima eiuſdem libri Archimedis. </s> |
|
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|
5032 <s xml:id="echoid-s2909" xml:space="preserve">Sed trian-<lb/> |
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|
5033 <anchor type="note" xlink:label="note-0116-03a" xlink:href="note-0116-03"/> |
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|
5034 guli b c d centrum grauitatis eſt in linea c f. </s> |
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5035 <s xml:id="echoid-s2910" xml:space="preserve">ergo in eadem <lb/>linea c h eſt centrum grauitatis trapezij a b d e, & </s> |
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5036 <s xml:id="echoid-s2911" xml:space="preserve">trian-<lb/>guli b c d: </s> |
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|
5037 <s xml:id="echoid-s2912" xml:space="preserve">hoc eſt pentagoni ipſius centrum & </s> |
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5038 <s xml:id="echoid-s2913" xml:space="preserve">centrum <lb/>circuli. </s> |
|
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|
5039 <s xml:id="echoid-s2914" xml:space="preserve">Rurſus ſi iuncta a d, bifariamq́; </s> |
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5040 <s xml:id="echoid-s2915" xml:space="preserve">ſecta in k, duca-<lb/>tur e k l: </s> |
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5041 <s xml:id="echoid-s2916" xml:space="preserve">demonſtrabimus in ipſa utrumque centrum in <lb/>eſſe. </s> |
|
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|
5042 <s xml:id="echoid-s2917" xml:space="preserve">Sequitur ergo, ut punctum, in quo lineæ c g, e l con-<lb/>ueniunt, idem ſit centrum circuli, & </s> |
|
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5043 <s xml:id="echoid-s2918" xml:space="preserve">centrum grauitatis <lb/>pentagoni.</s> |
|
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5044 <s xml:id="echoid-s2919" xml:space="preserve"/> |
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|
5045 </p> |
|
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|
5046 <div xml:id="echoid-div194" type="float" level="2" n="3"> |
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|
5047 <figure xlink:label="fig-0116-01" xlink:href="fig-0116-01a"> |
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|
5048 <image file="0116-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0116-01"/> |
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|
5049 </figure> |
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|
5050 <note position="left" xlink:label="note-0116-01" xlink:href="note-0116-01a" xml:space="preserve">4. Primi.</note> |
|
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|
5051 <note position="left" xlink:label="note-0116-02" xlink:href="note-0116-02a" xml:space="preserve">08. primi.</note> |
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|
5052 <note position="left" xlink:label="note-0116-03" xlink:href="note-0116-03a" xml:space="preserve">13. Archi-<lb/>medis.</note> |
|
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|
5053 </div> |
|
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|
5054 <p> |
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|
5055 <s xml:id="echoid-s2920" xml:space="preserve">Sit hexagonum a b c d e f æquilaterum, & </s> |
|
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|
5056 <s xml:id="echoid-s2921" xml:space="preserve">æquiangulum <lb/>in circulo deſignatum: </s> |
|
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|
5057 <s xml:id="echoid-s2922" xml:space="preserve">iunganturq́; </s> |
|
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|
5058 <s xml:id="echoid-s2923" xml:space="preserve">b d, a c: </s> |
|
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|
5059 <s xml:id="echoid-s2924" xml:space="preserve">& </s> |
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|
5060 <s xml:id="echoid-s2925" xml:space="preserve">bifariam ſe- |
|
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|
5061 <pb o="3" file="0117" n="117" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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|
5062 cta b d in g puncto, ducatur c g; </s> |
|
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|
5063 <s xml:id="echoid-s2926" xml:space="preserve">& </s> |
|
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|
5064 <s xml:id="echoid-s2927" xml:space="preserve">protrahatur ad circuli <lb/>uſque circumferentiam; </s> |
|
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|
5065 <s xml:id="echoid-s2928" xml:space="preserve">quæ ſecet a e in h. </s> |
|
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|
5066 <s xml:id="echoid-s2929" xml:space="preserve">Similiter conclu <lb/>demus c g per centrum circuli tranſire: </s> |
|
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|
5067 <s xml:id="echoid-s2930" xml:space="preserve">& </s> |
|
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|
5068 <s xml:id="echoid-s2931" xml:space="preserve">bifariam ſecare <lb/>lineam a e; </s> |
|
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|
5069 <s xml:id="echoid-s2932" xml:space="preserve">itemq́; </s> |
|
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|
5070 <s xml:id="echoid-s2933" xml:space="preserve">lineas b d, a e inter ſe æquidiſtantes eſſe. <lb/></s> |
|
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|
5071 <s xml:id="echoid-s2934" xml:space="preserve">Cumigitur c g per centrum circuli tranſeat; </s> |
|
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|
5072 <s xml:id="echoid-s2935" xml:space="preserve">& </s> |
|
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|
5073 <s xml:id="echoid-s2936" xml:space="preserve">ad punctũ <lb/>f perueniat neceſſe eſt: </s> |
|
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|
5074 <s xml:id="echoid-s2937" xml:space="preserve">quòd c d e f ſit dimidium circumfe <lb/>rentiæ circuli. </s> |
|
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|
5075 <s xml:id="echoid-s2938" xml:space="preserve">Quare in eadem <lb/> |
|
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|
5076 <anchor type="figure" xlink:label="fig-0117-01a" xlink:href="fig-0117-01"/> |
|
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|
5077 diametro c f erunt centra gra <lb/> |
|
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|
5078 <anchor type="note" xlink:label="note-0117-01a" xlink:href="note-0117-01"/> |
|
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|
5079 uitatis triangulorum b c d, <lb/>a f e, & </s> |
|
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|
5080 <s xml:id="echoid-s2939" xml:space="preserve">quadrilateri a b d e, ex <lb/> |
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5081 <anchor type="note" xlink:label="note-0117-02a" xlink:href="note-0117-02"/> |
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5082 quibus conſtat hexagonum a b <lb/>c d e f. </s> |
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5083 <s xml:id="echoid-s2940" xml:space="preserve">perſpicuum eſt igitur in <lb/>ipſa c f eſſe circuli centrum, & </s> |
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5084 <s xml:id="echoid-s2941" xml:space="preserve"><lb/>centrum grauitatis hexagoni. <lb/></s> |
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5085 <s xml:id="echoid-s2942" xml:space="preserve">Rurſus ducta altera diametro <lb/>a d, eiſdem rationibus oſtende-<lb/>mus in ipſa utrumque cẽtrum <lb/>ineſſe. </s> |
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5086 <s xml:id="echoid-s2943" xml:space="preserve">Centrum ergo grauita-<lb/>tis hexagoni, & </s> |
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5087 <s xml:id="echoid-s2944" xml:space="preserve">centrum circuli idem erit.</s> |
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5088 <s xml:id="echoid-s2945" xml:space="preserve"/> |
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5089 </p> |
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5090 <div xml:id="echoid-div195" type="float" level="2" n="4"> |
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5091 <figure xlink:label="fig-0117-01" xlink:href="fig-0117-01a"> |
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5092 <image file="0117-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0117-01"/> |
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5093 </figure> |
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5094 <note position="right" xlink:label="note-0117-01" xlink:href="note-0117-01a" xml:space="preserve">13. Archi <lb/>medis.</note> |
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5095 <note position="right" xlink:label="note-0117-02" xlink:href="note-0117-02a" xml:space="preserve">9. @iuſdé.</note> |
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5096 </div> |
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5097 <p> |
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5098 <s xml:id="echoid-s2946" xml:space="preserve">Sit heptagonum a b c d e f g æquilaterum atque æquian <lb/>gulum in circulo deſcriptum: <lb/></s> |
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5099 <s xml:id="echoid-s2947" xml:space="preserve"> |
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5100 <anchor type="figure" xlink:label="fig-0117-02a" xlink:href="fig-0117-02"/> |
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5101 & </s> |
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5102 <s xml:id="echoid-s2948" xml:space="preserve">iungantur c e, b f, a g: </s> |
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5103 <s xml:id="echoid-s2949" xml:space="preserve">di-<lb/>uiſa autem c e bifariam in pũ <lb/>cto h: </s> |
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5104 <s xml:id="echoid-s2950" xml:space="preserve">& </s> |
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5105 <s xml:id="echoid-s2951" xml:space="preserve">iuncta d h produca-<lb/>tur in k. </s> |
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5106 <s xml:id="echoid-s2952" xml:space="preserve">non aliter demon-<lb/>ſtrabimus in linea d k eſſe cen <lb/>trum circuli, & </s> |
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5107 <s xml:id="echoid-s2953" xml:space="preserve">centrum gra-<lb/>uitatis trianguli c d e, & </s> |
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5108 <s xml:id="echoid-s2954" xml:space="preserve">tra-<lb/>peziorum b c e f, a b f g, hoc <lb/>eſt centrum totius heptago-<lb/>ni: </s> |
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5109 <s xml:id="echoid-s2955" xml:space="preserve">& </s> |
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5110 <s xml:id="echoid-s2956" xml:space="preserve">rurſus eadem centra in <lb/>alia diametro cl ſimiliter du-<lb/>cta contineri. </s> |
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5111 <s xml:id="echoid-s2957" xml:space="preserve">Quare & </s> |
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5112 <s xml:id="echoid-s2958" xml:space="preserve">centrum grauitatis heptagoni, & </s> |
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5113 <s xml:id="echoid-s2959" xml:space="preserve"><lb/>centrum circuli in idem punctum conucniunt. </s> |
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5114 <s xml:id="echoid-s2960" xml:space="preserve">Eodem mo |
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5115 <pb file="0118" n="118" rhead="FED. COMMANDINI"/> |
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5116 do in reliquis figuris æquilateris, & </s> |
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5117 <s xml:id="echoid-s2961" xml:space="preserve">æquiangulis, quæ in cir-<lb/>culo deſcribuntur, probabimus cẽtrum grauitatis earum, <lb/>& </s> |
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5118 <s xml:id="echoid-s2962" xml:space="preserve">centrum circuli idem eſſe. </s> |
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5119 <s xml:id="echoid-s2963" xml:space="preserve">quod quidem demonſtrare <lb/>oportebat.</s> |
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5120 <s xml:id="echoid-s2964" xml:space="preserve"/> |
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5121 </p> |
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5122 <div xml:id="echoid-div196" type="float" level="2" n="5"> |
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5123 <figure xlink:label="fig-0117-02" xlink:href="fig-0117-02a"> |
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5124 <image file="0117-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0117-02"/> |
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5125 </figure> |
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5126 </div> |
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5127 <p> |
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5128 <s xml:id="echoid-s2965" xml:space="preserve">Ex quibus apparet cuiuslibet figuræ rectilineæ <lb/>in circulo plane deſcriptæ centrum grauitatis idẽ <lb/>eſſe, quod & </s> |
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5129 <s xml:id="echoid-s2966" xml:space="preserve">circuli centrum.</s> |
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5130 <s xml:id="echoid-s2967" xml:space="preserve"/> |
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|
5131 </p> |
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5132 <p> |
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5133 <s xml:id="echoid-s2968" xml:space="preserve">Figuram in circulo plane deſcriptam appella-<lb/> |
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5134 <anchor type="note" xlink:label="note-0118-01a" xlink:href="note-0118-01"/> |
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5135 mus, cuiuſmodi eſt ea, quæ in duodecimo elemen <lb/>torum libro, propoſitione ſecunda deſcribitur. <lb/></s> |
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5136 <s xml:id="echoid-s2969" xml:space="preserve">ex æqualibus enim lateribus, & </s> |
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5137 <s xml:id="echoid-s2970" xml:space="preserve">angulis conſtare <lb/>perſpicuum eſt.</s> |
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5138 <s xml:id="echoid-s2971" xml:space="preserve"/> |
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5139 </p> |
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5140 <div xml:id="echoid-div197" type="float" level="2" n="6"> |
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5141 <note position="left" xlink:label="note-0118-01" xlink:href="note-0118-01a" xml:space="preserve">γνωρ@ μω@</note> |
|
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|
5142 </div> |
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|
5143 </div> |
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|
5144 <div xml:id="echoid-div199" type="section" level="1" n="65"> |
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5145 <head xml:id="echoid-head72" xml:space="preserve">THEOREMA II. PROPOSITIO II.</head> |
|
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|
5146 <p> |
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5147 <s xml:id="echoid-s2972" xml:space="preserve">Omnis figuræ rectilineæ in ellipſi plane deſcri-<lb/>ptæ centrum grauitatis eſt idem, quod ellipſis <lb/>centrum.</s> |
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5148 <s xml:id="echoid-s2973" xml:space="preserve"/> |
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|
5149 </p> |
|
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|
5150 <p> |
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5151 <s xml:id="echoid-s2974" xml:space="preserve">Quo modo figura rectilinea in ellipſi plane deſcribatur, <lb/>docuimus in commentarijs in quintam propoſitionem li-<lb/>bri Archimedis de conoidibus, & </s> |
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5152 <s xml:id="echoid-s2975" xml:space="preserve">ſphæroidibus.</s> |
|
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|
5153 <s xml:id="echoid-s2976" xml:space="preserve"/> |
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|
5154 </p> |
|
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diff
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|
5155 <p> |
|
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|
5156 <s xml:id="echoid-s2977" xml:space="preserve">Sit ellipſis a b c d, cuius maior axis a c, minor b d: </s> |
|
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|
5157 <s xml:id="echoid-s2978" xml:space="preserve">iun-<lb/>ganturq́; </s> |
|
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diff
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|
5158 <s xml:id="echoid-s2979" xml:space="preserve">a b, b c, c d, d a: </s> |
|
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diff
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|
5159 <s xml:id="echoid-s2980" xml:space="preserve">& </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff
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|
5160 <s xml:id="echoid-s2981" xml:space="preserve">bifariam diuidantur in pun-<lb/>ctis e f g h. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5161 <s xml:id="echoid-s2982" xml:space="preserve">à centro autem, quod ſit k ductæ lineæ k e, k f, <lb/>k g, k h uſque ad ſectionem in puncta l m n o protrahan-<lb/>tur: </s> |
|
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|
5162 <s xml:id="echoid-s2983" xml:space="preserve">& </s> |
|
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parents:
diff
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|
5163 <s xml:id="echoid-s2984" xml:space="preserve">iungantur l m, m n, n o, o l, ita ut a c ſecet li-<lb/>neas l o, m n, in z φ punctis, & </s> |
|
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diff
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|
5164 <s xml:id="echoid-s2985" xml:space="preserve">b d ſecet l m, o n in χ ψ. <lb/></s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff
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|
5165 <s xml:id="echoid-s2986" xml:space="preserve">erunt l k, k n linea una, itemq́ue linea unaipſæ m k, k o: </s> |
|
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5166 <s xml:id="echoid-s2987" xml:space="preserve"><lb/>& </s> |
|
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diff
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|
5167 <s xml:id="echoid-s2988" xml:space="preserve">lineæ b a, c d æquidiſtabunt lineæ m o: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5168 <s xml:id="echoid-s2989" xml:space="preserve">& </s> |
|
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parents:
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|
5169 <s xml:id="echoid-s2990" xml:space="preserve">b c, a d ipſi <lb/>l n. </s> |
|
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5170 <s xml:id="echoid-s2991" xml:space="preserve">rurſus l o, m n axi b d æquidiſtabunt: </s> |
|
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|
5171 <s xml:id="echoid-s2992" xml:space="preserve">& </s> |
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|
5172 <s xml:id="echoid-s2993" xml:space="preserve">l m, |
|
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|
5173 <pb o="4" file="0119" n="119" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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|
5174 o n ipſi a c. </s> |
|
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|
5175 <s xml:id="echoid-s2994" xml:space="preserve">Quoniam enim triangulorum a b k, a d k, latus <lb/>b k eſt æquale lateri k d, & </s> |
|
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5176 <s xml:id="echoid-s2995" xml:space="preserve">a k utrique commune; </s> |
|
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|
5177 <s xml:id="echoid-s2996" xml:space="preserve">anguliq́; <lb/></s> |
|
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|
5178 <s xml:id="echoid-s2997" xml:space="preserve">ad k recti baſis a b baſi a d; </s> |
|
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|
5179 <s xml:id="echoid-s2998" xml:space="preserve">& </s> |
|
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|
5180 <s xml:id="echoid-s2999" xml:space="preserve">reliqui anguli reliquis an-<lb/> |
|
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|
5181 <anchor type="note" xlink:label="note-0119-01a" xlink:href="note-0119-01"/> |
|
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|
5182 gulis æquales erunt. </s> |
|
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|
5183 <s xml:id="echoid-s3000" xml:space="preserve">eadem quoqueratione oſtendetur b c <lb/>æqualis c d; </s> |
|
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|
5184 <s xml:id="echoid-s3001" xml:space="preserve">& </s> |
|
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|
5185 <s xml:id="echoid-s3002" xml:space="preserve">a b ipſi <lb/> |
|
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diff
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|
5186 <anchor type="figure" xlink:label="fig-0119-01a" xlink:href="fig-0119-01"/> |
|
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|
5187 b c. </s> |
|
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diff
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|
5188 <s xml:id="echoid-s3003" xml:space="preserve">quare omnes a b, <lb/>b c, c d, d a ſunt æqua-<lb/>les. </s> |
|
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|
5189 <s xml:id="echoid-s3004" xml:space="preserve">& </s> |
|
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diff
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|
5190 <s xml:id="echoid-s3005" xml:space="preserve">quoniam anguli <lb/>ad a æquales ſunt angu <lb/>lis ad c; </s> |
|
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|
5191 <s xml:id="echoid-s3006" xml:space="preserve">erunt anguli b <lb/>a c, a c d coalterni inter <lb/>ſe æquales; </s> |
|
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|
5192 <s xml:id="echoid-s3007" xml:space="preserve">itemq́; </s> |
|
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|
5193 <s xml:id="echoid-s3008" xml:space="preserve">d a c, <lb/>a c b. </s> |
|
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|
5194 <s xml:id="echoid-s3009" xml:space="preserve">ergo c d ipſi b a; <lb/></s> |
|
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|
5195 <s xml:id="echoid-s3010" xml:space="preserve">& </s> |
|
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|
5196 <s xml:id="echoid-s3011" xml:space="preserve">a d ipſi b c æquidi-<lb/>ſtat. </s> |
|
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|
5197 <s xml:id="echoid-s3012" xml:space="preserve">Atuero cum lineæ <lb/>a b, c d inter ſe æquidi-<lb/>ſtantes bifariam ſecen-<lb/>tur in punctis e g; </s> |
|
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|
5198 <s xml:id="echoid-s3013" xml:space="preserve">erit li <lb/>nea l e k g n diameter ſe <lb/>ctionis, & </s> |
|
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|
5199 <s xml:id="echoid-s3014" xml:space="preserve">linea una, ex <lb/>demonſtratis in uigeſi-<lb/>ma octaua ſecundi coni <lb/>corum. </s> |
|
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|
5200 <s xml:id="echoid-s3015" xml:space="preserve">Et eadem ratione linea una m f k h o. </s> |
|
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|
5201 <s xml:id="echoid-s3016" xml:space="preserve">Sunt autẽ a d, <lb/>b c inter ſe ſe æquales, & </s> |
|
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|
5202 <s xml:id="echoid-s3017" xml:space="preserve">æquidiſtantes. </s> |
|
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|
5203 <s xml:id="echoid-s3018" xml:space="preserve">quare & </s> |
|
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|
5204 <s xml:id="echoid-s3019" xml:space="preserve">earum di-<lb/>midiæ a h, b f; </s> |
|
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|
5205 <s xml:id="echoid-s3020" xml:space="preserve">itemq́; </s> |
|
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|
5206 <s xml:id="echoid-s3021" xml:space="preserve">h d, f e; </s> |
|
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|
5207 <s xml:id="echoid-s3022" xml:space="preserve">& </s> |
|
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|
5208 <s xml:id="echoid-s3023" xml:space="preserve">quæ ipſas coniunguntrectæ <lb/> |
|
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|
5209 <anchor type="note" xlink:label="note-0119-02a" xlink:href="note-0119-02"/> |
|
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|
5210 lineæ æquales, & </s> |
|
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|
5211 <s xml:id="echoid-s3024" xml:space="preserve">æquidiſtantes erunt. </s> |
|
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|
5212 <s xml:id="echoid-s3025" xml:space="preserve">æquidiſtãt igitur b a, <lb/>c d diametro m o: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5213 <s xml:id="echoid-s3026" xml:space="preserve">& </s> |
|
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|
5214 <s xml:id="echoid-s3027" xml:space="preserve">pariter a d, b c ipſi l n æquidiſtare o-<lb/>ſtendemus. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5215 <s xml:id="echoid-s3028" xml:space="preserve">Si igitur manẽte diametro a c intelligatur a b c <lb/>portio ellipſis ad portionem a d c moueri, cum primum b <lb/>applicuerit ad d, cõgruet tota portio toti portioni, lineaq́; <lb/></s> |
|
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|
5216 <s xml:id="echoid-s3029" xml:space="preserve">b a lineæ a d; </s> |
|
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|
5217 <s xml:id="echoid-s3030" xml:space="preserve">& </s> |
|
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changeset
|
5218 <s xml:id="echoid-s3031" xml:space="preserve">b c ipſi c d congruet: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
5219 <s xml:id="echoid-s3032" xml:space="preserve">punctum uero e ca-<lb/>det in h; </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5220 <s xml:id="echoid-s3033" xml:space="preserve">f in g: </s> |
|
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|
5221 <s xml:id="echoid-s3034" xml:space="preserve">& </s> |
|
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changeset
|
5222 <s xml:id="echoid-s3035" xml:space="preserve">linea k e in lineam k h: </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5223 <s xml:id="echoid-s3036" xml:space="preserve">& </s> |
|
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|
5224 <s xml:id="echoid-s3037" xml:space="preserve">k f in k g. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
5225 <s xml:id="echoid-s3038" xml:space="preserve">qua <lb/>re & </s> |
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5226 <s xml:id="echoid-s3039" xml:space="preserve">el in h o, et fm in g n. </s> |
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|
5227 <s xml:id="echoid-s3040" xml:space="preserve">Atipſa lz in z o; </s> |
|
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|
5228 <s xml:id="echoid-s3041" xml:space="preserve">et m φ in φ n <lb/>cadet. </s> |
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|
5229 <s xml:id="echoid-s3042" xml:space="preserve">congruet igitur triangulum l k z triangulo o k z: </s> |
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|
5230 <s xml:id="echoid-s3043" xml:space="preserve">et |
|
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diff
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|
5231 <pb file="0120" n="120" rhead="FED. COMMANDINI"/> |
|
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diff
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|
5232 triangulum m k φ triangulo n k φ. </s> |
|
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5233 <s xml:id="echoid-s3044" xml:space="preserve">ergo anguli l z k, o z k, <lb/>m φ k, n φ k æquales ſunt, ac recti. </s> |
|
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5234 <s xml:id="echoid-s3045" xml:space="preserve">quòd cum etiam recti <lb/>ſint, qui ad k; </s> |
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|
5235 <s xml:id="echoid-s3046" xml:space="preserve">æquidiſtabunt lineæ l o, m n axi b d. </s> |
|
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|
5236 <s xml:id="echoid-s3047" xml:space="preserve">& </s> |
|
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diff
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|
5237 <s xml:id="echoid-s3048" xml:space="preserve">ita. <lb/></s> |
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|
5238 <s xml:id="echoid-s3049" xml:space="preserve"> |
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diff
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|
5239 <anchor type="note" xlink:label="note-0120-01a" xlink:href="note-0120-01"/> |
|
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|
5240 demonſtrabuntur l m, o n ipſi a c æquidiſtare. </s> |
|
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|
5241 <s xml:id="echoid-s3050" xml:space="preserve">Rurſus ſi <lb/>iungantur a l, l b, b m, m c, c n, n d, d o, o a: </s> |
|
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5242 <s xml:id="echoid-s3051" xml:space="preserve">& </s> |
|
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diff
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|
5243 <s xml:id="echoid-s3052" xml:space="preserve">bifariam di <lb/>uidantur: </s> |
|
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|
5244 <s xml:id="echoid-s3053" xml:space="preserve">à centro autem k ad diuiſiones ductæ lineæ pro-<lb/>trahantur uſque ad ſectionem in puncta p q r s t u x y: </s> |
|
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|
5245 <s xml:id="echoid-s3054" xml:space="preserve">& </s> |
|
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diff
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|
5246 <s xml:id="echoid-s3055" xml:space="preserve">po <lb/>ſtremo p y, q x, r u, s t, q r, p s, y t, x u coniungantur. </s> |
|
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|
5247 <s xml:id="echoid-s3056" xml:space="preserve">Simili-<lb/>ter oſtendemus lineas <lb/> |
|
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diff
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|
5248 <anchor type="figure" xlink:label="fig-0120-01a" xlink:href="fig-0120-01"/> |
|
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diff
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|
5249 p y, q x, r u, s t axi b d æ-<lb/>quidiſtantes eſſe: </s> |
|
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|
5250 <s xml:id="echoid-s3057" xml:space="preserve">& </s> |
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diff
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|
5251 <s xml:id="echoid-s3058" xml:space="preserve">q r, <lb/>p s, y t, x u æquidiſtan-<lb/>tesipſi a c. </s> |
|
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|
5252 <s xml:id="echoid-s3059" xml:space="preserve">Itaque dico <lb/>harum figurarum in el-<lb/>lipſi deſcriptarum cen-<lb/>trum grauitatis eſſe pũ-<lb/>ctum k, idem quod & </s> |
|
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|
5253 <s xml:id="echoid-s3060" xml:space="preserve">el <lb/>lipſis centrum. </s> |
|
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|
5254 <s xml:id="echoid-s3061" xml:space="preserve">quadri-<lb/>lateri enim a b c d cen-<lb/>trum eſt k, ex decima e-<lb/>iuſdem libri Archime-<lb/>dis, quippe cũ in eo om <lb/>nes diametri cõueniãt. <lb/></s> |
|
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|
5255 <s xml:id="echoid-s3062" xml:space="preserve">Sed in figura alb m c n <lb/> |
|
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|
5256 <anchor type="note" xlink:label="note-0120-02a" xlink:href="note-0120-02"/> |
|
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|
5257 d o, quoniam trianguli <lb/>alb centrum grauitatis <lb/> |
|
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|
5258 <anchor type="note" xlink:label="note-0120-03a" xlink:href="note-0120-03"/> |
|
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|
5259 eſt in linea l e: </s> |
|
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|
5260 <s xml:id="echoid-s3063" xml:space="preserve">trapezijq́; </s> |
|
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|
5261 <s xml:id="echoid-s3064" xml:space="preserve">a b m o centrum in linea e k: </s> |
|
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|
5262 <s xml:id="echoid-s3065" xml:space="preserve">trape <lb/>zij o m c d in k g: </s> |
|
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|
5263 <s xml:id="echoid-s3066" xml:space="preserve">& </s> |
|
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|
5264 <s xml:id="echoid-s3067" xml:space="preserve">trianguli c n d in ipſa g n: </s> |
|
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|
5265 <s xml:id="echoid-s3068" xml:space="preserve">erit magnitu <lb/>dinis ex his omnibus conſtantis, uidelicet totius figuræ cen <lb/>trum grauitatis in linea l n: </s> |
|
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|
5266 <s xml:id="echoid-s3069" xml:space="preserve">& </s> |
|
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|
5267 <s xml:id="echoid-s3070" xml:space="preserve">o b eandem cauſſam in linea <lb/>o m. </s> |
|
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|
5268 <s xml:id="echoid-s3071" xml:space="preserve">eſt enim trianguli a o d centrum in linea o h: </s> |
|
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|
5269 <s xml:id="echoid-s3072" xml:space="preserve">trapezij <lb/>a l n d in h k: </s> |
|
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|
5270 <s xml:id="echoid-s3073" xml:space="preserve">trapezij l b c n in k f: </s> |
|
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|
5271 <s xml:id="echoid-s3074" xml:space="preserve">& </s> |
|
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|
5272 <s xml:id="echoid-s3075" xml:space="preserve">trianguli b m c in fm. <lb/></s> |
|
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|
5273 <s xml:id="echoid-s3076" xml:space="preserve">cum ergo figuræ a l b m c n d o centrum grauitatis ſit in li-<lb/>nea l n, & </s> |
|
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|
5274 <s xml:id="echoid-s3077" xml:space="preserve">in linea o m; </s> |
|
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|
5275 <s xml:id="echoid-s3078" xml:space="preserve">erit centrum ipſius punctum k, in |
|
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|
5276 <pb o="5" file="0121" n="121" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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|
5277 quo ſcilicet ln, om conueniunt. </s> |
|
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|
5278 <s xml:id="echoid-s3079" xml:space="preserve">Poſtremo in figura <lb/>a p l q b r m s c t n u d x o y centrum grauitatis trian <lb/>guli pay, & </s> |
|
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|
5279 <s xml:id="echoid-s3080" xml:space="preserve">trapezii ploy eſtin linea a z: </s> |
|
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|
5280 <s xml:id="echoid-s3081" xml:space="preserve">trapeziorum <lb/>uero lqxo, q b d x centrum eſtin linea z k: </s> |
|
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|
5281 <s xml:id="echoid-s3082" xml:space="preserve">& </s> |
|
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changeset
|
5282 <s xml:id="echoid-s3083" xml:space="preserve">trapeziorũ <lb/>b r u d, r m n u in k φ: </s> |
|
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|
5283 <s xml:id="echoid-s3084" xml:space="preserve">& </s> |
|
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changeset
|
5284 <s xml:id="echoid-s3085" xml:space="preserve">denique trapezii m s t n; </s> |
|
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|
5285 <s xml:id="echoid-s3086" xml:space="preserve">& </s> |
|
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changeset
|
5286 <s xml:id="echoid-s3087" xml:space="preserve">triangu <lb/>li s c t in φ c. </s> |
|
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|
5287 <s xml:id="echoid-s3088" xml:space="preserve">quare magnitudinis ex his compoſitæ centrū <lb/>in linea a c conſiſtit. </s> |
|
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|
5288 <s xml:id="echoid-s3089" xml:space="preserve">Rurſus trianguli q b r, & </s> |
|
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|
5289 <s xml:id="echoid-s3090" xml:space="preserve">trapezii q l <lb/>m r centrum eſt in linea b χ: </s> |
|
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changeset
|
5290 <s xml:id="echoid-s3091" xml:space="preserve">trapeziorum l p s m, p a c s, <lb/>a y t c, y o n t in linea χ φ: </s> |
|
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|
5291 <s xml:id="echoid-s3092" xml:space="preserve">trapeziiq; </s> |
|
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|
5292 <s xml:id="echoid-s3093" xml:space="preserve">o x u n, & </s> |
|
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|
5293 <s xml:id="echoid-s3094" xml:space="preserve">trianguli <lb/>x d u centrum in ψ d. </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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changeset
|
5294 <s xml:id="echoid-s3095" xml:space="preserve">totius ergo magnitudinis centrum <lb/>eſtin linea b d. </s> |
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5295 <s xml:id="echoid-s3096" xml:space="preserve">ex quo ſequitur, centrum grauitatis figuræ <lb/>a p l q b r m s c t n u d x o y eſſe punctū _K_, lineis ſcilicet a c, <lb/>b d commune, quæ omnia demonſtrare oportebat.</s> |
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5296 <s xml:id="echoid-s3097" xml:space="preserve"/> |
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5297 </p> |
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5298 <div xml:id="echoid-div199" type="float" level="2" n="1"> |
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5299 <note position="right" xlink:label="note-0119-01" xlink:href="note-0119-01a" xml:space="preserve">8. primi</note> |
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5300 <figure xlink:label="fig-0119-01" xlink:href="fig-0119-01a"> |
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5301 <image file="0119-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0119-01"/> |
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5302 </figure> |
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5303 <note position="right" xlink:label="note-0119-02" xlink:href="note-0119-02a" xml:space="preserve">33. primit</note> |
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5304 <note position="left" xlink:label="note-0120-01" xlink:href="note-0120-01a" xml:space="preserve">28. primi.</note> |
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5305 <figure xlink:label="fig-0120-01" xlink:href="fig-0120-01a"> |
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5306 <image file="0120-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0120-01"/> |
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5307 </figure> |
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5308 <note position="left" xlink:label="note-0120-02" xlink:href="note-0120-02a" xml:space="preserve">13. Archi <lb/>medis.</note> |
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5309 <note position="left" xlink:label="note-0120-03" xlink:href="note-0120-03a" xml:space="preserve">Vltima.</note> |
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5310 </div> |
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5311 </div> |
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5312 <div xml:id="echoid-div201" type="section" level="1" n="66"> |
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5313 <head xml:id="echoid-head73" xml:space="preserve">THE OREMA III. PROPOSITIO III.</head> |
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5314 <p> |
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5315 <s xml:id="echoid-s3098" xml:space="preserve">Cuiuslibet portio-<lb/> |
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5316 <anchor type="figure" xlink:label="fig-0121-01a" xlink:href="fig-0121-01"/> |
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5317 nis circuli, & </s> |
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5318 <s xml:id="echoid-s3099" xml:space="preserve">ellipſis, <lb/>quæ dimidia non ſit <lb/>maior, centrum graui <lb/>tatis in portionis dia-<lb/>metro conſiſtit.</s> |
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5319 <s xml:id="echoid-s3100" xml:space="preserve"/> |
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5320 </p> |
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5321 <div xml:id="echoid-div201" type="float" level="2" n="1"> |
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5322 <figure xlink:label="fig-0121-01" xlink:href="fig-0121-01a"> |
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5323 <image file="0121-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0121-01"/> |
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5324 </figure> |
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5325 </div> |
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5326 <p> |
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5327 <s xml:id="echoid-s3101" xml:space="preserve">HOC eodem prorſus <lb/>modo demonſtrabitur, <lb/>quo in libro de centro gra <lb/>uitatis planorum ab Ar-<lb/>chimede demonſtratũ eſt, <lb/>in portione cõtenta recta <lb/>linea, & </s> |
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5328 <s xml:id="echoid-s3102" xml:space="preserve">rectanguli coni ſe <lb/>ctione grauitatis cẽtrum <lb/>eſſe in diametro portio-<lb/>nis. </s> |
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5329 <s xml:id="echoid-s3103" xml:space="preserve">Etita demonſtrari po <lb/> |
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5330 <anchor type="handwritten" xlink:label="hd-0121-02a" xlink:href="hd-0121-02"/> |
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5331 <pb file="0122" n="122" rhead="FED. COMMANDINI"/> |
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5332 teſt in portione, quæ recta linea & </s> |
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5333 <s xml:id="echoid-s3104" xml:space="preserve">obtuſianguli coni ſe-<lb/>ctione, ſeu hyperbola continetur.</s> |
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5334 <s xml:id="echoid-s3105" xml:space="preserve"/> |
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5335 </p> |
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5336 <div xml:id="echoid-div202" type="float" level="2" n="2"> |
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5337 <handwritten xlink:label="hd-0121-02" xlink:href="hd-0121-02a"/> |
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5338 </div> |
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5339 </div> |
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5340 <div xml:id="echoid-div204" type="section" level="1" n="67"> |
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5341 <head xml:id="echoid-head74" xml:space="preserve">THE OREMA IIII. PROPOSITIO IIII.</head> |
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5342 <p> |
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5343 <s xml:id="echoid-s3106" xml:space="preserve"><emph style="sc">In</emph> circulo & </s> |
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5344 <s xml:id="echoid-s3107" xml:space="preserve">ellipſiidem eſt figuræ & </s> |
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5345 <s xml:id="echoid-s3108" xml:space="preserve">graui-<lb/>tatis centrum.</s> |
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5346 <s xml:id="echoid-s3109" xml:space="preserve"/> |
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5347 </p> |
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5348 <p> |
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5349 <s xml:id="echoid-s3110" xml:space="preserve">SIT circulus, uel ellipſis, cuius centrum a. </s> |
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5350 <s xml:id="echoid-s3111" xml:space="preserve">Dico a gra-<lb/>uitatis quoque centrum eſſe. </s> |
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5351 <s xml:id="echoid-s3112" xml:space="preserve">Si enim fieri poteſt, ſit b cen-<lb/>trum grauitatis: </s> |
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5352 <s xml:id="echoid-s3113" xml:space="preserve">& </s> |
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5353 <s xml:id="echoid-s3114" xml:space="preserve">iuncta a b extra figuram in c produca <lb/>tur: </s> |
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5354 <s xml:id="echoid-s3115" xml:space="preserve">quam uero proportionem habetlinea c a ad a b, ha-<lb/>beat circulus a ad alium circulum, in quo d; </s> |
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5355 <s xml:id="echoid-s3116" xml:space="preserve">uel ellipſis ad <lb/>aliam ellipſim: </s> |
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5356 <s xml:id="echoid-s3117" xml:space="preserve">& </s> |
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5357 <s xml:id="echoid-s3118" xml:space="preserve">in circulo, uel ellipſi ſigura rectilinea pla-<lb/>ne deſcribatur adeo, ut tandem relinquantur portiones <lb/>quædam minores circulo, uel ellipſid; </s> |
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5358 <s xml:id="echoid-s3119" xml:space="preserve">quæ figura ſit e f g <lb/>h _k_ l m n. </s> |
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5359 <s xml:id="echoid-s3120" xml:space="preserve">Illud uero in circulo fieri poſſe ex duodecimo <lb/>elementorum libro, propoſitione ſecunda manifeſte con-<lb/>ſtat; </s> |
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5360 <s xml:id="echoid-s3121" xml:space="preserve">at in ellipſi nos demonſtra-<lb/> |
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5361 <anchor type="figure" xlink:label="fig-0122-01a" xlink:href="fig-0122-01"/> |
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5362 uinius in commentariis in quin-<lb/>tam propoſitionem Archimedis <lb/>de conoidibus, & </s> |
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5363 <s xml:id="echoid-s3122" xml:space="preserve">ſphæroidibus. <lb/></s> |
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5364 <s xml:id="echoid-s3123" xml:space="preserve">erit igitur a centrum grauitatis <lb/>ipſius figuræ, quod proxime oſtē <lb/>dimus. </s> |
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5365 <s xml:id="echoid-s3124" xml:space="preserve">Itaque quoniam circulus <lb/>a ad circulum d; </s> |
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5366 <s xml:id="echoid-s3125" xml:space="preserve">uel ellipſis a ad <lb/>ellipſim d eandem proportionē <lb/>habet, quam linea c a ad a b: </s> |
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5367 <s xml:id="echoid-s3126" xml:space="preserve"><lb/>portiones uero ſunt minores cir <lb/> |
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5368 <anchor type="note" xlink:label="note-0122-01a" xlink:href="note-0122-01"/> |
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5369 culo uel ellipſi d: </s> |
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5370 <s xml:id="echoid-s3127" xml:space="preserve">habebit circu-<lb/>lus, uel ellipſis ad portiones ma-<lb/>iorem proportionem, quàm c a <lb/> |
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5371 <anchor type="note" xlink:label="note-0122-02a" xlink:href="note-0122-02"/> |
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5372 ad a b: </s> |
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5373 <s xml:id="echoid-s3128" xml:space="preserve">& </s> |
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5374 <s xml:id="echoid-s3129" xml:space="preserve">diuidendo figura recti-<lb/>linea e f g h _k_ l m n ad portiones |
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5375 <pb o="6" file="0123" n="123" rhead="DE CENTRO GRAVIT. SOLID."/> |
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5376 habebit maiorem proportionẽ, <lb/> |
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5377 <anchor type="figure" xlink:label="fig-0123-01a" xlink:href="fig-0123-01"/> |
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5378 quam c b ad b a. </s> |
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5379 <s xml:id="echoid-s3130" xml:space="preserve">fiat o b ad b a, <lb/>ut figura rectilinea ad portio-<lb/>nes. </s> |
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5380 <s xml:id="echoid-s3131" xml:space="preserve">cum igitur à circulo, uel el-<lb/>lipſi, cuius grauitatis centrum <lb/>eſt b, auferatur figura rectilinea <lb/>e f g h k l m n, cuius centrum a; <lb/></s> |
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5381 <s xml:id="echoid-s3132" xml:space="preserve">reliquæ magnitudinis ex portio <lb/> |
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5382 <anchor type="note" xlink:label="note-0123-01a" xlink:href="note-0123-01"/> |
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5383 nibus compoſitæ centrum graui <lb/>tatis erit in linea a b producta, <lb/>& </s> |
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5384 <s xml:id="echoid-s3133" xml:space="preserve">in puncto o, extra figuram po <lb/>ſito. </s> |
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5385 <s xml:id="echoid-s3134" xml:space="preserve">quod quidem fieri nullo mo <lb/>do poſſe perſpicuum eſt. </s> |
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5386 <s xml:id="echoid-s3135" xml:space="preserve">ſequi-<lb/>tur ergo, ut circuli & </s> |
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5387 <s xml:id="echoid-s3136" xml:space="preserve">ellipſis cen <lb/>trum grauitatis ſit punctum a, <lb/>idem quod figuræ centrum.</s> |
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5388 <s xml:id="echoid-s3137" xml:space="preserve"/> |
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5389 </p> |
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5390 <div xml:id="echoid-div204" type="float" level="2" n="1"> |
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5391 <figure xlink:label="fig-0122-01" xlink:href="fig-0122-01a"> |
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5392 <image file="0122-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0122-01"/> |
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5393 </figure> |
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5394 <note position="left" xlink:label="note-0122-01" xlink:href="note-0122-01a" xml:space="preserve">8. quinti.</note> |
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5395 <note position="left" xlink:label="note-0122-02" xlink:href="note-0122-02a" xml:space="preserve">19. quinti <lb/>apud Cã <lb/>panum.</note> |
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5396 <figure xlink:label="fig-0123-01" xlink:href="fig-0123-01a"> |
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5397 <image file="0123-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0123-01"/> |
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5398 </figure> |
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5399 <note position="right" xlink:label="note-0123-01" xlink:href="note-0123-01a" xml:space="preserve">8. Archi-<lb/>medis.</note> |
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5400 </div> |
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5401 </div> |
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5402 <div xml:id="echoid-div206" type="section" level="1" n="68"> |
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5403 <head xml:id="echoid-head75" xml:space="preserve">ALITER.</head> |
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5404 <p> |
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5405 <s xml:id="echoid-s3138" xml:space="preserve">Sit circulus, uel ellipſis a b c d, <lb/>cuius diameter d b, & </s> |
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5406 <s xml:id="echoid-s3139" xml:space="preserve">centrum e: </s> |
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5407 <s xml:id="echoid-s3140" xml:space="preserve">ducaturq; </s> |
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5408 <s xml:id="echoid-s3141" xml:space="preserve">per e recta li <lb/>nea a c, ſecans ipſam d b adrectos angulos. </s> |
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5409 <s xml:id="echoid-s3142" xml:space="preserve">erunt a d c, <lb/>a b c circuli, uel ellipſis dimidiæ portiones. </s> |
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5410 <s xml:id="echoid-s3143" xml:space="preserve">Itaque quo-<lb/>niam por <lb/> |
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5411 <anchor type="figure" xlink:label="fig-0123-02a" xlink:href="fig-0123-02"/> |
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5412 tiõis a d c <lb/>cétrū gra-<lb/>uitatis eſt <lb/>in diame-<lb/>tro d e: </s> |
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5413 <s xml:id="echoid-s3144" xml:space="preserve">& </s> |
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5414 <s xml:id="echoid-s3145" xml:space="preserve"><lb/>portionis <lb/>a b c cen-<lb/>trum eſt ĩ <lb/>ipſa e b: </s> |
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5415 <s xml:id="echoid-s3146" xml:space="preserve">to <lb/>tius circu <lb/>li, uel ellipſis grauitatis centrum eritin diametro d b. <lb/></s> |
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5416 <s xml:id="echoid-s3147" xml:space="preserve">Sit autem portionis a d c cẽtrum grauitatis f: </s> |
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5417 <s xml:id="echoid-s3148" xml:space="preserve">& </s> |
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5418 <s xml:id="echoid-s3149" xml:space="preserve">ſumatur |
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5419 <pb file="0124" n="124" rhead="FED. COMMANDINI"/> |
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5420 in linea e b punctũ g, it aut ſit g e æqualis e f. </s> |
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5421 <s xml:id="echoid-s3150" xml:space="preserve">erit g por-<lb/>tionis a b c centrum. </s> |
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5422 <s xml:id="echoid-s3151" xml:space="preserve">nam ſi hæ portiones, quæ æquales <lb/>& </s> |
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5423 <s xml:id="echoid-s3152" xml:space="preserve">ſimiles ſunt, inter ſe ſe aptentur, ita ut b e cadat in d e, <lb/>& </s> |
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5424 <s xml:id="echoid-s3153" xml:space="preserve">punctum b in d cadet, & </s> |
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5425 <s xml:id="echoid-s3154" xml:space="preserve">g in f: </s> |
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5426 <s xml:id="echoid-s3155" xml:space="preserve">figuris autem æquali-<lb/>bus, & </s> |
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5427 <s xml:id="echoid-s3156" xml:space="preserve">ſimilibus inter ſe aptatis, centra quoque grauitatis <lb/>ipſarum inter ſe aptata erunt, ex quinta petitione Archi-<lb/>medis in libro de centro grauitatis planorum. </s> |
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5428 <s xml:id="echoid-s3157" xml:space="preserve">Quare cum <lb/>portionis a d c centrum grauitatis ſit ſ: </s> |
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5429 <s xml:id="echoid-s3158" xml:space="preserve">& </s> |
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5430 <s xml:id="echoid-s3159" xml:space="preserve">portionis <lb/>a b c centrum g: </s> |
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5431 <s xml:id="echoid-s3160" xml:space="preserve">magnitudinis; </s> |
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5432 <s xml:id="echoid-s3161" xml:space="preserve">quæ ex utriſque efficitur: <lb/></s> |
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5433 <s xml:id="echoid-s3162" xml:space="preserve">hoc eſt circuli uel ellipſis grauitatis centrum in medio li-<lb/>neæ f g, quod eſt e, conſiſtet, ex quarta propoſitione eiuſ-<lb/>dem libri Archimedis. </s> |
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5434 <s xml:id="echoid-s3163" xml:space="preserve">ergo circuli, uel ellipſis centrum <lb/>grauitatis eſt idem, quod figuræ centrum. </s> |
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5435 <s xml:id="echoid-s3164" xml:space="preserve">atque illud eſt, <lb/>quod demonſtrare oportebat.</s> |
|
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5436 <s xml:id="echoid-s3165" xml:space="preserve"/> |
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|
5437 </p> |
|
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5438 <div xml:id="echoid-div206" type="float" level="2" n="1"> |
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5439 <figure xlink:label="fig-0123-02" xlink:href="fig-0123-02a"> |
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5440 <image file="0123-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0123-02"/> |
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5441 </figure> |
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5442 </div> |
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5443 <p> |
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5444 <s xml:id="echoid-s3166" xml:space="preserve">Ex quibus ſequitur portionis circuli, uel ellip-<lb/>ſis, quæ dimidia maior ſit, centrum grauitatis in <lb/>diametro quoque ipſius conſiſtere.</s> |
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5445 <s xml:id="echoid-s3167" xml:space="preserve"/> |
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5446 </p> |
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5447 <figure> |
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5448 <image file="0124-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0124-01"/> |
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5449 </figure> |
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5450 <p> |
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5451 <s xml:id="echoid-s3168" xml:space="preserve">Sit enim maior portio a b c, cu_i_us diameter b d, & </s> |
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5452 <s xml:id="echoid-s3169" xml:space="preserve">com-<lb/>pleatur circulus, uel ellipſis, ut portio reliqua ſit a e c, dia |
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5453 <pb o="7" file="0125" n="125" rhead="DE CENTRO GRAVIT. SOLID."/> |
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5454 metrum habens e d. </s> |
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5455 <s xml:id="echoid-s3170" xml:space="preserve">Quoniam igitur circuli uel ellipſis <lb/>a e c b grauitatis centrum eſt in diametro b e, & </s> |
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5456 <s xml:id="echoid-s3171" xml:space="preserve">portio-<lb/>nis a e c centrum in linea e d: </s> |
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5457 <s xml:id="echoid-s3172" xml:space="preserve">reliquæ portionis, uidelicet <lb/>a b c centrum grauitatis in ipſa b d conſiſtat neceſſe eſt, ex <lb/>octaua propoſitione eiuſdem.</s> |
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5458 <s xml:id="echoid-s3173" xml:space="preserve"/> |
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5459 </p> |
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5460 </div> |
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5461 <div xml:id="echoid-div208" type="section" level="1" n="69"> |
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5462 <head xml:id="echoid-head76" xml:space="preserve">THEOREMA V. PROPOSITIO V.</head> |
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5463 <p> |
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5464 <s xml:id="echoid-s3174" xml:space="preserve">SI priſma ſecetur plano oppoſitis planis æqui <lb/>diſtante, ſectio erit figura æqualis & </s> |
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5465 <s xml:id="echoid-s3175" xml:space="preserve">ſimilis ei, <lb/>quæ eſt oppoſitorum planorum, centrum graui <lb/>tatis in axe habens.</s> |
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5466 <s xml:id="echoid-s3176" xml:space="preserve"/> |
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5467 </p> |
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5468 <p> |
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5469 <s xml:id="echoid-s3177" xml:space="preserve">Sit priſma, in quo plana oppoſita ſint triangula a b c, <lb/>d e f; </s> |
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5470 <s xml:id="echoid-s3178" xml:space="preserve">axis g h: </s> |
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5471 <s xml:id="echoid-s3179" xml:space="preserve">& </s> |
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5472 <s xml:id="echoid-s3180" xml:space="preserve">ſecetur plano iam dictis planis æquidiſtã <lb/>te; </s> |
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5473 <s xml:id="echoid-s3181" xml:space="preserve">quod faciat ſectionem <emph style="sc">K</emph> l m; </s> |
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5474 <s xml:id="echoid-s3182" xml:space="preserve">& </s> |
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5475 <s xml:id="echoid-s3183" xml:space="preserve">axi in pũcto n occurrat. <lb/></s> |
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5476 <s xml:id="echoid-s3184" xml:space="preserve">Dico _k_ l m triangulum æquale eſſe, & </s> |
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5477 <s xml:id="echoid-s3185" xml:space="preserve">ſimile triangulis a b c <lb/>d e f; </s> |
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5478 <s xml:id="echoid-s3186" xml:space="preserve">atque eius grauitatis centrum eſſe punctum n. </s> |
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5479 <s xml:id="echoid-s3187" xml:space="preserve">Quo-<lb/>niam enim plana a b c <lb/> |
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5480 <anchor type="figure" xlink:label="fig-0125-01a" xlink:href="fig-0125-01"/> |
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5481 K l m æquidiſtantia ſecã <lb/> |
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5482 <anchor type="note" xlink:label="note-0125-01a" xlink:href="note-0125-01"/> |
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5483 tur a plano a e; </s> |
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5484 <s xml:id="echoid-s3188" xml:space="preserve">rectæ li-<lb/>neæ a b, K l, quæ ſunt ip <lb/>ſorum cõmunes ſectio-<lb/>nes inter ſe ſe æquidi-<lb/>ſtant. </s> |
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5485 <s xml:id="echoid-s3189" xml:space="preserve">Sed æquidiſtant <lb/>a d, b e; </s> |
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5486 <s xml:id="echoid-s3190" xml:space="preserve">cum a e ſit para <lb/>lelogrammum, ex priſ-<lb/>matis diffinitione. </s> |
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5487 <s xml:id="echoid-s3191" xml:space="preserve">ergo <lb/>& </s> |
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5488 <s xml:id="echoid-s3192" xml:space="preserve">al parallelogrammũ <lb/>erit; </s> |
