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Colored diff for /texts/archimedes/xml/stevi_weegc_085_nl_1586.xml between version 1.10 and 1.11

version 1.10, 2004/07/13 09:49:50

version 1.11, 2006/03/03 22:25:09

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<margin.target id="marg6"/>Effecta.

</s>

<s> Waerom sal yemant mueghen segghen, dit niet bestelt duer leegher (naer de ghebruyck) totten Hoochsten vrie toeganck hebbende? Yghelick, om dattet t'onghehoort is, soude vreesen niet alleen met een lacherlick * voorstel te verschijnen, maer selfs oock belacht te worden: Nu op dat der spotters schamp tot ghetuych haerder onwetenheyt strecke, wy hebbent duer t'ghene willen versoucken, dat voor den verstandighen alsulcx ende meerder veruaet. </s><s>daerafmen wyder en breeder soude connen segghen; Maer want ons einde tot Saken streckt, niet tot Woorden, sullen dese verlatende en die verwachtende, uwe K. M. in alle ootmoedighe eerbieding veel ghelucx wenschen.</s><s> Uyt Leyden in Oogstmaent des I586 Iaers. </s>

<pb/>

Line 155

<margin.target id="marg18"/>Proportionis.

</s>

<s> Maer soo sy vraghen wat sulcke woorden te bedien hebben, men mach antwoorden, dattet de opening is van t'ghene tot nochtoe den voorighen sterflicken seerbegheerde verborghentheden gheweest sijn, streckende tot groot voordeel van t'menschelick gheslacht, want hoewel yder lichaem in sijn eyghen plaets licht noch swaer en is, nochtans t'ghewicht des lochts is duer sulcx nu volcomelick ghelijck van ander stoffen, metgaders ettelicke sijn noytbekende <arrow.to.target n="marg19"/>ancleuinghen, openbaer gheworden, soo de <arrow.to.target n="marg20"/>daet van dies ende meer anderen cortelick betuyghen sal. </s><s>Laetter maer cloeclick anuallen, want hebbender Reuchlinus, Valla, Erasmus, Barbarus, Picus, Politianus, &c. </s><s>me duer gherocht, die maer Latijn en beschermden, Sghelijcx de Françoysen, wiens <arrow.to.target n="marg21"/>strijtredenen ende talens <arrow.to.target n="marg22"/>stof ons kennelick ghenouch sijn, wat sullen wy die het (O weerdighen <arrow.to.target n="marg23"/>grondt!) DVYTSCH voorstaen? Seker niet alleen de spraeck ophelpen, noch ons seluen voorderen, maer oock ander volcken, welcke alsdan niet alleen huer wooninghen ende lichamen met der Duytschen constighe wercken vercieren sullen, maer oock haren gheest met wetenschap, want de Consten welcke ander met haer eyghen woorden niet uyten en connen, die sal den ghemeenen man alhier duer de beghinselen grondelick mueghen verstaen, ende door sijn ingheboren gheneghentheyt tot de selue, die tot yder volcx baet al andersins connen voorderen dant den anderen mueghelick is.</s>

<s> Maer soo sy vraghen wat sulcke woorden te bedien hebben, men mach antwoorden, dattet de opening is van t'ghene tot nochtoe den voorighen sterflicken seerbegheerde verborghentheden gheweest sijn, streckende tot groot voordeel van t'menschelick gheslacht, want hoewel yder lichaem in sijn eyghen plaets licht noch swaer en is, nochtans t'ghewicht des lochts is duer sulcx nu volcomelick ghelijck van ander stoffen, metgaders ettelicke sijn noytbekende <arrow.to.target n="marg19"/>ancleuinghen, openbaer gheworden, soo de <arrow.to.target n="marg20"/>daet van dies ende meer anderen cortelick betuyghen sal. </s><s>Laetter maer cloeclick anuallen, want hebbender Reuchlinus, Valla, Erasmus, Barbarus, Picus, Politianus, &c. me duer gherocht, die maer Latijn en beschermden, Sghelijcx de Françoysen, wiens <arrow.to.target n="marg21"/>strijtredenen ende talens <arrow.to.target n="marg22"/>stof ons kennelick ghenouch sijn, wat sullen wy die het (O weerdighen <arrow.to.target n="marg23"/>grondt!) DVYTSCH voorstaen? Seker niet alleen de spraeck ophelpen, noch ons seluen voorderen, maer oock ander volcken, welcke alsdan niet alleen huer wooninghen ende lichamen met der Duytschen constighe wercken vercieren sullen, maer oock haren gheest met wetenschap, want de Consten welcke ander met haer eyghen woorden niet uyten en connen, die sal den ghemeenen man alhier duer de beghinselen grondelick mueghen verstaen, ende door sijn ingheboren gheneghentheyt tot de selue, die tot yder volcx baet al andersins connen voorderen dant den anderen mueghelick is.</s>

<s>

<margin.target id="marg19"/>Adiunctis.

<margin.target id="marg20"/>Effectus.

Line 323

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

<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, <arrow.to.target n="marg48"/>Lancworpighe Clooten, 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 <arrow.to.target n="marg49"/>naelden, 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"/>LAET<emph.end/> ABC een cloot sijn, diens stof ouer al eueswaer <figure/> 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, <arrow.to.target n="marg48"/>Lancworpighe Clooten, 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 <arrow.to.target n="marg49"/>naelden, 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 11. Bepaling verclaert worden.</s>

Line 356

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

<s> <emph type="other"/>LAETA<emph.end/> ende B twee lichamen wesen, ende haer swaerheydts middellinien CD ende EF, tusschen de welcke ghetrocken sijn, eenighe linien soot valt als GH, AB, IK, yder van dien, ende alle ander alsoo begrepen tusschen twee swaerheydts middellinien, noemen wy den Balck der swaerheden <emph type="other"/>AB<emph.end/>. </s>

<s> <emph type="other"/>LAET A<emph.end/> ende B twee lichamen wesen, ende <figure/> haer swaerheydts middellinien CD ende EF, tusschen de welcke ghetrocken sijn, eenighe linien soot valt als GH, AB, IK, yder van dien, ende alle ander alsoo begrepen tusschen twee swaerheydts middellinien, noemen wy den Balck der swaerheden <emph type="other"/>AB<emph.end/>. </s>

<s><emph type="other"/>VIII. BEPALING<emph.end/>.</s>

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

<s> <emph type="other"/>LAET<emph.end/> AB, twee lichamen wesen, diens balck sy CD, welcke ghedeelt is in E, met de swaerheydts middellini FG, daer de twee swaerheden euestaltwichtich an hanghen; de twee deelen des balcx als EC ende ED worden Ermen ghenoemt. </s>

<s> <emph type="other"/>LAET<emph.end/> AB, twee lichamen wesen, diens balck <figure/> sy CD, welcke ghedeelt is in E, met de swaerheydts middellini FG, daer de twee swaerheden euestaltwichtich an hanghen; de twee deelen des balcx als EC ende ED worden Ermen ghenoemt. </s>

<s><emph type="other"/>IX. BEPALING<emph.end/>.</s>

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

<s> <emph type="other"/>LAET<emph.end/> den pilaer A, een swaerheyt wesen, diens lini daer sy alsoo by ghehouden wort sy BC, ende tpunt daer sy op rust D, ende E, sy t'ghewicht dat tlichaem A in die ghestalt houdt. </s>

<s> <emph type="other"/>LAET<emph.end/> den pilaer <figure/> A, een swaerheyt wesen, diens lini daer sy alsoo by ghehouden wort sy BC, ende tpunt daer sy op rust D, ende E, sy t'ghewicht dat tlichaem A in die ghestalt houdt. </s>

<s> Wy noemen E der eerste ende tweede Form Hefwicht, overmidts tselve wicht, het lichaem A verheft, oft in die verheven ghestalt houdt. </s>

