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version 1.121, 2003/09/11 19:26:20

version 1.133, 2003/09/12 21:17:24

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<cvs_file>stevi_weegc_01_nl_1586</cvs_file>

<cvs_version/>

</info> <text> <front> </front> <body>

</info><text><front><section>

<chap>

<pb/><s>NOT_TRANSCRIBED</s>

<pb/>

DE <lb/>

BEGHINSELEN <lb/>

DER WEEGHCONST <lb/>

BESCHREIVEN DVER <lb/>

SIMON STEVIN<emph.end/> <lb/>

van Brugghe.</s>

<s>

<emph type="other"/>Tot Leyden<emph.end/>, <lb/>

Inde Druckerye van Christoffel Plantijn <lb/>

By Francoys van Raphelinghen. <lb/>

MDLXXXVI

</s>

<pb/><s>NOT_TRANSCRIBED</s>

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<pb/>

<s> ... </s><s>dattet voormael een seer wys, gheleert, ende ouertreflick Gheslacht is gheweest, als vooren bethoont is, daertoe ghecommen sijnde met langher tijdt, duer veel eruaringhen. </s><s>Ende soo wy van dese voorghanghers weerdighe navolghers willen gheacht sijn, en sullen niet duer een beestelick ghetuych van ondancbaerheyt, so groote gauen ons naghelaten, duer onwetenheyt versmaen, noch, den lasteraers diese niet en kennen, sottelick gheloouen, noch verlatende den spieghel der talen, ons dickmael behaghen in haer leelick schrapsel van schuym der vuyllicheyt; maer sullen ter contrari die clouclick beschermen, niet met ydel woorden als d'hare sijn, noch na t'onuerstandt van hemlien die de goetheyt der Saken in haer talen beschreuen, onbescheydelick de goetheyt der talen meenen te wesen; maer ghelijck t'gout duer t'vier beproeft wort, alsoo salmen haer weerdicheyt duer de daet <arrow.to.target n="marg12"/> bethoonen: Welcke sal die sijn? dese, neemt voor grondt <arrow.to.target n="marg13"/> t'ghene in al d'ander spraken tot noch toe der Naturen diepe verholentheden sijn, welcke sy niet ter deghe bedien en connen, als dat sy v (onder duysentich anderen daer het Duytsch vol af is) dit na segghen: <arrow.to.target n="marg14"/> <arrow.to.target n="marg15"/>; ende diet soo doen connen, belooftse vrielijck een koeck; Ia dat sy t'auent op sullen blijuen (voor kinderen dienen doch kinder prijsen <arrow.to.target n="marg16"/>) ende ick verseker v dat ghyder sonder schade sult afcommen, want het is blijckelijck ghenouch wat sy hier in vermueghen, te weten voor dese woorden langhe redenen te stellen, die t'onderscheyt der palen <arrow.to.target n="marg17"/>, ende de form der eueredenheyt <arrow.to.target n="marg18"/> oueral seer <pb/>verduysteren. </s>

<s> ... </s><s>dattet voormael een seer wys, gheleert, ende ouertreflick Gheslacht is gheweest, als vooren bethoont is, daertoe ghecommen sijnde met langher tijdt, duer veel eruaringhen. </s><s>Ende soo wy van dese voorghanghers weerdighe navolghers willen gheacht sijn, en sullen niet duer een beestelick ghetuych van ondancbaerheyt, so groote gauen ons naghelaten, duer onwetenheyt versmaen, noch, den lasteraers diese niet en kennen, sottelick gheloouen, noch verlatende den spieghel der talen, ons dickmael behaghen in haer leelick schrapsel van schuym der vuyllicheyt; maer sullen ter contrari die clouclick beschermen, niet met ydel woorden als d'hare sijn, noch na t'onuerstandt van hemlien die de goetheyt der Saken in haer talen beschreuen, onbescheydelick de goetheyt der talen meenen te wesen; maer ghelijck t'gout duer t'vier beproeft wort, alsoo salmen haer weerdicheyt duer de daet <arrow.to.target n="marg12"/> bethoonen: <emph.end/><emph.end/><emph.end/>Welcke sal die sijn? dese, neemt voor grondt <arrow.to.target n="marg13"/> t'ghene in al d'ander spraken tot noch toe der Naturen diepe verholentheden sijn, welcke sy niet ter deghe bedien en connen, als dat sy v (onder duysentich anderen daer het Duytsch vol af is) dit na segghen: <arrow.to.target n="marg14"/> <arrow.to.target n="marg15"/>; ende diet soo doen connen, belooftse vrielijck een koeck; Ia dat sy t'auent op sullen blijuen (voor kinderen dienen doch kinder prijsen <arrow.to.target n="marg16"/>) ende ick verseker v dat ghyder sonder schade sult afcommen, want het is blijckelijck ghenouch wat sy hier in vermueghen, te weten voor dese woorden langhe redenen te stellen, die t'onderscheyt der palen <arrow.to.target n="marg17"/>, ende de form der eueredenheyt <arrow.to.target n="marg18"/> oueral seer <pb/>verduysteren. </s>

<s>

<margin.target id="marg12"/>Effectum.

<margin.target id="marg13"/>Subiecto.

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<s> Dit is t'ghene wy vande weerdicheyt der Duytsche tael voorghenomen hadden te verclaren; Inde selue sullen wy de WEEGHCONST, die de wonderlicste der is, eerst tot Constens form laten commen, als spraeck die der Natueren eyghenschappen grondelicxt beteeckenen can, ende als bequaemste wit, daer al d'ander die willen, tot yder ghemeentens grootste nut, na micken, ende haer bepalinghen <arrow.to.target n="marg24"/>, daer inde Consten veel an gheleghen is, na rechten mueghen. </s><s>Oock by aldien der Duytschen vliet daerin alsoo vermeerderde, ghelijct de reden wel eyscht, t'selue soude ons voornemen verstercken om met ander angheuanghen voort te varen: Doch soo de contrarie gheschiede, ick can my vernoughen in een eerlick voornemen mijn goede wille te verclaren, welcke in haer beroup tot yders dienst gheeyghent is.</s>

<s> Dit is t'ghene wy vande weerdicheyt der Duytsche tael voorghenomen hadden te verclaren; Inde selue sullen wy de WEEGHCONST, die de wonderlicste der <emph.end/><emph.end/><emph.end/>is, eerst tot Constens form laten commen, als spraeck die der Natueren eyghenschappen grondelicxt beteeckenen can, ende als bequaemste wit, daer al d'ander die willen, tot yder ghemeentens grootste nut, na micken, ende haer bepalinghen <arrow.to.target n="marg24"/>, daer inde Consten veel an gheleghen is, na rechten mueghen. </s><s>Oock by aldien der Duytschen vliet daerin alsoo vermeerderde, ghelijct de reden wel eyscht, t'selue soude ons voornemen verstercken om met ander angheuanghen voort te varen: Doch soo de contrarie gheschiede, ick can my vernoughen in een eerlick voornemen mijn goede wille te verclaren, welcke in haer beroup tot yders dienst gheeyghent is.</s>

<pb/>

<s>

<margin.target id="marg24"/>Definitiones.

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<s>DE Beghinselen der Weeghconst, welcke van swaerheyt sijn duer t'ghedacht van natuerlicke stof gheweert, sullen in twee boucken begrepen worden. </s><s>Des eersten boucx eerste deel sal van 14 * bepalinghen wesen: T'ander van 28 * voorstellen, van de ghedaenten der ghewichten, die tweederhande sijn, als Rechtwichten, ende Scheefwichten. </s><s>Der Rechtwichten sijn twee * afcomsten, te weten Rechtdaelwichten, ende Rechthefwichten, beschreuen inde achtien eerste voorstellen. </s><s>Der Scheefwichten sijn oock twee afcomsten, als Scheefdaelwichten, ende Scheefhefwichten, verclaert inde rest der voorstellen.</s>

<s>DE Beghinselen der Weeghconst, welcke van swaerheyt sijn duer <emph.end/><emph.end/><emph.end/>t'ghedacht van natuerlicke stof gheweert, sullen in twee boucken begrepen worden. </s><s>Des eersten boucx eerste deel sal van 14 * bepalinghen wesen: T'ander van 28 * voorstellen, van de ghedaenten der ghewichten, die tweederhande sijn, als Rechtwichten, ende Scheefwichten. </s><s>Der Rechtwichten sijn twee * afcomsten, te weten Rechtdaelwichten, ende Rechthefwichten, beschreuen inde achtien eerste voorstellen. </s><s>Der Scheefwichten sijn oock twee afcomsten, als Scheefdaelwichten, ende Scheefhefwichten, verclaert inde rest der voorstellen.</s>

<s>

<margin.target id="marg26"/>Definitionib.Propositio<lb/>nibus. Species.

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<s>

<margin.target id="marg42"/>.

</s>

</section></front><body><chap>

<pb/>

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<s><emph type="other"/>VERCLARING<emph.end/>.</s>

<s> <emph type="other"/>GHELIICK<emph.end/> de Meetconst <arrow.to.target n="marg44"/> ansiet der formen grootheden niet hare swaerheden, houdende die alleenelick voor ofte oneuen, diens grootheden euen ofte oneuen sijn; Alsoo ansiet ter contrarie de Weegconst haer swaerheden, niet haer grootheden, houdende die voor euen ende oneuen, diens ghewichten euen ofte oneuen sijn: Ende ghelijck diens voornamelicke wercking bestaet int ondersoucken der Redenen Eueredenheden, ende Ghedaenten <arrow.to.target n="marg45"/> haerder grootheden, Also desens int ondersoucken der Redenen Eueredenheden, ende ghedaenten haerder swaerheden ofte ghewichten, welcker bescriuing t'voornemen is deses handels.</s>

<s> <emph type="other"/>GHELIICK<emph.end/> de Meetconst <arrow.to.target n="marg44"/> ansiet der formen grootheden niet hare swaerheden, houdende die alleenelick voor euen ofte oneuen, diens grootheden euen ofte oneuen sijn; Alsoo ansiet ter contrarie de Weegconst haer swaerheden, niet haer grootheden, houdende die voor euen ende oneuen, diens ghewichten euen ofte oneuen sijn: Ende ghelijck diens voornamelicke wercking bestaet int ondersoucken der Redenen Eueredenheden, ende Ghedaenten <arrow.to.target n="marg45"/> haerder grootheden, Also desens int ondersoucken der Redenen Eueredenheden, ende ghedaenten haerder swaerheden ofte ghewichten, welcker bescriuing t'voornemen is deses handels.</s>

<s>

<margin.target id="marg44"/>Geometria

<margin.target id="marg45"/>* Rationum, proportionum & qualitatum.

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<s><emph type="other"/>VERCLARING<emph.end/>.</s>

<s> <emph type="other"/>DE<emph.end/> swaerheydt ofte lichticheydt die wy ghemeenelick segghen een lichaem te hebben, en is niet sijn eyghen wesentlicke ghedaente, maer veroirsaect uyt sijn ghemeenschap met een ander (wiens breeder verclaring wy gheschict hebben) want veel Stoffen <arrow.to.target n="marg46"/> die swaer sijn inde locht, worden licht beuonden int water, ende de lichte inde locht, sijn el<pb pagenum="2"/>ders swaer; daerom als wy segghen een haudt te weghen hondert pondt, wy verstaen daer by de macht sijnder daling in gestelde plaets, dat is in dien Grondt <arrow.to.target n="marg47"/> daert in gheweghen was. </s><s><emph type="other"/>DOOR<emph.end/> t verkerde deser bepaling is te verstaen, dat lichticheyt eens lichaems de macht is sijnder rijsing, maer in ghestelde plaets, want eyghentlick is alle lichaem swaer.</s>

<s> <emph type="other"/>DE<emph.end/> swaerheydt ofte lichticheydt die wy ghemeenelick segghen een lichaem te hebben, en is niet sijn eyghen wesentlicke ghedaente, maer veroirsaect uyt sijn ghemeenschap met een ander (wiens breeder verclaring wy elder gheschict hebben) want veel Stoffen <arrow.to.target n="marg46"/> die swaer sijn inde locht, worden licht beuonden int water, ende de lichte inde locht, sijn el<pb pagenum="2"/>ders swaer; daerom als wy segghen een haudt te weghen hondert pondt, wy verstaen daer by de macht sijnder daling in gestelde plaets, dat is in dien Grondt <arrow.to.target n="marg47"/> daert in gheweghen was. </s><s><emph type="other"/>DOOR<emph.end/> t verkerde deser bepaling is te verstaen, dat lichticheyt eens lichaems de macht is sijnder rijsing, maer in ghestelde plaets, want eyghentlick is alle lichaem swaer.</s>

<s>

<margin.target id="marg46"/>Materia.

</s>

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<s> <emph type="other"/>LAET<emph.end/> ABC een cloot sijn, diens stof ouer al eueswaer is, welcke wy met haer middelpunt D door ons ghedacht nemen te hanghen ande lini ED; Ende is kennelick dat dien cloot ghekeert wordende, sal houden alle ghestalt diemen haer gheeft, want soomen B keerde daer A is, B sal daer blijuen, ende voort yder deel op sijn plaets, want soo dat niet en gheschiede, de stof soude an deen sijde swaerder sijn als an d'ander, twelck teghen tghestelde waer. </s><s>D dan naer luyt deser bepaling is Swaerheydts middelpunt des cloots ABC; Ende alsoo salmen verstaen dat binnen alle lichamen soo wel ongeschicter form ende van stof oneenuaerdigher swaerheydt als gheschicter ende eenvaerdigher, is eenich sulcken punt, waer an tlichaem also hanghende, alle ghestalt houdt diemen hem gheeft, welck punt ghenoemt wort sijn Swaerheydts middelpunt. </s><s> Ende op dattet door eenighe sijne eyghenschappen kennelicker sy, sullender noch dit toe segghen: Het swaerheydts middelpunt der oirdentlicke lichamen als Pilaren, Clooten, Lancworpighe Clooten <arrow.to.target n="marg48"/>, der Vijf gheschicte lichamen, &c. ouer al eue<pb pagenum="3"/>wichtigher Stof sijnde, is tselue der Form ofte grootheydt, datmen anders Meetconstich middelpunt noemt. </s><s>Maer die niet ouer al euewichtigher Stof en sijn, en hebben dese twee punten niet nootsaeckelick tot een selfde plaets. </s><s>Wat de naelden <arrow.to.target n="marg49"/>, ende ongheschicte lichamen belangt, sy en hebben gheen formens ofte grootheydts middelpunt, maer alleen des swaerheydts. </s><s>Het ghebuert oock in veel lichamen als Rynghen, Haecken, Beckens, ende dier ghelijcke, dat haer swaerheydts middelpunt niet en valt inde stof des lichaems, maer binnen tlichaem uyt de stof. </s>

<s><emph type="other"/>DAER<emph.end/> wort inde bepaling gheseydt <arrow.to.target n="marg50"/> reden datmen int bepalen moet nemen, tghene den aert van tbepaelde best verclaert, twelck Pappus (daer hy int 8 bouck het swaerheydts middelpunt bepaelt) door tghedacht oock bequamelick ghedaen heeft. </s><s>Men soudet oock mueghen aldus bepalen: <arrow.to.target n="marg51"/> Wat Euestaltwichticheyt is sal door de Bepaling verclaert worden.</s>

<s>

<margin.target id="marg48"/>* Sphæroidalium.

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<s><emph type="other"/>VERCLARING<emph.end/>.</s>

<s> <emph type="other"/>DE<emph.end/> reden is dese; Laet ABCD den eertscloot sijn, wiens middelpunt E, ende sichteinder <arrow.to.target n="marg70"/> AC, ende FG een balck, euewydich vanden sichteinder AC, diens balcx even ermen HF, HG, ende euen swaerheden daer an I, K; alwaer het blijct, dat de hanghende linien FI, ende GK, gheen euewydighe en sijn, maer onder naerder malcander dan bouen; Laet daer naer den balc FG ghekeert worden op t'vastpunt H, alsoo dat G comme daer nu L is, ende F daer M, ende K sal commen daer nu N, ende I daer nu O is, ende den houck LME is naerder den rechthouck dan MLE, waer duer O (als in het volghende voorstel blijcken sal) naer de ghestalt swaerder is dan N. </s><s>Vyt desen volght oock dat onder alle lichamelicke formen die inde natuer bestaen, so en isser gheen ander, wisconstelick <arrow.to.target n="marg71"/> sprekende, dan den cloot, an wiens swaerheydts middelpunt het lichaem door ons ghedacht hanghende, alle ghestalt houdt diemen hem gheeft; Ofte door t'welck alle plat, t'lichaem deelt <pb pagenum="10"/>in euestaltwichtighe deelen, maer om de oneindelicke verscheyden ghestalten, sullender oneindelicke verscheyden swaerheyts middelpunten in sijn. </s><s>Oock en soude (teghen t'volghende voorstel) de swaerste swaerheyt niet sulcken reden hebben tot de lichtste, als den langsten erm tot den cortsten, maer d'eene soude naer de ghestalt swaerder sijn, om dat haer houck plomper ende den rechthouck naerder is dan des anders houck. </s><s>Maer om t'selue by voorbeelt te verclaren, laet AB den cortsten erm sijn, diens ghewicht C, ende AD den langsten erm, diens ghewicht E in sulcken reden sy tot tghewicht C, als AB tot AD, ende F sy t'sweerelts middelpunt; Alwaer blijct dat den houck FBA plomper ende den rechthouck naerder is, dan den houck ADF, waer uyt volght (door tvoornoemde 22 voorstel) dat C naer de ghestalt swaerder sal sijn dan E. </s>

<s> <emph type="other"/>DE<emph.end/> reden is dese; Laet ABCD den eertscloot sijn, wiens middelpunt E, ende sichteinder <arrow.to.target n="marg70"/> AC, ende FG een balck, euewydich vanden sichteinder AC, diens balcx even ermen HF, HG, ende euen swaerheden daer an I, K; alwaer het blijct, dat de hanghende linien FI, ende GK, gheen euewydighe en sijn, maer onder naerder malcander dan bouen; Laet daer naer den balc FG ghekeert worden op t'vastpunt H, alsoo dat G comme daer nu L is, ende F daer M, ende K sal commen daer nu N, ende I daer nu O is, ende den houck LME is naerder den rechthouck dan MLE, waer duer O (als in het volghende 22. voorstel blijcken sal) naer de ghestalt swaerder is dan N. </s><s>Vyt desen volght oock dat onder alle lichamelicke formen die inde natuer bestaen, so en isser gheen ander, wisconstelick <arrow.to.target n="marg71"/> sprekende, dan den cloot, an wiens swaerheydts middelpunt het lichaem door ons ghedacht hanghende, alle ghestalt houdt diemen hem gheeft; Ofte door t'welck alle plat, t'lichaem deelt <pb pagenum="10"/>in euestaltwichtighe deelen, maer om de oneindelicke verscheyden ghestalten, sullender oneindelicke verscheyden swaerheyts middelpunten in sijn. </s><s>Oock en soude (teghen t'volghende I. voorstel) de swaerste swaerheyt niet sulcken reden hebben tot de lichtste, als den langsten erm tot den cortsten, maer d'eene soude naer de ghestalt swaerder sijn, om dat haer houck plomper ende den rechthouck naerder is dan des anders houck. </s><s>Maer om t'selue by voorbeelt te verclaren, laet AB den cortsten erm sijn, diens ghewicht C, ende AD den langsten erm, diens ghewicht E in sulcken reden sy tot tghewicht C, als AB tot AD, ende F sy t'sweerelts middelpunt; Alwaer blijct dat den houck FBA plomper ende den rechthouck naerder is, dan den houck ADF, waer uyt volght (door tvoornoemde 22 voorstel) dat C naer de ghestalt swaerder sal sijn dan E. </s>

<s>

<margin.target id="marg70"/>Horizon.

<margin.target id="marg71"/>Mathematice.