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5489 <s xml:id="echoid-s3193" xml:space="preserve">& </s> |
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5490 <s xml:id="echoid-s3194" xml:space="preserve">propterea linea <lb/> |
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5491 <anchor type="note" xlink:label="note-0125-02a" xlink:href="note-0125-02"/> |
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5492 _k_l, ipſi a b æqualis. </s> |
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5493 <s xml:id="echoid-s3195" xml:space="preserve">Si-<lb/>militer demonſtrabitur <lb/>l m æquidiſtans, & </s> |
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5494 <s xml:id="echoid-s3196" xml:space="preserve">æqua <lb/>lis b c; </s> |
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5495 <s xml:id="echoid-s3197" xml:space="preserve">& </s> |
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5496 <s xml:id="echoid-s3198" xml:space="preserve">m <emph style="sc">K</emph> ipſi c a.</s> |
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5497 <s xml:id="echoid-s3199" xml:space="preserve"/> |
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5498 </p> |
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5499 <div xml:id="echoid-div208" type="float" level="2" n="1"> |
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5500 <figure xlink:label="fig-0125-01" xlink:href="fig-0125-01a"> |
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5501 <image file="0125-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0125-01"/> |
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5502 </figure> |
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5503 <note position="right" xlink:label="note-0125-01" xlink:href="note-0125-01a" xml:space="preserve">16. unde-<lb/>cimi.</note> |
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5504 <note position="right" xlink:label="note-0125-02" xlink:href="note-0125-02a" xml:space="preserve">34. prim@</note> |
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5505 </div> |
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5506 <pb file="0126" n="126" rhead="FED. COMMANDINI"/> |
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5507 <p> |
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5508 <s xml:id="echoid-s3200" xml:space="preserve">Itaque quoniam duæ lineæ K l, l m ſe ſe tangentes, duabus <lb/>lineis ſe ſe tangentibus a b, b c æquidiſtant; </s> |
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5509 <s xml:id="echoid-s3201" xml:space="preserve">nec ſunt in eo-<lb/>dem plano: </s> |
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5510 <s xml:id="echoid-s3202" xml:space="preserve">angulus <emph style="sc">K</emph> l m æqualis eſt angulo a b c: </s> |
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5511 <s xml:id="echoid-s3203" xml:space="preserve">& </s> |
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|
5512 <s xml:id="echoid-s3204" xml:space="preserve">ita an <lb/> |
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|
5513 <anchor type="note" xlink:label="note-0126-01a" xlink:href="note-0126-01"/> |
|
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|
5514 gulus l m <emph style="sc">K</emph>, angulo b c a, & </s> |
|
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|
5515 <s xml:id="echoid-s3205" xml:space="preserve">m <emph style="sc">K</emph> lipſi c a b æqualis prob abi <lb/>tur. </s> |
|
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|
5516 <s xml:id="echoid-s3206" xml:space="preserve">triangulum ergo <emph style="sc">K</emph> l m eſt æquale, & </s> |
|
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|
5517 <s xml:id="echoid-s3207" xml:space="preserve">ſimile triang ulo <lb/>a b c. </s> |
|
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|
5518 <s xml:id="echoid-s3208" xml:space="preserve">quare & </s> |
|
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|
5519 <s xml:id="echoid-s3209" xml:space="preserve">triangulo d e f. </s> |
|
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5520 <s xml:id="echoid-s3210" xml:space="preserve">Ducatur linea c g o, & </s> |
|
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5521 <s xml:id="echoid-s3211" xml:space="preserve">per ip <lb/>ſam, & </s> |
|
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|
5522 <s xml:id="echoid-s3212" xml:space="preserve">per c f ducatur planum ſecans priſma, cuius & </s> |
|
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|
5523 <s xml:id="echoid-s3213" xml:space="preserve">paral <lb/>lelogrammi a e communis ſectio ſit o p q. </s> |
|
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|
5524 <s xml:id="echoid-s3214" xml:space="preserve">tranſibit linea <lb/>f q per h, & </s> |
|
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|
5525 <s xml:id="echoid-s3215" xml:space="preserve">m p per n. </s> |
|
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|
5526 <s xml:id="echoid-s3216" xml:space="preserve">nam cum plana æquidiſtantia ſecen <lb/>tur à plano c q, communes eorum ſectiones c g o, m p, f q <lb/>ſibi ipſis æquidiſtabunt. </s> |
|
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|
5527 <s xml:id="echoid-s3217" xml:space="preserve">Sed & </s> |
|
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diff
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|
5528 <s xml:id="echoid-s3218" xml:space="preserve">æquidiſtant a b, <emph style="sc">K</emph> l, d e. </s> |
|
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|
5529 <s xml:id="echoid-s3219" xml:space="preserve">an-<lb/>guli ergo a o c, <emph style="sc">K</emph> p m, d q f inter ſe æquales ſunt: </s> |
|
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|
5530 <s xml:id="echoid-s3220" xml:space="preserve">& </s> |
|
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|
5531 <s xml:id="echoid-s3221" xml:space="preserve">ſunt <lb/> |
|
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|
5532 <anchor type="note" xlink:label="note-0126-02a" xlink:href="note-0126-02"/> |
|
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|
5533 æquales qui ad puncta a k d conſtituuntur. </s> |
|
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|
5534 <s xml:id="echoid-s3222" xml:space="preserve">quare & </s> |
|
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|
5535 <s xml:id="echoid-s3223" xml:space="preserve">reliqui <lb/>reliquis æquales; </s> |
|
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|
5536 <s xml:id="echoid-s3224" xml:space="preserve">& </s> |
|
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|
5537 <s xml:id="echoid-s3225" xml:space="preserve">triangula a c o, _K_ m p, d f q inter ſe ſimi <lb/>lia erunt. </s> |
|
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|
5538 <s xml:id="echoid-s3226" xml:space="preserve">Vtigitur ca ad a o, ita fd ad d q: </s> |
|
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|
5539 <s xml:id="echoid-s3227" xml:space="preserve">& </s> |
|
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|
5540 <s xml:id="echoid-s3228" xml:space="preserve">permutando <lb/> |
|
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|
5541 <anchor type="note" xlink:label="note-0126-03a" xlink:href="note-0126-03"/> |
|
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|
5542 ut c a ad fd, ita a o ad d q. </s> |
|
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|
5543 <s xml:id="echoid-s3229" xml:space="preserve">eſt autem c a æqualis fd. </s> |
|
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|
5544 <s xml:id="echoid-s3230" xml:space="preserve">ergo & </s> |
|
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|
5545 <s xml:id="echoid-s3231" xml:space="preserve"><lb/>a o ipſi d q. </s> |
|
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|
5546 <s xml:id="echoid-s3232" xml:space="preserve">eadem quoque ratione & </s> |
|
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|
5547 <s xml:id="echoid-s3233" xml:space="preserve">a o ipſi _K_ p æqualis <lb/>demonſtrabitur. </s> |
|
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|
5548 <s xml:id="echoid-s3234" xml:space="preserve">Itaque ſi triangula, a b c, d e f æqualia & </s> |
|
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5549 <s xml:id="echoid-s3235" xml:space="preserve"><lb/>ſimilia inter ſe aptétur, <lb/> |
|
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|
5550 <anchor type="figure" xlink:label="fig-0126-01a" xlink:href="fig-0126-01"/> |
|
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|
5551 cadet linea f q in lineam <lb/>c g o. </s> |
|
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|
5552 <s xml:id="echoid-s3236" xml:space="preserve">Sed & </s> |
|
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|
5553 <s xml:id="echoid-s3237" xml:space="preserve">centrũ gra <lb/> |
|
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|
5554 <anchor type="note" xlink:label="note-0126-04a" xlink:href="note-0126-04"/> |
|
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|
5555 uitatis h in g centrũ ca-<lb/>det. </s> |
|
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|
5556 <s xml:id="echoid-s3238" xml:space="preserve">trãſibit igitur linea <lb/>f q per h: </s> |
|
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|
5557 <s xml:id="echoid-s3239" xml:space="preserve">& </s> |
|
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|
5558 <s xml:id="echoid-s3240" xml:space="preserve">planum per <lb/>c o & </s> |
|
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|
5559 <s xml:id="echoid-s3241" xml:space="preserve">c f ductũ per axẽ <lb/>g h ducetur: </s> |
|
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|
5560 <s xml:id="echoid-s3242" xml:space="preserve">idcircoq; </s> |
|
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|
5561 <s xml:id="echoid-s3243" xml:space="preserve">li <lb/>neam m p etiã per n trã <lb/>ſire neceſſe erit. </s> |
|
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|
5562 <s xml:id="echoid-s3244" xml:space="preserve">Quo-<lb/>niam ergo ſh, c g æqua-<lb/>les ſunt, & </s> |
|
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|
5563 <s xml:id="echoid-s3245" xml:space="preserve">æquidiſtãtes: <lb/></s> |
|
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|
5564 <s xml:id="echoid-s3246" xml:space="preserve">itemq; </s> |
|
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|
5565 <s xml:id="echoid-s3247" xml:space="preserve">h q, g o; </s> |
|
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|
5566 <s xml:id="echoid-s3248" xml:space="preserve">rectæ li-<lb/>neæ, quæ ipſas cónectũt <lb/>c m f, g n h, o p q æqua-<lb/>les & </s> |
|
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|
5567 <s xml:id="echoid-s3249" xml:space="preserve">æquidiſtãtes erũt.</s> |
|
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|
5568 <s xml:id="echoid-s3250" xml:space="preserve"> |
|
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|
5569 <pb o="8" file="0127" n="127" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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changeset
|
5570 æquidiſtant autem c g o, m n p. </s> |
|
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|
5571 <s xml:id="echoid-s3251" xml:space="preserve">ergo parallelogrãma ſunt <lb/>o n, g m, & </s> |
|
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moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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|
5572 <s xml:id="echoid-s3252" xml:space="preserve">linea m n æqualis c g; </s> |
|
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|
5573 <s xml:id="echoid-s3253" xml:space="preserve">& </s> |
|
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|
5574 <s xml:id="echoid-s3254" xml:space="preserve">n p ipſi g o. </s> |
|
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|
5575 <s xml:id="echoid-s3255" xml:space="preserve">aptatis igi-<lb/>tur <emph style="sc">K</emph> l m, a b c triãgulis, quæ æqualia & </s> |
|
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|
5576 <s xml:id="echoid-s3256" xml:space="preserve">ſimilia sũt; </s> |
|
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|
5577 <s xml:id="echoid-s3257" xml:space="preserve">linea m p <lb/>in c o, & </s> |
|
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|
5578 <s xml:id="echoid-s3258" xml:space="preserve">punctum n in g cadet. </s> |
|
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|
5579 <s xml:id="echoid-s3259" xml:space="preserve">Quòd cũ g ſit centrum gra-<lb/>uitatis trianguli a b c, & </s> |
|
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|
5580 <s xml:id="echoid-s3260" xml:space="preserve">n trianguli <emph style="sc">K</emph> l m grauitatis cen-<lb/>trum erit id, quod demonſtrandum relinquebatur. </s> |
|
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|
5581 <s xml:id="echoid-s3261" xml:space="preserve">Simili <lb/>ratione idem contingere demonſtrabimus in aliis priſma-<lb/>tibus, ſiue quadrilatera, ſiue plurilatera habeant plana, <lb/>quæ opponuntur.</s> |
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5582 <s xml:id="echoid-s3262" xml:space="preserve"/> |
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5583 </p> |
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5584 <div xml:id="echoid-div209" type="float" level="2" n="2"> |
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5585 <note position="left" xlink:label="note-0126-01" xlink:href="note-0126-01a" xml:space="preserve">10. unde <lb/>cimi</note> |
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5586 <note position="left" xlink:label="note-0126-02" xlink:href="note-0126-02a" xml:space="preserve">10. unde-<lb/>cimi</note> |
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5587 <note position="left" xlink:label="note-0126-03" xlink:href="note-0126-03a" xml:space="preserve">4. ſexti</note> |
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5588 <figure xlink:label="fig-0126-01" xlink:href="fig-0126-01a"> |
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5589 <image file="0126-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0126-01"/> |
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5590 </figure> |
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5591 <note position="left" xlink:label="note-0126-04" xlink:href="note-0126-04a" xml:space="preserve">per 5. pe-<lb/>titionem <lb/>Archime <lb/>dis.</note> |
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5592 </div> |
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5593 </div> |
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5594 <div xml:id="echoid-div211" type="section" level="1" n="70"> |
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5595 <head xml:id="echoid-head77" xml:space="preserve">COROLLARIVM.</head> |
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5596 <p> |
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5597 <s xml:id="echoid-s3263" xml:space="preserve">Exiam demonſtratis perſpicue apparet, cuius <lb/>Iibet priſmatis axem, parallelogrammorum lat eri <lb/>bus, quæ ab oppoſitis planis ducũtur æquidiſtare.</s> |
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5598 <s xml:id="echoid-s3264" xml:space="preserve"/> |
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5599 </p> |
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5600 </div> |
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5601 <div xml:id="echoid-div212" type="section" level="1" n="71"> |
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5602 <head xml:id="echoid-head78" xml:space="preserve">THEOREMA VI. PROPOSITIO VI.</head> |
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5603 <p> |
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5604 <s xml:id="echoid-s3265" xml:space="preserve">Cuiuslibet priſmatis centrum grauitatis eſt in <lb/>plano, quod oppoſitis planis æquidiſtans, reli-<lb/>quorum planorum latera bifariam diuidit.</s> |
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5605 <s xml:id="echoid-s3266" xml:space="preserve"/> |
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5606 </p> |
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5607 <p> |
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5608 <s xml:id="echoid-s3267" xml:space="preserve">Sit priſma, in quo plana, quæ opponuntur ſint trian-<lb/>gula a c e, b d f: </s> |
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5609 <s xml:id="echoid-s3268" xml:space="preserve">& </s> |
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5610 <s xml:id="echoid-s3269" xml:space="preserve">parallelogrammorum latera a b, c d, <lb/>e f bifariam diuidãtur in punctis g h _K_: </s> |
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5611 <s xml:id="echoid-s3270" xml:space="preserve">per diuiſiones au-<lb/>tem planum ducatur; </s> |
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5612 <s xml:id="echoid-s3271" xml:space="preserve">cuius ſectio figura g h _K_. </s> |
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5613 <s xml:id="echoid-s3272" xml:space="preserve">eritlinea <lb/> |
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5614 <anchor type="note" xlink:label="note-0127-01a" xlink:href="note-0127-01"/> |
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5615 g h æquidiſtans lineis a c, b d & </s> |
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5616 <s xml:id="echoid-s3273" xml:space="preserve">h k ipſis c e, d f. </s> |
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5617 <s xml:id="echoid-s3274" xml:space="preserve">quare ex <lb/>decimaquinta undecimi elementorum, planum illud pla <lb/>nis a c e, b d f æquidiſtabit, & </s> |
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5618 <s xml:id="echoid-s3275" xml:space="preserve">ſaciet ſectionem figu-<lb/> |
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5619 <anchor type="note" xlink:label="note-0127-02a" xlink:href="note-0127-02"/> |
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5620 ram ipſis æqualem, & </s> |
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5621 <s xml:id="echoid-s3276" xml:space="preserve">ſimilem, ut proxime demonſtra-<lb/>uimus. </s> |
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5622 <s xml:id="echoid-s3277" xml:space="preserve">Dico centrum grauitatis priſmatis eſſe in plano <lb/>g h <emph style="sc">K</emph>. </s> |
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5623 <s xml:id="echoid-s3278" xml:space="preserve">Si enim fieri poteſt, ſit eius centrum l: </s> |
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5624 <s xml:id="echoid-s3279" xml:space="preserve">& </s> |
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5625 <s xml:id="echoid-s3280" xml:space="preserve">ducatur <lb/>l m uſque ad planum g h <emph style="sc">K</emph>, quæ ipſi a b æquidiſtet.</s> |
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5626 <s xml:id="echoid-s3281" xml:space="preserve"> |
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5627 <pb file="0128" n="128" rhead="FED. COMMANDINI"/> |
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5628 ergo linea a g continenter in duas partes æquales diui-<lb/> |
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5629 <anchor type="note" xlink:label="note-0128-01a" xlink:href="note-0128-01"/> |
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5630 ſa, relinquetur tãdem pars aliqua n g, quæ minor eritl m. <lb/></s> |
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5631 <s xml:id="echoid-s3282" xml:space="preserve">Vtraque uero linearum a g, g b diuidatur in partes æqua-<lb/>les ipſi n g: </s> |
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5632 <s xml:id="echoid-s3283" xml:space="preserve">& </s> |
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5633 <s xml:id="echoid-s3284" xml:space="preserve">per puncta diuiſionum plana oppoſitis pla-<lb/> |
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5634 <anchor type="note" xlink:label="note-0128-02a" xlink:href="note-0128-02"/> |
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5635 nis æquidiſtantia ducantur. </s> |
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5636 <s xml:id="echoid-s3285" xml:space="preserve">erunt ſectiones figuræ æqua-<lb/>les, ac ſimiles ipſis a c e, b d f: </s> |
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5637 <s xml:id="echoid-s3286" xml:space="preserve">& </s> |
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5638 <s xml:id="echoid-s3287" xml:space="preserve">totum priſma diuiſum erit <lb/>in priſmata æqualia, & </s> |
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5639 <s xml:id="echoid-s3288" xml:space="preserve">ſimilia: </s> |
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5640 <s xml:id="echoid-s3289" xml:space="preserve">quæ cum inter ſe congruãt; <lb/></s> |
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5641 <s xml:id="echoid-s3290" xml:space="preserve">& </s> |
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5642 <s xml:id="echoid-s3291" xml:space="preserve">grauitatis centra ſibi ipſis congruentia, reſpondentiaq; </s> |
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5643 <s xml:id="echoid-s3292" xml:space="preserve"><lb/>habebunt. </s> |
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5644 <s xml:id="echoid-s3293" xml:space="preserve">Itaq: </s> |
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5645 <s xml:id="echoid-s3294" xml:space="preserve"><lb/> |
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5646 <anchor type="figure" xlink:label="fig-0128-01a" xlink:href="fig-0128-01"/> |
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5647 ſunt magnitudi-<lb/>nes quædã æqua-<lb/>les ipſi n h, & </s> |
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5648 <s xml:id="echoid-s3295" xml:space="preserve">nu-<lb/>mero pares, qua-<lb/>rum centra gra-<lb/>uitatis in eadẽ re <lb/>cta linea conſti-<lb/>tuuntur: </s> |
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5649 <s xml:id="echoid-s3296" xml:space="preserve">duæ ue-<lb/>ro mediæ æqua-<lb/>les ſunt: </s> |
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5650 <s xml:id="echoid-s3297" xml:space="preserve">& </s> |
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5651 <s xml:id="echoid-s3298" xml:space="preserve">quæ ex <lb/>utraque parte i-<lb/>pſarum ſimili --<lb/>ter æquales: </s> |
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5652 <s xml:id="echoid-s3299" xml:space="preserve">& </s> |
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5653 <s xml:id="echoid-s3300" xml:space="preserve">æ-<lb/>quales rectæ li-<lb/>neæ, quæ inter <lb/>grauitatis centra <lb/>interiiciuntur. <lb/></s> |
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5654 <s xml:id="echoid-s3301" xml:space="preserve">quare ex corolla-<lb/>rio quintæ pro-<lb/>poſitionis primi <lb/>libri Archimedis <lb/>de centro graui-<lb/>tatis planorum; </s> |
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5655 <s xml:id="echoid-s3302" xml:space="preserve">magnitudinis ex his omnibus compoſitæ <lb/>centrum grauitatis eſt in medio lineæ, quæ magnitudi-<lb/>num mediarum centra coniungit. </s> |
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5656 <s xml:id="echoid-s3303" xml:space="preserve">at qui non ita res ha- |
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5657 <pb o="9" file="0129" n="129" rhead="DE CENTRO GRAVIT. SOLID."/> |
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5658 bet, ſi quidem 1 extra medias magnitudines poſitum eſt. <lb/></s> |
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5659 <s xml:id="echoid-s3304" xml:space="preserve">Conſtatigitur centrum grauitatis priſmatis eſſe in plano <lb/> |
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5660 <anchor type="figure" xlink:label="fig-0129-01a" xlink:href="fig-0129-01"/> |
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5661 g h k, quod nos demonſtrandum propoſuimus. </s> |
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5662 <s xml:id="echoid-s3305" xml:space="preserve">At ſi op-<lb/>poſita plana in priſmate ſint quadrilatera, uel plurilatera, <lb/>eadem erit in omnibus demonſtratio.</s> |
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5663 <s xml:id="echoid-s3306" xml:space="preserve"/> |
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5664 </p> |
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5665 <div xml:id="echoid-div212" type="float" level="2" n="1"> |
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5666 <note position="right" xlink:label="note-0127-01" xlink:href="note-0127-01a" xml:space="preserve">33. primi</note> |
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5667 <note position="right" xlink:label="note-0127-02" xlink:href="note-0127-02a" xml:space="preserve">5. huius</note> |
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5668 <note position="left" xlink:label="note-0128-01" xlink:href="note-0128-01a" xml:space="preserve">1. decimi</note> |
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5669 <note position="left" xlink:label="note-0128-02" xlink:href="note-0128-02a" xml:space="preserve">5 huius</note> |
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5670 <figure xlink:label="fig-0128-01" xlink:href="fig-0128-01a"> |
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5671 <image file="0128-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0128-01"/> |
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5672 </figure> |
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5673 <figure xlink:label="fig-0129-01" xlink:href="fig-0129-01a"> |
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5674 <image file="0129-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0129-01"/> |
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5675 </figure> |
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5676 </div> |
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5677 </div> |
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5678 <div xml:id="echoid-div214" type="section" level="1" n="72"> |
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5679 <head xml:id="echoid-head79" xml:space="preserve">THE OREMA VII. PROPOSITIO VII.</head> |
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5680 <p> |
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5681 <s xml:id="echoid-s3307" xml:space="preserve">Cuiuslibet cylindri, & </s> |
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5682 <s xml:id="echoid-s3308" xml:space="preserve">cuiuslibet cylindri por <lb/>tionis centrum grauitatis eſt in plano, quod baſi-<lb/>bus æquidiſtans, parallelogrammi per axem late-<lb/>ra bifariam ſecat.</s> |
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5683 <s xml:id="echoid-s3309" xml:space="preserve"/> |
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5684 </p> |
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5685 <pb file="0130" n="130" rhead="FED. COMMANDINI"/> |
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5686 <p> |
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5687 <s xml:id="echoid-s3310" xml:space="preserve">SIT cylindrus, uel cylindri po rtio a c: </s> |
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5688 <s xml:id="echoid-s3311" xml:space="preserve">& </s> |
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5689 <s xml:id="echoid-s3312" xml:space="preserve">plano per a-<lb/>xem ducto ſecetur; </s> |
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5690 <s xml:id="echoid-s3313" xml:space="preserve">cuius ſectio ſit parallelogrammum a b <lb/>c d: </s> |
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5691 <s xml:id="echoid-s3314" xml:space="preserve">& </s> |
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5692 <s xml:id="echoid-s3315" xml:space="preserve">bifariam diuiſis a d, b c parallelogrammi lateribus, <lb/>per diuiſionum puncta e f planum baſi æquidiſtans duca-<lb/>tur; </s> |
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5693 <s xml:id="echoid-s3316" xml:space="preserve">quod faciet ſectionem, in cy lindro quidem circulum <lb/>æqualem iis, qui ſunt in baſibus, ut demonſtrauit Serenus <lb/>in libro cylindricorum, propoſitione quinta: </s> |
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5694 <s xml:id="echoid-s3317" xml:space="preserve">in cylindri <lb/>uero portione ellipſim æqualem, & </s> |
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5695 <s xml:id="echoid-s3318" xml:space="preserve">ſimilem eis, quæ ſunt <lb/>in oppoſitis planis, quod nos <lb/> |
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5696 <anchor type="figure" xlink:label="fig-0130-01a" xlink:href="fig-0130-01"/> |
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5697 demonſtrauimus in commen <lb/>tariis in librum Archimedis <lb/>de conoidibus, & </s> |
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5698 <s xml:id="echoid-s3319" xml:space="preserve">ſphæroidi-<lb/>bus. </s> |
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5699 <s xml:id="echoid-s3320" xml:space="preserve">Dico centrum grauita-<lb/>tis cylindri, uel cylindri por-<lb/>tionis eſſe in plano e f. </s> |
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5700 <s xml:id="echoid-s3321" xml:space="preserve">Si enĩ <lb/>fieri poteſt, fit centrum g: </s> |
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5701 <s xml:id="echoid-s3322" xml:space="preserve">& </s> |
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5702 <s xml:id="echoid-s3323" xml:space="preserve"><lb/>ducatur g h ipſi a d æquidi-<lb/>ſtans, uſque ad e f planum. <lb/></s> |
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5703 <s xml:id="echoid-s3324" xml:space="preserve">Itaque linea a e continenter <lb/>diuiſa bifariam, erit tandem <lb/>pars aliqua ipſius k e, minor <lb/>g h. </s> |
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5704 <s xml:id="echoid-s3325" xml:space="preserve">Diuidantur ergo lineæ <lb/>a e, e d in partes æquales ipſi <lb/>k e: </s> |
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5705 <s xml:id="echoid-s3326" xml:space="preserve">& </s> |
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5706 <s xml:id="echoid-s3327" xml:space="preserve">per diuiſiones plana ba <lb/>ſibus æquidiſtantia ducãtur. </s> |
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5707 <s xml:id="echoid-s3328" xml:space="preserve"><lb/>erunt iam ſectiones, figuræ æ-<lb/>quales, & </s> |
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5708 <s xml:id="echoid-s3329" xml:space="preserve">ſimiles eis, quæ ſunt <lb/>in baſibus: </s> |
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5709 <s xml:id="echoid-s3330" xml:space="preserve">atque erit cylindrus in cylindros diuiſus: </s> |
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5710 <s xml:id="echoid-s3331" xml:space="preserve">& </s> |
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5711 <s xml:id="echoid-s3332" xml:space="preserve">cy <lb/>lindri portio in portiones æquales, & </s> |
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5712 <s xml:id="echoid-s3333" xml:space="preserve">ſimiles ipſi k f. </s> |
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5713 <s xml:id="echoid-s3334" xml:space="preserve">reli-<lb/>qua ſimiliter, ut ſuperius in priſmate concludentur.</s> |
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5714 <s xml:id="echoid-s3335" xml:space="preserve"/> |
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|
5715 </p> |
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5716 <div xml:id="echoid-div214" type="float" level="2" n="1"> |
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5717 <figure xlink:label="fig-0130-01" xlink:href="fig-0130-01a"> |
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5718 <image file="0130-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0130-01"/> |
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5719 </figure> |
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5720 </div> |
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5721 <pb o="10" file="0131" n="131" rhead="DE CENTRO GRA VIT. SOLID."/> |
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5722 <figure> |
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5723 <image file="0131-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0131-01"/> |
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5724 </figure> |
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5725 </div> |
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5726 <div xml:id="echoid-div216" type="section" level="1" n="73"> |
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5727 <head xml:id="echoid-head80" xml:space="preserve">THE OREMA VIII. PROPOSITIO VIII.</head> |
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5728 <p> |
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5729 <s xml:id="echoid-s3336" xml:space="preserve">Cuiuslibet priſmatis, & </s> |
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5730 <s xml:id="echoid-s3337" xml:space="preserve">cuiuslibet cylindri, uel <lb/>cylindri portionis grauitatis centrum in medio <lb/>ipſius axis conſiſtit.</s> |
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5731 <s xml:id="echoid-s3338" xml:space="preserve"/> |
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5732 </p> |
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5733 <p> |
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5734 <s xml:id="echoid-s3339" xml:space="preserve">Sit primum a f priſma æ quidiſtantibus planis contentũ, <lb/>quod ſolidum parallelepipedum appellatur: </s> |
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5735 <s xml:id="echoid-s3340" xml:space="preserve">& </s> |
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5736 <s xml:id="echoid-s3341" xml:space="preserve">oppoſito-<lb/>rum planorum c f, a h, d a, f g latera bifariam diuidantur in <lb/>punctis k l m n o p q r s t u x: </s> |
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5737 <s xml:id="echoid-s3342" xml:space="preserve">& </s> |
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5738 <s xml:id="echoid-s3343" xml:space="preserve">per diuiſiones ducantur <lb/>plana κ n, o r, s x. </s> |
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5739 <s xml:id="echoid-s3344" xml:space="preserve">communes autem eorum planorum ſe-<lb/>ctiones ſint lineæ y z, θ φ, χ ψ: </s> |
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5740 <s xml:id="echoid-s3345" xml:space="preserve">quæ in puncto ω conueniãt. <lb/></s> |
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5741 <s xml:id="echoid-s3346" xml:space="preserve">erit ex decima eiuſdem libri Archimedis parallelogrammi <lb/>c f centrum grauitatis punctum y; </s> |
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5742 <s xml:id="echoid-s3347" xml:space="preserve">parallelogrammi a h |
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5743 <pb file="0132" n="132" rhead="FED. COMMANDINI"/> |
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5744 centrum z: </s> |
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5745 <s xml:id="echoid-s3348" xml:space="preserve">parallelogram mi a d, θ: </s> |
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5746 <s xml:id="echoid-s3349" xml:space="preserve">parallelogrammi f g, φ: <lb/></s> |
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5747 <s xml:id="echoid-s3350" xml:space="preserve">parallelogrammi d h, χ: </s> |
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5748 <s xml:id="echoid-s3351" xml:space="preserve">& </s> |
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5749 <s xml:id="echoid-s3352" xml:space="preserve"><lb/> |
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5750 <anchor type="figure" xlink:label="fig-0132-01a" xlink:href="fig-0132-01"/> |
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5751 parallelogrammi c g centrũ <lb/>ψ: </s> |
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5752 <s xml:id="echoid-s3353" xml:space="preserve">atque erit ω punctum me <lb/>dium uniuſcuiuſque axis, ui <lb/>delicet eius lineæ, quæ oppo <lb/>ſitorum planorũ centra con <lb/>iungit. </s> |
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5753 <s xml:id="echoid-s3354" xml:space="preserve">Dico ω centrum effe <lb/>grauitatis ipſius ſolidi. </s> |
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5754 <s xml:id="echoid-s3355" xml:space="preserve">eſt <lb/>enim, ut demonſtrauimus, <lb/> |
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5755 <anchor type="note" xlink:label="note-0132-01a" xlink:href="note-0132-01"/> |
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5756 ſolidi a f centrum grauitatis <lb/>in plano K n; </s> |
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5757 <s xml:id="echoid-s3356" xml:space="preserve">quod oppoſi-<lb/>tis planis a d, g f æ quidiſtans <lb/>reliquorum planorum late-<lb/>ra biſariam diuidit: </s> |
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5758 <s xml:id="echoid-s3357" xml:space="preserve">& </s> |
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5759 <s xml:id="echoid-s3358" xml:space="preserve">fimili <lb/>rationeidem centrum eſt in plano o r, æ quidiſtante planis <lb/>a e, b f oppo ſitis. </s> |
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5760 <s xml:id="echoid-s3359" xml:space="preserve">ergo in communi ipſorum fectione: </s> |
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5761 <s xml:id="echoid-s3360" xml:space="preserve">ui-<lb/>delicet in linea y z. </s> |
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5762 <s xml:id="echoid-s3361" xml:space="preserve">Sed eſt etiam in plano t u, quod quidẽ <lb/>y z ſecat in ω. </s> |
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5763 <s xml:id="echoid-s3362" xml:space="preserve">Conſtat igitur centrum grauitatis ſolidi eſſe <lb/>punctum ω, medium ſcilicet axium, hoc eſt linearum, quæ <lb/>planorum oppoſitorum centra coniungunt.</s> |
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5764 <s xml:id="echoid-s3363" xml:space="preserve"/> |
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|
5765 </p> |
|
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5766 <div xml:id="echoid-div216" type="float" level="2" n="1"> |
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5767 <figure xlink:label="fig-0132-01" xlink:href="fig-0132-01a"> |
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5768 <image file="0132-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0132-01"/> |
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5769 </figure> |
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5770 <note position="left" xlink:label="note-0132-01" xlink:href="note-0132-01a" xml:space="preserve">6. huius</note> |
|
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|
5771 </div> |
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|
5772 <p> |
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5773 <s xml:id="echoid-s3364" xml:space="preserve">Sit aliud prima a f; </s> |
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5774 <s xml:id="echoid-s3365" xml:space="preserve">& </s> |
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5775 <s xml:id="echoid-s3366" xml:space="preserve">in eo plana, quæ opponuntur, tri-<lb/>angula a b c, d e f: </s> |
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5776 <s xml:id="echoid-s3367" xml:space="preserve">diuiſisq; </s> |
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5777 <s xml:id="echoid-s3368" xml:space="preserve">bifariam parallelogrammorum <lb/>lateribus a d, b e, c f in punctis g h κ, per diuiſiones planũ <lb/>ducatur, quod oppoſitis planis æ quidiſtans faciet ſe ctionẽ <lb/>triangulum g h k æ quale, & </s> |
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5778 <s xml:id="echoid-s3369" xml:space="preserve">ſimile ipſis a b c, d e f. </s> |
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5779 <s xml:id="echoid-s3370" xml:space="preserve">Rurſus <lb/>diuidatur a b bifariam in l: </s> |
|
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5780 <s xml:id="echoid-s3371" xml:space="preserve">& </s> |
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5781 <s xml:id="echoid-s3372" xml:space="preserve">iuncta c l per ipſam, & </s> |
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5782 <s xml:id="echoid-s3373" xml:space="preserve">per <lb/>c _K_ f planum ducatur priſma ſecans, cuius, & </s> |
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5783 <s xml:id="echoid-s3374" xml:space="preserve">parallelogrã <lb/>mi a e communis ſcctio ſit l m n. </s> |
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5784 <s xml:id="echoid-s3375" xml:space="preserve">diuidet pun ctum m li-<lb/>neam g h bifariam; </s> |
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|
5785 <s xml:id="echoid-s3376" xml:space="preserve">& </s> |
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5786 <s xml:id="echoid-s3377" xml:space="preserve">ita n diuidet lineam d e: </s> |
|
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5787 <s xml:id="echoid-s3378" xml:space="preserve">quoniam <lb/>triangula a c l, g k m, d f n æ qualia ſunt, & </s> |
|
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5788 <s xml:id="echoid-s3379" xml:space="preserve">ſimilia, ut ſu pra <lb/> |
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|
5789 <anchor type="note" xlink:label="note-0132-02a" xlink:href="note-0132-02"/> |
|
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|
5790 demonſtrauimus. </s> |
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5791 <s xml:id="echoid-s3380" xml:space="preserve">Iam ex iis, quæ tradita ſunt, conſtat cen <lb/>trum greuitatis priſmatis in plano g h k contineri. </s> |
|
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5792 <s xml:id="echoid-s3381" xml:space="preserve">Dico <lb/>ipſum eſſe in linea k m. </s> |
|
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|
5793 <s xml:id="echoid-s3382" xml:space="preserve">Si enim fieri poteſt, ſit o centrum;</s> |
|
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5794 <s xml:id="echoid-s3383" xml:space="preserve"> |
|
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5795 <pb o="11" file="0133" n="133" rhead="DE CENTRO GRA VIT. SOLID."/> |
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5796 & </s> |
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5797 <s xml:id="echoid-s3384" xml:space="preserve">per o ducatur o p ad k m ipſi h g æquidiſtans. </s> |
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5798 <s xml:id="echoid-s3385" xml:space="preserve">Itaque li <lb/>nea h m bifariã uſque eò diuidatur, quoad reliqua ſit pars <lb/>quædam q m, minor o p. </s> |
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5799 <s xml:id="echoid-s3386" xml:space="preserve">deinde h m, m g diuidantur in <lb/>partes æ quales ipſi m q: </s> |
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5800 <s xml:id="echoid-s3387" xml:space="preserve">& </s> |
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5801 <s xml:id="echoid-s3388" xml:space="preserve">per diuiſiones lineæ ipſi m K <lb/>æ quidiſtantes ducantur. </s> |
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5802 <s xml:id="echoid-s3389" xml:space="preserve">puncta uero, in quibus hæ trian-<lb/>gulorum latera ſecant, coniungantur ductis lineis r s, t u, <lb/> |
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5803 <anchor type="figure" xlink:label="fig-0133-01a" xlink:href="fig-0133-01"/> |
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5804 x y; </s> |
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5805 <s xml:id="echoid-s3390" xml:space="preserve">quæ baſi g h æquidiſtabunt. </s> |
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5806 <s xml:id="echoid-s3391" xml:space="preserve">Quoniam enim lineæ g z, <lb/>h α ſunt æ quales: </s> |
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5807 <s xml:id="echoid-s3392" xml:space="preserve">itemq; </s> |
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5808 <s xml:id="echoid-s3393" xml:space="preserve">æquales g m, m h: </s> |
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5809 <s xml:id="echoid-s3394" xml:space="preserve">ut m g ad g z, <lb/>ita erit m h, ad h α: </s> |
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5810 <s xml:id="echoid-s3395" xml:space="preserve">& </s> |
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5811 <s xml:id="echoid-s3396" xml:space="preserve">diuidendo, ut m z ad z g, ita m α ad <lb/>α h. </s> |
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5812 <s xml:id="echoid-s3397" xml:space="preserve">Sed ut m z ad z g, ita k r ad r g: </s> |
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5813 <s xml:id="echoid-s3398" xml:space="preserve">& </s> |
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5814 <s xml:id="echoid-s3399" xml:space="preserve">ut m α ad α h, ita k s <lb/> |
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5815 <anchor type="note" xlink:label="note-0133-01a" xlink:href="note-0133-01"/> |
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5816 ad s h. </s> |
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5817 <s xml:id="echoid-s3400" xml:space="preserve">quare ut κ r ad r g, ita k s ad s h. </s> |
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5818 <s xml:id="echoid-s3401" xml:space="preserve">æ quidiſtant igitur <lb/> |
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5819 <anchor type="note" xlink:label="note-0133-02a" xlink:href="note-0133-02"/> |
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5820 inter ſe ſe r s, g h. </s> |
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5821 <s xml:id="echoid-s3402" xml:space="preserve">eadem quoque ratione demonſtrabimus <lb/> |
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5822 <anchor type="note" xlink:label="note-0133-03a" xlink:href="note-0133-03"/> |
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5823 <pb file="0134" n="134" rhead="FED. COMMANDINI"/> |
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5824 t u, x y ipſi g h æquidiſtare. </s> |
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5825 <s xml:id="echoid-s3403" xml:space="preserve">Et quoniam triangula, quæ <lb/>fiunt à lineis K y, y u, u s, s h æqualia ſuntinter ſe, & </s> |
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5826 <s xml:id="echoid-s3404" xml:space="preserve">ſimilia <lb/>triangulo K m h: </s> |
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5827 <s xml:id="echoid-s3405" xml:space="preserve">habebit triangulum K m h ad triangulũ <lb/> |
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5828 <anchor type="note" xlink:label="note-0134-01a" xlink:href="note-0134-01"/> |
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5829 K δ y duplam proportionem eius, quæ eſt lineæ k h ad K y. <lb/></s> |
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5830 <s xml:id="echoid-s3406" xml:space="preserve">ſed _K_ h poſita eſt quadrupla ipſius k y. </s> |
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5831 <s xml:id="echoid-s3407" xml:space="preserve">ergo triangulum <lb/>κ m h ad triangulum _K_ δ y eãdem proportionem habebit, <lb/>quam ſexdecim ad unũ & </s> |
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5832 <s xml:id="echoid-s3408" xml:space="preserve">ad quatuor triangula k δ y, y u, <lb/>u s, s α h habebit eandem, quam fexdecim ad quatuor, hoc <lb/>eſt quam h K ad κ y: </s> |
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5833 <s xml:id="echoid-s3409" xml:space="preserve">& </s> |
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5834 <s xml:id="echoid-s3410" xml:space="preserve">ſimiliter eandem habere demonſtra <lb/>bitur trian-<lb/> |
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5835 <anchor type="figure" xlink:label="fig-0134-01a" xlink:href="fig-0134-01"/> |
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5836 gulum κ m g <lb/>ad quatuor <lb/>triãgula K δ <lb/>x, x γ t, t β r, <lb/>r z g. </s> |
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5837 <s xml:id="echoid-s3411" xml:space="preserve">quare <lb/> |
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5838 <anchor type="note" xlink:label="note-0134-02a" xlink:href="note-0134-02"/> |
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5839 totum trian <lb/>gulum K g h <lb/>ad omnia tri <lb/>angula g z r, <lb/>r β t, t γ x, x δ <lb/>_K_, K δ y, y u, <lb/>u s, s α h ita <lb/>erit, ut h κ a d <lb/>k y, hoc eſt <lb/>ut h m ad m <lb/>q. </s> |
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5840 <s xml:id="echoid-s3412" xml:space="preserve">Si igitur in <lb/>triangulis a b c, d e f deſcribantur figuræ ſimiles ei, quæ de-<lb/>ſcripta eſt in g h K triangulo: </s> |
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5841 <s xml:id="echoid-s3413" xml:space="preserve">& </s> |
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5842 <s xml:id="echoid-s3414" xml:space="preserve">per lineas ſibi reſp onden-<lb/>tes plana ducantur: </s> |
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5843 <s xml:id="echoid-s3415" xml:space="preserve">totum priſma a f diuiſum eritin tria <lb/>ſolida parallelepipeda y γ, u β, s z, quorum baſes ſunt æ qua <lb/>les & </s> |
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5844 <s xml:id="echoid-s3416" xml:space="preserve">ſimiles ipſis parallelogrammis y γ, u β, s z: </s> |
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5845 <s xml:id="echoid-s3417" xml:space="preserve">& </s> |
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5846 <s xml:id="echoid-s3418" xml:space="preserve">in octo <lb/>priſmata g z r, r β t, t γ x, x δ K, κ δ y, y u, u s, s α h: </s> |
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5847 <s xml:id="echoid-s3419" xml:space="preserve">quorum <lb/>item baſes æquales, & </s> |
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5848 <s xml:id="echoid-s3420" xml:space="preserve">ſimiles ſunt dictis triangulis; </s> |
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5849 <s xml:id="echoid-s3421" xml:space="preserve">altitu-<lb/>do autem in omnibus, totius priſmatis altitudini æ qualis.</s> |
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5850 <s xml:id="echoid-s3422" xml:space="preserve"> |
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5851 <pb o="12" file="0135" n="135" rhead="DE CENTRO GRA VIT. SOLID."/> |
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5852 Itaque ſolidi parallelepipedi y γ centrum grauitatis eſt in <lb/>linea δ: </s> |
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5853 <s xml:id="echoid-s3423" xml:space="preserve">ſolidi u β centrum eſt in linea ε η: </s> |
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5854 <s xml:id="echoid-s3424" xml:space="preserve">& </s> |
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5855 <s xml:id="echoid-s3425" xml:space="preserve">ſolidi s z in li <lb/>nea η m, quæ quidem lineæ axes ſunt, cum planorum oppo <lb/>ſitorum centra coniungant. </s> |
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5856 <s xml:id="echoid-s3426" xml:space="preserve">ergo magnitudinis ex his ſoli <lb/>dis compoſitæ centrum grauitatis eſt in linea δ m, quod ſit <lb/>θ; </s> |
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5857 <s xml:id="echoid-s3427" xml:space="preserve">& </s> |
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5858 <s xml:id="echoid-s3428" xml:space="preserve">iuncta θ o producatur: </s> |
|
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5859 <s xml:id="echoid-s3429" xml:space="preserve">à puncto autem h ducatur h μ <lb/>ipſi m κ æquidiſtans, quæ cum θ o in μ conueniat. </s> |
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5860 <s xml:id="echoid-s3430" xml:space="preserve">triangu <lb/>lum igitur g h κ ad omnia triangula g z r, r β t, t γ x, x δ k, <lb/>κ δ y, y u, u s, s α h eandem habet proportionem, quam h m <lb/>ad m q; </s> |
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5861 <s xml:id="echoid-s3431" xml:space="preserve">hoc eſt, quam μ θ ad θ λ: </s> |
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|
5862 <s xml:id="echoid-s3432" xml:space="preserve">nam ſi h m, μ θ produci in <lb/>telligantur, quouſque coeant; </s> |
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5863 <s xml:id="echoid-s3433" xml:space="preserve">erit ob linearum q y, m k æ-<lb/>quidiſtantiam, ut h q ad q m, ita μ λ ad ad λ θ: </s> |
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5864 <s xml:id="echoid-s3434" xml:space="preserve">& </s> |
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5865 <s xml:id="echoid-s3435" xml:space="preserve">componen <lb/>do, ut h m ad m q, ita μ θ ad θ λ. </s> |
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5866 <s xml:id="echoid-s3436" xml:space="preserve">linea uero θ o maior eſt, <lb/>quàm θ λ: </s> |
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5867 <s xml:id="echoid-s3437" xml:space="preserve">habebit igitur μ θ ad θ λ maiorem proportio-<lb/> |
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5868 <anchor type="note" xlink:label="note-0135-01a" xlink:href="note-0135-01"/> |
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5869 nem, quàm ad θ o. </s> |
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5870 <s xml:id="echoid-s3438" xml:space="preserve">quare triangulum etiam g h k ad omnia <lb/>iam dicta triangula maiorem proportionẽ habebit, quàm <lb/>μ θ ad θ o. </s> |
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5871 <s xml:id="echoid-s3439" xml:space="preserve">ſed ut triangulũ g h k ad omnia triangula, ita to-<lb/>tũ priſma a f ad omnia priſmata g z r, r β t, t γ x, x δ k, k δ y, <lb/>y u, u s, s α h: </s> |
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5872 <s xml:id="echoid-s3440" xml:space="preserve">quoniam enim ſolida parallelepipeda æque al <lb/>ta, eandem inter ſe proportionem habent, quam baſes; </s> |
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5873 <s xml:id="echoid-s3441" xml:space="preserve">ut <lb/>ex trigeſimaſecunda undecimi elementorum conſtat. </s> |
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5874 <s xml:id="echoid-s3442" xml:space="preserve">ſunt <lb/> |
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5875 <anchor type="note" xlink:label="note-0135-02a" xlink:href="note-0135-02"/> |
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5876 autem ſolida parallelepipeda priſmatum triangulares ba-<lb/>ſes habentium dupla: </s> |
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5877 <s xml:id="echoid-s3443" xml:space="preserve">ſequitur, ut etiam huiuſmodi priſ-<lb/> |
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5878 <anchor type="note" xlink:label="note-0135-03a" xlink:href="note-0135-03"/> |
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5879 matainter ſe ſint, ſicut eorum baſes. </s> |
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5880 <s xml:id="echoid-s3444" xml:space="preserve">ergo totum priſma ad <lb/>omnia priſmata maiorem proportionem habet, quam μ θ <lb/>ad θ o: </s> |
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5881 <s xml:id="echoid-s3445" xml:space="preserve">& </s> |
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5882 <s xml:id="echoid-s3446" xml:space="preserve">diuidendo ſolida parallelepipeda y γ, u β, s z ad o-<lb/> |
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5883 <anchor type="note" xlink:label="note-0135-04a" xlink:href="note-0135-04"/> |
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5884 mnia prifmata proportionem habent maiorem, quàm μ o <lb/>ad o θ. </s> |
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5885 <s xml:id="echoid-s3447" xml:space="preserve">fiat @ o ad o θ, ut folida parallelepipeda y γ, u β, s z ad <lb/>omnia priſmata. </s> |
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5886 <s xml:id="echoid-s3448" xml:space="preserve">Itaque cum à priſmate a f, cuius cẽtrum <lb/>grauitatis eſt o, auferatur magnitudo ex ſolidis parallelepi <lb/>pedis y γ, u β, s z conſtans: </s> |
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5887 <s xml:id="echoid-s3449" xml:space="preserve">atque ipfius grauitatis centrum <lb/>ſit θ: </s> |
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5888 <s xml:id="echoid-s3450" xml:space="preserve">reliquæ magnitudinis, quæ ex omnibus priſmatibus <lb/>conſtat, grauitatis centrum erit in linea θ o producta: </s> |
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5889 <s xml:id="echoid-s3451" xml:space="preserve">& </s> |
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5890 <s xml:id="echoid-s3452" xml:space="preserve"><lb/>in puncto ν, ex o ctaua propoſitione eiuſdem libri Archi- |
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5891 <pb file="0136" n="136" rhead="FED. COMMANDINI"/> |
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5892 medis. </s> |
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5893 <s xml:id="echoid-s3453" xml:space="preserve">ergo punctum v extra p riſima a f poſitum, centrũ <lb/>erit magnitudinis cõpoſitæ e x omnibus priſmatibus g z r, <lb/>r β t, t γ x, x δ k, k δ y, y u, u s, s α h, quod fieri nullo modo po <lb/>teſt. </s> |
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5894 <s xml:id="echoid-s3454" xml:space="preserve">eſt enim ex diſſinitione centrum grauitatis ſolidæ figu <lb/>ræ intra ipſam poſitum, non extra. </s> |
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5895 <s xml:id="echoid-s3455" xml:space="preserve">quare relinquitur, ut cẽ <lb/>trum grauitatis priſmatis ſit in linea K m. </s> |
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5896 <s xml:id="echoid-s3456" xml:space="preserve">Rurſus b c bifa-<lb/>riam in ξ diuidatur: </s> |
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5897 <s xml:id="echoid-s3457" xml:space="preserve">& </s> |
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5898 <s xml:id="echoid-s3458" xml:space="preserve">ducta a ξ, per ipſam, & </s> |
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5899 <s xml:id="echoid-s3459" xml:space="preserve">per lineam <lb/>a g d plan um ducatur; </s> |
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5900 <s xml:id="echoid-s3460" xml:space="preserve">quod priſma ſecet: </s> |
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5901 <s xml:id="echoid-s3461" xml:space="preserve">faciatq; </s> |
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5902 <s xml:id="echoid-s3462" xml:space="preserve">in paral <lb/>lelogrammo b f ſectionem ξ π di uidet punctum π lineam <lb/>quoque c f bifariam: </s> |
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5903 <s xml:id="echoid-s3463" xml:space="preserve">& </s> |
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5904 <s xml:id="echoid-s3464" xml:space="preserve">erit p lani eius, & </s> |
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5905 <s xml:id="echoid-s3465" xml:space="preserve">trianguli g h K <lb/>communis ſectio g u; </s> |
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5906 <s xml:id="echoid-s3466" xml:space="preserve">quòd p ũctum u in inedio lineæ h K <lb/> |
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5907 <anchor type="figure" xlink:label="fig-0136-01a" xlink:href="fig-0136-01"/> |
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5908 poſitum ſi t. </s> |
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5909 <s xml:id="echoid-s3467" xml:space="preserve">Similiter demonſtrabimus centrum grauita-<lb/>tis priſm atis in ipſa g u ineſſe. </s> |
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5910 <s xml:id="echoid-s3468" xml:space="preserve">ſit autem planorum c f n l, <lb/>a d π ξ communis ſectio linea ρ ο τ quæ quidem priſmatis <lb/>axis erit, cum tranſeat per centra grauitatis triangulorum <lb/>a b c, g h k, d e f, ex quartadecima eiuſdem. </s> |
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5911 <s xml:id="echoid-s3469" xml:space="preserve">ergo centrum <lb/>grauitatis pri ſmatis a f eſt punctum σ, centrum ſcilicet |
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5912 <pb o="13" file="0137" n="137" rhead="DE CENTRO GRAVIT. SOLID."/> |
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5913 trianguli g h K, & </s> |
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5914 <s xml:id="echoid-s3470" xml:space="preserve">ipſius ρ τ axis medium.</s> |
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5915 <s xml:id="echoid-s3471" xml:space="preserve"/> |
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5916 </p> |
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5917 <div xml:id="echoid-div217" type="float" level="2" n="2"> |
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5918 <note position="left" xlink:label="note-0132-02" xlink:href="note-0132-02a" xml:space="preserve">5. huius</note> |
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5919 <figure xlink:label="fig-0133-01" xlink:href="fig-0133-01a"> |
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5920 <image file="0133-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0133-01"/> |
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5921 </figure> |
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5922 <note position="right" xlink:label="note-0133-01" xlink:href="note-0133-01a" xml:space="preserve">2. ſexti.</note> |
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5923 <note position="right" xlink:label="note-0133-02" xlink:href="note-0133-02a" xml:space="preserve">I1. quinti</note> |
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5924 <note position="right" xlink:label="note-0133-03" xlink:href="note-0133-03a" xml:space="preserve">2. ſexti.</note> |
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5925 <note position="left" xlink:label="note-0134-01" xlink:href="note-0134-01a" xml:space="preserve">19. ſexti</note> |
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5926 <figure xlink:label="fig-0134-01" xlink:href="fig-0134-01a"> |
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5927 <image file="0134-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0134-01"/> |
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5928 </figure> |
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5929 <note position="left" xlink:label="note-0134-02" xlink:href="note-0134-02a" xml:space="preserve">2. uel 121 <lb/>quinti.</note> |
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5930 <note position="right" xlink:label="note-0135-01" xlink:href="note-0135-01a" xml:space="preserve">8. quinti.</note> |
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5931 <note position="right" xlink:label="note-0135-02" xlink:href="note-0135-02a" xml:space="preserve">28. unde <lb/>cimi</note> |
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5932 <note position="right" xlink:label="note-0135-03" xlink:href="note-0135-03a" xml:space="preserve">15. quinti</note> |
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5933 <note position="right" xlink:label="note-0135-04" xlink:href="note-0135-04a" xml:space="preserve">19. quinti <lb/>apud Cã <lb/>panum.</note> |