Line 408

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

<s> <emph type="other"/>ALS<emph.end/> de rechte lini CB der I2. bepaling noemen wy inde 1. ende 2. form Heflini, maer inde 3. ende 4. Daellini. </s><s>Ende sulcke linien (die bouen de voorgaende ons oock euewidich vanden sichtender connen ontmoeten) in t'ghemeen Trecklini.</s>

<s> <emph type="other"/>ALS<emph.end/> de rechte lini CB der 12. bepaling noemen wy inde 1. ende 2. form Heflini, maer inde 3. ende 4. Daellini. </s><s>Ende sulcke linien (die bouen de voorgaende ons oock euewidich vanden sichtender connen ontmoeten) in t'ghemeen Trecklini.</s>

<s><emph type="other"/>XIIII. BEPALING<emph.end/>.</s>

Line 431

<margin.target id="marg56"/>, wy en souden daer af hier niet segghen, ouermits siin eyghen plaets inde <margin.target id="marg57"/>Sterconst is; Maer want den ongheuallighen slaep des Spieghels der talen sulcx niet toeghelaten en heeft, oock dat dit woordt hier naer dickmael sal ghenoemt worden, sullen dat verclaren, doch niet als wesentlicke bepaling deses boucx, om de redenen als vooren, Aldus:

Sichteinder is <pb pagenum="7"/>des weerelts grootste rondt, dat haer sienlick deel scheydt van het onsienlick: <arrow.to.target n="marg58"/>{Parallela.}<arrow.to.target n="marg59"/>{Metaphorice.}<arrow.to.target n="marg60"/>{Mathematice.}<arrow.to.target n="marg61"/></s>

<s>

<margin.target id="marg58"/>Dat is, onder veel ronden die inde Sterconst bepaelt worden, soo isser een het aldermerckelicste, scheydende ooghenschynlick den oppersten haluen weereltcloot vanden ondersten, ende in ons ansien den hemel met sijn omtreck naeckende, t'welck volcommentlicxst schijnt vande hoochste plaets eender contreyen, ofte op een water daer hem nerghens landt en vertoocht; Ende ouermits ons ghesicht langs der eerden ofte langs het water niet voorder strecken en can dan tot diens rondts voornoemden omtreck, ende daer in eindet, soo wort dat rondt ghenoemt den Sichteinder, dat is den Einder van t'ghesicht. Ende alle platten die op t'eertrick vanden Sichteinder <margin.target id="marg59"/>euewydich siin (welcke by ons ghemeenelick gheseyt worden op waterpas te ligghen) worden <margin.target id="marg60"/>lijckspreuckelick oock sichteinders ghenoemt. Ick seg lijckspreucklick want eyghentlick ofte <margin.target id="marg61"/>wisconstelick en isser gheen ander, dan dat door des weerelts middelpunt strect.

<margin.target id="marg58"/>Dat is, onder veel ronden die inde Sterconst bepaelt worden, soo isser een het aldermerckelicste, scheydende ooghenschynlick den oppersten haluen weereltcloot vanden ondersten, ende in ons ansien den hemel met sijn omtreck naeckende, t'welck volcommentlicxst schijnt vande hoochste plaets eender contreyen, ofte op een water daer hem nerghens landt en vertoocht; Ende ouermits ons ghesicht langs der eerden ofte langs het water niet voorder strecken en can dan tot diens rondts voornoemden omtreck, ende daer in eindet, soo wort dat rondt ghenoemt den Sichteinder, dat is den Einder van t'ghesicht. </s><s>Ende alle platten die op t'eertrick vanden Sichteinder <margin.target id="marg59"/>euewydich siin (welcke by ons ghemeenelick gheseyt worden op waterpas te ligghen) worden <margin.target id="marg60"/>lijckspreuckelick oock sichteinders ghenoemt. </s><s>Ick seg lijckspreucklick want eyghentlick ofte <margin.target id="marg61"/>wisconstelick en isser gheen ander, dan dat door des weerelts middelpunt strect.

</s>

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

Line 440

<s>

<margin.target id="marg62"/>form vanden Weegconstighen Pilaer

<margin.target id="marg63"/>, is de selue der Meetconst

<margin.target id="marg64"/>, maer wy nemen hier siin stof eenuaerdigher swaerheyt te wesen, ende siin grondt ende decsel viercanten. Wat de ghemeene constwoorden belangt int Latijn aldus ghebruyckt.

<margin.target id="marg64"/>, maer wy nemen hier siin stof eenuaerdigher swaerheyt te wesen, ende siin grondt ende decsel viercanten. </s><s>Wat de ghemeene constwoorden belangt int Latijn aldus ghebruyckt.

</s>

Line 454

<s> <emph type="other"/>WELKE<emph.end/> <arrow.to.target n="marg66"/>

<margin.target id="marg66"/>Latijnsche met eenighe ander dieder by mueghen vallen wy tot meerder claerheyt, somwylen inden cant sullen scriven neven haer duytsche. Dese drie letteren v. b. E. altemet inde cant ghestelt beteeckenen om cortheydt, voorstel, bouck, Euclides, als 2 v. 6. b. E. dat is te segghen het 2. voorstel des 6. boucx van Euclides.

<margin.target id="marg66"/>Latijnsche met eenighe ander dieder by mueghen vallen wy tot meerder claerheyt, somwylen inden cant sullen scriven neven haer duytsche. </s><s>Dese drie letteren v. b. E. altemet inde cant ghestelt beteeckenen om cortheydt, voorstel, bouck, Euclides, als 2 v. 6. b. E. dat is te segghen het 2. voorstel des 6. boucx van Euclides.

</s>

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

Line 490

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

<s> <emph type="other"/>ALS<emph.end/> wesende AB een pilaer diens as CD, ende de selue doorsneen met eenich plat als EFGH, datmen door t'bescreuen plat EFGH, al de rest acherghelaten, verstaen sal den ghegheuen pilaer. </s>

<s> <emph type="other"/>ALS<emph.end/> wesende AB een pilaer diens <figure/> as CD, ende de selue doorsneen met eenich plat als EFGH, datmen door t'bescreuen plat EFGH, al de rest acherghelaten, verstaen sal den ghegheuen pilaer. </s>

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

Line 501

<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 <arrow.to.target n="marg70"/>sichteinder 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, <arrow.to.target n="marg71"/>wisconstelick 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 1. 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 <figure/> E, ende <arrow.to.target n="marg70"/>sichteinder 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, <arrow.to.target n="marg71"/>wisconstelick 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, <figure/> sullender oneindelicke verscheyden swaerheyts middelpunten in sijn. </s><s>Oock en soude (teghen t'volghende 1. 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 <emph type="other"/>WEEGDAET<emph.end/> nochtans niet voorderlicker, so ist beter ghelaten. </s>

<s> Alle dese ongheuallen spruyten daer uyt, dat <figure/> 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/>

Line 524

<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 <arrow.to.target n="marg75"/>platten 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 7. 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 8. 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 <figure/> 6 , welcke ghedeelt sy in 6 euen deelen, door <arrow.to.target n="marg75"/>platten 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 7. 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 8. 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.