</s>

<s> Alle dese ongheuallen spruyten daer uyt, dat FE met GE in d'eerste form, ofte BF met DF der tweede form, gheen evewydighe linien en sijn: Maer ouermits dat verschil in alle t'ghene de menschen weghen, onbemerckelick is, want den balck soude al veel milen lanck moeten sijn eer hem dat soude connen openbaren, soo begheeren wy datse voor euewydighe ghehouden worden. </s><s>Wel is waer dat wy die ansiende voor t'ghene sy sijn, volcommelick souden connen wercken na haerlieder ghedaente, maer want dat moeyelicker soude wesen, ende tot de saeck, dat is de nochtans niet voorderlicker, so ist beter ghelaten. </s>

<s> Alle dese ongheuallen spruyten daer uyt, dat FE met GE in d'eerste form, ofte BF met DF der tweede form, gheen evewydighe linien en sijn: Maer ouermits dat verschil in alle t'ghene de menschen weghen, onbemerckelick is, want den balck soude al veel milen lanck moeten sijn eer hem dat soude connen openbaren, soo begheeren wy datse voor euewydighe ghehouden worden. </s><s>Wel is waer dat wy die ansiende voor t'ghene sy sijn, volcommelick souden connen wercken na haerlieder ghedaente, maer want dat moeyelicker soude wesen, ende tot de saeck, dat is de <emph type="other"/>WEEGDAET<emph.end/> nochtans niet voorderlicker, so ist beter ghelaten. </s>

<s><emph type="other"/>HET ANDER DEEL<emph.end/> <lb/>

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<s><emph type="other"/>I. VOORBEELT<emph.end/>.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s><arrow.to.target n="marg74"/> Laet ABCD een pilaer sijn weghende 6 . </s><s>welcke ghedeelt sy in 6 euen deelen, door platten <arrow.to.target n="marg75"/> euewydich van sijn grondt AD, als EF, GH, IK, LM, NO, sniende den as PQ in R, S, T, V, X: Laet ons nu nemen LMDA voor de swaerste swaerheydt, wiens swaerheyts middelpunt is S, ende LMCB voor de lichtste swaerheydt, wiens swaerheyts middelpunt is X, ende SX is dier deelen balck door de bepaling, ende T is t'swaerheyts middelpunt des heelen pilaers, ende TI d'hanthaef, waer an LMDA ende LMCB evestaltwichtich hangen, ende TX is den langsten erm, ende TS den cortsten door de bepaling. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s><arrow.to.target n="marg76"/> wy moeten bewysen dat ghelijck de swaerste swaerheydt LMDA, tot de lichtste LMCB, also den langsten erm TX, tot den cortsten TS. </s><s><emph type="other"/>TBEWIIS<emph.end/>. </s><s><arrow.to.target n="marg77"/> De swaerste swaerheydt LMDA weeght 4 lb, ende de lichtste LMCB 2 lb, ende den langsten erm TX heeft sulcken reden tot de cortste TS, ghelijck 2 tot 1 door t'ghegheven: Maer ghelijck 4 tot 2, alsoo 2 tot 1, ghelijck dan de swaerste swaerheydt LMDA, tot de lichtste LMCB, also den langsten erm TX, tot den cortsten TS.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s><arrow.to.target n="marg74"/> Laet ABCD een pilaer sijn weghende 6 . </s><s>welcke ghedeelt sy in 6 euen deelen, door platten <arrow.to.target n="marg75"/> euewydich van sijn grondt AD, als EF, GH, IK, LM, NO, sniende den as PQ in R, S, T, V, X: Laet ons nu nemen LMDA voor de swaerste swaerheydt, wiens swaerheyts middelpunt is S, ende LMCB voor de lichtste swaerheydt, wiens swaerheyts middelpunt is X, ende SX is dier deelen balck door de <emph.end/><emph.end/><emph.end/>bepaling, ende T is t'swaerheyts middelpunt des heelen pilaers, ende TI d'hanthaef, waer an LMDA ende LMCB evestaltwichtich hangen, ende TX is den langsten erm, ende TS den cortsten door de <emph.end/><emph.end/><emph.end/>bepaling. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s><arrow.to.target n="marg76"/> wy moeten bewysen dat ghelijck de swaerste swaerheydt LMDA, tot de lichtste LMCB, also den langsten erm TX, tot den cortsten TS. </s><s><emph type="other"/>TBEWIIS<emph.end/>. </s><s><arrow.to.target n="marg77"/> De swaerste swaerheydt LMDA weeght 4 lb, ende de lichtste LMCB 2 lb, ende den langsten erm TX heeft sulcken reden tot de cortste TS, ghelijck 2 tot 1 door t'ghegheven: Maer ghelijck 4 tot 2, alsoo 2 tot 1, ghelijck dan de swaerste swaerheydt LMDA, tot de lichtste LMCB, also den langsten erm TX, tot den cortsten TS.</s>

<s>

<margin.target id="marg74"/>Datum.

<margin.target id="marg75"/>Plana parallela.

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</s>

<s><emph type="other"/>NV<emph.end/> mocht yemant segghen, ghy hebt dat voorstel wel bewesen in deelen die tsamen een heel pilaer maken eenvaerdigher swaerheyt, maer wie weet of dat also plaets sal houden in allen anderen verscheyden deelen van ongheschicter form, ende oneueswaerder stof, daerom sullen wy de ghemeenheydt des voorstels aldus bethoonen: Laet ons achten dat den balck KL der I ghestalt hier bouen, in haer plaets bliue, ende dat het stick EFDA neerghetrocken wordt, ende dat het blyue hanghende met een lini uyt sijn swaerheydts middelpunt an tpunt K, ende dat insghelijcx oock neerghetrocken sy het ander stick EFCB, ende dat het blijue hanghende by sijn swaerheydts middelpunt an <pb pagenum="13"/>tpunt L, ende dat EFCB niet en ghenake an EFDA, ende haer ghestalt sy dan soo dees form uytwyst. </s><s>Nu doen het lichaem in d'eerste ghestalt hinck ande hanthaef MN, alsdoen was EFDA euestaltwichtich met EFCB; Maer tghewicht EFDA in dees tweede ghestalt neerghetrocken synde, en brengt an KL gheen meerder noch minder swaerheyt dan in d'eerste ghestalt door de begheerte. </s><s>Sghelijcx en brengt tghewicht EFCB der tweede ghestalt, an LK gheen meerder swaerheydt dan in d'eerste ghestalt, daerom de ghewichten der tweede ghestalt sijn an KL de selfde die sy in d'eerste waren, daerom oock de balck KL blijft noch inde selue eerste ghestalt, waer duer EFDA noch euestaltwichtich blijft met EFCB. </s><s>De sticken dan des pilaers blijuen soo wel euestaltwichtich verscheyden, als doen sy an malcanderen waren, ende de ermen oock inde selue reden. </s>

<s><emph type="other"/>NV<emph.end/> mocht yemant segghen, ghy hebt dat voorstel wel bewesen in deelen die tsamen een heel pilaer maken eenvaerdigher swaerheyt, maer wie weet of dat also plaets sal houden in allen anderen verscheyden deelen van ongheschicter form, ende oneueswaerder stof, daerom sullen wy de ghemeenheydt des voorstels aldus bethoonen: Laet ons achten dat den balck KL der I ghestalt hier bouen, in haer plaets bliue, ende dat het stick EFDA neerghetrocken wordt, ende dat het blyue hanghende met een lini uyt sijn swaerheydts middelpunt an tpunt K, ende dat insghelijcx oock neerghetrocken sy het ander stick EFCB, ende dat het blijue hanghende by sijn swaerheydts middelpunt an <pb pagenum="13"/>tpunt L, ende dat EFCB niet en ghenake an EFDA, ende haer ghestalt sy dan soo dees form uytwyst. </s><s>Nu doen het lichaem in d'eerste ghestalt hinck ande hanthaef MN, alsdoen was EFDA euestaltwichtich met EFCB; Maer tghewicht EFDA in dees tweede ghestalt neerghetrocken synde, en brengt an KL gheen meerder noch minder swaerheyt dan in d'eerste ghestalt door de 3. begheerte. </s><s>Sghelijcx en brengt tghewicht EFCB der tweede ghestalt, an LK gheen meerder swaerheydt dan in d'eerste ghestalt, daerom de ghewichten der tweede ghestalt sijn an KL de selfde die sy in d'eerste waren, daerom oock de balck KL blijft noch inde selue eerste ghestalt, waer duer EFDA noch euestaltwichtich blijft met EFCB. </s><s>De sticken dan des pilaers blijuen soo wel euestaltwichtich verscheyden, als doen sy an malcanderen waren, ende de ermen oock inde selue reden. </s>

<s><emph type="other"/>DIT<emph.end/> so synde, laet ons de lichamen EFDA ende EFCB der tweede ghestalt ander formen gheuen, die also duwende (neemt dat de stof sy van was, cleye, ofte yet soodanich t'welck sulcx lijde) dat EFDA der tweede ghestalt, sy EFDA deser derde ghestalt, ende dat EFCB der tweede ghestalt, sy EFCB deser derde ghestalt; Ende is openbaer dat KL noch in haer selue ghestalt sal blyuen, ende de ermen ML, MK, inde selue reden, ende veruolgens EFDA noch euestaltwichtich met EFCB, want dees verandering der form (al de stof bliuende) en veroirsaect gheen verandering des ghewichts.</s>

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<s><emph type="other"/>III. VOORBEELT<emph.end/>.</s>

<s> <emph type="other"/>MEN<emph.end/> can tvoorgaende oock bethoonen, blyuende de twee swaerheden hanghende an eenen lichamelicken balck, in deser voughen: Laet den pilaer ABCD ghesneen sijn in twee deelen, met een plat door den as EF, ende den as des ondersten deels EC sy GH, ende EC sy doorsneen met een plat IK euewydich vanden grondt ED, sniende den as GH in L, ende het swaerheydts middelpunt van het deel IKDE sy M int middel van GL, ende van het deel IKCF sy N int middel van LH, ende des heels ABCD sy O in middel van EF, ende OP sy swaerheydts middellini des heels ABCD, ende MQ van IKDE, ende NR van IKCF. </s><s> Dit soo sijnde tis kennelick dat des heels pilaers rechter sijde, euewichtich is teghen haer slincker. </s>

<s><emph type="other"/>LAET<emph.end/> ons nu het onderste deel EFCD neertrecken, also dat het blyue hanghende ande linien ML [MQ] ende NR, als hier neuens. </s><s>Ende is openbaer dat den lichamelicken balc ABFE noch in haer eerste ghestalt sal blyuen. </s><s>Laet ons nu achten dat het deel IKDE, ghesneen sy van IKCF, ende dat elck deel vallen mach daert wil, maer sy hanghen an haer swaerheyts <pb pagenum="15"/>middelpunten M, N, sy houden dan haer eerste ghegheuen ghestalt door de bepaling, daerom ABFE blijft oock noch in sijn eerste ghedaente. </s>

<s><emph type="other"/>LAET<emph.end/> ons nu het onderste deel EFCD neertrecken, also dat het blyue hanghende ande linien ML [MQ] ende NR, als hier neuens. </s><s>Ende is openbaer dat den lichamelicken balc ABFE noch in haer eerste ghestalt sal blyuen. </s><s>Laet ons nu achten dat het deel IKDE, ghesneen sy van IKCF, ende dat elck deel vallen mach daert wil, maer sy hanghen an haer swaerheyts <pb pagenum="15"/>middelpunten M, N, sy houden dan haer eerste ghegheuen ghestalt door de 4. bepaling, daerom ABFE blijft oock noch in sijn eerste ghedaente. </s>

<s> Maer IKDE, sulcken reden te hebben tot IKCF, als den erm OR, tot den erm OQ, is vooren beprouft; Inder voughen dat tghene eerst betoocht was anden weegconstighen balck (dat is een lini) sulcx hebben wy hier verclaert an een lichamelicken. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s><arrow.to.target n="marg85"/> Wesende dan twee euestaltwichtighe swaerheden, de swaerste heeft sulcken reden tot de lichtste (van wat stof ofte form oock de lichamen sijn) als den langsten erm tot den cortsten, twelck wy bewysen moesten.</s>

<s>

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<s><emph type="other"/>I. VOORBEELT<emph.end/>.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet d'een swaerheyt A sijn weghende 3 , hanghende an C, d'ander B van 1 hanghende an D; ende CD si balck. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet d'een swaerheyt A sijn weghende 3 , hanghende an C, d'ander B van 1 lb hanghende an D; ende CD si balck. </s>

<s><emph type="other"/>TBEGHEERDE<emph.end/> Wy moeten haer hanthaef vinden. </s><s> <emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal CD also deelen, dat haer meeste stick naest de swaerheydts middellini van de minste swaerheydt, sulcken reden hebbe tot het minste stick, ghelijck de meeste swaerheyt tot de minste, twelck sy in E, te weten dat ED sulcken reden hebbe tot EC, als 3 lb van A, tot 1 lb van B. </s><s> Ick seg dat de hanghende door E, als EF, d'hanthaef is.</s>

<s><emph type="other"/>II. VOORBEELT<emph.end/>.</s>

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<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD wederom den pilaer sijn, ghedeelt als vooren, hanghende Y 6 lb an X, ende Z 24 lb an R, ende Æ 12 lb an Q. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten d'handthaef vinden. </s><s> <emph type="other"/>TWERCK<emph.end/>. </s><s>De swaerheydts middellini van ABCDYZ is SG door het 4 voorbeelt, ende van Æ is QB, ende SQ is balck: de selue salmen in tween deelen, also dat de sticken de reden hebben als 36 lb vanden pilaer met Y <pb pagenum="18"/>ende Z, tot 12 lb van Æ, te weten tcortste stick naer de swaerheydts middellini van tswaerste deel, twelck vallen sal in T, inder voughen dat TI de begheerde handthaef sal sijn. </s>

<s>Ende soomen noch hinghe an P 24 lb, d'handthaef soude SG sijn, ende so voorts met allen anderen swaerheden diemen anden pilaer soude mueghen hanghen. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>De swaerste swaerheydt A int eerste voorbeelt, heeft sulcken reden tot de lichtste B, als den langsten erm ED, tot den cortsten EC, daerom EF door de bepaling is d'hanthaef. </s><s>Sghelijcx sal oock tbewys sijn van al dander voorbeelden, twelck wy om de cortheydt achterlaten. </s>

<s>Ende soomen noch hinghe an P 24 lb, d'handthaef soude SG sijn, ende so voorts met allen anderen swaerheden diemen anden pilaer soude mueghen hanghen. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>De swaerste swaerheydt A int eerste voorbeelt, heeft sulcken reden tot de lichtste B, als den langsten erm ED, tot den cortsten EC, daerom EF door de 9. bepaling is d'hanthaef. </s><s>Sghelijcx sal oock tbewys sijn van al dander voorbeelden, twelck wy om de cortheydt achterlaten. </s>

<s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen bekende swaerheden, wy hebben haer handthaef gheuonden naer den eysch. </s>

<s><emph type="other"/>MERCKT<emph.end/>. </s>

<s> <emph type="other"/>SOOMEN<emph.end/> <arrow.to.target n="marg87"/></s><s>

<margin.target id="marg87"/>tghewicht Y des 2 voorbeelts verswaerde van 1 lb, ende datmen an V hinghe 1 lb, inder voughen dat haer ghestalt dan waer als hier onder, Tis kennelick uyt het voorgaende dat XN noch handthaef blijft, ende alles an haer euestaltwichtich hangt. Tselue sal XN oock blijuen, soomen Z 1 lb hangt an T, ende dat Y doe 14 lb, ofte Z 1 lb an S, ende dat Y doe 15 lb, ofte Z 1 lb an R, ende dat Y doe 16 lb, ofte Z 1 lb an P, ende dat Y doe 17 lb, ende soo oirdentlick voort by aldien den pilaer langher waer; te weten, verswarende Y altijt van 1 lb, voor elcke langde als XV, daermen Z voorder an verschuyft. Waer uyt de Ghedaenten { Qualitates. } des s bekent siin, als inde Weegdaet breeder daer af sal ghehandelt worden.

<margin.target id="marg87"/>tghewicht Y des 2 voorbeelts verswaerde van 1 lb, ende datmen an V hinghe 1 lb, inder voughen dat haer ghestalt dan waer als hier onder, Tis kennelick uyt het voorgaende dat XN noch handthaef blijft, ende alles an haer euestaltwichtich hangt. Tselue sal XN oock blijuen, soomen Z 1 lb hangt an T, ende dat Y doe 14 lb, ofte Z 1 lb an S, ende dat Y doe 15 lb, ofte Z 1 lb an R, ende dat Y doe 16 lb, ofte Z 1 lb an P, ende dat Y doe 17 lb, ende soo oirdentlick voort by aldien den pilaer langher waer; 1. te weten, verswarende Y altijt van 1 lb, voor elcke langde als XV, daermen Z voorder an verschuyft. Waer uyt de Ghedaenten { Qualitates. } des s bekent siin, als inde Weegdaet breeder daer af sal ghehandelt worden.

</s>

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<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen twee bekende euestaltwichtighe swaerheden met de langde van d'eenen erm: de langde des anderen erms te vinden. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A ende B twee euestaltwichtighe swaerheden sijn, welcker A hanghende an C weeght 3 lb, ende B hanghende an D 1 lb, ende de langde des erms DE sy 6 voeten.</s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><pb pagenum="20"/><s>Wy moeten de langde des anderen erms vinden. </s><s> <emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal segghen A 3 lb, gheeft B 1 lb, wat DE 6 voeten ? comt voor EC 2 voeten. </s><s>Ende der ghelijcke voorbeelden mochten wy stellen op de formen der voorbeelden des voorstels, ten waer die duer tvoorgaende kennelick ghenouch sijn. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet EC, soot mueghelick waer, langher sijn dan 2 voeten; den langsten erm sal dan minder reden hebben tot den corsten, dan de swaerste swaerheyt tot de lichtste, twelck teghen het eerste voorstel is, EC dan en is niet langher dan 2 voeten; Sghelijcx salmense oock bewysen niet corter te sijn, sy is dan effen van twee voeten, twelck wy bewysen moesten. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A ende B twee euestaltwichtighe swaerheden sijn, welcker A hanghende an C weeght 3 lb, ende B hanghende an D 1 lb, ende de langde des erms DE sy 6 voeten.</s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><pb pagenum="20"/><s>Wy moeten de langde des anderen erms vinden. </s><s> <emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal segghen A 3 lb, gheeft B 1 lb, wat DE 6 voeten ? comt voor EC 2 voeten. </s><s>Ende der ghelijcke voorbeelden mochten wy stellen op de formen der voorbeelden des 2. voorstels, ten waer die duer tvoorgaende kennelick ghenouch sijn. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet EC, soot mueghelick waer, langher sijn dan 2 voeten; den langsten erm sal dan minder reden hebben tot den corsten, dan de swaerste swaerheyt tot de lichtste, twelck teghen het eerste voorstel is, EC dan en is niet langher dan 2 voeten; Sghelijcx salmense oock bewysen niet corter te sijn, sy is dan effen van twee voeten, twelck wy bewysen moesten. </s>

<s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen twee euestaltwichtighe swaerheden met de langde van d'eenen erm, wy hebben de langde des anderen erms gheuonden, naer den eysch.</s>

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<s> Ende van het tweede punt I (van het tweede om dat 2 het ander der ghegeuen ghetalen is) salmen den pilaer ophangen byde swaerheyts, middellini IN; Daer naer salmen an tvijfde punt M een ghewicht hanghen als O, euen so swaer dat alles in euestaltwichticheyt sy, twelck soo wesende, ick seg dat tghewicht van O, in sulcken reden is tot tghewicht des pilaers, als 2 tot 3, ofte dat O euen is ande 2/3 des pilaers. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>G is swaerheydts middelpunt <arrow.to.target n="marg90"/> des pilaers ABCD, ende MP swaerheyts middellini van O, daerom ghelijck den erm IG tot den erm IM, alsoo O tot den pilaer door het voorstel, maer IG heeft sulcken reden tot IM, als 2 tot 3, daerom O heeft sulcken reden tot den pilaer, als 2 tot 3, twelck wy bewysen moesten. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een pilaer, wy hebben gheuonden een ghewicht in ghestelde reden tot des pilaers ghewicht, naer den eysch. </s><s><emph type="other"/>MERCKT<emph.end/>. </s><s>Wy souden oock mueghen voorbeelden stellen met Redenen van onmetelicke palen <arrow.to.target n="marg91"/>, maer sulcx is openbaer ghenouch door tvoorgaende, metgaders tghene wy vande onmetelicke grootheden elders ghescreuen hebben. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>G is swaerheydts middelpunt <arrow.to.target n="marg90"/> des pilaers ABCD, ende MP swaerheyts middellini van O, daerom ghelijck den erm IG tot den erm IM, alsoo O tot den pilaer door het 1. voorstel, maer IG heeft sulcken reden tot IM, als 2 tot 3, daerom O heeft sulcken reden tot den pilaer, als 2 tot 3, twelck wy bewysen moesten. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een pilaer, wy hebben gheuonden een ghewicht in ghestelde reden tot des pilaers ghewicht, naer den eysch. </s><s><emph type="other"/>MERCKT<emph.end/>. </s><s>Wy souden oock mueghen voorbeelden stellen met Redenen van onmetelicke palen <arrow.to.target n="marg91"/>, maer sulcx is openbaer ghenouch door tvoorgaende, metgaders tghene wy vande onmetelicke grootheden elders ghescreuen hebben. </s>

<s>

<margin.target id="marg90"/>Centrum grauitatis.

<margin.target id="marg91"/>Incommensurabilium terminorum.

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</s>

<s> <emph type="other"/>DES<emph.end/> Bewysredens <arrow.to.target n="marg97"/> tweede voorstel is daer uyt openbaer, dat het punt gheen grootheyt en is, ende veruolghens gheen grondt: wel is waer dat wy dickmael nemen door tghestelde een lichaem alsoo te rusten, maer metter daet en connen wy dat niet te weeg brenghen. </s><s>Inder voughen dat hoewel den as IK euewydich ghestelt is vanden sichteinder LM, soo sal nochtans den pilaer (tpunt G vast blyuende) omkeeren ouer die sijde daer hy eerst beghint. </s><s>Maer dat hy solang keeren sal tot dat sijn swaerheydts middelpunt inde swaerheydts middellini is, is door het voorstel openbaer. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan den pilaer ghesneen, &c.</s>

<s> <emph type="other"/>DES<emph.end/> Bewysredens <arrow.to.target n="marg97"/> tweede voorstel is daer uyt openbaer, dat het punt gheen grootheyt en is, ende veruolghens gheen grondt: wel is waer dat wy dickmael nemen door tghestelde een lichaem alsoo te rusten, maer metter daet en connen wy dat niet te weeg brenghen. </s><s>Inder voughen dat hoewel den as IK euewydich ghestelt is vanden sichteinder LM, soo sal nochtans den pilaer (tpunt G vast blyuende) omkeeren ouer die sijde daer hy eerst beghint. </s><s>Maer dat hy solang keeren sal tot dat sijn swaerheydts middelpunt inde swaerheydts middellini is, is door het 6. voorstel openbaer. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan den pilaer ghesneen, &c.</s>

<s>

<margin.target id="marg97"/>Syllogismi.

