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Removing DESpecs directory which deserted to git
author Klaus Thoden <kthoden@mpiwg-berlin.mpg.de>
date Wed, 29 Nov 2017 16:55:37 +0100
parents 22d6a63640c6
children
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1 <pb>
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2 <head>FEDERICI
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3 COMMANDINI
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4 VRBINATIS</head>
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5 <head>LIBER DE CENTRO
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6 GRAVITATIS
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7 SOLIDORVM.</head>
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8 <fig>
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9 <head>CVM PRIVILEGIO IN ANNOS X.</head>
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10 <head>BONONIAE,</head>
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11 <head>Ex Officina Alexandri Benacii.</head>
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12 <head>M D LXV.</head>
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13 <pb>
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14 <head>ALEXANDRO FARNESIO
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15 CARDINALI AMPLISSIMO.
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16 ET OPTIMO.</head>
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17 <P>Cvm mult&aelig; res in mathematicis
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18 di$ciplinis nequaquam $atis ad-
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19 huc explicat&aelig; $int, tum perdif-
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20 ficilis, &amp; perob$cura qu&aelig;$tio
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21 e$t de centro grauitatis corpo-
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22 rum $olidorum; qu&aelig;, &amp; ad co-
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23 gno$cendum pulcherrima e$t,
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24 &amp; ad multa, qu&aelig; &agrave; mathematicis proponuntur, pr&aelig;-
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25 clare intelligenda maximum affert adiumentum. de
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26 qua neminem ex mathematicis, neque no$tra, neque
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27 patrum no$trorum memoria $criptum reliqui$$e $ci-
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28 mus. &amp; quamuis in earum monumentis literarum n&otilde;
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29 nulla reperiantur, ex quibus in hanc $ententiam addu
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30 ci po$$umus, vt exi$timemus hanc rem ab ij$d&etilde; vber-
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31 rime tractatam e$$e; tamen ne$cio quo fato adhuc
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32 in eiu$modi librorum ignoratione ver$amur. Archi-
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33 medes quidem mathematicor&utilde; princeps in libello,
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34 cuius in$criptio e$t, <G>ke/ttraba/rwg ipipe/dwg,</G> de centro pla-
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35 norum copio$i$sime, atque acuti$sime con$crip$it: &amp;
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36 in co explicando fumm&atilde; ingenii, &amp; $cienti&aelig; glori&atilde; e$t
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37 c&otilde;$ecutus. Sed de cognitione c&etilde;tri grauitatis corpor&utilde;
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38 $olidor&utilde; nulla in eius libris litera inuenitur. non mul
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39 tos abhinc annos MARC<*>LLVS <*>I. PONT. MAX<*>
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40 <pb>
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41 cum adhuc Cardinalis e$$et, mihi, qu&aelig; $ua erat hu-
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42 manitas, libros ciu$dem Archimedis de ijs, qu&aelig; ve-
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43 huntur in aqua, latine redditos dono dedit. hos cum
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44 ego, ut aliorum $tudia incitarem, emend&atilde;dos, &amp; c&otilde;-
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45 mentariis illu$trandos $u$cepi$$em, animaduerti dubi
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46 tari non po$$e, quin Archimedes vel de hac materia
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47 $crip$i$$et, vel aliorum mathematicorum $cripta per-
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48 legi$$et. nam in iis tum alia nonnulla, tum maxime
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49 illam propo$itionem, ut cuidentem, &amp; ali&agrave;s proba-
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50 tam a$$umit, Centr&utilde; grauitatis in portionibus conoi
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51 dis rectanguli axem ita diuidere, vt pars, qu&aelig; ad verti
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52 cem terminatur, alterius partis, qu&aelig; ad ba$im dupla
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53 $it. Verum h&aelig;c ad cam partem mathematicarum
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54 di$ciplinarum pr&aelig;cipue refertur, in qua de centro
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55 grauitatis corporum $olidorum tractatur. non e$t au
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56 tem con$entaneum Archimedem illum admirabilem
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57 virum hanc propo$itionem $ibi argumentis con-
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58 firmandam exi$timaturum non fui$$e, ni$i eam vel
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59 aliis in locis probaui$$et, vel ab aliis probatam e$$e
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60 comperi$$et. quamobrem nequid in iis libris intel-
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61 ligendis de$iderari po$$et, $tatui hanc etiam partem
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62 vel &agrave; veteribus pr&aelig;termi$$am, vel tractatam quidem,
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63 fed in tenebris iacentem, non intactam relinquere;
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64 atque ex a$sidua mathematicorum, pr&aelig;$ertim Archi-
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65 medis lectione, qu&aelig; mihi in mentem venerunt, ea in
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66 medium afferre; ut centri grauitatis corporum $oli-
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67 dorum, $i non perfectam, at certe aliquam noti-
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68 <pb>
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69 tiam haberemus. Qucm meum laborem n&otilde; mathe-
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70 maticis $olum, verum iis etiam, qui natur&aelig; ob$curi-
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71 tate delectantur, n&otilde; iniucundam fore $peraui: multa
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72 enim <G>problh/mxta</G> cognitione digni$sima, qu&aelig; ad vtr&atilde;-
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73 que $cientiam attinent, $e$e legentibus obtuli$$ent.
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74 neque id vlli mirandum videri debet. vt enim in cor-
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75 poribus no$tris omnia membra, ex quibus certa qu&aelig;
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76 dam officia na$cuntur, diuino quodam ordine inter
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77 $e implicata, &amp; colligata $unt: in iis&qacute;; admirabilis il-
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78 la con$piratio, quam <G>tn/mpnoixn</G> gr&aelig;ci vocant, eluce$cit,
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79 ita tres ill&aelig; Philo$ophi&aelig; (ut Ari$totelis verbo vtar)
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80 qu&aelig; veritatem $olam propo$itam habent, licet qui-
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81 bu$dam qua$i finibus $uis regantur: tamen ear&utilde; vna-
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82 qu&aelig;que per $e ip$am quodammodo imperfecta e$t:
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83 neque altera $ine alterius auxilio plene comprehen-
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84 di pote$t. complures pr&aelig;terca mathematicorum no-
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85 di ante hac explicatu difficillimi nullo negotio expe
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86 diti e$$ent: atque (ut vno verbo complectar) ni$i
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87 mea vaide amo, tractationem hanc meam $tudio$is
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88 non mediocrem vtilitatem, &amp; magnam volupta-
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89 tem allaturam e$$e mihi per$ua$i. cum autem ad hoc
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90 $cribendum aggre$$us e$sem, allatus e$t ad me liber
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91 Franci$ci Maurolici Me$$anen$is, in quo vir ille do-
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92 cti$simus, &amp; in iis di$ciplinis exercitati$simus af-
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93 firmabat $e de centro grauitatis corporum $olido-
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94 rum co<*>$crip$i$$e. cum hoc intellexi$$em, $u$tinui
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95 me pauli$per: tacitus'que expectaui, dum opus cla-
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96 <pb>
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97 ris$imi uiri, quem $emper honoris cau$$a nomino,
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98 in lucem proferretur: mihi enim exploratis$imum
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99 erat: Franci$cum Maurolicum multo doctius, &amp;
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100 exqui$itius hoc di$ciplinarum genus $criptis $uis tra
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101 diturum. $ed cum id tardius fieret, hoc e$t, ut ego
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102 interpretor, diligentius, mihi diutius hac $criptione
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103 non $uper$edendum e$$e duxi, pr&aelig;$ertim cum iam li-
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104 bri Archimedis de iis, qu&aelig; uehuntur in aqua, opera
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105 mea illu$trati typis excud&etilde;di e$$ent. nec me alia cau$
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106 $a impuli$$et, ut de centro grauitatis corporum $oli-
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107 dorum $criberem, ni$i ut hac etiam ratione lux eis
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108 qu&agrave;m maxime fieri po$let afferretur. atq; id e&ograve; mihi
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109 faciendum exi$timaui, qu&ograve;d in $pem ueniebam fore,
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110 ut cum ego ex omnibus mathematicis primus, hanc
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111 materiam explicandam $u$cepi$$em; $i quid errati for
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112 te &agrave; me commi$$um e$$et, boni uiri potius id me&aelig; de
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113 $tudio$is hominibus bene mer&etilde;di cupiditati, qu&agrave;m
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114 arroganti&aelig; a$criberent. re$tabat ut con$iderarem, cui
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115 potis$imum ex principibus uiris contemplationem
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116 hanc, nunc primum memori&aelig;, ac literis proditam de
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117 dicarem. harum mearum cogitationum $umma fa-
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118 cta, exi$timaui nemini conuenientius de centro graui
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119 tatis corporum opus dicari oportere, qu&agrave;m ALE-
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120 XANDRO FARNESIO grauis$imo, ac prudentis$i-
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121 mo Cardinali, quo in uiro $umma fortuna $emper c&utilde;
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122 $umma uirtute certauit. quid enim maxime in te ad-
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123 mirati debeant homines, ob$curum e$t; u$um'ne re-
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124 <pb>
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125 rum, qui pueriti&aelig; tempus extremum principium ha||
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126 bui$ti, &amp; imperior&utilde;, &amp; ad Reges, &amp; Imperatores ho-
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127 norificenti$simarum legationum; an excellentiam
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128 in omni genere literarum, qui vix adole$c&etilde;tulus, qu&aelig;
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129 homines iam confirmata &aelig;tate $ummo $tudio, diu-
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130 turnis&qacute;; laboribus didicerunt, $cientia, &amp; cognitione
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131 comprehendi$ti: an con$ilium, &amp; $apientiam in re-
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132 gendis, &amp; gubern&atilde;dis Ciuitatibus, cuius graui$sim&aelig;
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133 $ententi&aelig; in $ancti$simo Reip. Chr$tian&aelig; con$ilio di-
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134 ct&aelig;, potius diuina oracula, qu&agrave;m $ententi&aelig; habit&aelig;
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135 $unt, &amp; habentur. pr&aelig;termitto liberalitatem, &amp; mu-
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136 nificentiam tuam, quam in $tudio$i$simo quoque ho
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137 ne$tando quotidie magis o$tendis, ne videar auribus
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138 tuis potius, qu&agrave;m veritati $eruire. quamuis &agrave; te in tot
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139 pr&aelig;claros viros tanta beneficia collata $unt, &amp; confe-
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140 r&utilde;tur, vt omnibus te$tatum $it, nihil tibi e$$e charius,
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141 nihil iucundius, qu&agrave;m eximia tua liberalitate homi-
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142 nes ad amplexandam virtutem, licet currentes incita-
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143 re. nihil dico de ceteris virtutibus tuis, qu&aelig; tant&aelig;
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144 $unt, quant&aelig; ne cogitatione quidem comprehendi
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145 po$$unt. Quamobrem hac pr&aelig;cipue de cau$$a te hu-
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146 ius me&aelig; lucubrationis patronum e$$e volui, quam ea,
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147 qua $oles, humanitate accipies. te enim $emper ob
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148 diuinas virtutes tuas colui, &amp; ob$eruaui: nihil&qacute;; mi-
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149 hi fuit optatius; qu&agrave;m tibi per$pectum e$$e meum
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150 erga te animum; $ingularem&qacute;; ob$eruantiam. c&oelig;-
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151 lum igitur digito attingam, $i po$t graui$simas oc-
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152 <pb>
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153 cupationes tuas legendo Federici tui libro aliquid
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154 impertiri temporis non grauaberis: cum&qacute;; in iis, qui
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155 tibi $emper addicti erunt, numerare. Vale.</P>
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156 <P>Federicus Commandinus.</P>
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157 <p n=>1</p>
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158 <head>FEDERICI COMMANDINI
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159 VRBINATIS LIBER DE CENTRO
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160 GRAVITATIS SOLIDORVM.</head>
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161 <head>DIFFINITIONES.</head>
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162 <marg>1</marg>
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163 <P>Centrvm grauitatis, Pappus
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164 Alexandrinus in octauo ma-
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165 thematicarum collectionum
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166 libro ita diffiniuit.</P>
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167 <P><G>le/gomen de/ ne/ntron ba/rons i/a/<*>ton da/
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168 matos <*>einai <*>mei<*>on ti aei/menon e/nto/s, a/f<*>
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169 on<*> kat'e/po<*>nian a\rtn<*>e/n to/ ba/ros n/mer<*>
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170 fero/menon, nai\ fula/ssei to/n i/x a/r<*>sri-
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171 sin, on\ mn\ peritrep o/menon i/nt<*> fora<*>.</G> hoc e$t,</P>
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172 <P>Dicimus autem centrum grauitatis uniu$cu-
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173 iu$que corporis punctum quoddam intra po$i-
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174 tum, &agrave; quo $i graue appen$um mente concipia-
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175 tur, dum fertur quie$cit; &amp; $eruat eam, quam in
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176 principio habebat po$itionem: neque in ip$a la-
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177 tione circumuertitur.</P>
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178 <P>Po$$umus etiam hoc modo diffinire.</P>
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179 <P>Centrum grauitatis uniu$cuiu$que $olid&aelig; figu
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180 r&aelig; e$t punctum illud intra po$itum, circa quod
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181 undique partes &aelig;qualium momentorum con$i-
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182 $tunt. $i enim per tale centrum ducatur planum
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183 figuram quomodocunque $ecans $emper in par-
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184 <foot>A</foot>
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185 <pb>
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186 tes &aelig;queponderantes ip$am diuidet.</P>
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187 <marg>2</marg>
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188 <P>Pri$matis, cylindri, &amp; portionis cylindri axem
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189 appello rectam lineam, qu&aelig; oppo$itorum plano-
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190 rum centra grauitatis coniungit.</P>
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191 <marg>3</marg>
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192 <P>Pyramidis, coni, &amp; portionis coni axem dico li
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193 neam, qu&aelig; &agrave; uertice ad centrum grauitatis ba$is
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194 perducitur.</P>
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195 <marg>4</marg>
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196 <P>Si pyramis, conus, portio coni, uel conoidis $e-
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197 cetur plano ba$i &aelig;quidi$tante, pars, qu&aelig; e$t ad ba-
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198 $im, fru$tum pyramidis, coni, portionis coni, uel
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199 conoidis dicetur; quorum plana &aelig;quidi$tantia,
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200 qu&aelig; opponuntur $imilia $unt, &amp; in&aelig;qualia: axes
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201 uero $unt axium figurarum partes, qu&aelig; in ip$is
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202 comprehenduntur.</P>
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203 <head>PETITIONES.</head>
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204 <marg>1</marg>
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205 <P>Solidarum figurarum fimilium centra grauita-
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206 tis $imiliter $unt po$ita.</P>
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207 <marg>2</marg>
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208 <P>Solidis figuris $imilibus, &amp; &aelig;qualibus inter $e
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209 aptatis, centra quoque grauitatis ip$arum inter $e
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210 aptata crunt.</P>
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211 <head>THEOREMA I. PROPOSITIO I.</head>
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212 <P>Omnis figur&aelig; rectiline&aelig; in circulo de$cript&aelig;,
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213 qu&aelig; &aelig;qualibus lateribus, &amp; angulis contine-
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214 <*>
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215 <p n=>2</p>
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216 tur, centrum grauitatis e$t idem, quod circuli cen
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217 trum.</P>
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218 <P>Sit primo triangulum &aelig;quilaterum a b c in circulo de-
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219 $criptum: &amp; diui$a a c bi$ariam in d, ducatur b d. erit in li-
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220 nea b d centrum grauitatis tri&atilde;guli a b c, ex tertia decima
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221 primi libri Archimedis de centro grauitatis planorum. Et
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222 <fig>
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223 quoniam linea a b e$t &aelig;qualis
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224 line&aelig; b c; &amp; a d ip$i d c; e$t&qacute;;
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225 b d utrique communis: trian-
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226 <marg><*></marg>
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227 gulum a b d &aelig;quale erit trian
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228 gulo c b d: &amp; anguli angulis &aelig;-
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229 quales, qui &aelig;qualibus lateri-
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230 <marg><*></marg>
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231 bus $ubtenduntur. ergo angu
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232 li ad d utriq; recti $unt. qu&ograve;d
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233 cum linea b d $ecet a e bifa-
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234 <marg><*></marg>
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235 riam, &amp; ad angulos rectos; in
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236 ip$a b d e$t centrum circuli.
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237 quare in eadem b d linea erit
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238 centrum grauitatis trianguli, &amp; circuli centrum. Similiter
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239 diui$a a b bi$ariam in e, &amp; ducta c e, o$tendetur in ip$a utr&utilde;
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240 que centrum contineri. ergo ea erunt in puncto, in quo li-
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241 ne&aelig; b d, c e conueniunt. trianguli igitur a b c centrum gra
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242 uitatis e$t idem, quod circuli centrum.</P>
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243 <fig>
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244 <P>Sit quadratum a b c d in cir-
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245 culo de$criptum: &amp; ducantur
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246 a c, b d, qu&aelig; conueniant in e. er-
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247 go punctum e e$t centrum gra
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248 uitatis quadrati, ex decima eiu$
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249 dem libri Archimedis. Sed cum
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250 omnes anguli ad a b c d recti
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251 <marg>3<*></marg>
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252 $int; erit a b c femicirculus:
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253 item&qacute;; b c d: &amp; propterea li-
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254 ne&aelig; a c, b d diametri circuli:
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255 <pb>
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256 qu&aelig; quidem in centro conucniunt. idem igitur e$t centrum
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257 grauitatis quadrati, &amp; circuli centrum.</P>
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258 <P>Sit pentagonum &aelig;quilaterum, &amp; &aelig;quiangulum in circu-
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259 <fig>
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260 lo de$criptum a b c d e. &amp; iun-
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261 cta b d, bifariam&qacute;; in f diui$a,
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262 ducatur c f, &amp; producatur ad
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263 circuli circumferentiam in g;
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264 qu&aelig; lineam a e in h $ecet: de-
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265 indeiungantur a c, c c. Eodem
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266 modo, quo $upra demon$tra-
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267 bimus angulum b c f &aelig;qualem
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268 e$$e. angulo d c f; &amp; angulos
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269 ad f utro$que rectos: &amp; idcir-
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270 co lineam c f g per circuli cen
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271 trum tran$ire. Quoniam igi-
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272 tur latera c b, b a, &amp; c d, d e &aelig;qualia $unt; &amp; &aelig;quales anguli
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273 <marg>4. Primi.</marg>
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274 c b a, c d e: erit ba$is c a ba$i: c e, &amp; angulus b c a angulo
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275 d c e &aelig;qualis. ergo &amp; reliquus a c h, reliquo e c h. e$t au-
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276 tem c h utrique triangulo a c h, e c h communis. quare
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277 ba$is a h &aelig;qualis e$t ba$i h c: &amp; anguli, qui ad h recti: $unt&qacute;;
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278 <marg>28. primi.</marg>
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279 recti, qui ad f. ergo line&aelig; a e, b d inter $e $e &aelig;quidi$tant.
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280 Itaque cum trapezij a b d e latera b d, a e &aelig;quidi$tantia &agrave; li
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281 nea fh bifariam diuidantur; centrum grauitatis ip$ius erit
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282 <marg>13. Archi-
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283 medis.</marg>
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284 in linea fh, ex ultima eiu$dem libri Archimedis. Sed trian-
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285 guli b c d centrum grauitatis e$t in linea c f. ergo in eadem
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286 linea c h e$t centrum grauitatis trapezij a b d e, &amp; trian-
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287 guli b c d: hoc e$t pentagoni ip$ius centrum: &amp; centrum
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288 circuli. Rur$us $i iuncta a d, bifariam&qacute;; $ecta in k, duca-
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289 tur e k l: demon$trabimus in ip$a utrumque centrum in
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290 e$$e. Sequitur ergo, ut punctum, in quo line&aelig; c g, e l con-
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291 ueniunt, idem $it centrum circuli, &amp; centrum grauitatis
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292 pentagoni.</P>
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293 <P>Sit hexagonum a b c d e f &aelig;quilaterum, &amp; &aelig;quiangulum
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294 in circulo de$ignatum: iungantur&qacute;; b d, a e: &amp; bifariam $e-
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295 <foot><*></foot>
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296 <p n=>3</p>
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297 cta b d in g puncto, ducatur c g; &amp; protrahatur ad circuli
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298 u$que circumferentiam; qu&aelig; $ecet a e in h. Similiter conclu
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299 demus c g per centrum circuli tran$ire: &amp; bifariam $ecate
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300 lineam a e; item&qacute;; lineas b d, a e inter $e &aelig;quidi$tantes e$$e.
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301 Cumigitur c g per centrum circuli tran$eat; &amp; ad punct&utilde;
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302 f perueniat nect$$e e$t: qu&ograve;d c d e f $it dimidium circumfe
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303 <fig>
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304 <marg>13<*>
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305 m<*>
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306 9.</marg>
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307 renti&aelig; circuli. Quare in eadem
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308 diametro c f erunt centra gra
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309 uitatis triangulorum b c d,
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310 a f e, &amp; quadrilateri a b d e, ex
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311 quibus con$tat hexagonum a b
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312 c d e f. per$picuum e$tigitur in
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313 ip$a c f e$$e circuli centrum, &amp;
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314 centrum grauitatis hexagoni.
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315 Rur$us ducta altera diamctro
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316 a d, ei$dem rationibus o$tende-
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317 mus in ip$a utrumque c&etilde;trum
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318 ine$$e. Centrum ergo grauita-
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319 tis hexagoni, &amp; centrum circuli idem erit.</P>
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320 <P>Sit heptagonum a b c d e f g &aelig;quilaterum atque &aelig;quian||
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321 <fig>
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322 gulum in circulo de$criptum:
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323 &amp; iungantur c e, b f, a g: di-
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324 ui$a autem c e bifariam in p&utilde;
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325 cto h: &amp; iuncta d h produca-
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326 tur in k. non aliter demon-
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327 $trabimus in linea d k e$$e cen
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328 trum circuli, &amp; centrum gra-
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329 uitatis trianguli c d e, &amp; tra-
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330 peziorum b c e f, a b f g, hoc
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331 e$t centrum totius heptago-
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332 ni: &amp; rur$us eadem centra in
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333 alia diametro c l $imiliter du-
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334 cta contineri. Quare &amp; centrum grauitatis heptagoni, &amp;
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335 centrum circuli in idem punctum conueniunt. Eodem mo
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336 <pb>
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337 do in reliquis figuris &aelig;quilateris, &amp; &aelig;quiangulis, qu&aelig; in cir-
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338 culo de$cribuntur, probabimus c&etilde;trum grauitatis carun<*>,
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339 &amp; centrum circuli idem e$$e. quod quidem demon$trare
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340 oportebat.</P>
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341 <P>Ex quibus apparet cuiuslibet figur&aelig; rectilinc&aelig;
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342 in circulo plane de$cript&aelig; centrum grauitatis id&etilde;
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343 e$$e, quod &amp; circuli centrum.</P>
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344 <marg><G><*>wr&iacute;mw</G><*></marg>
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345 <P>Figuram in circulo plane de$criptam appella-
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346 mus, cuiu$modi e$t ea, qu&aelig; in duodecimo elemen
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347 torum libro, propo$itione $ecunda de$cribitur.
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348 ex &aelig;qualibus enim lateribus, &amp; angulis con$tare
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349 per$picuum e$t.</P>
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350 <head>THEOREMA II, PROPOSITIO II.</head>
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351 <P>Omnis figur&aelig; rectiline&aelig; in ellip$i plane de$cri-
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352 pt&aelig; centrum grauitatis e$t idem, quod ellip$is
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353 centrum.</P>
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354 <P>Quo modo figura rectilinea in ellip$i plane de$cribatur,
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355 docuimus in commentarijs in quintam propo$itionem li-
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356 bri Archimedis de conoidibus, &amp; $ph&aelig;roidibus.</P>
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357 <P>Sit ellip$is a b c d, cuius maior axis a c, minor b d: iun-
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358 gantur&qacute;; a b, b c, c d, d a: &amp; bifariam diuidantur in pun-
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359 ctis e f g h. &agrave; centro autem, quod $it k duct&aelig; line&aelig; k e, k f,
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360 k g, k h u$que ad $ectionem in puncta l m n o protrahan-
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361 tur: &amp; iungantur l m, m n, n o, o l, ita ut a c $ecet li-
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362 neas l o, m n, in z <G>f</G> punctis; &amp; b d $ecet l m, o n in <G>x y.</G>
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363 erunt l k, k n linea una, item&qacute;ue linea unaip$&aelig; m k, k o:
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364 &amp; line&aelig; b a, c d &aelig;quidi$tabunt line&aelig; m o: &amp; b c, a d ip$i
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365 l n. rur$us l o, m n axi b d &aelig;quidi$tabunt: &amp; l m,
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366 <foot>o n</foot>
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367 <p n=>4</p>
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368 o n ip$i a c. Quoniam enim triangulorum a b k, a d k, latus
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369 b k e$t &aelig;quale lateri k d, &amp; a k utrique commune; anguli&qacute;;
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370 <marg>8.<*></marg>
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371 ad k recti. ba$is a b ba$i a d; &amp; reliqui anguli reliquis an-
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372 gulis &aelig;quales erunt. eadem quoqueratione o$tendetur b c
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373 <fig>
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374 &aelig;qualis c d; &amp; a b ip$i
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375 b c. quare omnes a b,
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376 b c, c d, d a $unt &aelig;qua-
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377 les. &amp; quoniam anguli
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378 ad a &aelig;quales $unt angu
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379 lis ad c; erunt anguli b
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380 a c, a c d coalterni inter
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381 $e &aelig;quales; item&qacute;; d a c,
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382 a c b. ergo c d ip$i b a;
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383 &amp; a d ip$i b c &aelig;quidi-
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384 $tat. Atuero cum line&aelig;
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385 a b, c d inter $e &aelig;quidi-
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386 $tantes bifariam $ecen-
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387 tur in punctis e g; erit li
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388 nea l e k g n diameter $e
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389 ctionis, &amp; linea una, ex
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390 demon$tratis in uige$i-
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391 maoctaua $ecundi coni
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392 corum. Et eadem ratione linea una m f k h o. Sunt aut&etilde; a d,
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393 b c inter $e $e &aelig;quales, &amp; &aelig;quidi$tantes. quare &amp; earum di-
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394 <marg>33<*></marg>
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395 midi&aelig; a h, b f; item&qacute;; h d, f e; &amp; qu&aelig; ip$as coniungunt rect&aelig;
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396 line&aelig; &aelig;quales, &amp; &aelig;quidi$tantes erunt. &aelig;quidi$t&atilde;t igitur b a,
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397 c d diametro m o: &amp; pariter a d, b c ip$i l n &aelig;quidi$tare o-
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398 $tendemus. Si igitur man&etilde;te diametro a c intelligatur a b c
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399 portio ellip$is ad portionem a d c moueri, cum primum b
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400 applicuerit ad d, c&otilde;gruet tota portio toti portioni, linea&qacute;;
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401 b a line&aelig; a d; &amp; b c ip$i c d congruet: punctum uero e ca-
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402 det in h; f in g: &amp; linea k e in lineam k h: &amp; k f in k g. qua
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403 re &amp; e l in h o, et f m in g n. Atip$a l z in z o; et m <G>f</G> in <G>f</G> n
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404 cadet. congruet igitur triangulum l k z triangulo o k z: et
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405 <pb>
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406 triangulum m k <G>f</G> triangulo n k <G>f.</G> ergo anguli l z k, o z k,
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407 m <G>f</G> k, n <G>f</G> k &aelig;quales $unt, ac recti. qu&ograve;d cum etram recti
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408 <marg>28. primi.</marg>
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409 $int, qui ad k; &aelig;quidi$tabunt line&aelig; l o, m n axi b d. &amp; ita
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410 denion$trabuntur l m, o n ip$i a c &aelig;quidi$tare. Rurfus $i
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411 iungantur a l, l b, b m, m c, c n, n d, d o, o a: &amp; bitariam di
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412 uidantur: &agrave; centro autem k ad diui$iones duct&aelig; line&aelig; pro-
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413 trahantur u$que ad $ectionem in puncta p q r s t u x y: &amp; po
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414 ftremo p y, q x, r u, s t, q r, p s, y t, x u coniungantur. Simili-
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415 <fig>
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416 ter o$tendemus lineas
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417 p y, q x, r u, s t axi b d &aelig;-
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418 quidi$tantes e$$e: &amp; q r,
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419 p s, y t, x u &aelig;quidi$tan-
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420 tesip$i a c. Itaque dico
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421 harum figurarum in el-
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422 lip$i de$criptarum cen-
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423 trum grauitatis e$$e p&utilde;-
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424 ctum k, idem quod &amp; el
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425 lip$is centrum. quadri-
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426 lateri enim a b c d cen-
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427 trum e$t k, ex decima e-
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428 iu$dem libri Archime-
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429 dis, quippe c&utilde; in eo om
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430 nes diametri c&otilde;ueni&atilde;t.