Line 539

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

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD wederom een pilaer sijn, ghedeelt <pb pagenum="12"/>met een plat euewydich van AD, als EF, sniende den as GH, waert sy in I, ende het swaerheyts middelpunt van het deel EFDA sy K, int middel van GI, ende van het deel EFCB, sy L int middel van IH, ende des heels ABCD sy M int middel van GH, ende MN sal der deelen EFDA ende EFCB handthaef sijn, daer an sy euestaltwichtich hanghen.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD wederom een pilaer sijn, ghedeelt <pb pagenum="12"/>met een plat euewydich van AD, als EF, sniende den as GH, waert sy <figure/> in I, ende het swaerheyts middelpunt van het deel EFDA sy K, int middel van GI, ende van het deel EFCB, sy L int middel van IH, ende des heels ABCD sy M int middel van GH, ende MN sal der deelen EFDA ende EFCB handthaef sijn, daer an sy euestaltwichtich hanghen.</s>

<s>

Line 554

<margin.target id="marg80"/>Alternam proportionem

</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 1. 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"/>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 1. 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 <figure/> 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>

<s><emph type="other"/>DIT<emph.end/> so synde, laet ons de lichamen EFDA ende EFCB der tweede <figure/> 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>

<s>

Line 568

<margin.target id="marg83"/>3. Ghestalt.

</s>

<s> <emph type="other"/>LAET<emph.end/> ons ten laetsten weeren EFDA der derde ghestalt ende hanghen in diens plaets een lichaem van loot des selfden ghewichts, ende inde plaets van EFCB een hauten lichaem des seluen ghe<pb pagenum="14"/>wichts, wiens vierde ghestalt alsdan sy als hier neuens. </s><s>Ende is kennelick dat KL noch inde selue ghestalt sal blyuen, ende veruolghens EFDA noch euestaltwichtich met EFCB, ende de ermen noch inde selue reden.</s>

<s> <emph type="other"/>LAET<emph.end/> ons ten laetsten weeren EFDA der derde ghestalt ende hanghen in diens plaets een lichaem van loot des selfden ghewichts, ende inde plaets van EFCB een hauten lichaem des seluen ghe<pb pagenum="14"/>wichts, wiens vierde ghestalt alsdan sy <figure/> als hier neuens. </s><s>Ende is kennelick dat KL noch inde selue ghestalt sal blyuen, ende veruolghens EFDA noch euestaltwichtich met EFCB, ende de ermen noch inde selue reden.</s>

<s>

Line 577

<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"/>MEN<emph.end/> can tvoorgaende oock bethoonen, <figure/> 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 4. bepaling, daerom ABFE blijft oock noch in sijn eerste ghedaente. </s>

<s><emph type="other"/>LAET<emph.end/> ons nu het onderste <figure/> 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>

Line 598

<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 lb 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 <figure/> 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>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet d'een swaerheydt sijn den pilaer ABCD weghende 6 lb, ghedeelt als den pilaer int beghin des eersten voorstels; Ende an Q hanghe een ghewicht Y van 12 lb. </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 des pilaers is IT, ende van tghewicht Y is BQ, ende TQ is balck, <pb pagenum="16"/>de selue salmen in tween deelen, alsoo dat de sticken de reden hebben als 12 lb van Y, tot 6 lb vanden pilaer, weluerstaende tcortste stick naer de swaerheydts middellini vande swaerste swaerheydt Y, twelck vallen sal in X, inder voughen dat NX de begheerde hanthaef is.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet d'een swaerheydt sijn den pilaer ABCD weghende 6 lb, ghedeelt als den pilaer int beghin des eersten voorstels; Ende an Q hanghe een ghewicht Y van 12 lb. </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 des pilaers is IT, ende van tghewicht Y is BQ, ende TQ is balck, <pb pagenum="16"/>de selue salmen in tween deelen, <figure/> alsoo dat de sticken de reden hebben als 12 lb van Y, tot 6 lb vanden pilaer, weluerstaende tcortste stick naer de swaerheydts middellini vande swaerste swaerheydt Y, twelck vallen sal in X, inder voughen dat NX de begheerde hanthaef is.</s>

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

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD wederom den pilaer sijn, ghedeelt als vooren, hanghende nu Y 6 lb an X. </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 des pilaers is IT, ende van Y is NX, ende TX is balck: de selue salmen in tween deelen, alsoo dat de sticken de reden hebben als 6 lb van Y, tot 6 lb des pilaers, twelck vallen sal in V, indervoughen dat VL de begheerde handthaef sijn sal.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD wederom den pilaer sijn, ghedeelt <figure/> als vooren, hanghende nu Y 6 lb an X. </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 des pilaers is IT, ende van Y is NX, ende TX is balck: de selue salmen in tween deelen, <figure/> alsoo dat de sticken de reden hebben als 6 lb van Y, tot 6 lb des pilaers, twelck vallen sal in V, indervoughen dat VL de begheerde handthaef sijn sal.</s>

<s><emph type="other"/>TVOORNOEMDE WERCK<emph.end/></s>

Line 617

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

<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. </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 <pb pagenum="17"/>swaerheydts middellini van ABCDY, is LV door het 3. voorbeelt, ende van Z is RE, daerom is RV balck: de selue salmen in tween deelen, alsoo dat de stucken de reden hebben als 12 lb van ABCDY, tot 24 lb van Z: wel verstaende tcortste stic naer de swaerheyts middellini van tswaerste deel, twelck vallen sal in S, inder voughen dat SG de begheerde handthaef sijn sal. </s>

<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. </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 <pb pagenum="17"/>swaerheydts middellini van <figure/> ABCDY, is LV door het 3. voorbeelt, ende van Z is RE, daerom is RV balck: de selue salmen in tween deelen, alsoo dat de stucken de reden hebben als 12 lb van ABCDY, tot 24 lb van Z: wel verstaende tcortste stic naer de swaerheyts middellini van tswaerste deel, twelck vallen sal in S, inder voughen dat SG de begheerde handthaef sijn sal. </s>

<s><emph type="other"/>TVOORNOEMDE WERCK OP<emph.end/></s>

Line 633

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

<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> <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 <figure/> 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 9. bepaling is d'hanthaef. </s><s>Sghelijcx sal oock tbewys sijn van al dander voorbeelden, twelck wy om de cortheydt achterlaten. </s>

Line 642

<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; 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.

<margin.target id="marg87"/>tghewicht Y des 2. voorbeelts verswaerde van 1 lb, ende datmen <figure/> 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. </s><s>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. </s><s>Waer uyt de Ghedaenten { Qualitates. } des s bekent siin, als inde Weegdaet breeder daer af sal ghehandelt worden.

</s>

<s><emph type="other"/>II. EYSCH. <emph.end/> <emph type="other"/>III. VOORSTEL<emph.end/>.</s>

Line 651

<s>I. <emph type="other"/>VOORBEELT<emph.end/>.</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, maer B hanghende an D is onbekent, ende EF sy d'handthaef. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten tghewicht van B bekent maken. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal ondersoucken wat reden den erm ED heeft, tot den erm EC, wort beuonden, neem ick, als van 3 tot 1, daerom seg ick, ED 3, gheeft EC 1, wat A 3 lb ? comt voor B 1 lb. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A ende B twee euestaltwichtighe swaerheden <figure/> sijn, welcker A hanghende an C weeght 3 lb, maer B hanghende an D is onbekent, ende EF sy d'handthaef. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten tghewicht van B bekent maken. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal ondersoucken wat reden den erm ED heeft, tot den erm EC, wort beuonden, neem ick, als van 3 tot 1, daerom seg ick, ED 3, gheeft EC 1, wat A 3 lb ? comt voor B 1 lb. </s>

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

Line 661

<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 2. voorstels, ten waer die duer tvoorgaende kennelick ghenouch sijn. </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 <figure/> 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>

Line 671

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een pilaer: te vinden een ghewicht in ghestelde reden tot des pilaers ghewicht. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer wesen, diens as EF, ende haer <arrow.to.target n="marg88"/>middelpunt G, ende de ghestelde reden sy van 2 tot 3.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer wesen, diens as EF, <figure/> ende haer <arrow.to.target n="marg88"/>middelpunt G, ende de ghestelde reden sy van 2 tot 3.</s>