</s>

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<s> <emph type="other"/>SOO<emph.end/> yemant thinhondt [t inhoudt] der voorgaende drie voorstellen door eenighe eruaring wilde sien, hy mocht nemen een reghel van houdt ofte ander stof eenvaerdigher dickte ende swaerheyt, als ABCD, teeckenende de punten E, F, G, H, inde middelen der linien AB, BC, CD, DA, treckende EG, ende HF, malcander sniende in I,maeckende daer naer een seer cleen gaetken en I, ende daer bouen een gaetken als K, ende onder I een gaetken als L. Ende stekende een naelde door tgatken K, die vrielick daerin drayen mach, d'eruaring sal bethoonen dat HF altijdt euewydich sal blijuen vanden sichteinder. Maer de naelde in I stekende, de reghel sal daerop alle ghestalt houden diemen haer gheeft. Ende de naelde in L ghesteken, alles sal omkeeren ouer <pb pagenum="25"/>die syde daert eerst beghint, tot dat I is in haer swaerheyts middellini, waer af d'oirsaeck inde voornoemde , 7, 8, voorstellen Wisconstlick blijckt.<arrow.to.target n="marg101"/></s>

<s> <emph type="other"/>SOO<emph.end/> yemant thinhondt [t inhoudt] der voorgaende drie voorstellen door eenighe eruaring wilde sien, hy mocht nemen een reghel van houdt ofte ander stof eenvaerdigher dickte ende swaerheyt, als ABCD, teeckenende de punten E, F, G, H, inde middelen der linien AB, BC, CD, DA, treckende EG, ende HF, malcander sniende in I,maeckende daer naer een seer cleen gaetken en I, ende daer bouen een gaetken als K, ende onder I een gaetken als L. Ende stekende een naelde door tgatken K, die vrielick daerin drayen mach, d'eruaring sal bethoonen dat HF altijdt euewydich sal blijuen vanden sichteinder. Maer de naelde in I stekende, de reghel sal daerop alle ghestalt houden diemen haer gheeft. Ende de naelde in L ghesteken, alles sal omkeeren ouer <pb pagenum="25"/>die syde daert eerst beghint, tot dat I is in haer swaerheyts middellini, waer af d'oirsaeck inde voornoemde 6., 7., 8., voorstellen <arrow.to.target n="marg101"/>Wisconstlick blijckt.</s>

<s>

<margin.target id="marg101"/>Mathematicè.

</s>

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<s><emph type="other"/>D'HANTHAEF<emph.end/> oneindelick voortghetrocken, deelt alle balcken tweer swaerheden in haer ermen. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet AB twee swaerheden sijn ende haer middellinien CD, EF, ende haer CE, ende d'hanthaef GH, inder voughen dat CG is tot GE, als de swaerheydt B tot A, Laet IK noch een balck wesen, oneuewydich van CE, ende laet GH oneindelick voortghetrocken worden naer L, sniende den balck IK in M. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet AB twee swaerheden sijn ende haer middellinien CD, EF, ende haer balck CE, ende d'hanthaef GH, inder voughen dat CG is tot GE, als de swaerheydt B tot A, Laet IK noch een balck wesen, oneuewydich van CE, ende laet GH oneindelick voortghetrocken worden naer L, sniende den balck IK in M. </s>

<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat IM ende MK, oock de ermen sijn der swaerheden AB; dat is ghelijck B tot A, alsoo MI tot MK. </s><s><emph type="other"/>TBEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden CN, euewydich van IK, sniende HL in O. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ghelijck CG tot GE, also CO tot ON <arrow.to.target n="marg102"/>, Maer CO is euen an IM, ende ON an MK <arrow.to.target n="marg103"/>, daerom ghelijck CG tot GE, alsoo IM tot MK, maer ghelijck B tot A, also CG tot GE, door tghegheuen, daerom ghelijck B tot A, also MI tot MK, tselfde sal also bewesen worden van allen balcken tusschen CD ende EF, als PQ, doorsneen in R, ende allen anderen diemen soude mueghen trecken. </s>

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<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 4 lb, ende ghesneen door sijn swaerheydts middelpunt E, met een plat FG euewydich vanden grondt AD, ende laet H vastpunt wesen beneden tmiddelpunt E int middel van EG; Ende anden pilaer twee ghewichten hanghen als I, K, elck weghende 4 lb, welcker middellinien vastpunten sijn D, C, ende laet LM den as, ende NO sichteinder wesen. </s>

<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten vinden of den as LM euewydich sal connen blijuen vanden sichteinder NO; ofte alle gestalt houden diemen haer gheeft; Ofte ommekeeren tot dat haer swaerheydts middelpunt E is inde swaerheyts middellini door H, welcke verscheydenheden vallen connen naer de reden der swaerheyt des pilaers, tot de ghewichten dieder anhangen. </s>

<s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal trecken door E de swaerheyts middellini PQ des pilaers, daer naer door <pb pagenum="27"/>G de swaerheyts middellini RS der ghewichten I, K, ende EG sal balck sijn, daer naer salmen sien door het voorstel waer tvastpunt der hanthaef valt: want commet onder H, soo keert LM tot sy euewydich blijft vanden sichteinder NO; Maer commet in H, sy houdt alle ghestalt die men haer gheeft; Commet bouen H, alles keert om. </s><s> Maer den pilaer weeght 4 lb, ende I, K, elck 4 lb, tsamen 8 lb door tghegheuen, daerom ghedeelt EG in T, alsoo dat ET, sulcken reden heb tot TG, als 8 tot 4: Ick seg dat LM keeren sal (ouermits T onder H comt) tot sy euewydich is vanden sichteinder. </s><s>Laet nu den pilaer weghen 4 lb, ende I en K elck 2 lb, tsamen 4 lb, daerom ghedeelt EG in H (welcke H tmiddel van EG is door tghegheuen) alsoo dat EH sulcken reden heb tot HG, als 4 tot 4: ick seg dat LM (ouermits het in H viel) alle ghestalt sal houden diemen haer gheeft. </s><s>Laet nu den pilaer weghen 4 lb, ende I, K elck I lb, tsamen 2 lb, daerom ghedeelt EG inV, also dat EV sulcken reden hebbe tot VG, als 2 tot 4, Ick seg dat den pilaer met al de rest omkeeren sal (ouermitds V bouen H comt) tot dat H is in haer swaerheydts middellini. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ten eersten I en K elck 4 lb weghende, dat dan LM keert tot sy euewydich is vanden sichteinder, blijct aldus: De hanghende door T ghelijck TX, is swaerheyts middellini des heels, daerom die latende, ende hanghende tgheheel ande hanghende <arrow.to.target n="marg107"/> door H, als HY (welcke H ons ghegheuen vastpunt is) so sal de sijde naer BCK, swaerder sijn dan naer ADI, daerom oock sal de sijde BCK neerdalen, tot dat H inde swaerheydts middellini is des heels, ende dan sal LM euewydich sijn vanden sichteinder NO. </s>

<s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal trecken door E de swaerheyts middellini PQ des pilaers, daer naer door <pb pagenum="27"/>G de swaerheyts middellini RS der ghewichten I, K, ende EG sal balck sijn, daer naer salmen sien door het 2. voorstel waer tvastpunt der hanthaef valt: want commet onder H, soo keert LM tot sy euewydich blijft vanden sichteinder NO; Maer commet in H, sy houdt alle ghestalt die men haer gheeft; Commet bouen H, alles keert om. </s><s> Maer den pilaer weeght 4 lb, ende I, K, elck 4 lb, tsamen 8 lb door tghegheuen, daerom ghedeelt EG in T, alsoo dat ET, sulcken reden heb tot TG, als 8 tot 4: Ick seg dat LM keeren sal (ouermits T onder H comt) tot sy euewydich is vanden sichteinder. </s><s>Laet nu den pilaer weghen 4 lb, ende I en K elck 2 lb, tsamen 4 lb, daerom ghedeelt EG in H (welcke H tmiddel van EG is door tghegheuen) alsoo dat EH sulcken reden heb tot HG, als 4 tot 4: ick seg dat LM (ouermits het in H viel) alle ghestalt sal houden diemen haer gheeft. </s><s>Laet nu den pilaer weghen 4 lb, ende I, K elck I lb, tsamen 2 lb, daerom ghedeelt EG inV, also dat EV sulcken reden hebbe tot VG, als 2 tot 4, Ick seg dat den pilaer met al de rest omkeeren sal (ouermitds V bouen H comt) tot dat H is in haer swaerheydts middellini. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ten eersten I en K elck 4 lb weghende, dat dan LM keert tot sy euewydich is vanden sichteinder, blijct aldus: De hanghende door T ghelijck TX, is swaerheyts middellini des heels, daerom die latende, ende hanghende tgheheel ande hanghende <arrow.to.target n="marg107"/> door H, als HY (welcke H ons ghegheuen vastpunt is) so sal de sijde naer BCK, swaerder sijn dan naer ADI, daerom oock sal de sijde BCK neerdalen, tot dat H inde swaerheydts middellini is des heels, ende dan sal LM euewydich sijn vanden sichteinder NO. </s>

<s>Ten tweeden I, K, elck 2 lb weghende, dat dan LM alle ghestalt houdt, wordt aldus bethoont: Laet ons achten dat I ende K opgheschorst sijn, alsoo dat D tswaerheydts middelpunt sy van I, ende C van K, ende door de begheerte sy en sullen anden pilaer gheen oirsaeck van verandering der swaerheydt wesen; Twelck soo sijnde, H is tswaerheyts middelpunt van soodanighen lichaem vergaert uyt den pilaer ende de twee ghewichten IK, ende door de bepaling tsal daer op alle ghestalt houden diemen hem gheeft, tselfde sal also bewesen worden in alle ghestalten daermen LM in soude connen stellen. </s>

<s>Ten tweeden I, K, elck 2 lb weghende, dat dan LM alle ghestalt houdt, wordt aldus bethoont: Laet ons achten dat I ende K opgheschorst sijn, alsoo dat D tswaerheydts middelpunt sy van I, ende C van K, ende door de 3. begheerte sy en sullen anden pilaer gheen oirsaeck van verandering der swaerheydt wesen; Twelck soo sijnde, H is tswaerheyts middelpunt van soodanighen lichaem vergaert uyt den pilaer ende de twee ghewichten IK, ende door de 4 bepaling tsal daer op alle ghestalt houden diemen hem gheeft, tselfde sal also bewesen worden in alle ghestalten daermen LM in soude connen stellen. </s>

<s>Ten laetsten I, K, elck I lb weghende, dat dan alles omkeert, wort aldus bethoont: De hanghende door V ghelijck VZ, is swaerheyts middellini des heels, daerom die latende, ende hanghende tgheheel ande hanghende HY door H ghegheuen vastpunt, so sal de sijde naer ADI, swaerder sijn dan naer BCK, daerom oock sal de sijde ADI neerdalen, tot dat H inde swaerheyts middellini is des heels, ende ofmen schoon LM (alles op tvastpunt H draeyende) euewydich stelde vanden sichteinder NO, sy en can so niet blyuen door het voorstel, maer alles sal omkeeren, twelck wy bewysen moesten. </s>

<s>Ten laetsten I, K, elck I lb weghende, dat dan alles omkeert, wort aldus bethoont: De hanghende door V ghelijck VZ, is swaerheyts middellini des heels, daerom die latende, ende hanghende tgheheel ande hanghende HY door H ghegheuen vastpunt, so sal de sijde naer ADI, swaerder sijn dan naer BCK, daerom oock sal de sijde ADI neerdalen, tot dat H inde swaerheyts middellini is des heels, ende ofmen schoon LM (alles op tvastpunt H draeyende) euewydich stelde vanden sichteinder NO, sy en can so niet blyuen door het 8 voorstel, maer alles sal omkeeren, twelck wy bewysen moesten. </s>

<s>

<margin.target id="marg107"/>Perpendicularem.

</s>

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<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn, weghende I0 lb, diens swaerheydts middelpunt E, ende laet de ghewichten daer an hanghende wesen <emph type="other"/>FI<emph.end/> lb, diens vastpunt G, ende H 4 lb, wiens vastpunt I.</s><s> <emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten het vastpunt vinden daerop sy alle ghestalt houden diemen haer gheeft.</s><s> <emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal <pb pagenum="29"/>trecken GI balck der ghewichten FH, daer naer salmen vinden haer ermen door het voorstel, dat is ghelijck FI lb, tot H 4 lb, also den erm KI, tot KG, daer naer salmen trecken EK balck des pilaers ter eender, ende der ghewichten FH ter ander sijden, de selue EK ghedeelt in L, also dat den erm EL sulcken reden hebbe tot LK, als 5 lb van FH, tot I0 lb des pilaers, L sal tbegheerde punt sijn op twelck sy alle ghestalt sullen houden diemen haer gheeft, waer af tbewys openbaer is door het voorstel.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn, weghende I0 lb, diens swaerheydts middelpunt E, ende laet de ghewichten daer an hanghende wesen <emph type="other"/>FI<emph.end/> lb, diens vastpunt G, ende H 4 lb, wiens vastpunt I.</s><s> <emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten het vastpunt vinden daerop sy alle ghestalt houden diemen haer gheeft.</s><s> <emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal <pb pagenum="29"/>trecken GI balck der ghewichten FH, daer naer salmen vinden haer ermen door het 2. voorstel, dat is ghelijck FI lb, tot H 4 lb, also den erm KI, tot KG, daer naer salmen trecken EK balck des pilaers ter eender, ende der ghewichten FH ter ander sijden, de selue EK ghedeelt in L, also dat den erm EL sulcken reden hebbe tot LK, als 5 lb van FH, tot I0 lb des pilaers, L sal tbegheerde punt sijn op twelck sy alle ghestalt sullen houden diemen haer gheeft, waer af tbewys openbaer is door het 7. voorstel.</s>

<s><emph type="other"/>VII. EYSCH<emph.end/>. <emph type="other"/>XII. VOORSTEL<emph.end/>.</s>

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<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, diens vastpunt E, ende handthaef EF, ende twee ghewichten G, H, elck 3 lb weghende, welcker vastpunten C, D; ende IK, sy as, euewydich vanden sichteinder LM, ende D sy tpunt voor de begheer<pb pagenum="30"/>de plaets.</s>

<s> Daer naer wort den as IK (alles draeyende op E) verheuen als inde tweede form. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten een ghewicht an D vinden, dat den as IK in die ghestalt houde. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal vinden door het voorstel, tvastpunt daer op den as alle ghestalt houde diemen haer gheeft twelck N sy: Daer naer salmen trecken DN, ende de hanghende <arrow.to.target n="marg112"/> EO, sniende ND in O, daer naer salmen sien wat reden NO heeft tot OD, ick neme als van I tot 2, daerom hanghe ick an D een ghewicht P van 6 lb, te weten in sulcken reden tot den pilaer met de twee ghewichten G, I, al tsamen I2 lb, als van I tot 2; Ick seg P 6 lb, te wesen het begheerde ghewicht. </s>

<s> Daer naer wort den as IK (alles draeyende op E) verheuen als inde tweede form. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten een ghewicht an D vinden, dat den as IK in die ghestalt houde. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal vinden door het 11. voorstel, tvastpunt daer op den as alle ghestalt houde diemen haer gheeft twelck N sy: Daer naer salmen trecken DN, ende de hanghende <arrow.to.target n="marg112"/> EO, sniende ND in O, daer naer salmen sien wat reden NO heeft tot OD, ick neme als van I tot 2, daerom hanghe ick an D een ghewicht P van 6 lb, te weten in sulcken reden tot den pilaer met de twee ghewichten G, I, al tsamen I2 lb, als van I tot 2; Ick seg P 6 lb, te wesen het begheerde ghewicht. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Tswaerste ghewicht I2 lb des erms ON, heeft sulcken reden tot het lichtste 6 lb des erms OD, ghelijck den langsten erm OD, tot den cortsten ON; Daerom hanghet al euestaltwichtich ande handthaef EF door het voorstel. </s><s>Ende veruolghens den as IK blijft in haer ghegheuen ghestalt.</s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Tswaerste ghewicht I2 lb des erms ON, heeft sulcken reden tot het lichtste 6 lb des erms OD, ghelijck den langsten erm OD, tot den cortsten ON; Daerom hanghet al euestaltwichtich ande handthaef EF door het 1. voorstel. </s><s>Ende veruolghens den as IK blijft in haer ghegheuen ghestalt.</s>

<s>

<margin.target id="marg112"/>Perpendicularem.

</s>

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<pb pagenum="31"/><s><emph type="other"/>TWERCK<emph.end/>.</s>

<s> Men sal vindendoor het voorstel tvastpunt daerop tghegheuen alle ghestalt houdt diemen hem gheeft, twelck Q sy, daer naer salmen trecken PQ, ende de hanghende <arrow.to.target n="marg114"/> ER, sniende PQ in R: siende daer naer wat reden RQ heeft tot RP, ick neem als van I tot 2, so hang ick an P een ghewicht van 4 1/2 lb, te weten in sulcken reden tot den pilaer met de twee ghewichten G, I, al tsamen 9 lb, als van I tot 2; Ick seg S 4 1/2 lb te wesen het begheerde ghewicht. </s>

<s> Men sal vindendoor het 11. voorstel tvastpunt daerop tghegheuen alle ghestalt houdt diemen hem gheeft, twelck Q sy, daer naer salmen trecken PQ, ende de hanghende <arrow.to.target n="marg114"/> ER, sniende PQ in R: siende daer naer wat reden RQ heeft tot RP, ick neem als van I tot 2, so hang ick an P een ghewicht van 4 1/2 lb, te weten in sulcken reden tot den pilaer met de twee ghewichten G, I, al tsamen 9 lb, als van I tot 2; Ick seg S 4 1/2 lb te wesen het begheerde ghewicht. </s>

<s>

<margin.target id="marg114"/>Perpendicularem.

</s>

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<s><emph type="other"/>TBEWYS<emph.end/>.</s>

<s> Tswaerste ghewicht 9 lb des erms RQ, heeft sulcken reden tot het lichtste ghewicht 4 1/2 lb des erms RP, ghelijck den langsten erm RP, tot den cortsten RQ, daerom hanghet al euestaltwichtich ande handthaef EF door het voorstel, ende veruolghens den as LM blijft in haer ghegheuen ghestalt, twelck wy bewysen moesten.</s>

<s> Tswaerste ghewicht 9 lb des erms RQ, heeft sulcken reden tot het lichtste ghewicht 4 1/2 lb des erms RP, ghelijck den langsten erm RP, tot den cortsten RQ, daerom hanghet al euestaltwichtich ande handthaef EF door het 1. voorstel, ende veruolghens den as LM blijft in haer ghegheuen ghestalt, twelck wy bewysen moesten.</s>

<s><emph type="other"/>TBESLVYT<emph.end/>.</s>

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<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een pilaer, ende twee punten inden as, t'een vast t'ander int langste deel verroerlick: Te vinden een rechthefwicht an tverroerlick, dat den pilaer in sijn ghegheuen standt houde </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn, weghende 6 lb, <pb pagenum="34"/>ende die ghedeelt als int beghin des voorstels, ende vastpunt sy R, ende roerlick V, int langste deel des as RQ, want int cortste RP ist onmueghelick dat eenich rechthefwicht den as in haer ghegheuen stant houde. </s><s> <emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten een rechthefwicht an V vinden, dat den pilaer in die standt houde. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal de lini QR voorttrecken tot in Y, also dat RY euen sy an RV: Daer naer salmen vinden tghewicht Z an Y, euestaltwichtich met den pilaer, tselue (ghedenckende dat R vastpunt is) sal van 4 lb wesen door het voorstel; Ick seg daerom dat het begheerde rechthefwicht twelck Æ sy, van 4 lb sal wesen. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn, weghende 6 lb, <pb pagenum="34"/>ende die ghedeelt als int beghin des 1. voorstels, ende vastpunt sy R, ende roerlick V, int langste deel des as RQ, want int cortste RP ist onmueghelick dat eenich rechthefwicht den as in haer ghegheuen stant houde. </s><s> <emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten een rechthefwicht an V vinden, dat den pilaer in die standt houde. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal de lini QR voorttrecken tot in Y, also dat RY euen sy an RV: Daer naer salmen vinden tghewicht Z an Y, euestaltwichtich met den pilaer, tselue (ghedenckende dat R vastpunt is) sal van 4 lb wesen door het 3. voorstel; Ick seg daerom dat het begheerde rechthefwicht twelck Æ sy, van 4 lb sal wesen. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ouermidts den erm RV des rechthefwichts Æ, euen is anden erm RY des ghewichts Z, ende Æ euen an Z, soo is de ghewelt Æ euen an de ghewelt van Z door het voorstel. </s><s>Maer de ghewelt van Z is (Æ gheweert sijnde) den pilaer in die standt te houden, de ghewelt dan van Æ (Z gheweert sijnde) is oock den pilaer in die standt te houden, twelck wy bewysen moesten. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een pilaer, ende twee punten inden as, t'een vast, t'ander int langste deel verroerlick: Wy hebben gheuonden een rechthefwicht an tverroerlick, dat den pilaer in sijn ghegheuen standt houdt naer den eysch. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ouermidts den erm RV des rechthefwichts Æ, euen is anden erm RY des ghewichts Z, ende Æ euen an Z, soo is de ghewelt Æ euen an de ghewelt van Z door het 13. voorstel. </s><s>Maer de ghewelt van Z is (Æ gheweert sijnde) den pilaer in die standt te houden, de ghewelt dan van Æ (Z gheweert sijnde) is oock den pilaer in die standt te houden, twelck wy bewysen moesten. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een pilaer, ende twee punten inden as, t'een vast, t'ander int langste deel verroerlick: Wy hebben gheuonden een rechthefwicht an tverroerlick, dat den pilaer in sijn ghegheuen standt houdt naer den eysch. </s>

<s><emph type="other"/>MERCKT<emph.end/>.</s>

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<s> <emph type="other"/>ANGHESIEN<emph.end/> den heelen pilaer door tghestelde 6 lb weeght, waer af Æ de 4 lb verheft, so volgt nootsaeckelick datter opt punt R, dat is op tsop des keghels &OElig;, 2 lb rusten. </s>

<s> <emph type="other"/>OFTE<emph.end/> soomen an R een rechthefwicht vougde, inde plaets des keghels &OElig;, als hier neuen, dat sal weghen 2 lb. </s>

<s> <emph type="other"/>OFTE<emph.end/> soomen an R een rechthefwicht 4 vougde, inde plaets des keghels OE, als hier neuen, dat <emph.end/><emph.end/><emph.end/>sal weghen 2 lb. </s>

<s> <emph type="other"/>OFTE<emph.end/> soomen an V een keghel Φ vougde, inde plaets des rechthefwichts Æ, als hier neuen, dat op den keghel &OElig; rusten sal 2 lb, ende op den keghel Φ 4 lb. </s>

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<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dattet rechthefwicht Æ den pilaer oock in die ghegheuen ghestalt houdt. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons weeren Æ ende anhanghen Z 4 lb, ende door het voorstel den pilaer sal in die ghestalt bliuen: Maer Æ doet by V soo grooten ghewelt anden pilaer als Z by Y door het I3 voorstel, daerom gheweert Z, ende Æ anghehanghen, soo sal Æ den pilaer oock in die ghestalt houden. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan twee punten inden as des pilaers t'een vast tander verroerlick, trechthefwicht an tverroerlick, dat den pilaer in een ghestalt houdt, sal hem in alle ghestalten houden, twelck wy bewysen moesten.</s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons weeren Æ ende anhanghen Z 4 lb, ende door het 10. voorstel den pilaer sal in die ghestalt bliuen: Maer Æ doet by V soo grooten ghewelt anden pilaer als Z by Y door het I3 voorstel, daerom gheweert Z, ende Æ anghehanghen, soo sal Æ den pilaer oock in die ghestalt houden. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan twee punten inden as des pilaers t'een vast tander verroerlick, trechthefwicht an tverroerlick, dat den pilaer in een ghestalt houdt, sal hem in alle ghestalten houden, twelck wy bewysen moesten.</s>

<s><emph type="other"/>IX VERTOOCH<emph.end/>. <emph type="other"/>XVII VOORSTEL<emph.end/>.</s>

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<s><emph type="other"/>VERVOLGH<emph.end/>. </s>

<s> <emph type="other"/>SOO<emph.end/> de twee punten daer den pilaer op rust, waren inde hanghende <arrow.to.target n="marg121"/> linien door R en V, de selue ghewichten die hier vooren op elck rustende punt waren, soudender nu oock op sijn. </s><s>Laet by boorbeelt door de punten R, V, hanghende linien ghetrocken worden, ende punten inde selue ghestelt als Y , Ghenomen nu dat Y ende de punten sijn daer den pilaer op rust, tis kennelick dat op Y rusten sal 2 lb, ende op 4 lb, waer uyt alsulcken vertooch openbaer is.</s>

<s> <emph type="other"/>SOO<emph.end/> de twee punten daer den pilaer op rust, waren inde hanghende <arrow.to.target n="marg121"/> linien door R en V, de selue ghewichten die hier vooren op elck rustende punt waren, soudender nu oock op sijn. </s><s>Laet by boorbeelt door de punten R, V, hanghende linien ghetrocken worden, ende punten inde selue ghestelt als Y <emph.end/><emph.end/><emph.end/>, Ghenomen nu dat Y ende <emph.end/><emph.end/><emph.end/>de punten sijn daer den pilaer op rust, tis kennelick dat op Y rusten sal 2 lb, ende op <emph.end/><emph.end/><emph.end/>4 lb, waer uyt alsulcken vertooch openbaer is.</s>

<s>

<margin.target id="marg121"/>Perpendicularibus.