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431 Sedin figura a l b m c n
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432 <marg>13. Archi
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433 medis.</marg>
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434 do, quoniam trianguli
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435 a l b centrum grauitatis
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436 <marg>Vltima.</marg>
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437 e$t in linea l e: trapezij&qacute;; a b m o centrum in linea e k: trape
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438 zij o m c d in k g: &amp; trianguli c n d in ip$a g n: erit magnitu
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439 dinis ex his omnibus con$tantis, uidelicet totius figur&aelig; cen
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440 trum grauitatis in linea l n: &amp; o b candem cau$$am in linea
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441 o m. e$t enim trianguli a o d centrum in linea o h: trapezij
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442 a l n d in h k: trapezij l b c n in k f: &amp; trianguli b m c in fm.
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443 cum ergo figur&aelig; a l b m c n d o centrum grauitatis $it in li-
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444 nea l n, &amp; in linea o m; erit centrum ip$ius punctum k, in
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445 <p n=>5</p>
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446 quo $cilicet l n, o m conueniunt. Po$tremo in figura
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447 a p l q b r m s c t n u d x o y centrum grauitatis trian
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448 guli p a y, &amp; trapezii p l o y e$t in linea a z: trapeziorum
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449 uero l q x o, q b d x centrum e$t in linea z k: &amp; trapezior&utilde;
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450 b r u d, r m n u in k <G>f:</G> &amp; denique trapezii m s t n; &amp; triangu
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451 li s c t in <G>f</G> c. quare magnitudinis ex his compo$it&aelig; centr&utilde;
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452 in linea a c con$i$tit. Rur$us trianguli q b r, &amp; trapezii q l
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453 m r centrum e$t in linea b <G>x.</G> trapeziorum l p s m, p a c s,
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454 a y t c, y o n t in linea <G>x f:</G> trapeziiq; o x u n, &amp; trianguli
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455 x d u centrum in <G>y</G> d. totius ergo magnitudinis centrum
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456 e$t in linea b d. ex quo $equitur, centrum grauitatis figur&aelig;
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457 a p l q b r m s c t n u d x o y e$$e punct&utilde; K, lineis $cilicet a c,
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458 b d commune, qu&aelig; omnia demon$trare oportebat.</P>
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459 <head>THEOREMA III. PROPOSITIO III.</head>
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460 <P>Cuiuslibet portio-
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461 nis circuli, &amp; ellip$is,
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462 qu&aelig; dimidia non $it
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463 maior, centrum graui
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464 tatis in portionis dia-
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465 metro con$i$tit.</P>
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466 <fig>
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467 <P>H O C eodem pror$us
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468 modo demon$trabitur,
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469 quo in libro de centro gra
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470 uitatis planorum ab Ar-
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471 chimede demon$trat&utilde; e$t,
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472 in portione c&otilde;tenta recta
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473 linea, &amp; rectanguli coni $e
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474 ctione grauitatis c&etilde;trum
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475 e$$e in diametro portio-
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476 nis. Etita demon$trari po
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477 <fig>
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478 <foot>B</foot>
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479 <pb>
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480 te$t in portione, qu&aelig; recta linea &amp; obtu$ianguli coni $e-
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481 ctione, $eu hyperbola continetur.</P>
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482 <head>THEOREMA IIII. PROPOSITIO IIII.</head>
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483 <P>IN circulo &amp; ellip$i idem e$t figur&aelig; &amp; graui-
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484 tatis centrum.</P>
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485 <P>SIT circulus, uel ellip$is, cuius centrum a. Dico a gra-
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486 uitatis quoque centrum e$$e. Si enim fieri pote$t, $it b cen-
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487 trum grauitatis: &amp; iuncta a b extra figuram in c produca
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488 tur: quam uero proportionem habet linea c a ad a b, ha-
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489 beat circulus a ad alium circulum, in quo d; uel ellip$is ad
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490 aliam ellip$im: &amp; in circulo, uel ellip$i $igura rectilinea pla-
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491 ne de$cribatur adco, ut tandem relinquantur portiones
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492 qu&aelig;dam minores circulo, uel ellip$i d; qu&aelig; figura $it e f g
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493 h k l m n. Illud uero in circulo fieri po$$e ex duodecimo
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494 elementorum libro, propo$itione $ecunda manife$te con-
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495 <fig>
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496 ftat; at in ellip$i nos demon$tra-
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497 uinms in commentariis in quin-
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498 tam propo$itionem Archimedis
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499 de conoidibus, &amp; $ph&aelig;roidibus.
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500 erit igitur a centrum grauitatis
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501 ip$ius figur&aelig;, quod proxime o$t&etilde;
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502 dimus. Itaque quoniam circulus
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503 a ad circulum d, uel ellip$is a ad
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504 ellip$im d candem proportion&etilde;
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505 habet, quam linea c a ad a b:
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506 portiones uero $unt minores cir
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507 <marg>8. quinti</marg>
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508 culo uel ellip$i d: habebit circu-
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509 lus, uel ellip$is ad portiones ma-
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510 iorem proportionem, qu&agrave;m c a
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511 <marg>19. quinti
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512 apud C&atilde;
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513 panum.</marg>
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514 ad a b: &amp; diuidendo figura recti-
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515 linea e f g h k l m n ad portiones
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516 <foot>habebit</foot>
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517 <p n=>6</p>
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518 <fig>
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519 habebit maiorem proportion&etilde;,
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520 quam c b ad b a. fiat o b ad b a,
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521 ut figura rectilinea ad portio-
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522 nes. cum igitur &agrave; circulo, uel el-
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523 lip$i, cuius grauitatis centrum
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524 e$t b, auferatur figura rectilinea
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525 e f g h k l m n, cuius centrum a;
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526 reliqu&aelig; magnitudinis ex portio
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527 <marg>8.<*>
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528 m<*></marg>
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529 nibus compo$it&aelig; centrum graui
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530 tatis erit in linea a b producta,
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531 &amp; in puncto o, extra figuram po
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532 $ito. quod quidem fieri nullo mo
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533 do po$$e per$picuum e$t. $equi-
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534 tur ergo, ut circuli &amp; ellip$is cen
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535 trum grauitatis $it punctum a,
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536 idem quod figur&aelig; centrum.</P>
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537 <head>ALITER.</head>
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538 <P>Sit circulus, uel ellip$is a b c d,
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539 cuius diameter d b, &amp; centrum e: ducaturq; per e rectall
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540 nea a c, $ecans ip$am d b ad rectos angulos. erunt a d c,
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541 a b c circuli, uel ellip$is dimidi&aelig; portiones. Itaque quo-
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542 <fig>
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543 niam por
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544 ti&otilde;is a d c
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545 c&etilde;tr&utilde; gra-
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546 uitatis e$t
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547 in diame-
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548 tro d e: &amp;
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549 portionis
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550 a b c cen-
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551 trum e$t &itilde;
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552 ip$a e b: to
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553 tius circu
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554 li, uel ellip$is grauitatis centrum erit in diametro d b.
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555 Sit autem portionis a d c c&etilde;trum grauitatis f: &amp; $umatur
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556 <foot>B <*></foot>
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557 <pb>
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558 in linea e b punct&utilde; g, itaut fit g e &aelig;qualis e f. erit g por-
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559 tionis a b c centrum. nam $i h&aelig; portiones, qu&aelig; &aelig;quales
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560 &amp; $imiles $unt, inter $e $e aptentur, ita ut b e cadat in d e,
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561 &amp; punctum b in d cadet, &amp; g in f: figuris autem &aelig;quali-
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562 bus, &amp; $imilibus inter $e aptatis, centra quoque grauitatis
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563 ip$arum inter $e aptata crunt, ex quinta petitione Archi-
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564 medis in libro de centro grauitatis planorum. Quare cum
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565 portionis a d c centrum grauitatis $it f: &amp; portionis
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566 a b c centrum g: magnitudinis; qu&aelig; ex utri$que efficitur:
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567 hoc e$t circuli uel ellip$is grauitatis centrum in medio li-
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568 ne&aelig; f g, quod e$t e, con$iftet, ex quarta propo$itione eiu$-
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569 dem libri Archimedis. ergo circuli, uel ellip$is centrum
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570 grauitatis e$t idem, quod figur&aelig; centrum. atque illud e$t,
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571 quod demon$trare oportebat.</P>
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572 <P>Ex quibus $equitur portionis circuli, uel ellip-
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573 $is, qu&aelig; dimidia maior $it, centrum grauitatis in
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574 diametro quoque ip$ius con$i$tere.</P>
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575 <fig>
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576 <P>Sit enim maior portio a b c, cu<I>i</I>us diameter b d, &amp; com-
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577 <*>
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578 <p n=>7</p>
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579 metrum habens c d. Quoniam igitur circuli uel ellip$is
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580 a e c b grauitatis centrum e$t in diametro b e, &amp; portio-
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581 nis a e c centrum in linea e d: reliqu&aelig; portionis, uidelicet
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582 a b c centrum grauitatis in ip$a b d con$i$tat nece$$e e$t, ex
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583 octaua propo$itione eiu$dem.</P>
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584 <head>THEOREMA V. PROPOSITIO V.</head>
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585 <P>SI pri$ma $ecetur plano oppo$itis planis &aelig;qui
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586 di$tante, $ectio erit figura &aelig;qualis &amp; $imilis ei,
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587 qu&aelig; e$t oppo$itorum planorum, centrum graui
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588 tatis in axe habens.</P>
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589 <P>Sit pri$ina, in quo plana oppo$ita $int triangula a b c,
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590 d e f; axis g h: &amp; $ecetur plano iam dictis planis &aelig;quidi$t&atilde;
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591 te; quod faciat $ectionem k l m; &amp; axi in p&utilde;cto n occurrat.
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592 Dico k l m trian gulum &aelig;quale e$$e, &amp; fimile triangulis a b c
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593 d e f; atque eius grauitatis centrum e$$e punctum n. Quo-
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594 <fig>
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595 niam enim plana a b c
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596 K l m &aelig;quidi$tantia $ec&atilde;
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597 <marg><*>
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598 ci<*></marg>
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599 tur a plano a e; rect&aelig; li-
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600 ne&aelig; a b, K l, qu&aelig; $unt ip
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601 $orum c&otilde;munes $ectio-
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602 nes inter $e $e &aelig;quidi-
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603 $tant. Sed &aelig;quidi$tant
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604 a d, b e; cum a e $it para
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605 lelogrammum, ex pri$-
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606 matis diffinitione. ergo
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607 &amp; al parallelogramm&utilde;
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608 erit; &amp; propterea linea
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609 <marg><*></marg>
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610 k l, ip$i a b &aelig;qualis. Si-
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611 militer demon$trabitur
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612 l m &aelig;quidi$tans, &amp; &aelig;qua
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613 <*>
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614 <pb>
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615 Itaque quoniam du&aelig; line&aelig; K l, l m $e $e tangentes, duab us
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616 lineis $e $e tangentibus a b, b c &aelig;quidi$tant; nec $unt in e o-
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617 dem plano: angulus k l m &aelig;qualis e$t angulo a b c: &amp; ita an
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618 <marg>10. unde
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619 cimi</marg>
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620 gulus l m k, angulo b c a, &amp; m k lip$i c a b &aelig;qualis prob abi
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621 tur. triangulum ergo k l m e$t &aelig;quale, &amp; $imile triang ulo
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622 a b c. quare &amp; triangulo d e f. Ducatur linea c g o, &amp; per ip
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623 $am, &amp; per c f ducatur planum $ecans pri$ma; cuius &amp; paral
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624 lelogram<*> a e communis $ectio $it o p q. tran$ibit linea
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625 fq per h, &amp; m p per n. nam cum plana &aelig;quidi$tantia $ecen
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626 tur &agrave; plano c q, communes eorum $ectiones c g o, m p, f q
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627 $ibi ip$is &aelig;quidi$tabunt. Sed &amp; &aelig;quidi$tant a b, k l, d e. an-
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628 <marg>10. unde-
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629 cimi</marg>
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630 guli ergo a o c, k p m, d q f inter $e &aelig;quales $unt: &amp; $unt
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631 &aelig;quales qui ad puncta a k d con$tituuntur. quare &amp; reliqui
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632 reliquis &aelig;quales; &amp; triangula a c o, K m p, d f q inter $e $imi
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633 <marg>4. $exti</marg>
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634 lia erunt. Vtigitur c a ad a o, ita fd ad d q: &amp; permutando
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635 ut c a ad f d, ita a o ad d q. e$t autem c a &aelig;qualis f d. ergo &amp;
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636 a o ip$i d q. eadem quoque ratione &amp; a o ip$i K p &aelig;qualis
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637 demon$trabitur. Itaque $i triangula, a b c, d e f &aelig;qualia &amp;
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638 <fig>
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639 $imilia inter $e apt&etilde;tur,
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640 cadet linea f q in lineam
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641 <marg>pe<*> 5. pe-
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642 titionem
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643 Archime
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644 dis.</marg>
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645 c g o. Sed &amp; centr&utilde; gra
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646 uitatis h in g centr&utilde; ca-
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647 det. tr&atilde;$ibit igitur linea
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648 f q per h: &amp; planum per
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649 c o &amp; c f duct&utilde; per ax&etilde;
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650 g h ducetur: idcircoq; li
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651 neam m p eti&atilde; per n tr&atilde;
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652 $ire nece$$e erit. Quo-
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653 niam ergo fh, c g &aelig;qua-
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654 les $unt, &amp; &aelig;quidi$t&atilde;tes:
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655 itemq; h q, g o; rect&aelig; li-
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656 ne&aelig;, qu&aelig; ip$as c&otilde;nect&utilde;t
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657 c m f, g n h, o p q &aelig;qua-
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658 <*>
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659 <p n=>8</p>
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660 &aelig;quidi$tant autem c g o, m n p. ergo parallelogr&atilde;ma $unt
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661 o n, g m, &amp; linea m n &aelig;qualis c g; &amp; n p ip$i g o. aptatis igi-
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662 tur k l m, a b c tri&atilde;gulis, qu&aelig; &aelig;qualia &amp; $imilia s&utilde;t; linea m p
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663 in c o, &amp; punctum n in g cadet. Qu&ograve;d c&utilde; g $it centrum gra-
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664 uitatis trianguli a b c, &amp; n trianguli k l m grauitatis cen-
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665 trum erit id, quod demon$trandum relinquebatur. Simili
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666 ratione idem contingere demon$trabimus in aliis pri$ma-
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667 tibus, $iue quadrilatera, $iue plurilatera habeant plana,
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668 qu&aelig; opponuntur.</P>
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669 <head>COROLLARIVM.</head>
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670 <P>Exiam demon$tratis per$picue apparet, cuius
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671 Iibet pri$matis axem, parallelogrammorum lat eri
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672 bus, qu&aelig; ab oppo$itis planis duc&utilde;tur &aelig;quidi$tare.</P>
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673 <head>THE OREMA VI. PROPOSITIO VI.</head>
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674 <P>Cuiuslibet pri$matis centrum grauitatis e$t in
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675 plano, quod oppo$itis planis &aelig;quidi$tans, reli-
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676 quorum planorum latera bifariam diuidit.</P>
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677 <P>Sit pri$ma, in quo plana, qu&aelig; opponuntur $int trian-
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678 gula a c e, b d f: &amp; parallelogrammorum latera a b, c d,
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679 e f bifariam diuid&atilde;tur in punctis g h k: per diui$iones au-
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680 <marg>33.<*></marg>
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681 tem planum ducatur; cuius $ectio figura g h K. erit linea
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682 g h &aelig;quidi$tans lineis a c, b d &amp; h k ip$is c e, d f. quare ex
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683 decimaquinta undecimi elementorum, planum illud pla
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684 nis a c e, b d f &aelig;quidi$tabit, &amp; faciet $ectionem figu-
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685 <marg>5. b<*></marg>
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686 ram ip$is &aelig;qualem, &amp; $imilem, ut proxime demon$tra-
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687 uimus. Dico centrum grauitatis pri$matis e$$e in plano
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688 g h k. Si enim fieri pote$t, $it eius centrum l: &amp; ducatur
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689 l m u$que ad planum g h k, qu&aelig; ip$i a b &aelig;quidi$tet.
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690 <pb>
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691 <marg>1. decimi</marg>
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692 ergo linea a g continenter in duas partes &aelig;quales diui-
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693 $a, relinquetur t&atilde;dem pars aliqua n g, qu&aelig; minor erit l m.
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694 Vtraque uero linearum a g, g b diuidatur in partes &aelig;qua-
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695 les ip$i n g: &amp; per puncta diui$ionum plana oppo$itis pla-
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696 <marg>5 huius</marg>
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697 nis &aelig;quidi$tantia ducantur. erunt $ectiones figur&aelig; &aelig;qua-
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698 les, ac $imiles ip$is a c e, b d f: &amp; totum pri$ma diui$um erit
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699 in pri$mata &aelig;qualia, &amp; $imilia: qu&aelig; cum inter $e congru&atilde;t;
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700 &amp; grauitatis centra $ibi ip$is congruentia, re$pondentiaq;
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701 <fig>
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702 habebunt. Itaq:
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703 $unt magnitudi-
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704 nes qu&aelig;d&atilde; &aelig;qua-
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705 les ip$i n h, &amp; nu-
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706 mero pares, qua-
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707 rum centra gra-
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708 uitatis in ead&etilde;re
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709 cta linea con$ti-
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710 tuuntur: du&aelig; ue-
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711 ro medi&aelig; &aelig;qua-
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712 les $unt: &amp; qu&aelig; ex
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713 utraque parte i-
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714 p$arum $imili- -
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715 ter &aelig;quales: &amp; &aelig;-
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716 quales rect&aelig; li-
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717 ne&aelig;, qu&aelig; inter
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718 grauitatis centra
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719 interiiciuntur.
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720 quare ex corolla-
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721 rio quint&aelig; pro-
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722 po$itionis primi
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723 libri Archimedis
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724 d e centro graui-
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725 tatis planorum; magnitudinis ex his omnibus compo$it&aelig;
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726 centrum grauitatis e$t in medio line&aelig;, qu&aelig; magnitudi-
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727 num mediarum centra coniungit. at qui non ita res ha-
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728 <foot>bet,</foot>
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729 <p n=>9</p>
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730 bet, $i quidem l extra medias magnitudines po$itum e$t.
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731 Con$tat igitur centrum granitatis pri$matis e$$e in plano
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732 <fig>
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733 g h k, quod nos demon$trandum propo$uimus. At$i op-
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734 po$ita plana in pri$mate $int quadrilatera, uel plurilatera,
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735 eadem erit in omnibus demon$tratio.</P>
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736 <head>THEOREMA VII. PROPOSITIO VII.</head>
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737 <P>Cuiuslibet cylindri, &amp; cuiuslibet cylindri por
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738 tionis centrum grauitatis e$t in plano, quod ba$i-
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739 bus &aelig;quidi$tans, parallelogrammi per axem late-
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740 ra bifariam $ecat.</P>
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741 <pb>
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742 <P>SIT cylindrus, uel cylindri portio a c: &amp; plano per a<*>
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743 xem ducto $ecetur; cuius fectio $it parallelogrammum ab
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744 c d: &amp; bifariam diui$is a d, b c parallelogrammi lateribus,
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745 per diui$ionum puncta e f planum ba$i &aelig;quidi$tans duca-
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746 tur; quod faciet $ectionem, in cylindro quidem circulum
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747 &aelig;qualem iis, qui $unt in ba$ibus, ut demon$trauit Serenus
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748 in libro cylindricorum, propo $itione quin ta: in cylindr
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749 uero portione ellip$im &aelig;qualem, &amp; $imilem eis, qu&aelig; $unt
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750 <fig>
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751 in oppo$itis planis, quod nos
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752 demon$trauimus in commen
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753 tariis in librum Archimedis
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754 de conoidibus, &amp; $ph&aelig;roidi-
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755 bus. Dico centrum grauita-
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756 tis cylindri, uel cylindri por-
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757 tionis e$$e in plano e f. Si en&itilde;
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758 fieri pote$t, fit centrum g: &amp;
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759 ducatur g h ip$i a d &aelig;quidi-
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760 $tans, u$que ad e f planum.
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761 Itaque linea a e continenter
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762 diui$a bifariam, erit tandem
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763 pars aliqua ip$ius k e, minor
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764 g h. Diuidantur ergo line&aelig;
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765 a e, e d in partes &aelig;quales ip$i
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766 k e: &amp; per diui$iones plana ba
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767 $ibus &aelig;quidi$tantia duc&atilde;tur.
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768 eruntiam $ectiones, figur&aelig; &aelig;-
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769 quales, &amp; $imiles eis, qu&aelig; $unt
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770 in ba$ibus: atque erit cylindrus in cylindros diui$us: &amp; cy
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771 lindri portio in portiones &aelig;quales, &amp; $imiles ip$i k f. reli-
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772 qua $imiliter, ut $uperius in pri$mate concludentur.</P>
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773 <P><*></P>
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774 <p n=>10</p>
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775 <fig>
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776 <head>THEOREMA VIII. PROPOSITIO VIII.</head>
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777 <P>Cuiuslibet pri$matis, &amp; cuiuslibet cylindri, uel
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778 cylindri portionis grauitatis centrum in medio
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779 ip$ius axis con$i$tit.</P>
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780 <P>Sit primum a f pri$ma &aelig;quidi$tantibus planis contcnt&utilde;,
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781 quod $olidum parallelepipedum appellatur: &amp; oppo$ito-
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782 rum planorum c f, a h, d a, f g latera bifariam diuidantur in
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783 punctis k l m n o p q r s t u x: &amp; per diui$iones ducantur
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784 plana k n, o r, s x. communes autem eorum planorum $e-
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785 ctiones $int line&aelig; y z, <G>q f, x y:</G> qu&aelig; in puncto <G>w</G> conueni&atilde;t.
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786 erit ex decima eiu$dem libri Archimedis parallelogrammi
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787 c f centrum grauitatis punctum y; parallelogrammi a h
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788 <foot>C 2</foot>
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789 <pb>
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790 centrum z: paraliclogram mi <*> d, <G>q:</G> parallclogrammi f g, <G><*>:</G>
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791 <fig>
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792 parallclogrammi d h, <G>x:</G> &amp;
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793 parallclogrammi c g centr&utilde;
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794 <G>y:</G> atque erit <G>w</G> punctum me
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795 dium uniu$cuiu$que axis, ui
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796 delicet eius line&aelig; qu&aelig; oppo
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797 $itorum planor&utilde; centra con
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798 iungit. Dico <G>w</G> centrum e$$e
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799 grauitatis ip$ius $olidi. e$t
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800 <marg>6 huius</marg>
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801 enim, ut demon$trauimus,
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802 $olidi a f centrum grauitatis
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803 in plano K n; quod oppo$i-
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804 tis planis a d, g f &aelig;quidi$tans
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805 reliquorum planorum late-
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806 ra bifariam diuidit: &amp; $imili
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807 ratione idem centrum e$t in plano o r, &aelig;quidi$tante planis
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808 a e, b f oppo$itis. ergo in communi ip$orum $ectione: ui-
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809 delicet in linea y z. Sed e$t etiam in plano t u, quod quid&etilde;
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810 y z $ecatin <G>w.</G> Con$tatigitur centrum grauitatis $olidi e$$e
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811 punctum <G>w,</G> medium $cilicet axium, hoc e$t linearum, qu&aelig;
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812 planorum oppo$itorum centra coniungunt.</P>
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813 <P>Sit aliud prima a f; &amp; in eo plana, qu&aelig; opponuntur, tri-
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814 angula a b c, d e f: diui$isq; bifariam parallelogrammorum
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815 lateribus a d, b e, c f in punctis g h k, per diui$iones plan&utilde;
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816 ducatur, quod oppo$itis planis &aelig;quidi$tans faciet $ection&etilde;
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817 triangulum g h x &aelig;quale, &amp; $imile ip$is a b c, d e f. Rur$us
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818 diuidatur a b bi$ariam in l: &amp; iuncta c l per ip$am, &amp; per
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819 c K f planum ducatur pri$ma $ecans, cuius, &amp; parallelogr&atilde;
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820 mi a e communis $ectio $it l m n. diuidet punctum m li-
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821 neam g h bifariam; &amp; ita n diuidet lineam d e: quoniam
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822 <marg>5.huius</marg>
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823 triangula a c l, g <G>k</G> m, d f n &aelig;qualia $unt, &amp; $imilia, ut $upra
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824 demon$trauimus. Iam ex iis, qu&aelig; tradita $unt, con$tat cen
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825 trum greuitatis pri$matis in plano g h <G>k</G> contineri. Dico
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diff changeset
826 ip$um e$$e in linea k m. Si enim fieri potc$t, $it o centrum;
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827 <foot>&amp; per</foot>
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828 <p n=>11</p>
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829 &amp; per o ducatur o p ad k m ip$i h g &aelig;quidi$tans. Itaquc li
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830 nea h m bifari&atilde; u$que c&ograve; diuidatur, quoad reliqua $it pars
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831 qu&aelig;dam q m, minor o p. deinde h m, m g diuidantur in
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832 partes &aelig;quales ip$i m q: &amp; per diui$iones line&aelig; ip$i m K
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833 &aelig;quidi$tantes ducantur. puncta uero, in quibus h&aelig; trian-
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834 gulorum latera $ecant, coniungantur ductis lineis r s, t u,
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835 <fig>
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836 x y; qu&aelig; ba$i g h &aelig;quidi$tabunt. Quoniam enim line&aelig; g z,
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837 h <G>a</G> $unt &aelig;quales: itemq; &aelig;quales g m, m h: ut m g ad g z,
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838 ita erit m h, ad h <G>a:</G> &amp; diuidendo, ut m z ad z g, ita m <G>a</G> ad
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839 <marg>2. $e<*>
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840 . q<*></marg>
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841 <G>a</G> h. Sed ut m z ad z g, ita k r ad r g: &amp; ut m <G>a</G> ad <G>a</G> h, ita k s
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842 ad s h. quare ut k r ad r g, ita <G>k</G> s ad s h. &aelig;quidi$tant igitur
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843 <marg>2. $e<*></marg>
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844 inter $e $e r s, g h. eadem quoque ratione demon$trabimus
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845 <pb>
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846 t u, x y ip$i g h &aelig;quidi$tare. Et quoniam triangula, qu&aelig;
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847 fiunt &agrave; lineis K y, y u, u s, s h &aelig;qualiz funt inter $e, &amp; $imilia
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848 <marg><*>. $exti</marg>
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849 triangulo K m h: habebit triangulum K m h ad triangul&utilde;
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850 K <G>d</G> y duplam proportioncm eius, qu&aelig; e$t line&aelig; <G>k</G> h ad K y.