<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten een ghewicht vinden in sulcken reden tot den pilaer, als van 2 tot 3, dat is euen an sijn 2/3. </s><s> <emph type="other"/>MERCKT<emph.end/>. </s><s>Ghelijck de <arrow.to.target n="marg89"/>Meetconstighe ende Telconstighe voorstellen verscheyden werckinghen hebben, alsoo oock de Weegconst, want men soude vanden pilaer een stuck connen snien in sulcken reden tot den heelen pilaer, als van 2 tot 3. </s><s>Oft andersins om den pilaer heel te laten, men mocht hem teghen ander stof weghen, daer af nemende de /, maer wy willent Weegconstlicker doen in deser voughen. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal van tmiddelpunt G af, naer F, teeckenen eenighe vijf punten (te weten 5 voor de somme der <pb pagenum="21"/>ghegheuen palen 2. </s><s>3) als H, I, K, L, M, van malcanderen euewyt; </s>

<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten een ghewicht vinden in sulcken reden tot den pilaer, als van 2 tot 3, dat is euen an sijn 2/3. </s><s> <emph type="other"/>MERCKT<emph.end/>. </s><s>Ghelijck de <arrow.to.target n="marg89"/>Meetconstighe ende Telconstighe voorstellen verscheyden werckinghen hebben, alsoo oock de Weegconst, want men soude vanden pilaer een stuck connen snien in sulcken reden tot den heelen pilaer, als van 2 tot 3. </s><s>Oft andersins om den pilaer heel te laten, men mocht hem teghen ander stof weghen, daer af nemende de /, maer wy willent Weegconstlicker doen in deser voughen. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal van tmiddelpunt G af, naer F, teeckenen eenighe vijf punten (te weten 5 voor de somme der <pb pagenum="21"/>ghegheuen palen 2. 3) als H, I, K, L, M, van malcanderen euewyt; </s>

<s>

<margin.target id="marg88"/>Centrum.

<margin.target id="marg89"/>Geometrica & Arithmetica propositiones.

Line 694

<margin.target id="marg92"/>Horizon.

</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn, ghesneen door sijn swaerheydts middelpunt met een plat FG, euewydich vanden grondt AD, ende laet H vastpunt inde swaerheydts middellini IG wesen, bouen het swaerheyts middelpunt E, ende KL sy as, ende MN sichteinder. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn, ghesneen door sijn <figure/> swaerheydts middelpunt met een plat FG, euewydich vanden grondt AD, ende laet H vastpunt inde swaerheydts middellini IG wesen, bouen het swaerheyts middelpunt E, ende KL sy as, ende MN sichteinder. </s>

<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat den as KL euewydich blijft vanden sichteinder. </s><s>MN. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet KL soot mueghelijck waer, oneuewydich sijn vanden sichteinder MN, als in <pb pagenum="22"/>dees tweede form, ende laet IH voortghetrocken worden tot in O, sniende AB in P, ende laet het stuck des pilaers POCB alsoo euewichtich blijuen hanghen teghen PODA, maer dat is grooter ende swaerder dan dit (want FGDA, is euen an FGCB, ende minder is den driehouck FHI ghesneen van FGCB, dan de driehouck OHG ghesneen van FGDA, daerom, &c.) het swaerder dan sal euewichtich sijn an een lichter twelck ongheschict is, KL dan blijft euewydich vanden sichteinder MN, als in d'eerste form. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet KL soot mueghelijck waer, oneuewydich sijn vanden sichteinder MN, als in <pb pagenum="22"/>dees tweede form, ende laet IH voortghetrocken worden tot in O,<figure/> sniende AB in P, ende laet het stuck des pilaers POCB alsoo euewichtich blijuen hanghen teghen PODA, maer dat is grooter ende swaerder dan dit (want FGDA, is euen an FGCB, ende minder is den driehouck FHI ghesneen van FGCB, dan de driehouck OHG ghesneen van FGDA, daerom, &c.) het swaerder dan sal euewichtich sijn an een lichter twelck ongheschict is, KL dan blijft euewydich vanden sichteinder MN, als in d'eerste form. </s>

<s>Tis oock te anmercken als voor ghemeenen Weegconstighen Reghel, dat</s>

Line 710

<s> <emph type="other"/>WESENDE<emph.end/> tvastpunt het swaerheydts middelpunt des hanghenden pilaers, hy houdt alle gestalt diemen hem gheeft. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer wesen, diens swaerheydts middelpunt E vast sy, daer by hanghende ande lini EF, ende den as GH sy euewydich vanden sichteinder IK. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet <figure/> ABCD een pilaer wesen, diens swaerheydts middelpunt E vast sy, daer by hanghende ande lini EF, ende den as GH sy euewydich vanden sichteinder IK. </s>

<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat den pilaer ABCD alle ghestalt houdt diemen hem gheeft. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons den ghegheuen pilaer (tpunt E vast blijuende) een ander ghestalt gheuen dan d'eerste, als in dees tweede form, <pb pagenum="23"/>ende laet FE voortghetrocken worden tot in L, sniende AB in M, ende en laet den pilaer soot mueghelick waer niet in die ghestalt blijuen, dan het stick MLDA, ofte MLCB neervallen; Maer dees twee deelen sijn ghelijck euegroot, ende daerom oock eueswaer, het eene dan van euewichtighe sal swaerder sijn dan t'ander, twelc ongheschickt is: Den pilaer dan blijft in die ghestalt, ende sghelijcx in allen anderen diemen hem soude mueghen gheuen. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan tvastpunt het swaerheydts middelpunt des pilaers, hy houdt alle ghestalt diemen hem gheeft, twelck wy bewysen moesten. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons den ghegheuen pilaer (tpunt E vast blijuende) een ander ghestalt gheuen dan d'eerste, als in dees tweede form, <pb pagenum="23"/>ende laet FE voortghetrocken <figure/> worden tot in L, sniende AB in M, ende en laet den pilaer soot mueghelick waer niet in die ghestalt blijuen, dan het stick MLDA, ofte MLCB neervallen; Maer dees twee deelen sijn ghelijck euegroot, ende daerom oock eueswaer, het eene dan van euewichtighe sal swaerder sijn dan t'ander, twelc ongheschickt is: Den pilaer dan blijft in die ghestalt, ende sghelijcx in allen anderen diemen hem soude mueghen gheuen. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan tvastpunt het swaerheydts middelpunt des pilaers, hy houdt alle ghestalt diemen hem gheeft, twelck wy bewysen moesten. </s>

<s><emph type="other"/>IIII VERTOOCH<emph.end/>. <emph type="other"/>VIII VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> den pilaer ghesneen door sijn swaerheyts middelpunt, met een plat euewydich vanden gront, ende wesende tvastpunt in dat plat beneden het swaerheydts middelpunt: Den pilaer (natuerlick verstaen) keert om tot dat sijn swaerheydts middelpunt is in sijn swaerheydts middellini. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer wesen, ghesneen door sijn swaerheyts middelpunt E, met een <arrow.to.target n="marg94"/>plat FG euewydich vanden grondt AD, ende laet G vastpunt sijn, beneden tswaerheydts middelpunt E, met welck punt G den pilaer ligt ofte rust op tpunt des pins H, ende IK sy as, euewydich vanden <arrow.to.target n="marg95"/>sichteinder LM. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet <figure/> ABCD een pilaer wesen, ghesneen door sijn swaerheyts middelpunt E, met een <arrow.to.target n="marg94"/>plat FG euewydich vanden grondt AD, ende laet G vastpunt sijn, beneden tswaerheydts middelpunt E, met welck punt G den pilaer ligt ofte rust op tpunt des pins H, ende IK sy as, euewydich vanden <arrow.to.target n="marg95"/>sichteinder LM. </s>

<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat den pilaer omkeeren sal, tot dat sijn swaerheydts middelpunt is in sijn swaerheyts middellini. </s><pb pagenum="24"/><s>maer dit natuerlick verstaen, want <arrow.to.target n="marg96"/>Wisconstelick ghenomen soo can sy daer op rusten. </s>

<s>

Line 741

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

<s> <emph type="other"/>YEMANT<emph.end/> mocht hier noch de verclaring begheeren des verschils tusschen hanghen en ligghen, waer op d'antwoort is dat wy een lichaem voor hanghende houden, als siin swaerheyts middelpunt is onder, oft int tghenaecsel daert op rust; Maer tswaerheyts middelpunt daer bouen siinde, alsdan houden wijt voor ligghen, staen, oft sitten; Ligghen, als de langste siide des lichaems haer strect langs den sichteinder: Staen, als sy daer op rechthouckich is; daerom ist oock dat wy den teerlinck (ouermits siin siiden al euen lanck siin) soo eyghentlick segghen te staen als te ligghen, ende te ligghen als te staen. Sitten is wat tusschen ligghen en staen.