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5934 <figure xlink:label="fig-0136-01" xlink:href="fig-0136-01a"> |
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5935 <image file="0136-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0136-01"/> |
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5936 </figure> |
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5937 </div> |
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5938 <p> |
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5939 <s xml:id="echoid-s3472" xml:space="preserve">Sit priſma a g, cuius oppoſita plana ſint quadrilatera <lb/>a b c d, e f g h: </s> |
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5940 <s xml:id="echoid-s3473" xml:space="preserve">ſecenturq; </s> |
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5941 <s xml:id="echoid-s3474" xml:space="preserve">a e, b f, c g, d h bifariam: </s> |
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5942 <s xml:id="echoid-s3475" xml:space="preserve">& </s> |
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5943 <s xml:id="echoid-s3476" xml:space="preserve">per di-<lb/>uiſiones planum ducatur; </s> |
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5944 <s xml:id="echoid-s3477" xml:space="preserve">quod ſectionem faciat quadrila-<lb/>terum _K_ l m n. </s> |
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5945 <s xml:id="echoid-s3478" xml:space="preserve">Deinde iuncta a c per lineas a c, a e ducatur <lb/>planum ſecãs priſma, quod ipſum diuidet in duo priſmata <lb/>triangulares baſes habentia a b c e f g, a d c e h g. </s> |
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5946 <s xml:id="echoid-s3479" xml:space="preserve">Sint autẽ <lb/>triangulorum a b c, e f g gra-<lb/> |
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5947 <anchor type="figure" xlink:label="fig-0137-01a" xlink:href="fig-0137-01"/> |
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5948 uitatis centra o p: </s> |
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5949 <s xml:id="echoid-s3480" xml:space="preserve">& </s> |
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5950 <s xml:id="echoid-s3481" xml:space="preserve">triangu-<lb/>lorum a d c, e h g centra q r: <lb/></s> |
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5951 <s xml:id="echoid-s3482" xml:space="preserve">iunganturq; </s> |
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5952 <s xml:id="echoid-s3483" xml:space="preserve">o p, q r; </s> |
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5953 <s xml:id="echoid-s3484" xml:space="preserve">quæ pla-<lb/>no _k_ l m n occurrant in pun-<lb/>ctis s t. </s> |
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5954 <s xml:id="echoid-s3485" xml:space="preserve">erit ex iis, quæ demon <lb/>ſtrauimus, punctum s grauita <lb/>tis centrum trianguli k l m; </s> |
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5955 <s xml:id="echoid-s3486" xml:space="preserve">& </s> |
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5956 <s xml:id="echoid-s3487" xml:space="preserve"><lb/>ipſius priſmatis a b c e f g: </s> |
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5957 <s xml:id="echoid-s3488" xml:space="preserve">pun <lb/>ctum uero t centrum grauita <lb/>tis trianguli _K_ n m, & </s> |
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5958 <s xml:id="echoid-s3489" xml:space="preserve">priſma-<lb/>tis a d c, e h g. </s> |
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5959 <s xml:id="echoid-s3490" xml:space="preserve">iunctis igitur <lb/>o q, p r, s t, erit in linea o q cẽ <lb/>trum grauitatis quadrilateri <lb/>a b c d, quod ſit u: </s> |
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5960 <s xml:id="echoid-s3491" xml:space="preserve">& </s> |
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5961 <s xml:id="echoid-s3492" xml:space="preserve">in linea <lb/>p r cẽtrum quadrilateri e f g h <lb/>ſit autem x. </s> |
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5962 <s xml:id="echoid-s3493" xml:space="preserve">deniqueiungatur <lb/>u x, quæ ſecet lineam ſ t in y. </s> |
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5963 <s xml:id="echoid-s3494" xml:space="preserve">ſe <lb/>cabit enim cum ſint in eodem <lb/> |
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5964 <anchor type="note" xlink:label="note-0137-01a" xlink:href="note-0137-01"/> |
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5965 plano: </s> |
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5966 <s xml:id="echoid-s3495" xml:space="preserve">atq; </s> |
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5967 <s xml:id="echoid-s3496" xml:space="preserve">erit y grauitatis centrum quadril ateri _K_ lm n. <lb/></s> |
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5968 <s xml:id="echoid-s3497" xml:space="preserve">Dico idem punctum y centrum quoque gra uitatis eſſe to-<lb/>tius priſmatis. </s> |
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5969 <s xml:id="echoid-s3498" xml:space="preserve">Quoniam enim quadri lateri k lm n graui-<lb/>tatis centrum eſt y: </s> |
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5970 <s xml:id="echoid-s3499" xml:space="preserve">linea s y ad y t eandem proportionem <lb/>habebit, quam triangulum k n m ad triangulum k lm, ex 8 <lb/>Archimedis de centro grauitatis planorum. </s> |
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5971 <s xml:id="echoid-s3500" xml:space="preserve">Vtautem triã <lb/>gulum k n m ad ipſum k l m, hoc eſt ut triangulum a d c ad <lb/>triangulum a b c, æqualia enim ſunt, ita priſina a d c e h g |
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5972 <pb file="0138" n="138" rhead="FED. COMMANDINI"/> |
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|
5973 ad priſma a b c e f g. </s> |
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5974 <s xml:id="echoid-s3501" xml:space="preserve">quare linea s y ad y t eandem propor-<lb/>tionem habet, quam priſma a d c e h g ad priſma a b c e f g. <lb/></s> |
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5975 <s xml:id="echoid-s3502" xml:space="preserve">Sed priſmatis a b c e f g centrum grauitatis eſts: </s> |
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5976 <s xml:id="echoid-s3503" xml:space="preserve">& </s> |
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5977 <s xml:id="echoid-s3504" xml:space="preserve">priſma-<lb/>tis a d c e h g centrum t. </s> |
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5978 <s xml:id="echoid-s3505" xml:space="preserve">magnitudinis igitur ex his compo <lb/>ſitæ, hoc eſt totius priſmatis a g centrum grauitatis eſt pun <lb/>ctum y; </s> |
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5979 <s xml:id="echoid-s3506" xml:space="preserve">medium ſcilicet axis u x, qui oppoſitorum plano-<lb/>rum centra coniungit.</s> |
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5980 <s xml:id="echoid-s3507" xml:space="preserve"/> |
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|
5981 </p> |
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5982 <div xml:id="echoid-div218" type="float" level="2" n="3"> |
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5983 <figure xlink:label="fig-0137-01" xlink:href="fig-0137-01a"> |
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5984 <image file="0137-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0137-01"/> |
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|
5985 </figure> |
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5986 <note position="right" xlink:label="note-0137-01" xlink:href="note-0137-01a" xml:space="preserve">5. huius.</note> |
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|
5987 </div> |
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|
5988 <p> |
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5989 <s xml:id="echoid-s3508" xml:space="preserve">Rurſus ſit priſma baſim habens pentagonum a b c d e: <lb/></s> |
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5990 <s xml:id="echoid-s3509" xml:space="preserve">& </s> |
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5991 <s xml:id="echoid-s3510" xml:space="preserve">quod ei opponitur ſit f g h _K_ l: </s> |
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5992 <s xml:id="echoid-s3511" xml:space="preserve">ſec enturq; </s> |
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5993 <s xml:id="echoid-s3512" xml:space="preserve">a f, b g, c h, <lb/>d _k_, el bifariam: </s> |
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|
5994 <s xml:id="echoid-s3513" xml:space="preserve">& </s> |
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5995 <s xml:id="echoid-s3514" xml:space="preserve">per diuiſiones ducto plano, ſectio ſit pẽ <lb/>tagonũ m n o p q. </s> |
|
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|
5996 <s xml:id="echoid-s3515" xml:space="preserve">deinde iuncta e b per lineas le, e b aliud <lb/>planum ducatur, diuidẽs priſ <lb/> |
|
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|
5997 <anchor type="figure" xlink:label="fig-0138-01a" xlink:href="fig-0138-01"/> |
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|
5998 ma a k in duo priſmata, in priſ <lb/>ma ſcilicet al, cuius plana op-<lb/>poſita ſint triangula a b e f g l: <lb/></s> |
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5999 <s xml:id="echoid-s3516" xml:space="preserve">& </s> |
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6000 <s xml:id="echoid-s3517" xml:space="preserve">in prima b _k_ cuius plana op <lb/>poſita ſint quadrilatera b c d e <lb/>g h _k_ l. </s> |
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6001 <s xml:id="echoid-s3518" xml:space="preserve">Sint autem triangulo-<lb/>rum a b e, f g l centra grauita <lb/>tis puncta r ſ: </s> |
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6002 <s xml:id="echoid-s3519" xml:space="preserve">& </s> |
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6003 <s xml:id="echoid-s3520" xml:space="preserve">b c d e, g h _k_ l <lb/>quadrilaterorum centra tu: </s> |
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6004 <s xml:id="echoid-s3521" xml:space="preserve"><lb/>iunganturq; </s> |
|
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6005 <s xml:id="echoid-s3522" xml:space="preserve">r s, t u o ccurren-<lb/>tes plano m n o p q in punctis <lb/>x y. </s> |
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6006 <s xml:id="echoid-s3523" xml:space="preserve">& </s> |
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6007 <s xml:id="echoid-s3524" xml:space="preserve">itidem iungãtur r t, ſu, <lb/>x y. </s> |
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6008 <s xml:id="echoid-s3525" xml:space="preserve">erit in linea r t cẽtrum gra <lb/>uitatis pentagoni a b c d e; </s> |
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6009 <s xml:id="echoid-s3526" xml:space="preserve"><lb/>quod ſit z: </s> |
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6010 <s xml:id="echoid-s3527" xml:space="preserve">& </s> |
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6011 <s xml:id="echoid-s3528" xml:space="preserve">in linea ſu cen-<lb/>trum pentagoni f g h k l: </s> |
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6012 <s xml:id="echoid-s3529" xml:space="preserve">ſit au <lb/>tem χ: </s> |
|
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6013 <s xml:id="echoid-s3530" xml:space="preserve">& </s> |
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6014 <s xml:id="echoid-s3531" xml:space="preserve">ducatur z χ, quæ di-<lb/>cto plano in χ occurrat. </s> |
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6015 <s xml:id="echoid-s3532" xml:space="preserve">Itaq; </s> |
|
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6016 <s xml:id="echoid-s3533" xml:space="preserve"><lb/>punctum x eſt centrum graui <lb/>tatis trianguli m n q, ac priſ-<lb/>matis al: </s> |
|
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6017 <s xml:id="echoid-s3534" xml:space="preserve">& </s> |
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6018 <s xml:id="echoid-s3535" xml:space="preserve">y grauitatis centrum quadrilateri n o p q, ac <lb/>priſmatis b k. </s> |
|
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6019 <s xml:id="echoid-s3536" xml:space="preserve">quare y centrum erit pentagoni m n o p q. </s> |
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6020 <s xml:id="echoid-s3537" xml:space="preserve">&</s> |
|
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6021 <s xml:id="echoid-s3538" xml:space="preserve"> |
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|
6022 <pb o="14" file="0139" n="139" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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6023 ſimiliter demonſtrabitur totius priſmatis a _K_ grauitatis eſ <lb/>ſe centrum. </s> |
|
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6024 <s xml:id="echoid-s3539" xml:space="preserve">Simili ratione & </s> |
|
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|
6025 <s xml:id="echoid-s3540" xml:space="preserve">in aliis priſinatibus illud <lb/>idem ſacile demonſtrabitur. </s> |
|
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|
6026 <s xml:id="echoid-s3541" xml:space="preserve">Quo autem pacto in omni <lb/>figura rectilinea centrum grauitatis inueniatur, do cuimus <lb/>in commentariis in ſextam propoſitionem Archimedis de <lb/>quadratura parabolæ.</s> |
|
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|
6027 <s xml:id="echoid-s3542" xml:space="preserve"/> |
|
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|
6028 </p> |
|
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|
6029 <div xml:id="echoid-div219" type="float" level="2" n="4"> |
|
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|
6030 <figure xlink:label="fig-0138-01" xlink:href="fig-0138-01a"> |
|
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6031 <image file="0138-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0138-01"/> |
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|
6032 </figure> |
|
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|
6033 </div> |
|
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6034 <p> |
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6035 <s xml:id="echoid-s3543" xml:space="preserve">Sit cylindrus, uel cylindri portio c e cuius axis a b: </s> |
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6036 <s xml:id="echoid-s3544" xml:space="preserve">ſece-<lb/>turq, plano per axem ducto; </s> |
|
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6037 <s xml:id="echoid-s3545" xml:space="preserve">quod ſectionem faciat paral-<lb/>lelo grammum c d e f: </s> |
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6038 <s xml:id="echoid-s3546" xml:space="preserve">& </s> |
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6039 <s xml:id="echoid-s3547" xml:space="preserve">diuiſis c f, d e bifariam in punctis <lb/> |
|
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6040 <anchor type="figure" xlink:label="fig-0139-01a" xlink:href="fig-0139-01"/> |
|
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6041 g h, per ea ducatur planum baſi æquidiſtans. </s> |
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6042 <s xml:id="echoid-s3548" xml:space="preserve">erit ſectio g h <lb/>circulus, uel ellipſis, centrum habens in axe; </s> |
|
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6043 <s xml:id="echoid-s3549" xml:space="preserve">quod ſit K: </s> |
|
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|
6044 <s xml:id="echoid-s3550" xml:space="preserve">at-<lb/> |
|
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|
6045 <anchor type="note" xlink:label="note-0139-01a" xlink:href="note-0139-01"/> |
|
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6046 que erunt ex iis, quæ demonſtrauimus, centra grauitatis <lb/>planorum oppoſitorum puncta a b: </s> |
|
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6047 <s xml:id="echoid-s3551" xml:space="preserve">& </s> |
|
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|
6048 <s xml:id="echoid-s3552" xml:space="preserve">plani g h ipſum _k_. </s> |
|
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6049 <s xml:id="echoid-s3553" xml:space="preserve">in <lb/>quo quidem plano eſt centrum grauitatis cylindri, uel cy-<lb/>lindri portionis. </s> |
|
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|
6050 <s xml:id="echoid-s3554" xml:space="preserve">Dico punctum K cylindri quoque, uel cy <lb/>lindri portionis grauitatis centrum eſſe. </s> |
|
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|
6051 <s xml:id="echoid-s3555" xml:space="preserve">Si enim fieri po-<lb/>teſt, ſitl centrum: </s> |
|
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|
6052 <s xml:id="echoid-s3556" xml:space="preserve">ducaturq; </s> |
|
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|
6053 <s xml:id="echoid-s3557" xml:space="preserve">k l, & </s> |
|
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|
6054 <s xml:id="echoid-s3558" xml:space="preserve">extra figuram in m pro-<lb/>ducatur. </s> |
|
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|
6055 <s xml:id="echoid-s3559" xml:space="preserve">quam uero proportionem habet linea m K ad _k_ l |
|
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|
6056 <pb file="0140" n="140" rhead="FED. COMMANDINI"/> |
|
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|
6057 habeat circulus, uel ellipſis g h ad aliud ſpacium, in quo u: <lb/></s> |
|
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|
6058 <s xml:id="echoid-s3560" xml:space="preserve">& </s> |
|
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|
6059 <s xml:id="echoid-s3561" xml:space="preserve">in circulo, uel ellipſi plane deſcribatur rectilinea figura, <lb/>ita ut tãdem relinquãtur portiones minores ſpacio u, quæ <lb/>ſit o p g q r s h t: </s> |
|
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|
6060 <s xml:id="echoid-s3562" xml:space="preserve">deſcriptaq; </s> |
|
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|
6061 <s xml:id="echoid-s3563" xml:space="preserve">ſimili figura in oppoſitis pla-<lb/>nis c d, f e, per lineas ſibi ipſis reſpondentes plana ducãtur. </s> |
|
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|
6062 <s xml:id="echoid-s3564" xml:space="preserve"><lb/>Itaque cylindrus, uel cylindri portio diuiditur in priſma, <lb/>cuius quidem baſis eſt figura rectilinea iam dicta, centrum <lb/>que grauitatis punctum K: </s> |
|
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|
6063 <s xml:id="echoid-s3565" xml:space="preserve">& </s> |
|
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|
6064 <s xml:id="echoid-s3566" xml:space="preserve">in multa ſolida, quæ pro baſi <lb/>bus habent relictas portiones, quas nos ſolidas portiones <lb/>appellabimus. </s> |
|
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|
6065 <s xml:id="echoid-s3567" xml:space="preserve">cum igitur portiones ſint minores ſpacio <lb/>u, circulus, uel ellipſis g h ad portiones maiorem propor-<lb/>tionem habebit, quàm linea m k ad K l. </s> |
|
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|
6066 <s xml:id="echoid-s3568" xml:space="preserve">fiat n k ad K l, ut <lb/>circulus uel ellipſis g h ad ipſas portiones. </s> |
|
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|
6067 <s xml:id="echoid-s3569" xml:space="preserve">Sed ut circulus <lb/>uel ellipſis g h ad figuram rectilineam in ipſa deſcri-<lb/>ptam, ita eſt cylindrus uel cylindri portio c e ad priſma, <lb/>quod rectilineam figuram pro baſi habet, & </s> |
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6068 <s xml:id="echoid-s3570" xml:space="preserve">altitudinem <lb/>æqualem; </s> |
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6069 <s xml:id="echoid-s3571" xml:space="preserve">id, quod infra demonſtrabitur, ergo per conuer <lb/>ſionem rationis, ut circulus, uel ellipſis g h ad portiones re <lb/>lictas, ita cylindrus, uel cylindri portio c e ad ſolidas por-<lb/>tiones, quare cylindrus uel cylindri portio ad ſolidas por-<lb/>tiones eandem proportionem habet, quam linea n k a d _k_ <lb/>& </s> |
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6070 <s xml:id="echoid-s3572" xml:space="preserve">diuidendo priſma, cuius baſis eſt rectilinea figura ad ſo-<lb/>lidas portiones eandem proportionem habet, quam n lad <lb/>1 _k_. </s> |
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6071 <s xml:id="echoid-s3573" xml:space="preserve">& </s> |
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6072 <s xml:id="echoid-s3574" xml:space="preserve">quoniam a cylindro uel cylindri portione, cuius gra-<lb/>uitatis centrum eſt l, aufertur priſma baſim habens rectili-<lb/>neam figurã, cuius centrũ grauitatis eſt _K_: </s> |
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6073 <s xml:id="echoid-s3575" xml:space="preserve">reſiduæ magnitu <lb/>dinis ex ſolidis portionibus cõpoſitæ grauitatis cẽtrũ erit <lb/>in linea k l protracta, & </s> |
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6074 <s xml:id="echoid-s3576" xml:space="preserve">in puncto n; </s> |
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6075 <s xml:id="echoid-s3577" xml:space="preserve">quod eſt abſurdū. </s> |
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6076 <s xml:id="echoid-s3578" xml:space="preserve">relin <lb/>quitur ergo, ut cẽtrum grauitatis cylindri; </s> |
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6077 <s xml:id="echoid-s3579" xml:space="preserve">uel cylin dri por <lb/>tionis ſit punctũ k. </s> |
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6078 <s xml:id="echoid-s3580" xml:space="preserve">quæ omnia demonſtrãda propoſuimus.</s> |
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6079 <s xml:id="echoid-s3581" xml:space="preserve"/> |
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6080 </p> |
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6081 <div xml:id="echoid-div220" type="float" level="2" n="5"> |
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6082 <figure xlink:label="fig-0139-01" xlink:href="fig-0139-01a"> |
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6083 <image file="0139-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0139-01"/> |
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6084 </figure> |
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6085 <note position="right" xlink:label="note-0139-01" xlink:href="note-0139-01a" xml:space="preserve">4. huius.</note> |
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6086 </div> |
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6087 <p> |
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6088 <s xml:id="echoid-s3582" xml:space="preserve">At uero cylindrum, uel cylindri portionẽ ce <lb/>ad priſma, cuius baſis eſt rectilinea figura in ſpa-<lb/>cio g h deſcripta, & </s> |
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6089 <s xml:id="echoid-s3583" xml:space="preserve">altitudo æqualis; </s> |
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6090 <s xml:id="echoid-s3584" xml:space="preserve">eandem ha- |
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6091 <pb o="15" file="0141" n="141" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6092 bere proportionem, quam ſpacium g h ad dictã <lb/>figuram, hoc modo demonſtrabimus.</s> |
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6093 <s xml:id="echoid-s3585" xml:space="preserve"/> |
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6094 </p> |
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6095 <p> |
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6096 <s xml:id="echoid-s3586" xml:space="preserve">Intelligatur circulus, uel ellipſis x æqualis figuræ rectili-<lb/>neæ in g h ſpacio deſcriptæ: </s> |
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6097 <s xml:id="echoid-s3587" xml:space="preserve">& </s> |
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6098 <s xml:id="echoid-s3588" xml:space="preserve">ab x conſtituatur conus, uel <lb/> |
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6099 <anchor type="figure" xlink:label="fig-0141-01a" xlink:href="fig-0141-01"/> |
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6100 coni portio, altitudinẽ habens eandẽ, quã cylindrus uel cy <lb/>lindri portio c e. </s> |
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6101 <s xml:id="echoid-s3589" xml:space="preserve">Sit deinde rectilinea figura, in quay eade, <lb/>quæ in ſpacio g h deſcripta eſt: </s> |
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6102 <s xml:id="echoid-s3590" xml:space="preserve">& </s> |
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6103 <s xml:id="echoid-s3591" xml:space="preserve">ab hac pyramis æquealta <lb/>conſtituatur. </s> |
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6104 <s xml:id="echoid-s3592" xml:space="preserve">Dico conũ uel coni portionẽ x pyramidiy æ-<lb/>qualẽ eſſe. </s> |
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6105 <s xml:id="echoid-s3593" xml:space="preserve">niſi enim ſit æqualis, uel maior, uel minor erit.</s> |
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6106 <s xml:id="echoid-s3594" xml:space="preserve"/> |
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6107 </p> |
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6108 <div xml:id="echoid-div221" type="float" level="2" n="6"> |
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6109 <figure xlink:label="fig-0141-01" xlink:href="fig-0141-01a"> |
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6110 <image file="0141-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0141-01"/> |
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6111 </figure> |
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6112 </div> |
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6113 <p> |
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6114 <s xml:id="echoid-s3595" xml:space="preserve">Sit primum maior, et exuperet ſolido z. </s> |
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6115 <s xml:id="echoid-s3596" xml:space="preserve">Itaque in circu <lb/>lo, uel ellipſi x deſcribatur figura rectilinea; </s> |
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6116 <s xml:id="echoid-s3597" xml:space="preserve">& </s> |
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6117 <s xml:id="echoid-s3598" xml:space="preserve">in ea pyra-<lb/>mis eandem, quam conus, uel coni portio altitudinem ha-<lb/>bens, ita ut portiones relictæ minores ſint ſolido z, quem-<lb/>admodum docetur in duodecimo libro elementorum pro <lb/>poſitione undecima. </s> |
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6118 <s xml:id="echoid-s3599" xml:space="preserve">erit pyramis x adhuc pyramide y ma <lb/>ior. </s> |
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6119 <s xml:id="echoid-s3600" xml:space="preserve">& </s> |
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6120 <s xml:id="echoid-s3601" xml:space="preserve">quoniam piramides æque altæ inter ſe ſunt, ſicuti ba <lb/> |
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6121 <anchor type="note" xlink:label="note-0141-01a" xlink:href="note-0141-01"/> |
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6122 ſes; </s> |
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6123 <s xml:id="echoid-s3602" xml:space="preserve">pyramis x ad piramidem y eandem proportionem ha-<lb/>bet, quàm figura rectilinea x ad figuram y. </s> |
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6124 <s xml:id="echoid-s3603" xml:space="preserve">Sed ſigura recti |
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6125 <pb file="0142" n="142" rhead="FED. COMMANDINI"/> |
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6126 <anchor type="figure" xlink:label="fig-0142-01a" xlink:href="fig-0142-01"/> |
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6127 linea x cum ſit minor circulo, uel ellipſi, eſt etiam minor fi-<lb/>gura rectilinea y. </s> |
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6128 <s xml:id="echoid-s3604" xml:space="preserve">ergo pyramis x pyramide y minor erit. <lb/></s> |
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6129 <s xml:id="echoid-s3605" xml:space="preserve">Sed & </s> |
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6130 <s xml:id="echoid-s3606" xml:space="preserve">maior; </s> |
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6131 <s xml:id="echoid-s3607" xml:space="preserve">quod fieri nõ poteſt. </s> |
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6132 <s xml:id="echoid-s3608" xml:space="preserve">At ſi conus, uel coni por <lb/>tio x ponatur minor pyramide y: </s> |
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6133 <s xml:id="echoid-s3609" xml:space="preserve">ſit alter conus æque al-<lb/>tus, uel altera coni portio χ ipſi pyramidi y æqualis. </s> |
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6134 <s xml:id="echoid-s3610" xml:space="preserve">erit <lb/>eius baſis circulus, uel ellipſis maior circulo, uel ellipſi x, <lb/>quorum exceſſus ſit ſpacium ω. </s> |
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6135 <s xml:id="echoid-s3611" xml:space="preserve">Siigitur in circulo, uel elli-<lb/>pſi χ figura rectilinea deſcribatur, ita ut portiones relictæ <lb/>ſint ω ſpacio minores, eiuſinodi figura adhuc maior erit cir <lb/>culo, uel ellipſi x, hoc eſt figura rectilinea _y_. </s> |
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6136 <s xml:id="echoid-s3612" xml:space="preserve">& </s> |
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6137 <s xml:id="echoid-s3613" xml:space="preserve">p_y_ramis in <lb/>ea conſtituta minor cono, uel coni portione χ, hoc eſt mi-<lb/>nor p_y_ramide_y_. </s> |
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6138 <s xml:id="echoid-s3614" xml:space="preserve">eſt ergo ut χ figura rectilinea ad figuram <lb/>rectilineam _y_, ita pyramis χ ad pyramidem _y_. </s> |
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6139 <s xml:id="echoid-s3615" xml:space="preserve">quare cum <lb/>figura rectilinea χ ſit maior figura_y_: </s> |
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6140 <s xml:id="echoid-s3616" xml:space="preserve">erit & </s> |
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6141 <s xml:id="echoid-s3617" xml:space="preserve">p_y_ramis χ p_y_-<lb/>ramide_y_ maior. </s> |
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6142 <s xml:id="echoid-s3618" xml:space="preserve">ſed erat minor; </s> |
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6143 <s xml:id="echoid-s3619" xml:space="preserve">quod rurſus fieri non po-<lb/>teſt. </s> |
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6144 <s xml:id="echoid-s3620" xml:space="preserve">non eſt igitur conus, uel coni portio x neque maior, <lb/>neque minor p_y_ramide_y_. </s> |
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6145 <s xml:id="echoid-s3621" xml:space="preserve">ergo ipſi neceſſario eſt æqualis. </s> |
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6146 <s xml:id="echoid-s3622" xml:space="preserve"><lb/>Itaque quoniam ut conus ad conum, uel coni portio ad co |
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6147 <pb o="15" file="0143" n="143" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6148 <anchor type="figure" xlink:label="fig-0143-01a" xlink:href="fig-0143-01"/> |
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6149 ni portionem, ita eſt c_y_lindrus ad c_y_lindrum, uel c_y_lin-<lb/>dri portio ad c_y_lindri portionem: </s> |
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6150 <s xml:id="echoid-s3623" xml:space="preserve">& </s> |
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6151 <s xml:id="echoid-s3624" xml:space="preserve">ut p_y_ramis ad p_y_ra-<lb/>midem, ita priſma ad priſma, cum eadem ſit baſis, & </s> |
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6152 <s xml:id="echoid-s3625" xml:space="preserve">æqua <lb/>lis altitudo; </s> |
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6153 <s xml:id="echoid-s3626" xml:space="preserve">erit c_y_lindrus uel c_y_lindri portio x priſma-<lb/>ti _y_ æqualis. </s> |
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6154 <s xml:id="echoid-s3627" xml:space="preserve">eftq; </s> |
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6155 <s xml:id="echoid-s3628" xml:space="preserve">ut ſpacium g h ad ſpacium x, ita c_y_lin-<lb/>drus, uel c_y_lindri portio c e ad c_y_lindrum, uel c_y_lindri por-<lb/>tionem x. </s> |
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6156 <s xml:id="echoid-s3629" xml:space="preserve">Conſtatigitur c_y_lindrum uel c_y_lindri portionẽ <lb/>c e, ad priſina_y_, quippe cuius baſis eſt figura rectilinea in <lb/> |
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6157 <anchor type="note" xlink:label="note-0143-01a" xlink:href="note-0143-01"/> |
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6158 ſpacio g h deſcripta, eandem proportionem habere, quam <lb/>ſpacium g h habet ad ſpacium x, hoc eſt ad dictam figuram. <lb/></s> |
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6159 <s xml:id="echoid-s3630" xml:space="preserve">quod demonſtrandum fuerat.</s> |
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6160 <s xml:id="echoid-s3631" xml:space="preserve"/> |
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6161 </p> |
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6162 <div xml:id="echoid-div222" type="float" level="2" n="7"> |
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6163 <note position="right" xlink:label="note-0141-01" xlink:href="note-0141-01a" xml:space="preserve">6. duode-<lb/>cimi.</note> |
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6164 <figure xlink:label="fig-0142-01" xlink:href="fig-0142-01a"> |
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6165 <image file="0142-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0142-01"/> |
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6166 </figure> |
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6167 <figure xlink:label="fig-0143-01" xlink:href="fig-0143-01a"> |
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6168 <image file="0143-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0143-01"/> |
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6169 </figure> |
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6170 <note position="right" xlink:label="note-0143-01" xlink:href="note-0143-01a" xml:space="preserve">7. quinti</note> |
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6171 </div> |
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6172 </div> |
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6173 <div xml:id="echoid-div224" type="section" level="1" n="74"> |
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6174 <head xml:id="echoid-head81" xml:space="preserve">THE OREMA IX. PROPOSITIO IX.</head> |
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6175 <p> |
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6176 <s xml:id="echoid-s3632" xml:space="preserve">Si pyramis ſecetur plano baſi æquidiſtante; </s> |
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6177 <s xml:id="echoid-s3633" xml:space="preserve">ſe-<lb/>ctio erit figura ſimilis ei, quæ eſt baſis, centrum <lb/>grauitatis in axe habens.</s> |
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6178 <s xml:id="echoid-s3634" xml:space="preserve"/> |
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6179 </p> |
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6180 <pb o="16" file="0144" n="144" rhead="FED. COMMANDINI"/> |
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6181 <p> |
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6182 <s xml:id="echoid-s3635" xml:space="preserve">SIT pyramis, cuius baſis triangulum a b c; </s> |
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6183 <s xml:id="echoid-s3636" xml:space="preserve">axis d e: </s> |
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6184 <s xml:id="echoid-s3637" xml:space="preserve">& </s> |
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6185 <s xml:id="echoid-s3638" xml:space="preserve"><lb/>ſecetur plano baſi æquidiſtante; </s> |
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6186 <s xml:id="echoid-s3639" xml:space="preserve">quod ſectionẽ faciat f g h; <lb/></s> |
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6187 <s xml:id="echoid-s3640" xml:space="preserve">occurratq; </s> |
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6188 <s xml:id="echoid-s3641" xml:space="preserve">axi in puncto k. </s> |
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6189 <s xml:id="echoid-s3642" xml:space="preserve">Dico f g h triangulum eſſe, ipſi <lb/>a b c ſimile; </s> |
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6190 <s xml:id="echoid-s3643" xml:space="preserve">cuius grauitatis centrum eſt K. </s> |
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6191 <s xml:id="echoid-s3644" xml:space="preserve">Quoniã enim <lb/> |
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6192 <anchor type="note" xlink:label="note-0144-01a" xlink:href="note-0144-01"/> |
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6193 duo plana æquidiſtantia a b c, f g h ſecantur à plano a b d; <lb/></s> |
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6194 <s xml:id="echoid-s3645" xml:space="preserve">communes eorum ſectiones a b, f g æquidiſtantes erunt: </s> |
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6195 <s xml:id="echoid-s3646" xml:space="preserve">& </s> |
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6196 <s xml:id="echoid-s3647" xml:space="preserve"><lb/>eadem ratione æquidiſtantes ipſæ b c, g h: </s> |
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6197 <s xml:id="echoid-s3648" xml:space="preserve">& </s> |
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6198 <s xml:id="echoid-s3649" xml:space="preserve">c a, h f. </s> |
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6199 <s xml:id="echoid-s3650" xml:space="preserve">Quòd <lb/>cum duæ lineæ f g, g h, duabus a b, b c æquidiſtent, nec <lb/>ſintin eodem plano; </s> |
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6200 <s xml:id="echoid-s3651" xml:space="preserve">angulus ad g æqualis eſt angulo ad <lb/> |
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6201 <anchor type="note" xlink:label="note-0144-02a" xlink:href="note-0144-02"/> |
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6202 b: </s> |
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6203 <s xml:id="echoid-s3652" xml:space="preserve">& </s> |
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6204 <s xml:id="echoid-s3653" xml:space="preserve">ſimiliter angulus ad h angulo ad c: </s> |
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6205 <s xml:id="echoid-s3654" xml:space="preserve">angulusq; </s> |
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6206 <s xml:id="echoid-s3655" xml:space="preserve">ad f ei, <lb/>qui ad a eſt æqualis. </s> |
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6207 <s xml:id="echoid-s3656" xml:space="preserve">triangulum igitur f g h ſimile eſt tri-<lb/>angulo a b c. </s> |
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6208 <s xml:id="echoid-s3657" xml:space="preserve">At uero punctum k centrum eſſe grauita-<lb/>tis trianguli f g h hoc modo oſtendemus. </s> |
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6209 <s xml:id="echoid-s3658" xml:space="preserve">Ducantur pla-<lb/>na per axem, & </s> |
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6210 <s xml:id="echoid-s3659" xml:space="preserve">per lineas d a, d b, d c: </s> |
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6211 <s xml:id="echoid-s3660" xml:space="preserve">erunt communes ſe-<lb/> |
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6212 <anchor type="note" xlink:label="note-0144-03a" xlink:href="note-0144-03"/> |
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6213 ctiones f K, a e æquidiſtantes: </s> |
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6214 <s xml:id="echoid-s3661" xml:space="preserve">pariterq; </s> |
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6215 <s xml:id="echoid-s3662" xml:space="preserve">k g, e b; </s> |
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6216 <s xml:id="echoid-s3663" xml:space="preserve">& </s> |
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6217 <s xml:id="echoid-s3664" xml:space="preserve">k h, e c: <lb/></s> |
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6218 <s xml:id="echoid-s3665" xml:space="preserve">quare angulus k f h angulo e a c; </s> |
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6219 <s xml:id="echoid-s3666" xml:space="preserve">& </s> |
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6220 <s xml:id="echoid-s3667" xml:space="preserve">angulus k f g ipſi e a b <lb/> |
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6221 <anchor type="note" xlink:label="note-0144-04a" xlink:href="note-0144-04"/> |
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6222 eſt æqualis. </s> |
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6223 <s xml:id="echoid-s3668" xml:space="preserve">Eadem ratione <lb/> |
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6224 <anchor type="figure" xlink:label="fig-0144-01a" xlink:href="fig-0144-01"/> |
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6225 anguli ad g angulis ad b: </s> |
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6226 <s xml:id="echoid-s3669" xml:space="preserve">& </s> |
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6227 <s xml:id="echoid-s3670" xml:space="preserve"><lb/>anguli ad h iis, qui ad c æ-<lb/>quales erunt. </s> |
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6228 <s xml:id="echoid-s3671" xml:space="preserve">ergo puncta <lb/>e _K_ in triangulis a b c, f g h <lb/>ſimiliter ſunt poſita, per ſe-<lb/>xtam poſitionem Archime-<lb/>dis in libro de centro graui-<lb/>tatis planorum. </s> |
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6229 <s xml:id="echoid-s3672" xml:space="preserve">Sed cum e <lb/>ſit centrum grauitatis trian <lb/>guli a b c, erit ex undecíma <lb/>propoſitione eiuſdem libri, <lb/>& </s> |
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6230 <s xml:id="echoid-s3673" xml:space="preserve">K trianguli f g h grauita <lb/>tis centrum. </s> |
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6231 <s xml:id="echoid-s3674" xml:space="preserve">id quod demonſtrare oportebat. </s> |
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6232 <s xml:id="echoid-s3675" xml:space="preserve">Non aliter <lb/>in ceteris pyramidibus, quod propoſitum eſt demonſtra-<lb/>bitur.</s> |
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6233 <s xml:id="echoid-s3676" xml:space="preserve"/> |
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6234 </p> |
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6235 <div xml:id="echoid-div224" type="float" level="2" n="1"> |
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6236 <note position="left" xlink:label="note-0144-01" xlink:href="note-0144-01a" xml:space="preserve">16. unde <lb/>cimi</note> |
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6237 <note position="left" xlink:label="note-0144-02" xlink:href="note-0144-02a" xml:space="preserve">10. undeci <lb/>mi.</note> |
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6238 <note position="left" xlink:label="note-0144-03" xlink:href="note-0144-03a" xml:space="preserve">16. unde-<lb/>cimi</note> |
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6239 <note position="left" xlink:label="note-0144-04" xlink:href="note-0144-04a" xml:space="preserve">10. unde-<lb/>cimi</note> |
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6240 <figure xlink:label="fig-0144-01" xlink:href="fig-0144-01a"> |
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6241 <image file="0144-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0144-01"/> |
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6242 </figure> |
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6243 </div> |
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6244 <pb o="17" file="0145" n="145" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6245 </div> |
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6246 <div xml:id="echoid-div226" type="section" level="1" n="75"> |
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6247 <head xml:id="echoid-head82" xml:space="preserve">PROBLEMA I. PROPOSITIO X.</head> |
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6248 <p> |
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6249 <s xml:id="echoid-s3677" xml:space="preserve"><emph style="sc">Data</emph> qualibet pyramide, fieri poteſt, ut fi-<lb/>gura ſolida in ipſa in ſcribatur, & </s> |
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6250 <s xml:id="echoid-s3678" xml:space="preserve">altera circũſcri-<lb/>batur ex priſmatibus æqualem aItitudinem ha-<lb/>bẽtibus, ita ut cir cumſcripta inſcriptam excedat <lb/>magnitudine, quæ minor ſit quacũque ſolida ma <lb/>gnitudine propoſita.</s> |
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6251 <s xml:id="echoid-s3679" xml:space="preserve"/> |
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6252 </p> |
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6253 <figure> |
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6254 <image file="0145-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0145-01"/> |
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6255 </figure> |
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6256 <p> |
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6257 <s xml:id="echoid-s3680" xml:space="preserve">Sit pyramis, cuius baſis <lb/>triangulũ a b c; </s> |
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6258 <s xml:id="echoid-s3681" xml:space="preserve">axis d e. <lb/></s> |
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6259 <s xml:id="echoid-s3682" xml:space="preserve">Sitq; </s> |
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6260 <s xml:id="echoid-s3683" xml:space="preserve">priſma, quod eandẽ <lb/>baſim habeat, & </s> |
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6261 <s xml:id="echoid-s3684" xml:space="preserve">axem eun <lb/>dem. </s> |
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6262 <s xml:id="echoid-s3685" xml:space="preserve">Itaque hoc priſma-<lb/>te continenter ſecto bifa-<lb/>riam, plano baſi æquidiftã <lb/>te, relinquetur tãdem priſ <lb/>ma quoddam minus pro-<lb/>poſita magnitudine: </s> |
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6263 <s xml:id="echoid-s3686" xml:space="preserve">quod <lb/>quidem baſim eandem ha <lb/>beat, quam pyramis, & </s> |
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6264 <s xml:id="echoid-s3687" xml:space="preserve">a-<lb/>xem e f. </s> |
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6265 <s xml:id="echoid-s3688" xml:space="preserve">diuidatur d e in <lb/>partes æquales ipſi e f in <lb/>punctis g h k l m n: </s> |
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6266 <s xml:id="echoid-s3689" xml:space="preserve">& </s> |
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6267 <s xml:id="echoid-s3690" xml:space="preserve">per <lb/>diuiſiones plana ducãtur: </s> |
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6268 <s xml:id="echoid-s3691" xml:space="preserve"><lb/>quæ baſibus æquidiſtent, <lb/>erunt ſectiones, triangula <lb/>ipſi a b c ſimilia, ut proxi-<lb/>me oſtendimus. </s> |
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6269 <s xml:id="echoid-s3692" xml:space="preserve">ab uno <lb/>quoque autẽ horum trian <lb/>gulorum duo priſmata cõ <lb/>ſtruantur; </s> |
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6270 <s xml:id="echoid-s3693" xml:space="preserve">unum quidem <lb/>ad partes e; </s> |
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6271 <s xml:id="echoid-s3694" xml:space="preserve">alterum ad |
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6272 <pb file="0146" n="146" rhead="FED. COMMANDINI"/> |
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6273 partes d. </s> |
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6274 <s xml:id="echoid-s3695" xml:space="preserve">in pyramide igitur inſcripta erit quædam figura, <lb/>ex priſinatibus æqualem altitudinem habentibus cóſtans, <lb/>ad partes e: </s> |
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6275 <s xml:id="echoid-s3696" xml:space="preserve">& </s> |
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6276 <s xml:id="echoid-s3697" xml:space="preserve">altera circumſcripta ad partes d. </s> |
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6277 <s xml:id="echoid-s3698" xml:space="preserve">Sed unum-<lb/>quodque eorum priſmatum, quæ in figura inſcripta conti-<lb/>nentur, æquale eſt priſmati, quod ab eodem fit triangulo in <lb/>figura circumſcripta: </s> |
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6278 <s xml:id="echoid-s3699" xml:space="preserve">nam priſma p q priſmati p o eſt æ-<lb/>quale; </s> |
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6279 <s xml:id="echoid-s3700" xml:space="preserve">priſma s t æquale priſmati s r; </s> |
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6280 <s xml:id="echoid-s3701" xml:space="preserve">priſma x y priſmati <lb/>x u; </s> |
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6281 <s xml:id="echoid-s3702" xml:space="preserve">priſma η θ priſinati η z; </s> |
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6282 <s xml:id="echoid-s3703" xml:space="preserve">priſina μ ν priſmati μ λ; </s> |
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6283 <s xml:id="echoid-s3704" xml:space="preserve">priſ-<lb/>ma ρ σ priſmati ρ π; </s> |
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6284 <s xml:id="echoid-s3705" xml:space="preserve">& </s> |
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6285 <s xml:id="echoid-s3706" xml:space="preserve">priſma φ χ priſinati φ τ æquale. </s> |
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6286 <s xml:id="echoid-s3707" xml:space="preserve">re-<lb/>linquitur ergo, ut circumſcripta figura exuperet inſcriptã <lb/>priſmate, quod baſim habet a b c triangulum, & </s> |
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6287 <s xml:id="echoid-s3708" xml:space="preserve">axem e f. <lb/></s> |
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6288 <s xml:id="echoid-s3709" xml:space="preserve">Illud uero minus eſt ſolida magnitudine propoſita. </s> |
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6289 <s xml:id="echoid-s3710" xml:space="preserve">Eadȩ <lb/>ratione inſcribetur, & </s> |
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6290 <s xml:id="echoid-s3711" xml:space="preserve">circumſcribetur ſolida figura in py-<lb/>ramide, quæ quadrilateram, uel plurilaterã baſim habeat.</s> |
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6291 <s xml:id="echoid-s3712" xml:space="preserve"/> |
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6292 </p> |
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6293 </div> |
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6294 <div xml:id="echoid-div227" type="section" level="1" n="76"> |
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6295 <head xml:id="echoid-head83" xml:space="preserve">PROBLEMA II. PROPOSITIO XI.</head> |
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6296 <p> |
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6297 <s xml:id="echoid-s3713" xml:space="preserve"><emph style="sc">Dato</emph> cono, fieri poteſt, ut figura ſolida in-<lb/>ſcribatur, & </s> |
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6298 <s xml:id="echoid-s3714" xml:space="preserve">altera circumſcribatur ex cylindris <lb/>æqualem habentibus altitudinem, ita ut circum-<lb/>ſcripta ſuperet inſcriptam, magnitudine, quæ ſo-<lb/>lida magnitudine propoſita ſit minor.</s> |
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6299 <s xml:id="echoid-s3715" xml:space="preserve"/> |
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6300 </p> |
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6301 <p> |
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6302 <s xml:id="echoid-s3716" xml:space="preserve">SIT conus, cuius axis b d: </s> |
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6303 <s xml:id="echoid-s3717" xml:space="preserve">& </s> |
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6304 <s xml:id="echoid-s3718" xml:space="preserve">ſecetur plano per axem <lb/>ducto, ut ſectio ſit triangulum a b c: </s> |
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6305 <s xml:id="echoid-s3719" xml:space="preserve">intelligaturq; </s> |
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6306 <s xml:id="echoid-s3720" xml:space="preserve">cylin-<lb/>drus, qui baſim eandem, & </s> |
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6307 <s xml:id="echoid-s3721" xml:space="preserve">eundem axem habeat. </s> |
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6308 <s xml:id="echoid-s3722" xml:space="preserve">Hoc igi-<lb/>tur cylindro continenter bifariam ſecto, relinquetur cylin <lb/>drus minor ſolida magnitudine propoſita. </s> |
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6309 <s xml:id="echoid-s3723" xml:space="preserve">Sit autem is cy <lb/>lindrus, qui baſim habet circulum circa diametrum a c, & </s> |
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6310 <s xml:id="echoid-s3724" xml:space="preserve"><lb/>axem d e. </s> |
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6311 <s xml:id="echoid-s3725" xml:space="preserve">Itaque diuidatur b d in partes æquales ipſi d e <lb/>in punctis f g h _K_lm: </s> |
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6312 <s xml:id="echoid-s3726" xml:space="preserve">& </s> |
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6313 <s xml:id="echoid-s3727" xml:space="preserve">per ea ducantur plana conum ſe-<lb/>cantia; </s> |
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6314 <s xml:id="echoid-s3728" xml:space="preserve">quæ baſi æquidiſtent. </s> |
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6315 <s xml:id="echoid-s3729" xml:space="preserve">erunt ſectiones circuli, cen-<lb/>tra in axi habentes, ut in primo libro conicorum, propoſi- |
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6316 <pb o="18" file="0147" n="147" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6317 tione quarta Apollonius demonſtrauit. </s> |
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6318 <s xml:id="echoid-s3730" xml:space="preserve">Si igitur à ſingu-<lb/>lis horum circulorum, duo cylindri fiant; </s> |
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6319 <s xml:id="echoid-s3731" xml:space="preserve">unus quidem ad <lb/>baſis partes; </s> |
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6320 <s xml:id="echoid-s3732" xml:space="preserve">alter ad partes uerticis: </s> |
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6321 <s xml:id="echoid-s3733" xml:space="preserve">inſcripta erit in co-<lb/>no ſolida quædam figura, & </s> |
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6322 <s xml:id="echoid-s3734" xml:space="preserve">altera circumſcripta ex cylin-<lb/>dris æqualem altitudinem habentibus conſtans; </s> |
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6323 <s xml:id="echoid-s3735" xml:space="preserve">quorum <lb/>unuſquiſque, qui in <lb/> |
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6324 <anchor type="figure" xlink:label="fig-0147-01a" xlink:href="fig-0147-01"/> |
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6325 figura inſcripta con-<lb/>tinetur æqualis eſt ei, <lb/>qui ab eodem fit cir-<lb/>culo in figura circũ-<lb/>ſcripta. </s> |
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6326 <s xml:id="echoid-s3736" xml:space="preserve">Itaque cylin <lb/>drus o p æqualis eſt <lb/>cylindro o n; </s> |
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6327 <s xml:id="echoid-s3737" xml:space="preserve">cylin-<lb/>drus r s cylĩdro r q; <lb/></s> |
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6328 <s xml:id="echoid-s3738" xml:space="preserve">cylindrus u x cylin-<lb/>dro u t cſt æqualis; </s> |
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6329 <s xml:id="echoid-s3739" xml:space="preserve"><lb/>& </s> |
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6330 <s xml:id="echoid-s3740" xml:space="preserve">alii aliis ſimiliter. </s> |
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6331 <s xml:id="echoid-s3741" xml:space="preserve"><lb/>quare conſtat circũ-<lb/>ſcriptam figuram ſu-<lb/>perare inſcriptam cy <lb/>lindro, cuius baſis eſt <lb/>circulus circa diametrum a c, & </s> |
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6332 <s xml:id="echoid-s3742" xml:space="preserve">axis d e. </s> |
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6333 <s xml:id="echoid-s3743" xml:space="preserve">atque hic eſtmi-<lb/>nor ſolida magnitudine propoſita.</s> |
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6334 <s xml:id="echoid-s3744" xml:space="preserve"/> |
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6335 </p> |
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6336 <div xml:id="echoid-div227" type="float" level="2" n="1"> |
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6337 <figure xlink:label="fig-0147-01" xlink:href="fig-0147-01a"> |
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6338 <image file="0147-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0147-01"/> |
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6339 </figure> |
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6340 </div> |
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6341 </div> |
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6342 <div xml:id="echoid-div229" type="section" level="1" n="77"> |
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6343 <head xml:id="echoid-head84" xml:space="preserve">PROBLEMA III. PROPOSITIO XII.</head> |
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6344 <p> |
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6345 <s xml:id="echoid-s3745" xml:space="preserve"><emph style="sc">Data</emph> coni portione, poteſt ſolida quædam <lb/>figura inſcribi, & </s> |
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6346 <s xml:id="echoid-s3746" xml:space="preserve">altera circumſcribi ex cylindri <lb/>portionibus æqualem altitudinem habentibus; <lb/></s> |
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6347 <s xml:id="echoid-s3747" xml:space="preserve">ita ut circumſcripta inſcriptam exuperet, magni <lb/>tudine, quæ minor ſit ſolida magnitudine pro-<lb/>poſita.</s> |
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6348 <s xml:id="echoid-s3748" xml:space="preserve"/> |
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6349 </p> |
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6350 <pb file="0148" n="148" rhead="FED. COMMANDINI"/> |
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6351 <p> |
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6352 <s xml:id="echoid-s3749" xml:space="preserve">Figuram ciuſmodi, & </s> |
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6353 <s xml:id="echoid-s3750" xml:space="preserve">inſcribemus, & </s> |
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6354 <s xml:id="echoid-s3751" xml:space="preserve">circũſcribemus, ita <lb/>ut in cono dictum eſt.</s> |
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6355 <s xml:id="echoid-s3752" xml:space="preserve"/> |
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6356 </p> |
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6357 <figure> |
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6358 <image file="0148-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0148-01"/> |
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6359 </figure> |
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6360 </div> |
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6361 <div xml:id="echoid-div230" type="section" level="1" n="78"> |
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6362 <head xml:id="echoid-head85" xml:space="preserve">PROBLEMA IIII. PROPOSITIO XIII.</head> |
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6363 <p> |
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6364 <s xml:id="echoid-s3753" xml:space="preserve"><emph style="sc">Data</emph> ſphæræ portione, quæ dimidia ſphæ-<lb/>ra maior non ſit, poteſt ſolida quædam portio in-<lb/>ſcribi & </s> |
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6365 <s xml:id="echoid-s3754" xml:space="preserve">altera circumſcribi ex cylindris æqualem <lb/>altitudinem habentibus, ita ut circumſcripta in-<lb/>ſcriptam excedat magnitudine, quæ ſolida ma-<lb/>gnitudine propoſita ſit minor.</s> |
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6366 <s xml:id="echoid-s3755" xml:space="preserve"/> |
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6367 </p> |
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6368 <p> |
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6369 <s xml:id="echoid-s3756" xml:space="preserve">HOC etiam eodem prorſus modo fiet: </s> |
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6370 <s xml:id="echoid-s3757" xml:space="preserve">atque ut ab <lb/>Archimede traditum eſt in conoidum, & </s> |
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6371 <s xml:id="echoid-s3758" xml:space="preserve">ſphæroidum por <lb/>tionibus, propofitione uigeſimaprima libri de conoidi-<lb/>bus, & </s> |
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6372 <s xml:id="echoid-s3759" xml:space="preserve">ſphæroidibus.</s> |
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6373 <s xml:id="echoid-s3760" xml:space="preserve"/> |
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6374 </p> |
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6375 <pb o="19" file="0149" n="149" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6376 <figure> |
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6377 <image file="0149-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0149-01"/> |
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6378 </figure> |
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6379 </div> |
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6380 <div xml:id="echoid-div231" type="section" level="1" n="79"> |
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6381 <head xml:id="echoid-head86" xml:space="preserve">THEOREMA X. PROPOSITIO XIIII.</head> |
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6382 <p> |
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6383 <s xml:id="echoid-s3761" xml:space="preserve">Cuiuslibet pyramidis, & </s> |
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6384 <s xml:id="echoid-s3762" xml:space="preserve">cuiuslibet coni, uel <lb/>coni portionis, centrum grauitatis in axe cõſiſtit.</s> |