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851 $ed K h po$ita e$t quadrupla ip$ius k y. ergo triangulum
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852 k m h ad triangulum K <G>d</G> y e&atilde;dem proportionem habcbit,
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853 quam $exdecim ad un&utilde;: &amp; ad quatuor triangula k <G>d</G> y, y u,
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854 u s, s <G>a</G> h habebit eandem, quam $exdecim ad quatuor, hoc
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855 e$t quam h K ad k y: &amp; $imiliter eandem habere demon$tra
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856 <fig>
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857 bitur trian-
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858 gulum k m g
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859 ad quatuor
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860 tri&atilde;gula K <G>d</G>
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861 x, x <G>g</G> t, t <G>b</G> r,
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862 <marg>uel 127
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863 utnti.</marg>
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864 r z g. quare
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865 totum trian
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866 gulum K g h
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867 ad omnia tri
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868 angula g z r,
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869 r <G>b</G> t, t <G>g</G> x, x <G>d</G>
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870 K, K <G>d</G> y, y u,
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871 u s, s <G>a</G> h ita
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872 erit, ut h k ad
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873 k y, hoc e$t
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874 ut h m ad m
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875 q. Siigitur in
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876 triangulis a b c, d e f de$cribantur figur&aelig; $imiles ei, qu&aelig; de-
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877 $cripta e$t in g h K triangulo: &amp; per lineas $ibi re$ponden-
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878 tes plana ducantur: totum pri$ma a f diui$um erit in tria
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879 $olida parallelepipeda y <G>g,</G> u <G>b,</G> s z, quorum ba$es $unt &aelig;qua
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880 les &amp; $imiles ip$is parallelogrammis y <G>g,</G> u <G>b,</G> s z: &amp; in octo
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881 pri$mata g z r, r <G>b</G> t, t <G>g</G> x, x <G>d</G> K, k <G>d</G> y, y u, u s, s <G>a</G> h: quorum
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882 item ba$es &aelig;quales, &amp; $imiles $unt dictis triangulis; altitu-
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883 <*>
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884 <p n=>12</p>
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885 Itaque $olidi parallelepipedi y <G>g</G> centrum grauitatis e$t in
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886 linea <G>d <*>:</G> $olidi <G><*> b</G> centrum e$t in linea <G><*>:</G> &amp; $olidi s z in li
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887 nea <*> m, qu&aelig; quidem line&aelig; axes $unt, cum planorum oppo
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888 $itorum centra coniungant. ergo magnitudinis ex his $oli
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889 dis compo$it&aelig; centrum grauitatis e$t in linea <G>d</G> m, quod $it
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890 <G>q;</G> &amp; iuncta <G>q</G> o producatur: &agrave; puncto autem h ducatur h <G>a</G>
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891 ip$i m k &aelig;quidi$tans, qu&aelig; cum <G>q</G> o in <G>m</G> conueniat. triangu
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892 lum igitur g h k ad omnia triangula g z r, r <G>b</G> t, t <G>g</G> x, x <G>d</G> k,
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893 k <G>d</G> y, y u, u s, s <G>a</G> h eandem habet proportionem, quam h m
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894 ad m q; hoc e$t, quam <G>m q</G> <*> <G>q l:</G> nam $i h m, <G>y q</G> produci in
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895 telligantur, quou$que coeant; erit ob linearum q y, m k &aelig;<*>
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896 quidi$tantiam, ut h q ad q m, ita <G>m l</G> ad ad <G>l q:</G> &amp; componen
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897 do, ut h m ad m q, ita <G>m q</G> ad <G>q l.</G> linea uero <G>q</G> o maior e$t,
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898 <marg>3.<*></marg>
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899 qu&agrave;m <G>q l:</G> habebit igitur <G>m q</G> ad <G>q l</G> maiorem proportio-
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900 nem, qu&agrave;m ad <G>q</G> o. quare triangulum ctiam g h <G>k</G> ad omnia
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901 iam dicta triangula maiorem proportion&etilde; habebit, qu&agrave;m
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902 <G>m q</G> ad <G>q</G> o. $ed ut triangul&utilde; g h <G>k</G> ad omnia triangula, ita to-
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903 t&utilde; pri$ma a fad omnia pri$mata g z r, r <G>b</G> t, t <G>g</G> x, x <G>d k, k d</G> y,
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904 y u, u s, s <G>a</G> h: quoniam enim $olida parallelepipeda &aelig;que al
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905 ta, eandem inter $e proportionem habent, quam ba$es; ut
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906 ex trige$ima$ecunda undecimi elementorum con$tat. $unt
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907 <marg>28.<*>
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908 cin<*></marg>
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909 autem $olida parallelepipeda pri$matum triangulares ba-
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910 <marg>15.<*></marg>
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911 $es habentium dupla: $equitur, ut etiam huiu$modi pri$-
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912 mata inter $e $int, $icut eorum ba$es. ergo totum pri$ma ad
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913 omnia pri$mata maiorem proportionem habet, quam <G>m q</G>
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914 <marg>19.<*>
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915 apu<*>
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916 pan<*></marg>
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917 ad <G>q</G> o: &amp; diuidendo $olida parallelepipeda y <G>g,</G> u <G>b,</G> s z ad o-
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918 mnia pri$mata proportionem habent maiorem, qu&agrave;m <G>m</G> o
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919 ad o <*>. fiat <G>n</G> o ad o <G>q,</G> ut $olida parallelepipeda y <G>g,</G> u <G>b,</G> s z ad
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920 omnia pri$mata. Itaque cum &agrave; pri$mate a f, cuius c&etilde;trum
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921 grauitatis e$t o, auferatur magnitudo ex $olidis parallelepi
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922 pedis y <G>g,</G> u <G>b,</G> s z con$tans: atque ipfius grauitatis centrum
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923 $it <G>q:</G> reliqu&aelig; magnitudinis, qu&aelig; ex omnibus pri$matibus
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924 con$tat, grauitatis centrum erit in linea <G>q</G> o producta: &amp;
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925 <*>
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926 <pb>
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927 medis. ergo punctum <G>n</G> extra p ri$ma a f po$itum, centr&utilde;
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928 erit magnitudinis c&otilde;po$it&aelig; e x omnibus pri$matibus g z r,
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929 r <G>b</G> t, t <G>g</G> x, x <G>d</G> k, k <G>d</G> y, y u, u s, s <G>a</G> h, quod fieri nullo modo po
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930 te$t. e$t enim ex diffinitione centr um grauitatis $olid&aelig; figu
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931 r&aelig; intra ip$am po$itum, non extra. quare relinquitur, ut c&etilde;
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932 trum grauitatis pri$matis $it in linea K m. Rur$us b c bifa-
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933 riam in diuidatur: &amp; ducta a <G>x,</G> per ip$am, &amp; per lineam
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934 a g d plan um ducatur; quod pri$ma $ecet: faciatq; in paral
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935 lelogrammo b f $ectionem <G>x p</G> diuidet punctum <G>p</G> lineam
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936 quoque c f bifariam: &amp; erit p lani eius, &amp; trianguli g h K
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937 communis $ectio g u; qu&ograve;d p&utilde;ctum u in medio line&aelig; h K
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938 <fig>
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939 po$itum $i t. Similiter demon$trabimus centrum grauita-
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940 tis pri$m atis in ip$a g u ine$$e. $it autem planorum c f n l,
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diff changeset
941 a d <G>p x</G> communis $ectio linea <G>r s t;</G> qu&aelig; quidem pri$matis
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942 axis erit, cum tran$eat per centra grauitatis triangulorum
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943 a b c, g h <G>k,</G> d e f, ex quartadecima eiu$dem. ergo centrum
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diff changeset
944 grauitatis pri$matis a f e$t punctum <G>s,</G> centrum $cilicet
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945 <p n=>13</p>
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946 trianguli g h K, &amp; ip$ius <G>r t</G> axis medium.</P>
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947 <P>Sit pri$ma a g, cuius oppo$ita plana $int quadrilatera
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948 a b c d, e f g h: $ecenturq; a c, b f, c g, d h bifariam: &amp; per di-
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949 ui$iones planum ducatur; quod $ectionem faciat quadrila-
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950 terum K l m n. Deinde iuncta a c per lineas a c, a e ducatur
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diff changeset
951 planum $ec&atilde;s pri$ma, quod ip$um diuidet in duo pri$mata
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952 triangulares ba$es habentia a b c e f g, a d c e h g. Sint aut&etilde;
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953 <fig>
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954 triangulorum a b c, e f g gra-
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955 uitatis centra o p: &amp; triangu-
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956 lorum a d c, e h g centra q r:
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957 iunganturq; o p, q r; qu&aelig; pla-
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958 no k l m n occurrant in pun-
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959 ctis s t. erit ex iis, qu&aelig; demon
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960 $trauimus, punctum s grauita
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961 tis centrum trianguli <G>k</G> l m; &amp;
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962 ip$ius pri$matis a b c e f g: pun
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963 ctum uero t centrum grauita
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964 tis trianguli K n m, &amp; pri$ma-
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965 tis a d c, e h g. iunctis igitur
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966 o q, p r, s t, erit in linea o q c&etilde;
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967 trum grauitatis quadrilateri
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968 a b c d, quod $it u: &amp; in linea
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969 p r c&etilde;trum quadrilateri e f g h
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970 $it autem x. denique iungatur
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971 u x, qu&aelig; $ecet lineam $ t in y. $e||
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972 cabit enim cum $int in eodem
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973 <marg>5.h<*></marg>
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974 plano: atq; erit y grauitatis centrum quadrilateri K l m n.
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975 Dico idem punctum y centrum quoque gra uitatis e$$e to-
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976 tius pri$matis. Quoniam enim quadrilateri k l m n graui-
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977 tatis centrum e$t y: linea s y ad y t ean dem proportionem
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978 habebit, quam triangulum <G>k</G> n m ad triangulum k l m, cx 8
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979 Archimedis de centro grauitatis planorum. Vt autem tri&atilde;
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980 gulum <G>k</G> n m ad ip$um <G>k</G> l m, hoc e$t ut triangulum a d c ad
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981 triangulum a b c, &aelig;qualia enim $unt, ita pri$ma a d c e h g
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982 <foot>D</foot>
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983 <pb>
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984 ad pri$ma a b c e f g. quare linea s y ad y t eandem propor-
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985 tionem habet, quam pri$ma a d c e h g ad pri$ma a b c e f g.
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986 Sed pri$matis a b c e f g centrum grauitatis e$t s: &amp; pri$ma-
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987 tis a d c e h g centrum t. magnitudinis igitur ex his compo
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988 $it&aelig; hoc e$t totius pri$matis a g centrum grauitatis e$t pun
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989 ctum y; medium $cilicet axis u x, qui oppo$itorum plano-
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990 rum centra coniungit.</P>
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991 <P>Rur$us $it pri$ma ba$im habens pentagonum a b c d e:
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992 &amp; quod ei opponitur $it f g h K l: $ec enturq; a f, b g, c h,
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993 d k, e l bi$ariam: &amp; per diui$iones ducto plano, $ectio $it p&etilde;
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994 tagon&utilde; m n o p q. deinde iuncta e b per lineas l e, e b aliud
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995 <fig>
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996 planum ducatur, diuid&etilde;s pri$
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997 ma a k in duo pri$mata; in pri$
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998 ma $cilicet al, cuius plana op-
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999 po$ita $int triangula a b e f g l:
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1000 &amp; in prima b k cuius plana op
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1001 po$ita $int quadrilatera b c d e
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1002 g h k l. Sint autem triangulo-
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1003 rum a b e, f g l centra grauita
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1004 tis puncta r $: &amp; b c d e, g h k l
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1005 quadrilaterorum centra t u:
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1006 iunganturq; r s, t u occurren-
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1007 tes plano m n o p q in punctis
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1008 x y. &amp; itidem iung&atilde;tur r t, $u,
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1009 x y. erit in lincar t c&etilde;trum gra
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1010 uitatis pentagoni a b c d e;
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1011 quod $it z: &amp; in linea $u cen-
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1012 trum pentagoni f g h <G>k</G> l:$it au
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1013 tem <G>x:</G> &amp; ducatur z <G>x,</G> qu&aelig; di-
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1014 cto plano in <G>y</G> occurrat. Itaq;
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1015 punctum x e$t centrum graui
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1016 tatis trianguli m n q, ac pri$-
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1017 matis a l: &amp; y grauitatis centrum quadrilateri n o p q, ac
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1018 pri$matis b k. quare y centrum erit pentagoni m n o p q. &amp;
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1019 <foot>fimiliter</foot>
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1020 <p n=>14</p>
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1021 $imiliter demon$irabitur totius pri$matis a K grauitatis ef
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1022 $e centrum. Simili ratione &amp; in aliis pri$matibus illud
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1023 idem facile demon$trabitur. Quo autem pacto in omni
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1024 figura rectilinea centrum grauitatis inueniatur, docuimus
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1025 in commentariis in $extam propo$itionem Archimedis de
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1026 quadratura parabol&aelig;.</P>
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1027 <P>Sit cylindrus, uel cylindri portio c e cuius axis a b: $ece-
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1028 turq, plano per axem ducto; quod $ectionem faciat paral-
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1029 lelogrammum c d e f: &amp; diui$is c f, d e bifariam in punctis
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1030 <fig>
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1031 g h, per ea ducatur planum ba$i &aelig;quidi$tans. erit $ectio g h
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1032 circulus, uel ellip$is, centrum habens in axe; quod $it K at-
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1033 <marg>4. hu<*></marg>
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1034 que erunt ex iis, qu&aelig; demon$trauimus, centra grauitatis
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1035 planorum oppo$itorum puncta a b: &amp; plani g h ip$um k in
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1036 quo quidem plano e$t centrum grauitatis cylindri, uel cy-
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1037 lindri portionis. Dico punctum K cylindri quoque, uel cy
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1038 lindri portionis grauitatis centrum e$$e. Si enim fieri po-
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1039 te$t, $it l centrum: ducaturq; k l, &amp; extra figuram in m pro-
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1040 ducatur. quam ucro proportionem habet linea m K ad k l
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1041 <foot><*></foot>
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1042 <pb>
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1043 habcat circulus, uel ellip$is g h ad aliud $pacium, in quo u:
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1044 &amp; in cit culo, uel ellip$i plane de$cribatur rectilinea figura,
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1045 ita ut t&atilde;dem relinqu&atilde;tur portiones minores $pacio u, qu&aelig;
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1046 $it o p g q r s h t: de$criptaq; $imili figura in oppo$itis pla-
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1047 nis c d, f e, per lineas $ibi ip$is re$pondentes plana duc&atilde;tur.
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1048 Itaque cylindrus, uel cylindri portio diuiditur in pri$ma,
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1049 cuius quidem ba$is e$t figura rectilinea iam dicta, centrum
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1050 que grauitatis punctuni K: &amp; in multa $olida, qa&aelig; pro ba$i
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1051 bus habent relictas portiones, quas nos $olidas portiones
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1052 appellabimus. cum igitur portiones $int minores $pacio
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1053 u, circulus, uel ellip$is g h ad portiones maiorem propor-
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1054 tionem habebit, qu&agrave;m linea m k ad K l. fiat n <G>k</G> ad K l, ut
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1055 circulus uel ellip$is g h ad ip$as portiones. Sed ut circulus
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1056 uel ellip$is g h ad figuram rectilincam in ip$a de$cri-
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1057 ptam, ita e$t cylindrus uel cylindri portio c c ad pri$ma,
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1058 quod rectilineam figuram pro ba$i habet, &amp; altitudinem
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1059 &aelig;qualem; id, quod infra demon$trabitur. crgo per conuer
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1060 $ionem rationis, ut circulus, uel ellip$is g h ad portioncs re
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1061 lictas, ita cylindrus, uel cylindri portio c e ad $olidas por-
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1062 tiones, quate cylindrus uel cylindri portio ad $olidas por-
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1063 tiones eandem proportionem habet, quam linea n <G>k</G> ad k
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1064 &amp; diuidendo pri$ma, cuius ba$is e$t rectilinea figura ad $o-
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1065 lidas portiones candem proportionem habet, quam n l ad
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1066 l k &amp; quoniam a cylindro uel cylindri portione, cuius gra-
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1067 uitatis centrum e$t l, aufertur pri$ma ba$im habens rectili-
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1068 neam figur&atilde;, cuius centr&utilde; grauitatis e$t K: re$idu&aelig; magnitu
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1069 dinis ex $olidis portionibus c&otilde;po$it&aelig; grauitatis c&etilde;tr&utilde; crit
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1070 in linea k l protracta, &amp; in puncto n; quod e$t ab$urd&utilde;. relin
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1071 quitur ergo, ut c&etilde;trum grauitatis cylindri; uel cylindri por
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1072 tionis $it punct&utilde; k. qu&aelig; omnia demon$tr&atilde;da propo$uimus.</P>
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1073 <P>At uero cylindrum, uel cylindri portion&etilde; ce
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1074 ad pri$ma, cuius ba$is e$t rectilinea figura in $pa-
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1075 cio g h de$cripta, &amp; altitudo &aelig;qualis; eandem ha-
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1076 <foot>bere</foot>
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1077 <p n=>15</p>
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1078 here proportionem, quam $pacium g h ad dict&atilde;
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1079 figuram, hoc modo demon$trabimus.</P>
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1080 <P>Intelligatur circulus, uel ellip$is x &aelig;qualis figur&aelig; rectili-
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1081 ne&aelig; in g h $pacio de$cript&aelig;. &amp; ab x con$tituatur conus, uel
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1082 <fig>
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1083 coni portio, altitudin&etilde; habens eand&etilde;, qu&atilde; cylindrus uel cy
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1084 lindri portio c e. Sit deinde rectilinea figura, in qua y eade,
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1085 qu&aelig; in $pacio g h de$cripta e$t: &amp; ab hac pyramis &aelig;qucalta
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1086 con$tituatur. Dico con&utilde; uel coni portione x pyramidi y &aelig;-
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1087 qual&etilde; e$$e. ni$i enim $it &aelig;qualis, uel maior, uel minor crit.</P>
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1088 <P>Sit primum maior, et exuperet $olido z. Itaque in circu
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1089 lo, uel ellip$i x de$cribatur figura rectilinea; &amp; in ea pyra-
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1090 mis candem, quam conus, uel coni portio altitudinem ha-
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1091 bens, ita ut portiones relict&aelig; minores $int $olido z, quem-
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1092 admodum docetur in duodecimo libro elementorum pro
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1093 po$itione undecima. erit pyramis x adhuc pyramide y ma
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1094 ior. &amp; quoniam piramides &aelig;que alt&aelig; inter $e $unt, $icuti ba
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1095 <marg>6. duo<*>
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1096 m.<*></marg>
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1097 $es; pyramis x ad piramidem y eandem proportionem ha-
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1098 bet, qu&agrave;m figura rectilinea x ad figuram y. Sed figura recti
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1099 <pb>
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1100 <fig>
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1101 linea x cum $it minor circulo, uel cllip$i, e$t etiam minor fi-
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1102 gura rectilinca y. ergo pyramis x pyramide y minor erit.
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1103 Sed &amp; maior; quod ficri n&otilde; pote$t. At $i conus, uel coni por
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1104 tio x ponatur minor pyramide y: $it alter conus &aelig;que al-
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1105 tus, uel altera coni portio X ip$i pyramidi y &aelig;qualis. crit
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1106 eius ba$is circulus, uel ellip$is maior circulo, uel ellip$i x,
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1107 quorum exce$$us $it $pacium <G>w.</G> Si igitur in circulo, uel eili-
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1108 p$i X figura rectilinea de$cribatur, ita ut portiones relict&aelig;
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1109 $int <G>w</G> $pacio minores, ciu$modi figura adhuc maior erit cir
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1110 culo, uel ellip$i x, hoc e$t figura rectilinea y. &amp; pyramis in
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1111 ca con$tituta minor cono, uel coni portione X, hoc e$t mi-
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1112 nor pyramide y. e$t ergo ut X figura rectilinea ad figuram
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1113 rectilineam y, ita pyramis X ad pyramidem y. quare cum
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1114 figura rectilinea X $it maior figura y: erit &amp; pyramis X py-
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1115 ramide y maior. $ed erat minor; quod rur$us fieri non po-
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1116 te$t. non e$t igitur conus, uel coni portio x neque maior,
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1117 neque minor pyramide y. ergo ip$i nece$$ario e$t &aelig;qualis.
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1118 Itaque quoniam ut conus ad conum, uel coni portio ad co
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1119 <foot>ni</foot>
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1120 <p n=>16</p>
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1121 <fig>
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1122 ni portionem, ita e$t cylindrus ad cylindrum, uel cylin-
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1123 dri portio ad cylindri portionem: &amp; ut pyramis ad pyra-
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1124 midem, ita pri$ma ad pri$ma, cum eadem $it ba$is, &amp; &aelig;qua
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1125 lis altitudo; crit cylindrus uel cylindri portio x pri$ma-
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1126 ti y &aelig;qualis. e$tq; ut $pacium g h ad $pacium x, ita cylin-
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1127 drus, uel cylindri portio c e ad cylindrum, uel cylindri por-
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1128 tionem x. Con$tat igitur cylindrum uel cylindri portion&etilde;
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1129 c e, ad pri$ma y, quippe cuius ba$is e$t figura rectilinea in
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1130 <marg>7. q<*></marg>
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1131 $pacio g h de$cripta, eandem proportionem habere, quam
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1132 $pacium g h habet ad $pacium x, hoc e$t ad dictam figuram.
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1133 quod demon$trandum fuerat.</P>
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1134 <head>THEOREMA IX. PROPOSITIO IX.</head>
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1135 <P>Si pyramis $ecetur plano ba$i &aelig;quidi$tante; $e-
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1136 ctio erit figura $imilis ei, qu&aelig; e$t ba$is, centrum
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1137 grauitatis in axe habens.</P>
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1138 <pb>
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1139 <P>SIT pyramis, cuius ba$is triangulum a b c; axis d c: &amp;
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1140 $ecetur plano ba$i &aelig;quidi$tante; quod $ection&etilde; $aciat f g h;
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1141 occurratq; axi in puncto <G>k.</G> Dico f g h triangulum e$$e, ip$i
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1142 a b c $imile; cuius grauitatis centrum e$t K. Quoni&atilde; enim
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1143 duo plana &aelig;quidi$tantia a b c, f g h $ecantur &agrave; plano a b d;
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1144 communes corum $ectiones a b, f g &aelig;quidi$tantes erunt: &amp;
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1145 cadem ratione &aelig;quidi$tantes ip$&aelig; b c, g h: &amp; c a, h f. Qu&ograve;d
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1146 cum du&aelig; line&aelig; f g, g h, duabus a b, b c &aelig;quidi$tent, nec
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1147 fintin eodem plano; angulus ad g &aelig;qualis e$t angulo ad
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1148 b. &amp; $imiliter angulus ad h angulo ad c: angulusq; ad f ci,
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1149 qui ad a e$t &aelig;qualis. triangulum igitur f g h $imile e$t tri-
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1150 angulo a b c. Atuero punctum <G>k</G> centrum e$$e grauita-
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1151 tis trianguli f g h hoc modo o$tendemus. Ducantur pla-
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1152 na per axem, &amp; per lineas d a, d b, d c: erunt communes $e-
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1153 ctiones f K, a e &aelig;quidi$tantes: pariterq; <G>k</G> g, e b; &amp; <G>k</G> h, e c:
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1154 quare angulus <G>k</G> f h angulo e a c; &amp; angulus <G>k</G> f g ip$i e a b
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1155 <fig>
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1156 e$t &aelig;qualis. Eadem ratione
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1157 anguli ad g angulis ad b: &amp;
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1158 anguli ad h iis, qui ad c &aelig;-
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1159 quales erunt. ergo puncta
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1160 e K in triangulis a b c, f g h
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1161 $imiliter $unt po$ita, per $e-
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1162 xtam po$itionem Archime-
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1163 dis in libro de centro graui-
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1164 tatis planorum. Sed cum e
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1165 $it centrum grauitatis trian
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1166 guli a b c, erit ex undecima
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1167 propo$itione eiu$dem libri,
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1168 &amp; K trianguli f g h grauita
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1169 tis centrum. id quod demon$trare oportebat. Non aliter
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1170 in ceteris pyramidibus, quod propo$itum e$t demon$tra-
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1171 <*></P>
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1172 <p n=>17</p>
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1173 <head>PROBLEMA I. PROPOSITIO X.</head>
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1174 <P>DATA qualibet pyramide, fieri pote$t, ut fi-
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1175 gura $olida in ip$a in $cribatur, &amp; altera circ&utilde;$cri-
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1176 batur ex pri$matibus &aelig;qualem altitudinem ha-
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1177 b&etilde;tibus, ita ut circum$cripta in$criptam excedat
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1178 magnitudine, qu&aelig; minor $it quac&utilde;que $olida ma
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1179 gnitudine propo$ita.</P>
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1180 <fig>
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1181 <P>Sit pyramis, cuius ba$is
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1182 triangul&utilde; a b c; axis d e.
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1183 Sitq; pri$ma, quod eand&etilde;
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1184 ba$im habeat, &amp; axem eun
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1185 dem. Itaque hoc pri$ma-
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1186 te continenter $ecto bifa-
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1187 riam, plano ba$i &aelig;quidi$t&atilde;
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1188 te, <*>relinquetur t&atilde;dem pri$
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1189 ma quoddam minus pro-
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1190 po$ita magnitudine: quod
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1191 quidem ba$im eandem ha
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1192 beat, quam pyramis, &amp; a-
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1193 xem e f. diuidatur d e in
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1194 partes &aelig;quales ip$i e f in
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1195 punctis g h <G>k</G> l m n: &amp; per
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1196 diui$iones plana duc&atilde;tur:
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1197 qu&aelig; ba$ibus &aelig;quidi$tent,
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1198 erunt $ectiones, triangula
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1199 ip$i a b c $imilia, ut proxi-
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1200 me o$tendimus. ab uno
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1201 quoque aut&etilde; horum trian
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1202 gulorum duo pri$mata c&otilde;
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1203 $truantur; unum quidem
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1204 ad partes e; alterum ad
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1205 <foot><*></foot>
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1206 <pb>
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1207 partes d. in pyramide igitur in$cripta erit qu&aelig;dam figura,
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1208 ex pri$matibus &aelig;qualem altitudinem habentibus c&otilde;$tans,
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1209 ad partes e: &amp; altera circum$cripta ad partes d. Sed unum-
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1210 quodque eorum pri$matum, qu&aelig; in figura in$cripta conti-
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1211 nentur, &aelig;quale e$t pri$mati, quod ab eodem fit triangulo in
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1212 figura circum&iacute;cripta: nam pri$ma p q pri$mati p o e$t &aelig;-
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1213 quale; pri$ma s t &aelig;quale pri$mati s r; pri$ma x y pri$mati
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diff changeset
1214 x u; pri$ma <G><*> q</G> pri$mati <G><*></G> z; pri$ma <G>m <*></G> pri$mati <G>m l;</G> pri$-
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diff changeset
1215 ma <G>r s</G> pri$mati <G>r p;</G> &amp; pri$ma <G>f x</G> pri$mati <G>f t</G> &aelig;quale. re-
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diff changeset
1216 linquitur ergo, ut circum$cripta figura exuperet in$cript&atilde;
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diff changeset
1217 pri$mate, quod ba$im habet a b c triangulum, &amp; axem e f.