</s>

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<s> <emph type="other"/>RVSTENDE<emph.end/> een pilaer op eenighe twee punten, ghelijck het asstick tusschen tswaerheyts middelpunt ende de hanghende door t'slinckerpunt, tottet asstick tusschen t'swaerheydts middelpunt ende de hanghende door trechterpunt, also tghewicht des pilaers rustende op trechterpunt, tottet ghewicht rustende op t'slinckerpunt. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer wesen, diens as EF, ende swaerheydts middelpunt G, ende de twee punten daer d'een pilaer op rust HI, waer duer ghetrocken sijn de hanghende linien KL, MN, sniende den as in O, P; Ick seg dat ghelijck GO tot GP, alsoo de swaerheydt rustende op tpunt I, tot de swaerheydt rustende op H, waer af tbewys openbaer is door tvervolgh des voorgaenden I7 voorstels, </s>

<s> nochtans om alhier wat breder vande nootsakelicheyt te segghen, so laet ons achten al of H ter plaets van O waer, twelck soo ghenomen tghewicht alsdan op H rustende, heeft sulcken reden tottet ghewicht op P rustende, ghelijck GP, tot GO, duer het I7 voorstel; Laet ons voort nemen dattet tpunt H vast blijuende, den pilaer in haer ghegheuen ghestalt neerghetrocken worde, soo verre als van H tot O, ende duer de begheerte, de swaerheydt an H rustende blijft de selue. </s><s>Sghelijcx salmen bethoonen de swaerheyt dieder op P rust, oock te rusten op I, daerom ghelijck GO tot GP, also de swaerheyt rustende op I, tot de swaerheyt rustende op H. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Rustende dan een pilaer op eenighe twee punten, &c. </s>

<s> nochtans om alhier wat breder vande nootsakelicheyt te segghen, so laet ons achten al of H ter plaets van O waer, twelck soo ghenomen tghewicht alsdan op H rustende, heeft sulcken reden tottet ghewicht op P rustende, ghelijck GP, tot GO, duer het I7. voorstel; Laet ons voort nemen dattet tpunt H vast blijuende, den pilaer in haer ghegheuen ghestalt neerghetrocken worde, soo verre als van H tot O, ende duer de 3. begheerte, de swaerheydt an H rustende blijft de selue. </s><s>Sghelijcx salmen bethoonen de swaerheyt dieder op P rust, oock te rusten op I, daerom ghelijck GO tot GP, also de swaerheyt rustende op I, tot de swaerheyt rustende op H. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Rustende dan een pilaer op eenighe twee punten, &c. </s>

<s><emph type="other"/>VERVOLGH<emph.end/>. </s>

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<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck wesen diens plat sy rechthouckich op den sichteinder, ende den grondt AC euewydich vanden sichteinder, ende op de sijde AB, die dobbel sy an BC, ligghe een cloot D, ende op de sijde BC een cloot E, euewichtich ende euegroot met den cloot D. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck de sijde AB 2, tot BCI, alsoo t'staltwicht des cloots E, tottet staltwicht des cloots D. </s>

<s> <emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons maecken rondtom den driehouck ABC eenen crans van veerthien clooten, euegroot, euewichtich, ende euewijt van malcanderen, als E, F, G, H, I, K, L, M, N, O, P, Q, R, D, al ghesnoert an een lini, streckende door haer middelpunten <arrow.to.target n="marg126"/>, also dat sy op die middelpunten drayen mueghen; Datter oock twee clooten passen op de sijde BC, ende vier op BA, dat is ghelijck lini tot lini, also clooten tot clooten; laet oock an S, T, V, drie vastpunten staen, ouer welcke de lini ofte t'snoer der clooten slieren mach, also dat de twee deelen des snoers die bouen den driehouck staen, euewydich <arrow.to.target n="marg127"/> sijn vande sijden AB, BC; Inder voughen dat alsmen den crans an d'een ofte d'ander sijde neertrect, soo rollen de clooten op de linien AB, BC. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Soo t'staltwicht der vier clooten D, R, Q, P, niet euen en waer met het staltwicht der twee clooten E, F, t'een of t'ander sal swaerder sijn, latet wesen (soot mueghelick waer) der vier D, R, Q. </s><s>P; Maer de vier clooten O, N, M, L, sijn euewichtich met de vier clooten G, H, I, K, de sijde dan der acht clooten D, R, Q, P, O, N, M, L, is swaerder na de ghestalt dan de sijde der ses clooten E, F, G, H, I, K: maer want het swaerste altijdt het lichtste ouerweeght, de acht clooten sullen neerwaert rollen, ende d'ander ses rijsen: </s><s> Latet soo wesen, ende D <pb pagenum="42"/>sy gheuallen daer nu O is, ende E, F, G, H, sullen sijn daer nu P, Q, R, D, ende I, K, daer nu E, F, sijn. </s><s>Maer dit soo wesende, den crans der clooten sal alsulcken ghestalt hebben als sy te vooren dede, ende sullen om de selue redenen de acht clooten ter slincker sijde wederom staltwichtigher sijn dan de ses clooten ter rechter, waer duer de acht clooten wederom neer sullen rollen, ende d'ander ses rijsen, welcke valling ter eender, ende rijsing ter ander, om dat de reden altijdt de selue is, altijdt ghedueren sal, ende de clooten sullen uyt haer seluen een maken, t'welck valsch is. </s><s> Het deel dan des crans D, R, Q, P, O, N, M, L, is euestaltwichtich met het deel E, F, G, H, I, K: Maer van sulke euewichtighe ghetrocken euewichtighe, de resten sijn euewichtich, laet ons dan van dat deel trecken de vier clooten O, N, M, L, ende van dit de vier clooten G, H, I, K, (welcke euen sijn ande voornoemde O, N, M, L,) de resten D, R, Q, P, ende E, F, sullen euestaltwichtich sijn, Maer wesende dese twee euestaltwichtich met die vier, E sal tweemael staltswaerder sijn als D. </s><s>Ghelijck dan de lini BA 2, tot de lini BCI, also t'staltwicht des cloots E, tottet staltwicht des cloots D.</s>

<s> <emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons maecken rondtom den driehouck ABC eenen crans van veerthien clooten, euegroot, euewichtich, ende euewijt van malcanderen, als E, F, G, H, I, K, L, M, N, O, P, Q, R, D, al ghesnoert an een lini, streckende door haer middelpunten <arrow.to.target n="marg126"/>, also dat sy op die middelpunten drayen mueghen; Datter oock twee clooten passen op de sijde BC, ende vier op BA, dat is ghelijck lini tot lini, also clooten tot clooten; laet oock an S, T, V, drie vastpunten staen, ouer welcke de lini ofte t'snoer der clooten slieren mach, also dat de twee deelen des snoers die bouen den driehouck staen, euewydich <arrow.to.target n="marg127"/> sijn vande sijden AB, BC; Inder voughen dat alsmen den crans an d'een ofte d'ander sijde neertrect, soo rollen de clooten op de linien AB, BC. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Soo t'staltwicht der vier clooten D, R, Q, P, niet euen en waer met het staltwicht der twee clooten E, F, t'een of t'ander sal swaerder sijn, latet wesen (soot mueghelick waer) der vier D, R, Q. </s><s>P; Maer de vier clooten O, N, M, L, sijn euewichtich met de vier clooten G, H, I, K, de sijde dan der acht clooten D, R, Q, P, O, N, M, L, is swaerder na de ghestalt dan de sijde der ses clooten E, F, G, H, I, K: maer want het swaerste altijdt het lichtste ouerweeght, de acht clooten sullen neerwaert rollen, ende d'ander ses rijsen: </s><s> Latet soo wesen, ende D <pb pagenum="42"/>sy gheuallen daer nu O is, ende E, F, G, H, sullen sijn daer nu P, Q, R, D, ende I, K, daer nu E, F, sijn. </s><s>Maer dit soo wesende, den crans der clooten sal alsulcken ghestalt hebben als sy te vooren dede, ende sullen om de selue redenen de acht clooten ter slincker sijde wederom staltwichtigher sijn dan de ses clooten ter rechter, waer duer de acht clooten wederom neer sullen rollen, ende d'ander ses rijsen, welcke valling ter eender, ende rijsing ter ander, om dat de reden altijdt de selue is, altijdt ghedueren sal, ende de clooten sullen uyt haer seluen een eeuwich roersel maken, t'welck valsch is. </s><s> Het deel dan des crans D, R, Q, P, O, N, M, L, is euestaltwichtich met het deel E, F, G, H, I, K: Maer van sulke euewichtighe ghetrocken euewichtighe, de resten sijn euewichtich, laet ons dan van dat deel trecken de vier clooten O, N, M, L, ende van dit de vier clooten G, H, I, K, (welcke euen sijn ande voornoemde O, N, M, L,) de resten D, R, Q, P, ende E, F, sullen euestaltwichtich sijn, Maer wesende dese twee euestaltwichtich met die vier, E sal tweemael staltswaerder sijn als D. </s><s>Ghelijck dan de lini BA 2, tot de lini BCI, also t'staltwicht des cloots E, tottet staltwicht des cloots D.</s>

<s>

<margin.target id="marg126"/>Centra.

<margin.target id="marg127"/>Parallela.

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<s><emph type="other"/>V. VERVOLGH<emph.end/>.</s>

<s><emph type="other"/>LAET<emph.end/> ons trecken de hanghende lini uyt het middelpunt <arrow.to.target n="marg129"/> des pilaers D als DK, sniende de sijde des pilaers in L, t'welck soo sijnde, den driehouck LDI is ghelijck an den driehouck ABC, want de houcken ACB ende LID sijn recht, ende LD is euewydich van BC ende DI van AB: Daerom ghelijck AB tot BC, alsoo <pb pagenum="44"/>LD tot DI; </s><s> Maer ghelijck AB tot BC, alsoo den Pilaer tot t'ghewicht E door het 4 veruolg, daerom ghelijck LD tot DI, also den pilaer tot E. </s><s>Laet ons nu ande lini KD voughen t rechthefwicht M met den pilaer euestaltwichtig, t'selue ghewicht M sal met den pilaer euewichtich sijn door het voorstel: Daerom ghelijck LD tot DI, also M tot E.</s>

<s><emph type="other"/>LAET<emph.end/> ons trecken de hanghende lini uyt het middelpunt <arrow.to.target n="marg129"/> des pilaers D als DK, sniende de sijde des pilaers in L, t'welck soo sijnde, den driehouck LDI is ghelijck an den driehouck ABC, want de houcken ACB ende LID sijn recht, ende LD is euewydich van BC ende DI van AB: Daerom ghelijck AB tot BC, alsoo <pb pagenum="44"/>LD tot DI; </s><s> Maer ghelijck AB tot BC, alsoo den Pilaer tot t'ghewicht E door het 4. veruolg, daerom ghelijck LD tot DI, also den pilaer tot E. </s><s>Laet ons nu ande lini KD voughen t rechthefwicht M met den pilaer euestaltwichtig, t'selue ghewicht M sal met den pilaer euewichtich sijn door het 14. voorstel: Daerom ghelijck LD tot DI, also M tot E.</s>

<s>

<margin.target id="marg129"/>Centro.

</s>

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<s><emph type="other"/>ENDE<emph.end/> dit niet alleen van clooten maer van ander lichamen slierende, ofte rollende, op punten ofte linien als hier onder (daerwy eyghentlicker af handelen sullen inde ) alwaer wy noch segghen ghelijck CE tot EO, also t'ghewicht des lichaems D tottet ghewicht P. </s>

<s> <emph type="other"/>WAER<emph.end/> uyt oock blijckt, dat wesende de lini AB euewydich vanden sichteinder <arrow.to.target n="marg133"/> als hier neuens, dat CE ende CO dan in een selfde lini sullen vallen, waer duer tusschen E en O gheen langde en sal sijn, ende veruolghens CE en sal tot EO gheen reden hebben, daermen by verstaen sal dat een swaerheydt inde plaets van P hoe cleen sy mocht wesen, en sal niet euestaltwichtich connen sijn teghen t'lichaem D, maer salt (wisconstelick <arrow.to.target n="marg134"/> verstaende) voorttrecken hoe swaer het sy: Waer uyt volght, dat alle swaerheden voortghetrocken langs den sichteinder, als schepen int water, waghens langs t'platte landt, &c. </s><s>en behouuen gheen tot haer verroersel, meer dan de omstaende verhindernissen en veroirsaecken, als Water, Locht, Naecsel der assen, teghen de bussen, naecsel der rayers teghen de straet, ende dierghelijcke. </s>

<s> <emph type="other"/>WAER<emph.end/> uyt oock blijckt, dat wesende de lini AB euewydich vanden sichteinder <arrow.to.target n="marg133"/> als hier neuens, dat CE ende CO dan in een selfde lini sullen vallen, waer duer tusschen E en O gheen langde en sal sijn, ende veruolghens CE en sal tot EO gheen reden hebben, daermen by verstaen sal dat een swaerheydt inde plaets van P hoe cleen sy mocht wesen, en sal niet euestaltwichtich connen sijn teghen t'lichaem D, maer salt (wisconstelick <arrow.to.target n="marg134"/> verstaende) voorttrecken hoe swaer het sy: Waer uyt volght, dat alle swaerheden voortghetrocken langs den sichteinder, als schepen int water, waghens langs t'platte landt, &c. </s><s>en behouuen gheen <emph.end/><emph.end/><emph.end/>tot haer verroersel, meer dan de omstaende verhindernissen en veroirsaecken, als Water, Locht, Naecsel der assen, teghen de bussen, naecsel der rayers teghen de straet, ende dierghelijcke. </s>

<s>

<margin.target id="marg133"/>Horizonte.

<margin.target id="marg134"/>Mathematicè.

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<s><emph type="other"/>MAER<emph.end/> anghesien den driehouck ABN int veruolg, tot dese eueredenheyt <arrow.to.target n="marg135"/> niet en gheeft noch en neemt, laet ons hem weeren, ansiende G voor vastpunt des pilaers rustende op een pin als hier neuen, ende sullen noch segghen ghelijck LD tot DO, also M tot P. </s>

<s><emph type="other"/>MAER<emph.end/> anghesien den driehouck ABN int 6. veruolg, tot dese eueredenheyt <arrow.to.target n="marg135"/> niet en gheeft noch en neemt, laet ons hem weeren, ansiende G voor vastpunt des pilaers rustende op een pin als hier neuen, ende sullen noch segghen ghelijck LD tot DO, also M tot P. </s>

<s>

<margin.target id="marg135"/>Proportionem

</s>

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<s><emph type="other"/>VII. VERVOLGH<emph.end/>.</s>

<s><emph type="other"/>MAER<emph.end/> op dat nu blijcke dese eueredenheydt <arrow.to.target n="marg136"/> niet alleen also te bestaen inde pilaren alwaer de rechtheflini als DL, comt uyt t'middelpunt des pilaers, ende diens vastpunt is des assens uyterste, als hier vooren G int 6 vervolg; Soo laet ABC een driehouck sijn, wiens sijde AB dobbel is an BC, ende BC sy hanghende <arrow.to.target n="marg137"/> op AC: Ende laet DE een pilaer sijn diens as FG rechthouckich op AB, ende sniende AB in t'punt H, ende I sy eenich ander punt inden seluen as; </s>

<s><emph type="other"/>MAER<emph.end/> op dat nu blijcke dese eueredenheydt <arrow.to.target n="marg136"/> niet alleen also te bestaen inde pilaren alwaer de rechtheflini als DL, comt uyt t'middelpunt des pilaers, ende diens vastpunt is des assens uyterste, als hier vooren G int 6. vervolg; Soo laet ABC een driehouck sijn, wiens sijde AB dobbel is an BC, ende BC sy hanghende <arrow.to.target n="marg137"/> op AC: Ende laet DE een pilaer sijn diens as FG rechthouckich op AB, ende sniende AB in t'punt H, ende I sy eenich ander punt inden seluen as; </s>

<s>

<margin.target id="marg136"/>Proportionem.

<margin.target id="marg137"/>Perpendicularis.

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<s> Laet oock KL een ander pilaer sijn, euen ende ghelijck anden pilaer DE, wiens as MN, ende O een punt des as naeckende BC, ende van ghelijcke ghestalt in sijn pilaer, als H inden pilaer DE; Laet oock P een ander punt sijn van sulcker ghestalt inden pilaer KL, als I inden pilaer DE; Ende laet Q een vastpunt sijn daer ouer de lini IQP slieren mach, alsoo dat de lini IQ euewydich <arrow.to.target n="marg138"/> sy van AB, ende QP euewydich van BC. </s><s> Ende om de redenen die int voorstel vande I4 clooten verclaert sijn (t'welck wy hier duer soodanighe veel slierende pilaren oock souden connen bewysen, maer want sulcx uyt t'voorgaende kennelick is, wy slaent ouer) het staltwicht des pilaers KL, sal dobbel sijn an t'staltwicht des pilaers DE.</s>

<s> Laet oock KL een ander pilaer sijn, euen ende ghelijck anden pilaer DE, wiens as MN, ende O een punt des as naeckende BC, ende van ghelijcke ghestalt in sijn pilaer, als H inden pilaer DE; Laet oock P een ander punt sijn van sulcker ghestalt inden pilaer KL, als I inden pilaer DE; Ende laet Q een vastpunt sijn daer ouer de lini IQP slieren mach, alsoo dat de lini IQ euewydich <arrow.to.target n="marg138"/> sy van AB, ende QP euewydich van BC. </s><s> Ende om de redenen die int 19. voorstel vande I4 clooten verclaert sijn (t'welck wy hier duer soodanighe veel slierende pilaren oock souden connen bewysen, maer want sulcx uyt t'voorgaende kennelick is, wy slaent ouer) het staltwicht des pilaers KL, sal dobbel sijn an t'staltwicht des pilaers DE.</s>

<s>

<margin.target id="marg138"/>Parallela.