</s>

<s><emph type="other"/>II. MERCK<emph.end/>.</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> <emph type="other"/>SOO<emph.end/> yemant thinhondt [t inhoudt] der voorgaende drie voorstellen door eenighe eruaring <figure/> 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. </s><s>Ende stekende een naelde door tgatken K, die vrielick daerin drayen mach, d'eruaring sal bethoonen dat HF altijdt euewydich sal blijuen vanden sichteinder. </s><s>Maer de naelde in I stekende, de reghel sal daerop alle ghestalt houden diemen haer gheeft. </s><s>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>

Line 755

<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 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"/>TGHEGHEVEN<emph.end/>. </s><s>Laet AB twee swaerheden sijn ende haer middellinien <figure/> 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>

Line 779

<s><emph type="other"/>5. EYSCH<emph.end/>. <emph type="other"/>I0. VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>5. EYSCH<emph.end/>. <emph type="other"/>10. VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een vastpunt des bekenden pilaers, ende bekende euestaltwichtighe swaerheden an hem hanghende: Te vinden of den as euewydich sal blijuen vanden <arrow.to.target n="marg106"/>sichteinder, oft alle ghestalt houden diemen hem gheeft, ofte omkeeren tot dat sijn swaerheydts middelpunt is in sijn swaerheyts middellini. </s>

<s>

<margin.target id="marg106"/>Horizonte.

</s>

<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"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 4 lb, ende <figure/> 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 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 <arrow.to.target n="marg107"/>hanghende 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 1 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 <arrow.to.target n="marg107"/>hanghende 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 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 8 voorstel, maer alles sal omkeeren, twelck wy bewysen moesten. </s>

<s>Ten laetsten I, K, elck 1 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>

Line 807

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

<s> <emph type="other"/>SOO<emph.end/> tweer euewychten als A, B, vastpunten C, D, waren in des pilaers as, euewyt van tmiddelpunt E, als in dees form, tis kennelick door het tweede deel des bewys van het I0 voorstel, dat E tbegheerde punt soude siin, maer wy sullen tvoorbeelt van ongheschicter ghestalt gheuen.

<s> <emph type="other"/>SOO<emph.end/> tweer euewychten als A, B, vastpunten C, D, <figure/> waren in des pilaers as, euewyt van tmiddelpunt E, als in dees form, tis kennelick door het tweede deel des bewys van het 10. voorstel, dat E tbegheerde punt soude siin, maer wy sullen tvoorbeelt van ongheschicter ghestalt gheuen.

</s>

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

openbaer dat wesende de twee vastpunten der ghewichten als CD, ende tvastpunt des handthaefs als E, alle drie in een rechte lini als hier bouen, ende an CD euen ghewichten ghehanghen, soo groot ofte cleen alst valt: E sal altijt tvastpunt blijuen, daer sy alle ghestalt op houden diemen haer gheeft. Maer soo die drie punten als CED in een rechte lini wesende C ende D niet euewyt en waren van E, ende datmen an haer ghewichten hinghe <arrow.to.target n="marg109"/>euerednich met de ermen, dat E noch altijt tvastpunt sal blijuen daer sy alle ghestalt op houden diemen haer gheeft.</s>

openbaer dat wesende de twee vastpunten der ghewichten als CD, ende tvastpunt des handthaefs als E, alle drie in een rechte lini als hier bouen, ende an CD euen ghewichten ghehanghen, soo groot ofte cleen alst valt: E sal altijt tvastpunt blijuen, daer sy alle ghestalt op houden diemen haer gheeft. </s><s>Maer soo die drie punten als CED in een rechte lini wesende C ende D niet euewyt en waren van E, ende datmen an haer ghewichten hinghe <arrow.to.target n="marg109"/>euerednich met de ermen, dat E noch altijt tvastpunt sal blijuen daer sy alle ghestalt op houden diemen haer gheeft.</s>

<s>

<margin.target id="marg109"/>Proportionales.</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"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn, weghende 10 lb, diens swaerheydts middelpunt E, ende laet de ghewichten daer an hanghende wesen <emph type="other"/>F 1<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 <figure/> der ghewichten FH, daer naer salmen vinden haer ermen door het 2. voorstel, dat is ghelijck F 1 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 10 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>

Line 828

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

<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> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD <figure/> 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 11. voorstel, tvastpunt daer op den as alle ghestalt houde diemen haer gheeft twelck N sy: Daer naer salmen trecken DN, ende de <arrow.to.target n="marg112"/>hanghende EO, sniende ND in O, daer naer salmen sien wat reden NO heeft tot OD, ick neme als van 1 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 1 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 <figure/> 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 <arrow.to.target n="marg112"/>hanghende EO, sniende ND in O, daer naer salmen sien wat reden NO heeft tot OD, ick neme als van 1 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 12 lb, als van 1 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 1. 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 12 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>

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

<s> <emph type="other"/>LAETA<emph.end/>BCD een pilaer sijn weghende 6 lb, diens vastpunt E, ende handthaef EF, ende G een ghewicht van 2 lb, diens vastpunt H, ende I een ghewicht van I lb, diens vastpunt K, ende den as LM sy euewydich vanden <arrow.to.target n="marg113"/>sichteinder NO, ende P sy een punt inden pilaer voor de begheerde plaets. </s><s> Daer naer werdt den as LM (alles draeyende op E) verheuen, als in de tweede form.</s>

<s> <emph type="other"/>LAETA<emph.end/>BCD een pilaer sijn weghende 6 lb, diens vastpunt E, ende handthaef EF, ende G een ghewicht van 2 lb, diens vastpunt H, ende I een ghewicht van 1 lb, diens vastpunt K, ende den as LM sy euewydich vanden <arrow.to.target n="marg113"/>sichteinder NO, ende P sy een punt inden pilaer voor de begheerde plaets. </s><s> Daer naer werdt den as LM (alles draeyende op E) verheuen, als in de tweede form.</s>

<s>

<margin.target id="marg113"/>Horizonte.