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diff changeset
1218 Illud uero minus e$t $olida magnitudine propo$ita. Ead&etilde;
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diff changeset
1219 ratione in$cribetur, &amp; circum$cribetur $olida figura in py-
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diff changeset
1220 ramide, qu&aelig; quadrilateram, uel plurilater&atilde; ba$im habeat.</P>
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diff changeset
1221 <head>PROBLEMA II. PROPOSITIO XI.</head>
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1222 <P>DATO cono, fieri pote$t, ut figura $olida in-
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diff changeset
1223 $cribatur, &amp; altera circum$cribatur ex cylindris
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1224 &aelig;qualem habentibus altitudinem, ita ut circum-
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diff changeset
1225 $cripta $uperet in$criptam, magnitudine, qu&aelig; $o-
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diff changeset
1226 lida magnitudine propo$ita $it minor.</P>
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diff changeset
1227 <P>SIT conus, cuius axis b d: &amp; $ecetur plano per axem
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diff changeset
1228 ducto, 'ut$ectio $it triangulum a b c: intelligaturq; cylin-
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diff changeset
1229 drus, qui ba$im eandem, &amp; eundem axem habeat. Hocigi-
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diff changeset
1230 tur cylindro continenter bifariam $ecto, relinquetur cylin
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diff changeset
1231 drus minor $olida magnitudine propo$ita. Sit autem is cy
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diff changeset
1232 lindrus, qui ba$im habet circulum circa diametrum a c, &amp;
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diff changeset
1233 axem d e. Itaque diuidatur b d in partes &aelig;quales ip$i d e
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1234 in punctis f g h K l m: &amp; per ea ducantur plana conum $e-
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diff changeset
1235 cantia; qu&aelig; ba$i &aelig;quidi$tent. erunt $ectiones circuli, cen-
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diff changeset
1236 train axi habentes, ut in primo libro conicorum, propo$i-
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diff changeset
1237 <p n=>18</p>
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diff changeset
1238 tione quarta Apollonius demon$trauit. Si igitur &agrave; $ingu-
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diff changeset
1239 lis horum circulorum, duo cylindri fiant; unus quidem ad
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1240 ba$is partes; alter ad partes uerticis: in$cripta erit in co-
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diff changeset
1241 no $olida qu&aelig;dam figura, &amp; altera circum$cripta ex cylin-
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1242 dris &aelig;qualem altitudinem habentibus con$tans; quorum
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1243 <fig>
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diff changeset
1244 unu$qui$que, qui in
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diff changeset
1245 figura in$cripta con-
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1246 tinetur &aelig;qualis e$t ei,
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1247 qui ab eodem fit cir-
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1248 culo in figura circ&utilde;-
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1249 $cripta. Itaque cylin
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1250 drus o p &aelig;qualis e$t
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1251 cylindro o n; cylin-
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1252 drus r s cyl&itilde;dro r q;
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1253 cylindrus u x cylin-
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1254 dro u t e$t &aelig;qualis;
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1255 &amp; alii aliis $imiliter.
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1256 quare con$tat circ&utilde;-
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1257 $criptam figuram $u-
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1258 perare in$criptam cy
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1259 lindro, cuius ba$is e$t
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1260 circulus circa diametrum a c, &amp; axis d e. atque hic e$t mi-
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1261 nor $olida magnitudine propo$ita.</P>
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1262 <head>PROBLEMA III. PROPOSITIO XII.</head>
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1263 <P>DATA coni portione, pote$t $olida qu&aelig;dam
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1264 figura in$cribi, &amp; altera circum$cribi ex cylindri
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1265 portionibus &aelig;qualem altitudinem habentibus;
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parents:
diff changeset
1266 ita ut circum$cripta in$criptam exuperet, magni
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1267 tudine, qu&aelig; minor fit $olida magnitudine pro-
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1268 po$ita.</P>
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1269 <foot>E 2</foot>
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1270 <pb>
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1271 <P>Figuram ciu$modi, &amp; in$cribemus, &amp; circ&utilde;$cribemus, ita
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1272 ut in cono dictum e$t.</P>
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1273 <fig>
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1274 <head>PROBLEMA IIII. PROPOSITIO XIII.</head>
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1275 <P>DATA $ph&aelig;r&aelig; portione, qu&aelig; dimidia $ph&aelig;-
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1276 ra maior non $it, pote$t $olida qu&aelig;dam portio in-
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diff changeset
1277 $cribi &amp; altera circum$cribi ex cylindris &aelig;qualem
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1278 altitudinem habentibus, ita ut circum$cripta in-
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1279 $criptam excedat magnitudine, qu&aelig; $olida ma-
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parents:
diff changeset
1280 gnitudine propo$ita $it minor.</P>
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1281 <P>HOC etiam codem pror$us modo $iet: atque ut ab
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1282 Archimedc traditum e$t in conoidum, &amp; $ph&aelig;roidum por
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1283 tionibus, propo$itione ulge$imaprima libri de conoidi-
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parents:
diff changeset
1284 bus, &amp; $ph&aelig;roidibus.</P>
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1285 <foot>THEO</foot>
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1286 <p n=>19</p>
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1287 <fig>
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1288 <head>THEOREMA X. PROPOSITIO XIIII.</head>
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1289 <P>Cuiuslibet pyramidis, &amp; cuiuslibet coni, uel
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1290 coni portionis, centrum grauitatis in axe c&otilde;$i$tit.</P>
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1291 <P>SIT pyramis, cuius ba$is triangulum a b c: &amp; axis d e.
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1292 Dico in linea d e ip$ius grauitatis centrum ine$$e. Si enim
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diff changeset
1293 fieri pote$t, $it centrum f: &amp; ab f ducatur ad ba$im pyrami
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1294 dis linea f g, axi &aelig;quidi$tans: iunctaq; e g ad latera trian-
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1295 guli a b c producatur in h. quam uero proportionem ha-
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1296 bet linea h e ad e g, habeat pyramis ad aliud $olidum, in
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1297 quo K: in$cribaturq; in pyramide $olida figura, &amp; altera cir
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1298 cum$cribatur ex pri$matibus &aelig;qualem habentibus altitu-
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1299 dinem, ita ut circum$cripta in$criptam exuperet magnitu-
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1300 dine, qu&aelig; $olido k $it minor. Et quoniam in pyramide pla
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1301 num ba$i &aelig;quidi$tans ductum $ectionem facit figuram $i-
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1302 milem ei, qu&aelig; e$t ba$is; centrumq; grauitatis in axe haben
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1303 tem: erit pri$matis s t grauitatis centr&utilde; in linear q; pri$-
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1304 matis u x centrum in linea q p, pri$inatis y z in linea p o;
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1305 pri$matis <G>h q</G> in linea o n; pri$matis <G>l <*></G> in linea n m; pri$-
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1306 matis <G>n p</G> in m l; &amp; denique pri$matis <G>p s</G> in l e. quare to-
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1307 <pb>
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1308 da $igura, &amp; altera circum$cribatur ex cylindris, uel cylin-
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1309 dri portionibus, $icuti dictum e$t, ita ut exce$$us, quo $igu-
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1310 ra circum$cripta in$criptam $uperat, $it $olido g minor.
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1311 Itaque centrum grauitatis cylindri, uel cylindri portionit
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1312 q r e$t in linea p o; cylindri, uel cylindri portionis s t cen-
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1313 trum in linea o n; centrum u x in linea n m; y z in m b; <G>n q</G>
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1314 in l k; <G>l m</G> in K h; &amp; denique <G>v p</G> centrum in h d. ergo figu-
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1315 <fig>
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1316 r&aelig; in$cript&aelig; centrum e$t in linea p d. Sit autem <G>r</G>: &amp; iun-
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1317 cta <G>r</G> e protendatur, ut cum linea, qu&aelig; &agrave; p&utilde;cto c ducta $ue-
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1318 rit axi &aelig;quidi$tans, conueniat in <G>s.</G> erit <G>s r</G> ad <G>r</G> e, ut c d
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1319 ad d f: &amp; conus, $eu coni portio ad exce$$um, quo circum-
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1320 $cripta figura in$criptam $uperat, habebit maiorem pro-
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1321 portionem, qu&agrave;m <G>t r</G> ad <G>r</G> e. ergo ad partem exce$$us, qu&aelig;
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1322 intra ip$ius $uperficiem comprehenditur, multo m aiorem
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1323 proportionem habebit. habeat eam, quam <G>t r</G> ad <G>r</G> e. erit
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1324 <foot>diuidendo</foot>
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1325 <p n=>21</p>
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1326 diuidendo figura $olida in$cripta ad dictam exce$$us par-
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1327 tem, ut <G>t e</G> ad c <G>p.</G> &amp; quoniam &agrave; cono, $cu coni portione,
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1328 cuius grauitatis centrum e$t e, au$ertur figura in$cripta,
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1329 cuius centrum <G>r:</G> re$idu&aelig; magnitudinis compo$it&aelig; cx par
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1330 te exce$$us, qu&aelig; intra coni, uel coni portionis $uper$iciem
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1331 continetur, centrum grauitatis erit in linea e protracta,
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1332 atque in puncto t. quod c$t ab$urdum. c&otilde;$tat ergo centr&utilde;
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1333 grauitatis coni, uel coni portionis, e$$e in axc b d: quod de
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1334 mon$trandum propo$uimus.</P>
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1335 <head>THEOREMA XI. PROPOSITIO XV.</head>
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1336 <P>Cuiuslibet portionis $ph&aelig;r&aelig; uel $ph&aelig;roidis,
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1337 qu&aelig; dimidia maior non $it:item&qacute;; cuiuslibet por
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1338 tionis conoidis, uel ab$ci$$&aelig; plano ad axem recto,
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1339 uel non recto, centrum grauitatis in axe con-
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1340 $i$tit.</P>
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1341 <P>Demon$tratio $imilis erit ei, quam $upra in cono, uel co
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1342 ni portione attulimus, ne toties eadem fru$tra iterentur.</P>
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1343 <fig>
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1344 <foot>F</foot>
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1345 <pb>
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1346 <head>THEOREMA XII. PROPOSITIO XVI.</head>
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1347 <P>In $ph&aelig;ra, &amp; $ph&aelig;roidc idem e$t grauitatis, &amp;
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1348 figur&aelig; centrum.</P>
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1349 <P>Secetur $ph&aelig;ra, uel $ph&aelig;roides piano per axem ducto;
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1350 quod $ectionem $aciat circulum, uel cllip$im a b c d, cuius
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1351 diameter, &amp; $ph&aelig;r&aelig;, uel $ph&aelig;roidis axis d b; &amp; centrum e.
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1352 Dico e grauitatis etiam centrum e$$e. $ecetur enim altero
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1353 plano per e, ad planum $ecans recto, cuius $ectio $it circu-
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1354 lus cir ca diametrum a &ccedil;. erunt a d c, a b c dimidi&aelig; portio-
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1355 nes $ph&aelig;r&aelig;, uel $ph&aelig;roidis. &amp; quoniam portionis a d c gra
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1356 uitatis centrum e$i in linead, &amp; centrum portionis a b c in
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1357 ip$a b c; totius $ph&aelig;r&aelig;, uel $ph&aelig;roidis grauitatis centrum
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1358 in a<*> c d b con$i$iet, Qu&ograve;d $i portionis a d c centrum grau&itilde;
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1359 tatis ponatur e$$c f &amp; fiatip$i f e &aelig;qualis e g. punct&utilde; g por
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1360 <fig>
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1361 <marg>per 2. pe-
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1362 tition<*>m</marg>
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1363 tionis a b c centrum crit. $olidis cnim figuris $imilibus &amp;
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1364 &aelig;q<*> alibus inter $e aptatis, &amp; c. ntra g. auitatis ip$arum in-
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1365 <marg>4 Arcl -
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1366 medis.</marg>
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1367 ter <*>e aptentur nece$le e$t. ex quo fit, ut magnitudinis, qu&aelig;
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1368 ex utilique c&otilde;$lat, hoc e$t ip$ius $ph&aelig;r&aelig;, uel $ph&aelig;roidis gra
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1369 uitatis centrum $it in medio linc&aelig; f g uid li<*> t in c. Sph&aelig;-
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1370 r&aelig; igitur, uel $ph&aelig;roidis grauitatis centrum e$tidem, quod
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1371 centrum figur&aelig;.</P>
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1372 <foot>Ex</foot>
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1373 <p n=>22</p>
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1374 <P>Ex demon$tratis per$picue apparet, portioni
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1375 $ph&aelig;r&aelig; uel $ph&aelig;roidis, qu&aelig; dimidia maier e$t, c&etilde;
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1376 trum grauitatis in axe con$i$tere.</P>
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1377 <fig>
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1378 <P>Data cnim
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1379 qualibet maio
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1380 ri porti&otilde;e, quo
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1381 ni&atilde; totius $ph&aelig;
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1382 r&aelig;, uel $ph&aelig;roi
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1383 dis grauitatis
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1384 centrum e$t in
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1385 axe; e$t autem
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1386 &amp; in axe cen-
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1387 trum portio-
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1388 nis minoris:
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1389 reliqu&aelig; portionis uidelicet maioris centrum in axe nece$-
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1390 $ario con$i$tet.</P>
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1391 <head>THEOREMA XIII. PROPOSITIO XVII.</head>
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1392 <fig>
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1393 <P>Cuiuslibet pyramidis tri&atilde;
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1394 gularem ba$im hab&etilde;tis gra
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1395 uitatis centrum e$t in pun-
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1396 cto, in quo ip$ius axes con-
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1397 ueniunt.</P>
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1398 <P>Sit pyramis, cuius ba$is trian
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1399 gulum a b c, axis d e: $itq; trian
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1400 guli b d c grauitatis centrum f:
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1401 &amp; iungatur a $. crit &amp; a faxis eiu$
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1402 dem pyramidis ex tertia diffini-
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1403 tione huius. Itaque quoniam centrum grauitatis e$t in
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1404 axe d e; e$t autem &amp; in axe a f; &qgrave;uod proxime demon$traui
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1405 <pb>
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1406 mus: erit utique grauitatis centrum pyramidis punctum
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1407 g. in quo $cilicet ip$i axes conueniunt.</P>
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1408 <head>THEOREMA XIIII. PROPOSITIO XVIII.</head>
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1409 <P>SI $olidum parallelepipedum $ecetur plano
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1410 ba$ibus &aelig;quidi$tante; erir $olidum ad $olidum,
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1411 $icut altitudo ad altitudinem, uel $icut axis ad
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1412 axem.</P>
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1413 <fig>
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1414 <P>Sit $olidum parallelepipe
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1415 dum a b c d e f g h, cuius axis
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1416 k l: $eceturq; plano ba$ibus
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1417 &aelig;quidi$tante, quod faciat
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1418 $ectionem m n o p; &amp; axi in
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1419 &ptilde;uncto q occurrat. Dico
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1420 $olidum g m ad $olidum m c
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1421 eam proportionem habere,
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1422 quam altitudo $olidi g m ha-
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1423 bet ad $olidi m c altitudi-
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1424 nem; uel quam axis <G>k</G> q ad
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1425 axem q l. Si enim axis K l ad
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1426 ba$is planum $it perpendicu
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1427 laris, &amp; linea g c, qu&aelig; ex quin
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1428 ta huius ip$i k l &aelig;quidi$tat,
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1429 perpendicularis erit ad id&etilde;
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1430 planum, &amp; $olidi altitudi-
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1431 <marg><*> undeci
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1432 <*>.</marg>
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1433 nem dimetietur. Itaque $o-
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1434 lidum gm ad $olidum m c
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1435 eam proportionem habet,
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1436 quam parallelogram n&utilde; g n
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1437 ad parallelo grammum n c,
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1438 <marg>$ext&itilde;.</marg>
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1439 hoce$t quam linea g o, qu&aelig;
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1440 <foot>e$t</foot>
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1441 <p n=>23</p>
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1442 e$t $olidi g m altitudo ad o e altitudinem $olidi m c, uel qu&atilde;
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1443 axis k q ad q l axem. Si uero axis k l non $it perpendicularis
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1444 ad planum ba$is; ducatur a puncto <G>k</G> ad idem planum per
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1445 pendicularis <G>k</G> r, occurr&etilde;s plano m n o p in s. $imiliter de-
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1446 m&otilde;$trabimus $olidum g m ad $olid&utilde; m c ita e$$e, ut axis k q
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1447 ad axem q l. Sed ut K q ad q l, ita <G>k</G> s altitudo ad altitudi-
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1448 <marg>17.<*>
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1449 cin<*></marg>
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1450 nem s r; nam line&aelig; K l, K r &agrave; planis &aelig;quidi$tantibus in ea$-
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1451 dem proportiones $ecantur. ergo $olidum g m ad $olidum
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1452 m c eand&etilde; proportionem habet, quam altitudo ad altitu||
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1453 din&etilde;, uel quam axis ad axem. quod dem&otilde;$trare oportebat.</P>
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1454 <head>THEOREMA XV. PROPOSITIO XIX.</head>
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1455 <P>Solida parallelepipeda in eadem ba$i, uel in
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1456 &aelig;qualibus ba$ibus con$tituta eam inter $e propor
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1457 tionem habent, quam altitudines: &amp; $i axes ip$o-
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1458 rum cum ba$ibus &aelig;quales angulos contineant,
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1459 eam quoque, quam axes proportionem habeb&utilde;t.</P>
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1460 <P>Sint $olida parallelepipeda in ead&etilde; ba$i c&otilde;$tituta a b c d,
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1461 a b e f: &amp; $it $olidi a b c d altitudo minor: producatur au-
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1462 tem planum c d adeo, ut$olidum a b e f $ecet; cuius $ectio
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1463 <fig>
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1464 <marg>29.1<*>
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1465 cim<*></marg>
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1466 $it gh. cr&utilde;t $oli
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1467 da a b c d, a b g h
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1468 in eadem ba$i,
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1469 &amp; &aelig;quali altitu||
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1470 dine inter $e &aelig;-
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1471 qualia. Quoni&atilde;
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1472 igitur $olidum
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1473 a b e f $ecatur
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1474 plano ba$ibus
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1475 &aelig;quidi$t&atilde;te, erit
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1476 <marg>18.<*></marg>
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1477 $olidum g h e f
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1478 adip$um a b g h
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1479 <pb>
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1480 ut altitudo ad altitudinem: &amp; componendo conuertendo
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1481 <marg>7. quinti.</marg>
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1482 que $olidum a b g h, hoc e$t $olidum a b c d ip$i &aelig;quale, ad-
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1483 $olidum a b e f, ut altitudo $olidi a b c d ad $olidi a b e f al-
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1484 titudinem.</P>
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1485 <P>Sint $olida parallelcpipeda a b, c d in &aelig;qualibus ba$ibus
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1486 con$tituta: $itq; b e altitudo $olidi a b: &amp; $olidi c d altitudo
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1487 d f; qu&aelig; quidem maior $it, qu&agrave;m b e. Dico $olidum a b ad
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1488 $olidum c d eandem habere proportionem, quam b e ad
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1489 d f. ab$cindatur enim &agrave; linea d f &aelig;qualis ip$i b e, qu&aelig; $it g f:
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1490 &amp; per g ducatur planum $ecans $olidum c d; quod ba$ibus
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1491 &aelig;quidi$tet, faciatq; $ection&etilde; h. K. crunt $olida a b, c <G>k</G> &aelig;que
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1492 <marg>31. unde
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1493 cimi</marg>
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1494 <fig>
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1495 alta inter
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1496 $e &aelig;qualia
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1497 c&utilde; &aelig;qua-
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1498 les ba$es
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1499 habeant.
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1500 <marg>18. huius</marg>
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1501 Sed $olid&utilde;
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1502 h d ad $oli
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1503 dum c K
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1504 e$t, ut alti
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1505 tudo d g
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1506 ad g falti-
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1507 tudin&etilde;; $e
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1508 catur enim $olidum c d plano ba$i
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1509 <fig>
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1510 bus &aelig;quidi$tante: &amp; rur$us c&otilde;po-
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1511 nende, conuertendoq; $olid&utilde; c k
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1512 <marg>7. quinti.</marg>
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1513 ad $olidum c d, ut g f ad f d. ergo
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1514 $olidum a b, quod e$t &aelig;quale ip$i
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1515 c <G>k</G> ad $olidum c d eam proportio
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1516 nem habet, quam altitudo g f, hoc
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1517 e$t b e ad d f altitudinem.</P>
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1518 <P>Sint deinde $olida parallelepipe
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1519 da a b, a c in cadem ba$i; quorum
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1520 axes d e, $ e cum ip$a &aelig;quales angu
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1521 <foot>los</foot>
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1522 <p n=>24</p>
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1523 los contineant. Dico $olidum a b ad $olidum a c e idem ha
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1524 bere proportionem, quam axis d e ad axem e f. Si enini
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1525 axes in eadem recta linea fuerint con$tituti, b&aelig;c dao lo'i-
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diff changeset
1526 da, in unum, atque idem $olidum conuenient. quare <*>x
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1527 iis, qu&aelig; proxime tradita $unt, habebit $olidum a b ad $o-
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diff changeset
1528 lidum a c candem proportionem, quam axis d e ad e f
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1529 axem. Si uero axes non $int in eadem recta linea, demittan
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diff changeset
1530 tur a punctis d, $ perpendiculares ad ba$is planum, d g, fh:
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1531 &amp; jungantur e g, e h. Quoniam igitur axes cum ba$ibus
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1532 &aelig;quales angulos continent, erit d e g angulus &aelig;qualis an-
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1533 <fig>
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diff changeset
1534 gulo f e h: &amp; $unt
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1535 anguli ad g h re-
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1536 cti, quare &amp; re-
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1537 liquus e d g &aelig;qua
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1538 lis erit reliquo
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1539 e fh: &amp; triangu-
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1540 lum d e g tri&atilde;gu-
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1541 lo f e h $imile. er-
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1542 go g d ad d e e$t',
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1543 ut h f ad fe: &amp; per
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1544 mutando g d ad
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1545 h f, ut d e ad c f.
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1546 Sed $olidum a b
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1547 ad $olidum a c
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1548 candem propor-
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1549 tionem habet,
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1550 quam d g altitu-
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1551 do ad altitudin&etilde;
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1552 f h. ergo &amp; ean-
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1553 d&etilde; habebit, qu&atilde;
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1554 axis d e a l e <*> ax&etilde;</P>
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1555 <P>Po$tremo $int
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1556 $olidi paral le pi
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parents:
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1557 peda a b, c d in
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1558 <pb>
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1559 &aelig;qualibus ba$ibus, quorum axes cum ba$ibus &aelig;quales an
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1560 gulos faciant. Dico $olidum a b ad$olid&utilde; c d ita e$$e, ut axis
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1561 e f ad axem g h: nam $i axes ad planum ba$is recti $int, il-
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parents:
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1562 lud per$picue con$tat: quoniam eadem linea, &amp; axem &amp; $oli
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1563 di altitudinem determinabit. Si uero $int inclinati, &agrave; pun-
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1564 ctis e g ad $ubiectum planum perpendiculares ducantur
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1565 e k, g l: &amp; iungantur fk, h l. rur$us quoniam axes cum ba
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1566 $ibus &aelig;quales faciunt angulos, eodem modo demon$trabi
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1567 tur, triangulum e f K triangulo g h l $imile e$$e: &amp; e <G>k</G> ad g l,
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1568 ut e f ad g h. Solidum autem a b ad $olidum c d e$t, ut
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1569 e K ad g l. ergo &amp; ut axis e f ad axem g h. qu&aelig; omnia de
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1570 mon$trarc oportebat.</P>
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1571 <P>Ex iis qu&aelig; demon$trata $unt, facile con$tare
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1572 pote$t, pri$mata omnia &amp; pyramides, qu&aelig; trian-
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1573 gulares ba$es habent, $iue in ei$dem, $iue in &aelig;qua
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1574 <marg>15. quinti</marg>
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1575 libus ba$ibus con$tituantur, eandem proportio-
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1576 nem habere, quam altitudines: &amp; $i axes cum ba
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parents:
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1577 $ibus &aelig;quales angulos contineant, $imiliter ean-
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1578 dem, quam axes, habere proportionem: $unt
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parents:
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1579 <marg>28. unde-
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parents:
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1580 cimi.</marg>
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parents:
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1581 enim $olida parallelepipeda pri$matum triangula
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parents:
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1582 <marg>7. duode-
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1583 cimi.</marg>
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1584 res ba$es habenti&utilde; dupla; &amp; pyramidum $extupla.</P>
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1585 <head>THEOREMA XVI. PROPOSITIO XX.</head>
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1586 <P>Pri$mata omnia &amp; pyramides, qu&aelig; in ei$dem,
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1587 uel &aelig;qualibus ba$ibus con$tituuntur, eam inter
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parents:
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1588 $e proportionem habent, quam altitudines: &amp; $i
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1589 axes cum ba$ibus faciant angulos &aelig;quales, eam
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1590 etiam, quam axes habent proportionem.</P>
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1591 <foot>Sint</foot>
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1592 <p n=>25</p>
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1593 <P>Sint duo pri$mata a e, a f, quorum eadem ba$is quadri-
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1594 latera a b c d: $itq; pri$matis a e altitudo e g; &amp; pri$matis
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1595 a f altitudo f h. Dico pri$ma a e ad pri$ma a f eam habere
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1596 proportionem, quam e g ad f h. iungatur enim a c: &amp; in
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1597 unoquoque pri$mate duo pri$mata intelligantur, quorum
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1598 <fig>
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1599 ba$es $int triangu
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1600 la a b c, a c d. habe
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1601 bunt duo pri$ma-
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1602 te in eadem ba$i
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1603 a b c con$tituta,
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1604 proportionem e&atilde;
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1605 dem, quam ip$o-
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1606 rum altitudines e
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1607 g, f h, ex iam de-
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1608 mon$tratis. &amp; $i-
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1609 militer alia duo,
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1610 qu&aelig; $unt in ba$i a
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1611 <marg><*>2. qu<*></marg>
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1612 c d. quare totum pri$ma a e ad pri$ma a f eandem propor
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1613 tionem habebit, quam altitudo e g ad f h altitudinem.
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1614 Qu&ograve;d cum pri$mata $int pyramidum tripla, &amp; ip$&aelig; pyrami
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1615 des, quarum eadem e$t ba$is quadrilatera, &amp; altitudo pri$-
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1616 matum altitudini &aelig;qualis, eam inter $e proportionem ha-
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1617 bebunt, quam altitudines.</P>
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1618 <P>Si uero pri$mata ba$es &aelig;quales habeant, n&otilde; ea$dem, $int
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1619 duo eiu$modi pri$mata a e, f l: &amp; $it ba$is pri$matis a e qua
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diff changeset
1620 drilaterum a b c d; &amp; pri$matis f l quadrilaterum f g h k.
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1621 Dico pri$ma a e ad pri$ma f l ita e$$e, ut altitudo illius ad
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1622 huius altitudinem. nam $i altitudo $it eadem, intellig&atilde;tur
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1623 <marg>6. du<*>
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1624 cimi<*></marg>
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1625 du&aelig; pyramides a b c d e, f g h k l. qu&aelig; &itilde;tcr$e &aelig;quales er&utilde;t,
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1626 cum &aelig;quales ba$es, &amp; altitudinem eandem habeant. quare
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1627 <marg>15. qu<*></marg>
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1628 &amp; pri$mata a e, f l, qu&aelig; $unt har&utilde; pyramidum tripla, &aelig;qua-
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1629 lia $int nece$$e e$t. ex quibus per$picue con$tat propo$it&utilde;.
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1630 Si uero altitudo pri$matis f l $it maior, &agrave; pri$mate f l ab-
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diff changeset
1631 $cindatur pri$ma fm, quod &aelig;que altum $it, atq; ip$um a e.