</s>

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<s> Laet AB een pilaer sijn, diens as CD, ende vastpunt E, ende t'verroerlick punt F, ende t'scheefhefwicht dat hem in die ghestalt houdt sy G, diens scheefheflini FH, ende FI sy rechtheflini, diens rechthefwicht K. </s><s>Laet LM oock een pilaer sijn, euen ende ghelijck an den pilaer AB, wiens as sy NO, ende vastpunt E, ende verroerlick punt P, also dat EN euen sy an ED, ende EF an EP, ende t'scheefhefwicht Q sy euen an G, ende sijn scheefheflini sy PR, euewydich <arrow.to.target n="marg140"/> van FH, ende trechthefwicht S sy euen an K, ende sijn rechtheflini sy PT. </s><s>Dit soo sijnde laet ons vergaren de twee pilaren AB ende LM, ansiende AM voor een heel pilaer, wiens swaerheyts middelpunt <arrow.to.target n="marg141"/> ende vastpunt sal E sijn door t'ghestelde. </s><s>Laet ons nu weeren de ghewichten K, G, S, Q, ende den pilaer AM sal op E alle ghestalt houden diemen hem gheeft door het voorstel, hy sal dan soo blijuen, ende den pilaer AB sal alsoo euewichtich blijuen teghen den pilaer LM. </s><s>Laet ons nu de ghewichten QG weder andoen, hanghende euewichtighe van ghelijcke ghestalt, an euewichtighe, ende door het voorstel, Q sal anden pilaer AM euen sulcken macht doen als G: Ende veruolghens Q doet sulcken macht an huer pilaer LM, als G an huer pilaer AB; maer de macht van G is AB in die ghestalt te houden door het 6 vervolg, de macht dan van Q is oock LM in die ghestalt te houden. </s><s>Insghelijcx soo is oock de macht van K, den pilaer AB in die ghestalt te houden, daerom oock is de macht van S den pilaer LM in die ghestalt te houden; Nu ghelijck IF tot FH, also K tot G door het 8 veruolg, Maer TP. </s><s>is euen an IF, ende PR an FH, ende S an K, ende Q an G, ghelijck dan TP tot PR, alsoo S tot Q. </s><s> Dese eueredenheydt dan, als wy gheseyt hebben, is so wel inde voorbeelden alwaer t'roerende punt P leegher is dan t'vastpunt E, ende alwaer de scheefheflini PR afwyct vande sijde des vastpunts E, als daert hoogher is, ende daer de scheefheflini helde naer t'vastpunt. </s>

<s> Laet AB een pilaer sijn, diens as CD, ende vastpunt E, ende t'verroerlick punt F, ende t'scheefhefwicht dat hem in die ghestalt houdt sy G, diens scheefheflini FH, ende FI sy rechtheflini, diens rechthefwicht K. </s><s>Laet LM oock een pilaer sijn, euen ende ghelijck an den pilaer AB, wiens as sy NO, ende vastpunt E, ende verroerlick punt P, also dat EN euen sy an ED, ende EF an EP, ende t'scheefhefwicht Q sy euen an G, ende sijn scheefheflini sy PR, euewydich <arrow.to.target n="marg140"/> van FH, ende trechthefwicht S sy euen an K, ende sijn rechtheflini sy PT. </s><s>Dit soo sijnde laet ons vergaren de twee pilaren AB ende LM, ansiende AM voor een heel pilaer, wiens swaerheyts middelpunt <arrow.to.target n="marg141"/> ende vastpunt sal E sijn door t'ghestelde. </s><s>Laet ons nu weeren de ghewichten K, G, S, Q, ende den pilaer AM sal op E alle ghestalt houden diemen hem gheeft door het 7. voorstel, hy sal dan soo blijuen, ende den pilaer AB sal alsoo euewichtich blijuen teghen den pilaer LM. </s><s>Laet ons nu de ghewichten QG weder andoen, hanghende euewichtighe van ghelijcke ghestalt, an euewichtighe, ende door het 13. voorstel, Q sal anden pilaer AM euen sulcken macht doen als G: Ende veruolghens Q doet sulcken macht an huer pilaer LM, als G an huer pilaer AB; maer de macht van G is AB in die ghestalt te houden door het 6 vervolg, de macht dan van Q is oock LM in die ghestalt te houden. </s><s>Insghelijcx soo is oock de macht van K, den pilaer AB in die ghestalt te houden, daerom oock is de macht van S den pilaer LM in die ghestalt te houden; Nu ghelijck IF tot FH, also K tot G door het 8 veruolg, Maer TP. </s><s>is euen an IF, ende PR an FH, ende S an K, ende Q an G, ghelijck dan TP tot PR, alsoo S tot Q. </s><s> Dese eueredenheydt dan, als wy gheseyt hebben, is so wel inde voorbeelden alwaer t'roerende punt P leegher is dan t'vastpunt E, ende alwaer de scheefheflini PR afwyct vande sijde des vastpunts E, als daert hoogher is, ende daer de scheefheflini helde naer t'vastpunt. </s>

<s>

<margin.target id="marg140"/>Parallela.

<margin.target id="marg141"/>Centrum grauitatis.

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<s><emph type="other"/>XII. VERVOLGH<emph.end/>.</s>

<s><emph type="other"/>MAER<emph.end/> dat dese eueredenheydt <arrow.to.target n="marg142"/> oock bestaet inde ghestalt daer den as euewydich is vanden sichteinder <arrow.to.target n="marg143"/>, wort aldus bethoont: Laet AB een pilaer sijn, diens as CD euewydich <arrow.to.target n="marg144"/> sy vande sichteinder, ende t'vastpunt daer in E, ende t'roerlick punt F, ende G t'scheefhefwicht dat den pilaer in die ghestalt houdt, wiens scheefheflini FH, ende I trechthefwicht dat den pilaer oock in die ghestalt houdt, wiens rechtheflini FK; T'welck soo sijnde, Laet KF tot FH een ander reden hebben (soot mueghelick waer) dan I tot G, By voorbeelt KF sy tot FH, als I tot 2, maer I tot G, als 3 tot 7. </s><s>Dit so ghenomen, laet ons den pilaer der eerste form neerduwen, ofte der tweeder form oplichten, tot dat KF sulcken reden hebbe tot FH, als 3 tot 7, ende alsdan sal G oock euestaltwichtich sijn teghen den pilaer door de voorgaende vervolghen; Inder voughen dat den pilaer hoogher ende leegher verheuen, sal teghen G euestaltwichtich blijuen, t'welck openbaer onmueghelick is, als oock wisconstlick <arrow.to.target n="marg145"/> sal blijcken door t'volghende voorstel. </s><s>KF dan en heeft tot FH gheen ander reden dan I tot G.</s>

<s><emph type="other"/>MAER<emph.end/> dat dese eueredenheydt <arrow.to.target n="marg142"/> oock bestaet inde ghestalt daer den as euewydich is vanden sichteinder <arrow.to.target n="marg143"/>, wort aldus bethoont: Laet AB een pilaer sijn, diens as CD euewydich <arrow.to.target n="marg144"/> sy vande sichteinder, ende t'vastpunt daer in E, ende t'roerlick punt F, ende G t'scheefhefwicht dat den pilaer in die ghestalt houdt, wiens scheefheflini FH, ende I trechthefwicht dat den pilaer oock in die ghestalt houdt, wiens rechtheflini FK; T'welck soo sijnde, Laet KF tot FH een ander reden hebben (soot mueghelick waer) dan I tot G, By voorbeelt KF sy tot FH, als I tot 2, maer I tot G, als 3 tot 7. </s><s>Dit so ghenomen, laet ons den pilaer der eerste form neerduwen, ofte der tweeder form oplichten, tot dat KF sulcken reden hebbe tot FH, als 3 tot 7, ende alsdan sal G oock euestaltwichtich sijn teghen den pilaer door de voorgaende vervolghen; Inder voughen dat den pilaer hoogher ende leegher verheuen, sal teghen G euestaltwichtich blijuen, t'welck openbaer onmueghelick is, als oock wisconstlick <arrow.to.target n="marg145"/> sal blijcken door t'volghende 22 voorstel. </s><s>KF dan en heeft tot FH gheen ander reden dan I tot G.</s>

<s>

<margin.target id="marg142"/>Proportio.

<margin.target id="marg143"/>Horizonte.

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<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn, diens as CD, ende vastpunt E, ende roerlick punt F, daer an den pilaer door trechtdaelwicht G in die ghestalt ghehouden wort, daer an oock den pilaer door t'scheefdaelwicht H (welverstaende G gheweert sijnde) in die ghestalt ghehouden wordt, ende de rechtdaellini snie de sijde des pilaers in I, maer de scheefdaellini snie de selue sijde in K. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck de rechtdaellini IF tot de scheefdaellini FK, alsoo t'rechtdaelwicht G tot het scheefdaelwicht H. </s><s><emph type="other"/>TBEREYTSEL<emph.end/>. </s><s>Laet ons teeckenen t'punt L, alsoo dat EL euen sy an EF, ende voughen an t'punt L t'rechthefwicht M, dat den pilaer in die ghestalt can houden, diens rechtheflini LN: Insghelijcx t'scheefhefwicht O, dat den pilaer oock in die ghestalt can houden, wiens scheefheflini LP euewydich sy met FK. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ghelijck NL tot LP, also M tot O, duer het voorstel maer de macht van G is anden pilaer euen met de macht van M, ende de macht van H met die van O duer het voorstel, ende IF is euen an LN, ende FK, an LP; Daerom ghelijck de rechtdaellini IF tot de scheefdaellini FK, alsoo t'rechtdaelwicht G tottet scheefdaelwicht H, Sghelijcx sal oock t'bewys sijn van alle d'ander ghestalten als inde formen hier na volghende.</s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ghelijck NL tot LP, also M tot O, duer het 20. voorstel maer de macht van G is anden pilaer euen met de macht van M, ende de macht van H met die van O duer het 13. voorstel, ende IF is euen an LN, ende FK, an LP; Daerom ghelijck de rechtdaellini IF tot de scheefdaellini FK, alsoo t'rechtdaelwicht G tottet scheefdaelwicht H, Sghelijcx sal oock t'bewys sijn van alle d'ander ghestalten als inde formen hier na volghende.</s>

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<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een bekenden pilaer, met een vastpunt inden as, ende een roerlick punt, an t'welck eenich onbekent treckwicht den pilaer in ghegheuen ghestalt houdt: Dat treckwicht bekent te maken. </s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, ende ghedeelt als int voorstel, ende t'vastpunt sy X, ende het roerende punt S, an t'welck gheuoecht sy een onbekent scheefhefwicht Y, met den pilaer euestaltwichtich, ende sijn scheefheflini snie de sijde des pilaers AB in &OElig;. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten dat onbekende scheefhefwicht Y bekent maken. </s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, ende ghedeelt als int 1. voorstel, ende t'vastpunt sy X, ende het roerende punt S, an t'welck gheuoecht sy een onbekent scheefhefwicht Y, met den pilaer euestaltwichtich, ende sijn scheefheflini snie de sijde des pilaers AB in &OElig;. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten dat onbekende scheefhefwicht Y bekent maken. </s>

<s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal sien wat rechthefwicht an S den pilaer in die ghestalt soude houden, wort beuonden door voorstel, van 4 lb, daer naer salmen ondersoucken wat reden eenighe hanghende lini <arrow.to.target n="marg146"/> als ZÆ, heeft tot Z&OElig;, ick neme als van 2 tot I, daer uyt seg ick 2 gheeft I, wat t'rechthefwicht van 4 lb ? comt voor Y 2 lb, t'welck ick seg sijn waer ghewicht te sijne. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de hanghende door S welcke sy AS. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ghelijck AS tot S&OElig;, also t'rechthefwicht tottet scheefhefwicht Y door het 20 voorstel, maer den driehouck &OElig;ZB, is ghelijck anden driehouck &OElig;SA, welcker lijckstandighe linien <arrow.to.target n="marg147"/> sijn &OElig;Z met &OElig;S, ende ZÆ met SA: Daerom ghelijck AS tot S&OElig;, alsoo ÆZ tot Z&OElig;, ende vervolghens ghelijck ÆZ 2, tot Z&OElig; I, also t'rechthefwicht 4 lb tot Y, daerom Y weghende 2 lb is bekent ghemaect, t'welck wy bewysen moesten. </s><s>Ende sghelijcx sal den voortganck sijn in allen anderen voorbeelden. </s>

<s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal sien wat rechthefwicht an S den pilaer in die ghestalt soude houden, wort beuonden door 14. voorstel, van 4 lb, daer naer salmen ondersoucken wat reden eenighe hanghende lini <arrow.to.target n="marg146"/> als ZÆ, heeft tot Z&OElig;, ick neme als van 2 tot I, daer uyt seg ick 2 gheeft I, wat t'rechthefwicht van 4 lb ? comt voor Y 2 lb, t'welck ick seg sijn waer ghewicht te sijne. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de hanghende door S welcke sy AS. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ghelijck AS tot S&OElig;, also t'rechthefwicht tottet scheefhefwicht Y door het 20 voorstel, maer den driehouck &OElig;ZB, is ghelijck anden driehouck &OElig;SA, welcker lijckstandighe linien <arrow.to.target n="marg147"/> sijn &OElig;Z met &OElig;S, ende ZÆ met SA: Daerom ghelijck AS tot S&OElig;, alsoo ÆZ tot Z&OElig;, ende vervolghens ghelijck ÆZ 2, tot Z&OElig; I, also t'rechthefwicht 4 lb tot Y, daerom Y weghende 2 lb is bekent ghemaect, t'welck wy bewysen moesten. </s><s>Ende sghelijcx sal den voortganck sijn in allen anderen voorbeelden. </s>

<s>

<margin.target id="marg146"/>Perpendicularis.

<margin.target id="marg147"/>Homologa.

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<s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet an t'punt F gheuoucht worden t'rechthefwicht L dat den pilaer oock in die ghestalt can houden, ende sijn rechtheflini sy FM. </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Want de linien FH, FK, sijn tusschen de euewydighe <arrow.to.target n="marg152"/> HK, CD, ende dat den houck HFC, euen is (door t'ghegheuen) an den houck KFD, so sijn FH ende FK euen. </s><s>waer uyt volght dat ghelijck MF tot FH, alsoo MF tot FK, Maer ghelijck MF tot FH, alsoo L tot G, daerom oock ghelijck MF tot FK, also L tot G; maer I is euen an G door t'ghestelde, ghelijck dan MF tot FK, alsoo L tot I. </s><s>T'welck so sijnde, I houdt den pilaer in die ghestalt door het voorstel. </s><s>Sghelijcx sal oock t'bewijs sijn in alle anderen voorbeelden. </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Want de linien FH, FK, sijn tusschen de euewydighe <arrow.to.target n="marg152"/> HK, CD, ende dat den houck HFC, euen is (door t'ghegheuen) an den houck KFD, so sijn FH ende FK euen. </s><s>waer uyt volght dat ghelijck MF tot FH, alsoo MF tot FK, Maer ghelijck MF tot FH, alsoo L tot G, daerom oock ghelijck MF tot FK, also L tot G; maer I is euen an G door t'ghestelde, ghelijck dan MF tot FK, alsoo L tot I. </s><s>T'welck so sijnde, I houdt den pilaer in die ghestalt door het 20 voorstel. </s><s>Sghelijcx sal oock t'bewijs sijn in alle anderen voorbeelden. </s>

<s>

<margin.target id="marg152"/>Parallelas.

</s>

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<s><emph type="other"/>II. VOORBEELT<emph.end/>. </s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn hanghende ande oneuewydighe linien CD, EF, welcke voortghetrocken sijn tot G, H, sniende malcanderen in I. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dattet punt I, inde swaerheyts middellini is des pilaers AB. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons DG ende FH ansien voor stylen ofte stiue linien daer den pilaer op rust, welcke door de begheerte niet en breken noch en buyghen; der seluer ghewelt is euen ande ghewelt der linien CD, EF, want ghelijck dese den pilaer in sijn ghegheuen standt houden alsoo oock die. </s><s>Ende wat punten wy inde linien DG, FH voor uytersten nemen, den pilaer <pb pagenum="58"/>houdt daer op sijn ghegheuen standt. </s><s> Laet ons nemen I, ghemeen uyterste punt van d'een en d'ander lini; den pilaer dan houdt daer op (Wisconstlick <arrow.to.target n="marg159"/> verstaende) sijn ghegheuen standt, maer rustende den pilaer op t'punt I, soo is de hanghende door I des pilaers swaerheydts middellini, inde welcke I is.</s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn hanghende ande oneuewydighe linien CD, EF, welcke voortghetrocken sijn tot G, H, sniende malcanderen in I. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dattet punt I, inde swaerheyts middellini is des pilaers AB. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons DG ende FH ansien voor stylen ofte stiue linien daer den pilaer op rust, welcke door de 2. begheerte niet en breken noch en buyghen; der seluer ghewelt is euen ande ghewelt der linien CD, EF, want ghelijck dese den pilaer in sijn ghegheuen standt houden alsoo oock die. </s><s>Ende wat punten wy inde linien DG, FH voor uytersten nemen, den pilaer <pb pagenum="58"/>houdt daer op sijn ghegheuen standt. </s><s> Laet ons nemen I, ghemeen uyterste punt van d'een en d'ander lini; den pilaer dan houdt daer op (Wisconstlick <arrow.to.target n="marg159"/> verstaende) sijn ghegheuen standt, maer rustende den pilaer op t'punt I, soo is de hanghende door I des pilaers swaerheydts middellini, inde welcke I is.</s>

<s>

<margin.target id="marg159"/>Mathematicè.

</s>

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<s> <emph type="other"/>HANGHENDE<emph.end/> een pilaer euestaltwichtich teghen twee scheefhefwichten: Ghelijck scheefheflini tot rechtheflini, alsoo elck scheefhefwicht tot sijn rechthefwicht. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn wiens as CD, ende twee punten daer in E, F, welcker scheefhefwichten die hem in die standt houden sijn G, H, ende rechthefwichten I, K, ende scheefheflinien EL, FM, ende rechtheflinien EN, FO. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck LE tot EN, alsoo G tot I, ende ghelijck MF tot FO, alsoo H tot K. </s><s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons F ansien voor vastpunt, ende E voor t'roerlick, daerom (door het voorstel) ghelijck LE <pb pagenum="61"/>tot EN, alsoo G tot I. </s><s>Laet ons ten tweeden E ansien voor vastpunt, ende F voor t'roerlick; Daerom (door t'voornoemde 20 voorstel) ghelijck MF tot FO, alsoo H tot K. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn wiens as CD, ende twee punten daer in E, F, welcker scheefhefwichten die hem in die standt houden sijn G, H, ende rechthefwichten I, K, ende scheefheflinien EL, FM, ende rechtheflinien EN, FO. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck LE tot EN, alsoo G tot I, ende ghelijck MF tot FO, alsoo H tot K. </s><s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons F ansien voor vastpunt, ende E voor t'roerlick, daerom (door het 20. voorstel) ghelijck LE <pb pagenum="61"/>tot EN, alsoo G tot I. </s><s>Laet ons ten tweeden E ansien voor vastpunt, ende F voor t'roerlick; Daerom (door t'voornoemde 20 voorstel) ghelijck MF tot FO, alsoo H tot K. </s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Hanghende dan een pilaer euestaltwichtich teghen twee scheefhefwichten: Ghelijc scheefheflini tot rechtheflini, alsoo elck scheefhefwicht tot sijn rechthefwicht, t'welck wy bewysen moesten.</s>

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<margin.target id="marg162"/>Proportiones.

</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ons t'voorbeelt nemen der eueredenheydt des voorstels aldus: Het sy een pilaer AB, diens as CD, ende swaerheyts middelpunt E, ende vastpunt daer in F, ende roerlick punt G, an t'welc gheuoucht sy een scheefhefwicht H, dat den pilaer in die ghestalt houde, diens scheefheflini GI. </s><s>Daer naer trechthefwicht K, dat den pilaer oock in die ghestalt houde, diens rechtheflini GL, alwaer wy segghen, ghelijck IG tot GL, also H tot K. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat dese eueredenheydt niet alleenlick also en bestaet in t'lichaem AB een pilaer sijnde, maer van sulcke form alst valt.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ons t'voorbeelt nemen der eueredenheydt des 20. voorstels aldus: Het sy een pilaer AB, diens as CD, ende swaerheyts middelpunt E, ende vastpunt daer in F, ende roerlick punt G, an t'welc gheuoucht sy een scheefhefwicht H, dat den pilaer in die ghestalt houde, diens scheefheflini GI. </s><s>Daer naer trechthefwicht K, dat den pilaer oock in die ghestalt houde, diens rechtheflini GL, alwaer wy segghen, ghelijck IG tot GL, also H tot K. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat dese eueredenheydt niet alleenlick also en bestaet in t'lichaem AB een pilaer sijnde, maer van sulcke form alst valt.</s>

<s> <emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons den pilaer AB (blijuende de linien FG ende IL op haer plaetsen) neertrecken, alsoo dat hy bliue hanghende an sijn swaerheyts middelpunt <arrow.to.target n="marg163"/> E, wiens ghestalt dan sy als hier neuens. </s><s>Ende door de begheerte den pilaer en veroirsaect op de punten F, G, gheen ander swaerheydt dan d'eerste; ende alles blijft noch euestaltwichtich, ende ghelijck IG tot GL, alsoo noch H tot K.</s>

<s> <emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons den pilaer AB (blijuende de linien FG ende IL op haer plaetsen) neertrecken, alsoo dat hy bliue hanghende an sijn swaerheyts middelpunt <arrow.to.target n="marg163"/> E, wiens ghestalt dan sy als hier neuens. </s><s>Ende door de 3. begheerte den pilaer en veroirsaect op de punten F, G, gheen ander swaerheydt dan d'eerste; ende alles blijft noch euestaltwichtich, ende ghelijck IG tot GL, alsoo noch H tot K.</s>

<s>

<margin.target id="marg163"/>Centrum grauitatis.

</s>

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<s> <emph type="other"/>LAET<emph.end/> nu de form des pilaers (al de stoff bliuende) verandert worden in eenighe ander ongheschickte form, als AB hier neuens, diens swaerheydts middelpunt E sy, ende een rechte lini daer duer CD (welcke vinding des swaerheyts middelpunts ende rechter linien inde verclaert sal worden, werckelick, niet Wisconstelick <arrow.to.target n="marg164"/>) ende alles blijft noch euestaltwichtich, ende ghelijck IG tot GL, also noch H tot K. </s>

<s> <emph type="other"/>LAET <emph.end/>nu de form des pilaers (al de stoff bliuende) verandert worden in eenighe ander ongheschickte form, als AB hier neuens, diens swaerheydts middelpunt E sy, ende een rechte lini daer duer CD (welcke vinding des swaerheyts middelpunts ende rechter linien inde Weeghdaet verclaert sal worden, werckelick, niet Wisconstelick <arrow.to.target n="marg164"/>) ende alles blijft noch euestaltwichtich, ende ghelijck IG tot GL, also noch H tot K. </s>

<s>

<margin.target id="marg164"/>Mechanicè non Mathematicè.

</s>

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<s><emph type="other"/>HET TWEEDE BOVCK <lb/>

VANDE BEGHINSELEN <arrow.to.target n="marg166"/><lb/>

VANDE <arrow.to.target n="marg166"/>BEGHINSELEN <lb/>

DER WEEGHCONST, DWELCK IS <lb/>

VANDE VINDING DER SWAER<lb/>

HEYDTS MIDDELPVNTEN, <lb/>

HEYDTS MIDDELPVNTEN<emph.end/>, <lb/>

Beschreuen door Simon Steuin.<emph.end/> </s>

Beschreuen door Simon Steuin. </s>

<s>

<margin.target id="marg166"/>Elementis.