</s>

Line 849

<pb pagenum="31"/><s><emph type="other"/>TWERCK<emph.end/>.</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 <arrow.to.target n="marg114"/>hanghende ER, sniende PQ in R: siende daer naer wat reden RQ heeft tot RP, ick neem als van 1 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 vinden door het <figure/> 11.voorstel tvastpunt daerop tghegheuen alle ghestalt houdt diemen hem gheeft, twelck Q sy, daer naer salmen trecken PQ, ende de <arrow.to.target n="marg114"/>hanghende ER, sniende PQ in R: siende daer naer wat reden RQ heeft tot RP, ick neem als van 1 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 1 tot 2; Ick seg S 4 1/2 lb te wesen het begheerde ghewicht. </s>

<s>

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

</s>

<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 1. voorstel, ende veruolghens den as LM blijft in haer ghegheuen ghestalt, twelck wy bewysen moesten.</s>

<s> Tswaerste ghewicht 9 lb <figure/> 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>

Line 868

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

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A des balcx BC vastpunt, ende AB met AC twee euen ermen sijn, ende an B hanghe het rechtreckwicht D, ende an C sy het rechthefwicht E, euewichtich an D, ende sijn balck sy FG, diens vastpunt H, ende euen ermen HF, HG, ende den houck ABI, sy euen anden houck ACF. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat het rechtdaelwicht D, ende trechthefwicht E, ande euen ermen AB, AC, euen ghewelden doen. </s><s><emph type="other"/>TBEREYTSEL<emph.end/>. </s><s>Laet an C een ghewicht K hanghen, euen an D. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons weeren E, ende is blijckelijck dat de macht van D is de ermen AB, AC, in die ghegheuen ghestalt te houden, want D is euen an K, ende AB an AC. </s><s>Laet nu D weeren, ende E wederom anhanghen, ende de macht van E is oock de ermen AB, AC, in die ghegheuen ghestalt te houden, want K is euen an E, ende HF an HG, daerom E ende D doen an,euen ermen AB,AC, euen ghewelden. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A des balcx BC vastpunt, ende AB met <figure/> AC twee euen ermen sijn, ende an B hanghe het rechtreckwicht D, ende an C sy het rechthefwicht E, euewichtich an D, ende sijn balck sy FG, diens vastpunt H, ende euen ermen HF, HG, ende den houck ABI, sy euen anden houck ACF. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat het rechtdaelwicht D, ende trechthefwicht E, ande euen ermen AB, AC, euen ghewelden doen. </s><s><emph type="other"/>TBEREYTSEL<emph.end/>. </s><s>Laet an C een ghewicht K hanghen, euen an D. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons weeren E, ende is blijckelijck dat de macht van D is de ermen AB, AC, in die ghegheuen ghestalt te houden, want D is euen an K, ende AB an AC. </s><s>Laet nu D weeren, ende E wederom anhanghen, ende de macht van E is oock de ermen AB, AC, in die ghegheuen ghestalt te houden, want K is euen an E, ende HF an HG, daerom E ende D doen an,euen ermen AB,AC, euen ghewelden. </s>

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

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A des handthaefs vastpunt, ende AB met AC twee euen ermen sijn, ende an B hanghe tscheefdaelwicht D, diens scheefdaellini BE, ende an C sy tscheefhefwicht F, euen an D, ende sijn scheefheflini sy CG, ende den houck ABE, sy euen anden houck ACG. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A des handthaefs vastpunt, ende AB <figure/> met AC twee euen ermen sijn, ende an B hanghe tscheefdaelwicht D, diens scheefdaellini BE, ende an C sy tscheefhefwicht F, euen an D, ende sijn scheefheflini sy CG, ende den houck ABE, sy euen anden houck ACG. </s>

<s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat het scheefdaelwicht D, ende tscheefhefwicht F, ande euen ermen AB, AC, euen ghewelden doen. </s><s><emph type="other"/>TBEREYTSEL<emph.end/>. </s><s>Laet an C een scheefdaelwicht H <pb pagenum="33"/>hanghen euen an D, diens scheefheflini CI, <arrow.to.target n="marg115"/>euewydich sy van BE, ende CB sy wat voortghetrocken tot in K. </s><s> <emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons weeren F, ende is kennelick dat de macht van D teghen H, is de ermen AB, AC, in die ghegheuen ghestalt te houden, want D is euen an H, ende den erm AB, an AC, ende den houck ACI, anden houck KBE. </s><s>Laet nu D weeren, ende F wederom anhanghen, ende de macht van F is oock de ermen AB, AC, in die ghegheuen ghestalt te houden, ouermidts H euen is an F.</s>

<s>

Line 881

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

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A des handthaefs vastpunt, ende AB met AC twee euen ermen sijn, ende an B hanghe het scheefdaelwicht D, diens scheefdaellini BE, ende an C sy het scheefhefwicht F, euen an D, diens scheefheflini sy CG, ende den houck KCG, sy euen anden houck KBE. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat het scheefdaelwicht D, ende het scheefhefwicht F, ande euen ermen AB, AC, euen ghewelden doen. </s><s><emph type="other"/>TBEREYTSEL<emph.end/>. </s><s>Laet an C een scheefdaelwicht H hanghen euen an D, diens scheefdaellini CI, also dat den houck ACI, euen sy anden houck ABE. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons weeren F, ende is kennelick dat de macht van D teghen H, is de ermen AB, AC, in die ghegheuen ghestalt te houden, want D is euen an H, ende den erm AB an AC, ende den houck ACI, anden houck ABE. </s><s>Laet nu D weeren, ende F wederom anhanghen, ende de macht van F is oock de ermen AB, AC, in die ghegheuen ghestalt te houden, ouermits H euen is an F. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet A des handthaefs vastpunt, ende AB met AC <figure/> twee euen ermen sijn, ende an B hanghe het scheefdaelwicht D, diens scheefdaellini BE, ende an C sy het scheefhefwicht F, euen an D, diens scheefheflini sy CG, ende den houck KCG, sy euen anden houck KBE. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat het scheefdaelwicht D, ende het scheefhefwicht F, ande euen ermen AB, AC, euen ghewelden doen. </s><s><emph type="other"/>TBEREYTSEL<emph.end/>. </s><s>Laet an C een scheefdaelwicht H hanghen euen an D, diens scheefdaellini CI, also dat den houck ACI, euen sy anden houck ABE. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons weeren F, ende is kennelick dat de macht van D teghen H, is de ermen AB, AC, in die ghegheuen ghestalt te houden, want D is euen an H, ende den erm AB an AC, ende den houck ACI, anden houck ABE. </s><s>Laet nu D weeren, ende F wederom anhanghen, ende de macht van F is oock de ermen AB, AC, in die ghegheuen ghestalt te houden, ouermits H euen is an F. </s>

<s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Een daelwicht dan ende een hefwicht an hem euen, doen met euen houcken an euen ermen euen ghewelden, twelck wy bewysen moesten. </s>

Line 889

<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 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"/>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, <figure/> 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 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>

<s> <emph type="other"/>MEN<emph.end/> soude oock mueghen segghen metten cortsten VR 3, gheeft RT 2, wat den pilaer 6 lb ? comt voor Æ 4 lb als vooren, waer af de reden int volghende I5 voorstel blijken sal.

</s>

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

<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 Π vougde, inde plaets des keghels OE, als hier neuen, dat Π sal weghen 2 lb. </s>

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

<s> <emph type="other"/>OFTE<emph.end/> soomen an V een keghel 4 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>

<s> <emph type="other"/>OFTE<emph.end/> soomen an V een <figure/> keghel 4 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>

<s> <emph type="other"/>OFTE<emph.end/> soomen den pilaer ophinghe an twee euewidighe <arrow.to.target n="marg117"/> linien &OElig;R, ende ΦV, als hier neuen, dat ande lini &OElig;R hanghen sal 2 lb, ende ande lini ΦV 4 lb.</s>

<s> <emph type="other"/>OFTE<emph.end/> soomen den pilaer <figure/> ophinghe an twee euewidighe <arrow.to.target n="marg117"/> linien &OElig;R, ende ΦV, als hier neuen, dat ande lini &OElig;R hanghen sal 2 lb, ende ande lini ΦV 4 lb.</s>

<s>

<margin.target id="marg117"/>Parallelæs.