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1632 <foot>G</foot>
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1633 <pb>
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1634 <fig>
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1635 erunt e&aelig;dem ra-
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1636 tione pri$mata a
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1637 e, f m inter $e &aelig;-
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1638 qualia. quare $i-
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1639 militer demon-
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1640 $trabitur pri$ma
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1641 f m ad pri$ma f l
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1642 eandem habere
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1643 proportionem,
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1644 quam pri$matis
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1645 f m altitudo ad
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1646 altitudinem ip-
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1647 $ius f l. ergo &amp; pri$ma a e ad pri$ma f l eandem propor-
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1648 tionem habebit, quam altitudo ad altitudinem. $equitur
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1649 igitur ut &amp; pyramides, qu&aelig; in &aelig;qualibus ba$ibus con$titu&utilde;
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1650 tur, candem inter $e $e, quam altitudines, proportionem
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1651 habeant.</P>
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1652 <fig>
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1653 <P>Sint deinde pri$mata a e, a f in eadem ba$i a b c d; quor&utilde;
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1654 axes cum ba$ibus &aelig;quales angulos contineant: &amp; $it pri$-
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1655 <foot>matis</foot>
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1656 <p n=>26</p>
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1657 matis a e axis g h; &amp; pri$matis a f axis l h. Dico pri$ma
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1658 a e ad pri$ma a f eam proportionem habere, quam g h ad
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1659 h l. ducantur &agrave; punctis g l perpendiculares ad ba$is pla-
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1660 <fig>
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1661 num g K, l m: &amp; iungantur k h,
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1662 h m. Itaque quoniam anguli g h
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1663 k, l h m $unt &aelig;quales, $imiliter ut
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1664 $upra demon$trabimus, triangu-
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1665 la g h K, l h m $imilia e$$e; &amp; ut g
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1666 K ad l m, ita g h ad h l. habet au
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1667 tem pri$ma a e ad pri$ma a f ean
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1668 dem proportionem, quam altitu||
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1669 do g K ad altitudinem l m, $icuti
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1670 demon$tratum e$t. ergo &amp; can-
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1671 dem habebit, quam g h, ad h l. py
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1672 ramis igitur a b c d g ad pyrami-
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1673 dem a b c d l eandem proportio-
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1674 nem habebit, quam axis g h ad h l axem.</P>
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1675 <fig>
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1676 <P>Denique $int pri$mata a e, <G>k</G> o in &aelig;qualibus ba$ibus a b
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1677 c d, k l m n con$tituta; quorum axes cum ba$ibus &aelig;quales
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1678 faciant angulos: $itq; pri$matis a e axis f g, &amp; altitudo f h:
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1679 pri$matis autem k o axis p q, &amp; altitudo p r. Dico pri$ma
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1680 a e ad pri$ma <G>k</G> o ita e$$e, ut f g ad p q. iunctis enim g h,
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1681 <foot>G 2</foot>
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1682 <pb>
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1683 q r, eodem, quo $upra, modo o$tendemns f g ad p q, ut f h
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1684 ad p r. $ed pri$ma a e ad ip$um <G>k</G> o e$t, ut f h ad p r. ergo
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1685 &amp; ut f g axis ad axem p q. ex quibus $it, ut pyramis a b c d f
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1686 <fig>
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1687 ad pyrami-
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1688 d&etilde; <G>k</G> l m n p
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1689 eandem ha
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1690 beat pro -
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1691 portion&etilde;,
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1692 qu&atilde; axis ad
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1693 ax&etilde;. quod
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1694 demon$tr&atilde;
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1695 d&utilde; $uerat.</P>
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1696 <P>Simili ra
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1697 tione in a-
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1698 liis pri$ma-
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1699 tibus &amp; py
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1700 ramidibus eadem demon$trabuntur.</P>
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1701 <head>THEOREMA XVII. PROPOSITIO XXI.</head>
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1702 <P>Pri$mata omnia, &amp; pyramides inter $e propor
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1703 tionem habent compo$itam ex proportione ba-
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1704 $ium, &amp; proportione altitudinum.</P>
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1705 <P>Sint duo pri$mata a e, g m: $itq; pri$matis a e ba$is qua
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1706 drilaterum a b c d, &amp; altitudo e f: pri$matis uero g m ba-
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diff changeset
1707 $is quadrilaterum g h K l, &amp; altitudo m n. Dico pri$ma a e
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1708 ad pri$ma g m proportionem habere compo$itam ex pro
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1709 portione ba$is a b c d ad ba$im g h k l, &amp; ex proportione
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1710 altitudinis e f, ad altitudinem m n.</P>
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1711 <P>Sint enim primum e f, m n &aelig;quales: &amp; ut ba$is a b c d
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1712 ad ba$im g h <G>k</G> l, ita fiat linea, in qua o ad lineam, in qua p:
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1713 ut autem e f ad m n, ita linea p ad lineam q. erunt line&aelig;
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1714 p q inter $e &aelig;quales. Itaque pri$ma a e ad pri$ma g m e&atilde;
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1715 <foot>pro</foot>
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1716 <p n=>27</p>
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1717 proportionem habet, quam ba$is a b c d ad ba$im g h <G>k</G> l:
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1718 $i enim intelligantur du&aelig; pyramides a b c d e, g h k l m, ha-
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1719 bebunt h&aelig; inter $e proportionem eandem, quam ip$ar um
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1720 ba$es ex $exta duodecimi elementorum. Sed ut ba$is a b c d
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1721 ad g h K l ba$im, ita linea o ad lineam p; hoc e$t ad lineam q
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1722 ei &aelig;qualem. ergo pri$ma a e ad pri$ma g m e$t, ut linea o
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1723 ad lineam q. proportio autem o ad q copo$ita e$t ex pro-
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1724 portione o ad p, &amp; ex proportione p ad q. quare pri$ma
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1725 a e ad pri$ma g m, &amp; idcirco pyramis a b c d e, ad pyrami-
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1726 dem g h K l m proportionem habet ex ei$dem proportio-
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1727 nibus compo$itam, uidelicet ex proportione ba$is a b c d
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1728 ad ba$im g h K l, &amp; ex proportione altitudinis e f ad m n al
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diff changeset
1729 titudinem. Qu&ograve;d $i line&aelig; e f, m n in&aelig;quales ponantur, $it
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1730 e f minor: &amp; ut e f ad m n, ita fiat linea p ad lineam u: de
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1731 <fig>
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1732 inde ab ip$a m n ab$cindatur r n &aelig;qualis e f: &amp; per r duca-
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1733 tur planum, quod oppo$itis planis &aelig;quidi$tans faciat $e-
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diff changeset
1734 ctionem s t. erit pri$ma a e, ad pri$ma g t, ut ba$is a b c d
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1735 ad ba$im g h k l; hoc e$t ut o ad p: ut autem pri$ma g t ad
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1736 <marg>20. h<*></marg>
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1737 pri$ma g m, ita altitudo r n; hoc e$te f ad altitudine m n;
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1738 uidelicet linea p ad lineam u. ergo ex &aelig;quali pri$ma a e ad
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1739 pri$ma g m e$t, ut linea o ad ip$am u. Sed proportio o ad
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1740 u c&otilde;po$ita e$t ex proportione o ad p, qu&aelig; e$t ba$is a b c d
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1741 ad ba$im g h <G>k</G> l; &amp; ex proportione p ad u, qu&aelig; e$t altitudi-
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1742 nis e f ad altitudinem m n. pri$ma igitur a e ad pri$ma g m
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1743 <foot><*></foot>
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1744 <pb>
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1745 compo$itam proportionem habet ex proportione ba$i&utilde;,
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1746 &amp; proportione altitudinum. Quare &amp; pyramis, cuius ba-
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1747 $is e$t quadrilaterum a b c d, &amp; altitudo e f ad pyramidem,
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1748 <fig>
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1749 cuius ba$is quadrilaterum g h K l, &amp; altitudo m n, compo$i
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1750 tam habet proportionem ex proportione ba$ium a b c d,
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1751 g h k l, &amp; ex proportione altitudinum e f, m n. quod qui-
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1752 dem demon$tra$$e oportebat.</P>
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1753 <P>Ex iam demon$tratis per$picuum e$t, pri$ma
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1754 ta omnia, &amp; pyramides, in quibus axes cum ba$i-
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1755 bus &aelig;quales angulos continent, proportionem
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1756 habere compo$itam ex ba$ium proportione, &amp;
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1757 proportione axium. demon$ttatum e$t enim, a-
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1758 xes inter $e eandem proportionem habere, quam
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1759 ip$&aelig; altitudines.</P>
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1760 <head>THEOREMA XVIII. PROPOSITIO XXII.</head>
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1761 <P>CVIVSLIBEt pyramidis, &amp; cuiuslibet coni,
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1762 <foot>uel</foot>
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1763 <p n=>28</p>
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1764 uel coni portionis axis &agrave; centro grauitatis ita diui
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1765 ditur, ut pars, qu&aelig; terminatur ad uerticem reli-
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1766 qu&aelig; partis, qu&aelig; ad ba$im, $it tripla.</P>
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1767 <P>Sit pyramis, cuius ba$is triangulum a b c; axis d e; &amp; gra
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1768 uitatis centrum K. Dico lineam d k ip$ius K e triplam e$$e.
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1769 trianguli enim b d c centrum grauitatis $it punctum f; tri&atilde;
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1770 guli a d c centr&utilde; g; &amp; trianguli a d b $it h: &amp; iungantur a f,
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1771 b g, c h. Quoniam igitur centr&utilde; grauitatis pyramidis in axe
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1772 <marg>17<*></marg>
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1773 c&otilde;$i$tit: $untq; d e, a f, b g, c h eiu$d&etilde; pyramidis axes: conue
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1774 nient omnes in id&etilde; punct&utilde; k, quod e$t grauitatis centrum.
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1775 Itaque animo concipiamus hanc pyramidem diui$am in
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1776 quatuor pyramides, quarum ba$es $int ip$a pyramidis
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1777 <marg><I>ucrfex</I></marg>
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1778 <fig>
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1779 triangula; &amp; <U>axis</U> pun-
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1780 ctum k qu&aelig; quidem py-
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1781 ramides inter $e &aelig;quales
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1782 $unt, ut dem&otilde;$trabitur.
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1783 Ducatur en&itilde; per lineas
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1784 d c, d e planum $ec&atilde;s, ut
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1785 $it ip$ius, &amp; ba$is a b c c&otilde;
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1786 munis $ectio recta linea
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1787 c e l: eiu$d&etilde; uero &amp; tri&atilde;-
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1788 guli a d b $it linea d h l.
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1789 erit linea al &aelig;qualis ip$i
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1790 l b: nam centrum graui-
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1791 tatis trianguli con$i$tit
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1792 in linea, qu&aelig; ab angulo
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1793 ad dimidiam ba$im per-
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1794 ducitur, ex tertia deci-
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1795 ma Archimedis. quare
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1796 <marg><*></marg>
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1797 triangulum a c l &aelig;quale
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1798 e$t triangulo b c l: &amp; propterea pyramis, cuius ba$is trian-
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1799 gulum a c l, uer tex d, e$t &aelig;qualis pyramidi, cuius ba$is b c l
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1800 <marg>5. d<*>
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1801 cim<*></marg>
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1802 triangulum, &amp; idem uertex. pyramides enim, qu&aelig; ab eod&etilde;
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1803 <pb>
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1804 $unt uertice, eandem proportionem habent, quam ip$ar&utilde;
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1805 ba$es. eadem ratione pyramis a c l k pyramidi b c l <G>k:</G> &amp; py
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1806 ramis a d l k ip$i b d l <G>k</G> pyramidi &aelig;qualis erit. Itaque $i a py
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1807 ramide a c l d auferantur pyramides a c l k, a d l k: &amp; &agrave; pyra
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1808 mide b c l d aufer&atilde;tur pyramides b c l <G>k,</G> d b l K: qu&aelig; relin-
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1809 quuntur erunt &aelig;qualia. &aelig;qualis igitur e$t pyramis a c d <G>k</G>
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1810 pyramidi b c d K. Rur$us $i per lineas a d, d e ducatur pla-
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1811 num quod pyramidem $ccet: $itq; eius &amp; ba$is communis
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1812 $ectio a e m: $imiliter o$tendetur pyramis a b d K &aelig;qualis
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1813 pyramidi a c d k. ducto denique alio plano per lineas c a,
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1814 a f: ut eius, &amp; trianguli c d b communis $ectio $it c fn, py-
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1815 ramis a b c k pyramidi a c d <G>k</G> &aelig;qualis demon$trabitur. c&utilde;
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1816 ergo tres pyramides b c d k, a b d k, a b c k uni, &amp; eidem py
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1817 ramidi a c d k $int &aelig;quales, omnes inter $e $e &aelig;quales er&utilde;t.
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1818 Sed ut pyramis a b c d ad pyramidem a b c <G>k,</G> ita d e axis ad
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1819 axem <G>k</G> e, ex uige$ima propo$itione huius: $unt enim h&aelig;
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1820 pyramides in eadem ba$i, &amp; axes cum ba$ibus &aelig;quales con
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1821 tinent angulos, qu&ograve;d in eadem recta linea con$tituantur.
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1822 quare diuidendo, ut tres pyramides a c d k, b c d K, a b d K
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1823 ad pyramidem a b c K, ita d k ad K e. con$tat igitur lineam
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1824 d K ip$ius K e triplam e$$e. $ed &amp; a <G>k</G> tripla e$t K f: itemque
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1825 b K ip$ius K g: &amp; c <G>k</G> ip$ius <G>k</G> l tripla. quod eodem modo
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1826 demon$trabimus.</P>
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1827 <P>Sit pyramis, cuius ba$is quadrilaterum a b c d; axis e f:
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1828 &amp; diuidatur e fin g, ita ut e g ip$ius g f $it tripla. Dico cen-
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1829 trum grauitatis pyramidis e$$e punctum g. ducatur enim
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1830 linea b d diuidens ba$im in duo triangula a b d, b c d: ex
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1831 quibus intellig&atilde;tur c&otilde;$titui du&aelig; pyramides a b d e, b c d e:
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1832 $itque pyramidis a b d e axis e h; &amp; pyramidis b c d e axis
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1833 e K: &amp; iungatur h K, qu&aelig; per f tran$ibit: e$t enim in ip$a h K
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1834 centrum grauitatis magnitudinis compo$it&aelig; ex triangulis
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1835 a b d, b c d, hoc e$t ip$ius quadrilateri. Itaque centrum gra
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1836 uitatis pyramidis a b d e $it punctum l: &amp; pyramidis b c d e
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1837 <marg>2. fexti.</marg>
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1838 $it m. ducta igitur l m ip$i h m line&aelig; &aelig;quidi$tabit. nam el ad
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1839 <foot>lh</foot>
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1840 <p n=>29</p>
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1841 lbeandem habet proportionem, quam e m ad m k, uideli-
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1842 cet triplam. quare lineal m ip$am e f $ecabit in punctog:
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1843 etenim e g ad g f e$t, ut el adlh. pr&aelig;terea quoniam h k, l m
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1844 &aelig;quidi$tant, erunt triangula h e f, l e g $imilia: itemq; inter
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1845 $e $imilia fe <G>k,</G> gem: &amp; ut e fad e g, itah fad l g: &amp; ita f K ad
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1846 gm. ergo uth fadl g, ita f <G>k</G> ad g m: &amp; permutando uth f
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1847 ad f K, ital g ad gm. $ed cum h $it centrum trianguli a b d;
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1848 &amp; <G>k</G> tri&atilde;guli b c d punct&utilde; uero f totius quadrilateri a b c d
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1849 centrum: erit ex 8. Archimedis de centro grauitatis plano
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1850 rum h fad f <G>k,</G> ut triangulum b c d ad triangulum a b d: ut,
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1851 autem bcd triangulum ad triangulum a b d, ita pyramis
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1852 <fig>
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1853 b c d e ad pyramidem a b d e. ergo
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1854 linea lg ad gm erit, ut pyramis
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1855 b c d e ad pyramid&etilde; a b d e. ex quo
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1856 $equitur, ut totius pyramidis
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1857 a b c d e punctum g $it grauitatis
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1858 centrum. Rur$us $it pyramis ba-
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1859 $im habens pentagonum a b c d e:
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1860 &amp; axem f g: diuidaturq; axis in p&utilde;
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1861 cto h, ita ut fh ad h g triplam habe
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1862 at proportionem. Dico h grauita-
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1863 tis centr&utilde; e$$e pyramidis a b c d e f.
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1864 iungatur enim e b: intelligaturq;
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1865 pyramis, cuius uertex f, &amp; ba$is
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1866 triangulum a b e: &amp; alia pyramis
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1867 intelligatur eundem uerticem ha-
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1868 bens, &amp; ba$im b c d e quadrilater&utilde;:
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1869 $it autem pyramidis a b e faxis f <G>k,</G>
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1870 &amp; grauitatis centrum l: &amp; pyrami
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1871 dis b c d e faxis f m, &amp; centrum gra
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1872 <*> iunganturq; <G>k</G> m, ln;
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1873 qu&aelig; per puncta g h tran$ibunt.
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1874 Rur$us eodemmodo, quo $up ra,
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1875 demon$trabimus lineas K g m, l h n $ibiip$is &aelig;quidi$tare:
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1876 <foot>H</foot>
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1877 <pb>
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1878 &amp; denique punctum h pyramidis a b c d e f grauitatis e$$e
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1879 centrum, &amp; ita in alils.</P>
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1880 <P>Sit conus, uel coni portio axem habens b d: $eceturque
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1881 plano per axem, quod $ectionem faciat triangulum a b c:
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1882 &amp; b d axis diuidatur in c, ita ut be ip$ius ed $it tripla.
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1883 Dico punctum e coni, uel coni portionis, grauitatis
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1884 e$$e centrum. Sienim fieri pote$t, $itcentrum f: &amp; pro-
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1885 ducatur e f extra figuram in g. quam uero proportionem
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1886 habet g e ad e f, habeat ba$is coni, uelconi portionis, hoc
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1887 e$t circulus, uel ellip$is circa diametrum ac ad aliud $pa-
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1888 cium, in quo h. Itaque in circulo, uel ellip$i plane de$cri-
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1889 batur rectilinea figura a x l m c n o p, ita ut qu&aelig; relinqu&utilde;-
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1890 tur portiones $int minores $pacio h: &amp; intelligatur pyra-
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1891 mis ba$im habens rectilineam figuram a K l m c n o p, &amp;
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1892 axem b d; cuius quidem grauitatis centrum erit punctum
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1893 e, utiam demon$trauimus. Et quoniam portiones $unt
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1894 minores $pacio h, circulus, uel ellip$is ad portiones ma-
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1895 <fig>
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1896 iorem proportionem habet, quam g e ad e f. $ed ut circu-
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1897 lus, uel ellip$is ad figuram rectilineam $ibi in$criptam, ita
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1898 conus, uel coni portio ad pyramidem, qu&aelig; figuram rectili-
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1899 neam pro ba$i habet; &amp; altitudinem &aelig;qualem: etenim $u-
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1900 <foot>pra</foot>
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1901 <p n=>30</p>
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1902 <marg>8 h<*></marg>
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1903 pra demon$tratum e$t, ita e$$e cylindrum, uel cylindri por-
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1904 tionem ad pri$ma, cuius ba$is rectilinea figura, &amp; &aelig;qua-
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1905 lis altitudo. ergo per conuer$ionem rationis, ut circulus,
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1906 uel ellip$is ad portiones, ita conus, uel coni portio ad por-
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1907 tiones $olidas. quare conus uel coni portio ad portiones
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1908 $olidas maiorem habet proportionem, quam g e ad e f: &amp;
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1909 diuidendo, pyramis ad portiones $olidas maiorem pro-
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1910 portionem habet, quam g f ad f e. fiatigitur q f ad f e
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1911 ut pyramis ad dictas portiones. Itaque quoniam a cono
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1912 uel coni portione, cuius grauitatis centrum e$t f, aufer-
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1913 tur pyramis, cuius centrum e; reliqu&aelig; magnitudinis,
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1914 qu&aelig; ex $olidis portionibus con$tat, centrum grauitatis
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1915 erit in linea e f protracta, &amp; in puncto q. quod fieri
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1916 non pote$t: e$t enim centrum grauitatis intra. Con$tat
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1917 igitur coni, uel coni portionis grauitatis centrum e$$e pun
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1918 ctum e. qu&aelig; omnia demon$trare oportebat.</P>
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1919 <head>THEOREMA XIX. PROPOSITIO XXIII.</head>
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1920 <P>QVODLIBET fru$tum &agrave; pyramide, qu&aelig;
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1921 triangularem ba$im habeat, ab$ci$$um, diuiditur
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1922 in tres pyramides proportionales, in ea proportio
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1923 ne, qu&aelig; e$t lateris maioris ba$is ad latus minoris
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1924 ip$i re$pondens.</P>
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1925 <P>Hoc demon$trauit Leonardus Pi$anus in libro, qui de-
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1926 praxi geometri&aelig; in$cribitur. Sed quoniam is adhuc im-
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1927 pre$$us non e$t, nos ip$ius demon$trationem breuiter
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1928 per$tringemus, rem ip$am $ecuti, non uerba. Sit fru-
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diff changeset
1929 $tum pyramidis a b c d e f, cuius maior ba$is triangulum
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1930 a b c, minor d e f: &amp; iunctis ae, cc, cd, per, line-
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1931 as a e, e c ducatur planum $ecans fru$tum: itemque per
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1932 lineas e c, c d; &amp; per cd, da alia plana ducantur, qu&aelig;
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1933 diuident fru$tum in trcs pyramides a b c e, a d c e, d e f c.
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1934 <foot>H 2</foot>
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1935 <pb>
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1936 Dico eas proportionales e$$e in proportione, qu&aelig; e$t la-
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1937 teris a b adlatus d e, itaut earum maior $it a b c e, me-
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1938 dia a d c e, &amp; minor d e f c. Quoniam enim line&aelig; d e,
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1939 a b &aelig;quidi$tant; &amp; interip$as $unt triangula a b e, a d e;
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1940 <marg>1. $exti.</marg>
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1941 <fig>
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1942 erit triangulum a b e
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1943 ad triangulum a b e,
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diff changeset
1944 utlinea a b ad lineam
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1945 d e. ut autem triangu
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1946 lum a b e ad triangu-
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1947 <marg>5. duodeci
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1948 mi.</marg>
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1949 lum a b e, ita pyramis
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1950 a b e c ad pyramidem
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1951 a d e c: habent enim
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1952 altitudinem eandem,
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1953 qu&aelig; e$t&agrave; puncto c ad
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1954 planum, in quo qua-
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1955 <marg>11. quinti.</marg>
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1956 drilaterum a b e d. er-
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1957 go ut a b ad d e, ita pyramis a b e c ad pyramidem a d e c.
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1958 Rur$us quoniam &aelig;quidi$tantes $unt a c, d f; erit eadem
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1959 <marg>4 $exti.</marg>
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1960 ratione pyramis a d c e ad pyramidem c d fe, ut ac ad
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1961 d f. Sed ut a c a l d f, ita a b ad d e, quoniam triangula
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1962 a b c, d e f $imilia $unt, ex nona huius. quare ut pyramis
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1963 a b c e ad pyramidem a b c e, ita pyramis a d c e ad ip$an<*>
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1964 d e f c. fru$tum igitur a b c d e f diuiditur in tres pyramides
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1965 proportionales in ea proportione, qu&aelig; e$t lateris a b ad d e
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1966 latus, &amp; earum maior e$t c a b e, media a d c e, &amp; minor
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1967 d e f c. quod demon$trare oportebat.</P>
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1968 <head>PROBLEMA V. PROPOSITIO XXIIII.</head>
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1969 <P>QVODLIBET fru$tum pyramidis, uel coni,
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1970 uel coni portionis, plano ba$i &aelig;quidi$tanti ita $e-
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1971 care, ut $ectio $it proportionalis inter maiorem,
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1972 &amp; minorem ba$im.</P>
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1973 <foot>Sit</foot>
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1974 <p n=>31</p>
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1975 <P>SIT fru$tum pyramidis a e, cuius maior ba$is triangu-
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1976 lum a b c, minor d e f: &amp; oporteat ip$um plano, quod ba$i
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1977 &aelig;quidi$tet, ita $ecare, ut $ectio $it proportionalis inter tri&atilde;
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diff changeset
1978 gula a b c, d e f. Inueniatur inter lineas a b, d e media pro-
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1979 portionalis, qu&aelig; $it b g: &amp; &agrave; puncto g erigatur g h &aelig;quidi-
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parents:
diff changeset
1980 $tans b e, $ecansq; a d in h: deinde per h ducatur planum
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1981 ba$ibus &aelig;quidi$tans, cuius $ectio $it triangulum h k l. Dico
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1982 triangulum h K l proportionale e$$e inter triangula a b c,
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1983 <fig>
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1984 d e f, hoc e$t triangulum a b c ad
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1985 triangulum h K l eandem habere
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1986 proportionem, quam tri&atilde;gulum
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1987 h K l ad ip$um d e f. Quoni&atilde; enim
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1988 <marg>16.<*>
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1989 cim<*></marg>
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1990 line&aelig; a b, h K &aelig;quidi$tantium pla
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1991 norum $ectiones inter $e &aelig;quidi-
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1992 $tant: atque &aelig;quidi$tant b k, gh:
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1993 <marg>34. l<*></marg>
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1994 linea h k ip$i g b e$t &aelig;qualis: &amp; pro
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1995 pterea proportionalis inter a b,
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1996 d e. quare ut a b ad h K, ita e$t h <G>k</G>
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1997 ad de. fiat ut h k ad d e, ita d e
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1998 ad aliam lineam, in qua $it m. erit
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1999 ex &aelig;quali ut a b ad d e, ita h k ad
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2000 <marg>9<*> hu<*>
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2001 coin<*></marg>
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2002 m. Et quoniam triangula a b c,
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2003 h K l, d e f $imilia $unt; triangul&utilde;
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2004 <marg>20. $<*></marg>
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2005 a b c ad triangulum h <G>k</G> l e$t, ut li-
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diff changeset
2006 nea a b ad lineam d e: triangul&utilde;
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2007 <marg>11. q<*></marg>
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2008 autem h <G>k</G> l adip$um d e f e$t, ut h k ad m. ergo triangulum
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parents:
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2009 a b c ad triangulum h k l eandem proportionem habet,
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2010 quam triangulum h K l ad ip$um d e f. Eodem modo in a-
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2011 liis fru$tis pyramidis idem demon$trabitur.</P>
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diff changeset
2012 <P>Sit fru$tum coni, uel coni portionis a d: &amp; $ecetur plano
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2013 per axem, cuius $ectio $it a b c d, ita ut maior ip$ius ba$is $it
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2014 circulus, uel ellip$is circa diametrum a b; minor circa c d.
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2015 Rur$us inter lineas a b, c d inueniatur proportionalis b e:
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2016 &amp; ab e ducta e f &aelig;quidi$tante b d, qu&aelig; lineam c a in f $ecet,
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2017 <pb>
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2018 per f planum ba$ibus &aelig;quidi$tans ducatur, ut $it $ectio cir
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2019 culus, uel ellip$is circa diametrum f g. Dico $ectionem a b
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2020 ad $ectionem f g eandem proportionem habere, quam f g
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2021 ad ip$am c d. Simili enim ratione, qua $upra, demon$trabi-
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diff changeset
2022 tur quadratum a b ad quadratum f g ita e$$e, ut quadrat&utilde;
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2023 <marg>2. duode
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2024 <*>imi</marg>
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2025 f g ad c d quadratum. Sed circuli inter $e candem propor-
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diff changeset
2026 tionem habent, quam diametrorum quadrata. ellip$es au-
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diff changeset
2027 tem circa a b, f g, c d, qu&aelig; $imiles $unt, ut o$tendimus in c&otilde;-
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diff changeset
2028 mentariis in principium libri Archimedis de conoidibus,
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2029 &amp; $ph&aelig;roidibus, eam hab&etilde;t proportionem, quam quadrar
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diff changeset
2030 ta diametrorum, qu&aelig; eiu$dem rationis $unt, ex corollaio-
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2031 <fig>
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diff changeset
2032 $eptim&aelig; propo$itionis eiu$dem li-
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2033 bri. ellip$es enim nunc appello ip-
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diff changeset
2034 $a $pacia ellip$ibus contenta. ergo
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diff changeset
2035 circulus, uel ellip$is a b ad circul&utilde;,
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parents:
diff changeset
2036 uel ellip$im f g eam proportionem
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diff changeset
2037 habet, quam circulus, uel ellip$is
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2038 f g ad circulum uel ellip$im c d.