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<s> <emph type="other"/>WY<emph.end/> hebben in t'eerste bouck tot het beschriuen der wichtighe ghedaenten, ghenomen een pilaer (voldoende aldaer het voornemen) diens swaerheyts middelpunt door ghemeene wetenschap bekent is, maer in veel ander lichamen en ghebueret niet also; wel is waer dattet door een corte ghemeene reghel in allen werckelick te vinden is, so door t'eerste voorstel der <arrow.to.target n="marg168"/> blijcken sal, maer met de Wisconstighe <arrow.to.target n="marg169"/> vinding ist anders ghestelt; Daer af heeft eerst gheschreuen Archimedes in platten, ende naer hem Frederic Commandin in lichamen: Wy sullen tottet een en t'ander (ouermits het een afcoemst <arrow.to.target n="marg170"/> van beghinselen is, byde voorgaende wel dienende, ende tottet volghende, so wel <emph type="other"/>WATERWICHT<emph.end/>, als <emph type="other"/>WEEGHDAET<emph.end/>, seer noodich) het onse voughen, ende alles naer onse oirden verspreyden, daer af beschrijuende der Beghinselen tweede bouck. </s>

<s> <emph type="other"/>WY<emph.end/> hebben in t'eerste bouck tot het beschriuen der wichtighe ghedaenten, ghenomen een pilaer (voldoende aldaer het voornemen) diens swaerheyts middelpunt door ghemeene wetenschap bekent is, maer in veel ander lichamen en ghebueret niet also; wel is waer dattet door een corte ghemeene reghel in allen werckelick te vinden is, so door t'eerste voorstel der <emph.end/><emph.end/><emph.end/><arrow.to.target n="marg168"/> blijcken sal, maer met de Wisconstighe <arrow.to.target n="marg169"/> vinding ist anders ghestelt; Daer af heeft eerst gheschreuen Archimedes in platten, ende naer hem Frederic Commandin in lichamen: Wy sullen tottet een en t'ander (ouermits het een afcoemst <arrow.to.target n="marg170"/> van beghinselen is, byde voorgaende wel dienende, ende tottet volghende, so wel <emph type="other"/>WATERWICHT<emph.end/>, als <emph type="other"/>WEEGHDAET<emph.end/>, seer noodich) het onse voughen, ende alles naer onse oirden verspreyden, daer af beschrijuende der Beghinselen tweede bouck. </s>

<s>

<margin.target id="marg168"/>Praxis.

<margin.target id="marg169"/>Mathematica.

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<s><emph type="other"/>VI. VOORBEELT.<emph.end/></s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. Laet ABCDEF een ongheschickt seshouck sijn. <emph type="other"/>T'BEGHEERDE<emph.end/>. Wy moeten sijn swaerheydts middelpunt vinden.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. <emph.end/><emph.end/><emph.end/>Laet ABCDEF een ongheschickt seshouck sijn. <emph type="other"/>T'BEGHEERDE<emph.end/>. <emph.end/><emph.end/><emph.end/>Wy moeten sijn swaerheydts middelpunt vinden.</s>

<s> <emph type="other"/>T'WERCK<emph.end/>.</s><s> Men sal den seshouck deelen in vier driehoucken, met eenighe linien als AC, AD, FD, vindende daer naer het swaerheyts middelpunt des vierhoucx ADCB door het 4. voorbeelt, t'welck G sy, ende des vierhoucx ADEF, t'welck H sy, ende de lini HG is balck. </s><s>Daernaer ghetrocken BI ende DK rechthouckich op AC, insghelijcx AL ende EM beyde rechthouckich op FD, men sal der drie linien welcker eerste FD, de tweede AC, de derde BI met KD, vinden de vierde euerednighe, welcke NO sy, deelende den balck HG in P, also dat den erm PG, sulcken reden hebbe tot den erm PH, ghelijck AL met EM, tot NO; Ick seg dat P het begheerde swaerheyts middelpunt is.

<s> <emph.end/><emph.end/><emph.end/><emph type="other"/>T'WERCK<emph.end/>.</s><s> <emph.end/><emph.end/><emph.end/>Men sal den seshouck deelen in vier driehoucken, met eenighe linien als AC, AD, FD, vindende daer naer het swaerheyts middelpunt des vierhoucx ADCB door het 4. voorbeelt, t'welck G sy, ende des vierhoucx ADEF, t'welck H sy, ende de lini HG is balck. </s><s>Daernaer ghetrocken BI ende DK rechthouckich op AC, insghelijcx AL ende EM beyde rechthouckich op FD, men sal der drie linien welcker eerste FD, de tweede AC, de derde BI met KD, vinden de vierde euerednighe, welcke NO sy, deelende den balck HG in P, also dat den erm PG, sulcken reden hebbe tot den erm PH, ghelijck AL met EM, tot NO; Ick seg dat P het begheerde swaerheyts middelpunt is.

</s><s>Ende soo salmen voort mueghen varen met ander veelhouckeghe platten.

</s>

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<margin.target id="marg199"/>I. v. 6. B. E.

</s>

<s> T'vierde voorbeelt is openbaer als wy bewesen hebben dat ghelijck DG, tot HB, alsoo den driehouck ACD, tot ACB in deser voughen: Nemende AC voor hoochde, ende DG ende HB voor gronden, soo heeft den rechthouck begrepen onder AC ende DG, sulcken reden tot den rechthouck onder AC ende HB, ghelijck DG tot HB <arrow.to.target n="marg200"/>; Maer ghelijck dien rechthouck tot desen, alsoo den driehouck ACD tot ACB, want elck driehouck is sijn rechthoucx helft <arrow.to.target n="marg201"/>, ghelijck dan DG tot HB, alsoo den driehouck ACD tot ACB. </s>

<s> T'vierde voorbeelt is openbaer als wy bewesen hebben dat ghelijck DG, tot HB, alsoo den driehouck ACD, tot ACB in deser voughen: <emph.end/><emph.end/><emph.end/>Nemende AC voor hoochde, ende DG ende HB voor gronden, soo heeft den rechthouck begrepen onder AC ende DG, sulcken reden tot den rechthouck onder AC ende HB, ghelijck DG tot HB <arrow.to.target n="marg200"/>; Maer ghelijck dien rechthouck tot desen, alsoo den driehouck ACD tot ACB, want elck driehouck is sijn rechthoucx helft <arrow.to.target n="marg201"/>, ghelijck dan DG tot HB, alsoo den driehouck ACD tot ACB. </s>

<s>

<margin.target id="marg200"/>I. v. 6. B. E.

<margin.target id="marg201"/>4I. v. I. B. E.

</s>

<s>Des 5. voorbeelts bewys sal oock claer sijn als wy bewesen hebben, dat ghelijck EK met IC tot LM, alsoo den vierhouck ACDE tot den driehouck ACB, aldus: Anghesien LM vierde euerednighe is der drie AD, AC, HB, de rechthouck begrepen onder AD ende LM, sal euen sijn <arrow.to.target n="marg202"/> an den rechthouck begrepen onder AC ende HB, Laet ons nu EK, IC, LM, ansien voor gronden, wiens ghemeene hoochde AD; </s><s> Maer <arrow.to.target n="marg203"/> ghelijck die gronden <pb pagenum="74"/>tot malcanderen, alsoo de rechthoucken begrepen onder haer ende hare ghemeene hoochde, daerom oock ghelijck de twee gronden EK, IC, tot den grondt LM, alsoo dier gronden rechthoucken tot deses grondts rechthouck; maer die twee rechthoucken sijn elck het dobbel haers driehoucx; Ghelijck dan EK met IC tot LM, also het dobbel vanden vierhouck ACDE tot den rechthouck begrepen onder AD ende LM: Maer desen is euen an den rechthouck begrepen onder AC ende HB als vooren betoocht is, ende de selue rechthouck begrepen onder AC ende HB is het dobbel des driehoucx ACB, daerom ghelijck EK met IC tot LM, alsoo het dobbel des vierhoucx ACDE tot het dobbel des driehoucx ACB, ende veruolghens ghelijck EK met IC tot LM, alsoo den vierhouck ACDE tot den driehouck ACB, waer uyt de reste openbaer is. </s><s>T'bewys van het 6 voorbeelt is duer dit oock kennelick ghenouch. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een rechthouckich plat: Wy hebben sijn swaerheydts middelpunt gheuonden naer den eysch.</s>

<s>Des 5. voorbeelts bewys sal oock claer sijn als wy bewesen hebben, dat ghelijck EK met IC tot LM, alsoo den vierhouck ACDE tot den driehouck ACB, aldus: <emph.end/><emph.end/><emph.end/>Anghesien LM vierde euerednighe is der drie AD, AC, HB, de rechthouck begrepen onder AD ende LM, sal euen sijn <arrow.to.target n="marg202"/> an den rechthouck begrepen onder AC ende HB, Laet ons nu EK, IC, LM, ansien voor gronden, wiens ghemeene hoochde AD; </s><s> Maer <arrow.to.target n="marg203"/> ghelijck die gronden <pb pagenum="74"/>tot malcanderen, alsoo de rechthoucken begrepen onder haer ende hare ghemeene hoochde, daerom oock ghelijck de twee gronden EK, IC, tot den grondt LM, alsoo dier gronden rechthoucken tot deses grondts rechthouck; maer die twee rechthoucken sijn elck het dobbel haers driehoucx; Ghelijck dan EK met IC tot LM, also het dobbel vanden vierhouck ACDE tot den rechthouck begrepen onder AD ende LM: Maer desen is euen an den rechthouck begrepen onder AC ende HB als vooren betoocht is, ende de selue rechthouck begrepen onder AC ende HB is het dobbel des driehoucx ACB, daerom ghelijck EK met IC tot LM, alsoo het dobbel des vierhoucx ACDE tot het dobbel des driehoucx ACB, ende veruolghens ghelijck EK met IC tot LM, alsoo den vierhouck ACDE tot den driehouck ACB, waer uyt de reste openbaer is. </s><s>T'bewys van het 6 voorbeelt is duer dit oock kennelick ghenouch. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een rechthouckich plat: Wy hebben sijn swaerheydts middelpunt gheuonden naer den eysch.</s>

<s>

<margin.target id="marg202"/>I6. v. 6. B. E.

<margin.target id="marg203"/>I. v. 6. B. E.

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</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een vierhouck sijn, diens twee euewydighe sijden AB ende DC, ende de lini uyt E middel van AB, tot F middel van DC, sy EF. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat t'swaerheyts middelpunt des vierhoucx ABCD inde lini EF is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet de drie linien DA, FE, CB, voortghetrocken worden, welcke om de eueredenheyt <arrow.to.target n="marg207"/> der linien AE, EB, DF, FC, vergaren sullen in een selfde punt t'welck G sy. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons den driehouck GDC ophanghen byde lini GF, ende het deel GFC sal euestaltwichtich sijn, teghen GFD door het voorstel, waer deur oock t'swaerheyts mid<pb pagenum="75"/>delpunt des driehoucx GDC inde lini GF is.</s><s> Maer den driehouck GEB, is oock euestaltwichtich teghen den driehouck GEA, daerom van euestaltwychtighe ghetrocken euestaltwichtighe, de resten als de vierhoucken EFDA, EFCB, sullen noch euestaltwichtich blijuen, ende haer swaerheyts middelpunt noch inde lini GF, maer niet uyt de form in EG; Nootsaecklick dan in EF. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Het swaerheydts middelpunt dan des vierhoucx met twee euewydighe sijden, is inde lini tusschen dier sijden middelpunten, t'welck wy bewysen moesten.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een vierhouck sijn, diens twee euewydighe sijden AB ende DC, ende de lini uyt E middel van AB, tot F middel van DC, sy EF. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat t'swaerheyts middelpunt des vierhoucx ABCD inde lini EF is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet de drie linien DA, FE, CB, voortghetrocken worden, welcke om de eueredenheyt <arrow.to.target n="marg207"/> der linien AE, EB, DF, FC, vergaren sullen in een selfde punt t'welck G sy. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons den driehouck GDC ophanghen byde lini GF, ende het deel GFC sal euestaltwichtich sijn, teghen GFD door het <emph.end/><emph.end/><emph.end/>voorstel, waer deur oock t'swaerheyts mid<pb pagenum="75"/>delpunt des driehoucx GDC inde lini GF is.</s><s> Maer den driehouck GEB, is oock euestaltwichtich teghen den driehouck GEA, daerom van euestaltwychtighe ghetrocken euestaltwichtighe, de resten als de vierhoucken EFDA, EFCB, sullen noch euestaltwichtich blijuen, ende haer swaerheyts middelpunt noch inde lini GF, maer niet uyt de form in EG; Nootsaecklick dan in EF. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Het swaerheydts middelpunt dan des vierhoucx met twee euewydighe sijden, is inde lini tusschen dier sijden middelpunten, t'welck wy bewysen moesten.</s>

<s>

<margin.target id="marg207"/>Proportionem.

</s>

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<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een vierhouck wesen met twee euewydighe sijden AB, DC, ende de lini tusschen haer middelpunten sy EF, ende t'swaerheydts middelpunt sy G. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck tweemael DC met eenmael AB, tot tweemael AB met eenmael DC. </s><s>also GE tot GF. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden DB, ende ghedeelt in drie euen deelen met de punten H, I, ende door de selue ghetrocken worden KL, ende MN, euewydich van DC, sniende EF in O en P. </s><s>Daer naer de lini DE, sniende MI in Q, Ende BF sniende KL in R, Ende ten laetsten RQ. </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Anghesien het swaerheydts middelpunt des driehoucx BDC, is in BF, duer het voorstel, ende oock in HL duer het voorstel, soo is R, sijn swaerheyts middelpunt, ende om de selue reden is Q swaerheyts middelpunt des driehoucx ABD, ende QR is dier driehoucken balck, inden welcken haer beyder, dat is des vierhoucx ABCD, swaerheyts <pb pagenum="76"/>middelpunt is, t'selue is oock in EF duer het voorstel, daerom G is t'swaerheyts middelpunt des vierhoucx ABCD. </s><s>Maer want de twee driehoucken CDB ende ABD sijn tusschen twee euewydighe AB ende DC, so sijn sy inde reden van haer gronden <arrow.to.target n="marg208"/>, dat is ghelijck den driehouck CDB tot ABD, alsoo DC tot AB: Maer ghelijck den driehouck CDB tot ADB, also den erm GQ tot GR duer het voorstel des I boucx, ghelijck dan DC tot AB, alsoo GQ tot GR; maer ghelijck GQ tot GR, alsoo PG tot GO (want sy tusschen de euewydeghe MN, KL sijn) ghelijck dan DC tot AB, alsoo GP tot PO [GO], daerom oock ghelijck tweemael DC met eenmael AB, tot tweemael AB met eenmael DC, also tweemael GP met eenmael GO, tot tweemael GO met eenmael GP. </s><s> Maer GE is euen an tweemael GP met eenmael GO, ende GF is euen an tweemael GO met eenmael GP, daerom ghelijck tweemael DC met eenmael AB, tot tweemael AB met eenmael DC, alsoo GE tot GF. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Het swaerheyts middelpunt dan des vierhoucx met twee, &c.</s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Anghesien het swaerheydts middelpunt des driehoucx BDC, is in BF, duer het <emph.end/><emph.end/><emph.end/>voorstel, ende oock in HL duer het <emph.end/><emph.end/><emph.end/>voorstel, soo is R, sijn swaerheyts middelpunt, ende om de selue reden is Q swaerheyts middelpunt des driehoucx ABD, ende QR is dier driehoucken balck, inden welcken haer beyder, dat is des vierhoucx ABCD, swaerheyts <pb pagenum="76"/>middelpunt is, t'selue is oock in EF duer het <emph.end/><emph.end/><emph.end/>voorstel, daerom G is t'swaerheyts middelpunt des vierhoucx ABCD. </s><s>Maer want de twee driehoucken CDB ende ABD sijn tusschen twee euewydighe AB ende DC, so sijn sy inde reden van haer gronden <arrow.to.target n="marg208"/>, dat is ghelijck den driehouck CDB tot ABD, alsoo DC tot AB: <emph.end/><emph.end/><emph.end/>Maer ghelijck den driehouck CDB tot ADB, also den erm GQ tot GR duer het <emph.end/><emph.end/><emph.end/>voorstel des I boucx, ghelijck dan DC tot AB, alsoo GQ tot GR; maer ghelijck GQ tot GR, alsoo PG tot GO (want sy tusschen de euewydeghe MN, KL sijn) ghelijck dan DC tot AB, alsoo GP tot PO [GO], daerom oock ghelijck tweemael DC met eenmael AB, tot tweemael AB met eenmael DC, also tweemael GP met eenmael GO, tot tweemael GO met eenmael GP. </s><s> Maer GE is euen an tweemael GP met eenmael GO, ende GF is euen an tweemael GO met eenmael GP, daerom ghelijck tweemael DC met eenmael AB, tot tweemael AB met eenmael DC, alsoo GE tot GF. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Het swaerheyts middelpunt dan des vierhoucx met twee, &c.</s>

<s>

<margin.target id="marg208"/>I. v. 6. B. E.

</s>

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<s>I. <emph type="other"/>VOORBEELT<emph.end/>.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een rechtlinich plat wesen, diens swaerheyts middelpunt E, ende BDA deel des plats, wiens swaerheyts middelpunt F. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten t'swaerheyts middelpunt vinden des ander deels BDC. </s><s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal trecken FE tot in G, alsoo dat FE sulcken reden hebbe tot EG, als t'stick BDC tottet stick BDA: Ick seg dat G t'begheerde swaerheyts middelpunt is des ander deels BDC. </s><s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Anghesien t'swaerheyts middelpunt van BDA is F, ende des heels ABCD is E, soo moet t'swaerheyts middelpunt des ander deels BDC sijn in de rechte FE oneindelick voortghetrocken. </s><s>Want soot mueghelick waer, latet daer buyten wesen als H, ende laet ons trecken FH, het swaerheyts middelpunt dan des heels sal in FH sijn, maer dat is teghen t'ghestelde <arrow.to.target n="marg209"/>, wantet E is; Ten is dan niet buyten FE oneindelick voortghetrocken maer daer in. </s><s> Latet nu wesen <pb pagenum="77"/>(soot mueghelick waer) tusschen de punten EG als I; Maer den langsten erm EF sal dan meerder reden hebben tot den cortsten EI, dan de swaerste swaerheyt BDC tot de lichtste BDA, twelck teghen het voorstel des I boucx waer. </s><s>Ten is dan tusschen EG niet: Sghelijcx salmen oock bethoonen dattet bouen G niet en is. </s><s>Tis dan nootsaecklick G, t'welck wy bewysen moesten.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een rechtlinich plat wesen, diens swaerheyts middelpunt E, ende BDA deel des plats, wiens swaerheyts middelpunt F. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten t'swaerheyts middelpunt vinden des ander deels BDC. </s><s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal trecken FE tot in G, alsoo dat FE sulcken reden hebbe tot EG, als t'stick BDC tottet stick BDA: Ick seg dat G t'begheerde swaerheyts middelpunt is des ander deels BDC. </s><s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Anghesien t'swaerheyts middelpunt van BDA is F, ende des heels ABCD is E, soo moet t'swaerheyts middelpunt des ander deels BDC sijn in de rechte FE oneindelick voortghetrocken. </s><s>Want soot mueghelick waer, latet daer buyten wesen als H, ende laet ons trecken FH, het swaerheyts middelpunt dan des heels sal in FH sijn, maer dat is teghen t'ghestelde <arrow.to.target n="marg209"/>, wantet E is; <emph.end/><emph.end/><emph.end/>Ten is dan niet buyten FE oneindelick voortghetrocken maer daer in. </s><s> Latet nu wesen <pb pagenum="77"/>(soot mueghelick waer) tusschen de punten EG als I; Maer den langsten erm EF sal dan meerder reden hebben tot den cortsten EI, dan de swaerste swaerheyt BDC tot de lichtste BDA, twelck teghen het <emph.end/><emph.end/><emph.end/>voorstel des I boucx waer. </s><s>Ten is dan tusschen EG niet: Sghelijcx salmen oock bethoonen dattet bouen G niet en is. </s><s>Tis dan nootsaecklick G, t'welck wy bewysen moesten.</s>

<s>

<margin.target id="marg209"/>Hypothesin.