</s>

Line 915

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

<s> <emph type="other"/>SO<emph.end/> anden pilaer (tpunt R vast sijnde als vooren) eenich gewicht ofte gewichten hingen, trechthefwicht sal oock bekent worden. </s><s>Laet by voorbeelt an X hanghen 6 lb, so sal Z moeten weghen I2 lb door het 3. voorstel, ende vervolghens Æ I2 lb. </s>

<s> <emph type="other"/>SO<emph.end/> anden pilaer <figure/> (tpunt R vast sijnde als vooren) eenich gewicht ofte gewichten hingen, trechthefwicht sal oock bekent worden. </s><s>Laet by voorbeelt an X hanghen 6 lb, so sal Z moeten weghen 12 lb door het 3. voorstel, ende vervolghens Æ 12 lb. </s>

<s><emph type="other"/>VII VERTOOCH<emph.end/>. <emph type="other"/>XV VOORSTEL<emph.end/>.</s>

Line 923

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

<s> <emph type="other"/>LAET<emph.end/> ons nemen de formen des I4 voorstels, al waer blijct dat ghelijck Æ 4 lb, tot tghewicht des pilaers 6 lb, alsoo TR tot RV. </s><s>Maer om d'oirsaeck hier af <arrow.to.target n="marg118"/>Wisconstelick te verclaren, soo is te weten dat ghelijck t'ghewicht Z, tottet ghewicht des pilaers, alsoo RT tot RY door het I voorstel; Maer Æ is euen an Z, ende RV is euen an RY door tghegeuen, ghelijck dan Æ tot den pilaer, alsoo TR tot RV. </s>

<s> <emph type="other"/>LAET<emph.end/> ons nemen de formen des 14 voorstels, al waer blijct dat ghelijck Æ 4 lb, tot tghewicht des pilaers 6 lb, alsoo TR tot RV. </s><s>Maer om d'oirsaeck hier af <arrow.to.target n="marg118"/>Wisconstelick te verclaren, soo is te weten dat ghelijck t'ghewicht Z, tottet ghewicht des pilaers, alsoo RT tot RY door het I voorstel; Maer Æ is euen an Z, ende RV is euen an RY door tghegeuen, ghelijck dan Æ tot den pilaer, alsoo TR tot RV. </s>

<s>

<margin.target id="marg118"/>Mathematicè.

</s>

Line 934

<s> <emph type="other"/>WESENDE<emph.end/> twee punten inden as des pilaers, t'een vast t'ander verroerlick: Trechthefwicht an tverroerlick dat den pilaer in een ghestalt houdt, sal hem in alle ghestalten houden. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ons den pilaer met sijn ghewichten des 14. voorstels wat verkeeren op tvastpunt R, ende dat Æ 4 lb noch sy rechthefwicht, alsoo dat dan alles van gestalt sy als hier neuen. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ons den pilaer met sijn ghewichten des 14. <figure/> voorstels wat verkeeren op tvastpunt R, ende dat Æ 4 lb noch sy rechthefwicht, alsoo dat dan alles van gestalt sy als hier neuen. </s>

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

Line 944

<s> <emph type="other"/>RVSTENDE<emph.end/> een pilaer op twee punten inden as: Ghelijck het asstick tusschen t'swaerheyts middelpunt ende tslinckerpunt, tottet asstick tusschen tswaerheyts middelpunt ende trechterpunt, alsoo tghewicht des pilaers rustende op trechterpunt, tottet ghewicht rustende op tslinckerpunt. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, ghedeelt als int 1. voorstel, rustende met de twee punten R, V, op de punten van &OElig;, Æ. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck het asstick TR, tottet asstick TV, also tghewicht rustende mettet punt V op tpunt van Æ, tottet tghewicht rustende mettet tpunt R op tpunt van &OElig;. </s><s> <emph type="other"/>TBEWYS<emph.end/>. </s><s>TR is dobbel an TV door tghewstelde, <pb pagenum="38"/>ende opt tpunt van Æ rust 4 lb, ende van &OElig; 2 lb door 1. veruolgh des 14. voorstels, maer 4 lb is tot 2 lb oock dobbel, ghelijck dan TR tot TV, alsoo tghewicht rustende op tpunt van Æ, tot tghewicht rustende op tpunt van &OElig;.</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, ghedeelt als int 1. voorstel, rustende met de twee punten R, V, op de punten van &OElig;, Æ. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck het asstick TR, tottet asstick TV, also tghewicht rustende mettet punt V op tpunt van Æ, tottet tghewicht rustende mettet tpunt R op tpunt van &OElig;. </s><s> <emph type="other"/>TBEWYS<emph.end/>. </s><s>TR is dobbel an TV door tghewstelde, <pb pagenum="38"/>ende opt tpunt van Æ rust 4 <figure/> lb, ende van &OElig; 2 lb door 1. veruolgh des 14. voorstels, maer 4 lb is tot 2 lb oock dobbel, ghelijck dan TR tot TV, alsoo tghewicht rustende op tpunt van Æ, tot tghewicht rustende op tpunt van &OElig;.</s>

<s> Maer om tghemeen nootsaeckelick veruolgh in allen te bewysen, laet ons voorttrecken VR tot in Z, also dat RZ euen sy an RV; aensiende daer naer R voor tvastpunt, so sal an Z moeten hanghen Π 4 lb, om de pilaer in die ghestalt te houden door het 3. voorstel. </s><s>Maer tghene an V den pilaer in die ghestalt houdt als Æ, doet daer an alsulcken gheweldt als Π, door het 13 voorstel; An Æ dan rust een ghewicht euen an Π. </s><s>Laet ons insghelicx voorttrecken, RV tot in Φ, also dat VΦ euen sy an VR, ansiende daer naer V voor vastpunt, soo sal an Φ moeten hanghen Δ 2 lb, om den pilaer in die ghestalt te houden door het 3. voorstel, maer tghene an R den pilaer in die ghestalt houdt als &OElig;, doet daeran alsulcke ghewelt als Δ door het 13 voorstel, An &OElig; dan rust een ghewicht euen an Δ. </s><s>Nu anghesien Π euestaltwichtich is teghen den pilaer op tghemeen vastpunt R, so heeft den erm TR, sulcken reden tot den erm RZ, als Π tot den pilaer door 1. voorstel. </s><s>Insghelijcx nemende V voor tvastpunt, soo heeft den erm TV sulcken reden tot den erm VΦ, als Δ tot den pilaer, maer RZ is altijt euen an VΦ: Wy hebben hier dan twee <arrow.to.target n="marg119"/>eueredenheden elck van vier <arrow.to.target n="marg120"/>palen, welcker tweede palen an malcanderen euen sijn, ende welcker laeste palen an malcanderen oock euen sijn. </s><s> Maer alle twee eueredenheden elck van vier palen, welcker tweede palen an mal<pb pagenum="39"/>cander euen sijn, ende welcker laetste palen an malcander oock euen sijn, die hebben dander palen oock euerednich, daerom ghelijck TR tot TV, alsoo Π tot Δ; maer Π is euen an tghewicht des pilaers rustende met tpunt V op tpunt Æ, ende tghewicht Δ is euen an t'ghewicht des pilaers rustende met tpunt R op tpunt van &OElig;, daerom ghelijck TR tot TV, also tghewicht rustende mettet tpunt V op tpunt van Æ, tottet ghewicht rustende mettet punt R op tpunt van &OElig;. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Rustende dan een pilaer op twee punten inden as, &c. </s>