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2039 quod quidem faciendum propo-
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2040 $uimus.</P>
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2041 <head>THEOREMA XX. PROPOSITIO XXV.</head>
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2042 <P>QVODLIBET fru$tum pyramidis, uel coni,
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2043 uel coni portionis ad pyramidem, uel conum, uel
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diff changeset
2044 coni portionem, cuius ba$is eadem e$t, &amp; &aelig;qualis
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diff changeset
2045 altitudo, eandem proportion&etilde; habet, quam utr&aelig;
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2046 que ba$es, maior, &amp; minor $imul $umpt&aelig; vn&agrave; c&utilde;
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2047 ca, qu&aelig; inter ip$as $it proportionalis, ad ba$im ma
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2048 iorem.</P>
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2049 <foot>Sit</foot>
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2050 <p n=>32</p>
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2051 <P>SIT fru$t&utilde; pyramidis, uel coni, uel coni portionis a d,
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2052 cuius maior ba$is a b, minor c d. &amp; $ecetur altero plano
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2053 ba$i &aelig;quidi$tante, ita ut $ectio e f $it proportionalis inter
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2054 ba$es a b, c d. con$tituatur aut&etilde; pyramis, uel conus, uel co-
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2055 ni portio a g b, cuius ba$is $it eadem, qu&aelig; ba$is maior fru-
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diff changeset
2056 <fig>
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parents:
diff changeset
2057 $ti, &amp; altitudo &aelig;qualis. Di-
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diff changeset
2058 co fru$tum a d ad pyrami-
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parents:
diff changeset
2059 dem, uel conum, uel coni
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parents:
diff changeset
2060 portionem a g b eandem
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parents:
diff changeset
2061 proportion&etilde; habere, qu&atilde;
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parents:
diff changeset
2062 utr&aelig;que ba$es, a b, c d un&agrave;
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parents:
diff changeset
2063 cum e f ad ba$im a b. e$t
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parents:
diff changeset
2064 enim fru$tum a d &aelig;quale
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parents:
diff changeset
2065 pyramidi, uel cono, uel co-
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parents:
diff changeset
2066 ni portioni, cuius ba$is ex
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parents:
diff changeset
2067 tribus ba$ibus a b, e f, c d
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diff changeset
2068 con$tat; &amp; altitudo ip$ius
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parents:
diff changeset
2069 altitudini e$t &aelig;qualis: quod mox o$tendemus. Sed pyrami
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diff changeset
2070 <fig>
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parents:
diff changeset
2071 des, coni, uel coni porti&otilde;es,
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parents:
diff changeset
2072 qu&aelig; $unt &aelig;quali altitudine,
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diff changeset
2073 e&atilde;dem inter $e, quam ba$es,
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parents:
diff changeset
2074 proportionem habent, $icu-
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parents:
diff changeset
2075 ti demon$tratum e$t, partim
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parents:
diff changeset
2076 <marg>6. 11.<*>
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parents:
diff changeset
2077 decin<*></marg>
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parents:
diff changeset
2078 ab Euclide in duodecimo li-
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parents:
diff changeset
2079 bro elementorum, partim &agrave;
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parents:
diff changeset
2080 nobis in c&otilde;mentariis in un-
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parents:
diff changeset
2081 decimam propo$ition&etilde; Ar-
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parents:
diff changeset
2082 chimedis de conoidibus, &amp;
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parents:
diff changeset
2083 $ph&aelig;roidibus. quare pyra-
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parents:
diff changeset
2084 mis, uel conus, uel coni por-
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parents:
diff changeset
2085 tio, cuius ba$is e$t tribus illis
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parents:
diff changeset
2086 ba$ibus &aelig;qualis ad a g b eam
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parents:
diff changeset
2087 habet proportionem, quam
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parents:
diff changeset
2088 ba$es a b, e f, c d ad a b ba$im. Fru$tum igitur a d ad a g b
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
2089 <foot><*></foot>
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parents:
diff changeset
2090 <pb>
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parents:
diff changeset
2091 pyramidem, uel conuni, uel coni portionem candem pro-
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
2092 portionem habet, quam ba$es a b, c d un&agrave; cum e f ad ba-
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
2093 $im a b. quod demon$trare uolebamus.</P>
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parents:
diff changeset
2094 <P>Fru$tum uero a d &aelig;quale e$$e pyramidi, uel co
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parents:
diff changeset
2095 no, uel coni portioni, cuius ba$is con$tat ex ba$i-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2096 bus a b, c d, e f, &amp; altitudo fru$ti altitudini e$t &aelig;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2097 qualis, hoc modo o$tendemus.</P>
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
2098 <P>Sit fru$tum pyramidis a b c d e f, cuius maior ba$is trian-
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parents:
diff changeset
2099 gulum a b c; minor d e f: &amp; $ecetur plano ba$ibus &aelig;quidi-
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
2100 $tante, quod $ectionem faciat triangulum g h <G>k</G> inter trian-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2101 gula a b c, d e f proportionale. Iam ex iis, qu&aelig; demon$trata
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
2102 $unt in 23. huius, patet fru$tum a b c d e f diuidi in tres pyra
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casties
parents:
diff changeset
2103 mides proportionales; &amp; earum maiorem e$$e pyramid&etilde;
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
2104 a b c d minor&etilde; uero d e f b. ergo pyramis &agrave; triangulo g h k
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
2105 con$tituta, qu&aelig; altitudinem habeat fru$ti altitudini &aelig;qua-
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parents:
diff changeset
2106 lem, proportionalis e$t inter pyramides a b c d, d e f b: &amp;
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casties
parents:
diff changeset
2107 idcirco fru$tum a b c d e f tribus dictis pyramidibus &aelig;qua
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
2108 <fig>
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parents:
diff changeset
2109 le erit. Itaque $i intelligatur alia pyra-
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
2110 mis &aelig;que alta, qu&aelig; ba$im habeat ex tri
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
2111 bus ba$ibus a b c, d e f, g h k con$tan-
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
2112 tem; per$picuum e$t ip$am ei$dem py-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2113 ramidibus, &amp; propterea ip$i fru$to &aelig;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2114 qualem e$$e.</P>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2115 <P>Rur$us $it fru$tum pyramidis a g, cu
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
2116 ius maior ba$is quadrilaterum a b c d,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2117 minor e f g h: &amp; $ecetur plano ba$i-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2118 bus &aelig;quidi$tante, ita ut fiat $ectio qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2119 drilaterum K l m n, quod $it proportio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2120 nale inter quadrilatera a b c d, e f g h. Dico pyramidem,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2121 cuius ba$is $it &aelig;qualis tribus quadrilateris a b c d, k l m n,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2122 e f g h, &amp; altitudo &aelig;qualis altitudini fru$ti, ip$i fru$to a g
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2123 &aelig;qualem e$$e. Ducatur enim planum per lincas f b, h d,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2124 <foot>quod</foot>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2125 <p n=>33</p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2126 quod diuidat fru$tum in duo fru$ta triangulares ba$es ha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2127 bentia, uidelicet in fru$tum a b d e f h, &amp; in fru$t&utilde; b c d f g h.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2128 erit triangulum <G>k</G> l n proportionale inter triangula a b d,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2129 e f h: &amp; triangulum l m n proportionale inter b c d, f g h.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2130 $ed pyramis &aelig;que alta, cuius ba$is con$tat ex tribus trian-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2131 <fig>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2132 gulis a b d, k l n, e f h, demon$trata
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2133 e$t fru$to a b d c f h &aelig;qualis. &amp; $i-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2134 militer pyramis, cuius ba$is con-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2135 $tat ex triangulis b c d, l m n, f g h
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2136 &aelig;qualis fru$to b c d f g h: compo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2137 nuntur autem tria quadrilatera a
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2138 b c d, k l m n, e f g h &egrave; $ex triangu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2139 lis iam dictis. pyramis igitur ba-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2140 $im habens &aelig;qualem tribus qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2141 drilateris, &amp; altitudinem eandem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2142 ip$i fru$to a g e$t &aelig;qualis. Eodem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2143 modo illud dem&otilde;$trabitur in aliis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2144 eiu$modi fru$tis.</P>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2145 <P>Sit fru$tum coni, uel coni portionis a d; cuius maior ba-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2146 $is circulus, uel ellip$is circa diametrum a b; minor circa
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2147 c d: &amp; $ecetur plano, quod ba$ibus &aelig;quidi$tet, faciatq; $e-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2148 ctionem circulum, uel ellip$im circa diametrum e f, ita ut
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2149 inter circulos, uel ellip$es a b, c d $it proportionalis. Dico
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2150 conum, uel coni portionem, cuius ba$is e$t &aelig;qualis tribus
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2151 circulis, uel tribus ellip$ibus a b, e f, c d; &amp; altitudo eadem,
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2152 qu&aelig; fru$ti a d, ip$i fru$to &aelig;qualem e$$e. producatur enim
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2153 fru$ti $uperficies quou$que coeat in unum punctum, quod
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2154 $it g: &amp; coni, uel coni portionis a g b axis $it g h, occurrens
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2155 planis a b, e f, c d in punctis h k l: circa circulum uero de-
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2156 $cribatur quadratum m n o p, &amp; circa ellip$im rectangul&utilde;
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2157 m n o p, quod ex ip$ius diametris con$tat: iunctisq; g m,
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2158 g n, g o, g p, ex eodem uertice intelligatur pyramis ba$im
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2159 habens dictum quadratum, uel rectangulum: &amp; plana in
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2160 quibus $unt circuli, uel ellip$es e f, c d u$que ad eius latera
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2161 <foot>I</foot>
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2162 <pb>
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2163 preducantur. Quoniam igitur pyramis $ecatur planis ba$i
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2164 <marg>9 huius</marg>
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2165 &aelig;quidi$tantibus, $ectiones $imiles erunt: atque erunt qua-
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2166 drata, uel rectangula circa circulos, uel ellip$es de$cripta,
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2167 quemadmodum &amp; in ip$a ba$i. Sed cum circuli inter $e e&atilde;
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2168 <marg>2. duode-
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2169 cnni.</marg>
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2170 proportionem habeant, quam diametrorum quadrata:
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2171 itemq; ellip$es eam quam rectangula ex ip$arum diametris
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2172 <marg>7. de co-
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2173 noidibus
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2174 &amp; $ph&aelig;-
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2175 roidibus</marg>
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2176 con$tantia: &amp; $it circulus, uel ellip$is circa diametrum e f
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2177 <fig>
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2178 proportionalis inter circulos, uel ellip$es a b, c d; erit re-
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2179 ctangulum e f etiam inter rectangula a b, c d proportio-
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2180 nale: per rectangulum enim nunc breuitatis cau$a eti&atilde; ip-
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2181 $um quadratum intelligemus. quare ex iis, qu&aelig; proxime
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2182 dicta $unt, pyramis ba$im habens &aelig;qualem dictis rectangu
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diff changeset
2183 lis, &amp; altitudinem eandem, quam fru$tum a d, ip$i fru$to &agrave;
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diff changeset
2184 pyramide ab$ci$$o &aelig;qualis probabitur. ut autem rectangu
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2185 lum c d ad rectangul&utilde; e f, ita circulus, uel ellip$is c d ad e f
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2186 circulum, uel ellip$im: componendoq; ut rectangula c d,
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2187 e f, ad e f rectangulum, ita circuli, uel ellip$es e d, e f, ad e f:
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diff changeset
2188 &amp; ut rectangulum e f ad rectangulum a b, ita circulus, uel
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diff changeset
2189 ellip$is e f ad a b circulum, uel ellip$im. ergo ex &aelig;quali, &amp;
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diff changeset
2190 componendo, ut rect&atilde;gula c d, e f, a b ad ip$um a b, ita cir-
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2191 <foot>culi,</foot>
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2192 <p n=>34</p>
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diff changeset
2193 culi, uel ellip$es c d, e f a b ad circulum, uel ellip$im a b. In-
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diff changeset
2194 telligatur pyramis q ba$im habens &aelig;qualem tribus rectan
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diff changeset
2195 gulis a b, e f, c d; &amp; altitudinem e&atilde;dem, quam fru$tum a d.
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diff changeset
2196 intelligatur ctiam conus, uel coni portio q, eadem altitudi||
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2197 ne, cuius ba$is $it tribus circulis, uel tribus ellip$ibus a b,
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2198 e f, c d &aelig;qualis. po$tremo intelligatur pyramis a l b, cuius.
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diff changeset
2199 ba$is $it rectangulum m n o p, &amp; altitudo eadem, qu&aelig; fru-
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2200 $ti: itemq, intelligatur conus, uel coni portio a l b, cuius
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2201 ba$is circulus, uel ellip$is circa diametrum a b, &amp; eadem al
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2202 <marg>6. II. duo
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2203 decimi</marg>
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2204 titudo. utigitur rectangula a b, e f, c d ad rectangulum a b,
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2205 ita pyramis q ad pyramidem a l b; &amp; ut circuli, uel ellip-
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2206 $es a b, e f, c d ad a b circulum, uel ellip$im, ita conus, uel co
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2207 ni portio q ad conum, uel coni portionem a l b. conus
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2208 igitur, uel coni portio q ad conum, uel coni portionem
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2209 a l b e$t, ut pyramis q ad pyramidem a l b. $ed pyramis
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2210 a l b ad pyramidem a g b e$t, ut altitudo ad altitudinem, ex
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2211 20. huius: &amp; ita e$t conus, uel coni portio al b ad conum,
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2212 uel coni portionem a g b ex 14. duodccimi elementorum,
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2213 &amp; ex iis, qu&aelig; nos demon$trauimus in commentariis in un-
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2214 decimam de conoidibus, &amp; $ph&aelig;roidibus, propo$itione
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2215 quarta. pyramis autem a g b ad pyramidem c g d propor-
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2216 tionem habet compo$itam ex proportione ba$ium &amp; pro
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2217 portione altitudinum, ex uige$ima prima huius: &amp; $imili-
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2218 ter conus, uel coni portio a g b ad conum, uel coni portio-
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2219 nem c g d proportionem habet compo$it&atilde; ex ei$dem pro-
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diff changeset
2220 port&iacute;onibus, per ea, qu&aelig; in dictis commentariis demon-
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2221 $trauimus, propo$itione quinta, &amp; $exta: altitudo enim in<*>
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2222 utri$que eadem e$t, &amp; ba$es inter $e $e eandem habent pro-
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2223 portionem. ergo ut pyramis a g b ad pyramidem c g d, ita
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2224 e$t conus, uel coni portio a g b ad a g d conum, uel coni
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2225 portionem: &amp; per conuer$ion&etilde; rationis, ut pyramis a g b
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2226 ad $ru$t&utilde; &agrave; pyramide ab$ci$$um, ita conus uel coni portio
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2227 a g b ad fru$tum a d. ex &aelig;quali igitur, ut pyramis q ad fru-
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2228 $tum &agrave; pyramide ab$ci$$um, ita conus uel coni portio q ad
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2229 <foot>I 2</foot>
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2230 <pb>
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diff changeset
2231 fru$tum a d. Sed pyramis q &aelig;qualis e$t fru$to &agrave; pyramide
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2232 ab$ci$$o, ut demon$trauimus. ergo &amp; conus, uel coni por-
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diff changeset
2233 tio q, cuius ba$is ex tribus circulis, uel ellip$ibus a b, e f, c d
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2234 con$tat, &amp; altitudo eadem, qu&aelig; fru$ti: ip$i fru$to a d e$t &aelig;-
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2235 qualis. atque illud e$t, quod demon$trare oportebat.</P>
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2236 <head>THEOREMA XXI. PROPOSITIO XXVI.</head>
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2237 <P>CVIVSLIBET fru$ti &agrave; pyramide, uel cono,
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2238 uel coni portione ab$cis$i, centrum grauitatis e$t
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2239 in axe, ita ut eo primum in duas portiones diui-
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diff changeset
2240 $o, portio $uperior, qu&aelig; minorem ba$im attingit
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2241 ad portionem reliquam eam habeat proportio-
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diff changeset
2242 nem, quam duplum lateris, uel diametri maioris
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diff changeset
2243 ba$is, vn&agrave; cum latere, uel diametro minoris, ip$i
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diff changeset
2244 re$pondente, habet ad duplum lateris, uel diame-
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diff changeset
2245 tri minoris ba$is vn&agrave; c&utilde; latere, uel diametro ma-
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parents:
diff changeset
2246 ioris: deinde &agrave; puncto diui$ionis quarta parte $u
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parents:
diff changeset
2247 perioris portionis in ip$a $umpta: &amp; rur$us ab in-
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2248 ferioris portionis termino, qui e$t ad ba$im maio
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parents:
diff changeset
2249 rem, $umpta quarta parte totius axis: centrum $it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2250 in linea, qu&aelig; his finibus continetur, atque in eo li
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parents:
diff changeset
2251 ne&aelig; puncto, quo $ic diuiditur, ut tota linea ad par
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parents:
diff changeset
2252 tem propinquiorem minori ba$i, e&atilde;dem propor-
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parents:
diff changeset
2253 tionem habeat, quam fru$tum ad pyramid&etilde;, uel
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parents:
diff changeset
2254 conum, uel coni portionem, cuius ba$is $it ea-
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parents:
diff changeset
2255 dem, qu&aelig; ba$is maior, &amp; altitudo fru$ti altitudini
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parents:
diff changeset
2256 &aelig;qualis.</P>
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parents:
diff changeset
2257 <foot>Sit</foot>
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diff changeset
2258 <p n=>35</p>
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diff changeset
2259 <P>Sit fru$tum a e a pyramide, qu&aelig; triangularem ba$im ha-
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parents:
diff changeset
2260 beat ab$ci$$um: cuius maior ba$is triangulum a b c, minor
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parents:
diff changeset
2261 d e f; &amp; axis g h. ducto autem plano per axem &amp; per line&atilde;
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parents:
diff changeset
2262 d a, quod $ectionem faciat d a <G>k</G> l quadrilaterum; puncta
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
2263 K l lineas b c, e f bifariam $ecabunt. nam cum g h $it axis
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
2264 fru$ti: erit h centrum grauitatis trianguli a b c: &amp; g
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
2265 <fig>
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parents:
diff changeset
2266 <marg>3. diffi. hu
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diff changeset
2267 ius.</marg>
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2268 centrum trianguli d e f: cen-
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parents:
diff changeset
2269 trum uero cuiuslibet triangu
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2270 li e$t in recta linea, qu&aelig; ab an-
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2271 gulo ip$ius ad dimidi&atilde; ba$im
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diff changeset
2272 ducitur ex decimatertia primi
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diff changeset
2273 libri Archimedis de c&etilde;tro gra
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parents:
diff changeset
2274 <marg>Vltima e-
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parents:
diff changeset
2275 iu$d&etilde; libri
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parents:
diff changeset
2276 Archime-
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parents:
diff changeset
2277 dis.</marg>
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parents:
diff changeset
2278 uitatis planorum. quare cen-
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parents:
diff changeset
2279 tr&utilde; grauitatis trapezii b c f e
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parents:
diff changeset
2280 e$t in linea K l, quod $it m: &amp; &agrave;
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parents:
diff changeset
2281 puncto m ad axem ducta m n
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
2282 ip$i a k, uel d l &aelig;quidi$tante;
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parents:
diff changeset
2283 erit axis g h diui$us in portio-
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diff changeset
2284 nes g n, n h, quas diximus: ean
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parents:
diff changeset
2285 dem enim proportionem ha-
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parents:
diff changeset
2286 bet g n ad n h, qu&atilde; l m ad m k.
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diff changeset
2287 At l m ad m K habet eam, qu&atilde;
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
2288 duplum lateris maioris ba$is
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diff changeset
2289 b c una cum latere minoris e f
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
2290 ad duplum lateris e f un&agrave; cum
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
2291 latere b c, ex ultima eiu$dem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2292 libri Archimedis. Itaque &agrave; li-
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diff changeset
2293 nea n g ab$cindatur, quarta
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
2294 pars, qu&aelig; fit n p: &amp; ab axe h g ab$cindatur itidem
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
2295 quarta pars h o: &amp; quam proportionem habet fru$tum ad
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
2296 pyramidem, cuius maior ba$is e$t triangulum a b c, &amp; alti-
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parents:
diff changeset
2297 tudo ip$i &aelig;qualis; habeat o p ad p q. Dico centrum graui-
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parents:
diff changeset
2298 tatis fru$ti e$$e in linea p o, &amp; in puncto q. namque ip$um
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parents:
diff changeset
2299 e$$e in linea g h manife$te con$tat. protractis enim fru$ti pla
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parents:
diff changeset
2300 <pb>
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parents:
diff changeset
2301 nis, quou$que in unum punctum r conueniant; erit pyra-
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parents:
diff changeset
2302 midis a b c r, &amp; pyramidis d e f r grauitatis centrum in li-
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parents:
diff changeset
2303 nca r h. ergo &amp; reliqu&aelig; magnitudinis, uidelicet fru$ti cen-
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
2304 trum in eadem linea nece$lario comperietur. Iungantur
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parents:
diff changeset
2305 d b, d c, d h, d m: &amp; per lineas d b, d c ducto altero plano
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2306 intelligatur fru$tum in duas pyramides diui$um: in pyra-
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parents:
diff changeset
2307 midem quidem, cuius ba$is e$t triangulum a b c, uertex d:
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
2308 &amp; in eam, cuius idem uertex, &amp; ba$is trapezium b c f e. erit
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
2309 igitur pyramidis a b c d axis d h, &amp; pyramidis b c f e d axis
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parents:
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2310 d m: atque erunt tres axes gh, d h, d m in eodem plano
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parents:
diff changeset
2311 d a K l. ducatur pr&aelig;terea per o linea $t ip$i a K &aelig;quidi$t&atilde;s,
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parents:
diff changeset
2312 qu&aelig; lineam d h in u $ecet: per p uero ducatur x y &aelig;quidi-
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
2313 <fig>
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parents:
diff changeset
2314 $tans eidem, $ecansque d m in
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parents:
diff changeset
2315 z: &amp; iungatur z u, qu&aelig; $ecet
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
2316 g h in <G>*f.</G> tran$ibit ea per q: &amp;
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parents:
diff changeset
2317 erunt <G>*f</G> q unum, atque idem
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parents:
diff changeset
2318 punctum; ut inferius appare-
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parents:
diff changeset
2319 bit. Quoniam igitur linea u o
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parents:
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2320 <marg>2. $exti.</marg>
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2321 &aelig;quidi$tat ip$i d g, erit d u ad
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parents:
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2322 u h, ut g o ad o h. Sed g o tri-
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parents:
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2323 pla e$t o h. quare &amp; d u ip$ius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2324 u h e$t tripla: &amp; ideo pyrami-
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parents:
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2325 dis a b c d centrum grauitatis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2326 erit punctum u. Rur$us quo-
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parents:
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2327 niam z y ip$i d l &aelig;quidi$tat, d z
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2328 ad z m e$t, ut l y ad y m: e$tque
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2329 l y ad y m, ut g p ad p n. ergo
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diff changeset
2330 d z ad z m e$t, ut g p ad p n.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2331 Qu&ograve;d cum g p $it tripla p n;
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parents:
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2332 erit etiam d z ip$ius z m tri-
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parents:
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2333 pla. atque ob candem cau$-
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parents:
diff changeset
2334 $am punctuniz e$t centr&utilde; gra-
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diff changeset
2335 uitatis pyramidis b c f e d. iun
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diff changeset
2336 ctaigitur z u, in ea erit c&etilde;trum
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diff changeset
2337 <foot>gra-</foot>
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diff changeset
2338 <p n=>36</p>
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diff changeset
2339 grauitatis magnitudinis, qu&aelig; ex utri$que pyramidibus c&otilde;
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diff changeset
2340 $tat; hoc e$t ip$ius fru$ti. Sed fru$ti centrum e$t etiam in a-
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diff changeset
2341 xe g h. ergo in puncto <G>*f,</G> in quo line&aelig; z u, g h conu<*>niunt.
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diff changeset
2342 <marg><*>. primi
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diff changeset
2343 libri Ar-
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diff changeset
2344 chunedis
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diff changeset
2345 de c&etilde;tro
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2346 gr<*>u ta-
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diff changeset
2347 tis plano
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diff changeset
2348 tu<*>n</marg>
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diff changeset
2349 Itaque u <*> ad <G>*f</G> z eam proportionem habet, quam pyramis
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diff changeset
2350 b c f e d ad pyramidem a b c d. &amp; componendo u z ad z <G>*f</G>
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diff changeset
2351 eam habet, quam fru$tum ad pyramidem a b c d. Vtuero
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diff changeset
2352 u z ad z <*>, ita o p ad p <G>*f</G> ob $imilitudinem triangulorum,
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diff changeset
2353 u o <*>, z p <G>*f.</G> quare o p ad p <G>*f</G> e$t ut fru$tum ad pyramidem
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diff changeset
2354 a b c d. $ed ita erat o p ad p q. &aelig;quales igitur $unt p <*>, p q: <*>
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diff changeset
2355 <marg>7. quinti.</marg>
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diff changeset
2356 q <G>*f</G> unum atque idem pun ctum. ex quibus $equitur lineam.
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diff changeset
2357 z u $ecare o p in q: &amp; propterea p&utilde;ctum q ip$ius fru$ti gra-
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2358 uitatis centrum e$$e.</P>
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diff changeset
2359 <P>Sit fru$tum a g &agrave; pyramide, qu&aelig; quadrangularem ba$im
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diff changeset
2360 habeat ab$ci$$um, cuius maior ba$is a b c d, minor e f g h,
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diff changeset
2361 &amp; axis <G>k</G> l. diuidatur autem prim&utilde; k l, ita ut quam propor-
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diff changeset
2362 tionem habet duplum lateris a b un&agrave; cum latere e f ad du
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diff changeset
2363 plum lateris e f un&agrave; cum a b; habeat k m ad m l. deinde &agrave;
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diff changeset
2364 p&utilde;cto m ad k $umatur quarta pars ip$ius m <G>k,</G> qu&aelig; $it m n.
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diff changeset
2365 &amp; rur$us ab l $umatur quarta pars totius axis l k, qu&aelig; $it
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diff changeset
2366 l o. po$tremo fiat o n ad n p, ut fru$tum a g ad pyramid&etilde;,
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diff changeset
2367 cuius ba$is $it eadem, qu&aelig; fru$ti, &amp; altitudo &aelig;qualis. Dico
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diff changeset
2368 punctum p fru$ti a g grauitatis centrum e$$e. ducantur
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diff changeset
2369 enim a c, e g: &amp; intelligantur duo fru$ta triangulares ba-
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diff changeset
2370 $es habentia, quorum alterum l f ex ba$ibus a b c, e f g c&otilde;-
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diff changeset
2371 $tet; alterum l h ex ba$ibus a c d, e g h. Sitq; fru$ti l f axis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2372 q r; in quo grauitatis centrum s: fru$ti uero l h axis t u, &amp;
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diff changeset
2373 x grauitatis centrum: deinde iungantur u r, t q, x s. tran$i-
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2374 bit u r per l: quoniam l e$t centrum grauitatis quadran-
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diff changeset
2375 guli a b c d: &amp; puncta r u grauitatis centra triangulorum
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diff changeset
2376 a b c, a c d; in qu&aelig; quadrangulum ip$um diuiditur. eadem
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2377 quoque ratione t q per punctum k tran$ibit. At uero pro
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2378 portiones, ex quibus fru$torum grauitatis centra inquiri-
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2379 mus, e&aelig;dem $unt in toto fru$to a g, &amp; in fru$tis l f, l h. Sunt
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diff changeset
2380 enim per octauam huius quadrilatera a b c d, e f g h $imilia:
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2381 <pb>
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diff changeset
2382 itemq; $imilia triangula a b c, e f g: &amp; a c d, e g h. idcir-
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2383 coq; latera $ibi ip$is re$pondentia eandem inter $e$e pro-
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diff changeset
2384 portionem $eruant. Vt igitur duplum lateris a b un&agrave;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2385 cum latere e f ad duplum lateris e f un&agrave; cum a b, ita e$t
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2386 <fig>
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parents:
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2387 duplum a d late-
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2388 ris una cum late-
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2389 re e h ad duplum
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2390 e h un&agrave; cum a d:
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2391 &amp; ita in aliis.