</s>

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<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een brantsne sijn diens middellini AD. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat t'swaerheyts middelpunt inde lini AD is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de linien EF, GH, IK, euewydighe van BC, ende sniende AD in L, M, N, daer naer EO, GP, IQ, KR, HS, FT, euewydighe van AD. </s>

<s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ouermidts EF euewydighe is van BC, ende EO, FT, van LD, soo sal EFTO euewydich vierhouck sijn, wiens EL euen is met LF, oock met OD ende DT, waer duer t'swaerheyts middelpunt van EFTO, in DL is duer het voorstel, Ende om de selue reden sal t'swaerheyts middelpunt des euewydich vierhoucx GHSP in LM wesen, ende van IKRQ in MN, ende veruolghens t'swaerheyts middelpunt der form IKRHSFTOEPGQ, ghemaeckt vande voornoemde drie vierhoucken sal inde lini ND oft AD sijn. </s><s>Maer hoe datter sulcke vierhoucken meer gheschreuen worden, hoe dattet verschil des brandtsnees ABC, ende der binnenschreuen form van die vierhoucken vergaert, minder is, wy connen dan door dat oneindelick naerderen sulck een form binnen de brandtsne stellen, dattet verschil tusschen haer ende de brantsne, minder sy dan eenich ghegheuen plat hoe cleen het sy, waer uyt volght, dat stellende AD als swaerheyts middellini, so sal t'staltwicht des deels ADC, min verschillen van t'staltwicht des deels ADB, dan eenich plat <pb pagenum="79"/>datmen soude connen gheuen, hoe cleen het sy, waer uyt ick aldus strije: </s>

<s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ouermidts EF euewydighe is van BC, ende EO, FT, van LD, soo sal EFTO euewydich vierhouck sijn, wiens EL euen is met LF, oock met OD ende DT, waer duer t'swaerheyts middelpunt van EFTO, in DL is duer het <emph.end/><emph.end/><emph.end/>voorstel, Ende om de selue reden sal t'swaerheyts middelpunt des euewydich vierhoucx GHSP in LM wesen, ende van IKRQ in MN, ende veruolghens t'swaerheyts middelpunt der form IKRHSFTOEPGQ, ghemaeckt vande voornoemde drie vierhoucken sal inde lini ND oft AD sijn. </s><s>Maer hoe datter sulcke vierhoucken meer gheschreuen worden, hoe dattet verschil des brandtsnees ABC, ende der binnenschreuen form van die vierhoucken vergaert, minder is, wy connen dan door dat oneindelick naerderen sulck een form binnen de brandtsne stellen, dattet verschil tusschen haer ende de brantsne, minder sy dan eenich ghegheuen plat hoe cleen het sy, waer uyt volght, dat stellende AD als swaerheyts middellini, so sal t'staltwicht des deels ADC, min verschillen van t'staltwicht des deels ADB, dan eenich plat <pb pagenum="79"/>datmen soude connen gheuen, hoe cleen het sy, waer uyt ick aldus strije: </s>

<s> A. Neuen alle verschillende staltswaerheden, can een swaerheyt ghestelt worden minder dan haer verschil; </s>

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<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brantsne sijn, diens middellini AD. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten haer swaerheyts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal de middellini AD, deelen in E, alsoo dat AE tot ED de reden hebbe van 3 tot 2: Ick seg dat E t'begheerde swaerheyts middelpunt is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden de rechte linien AB, ende AC, ende de selue ghedeelt in haer middelen, F, G, ende ghetrocken worden FG sniende AD in H, daer naer FI ende GK euewydighe van AD, ende laet ghestelt worden t'punt L in IF, inder voughen dat IL sy tot LF, als AE tot ED: Laet oock ghestelt worden t'punt M in KG, alsoo dat MG euen sy an LF, ende laet ghetrocken worden LM sniende AD in N, ende IK sniende AD in O, ende laet IF voortghetrocken worden tot Q, inden grondt BC, ende laet ghestelt worden t'punt P, alsoo dat AP dobbel sy an PD, ende P sal swaerheyts middelpunt sijn des driehoucx ABC, ende want L, M, als swaerheyts middelpunten ghestelt sijn der brantsnekens ABI, ende ACK, soo sal N swaerheyts middelpunt sijn dier twee brandtsnekens, daerom ghedeelt den balck PN, also dat d'een erm sulcken reden hebbe tot d'ander, als den driehouck ABC tot die twee brantsnekens, wy sullen t'begheerde hebben; maar de heel brantsne heeft sulcken reden tot den driehouck ABC als 4 tot 3 (duer het 24 voorstel vande viercanting der brantsne van Archimed.) daerom den driehouck ABC heeft sulcken reden tot de twee brantsnekens, als 3 tot I; Ghedeelt dan PN alsoo dat het opperste stick, drievoudich sy tot het onderste, wy sullen t'swaerheyts middelpunt des heels hebben. </s><s>Ist dan dat wy bethoonen t'selue, te vallen in E (welcke E duer t'werck soo staet dat AE is tot ED inde reden van 3 tot 2) so is E het ware swaerheyts middelpunt. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brantsne sijn, diens middellini AD. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten haer swaerheyts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal de middellini AD, deelen in E, alsoo dat AE tot ED de reden hebbe van 3 tot 2: Ick seg dat E t'begheerde swaerheyts middelpunt is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden de rechte linien AB, ende AC, ende de selue ghedeelt in haer middelen, F, G, ende ghetrocken worden FG sniende AD in H, daer naer FI ende GK euewydighe van AD, ende laet ghestelt worden t'punt L in IF, inder voughen dat IL sy tot LF, als AE tot ED: Laet oock ghestelt worden t'punt M in KG, alsoo dat MG euen sy an LF, ende laet ghetrocken worden LM sniende AD in N, ende IK sniende AD in O, ende laet IF voortghetrocken worden tot Q, inden grondt BC, ende laet ghestelt worden t'punt P, alsoo dat AP dobbel sy an PD, ende P sal swaerheyts middelpunt sijn des driehoucx ABC, ende want L, M, als swaerheyts middelpunten ghestelt sijn der brantsnekens ABI, ende ACK, soo sal N swaerheyts middelpunt sijn dier twee brandtsnekens, <emph.end/><emph.end/><emph.end/>daerom ghedeelt den balck PN, also dat d'een erm sulcken reden hebbe tot d'ander, als den driehouck ABC tot die twee brantsnekens, wy sullen t'begheerde hebben; maar de heel brantsne heeft sulcken reden tot den driehouck ABC als 4 tot 3 (duer het 24 voorstel vande viercanting der brantsne van Archimed.) daerom den driehouck ABC heeft sulcken reden tot de twee brantsnekens, als 3 tot I; Ghedeelt dan PN alsoo dat het opperste stick, drievoudich sy tot het onderste, wy sullen t'swaerheyts middelpunt des heels hebben. </s><s>Ist dan dat wy bethoonen t'selue, te vallen in E (welcke E duer t'werck soo staet dat AE is tot ED inde reden van 3 tot 2) so is E het ware swaerheyts middelpunt. </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>AO ende OH soo wy verclaert hebben int II voorstel, sijn elck 1/4 van AD, Maer ghelijck 3 tot 2, alsoo AE tot ED, ende IL tot LF, ende ON tot NH, daerom ghedeelt OH 1/4, in sulcken reden <pb pagenum="82"/>als 3 tot 2, so sal t'stick NH doen 1/10 van AD,</s><s> daer toeghedaen 1/2 voor HD, doet voor ND 3/5, daer af ghetrocken PD 1/3 rest voor NP 4/15, de selue is duer t'bereytsel ghedeelt in E, alsoo dat NE is tot EP, als 3 tot I, daerom EP doet 1/15, daer toe ghedaen PD 1/3, comt voor ED 2/5 van AD: Maer wesende ED 2/5, so sal EA doen 3/5, daerom AE heeft sulcken reden tot ED, als 3 tot 2, ende veruolghens E is t'swaerheyts middelpunt des brantsnees ABC, t'welck wy bewysen moesten. </s>

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<s> <emph type="other"/>YDER<emph.end/> lichaems formens middelpunt, is oock sijn swaerheyts middelpunt. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een viergrondich <arrow.to.target n="marg246"/> wesen, diens formens middelpunt E sy, ende den as van A duer E, tot in F, middelpunt des driehoucx BCD, sy AF. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat E oock is sijn swaerheyts middelpunt. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons t'lichaem ophanghen byde lini AF, maer het viergrondich bestaet uyt vier euen ende ghelijcke naelden een selfder ghestalt, wiens ghemeene sop E, daerom AF is des lichaems swaerheyts middellini, ende om de selue reden sal de lini CE oock des swaerheyts middellini sijn: E dan is oock het swaerheyts <pb pagenum="84"/>middelpunt.</s><s> Sghelijcx sal oock t'bewys sijn van allen lichamen hebbende middelpunten der form, soo wel vermeerde ende ghecorte gheschicte lichamen, als gheschicte, want soomense ophangt byde middellinien deur eenighen lichamelicken houck, ofte duer het middelpunt haerder gronden ende des formens middelpunt, soo hebben al de naelden (wiens ghemeene sop het formens middelpunt, ende gronden de platten des lichaems sijn) tot allen sijden ghelijcke ghestalt, daerom oock duer ghemeene wetenschap, ende duer de begheerte des I boucx, alles hangt an die lini euewichtich, ende veruolghens de sne sulcker twee swaerheyts middellinien malcander sniende in des formens middelpunt, is ock het sawerheyts middelpunt. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder lichaems formens middelpunt dan, is oock sijn swaerheyts middelpunt.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een viergrondich <arrow.to.target n="marg246"/> wesen, diens formens middelpunt E sy, ende den as van A duer E, tot in F, middelpunt des driehoucx BCD, sy AF. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat E oock is sijn swaerheyts middelpunt. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons t'lichaem ophanghen byde lini AF, maer het viergrondich bestaet uyt vier euen ende ghelijcke naelden een selfder ghestalt, wiens ghemeene sop E, daerom AF is des lichaems swaerheyts middellini, ende om de selue reden sal de lini CE oock des swaerheyts middellini sijn: E dan is oock het swaerheyts <pb pagenum="84"/>middelpunt.</s><s> Sghelijcx sal oock t'bewys sijn van allen lichamen hebbende middelpunten der form, soo wel vermeerde ende ghecorte gheschicte lichamen, als gheschicte, want soomense ophangt byde middellinien deur eenighen lichamelicken houck, ofte duer het middelpunt haerder gronden ende des formens middelpunt, soo hebben al de naelden (wiens ghemeene sop het formens middelpunt, ende gronden de platten des lichaems sijn) tot allen sijden ghelijcke ghestalt, daerom oock duer ghemeene wetenschap, ende duer de <emph.end/><emph.end/><emph.end/>begheerte des I boucx, alles hangt an die lini euewichtich, ende veruolghens de sne sulcker twee swaerheyts middellinien malcander sniende in des formens middelpunt, is ock het sawerheyts middelpunt. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder lichaems formens middelpunt dan, is oock sijn swaerheyts middelpunt.</s>

<s>

<margin.target id="marg246"/>Tetraedron.

</s>

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<s><emph type="other"/>I. VOORBEELT<emph.end/>.</s>

<s> <emph type="other"/>T'GHEGHE<emph.end/>. </s><s>Laet AB een driehouckich pilaer sijn diens grondt ACD. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat sijn swaerheyts middelpunt int middel vanden as is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken van D tot E int middel van AC de lini DE: Daer naer FG ende HI euewydighe van AC, sniende DE inde punten K, L, daer naer de linien FM, HN, IO, GP, euewydighe met DE, daer naer van A tot Q int middel der sijde DC, de lini AQ: Laet sghelijcx oock beschreuen worden het decsel, ende laet ons den pilaer doorsnien met een plat <arrow.to.target n="marg247"/> RS euewidich met den grondt ADC, ende S sy int middel van CB. </s>

<s> <emph type="other"/>T'GHEGHE<emph.end/>. </s><s>Laet AB een driehouckich pilaer sijn diens grondt ACD. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat sijn swaerheyts middelpunt int middel vanden as is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken van D tot E int middel van AC de lini DE: Daer naer FG ende HI euewydighe van AC, sniende DE inde punten K, L, daer naer de linien FM, HN, IO, GP, euewydighe met DE, daer naer van A tot Q int middel der sijde DC, de lini AQ: <emph.end/><emph.end/><emph.end/>Laet sghelijcx oock beschreuen worden het decsel, ende laet ons den pilaer doorsnien met een plat <arrow.to.target n="marg247"/> RS euewidich met den grondt ADC, ende S sy int middel van CB. </s>

<s>

<margin.target id="marg247"/>Plano.

</s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>T'plat ghetrocken duer DE, ende duer haer lijckstandighe <arrow.to.target n="marg248"/> int decksel, deelt den binneschreuen pilaer uyt die twee vierhouckighe pilaren vergaert, in twee euen ende ghelijcke deelen, ende van ghelijcke ghestalt; het doorsnijt dan dier binneschreuen pilaers swaerheydts middelpunt. </s><s>Maer hoe datter sulcke vierhouckighe pilaren meer beschreuen sijn inde ghegheuen driehouckighen, hoe dat dese min verschilt van die; wy connen dan duer dat oneindelick <pb pagenum="85"/>naerderen sulck een form binnen de ghegheuen pilaer beschrijuen, dat haer verschil vande binnescreuen minder sal wesen dan eenich ghegheuen lichaem hoe cleen het sy, waer uyt volght dat het staltwicht des deels DECB ouer d'een sijde des plats, min verschillen sal van t'staltwicht des deels ouer d'ander sijde des plats, dan eenich lichaem datmen soude connen gheuen hoe cleen het sy, waer uyt ick aldus strie: </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>T'plat ghetrocken duer DE, ende duer haer lijckstandighe <arrow.to.target n="marg248"/> int decksel, deelt den binneschreuen pilaer uyt die twee vierhouckighe pilaren vergaert, in twee euen ende ghelijcke deelen, ende van ghelijcke ghestalt; het doorsnijt dan dier binneschreuen pilaers swaerheydts middelpunt. </s><s>Maer hoe datter sulcke vierhouckighe pilaren meer beschreuen sijn inde ghegheuen driehouckighen, hoe dat dese min verschilt van die; wy connen dan duer dat oneindelick <pb pagenum="85"/>naerderen <emph.end/><emph.end/><emph.end/>sulck een form binnen de ghegheuen pilaer beschrijuen, dat haer verschil vande binnescreuen minder sal wesen dan eenich ghegheuen lichaem hoe cleen het sy, waer uyt volght dat het staltwicht des deels DECB ouer d'een sijde des plats, min verschillen sal van t'staltwicht des deels ouer d'ander sijde des plats, dan eenich lichaem datmen soude connen gheuen hoe cleen het sy, waer uyt ick aldus strie: </s>

<s>

<margin.target id="marg248"/>Homologam.

</s>

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<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een naelde sijn, diens grondt den driehouck BCD, wiens swaerheyts middelpunt E, ende den as AE. </s>

<s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat des naeldens swaerheyts middelpunt inden as AE is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons de naelde snien met een plat FGH euewydich met BCD, ende sniende den as AE in I: Laet oock ghetrocken worden FK, GL, HM, euewydich vanden as AE, alsoo dat de punten K, L, M, int plat sijn des driehoucks BCD, inder voughen dat FGHKLM een pilaer is, wiens grondt IKL euen ende ghelijck is an het decksel FGH, ende ghelijck anden grondt BCD; Daer naer ghelijck de naelde doorsneen was met FGH, laetse noch eenmal alsoo doorsneen sijn met het plat NOP, sniende den as in Q, ende daer uyt oock alsoo ghemaect den pilaer NOPRST, te weten NR, OS, PT, euewydich vanden as AE, ende de punten R, S, T, int plat FGH. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Anghesien de driehoucken NOP, RST, FGH, KLM, alle ghelijck sijn anden driehouck BCD, ende dat haer punten Q, I, E, in haer sulcken stant hebben als E inden driehouck BCD, ende dat E des driehoucx BCD swaerheyts middelpunt is, soo sijn oock die Q, I, E, haer driehouckens swaerheyts middelpunten, waer duer IE as is des pilaers FGHKLM, in wiens middel huer swaerheyts middelpunt is duer het voorstel. </s><s>Sghelijcx is QI oock as des pilaers NOPRST, in wiens middel huer swaerheyts middelpunt is, ende vervolgens het swaerheyts middelpunt des lichaems uyt die twee pilaren vergaert is in QE, daerom oock in AE; Maer hoe datter inde naelde sulcke pilaren meer beschreuen worden, hoe dattet verschil der naelde ende der <pb pagenum="87"/>binneschreuen form van sulcke pilaren vergaert, minder is, </s><s> blijuende nochtan het swaerheyts middelpunt der binneschreuen form altijt inden as AE; Wy connen dan duer dat oneindelick naerderen sulcken form binnen de naelde stellen, dattet verschil tusschen haer ende de naelde, minder sal wesen dan eenich ghegheuen lichaem hoe cleen het sy, waer uyt volght dat stellende AE voor swaerheyts middellini, der naelde, soo sal het staltwicht van d'een sijde tot d'ander, min verschillen dan eenighe swaerheyt diemen soude connen gheuen, waer uyt ick aldus strie: </s>

<s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat des naeldens swaerheyts middelpunt inden as AE is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons de naelde snien met een plat FGH euewydich met BCD, ende sniende den as AE in I: Laet oock ghetrocken worden FK, GL, HM, euewydich vanden as AE, alsoo dat de punten K, L, M, int plat sijn des driehoucks BCD, inder voughen dat FGHKLM een pilaer is, wiens grondt IKL euen ende ghelijck is an het decksel FGH, ende ghelijck anden grondt BCD; <emph.end/><emph.end/><emph.end/>Daer naer ghelijck de naelde doorsneen was met FGH, laetse noch eenmal alsoo doorsneen sijn met het plat NOP, sniende den as in Q, ende daer uyt oock alsoo ghemaect den pilaer NOPRST, te weten NR, OS, PT, euewydich vanden as AE, ende de punten R, S, T, int plat FGH. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Anghesien de driehoucken NOP, RST, FGH, KLM, alle ghelijck sijn anden driehouck BCD, ende dat haer punten Q, I, E, in haer sulcken stant hebben als E inden driehouck BCD, ende dat E des driehoucx BCD swaerheyts middelpunt is, soo sijn oock die Q, I, E, haer driehouckens swaerheyts middelpunten, waer duer IE as is des pilaers FGHKLM, in wiens middel huer swaerheyts middelpunt is duer het <emph.end/><emph.end/><emph.end/>voorstel. </s><s>Sghelijcx is QI oock as des pilaers NOPRST, in wiens middel huer swaerheyts middelpunt is, ende vervolgens het swaerheyts middelpunt des lichaems uyt die twee pilaren vergaert is in QE, daerom oock in AE; <emph.end/><emph.end/><emph.end/>Maer hoe datter inde naelde sulcke pilaren meer beschreuen worden, hoe dattet verschil der naelde ende der <pb pagenum="87"/>binneschreuen form van sulcke pilaren vergaert, minder is, </s><s> blijuende nochtan het swaerheyts middelpunt der binneschreuen form altijt inden as AE; Wy connen dan duer dat oneindelick naerderen sulcken form binnen de naelde stellen, dattet verschil tusschen haer ende de naelde, minder sal wesen dan eenich ghegheuen lichaem hoe cleen het sy, waer uyt volght dat stellende AE voor swaerheyts middellini, der naelde, soo sal het staltwicht van d'een sijde tot d'ander, min verschillen dan eenighe swaerheyt diemen soude connen gheuen, waer uyt ick aldus strie: </s>

<s> A. Neuen alle verschillende staltswaerheden, can een swaerheyt ghestelt worden minder dan haer verschil; </s>

<s>O. Neuen dese staltswaerheden en can gheen swaerheyt ghestelt worden minder dan haer verschil; </s>

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<margin.target id="marg259"/>Door t'verkeerde des 12 cap. 1. lib. Almag. Ptol.

</s>

<s> <emph type="other"/>MAER<emph.end/> soo des ghegheuen naeldens grondt een vierhouck waer, t'voorstel sal in die oock also bewesen worden: Laet by voorbeelt ABCDE een naelde wesen, wiens grondt een vierhouck BCDE, ende de as AF sy. </s><s>Nu dese vierhouckighe naelde ghedeelt in twee driehoekighe, wiens gronden ECB, ende ECD, diens assen AG, ende AH, wiens swaerheyts middelpunten I, K, des heelen naeldens swaerheyts middelpunt sal inde lini IK wesen, tis oock in AF duer het voorstel, tis dan L: Maer want AGH een driehouck is, ende IK euewydich van GH (want IG is t'vierendeel van GA, ende HK t'vierendeel van HA daerom &c.) soo sal AL sulcken reden hebben tot LF <arrow.to.target n="marg263"/>, als AI, tot IG, dat is drieuoudich. </s><s>Sghelijcx sal oock t'bewys sijn in alle naelde met veelsidighen grondt.</s>

<s> <emph.end/><emph.end/><emph.end/><emph type="other"/>MAER<emph.end/> soo des ghegheuen naeldens grondt een vierhouck waer, t'voorstel sal in die oock also bewesen worden: Laet by voorbeelt ABCDE een naelde wesen, wiens grondt een vierhouck BCDE, ende de as AF sy. </s><s>Nu dese vierhouckighe naelde ghedeelt in twee driehoekighe, wiens gronden ECB, ende ECD, diens assen AG, ende AH, wiens swaerheyts middelpunten I, K, des heelen naeldens swaerheyts middelpunt sal inde lini IK wesen, tis oock in AF duer het <emph.end/><emph.end/><emph.end/>voorstel, tis dan L: Maer want AGH een driehouck is, ende IK euewydich van GH (want IG is t'vierendeel van GA, ende HK t'vierendeel van HA daerom &c.) soo sal AL sulcken reden hebben tot LF <arrow.to.target n="marg263"/>, als AI, tot IG, dat is drieuoudich. </s><s>Sghelijcx sal oock t'bewys sijn in alle naelde met veelsidighen grondt.</s>

<s>

<margin.target id="marg263"/>2. v. 6. B. E.