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6385 <s xml:id="echoid-s3763" xml:space="preserve"/> |
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6386 </p> |
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6387 <p> |
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6388 <s xml:id="echoid-s3764" xml:space="preserve">SIT pyramis, cuius baſis triangulum a b c: </s> |
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6389 <s xml:id="echoid-s3765" xml:space="preserve">& </s> |
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6390 <s xml:id="echoid-s3766" xml:space="preserve">axis d e. <lb/></s> |
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6391 <s xml:id="echoid-s3767" xml:space="preserve">Dico in linea d e ipſius grauitatis centrum ineſſe. </s> |
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6392 <s xml:id="echoid-s3768" xml:space="preserve">Si enim <lb/>fieri poteſt, ſit centrum f: </s> |
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6393 <s xml:id="echoid-s3769" xml:space="preserve">& </s> |
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6394 <s xml:id="echoid-s3770" xml:space="preserve">ab f ducatur ad baſim pyrami <lb/>dis linea f g, axi æquidiſtans: </s> |
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6395 <s xml:id="echoid-s3771" xml:space="preserve">iunctaq; </s> |
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6396 <s xml:id="echoid-s3772" xml:space="preserve">e g ad latera trian-<lb/>guli a b c producatur in h. </s> |
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6397 <s xml:id="echoid-s3773" xml:space="preserve">quam uero proportionem ha-<lb/>bet linea h e ad e g, habeat pyramis ad aliud ſolidum, in <lb/>quo K: </s> |
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6398 <s xml:id="echoid-s3774" xml:space="preserve">inſcribaturq; </s> |
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6399 <s xml:id="echoid-s3775" xml:space="preserve">in pyramide ſolida figura, & </s> |
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6400 <s xml:id="echoid-s3776" xml:space="preserve">altera cir <lb/>cumſcribatur ex priſmatibus æqualem habentibus altitu-<lb/>dinem, ita ut circumſcripta inſcriptam exuperet magnitu-<lb/>dine, quæ ſolido _k_ ſit minor. </s> |
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6401 <s xml:id="echoid-s3777" xml:space="preserve">Et quoniam in pyramide pla <lb/>num baſi æquidiſtans ductum ſectionem facit figuram ſi-<lb/>milem ei, quæ eſt baſis; </s> |
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6402 <s xml:id="echoid-s3778" xml:space="preserve">centrumq; </s> |
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6403 <s xml:id="echoid-s3779" xml:space="preserve">grauitatis in axe haben <lb/>tem: </s> |
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6404 <s xml:id="echoid-s3780" xml:space="preserve">erit priſmatis s t grauitatis centrũ in linear q; </s> |
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6405 <s xml:id="echoid-s3781" xml:space="preserve">priſ-<lb/>matis u x centrum in linea q p; </s> |
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6406 <s xml:id="echoid-s3782" xml:space="preserve">priſmatis y z in linea p o; </s> |
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6407 <s xml:id="echoid-s3783" xml:space="preserve"><lb/>priſmatis η θ in l_i_nea o n; </s> |
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6408 <s xml:id="echoid-s3784" xml:space="preserve">priſmatis λ μ in linea n m; </s> |
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6409 <s xml:id="echoid-s3785" xml:space="preserve">priſ-<lb/>matis ν π in m l; </s> |
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6410 <s xml:id="echoid-s3786" xml:space="preserve">& </s> |
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6411 <s xml:id="echoid-s3787" xml:space="preserve">denique priſmatis ρ σ in l e. </s> |
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6412 <s xml:id="echoid-s3788" xml:space="preserve">quare to- |
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6413 <pb file="0150" n="150" rhead="FED. COMMANDINI"/> |
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6414 tius figuræ inſcriptæ centrum grauitatis eſt in linea r e: <lb/></s> |
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6415 <s xml:id="echoid-s3789" xml:space="preserve">quod ſit τ: </s> |
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6416 <s xml:id="echoid-s3790" xml:space="preserve">iũ-<lb/> |
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6417 <anchor type="figure" xlink:label="fig-0150-01a" xlink:href="fig-0150-01"/> |
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6418 ctaque τ f, & </s> |
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6419 <s xml:id="echoid-s3791" xml:space="preserve"><lb/>producta, à <lb/>puncto h du-<lb/>catur linea a-<lb/>xi pyramidis <lb/>æquidiſtans, <lb/>quæ cũ linea <lb/>τ f conueniat <lb/>in φ. </s> |
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6420 <s xml:id="echoid-s3792" xml:space="preserve">habebit <lb/>φ τ ad τ f ean-<lb/>dem propor-<lb/>tionem, quã <lb/>h e ad e g. <lb/></s> |
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6421 <s xml:id="echoid-s3793" xml:space="preserve">Quoniam igi <lb/>tur exceſſus, <lb/>quo circũſcri <lb/>pta figura in-<lb/>ſcriptam ſupe <lb/>rat, minor eſt <lb/>ſolido <emph style="sc">K</emph>; </s> |
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6422 <s xml:id="echoid-s3794" xml:space="preserve">py-<lb/>ramis ad eun-<lb/>dẽ exceſsũ ma <lb/>ioré propor-<lb/>tionȩ habet, <lb/>quàm ad _K_ ſo <lb/>lidum: </s> |
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6423 <s xml:id="echoid-s3795" xml:space="preserve">uideli <lb/>cet maiorem, <lb/>quàm linea h <lb/>e ad e g; </s> |
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6424 <s xml:id="echoid-s3796" xml:space="preserve">hoc <lb/>eſt quam φ τ <lb/>ad τ f: </s> |
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6425 <s xml:id="echoid-s3797" xml:space="preserve">& </s> |
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6426 <s xml:id="echoid-s3798" xml:space="preserve">propterea multo maiorem habet ad partem ex-<lb/>ceſſus, quæ intra pyramidem comprehenditur. </s> |
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6427 <s xml:id="echoid-s3799" xml:space="preserve">Itaque ha- |
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6428 <pb o="20" file="0151" n="151" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6429 beat eam, quam χ τ ad τ f. </s> |
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6430 <s xml:id="echoid-s3800" xml:space="preserve">erit diuidendo ut χ f ad f τ, ita fi <lb/>gura ſolida inſcripta ad partem exceſſus, quæ eſtintra pyra <lb/>midem. </s> |
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6431 <s xml:id="echoid-s3801" xml:space="preserve">Cum ergo à pyramide, cuius grauitatis cẽtrum eſt <lb/>punctum f, ſolida figura inſcripta auferatur, cuius centrũ <lb/>τ: </s> |
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6432 <s xml:id="echoid-s3802" xml:space="preserve">reliquæ magnitudinis conſtantis ex parte exceſſus, quæ <lb/>eſtintra pyramidem, centrum grauitatis erit in linea τ f <lb/>producta, & </s> |
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6433 <s xml:id="echoid-s3803" xml:space="preserve">in puncto χ. </s> |
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6434 <s xml:id="echoid-s3804" xml:space="preserve">quod fieri non poteſt. </s> |
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6435 <s xml:id="echoid-s3805" xml:space="preserve">Sequitur <lb/>igitur, ut centrum grauitatis pyramidis in linea d e; </s> |
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6436 <s xml:id="echoid-s3806" xml:space="preserve">hoc <lb/>eſt in eius axe conſiſtat.</s> |
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6437 <s xml:id="echoid-s3807" xml:space="preserve"/> |
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6438 </p> |
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6439 <div xml:id="echoid-div231" type="float" level="2" n="1"> |
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6440 <figure xlink:label="fig-0150-01" xlink:href="fig-0150-01a"> |
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6441 <image file="0150-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0150-01"/> |
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6442 </figure> |
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6443 </div> |
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6444 <p> |
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6445 <s xml:id="echoid-s3808" xml:space="preserve">Sit conus, uel coni portio, cuius axis b d: </s> |
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6446 <s xml:id="echoid-s3809" xml:space="preserve">& </s> |
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6447 <s xml:id="echoid-s3810" xml:space="preserve">ſecetur plano <lb/>per axem, ut ſectio ſit triangulum a b c. </s> |
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6448 <s xml:id="echoid-s3811" xml:space="preserve">Dico centrum gra <lb/>uitatis ipſius eſſe in linea b d. </s> |
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6449 <s xml:id="echoid-s3812" xml:space="preserve">Sit enim, ſi fieri poteſt, centrũ <lb/> |
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6450 <anchor type="figure" xlink:label="fig-0151-01a" xlink:href="fig-0151-01"/> |
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6451 e: </s> |
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6452 <s xml:id="echoid-s3813" xml:space="preserve">perq; </s> |
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6453 <s xml:id="echoid-s3814" xml:space="preserve">e ducatur e f axi æquidiſtans: </s> |
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6454 <s xml:id="echoid-s3815" xml:space="preserve">& </s> |
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6455 <s xml:id="echoid-s3816" xml:space="preserve">quam propor-<lb/>tionem habet c d ad d f, habeat conus, uel coni portio ad <lb/>ſolidum g. </s> |
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6456 <s xml:id="echoid-s3817" xml:space="preserve">inſcribatur ergo in cono, uel coni portione ſoli |
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6457 <pb file="0152" n="152" rhead="FED. COMMANDINI"/> |
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6458 da figura, & </s> |
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6459 <s xml:id="echoid-s3818" xml:space="preserve">altera circumſcribatur ex cylindris, uel cylin-<lb/>dri portionibus, ſicuti dictum eſt, ita ut exceſſus, quo figu-<lb/>ra circumſcripta inſcriptam ſuperat, ſit ſolido g minor. <lb/></s> |
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6460 <s xml:id="echoid-s3819" xml:space="preserve">Itaque centrum grauitatis cylindri, uel cylindri portionis <lb/>q r eſt in linea p o; </s> |
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6461 <s xml:id="echoid-s3820" xml:space="preserve">cylindri, uel cylindri portionis st cen-<lb/>trum in linea on; </s> |
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6462 <s xml:id="echoid-s3821" xml:space="preserve">centrum u x in linea n m; </s> |
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6463 <s xml:id="echoid-s3822" xml:space="preserve">y z in m b; </s> |
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6464 <s xml:id="echoid-s3823" xml:space="preserve">η @ <lb/>in 1k; </s> |
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6465 <s xml:id="echoid-s3824" xml:space="preserve">λ μ in K h; </s> |
|
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6466 <s xml:id="echoid-s3825" xml:space="preserve">& </s> |
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6467 <s xml:id="echoid-s3826" xml:space="preserve">denique ν π centrum in h d. </s> |
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6468 <s xml:id="echoid-s3827" xml:space="preserve">ergo figu-<lb/> |
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6469 <anchor type="figure" xlink:label="fig-0152-01a" xlink:href="fig-0152-01"/> |
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6470 ræ inſcriptæ centrum eſt in linea p d. </s> |
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6471 <s xml:id="echoid-s3828" xml:space="preserve">Sitautem ρ: </s> |
|
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6472 <s xml:id="echoid-s3829" xml:space="preserve">& </s> |
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6473 <s xml:id="echoid-s3830" xml:space="preserve">iun-<lb/>cta ρ e protendatur, ut cum linea, quæ à pũctoc ducta fue-<lb/>rit axi æquidiſtans, conueniat in σ. </s> |
|
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6474 <s xml:id="echoid-s3831" xml:space="preserve">erit σ ζ ad ρ e, ut c d <lb/>ad d f: </s> |
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6475 <s xml:id="echoid-s3832" xml:space="preserve">& </s> |
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6476 <s xml:id="echoid-s3833" xml:space="preserve">conus, ſeu coni portio ad exceſſum, quo circum-<lb/>ſcripta figura inſcriptam ſuperat, habebit maiorem pro-<lb/>portionem, quàm σ ζ ad ρ e. </s> |
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6477 <s xml:id="echoid-s3834" xml:space="preserve">ergo ad partem exceſſus, quæ <lb/>intra ipſius ſuperficiem comprehenditur, multo maiorem <lb/>proportionem habebit. </s> |
|
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6478 <s xml:id="echoid-s3835" xml:space="preserve">habeat eam, quam τ ρ ad ρ e. </s> |
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|
6479 <s xml:id="echoid-s3836" xml:space="preserve">erit |
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6480 <pb o="21" file="0153" n="153" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6481 diuidendo figura ſolida inſcripta ad dictam exceſſus par-<lb/>tem, ut τ e ad e ρ. </s> |
|
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6482 <s xml:id="echoid-s3837" xml:space="preserve">& </s> |
|
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6483 <s xml:id="echoid-s3838" xml:space="preserve">quoniam à cono, ſeu coni portione, <lb/>cuius grauitatis centrum eſt e, aufertur figura inſcripta, <lb/>cuius centrum ρ: </s> |
|
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|
6484 <s xml:id="echoid-s3839" xml:space="preserve">reſiduæ magnitudinis compoſitæ ex par <lb/>te exceſſus, quæ intra coni, uel coni portionis ſuperficiem <lb/>continetur, centrum grauitatis erit in linea ζ e protracta, <lb/>atque in puncto τ. </s> |
|
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|
6485 <s xml:id="echoid-s3840" xml:space="preserve">quod eſt abſurdum. </s> |
|
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6486 <s xml:id="echoid-s3841" xml:space="preserve">cõſtat ergo centrũ <lb/>grauitatis coni, uel coni portionis, eſſe in axe b d: </s> |
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6487 <s xml:id="echoid-s3842" xml:space="preserve">quod de <lb/>monſcrandum propoſuimus.</s> |
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6488 <s xml:id="echoid-s3843" xml:space="preserve"/> |
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6489 </p> |
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6490 <div xml:id="echoid-div232" type="float" level="2" n="2"> |
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6491 <figure xlink:label="fig-0151-01" xlink:href="fig-0151-01a"> |
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6492 <image file="0151-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0151-01"/> |
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6493 </figure> |
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6494 <figure xlink:label="fig-0152-01" xlink:href="fig-0152-01a"> |
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6495 <image file="0152-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0152-01"/> |
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6496 </figure> |
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6497 </div> |
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6498 </div> |
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6499 <div xml:id="echoid-div234" type="section" level="1" n="80"> |
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6500 <head xml:id="echoid-head87" xml:space="preserve">THE OREMA XI. PROPOSITIO XV.</head> |
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6501 <p> |
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6502 <s xml:id="echoid-s3844" xml:space="preserve">Cuiuslibet portionis ſphæræ uel ſphæroidis, <lb/>quæ dimidia maior non ſit: </s> |
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6503 <s xml:id="echoid-s3845" xml:space="preserve">itemq́; </s> |
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6504 <s xml:id="echoid-s3846" xml:space="preserve">cuiuslibet por <lb/>tionis conoidis, uel abſciſſæ plano ad axem recto, <lb/>uel non recto, centrum grauitatis in axe con-<lb/>ſiſtit.</s> |
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6505 <s xml:id="echoid-s3847" xml:space="preserve"/> |
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6506 </p> |
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6507 <p> |
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6508 <s xml:id="echoid-s3848" xml:space="preserve">Demonſtratio ſimilis erit ei, quam ſupra in cono, uel co <lb/>ni portione attulimus, ne toties eadem fruſtra iterentur.</s> |
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6509 <s xml:id="echoid-s3849" xml:space="preserve"/> |
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6510 </p> |
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6511 <figure> |
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6512 <image file="0153-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0153-01"/> |
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6513 </figure> |
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6514 <pb file="0154" n="154" rhead="FED. COMMANDINI"/> |
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6515 </div> |
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6516 <div xml:id="echoid-div235" type="section" level="1" n="81"> |
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6517 <head xml:id="echoid-head88" xml:space="preserve">THE OREMA XII. PROPOSITIO XVI.</head> |
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6518 <p> |
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6519 <s xml:id="echoid-s3850" xml:space="preserve">In ſphæra, & </s> |
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6520 <s xml:id="echoid-s3851" xml:space="preserve">ſphæroide idem eſt grauitatis, & </s> |
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6521 <s xml:id="echoid-s3852" xml:space="preserve"><lb/>figuræ centrum.</s> |
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6522 <s xml:id="echoid-s3853" xml:space="preserve"/> |
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6523 </p> |
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6524 <p> |
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6525 <s xml:id="echoid-s3854" xml:space="preserve">Secetur ſphæra, uel ſphæroid<gap/>no per axem ducto; <lb/></s> |
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6526 <s xml:id="echoid-s3855" xml:space="preserve">quod ſectionem faciat circulum, <gap/>ellipſim a b c d, cuius <lb/>diameter, & </s> |
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6527 <s xml:id="echoid-s3856" xml:space="preserve">ſphæræ, uelſphæroidis axis d b; </s> |
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6528 <s xml:id="echoid-s3857" xml:space="preserve">& </s> |
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6529 <s xml:id="echoid-s3858" xml:space="preserve">centrume. </s> |
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6530 <s xml:id="echoid-s3859" xml:space="preserve"><lb/>Dico e grauitatis etiam centrum eſſe. </s> |
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6531 <s xml:id="echoid-s3860" xml:space="preserve">ſecetur enim altero <lb/>plano per e, ad planum ſecans recto, cuius fectio ſit circu-<lb/>lus circa diametrum a c. </s> |
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6532 <s xml:id="echoid-s3861" xml:space="preserve">erunt a d c, a b c dimidiæ portio-<lb/>nes ſphæræ, uel fphæroidis. </s> |
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6533 <s xml:id="echoid-s3862" xml:space="preserve">& </s> |
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6534 <s xml:id="echoid-s3863" xml:space="preserve">quoniam portionis a d c gra <lb/>uitatis centrum eſt in linea d, & </s> |
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6535 <s xml:id="echoid-s3864" xml:space="preserve">centrum portionis a b c in <lb/>ipſa b e; </s> |
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6536 <s xml:id="echoid-s3865" xml:space="preserve">totius ſphæræ, uel ſphæroidis grauitatis centrum <lb/>in axe d b conſiſtet. </s> |
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6537 <s xml:id="echoid-s3866" xml:space="preserve">Quòd ſi portionis a d c centrum graui <lb/>tatis ponatur eſſe f. </s> |
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6538 <s xml:id="echoid-s3867" xml:space="preserve">& </s> |
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6539 <s xml:id="echoid-s3868" xml:space="preserve">fiat ipſi f e æqualis e g: </s> |
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6540 <s xml:id="echoid-s3869" xml:space="preserve">punctũ g por <lb/> |
|
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6541 <anchor type="figure" xlink:label="fig-0154-01a" xlink:href="fig-0154-01"/> |
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6542 tionis a b c centrum erit. </s> |
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6543 <s xml:id="echoid-s3870" xml:space="preserve">ſolidis enim figuris ſimilibus & </s> |
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6544 <s xml:id="echoid-s3871" xml:space="preserve"><lb/> |
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6545 <anchor type="note" xlink:label="note-0154-01a" xlink:href="note-0154-01"/> |
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|
6546 æqualibus inter ſe aptatis, & </s> |
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6547 <s xml:id="echoid-s3872" xml:space="preserve">centra grauitatis ipſarum in-<lb/>ter fe aptentur neceſſe eſt. </s> |
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6548 <s xml:id="echoid-s3873" xml:space="preserve">ex quo fit, ut magnitudinis, quæ <lb/> |
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6549 <anchor type="note" xlink:label="note-0154-02a" xlink:href="note-0154-02"/> |
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6550 ex utriſque cõſtat, hoc eſt ipſius ſphæræ, uel ſphæroidis gra <lb/>uitatis centrum ſitin medio lineæ f g, uidelicet in e. </s> |
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6551 <s xml:id="echoid-s3874" xml:space="preserve">Sphæ-<lb/>ræ igitur, uel ſphæroidis grauitatis centrum eſtidem, quod <lb/>centrum figuræ.</s> |
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6552 <s xml:id="echoid-s3875" xml:space="preserve"/> |
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|
6553 </p> |
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|
6554 <div xml:id="echoid-div235" type="float" level="2" n="1"> |
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6555 <figure xlink:label="fig-0154-01" xlink:href="fig-0154-01a"> |
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6556 <image file="0154-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0154-01"/> |
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|
6557 </figure> |
|
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|
6558 <note position="left" xlink:label="note-0154-01" xlink:href="note-0154-01a" xml:space="preserve">per 2. pe-<lb/>titionem</note> |
|
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|
6559 <note position="left" xlink:label="note-0154-02" xlink:href="note-0154-02a" xml:space="preserve">4 Arch-<lb/>medis.</note> |
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6560 </div> |
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6561 <pb o="22" file="0155" n="155" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6562 <p> |
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6563 <s xml:id="echoid-s3876" xml:space="preserve">Ex demonſtratis perſpicue apparet, portioni <lb/>ſphæræ uel ſphæroidis, quæ dimidia maior eſt, cẽ <lb/>trum grauitatis in axe conſiſtere.</s> |
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6564 <s xml:id="echoid-s3877" xml:space="preserve"/> |
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6565 </p> |
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6566 <p> |
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6567 <s xml:id="echoid-s3878" xml:space="preserve">Data enim <lb/> |
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6568 <anchor type="figure" xlink:label="fig-0155-01a" xlink:href="fig-0155-01"/> |
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6569 qualibet maio <lb/>ri portiõe, quo <lb/>niã totius ſphæ <lb/>ræ, uel ſphæroi <lb/>dis grauitatis <lb/>centrum eſt in <lb/>axe; </s> |
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6570 <s xml:id="echoid-s3879" xml:space="preserve">eſt autem <lb/>& </s> |
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6571 <s xml:id="echoid-s3880" xml:space="preserve">in axe cen-<lb/>trum portio-<lb/>nis minoris: <lb/></s> |
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6572 <s xml:id="echoid-s3881" xml:space="preserve">reliquæ portionis uidelicet maioris centrum in axe neceſ-<lb/>ſario conſiſtet.</s> |
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6573 <s xml:id="echoid-s3882" xml:space="preserve"/> |
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6574 </p> |
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6575 <div xml:id="echoid-div236" type="float" level="2" n="2"> |
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6576 <figure xlink:label="fig-0155-01" xlink:href="fig-0155-01a"> |
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6577 <image file="0155-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0155-01"/> |
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6578 </figure> |
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6579 </div> |
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6580 </div> |
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6581 <div xml:id="echoid-div238" type="section" level="1" n="82"> |
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6582 <head xml:id="echoid-head89" xml:space="preserve">THE OREMA XIII. PROPOSITIO XVII.</head> |
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6583 <p> |
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6584 <s xml:id="echoid-s3883" xml:space="preserve">Cuiuslibet pyramidis triã <lb/> |
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6585 <anchor type="figure" xlink:label="fig-0155-02a" xlink:href="fig-0155-02"/> |
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6586 gularem baſim habẽtis gra <lb/>uitatis centrum eſt in pun-<lb/>cto, in quo ipſius axes con-<lb/>ueniunt.</s> |
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6587 <s xml:id="echoid-s3884" xml:space="preserve"/> |
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6588 </p> |
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6589 <div xml:id="echoid-div238" type="float" level="2" n="1"> |
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6590 <figure xlink:label="fig-0155-02" xlink:href="fig-0155-02a"> |
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6591 <image file="0155-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0155-02"/> |
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6592 </figure> |
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6593 </div> |
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6594 <p> |
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6595 <s xml:id="echoid-s3885" xml:space="preserve">Sit pyramis, cuius baſis trian <lb/>gulum a b c, axis d e: </s> |
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6596 <s xml:id="echoid-s3886" xml:space="preserve">ſitq; </s> |
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6597 <s xml:id="echoid-s3887" xml:space="preserve">trian <lb/>guli b d c grauitatis centrum f: <lb/></s> |
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6598 <s xml:id="echoid-s3888" xml:space="preserve">& </s> |
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6599 <s xml:id="echoid-s3889" xml:space="preserve">iungatur a f. </s> |
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6600 <s xml:id="echoid-s3890" xml:space="preserve">erit & </s> |
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6601 <s xml:id="echoid-s3891" xml:space="preserve">a faxis eiuſ <lb/>dem pyramidis ex tertia diffini-<lb/>tione huius. </s> |
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6602 <s xml:id="echoid-s3892" xml:space="preserve">Itaque quoniam centrum grauitatis eſt in <lb/>axe d e; </s> |
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6603 <s xml:id="echoid-s3893" xml:space="preserve">eſt autem & </s> |
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6604 <s xml:id="echoid-s3894" xml:space="preserve">in axe a f; </s> |
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6605 <s xml:id="echoid-s3895" xml:space="preserve">quod proxime demonſtraui |
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6606 <pb file="0156" n="156" rhead="FED. COMMANDINI"/> |
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6607 mus: </s> |
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6608 <s xml:id="echoid-s3896" xml:space="preserve">erit utique grauitatis centrum pyramidis punctum <lb/>g: </s> |
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6609 <s xml:id="echoid-s3897" xml:space="preserve">in quo ſcilicet ipſi axes conueniunt.</s> |
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6610 <s xml:id="echoid-s3898" xml:space="preserve"/> |
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6611 </p> |
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6612 </div> |
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6613 <div xml:id="echoid-div240" type="section" level="1" n="83"> |
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6614 <head xml:id="echoid-head90" xml:space="preserve">THEOREMA XIIII. PROPOSITIO XVIII.</head> |
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6615 <p> |
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6616 <s xml:id="echoid-s3899" xml:space="preserve"><emph style="sc">Si</emph> ſolidum parallelepipedum ſecetur plano <lb/>baſibus æquidiſtante; </s> |
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6617 <s xml:id="echoid-s3900" xml:space="preserve">erit ſolidum ad ſolidum, <lb/>ſicut altitudo ad altitudinem, uel ſicut axisad <lb/>axem.</s> |
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6618 <s xml:id="echoid-s3901" xml:space="preserve"/> |
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6619 </p> |
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6620 <p> |
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6621 <s xml:id="echoid-s3902" xml:space="preserve">Sit ſolidum parallelepipe <lb/> |
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6622 <anchor type="figure" xlink:label="fig-0156-01a" xlink:href="fig-0156-01"/> |
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6623 dum a b c d e f g h, cuius axis <lb/>k 1: </s> |
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6624 <s xml:id="echoid-s3903" xml:space="preserve">ſeceturq; </s> |
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6625 <s xml:id="echoid-s3904" xml:space="preserve">plano baſibus <lb/>æquidiſtante, quod faciat <lb/>fectionem m n o p; </s> |
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6626 <s xml:id="echoid-s3905" xml:space="preserve">& </s> |
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6627 <s xml:id="echoid-s3906" xml:space="preserve">axi in <lb/>puncto q occurrat. </s> |
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6628 <s xml:id="echoid-s3907" xml:space="preserve">Dico <lb/>ſolidum g m ad ſolidum m c <lb/>eam proportionem habere, <lb/>quam altitudo ſolidi g m ha-<lb/>betad ſolidi m c altitudi-<lb/>nem; </s> |
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6629 <s xml:id="echoid-s3908" xml:space="preserve">uel quam axis k q ad <lb/>axem q l. </s> |
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6630 <s xml:id="echoid-s3909" xml:space="preserve">Sienim axis K l ad <lb/>baſis planum ſit perpendicu <lb/>laris, & </s> |
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6631 <s xml:id="echoid-s3910" xml:space="preserve">linea g c, quæ ex quin <lb/>ta huius ipſi k l æquidiſtat, <lb/>perpendicularis erit ad idẽ <lb/>planum, & </s> |
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6632 <s xml:id="echoid-s3911" xml:space="preserve">ſolidi altitudi-<lb/>nem dimetietur. </s> |
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6633 <s xml:id="echoid-s3912" xml:space="preserve">Itaqueſo-<lb/> |
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6634 <anchor type="note" xlink:label="note-0156-01a" xlink:href="note-0156-01"/> |
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6635 lidum g m ad ſolidum m c <lb/>eam proportionem habet, <lb/>quam parallelogrammũ g n <lb/>ad parallelogrammum n c, <lb/>hoc eſt quam linea g o, quæ <lb/> |
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6636 <anchor type="note" xlink:label="note-0156-02a" xlink:href="note-0156-02"/> |
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6637 <pb o="23" file="0157" n="157" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6638 eſtſolidi g m altitudo ad o e altitudinem ſolidi m c, uel quã <lb/>axis k q ad q l axem. </s> |
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6639 <s xml:id="echoid-s3913" xml:space="preserve">Si uero axis k l non ſit perpendicularis <lb/>ad planum baſis; </s> |
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6640 <s xml:id="echoid-s3914" xml:space="preserve">ducatur a puncto k ad idem planum per <lb/>pendicularis k r, occurrẽs plano m n o p in s. </s> |
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6641 <s xml:id="echoid-s3915" xml:space="preserve">ſimiliter de-<lb/>mõſtrabimus ſolidum g m ad ſoli<gap/> m c ita eſſe, ut axis k q <lb/>ad axem q l. </s> |
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6642 <s xml:id="echoid-s3916" xml:space="preserve">Sed ut K q ad q l, ita k s altitudo ad altitudi-<lb/>nem s r, nam lineæ K l, K r à planis æquidiſtantibus in eaſ-<lb/> |
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6643 <anchor type="note" xlink:label="note-0157-01a" xlink:href="note-0157-01"/> |
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6644 dem proportiones ſecantur. </s> |
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6645 <s xml:id="echoid-s3917" xml:space="preserve">ergo ſolidum g m ad ſolidum <lb/>m c eandẽ proportionem habet, quam altitudo ad altitu <lb/>dinẽ, uel quam axis ad axem. </s> |
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6646 <s xml:id="echoid-s3918" xml:space="preserve">quod demõſtrare oportebat.</s> |
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6647 <s xml:id="echoid-s3919" xml:space="preserve"/> |
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6648 </p> |
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6649 <div xml:id="echoid-div240" type="float" level="2" n="1"> |
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6650 <figure xlink:label="fig-0156-01" xlink:href="fig-0156-01a"> |
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6651 <image file="0156-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0156-01"/> |
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6652 </figure> |
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6653 <note position="left" xlink:label="note-0156-01" xlink:href="note-0156-01a" xml:space="preserve">2. undeci <lb/>mi.</note> |
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6654 <note position="left" xlink:label="note-0156-02" xlink:href="note-0156-02a" xml:space="preserve">i. ſexti.</note> |
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6655 <note position="right" xlink:label="note-0157-01" xlink:href="note-0157-01a" xml:space="preserve">17. unde-<lb/>cimi</note> |
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6656 </div> |
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6657 </div> |
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6658 <div xml:id="echoid-div242" type="section" level="1" n="84"> |
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6659 <head xml:id="echoid-head91" xml:space="preserve">THEOREMA XV. PROPOSITIO XIX.</head> |
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6660 <p> |
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6661 <s xml:id="echoid-s3920" xml:space="preserve">Solida parallelepipedain eadem baſi, uel in <lb/>æqualibus baſibus conſtituta eam inter ſe propor <lb/>tionem habent, quam altitudines: </s> |
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6662 <s xml:id="echoid-s3921" xml:space="preserve">& </s> |
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6663 <s xml:id="echoid-s3922" xml:space="preserve">ſi axes ipſo-<lb/>rum cum baſibus æquales angulos contineant, <lb/>eam quoque, quam axes proportionem habebũt.</s> |
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6664 <s xml:id="echoid-s3923" xml:space="preserve"/> |
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6665 </p> |
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6666 <p> |
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6667 <s xml:id="echoid-s3924" xml:space="preserve">Sint ſolida parallelepipeda in eadẽ baſi cõſtituta a b c d, <lb/>a b e f: </s> |
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6668 <s xml:id="echoid-s3925" xml:space="preserve">& </s> |
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6669 <s xml:id="echoid-s3926" xml:space="preserve">ſit ſolidi a b c d altitudo minor: </s> |
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6670 <s xml:id="echoid-s3927" xml:space="preserve">producatur au-<lb/>tem planum c d adeo, utſolidum a b e f ſecet; </s> |
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6671 <s xml:id="echoid-s3928" xml:space="preserve">cuius ſectio <lb/>ſit g h. </s> |
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6672 <s xml:id="echoid-s3929" xml:space="preserve">erũſoli <lb/> |
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6673 <anchor type="figure" xlink:label="fig-0157-01a" xlink:href="fig-0157-01"/> |
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6674 <anchor type="note" xlink:label="note-0157-02a" xlink:href="note-0157-02"/> |
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6675 da a b c d, a b g h <lb/>in eadem baſi, <lb/>& </s> |
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6676 <s xml:id="echoid-s3930" xml:space="preserve">æquali altitu <lb/>dine inter ſe æ-<lb/>qualia. </s> |
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6677 <s xml:id="echoid-s3931" xml:space="preserve">Quoniã <lb/>igitur ſolidum <lb/>a b e f ſecatur <lb/>plano baſibus <lb/>æquidiſtãte, erit <lb/>ſolidum g h e f <lb/> |
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6678 <anchor type="note" xlink:label="note-0157-03a" xlink:href="note-0157-03"/> |
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6679 adipſum a b g h |
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6680 <pb file="0158" n="158" rhead="FED. COMMANDINI"/> |
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6681 ut altitudo ad altitudinem & </s> |
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6682 <s xml:id="echoid-s3932" xml:space="preserve">componendo conuertendo <lb/>que ſolidum a b g h, hoc eſt ſolidum a b c d ipſi æquale, ad <lb/> |
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6683 <anchor type="note" xlink:label="note-0158-01a" xlink:href="note-0158-01"/> |
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6684 ſolidum a b e f, ut altitudo ſolidi a b c d ad ſolidi a b e f al-<lb/>titudinem.</s> |
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6685 <s xml:id="echoid-s3933" xml:space="preserve"/> |
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6686 </p> |
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6687 <div xml:id="echoid-div242" type="float" level="2" n="1"> |
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6688 <figure xlink:label="fig-0157-01" xlink:href="fig-0157-01a"> |
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6689 <image file="0157-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0157-01"/> |
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6690 </figure> |
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6691 <note position="right" xlink:label="note-0157-02" xlink:href="note-0157-02a" xml:space="preserve">29. unde-<lb/>cimi</note> |
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6692 <note position="right" xlink:label="note-0157-03" xlink:href="note-0157-03a" xml:space="preserve">18. huius</note> |
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6693 <note position="left" xlink:label="note-0158-01" xlink:href="note-0158-01a" xml:space="preserve">7. quinti.</note> |
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6694 </div> |
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6695 <p> |
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6696 <s xml:id="echoid-s3934" xml:space="preserve">Sint ſolida parallelepipeda a b, c d in æqualibus baſibus <lb/>conſtituta: </s> |
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6697 <s xml:id="echoid-s3935" xml:space="preserve">ſitq; </s> |
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6698 <s xml:id="echoid-s3936" xml:space="preserve">b e altitudo ſolidi a b: </s> |
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6699 <s xml:id="echoid-s3937" xml:space="preserve">& </s> |
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6700 <s xml:id="echoid-s3938" xml:space="preserve">ſolidi c d altitudo <lb/>d f; </s> |
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6701 <s xml:id="echoid-s3939" xml:space="preserve">quæ quidem maior ſit, quàm b e. </s> |
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6702 <s xml:id="echoid-s3940" xml:space="preserve">Dico ſolidum a b ad <lb/>ſolidum c d eandem habere proportionem, quam be ad <lb/>d f. </s> |
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6703 <s xml:id="echoid-s3941" xml:space="preserve">abſcindatur enim à linea d f æqualis ipſi b e, quæ ſit g f: <lb/></s> |
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6704 <s xml:id="echoid-s3942" xml:space="preserve">& </s> |
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6705 <s xml:id="echoid-s3943" xml:space="preserve">per g ducatur planum ſecans ſolidum c d; </s> |
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6706 <s xml:id="echoid-s3944" xml:space="preserve">quod baſibus <lb/>æquidiſtet, faciatq; </s> |
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6707 <s xml:id="echoid-s3945" xml:space="preserve">ſectionẽ h K. </s> |
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6708 <s xml:id="echoid-s3946" xml:space="preserve">erunt ſolida a b, c k æque <lb/> |
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6709 <anchor type="note" xlink:label="note-0158-02a" xlink:href="note-0158-02"/> |
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6710 alta inter <lb/> |
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6711 <anchor type="figure" xlink:label="fig-0158-01a" xlink:href="fig-0158-01"/> |
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6712 ſe æqualia <lb/>cũ æqua-<lb/>les baſes <lb/>habeant. <lb/></s> |
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6713 <s xml:id="echoid-s3947" xml:space="preserve">Sed ſolidũ <lb/> |
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6714 <anchor type="note" xlink:label="note-0158-03a" xlink:href="note-0158-03"/> |
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6715 h d ad ſoli <lb/>dum c _K_ <lb/>eſt, ut alti <lb/>tudo d g <lb/>ad g f alti-<lb/>tudinẽ ſe <lb/>catur enim ſolidum c d plano baſi <lb/> |
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6716 <anchor type="figure" xlink:label="fig-0158-02a" xlink:href="fig-0158-02"/> |
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6717 bus æquidiſtante: </s> |
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6718 <s xml:id="echoid-s3948" xml:space="preserve">& </s> |
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6719 <s xml:id="echoid-s3949" xml:space="preserve">rurſus cõpo-<lb/>nendo, conuertendoq; </s> |
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6720 <s xml:id="echoid-s3950" xml:space="preserve">ſolidũ c _k_ <lb/>ad ſolidum c d, ut g f ad fd. </s> |
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6721 <s xml:id="echoid-s3951" xml:space="preserve">ergo <lb/> |
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6722 <anchor type="note" xlink:label="note-0158-04a" xlink:href="note-0158-04"/> |
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6723 ſolidum a b, quod eſt æquale ipſi <lb/>c k ad ſolidum c d eam proportio <lb/>nem habet, quam altitudo g f, hoc <lb/>eſt b e ad d f altitudinem.</s> |
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6724 <s xml:id="echoid-s3952" xml:space="preserve"/> |
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6725 </p> |
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6726 <div xml:id="echoid-div243" type="float" level="2" n="2"> |
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6727 <note position="left" xlink:label="note-0158-02" xlink:href="note-0158-02a" xml:space="preserve">31. unde <lb/>cimi</note> |
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6728 <figure xlink:label="fig-0158-01" xlink:href="fig-0158-01a"> |
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6729 <image file="0158-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0158-01"/> |
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6730 </figure> |
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6731 <note position="left" xlink:label="note-0158-03" xlink:href="note-0158-03a" xml:space="preserve">18. huius</note> |
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6732 <figure xlink:label="fig-0158-02" xlink:href="fig-0158-02a"> |
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6733 <image file="0158-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0158-02"/> |
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6734 </figure> |
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6735 <note position="left" xlink:label="note-0158-04" xlink:href="note-0158-04a" xml:space="preserve">7. quinti.</note> |
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6736 </div> |
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6737 <p> |
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6738 <s xml:id="echoid-s3953" xml:space="preserve">Sint deinde ſolida parallelepipe <lb/>da a b, a c in eadem baſi; </s> |
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6739 <s xml:id="echoid-s3954" xml:space="preserve">quorum <lb/>axes d e, ſ e cum ipſa æquales angu |
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6740 <pb o="24" file="0159" n="159" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6741 los contineant. </s> |
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6742 <s xml:id="echoid-s3955" xml:space="preserve">Dico ſolidum a b ad ſolidum a c eãdem ha <lb/>bere proportionem, quam axis d e ad axem e f. </s> |
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6743 <s xml:id="echoid-s3956" xml:space="preserve">Sienim <lb/>axes in eadem recta linea fuerint conſtituti, hæc duo ſoli-<lb/>da, in unum, atque i @m ſolidum conuenient. </s> |
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6744 <s xml:id="echoid-s3957" xml:space="preserve">quare ex <lb/>iis, quæ proxime tradita ſunt, habebit ſolidum a b ad ſo-<lb/>lidum a c eandem proportionem, quam axis d e ad e f <lb/>axem. </s> |
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6745 <s xml:id="echoid-s3958" xml:space="preserve">Siuero axes non ſint in eadem recta linea, demittan <lb/>tur a punctis d, f perpendiculares ad baſis planum, d g, fh: <lb/></s> |
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6746 <s xml:id="echoid-s3959" xml:space="preserve">& </s> |
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6747 <s xml:id="echoid-s3960" xml:space="preserve">iungantur e g, e h. </s> |
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6748 <s xml:id="echoid-s3961" xml:space="preserve">Quoniam igitur axes cum baſibus <lb/>æquales angulos eontinent, erit d e g angulus æqualis an-<lb/>gulo f e h: </s> |
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6749 <s xml:id="echoid-s3962" xml:space="preserve">& </s> |
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6750 <s xml:id="echoid-s3963" xml:space="preserve">ſunt <lb/> |
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6751 <anchor type="figure" xlink:label="fig-0159-01a" xlink:href="fig-0159-01"/> |
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6752 anguli ad g h re-<lb/>cti, quare & </s> |
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6753 <s xml:id="echoid-s3964" xml:space="preserve">re-<lb/>liquus e d g æqua <lb/>lis erit reliquo <lb/>e fh: </s> |
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6754 <s xml:id="echoid-s3965" xml:space="preserve">& </s> |
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6755 <s xml:id="echoid-s3966" xml:space="preserve">triangu-<lb/>lum d e g triãgu-<lb/>lo f e h ſimile. </s> |
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6756 <s xml:id="echoid-s3967" xml:space="preserve">er-<lb/>go g d ad d e eſt, <lb/>ut h f ad f e: </s> |
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6757 <s xml:id="echoid-s3968" xml:space="preserve">& </s> |
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6758 <s xml:id="echoid-s3969" xml:space="preserve">per <lb/>mutando g d ad <lb/>h f, ut d e ad e f. <lb/></s> |
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6759 <s xml:id="echoid-s3970" xml:space="preserve">Sed ſolidum a b <lb/>ad ſolidum a c <lb/>eandem propor-<lb/>tionem habet, <lb/>quam d g altitu-<lb/>do ad altitudinẽ <lb/>f h. </s> |
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6760 <s xml:id="echoid-s3971" xml:space="preserve">ergo & </s> |
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6761 <s xml:id="echoid-s3972" xml:space="preserve">ean-<lb/>dẽ habebit, quã <lb/>axis d e a l e f axẽ</s> |
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6762 </p> |
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6763 <div xml:id="echoid-div244" type="float" level="2" n="3"> |
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6764 <figure xlink:label="fig-0159-01" xlink:href="fig-0159-01a"> |
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6765 <image file="0159-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0159-01"/> |
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6766 </figure> |
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6767 </div> |
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6768 <p> |
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6769 <s xml:id="echoid-s3973" xml:space="preserve">Poſtremo ſint <lb/>ſolida parallelepi <lb/>peda a b, c d in |
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6770 <pb file="0160" n="160" rhead="FED. COMMANDINI"/> |
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6771 æqualibus baſibus, quorum axes cum baſibus æquales an <lb/>gulos faciant. </s> |
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6772 <s xml:id="echoid-s3974" xml:space="preserve">Dico ſolidum a b adſolidũ c d ita eſſe, ut axis <lb/>e f ad axem g h: </s> |
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6773 <s xml:id="echoid-s3975" xml:space="preserve">nam ſi axes ad planum baſis recti ſint, il-<lb/>lud perſpicue conſtat: </s> |
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6774 <s xml:id="echoid-s3976" xml:space="preserve">quoniam eadem linea, & </s> |
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|
6775 <s xml:id="echoid-s3977" xml:space="preserve">axem & </s> |
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6776 <s xml:id="echoid-s3978" xml:space="preserve">ſoli <lb/>di altitudinem determinabit. </s> |
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6777 <s xml:id="echoid-s3979" xml:space="preserve">Si uero ſintinclinati, à pun-<lb/>ctis e g ad ſubiectum planum perpendiculares ducantur <lb/>e k, g l: </s> |
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6778 <s xml:id="echoid-s3980" xml:space="preserve">& </s> |
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6779 <s xml:id="echoid-s3981" xml:space="preserve">iungantur f_k_, h l. </s> |
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6780 <s xml:id="echoid-s3982" xml:space="preserve">rurſus quoniam axes cum ba <lb/>ſibus æquales faciunt angulos, eodem modo demonſtrabi <lb/>tur, triangulum e f K triangulo g h l ſimile eſſe: </s> |
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6781 <s xml:id="echoid-s3983" xml:space="preserve">& </s> |
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6782 <s xml:id="echoid-s3984" xml:space="preserve">e k ad g l, <lb/>ut e f ad g h. </s> |
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6783 <s xml:id="echoid-s3985" xml:space="preserve">Solidum autem a b ad ſolidum c d eſt, ut <lb/>e K ad g l. </s> |
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6784 <s xml:id="echoid-s3986" xml:space="preserve">ergo & </s> |
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6785 <s xml:id="echoid-s3987" xml:space="preserve">ut axis e f ad axem g h. </s> |
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6786 <s xml:id="echoid-s3988" xml:space="preserve">quæ omnia de <lb/>monſtrare oportebat.</s> |
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6787 <s xml:id="echoid-s3989" xml:space="preserve"/> |
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6788 </p> |
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6789 <p> |
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6790 <s xml:id="echoid-s3990" xml:space="preserve">Ex iis quæ demonſtrata ſunt, facile conſtare <lb/>poteſt, priſmata omnia & </s> |
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6791 <s xml:id="echoid-s3991" xml:space="preserve">pyramides, quæ trian-<lb/>gulares baſes habent, ſiue in eiſdem, ſiue in æqua <lb/>libus baſibus conſtituantur, eandem proportio-<lb/> |
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6792 <anchor type="note" xlink:label="note-0160-01a" xlink:href="note-0160-01"/> |
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6793 nem habere, quam altitudines: </s> |
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6794 <s xml:id="echoid-s3992" xml:space="preserve">& </s> |
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6795 <s xml:id="echoid-s3993" xml:space="preserve">ſi axes cum ba <lb/>ſibus æquales angulos contineant, ſimiliter ean-<lb/>dem, quam axes, habere proportionem: </s> |
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6796 <s xml:id="echoid-s3994" xml:space="preserve">ſunt <lb/> |
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6797 <anchor type="note" xlink:label="note-0160-02a" xlink:href="note-0160-02"/> |
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6798 enim ſolida parallelepipeda priſmatum triangula <lb/>res baſes habentiũ dupla; </s> |
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6799 <s xml:id="echoid-s3995" xml:space="preserve">& </s> |
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6800 <s xml:id="echoid-s3996" xml:space="preserve">pyramidum ſextupla.</s> |
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6801 <s xml:id="echoid-s3997" xml:space="preserve"/> |
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6802 </p> |
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6803 <div xml:id="echoid-div245" type="float" level="2" n="4"> |
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6804 <note position="left" xlink:label="note-0160-01" xlink:href="note-0160-01a" xml:space="preserve">15. quinti</note> |
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6805 <note position="left" xlink:label="note-0160-02" xlink:href="note-0160-02a" xml:space="preserve">28. unde-<lb/>cimi.</note> |
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6806 </div> |
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6807 <note position="left" xml:space="preserve">7. duode-<lb/>cimi.</note> |
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6808 </div> |
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6809 <div xml:id="echoid-div247" type="section" level="1" n="85"> |
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6810 <head xml:id="echoid-head92" xml:space="preserve">THE OREMA XVI. PROPOSITIO XX.</head> |
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6811 <p> |
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6812 <s xml:id="echoid-s3998" xml:space="preserve">Priſmata omnia & </s> |
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6813 <s xml:id="echoid-s3999" xml:space="preserve">pyramides, quæ in eiſdem, <lb/>uel æqualibus baſibus conſtituuntur, eam inter <lb/>ſe proportionem habent, quam altitudines: </s> |
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6814 <s xml:id="echoid-s4000" xml:space="preserve">& </s> |
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6815 <s xml:id="echoid-s4001" xml:space="preserve">ſi <lb/>axes cum baſibus faciant angulos æquales, eam <lb/>etiam, quam axes habent proportionem.</s> |
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6816 <s xml:id="echoid-s4002" xml:space="preserve"/> |
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6817 </p> |
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6818 <pb o="25" file="0161" n="161" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6819 <p> |
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6820 <s xml:id="echoid-s4003" xml:space="preserve">Sint duo priſmata a e, a f, quorum eadem baſis quadri-<lb/>latera a b c d: </s> |
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6821 <s xml:id="echoid-s4004" xml:space="preserve">ſitq; </s> |
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6822 <s xml:id="echoid-s4005" xml:space="preserve">priſmatis a e altitudo e g; </s> |
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6823 <s xml:id="echoid-s4006" xml:space="preserve">& </s> |
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6824 <s xml:id="echoid-s4007" xml:space="preserve">priſmatis <lb/>a f altitudo f h. </s> |
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6825 <s xml:id="echoid-s4008" xml:space="preserve">Dico priſma a e ad priſma a f eam habere <lb/>proportionem, quam e g ad f h. </s> |
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6826 <s xml:id="echoid-s4009" xml:space="preserve">iungatur enim a c: </s> |
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6827 <s xml:id="echoid-s4010" xml:space="preserve">& </s> |
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6828 <s xml:id="echoid-s4011" xml:space="preserve">in <lb/>unoquoque priſmate duo priſmata intelligantur, quorum <lb/>baſes ſint triangu <lb/> |
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6829 <anchor type="figure" xlink:label="fig-0161-01a" xlink:href="fig-0161-01"/> |
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6830 la a b c, a c d. </s> |
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6831 <s xml:id="echoid-s4012" xml:space="preserve">habe <lb/>bunt duo priſma-<lb/>te in eadem baſi <lb/>a b c conſtituta, <lb/>proportionem eã <lb/>dem, quam ipſo-<lb/>rum altitudines e <lb/>g, f h, exiam de-<lb/>monſtratis. </s> |
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6832 <s xml:id="echoid-s4013" xml:space="preserve">& </s> |
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6833 <s xml:id="echoid-s4014" xml:space="preserve">ſi-<lb/>militer alia duo, <lb/>quæ ſunt in baſi a <lb/>c d. </s> |
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6834 <s xml:id="echoid-s4015" xml:space="preserve">quare totum priſma a e ad priſma a f eandem propor <lb/> |
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6835 <anchor type="note" xlink:label="note-0161-01a" xlink:href="note-0161-01"/> |
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6836 tionem habebit, quam altitudo e g ad f h altitudinem. <lb/></s> |
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6837 <s xml:id="echoid-s4016" xml:space="preserve">Quòd cum priſmata ſint pyramidum tripla, & </s> |
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6838 <s xml:id="echoid-s4017" xml:space="preserve">ipſæ pyrami <lb/>des, quarum eadem eſt baſis quadrilatera, & </s> |
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6839 <s xml:id="echoid-s4018" xml:space="preserve">altitudo priſ-<lb/>matum altitudini æqualis, eam inter ſe proportionem ha-<lb/>bebunt, quam altitudines.</s> |
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6840 <s xml:id="echoid-s4019" xml:space="preserve"/> |
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6841 </p> |
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6842 <div xml:id="echoid-div247" type="float" level="2" n="1"> |
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6843 <figure xlink:label="fig-0161-01" xlink:href="fig-0161-01a"> |
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6844 <image file="0161-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0161-01"/> |
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6845 </figure> |
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6846 <note position="right" xlink:label="note-0161-01" xlink:href="note-0161-01a" xml:space="preserve">12. quinti</note> |
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6847 </div> |
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6848 <p> |
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6849 <s xml:id="echoid-s4020" xml:space="preserve">Si uero priſmata baſes æquales habeant, nõ eaſdem, ſint <lb/>duo eiuſmodi priſmata a e, f l: </s> |
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6850 <s xml:id="echoid-s4021" xml:space="preserve">& </s> |
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6851 <s xml:id="echoid-s4022" xml:space="preserve">ſit baſis priſmatis a e qua <lb/>drilaterum a b c d; </s> |
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6852 <s xml:id="echoid-s4023" xml:space="preserve">& </s> |
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6853 <s xml:id="echoid-s4024" xml:space="preserve">priſmatis f l quadrilaterum f g h k. <lb/></s> |
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6854 <s xml:id="echoid-s4025" xml:space="preserve">Dico priſma a e ad priſma f l ita eſſe, ut altitudo illius ad <lb/>huius altitudinem. </s> |