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diff changeset
2392 Rur$us fru$tum
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2393 a g ad pyramid&etilde;,
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2394 cuius eadem e$t
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2395 ba$is, &amp; &aelig;qualis
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2396 altitudo eandem
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2397 proportion&etilde; ha
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2398 bet, quam fru$t&utilde;
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2399 l f ad pyramid&etilde;,
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2400 qu&aelig; e$t ead&etilde; ba-
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2401 $i, &amp; &aelig;quali alti-
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2402 tudine: &amp; $imili-
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2403 ter quam l h fru-
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2404 $tum ad pyrami-
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2405 dem, qu&aelig; ex ea-
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2406 d&etilde; ba$i, &amp; &aelig;quali
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2407 altitudine con-
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2408 $tat. nam $i inter
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2409 ip$as ba$es me-
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2410 di&aelig; proportio-
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2411 nales con$tituan
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2412 tur, tres ba$es $imul $umpt&aelig; ad maiorem ba$im in om-
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2413 nibus codem modo $e habebunt. Vnde fit, ut axes K l,
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2414 q r, t u &agrave; punctis p s x in eandem proportionem $ecen-
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2415 <marg>a. $exti.</marg>
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2416 tur. ergo linea x s per p tran$ibit: &amp; line&aelig; r u, s x, q t in-
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2417 ter $e &aelig;quidi$tantes erunt. Itaque cum fru$ti a g latera pro-
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2418 <foot>ducta</foot>
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2419 <p n=>37</p>
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2420 ducta $uerint, ita ut in unum punctum y cocant, erunt tri&agrave;
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2421 gala u y l, x y p, t y k inter $e $imilia: &amp; $imilia etiam triangu
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2422 la l y r, p y s, k y q. quare ut in 19 huius, demon$trabitur
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parents:
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2423 x p, ad p s: itemq; t <G>k</G> ad k q candem habere proportion&etilde;,
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2424 quam u l ad l r. Sed ut u l ad l <*>, ita e$t triangulum a b c ad
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parents:
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2425 triangulum a c d: &amp; ut t k ad K q, ita triangulum e f g ad
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2426 triangulum e g h. Vt autem triangulum a b c ad triangu-
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parents:
diff changeset
2427 lum a c d, ita pyramis a b c y ad pyramidem a c d y. &amp; ut
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diff changeset
2428 triangulum e f g ad triangulum e g h, ita pyramis e f g y
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2429 ad pyramidem e g h y; ergo ut pyramis a b c y ad pyramid&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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diff changeset
2430 <marg>19. quinti</marg>
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casties
parents:
diff changeset
2431 a c d y, ita pyramis e f g y ad pyramidem e g h y. reliquum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2432 igitur fru$t&utilde; l f ad reliquum fru$t&utilde; l h e$t ut pyramis a b c y
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2433 ad pyramidem a c d y, hoc e$t ut u l ad l r, &amp; ut x p ad p s.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2434 Qu&ograve;d cum fru$ti l f centrum grauitatis $its: &amp; fru$ti l h $it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2435 <marg>8. Archi-
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casties
parents:
diff changeset
2436 medis.</marg>
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casties
parents:
diff changeset
2437 centrum x: con$tat punctum p totius fru$ti a g grauitatis
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parents:
diff changeset
2438 e$$e centrum. Eodem modo fret demon$tratio etiam in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2439 aliis pyramidibus.</P>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2440 <P>Sit fru$tum a d &agrave; cono, uel coni portione ab$ci$$um, eu-
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
2441 ius maior ba$is circulus, uel ellip$is circa diametrum a b;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2442 minor circa diametrum c d: &amp; axis e f. diuidatur aut&etilde; e f
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
2443 in g, ita ut e g ad g f eandem proportionem habeat, quam
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2444 duplum diametri a b un&agrave; cum diametro e d ad duplum c d
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
2445 un&agrave; cum a b. Sitq; g h quarta pars line&aelig; g e: &amp; $it $ K item
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2446 quarta pars totius f e axis. Rur$us quam proportionem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2447 habet fru$tum a d ad conum, uel coni portionem, in cad&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2448 ba$i, &amp; &aelig;quali altitudine, habeat linea K h ad h l. Dico pun-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2449 ctum l fru$ti a d grauitatis centrum e$$e. Si enini fieri po-
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
2450 te$t, $it m centrum: producaturq; l m extra fru$tum in n:
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
2451 &amp; ut n l ad l m, ita fiat circulus, uel ellip$is circa diametr&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2452 a b ad aliud $pacium, in quo $it o. Itaque in circulo, uel
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
2453 ellip$i circa diametrum a b rectilinea figura plane de$cri-
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
2454 batur, ita ut qu&aelig; relinquuntur portiones $int o $pacio mi-
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
2455 nores: &amp; intelligatur pyramis a p b, ba$im habens rectili-
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
2456 neam figuram in circulo, uel ellip$i a b de$criptam: &agrave; qua
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
2457 <foot><*></foot>
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parents:
diff changeset
2458 <pb>
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casties
parents:
diff changeset
2459 fru$tum pyramidis $it ab$ci$$um. erit ex iis qu&aelig; proxime
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2460 tradidimus, fru$ti pyramidis a d ceutrum grauitatis l. Quo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2461 niam igitur portiones $pacio o minores $unt; habebit cir
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2462 <fig>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2463 culus, uel ellip$is a b ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2464 portiones dictas maior&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2465 proportionem, qu&agrave;m n l
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
2466 ad l m. $ed ut circulus, uel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2467 ellip$is a b ad portiones,
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
2468 ita a p b conus, uel coni
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2469 portio ad $olidas portio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2470 nes, id quod $upra demon
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2471 $tratum e$t: &amp; ut circulus
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
2472 <marg>22. huius</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2473 uel ellip$is c d ad portio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2474 nes, qu&aelig; ip $i in$unt, ita co
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
2475 nus, uel coni portio c p d
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2476 ad $olidas ip$ius portio-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2477 nes. Qu&ograve;d cum figur&aelig; in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2478 circulis, uel ellip$ibus a b
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2479 c d de$cript&aelig; $imiles $int,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2480 erit proportio circuli, uel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2481 ellip$is a b ad $uas portio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2482 nes, ead&etilde;, qu&aelig; circuli uel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2483 ellip$is c d ad $uas. ergo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2484 conus, uel coni portio a p
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2485 b ad portiones $olidas e&atilde;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2486 dem habet proportion&etilde;,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2487 quam conus, uel coni por
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
2488 tio c p d ad $olidas ip$ius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2489 <marg>19. qu&iacute;nti</marg>
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
2490 portiones. reliquum igi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2491 tur coni, uel coni portionis fru$t&utilde;, $cilicet a d ad reliquas
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2492 portiones $olidas in ip$o contentas eandem proportion&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2493 habet, quam conus, uel coni portio a p b ad $olidas portio
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2494 nes: hoc e$t eandem, quam circulus, uel ellip$is a b ad por
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2495 tiones planas. quare fru$tum coni, uel coni portionis a d
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2496 <foot>ad</foot>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2497 <p n=>38</p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2498 ad portiones $olidas maiorem habet proportion&etilde;, qu&agrave;m
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2499 n l ad l m: &amp; diuidendo fru$tum pyramidis ad dictas por-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2500 tiones maiorem proportionem habet, qu&agrave;m n m ad m l.
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
2501 fiat igitur ut fru$tum pyramidis ad portiones, ita q m ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2502 m l. Itaque quoniam &agrave; fru$to coni, uel coni portionis a d,
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
2503 cuius grauitatis centrum e$tm, au$ertur fru$tum pyrami-
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
2504 dis habens centrum l; erit reliqu&aelig; magnitudinis, qu&aelig; ex
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2505 portionibus $olidis con$tat; grauitatis c&etilde;trum in linea l m
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2506 producta, atque in puncto q, extra figuram po$ito: quod
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2507 fieri nullo modo pote$t. relinquitur ergo, ut punctum l $it
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2508 fru$ti a d grauitatis centrum. quz omnia demon$tranda
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2509 proponebantur.</P>
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
2510 <head>THEOREMA XXII. PROPOSITIO XXVII.</head>
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
2511 <P>OMNIVM $olidorum in $ph&aelig;ra de$cripto-
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
2512 rum, qu&aelig; &aelig;qualibus, &amp; $imilibus ba$ibus conti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2513 nentur, centrum grauitatis e$t idem, quod $ph&aelig;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2514 r&aelig; centrum.</P>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2515 <P>Solida eiu$modi corpora regularia appellare $olent, de
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2516 quibus agitur in tribus ultimis libris elementorum: $unt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2517 autem numero quinque, tetrahedrum, uel pyramis, hexa-
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casties
parents:
diff changeset
2518 hedrum, uel cubus, octahedrum, dodecahedrum, &amp; ico$a-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
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diff changeset
2519 hedrum.</P>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2520 <P>Sit primo a b c d pyramis &itilde; $ph&aelig;ra de$cripta, cuius $ph&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2521 r&aelig; centrum $it e. Dico e pyramidis a b c d grauitatis e$$e
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casties
parents:
diff changeset
2522 centrum. Si enim iuncta d c producatur ad ba$im a b c in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2523 f; exiis, qu&aelig; demon$trauit Campanus in quartodecimo li
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casties
parents:
diff changeset
2524 bro elementorum, propo$itione decima quinta, &amp; decima
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casties
parents:
diff changeset
2525 feptima, erit f centrum circuli circa triangulum a b c de-
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casties
parents:
diff changeset
2526 fcripti: atque erit e f $exta pars ip$ius $ph&aelig;r&aelig; axis. quare
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casties
parents:
diff changeset
2527 ex prima huius con$tat trianguli a b c grauitatis centrum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2528 e$$e punctum f: &amp; idcirco lineam d f e$$e pyramidis axem.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2529 <foot>K 2 <*></foot>
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casties
parents:
diff changeset
2530 <pb>
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casties
parents:
diff changeset
2531 <fig>
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casties
parents:
diff changeset
2532 At cum e f $it $exta pars axis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2533 $ph&aelig;r&aelig;, crit d e tripla e f. ergo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2534 punctum e e$t grauitatis cen-
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casties
parents:
diff changeset
2535 trum ip$ius pyramidis: quod
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2536 in uige$ima $ecunda huius de-
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parents:
diff changeset
2537 mon$tratum $uit. Sed e e$t cen
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2538 trum $ph&aelig;r&aelig;. Sequitur igitur,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2539 ut centrum grauitatis pyrami-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2540 dis in $ph&aelig;ra de$cript&aelig; idem
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
2541 $it, quod ip$ius $ph&aelig;r&aelig; cen-
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parents:
diff changeset
2542 trum.</P>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2543 <P>Sit cubus in $ph&aelig;ra de$criptus a b, &amp; oppo$itorum pla-
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
2544 norum lateribus bifariam diui$is, per puncta diui$ionum
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parents:
diff changeset
2545 plana ducantur, ut communis ip$orum $ectio $it rectali-
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parents:
diff changeset
2546 nea c d. Itaque $i ducatur a b, $olidi $cilicet diameter, line&aelig;
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
2547 a b, c d ex trige$iman onaun decimi$e$e bifariam $ecabunt.
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
2548 <fig>
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casties
parents:
diff changeset
2549 $ecent autem in puncto e. erit,
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casties
parents:
diff changeset
2550 e centr&utilde; grauitatis $olidi a b,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2551 id quod demon$tratum e$t in
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
2552 octaua huius. Sed quoniam ab
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2553 e$t $ph&aelig;r&aelig; diametro &aelig;qualis,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2554 ut in decima quinta propo$i-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2555 tione tertii decimilibri elem&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2556 torum o$tenditur: punctum e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2557 $ph&aelig;r&aelig; quoque centrum erit.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2558 Cubi igitur in $ph&aelig;ra de$cri-
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
2559 pti grauitatis centrum idem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2560 e$t, quod centrum ip$ius $ph&aelig;r&aelig;.</P>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2561 <P>Sit octahedrum a b c d e f, in $ph&aelig;ra de$criptum, cuius
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2562 $ph&aelig;r&aelig; centrum $itg. Dico punctum g ip$ius octahedri
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2563 grauitatis centrum e$$e. Con$tat enim ex iis, qu&aelig; demon-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2564 $trata $unt &agrave; Campano in quinto decimo libro elemento-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2565 rum, propo$itione $extadecima eiu$modi $olidum diuidi
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
2566 in duas pyramides &aelig;quales, &amp; $imiles; uidelicet in pyrami-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2567 <foot>dem,</foot>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2568 <p n=>39</p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2569 dem, cuius ba$is e$t quadratum a b c d, &amp; altitudo e g: &amp;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2570 in pyramidem, cuius ead&etilde; ba$is, altitudoq; f g; ut $int e g,
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
2571 g f $emidiametri $ph&aelig;r&aelig;, &amp; linea una. C&utilde;igitur g $it $ph&aelig;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2572 r&aelig; centrum, erit etiam centrum circuli, qui circa quadrat&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2573 a b c d de$cribitur: &amp; propterea eiu$dem quadrati grauita
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2574 tis centrum: quod in prima propo$itione huius demon-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2575 $tratum e$t. quare pyramidis a b c d e axis erit e g: &amp; pyra
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2576 midis a b c d f axis f g. Itaque $ith centrum grauitatis py-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2577 ramidis a b c d e, &amp; pyramidis a b c d f centrum $it <I>K:</I> per-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2578 $picuum e$t ex uige$ima $ecunda propo$itione huius, line&atilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2579 <fig>
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casties
parents:
diff changeset
2580 c h triplam e$$e h g: c&otilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2581 ponendoq; e g ip$ius g
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2582 h quadruplam. &amp; ead&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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2583 ratione f g quadrupl&atilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2584 ip$ius g <G>k.</G> quod cum e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2585 g, g f $int &aelig;quales, &amp; h
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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2586 g, g <I>K</I> nece$$ario &aelig;qua-
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
2587 les erunt. ergo ex quar
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casties
parents:
diff changeset
2588 ta propo$itione primi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2589 libri Archimedis de c&etilde;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
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2590 tro grauitatis planor&utilde;,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2591 totius octahedri, quod
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2592 ex dictis pyramidibus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2593 con$tat, centrum graui
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casties
parents:
diff changeset
2594 tatis erit punctum g idem, quod ip$ius $ph&aelig;r&aelig; centrum.</P>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2595 <P>Sit ico$ahedrum a d de$criptum in $ph&aelig;ra, cuius centr&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2596 $it g. Dico g ip$ius ico$ahedri grauitatis e$$e centrum. Si
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2597 enim ab angulo a per g ducatur recta linea u$que ad $ph&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2598 r&aelig; $uperficiem; con$tat ex $exta decima propo$itione libri
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2599 tertii decimi elementorum, cadere eam in angulum ip$i a
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casties
parents:
diff changeset
2600 oppo$itum. cadat in d: $itq; una aliqua ba$is ico$ahedri tri-
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casties
parents:
diff changeset
2601 angulum a b c: &amp; iunct&aelig; b g, c g producantur, &amp; cadant in
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2602 angulos e f, ip$is b c oppo$itos. Itaque per triangula
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casties
parents:
diff changeset
2603 a b c<*> d e f ducantur plana $ph&aelig;ram $ecantia<*> erunt <*> $e-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2604 <pb>
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casties
parents:
diff changeset
2605 ctiones circuli ex prima propo$itione $ph&aelig;ricorum Theo
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2606 do$ii: unus quidem circa triangulum a b c de$criptus: al-
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casties
parents:
diff changeset
2607 ter uero circa d e f: &amp; quoniam triangula a b c, d e f &aelig;qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2608 lia $unt, &amp; $imilia; erunt ex prima, &amp; $ecunda propo$itione
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2609 duodecimi libri clementorum, circuli quoque inter $e $e
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2610 &aelig;quales. po$tremo a centro g ad circulum a b c perpendi
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casties
parents:
diff changeset
2611 cularis ducatur g h; &amp; alia perpendicularis ducatur ad cir
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casties
parents:
diff changeset
2612 culum d e f, qu&aelig; $it g k; &amp; iungantur a h, d <G>k.</G> per$picuum
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casties
parents:
diff changeset
2613 e$t ex corollario prim&aelig; $ph&aelig;ricorum Theodo$ii, punctum
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diff changeset
2614 h centrum e$$e circuli a b c, &amp; <G>k</G> centrum circuli d e f. Quo
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diff changeset
2615 niam igitur triangulorum g a h, g d K latus a g e$t &aelig;quale la
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diff changeset
2616 teri g d; $unt enim &agrave; centro $ph&aelig;r&aelig; ad $uperficiem: atque
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diff changeset
2617 e$t a h &aelig;quale d k: &amp; ex $exta propo$itione libri primi $ph&aelig;
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diff changeset
2618 ricorum Theodo$ii g h ip$i g K: triangulum g a h &aelig;quale
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diff changeset
2619 erit, &amp; $imile g d <G>k</G> triangulo: &amp; angulus a g h &aelig;qualis an-
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diff changeset
2620 <marg>13. primi</marg>
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diff changeset
2621 gulo d g <I>K.</I> $ed anguli a g h, h g d $unt &aelig;quales duobus re-
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diff changeset
2622 ctis. crgo &amp; ip$i h g d, d g <G>k</G> duobus rectis &aelig;quales erunt.
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diff changeset
2623 <marg>14. primi</marg>
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parents:
diff changeset
2624 &amp; idcirco h g, g <I>K</I> una, atque eadem erit linea. cum autem
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diff changeset
2625 <fig>
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parents:
diff changeset
2626 h $it centr&utilde; circuli, &amp; tri-
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diff changeset
2627 anguli a b c grauitatis cen
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parents:
diff changeset
2628 tr&utilde; probabitur exiis, qu&aelig;
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parents:
diff changeset
2629 in prima propo$itione hu
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parents:
diff changeset
2630 ius tradita $unt. quare g h
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parents:
diff changeset
2631 erit pyramidis a b c g axis.
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diff changeset
2632 &amp; ob eandem cau$$am g k
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diff changeset
2633 axis pyramidis d e f g. lta-
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diff changeset
2634 que centrum grauitatls py
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parents:
diff changeset
2635 ramidis a b c g $it p&utilde;ctum
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parents:
diff changeset
2636 l, &amp; pyramidis d e f g $it m.
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diff changeset
2637 Simillter ut $upra demon-
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parents:
diff changeset
2638 $trabimus m g, g linter $e &aelig;quales e$$e, &amp; punctum g graui
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parents:
diff changeset
2639 tatis centrum magnitudinis, qu&aelig; ex utri$que pyramidibus
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diff changeset
2640 con$tat. eodem modo demon$trabitur, quarumcunque
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diff changeset
2641 duarum pyramidum, qu&aelig; opponuntur, grauitatis centr&utilde;
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diff changeset
2642 <foot><*></foot>
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diff changeset
2643 <p n=>40</p>
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diff changeset
2644 e$$e punctum g. Sequitur ergo utico$ahedri centrum gra
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diff changeset
2645 uitatis $it idem, quod ip$ius $ph&aelig;r&aelig; centrum.</P>
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parents:
diff changeset
2646 <P>Sit dodecahedrum a f in $ph&aelig;ra de$ignatum, $itque $ph&aelig;
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diff changeset
2647 r&aelig; centrum m. Dico m centrum e$$e grauitatis ip$ius do-
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diff changeset
2648 decahedri. Sit enim pentagonum a b c d e una ex duode-
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parents:
diff changeset
2649 cim ba$ibus $olidi a f: &amp; iuncta a m producatur ad $ph&aelig;r&aelig;
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parents:
diff changeset
2650 $uperficiem. cadet in angulum ip$i a oppo$itum; quod col-
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parents:
diff changeset
2651 ligitur ex decima $eptima propo$iticne tertiidecimi libri
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parents:
diff changeset
2652 clementorum. cadat in f. at $i ab aliis angulis b c d e per c&etilde;
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parents:
diff changeset
2653 trum itidem line&aelig; ducantur ad $uperficiem $ph&aelig;r&aelig; in pun
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diff changeset
2654 cta g h <G>k</G> l; cadent h&aelig; in alios angulos ba$is, qu&aelig; ip$i a b c d
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2655 ba$i opponitur. tran$eant ergo per pentagona a b c d e,
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parents:
diff changeset
2656 f g h K l plana $ph&aelig;ram $ecantia, qu&aelig; facient $ectiones cir-
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parents:
diff changeset
2657 culos &aelig;quales inter $e $e: po$tea ducantur ex centro $ph&aelig;r&aelig;
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2658 <fig>
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parents:
diff changeset
2659 m perpendiculares ad pla-
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2660 na dictorum circulor&utilde;; ad
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parents:
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2661 circulum quidem a b c d e
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parents:
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2662 perpendicularis m n: &amp; ad
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parents:
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2663 circulum f g h K l ip$a m o,
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2664 <marg>corol. pri
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parents:
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2665 m&aelig; fph&aelig;
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parents:
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2666 ricorum
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parents:
diff changeset
2667 Theod.</marg>
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2668 erunt puncta n o circulor&utilde;
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2669 centra: &amp; line&aelig; m n, m o in
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2670 ter $e &aelig;quales: qu&ograve;d circu-
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parents:
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2671 <marg>6. primi
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parents:
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2672 ph&aelig;rico
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parents:
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2673 rum.</marg>
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2674 li &aelig;quales $int. Eodem mo
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parents:
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2675 do, quo $upra, demon$trabi
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2676 mus lineas m n, m o in un&atilde;
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2677 atque eandem lineam con-
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2678 uenire. ergo cum puncta n o $int centra circulorum, con-
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parents:
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2679 $tat ex prima huius &amp; pentagonor&utilde; grauitatis e$$e centra:
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parents:
diff changeset
2680 idcircoq; m n, m o pyramidum a b c d e m, f g h <I>K</I> l m axes.
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2681 ponatur a b c d e m pyramidis grauitatis centrum p: &amp; py
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2682 ramidis f g h <G>k</G> l m ip$um q centrum. erunt p m, m q &aelig;qua-
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parents:
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2683 les, &amp; punctum m grauitatis centrum magnitudinis, qu&aelig;
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2684 ex ip$is pyramidibus con$tat. eod&etilde; modo probabitur qua-
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parents:
diff changeset
2685 rumlibet pyramidum, qu&aelig; &egrave; regione opponuntur, centr&utilde;
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2686 <pb>
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parents:
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2687 grauitatis e$$e punctum m. patetigitur totius dodecahe-
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2688 dri, centrum grauitatis id&etilde; e$$e, quod &amp; $ph&aelig;r&aelig; ip$um com
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parents:
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2689 prehendentis centrum. qu&aelig; quidem omnia demon$tra$$e
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2690 oportebat.</P>
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2691 <head>PROBLEMA VI. PROPOSITIO XXVIII.</head>
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2692 <P>DATA qualibet portione conoidis rectangu
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2693 li, ab$ci$$a plano ad axem recto, uel non recto; fie-
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2694 ri pote$t, ut portio $olida in$cribatur, uel circum-
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2695 $cribatur ex cylindris, uel cylindri portionibus,
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2696 &aelig;qualem habentibus altitudinem, ita ut recta li-
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parents:
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2697 nea, qu&aelig; inter centrum grauitatis portionis, &amp;
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parents:
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2698 figur&aelig; in$cript&aelig;, uel circum$cript&aelig; interiicitur,
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diff changeset
2699 $it minor qualibet recta linea propo$ita.</P>
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2700 <P>Sit portio conoidis rectanguli a b c, cuius axis b d, gra-
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2701 uitatisq; centrum e: &amp; $it g recta linea propo$ita. quam ue
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2702 ro proportionem habet linea b e ad lineam g, eandem ha-
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2703 beat portio conoidis ad $olidum h: &amp; circum$cribatur por
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parents:
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2704 tioni figura, $icuti dictum e$t, ita ut portiones reliqu&aelig; $int
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diff changeset
2705 $olido h minores: cuius quidem figur&aelig; centrum grauitatis
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2706 $it punctum <G>k.</G> Dico line&atilde; k e minorem e$$e linea g propo-
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2707 $ita. ni$i enim $it minor, uel &aelig;qualis, uel maior erit. &amp; quo-
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2708 niam figura circum$cripta ad reliquas portiones maiorem
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2709 <marg>8. qu&iacute;nti.</marg>
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2710 proportionem habet, qu&agrave;m portio conoidis ad $olidum h;
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2711 hoc e$t maiorem, qu&agrave;m b c ad g: &amp; b e ad g non minorem
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2712 habet proportionem, qu&agrave;m ad k e, propterea quod k e non
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2713 ponitur minor ip$a g: habcbit figura circum$cripta ad por
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2714 tiones reliquas maiorem proportionem qu&agrave;m b e ad e k:
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2715 <marg>29. qu&iacute;nti
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2716 ex tradi-
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2717 tione C&atilde;-
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2718 l ani.</marg>
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2719 &amp; diuidendo portio conoidis ad reliquas portiones habe-
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2720 bit maiorem, qu&agrave;m b <G>k</G> ad K e. quare $i fiat ut portio co-
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2721 <foot>noidis</foot>
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2722 <p n=>41</p>
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2723 noidis ad portiones reliquas, ita alia linea, qu&aelig; $it l <G>k</G> ad
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2724 <G>k</G> e: erit l k maior, quam b k: &amp; ideo punctum l extra por-
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2725 <fig>
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2726 tionem cadet. Quoni&atilde;
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2727 igitur &agrave; figura circum-
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2728 $cripta, cuius grauitatis
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2729 centrum e$t k, aufertur
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2730 portio conoidis, cuius
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2731 centrum e. habetq; l K
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2732 ad K e eam proportio-
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2733 nem, quam portio co-
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2734 noidis ad reliquas por-
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2735 tiones; erit punctum l
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2736 extra portionem cad&etilde;s,
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2737 centrum magnitudinis
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2738 ex reliquis portionibus compo$it&aelig;. illud autem fieri nullo
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2739 modo pote$t. quare con$tat lineam k e ip$a g linea propo$i
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2740 ta minorem e$$e.</P>
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2741 <P>Rur$us in$cribatur portioni figura, uidelicet cylindr us
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2742 <fig>
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2743 m n, ut $it ip$ius altitudo
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2744 &aelig;qualis dimidio axis b d:
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2745 &amp; quam proportionem
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2746 habet b e ad g, habeat m n
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2747 cylindrus ad $olidum o.