</s>

<s> <emph type="other"/>MAER<emph.end/> de naelde een keghel sijnde, te weten dat den grondt waer een rondt ofte lanckrondt, t'selfde sal daerin oock alsoo bewesen <pb pagenum="89"/>worden, want het is duer t'voorgaende kennelick, dat alle veelhouckighe naelde in haer beschreuen, t'swaerheyts middelpunt alsoo sal hebben, dattet opperste deel drievoudich is teghen het onderste. </s><s>Maer hoe de naelde daer in beschreuen van meer houcken is, hoe die binneschreuen naeldens grootheyt vande ronde naelde min verschilt, daerom oock connen wy om het oneindelick naerderen, een binneschreuen setten, min verschillende vande veruatende, dan eenich ghegheuen lichaem hoe cleen het sy; Daerom oock de langde der plaets van diens swaerheyts middelpunt tot deses, corter soude moeten wesen dan eenighe langde die mueghelick is ghegheuen te worden, waer uyt ick aldus strie: </s>

<s>A. Neuen alle twee punten in verscheyden plaetsen staende, connen twee punten ghestelt worden die malcander naerder siin; </s>

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<s>O. Dese twee punten dan en staen in gheen verscheyden plaetsen. </s>

<emph type="other"/>T'BESLVYT<emph.end/><s>Het swaerheyts middelpunt dan van yder naelde, deelt den as alsoo, dat het stuck naer den houck drieuoudich is an t'ander.</s>

<s><emph type="other"/>T'BESLVYT<emph.end/></s><s>Het swaerheyts middelpunt dan van yder naelde, deelt den as alsoo, dat het stuck naer den houck drieuoudich is an t'ander.</s>

<s><emph type="other"/>VII EYSCH<emph.end/>. <emph type="other"/>XIX VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen t'swaerheyts middelpunt eens lichaems ende sijns deels, wiens reden an t'ander deel kennelick is: Het swaerheyts middelpunt des ander deels te vinden. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een lichaem sijn, diens swaerheyts middelpunt E, ende BDA deel des lichaems, wiens swaerheyts middelpunt F. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten t'swaerheyts middelpunt vinden des ander deels BCD. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal trecken FE tot in G, alsoo dat FE sulcken reden hebbe tot EG, als tstick BDC tottet stick BDA: Ick seg dat G t'begheerde swaerheyts middelpunt is, des ander sticx BDC; waer af t'bewys ghelijck sal sijn an t'bewys des voorstels. </s><s>Wy souden oock moghen voorbeelt setten van een heele cloot, wiens ander deel oock een cloot sy, maer sulcx is openbaer ghenouch duer het tweede voorbeelt des boueschreuen 9 voorstels in ronden. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen t'swaerheydts middelpunt eens lichaems ende sijns deels, wiens reden an t'ander deel <pb pagenum="90"/>kennelick is, wy hebben t'swaerheyts middelpunt des ander deels gheuonden, naer den eysch.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een lichaem sijn, diens swaerheyts middelpunt E, ende BDA deel des lichaems, wiens swaerheyts middelpunt F. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten t'swaerheyts middelpunt vinden des ander deels BCD. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal trecken FE tot in G, alsoo dat FE sulcken reden hebbe tot EG, als tstick BDC tottet stick BDA: Ick seg dat G t'begheerde swaerheyts middelpunt is, des ander sticx BDC; waer af t'bewys ghelijck sal sijn an t'bewys des <emph.end/><emph.end/><emph.end/>voorstels. </s><s>Wy souden oock moghen voorbeelt setten van een heele cloot, wiens ander deel oock een cloot sy, maer sulcx is openbaer ghenouch duer het tweede voorbeelt des boueschreuen 9 voorstels in ronden. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen t'swaerheydts middelpunt eens lichaems ende sijns deels, wiens reden an t'ander deel <pb pagenum="90"/>kennelick is, wy hebben t'swaerheyts middelpunt des ander deels gheuonden, naer den eysch.</s>

<s><emph type="other"/>VIII EYSCH<emph.end/>. <emph type="other"/>XX VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>VIII EYSCH<emph.end/>. <emph.end/><emph.end/><emph.end/><emph type="other"/>XX VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een ghecorte naelde <arrow.to.target n="marg267"/>: Huer swaerheyts middelpunt te vinden. </s>

<s>

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<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCDEF een ghecorte naelde sijn, diens decksel ABC, ende grondt DEF. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten huer swaerheyts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal de ghecorte naelde volmaken, daer an stellende het ghebrekende ABCG, vindende H swaerheyts middelpunt des driehoucx DEF, treckende den as GH, wiens punt inden driehouck ABC, sy I, daernaer salmen teeckenen K, alsoo dat GK drievoudich sy an KI: Insghelijcx L, also dat GL drievoudich sy an LH, teeckenende M, alsoo dat KL sulcken reden hebbe tot LM, ghelijck de ghecorte naelde ABCDEF, tot de naelde ABCG, Ick seg dat M t'begheerde swaerheyts middelpunt is. </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>L is swaerheyts middelpunt des heels, ende K des deels, ende ghelijck t'onderste deel tottet bouenste, also KL tot LM, Daerom M, door het voorstel des I boucx is t'begheerde swaerheyts middelpunt, t'welck wy bewysen moesten. </s><s>Sghelijcx sal oock den voortganck sijn in allen anderen ghecorte naelden. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een ghecorte naelde: Wy hebben huer swaerheyts middelpunt gheuonden, naer den eysch.</s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>L is swaerheyts middelpunt des heels, ende K des deels, ende ghelijck t'onderste deel tottet bouenste, also KL tot LM, Daerom M, door het <emph.end/><emph.end/><emph.end/>voorstel des I boucx is t'begheerde swaerheyts middelpunt, t'welck wy bewysen moesten. </s><s>Sghelijcx sal oock den voortganck sijn in allen anderen ghecorte naelden. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een ghecorte naelde: Wy hebben huer swaerheyts middelpunt gheuonden, naer den eysch.</s>

<s><emph type="other"/>IX. EYSCH<emph.end/>. <emph type="other"/>XXI. VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een platgrondich lichaem soodanigher form alst valt: Sijn swaerheyts middelpunt te vinden. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet A een ongheschickt platgrondich lichaem sijn, dat is omvanghen in platten so veel alst sy. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden. </s><s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal <pb pagenum="91"/>t'lichaem deelen inde nalden <arrow.to.target n="marg268"/> dieder ten weynichsten ende bequamelicxt uyt vallen willen. </s><s>Ten quaetsten commende men can als duer ghemeene reghel, alle platgrondich lichaem in soo veel naelden deelen alst platten heeft, stellende eenich punt int lichaem voor haer ghemeene sop. </s><s> Dit soo sijnde, men sal yder naeldens swaerheyts middelpunt vinden duer het voorstel. </s><s>Daer naer om te vinden t'ghemeene swaerheydts middelpunt van twee naelden, men sal tusschen haer swaerheyts middelpunten een balck trecken, die deelende in sulcken reden als haer twee naelden tot malcanderen sijn, weluerstaende t'cortste deel naer de swaerste naelde. </s><s>Ende inder selue voughen salmen daertoe vergaderen de derde naelde, ende alle d'ander, ende t'punt den balck alsoo ten laetsten deelende, sal t'begheerde swaerheyts middelpunt sijn, waer af t'bewys openbaer is. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet A een ongheschickt platgrondich lichaem sijn, dat is omvanghen in platten so veel alst sy. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden. </s><s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal <pb pagenum="91"/>t'lichaem deelen inde nalden <arrow.to.target n="marg268"/> dieder ten weynichsten ende bequamelicxt uyt vallen willen. </s><s>Ten quaetsten commende men can als duer ghemeene reghel, alle platgrondich lichaem in soo veel naelden deelen alst platten heeft, stellende eenich punt int lichaem voor haer ghemeene sop. </s><s> Dit soo sijnde, men sal yder naeldens swaerheyts middelpunt vinden duer het <emph.end/><emph.end/><emph.end/>voorstel. </s><s>Daer naer om te vinden t'ghemeene swaerheydts middelpunt van twee naelden, men sal tusschen haer swaerheyts middelpunten een balck trecken, die deelende in sulcken reden als haer twee naelden tot malcanderen sijn, weluerstaende t'cortste deel naer de swaerste naelde. </s><s>Ende inder selue voughen salmen daertoe vergaderen de derde naelde, ende alle d'ander, ende t'punt den balck alsoo ten laetsten deelende, sal t'begheerde swaerheyts middelpunt sijn, waer af t'bewys openbaer is. </s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een platgrondich lichaem soodanigher form alst valt, Wy hebben sijn swaerheyts middelpunt gheuonden, naer den heysch.</s>

<s>

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<margin.target id="marg270"/>Ellipses.

</s>

<s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Want LD halfmiddellini <arrow.to.target n="marg271"/> des lancrondts LM euen is an DM, oock an NI, ende IO, soo sal ID as sijn des pilaers EM, inde welcke diens pilaers swaerheyts middelpunt is: ende om de selue reden sal t'swaerheyts middelpunt des pilaers GO oock wesen in KI, ende veruolghens t'swaerheyts middelpunt des lichaems uyt die twee pilaren vergaert is in KD, daerom oock in AD. </s><s>Maer hoe datter sulcke pilaren inden brander meer beschreuen worden, hoe dattet verschil des branders ende der binneschreuen form van sulcke pilaren vergaert, minder is. </s><s>Wy connen dan duer dat oneindelick naerderen sulcken form binnen den brander stellen, dat huer verschil minder sal wesen, dan eenich ghegheuen lichaem hoe cleen het sy; Waer uyt volght dat stellende AD voor swaerheyts middellini des branders, soo sal t'staltwicht van d'een sijde tot d'ander, min verschillen dan eenighe swaerheyt diemen soude connen gheuen, waer uyt ick aldus strie: </s>

<s>

<margin.target id="marg271"/>Semidiameter.

</s>

<s>A. Neuen alle verschillende staltswaerheden, can een swaerheyt ghestelt worden minder dan haer verschil; </s>

<s>O. Neuen dese staltswaerheden van d'eene en dander siide des branders, en can gheen swaerheyt ghestelt worden minder dan haer verschil; </s>

<s>O. Dese staltswaerheden dan van d'eene ende dander siide des branders en verschillen niet. </s>

<s>Daerom AD is haer swaerheyts middellini. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yders branders swaerheyts middelpunt dan, is inden as; t'welck wy bewysen moesten.</s>

<s><emph type="other"/>X. EYSCH<emph.end/>. <emph type="other"/>XXIII. VOORSTEL<emph.end/>.</s>

<s>

<margin.target id="marg272"/>Neuen alle verschillende staltswaerheden, can een swaerheyt ghestelt worden minder dan haer verschil

<margin.target id="marg273"/>Neuen dese staltswaerheden van d'eene en dander siide des branders, en can gheen swaerheyt ghestelt worden minder dan haer verschil

<margin.target id="marg274"/>Dese staltswaerheden dan van d'eene ende dander siide des branders en verschillen niet

</s>

<s>X. <emph type="other"/>EYSCH. XXIII. VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een brander: Huer swaerheyts middelpunt te vinden. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brander wesen diens sop A, ende as AD sy. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal den as AD in E deelen, alsoo dat AE dobbel sy an ED, ende E sal t'begheerde swaerheyts middelpunt sijn; T'welck bewesen is duer Frederick Commandin int 29 voorstel, waer af den sin verclaert naer onse manier soodanich is. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet den brander doorsneen worden met een plat FG, euewydich vanden grondt BC, ende duer t'middel des as H, ende sniende de uytersten des branders in I, K, ende laet BCGF ende IKLM twee pilaren sijn, beschreuen omme den brander, wiens middelpunten N, O, ende IKPQ een pilaer binnen den brander, wiens swaerheyts middelpunt oock O sijn sal. </s><s> Nu want ghelijck AD tot AH, t'welck is als 2 tot I, alsoo <pb pagenum="93"/>t'rondt BC tottet rondt IK <arrow.to.target n="marg275"/>, soo sal den pilaer BG sulcken reden hebben tot den pilaer IL <arrow.to.target n="marg276"/>, als 2 tot 1, daerom laet BG weghen 2 lb, ende IL 1 lb: Maer huer swaerheyts middelpunten sijn N, O, de lini dan NO sal balck sijn de selue ghedeelt in huer ermen, dat is in R, alsoo dat NR dobbel sy an RO, soo sal R swaerheyts middelpunt sijn der twee ommeschreuen pilaren, ende O ist vande binneschreuen, ende R sal soo verre van E vallen, als O van E, te weten elck 1/12 van AD: Ende sulcx sal in alle anderen der ghelijcke voorbeelden oock alsoo gheschien. </s><s>Maer op dattet claerder sy, Wy sullender noch een besonder voorbeelt af beschrijuen aldus: </s>

<s> Nu want ghelijck AD tot AH, t'welck is als 2 tot I, alsoo <pb pagenum="93"/>t'rondt BC tottet rondt IK <arrow.to.target n="marg275"/>, soo sal den pilaer BG sulcken reden hebben tot den pilaer IL <arrow.to.target n="marg276"/>, als 2 tot I, daerom laet BG weghen 2 lb, ende I<emph type="other"/>LI<emph.end/> lb: Maer huer swaerheyts middelpunten sijn N, O, de lini dan NO sal balck sijn de selue ghedeelt in huer ermen, dat is in R, alsoo dat NR dobbel sy an RO, soo sal R swaerheyts middelpunt sijn der twee ommeschreuen pilaren, ende O ist vande binneschreuen, ende R sal soo verre van E vallen, als O van E, te weten elck I/I2 van AD: Ende sulcx sal in alle anderen der ghelijcke voorbeelden oock alsoo gheschien. </s><s>Maer op dattet claerder sy, Wy sullender noch een besonder voorbeelt af beschrijuen aldus: Laet ons den brander ABC noch eenmael snien duer de middelen van AH, ende HD, daer uyt teeckenende vier omschreuen, ende drie binneschreuen pilaren, als hier onder, alwaer AD des branders as is, ende der pilaren middelpunten sijn I, K, L, M, ende AE sy noch dobbel an ED als vooren. </s><s>Nu want ghelijck AD tot AN (t'welck is als 4 tot 3 <arrow.to.target n="marg277"/>) alsoo het rondt BC tottet rondt OP, soo sal den pilaer BF sulcken reden hebben tot den pilaer OQ, als 4 tot 3, ende om de selue oirsaeck sal BF sulcken reden hebben tot de derde diens middelpunt K, als 4 tot 2, ende tot den omschreuen pilaer wiens middelpunt I, als 4 tot I: Daerom laet d'onderste der omschreuen pilaren weghen 4 lb, d'ander 3 lb, de volghende 2 lb, de hoochste I lb: Laet oock om de selue reden de leegste der binneschreuen pilaren weghen 3 lb, d'ander 2 lb, de laetste I lb. </s><s>T'welck soo sijnde, ende anghesien alle de swaerheyts middelpunten ende der pilaren swaerheden bekent sijn, soo ist openbaer duer het voorstel des I boucx, dattet swaerheyts middelpunt der vier omschreuen pilaren sal vallen in L, also dat LE sal doen I/24 van AD, ende der drie binneschreuen pilaren sal vallen in S, also <pb pagenum="94"/>dat SE oock sal doen I/24 van AD. </s><s>Dees twee punten dan L ende S vallen wederom euen verre van E. </s>

<s>

<margin.target id="marg275"/>20. v. I. B. van Apollonius.

<margin.target id="marg275"/>20. v. 1. B. van Apollonius.

<margin.target id="marg276"/>I3. v. I2. B. E.

<margin.target id="marg276"/>13. v. 12. B. E.

<margin.target id="marg277"/>20. v. I. B. van Apollonius.

</s>

<s>Laet ons den brander ABC noch eenmael snien duer de middelen van AH, ende HD, daer uyt teeckenende vier omschreuen, ende drie binneschreuen pilaren, als hier onder, alwaer AD des branders as is, ende der pilaren middelpunten sijn I, K, L, M, ende AE sy noch dobbel an ED als vooren. </s><s>Nu want ghelijck AD tot AN (t'welck is als 4 tot 3 <arrow.to.target n="marg277"/>) alsoo het rondt BC tottet rondt OP, soo sal den pilaer BF sulcken reden hebben tot den pilaer OQ, als 4 tot 3, ende om de selue oirsaeck sal BF sulcken reden hebben tot de derde diens middelpunt K, als 4 tot 2, ende tot den omschreuen pilaer wiens middelpunt I, als 4 tot 1: Daerom laet d'onderste der omschreuen pilaren weghen 4 lb, d'ander 3 lb, de volghende 2 lb, de hoochste 1 lb: Laet oock om de selue reden de leegste der binneschreuen pilaren weghen 3 lb, d'ander 2 lb, de laetste 1 lb. </s><s>T'welck soo sijnde, ende anghesien alle de swaerheyts middelpunten ende der pilaren swaerheden bekent sijn, soo ist openbaer duer het <emph.end/><emph.end/><emph.end/>voorstel des 1. boucx, dattet swaerheyts middelpunt der vier omschreuen pilaren sal vallen in L, also dat LE sal doen 1/24 van AD, ende der drie binneschreuen pilaren sal vallen in S, alsoo <pb pagenum="94"/>dat SE oock sal doen 1/24 van AD. </s><s>Dees twee punten dan L ende S vallen wederom euen verre van E. </s>

<s>

<margin.target id="marg277"/>20. v. 1. B. van Apollonius.

</s>

<s> Maer soomen om den brander schreue sulcke acht pilaren, ende seuen daer binnen, men sal sulcke punten noch euewydich vinden van E; te weten elck 1/48 van AD. </s>

<s>Maer soomen om den brander schreue soodanighe sesthien pilaren, ende vijfthien daer binnen, men sal sulcke punten noch euewydich vinden van E, te weten elck 1/96 van AD: Inder voughen dat het verschil der volghende inschrijuing, altijt den helft is der voorgaende, daer af wy naer t'nootsaecklick veruolg in allen souden trachten, ten waer wy dat lieten om de cortheyt. </s>

<s>Dit soo sijnde E is t'swaerheyts middelpunt des ghegheuen branders: want latet (soot mueghelick waer) daer buyten sijn tusschen EL ofte ES, men sal dan duer de oneindelicke omschrijuing ende binneschrijuing veler pilaren, daer toe commen, dattet swaerheyts middelpunt des omschreuen forms, leegher sal commen dan des branders: ofte der binneschreuen form, hoogher dan des branders, t'welck ommueghelick is. </s><s>Ten is dan gheen ander punt dan E, t'welck wy bewysen moesten. </s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een brander, wy hebben sijn swaerheyts middelpunt gheuonden, naer den eysch.</s>

<s><emph type="other"/>MERCKT<emph.end/>.</s>

<s> <emph type="other"/>ANGHESIEN<emph.end/> des driehoucx lini vanden houck tot int middel der sijde, van t'swaerheyts middelpunt in sulcken reden ghedeelt wordt, als desen as des branders duer het <emph.end/><emph.end/><emph.end/>voorstel, soo volgt dat inden driehouck der ghelijcke ghedaenten sullen beuonden worden duer omschreuen ende binneschreuen vierhoucken, ghelijck hier vooren gheschiet is met omschreuen ende binneschreuen pilaren.</s>

<s> Maer soomen om den brander schreue sulcke acht pilaren, ende seuen daer binnen, men sal sulcke punten noch euewydich vinden van E; te weten elck I/48 van AD. </s><s>Maer soomen om den brander schreue soodanighe sesthien pilaren, ende vijfthien daer binnen, men sal sulcke punten noch euewydich vinden van E, te weten elck I/96 van AD: Inder voughen dat het verschil der volghende inschrijuing, altijt den helft is der voorgaende, daer af wy naer t'nootsaecklick veruolg in allen souden trachten, ten waer wy dat lieten om de cortheyt. </s><s>Dit soo sijnde E is t'swaerheyts middelpunt des ghegheuen branders: want latet (soot mueghelick waer) daer buyten sijn tusschen <emph type="other"/>EL<emph.end/> ofte <emph type="other"/>ES<emph.end/>, men sal dan duer de oneindelicke omschrijuing ende binneschrijuing veler pilaren, daer toe commen, dattet swaerheyts middelpunt des omschreuen forms, leegher sal commen dan des branders: ofte der binneschreuen form, hoogher dan des branders, t'welck ommueghelick is. </s><s>Ten is dan gheen ander punt dan E, t'welck wy bewysen moesten. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een brander, wy hebben sijn swaerheyts middelpunt gheuonden, naer den eysch.</s>

<s><emph type="other"/>XI. EYSCH<emph.end/>. <emph type="other"/>XXIIII. VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>MERCKT<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een ghecorten brander: Huer swaerheyts middelpunt te vinden. </s>

<s> <emph type="other"/>ANGHESIEN<emph.end/> des driehoucx lini vanden houck tot int middel der sijde, van t'swaerheyts middelpunt in sulcken reden ghedeelt wordt, als desen as des branders duer het voorstel, soo volgt dat inden driehouck der ghelijcke ghedaenten sullen beuonden worden duer omschreuen ende binneschreuen vierhoucken, ghelijck hier vooren gheschiet is met omschreuen ende binneschreuen pilaren.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ghecorten brander sijn, diens decsel AB, ende grondt DC, ende as EF. </s>

<s><emph type="other"/>XI. EYSCH. XXIIII. VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten huer swaerheyts middelpunt vinden. </s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een ghecorten brander: Huer swaerheyts middelpunt te vinden. </s>

<s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal den ghecorten brander volmaken, daer an stellende t'ghebrekende ABG, Daernaer salmen teeckenen H, alsoo dat GH dobble sy an HE, sghelijcx, I also dat GI dobbel sy an IF, daernaer K, alsoo dat IH sulcken reden hebbe tot IK, als den ghecorten brander ABCD, tottet branderken ABG: </s><s> Ick seg dat K t'be<pb pagenum="95"/>gheerde swaerheyts middelpunt is. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ghecorten brander sijn, diens decsel AB, ende grondt DC, ende as EF. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten huer swaerheyts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal den ghecorten brander volmaken, daer an stellende t'ghebrekende ABG, Daernaer salmen teeckenen H, alsoo dat GH dobble sy an HE, sghelijcx, I also dat GI dobbel sy an IF, daernaer K, alsoo dat IH sulcken reden hebbe tot IK, als den ghecorten brander ABCD, tottet branderken ABG: </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>I is swaerheyts middelpunt des heels DCG, ende H des deels ABG, ende ghelijck t'ander deel ABCD, tot dit deel ABG, alsoo HI tot IK duer t'werck, daerom K, duer het <emph.end/><emph.end/><emph.end/>voorstel, is t'begheerde swaerheyts middelpunt, t'welck wy bewysen moesten. </s>

<s> Ick seg dat K t'be<pb pagenum="95"/>gheerde swaerheyts middelpunt is. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>I is swaerheyts middelpunt des heels DCG, ende H des deels ABG, ende ghelijck t'ander deel ABCD, tot dit deel ABG, alsoo HI tot IK duer t'werck, daerom K, duer het voorstel, is t'begheerde swaerheyts middelpunt, t'welck wy bewysen moesten. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een ghecorten brander, wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een ghecorten brander, wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

<s> <emph type="other"/>EINDEDESTWEEDENBOVCX<emph.end/>. </s>

<s> <emph type="other"/>EINDE DES TWEEDEN BOVCK<emph.end/>. </s>

</chap></body><back/></text></archimedes>

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