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6855 <s xml:id="echoid-s4026" xml:space="preserve">nam ſi altitudo ſit eadem, intelligãtur <lb/>duæ pyramides a b c d e, f g h k l. </s> |
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6856 <s xml:id="echoid-s4027" xml:space="preserve">quæ ĩter ſe æquales erũt, <lb/> |
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6857 <anchor type="note" xlink:label="note-0161-02a" xlink:href="note-0161-02"/> |
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6858 cum æ quales baſes, & </s> |
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6859 <s xml:id="echoid-s4028" xml:space="preserve">altitudinem eandem habeant. </s> |
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6860 <s xml:id="echoid-s4029" xml:space="preserve">quare <lb/>& </s> |
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6861 <s xml:id="echoid-s4030" xml:space="preserve">priſmata a e, f l, quæ ſunt harù pyramidum tripla, æqua-<lb/> |
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6862 <anchor type="note" xlink:label="note-0161-03a" xlink:href="note-0161-03"/> |
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6863 lia ſint neceſſe eſt. </s> |
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6864 <s xml:id="echoid-s4031" xml:space="preserve">ex quibus perſpicue conſtat propoſitũ. <lb/></s> |
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6865 <s xml:id="echoid-s4032" xml:space="preserve">Si uero altitudo priſmatis f l ſit maior, à priſmate f l ab-<lb/>ſcindatur priſma fm, quod æque altum ſit, atq; </s> |
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6866 <s xml:id="echoid-s4033" xml:space="preserve">ipſum a e.</s> |
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6867 <s xml:id="echoid-s4034" xml:space="preserve"> |
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6868 <pb file="0162" n="162" rhead="FED. COMMANDINI"/> |
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6869 erunteadem ra-<lb/> |
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6870 <anchor type="figure" xlink:label="fig-0162-01a" xlink:href="fig-0162-01"/> |
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6871 tione priſmata a <lb/>e, f m inter ſe æ-<lb/>qualia. </s> |
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6872 <s xml:id="echoid-s4035" xml:space="preserve">quare ſi-<lb/>militer demon-<lb/>ſtrabitur priſma <lb/>f m ad priſma f l <lb/>eandem habere <lb/>proportionem, <lb/>quam priſmatis <lb/>f m altitudo ad <lb/>altitudinem ip-<lb/>ſius f l. </s> |
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6873 <s xml:id="echoid-s4036" xml:space="preserve">ergo & </s> |
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6874 <s xml:id="echoid-s4037" xml:space="preserve">priſma a e ad priſma f l eandem propor-<lb/>tionem habebit, quam altitudo ad altitudinem. </s> |
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6875 <s xml:id="echoid-s4038" xml:space="preserve">ſequitur <lb/>igitur ut & </s> |
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6876 <s xml:id="echoid-s4039" xml:space="preserve">pyramides, quæ in æqualibus baſibus conſtituũ <lb/>tur, eandem inter ſe ſe, quam altitudines, proportionem <lb/>habeant.</s> |
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6877 <s xml:id="echoid-s4040" xml:space="preserve"/> |
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|
6878 </p> |
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6879 <div xml:id="echoid-div248" type="float" level="2" n="2"> |
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6880 <note position="right" xlink:label="note-0161-02" xlink:href="note-0161-02a" xml:space="preserve">6. duode <lb/>cimi</note> |
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6881 <note position="right" xlink:label="note-0161-03" xlink:href="note-0161-03a" xml:space="preserve">15. quintĩ</note> |
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6882 <figure xlink:label="fig-0162-01" xlink:href="fig-0162-01a"> |
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6883 <image file="0162-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0162-01"/> |
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6884 </figure> |
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6885 </div> |
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6886 <figure> |
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6887 <image file="0162-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0162-02"/> |
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6888 </figure> |
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6889 <p> |
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6890 <s xml:id="echoid-s4041" xml:space="preserve">Sint deinde priſmata a e, a f in eadem baſi a b c d; </s> |
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6891 <s xml:id="echoid-s4042" xml:space="preserve">quorũ <lb/>axes cum baſibus æ quales angulos contineant: </s> |
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6892 <s xml:id="echoid-s4043" xml:space="preserve">& </s> |
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6893 <s xml:id="echoid-s4044" xml:space="preserve">ſit priſ- |
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6894 <pb o="26" file="0163" n="163" rhead="DE CENTRO GRAVIT. SOLID."/> |
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6895 matis a e axis g h; </s> |
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6896 <s xml:id="echoid-s4045" xml:space="preserve">& </s> |
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6897 <s xml:id="echoid-s4046" xml:space="preserve">priſmatis a f axis l h. </s> |
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6898 <s xml:id="echoid-s4047" xml:space="preserve">Dico priſma <lb/>a e ad priſma a f eam proportionem habere, quam g h ad <lb/>h l. </s> |
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6899 <s xml:id="echoid-s4048" xml:space="preserve">ducantur à punctis g l perpendiculares ad baſis pla-<lb/>num g K, l m: </s> |
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6900 <s xml:id="echoid-s4049" xml:space="preserve">& </s> |
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6901 <s xml:id="echoid-s4050" xml:space="preserve">iungantur k h, <lb/> |
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6902 <anchor type="figure" xlink:label="fig-0163-01a" xlink:href="fig-0163-01"/> |
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6903 h m. </s> |
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6904 <s xml:id="echoid-s4051" xml:space="preserve">Itaque quoniam anguli g h <lb/>k, l h m ſunt æquales, ſimiliter ut <lb/>ſupra demonſtrabimus, triangu-<lb/>la g h K, l h m ſimilia eſſe; </s> |
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6905 <s xml:id="echoid-s4052" xml:space="preserve">& </s> |
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6906 <s xml:id="echoid-s4053" xml:space="preserve">ut g <lb/>K adlm, ita g h ad h l. </s> |
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6907 <s xml:id="echoid-s4054" xml:space="preserve">habet au <lb/>tem priſma a e ad priſma a f ean <lb/>dem proportionem, quam altitu <lb/>do g k ad altitudinem l m, ſicuti <lb/>demonſtratum eſt. </s> |
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6908 <s xml:id="echoid-s4055" xml:space="preserve">ergo & </s> |
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6909 <s xml:id="echoid-s4056" xml:space="preserve">ean-<lb/>dem habebit, quam g h, ad h l. </s> |
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6910 <s xml:id="echoid-s4057" xml:space="preserve">py <lb/>ramis igitur a b c d g ad pyrami-<lb/>dem a b c d l eandem proportio-<lb/>nem habebit, quam axis g h ad h l axem.</s> |
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6911 <s xml:id="echoid-s4058" xml:space="preserve"/> |
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6912 </p> |
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6913 <div xml:id="echoid-div249" type="float" level="2" n="3"> |
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6914 <figure xlink:label="fig-0163-01" xlink:href="fig-0163-01a"> |
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6915 <image file="0163-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0163-01"/> |
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6916 </figure> |
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6917 </div> |
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6918 <figure> |
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6919 <image file="0163-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0163-02"/> |
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|
6920 </figure> |
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|
6921 <p> |
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|
6922 <s xml:id="echoid-s4059" xml:space="preserve">Denique ſint priſmata a e, k o in æqualibus baſibus a b <lb/>c d, k l m n conſtituta; </s> |
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6923 <s xml:id="echoid-s4060" xml:space="preserve">quorum axes cum baſibus æquales <lb/>faciant angulos: </s> |
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6924 <s xml:id="echoid-s4061" xml:space="preserve">ſitq; </s> |
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6925 <s xml:id="echoid-s4062" xml:space="preserve">priſmatis a e axis f g, & </s> |
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6926 <s xml:id="echoid-s4063" xml:space="preserve">altitudo f h: <lb/></s> |
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6927 <s xml:id="echoid-s4064" xml:space="preserve">priſmatis autem k o axis p q, & </s> |
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6928 <s xml:id="echoid-s4065" xml:space="preserve">altitudo p r. </s> |
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6929 <s xml:id="echoid-s4066" xml:space="preserve">Dico priſma <lb/>a e ad priſma k o ita eſſe, ut f g ad p q. </s> |
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6930 <s xml:id="echoid-s4067" xml:space="preserve">iunctis enim g h, |
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|
6931 <pb file="0164" n="164" rhead="FED. COMMANDINI"/> |
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6932 qr, eodem, quo ſupra, modo oſtendemns f g ad p q, ut f h <lb/>ad p r. </s> |
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|
6933 <s xml:id="echoid-s4068" xml:space="preserve">ſed priſma a e ad ipſum k o eſt, ut f h ad p r. </s> |
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6934 <s xml:id="echoid-s4069" xml:space="preserve">ergo <lb/>& </s> |
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6935 <s xml:id="echoid-s4070" xml:space="preserve">ut f g axis ad axem p q. </s> |
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6936 <s xml:id="echoid-s4071" xml:space="preserve">ex quibus fit, ut pyramis a b c d f <lb/>ad pyrami-<lb/> |
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6937 <anchor type="figure" xlink:label="fig-0164-01a" xlink:href="fig-0164-01"/> |
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|
6938 dẽ k l m n p <lb/>eandem-ha <lb/>beat pro-<lb/>portionẽ, <lb/>quãaxis ad <lb/>axẽ. </s> |
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6939 <s xml:id="echoid-s4072" xml:space="preserve">quod <lb/>demonſtrã <lb/>dũ fuerat.</s> |
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|
6940 <s xml:id="echoid-s4073" xml:space="preserve"/> |
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|
6941 </p> |
|
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|
6942 <div xml:id="echoid-div250" type="float" level="2" n="4"> |
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|
6943 <figure xlink:label="fig-0164-01" xlink:href="fig-0164-01a"> |
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|
6944 <image file="0164-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0164-01"/> |
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|
6945 </figure> |
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|
6946 </div> |
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|
6947 <p> |
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|
6948 <s xml:id="echoid-s4074" xml:space="preserve">Simili ra <lb/>tione in a-<lb/>liis priſma-<lb/>tibus & </s> |
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|
6949 <s xml:id="echoid-s4075" xml:space="preserve">py <lb/>ramidibus eadem demonſtrabuntur.</s> |
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|
6950 <s xml:id="echoid-s4076" xml:space="preserve"/> |
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|
6951 </p> |
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|
6952 </div> |
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|
6953 <div xml:id="echoid-div252" type="section" level="1" n="86"> |
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|
6954 <head xml:id="echoid-head93" xml:space="preserve">THEOREMA XVII. PROPOSITIO XXI.</head> |
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|
6955 <p> |
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|
6956 <s xml:id="echoid-s4077" xml:space="preserve">Priſmata omnia, & </s> |
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|
6957 <s xml:id="echoid-s4078" xml:space="preserve">pyramides inter ſe propor <lb/>tionem habent compoſitam ex proportione ba-<lb/>ſium, & </s> |
|
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|
6958 <s xml:id="echoid-s4079" xml:space="preserve">proportione altitudinum.</s> |
|
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|
6959 <s xml:id="echoid-s4080" xml:space="preserve"/> |
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|
6960 </p> |
|
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|
6961 <p> |
|
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|
6962 <s xml:id="echoid-s4081" xml:space="preserve">Sint duo priſmata a e, g m: </s> |
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|
6963 <s xml:id="echoid-s4082" xml:space="preserve">ſitq; </s> |
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|
6964 <s xml:id="echoid-s4083" xml:space="preserve">priſmatis a e baſis qua <lb/>drilaterum a b c d, & </s> |
|
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|
6965 <s xml:id="echoid-s4084" xml:space="preserve">altitudo e f: </s> |
|
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|
6966 <s xml:id="echoid-s4085" xml:space="preserve">priſmatis uero g m ba-<lb/>fis quadrilaterum g h K l, & </s> |
|
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|
6967 <s xml:id="echoid-s4086" xml:space="preserve">altitudo m n. </s> |
|
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|
6968 <s xml:id="echoid-s4087" xml:space="preserve">Dico priſma a e <lb/>ad priſma g m proportionem habere compoſitam ex pro <lb/>portione baſis a b c d ad baſim g h k l, & </s> |
|
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|
6969 <s xml:id="echoid-s4088" xml:space="preserve">ex proportione <lb/>altitudinis e f, ad altitudinem m n.</s> |
|
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|
6970 <s xml:id="echoid-s4089" xml:space="preserve"/> |
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|
6971 </p> |
|
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|
6972 <p> |
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|
6973 <s xml:id="echoid-s4090" xml:space="preserve">Sint enim primum e f, m n æquales: </s> |
|
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|
6974 <s xml:id="echoid-s4091" xml:space="preserve">& </s> |
|
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|
6975 <s xml:id="echoid-s4092" xml:space="preserve">ut baſis a b c d <lb/>ad baſim g h k l, ita fiat linea, in qua o ad lineam, in qua p: <lb/></s> |
|
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|
6976 <s xml:id="echoid-s4093" xml:space="preserve">ut autem e f ad m n, ita linea p ad lineam q. </s> |
|
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|
6977 <s xml:id="echoid-s4094" xml:space="preserve">erunt lineæ <lb/>p q inter ſe æquales. </s> |
|
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|
6978 <s xml:id="echoid-s4095" xml:space="preserve">Itaque priſma a e ad priſma g m eã |
|
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|
6979 <pb o="27" file="0165" n="165" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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|
6980 proportionem habet, quam baſis a b c d ad baſim g h k l: <lb/></s> |
|
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6981 <s xml:id="echoid-s4096" xml:space="preserve">ſi enim intelligantur duæ pyramides a b c d e, g h k l m, ha-<lb/>bebunt hæ inter ſe proportionem eandem, quam ipſarum <lb/>baſes ex ſexta duodecimi elementorum. </s> |
|
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|
6982 <s xml:id="echoid-s4097" xml:space="preserve">Sed ut baſis a b c d <lb/>ad g h K l baſim, ita linea o ad lineam p; </s> |
|
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|
6983 <s xml:id="echoid-s4098" xml:space="preserve">hoc eſt ad lineam q <lb/>ei æqualem. </s> |
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|
6984 <s xml:id="echoid-s4099" xml:space="preserve">ergo priſma a e ad priſma g m eſt, ut linea o <lb/>ad lineam q. </s> |
|
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|
6985 <s xml:id="echoid-s4100" xml:space="preserve">proportio autem o ad q cõpoſita eſt ex pro-<lb/>portione o ad p, & </s> |
|
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|
6986 <s xml:id="echoid-s4101" xml:space="preserve">ex proportione p ad q. </s> |
|
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|
6987 <s xml:id="echoid-s4102" xml:space="preserve">quare priſma <lb/>a e ad priſma g m, & </s> |
|
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|
6988 <s xml:id="echoid-s4103" xml:space="preserve">idcirco pyramis a b c d e, ad pyrami-<lb/>dem g h K l m proportionem habet ex eiſdem proportio-<lb/>nibus compoſitam, uidelicet ex proportione baſis a b c d <lb/>ad baſim g h _K_ l, & </s> |
|
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|
6989 <s xml:id="echoid-s4104" xml:space="preserve">ex proportione altitudinis e f ad m n al <lb/>titudinem. </s> |
|
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6990 <s xml:id="echoid-s4105" xml:space="preserve">Quòd ſi lineæ e f, m n inæquales ponantur, ſit <lb/>e f minor: </s> |
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6991 <s xml:id="echoid-s4106" xml:space="preserve">& </s> |
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6992 <s xml:id="echoid-s4107" xml:space="preserve">ut e f ad m n, ita fiat linea p ad lineam u: </s> |
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6993 <s xml:id="echoid-s4108" xml:space="preserve">de <lb/> |
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6994 <anchor type="figure" xlink:label="fig-0165-01a" xlink:href="fig-0165-01"/> |
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6995 inde ab ipſa m n abſcindatur r n æqualis e f: </s> |
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6996 <s xml:id="echoid-s4109" xml:space="preserve">& </s> |
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6997 <s xml:id="echoid-s4110" xml:space="preserve">per r duca-<lb/>tur planum, quod oppoſitis planis æquidiſtans faciat ſe-<lb/>ctionem s t. </s> |
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6998 <s xml:id="echoid-s4111" xml:space="preserve">erit priſma a e, ad priſma g t, ut baſis a b c d <lb/>ad baſim g h k l; </s> |
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6999 <s xml:id="echoid-s4112" xml:space="preserve">hoc eſt ut o ad p: </s> |
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7000 <s xml:id="echoid-s4113" xml:space="preserve">ut autem priſma g t ad <lb/>priſma g m, ita altitudo r n; </s> |
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7001 <s xml:id="echoid-s4114" xml:space="preserve">hoc eſt e f ad altitudinẽ m n; <lb/></s> |
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7002 <s xml:id="echoid-s4115" xml:space="preserve"> |
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7003 <anchor type="note" xlink:label="note-0165-01a" xlink:href="note-0165-01"/> |
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7004 uidelicet linea p ad lineam u. </s> |
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7005 <s xml:id="echoid-s4116" xml:space="preserve">ergo ex æquali priſma a e ad <lb/>priſma g m eſt, ut linea o ad ipſam u. </s> |
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7006 <s xml:id="echoid-s4117" xml:space="preserve">Sed proportio o ad <lb/>u cõpoſita eſt ex proportione o ad p, quæ eſt baſis a b c d <lb/>ad baſim g h k l; </s> |
|
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7007 <s xml:id="echoid-s4118" xml:space="preserve">& </s> |
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7008 <s xml:id="echoid-s4119" xml:space="preserve">ex proportione p ad u, quæ eſt altitudi-<lb/>nis e f ad altitudinem m n. </s> |
|
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7009 <s xml:id="echoid-s4120" xml:space="preserve">priſma igitur a e ad priſma g m |
|
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|
7010 <pb file="0166" n="166" rhead="FED. COMMANDINI"/> |
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|
7011 compoſitam proportionem habet ex proportione baſiũ, <lb/>& </s> |
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7012 <s xml:id="echoid-s4121" xml:space="preserve">proportione altitudinum. </s> |
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7013 <s xml:id="echoid-s4122" xml:space="preserve">Quare & </s> |
|
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|
7014 <s xml:id="echoid-s4123" xml:space="preserve">pyramis, cuius ba-<lb/>ſis eſt quadrilaterum a b c d, & </s> |
|
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7015 <s xml:id="echoid-s4124" xml:space="preserve">altitudo e f ad pyramidem, <lb/> |
|
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7016 <anchor type="figure" xlink:label="fig-0166-01a" xlink:href="fig-0166-01"/> |
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7017 cuius baſis quadrilaterum g h K l, & </s> |
|
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7018 <s xml:id="echoid-s4125" xml:space="preserve">altitudo m n, compoſi <lb/>tam habet proportionem ex proportione baſium a b c d, <lb/>g h k l, & </s> |
|
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7019 <s xml:id="echoid-s4126" xml:space="preserve">ex proportione altitudinum e f, m n. </s> |
|
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7020 <s xml:id="echoid-s4127" xml:space="preserve">quod qui-<lb/>dem demonſtraſſe oportebat.</s> |
|
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|
7021 <s xml:id="echoid-s4128" xml:space="preserve"/> |
|
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|
7022 </p> |
|
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|
7023 <div xml:id="echoid-div252" type="float" level="2" n="1"> |
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7024 <figure xlink:label="fig-0165-01" xlink:href="fig-0165-01a"> |
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|
7025 <image file="0165-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0165-01"/> |
|
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|
7026 </figure> |
|
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|
7027 <note position="right" xlink:label="note-0165-01" xlink:href="note-0165-01a" xml:space="preserve">20. huius</note> |
|
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7028 <figure xlink:label="fig-0166-01" xlink:href="fig-0166-01a"> |
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|
7029 <image file="0166-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0166-01"/> |
|
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|
7030 </figure> |
|
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|
7031 </div> |
|
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|
7032 <p> |
|
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|
7033 <s xml:id="echoid-s4129" xml:space="preserve">Ex iam demonſtratis perſpicuum eſt, priſma <lb/>ta omnia, & </s> |
|
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|
7034 <s xml:id="echoid-s4130" xml:space="preserve">pyramides, in quibus axes cum baſi-<lb/>bus æquales angulos continent, proportionem <lb/>habere compoſitam ex baſium proportione, & </s> |
|
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7035 <s xml:id="echoid-s4131" xml:space="preserve"><lb/>proportione axium. </s> |
|
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7036 <s xml:id="echoid-s4132" xml:space="preserve">demonſttatum eſt enim, a-<lb/>xes inter ſe eandem proportionem habere, quam <lb/>ipſæ altitudines.</s> |
|
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|
7037 <s xml:id="echoid-s4133" xml:space="preserve"/> |
|
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|
7038 </p> |
|
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|
7039 </div> |
|
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|
7040 <div xml:id="echoid-div254" type="section" level="1" n="87"> |
|
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|
7041 <head xml:id="echoid-head94" xml:space="preserve">THE OREMA XVIII. PROPOSITIO XXII.</head> |
|
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|
7042 <p> |
|
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|
7043 <s xml:id="echoid-s4134" xml:space="preserve"><emph style="sc">Cvivslibet</emph> pyramidis, & </s> |
|
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|
7044 <s xml:id="echoid-s4135" xml:space="preserve">cuiuslibet coni, |
|
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|
7045 <pb o="28" file="0167" n="167" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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|
7046 uel coni portionis axis à centro grauitatis ita diui <lb/>ditur, ut pars, quæ terminatur ad uerticem reli-<lb/>quæ partis, quæ ad baſim, ſit tripla.</s> |
|
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|
7047 <s xml:id="echoid-s4136" xml:space="preserve"/> |
|
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|
7048 </p> |
|
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|
7049 <p> |
|
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|
7050 <s xml:id="echoid-s4137" xml:space="preserve">Sit pyramis, cuius baſis triangulum a b c; </s> |
|
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|
7051 <s xml:id="echoid-s4138" xml:space="preserve">axis d e; </s> |
|
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|
7052 <s xml:id="echoid-s4139" xml:space="preserve">& </s> |
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|
7053 <s xml:id="echoid-s4140" xml:space="preserve">gra <lb/>uitatis centrum _K_. </s> |
|
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|
7054 <s xml:id="echoid-s4141" xml:space="preserve">Dico lineam d k ipſius _K_ e triplam eſſe. <lb/></s> |
|
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|
7055 <s xml:id="echoid-s4142" xml:space="preserve">trianguli enim b d c centrum grauitatis ſit punctum f; </s> |
|
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|
7056 <s xml:id="echoid-s4143" xml:space="preserve">triã <lb/>guli a d c centrũ g; </s> |
|
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|
7057 <s xml:id="echoid-s4144" xml:space="preserve">& </s> |
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|
7058 <s xml:id="echoid-s4145" xml:space="preserve">trianguli a d b ſit h: </s> |
|
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|
7059 <s xml:id="echoid-s4146" xml:space="preserve">& </s> |
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|
7060 <s xml:id="echoid-s4147" xml:space="preserve">iungantur a f, <lb/>b g, c h. </s> |
|
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|
7061 <s xml:id="echoid-s4148" xml:space="preserve">Quoniam igitur centrũ grauitatis pyramidis in axe <lb/>cõſiſtit: </s> |
|
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7062 <s xml:id="echoid-s4149" xml:space="preserve">ſuntq; </s> |
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7063 <s xml:id="echoid-s4150" xml:space="preserve">d e, a f, b g, c h eiuſdẽ pyramidis axes: </s> |
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7064 <s xml:id="echoid-s4151" xml:space="preserve">conue <lb/> |
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7065 <anchor type="note" xlink:label="note-0167-01a" xlink:href="note-0167-01"/> |
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7066 nient omnes in idẽ punctũ _k_, quod eſt grauitatis centrum. <lb/></s> |
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7067 <s xml:id="echoid-s4152" xml:space="preserve">Itaque animo concipiamus hanc pyramidem diuiſam in <lb/>quatuor pyramides, quarum baſes ſint ipſa pyramidis <lb/>triangula; </s> |
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7068 <s xml:id="echoid-s4153" xml:space="preserve">& </s> |
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7069 <s xml:id="echoid-s4154" xml:space="preserve">axis pun-<lb/> |
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7070 <anchor type="handwritten" xlink:label="hd-0167-01a" xlink:href="hd-0167-01"/> |
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7071 <anchor type="figure" xlink:label="fig-0167-01a" xlink:href="fig-0167-01"/> |
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7072 ctum k quæ quidem py-<lb/>ramides inter ſe æquales <lb/>ſunt, ut demõſtrabitur. <lb/></s> |
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7073 <s xml:id="echoid-s4155" xml:space="preserve">Ducatur enĩ per lineas <lb/>d c, d e planum ſecãs, ut <lb/>ſit ipſius, & </s> |
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7074 <s xml:id="echoid-s4156" xml:space="preserve">baſis a b c cõ <lb/>munis ſectio recta linea <lb/>c e l: </s> |
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7075 <s xml:id="echoid-s4157" xml:space="preserve">eiuſdẽ uero & </s> |
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7076 <s xml:id="echoid-s4158" xml:space="preserve">triã-<lb/>guli a d b ſitlinea d h l. </s> |
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7077 <s xml:id="echoid-s4159" xml:space="preserve"><lb/>erit linea a l æqualis ipſi <lb/>l b: </s> |
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7078 <s xml:id="echoid-s4160" xml:space="preserve">nam centrum graui-<lb/>tatis trianguli conſiſtit <lb/>in linea, quæ ab angulo <lb/>ad dimidiam baſim per-<lb/>ducitur, ex tertia deci-<lb/>ma Archimedis. </s> |
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7079 <s xml:id="echoid-s4161" xml:space="preserve">quare <lb/> |
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7080 <anchor type="note" xlink:label="note-0167-02a" xlink:href="note-0167-02"/> |
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7081 triangulum a c l æquale <lb/>eſt triangulo b c l: </s> |
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7082 <s xml:id="echoid-s4162" xml:space="preserve">& </s> |
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7083 <s xml:id="echoid-s4163" xml:space="preserve">propterea pyramis, cuius baſis trian-<lb/>gulum a c l, uertex d, eſt æqualis pyramidi, cuius baſis b c l <lb/>triangulum, & </s> |
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7084 <s xml:id="echoid-s4164" xml:space="preserve">idem uertex. </s> |
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7085 <s xml:id="echoid-s4165" xml:space="preserve">pyramides enim, quæ ab eodẽ <lb/> |
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7086 <anchor type="note" xlink:label="note-0167-03a" xlink:href="note-0167-03"/> |
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|
7087 <pb file="0168" n="168" rhead="FED. COMMANDINI"/> |
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7088 ſunt uertice, eandem proportionem habent, quam ipſarũ <lb/>baſes. </s> |
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7089 <s xml:id="echoid-s4166" xml:space="preserve">eadem ratione pyramis a c l k pyramidi b c l k: </s> |
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7090 <s xml:id="echoid-s4167" xml:space="preserve">& </s> |
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7091 <s xml:id="echoid-s4168" xml:space="preserve">py <lb/>ramis a d l k ipſi b d l k pyramidi æqualis erit. </s> |
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7092 <s xml:id="echoid-s4169" xml:space="preserve">Itaque ſi a py <lb/>ramide a c l d auferantur pyramides a clk, a d l k: </s> |
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7093 <s xml:id="echoid-s4170" xml:space="preserve">& </s> |
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7094 <s xml:id="echoid-s4171" xml:space="preserve">à pyra <lb/>mide b c l d auferãtur pyramides b c l k, d b l K: </s> |
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7095 <s xml:id="echoid-s4172" xml:space="preserve">quæ relin-<lb/>quuntur erunt æqualia. </s> |
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7096 <s xml:id="echoid-s4173" xml:space="preserve">æqualis igitur eſt pyramis a c d k <lb/>pyramidi b c d _K_. </s> |
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7097 <s xml:id="echoid-s4174" xml:space="preserve">Rurſus ſi per lineas a d, d e ducatur pla-<lb/>num quod pyramidem ſecet: </s> |
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7098 <s xml:id="echoid-s4175" xml:space="preserve">ſitq; </s> |
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7099 <s xml:id="echoid-s4176" xml:space="preserve">eius & </s> |
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7100 <s xml:id="echoid-s4177" xml:space="preserve">baſis communis <lb/>ſectio a e m: </s> |
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7101 <s xml:id="echoid-s4178" xml:space="preserve">ſimiliter oſtendetur pyramis a b d K æqualis <lb/>pyramidi a c d <emph style="sc">K</emph>. </s> |
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7102 <s xml:id="echoid-s4179" xml:space="preserve">ducto denique alio piano per lineas c a, <lb/>a f: </s> |
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7103 <s xml:id="echoid-s4180" xml:space="preserve">ut eius, & </s> |
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7104 <s xml:id="echoid-s4181" xml:space="preserve">trianguli c d b communis ſectio ſit c fn, py-<lb/>ramis a b c k pyramidi a c d <emph style="sc">K</emph> æqualis demonſtrabitur. </s> |
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7105 <s xml:id="echoid-s4182" xml:space="preserve">cũ <lb/>ergo tres pyramides b c d _k_, a b d k, a b c k uni, & </s> |
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7106 <s xml:id="echoid-s4183" xml:space="preserve">eidem py <lb/>ramidia c d k ſint æquales, omnes inter ſe ſe æquales erũt. <lb/></s> |
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7107 <s xml:id="echoid-s4184" xml:space="preserve">Sed ut pyramis a b c d ad pyramidem a b c k, ita d e axis ad <lb/>axem k e, ex uigeſima propoſitione huius: </s> |
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7108 <s xml:id="echoid-s4185" xml:space="preserve">ſunt enim hæ <lb/>pyramides in eadem baſi, & </s> |
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7109 <s xml:id="echoid-s4186" xml:space="preserve">axes cum baſibus æquales con <lb/>tinent angulos, quòd in eadem recta linea conſtituantur. </s> |
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7110 <s xml:id="echoid-s4187" xml:space="preserve"><lb/>quare diuidendo, ut tres pyramides a c d k, b c d _K_, a b d _K_ <lb/>ad pyramidem a b c _K_, ita d _k_ ad _K_ e. </s> |
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7111 <s xml:id="echoid-s4188" xml:space="preserve">conſtat igitur lineam <lb/>d K ipſius _K_ e triplam eſſe. </s> |
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7112 <s xml:id="echoid-s4189" xml:space="preserve">ſed & </s> |
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7113 <s xml:id="echoid-s4190" xml:space="preserve">a k tripla eſt K f: </s> |
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7114 <s xml:id="echoid-s4191" xml:space="preserve">itemque <lb/>b K ipſius _K_ g: </s> |
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7115 <s xml:id="echoid-s4192" xml:space="preserve">& </s> |
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7116 <s xml:id="echoid-s4193" xml:space="preserve">c <emph style="sc">K</emph> ipſius <emph style="sc">K</emph> l tripla. </s> |
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7117 <s xml:id="echoid-s4194" xml:space="preserve">quod eodem modo <lb/>demonſtrabimus.</s> |
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7118 <s xml:id="echoid-s4195" xml:space="preserve"/> |
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|
7119 </p> |
|
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|
7120 <div xml:id="echoid-div254" type="float" level="2" n="1"> |
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7121 <note position="right" xlink:label="note-0167-01" xlink:href="note-0167-01a" xml:space="preserve">17. huíus</note> |
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7122 <handwritten xlink:label="hd-0167-01" xlink:href="hd-0167-01a"/> |
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7123 <figure xlink:label="fig-0167-01" xlink:href="fig-0167-01a"> |
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7124 <image file="0167-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0167-01"/> |
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|
7125 </figure> |
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7126 <note position="right" xlink:label="note-0167-02" xlink:href="note-0167-02a" xml:space="preserve">1. ſexti.</note> |
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7127 <note position="right" xlink:label="note-0167-03" xlink:href="note-0167-03a" xml:space="preserve">5. duode-<lb/>cimi.</note> |
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|
7128 </div> |
|
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|
7129 <p> |
|
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|
7130 <s xml:id="echoid-s4196" xml:space="preserve">Sit pyramis, cuius baſis quadrilaterum a b c d; </s> |
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7131 <s xml:id="echoid-s4197" xml:space="preserve">axis e f: <lb/></s> |
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7132 <s xml:id="echoid-s4198" xml:space="preserve">& </s> |
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|
7133 <s xml:id="echoid-s4199" xml:space="preserve">diuidatur e fin g, ita ut e g ipſius g f ſit tripla. </s> |
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|
7134 <s xml:id="echoid-s4200" xml:space="preserve">Dico cen-<lb/>trum grauitatis pyramidis eſſe punctum g. </s> |
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7135 <s xml:id="echoid-s4201" xml:space="preserve">ducatur enim <lb/>linea b d diuidens baſim in duo triangula a b d, b c d: </s> |
|
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7136 <s xml:id="echoid-s4202" xml:space="preserve">ex <lb/>quibus intelligãtur cõſtitui duæ pyramides a b d e, b c d e: </s> |
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7137 <s xml:id="echoid-s4203" xml:space="preserve"><lb/>ſitque pyramidis a b d e axis e h; </s> |
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7138 <s xml:id="echoid-s4204" xml:space="preserve">& </s> |
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|
7139 <s xml:id="echoid-s4205" xml:space="preserve">pyramidis b c d e axis <lb/>e K: </s> |
|
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7140 <s xml:id="echoid-s4206" xml:space="preserve">& </s> |
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7141 <s xml:id="echoid-s4207" xml:space="preserve">iungatur h _K_, quæ per ftranſibit: </s> |
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7142 <s xml:id="echoid-s4208" xml:space="preserve">eſt enim in ipſa h K <lb/>centrum grauitatis magnitudinis compoſitæ ex triangulis <lb/>a b d, b c d, hoc eſt ipſius quadrilateri. </s> |
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7143 <s xml:id="echoid-s4209" xml:space="preserve">Itaque centrum gra <lb/>uitatis pyramidis a b d e ſit punctum l: </s> |
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7144 <s xml:id="echoid-s4210" xml:space="preserve">& </s> |
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|
7145 <s xml:id="echoid-s4211" xml:space="preserve">pyramidis b c d e <lb/>ſit m. </s> |
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7146 <s xml:id="echoid-s4212" xml:space="preserve">ductaigitur l m ipſi h m lineæ æquidiſtabit: </s> |
|
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7147 <s xml:id="echoid-s4213" xml:space="preserve">nam el ad <lb/> |
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7148 <anchor type="note" xlink:label="note-0168-01a" xlink:href="note-0168-01"/> |
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7149 <pb o="29" file="0169" n="169" rhead="DE CENTRO GRAVIT. SOLID."/> |
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7150 l h eandem habet proportionem, quam e m ad m k, uideli-<lb/>cet triplam. </s> |
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7151 <s xml:id="echoid-s4214" xml:space="preserve">quare linea l m ipſam e f ſecabit in puncto g: <lb/></s> |
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7152 <s xml:id="echoid-s4215" xml:space="preserve">etenim e g ad g f eſt, ut el ad l h. </s> |
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7153 <s xml:id="echoid-s4216" xml:space="preserve">præterea quoniam h k, l m <lb/>æquidiſtant, erunt triangula h e f, l e g ſimilia: </s> |
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7154 <s xml:id="echoid-s4217" xml:space="preserve">itemq; </s> |
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7155 <s xml:id="echoid-s4218" xml:space="preserve">inter <lb/>ſe ſimilia f e k, g e m: </s> |
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7156 <s xml:id="echoid-s4219" xml:space="preserve">& </s> |
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7157 <s xml:id="echoid-s4220" xml:space="preserve">ut e fad e g, ita h fad l g: </s> |
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7158 <s xml:id="echoid-s4221" xml:space="preserve">& </s> |
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|
7159 <s xml:id="echoid-s4222" xml:space="preserve">ita f _K_ ad <lb/>g m. </s> |
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7160 <s xml:id="echoid-s4223" xml:space="preserve">ergo uth fadlg, ita f k ad g m: </s> |
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7161 <s xml:id="echoid-s4224" xml:space="preserve">& </s> |
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7162 <s xml:id="echoid-s4225" xml:space="preserve">permutando uth f <lb/>ad f _K_, ita l g ad g m. </s> |
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7163 <s xml:id="echoid-s4226" xml:space="preserve">ſed cum h ſit centrum trianguli a b d; </s> |
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7164 <s xml:id="echoid-s4227" xml:space="preserve"><lb/>& </s> |
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7165 <s xml:id="echoid-s4228" xml:space="preserve">K triãguli b c d: </s> |
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7166 <s xml:id="echoid-s4229" xml:space="preserve">punctũ uero f totius quadrilateri a b c d <lb/>centrum: </s> |
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7167 <s xml:id="echoid-s4230" xml:space="preserve">erit ex 8. </s> |
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7168 <s xml:id="echoid-s4231" xml:space="preserve">Archimedis de centro grauitatis plano <lb/>rum h fad f <emph style="sc">K</emph>, ut triangulum b c d ad triangulum a b d: </s> |
|
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7169 <s xml:id="echoid-s4232" xml:space="preserve">ut <lb/>autem b c d triangulum ad triangulum a b d, ita pyramis <lb/>b c d e ad pyramidem a b d e. </s> |
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7170 <s xml:id="echoid-s4233" xml:space="preserve">ergo <lb/> |
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|
7171 <anchor type="figure" xlink:label="fig-0169-01a" xlink:href="fig-0169-01"/> |
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7172 linea lg ad g m erit, ut pyramis <lb/>b c d e ad pyramidé a b d e. </s> |
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7173 <s xml:id="echoid-s4234" xml:space="preserve">ex quo <lb/>ſequitur, ut totius pyramidis <lb/>a b c d e punctum g ſit grauitatis <lb/>centrum. </s> |
|
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7174 <s xml:id="echoid-s4235" xml:space="preserve">Rurſus ſit pyramis ba-<lb/>ſim habens pentagonum a b c d e: <lb/></s> |
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|
7175 <s xml:id="echoid-s4236" xml:space="preserve">& </s> |
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|
7176 <s xml:id="echoid-s4237" xml:space="preserve">axem f g: </s> |
|
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|
7177 <s xml:id="echoid-s4238" xml:space="preserve">diuidaturq; </s> |
|
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|
7178 <s xml:id="echoid-s4239" xml:space="preserve">axis in pũ <lb/>cto h, ita ut fh ad h g triplam habe <lb/>at proportionem. </s> |
|
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7179 <s xml:id="echoid-s4240" xml:space="preserve">Dico h grauita-<lb/>tis centrũ eſſe pyramidis a b c d e f. </s> |
|
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7180 <s xml:id="echoid-s4241" xml:space="preserve"><lb/>iungatur enim e b: </s> |
|
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7181 <s xml:id="echoid-s4242" xml:space="preserve">intelligaturq; </s> |
|
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7182 <s xml:id="echoid-s4243" xml:space="preserve"><lb/>pyramis, cuius uertex f, & </s> |
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|
7183 <s xml:id="echoid-s4244" xml:space="preserve">baſis <lb/>triangulum a b e: </s> |
|
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|
7184 <s xml:id="echoid-s4245" xml:space="preserve">& </s> |
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|
7185 <s xml:id="echoid-s4246" xml:space="preserve">alia pyramis <lb/>intelligatur eundem uerticem ha-<lb/>bens, & </s> |
|
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|
7186 <s xml:id="echoid-s4247" xml:space="preserve">baſim b c d e quadrilaterũ: </s> |
|
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|
7187 <s xml:id="echoid-s4248" xml:space="preserve"><lb/>ſit autem pyramidis a b e faxis f <emph style="sc">K</emph>, <lb/>& </s> |
|
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|
7188 <s xml:id="echoid-s4249" xml:space="preserve">grauitatis centrum l: </s> |
|
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|
7189 <s xml:id="echoid-s4250" xml:space="preserve">& </s> |
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|
7190 <s xml:id="echoid-s4251" xml:space="preserve">pyrami <lb/>dis b c d e faxis f m, & </s> |
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7191 <s xml:id="echoid-s4252" xml:space="preserve">centrum gra <lb/>uitatis n: </s> |
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|
7192 <s xml:id="echoid-s4253" xml:space="preserve">iunganturq; </s> |
|
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|
7193 <s xml:id="echoid-s4254" xml:space="preserve"><emph style="sc">K</emph> m, l n; </s> |
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|
7194 <s xml:id="echoid-s4255" xml:space="preserve"><lb/>quæ per puncta g h tranſibunt. </s> |
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|
7195 <s xml:id="echoid-s4256" xml:space="preserve"><lb/>Rurſus eodem modo, quo ſup ra, <lb/>demonſtrabimus lineas K g m, l h n ſibiipſis æ quidiſtare |
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|
7196 <pb file="0170" n="170" rhead="FED. COMMANDINI"/> |
|
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|
7197 & </s> |
|
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|
7198 <s xml:id="echoid-s4257" xml:space="preserve">denique punctum h pyramidis a b c d e f grauitatis eſſe <lb/>centrum, & </s> |
|
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|
7199 <s xml:id="echoid-s4258" xml:space="preserve">ita in aliis.</s> |
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7200 <s xml:id="echoid-s4259" xml:space="preserve"/> |
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7201 </p> |
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7202 <div xml:id="echoid-div255" type="float" level="2" n="2"> |
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7203 <note position="right" xlink:label="note-0168-01" xlink:href="note-0168-01a" xml:space="preserve">2. ſexti.</note> |
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7204 <figure xlink:label="fig-0169-01" xlink:href="fig-0169-01a"> |
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7205 <image file="0169-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0169-01"/> |
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7206 </figure> |
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7207 </div> |
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7208 <p> |
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7209 <s xml:id="echoid-s4260" xml:space="preserve">Sit conus, uel coni portio axem habens b d: </s> |
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7210 <s xml:id="echoid-s4261" xml:space="preserve">ſecetur que <lb/>plano per axem, quod ſectionem faciat triangulum a b c: <lb/></s> |
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7211 <s xml:id="echoid-s4262" xml:space="preserve">& </s> |
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7212 <s xml:id="echoid-s4263" xml:space="preserve">b d axis diuidatur in e, ita ut b e ipſius e d ſit tripla. </s> |
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7213 <s xml:id="echoid-s4264" xml:space="preserve"><lb/>Dico punctum e coni, uel coni portionis, grauitatis <lb/>eſſe centrum. </s> |
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7214 <s xml:id="echoid-s4265" xml:space="preserve">Sienim fieri poteſt, ſit centrum f: </s> |
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7215 <s xml:id="echoid-s4266" xml:space="preserve">& </s> |
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7216 <s xml:id="echoid-s4267" xml:space="preserve">pro-<lb/>ducatur e f extra figuram in g. </s> |
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7217 <s xml:id="echoid-s4268" xml:space="preserve">quam uero proportionem <lb/>habet g e ad e f, habeat baſis coni, uel coni portionis, hoc <lb/>eſt circulus, uel ellipſis circa diametrum a c ad aliud ſpa-<lb/>cium, in quo h. </s> |
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7218 <s xml:id="echoid-s4269" xml:space="preserve">Itaque in circulo, uel ellipſi plane deſcri-<lb/>batur rectilinea figura a k l m c n o p, ita ut quæ relinquũ-<lb/>tur portiones ſint minores ſpacio h: </s> |
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7219 <s xml:id="echoid-s4270" xml:space="preserve">& </s> |
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7220 <s xml:id="echoid-s4271" xml:space="preserve">intelligatur pyra-<lb/>mis baſim habens rectilineam figuram a K l m c n o p, & </s> |
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7221 <s xml:id="echoid-s4272" xml:space="preserve"><lb/>axem b d; </s> |
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7222 <s xml:id="echoid-s4273" xml:space="preserve">cuius quidem grauitatis centrum erit punctum <lb/>e, ut iam demonſtrauimus. </s> |
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7223 <s xml:id="echoid-s4274" xml:space="preserve">Et quoniam portiones ſunt <lb/>minores ſpacio h, circulus, uel ellipſis ad portiones ma-<lb/> |
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7224 <anchor type="figure" xlink:label="fig-0170-01a" xlink:href="fig-0170-01"/> |
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7225 iorem proportionem habet, quam g e a d e f. </s> |
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7226 <s xml:id="echoid-s4275" xml:space="preserve">ſed ut circu-<lb/>lus, uel ellipſis ad figuram rectilineam ſibi inſcriptam, ita <lb/>conus, uel coni portio ad pyramidem, quæ figuram rectili-<lb/>neam pro baſi habet; </s> |
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7227 <s xml:id="echoid-s4276" xml:space="preserve">& </s> |
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7228 <s xml:id="echoid-s4277" xml:space="preserve">altitudinem æqualem: </s> |
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7229 <s xml:id="echoid-s4278" xml:space="preserve">etenim ſu- |
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7230 <pb o="30" file="0171" n="171" rhead="DE CENTRO GRAVIT. SOLID."/> |
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7231 pra demonſtratum eſt, ita eſſe cylindrum, uel cylindri por-<lb/> |
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7232 <anchor type="note" xlink:label="note-0171-01a" xlink:href="note-0171-01"/> |
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7233 tionem ad priſina, cuius baſis rectilinea figura, & </s> |
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7234 <s xml:id="echoid-s4279" xml:space="preserve">æqua-<lb/>lis altitudo. </s> |
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7235 <s xml:id="echoid-s4280" xml:space="preserve">ergo per conuerſionem rationis, ut circulus, <lb/>uel ellipſis ad portiones, ita conus, uel coni portio ad por-<lb/>tiones ſolidas. </s> |
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7236 <s xml:id="echoid-s4281" xml:space="preserve">quare conus uel coni portio ad portiones <lb/>ſolidas maiorem habet proportionem, quam g e ad e f: </s> |
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7237 <s xml:id="echoid-s4282" xml:space="preserve">& </s> |
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7238 <s xml:id="echoid-s4283" xml:space="preserve"><lb/>diuidendo, pyramis ad portiones ſolidas maiorem pro-<lb/>portionem habet, quam g f ad f e. </s> |
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7239 <s xml:id="echoid-s4284" xml:space="preserve">ſiat igitur q f ad f e <lb/>ut pyramis ad dictas portiones. </s> |
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7240 <s xml:id="echoid-s4285" xml:space="preserve">Itaque quoniam à cono <lb/>uel coni portione, cuius grauitatis centrum eſt f, aufer-<lb/>tur pyramis, cuius centrum e; </s> |
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7241 <s xml:id="echoid-s4286" xml:space="preserve">reliquæ magnitudinis, <lb/>quæ ex ſolidis portionibus conſtat, centrum grauitatis <lb/>erit in linea e f protracta, & </s> |
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7242 <s xml:id="echoid-s4287" xml:space="preserve">in puncto q. </s> |
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7243 <s xml:id="echoid-s4288" xml:space="preserve">quod fieri <lb/>non poteft: </s> |
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7244 <s xml:id="echoid-s4289" xml:space="preserve">eſt enim centrum grauitatis intra. </s> |
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7245 <s xml:id="echoid-s4290" xml:space="preserve">Conſtat <lb/>igitur coni, uel coni portionis grauitatis centrum eſſe pun <lb/>ctum e. </s> |
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7246 <s xml:id="echoid-s4291" xml:space="preserve">quæ omnia demonſtrare oportebat.</s> |
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7247 <s xml:id="echoid-s4292" xml:space="preserve"/> |
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7248 </p> |
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7249 <div xml:id="echoid-div256" type="float" level="2" n="3"> |
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7250 <figure xlink:label="fig-0170-01" xlink:href="fig-0170-01a"> |
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7251 <image file="0170-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0170-01"/> |
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7252 </figure> |
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7253 <note position="right" xlink:label="note-0171-01" xlink:href="note-0171-01a" xml:space="preserve">8. huius</note> |
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7254 </div> |
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7255 </div> |
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7256 <div xml:id="echoid-div258" type="section" level="1" n="88"> |
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7257 <head xml:id="echoid-head95" xml:space="preserve">THEOREMA XIX. PROPOSITIO XXIII.</head> |
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7258 <p> |
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7259 <s xml:id="echoid-s4293" xml:space="preserve"><emph style="sc">Qvodlibet</emph> fruſtum à pyramide, quæ <lb/>triangularem baſim habeat, abſciſſum, diuiditur <lb/>in tres pyramides proportionales, in ea proportio <lb/>ne, quæ eſt lateris maioris baſis ad latus minoris <lb/>ipſi reſpondens.</s> |
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7260 <s xml:id="echoid-s4294" xml:space="preserve"/> |
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7261 </p> |
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7262 <p> |
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7263 <s xml:id="echoid-s4295" xml:space="preserve">Hoc demonſtrauit Leonardus Piſanus in libro, qui de-<lb/>praxi geometriæ inſcribitur. </s> |
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7264 <s xml:id="echoid-s4296" xml:space="preserve">Sed quoniam is adhucim-<lb/>preſſus non eſt, nos ipſius demonſtrationem breuíter <lb/>perſtringemus, rem ipſam ſecuti, non uerba. </s> |
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7265 <s xml:id="echoid-s4297" xml:space="preserve">Sit fru-<lb/>ſtum pyramidis a b c d e f, cuíus maior baſis triangulum <lb/>a b c, minor d e f: </s> |
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7266 <s xml:id="echoid-s4298" xml:space="preserve">& </s> |
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7267 <s xml:id="echoid-s4299" xml:space="preserve">iunctis a e, e c, c d, per line-<lb/>as a e, e c ducatur planum ſecans fruſtum: </s> |
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7268 <s xml:id="echoid-s4300" xml:space="preserve">itemque per <lb/>lineas e c, c d; </s> |
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7269 <s xml:id="echoid-s4301" xml:space="preserve">& </s> |
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7270 <s xml:id="echoid-s4302" xml:space="preserve">per c d, d a alia plana ducantur, quæ, <lb/>diuident fruſtum in tres pyramides a b c e, a d c e, d e f c.</s> |
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7271 <s xml:id="echoid-s4303" xml:space="preserve"> |
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7272 <pb file="0172" n="172" rhead="FED. COMMANDINI"/> |
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7273 Dico eas proportion ales eſſe in proportione, quæ eſt la-<lb/>teris a b adlatus d e, itaut earum maior ſit a b c e, me-<lb/>dia a d c e, & </s> |
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7274 <s xml:id="echoid-s4304" xml:space="preserve">minor d e f c. </s> |
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7275 <s xml:id="echoid-s4305" xml:space="preserve">Quoniam enim lineæ d e, <lb/>a b æquidiſtant; </s> |
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7276 <s xml:id="echoid-s4306" xml:space="preserve">& </s> |
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7277 <s xml:id="echoid-s4307" xml:space="preserve">interipſas ſunt triangula a b e, a d e; <lb/></s> |
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7278 <s xml:id="echoid-s4308" xml:space="preserve">erit triangulum a b e <lb/> |
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7279 <anchor type="figure" xlink:label="fig-0172-01a" xlink:href="fig-0172-01"/> |
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7280 <anchor type="note" xlink:label="note-0172-01a" xlink:href="note-0172-01"/> |
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7281 ad triangulum a d e, <lb/>ut linea a b ad lineam <lb/>d e. </s> |
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7282 <s xml:id="echoid-s4309" xml:space="preserve">ut autem triangu <lb/>lum a b e ad triangu-<lb/>lum a d e, ita pyramis <lb/> |
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7283 <anchor type="note" xlink:label="note-0172-02a" xlink:href="note-0172-02"/> |
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7284 a b e c ad pyramidem <lb/>a d e c: </s> |
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7285 <s xml:id="echoid-s4310" xml:space="preserve">habent enim <lb/>altitudinem eandem, <lb/>quæ eſt à puncto c ad <lb/>planum, in quo qua-<lb/>drilaterum a b e d. </s> |
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7286 <s xml:id="echoid-s4311" xml:space="preserve">er-<lb/> |
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7287 <anchor type="note" xlink:label="note-0172-03a" xlink:href="note-0172-03"/> |
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7288 go ut a b ad d e, ita pyramis a b e c ad pyramidem a d e c. <lb/></s> |
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7289 <s xml:id="echoid-s4312" xml:space="preserve">Rurſus quoniam æquidiſtantes ſunt a c, d f; </s> |
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7290 <s xml:id="echoid-s4313" xml:space="preserve">erit eadem <lb/>ratione pyramis a d c e ad pyramidem c d f e, ut a c ad <lb/> |
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7291 <anchor type="note" xlink:label="note-0172-04a" xlink:href="note-0172-04"/> |
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7292 d f. </s> |
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7293 <s xml:id="echoid-s4314" xml:space="preserve">Sed ut a c a l d f, ita a b ad d e, quoniam triangula <lb/>a b c, d e f ſimilia ſunt, ex nona huius. </s> |
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7294 <s xml:id="echoid-s4315" xml:space="preserve">quare ut pyramis <lb/>a b c e ad pyramidem a d c e, ita pyramis a d c e ad ipſam <lb/>d e f c. </s> |
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7295 <s xml:id="echoid-s4316" xml:space="preserve">fruſtum igitur a b c d e f diuiditur in tres pyramides <lb/>proportionales in ea proportione, quæ eſt lateris a b ad d e <lb/>latus, & </s> |
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7296 <s xml:id="echoid-s4317" xml:space="preserve">earum maior eſt c a b e, media a d c e, & </s> |
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7297 <s xml:id="echoid-s4318" xml:space="preserve">minor <lb/>d e f c. </s> |
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7298 <s xml:id="echoid-s4319" xml:space="preserve">quod demonſtrare oportebat.</s> |
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7299 <s xml:id="echoid-s4320" xml:space="preserve"/> |
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7300 </p> |
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7301 <div xml:id="echoid-div258" type="float" level="2" n="1"> |
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7302 <figure xlink:label="fig-0172-01" xlink:href="fig-0172-01a"> |