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2748 in$cribatur deinde eidem
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2749 alia figura, ita ut portio-
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2750 nes reliqu&aelig; $int $olido o
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2751 minores: &amp; centrum gra
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2752 uitatis figur&aelig; $it p. Dico
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2753 lineam p e ip$a g minor&etilde;
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2754 e$$e. $i enim n<*>n $it mi-
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2755 nor, codem, quo $upra modo demon$trabimus figuram in
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2756 $criptam ad reliquas portiones maiorem proportionem
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2757 habere, qu&agrave;m b e ad e p. &amp; $i fiat alia linea l e ad e p, ut e$t
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2758 figura in$cripta ad reliquas portiones, p&utilde;ctum l extra por
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2759 <foot>L</foot>
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2760 <pb>
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2761 tionem cadet: Itaque cum &agrave; portione conoidis, cuius gra-
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2762 uitatis centrum e auferatur in$cripta figura, centrum ha-
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2763 bens p: &amp; $itl e ad e p, ut figura in$cripta ad portiones reli
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2764 quas: erit magnitudinis, qu&aelig; ex reliquis portionibus con
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2765 $tat, centrum grauitatis punctum l, extra portionem ca-
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2766 dens. quod ficrinequit. ergo linca p e minor e$tip$a g li-
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2767 nea propo$ita.</P>
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2768 <P>Ex quibus per$picuum e$t centrum grauitatis
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2769 figur&aelig; in$cript&aelig;, &amp; circum$cript&aelig; eo magis acce
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2770 dere ad portionis centrum, quo pluribus cylin-
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2771 dris, uel cylindri portionibus con$tet: fiat&qacute;; figu
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2772 rain$cripta maior, &amp; circum$cripta minor. &amp;
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2773 quanquam continenter ad portionis centr&utilde; pro-
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2774 pius admoueatur: nunquam tamen ad ip$um per
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2775 ueniet. $equeretur enim figuram in$criptam, n&otilde;
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2776 $olum portioni, $ed etiam circum$cript&aelig; figur&aelig;
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2777 &aelig;qualem e$$e. quod e$t ab$urdum.</P>
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2778 <head>THEOREMA XXIII. PROPOSITIO XXIX.</head>
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2779 <P>CVIVSLIBET portionis conoidis rectangu-
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diff changeset
2780 li axis &agrave; c&etilde;tro grauitatis ita diuiditur, ut pars qu&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2781 terminatur ad uerticem, reliqu&aelig; partis, qu&aelig; ad ba
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2782 $im $it dupla.</P>
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parents:
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2783 <P>SIT portio conoidis rectanguli uel ab$ci$$a plano ad
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2784 axem recto, uel non recto: &amp; $ecta ip$a altero plano per ax<*>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2785 $it $uperficici $ectio a b crectanguli coni $ectio, uel parabo
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2786 le; plani ab$cindentis portionem $ectio $it recta linea a c:
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2787 axis portionis, &amp; $ectionis diameter b d. Sumatur autem
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2788 in linea b d punctum e, ita ut b e $itip$ius e d dupla. Dico
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2789 <foot>e por-</foot>
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2790 <p n=>42</p>
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2791 <fig>
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2792 e portionis a b
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2793 c grauitatis e$$e
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2794 centrum. Diui-
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2795 datur enim b d
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2796 bifariam in m:
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2797 &amp; rur$us d m, m
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parents:
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2798 b bifariam diui-
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2799 dantur in pun-
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2800 ctis n, o: in$cri-
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2801 baturq; portio-
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casties
parents:
diff changeset
2802 ni figura $olida,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2803 &amp; altera circum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2804 $cribatur ex cy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2805 lindris &aelig;qualem
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2806 altitudinem ha-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2807 bentibus, ut$u-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2808 perius dict&utilde; e$t'.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2809 Sit autem pri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2810 mum figura in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2811 $cripta cyl&itilde;drus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2812 f g: &amp; circ&utilde;$cri-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2813 pta ex cylindris
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2814 a h, K<*> con$tet.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2815 <marg>7. huius</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2816 punctum n erit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2817 centrum graui-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2818 tatis figur&aelig; in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2819 fcript&aelig;, medi&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2820 $cilicet ip$ius d
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2821 m axis: atq; id&etilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2822 erit centrum cy
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2823 lindri ah: &amp; cy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2824 lindri <G>k</G> l centr&utilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2825 o, axis b m me-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2826 dium. quare $i li
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2827 <foot>L 2</foot>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2828 <pb>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2829 <fig>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2830 neam o n ita di
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2831 ui$c<*>imus in p,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2832 ut qu&atilde; propor-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2833 tion&etilde; habet cy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2834 lindrus a h ad
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2835 cylindrum <G>k</G> l,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2836 habeat linea o p
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2837 <marg><*>imi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2838 <*> Ar-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2839 <*>dis</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2840 ad p n: centrum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2841 grauitatis toti-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2842 us figur&aelig; circ&utilde;-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2843 $cript&aelig; erit pun
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2844 <marg><*>o.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2845 <*>li.</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2846 ctum p. Sed cy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2847 lindri, qui $unt
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2848 &aelig;quali altitudi-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2849 ne, eandem in-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2850 ter $e $e, quam
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2851 ba$es propor&mdash;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2852 tionem habent:
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2853 e$tq; ut linea d b
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2854 ad b m, ita qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2855 drat&utilde; line&aelig; a d
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2856 ad quadrat&utilde; ip-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2857 $ius K m, ex uige
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2858 $ima primi libri
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2859 <marg><*>inti</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2860 conicor&utilde; &amp; ita
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2861 quadratum a c
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2862 ad quadrat&utilde; K
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2863 <marg><*>de-</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2864 g: hoc e$t circu-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2865 lus circa diame
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2866 trum a c ad cir-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2867 culum circa dia
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2868 metrum k g. du
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2869 pla e$t autem li-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2870 nca d b line&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2871 <foot>b m.</foot>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2872 <p n=>43</p>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2873 b m. crgo circulus a c circuli k g: &amp; idcirco cylindrus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2874 a h cylindri k. l duplus erit. quare &amp; linea o p dupla
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2875 ip$ius p n. Deinde in$cripta &amp; circum$cripta portioni
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2876 alia figura, ita ut in$cripta con$tituatur ex tribus cylin-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2877 dris q r, s g, t u: circum$cripta uero ex quatuor a x, y z,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2878 K <G>g, q l:</G> diuidantur b o, o m, m n, n d bifariam in punctis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2879 <G>m g p r.</G> Itaque cylindri <G>q l</G> centrum grauitatis e$t punctum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2880 <G>m:</G> &amp; cylindri <G>k <*></G> centrum <G>g.</G> ergo $i linea <G>m g</G> diuidatur in <G>s,</G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2881 ita ut <G>m s</G> ad <G>s g</G> proportion&etilde; ea habeat, quam cylindrus K <G><*></G>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2882 ad cylindrum <G>q l,</G> uidelicet quam quadratum <G>k</G> nr ad qua-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2883 <marg>20. primi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2884 conicor</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2885 dratum <*> o, hoc e$t, quam linea m b ad b o: erit <G>s</G> centrum
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2886 magnitudinis compo$it&aelig; ex cylindris <G>k g, q l.</G> &amp; cum linca
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2887 m b $it dupla b o, erit &amp; <G>m s</G> ip$ius <G>s g</G> dupla. pr&aelig;terea quo-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2888 niam cylindri y z centrum grauitatis e$t <G>p,</G> linea <G>s p</G> ita diui
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2889 $ain <G>t,</G> ut <G>s t</G> ad <G>t p</G> eam habeat proportionem, quam cylin
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2890 drus y z ad duos cylindros K <G>g, q l:</G> crit <G>t</G> centrum magnitu
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2891 dinis, qu&aelig; ex dictis tribus cylindris con$tat. cylindrus au-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2892 t&etilde; y z ad cylindrum <G>q l</G> e$t, utlinea n b ad b o, hoc e$t ut 3
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2893 ad 1: &amp; ad cylindrum k <G><*>,</G> ut n b ad b m, uidelicet ut 3 ad 2.
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2894 quare y z cyl&itilde;drus duobus cylindris k <G>g, q l</G> &aelig;qualis erit. &amp;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2895 propterea linea <G>s t</G> &aelig;qualis ip$i <G>t p.</G> denique cylindri a x
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2896 centrum grauitatis e$t punctum <G>r.</G> &amp; cum <G>t r</G> diui$a fuerit
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2897 in e&atilde; proportionem, quam habet cylindrus ax ad tres cy-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2898 lindros y z, k <G>g, q l:</G> erit in eo puncto centrum grauitatis
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2899 totius figur&aelig; circ&utilde;$cript&aelig;. Sed cylindrus a x ad ip$um y z
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diff changeset
2900 e$t ut linea d b ad b n: hoc e$t ut 4 ad 3: &amp; duo cylindri k <G>h
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diff changeset
2901 q l</G> cylindro y z $unt &aelig;quales. cylindrus igitur a x ad tres
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diff changeset
2902 iam dictos cylindros e$t ut 2 ad 3. Sed quoni&atilde; <G>m s</G> e$t dua-
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diff changeset
2903 rum partium, &amp; <G>s g</G> unius, qualium <G>m p</G> e$t $ex; erit <G>s p</G> par-
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diff changeset
2904 tium quatuor: proptereaq; <G>t p</G> duarum, &amp; <G>g p,</G> hoc e$t <G>p r</G>
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diff changeset
2905 trium. quare $equitur ut punctum <G>p</G> totius figur&aelig; circum
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parents:
diff changeset
2906 $cript&aelig; $it centrum. Itaque fiat <G>g u</G> ad <G>u p,</G> ut <G>m s</G> ad <G>s g.</G> &amp; <G>u r</G>
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parents:
diff changeset
2907 bifariam diuidatur in <G>f.</G> Similiter utin circum$cripta figu
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parents:
diff changeset
2908 ra o$tendetur centrum magnitudinis compo$it&aelig; ex cylin-
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diff changeset
2909 <pb>
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diff changeset
2910 <fig>
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parents:
diff changeset
2911 dris s g, tu e$$e
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parents:
diff changeset
2912 punctum <G>u:</G> &amp;
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parents:
diff changeset
2913 totius figur&aelig; in
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parents:
diff changeset
2914 $cript&aelig;, qu&aelig; c&otilde;-
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diff changeset
2915 $tat ex cylindris
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diff changeset
2916 q r, $ g, t u e$$e <G>f</G>
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diff changeset
2917 centrum. Sunt
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parents:
diff changeset
2918 enim hi cylindri
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diff changeset
2919 &aelig;quales &amp; $imi-
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diff changeset
2920 les cylindris y z,
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diff changeset
2921 K <G>h, q l,</G> figur&aelig;
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parents:
diff changeset
2922 circum$cript&aelig;.
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diff changeset
2923 Quoni&atilde; igitur
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diff changeset
2924 ut b e ad e d, ita
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diff changeset
2925 e$t o p ad p n;
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parents:
diff changeset
2926 utraq; enim u-
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parents:
diff changeset
2927 triu$que e$t du-
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diff changeset
2928 pla: erit compo
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diff changeset
2929 nendo, ut b d ad
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diff changeset
2930 d e, ita o n ad n
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parents:
diff changeset
2931 p; &amp; permutan
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parents:
diff changeset
2932 do, ut b d ad o
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diff changeset
2933 n, ita d e ad n p.
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parents:
diff changeset
2934 Sed b d dupla
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diff changeset
2935 e$t o n. ergo &amp;
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diff changeset
2936 e d ip$ius n p du
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diff changeset
2937 pla erit. qu&ograve;d $i
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diff changeset
2938 e d bifariam di-
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diff changeset
2939 uidatur &itilde; <G>x,</G> erit
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diff changeset
2940 <G>x</G> d, uel e <G>x</G> &aelig;-
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diff changeset
2941 qualis n p: &amp;
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parents:
diff changeset
2942 $ublata e n, qu&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2943 e$t c&otilde;munis u-
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2944 trique e <G>x,</G> p n,
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2945 <foot>relin-</foot>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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2946 <p n=>44</p>
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diff changeset
2947 relinquetur p e ip$i n <G>x</G> &aelig;qualis. cum autem b e $it dupla
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diff changeset
2948 e d, &amp; o p dupla p n, hoc e$tip$ius e <G>x,</G> &amp; reliquum, uideli-
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
2949 <marg>1<*> quinti</marg>
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diff changeset
2950 cet b o un&agrave; cum p e ip$ius reliqui <G>x</G> d duplnm erit. e$tque
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
2951 b o dupla <G>r</G> d. ergo p e, hoc e$t n <G>x</G> ip$ius <G>x r</G> dupla. $ed d n
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
2952 dupla e$t n <G>r.</G> reliquaigitur d <G>x</G> dupla reliqu&aelig; <G>x</G> n. $unt au-
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
2953 tem d <G>x,</G> p n inter $e &aelig;quales: itemq; &aelig;quales <G>x</G> n, p e. qua-
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
2954 re con$tat n p ip$ius p e duplam e$$e. &amp; idcirco p e ip$i e n
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2955 &aelig;qualem. Rur$us cum $it <G>m g</G> duplao <G><*>,</G> &amp; <G>m s</G> dupla <G>s g;</G> erit
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
2956 etiam reliqua <G>g s</G> reliqu&aelig; <G>s</G> o dupla. Eadem quoque ratione
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
2957 c&otilde;cludetur <G>p u</G> dupla <G>u</G> m. ergo ut <G>g s</G> ad <G>s</G> o, ita <G>p u</G> ad <G>u</G> m:
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
2958 componendoq;, &amp; permutando, ut <G>g</G> o ad <G>p</G> m, ita o <G>s</G> ad
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
2959 m <G>u:</G> &amp; $unt &aelig;quales <G>g</G> o, <G>p</G> m. quare &amp; o <G>s,</G> m <G>u</G> &aelig;quales. pr&aelig;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
2960 terea <G>s p</G> dupla e$t <G>p t,</G> &amp; <G>g p</G> ip$ius <G>p</G> m. reliquaigitur <G>s g</G> re
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
2961 liqu&aelig; m <G>t</G> dupla. atque erat <G>g s</G> dupla <G>s</G> o. ergo m <G>t, s</G> o &aelig;-
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
2962 quales $unt: &amp; ita &aelig;quales m <G>u,</G> n <G>f.</G> at o <G>s,</G> e$t &aelig;qualis
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
2963 m <G>u.</G> Sequitur igitur, ut omnes o <G>s,</G> m <G>t,</G> m <G>u,</G> n <G>f</G> in-
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
2964 ter $e $int &aelig;quales. Sed ut <G>r p</G> ad <G>p t,</G> hoc e$t ut 3 ad 2, ita n d
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
2965 ad d <G>x:</G> permut&atilde;doq; ut <G>r p</G> ad n d, ita <G>p t</G> ad d <G>x.</G> &amp; $&utilde;t &aelig;qua
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
diff changeset
2966 les <G>r p,</G> n d. ergo d <G>x,</G> hoc e$t n p, &amp; <G>p t</G> &aelig;quales. Sed etiam &aelig;-
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
2967 quales n <G>p, p</G> m. reliqua igitur <G>p</G> preliqu&aelig; m <G>t,</G> hoc e$t ip$i
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
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parents:
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2968 n <G>f</G> &aelig;qualis erit. quare dempta p <G>p</G> ex p e, &amp; <G>f</G> n dempta ex
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
2969 n e, relinquitur p e &aelig;qualis e <G>f.</G> Itaque <G>p, f</G> centra figurar&utilde;
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parents:
diff changeset
2970 $ecundo loco de$criptarum a primis centris p n &aelig;quali in-
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parents:
diff changeset
2971 teruallo recedunt. qu&ograve;d $i rur$us ali&aelig; figur&aelig; de$cribantur,
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parents:
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2972 codem modo demon$trabimus earum centra &aelig;qualiter ab
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2973 his recedere, &amp; ad portionis conoidis centrum propius ad
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parents:
diff changeset
2974 moueri. Ex quibus con$tat lineam <G>p f</G> &agrave; centro grauitatis
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parents:
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2975 portionis diuidi in partes &aelig;quales. Si enim fieri pote$t, non
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parents:
diff changeset
2976 $it centrum in puncto e, quod e$t line&aelig; <G>p f</G> medium: $ed in
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parents:
diff changeset
2977 <G>y:</G> &amp; ip$i <G>p y</G> &aelig;qualis fiat <G>f <*>.</G> Cumigitur in portione $olida
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2978 qu&aelig;dam figura in$cribi pos$it, ita utlinea, qu&aelig; inter cen-
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parents:
diff changeset
2979 trum grauitatis portionis, &amp; in$cript&aelig; figur&aelig; interiicitur,
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2980 qualibet linea propo$ita $it minor, quod proxime demon-
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parents:
diff changeset
2981 $trauimus: perueniet tandem <G>f</G> centrum in$cript&aelig; figur&aelig;
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parents:
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2982 <pb>
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2983 <fig>
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parents:
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2984 <p n=>45</p>
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casties
parents:
diff changeset
2985 ad punctum <G>w.</G> Sed quoniam <G>p</G> circum$criptaitidem alia
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diff changeset
2986 figura &aelig;quali interuallo ad portionis centrum accedit, ubi
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diff changeset
2987 primum <G>f</G> applicuerit $e ad <G>w,</G> &amp; <G>p</G> ad punct&utilde; <G>y,</G> hoc e$t ad
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diff changeset
2988 portionis centrum $e applicabit. quod fieri nullo modo
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diff changeset
2989 po$$e per$picuum e$t. non aliter idem ab$urdum $equetur,
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2990 fi ponamus centrum portionis recedere &agrave; medio ad par-
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diff changeset
2991 tes <G>w;</G> e$$et enim aliquando centrum figur&aelig; in$cript&aelig; idem
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diff changeset
2992 quod portionis centr&utilde;. ergo punctum e centrum erit gra
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diff changeset
2993 uitatis portionis a b c. quod demon$trare oportebat.</P>
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diff changeset
2994 <P>Quod autem $upra dem&otilde;$tratum e$t in portione conoi-
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diff changeset
2995 dis recta per figuras, qu&aelig; ex cylindris &aelig;qualem altitudi-
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2996 dinem habentibus con$tant, idem $imiliter demon$trabi-
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diff changeset
2997 mus per figuras ex cylindri portionibus con<*>antes in ea
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diff changeset
2998 portione, qu&aelig; plano non ad axem recto ab$cinditur. ut
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diff changeset
2999 enim tradidimus in commentariis in undecimam propo$i
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diff changeset
3000 tionem libri Archimedis de conoidibus &amp; $ph&aelig;roidibus.
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diff changeset
3001 portiones cylindri, qu&aelig; &aelig;quali $unt altitudine eam inter $e
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diff changeset
3002 $e proportionem habent, quam ip$arum ba$es: ba$es aut&etilde;
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diff changeset
3003 <marg>corol. <*>
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diff changeset
3004 de conoi-
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diff changeset
3005 dibus &amp;
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diff changeset
3006 $ph&aelig;roi-
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diff changeset
3007 dibus.</marg>
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diff changeset
3008 qu&aelig; $unt ellip$es $imiles eandem proportionem habere,
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diff changeset
3009 quam quadrata diametrorum eiu$dem rationis, ex corol-
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diff changeset
3010 lario $eptlm&aelig; propo$itionis libri de conoidibus, &amp; $ph&aelig;-
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diff changeset
3011 roidibus, manife$te apparet.</P>
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diff changeset
3012 <head>THEOREMA XXIIII. PROPOSITIO XXX.</head>
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diff changeset
3013 <P>Sr &agrave; portione conoidis rectanguli alia portio
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diff changeset
3014 ab$cindatur, plano ba$i &aelig;quidi$tante; habebit
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diff changeset
3015 portio tota ad eam, qu&aelig; ab$ci$$a e$t, duplam pro
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diff changeset
3016 portion em eius, qu&aelig; e$t ba$is maioris portionis
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diff changeset
3017 ad ba$i m minoris, uel qu&aelig; axis maioris ad axem
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
3018 minoris.</P>
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diff changeset
3019 <foot>M</foot>
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diff changeset
3020 <pb>
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parents:
diff changeset
3021 <P>ABSCINDATVR &agrave; portione conoidis rectanguli
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
3022 a b c alia portio e b f, plano ba$i &aelig;quidi$tante: &amp; eadem
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
3023 portio $ecetur alio plano per axem; ut $uperficiei $ectio $it
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
3024 parabole a b c: planor&utilde; portiones ab$cindentium rect&aelig;
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
3025 linc&aelig; a c, e f: axis autem portionis, &amp; $ectionis diameter
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
3026 b d; quam linea e fin puncto g $ecet. Dico portionem co-
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
3027 noidis a b c ad portionem e b f duplam proportionem ha-
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
3028 bere eius, qu&aelig; e$t ba$is a c ad ba$im e f; uel axis d b ad b <*>
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
3029 axem. Intelligantur enim duo coni, $eu coni portiones
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
3030 a b c, e b f, e&atilde;dem ba$im, quam portiones conoidis, &amp; &aelig;qua
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
3031 lem habentes altitudinem. &amp; quoniam a b c portio conoi
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
3032 dis fe$quialtera e$t coni, $eu portionis coni a b c; &amp; portio
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
3033 e b f coni feu portionis coni e b f e$t $e$quialtera, quod de-
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
3034 <fig>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3035 mon$trauit Archimedes in propo$itionibus 23, &amp; 24 libri
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
3036 de conoidibus, &amp; $ph&aelig;roidibus: erit conoidis portio ad
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
3037 conoidis portionem, ut conus ad conum, uel ut coni por-
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
3038 tio ad coni portionem. Sed conns, nel coni portio a b c ad
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
3039 conum, uel coni portionem e b f compo$itam proportio-
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
3040 nem habet ex proportione ba$is a c ad ba$im e f, &amp; ex pro-
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
3041 portione altitudinis coni, uel coni portionis a b c ad alti-
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
3042 tudinem ipfius e b f, ut nos demon$trauimus in com men-
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
3043 tariis in undecimam propo$itionem eiu$dem libri A rchi-
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parents:
diff changeset
3044 medis: altitudo autem ad altitudinem c$t, ut axis ad axem.
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
3045 quod quidem in conis rectis per$picuum e$t, in $calenis ue
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
3046 <foot><*></foot>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3047 <p n=>46</p>
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
3048 ro ita demon$trabitur. Ducatur &agrave; puncto b ad planum ba-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3049 $is a c perpendicularis linea b h, qu&aelig; ip$am e fin K $ecet.
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
3050 erit b h altitudo coni, uel coni portionis a b c: &amp; b K altitu||
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
3051 <marg>16. unde-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3052 cim<*></marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3053 do efg. Quod cum line&aelig; a c, e f inter $e &aelig;quidi$tent, $unt
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
3054 enim planorum &aelig;quidi$tantium $ectiones: habebit d b ad
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
3055 <marg>4 <*>.</marg>
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parents:
diff changeset
3056 b g proportionem eandem, quam h b ad b <G>k.</G> quare por-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3057 tio conoidis a b c ad portionem e f g proportionem habet
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
3058 compo$itam ex proportione ba$is a c ad ba$im e f; &amp; ex
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
3059 <marg>2. duode
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3060 cimi</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3061 proportione d b axis ad axem b g. Sed circulus, uel
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3062 ellip$is circa diametrum a c ad circulum, uel ellip$im
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
3063 <marg>7. de co-
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3064 noidibus
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3065 &amp; $ph&aelig;-
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
3066 roidibus</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3067 circa e f, e$t ut quadratum a c ad quadratum e f; hoc e$t ut
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
3068 quadrat&utilde; a d ad quadrat&utilde; e g. &amp; quadratum a d ad quadra
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
3069 tum e g e$t, ut linea d b ad lineam b g. circulus igitur, uel el
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
3070 <marg><*>. quinti</marg>
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casties
parents:
diff changeset
3071 lip$is circa diametrum a c ad circul&utilde;, uel el<*>$im circa e f,
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3072 <marg>20. p<*>mi
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3073 comcor&utilde;</marg>
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3074 hoc e$t ba$is ad ba$im eandem proportion<*>t, qu&atilde;
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3075 d b axis ad axem b g. ex quibus $equitur por<*>em a b c
22d6a63640c6 moved texts from SVN https://it-dev.mpiwg-berlin.mpg.de/svn/mpdl-project-content/trunk/texts/eXist/
casties
parents:
diff changeset
3076 ad portionem e b f habere proportionem duplam eius,
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casties
parents:
diff changeset
3077 qu&aelig; e$t ba$is a c ad ba$im e f: uel axis d b ad b g axem. quod
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3078 demon$trandum proponebatur.</P>
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3079 <head>THEOREMA XXV. PROPOSITIO XXXI.</head>
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3080 <P>Cuiuslibet fru$ti &agrave; portione rectanguli conoi
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3081 dis ab$cis$i, centrum grauitatis e$t in axe, ita ut
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3082 demptis primum &agrave; quadrato, quod fit ex diame-
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3083 tro maioris ba$is, tertia ip$ius parte, &amp; duabus
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3084 tertiis quadrati, quod fit ex diamerro ba$is mino-
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3085 ris: deinde &agrave; tertia parte quadrati maioris ba$is
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3086 rur$us dempta portione, ad quam reliquum qua
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3087 drati ba$is maioris un&agrave; cum dicta portione dupl&atilde;
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3088 proportionem habeat eius, qu&aelig; e$t quadrati ma-
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3089 <foot>M 2</foot>
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3090 <pb>
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3091 ioris ba$is ad quadratum minoris: centrum $it in
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3092 eo axis puncto, quo ita diuiditur ut pars, qu&aelig; mi
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3093 norem ba$im attingit ad alteram partem eandem
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3094 proportionem habeat, quam dempto quadrato
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3095 minoris ba$is &agrave; duabus tertiis quadrati maioris,
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3096 habet id, quod reliquum e$t un&agrave; cum portione &agrave;
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diff changeset
3097 tertia quadrati maioris parte dempta, ad reliqu&atilde;
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3098 eiu$dem terti&aelig; portionem.</P>
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3099 <P>SIT fru$tum &agrave; portione rectanguli conoidis ab$ci$$um
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3100 a b c d, cuius maior ba$is circulus, uel ellip$is circa diame-
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3101 trum b c, minor circa diametrum a d; &amp; axis e f. de$criba-
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3102 tur<*>atem portio conoidis, &agrave; quo illud ab$ci$$um e$t, &amp; pla-
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3103 <fig>
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3104 no per axem ducto $ecetur; ut $uperficiei $ectio $it parabo-
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3105 le b g c, cuius diameter, &amp; axis portionis g f: deinde g f diui
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3106 datur in puncto h, ita ut g h $it dupla h f: &amp; rur$us g e in ean
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3107 dem proportionem diuidatur: $itq; g k ip$ius k e dupla. I&atilde;
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3108 ex iis, qu&aelig; proxime demon$trauimus, con$tat centrum gra
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3109 uitatis portionis b g c e$$e h punctum: &amp; portionis a g c
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3110 punctum k. $umpto igitur infra h puncto l, ita ut <G>k</G> h ad h l
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3111 <foot>eam</foot>
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3112 <p n=>47</p>
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3113 eam proportionem habeat, quam a b c d fru$tum ad por-
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3114 tionem a g d; erit punctum l eius fru$ti grauitatis c&etilde;trum:
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3115 habebitq; componendo K l ad l h proportionem eandem,
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3116 <marg>20. 1. con<*>
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3117 corum.</marg>
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3118 quam portio conoidis b gc ad a g d portionem. Itaq; quo
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3119 niam quadratum b f ad quadratum a e, hoc e$t quadratum
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3120 b c ad quadratum a d e$t, ut linea fg ad ge: erunt du&aelig; ter-
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3121 ti&aelig; quadrati b c ad duas tertias quadrati a d, uth g ad g k:
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3122 &amp; $i &agrave; duabus tertiis quadrati b c dempt&aelig; fuerint du&aelig; ter-
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3123 ti&aelig; quadrati a d: erit diuid&etilde;do id, quod relinquitur ad duas
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3124 tertias quadrati a d, ut h k ad k g. Rur$us du&aelig; terti&aelig; quadra
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3125 ti a d ad duas tertias quadrati b c $unt, ut k g ad g h: &amp; du&aelig;
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3126 terti&aelig; quadrati b c ad terti&atilde; part&etilde; ip$ius, ut g h ad h f. ergo
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3127 ex &aelig;quali id, quod relinquitur ex duabus tertiis quadrati
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3128 b c, demptis ab ip$is quadrati a d duabus tertiis, ad terti&atilde;
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3129 partem quadrati b c, ut k h ad h f: &amp; ad portionem eiu$d&etilde;
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3130 terti&aelig; partis, ad quam un&agrave; cum ip$a portione, duplam pro
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3131 portionem habeat eius, qu&aelig; e$t quadrati b c ad quadrat&utilde;
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3132 a d, ut K l ad l h. habet enim K l ad l h candem proportio-
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3133 nem, quam conoidis portio b g c ad portionem a g d: por-
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3134 tio autem b g c ad portionem a g d duplam proportionem
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3135 habet eius, qu&aelig; e$t ba$is b c ad ba$im a d: hoc e$t quadrati
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3136 <marg>30 hui<*></marg>
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3137 b c ad quadratum a d; ut proxime demon$tratum e$t. quare
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3138 dempto a d quadrato &agrave; duabus tertiis quadrati b c, erit id,
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3139 quod relinquitur un&agrave; cum dicta portione terti&aelig; partis ad
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3140 reliquam eiu$dem portionem, ut e l ad l f. Cum igitur cen-
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3141 trum grauitatis fru$ti a b c d $it l, &agrave; quo axis e f in eam, qu&atilde;
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3142 diximus, proportionem diuidatur; con$tat uer&utilde; e$$e illud,
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3143 quod demon$trandum propo$uimus.</P>
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3144 <head>FINIS LIBRI DE CENTRO</head>
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3145 <head>GRAVITATIS SOLIDORVM.</head>
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3146 <P>Impre$$. Bononi&aelig; cum licentia Superiorum,
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3147 <*></P>