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7303 <image file="0172-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0172-01"/> |
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7304 </figure> |
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7305 <note position="left" xlink:label="note-0172-01" xlink:href="note-0172-01a" xml:space="preserve">1. ſextí.</note> |
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7306 <note position="left" xlink:label="note-0172-02" xlink:href="note-0172-02a" xml:space="preserve">5. duodeci <lb/>mi.</note> |
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7307 <note position="left" xlink:label="note-0172-03" xlink:href="note-0172-03a" xml:space="preserve">11. quinti.</note> |
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7308 <note position="left" xlink:label="note-0172-04" xlink:href="note-0172-04a" xml:space="preserve">4 ſexti.</note> |
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7309 </div> |
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7310 </div> |
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7311 <div xml:id="echoid-div260" type="section" level="1" n="89"> |
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7312 <head xml:id="echoid-head96" xml:space="preserve">PROBLEMA V. PROPOSITIO XXIIII.</head> |
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7313 <p> |
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7314 <s xml:id="echoid-s4321" xml:space="preserve"><emph style="sc">Qvodlibet</emph> fruſtum pyramidis, uel coni, <lb/>uel coni portionis, plano baſi æquidiſtanti ita ſe-<lb/>care, ut ſectio ſit proportionalis inter maiorem, <lb/>& </s> |
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7315 <s xml:id="echoid-s4322" xml:space="preserve">minorem baſim.</s> |
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7316 <s xml:id="echoid-s4323" xml:space="preserve"/> |
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7317 </p> |
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7318 <pb o="31" file="0173" n="173" rhead="DE CENTRO GRAVIT. SOLID."/> |
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7319 <p> |
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7320 <s xml:id="echoid-s4324" xml:space="preserve">SIT fruſtum pyramidis a e, cuius maior baſis triangu-<lb/>lum a b c, minor d e f: </s> |
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7321 <s xml:id="echoid-s4325" xml:space="preserve">& </s> |
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7322 <s xml:id="echoid-s4326" xml:space="preserve">oporteat ipſum plano, quod baſi <lb/>æquidiſtet, ita ſecare, ut ſectio ſit proportionalis inter triã <lb/>gula a b c, d e f. </s> |
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7323 <s xml:id="echoid-s4327" xml:space="preserve">Inueniatur inter lineas a b, d e media pro-<lb/>portionalis, quæ ſit b g: </s> |
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7324 <s xml:id="echoid-s4328" xml:space="preserve">& </s> |
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7325 <s xml:id="echoid-s4329" xml:space="preserve">à puncto g erigatur g h æquidi-<lb/>ſtans b e, ſecansq; </s> |
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7326 <s xml:id="echoid-s4330" xml:space="preserve">a d in h: </s> |
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7327 <s xml:id="echoid-s4331" xml:space="preserve">deinde per h ducatur planum <lb/>baſibus æ quidiſtans, cuius ſectio ſit triangulum h _k_ 1. </s> |
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7328 <s xml:id="echoid-s4332" xml:space="preserve">Dico <lb/>triangulum h K l proportionale eſſe inter triangula a b c, <lb/>d e f, hoc eſt triangulum a b c ad <lb/> |
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7329 <anchor type="figure" xlink:label="fig-0173-01a" xlink:href="fig-0173-01"/> |
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7330 triangulum h K l eandem habere <lb/>proportionem, quam triãgulum <lb/>h K l ad ipſum d e f. </s> |
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7331 <s xml:id="echoid-s4333" xml:space="preserve">Quoniã enim <lb/>lineæ a b, h K æquidiſtantium pla <lb/> |
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7332 <anchor type="note" xlink:label="note-0173-01a" xlink:href="note-0173-01"/> |
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7333 norum ſectiones inter ſe æquidi-<lb/>ſtant: </s> |
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7334 <s xml:id="echoid-s4334" xml:space="preserve">atque æquidiſtant b _k_, g h: <lb/></s> |
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7335 <s xml:id="echoid-s4335" xml:space="preserve">linea h _k_ ipſi g b eſt æqualis: </s> |
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7336 <s xml:id="echoid-s4336" xml:space="preserve">& </s> |
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7337 <s xml:id="echoid-s4337" xml:space="preserve">pro <lb/> |
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7338 <anchor type="note" xlink:label="note-0173-02a" xlink:href="note-0173-02"/> |
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7339 pterea proportionalis inter a b, <lb/>d e. </s> |
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7340 <s xml:id="echoid-s4338" xml:space="preserve">quare ut a b ad h K, ita eſt h <emph style="sc">K</emph> <lb/>ad d e. </s> |
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7341 <s xml:id="echoid-s4339" xml:space="preserve">fiat ut h k ad d e, ita d e <lb/>ad aliam lineam, in qua ſit m. </s> |
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7342 <s xml:id="echoid-s4340" xml:space="preserve">erit <lb/>ex æquali ut a b ad d e, ita h k ad <lb/>m. </s> |
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7343 <s xml:id="echoid-s4341" xml:space="preserve">Et quoniam triangula a b c, <lb/> |
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7344 <anchor type="note" xlink:label="note-0173-03a" xlink:href="note-0173-03"/> |
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7345 h K l, d e f ſimilia ſunt; </s> |
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7346 <s xml:id="echoid-s4342" xml:space="preserve">triangulū <lb/>a b c ad triangulum h k l eſt, ut li-<lb/> |
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7347 <anchor type="note" xlink:label="note-0173-04a" xlink:href="note-0173-04"/> |
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7348 nea a b ad lineam d e: </s> |
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7349 <s xml:id="echoid-s4343" xml:space="preserve">triangulũ <lb/>autem h k l ad ipſum d e f eſt, ut h _k_ ad m. </s> |
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7350 <s xml:id="echoid-s4344" xml:space="preserve">ergo tríangulum <lb/> |
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7351 <anchor type="note" xlink:label="note-0173-05a" xlink:href="note-0173-05"/> |
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7352 a b c ad triangulum h k l eandem proportionem habet, <lb/>quam triangulum h K l ad ipſum d e f. </s> |
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7353 <s xml:id="echoid-s4345" xml:space="preserve">Eodem modo in a-<lb/>liis fruſtis pyramidis idem demonſtrabitur.</s> |
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7354 <s xml:id="echoid-s4346" xml:space="preserve"/> |
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7355 </p> |
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7356 <div xml:id="echoid-div260" type="float" level="2" n="1"> |
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7357 <figure xlink:label="fig-0173-01" xlink:href="fig-0173-01a"> |
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7358 <image file="0173-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0173-01"/> |
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7359 </figure> |
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7360 <note position="right" xlink:label="note-0173-01" xlink:href="note-0173-01a" xml:space="preserve">16. unde <lb/>cimi</note> |
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7361 <note position="right" xlink:label="note-0173-02" xlink:href="note-0173-02a" xml:space="preserve">34. primi</note> |
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7362 <note position="right" xlink:label="note-0173-03" xlink:href="note-0173-03a" xml:space="preserve">9. huius <lb/>corol.</note> |
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7363 <note position="right" xlink:label="note-0173-04" xlink:href="note-0173-04a" xml:space="preserve">20. ſexti</note> |
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7364 <note position="right" xlink:label="note-0173-05" xlink:href="note-0173-05a" xml:space="preserve">11. quinti</note> |
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7365 </div> |
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7366 <p> |
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7367 <s xml:id="echoid-s4347" xml:space="preserve">Sit fruſtum coni, uel coni portionis a d: </s> |
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7368 <s xml:id="echoid-s4348" xml:space="preserve">& </s> |
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7369 <s xml:id="echoid-s4349" xml:space="preserve">ſecetur plano <lb/>per axem, cuius ſectio ſit a b c d, ita ut maior ipſius baſis ſit <lb/>circulus, uel ellipſis circa diametrum a b; </s> |
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7370 <s xml:id="echoid-s4350" xml:space="preserve">minor circa c d. <lb/></s> |
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7371 <s xml:id="echoid-s4351" xml:space="preserve">Rurſus inter lineas a b, c d inueniatur proportionalis b e: </s> |
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7372 <s xml:id="echoid-s4352" xml:space="preserve"><lb/>& </s> |
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7373 <s xml:id="echoid-s4353" xml:space="preserve">ab e ducta e ſ æquid_i_ſtante b d, quæ lineam c a in f ſecet, |
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7374 <pb file="0174" n="174" rhead="FED. COMMANDINI"/> |
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7375 per f planum baſibus æquidiſtans ducatur, ut ſit ſectio cir <lb/>culus, uel ellipſis circa diametrum f g. </s> |
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7376 <s xml:id="echoid-s4354" xml:space="preserve">Dico ſectionem a b <lb/>ad ſectionem f g eandem proportionem habere, quam f g <lb/>ad ipſam c d. </s> |
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7377 <s xml:id="echoid-s4355" xml:space="preserve">Simili enim ratione, qua ſupra, demonſtrabi-<lb/>tur quadratum a b ad quadratum f g ita eſſe, ut quadratũ <lb/>f g ad c d quadratum. </s> |
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7378 <s xml:id="echoid-s4356" xml:space="preserve">Sed circuli inter ſe eandem propor-<lb/> |
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7379 <anchor type="note" xlink:label="note-0174-01a" xlink:href="note-0174-01"/> |
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7380 tionem habent, quam diametrorum quadrata. </s> |
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7381 <s xml:id="echoid-s4357" xml:space="preserve">ellipſes au-<lb/>tem circa a b, f g, c d, quæ ſimiles ſunt, ut oſten dimus in cõ-<lb/>mentariis in principium libri Archimedis de conoidibus, <lb/>& </s> |
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7382 <s xml:id="echoid-s4358" xml:space="preserve">ſphæroidibus, eam habẽt proportionem, quam quadrar <lb/>ta diametrorum, quæ eiuſdem rationis ſunt, ex corollaio-<lb/>ſeptimæ propoſitionis eiuſdem li-<lb/> |
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7383 <anchor type="figure" xlink:label="fig-0174-01a" xlink:href="fig-0174-01"/> |
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7384 bri. </s> |
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7385 <s xml:id="echoid-s4359" xml:space="preserve">ellipſes enim nunc appello ip-<lb/>ſa ſpacia ellipſibus contenta. </s> |
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7386 <s xml:id="echoid-s4360" xml:space="preserve">ergo <lb/>circulus, uel ellipſis a b ad circulũ, <lb/>uel ellipſim f g eam proportionem <lb/>habet, quam circulus, uel ellipſis <lb/>f g ad circulum uel ellipſim c d. <lb/></s> |
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7387 <s xml:id="echoid-s4361" xml:space="preserve">quod quidem facienduni propo-<lb/>ſuimus.</s> |
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7388 <s xml:id="echoid-s4362" xml:space="preserve"/> |
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7389 </p> |
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7390 <div xml:id="echoid-div261" type="float" level="2" n="2"> |
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7391 <note position="left" xlink:label="note-0174-01" xlink:href="note-0174-01a" xml:space="preserve">2. duode <lb/>cimi</note> |
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7392 <figure xlink:label="fig-0174-01" xlink:href="fig-0174-01a"> |
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7393 <image file="0174-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0174-01"/> |
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7394 </figure> |
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7395 </div> |
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7396 </div> |
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7397 <div xml:id="echoid-div263" type="section" level="1" n="90"> |
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7398 <head xml:id="echoid-head97" xml:space="preserve">THEOREMA XX. PROPOSITIO XXV.</head> |
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7399 <p> |
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7400 <s xml:id="echoid-s4363" xml:space="preserve"><emph style="sc">Qvodlibet</emph> fruſtum pyramidis, uel coni, <lb/>uel coni portionis ad pyramidem, uel conum, uel <lb/>coni portionem, cuius baſis eadem eſt, & </s> |
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7401 <s xml:id="echoid-s4364" xml:space="preserve">æqualis <lb/>altitudo, eandem proportionẽ habet, quam utræ <lb/>que baſes, maior, & </s> |
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7402 <s xml:id="echoid-s4365" xml:space="preserve">minor ſimul ſumptæ vnà cũ <lb/>ea, quæ inter ipſas ſit proportionalis, ad baſim ma <lb/>iorem.</s> |
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7403 <s xml:id="echoid-s4366" xml:space="preserve"/> |
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7404 </p> |
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7405 <pb o="32" file="0175" n="175" rhead="DE CENTRO GRAVIT. SOLID."/> |
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7406 <p> |
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7407 <s xml:id="echoid-s4367" xml:space="preserve">SIT fruſtũ pyramidis, uel coni, uel coni portionis a d, <lb/>cuius maior baſis a b, minor c d. </s> |
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7408 <s xml:id="echoid-s4368" xml:space="preserve">& </s> |
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7409 <s xml:id="echoid-s4369" xml:space="preserve">ſecetur altero plano <lb/>baſi æquidiſtante, ita utſectio e f ſit proportionalis inter <lb/>baſes a b, c d. </s> |
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7410 <s xml:id="echoid-s4370" xml:space="preserve">conſtituatur autẽ pyramis, uel conus, uel co-<lb/>ni portio a g b, cuius baſis ſit eadem, quæ baſis maior fru-<lb/>ſti, & </s> |
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7411 <s xml:id="echoid-s4371" xml:space="preserve">altitudo æqualis. </s> |
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7412 <s xml:id="echoid-s4372" xml:space="preserve">Di-<lb/> |
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7413 <anchor type="figure" xlink:label="fig-0175-01a" xlink:href="fig-0175-01"/> |
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7414 co fruſtum a d ad pyrami-<lb/>dem, uel conum, uel coni <lb/>portionem a g b eandem <lb/>proportionẽ habere, quã <lb/>utræque baſes, a b, c d unà <lb/>cum e f ad baſim a b. </s> |
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7415 <s xml:id="echoid-s4373" xml:space="preserve">eſt <lb/>enim fruſtum a d æquale <lb/>pyramidi, uel cono, uel co-<lb/>ni portioni, cuius baſis ex <lb/>tribus baſibus a b, e f, c d <lb/>conſtat; </s> |
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7416 <s xml:id="echoid-s4374" xml:space="preserve">& </s> |
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7417 <s xml:id="echoid-s4375" xml:space="preserve">altitudo ipſius <lb/>altitudini eſt æqualis: </s> |
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7418 <s xml:id="echoid-s4376" xml:space="preserve">quod mox oſtendemus. </s> |
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7419 <s xml:id="echoid-s4377" xml:space="preserve">Sed pyrami <lb/>des, coni, uel coni portiões, <lb/> |
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7420 <anchor type="figure" xlink:label="fig-0175-02a" xlink:href="fig-0175-02"/> |
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7421 quæ ſunt æquali altitudine, <lb/>eãdem inter ſe, quam baſes, <lb/>proportionem habent, ſicu-<lb/>ti demonſtratum eſt, partim <lb/>ab Euclide in duodecimo li-<lb/> |
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7422 <anchor type="note" xlink:label="note-0175-01a" xlink:href="note-0175-01"/> |
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7423 bro elementorum, partim à <lb/>nobis in cõmentariis in un-<lb/>decimam propoſitionẽ Ar-<lb/>chimedis de conoidibus, & </s> |
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7424 <s xml:id="echoid-s4378" xml:space="preserve"><lb/>ſphæroidibus. </s> |
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7425 <s xml:id="echoid-s4379" xml:space="preserve">quare pyra-<lb/>mis, uel conus, uel coni por-<lb/>tio, cuius baſis eſt tribus illis <lb/>baſibus æqualis ad a g b eam <lb/>habet proportionem, quam <lb/>baſes a b, e f, c d ad ab bafim. </s> |
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7426 <s xml:id="echoid-s4380" xml:space="preserve">Fruſtum igitur a d ad a g b |
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7427 <pb file="0176" n="176" rhead="FED. COMMANDINI"/> |
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7428 pyramidem, uel conum, uel coni portionem candem pro-<lb/>portionem habet, quam baſes ab, cd unà cum e ſ ad ba-<lb/>ſim a b. </s> |
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7429 <s xml:id="echoid-s4381" xml:space="preserve">quod demonſtrare uolebamus.</s> |
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7430 <s xml:id="echoid-s4382" xml:space="preserve"/> |
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|
7431 </p> |
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7432 <div xml:id="echoid-div263" type="float" level="2" n="1"> |
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7433 <figure xlink:label="fig-0175-01" xlink:href="fig-0175-01a"> |
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7434 <image file="0175-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0175-01"/> |
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7435 </figure> |
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7436 <figure xlink:label="fig-0175-02" xlink:href="fig-0175-02a"> |
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7437 <image file="0175-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0175-02"/> |
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7438 </figure> |
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7439 <note position="right" xlink:label="note-0175-01" xlink:href="note-0175-01a" xml:space="preserve">6. 11. duo <lb/>decimi</note> |
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7440 </div> |
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7441 <p> |
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7442 <s xml:id="echoid-s4383" xml:space="preserve">Fruſtum uero a d æquale eſſe pyramidi, uel co <lb/>no, uel coni portioni, cuius baſis conſtat ex baſi-<lb/>bus a b, c d, e f, & </s> |
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7443 <s xml:id="echoid-s4384" xml:space="preserve">altitudo fruſti altitudini eſt æ-<lb/>qualis, hoc modo oſten demus.</s> |
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7444 <s xml:id="echoid-s4385" xml:space="preserve"/> |
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7445 </p> |
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7446 <p> |
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7447 <s xml:id="echoid-s4386" xml:space="preserve">Sit fruſtum pyramidis a b c d e f, cuius maior baſis trian-<lb/>gulum a b c; </s> |
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7448 <s xml:id="echoid-s4387" xml:space="preserve">minor d e f: </s> |
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7449 <s xml:id="echoid-s4388" xml:space="preserve">& </s> |
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7450 <s xml:id="echoid-s4389" xml:space="preserve">ſecetur plano baſibus æquidi-<lb/>ſtante, quod ſectionem faciat triangulum g h k inter trian-<lb/>gula a b c, d e f proportionale. </s> |
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7451 <s xml:id="echoid-s4390" xml:space="preserve">Iam ex iis, quæ demonſtrata <lb/>ſuntin 23. </s> |
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7452 <s xml:id="echoid-s4391" xml:space="preserve">huius, patet ſruſtum a b c d e f diuidi in tres pyra <lb/>mides proportionales; </s> |
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7453 <s xml:id="echoid-s4392" xml:space="preserve">& </s> |
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7454 <s xml:id="echoid-s4393" xml:space="preserve">earum maiorem eſſe pyramidẽ <lb/>a b c d minorẽ uero d e f b. </s> |
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7455 <s xml:id="echoid-s4394" xml:space="preserve">ergo pyramis à triangulo g h k <lb/>conſtituta, quæ altitudinem habeat ſruſti altitudini æqua-<lb/>lem, proportionalis eſtinter pyramides a b c d, d e f b: </s> |
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7456 <s xml:id="echoid-s4395" xml:space="preserve">& </s> |
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7457 <s xml:id="echoid-s4396" xml:space="preserve"><lb/>idcirco fruſtum a b c d e f tribus dictis pyramidibus æqua <lb/>le erit. </s> |
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7458 <s xml:id="echoid-s4397" xml:space="preserve">Itaque ſi intelligatur alia pyra-<lb/> |
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7459 <anchor type="figure" xlink:label="fig-0176-01a" xlink:href="fig-0176-01"/> |
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7460 mis æque alta, quæ baſim habeat ex tri <lb/>bus baſibus a b c, d e f, g h k conſtan-<lb/>tem; </s> |
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7461 <s xml:id="echoid-s4398" xml:space="preserve">perſpicuum eſtipſam eiſdem py-<lb/>ramidibus, & </s> |
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7462 <s xml:id="echoid-s4399" xml:space="preserve">propterea ipſi fruſto æ-<lb/>qualem eſſe.</s> |
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|
7463 <s xml:id="echoid-s4400" xml:space="preserve"/> |
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|
7464 </p> |
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|
7465 <div xml:id="echoid-div264" type="float" level="2" n="2"> |
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7466 <figure xlink:label="fig-0176-01" xlink:href="fig-0176-01a"> |
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7467 <image file="0176-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0176-01"/> |
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|
7468 </figure> |
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|
7469 </div> |
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7470 <p> |
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7471 <s xml:id="echoid-s4401" xml:space="preserve">Rurſus ſit ſruſtum pyramidis a g, cu <lb/>ius maior baſis quadrilaterum a b c d, <lb/>minor e f g h: </s> |
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7472 <s xml:id="echoid-s4402" xml:space="preserve">& </s> |
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7473 <s xml:id="echoid-s4403" xml:space="preserve">ſecetur plano baſi-<lb/>bus æquidiſtante, ita ut fiat ſectio qua-<lb/>drilaterum K lm n, quod ſit proportio <lb/>nale inter quadrilatera a b c d, e f g h. </s> |
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7474 <s xml:id="echoid-s4404" xml:space="preserve">Dico pyramidem, <lb/>cuius baſis ſit æqualis tribus quadrilateris a b c d, _k_ l m n, <lb/>e f g h, & </s> |
|
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7475 <s xml:id="echoid-s4405" xml:space="preserve">altitudo æqualis altitudini fruſti, ipſi fruſto a g <lb/>æqualem eſſe. </s> |
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7476 <s xml:id="echoid-s4406" xml:space="preserve">Ducatur enim planum per lineas f b, h d, |
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7477 <pb o="33" file="0177" n="177" rhead="DE CENTRO GRAVIT. SOLID."/> |
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7478 quod diuidat fruſtum in duo fruſta triangulares baſes ha-<lb/>bentia, uidelicet in fruſtum a b d e f h, & </s> |
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7479 <s xml:id="echoid-s4407" xml:space="preserve">in fruſtũ b c d f g h. <lb/></s> |
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7480 <s xml:id="echoid-s4408" xml:space="preserve">erit triangulum k l n proportionale inter triangula a b d, <lb/>e f h: </s> |
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7481 <s xml:id="echoid-s4409" xml:space="preserve">& </s> |
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7482 <s xml:id="echoid-s4410" xml:space="preserve">triangulum l m n proportionale inter b c d, f g h. </s> |
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7483 <s xml:id="echoid-s4411" xml:space="preserve"><lb/>ſed pyramis æque alta, cuius baſis conſtat ex tribus trian-<lb/>gulis a b d, k l n, e f h, demonſtrata <lb/> |
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7484 <anchor type="figure" xlink:label="fig-0177-01a" xlink:href="fig-0177-01"/> |
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7485 eſt ſruſto a b d e f h æqualis. </s> |
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7486 <s xml:id="echoid-s4412" xml:space="preserve">& </s> |
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7487 <s xml:id="echoid-s4413" xml:space="preserve">ſi-<lb/>militer pyramis, cuius baſis con-<lb/>ſtat ex triangulis b c d, l m n, f g h <lb/>æqualis fruſto b c d f g h: </s> |
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7488 <s xml:id="echoid-s4414" xml:space="preserve">compo-<lb/>nuntur autem tria quadrilatera a <lb/>b c d, _k_ l m n, e f g h è ſex triangu-<lb/>lis iam dictis. </s> |
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7489 <s xml:id="echoid-s4415" xml:space="preserve">pyramis igitur ba-<lb/>ſim habens æqualem tribus qua-<lb/>drilateris, & </s> |
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7490 <s xml:id="echoid-s4416" xml:space="preserve">altitudinem eandem <lb/>ipſi fruſto a g eſt æqualis. </s> |
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7491 <s xml:id="echoid-s4417" xml:space="preserve">Eodem <lb/>modo illud demõſtrabitur in aliis <lb/>eiuſmodi fruſtis.</s> |
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7492 <s xml:id="echoid-s4418" xml:space="preserve"/> |
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7493 </p> |
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7494 <div xml:id="echoid-div265" type="float" level="2" n="3"> |
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7495 <figure xlink:label="fig-0177-01" xlink:href="fig-0177-01a"> |
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7496 <image file="0177-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0177-01"/> |
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7497 </figure> |
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7498 </div> |
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7499 <p> |
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7500 <s xml:id="echoid-s4419" xml:space="preserve">Sit fruſtum coni, uel coni, uel coni portionis a d; </s> |
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7501 <s xml:id="echoid-s4420" xml:space="preserve">cuius maior ba-<lb/>ſis circulus, uel ellipſis circa diametrum a b; </s> |
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7502 <s xml:id="echoid-s4421" xml:space="preserve">minor circa <lb/>c d: </s> |
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7503 <s xml:id="echoid-s4422" xml:space="preserve">& </s> |
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7504 <s xml:id="echoid-s4423" xml:space="preserve">ſecetur plano, quod baſibus æquidiſtet, faciatq; </s> |
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7505 <s xml:id="echoid-s4424" xml:space="preserve">ſe-<lb/>ctionem circulum, uel ellipſim circa diametrum e f, ita ut <lb/>inter circulos, uel ellipſes a b, c d ſit proportionalis. </s> |
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7506 <s xml:id="echoid-s4425" xml:space="preserve">Dico <lb/>conum, uel coni portionem, cuius baſis eſt æqualis tribus <lb/>circulis, uel tribus ellipſibus a b, e f, c d; </s> |
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7507 <s xml:id="echoid-s4426" xml:space="preserve">& </s> |
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7508 <s xml:id="echoid-s4427" xml:space="preserve">altitudo eadem, <lb/>quæ fruſti a d, ipſi fruſto æqualem eſſe. </s> |
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7509 <s xml:id="echoid-s4428" xml:space="preserve">producatur enim <lb/>fruſti ſuperficies quouſque coeat in unum punctum, quod <lb/>ſit g: </s> |
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7510 <s xml:id="echoid-s4429" xml:space="preserve">& </s> |
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7511 <s xml:id="echoid-s4430" xml:space="preserve">coni, uel coni portionis a g b axis ſit g h, occurrens <lb/>planis a b, e f, c d in punctis h _k_ l: </s> |
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7512 <s xml:id="echoid-s4431" xml:space="preserve">circa circulum uero de-<lb/>ſcribatur quadratum m n o p, & </s> |
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7513 <s xml:id="echoid-s4432" xml:space="preserve">circa ellipſim rectangulũ <lb/>m n o p, quod ex ipſius diametris conſtat: </s> |
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7514 <s xml:id="echoid-s4433" xml:space="preserve">iunctisq; </s> |
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7515 <s xml:id="echoid-s4434" xml:space="preserve">g m, <lb/>g n, g o, g p, ex eodem uertice intelligatur pyramis baſim <lb/>habens dictum quadratum, uel rectangulum: </s> |
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7516 <s xml:id="echoid-s4435" xml:space="preserve">& </s> |
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7517 <s xml:id="echoid-s4436" xml:space="preserve">plana in <lb/>quibus ſunt circuli, uel ellipſes e f, c d uſque ad eius latera |
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7518 <pb file="0178" n="178" rhead="FED. COMMANDINI"/> |
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7519 producantur. </s> |
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7520 <s xml:id="echoid-s4437" xml:space="preserve">Quoniam igitur pyramis ſecatur planis bafi <lb/>æquidiſtantibus, ſectiones ſimiles erunt: </s> |
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7521 <s xml:id="echoid-s4438" xml:space="preserve">atque erunt qua-<lb/> |
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7522 <anchor type="note" xlink:label="note-0178-01a" xlink:href="note-0178-01"/> |
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7523 drata, uel rectangula circa circulos, uel ellipſes deſcripta, <lb/>quemadmodum & </s> |
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7524 <s xml:id="echoid-s4439" xml:space="preserve">in ipſa baſi. </s> |
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7525 <s xml:id="echoid-s4440" xml:space="preserve">Sed cum circuli inter ſe eã <lb/>proportionem habeant, quam diametrorum quadrata: <lb/></s> |
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7526 <s xml:id="echoid-s4441" xml:space="preserve"> |
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7527 <anchor type="note" xlink:label="note-0178-02a" xlink:href="note-0178-02"/> |
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7528 itemq; </s> |
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7529 <s xml:id="echoid-s4442" xml:space="preserve">ellipſes eam quam rectangula ex ipſarum diametris <lb/>conſtantia: </s> |
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7530 <s xml:id="echoid-s4443" xml:space="preserve">& </s> |
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7531 <s xml:id="echoid-s4444" xml:space="preserve">ſit circulus, uel ellipſis circa diametrum e f <lb/> |
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7532 <anchor type="figure" xlink:label="fig-0178-01a" xlink:href="fig-0178-01"/> |
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7533 <anchor type="note" xlink:label="note-0178-03a" xlink:href="note-0178-03"/> |
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7534 proportionalis inter circulos, uel ellipſes a b, c d; </s> |
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7535 <s xml:id="echoid-s4445" xml:space="preserve">erit re-<lb/>ctangulum e f etiam inter rectangula a b, c d proportio-<lb/>nale: </s> |
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7536 <s xml:id="echoid-s4446" xml:space="preserve">per rectangulum enim nunc breuitatis cauſa etiã ip-<lb/>ſum quadratum intelligemus. </s> |
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7537 <s xml:id="echoid-s4447" xml:space="preserve">quare ex iis, quæ proxime <lb/>dicta ſunt, pyramis baſim habens æqualem dictis rectangu <lb/>lis, & </s> |
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7538 <s xml:id="echoid-s4448" xml:space="preserve">altitudinem eandem, quam fruſtum a d, ipſi fruſto à <lb/>pyramide abſciſſo æqualis probabitur. </s> |
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7539 <s xml:id="echoid-s4449" xml:space="preserve">ut autem rectangu <lb/>lum c d ad rectangulũ e f, ita circulus, uel ellipſis c d a d e f <lb/>circulum, uel ellipſim: </s> |
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7540 <s xml:id="echoid-s4450" xml:space="preserve">componendoq; </s> |
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7541 <s xml:id="echoid-s4451" xml:space="preserve">ut rectangula c d, <lb/>e f, ad e f rectangulum, ita circuli, uel ellipſes e d, e f, ad e f: <lb/></s> |
|
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7542 <s xml:id="echoid-s4452" xml:space="preserve">& </s> |
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7543 <s xml:id="echoid-s4453" xml:space="preserve">ut rectangulum e f ad rectangulum a b, ita cir culus, uel <lb/>cllipſis e f ad a b circulum, uel ellipſim. </s> |
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7544 <s xml:id="echoid-s4454" xml:space="preserve">ergo ex æquali, & </s> |
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7545 <s xml:id="echoid-s4455" xml:space="preserve"><lb/>componendo, utrectãgula c d, e f, a b ad ipſum a b, ita cir- |
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7546 <pb o="34" file="0179" n="179" rhead="DE CENTRO GRAVIT. SOLID."/> |
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7547 culi, uel ellipſes c d, e ſ a b ad circulum, uel ellipſim a b. </s> |
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7548 <s xml:id="echoid-s4456" xml:space="preserve">In-<lb/>telligatur pyramis q baſim habens æqualem tribus rectan <lb/>gulis a b, e f, c d; </s> |
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7549 <s xml:id="echoid-s4457" xml:space="preserve">& </s> |
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7550 <s xml:id="echoid-s4458" xml:space="preserve">altitudinem eãdem, quam fruſtum a d. <lb/></s> |
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7551 <s xml:id="echoid-s4459" xml:space="preserve">intelligatur etiam conus, uel coni portio q, eadem altitudi <lb/>ne, cuius baſis ſit tribus circulis, uel tribus ellipſibus a b, <lb/>e f, c d æqualis. </s> |
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7552 <s xml:id="echoid-s4460" xml:space="preserve">poſtremo intelligatur pyramis a l b, cuius <lb/>baſis ſit rectangulum m n o p, & </s> |
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7553 <s xml:id="echoid-s4461" xml:space="preserve">altitudo eadem, quæ fru-<lb/>ſti: </s> |
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7554 <s xml:id="echoid-s4462" xml:space="preserve">itemq, intelligatur conus, uel coni portio a l b, cuius <lb/>baſis circulus, uel ellipſis circa diametrum a b, & </s> |
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7555 <s xml:id="echoid-s4463" xml:space="preserve">eadem al <lb/>titudo. </s> |
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7556 <s xml:id="echoid-s4464" xml:space="preserve">ut igitur rectangula a b, e f, c d ad rectangulum a b, <lb/> |
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7557 <anchor type="note" xlink:label="note-0179-01a" xlink:href="note-0179-01"/> |
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7558 ita pyramis q ad pyramidem a l b; </s> |
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7559 <s xml:id="echoid-s4465" xml:space="preserve">& </s> |
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7560 <s xml:id="echoid-s4466" xml:space="preserve">ut circuli, uel ellip-<lb/>ſes a b, e f, c d ad a b circulum, uel ellipſim, ita conus, uel co <lb/>ni portio q ad conum, uel coni portionem a l b. </s> |
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7561 <s xml:id="echoid-s4467" xml:space="preserve">conus <lb/>igitur, uel coni portio q ad conum, uel coni portionem <lb/>a l b eſt, ut pyramis q ad pyramidem a l b. </s> |
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7562 <s xml:id="echoid-s4468" xml:space="preserve">ſed pyramis <lb/>a l b ad pyramidem a g b eſt, ut altitudo ad altitudinem, ex <lb/>20. </s> |
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7563 <s xml:id="echoid-s4469" xml:space="preserve">huius: </s> |
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7564 <s xml:id="echoid-s4470" xml:space="preserve">& </s> |
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7565 <s xml:id="echoid-s4471" xml:space="preserve">ita eſt conus, uel coni portio al b ad conum, <lb/>uel coni portionem a g b ex 14. </s> |
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7566 <s xml:id="echoid-s4472" xml:space="preserve">duodecimi elementorum, <lb/>& </s> |
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7567 <s xml:id="echoid-s4473" xml:space="preserve">ex iis, quæ nos demonſtrauimus in commentariis in un-<lb/>decimam de conoidibus, & </s> |
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7568 <s xml:id="echoid-s4474" xml:space="preserve">ſphæroidibus, propoſitione <lb/>quarta. </s> |
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7569 <s xml:id="echoid-s4475" xml:space="preserve">pyramis autem a g b ad pyramidem c g d propor-<lb/>tionem habet compoſitam ex proportione baſium & </s> |
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7570 <s xml:id="echoid-s4476" xml:space="preserve">pro <lb/>portione altitudinum, ex uigeſima prima huius: </s> |
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7571 <s xml:id="echoid-s4477" xml:space="preserve">& </s> |
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7572 <s xml:id="echoid-s4478" xml:space="preserve">ſimili-<lb/>ter conus, uel coni portio a g b a d conum, uel coni portio-<lb/>nem c g d proportionem habet compoſitã ex eiſdem pro-<lb/>portionibus, per ea, quæ in dictis commentariis demon-<lb/>ſtrauimus, propoſitione quinta, & </s> |
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7573 <s xml:id="echoid-s4479" xml:space="preserve">ſexta: </s> |
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7574 <s xml:id="echoid-s4480" xml:space="preserve">altitudo enim in <lb/>utriſque eadem eſt, & </s> |
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7575 <s xml:id="echoid-s4481" xml:space="preserve">baſes inter ſe ſe eandem habent pro-<lb/>portionem. </s> |
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7576 <s xml:id="echoid-s4482" xml:space="preserve">ergo ut pyramis a g b ad pyramidem c g d, ita <lb/>eſt conus, uel coni portio a g b ad a g d conum, uel coni <lb/>portionem: </s> |
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7577 <s xml:id="echoid-s4483" xml:space="preserve">& </s> |
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7578 <s xml:id="echoid-s4484" xml:space="preserve">per conuerſionẽ rationis, ut pyramis a g b <lb/>ad fruſtū à pyramide abſciſſum, ita conus uel coni portio <lb/>a g b ad fruſtum a d. </s> |
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7579 <s xml:id="echoid-s4485" xml:space="preserve">ex æquali igitur, ut pyramis q ad fru-<lb/>ſtum à pyramide abſciſſum, ita conus uel coni portio q ad |
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7580 <pb file="0180" n="180" rhead="FED. COMMANDINI"/> |
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7581 fruſtum a d. </s> |
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7582 <s xml:id="echoid-s4486" xml:space="preserve">Sed pyramis q æqualis eſt fruſto à pyramide <lb/>abſciſſo, ut dem onſtrauimus. </s> |
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7583 <s xml:id="echoid-s4487" xml:space="preserve">ergo & </s> |
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7584 <s xml:id="echoid-s4488" xml:space="preserve">conus, uel coni por-<lb/>tio q, cuius baſis ex tribus circulis, uel ellipſibus a b, e f, c d <lb/>conſtat, & </s> |
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7585 <s xml:id="echoid-s4489" xml:space="preserve">altitudo eadem, quæ fruſti: </s> |
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7586 <s xml:id="echoid-s4490" xml:space="preserve">ipſi fruſto a d eſt æ-<lb/>qualis. </s> |
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7587 <s xml:id="echoid-s4491" xml:space="preserve">atque illud eſt, quod demonſtrare oportebat.</s> |
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7588 <s xml:id="echoid-s4492" xml:space="preserve"/> |
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7589 </p> |
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7590 <div xml:id="echoid-div266" type="float" level="2" n="4"> |
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7591 <note position="left" xlink:label="note-0178-01" xlink:href="note-0178-01a" xml:space="preserve">9. huius</note> |
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7592 <note position="left" xlink:label="note-0178-02" xlink:href="note-0178-02a" xml:space="preserve">2. duode-<lb/>cimi.</note> |
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7593 <figure xlink:label="fig-0178-01" xlink:href="fig-0178-01a"> |
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7594 <image file="0178-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0178-01"/> |
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7595 </figure> |
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7596 <note position="left" xlink:label="note-0178-03" xlink:href="note-0178-03a" xml:space="preserve">7. de co-<lb/>noidibus <lb/>& ſphæ-<lb/>roidibus</note> |
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7597 <note position="right" xlink:label="note-0179-01" xlink:href="note-0179-01a" xml:space="preserve">6. 11. duo <lb/>decimi</note> |
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7598 </div> |
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7599 </div> |
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7600 <div xml:id="echoid-div268" type="section" level="1" n="91"> |
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7601 <head xml:id="echoid-head98" xml:space="preserve">THEOREMA XXI. PROPOSITIO XXVI.</head> |
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7602 <p> |
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7603 <s xml:id="echoid-s4493" xml:space="preserve"><emph style="sc">Cvivslibet</emph> fruſti à pyramide, uel cono, <lb/>uel coni portione abſcisſi, centrum grauitatis eſt <lb/>in axe, ita ut eo primum in duas portiones diui-<lb/>ſo, portio ſuperior, quæ minorem baſim attingit <lb/>ad portionem reliquam eam habeat proportio-<lb/>nem, quam duplum lateris, uel diametri maioris <lb/>baſis, vnà cum latere, uel diametro minoris, ipſi <lb/>reſpondente, habet ad duplum lateris, uel diame-<lb/>tri minoris baſis vnà cũ latere, uel diametro ma-<lb/>ioris: </s> |
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7604 <s xml:id="echoid-s4494" xml:space="preserve">deinde à puncto diuiſionis quarta parte ſu <lb/>perioris portionis in ipſa ſumpta: </s> |
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7605 <s xml:id="echoid-s4495" xml:space="preserve">& </s> |
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7606 <s xml:id="echoid-s4496" xml:space="preserve">rurſus ab in-<lb/>ferioris portionis termino, qui eſt ad baſim maio <lb/>rem, ſumpta quarta parte totius axis: </s> |
|
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7607 <s xml:id="echoid-s4497" xml:space="preserve">centrum ſit <lb/>in linea, quæ his finibus continetur, atque in eo li <lb/>neæ puncto, quo ſic diuiditur, ut tota linea ad par <lb/>tem propinquiorem minori baſi, eãdem propor-<lb/>tionem habeat, quam fruſtum ad pyramidẽ, uel <lb/>conum, uel coni portionem, cuius baſis ſit ea-<lb/>dem, quæ baſis maior, & </s> |
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7608 <s xml:id="echoid-s4498" xml:space="preserve">altitudo fruſti altitudini <lb/>æqualis.</s> |
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7609 <s xml:id="echoid-s4499" xml:space="preserve"/> |
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|
7610 </p> |
|
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7611 <pb o="35" file="0181" n="181" rhead="DE CENTRO GRAVIT. SOLID."/> |
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7612 <p> |
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7613 <s xml:id="echoid-s4500" xml:space="preserve">Sit ſruſtum a e a pyramide, quæ triangularem baſim ha-<lb/>beat abſciſſum: </s> |
|
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7614 <s xml:id="echoid-s4501" xml:space="preserve">cuius maior baſis triangulum a b c, minor <lb/>d e f; </s> |
|
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7615 <s xml:id="echoid-s4502" xml:space="preserve">& </s> |
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7616 <s xml:id="echoid-s4503" xml:space="preserve">axis g h. </s> |
|
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7617 <s xml:id="echoid-s4504" xml:space="preserve">ducto autem plano per axem & </s> |
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7618 <s xml:id="echoid-s4505" xml:space="preserve">per lineã <lb/>d a, quod ſectionem faciat d a k l quadrilaterum; </s> |
|
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7619 <s xml:id="echoid-s4506" xml:space="preserve">puncta <lb/>K l lineas b c, e f bifariam ſecabunt. </s> |
|
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7620 <s xml:id="echoid-s4507" xml:space="preserve">nam cum g h ſit axis <lb/>ſruſti: </s> |
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7621 <s xml:id="echoid-s4508" xml:space="preserve">erit h centrum grauitatis trianguli a b c: </s> |
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7622 <s xml:id="echoid-s4509" xml:space="preserve">& </s> |
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7623 <s xml:id="echoid-s4510" xml:space="preserve">g <lb/>centrum trianguli d e f: </s> |
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7624 <s xml:id="echoid-s4511" xml:space="preserve">cen-<lb/> |
|
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7625 <anchor type="figure" xlink:label="fig-0181-01a" xlink:href="fig-0181-01"/> |
|
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7626 <anchor type="note" xlink:label="note-0181-01a" xlink:href="note-0181-01"/> |
|
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7627 trum uero cuiuslibet triangu <lb/>li eſt in recta linea, quæ ab an-<lb/>gulo ipſius ad dimidiã baſim <lb/>ducitur ex decimatertia primi <lb/>libri Archimedis de cẽtro gra <lb/>uitatis planorum. </s> |
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7628 <s xml:id="echoid-s4512" xml:space="preserve">quare cen-<lb/> |
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7629 <anchor type="note" xlink:label="note-0181-02a" xlink:href="note-0181-02"/> |
|
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|
7630 trũ grauitatis trapezii b c f e <lb/>eſt in linea _K_ l, quod ſit m: </s> |
|
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|
7631 <s xml:id="echoid-s4513" xml:space="preserve">& </s> |
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7632 <s xml:id="echoid-s4514" xml:space="preserve">à <lb/>puncto m ad axem ducta m n <lb/>ipſi a k, uel d l æquidiſtante; <lb/></s> |
|
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7633 <s xml:id="echoid-s4515" xml:space="preserve">erit axis g h diuiſus in portio-<lb/>nes g n, n h, quas diximus: </s> |
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7634 <s xml:id="echoid-s4516" xml:space="preserve">ean <lb/>dem enim proportionem ha-<lb/>bet g n ad n h, quã l m ad m _k_. </s> |
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7635 <s xml:id="echoid-s4517" xml:space="preserve"><lb/>At l m ad m K habet eam, quã <lb/>duplum lateris maioris baſis <lb/>b c una cum latere minoris e f <lb/>ad duplum lateris e f unà cum <lb/>later b c, ex ultima eiuſdem <lb/>libri Archimedis. </s> |
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7636 <s xml:id="echoid-s4518" xml:space="preserve">Itaque à li-<lb/>nea n g abſcindatur, quarta <lb/>pars, quæ ſit n p: </s> |
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7637 <s xml:id="echoid-s4519" xml:space="preserve">& </s> |
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7638 <s xml:id="echoid-s4520" xml:space="preserve">ab axe h g abſcindatur itidem <lb/>quarta pars h o: </s> |
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7639 <s xml:id="echoid-s4521" xml:space="preserve">& </s> |
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7640 <s xml:id="echoid-s4522" xml:space="preserve">quam proportionem habet fruſtum ad <lb/>pyramidem, cuius maior baſis eſt triangulum a b c, & </s> |
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7641 <s xml:id="echoid-s4523" xml:space="preserve">alti-<lb/>tudo ipſi æqualis; </s> |
|
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7642 <s xml:id="echoid-s4524" xml:space="preserve">habeat o p ad p q. </s> |
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7643 <s xml:id="echoid-s4525" xml:space="preserve">Dico centrum graui-<lb/>tatis fruſti eſſe in linea p o, & </s> |
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7644 <s xml:id="echoid-s4526" xml:space="preserve">in puncto q. </s> |
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7645 <s xml:id="echoid-s4527" xml:space="preserve">namque ipſum <lb/>eſſe in linea g h manifeſte conſtat. </s> |
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7646 <s xml:id="echoid-s4528" xml:space="preserve">protractis enim fruſti pla |
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|
7647 <pb file="0182" n="182" rhead="FED. COMMANDINI"/> |
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7648 nis, quouſque in unum punctum r conueniant; </s> |
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7649 <s xml:id="echoid-s4529" xml:space="preserve">erit pyra-<lb/>midis a b c r, & </s> |
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7650 <s xml:id="echoid-s4530" xml:space="preserve">pyramidis d e f r grauitatis centrum in li-<lb/>nea r h. </s> |
|
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7651 <s xml:id="echoid-s4531" xml:space="preserve">ergo & </s> |
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7652 <s xml:id="echoid-s4532" xml:space="preserve">reliquæ magnitudinis, uidelicet fruſti cen-<lb/>trum in eadem linea neceſſario comperietur. </s> |
|
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7653 <s xml:id="echoid-s4533" xml:space="preserve">Iungantur <lb/>d b, d c, d h, d m: </s> |
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7654 <s xml:id="echoid-s4534" xml:space="preserve">& </s> |
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7655 <s xml:id="echoid-s4535" xml:space="preserve">per lineas d b, d c ducto altero plano <lb/>intelligatur fruſtum in duas pyramides diuiſum: </s> |
|
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7656 <s xml:id="echoid-s4536" xml:space="preserve">in pyra-<lb/>midem quidem, cuius baſis eſt triangulum a b c, uertex d: <lb/></s> |
|
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7657 <s xml:id="echoid-s4537" xml:space="preserve">& </s> |
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7658 <s xml:id="echoid-s4538" xml:space="preserve">in eam, cuius idem uertex, & </s> |
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7659 <s xml:id="echoid-s4539" xml:space="preserve">baſis trapezium b c f e. </s> |
|
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7660 <s xml:id="echoid-s4540" xml:space="preserve">erit <lb/>igitur pyramidis a b c d axis d h, & </s> |
|
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7661 <s xml:id="echoid-s4541" xml:space="preserve">pyramidis b c f e d axis <lb/>d m: </s> |
|
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|
7662 <s xml:id="echoid-s4542" xml:space="preserve">atque erunt tres axes g h, d h, d m in eodem plano <lb/>d a K l. </s> |
|
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7663 <s xml:id="echoid-s4543" xml:space="preserve">ducatur præterea per o linea ſt ip ſi a K æquidiſtãs, <lb/>quæ lineam d h in u ſecet: </s> |
|
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|
7664 <s xml:id="echoid-s4544" xml:space="preserve">per p uero ducatur x y æquidi-<lb/>ſtans eidem, ſecansque d m in <lb/> |
|
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|
7665 <anchor type="figure" xlink:label="fig-0182-01a" xlink:href="fig-0182-01"/> |
|
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|
7666 z: </s> |
|
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|
7667 <s xml:id="echoid-s4545" xml:space="preserve">& </s> |
|
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|
7668 <s xml:id="echoid-s4546" xml:space="preserve">iungatur z u, quæ ſecet <lb/>g h in φ. </s> |
|
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|
7669 <s xml:id="echoid-s4547" xml:space="preserve">tranſibit ea per q: </s> |
|
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|
7670 <s xml:id="echoid-s4548" xml:space="preserve">& </s> |
|
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|
7671 <s xml:id="echoid-s4549" xml:space="preserve"><lb/>erunt φ q unum, atque idem <lb/>pun ctum; </s> |
|
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|
7672 <s xml:id="echoid-s4550" xml:space="preserve">ut inferius appare-<lb/>bit. </s> |
|
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|
7673 <s xml:id="echoid-s4551" xml:space="preserve">Quoniam igitur linea u o <lb/>æ quidiſtat ipſi d g, erit d u ad <lb/> |
|
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|
7674 <anchor type="note" xlink:label="note-0182-01a" xlink:href="note-0182-01"/> |
|
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7675 u h, ut g o ad o h. </s> |
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7676 <s xml:id="echoid-s4552" xml:space="preserve">Sed g o tri-<lb/>pla eſt o h. </s> |
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7677 <s xml:id="echoid-s4553" xml:space="preserve">quare & </s> |
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7678 <s xml:id="echoid-s4554" xml:space="preserve">d u ipſius <lb/>u h eſt tripla: </s> |
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7679 <s xml:id="echoid-s4555" xml:space="preserve">& </s> |
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7680 <s xml:id="echoid-s4556" xml:space="preserve">ideo pyrami-<lb/>dis a b c d centrum grauitatis <lb/>erit punctum 11. </s> |
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7681 <s xml:id="echoid-s4557" xml:space="preserve">Rurſus quo-<lb/>niam z y ipſi d l æquidiſtat, d z <lb/>a d z m eſt, utly ad y m: </s> |
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7682 <s xml:id="echoid-s4558" xml:space="preserve">eſtque <lb/>ly ad y m, ut g p ad p n. </s> |
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7683 <s xml:id="echoid-s4559" xml:space="preserve">ergo <lb/>d z ad z m eſt, ut g p ad p n. <lb/></s> |
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7684 <s xml:id="echoid-s4560" xml:space="preserve">Quòd cum g p ſit tripla p n; </s> |
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7685 <s xml:id="echoid-s4561" xml:space="preserve"><lb/>erit etiam d z ipſius z m tri-<lb/>pla. </s> |
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7686 <s xml:id="echoid-s4562" xml:space="preserve">atque ob eandem cauſ-<lb/>ſam punctum z eſt centrũ gra-<lb/>uitatis pyramidis b c f e d. </s> |
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7687 <s xml:id="echoid-s4563" xml:space="preserve">iun <lb/>ctaigitur z u, in ea erit cẽtrum |
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7688 <pb o="36" file="0183" n="183" rhead="DE CENTRO GRAVIT. SOLID."/> |
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7689 grauitatis magnitudinis, quæ ex utriſque pyramidibus cõ <lb/>ſtat; </s> |
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7690 <s xml:id="echoid-s4564" xml:space="preserve">hoc eſt ipſius fruſti. </s> |
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7691 <s xml:id="echoid-s4565" xml:space="preserve">Sed fruſti centrum eſt etiam in a-<lb/>xe g h. </s> |
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7692 <s xml:id="echoid-s4566" xml:space="preserve">ergo in puncto φ, in quo lineæ z u, g h conueniunt. <lb/></s> |
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7693 <s xml:id="echoid-s4567" xml:space="preserve">Itaque u φ ad φ z eam proportionem habet, quam pyramis <lb/> |
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7694 <anchor type="note" xlink:label="note-0183-01a" xlink:href="note-0183-01"/> |
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7695 b c f e d ad pyramidem a b c d. </s> |
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7696 <s xml:id="echoid-s4568" xml:space="preserve">& </s> |
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7697 <s xml:id="echoid-s4569" xml:space="preserve">componendo u z ad z φ <lb/>eam habet, quam fruſtum ad pyramidem a b c d. </s> |
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7698 <s xml:id="echoid-s4570" xml:space="preserve">Vtuero <lb/>u z ad z φ, ita o p ad p φ ob ſimilitudinem triangulorum, <lb/>u o φ, z p φ. </s> |
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7699 <s xml:id="echoid-s4571" xml:space="preserve">quare o p ad p φ eſt ut fruſtum ad pyramidem <lb/>a b c d. </s> |
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7700 <s xml:id="echoid-s4572" xml:space="preserve">ſed ita erat o p ad p q. </s> |
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7701 <s xml:id="echoid-s4573" xml:space="preserve">æquales igitur ſunt p φ, p q: </s> |
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7702 <s xml:id="echoid-s4574" xml:space="preserve">& </s> |
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7703 <s xml:id="echoid-s4575" xml:space="preserve"><lb/> |
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7704 <anchor type="note" xlink:label="note-0183-02a" xlink:href="note-0183-02"/> |
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7705 q φ unum atque idem punctum. </s> |
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7706 <s xml:id="echoid-s4576" xml:space="preserve">ex quibus ſequitur lineam <lb/>z u ſecare o p in q: </s> |
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7707 <s xml:id="echoid-s4577" xml:space="preserve">& </s> |
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7708 <s xml:id="echoid-s4578" xml:space="preserve">propterea pũctum q ipſius fruſti gra-<lb/>uitatis centrum eſſe.</s> |
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7709 <s xml:id="echoid-s4579" xml:space="preserve"/> |
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|
7710 </p> |
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7711 <div xml:id="echoid-div268" type="float" level="2" n="1"> |
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7712 <figure xlink:label="fig-0181-01" xlink:href="fig-0181-01a"> |
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|
7713 <image file="0181-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0181-01"/> |
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|
7714 </figure> |
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|
7715 <note position="right" xlink:label="note-0181-01" xlink:href="note-0181-01a" xml:space="preserve">3. diffi. hu <lb/>ius.</note> |
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7716 <note position="right" xlink:label="note-0181-02" xlink:href="note-0181-02a" xml:space="preserve">Vltima e-<lb/>auſdẽ libri <lb/>Archime-<lb/>dis.</note> |
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7717 <figure xlink:label="fig-0182-01" xlink:href="fig-0182-01a"> |
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|
7718 <image file="0182-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0182-01"/> |
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|
7719 </figure> |
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7720 <note position="left" xlink:label="note-0182-01" xlink:href="note-0182-01a" xml:space="preserve">2. ſexti.</note> |
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7721 <note position="right" xlink:label="note-0183-01" xlink:href="note-0183-01a" xml:space="preserve">8. prim I <lb/>libri Ar-<lb/>chimedis <lb/>de cẽtro <lb/>grauita-<lb/>tis plano <lb/>runi</note> |
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7722 <note position="right" xlink:label="note-0183-02" xlink:href="note-0183-02a" xml:space="preserve">7. quinti.</note> |
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|
7723 </div> |
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|
7724 <p> |
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7725 <s xml:id="echoid-s4580" xml:space="preserve">Sit fruſtum a g à pyramide, quæ quadrangularem baſim <lb/>habeat abſciſſum, cuius maior baſis a b c d, minor e f g h, <lb/>& </s> |
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7726 <s xml:id="echoid-s4581" xml:space="preserve">axis k l. </s> |
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7727 <s xml:id="echoid-s4582" xml:space="preserve">diuidatur autem primũ _k_ l, ita ut quam propor-<lb/>tionem habet duplum lateris a b unà cum latere e f ad du <lb/>plum lateris e f unà cum a b; </s> |
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7728 <s xml:id="echoid-s4583" xml:space="preserve">habeat k m ad m l. </s> |
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7729 <s xml:id="echoid-s4584" xml:space="preserve">deinde à <lb/>púcto m ad k ſumatur quarta pars ipſius m k, quæ ſit m n. <lb/></s> |
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7730 <s xml:id="echoid-s4585" xml:space="preserve">& </s> |
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7731 <s xml:id="echoid-s4586" xml:space="preserve">rurſus ab l ſumatur quarta pars totius axis l k, quæ ſit <lb/>l o. </s> |
|
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7732 <s xml:id="echoid-s4587" xml:space="preserve">poſtremo fiat o n ad n p, ut fruſtum a g ad pyramidẽ, <lb/>cuius baſis ſit eadem, quæ fruſti, & </s> |
|
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7733 <s xml:id="echoid-s4588" xml:space="preserve">altitudo æqualis. </s> |
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|
7734 <s xml:id="echoid-s4589" xml:space="preserve">Dico <lb/>punctum p fruſti a g grauitatis centrum eſſe. </s> |
|
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|
7735 <s xml:id="echoid-s4590" xml:space="preserve">ducantur <lb/>enim a c, e g: </s> |
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|
7736 <s xml:id="echoid-s4591" xml:space="preserve">& </s> |
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7737 <s xml:id="echoid-s4592" xml:space="preserve">intelligantur duo fruſta triangulares ba-<lb/>ſes habentia, quorum alterum l f ex baſibus a b c, e f g cõ-<lb/>ſtet; </s> |
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7738 <s xml:id="echoid-s4593" xml:space="preserve">alterum l h ex baſibus a c d, e g h. </s> |
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7739 <s xml:id="echoid-s4594" xml:space="preserve">Sitq; </s> |
|
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|
7740 <s xml:id="echoid-s4595" xml:space="preserve">fruſti l f axis <lb/>q r; </s> |
|
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|
7741 <s xml:id="echoid-s4596" xml:space="preserve">in quo grauitatis centrum s: </s> |
|
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|
7742 <s xml:id="echoid-s4597" xml:space="preserve">fruſti uero l h axis t u, & </s> |
|
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7743 <s xml:id="echoid-s4598" xml:space="preserve"><lb/>x grauitatis centrum: </s> |
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|
7744 <s xml:id="echoid-s4599" xml:space="preserve">deinde iungantur u r, t q, x s. </s> |
|
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|
7745 <s xml:id="echoid-s4600" xml:space="preserve">tranſi-<lb/>bit u r per l: </s> |
|
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|
7746 <s xml:id="echoid-s4601" xml:space="preserve">quoniam l eſt centrum grauitatis quadran-<lb/>guli a b c d: </s> |
|
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|
7747 <s xml:id="echoid-s4602" xml:space="preserve">& </s> |
|
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diff
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|
7748 <s xml:id="echoid-s4603" xml:space="preserve">puncta r u grauitatis centra triangulorum <lb/>a b c, a c d; </s> |
|
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|
7749 <s xml:id="echoid-s4604" xml:space="preserve">in quæ quadrangulum ipſum diuiditur. </s> |
|
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|
7750 <s xml:id="echoid-s4605" xml:space="preserve">eadem <lb/>quoque ratione t q per punctum _k_ tranſibit. </s> |
|
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|
7751 <s xml:id="echoid-s4606" xml:space="preserve">At uero pro <lb/>portiones, ex quibus fruſtorum grauitatis centra inquiri-<lb/>mus, eædem ſunt in toto ſruſto a g, & </s> |
|
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|
7752 <s xml:id="echoid-s4607" xml:space="preserve">in fruſtis l f, l h. </s> |
|
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diff
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|
7753 <s xml:id="echoid-s4608" xml:space="preserve">Sunt <lb/>enim per octauam huius quadrilatera a b c d, e f g h ſimilia:</s> |
|
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|
7754 <s xml:id="echoid-s4609" xml:space="preserve"> |
|
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|
7755 <pb file="0184" n="184" rhead="FED. COMMANDINI"/> |
|
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|
7756 itemq; </s> |
|
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|
7757 <s xml:id="echoid-s4610" xml:space="preserve">ſimilia triangula a b c, e f g; </s> |
|
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|
7758 <s xml:id="echoid-s4611" xml:space="preserve">& </s> |
|
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|
7759 <s xml:id="echoid-s4612" xml:space="preserve">a c d, e g h. </s> |
|
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|
7760 <s xml:id="echoid-s4613" xml:space="preserve">idcir-<lb/>coq; </s> |
|
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|
7761 <s xml:id="echoid-s4614" xml:space="preserve">latera ſibi ipſis reſpondentia eandem inter ſe ſe pro-<lb/>portionem ſeruant. </s> |
|
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|
7762 <s xml:id="echoid-s4615" xml:space="preserve">Vt igitur duplum lateris a b unà <lb/>cum latere e f ad duplum lateris e f unà cum a b, ita eſt <lb/>duplum a d late-<lb/> |
|
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|
7763 <anchor type="figure" xlink:label="fig-0184-01a" xlink:href="fig-0184-01"/> |
|
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changeset
|
7764 ris unà cum late-<lb/>re e h ad duplum <lb/>e h unà cum a d: <lb/></s> |
|
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|
7765 <s xml:id="echoid-s4616" xml:space="preserve">& </s> |
|
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|
7766 <s xml:id="echoid-s4617" xml:space="preserve">ita in aliis. </s> |
|
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|
7767 <s xml:id="echoid-s4618" xml:space="preserve"><lb/>Rurſus fruſtum <lb/>a g ad pyramidẽ, <lb/>cuius eadem eſt <lb/>bafis, & </s> |
|
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|
7768 <s xml:id="echoid-s4619" xml:space="preserve">æqualis <lb/>altitudo eandem <lb/>proportionẽ ha <lb/>bet, quam fruſtũ <lb/>l f ad pyramidẽ, <lb/>quæ eſt eadẽ ba-<lb/>ſi, & </s> |
|
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|
7769 <s xml:id="echoid-s4620" xml:space="preserve">æquali alti-<lb/>tudine: </s> |
|
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|
7770 <s xml:id="echoid-s4621" xml:space="preserve">& </s> |
|
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diff
changeset
|
7771 <s xml:id="echoid-s4622" xml:space="preserve">ſimili-<lb/>ter quam l h fru-<lb/>ſtum ad pyrami-<lb/>dem, quæ ex ea-<lb/>dẽ baſi, & </s> |
|
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|
7772 <s xml:id="echoid-s4623" xml:space="preserve">æquali <lb/>altitudine con-<lb/>ſtat. </s> |
|
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diff
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|
7773 <s xml:id="echoid-s4624" xml:space="preserve">nam ſi inter <lb/>ipſas baſes me-<lb/>diæ proportio-<lb/>nales conſtituan <lb/>tur, tres baſes ſimul ſumptæ ad maiorem baſim in om-<lb/>nibus eodem modo ſe habebunt. </s> |
|
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|
7774 <s xml:id="echoid-s4625" xml:space="preserve">Vnde fit, ut axes K l, <lb/>q r, t u à punctis p s x in eandem proportionem ſecen-<lb/>tur. </s> |
|
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diff
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|
7775 <s xml:id="echoid-s4626" xml:space="preserve">ergo linea x s per p tranſibit: </s> |
|
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|
7776 <s xml:id="echoid-s4627" xml:space="preserve">& </s> |
|
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|
7777 <s xml:id="echoid-s4628" xml:space="preserve">lineæ r u, s x, q t in-<lb/> |
|
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|
7778 <anchor type="note" xlink:label="note-0184-01a" xlink:href="note-0184-01"/> |
|
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|
7779 ter ſe æquidiſtantes erunt. </s> |
|
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|
7780 <s xml:id="echoid-s4629" xml:space="preserve">Itaque cum fruſti a g latera pro- |
|
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|
7781 <pb o="37" file="0185" n="185" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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|
7782 ducta fuerìnt, ira ut in unum punctum y coeant, erunt triã <lb/>gala u y l, x y p, t y _k_ inter ſe ſimilia: </s> |
|
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|
7783 <s xml:id="echoid-s4630" xml:space="preserve">& </s> |
|
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|
7784 <s xml:id="echoid-s4631" xml:space="preserve">ſimilia etiam triangu <lb/>la l y r, p y s, _k_ y q. </s> |
|
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|
7785 <s xml:id="echoid-s4632" xml:space="preserve">quare ut in 19 huius, demonſtrabitur <lb/>x p, ad p s: </s> |
|
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|
7786 <s xml:id="echoid-s4633" xml:space="preserve">itemq; </s> |
|
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diff
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|
7787 <s xml:id="echoid-s4634" xml:space="preserve">t k ad _k_ q èandem habere proportionẽ, <lb/>quam u l ad l r. </s> |
|
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|
7788 <s xml:id="echoid-s4635" xml:space="preserve">Sed ut u l ad l r, ita eſt triangulum a b c ad <lb/>triangulum a c d: </s> |
|
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|
7789 <s xml:id="echoid-s4636" xml:space="preserve">& </s> |
|
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changeset
|
7790 <s xml:id="echoid-s4637" xml:space="preserve">ut t k ad K q, ita triangulum e f g ad <lb/>triangulum e g h. </s> |
|
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|
7791 <s xml:id="echoid-s4638" xml:space="preserve">Vt autem triangulum a b c ad triangu-<lb/>lum a c d, ita pyramis a b c y ad pyramidem a c d y. </s> |
|
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|
7792 <s xml:id="echoid-s4639" xml:space="preserve">& </s> |
|
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diff
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|
7793 <s xml:id="echoid-s4640" xml:space="preserve">ut <lb/>triangulum e f g ad triangulum e g h, ita pyramis e f g y <lb/>ad pyramidem e g h y; </s> |
|
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|
7794 <s xml:id="echoid-s4641" xml:space="preserve">ergo ut pyramis a b c y ad pyramidẽ <lb/>a c d y, ita pyramis e f g y ad pyramidem e g h y. </s> |
|
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|
7795 <s xml:id="echoid-s4642" xml:space="preserve">reliquum <lb/> |
|
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|
7796 <anchor type="note" xlink:label="note-0185-01a" xlink:href="note-0185-01"/> |
|
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|
7797 igitur fruſtũ l f ad reliquum fruſtũ l h eſt ut pyramis a b c y <lb/>ad pyramidem a c d y, hoc eſt ut u l ad l r, & </s> |
|
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|
7798 <s xml:id="echoid-s4643" xml:space="preserve">ut x p ad p s. <lb/></s> |
|
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|
7799 <s xml:id="echoid-s4644" xml:space="preserve">Quòd cum fruſti l f centrum grauitatis ſit s: </s> |
|
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|
7800 <s xml:id="echoid-s4645" xml:space="preserve">& </s> |
|
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|
7801 <s xml:id="echoid-s4646" xml:space="preserve">fruſti l h ſit <lb/>centrum x: </s> |
|
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|
7802 <s xml:id="echoid-s4647" xml:space="preserve">conſtat punctum p totius fruſti a g grauitatis <lb/> |
|
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|
7803 <anchor type="note" xlink:label="note-0185-02a" xlink:href="note-0185-02"/> |
|
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changeset
|
7804 eſſe centrum. </s> |
|
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|
7805 <s xml:id="echoid-s4648" xml:space="preserve">Eodem modo fiet demonſtratio etiam in <lb/>aliis pyramidibus.</s> |
|
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|
7806 <s xml:id="echoid-s4649" xml:space="preserve"/> |
|
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changeset
|
7807 </p> |
|
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|
7808 <div xml:id="echoid-div269" type="float" level="2" n="2"> |
|
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|
7809 <figure xlink:label="fig-0184-01" xlink:href="fig-0184-01a"> |
|
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|
7810 <image file="0184-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0184-01"/> |
|
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|
7811 </figure> |
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7812 <note position="left" xlink:label="note-0184-01" xlink:href="note-0184-01a" xml:space="preserve">2. ſexti.</note> |
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7813 <note position="right" xlink:label="note-0185-01" xlink:href="note-0185-01a" xml:space="preserve">19. quinti</note> |
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7814 <note position="right" xlink:label="note-0185-02" xlink:href="note-0185-02a" xml:space="preserve">8. Archi-<lb/>medis.</note> |
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7815 </div> |
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7816 <p> |
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7817 <s xml:id="echoid-s4650" xml:space="preserve">Sit fruſtum a d à cono, uel coni portione abſciſſum, cu-<lb/>ius maior baſis circulus, uel ellipſis circa diametrum a b; <lb/></s> |
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7818 <s xml:id="echoid-s4651" xml:space="preserve">minor circa diametrum c d: </s> |
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7819 <s xml:id="echoid-s4652" xml:space="preserve">& </s> |
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7820 <s xml:id="echoid-s4653" xml:space="preserve">axis e f. </s> |
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7821 <s xml:id="echoid-s4654" xml:space="preserve">diuidatur autẽ e f <lb/>in g, ita ut e g ad g f eandem proportionem habeat, quam <lb/>duplum diametri a b unà cum diametro c d ad duplum c d <lb/>unà cum a b. </s> |
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7822 <s xml:id="echoid-s4655" xml:space="preserve">Sitq; </s> |
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7823 <s xml:id="echoid-s4656" xml:space="preserve">g h quarta pars lineæ g e: </s> |
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7824 <s xml:id="echoid-s4657" xml:space="preserve">& </s> |
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7825 <s xml:id="echoid-s4658" xml:space="preserve">ſit ſ K item <lb/>quarta pars totius f e axis. </s> |
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7826 <s xml:id="echoid-s4659" xml:space="preserve">Rurfus quam proportionem <lb/>habet fruſtum a d ad conum, uel coni portionem, in eadẽ <lb/>baſi, & </s> |
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7827 <s xml:id="echoid-s4660" xml:space="preserve">æquali altitudine, habeat linea _k_ h ad h l. </s> |
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7828 <s xml:id="echoid-s4661" xml:space="preserve">Dico pun-<lb/>ctum l fruſti a d grauitatis centrum eſſe. </s> |
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7829 <s xml:id="echoid-s4662" xml:space="preserve">Si enim fieri po-<lb/>teſt, ſit m centrum: </s> |
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7830 <s xml:id="echoid-s4663" xml:space="preserve">producaturq; </s> |
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7831 <s xml:id="echoid-s4664" xml:space="preserve">l m extra fruſtum in n: </s> |
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7832 <s xml:id="echoid-s4665" xml:space="preserve"><lb/>& </s> |
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7833 <s xml:id="echoid-s4666" xml:space="preserve">ut n l ad l m, ita fiat circulus, uel ellipſis circa diametrũ <lb/>a b ad aliud ſpacium, in quo ſit o. </s> |
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7834 <s xml:id="echoid-s4667" xml:space="preserve">Itaque in circulo, uel <lb/>ellipſi circa diametrum a b rectilinea figura plane deſcri-<lb/>batur, ita ut quæ relinquuntur portiones ſint o ſpacio mi-<lb/>nores: </s> |
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7835 <s xml:id="echoid-s4668" xml:space="preserve">& </s> |
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7836 <s xml:id="echoid-s4669" xml:space="preserve">inteiligatur pyramis a p b, baſim habens rectili-<lb/>neam figuram in circulo, uel ellipſi a b deſcriptam: </s> |
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7837 <s xml:id="echoid-s4670" xml:space="preserve">à qua |
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7838 <pb file="0186" n="186" rhead="FED. COMMANDINI"/> |
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7839 fruſtum pyramidis ſit abſciſſum. </s> |
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7840 <s xml:id="echoid-s4671" xml:space="preserve">erit ex iis quæ proxime <lb/>tradidimus, fruſti pyramidis a d centrum grauitatis l. </s> |
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7841 <s xml:id="echoid-s4672" xml:space="preserve">Quo <lb/>niam igitur portiones ſpacio o minores ſunt; </s> |
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7842 <s xml:id="echoid-s4673" xml:space="preserve">habebit cir <lb/>culus, uel ellipſis a b ad <lb/> |
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7843 <anchor type="figure" xlink:label="fig-0186-01a" xlink:href="fig-0186-01"/> |
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7844 portiones dictas maiorẽ <lb/>proportionem, quàm n l <lb/>ad lm. </s> |
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7845 <s xml:id="echoid-s4674" xml:space="preserve">ſed ut circulus, uel <lb/>ellipſis a b ad portiones, <lb/>ita a p b conus, uel coni <lb/>portio ad ſolidas portio-<lb/>nes, id quod ſupra demon <lb/>ſtratum eſt: </s> |
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7846 <s xml:id="echoid-s4675" xml:space="preserve">& </s> |
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7847 <s xml:id="echoid-s4676" xml:space="preserve">ut circulus <lb/>uel ellipſis c d ad portio-<lb/> |
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7848 <anchor type="note" xlink:label="note-0186-01a" xlink:href="note-0186-01"/> |
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7849 nes, quæ ipſi inſunt, ita co <lb/>nus, uel coni portio c p d <lb/>ad ſolidas ipſius portio-<lb/>nes. </s> |
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7850 <s xml:id="echoid-s4677" xml:space="preserve">Quòd cum figuræ in <lb/>circulis, uel ellipſibus a b <lb/>c d deſcriptæ ſimiles ſint, <lb/>erit proportio circuli, uel <lb/>ellipſis a b ad ſuas portio <lb/>nes, eadẽ, quæ circuli uel <lb/>ellipſis c d ad ſuas. </s> |
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7851 <s xml:id="echoid-s4678" xml:space="preserve">ergo <lb/>conus, uel coni portio a p <lb/>b ad portiones ſolidas eã-<lb/>dem habet proportionẽ, <lb/>quam conus, uel coni por <lb/>tio c p d ad ſolidas ipſius <lb/>portiones. </s> |
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7852 <s xml:id="echoid-s4679" xml:space="preserve">reliquum igi-<lb/> |
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7853 <anchor type="note" xlink:label="note-0186-02a" xlink:href="note-0186-02"/> |
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7854 tur coni, uel coni portionis fruſtũ, ſcilicet a d ad reliquas <lb/>portiones ſolidas in ipſo contentas eandem proportionẽ <lb/>habet, quam conus, uel coni portio a p b ad ſolidas portio <lb/>nes: </s> |
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7855 <s xml:id="echoid-s4680" xml:space="preserve">hoc eſt eandem, quam circulus, uel ellipſis a b ad por <lb/>tiones planas. </s> |
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7856 <s xml:id="echoid-s4681" xml:space="preserve">quare fruſtum coni, uel coni portionis a d |
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7857 <pb o="38" file="0187" n="187" rhead="DE CENTRO GRA VIT. SOLID."/> |
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7858 ad portiones ſolidas maiorem habet proportioné, quàm <lb/>n l ad l m: </s> |
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7859 <s xml:id="echoid-s4682" xml:space="preserve">& </s> |
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7860 <s xml:id="echoid-s4683" xml:space="preserve">diuidendo fruſtum pyramidis ad dictas por-<lb/>tiones maiorem proportionem habet, quàm n m ad m l. <lb/></s> |
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7861 <s xml:id="echoid-s4684" xml:space="preserve">fiat igitur ut fruſtum pyramidis ad portiones, ita q m ad <lb/>m l. </s> |
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7862 <s xml:id="echoid-s4685" xml:space="preserve">Itaque quoniam à fruſto coni, uel coni portionis a d, <lb/>cuius grauitatis centrum eſtm, aufertur fruſtum pyrami-<lb/>dis habens centruml; </s> |
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7863 <s xml:id="echoid-s4686" xml:space="preserve">erit reliquæ magnitudinis, quæ ex <lb/>portionibus ſolidis conſtat; </s> |
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7864 <s xml:id="echoid-s4687" xml:space="preserve">grauitatis cẽtrum in linea l m <lb/>producta, atque in puncto q, extra figuram poſito. </s> |
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7865 <s xml:id="echoid-s4688" xml:space="preserve">quod <lb/>fieri nullo modo poteſt. </s> |
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7866 <s xml:id="echoid-s4689" xml:space="preserve">relinquitur ergo, ut punctum l ſit <lb/>fruſti a d grauitatis centrum. </s> |
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7867 <s xml:id="echoid-s4690" xml:space="preserve">quæ omnia demonſtranda <lb/>proponebantur.</s> |
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7868 <s xml:id="echoid-s4691" xml:space="preserve"/> |
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|
7869 </p> |
|
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7870 <div xml:id="echoid-div270" type="float" level="2" n="3"> |
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7871 <figure xlink:label="fig-0186-01" xlink:href="fig-0186-01a"> |
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7872 <image file="0186-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0186-01"/> |
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|
7873 </figure> |
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7874 <note position="left" xlink:label="note-0186-01" xlink:href="note-0186-01a" xml:space="preserve">22. huius</note> |
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7875 <note position="left" xlink:label="note-0186-02" xlink:href="note-0186-02a" xml:space="preserve">19. quinti</note> |
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|
7876 </div> |
|
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|
7877 </div> |
|
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7878 <div xml:id="echoid-div272" type="section" level="1" n="92"> |
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7879 <head xml:id="echoid-head99" xml:space="preserve">THEOREMA XXII. PROPOSITIO XXVII.</head> |
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7880 <p> |
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7881 <s xml:id="echoid-s4692" xml:space="preserve"><emph style="sc">Omnivm</emph> ſolidorum in ſphæra deſcripto-<lb/>rum, quæ æqualibus, & </s> |
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7882 <s xml:id="echoid-s4693" xml:space="preserve">ſimilibus baſibus conti-<lb/>nentur, centrum grauitatis eſt idem, quod ſphæ-<lb/>ræ centrum.</s> |
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7883 <s xml:id="echoid-s4694" xml:space="preserve"/> |
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7884 </p> |
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7885 <p> |
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7886 <s xml:id="echoid-s4695" xml:space="preserve">Solida eiuſmodi corpora regularia appellare ſolent, de <lb/>quibus agitur in tribus ultimis libris elementorum: </s> |
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7887 <s xml:id="echoid-s4696" xml:space="preserve">ſunt <lb/>autem numero quinque, tetrahedrum, uel pyramis, hexa-<lb/>hedrum, uel cubus, octahedrum, dodecahedrum, & </s> |
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7888 <s xml:id="echoid-s4697" xml:space="preserve">icoſa-<lb/>hedrum.</s> |
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7889 <s xml:id="echoid-s4698" xml:space="preserve"/> |
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7890 </p> |
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7891 <p> |
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7892 <s xml:id="echoid-s4699" xml:space="preserve">Sit primo a b c d pyramis ĩ ſphæra deſcripta, cuíus ſphæ <lb/>ræ centrum ſit e. </s> |
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7893 <s xml:id="echoid-s4700" xml:space="preserve">Dico e pyramidis a b c d grauitatis eſſe <lb/>centrum. </s> |
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7894 <s xml:id="echoid-s4701" xml:space="preserve">Si enim iuncta d e producatur ad baſim a b c in <lb/>f; </s> |
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7895 <s xml:id="echoid-s4702" xml:space="preserve">ex iis, quæ demonſtrauit Campanus in quartodecimo li <lb/>bro elementorum, propoſitione decima quinta, & </s> |
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7896 <s xml:id="echoid-s4703" xml:space="preserve">decima <lb/>ſeptima, erit f centrum circuli circa triangulum a b c de-<lb/>ſcripti: </s> |
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7897 <s xml:id="echoid-s4704" xml:space="preserve">atque erit e f ſexta pars ipſius ſphæræ axis. </s> |
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7898 <s xml:id="echoid-s4705" xml:space="preserve">quare <lb/>ex prima huius conſtat trianguli a b c grauitatis centrum <lb/>eſſe punctum f: </s> |
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7899 <s xml:id="echoid-s4706" xml:space="preserve">& </s> |
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7900 <s xml:id="echoid-s4707" xml:space="preserve">idcirco lineam d f eſſe pyramidis axem.</s> |
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7901 <s xml:id="echoid-s4708" xml:space="preserve"> |
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7902 <pb file="0188" n="188" rhead="FED. COMMANDINI"/> |
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7903 At cum e f ſit ſexta pars axis <lb/> |
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7904 <anchor type="figure" xlink:label="fig-0188-01a" xlink:href="fig-0188-01"/> |
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7905 ſphæræ, crit d e tripla e f. </s> |
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7906 <s xml:id="echoid-s4709" xml:space="preserve">ergo <lb/>punctum e eſt grauitatis cen-<lb/>trum ipſius pyramidis: </s> |
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7907 <s xml:id="echoid-s4710" xml:space="preserve">quod <lb/>in uigeſima ſecunda huius de-<lb/>monſtratum fuit. </s> |
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7908 <s xml:id="echoid-s4711" xml:space="preserve">Sed e eſt cen <lb/>trum ſphæræ. </s> |
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7909 <s xml:id="echoid-s4712" xml:space="preserve">Sequitur igitur, <lb/>ut centrum grauitatis pyrami-<lb/>dis in ſphæra deſcriptæ idem <lb/>ſit, quod ipſius ſphæræ cen-<lb/>trum.</s> |
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7910 <s xml:id="echoid-s4713" xml:space="preserve"/> |
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|
7911 </p> |
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7912 <div xml:id="echoid-div272" type="float" level="2" n="1"> |
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7913 <figure xlink:label="fig-0188-01" xlink:href="fig-0188-01a"> |
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7914 <image file="0188-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0188-01"/> |
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7915 </figure> |
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7916 </div> |
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7917 <p> |
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7918 <s xml:id="echoid-s4714" xml:space="preserve">Sit cubus in ſphæra deſcriptus a b, & </s> |
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7919 <s xml:id="echoid-s4715" xml:space="preserve">oppoſitorum pla-<lb/>norum lateribus bifariam diuiſis, per puncta diuiſionum <lb/>plana ducantur, ut communis ipſorum ſectio ſit recta li-<lb/>nea c d. </s> |
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7920 <s xml:id="echoid-s4716" xml:space="preserve">Itaque ſi ducatur a b, ſolidi ſcilicet diameter, lineæ <lb/>a b, c d ex trigeſimanona undecimi ſeſe bifariam ſecabunt. <lb/></s> |
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7921 <s xml:id="echoid-s4717" xml:space="preserve">ſecent autem in puncto e. </s> |
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7922 <s xml:id="echoid-s4718" xml:space="preserve">erit <lb/> |
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7923 <anchor type="figure" xlink:label="fig-0188-02a" xlink:href="fig-0188-02"/> |
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7924 e centrũ grauitatis ſolidi a b, <lb/>id quod demonſtratum eſt in <lb/>octaua huius. </s> |
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7925 <s xml:id="echoid-s4719" xml:space="preserve">Sed quoniam ab <lb/>eſt ſphæræ diametro æqualis, <lb/>ut in decima quinta propoſi-<lb/>tione tertii decimi libri elemẽ <lb/>torum oſtenditur: </s> |
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7926 <s xml:id="echoid-s4720" xml:space="preserve">punctum e <lb/>ſphæræ quoque centrum erit. <lb/></s> |
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7927 <s xml:id="echoid-s4721" xml:space="preserve">Cubi igitur in ſphæra deſcri-<lb/>pti grauitatis centrum idem <lb/>eſt, quod centrum ipſius ſphæræ.</s> |
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7928 <s xml:id="echoid-s4722" xml:space="preserve"/> |
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|
7929 </p> |
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7930 <div xml:id="echoid-div273" type="float" level="2" n="2"> |
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7931 <figure xlink:label="fig-0188-02" xlink:href="fig-0188-02a"> |
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7932 <image file="0188-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0188-02"/> |
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|
7933 </figure> |
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7934 </div> |
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7935 <p> |
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7936 <s xml:id="echoid-s4723" xml:space="preserve">Sit octahedrum a b c d e f, in ſphæra deſcriptum, cuius <lb/>ſphæræ centrum ſit g. </s> |
|
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7937 <s xml:id="echoid-s4724" xml:space="preserve">Dico punctum g ipſius octahedri <lb/>grauitatis centrum eſſe. </s> |
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7938 <s xml:id="echoid-s4725" xml:space="preserve">Conſtat enim ex iis, quæ demon-<lb/>ſtrata ſunt à Campano in quinto decimo libro elemento-<lb/>rum, propoſitione ſextadecima eiuſimodi ſolidum diuidi <lb/>in duas pyramides æquales, & </s> |
|
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7939 <s xml:id="echoid-s4726" xml:space="preserve">ſimiles; </s> |
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7940 <s xml:id="echoid-s4727" xml:space="preserve">uidelicetin pyrami- |
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7941 <pb o="39" file="0189" n="189" rhead="DE CENTRO GRAVIT. SOLID."/> |
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|
7942 dem, cuius baſis eſt quadratum a b c d, & </s> |
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7943 <s xml:id="echoid-s4728" xml:space="preserve">altitudo e g: </s> |
|
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7944 <s xml:id="echoid-s4729" xml:space="preserve">& </s> |
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7945 <s xml:id="echoid-s4730" xml:space="preserve"><lb/>in pyramidem, cuius eadé baſis, altitudoq; </s> |
|
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7946 <s xml:id="echoid-s4731" xml:space="preserve">f g; </s> |
|
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|
7947 <s xml:id="echoid-s4732" xml:space="preserve">ut ſint e g, <lb/>g f ſemidiametri ſphæræ, & </s> |
|
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|
7948 <s xml:id="echoid-s4733" xml:space="preserve">linea una. </s> |
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7949 <s xml:id="echoid-s4734" xml:space="preserve">Cũigitur g ſit ſphæ-<lb/>ræ centrum, erit etiam centrum circuli, qui circa quadratũ <lb/>a b c d deſcribitur: </s> |
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7950 <s xml:id="echoid-s4735" xml:space="preserve">& </s> |
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|
7951 <s xml:id="echoid-s4736" xml:space="preserve">propterea eiuſdem quadrati grauita <lb/>tis centrum: </s> |
|
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|
7952 <s xml:id="echoid-s4737" xml:space="preserve">quod in prima propoſitione huius demon-<lb/>ſtratum eſt. </s> |
|
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|
7953 <s xml:id="echoid-s4738" xml:space="preserve">quare pyramidis a b c d e axis erit e g: </s> |
|
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|
7954 <s xml:id="echoid-s4739" xml:space="preserve">& </s> |
|
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|
7955 <s xml:id="echoid-s4740" xml:space="preserve">pyra <lb/>midis a b c d f axis f g. </s> |
|
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|
7956 <s xml:id="echoid-s4741" xml:space="preserve">Itaque ſit h centrum grauitatis py-<lb/>ramidis a b c d e, & </s> |
|
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|
7957 <s xml:id="echoid-s4742" xml:space="preserve">pyramidis a b c d f centrum ſit _K_: </s> |
|
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|
7958 <s xml:id="echoid-s4743" xml:space="preserve">per-<lb/>ſpicuum eſt ex uigeſima ſecunda propoſitione huius, lineã <lb/>e h triplam eſſe h g: </s> |
|
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|
7959 <s xml:id="echoid-s4744" xml:space="preserve">cõ <lb/> |
|
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|
7960 <anchor type="figure" xlink:label="fig-0189-01a" xlink:href="fig-0189-01"/> |
|
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|
7961 ponendoq; </s> |
|
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|
7962 <s xml:id="echoid-s4745" xml:space="preserve">e g ipſius g <lb/>h quadruplam. </s> |
|
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|
7963 <s xml:id="echoid-s4746" xml:space="preserve">& </s> |
|
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|
7964 <s xml:id="echoid-s4747" xml:space="preserve">eadẽ <lb/>ratione f g quadruplã <lb/>ipſius g k. </s> |
|
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|
7965 <s xml:id="echoid-s4748" xml:space="preserve">quod cum e <lb/>g, g f ſintæquales, & </s> |
|
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|
7966 <s xml:id="echoid-s4749" xml:space="preserve">h <lb/>g, g _k_ neceſſario æqua-<lb/>les erunt. </s> |
|
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|
7967 <s xml:id="echoid-s4750" xml:space="preserve">ergo ex quar <lb/>ta propoſitione primi <lb/>libri Archimedis de cẽ-<lb/>tro grauitatis planorũ, <lb/>totius octahedri, quod <lb/>ex dictis pyramidibus <lb/>conſtat, centrum graui <lb/>tatis erit punctum g idem, quodipſius ſphæræ centrum.</s> |
|
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|
7968 <s xml:id="echoid-s4751" xml:space="preserve"/> |
|
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|
7969 </p> |
|
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|
7970 <div xml:id="echoid-div274" type="float" level="2" n="3"> |
|
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|
7971 <figure xlink:label="fig-0189-01" xlink:href="fig-0189-01a"> |
|
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|
7972 <image file="0189-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0189-01"/> |
|
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|
7973 </figure> |
|
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|
7974 </div> |
|
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|
7975 <p> |
|
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|
7976 <s xml:id="echoid-s4752" xml:space="preserve">Sit icoſahedrum a d deſcriptum in ſphæra, cuius centrū <lb/>ſit g. </s> |
|
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|
7977 <s xml:id="echoid-s4753" xml:space="preserve">Dico g ipſius icoſahedri grauitatis eſſe centrum. </s> |
|
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|
7978 <s xml:id="echoid-s4754" xml:space="preserve">Si <lb/>enim ab angnlo a per g ducatur rectalinea uſque ad ſphæ <lb/>ræ ſuperficiem; </s> |
|
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|
7979 <s xml:id="echoid-s4755" xml:space="preserve">conſtat ex ſexta decima propoſitione libri <lb/>tertii decimi elementorum, cadere eam in angulum ipſi a <lb/>oppoſitum. </s> |
|
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|
7980 <s xml:id="echoid-s4756" xml:space="preserve">cadat in d: </s> |
|
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|
7981 <s xml:id="echoid-s4757" xml:space="preserve">ſitq; </s> |
|
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|
7982 <s xml:id="echoid-s4758" xml:space="preserve">una aliqua baſis icoſahedri tri-<lb/>angulum a b c: </s> |
|
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|
7983 <s xml:id="echoid-s4759" xml:space="preserve">& </s> |
|
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|
7984 <s xml:id="echoid-s4760" xml:space="preserve">iunctæ b g, c g producantur, & </s> |
|
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|
7985 <s xml:id="echoid-s4761" xml:space="preserve">cadant in <lb/>angulos e f, ipſis b c oppoſitos. </s> |
|
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|
7986 <s xml:id="echoid-s4762" xml:space="preserve">Itaque per triangula <lb/>a b c, d e f ducantur plana ſphæram ſecantia. </s> |
|
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|
7987 <s xml:id="echoid-s4763" xml:space="preserve">erunt hæ ſe- |
|
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|
7988 <pb file="0190" n="190" rhead="FED. COMMANDINI"/> |
|
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|
7989 ctiones circuli ex prima propofitione ſphæricorum Theo <lb/>doſii: </s> |
|
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|
7990 <s xml:id="echoid-s4764" xml:space="preserve">unus quidem circa triangulum a b c deſcriptus: </s> |
|
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|
7991 <s xml:id="echoid-s4765" xml:space="preserve">al-<lb/>ter uero circa d e f: </s> |
|
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|
7992 <s xml:id="echoid-s4766" xml:space="preserve">& </s> |
|
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|
7993 <s xml:id="echoid-s4767" xml:space="preserve">quoniam triangula a b c, d e f æqua-<lb/>lia ſunt, & </s> |
|
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|
7994 <s xml:id="echoid-s4768" xml:space="preserve">ſimilia; </s> |
|
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|
7995 <s xml:id="echoid-s4769" xml:space="preserve">erunt ex prima, & </s> |
|
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|
7996 <s xml:id="echoid-s4770" xml:space="preserve">ſecunda propoſitione <lb/>duodecimi libri elementorum, circuli quoque inter ſe ſe <lb/>æquales. </s> |
|
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|
7997 <s xml:id="echoid-s4771" xml:space="preserve">poſtremo a centro g ad circulum a b c perpendi <lb/>cularis ducatur g h; </s> |
|
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|
7998 <s xml:id="echoid-s4772" xml:space="preserve">& </s> |
|
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|
7999 <s xml:id="echoid-s4773" xml:space="preserve">alia perpendicularis ducatur ad cir <lb/>culum d e f, quæ ſit g _k_; </s> |
|
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|
8000 <s xml:id="echoid-s4774" xml:space="preserve">& </s> |
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|
8001 <s xml:id="echoid-s4775" xml:space="preserve">iungantur a h, d k. </s> |
|
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|
8002 <s xml:id="echoid-s4776" xml:space="preserve">perſpicuum <lb/>eſt ex corollario primæ ſphæricorum Theodoſii, punctum <lb/>h centrum eſſe circuli a b c, & </s> |
|
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|
8003 <s xml:id="echoid-s4777" xml:space="preserve">k centrum circuli d e f. </s> |
|
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|
8004 <s xml:id="echoid-s4778" xml:space="preserve">Quo <lb/>niam igitur triangulorum g a h, g d K latus a g eſt æquale la <lb/>teri g d; </s> |
|
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|
8005 <s xml:id="echoid-s4779" xml:space="preserve">ſunt enim à centro ſphæræ ad ſuperficiem: </s> |
|
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|
8006 <s xml:id="echoid-s4780" xml:space="preserve">atque <lb/>eſt a h æquale d k: </s> |
|
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|
8007 <s xml:id="echoid-s4781" xml:space="preserve">& </s> |
|
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|
8008 <s xml:id="echoid-s4782" xml:space="preserve">ex ſexta propoſitione libri primi ſphæ <lb/>ricorum Theodoſii g h ipſi g K: </s> |
|
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|
8009 <s xml:id="echoid-s4783" xml:space="preserve">triangulum g a h æquale <lb/>erit, & </s> |
|
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|
8010 <s xml:id="echoid-s4784" xml:space="preserve">ſimile g d k triangulo: </s> |
|
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|
8011 <s xml:id="echoid-s4785" xml:space="preserve">& </s> |
|
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|
8012 <s xml:id="echoid-s4786" xml:space="preserve">angulus a g h æqualis an-<lb/>gulo d g _K_. </s> |
|
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|
8013 <s xml:id="echoid-s4787" xml:space="preserve">ſed anguli a g h, h g d ſunt æquales duobus re-<lb/> |
|
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|
8014 <anchor type="note" xlink:label="note-0190-01a" xlink:href="note-0190-01"/> |
|
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|
8015 ctis. </s> |
|
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|
8016 <s xml:id="echoid-s4788" xml:space="preserve">ergo & </s> |
|
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|
8017 <s xml:id="echoid-s4789" xml:space="preserve">ipſi h g d, d g k duobus rectis æquales erunt. <lb/></s> |
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8018 <s xml:id="echoid-s4790" xml:space="preserve">& </s> |
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8019 <s xml:id="echoid-s4791" xml:space="preserve">idcirco h g, g _K_ una, atque eadem erit linea. </s> |
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8020 <s xml:id="echoid-s4792" xml:space="preserve">cum autem <lb/> |
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8021 <anchor type="note" xlink:label="note-0190-02a" xlink:href="note-0190-02"/> |
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|
8022 h ſit centrũ circuli, & </s> |
|
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|
8023 <s xml:id="echoid-s4793" xml:space="preserve">tri-<lb/> |
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|
8024 <anchor type="figure" xlink:label="fig-0190-01a" xlink:href="fig-0190-01"/> |
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|
8025 anguli a b c grauitatis cen <lb/>trũ probabitur ex iis, quæ <lb/>in prima propoſitione hu <lb/>ius tradita funt. </s> |
|
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8026 <s xml:id="echoid-s4794" xml:space="preserve">quare g h <lb/>erit pyramidis a b c g axis. <lb/></s> |
|
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8027 <s xml:id="echoid-s4795" xml:space="preserve">& </s> |
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|
8028 <s xml:id="echoid-s4796" xml:space="preserve">ob eandem cauſſam g k <lb/>axis pyramidis d e f g. </s> |
|
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|
8029 <s xml:id="echoid-s4797" xml:space="preserve">Ita-<lb/>que centrum grauitatis py <lb/>ramidis a b c g ſit púctum <lb/>l, & </s> |
|
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|
8030 <s xml:id="echoid-s4798" xml:space="preserve">pyramidis d e f g ſit m. </s> |
|
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|
8031 <s xml:id="echoid-s4799" xml:space="preserve"><lb/>Similiter ut ſupra demon-<lb/>ſtrabimus m g, g linter ſe æquales eſſe, & </s> |
|
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8032 <s xml:id="echoid-s4800" xml:space="preserve">punctum g graui <lb/>tatis centrum magnitudinis, quæ ex utriſque pyramidibus <lb/>conſtat. </s> |
|
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|
8033 <s xml:id="echoid-s4801" xml:space="preserve">eodem modo demonſtrabitur, quarumcunque <lb/>duarum pyramidum, quæ opponuntur, grauitatis centrũ |
|
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|
8034 <pb o="40" file="0191" n="191" rhead="DE CENTRO GRAVIT. SOLID."/> |
|
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|
8035 eſſe pun ctum g. </s> |
|
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|
8036 <s xml:id="echoid-s4802" xml:space="preserve">Sequitur ergo uticoſahedri centrum gra <lb/>uitatis fit idem, quodipſius ſphæræ centrum.</s> |
|
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|
8037 <s xml:id="echoid-s4803" xml:space="preserve"/> |
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|
8038 </p> |
|
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|
8039 <div xml:id="echoid-div275" type="float" level="2" n="4"> |
|
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|
8040 <note position="left" xlink:label="note-0190-01" xlink:href="note-0190-01a" xml:space="preserve">13. primi</note> |
|
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|
8041 <note position="left" xlink:label="note-0190-02" xlink:href="note-0190-02a" xml:space="preserve">14. primi</note> |
|
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|
8042 <figure xlink:label="fig-0190-01" xlink:href="fig-0190-01a"> |
|
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|
8043 <image file="0190-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0190-01"/> |
|
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|
8044 </figure> |
|
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|
8045 </div> |
|
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|
8046 <p> |
|
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|
8047 <s xml:id="echoid-s4804" xml:space="preserve">Sit dodecahedrum a ſin ſphæra deſignatum, ſitque ſphæ <lb/>ræ centrum m. </s> |
|
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|
8048 <s xml:id="echoid-s4805" xml:space="preserve">Dico m centrum eſſe grauitatis ipſius do-<lb/>decahedri. </s> |
|
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|
8049 <s xml:id="echoid-s4806" xml:space="preserve">Sit enim pentagonum a b c d e una ex duode-<lb/>cim baſibus ſolidi a f: </s> |
|
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|
8050 <s xml:id="echoid-s4807" xml:space="preserve">& </s> |
|
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|
8051 <s xml:id="echoid-s4808" xml:space="preserve">iuncta a m producatur ad ſphæræ <lb/>ſuperficiem. </s> |
|
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|
8052 <s xml:id="echoid-s4809" xml:space="preserve">cadetin angulum ipſi a oppoſitum; </s> |
|
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|
8053 <s xml:id="echoid-s4810" xml:space="preserve">quod col-<lb/>ligitur ex decima ſeptima propoſitione tertiidecimilibri <lb/>elementorum. </s> |
|
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|
8054 <s xml:id="echoid-s4811" xml:space="preserve">cadat in f. </s> |
|
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|
8055 <s xml:id="echoid-s4812" xml:space="preserve">at ſi ab aliis angulis b c d e per cẽ <lb/>trum itidem lineæ ducantur ad ſuperficiem ſphæræ in pun <lb/>cta g h k l; </s> |
|
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|
8056 <s xml:id="echoid-s4813" xml:space="preserve">cadent hæ in alios angulos baſis, quæ ipſi a b c d <lb/>baſi opponitur. </s> |
|
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|
8057 <s xml:id="echoid-s4814" xml:space="preserve">tranſeant ergo per pentagona a b c d e, <lb/>f g h K l plana ſphæram ſecantia, quæ facient ſectiones cir-<lb/>culos æquales inter ſe ſe poſtea ducantur ex centro ſphæræ <lb/>m perpen diculares ad pla-<lb/> |
|
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|
8058 <anchor type="figure" xlink:label="fig-0191-01a" xlink:href="fig-0191-01"/> |
|
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|
8059 na dictorum circulorũ; </s> |
|
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|
8060 <s xml:id="echoid-s4815" xml:space="preserve">ad <lb/>circulum quidem a b c d e <lb/>perpendicularis m n: </s> |
|
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|
8061 <s xml:id="echoid-s4816" xml:space="preserve">& </s> |
|
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|
8062 <s xml:id="echoid-s4817" xml:space="preserve">ad <lb/>circulum f g h K l ipſa m o, <lb/> |
|
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|
8063 <anchor type="note" xlink:label="note-0191-01a" xlink:href="note-0191-01"/> |
|
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|
8064 erunt puncta n o circulorũ <lb/>centra: </s> |
|
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|
8065 <s xml:id="echoid-s4818" xml:space="preserve">& </s> |
|
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|
8066 <s xml:id="echoid-s4819" xml:space="preserve">lineæ m n, m o in <lb/>ter ſe æquales: </s> |
|
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|
8067 <s xml:id="echoid-s4820" xml:space="preserve">quòd circu-<lb/>li æquales ſint. </s> |
|
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|
8068 <s xml:id="echoid-s4821" xml:space="preserve">Eodem mo <lb/> |
|
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|
8069 <anchor type="note" xlink:label="note-0191-02a" xlink:href="note-0191-02"/> |
|
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|
8070 do, quo ſupra, demonſtrabi <lb/>mus lineas m n, m o in unã <lb/>atque eandem lineam con-<lb/>uenire. </s> |
|
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|
8071 <s xml:id="echoid-s4822" xml:space="preserve">ergo cum puncta n o ſint centra circulorum, con-<lb/>ſtat ex prima huius & </s> |
|
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|
8072 <s xml:id="echoid-s4823" xml:space="preserve">pentagonorũ grauitatis eſſe centra: <lb/></s> |
|
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|
8073 <s xml:id="echoid-s4824" xml:space="preserve">idcircoq; </s> |
|
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|
8074 <s xml:id="echoid-s4825" xml:space="preserve">m n, m o pyramidum a b c d e m, ſ g h _K_ l m axes. </s> |
|
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|
8075 <s xml:id="echoid-s4826" xml:space="preserve"><lb/>ponatur a b c d e m pyramidis grauitatis centrum p: </s> |
|
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|
8076 <s xml:id="echoid-s4827" xml:space="preserve">& </s> |
|
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|
8077 <s xml:id="echoid-s4828" xml:space="preserve">py <lb/>ramidis f g h <emph style="sc">K</emph> l m ipſum q centrum. </s> |
|
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|
8078 <s xml:id="echoid-s4829" xml:space="preserve">erunt p m, m q æqua-<lb/>les, & </s> |
|
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|
8079 <s xml:id="echoid-s4830" xml:space="preserve">punctum m grauitatis centrum magnitudinis, quæ <lb/>ex ipſis pyramidibus conſtat. </s> |
|
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|
8080 <s xml:id="echoid-s4831" xml:space="preserve">eodẽ modo probabitur qua-<lb/>rumlibet pyramidum, quæ è regione opponuntur, centrũ |
|
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|
8081 <pb file="0192" n="192" rhead="FED. COMMANDINI"/> |
|
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|
8082 grauitatis eſſe punctum m. </s> |
|
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|
8083 <s xml:id="echoid-s4832" xml:space="preserve">patetigitur totius dodecahe-<lb/>dri, centrum grauitatis idẽ eſſe, quod & </s> |
|
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|
8084 <s xml:id="echoid-s4833" xml:space="preserve">ſphæræ ipſum com <lb/>prehendentis centrum. </s> |
|
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|
8085 <s xml:id="echoid-s4834" xml:space="preserve">quæ quidem omnia demonſtraſſe <lb/>oportebat.</s> |
|
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8086 <s xml:id="echoid-s4835" xml:space="preserve"/> |
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8087 </p> |
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8088 <div xml:id="echoid-div276" type="float" level="2" n="5"> |
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8089 <figure xlink:label="fig-0191-01" xlink:href="fig-0191-01a"> |
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8090 <image file="0191-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0191-01"/> |
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8091 </figure> |
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8092 <note position="right" xlink:label="note-0191-01" xlink:href="note-0191-01a" xml:space="preserve">corol. pri <lb/>mæ ſphæ <lb/>ricorum <lb/>Theod.</note> |
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8093 <note position="right" xlink:label="note-0191-02" xlink:href="note-0191-02a" xml:space="preserve">6. primi <lb/>phærico <lb/>rum.</note> |
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8094 </div> |
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8095 </div> |
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8096 <div xml:id="echoid-div278" type="section" level="1" n="93"> |
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8097 <head xml:id="echoid-head100" xml:space="preserve">PROBLEMA VI. PROPOSITIO XX VIII.</head> |
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8098 <p> |
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8099 <s xml:id="echoid-s4836" xml:space="preserve"><emph style="sc">Data</emph> qualibet portione conoidis rectangu <lb/>li, abſciſſa plano ad axem recto, uel non recto; </s> |
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8100 <s xml:id="echoid-s4837" xml:space="preserve">fie-<lb/>ri poteſt, ut portio ſolida inſcribatur, uel circum-<lb/>ſcribatur ex cylindris, uel cylindri portionibus, <lb/>æqualem habentibus altitudinem, ita ut recta li-<lb/>nea, quæ inter centrum grauitatis portionis, & </s> |
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8101 <s xml:id="echoid-s4838" xml:space="preserve"><lb/>figuræ inſcriptæ, uel circumſcriptæ interiicitur, <lb/>ſit minor qualibet recta linea propoſita.</s> |
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8102 <s xml:id="echoid-s4839" xml:space="preserve"/> |
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8103 </p> |
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8104 <p> |
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8105 <s xml:id="echoid-s4840" xml:space="preserve">Sit portio conoidis rectanguli a b c, cuius axis b d, gra-<lb/>uitatisq; </s> |
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8106 <s xml:id="echoid-s4841" xml:space="preserve">centrum e: </s> |
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8107 <s xml:id="echoid-s4842" xml:space="preserve">& </s> |
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8108 <s xml:id="echoid-s4843" xml:space="preserve">fit g recta linea propoſita. </s> |
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8109 <s xml:id="echoid-s4844" xml:space="preserve">quam ue <lb/>ro proportionem habet linea b e ad lineam g, eandem ha-<lb/>beat portio conoidis ad ſolidum h: </s> |
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8110 <s xml:id="echoid-s4845" xml:space="preserve">& </s> |
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8111 <s xml:id="echoid-s4846" xml:space="preserve">circumſcribatur por <lb/>tioni figura, ſicuti dictum eſt, ita ut portiones reliquæ ſint <lb/>ſolido h minores: </s> |
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8112 <s xml:id="echoid-s4847" xml:space="preserve">cuius quidem figuræ centrum grauitatis <lb/>ſit punctum <emph style="sc">K</emph>. </s> |
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8113 <s xml:id="echoid-s4848" xml:space="preserve">Dico lineã k e minorem eſſe linea g propo-<lb/>ſita. </s> |
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8114 <s xml:id="echoid-s4849" xml:space="preserve">niſi enim ſit minor, uel æqualis, uel maior erit. </s> |
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8115 <s xml:id="echoid-s4850" xml:space="preserve">& </s> |
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8116 <s xml:id="echoid-s4851" xml:space="preserve">quo-<lb/>niam figura circumſcripta ad reliquas portiones maiorem <lb/> |
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8117 <anchor type="note" xlink:label="note-0192-01a" xlink:href="note-0192-01"/> |
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8118 proportionem habet, quàm portio conoidis ad ſolidum h; <lb/></s> |
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8119 <s xml:id="echoid-s4852" xml:space="preserve">hoc eſt maiorem, quàm b c ad g: </s> |
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8120 <s xml:id="echoid-s4853" xml:space="preserve">& </s> |
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8121 <s xml:id="echoid-s4854" xml:space="preserve">b e ad g non minorem <lb/>habet proportionem, quàm ad _k_ e, propterea quod k e non <lb/>ponitur minor ipſa g: </s> |
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8122 <s xml:id="echoid-s4855" xml:space="preserve">habebit figura circumſcripta ad por <lb/>tiones reliquas maiorem proportionem quàm b e ad e k: </s> |
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8123 <s xml:id="echoid-s4856" xml:space="preserve"><lb/> |
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8124 <anchor type="note" xlink:label="note-0192-02a" xlink:href="note-0192-02"/> |
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8125 & </s> |
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8126 <s xml:id="echoid-s4857" xml:space="preserve">diuidendo portio conoidis ad reliquas portiones habe-<lb/>bit maiorem, quàm b <emph style="sc">K</emph> ad K e. </s> |
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8127 <s xml:id="echoid-s4858" xml:space="preserve">quare ſi fiat ut portio co- |
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8128 <pb o="41" file="0193" n="193" rhead="DE CENTRO GRAVIT. SOLID."/> |
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8129 noidis ad portiones reliquas, ita alia linea, quæ ſit 1 <emph style="sc">K</emph> ad <lb/>k e: </s> |
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8130 <s xml:id="echoid-s4859" xml:space="preserve">erit 1k maior, quam b k: </s> |
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8131 <s xml:id="echoid-s4860" xml:space="preserve">& </s> |
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8132 <s xml:id="echoid-s4861" xml:space="preserve">ideo punctum l extra por-<lb/>tionem cadet. </s> |
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8133 <s xml:id="echoid-s4862" xml:space="preserve">Quoniã <lb/> |
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8134 <anchor type="figure" xlink:label="fig-0193-01a" xlink:href="fig-0193-01"/> |
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8135 igitur à figura circum-<lb/>ſcripta, cuius grauitatis <lb/>centrum eſt k, aufertur <lb/>portio conoidis, cuius <lb/>centrum e. </s> |
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8136 <s xml:id="echoid-s4863" xml:space="preserve">habetq; </s> |
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8137 <s xml:id="echoid-s4864" xml:space="preserve">l K <lb/>ad K e eam proportio-<lb/>nem, quam portio co-<lb/>noidis ad reliquas por-<lb/>tiones; </s> |
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8138 <s xml:id="echoid-s4865" xml:space="preserve">erit punctum l <lb/>extra portionem cadẽs, <lb/>centrum magnitudinis <lb/>ex reliquis portionibus compoſitæ. </s> |
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8139 <s xml:id="echoid-s4866" xml:space="preserve">illud autem fieri nullo <lb/>modo poteſt. </s> |
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8140 <s xml:id="echoid-s4867" xml:space="preserve">quare conſtat lineam k e ipſa g linea propoſi <lb/>ta minorem eſſe.</s> |
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8141 <s xml:id="echoid-s4868" xml:space="preserve"/> |
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|
8142 </p> |
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8143 <div xml:id="echoid-div278" type="float" level="2" n="1"> |
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8144 <note position="left" xlink:label="note-0192-01" xlink:href="note-0192-01a" xml:space="preserve">8. quĭnti.</note> |
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8145 <note position="left" xlink:label="note-0192-02" xlink:href="note-0192-02a" xml:space="preserve">29. quĭnti <lb/>ex tradi-<lb/>tione Cã-<lb/>ſàni.</note> |
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8146 <figure xlink:label="fig-0193-01" xlink:href="fig-0193-01a"> |
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8147 <image file="0193-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0193-01"/> |
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8148 </figure> |
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8149 </div> |
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8150 <p> |
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8151 <s xml:id="echoid-s4869" xml:space="preserve">Rurfus inſcribatur portioni figura, uidelicet cylindr us <lb/>m n, ut ſit ipſius altitudo <lb/> |
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8152 <anchor type="figure" xlink:label="fig-0193-02a" xlink:href="fig-0193-02"/> |
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8153 æqualis dimidio axis b d: <lb/></s> |
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8154 <s xml:id="echoid-s4870" xml:space="preserve">& </s> |
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8155 <s xml:id="echoid-s4871" xml:space="preserve">quam proportionem <lb/>habet b e ad g, habeat m n <lb/>cylindrus ad ſolidum o. </s> |
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8156 <s xml:id="echoid-s4872" xml:space="preserve"><lb/>inſcrib itur deinde eidem <lb/>alia figura, ita ut portio-<lb/>nes reliquæ ſint ſolido o <lb/>minores: </s> |
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8157 <s xml:id="echoid-s4873" xml:space="preserve">& </s> |
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8158 <s xml:id="echoid-s4874" xml:space="preserve">centrum gra <lb/>uitatis figuræ ſit p. </s> |
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8159 <s xml:id="echoid-s4875" xml:space="preserve">Dico <lb/>lineam p e ipſa g minorẽ <lb/>eſſe. </s> |
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8160 <s xml:id="echoid-s4876" xml:space="preserve">ſi enim non ſit mi-<lb/>nor, eodem, quo ſupra modo demonſtrabimus figuram in <lb/>ſcriptam ad reliquas portiones maiorem proportionem <lb/>habere, quàm b e ad e p. </s> |
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8161 <s xml:id="echoid-s4877" xml:space="preserve">& </s> |
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8162 <s xml:id="echoid-s4878" xml:space="preserve">ſi fiat alia linea l e ad e p, ut eſt <lb/>figura inſcripta ad reliquas portiones, pũctum l extra por |
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8163 <pb file="0194" n="194" rhead="FED. COMMANDINI"/> |
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8164 tionem cadet: </s> |
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8165 <s xml:id="echoid-s4879" xml:space="preserve">Itaque cum à portione conoidis, cuius gra-<lb/>uitatis centrum e auferatur inſcripta figura, centrum ha-<lb/>bens p: </s> |
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8166 <s xml:id="echoid-s4880" xml:space="preserve">& </s> |
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8167 <s xml:id="echoid-s4881" xml:space="preserve">ſit l e ad e p, ut figura inſcripta ad portiones reli <lb/>quas: </s> |
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8168 <s xml:id="echoid-s4882" xml:space="preserve">erit magnitudinis, quæ ex reliquis portionibus con <lb/>ſtat, centrum grauitatis punctum l, extra portionem ca-<lb/>dens. </s> |
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8169 <s xml:id="echoid-s4883" xml:space="preserve">quod fieri nequit. </s> |
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8170 <s xml:id="echoid-s4884" xml:space="preserve">ergo linea p e minor eſt ip ſa g li-<lb/>nea propoſita.</s> |
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8171 <s xml:id="echoid-s4885" xml:space="preserve"/> |
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8172 </p> |
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8173 <div xml:id="echoid-div279" type="float" level="2" n="2"> |
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8174 <figure xlink:label="fig-0193-02" xlink:href="fig-0193-02a"> |
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8175 <image file="0193-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0193-02"/> |
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8176 </figure> |
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8177 </div> |
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8178 <p> |
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8179 <s xml:id="echoid-s4886" xml:space="preserve">Ex quibus perſpicuum eſt centrum grauitatis <lb/>figuræ inſcriptæ, & </s> |
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8180 <s xml:id="echoid-s4887" xml:space="preserve">circumſcriptæ eo magis acce <lb/>dere ad portionis centrum, quo pluribus cylin-<lb/>dris, uel cylindri portionibus conſtet: </s> |
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8181 <s xml:id="echoid-s4888" xml:space="preserve">fiatq́ figu <lb/>ra inſcripta maior, & </s> |
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8182 <s xml:id="echoid-s4889" xml:space="preserve">circumſcripta minor. </s> |
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8183 <s xml:id="echoid-s4890" xml:space="preserve">& </s> |
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8184 <s xml:id="echoid-s4891" xml:space="preserve"><lb/>quanquam continenter ad portionis centrū pro-<lb/>pius admoueatur nunquam tamen ad ipſum per <lb/>ueniet. </s> |
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8185 <s xml:id="echoid-s4892" xml:space="preserve">ſequeretur enim figuram inſcriptam, nó <lb/>ſolum portioni, ſed etiam circumſcriptæ figuræ <lb/>æqualem eſſe. </s> |
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8186 <s xml:id="echoid-s4893" xml:space="preserve">quod eſt abſurdum.</s> |
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8187 <s xml:id="echoid-s4894" xml:space="preserve"/> |
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8188 </p> |
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8189 </div> |
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8190 <div xml:id="echoid-div281" type="section" level="1" n="94"> |
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8191 <head xml:id="echoid-head101" xml:space="preserve">THE OREMA XXIII. PROPOSITIO XXIX.</head> |
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8192 <p> |
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8193 <s xml:id="echoid-s4895" xml:space="preserve"><emph style="sc">Cvivslibet</emph> portionis conoidis rectangu-<lb/>li axis à cẽtro grauitatis ita diuiditur, ut pars quæ <lb/>terminatur ad uerticem, reliquæ partis, quæ ad ba <lb/>ſim ſit dupla.</s> |
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8194 <s xml:id="echoid-s4896" xml:space="preserve"/> |
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8195 </p> |
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8196 <p> |
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8197 <s xml:id="echoid-s4897" xml:space="preserve">SIT portio conoidis rectanguli uel abſciſſa plano ad <lb/>axem recto, uel non recto: </s> |
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8198 <s xml:id="echoid-s4898" xml:space="preserve">& </s> |
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8199 <s xml:id="echoid-s4899" xml:space="preserve">ſecta ipſa altero plano per axé <lb/>ſit ſuperſiciei ſe ctio a b c r ectanguli coni ſectio, uel parabo <lb/>le; </s> |
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8200 <s xml:id="echoid-s4900" xml:space="preserve">plani abſcindentis portionem ſectio ſit recta linea a c: <lb/></s> |
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8201 <s xml:id="echoid-s4901" xml:space="preserve">axis portionis, & </s> |
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8202 <s xml:id="echoid-s4902" xml:space="preserve">ſectionis diameter b d. </s> |
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8203 <s xml:id="echoid-s4903" xml:space="preserve">Sumatur autem <lb/>in linea b d punctum e, ita ut b e ſit ipſius e d dupla. </s> |
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8204 <s xml:id="echoid-s4904" xml:space="preserve">Dico |
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8205 <pb o="42" file="0195" n="195" rhead="DE CENTRO GRAVIT. SOLID."/> |
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8206 e portionis a b <lb/> |
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8207 <anchor type="figure" xlink:label="fig-0195-01a" xlink:href="fig-0195-01"/> |
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8208 c grauitatis eſſe <lb/>centrum. </s> |
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8209 <s xml:id="echoid-s4905" xml:space="preserve">Diui-<lb/>datur enim b d <lb/>bifariam in m: <lb/></s> |
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8210 <s xml:id="echoid-s4906" xml:space="preserve">&</s> |
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8211 <s xml:id="echoid-s4907" xml:space="preserve">rurſus d m, m <lb/>b bifariam diui-<lb/>dantur in pun-<lb/>ctis n, o: </s> |
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8212 <s xml:id="echoid-s4908" xml:space="preserve">inſcri-<lb/>baturq; </s> |
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8213 <s xml:id="echoid-s4909" xml:space="preserve">portio-<lb/>ni figura ſolida, <lb/>& </s> |
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8214 <s xml:id="echoid-s4910" xml:space="preserve">altera circum <lb/>ſcribatur ex cy-<lb/>lindris æqualem <lb/>altitudinem ha-<lb/>bentibus, utſu-<lb/>perius dictũ eſt. </s> |
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8215 <s xml:id="echoid-s4911" xml:space="preserve"><lb/>Sit autem pri-<lb/>mum figura in-<lb/>ſcripta cylĩ drus <lb/>f g: </s> |
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8216 <s xml:id="echoid-s4912" xml:space="preserve">& </s> |
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8217 <s xml:id="echoid-s4913" xml:space="preserve">circũſcri-<lb/>pta ex cylindris <lb/>a h, K l conſtet. </s> |
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8218 <s xml:id="echoid-s4914" xml:space="preserve"><lb/>punctum n erit <lb/> |
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8219 <anchor type="note" xlink:label="note-0195-01a" xlink:href="note-0195-01"/> |
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8220 centrum graui-<lb/>tatis figuræ in-<lb/>fcriptæ, mediũ <lb/>ſcilicet ipſius d <lb/>m axis: </s> |
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8221 <s xml:id="echoid-s4915" xml:space="preserve">atq; </s> |
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8222 <s xml:id="echoid-s4916" xml:space="preserve">idẽ <lb/>erit centrum cy <lb/>lindri a h: </s> |
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8223 <s xml:id="echoid-s4917" xml:space="preserve">& </s> |
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8224 <s xml:id="echoid-s4918" xml:space="preserve">cy-<lb/>lindri <emph style="sc">K</emph> l centrũ <lb/>o, axis b m me-<lb/>dium, quare ſi li |
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|
8225 <pb file="0196" n="196" rhead="FED. COMMANDINI"/> |
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8226 neam on ita di <lb/> |
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8227 <anchor type="figure" xlink:label="fig-0196-01a" xlink:href="fig-0196-01"/> |
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8228 uiſerimus in p, <lb/>ut quã propor-<lb/>tionẽ habet cy-<lb/>lindrus a h ad <lb/>cylindrum <emph style="sc">K</emph> l, <lb/>habeat linea o p <lb/>ad p n: </s> |
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8229 <s xml:id="echoid-s4919" xml:space="preserve">centrum <lb/> |
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8230 <anchor type="note" xlink:label="note-0196-01a" xlink:href="note-0196-01"/> |
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8231 grauitatis toti-<lb/>us figuræ circũ-<lb/>ſcriptæ erit pun <lb/>ctum p. </s> |
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8232 <s xml:id="echoid-s4920" xml:space="preserve">Sed cy-<lb/> |
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8233 <anchor type="note" xlink:label="note-0196-02a" xlink:href="note-0196-02"/> |
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8234 lindri, qui ſunt <lb/>æquali altitudi-<lb/>ne, candem in-<lb/>ter ſe ſe, quam <lb/>baſes propor-<lb/>tionem habent: <lb/></s> |
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8235 <s xml:id="echoid-s4921" xml:space="preserve">eſtq; </s> |
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8236 <s xml:id="echoid-s4922" xml:space="preserve">ut linea d b <lb/>ad b m, ita qua-<lb/>dratũ lineæ a d <lb/>ad quadratũ ip-<lb/>ſius _K_ in, ex uige <lb/>ſima primi libri <lb/>conicorũ: </s> |
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8237 <s xml:id="echoid-s4923" xml:space="preserve">& </s> |
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8238 <s xml:id="echoid-s4924" xml:space="preserve">ita <lb/> |
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8239 <anchor type="note" xlink:label="note-0196-03a" xlink:href="note-0196-03"/> |
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8240 quadratum a c <lb/>ad quadratũ K <lb/>g: </s> |
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8241 <s xml:id="echoid-s4925" xml:space="preserve">hoc eſt circu-<lb/> |
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8242 <anchor type="note" xlink:label="note-0196-04a" xlink:href="note-0196-04"/> |
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8243 lus circa diame <lb/>trum a c ad cir-<lb/>culum circa dia <lb/>metrum k g. </s> |
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8244 <s xml:id="echoid-s4926" xml:space="preserve">du <lb/>pla eſt autem li-<lb/>nea d b lineæ |
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8245 <pb o="43" file="0197" n="197" rhead="DE CENTRO GRAVIT. SOLID."/> |
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8246 b m. </s> |
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8247 <s xml:id="echoid-s4927" xml:space="preserve">ergo circulus a c circuli _k_ g: </s> |
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8248 <s xml:id="echoid-s4928" xml:space="preserve">& </s> |
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8249 <s xml:id="echoid-s4929" xml:space="preserve">idcirco cylindrus <lb/>a h cylindri _k_ l duplus erit. </s> |
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8250 <s xml:id="echoid-s4930" xml:space="preserve">quare & </s> |
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8251 <s xml:id="echoid-s4931" xml:space="preserve">linea o p dupla <lb/>ipſius p n. </s> |
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8252 <s xml:id="echoid-s4932" xml:space="preserve">Deinde inſcripta & </s> |
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8253 <s xml:id="echoid-s4933" xml:space="preserve">circumſcripta portioni <lb/>alia figura, ita ut inſcripta conſtituatur ex tribus cylin-<lb/>dris q r, s g, tu: </s> |
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8254 <s xml:id="echoid-s4934" xml:space="preserve">circumſcripta uero ex quatuor a x, y z, <lb/>_K_ ν, θ λ: </s> |
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8255 <s xml:id="echoid-s4935" xml:space="preserve">diuidantur b o, o m, m n, n d bifariam in punctis <lb/>μ ν π ρ. </s> |
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8256 <s xml:id="echoid-s4936" xml:space="preserve">Itaque cylindri θ λ centrum grauitætis eſt punctum <lb/>μ: </s> |
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8257 <s xml:id="echoid-s4937" xml:space="preserve">& </s> |
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8258 <s xml:id="echoid-s4938" xml:space="preserve">cylindri <emph style="sc">K</emph> ν centrum ν. </s> |
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8259 <s xml:id="echoid-s4939" xml:space="preserve">ergo ſi linea μ ν diuidatur in σ, <lb/>ita ut μ σ ad σ ν proportionẽ eã habeat, quam cylindrus K ν <lb/>ad cylindrum θ λ, uidelicet quam quadratum <emph style="sc">K</emph> m ad qua-<lb/>dratum θ o, hoc eſt, quam linea m b ad b o: </s> |
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8260 <s xml:id="echoid-s4940" xml:space="preserve">erit σ centrum <lb/> |
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8261 <anchor type="note" xlink:label="note-0197-01a" xlink:href="note-0197-01"/> |
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8262 magnitudinis compoſitæ ex cylindris <emph style="sc">K</emph> ν, θ λ. </s> |
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8263 <s xml:id="echoid-s4941" xml:space="preserve">& </s> |
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8264 <s xml:id="echoid-s4942" xml:space="preserve">cum linea <lb/>m b ſit dupla b o, erit & </s> |
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8265 <s xml:id="echoid-s4943" xml:space="preserve">μ σ ipſius σ ν dupla. </s> |
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8266 <s xml:id="echoid-s4944" xml:space="preserve">præterea quo-<lb/>niam cylindri y z centrum grauitatis eſt π, linea σ π ita diui <lb/>ſa in τ, ut σ τ ad τ π eam habeat proportionem, quam cylin <lb/>drus y z ad duos cylindros K ν, θ λ: </s> |
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8267 <s xml:id="echoid-s4945" xml:space="preserve">erit τ centrum magnitu <lb/>dinis, quæ ex dictis tribus cylindris conſtat. </s> |
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8268 <s xml:id="echoid-s4946" xml:space="preserve">cylindrus au-<lb/>tẽ y z ad cylindrum θ λ eſt, ut linea n b ad b o, hoc eſt ut 3 <lb/>ad 1: </s> |
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8269 <s xml:id="echoid-s4947" xml:space="preserve">& </s> |
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8270 <s xml:id="echoid-s4948" xml:space="preserve">ad cylindrum k ν, ut n b ad b m, uidelicet ut 3 ad 2. <lb/></s> |
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8271 <s xml:id="echoid-s4949" xml:space="preserve">quare y z cylĩdrus duobus cylindris k ν, θ λ æqualis erit. </s> |
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8272 <s xml:id="echoid-s4950" xml:space="preserve">& </s> |
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8273 <s xml:id="echoid-s4951" xml:space="preserve"><lb/>propterea linea σ τ æqualis ipſi τ π. </s> |
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8274 <s xml:id="echoid-s4952" xml:space="preserve">denique cylindri a x <lb/>centrum grauitatis eſt punctum ρ. </s> |
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8275 <s xml:id="echoid-s4953" xml:space="preserve">& </s> |
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8276 <s xml:id="echoid-s4954" xml:space="preserve">cum τ ζ diuiſa fuerit <lb/>in eã proportionem, quam habet cylindrus a x ad tres cy-<lb/>lindros y z, _k_ ν, θ λ: </s> |
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8277 <s xml:id="echoid-s4955" xml:space="preserve">erit in eo puncto centrum grauitatis <lb/>totius figuræ circũſcriptæ. </s> |
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8278 <s xml:id="echoid-s4956" xml:space="preserve">Sed cylindrus a x ad ipſum y z <lb/>eſt ut linea d b ad b n: </s> |
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8279 <s xml:id="echoid-s4957" xml:space="preserve">hoc eſt ut 4 ad 3: </s> |
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8280 <s xml:id="echoid-s4958" xml:space="preserve">& </s> |
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8281 <s xml:id="echoid-s4959" xml:space="preserve">duo cylindri _k_ ν <lb/>θ λ cylindro y z ſunt æquales. </s> |
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8282 <s xml:id="echoid-s4960" xml:space="preserve">cylindrns igitur a x ad tres <lb/>iam dictos cylindros eſt ut 2 ad 3. </s> |
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8283 <s xml:id="echoid-s4961" xml:space="preserve">Sed quoniã μ σ eſt dua-<lb/>rum partium, & </s> |
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8284 <s xml:id="echoid-s4962" xml:space="preserve">σ ν unius, qualium μ π eſt ſex; </s> |
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8285 <s xml:id="echoid-s4963" xml:space="preserve">erit σ π par-<lb/>tium quatuor: </s> |
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8286 <s xml:id="echoid-s4964" xml:space="preserve">proptereaq; </s> |
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8287 <s xml:id="echoid-s4965" xml:space="preserve">τ π duarum, & </s> |
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8288 <s xml:id="echoid-s4966" xml:space="preserve">ν π, hoc eſt π ρ <lb/>trium. </s> |
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8289 <s xml:id="echoid-s4967" xml:space="preserve">quare ſequitur ut punctum π totius figuræ circum <lb/>ſcriptæ ſit centrum. </s> |
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8290 <s xml:id="echoid-s4968" xml:space="preserve">Itaque fiat ν υ ad υ π, ut μ σ ad σ ν. </s> |
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8291 <s xml:id="echoid-s4969" xml:space="preserve">& </s> |
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|
8292 <s xml:id="echoid-s4970" xml:space="preserve">υ ρ <lb/>bifariam diuidatur in φ. </s> |
|
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8293 <s xml:id="echoid-s4971" xml:space="preserve">Similiter ut in circumſcripta figu <lb/>ra oſtendetur centrum magnitudinis compoſitæ ex cylin- |
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|
8294 <pb file="0198" n="198" rhead="FED. COMMANDINI"/> |
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|
8295 dris s g, t u eſſe <lb/> |
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8296 <anchor type="figure" xlink:label="fig-0198-01a" xlink:href="fig-0198-01"/> |
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|
8297 punctum υ & </s> |
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8298 <s xml:id="echoid-s4972" xml:space="preserve"><lb/>totius figuræ in <lb/>ſcriptæ, quæ cõ-<lb/>ſtat ex cylindris <lb/>q r, ſ g, t u eſſe φ <lb/>centrum. </s> |
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8299 <s xml:id="echoid-s4973" xml:space="preserve">Sunt <lb/>enim hi cylindri <lb/>æquales & </s> |
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8300 <s xml:id="echoid-s4974" xml:space="preserve">ſimi-<lb/>les cylindris y z, <lb/>K ν, θ λ, figuræ <lb/>circumſcriptæ. <lb/></s> |
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|
8301 <s xml:id="echoid-s4975" xml:space="preserve">Quoniã igitur <lb/>ut b e ad e d, ita <lb/>eſt o p ad p n; </s> |
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8302 <s xml:id="echoid-s4976" xml:space="preserve"><lb/>utraq; </s> |
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|
8303 <s xml:id="echoid-s4977" xml:space="preserve">enim u-<lb/>triuſque eſt du-<lb/>pla: </s> |
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8304 <s xml:id="echoid-s4978" xml:space="preserve">erit compo <lb/>nendo, ut b d ad <lb/>d e, ita o n ad n <lb/>p; </s> |
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|
8305 <s xml:id="echoid-s4979" xml:space="preserve">& </s> |
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|
8306 <s xml:id="echoid-s4980" xml:space="preserve">permutan <lb/>do, ut b d ad o <lb/>n, ita d e ad n p. </s> |
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8307 <s xml:id="echoid-s4981" xml:space="preserve"><lb/>Sed b d dupla <lb/>eſt o n. </s> |
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|
8308 <s xml:id="echoid-s4982" xml:space="preserve">ergo & </s> |
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|
8309 <s xml:id="echoid-s4983" xml:space="preserve"><lb/>e d ipſius n p du <lb/>pla erit. </s> |
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8310 <s xml:id="echoid-s4984" xml:space="preserve">quòd ſi <lb/>e d bifariam di-<lb/>uidatur ĩ χ, erit <lb/>χ d, uel e χ æ-<lb/>qualis n p: </s> |
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|
8311 <s xml:id="echoid-s4985" xml:space="preserve">& </s> |
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|
8312 <s xml:id="echoid-s4986" xml:space="preserve"><lb/>ſublata e n, quæ <lb/>eſt cõmunis u-<lb/>trique e χ, p n, |
|
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|
8313 <pb o="44" file="0199" n="199" rhead="DE CENTRO GRAVIT. SOLID."/> |
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|
8314 relinquetur p e ipſi n χ æqualis. </s> |
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|
8315 <s xml:id="echoid-s4987" xml:space="preserve">cum autem b e ſit dupla <lb/>e d, & </s> |
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8316 <s xml:id="echoid-s4988" xml:space="preserve">o p dupla p n, hoc eſt ipſius e χ, & </s> |
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|
8317 <s xml:id="echoid-s4989" xml:space="preserve">reliquum, uideli-<lb/>cet b o unà cum p e ipſius reliqui χ d duplnm erit. </s> |
|
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|
8318 <s xml:id="echoid-s4990" xml:space="preserve">eſtque <lb/> |
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|
8319 <anchor type="note" xlink:label="note-0199-01a" xlink:href="note-0199-01"/> |
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|
8320 b o dupla ζ d. </s> |
|
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|
8321 <s xml:id="echoid-s4991" xml:space="preserve">ergo p e, hoc eſt n χ ipſius χ ρ dupla. </s> |
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|
8322 <s xml:id="echoid-s4992" xml:space="preserve">ſed d n <lb/>dupla eſt n ζ. </s> |
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|
8323 <s xml:id="echoid-s4993" xml:space="preserve">reliqua igitur d χ dupla reliquæ χ n. </s> |
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|
8324 <s xml:id="echoid-s4994" xml:space="preserve">ſunt au-<lb/>tem d χ, p n inter ſe æquales: </s> |
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|
8325 <s xml:id="echoid-s4995" xml:space="preserve">itemq; </s> |
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|
8326 <s xml:id="echoid-s4996" xml:space="preserve">æquales χ n, p e. </s> |
|
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|
8327 <s xml:id="echoid-s4997" xml:space="preserve">qua-<lb/>re conſtat n p ipſius p e duplam eſſe. </s> |
|
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|
8328 <s xml:id="echoid-s4998" xml:space="preserve">& </s> |
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|
8329 <s xml:id="echoid-s4999" xml:space="preserve">idcirco p e ipſi e n <lb/>æqualem. </s> |
|
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|
8330 <s xml:id="echoid-s5000" xml:space="preserve">Rurſus cum ſit μ ν dupla o ν, & </s> |
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|
8331 <s xml:id="echoid-s5001" xml:space="preserve">μ σ dupla σ ν; </s> |
|
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|
8332 <s xml:id="echoid-s5002" xml:space="preserve">erit <lb/>etiam reliqua ν σ o dupla. </s> |
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|
8333 <s xml:id="echoid-s5003" xml:space="preserve">Eadem quoque ratione <lb/>cõcludetur π υ dupla υ m. </s> |
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|
8334 <s xml:id="echoid-s5004" xml:space="preserve">ergo ut ν σ ad σ O, ita π υ ad υ m: <lb/></s> |
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|
8335 <s xml:id="echoid-s5005" xml:space="preserve">componendoq;</s> |
|
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|
8336 <s xml:id="echoid-s5006" xml:space="preserve">, & </s> |
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|
8337 <s xml:id="echoid-s5007" xml:space="preserve">permutando, ut υ o ad π m, ita o σ ad <lb/>m υ & </s> |
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|
8338 <s xml:id="echoid-s5008" xml:space="preserve">ſunt æquales ν o, π m. </s> |
|
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|
8339 <s xml:id="echoid-s5009" xml:space="preserve">quare & </s> |
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|
8340 <s xml:id="echoid-s5010" xml:space="preserve">o σ, m υ æquales. </s> |
|
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|
8341 <s xml:id="echoid-s5011" xml:space="preserve">præ <lb/>terea σ π dupla eſt π τ, & </s> |
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|
8342 <s xml:id="echoid-s5012" xml:space="preserve">ν π ipſius π m. </s> |
|
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|
8343 <s xml:id="echoid-s5013" xml:space="preserve">reliqua igitur σ ν re <lb/>liquæ m τ dupla. </s> |
|
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|
8344 <s xml:id="echoid-s5014" xml:space="preserve">atque erat ν σ dupla σ o. </s> |
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|
8345 <s xml:id="echoid-s5015" xml:space="preserve">ergo m τ, σ o æ-<lb/>quales ſunt: </s> |
|
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|
8346 <s xml:id="echoid-s5016" xml:space="preserve">& </s> |
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|
8347 <s xml:id="echoid-s5017" xml:space="preserve">ita æquales m υ, n φ. </s> |
|
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|
8348 <s xml:id="echoid-s5018" xml:space="preserve">at o σ, eſt æqualis <lb/>m υ. </s> |
|
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|
8349 <s xml:id="echoid-s5019" xml:space="preserve">Sequitur igitur, ut omnes o σ, m τ, m υ, n φ in-<lb/>ter ſe ſint æquales. </s> |
|
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8350 <s xml:id="echoid-s5020" xml:space="preserve">Sed ut ρ π ad π τ, hoc eſt ut 3 ad 2, ita n d <lb/>ad d χ: </s> |
|
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8351 <s xml:id="echoid-s5021" xml:space="preserve">permutãdoq; </s> |
|
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|
8352 <s xml:id="echoid-s5022" xml:space="preserve">ut ρ π ad n d, ita π τ ad d χ. </s> |
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|
8353 <s xml:id="echoid-s5023" xml:space="preserve">& </s> |
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|
8354 <s xml:id="echoid-s5024" xml:space="preserve">ſũt æqua <lb/>les ζ π, n d. </s> |
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|
8355 <s xml:id="echoid-s5025" xml:space="preserve">ergo d χ, hoc eſt n p, & </s> |
|
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|
8356 <s xml:id="echoid-s5026" xml:space="preserve">π τ æquales. </s> |
|
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|
8357 <s xml:id="echoid-s5027" xml:space="preserve">Sed etiam æ-<lb/>quales n π, π m. </s> |
|
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|
8358 <s xml:id="echoid-s5028" xml:space="preserve">reliqua igitur π p reliquæ m τ, hoc eſt ipſi <lb/>n φ æqualis erit. </s> |
|
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8359 <s xml:id="echoid-s5029" xml:space="preserve">quare dempta p π ex p e, & </s> |
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|
8360 <s xml:id="echoid-s5030" xml:space="preserve">φ n dempta ex <lb/>n e, relinquitur p e æqualis e φ. </s> |
|
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|
8361 <s xml:id="echoid-s5031" xml:space="preserve">Itaque π, ρ centra figurarũ <lb/>ſecundo loco deſcriptarum a primis centris p n æquali in-<lb/>teruallo recedunt. </s> |
|
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8362 <s xml:id="echoid-s5032" xml:space="preserve">quòd ſi rurſus aliæ figuræ deſcribantur, <lb/>eodem modo demonſtrabimus earum centra æqualiter ab <lb/>his recedere, & </s> |
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8363 <s xml:id="echoid-s5033" xml:space="preserve">ad portionis conoidis centrum propius ad <lb/>moueri. </s> |
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8364 <s xml:id="echoid-s5034" xml:space="preserve">Ex quibus conſtat lineam π φ à centro grauitatis <lb/>portionis diuidi in partes æquales. </s> |
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8365 <s xml:id="echoid-s5035" xml:space="preserve">Si enim fieri poteſt, non <lb/>ſit centrum in puncto e, quod eſt lineæ π φ medium: </s> |
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8366 <s xml:id="echoid-s5036" xml:space="preserve">ſed in <lb/>ψ: </s> |
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8367 <s xml:id="echoid-s5037" xml:space="preserve">& </s> |
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8368 <s xml:id="echoid-s5038" xml:space="preserve">ipſi π ψ æqualis fiat φ ω. </s> |
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8369 <s xml:id="echoid-s5039" xml:space="preserve">Cum igitur in portione ſolida <lb/>quædam figura inſcribi posſit, ita ut linea, quæ inter cen-<lb/>trum grauitatis portionis, & </s> |
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8370 <s xml:id="echoid-s5040" xml:space="preserve">inſcriptæ figuræ interiicitur, <lb/>qualibet linea propoſita ſit minor, quod proxime demon-<lb/>ſtrauimus: </s> |
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8371 <s xml:id="echoid-s5041" xml:space="preserve">perueniet tandem φ centrum inſcriptæ figuræ |
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8372 <pb file="0200" n="200" rhead="FED. COMMANDINI"/> |
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8373 <anchor type="figure" xlink:label="fig-0200-01a" xlink:href="fig-0200-01"/> |
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8374 <pb o="45" file="0201" n="201" rhead="DE CENTRO GRAVIT. SOLID."/> |
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8375 ad punctum ω. </s> |
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8376 <s xml:id="echoid-s5042" xml:space="preserve">Sed quoniam π circum ſcripta itidem alia <lb/>figura æquali interuallo ad portionis centrum accedit, ubi <lb/>primum φ applieuerit ſe ad ω, & </s> |
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8377 <s xml:id="echoid-s5043" xml:space="preserve">π ad punctũ ψ, hoc eſt ad <lb/>portionis centrum ſe applicabit. </s> |
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8378 <s xml:id="echoid-s5044" xml:space="preserve">quod fieri nullo modo <lb/>poſſe perſpicuum eſt. </s> |
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8379 <s xml:id="echoid-s5045" xml:space="preserve">non aliter idem abſurdum ſequetur, <lb/>ſi ponamus centrum portionis recedere à medio ad par-<lb/>tes ω; </s> |
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8380 <s xml:id="echoid-s5046" xml:space="preserve">eſſet enim aliquando centrum figuræ inſcriptæ idem <lb/>quod portionis centrũ. </s> |
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8381 <s xml:id="echoid-s5047" xml:space="preserve">ergo punctum e centrum erit gra <lb/>uitatis portionis a b c. </s> |
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8382 <s xml:id="echoid-s5048" xml:space="preserve">quod demonſtrare oportebat.</s> |
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8383 <s xml:id="echoid-s5049" xml:space="preserve"/> |
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8384 </p> |
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8385 <div xml:id="echoid-div281" type="float" level="2" n="1"> |
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8386 <figure xlink:label="fig-0195-01" xlink:href="fig-0195-01a"> |
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8387 <image file="0195-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0195-01"/> |
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8388 </figure> |
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8389 <note position="right" xlink:label="note-0195-01" xlink:href="note-0195-01a" xml:space="preserve">7. huius</note> |
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8390 <figure xlink:label="fig-0196-01" xlink:href="fig-0196-01a"> |
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8391 <image file="0196-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0196-01"/> |
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8392 </figure> |
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8393 <note position="left" xlink:label="note-0196-01" xlink:href="note-0196-01a" xml:space="preserve">8. primi <lb/>libri Ar-<lb/>chimedis</note> |
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8394 <note position="left" xlink:label="note-0196-02" xlink:href="note-0196-02a" xml:space="preserve">11. duo-<lb/>decimi.</note> |
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8395 <note position="left" xlink:label="note-0196-03" xlink:href="note-0196-03a" xml:space="preserve">15. quinti</note> |
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8396 <note position="left" xlink:label="note-0196-04" xlink:href="note-0196-04a" xml:space="preserve">2. duode-<lb/>cimi.</note> |
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8397 <note position="right" xlink:label="note-0197-01" xlink:href="note-0197-01a" xml:space="preserve">20. primi <lb/>conicorũ</note> |
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8398 <figure xlink:label="fig-0198-01" xlink:href="fig-0198-01a"> |
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8399 <image file="0198-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0198-01"/> |
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8400 </figure> |
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8401 <note position="right" xlink:label="note-0199-01" xlink:href="note-0199-01a" xml:space="preserve">19. quinti</note> |
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8402 <figure xlink:label="fig-0200-01" xlink:href="fig-0200-01a"> |
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8403 <image file="0200-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0200-01"/> |
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8404 </figure> |
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8405 </div> |
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8406 <p> |
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8407 <s xml:id="echoid-s5050" xml:space="preserve">Quod autem ſupra demõſtratum eſt in portione conoi-<lb/>dis recta per figuras, quæ ex cylindris æqualem altitudi-<lb/>dinem habentibus conſtant, idem ſimiliter demonſtrabi-<lb/>mus per figuras ex cylindri portionibus conſtantes in ea <lb/>portione, quæ plano non ad axem recto abſcinditur. </s> |
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8408 <s xml:id="echoid-s5051" xml:space="preserve">ut <lb/>enim tradidimus in commentariis in undecimam propoſi <lb/>tionem libri Archimedis de conoidibus & </s> |
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8409 <s xml:id="echoid-s5052" xml:space="preserve">ſphæroidibus. <lb/></s> |
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8410 <s xml:id="echoid-s5053" xml:space="preserve">portiones cylindri, quæ æquali ſunt altitudine eam inter ſe <lb/>ſe proportionem habent, quam ipſarum baſes; </s> |
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8411 <s xml:id="echoid-s5054" xml:space="preserve">baſes autẽ <lb/>quæ ſunt ellipſes ſimiles eandem proportionem habere, <lb/> |
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8412 <anchor type="note" xlink:label="note-0201-01a" xlink:href="note-0201-01"/> |
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8413 quam quadrata diametrorum eiuſdem rationis, ex corol-<lb/>lario ſeptimæ propoſitionis libri de conoidibus, & </s> |
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8414 <s xml:id="echoid-s5055" xml:space="preserve">ſphæ-<lb/>roidibus, manifeſte apparet.</s> |
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8415 <s xml:id="echoid-s5056" xml:space="preserve"/> |
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|
8416 </p> |
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8417 <div xml:id="echoid-div282" type="float" level="2" n="2"> |
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8418 <note position="right" xlink:label="note-0201-01" xlink:href="note-0201-01a" xml:space="preserve">corol. 15 <lb/>deconoi-<lb/>dibus & <lb/>ſphæroi-<lb/>dibus.</note> |
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8419 </div> |
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8420 </div> |
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8421 <div xml:id="echoid-div284" type="section" level="1" n="95"> |
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8422 <head xml:id="echoid-head102" xml:space="preserve">THEOREMA XXIIII. PROPOSITIO XXX.</head> |
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8423 <p> |
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8424 <s xml:id="echoid-s5057" xml:space="preserve">SI à portione conoidis rectanguli alia portio <lb/>abſcindatur, plano baſi æquidiſtante; </s> |
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8425 <s xml:id="echoid-s5058" xml:space="preserve">habebit <lb/>portio tota ad eam, quæ abſciſſa eſt, duplam pro <lb/>portio nem eius, quæ eſt baſis maioris portionis <lb/>ad baſi m minoris, uel quæ axis maioris ad axem <lb/>minoris.</s> |
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8426 <s xml:id="echoid-s5059" xml:space="preserve"/> |
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8427 </p> |
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8428 <pb file="0202" n="202" rhead="FED. COMMANDINI"/> |
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8429 <p> |
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8430 <s xml:id="echoid-s5060" xml:space="preserve">ABSCINDATVR à portione conoidis rectanguli <lb/>a b c alia portio e b f, plano baſi æquidiſtante: </s> |
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8431 <s xml:id="echoid-s5061" xml:space="preserve">& </s> |
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8432 <s xml:id="echoid-s5062" xml:space="preserve">eadem <lb/>portio ſecetur alio plano per axem; </s> |
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8433 <s xml:id="echoid-s5063" xml:space="preserve">ut ſuperficiei ſectio ſit <lb/>parabole a b c: </s> |
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8434 <s xml:id="echoid-s5064" xml:space="preserve">planorũ portiones abſcindentium rectæ <lb/>lineæ a c, e f: </s> |
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8435 <s xml:id="echoid-s5065" xml:space="preserve">axis autem portionis, & </s> |
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8436 <s xml:id="echoid-s5066" xml:space="preserve">ſectionis diameter <lb/>b d; </s> |
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8437 <s xml:id="echoid-s5067" xml:space="preserve">quam linea e fin puncto g ſecet. </s> |
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8438 <s xml:id="echoid-s5068" xml:space="preserve">Dico portionem co-<lb/>noidis a b c ad portionem e b f duplam proportionem ha-<lb/>bere eius, quæ eſt baſis a c ad baſim e f; </s> |
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8439 <s xml:id="echoid-s5069" xml:space="preserve">uel axis d b ad b g <lb/>axem. </s> |
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8440 <s xml:id="echoid-s5070" xml:space="preserve">Intelligantur enim duo coni, ſeu coni portiones <lb/>a b c, e b f, eãdem baſim, quam portiones conoidis, & </s> |
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8441 <s xml:id="echoid-s5071" xml:space="preserve">æqua <lb/>lem habentes altitudinem. </s> |
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8442 <s xml:id="echoid-s5072" xml:space="preserve">& </s> |
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8443 <s xml:id="echoid-s5073" xml:space="preserve">quoniam a b c portio conoi <lb/>dis ſeſquialtera eſt coni, ſeu portionis coni a b c; </s> |
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8444 <s xml:id="echoid-s5074" xml:space="preserve">& </s> |
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8445 <s xml:id="echoid-s5075" xml:space="preserve">portio <lb/>e b f coniſeu portionis coni e b feſt ſeſquialtera, quod de-<lb/> |
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8446 <anchor type="figure" xlink:label="fig-0202-01a" xlink:href="fig-0202-01"/> |
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8447 monſtrauit Archimedes in propoſitionibus 23, & </s> |
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8448 <s xml:id="echoid-s5076" xml:space="preserve">24 libri <lb/>de conoidibus, & </s> |
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8449 <s xml:id="echoid-s5077" xml:space="preserve">ſphæroidibus: </s> |
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8450 <s xml:id="echoid-s5078" xml:space="preserve">erit conoidis portio ad <lb/>conoidis portionem, ut conus ad conum, uel ut coni por-<lb/>tio ad coni portionem. </s> |
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8451 <s xml:id="echoid-s5079" xml:space="preserve">Sed conus, uel coni portio a b c ad <lb/>conum, uel coni portionem e b f compoſitam proportio-<lb/>nem habet ex proportione baſis a c ad baſim e f, & </s> |
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8452 <s xml:id="echoid-s5080" xml:space="preserve">ex pro-<lb/>portione altitudinis coni, uel coni portionis a b c ad alti-<lb/>tudinem ipſius e b f, ut nos demonſtrauimus in com men-<lb/>tariis in undecimam propoſitionem eiuſdem libri A rchi-<lb/>medis: </s> |
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8453 <s xml:id="echoid-s5081" xml:space="preserve">altitudo autem ad altitudinem eſt, ut axis ad axem. <lb/></s> |
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8454 <s xml:id="echoid-s5082" xml:space="preserve">quod quidem in conis rectis perſpicuum eſt, in ſcalenis ue |
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|
8455 <pb o="46" file="0203" n="203" rhead="DE CENTRO GRAVIT. SOLID."/> |
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|
8456 ro ita demonſtrabitur. </s> |
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8457 <s xml:id="echoid-s5083" xml:space="preserve">Ducatur à puncto b ad planum ba-<lb/>ſis a c perpendicularis linea b h, quæ ipſam e fin K ſecet. <lb/></s> |
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8458 <s xml:id="echoid-s5084" xml:space="preserve">erit b h altitudo coni, uel coni portionis a b c: </s> |
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8459 <s xml:id="echoid-s5085" xml:space="preserve">& </s> |
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8460 <s xml:id="echoid-s5086" xml:space="preserve">b K altitu <lb/> |
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8461 <anchor type="note" xlink:label="note-0203-01a" xlink:href="note-0203-01"/> |
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8462 do e f g. </s> |
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8463 <s xml:id="echoid-s5087" xml:space="preserve">Quod cum lineæ a c, e f inter ſe æ quidiſtent, ſunt <lb/>enim planorum æ quidiſtantium ſectiones: </s> |
|
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8464 <s xml:id="echoid-s5088" xml:space="preserve">habebit d b ad <lb/> |
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8465 <anchor type="note" xlink:label="note-0203-02a" xlink:href="note-0203-02"/> |
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8466 b g proportionem ean dem, quam h b ad b k. </s> |
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8467 <s xml:id="echoid-s5089" xml:space="preserve">quare por-<lb/>tio conoidis a b c ad portionem e f g proportionem habet <lb/>compoſitam ex proportione baſis a c ad baſim e f; </s> |
|
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8468 <s xml:id="echoid-s5090" xml:space="preserve">& </s> |
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8469 <s xml:id="echoid-s5091" xml:space="preserve">ex <lb/>proportione d b axis ad axem b g. </s> |
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8470 <s xml:id="echoid-s5092" xml:space="preserve">Sed circulus, uel <lb/> |
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8471 <anchor type="note" xlink:label="note-0203-03a" xlink:href="note-0203-03"/> |
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8472 ellipſis circa diametrum a c ad circulum, uel ellipſim <lb/> |
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8473 <anchor type="note" xlink:label="note-0203-04a" xlink:href="note-0203-04"/> |
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8474 circa e f, eſt ut quadratum a c ad quadratum e f; </s> |
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8475 <s xml:id="echoid-s5093" xml:space="preserve">hoc eſt ut <lb/>quadratũ a d ad quadratũ e g. </s> |
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8476 <s xml:id="echoid-s5094" xml:space="preserve">& </s> |
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8477 <s xml:id="echoid-s5095" xml:space="preserve">quadratum a d ad quadra <lb/>tum e g eſt, ut linea d b ad lineam b g. </s> |
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8478 <s xml:id="echoid-s5096" xml:space="preserve">circulus igitur, uel el <lb/>lipſis circa diametrum a c ad circulũ, uel ellipſim circa e f, <lb/> |
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8479 <anchor type="note" xlink:label="note-0203-05a" xlink:href="note-0203-05"/> |
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8480 hoc eſt baſis ad baſim eandem proportionem habet, quã <lb/> |
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8481 <anchor type="note" xlink:label="note-0203-06a" xlink:href="note-0203-06"/> |
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|
8482 d b axis ad axem b g. </s> |
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8483 <s xml:id="echoid-s5097" xml:space="preserve">ex quibus ſequitur portionem a b c <lb/>ad portionem e b f habere proportionem duplam eius, <lb/>quæ eſt baſis a c ad bafim e f: </s> |
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8484 <s xml:id="echoid-s5098" xml:space="preserve">uel axis d b ad b g axem. </s> |
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|
8485 <s xml:id="echoid-s5099" xml:space="preserve">quod <lb/>demonſtrandum proponebatur.</s> |
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8486 <s xml:id="echoid-s5100" xml:space="preserve"/> |
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|
8487 </p> |
|
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|
8488 <div xml:id="echoid-div284" type="float" level="2" n="1"> |
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|
8489 <figure xlink:label="fig-0202-01" xlink:href="fig-0202-01a"> |
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|
8490 <image file="0202-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0202-01"/> |
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|
8491 </figure> |
|
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|
8492 <note position="right" xlink:label="note-0203-01" xlink:href="note-0203-01a" xml:space="preserve">16. unde-<lb/>cimi.</note> |
|
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|
8493 <note position="right" xlink:label="note-0203-02" xlink:href="note-0203-02a" xml:space="preserve">4 ſexti.</note> |
|
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|
8494 <note position="right" xlink:label="note-0203-03" xlink:href="note-0203-03a" xml:space="preserve">2. duode <lb/>cimi</note> |
|
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8495 <note position="right" xlink:label="note-0203-04" xlink:href="note-0203-04a" xml:space="preserve">7. de co-<lb/>noidibus <lb/>& ſphæ-<lb/>roidibus</note> |
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|
8496 <note position="right" xlink:label="note-0203-05" xlink:href="note-0203-05a" xml:space="preserve">15. quinti</note> |
|
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|
8497 <note position="right" xlink:label="note-0203-06" xlink:href="note-0203-06a" xml:space="preserve">20. primi <lb/>conicorũ</note> |
|
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|
8498 </div> |
|
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8499 </div> |
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8500 <div xml:id="echoid-div286" type="section" level="1" n="96"> |
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8501 <head xml:id="echoid-head103" xml:space="preserve">THEOREMA XXV. PROPOSITIO XXXI.</head> |
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8502 <p> |
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8503 <s xml:id="echoid-s5101" xml:space="preserve">Cuiuslibet fruſti à portione rectanguli conoi <lb/>dis abſcisſi, centrum grauitatis eſt in axe, ita ut <lb/>demptis primum à quadrato, quod fit ex diame-<lb/>tro maioris baſis, tertia ipſius parte, & </s> |
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8504 <s xml:id="echoid-s5102" xml:space="preserve">duabus <lb/>tertiis quadrati, quod fit ex diametro baſis mino-<lb/>ris: </s> |
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8505 <s xml:id="echoid-s5103" xml:space="preserve">deinde à tertia parte quadrati maioris baſis <lb/>rurſus dempta portione, ad quam reliquum qua <lb/>drati baſis maioris unà cum dicta portione duplã <lb/>proportionem habeat eius, quæ eſt quadrati ma- |
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8506 <pb file="0204" n="204" rhead="FED. COMMANDINI"/> |
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8507 ioris baſis ad quadratum minoris: </s> |
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8508 <s xml:id="echoid-s5104" xml:space="preserve">centrum ſit in <lb/>eo axis puncto, quo ita diuiditur ut pars, quæ mi <lb/>norem baſim attingit ad alteram partem eandem <lb/>proportionem habeat, quam dempto quadrato <lb/>minoris baſis à duabus tertiis quadrati maioris, <lb/>habet id, quod reliquum eſt unà cum portione à <lb/>tertia quadrati maioris parte dempta, ad reliquà <lb/>eiuſdem tertiæ portionem.</s> |
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8509 <s xml:id="echoid-s5105" xml:space="preserve"/> |
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8510 </p> |
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8511 <p> |
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8512 <s xml:id="echoid-s5106" xml:space="preserve">SIT fruſtum à portione rectanguli conoidis abſciſſum <lb/>a b c d, cuius maior baſis circulus, uel ellipſis circa diame-<lb/>trum b c, minor circa diametrum a d; </s> |
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8513 <s xml:id="echoid-s5107" xml:space="preserve">& </s> |
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8514 <s xml:id="echoid-s5108" xml:space="preserve">axis e f. </s> |
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8515 <s xml:id="echoid-s5109" xml:space="preserve">deſcriba-<lb/>tur autem portio conoidis, à quo illud abſciſſum eſt, & </s> |
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8516 <s xml:id="echoid-s5110" xml:space="preserve">pla-<lb/> |
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8517 <anchor type="figure" xlink:label="fig-0204-01a" xlink:href="fig-0204-01"/> |
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8518 no per axem ducto ſecetur; </s> |
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8519 <s xml:id="echoid-s5111" xml:space="preserve">ut ſuperficiei ſectio ſit parabo-<lb/>le b g c, cuius diameter, & </s> |
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8520 <s xml:id="echoid-s5112" xml:space="preserve">axis portionis g f: </s> |
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8521 <s xml:id="echoid-s5113" xml:space="preserve">deinde g f diui <lb/>datur in puncto h, ita ut g h ſit dupla h f: </s> |
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8522 <s xml:id="echoid-s5114" xml:space="preserve">& </s> |
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8523 <s xml:id="echoid-s5115" xml:space="preserve">rurſus g e in ean <lb/>dem proportionem diuidatur: </s> |
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8524 <s xml:id="echoid-s5116" xml:space="preserve">ſitq; </s> |
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8525 <s xml:id="echoid-s5117" xml:space="preserve">g _k_ ipſius k e dupla. </s> |
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8526 <s xml:id="echoid-s5118" xml:space="preserve">Iã <lb/>ex iis, quæ proxime demonſtrauimus, conſtat centrum gra <lb/>uitatis portionis b g c eſſe h punctum: </s> |
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8527 <s xml:id="echoid-s5119" xml:space="preserve">& </s> |
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8528 <s xml:id="echoid-s5120" xml:space="preserve">portionis a g c <lb/>punctum k. </s> |
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8529 <s xml:id="echoid-s5121" xml:space="preserve">ſumpto igitur infra h punctol, ita ut k h ad h l |
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8530 <pb o="47" file="0205" n="205" rhead="DE CENTRO GRAVIT. SOLID."/> |
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8531 eani proportionem habeat, quam a b c d fruſtum ad por-<lb/>tionem a g d; </s> |
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8532 <s xml:id="echoid-s5122" xml:space="preserve">erit punctum l eius fruſti grauitatis cẽtrum: <lb/></s> |
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8533 <s xml:id="echoid-s5123" xml:space="preserve">habebitq; </s> |
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8534 <s xml:id="echoid-s5124" xml:space="preserve">componendo K l ad 1 h proportionem eandem, <lb/>quam portio conoidis b gc ad a g d portionem. </s> |
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8535 <s xml:id="echoid-s5125" xml:space="preserve">Itaq; </s> |
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8536 <s xml:id="echoid-s5126" xml:space="preserve">quo <lb/> |
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8537 <anchor type="note" xlink:label="note-0205-01a" xlink:href="note-0205-01"/> |
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8538 niam quadratum b f ad quadratum a e, hoc eſt quadratum <lb/>b c ad quadratum a d eſt, ut linea f g ad g e: </s> |
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8539 <s xml:id="echoid-s5127" xml:space="preserve">erunt duæ ter-<lb/>tiæ quadrati b c ad duas tertias quadrati a d, ut h g ad g _k_: <lb/></s> |
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8540 <s xml:id="echoid-s5128" xml:space="preserve">& </s> |
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8541 <s xml:id="echoid-s5129" xml:space="preserve">ſi à duabus tertiis quadrati b c demptæ fuerint duæ ter-<lb/>tiæ quadrati a d: </s> |
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8542 <s xml:id="echoid-s5130" xml:space="preserve">erit diuidẽdo id, quod relinquitur ad duas <lb/>tertias quadrati a d, ut h k ad k g. </s> |
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8543 <s xml:id="echoid-s5131" xml:space="preserve">Rurſus duæ tertiæ quadra <lb/>ti a d ad duas tertias quadrati b c ſunt, ut _k_ g ad g h: </s> |
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8544 <s xml:id="echoid-s5132" xml:space="preserve">& </s> |
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8545 <s xml:id="echoid-s5133" xml:space="preserve">duæ <lb/>tertiæ quadrati b c ad tertiã partẽ ipſius, ut g h ad h f. </s> |
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8546 <s xml:id="echoid-s5134" xml:space="preserve">ergo <lb/>ex æ quali id, quod relinquitur ex duabus tertiis quadrati <lb/>b c, demptis ab ipſis quadrati a d duabus tertiis, ad tertiã <lb/>partem quadrati b c, ut _k_ h ad h f: </s> |
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8547 <s xml:id="echoid-s5135" xml:space="preserve">& </s> |
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8548 <s xml:id="echoid-s5136" xml:space="preserve">ad portionem eiuſdẽ <lb/>tertiæ partis, ad quam unà cum ipſa portione, duplam pro <lb/>portionem habeat eius, quæ eſt quadrati b c ad quadratũ <lb/>a d, ut K 1 ad 1 h. </s> |
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8549 <s xml:id="echoid-s5137" xml:space="preserve">habet enim _K_l ad 1 h ean dem proportio-<lb/>nem, quam conoidis portio b g c ad portionem a g d: </s> |
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8550 <s xml:id="echoid-s5138" xml:space="preserve">por-<lb/>tio autem b g c ad portionem a g d duplam proportionem <lb/>habet eius, quæ eſt baſis b c ad baſim a d: </s> |
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8551 <s xml:id="echoid-s5139" xml:space="preserve">hoc eſt quadrati <lb/>b c ad quadratum a d; </s> |
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8552 <s xml:id="echoid-s5140" xml:space="preserve">ut proxime demonſtratum eſt. </s> |
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8553 <s xml:id="echoid-s5141" xml:space="preserve">quare <lb/> |
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8554 <anchor type="note" xlink:label="note-0205-02a" xlink:href="note-0205-02"/> |
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8555 dempto a d quadrato à duabus tertiis quadrati b c, erit id, <lb/>quod relin quitur unà cum dicta portione tertiæ partis ad <lb/>reliquam eiuſdem portionem, ut el ad 1 f. </s> |
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8556 <s xml:id="echoid-s5142" xml:space="preserve">Cum igitur cen-<lb/>trum grauitatis fruſti a b c d ſit l, à quo axis e f in eam, quã <lb/>diximus, proportionem diuidatur; </s> |
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8557 <s xml:id="echoid-s5143" xml:space="preserve">conſtat uerũ eſſe illud, <lb/>quod demonſtrandum propoſuimus.</s> |
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8558 <s xml:id="echoid-s5144" xml:space="preserve"/> |
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|
8559 </p> |
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8560 <div xml:id="echoid-div286" type="float" level="2" n="1"> |
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8561 <figure xlink:label="fig-0204-01" xlink:href="fig-0204-01a"> |
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8562 <image file="0204-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0204-01"/> |
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|
8563 </figure> |
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8564 <note position="right" xlink:label="note-0205-01" xlink:href="note-0205-01a" xml:space="preserve">20. I. coni <lb/>corum.</note> |
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8565 <note position="right" xlink:label="note-0205-02" xlink:href="note-0205-02a" xml:space="preserve">30. huius</note> |
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8566 </div> |
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8567 </div> |
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8568 <div xml:id="echoid-div288" type="section" level="1" n="97"> |
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8569 <head xml:id="echoid-head104" xml:space="preserve">FINIS LIBRI DE CENTRO <lb/>GRAVITATIS SOLIDORVM.</head> |
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8570 <p> |
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8571 <s xml:id="echoid-s5145" xml:space="preserve">Impreſſ. </s> |
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8572 <s xml:id="echoid-s5146" xml:space="preserve">Bononiæ cum licentia Superiorum.</s> |
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8573 <s xml:id="echoid-s5147" xml:space="preserve"/> |
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8574 </p> |
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8575 <pb file="0206" n="206"/> |
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8576 <pb file="0207" n="207"/> |
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8577 <pb file="0208" n="208"/> |
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8578 <figure> |
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8579 <image file="0208-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/4E7V2WGH/figures/0208-01"/> |
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8580 </figure> |
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8581 <pb file="0209" n="209"/> |
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8582 <pb file="0210" n="210"/> |
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8583 <pb file="0211" n="211"/> |
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8584 <pb file="0212" n="212"/> |
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8585 </div></text> |
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8586 </echo> |