<s> Maer om tghemeen nootsaeckelick <figure/> veruolgh in allen te bewysen, laet ons voorttrecken VR tot in Z, also dat RZ euen sy an RV; aensiende daer naer R voor tvastpunt, so sal an Z moeten hanghen Π 4 lb, om de pilaer in die ghestalt te houden door het 3. voorstel. </s><s>Maer tghene an V den pilaer in die ghestalt houdt als Æ, doet daer an alsulcken gheweldt als Π, door het 13 voorstel; An Æ dan rust een ghewicht euen an Π. </s><s>Laet ons insghelicx voorttrecken, RV tot in Φ, also dat VΦ euen sy an VR, ansiende daer naer V voor vastpunt, soo sal an Φ moeten hanghen Δ 2 lb, om den pilaer in die ghestalt te houden door het 3. voorstel, maer tghene an R den pilaer in die ghestalt houdt als &OElig;, doet daeran alsulcke ghewelt als Δ door het 13 voorstel, An &OElig; dan rust een ghewicht euen an Δ. </s><s>Nu anghesien Π euestaltwichtich is teghen den pilaer op tghemeen vastpunt R, so heeft den erm TR, sulcken reden tot den erm RZ, als Π tot den pilaer door 1. voorstel. </s><s>Insghelijcx nemende V voor tvastpunt, soo heeft den erm TV sulcken reden tot den erm VΦ, als Δ tot den pilaer, maer RZ is altijt euen an VΦ: Wy hebben hier dan twee <arrow.to.target n="marg119"/>eueredenheden elck van vier <arrow.to.target n="marg120"/>palen, welcker tweede palen an malcanderen euen sijn, ende welcker laeste palen an malcanderen oock euen sijn. </s><s> Maer alle twee eueredenheden elck van vier palen, welcker tweede palen an mal<pb pagenum="39"/>cander euen sijn, ende welcker laetste palen an malcander oock euen sijn, die hebben dander palen oock euerednich, daerom ghelijck TR tot TV, alsoo Π tot Δ; maer Π is euen an tghewicht des pilaers rustende met tpunt V op tpunt Æ, ende tghewicht Δ is euen an t'ghewicht des pilaers rustende met tpunt R op tpunt van &OElig;, daerom ghelijck TR tot TV, also tghewicht rustende mettet tpunt V op tpunt van Æ, tottet ghewicht rustende mettet punt R op tpunt van &OElig;. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Rustende dan een pilaer op twee punten inden as, &c. </s>

<s>

<margin.target id="marg119"/>Proportiones.

<margin.target id="marg120"/>Terminis.

Line 963

<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> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet <figure/> 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 17. 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 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> nochtans om alhier wat breeder <figure/> 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 17. 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>

Line 986

<s> Ghelijck des driehoucx rechter sijde tot de slincker, also t' des cloots op de slincker sijde, tottet staltwicht des cloots op de rechter sijde. </s>

<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 BC 1, alsoo t'staltwicht des cloots E, tottet staltwicht des cloots D. </s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck wesen diens plat sy <figure/> 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 BC 1, 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 <arrow.to.target n="marg126"/>middelpunten, 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, <arrow.to.target n="marg127"/>euewydich 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> <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 <arrow.to.target n="marg126"/>middelpunten, 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, <arrow.to.target n="marg127"/>euewydich 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, 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.

Line 998

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

<s><emph type="other"/>LAET<emph.end/> ABC een driehouck sijn als vooren, wiens sijde AB dobbel sy an BC, ende laet op AB ligghen een cloot D, ende op de sijde BC een cloot E euewichtich anden helft van D, ende an F sy een vastpunt daer ouer de lini DFE. </s><s>(te weten uyt het <arrow.to.target n="marg128"/>middelpunt des cloots D ouer F tot int middelpunt des cloots E) slieren mach, alsoo dat DF euewydich blijue van AB, ende FE van BC. </s><s>Dit also sijnde, anghesien de vier clooten P, Q, R, D, hier vooren, euestaltwichtich waren met de twee clooten E, F, so sal desen cloot D, euestaltwichtich sijn teghen de cloot E: want ghelijck die P, Q, R, D, tot E, F, also dese D tot E: Daerom ghelijck de lini AB, tot BC, alsoo den cloot D tot den cloot E.</s>

<s><emph type="other"/>LAET<emph.end/> ABC een driehouck sijn als vooren, wiens sijde AB dobbel <figure/> sy an BC, ende laet op AB ligghen een cloot D, ende op de sijde BC een cloot E euewichtich anden helft van D, ende an F sy een vastpunt daer ouer de lini DFE. (te weten uyt het <arrow.to.target n="marg128"/>middelpunt des cloots D ouer F tot int middelpunt des cloots E) slieren mach, alsoo dat DF euewydich blijue van AB, ende FE van BC. </s><s>Dit also sijnde, anghesien de vier clooten P, Q, R, D, hier vooren, euestaltwichtich waren met de twee clooten E, F, so sal desen cloot D, euestaltwichtich sijn teghen de cloot E: want ghelijck die P, Q, R, D, tot E, F, also dese D tot E: Daerom ghelijck de lini AB, tot BC, alsoo den cloot D tot den cloot E.</s>

<s>

<margin.target id="marg128"/>Centro.

</s>

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

<s> <emph type="other"/>LAET<emph.end/> ons nu d'een sijde des driehouckx als BC (ande welcke AB dobbel is) rechthouckich stellen op AC als hier neuen; Ende den cloot D die dobbel is an E, sal noch met E euestaltwichtich sijn, want ghelijck AB tot BC, also den cloot D tot den cloot E. </s>

<s> <emph type="other"/>LAET<emph.end/> ons nu d'een sijde des driehouckx <figure/> als BC (ande welcke AB dobbel is) rechthouckich stellen op AC als hier neuen; Ende den cloot D die dobbel is an E, sal noch met E euestaltwichtich sijn, want ghelijck AB tot BC, also den cloot D tot den cloot E. </s>

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

<s> <emph type="other"/>LAET<emph.end/> ons nu inde plaets van t'punt F, stellen een caterol als hier neuen, also dat de scheefheflini van D naer F euewydich blijue van AB, ende inde plaets vanden cloot E sy eenich wicht van form soot valt, maer euewichtich anden cloot E: t'selue is noch euestaltwichtich met D, Daerom ghelijck AB tot BC, alsoo noch den cloot D tottet ghewicht E. </s>

<s> <emph type="other"/>LAET<emph.end/> ons nu inde plaets <figure/> van t'punt F, stellen een caterol als hier neuen, also dat de scheefheflini van D naer F euewydich blijue van AB, ende inde plaets vanden cloot E sy eenich wicht van form soot valt, maer euewichtich anden cloot E: t'selue is noch euestaltwichtich met D, Daerom ghelijck AB tot BC, alsoo noch den cloot D tottet ghewicht E. </s>

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

<s> <emph type="other"/>ANGHESIEN<emph.end/> den cloot des 3. veruolgs naect de lini AB, in t'punt G, als vastpunt, soo sal den as GH rechthouckich sijn op AB {}; Daerom laet ons weren den cloot, ende stellen in die plaets den pilaer D euewichtich met den cloot, alsoo dat den as GH (diens vastpunt G) rechthouckich sy op AB, ende de scheefheflini tusschen DF noch euewydich van AB, ende sniende de sijde des pilaers in I, Als hier neuens. </s><s> Ende is openbaer dat ghelijck AB tot BC, (dat is dobbel als vooren) also den pilaer D tottet t'ghewicht E. </s>

<s> <emph type="other"/>ANGHESIEN<emph.end/> den cloot des 3. veruolgs naect de lini AB, in <figure/> t'punt G, als vastpunt, soo sal den as GH rechthouckich sijn op AB {}; Daerom laet ons weren den cloot, ende stellen in die plaets den pilaer D euewichtich met den cloot, alsoo dat den as GH (diens vastpunt G) rechthouckich sy op AB, ende de scheefheflini tusschen DF noch euewydich van AB, ende sniende de sijde des pilaers in I, Als hier neuens. </s><s> Ende is openbaer dat ghelijck AB tot BC, (dat is dobbel als vooren) also den pilaer D tottet t'ghewicht E. </s>

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

<s><emph type="other"/>LAET<emph.end/> ons trecken de hanghende lini uyt het <arrow.to.target n="marg129"/>middelpunt 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><emph type="other"/>LAET<emph.end/> ons trecken <figure/> de hanghende lini uyt het <arrow.to.target n="marg129"/>middelpunt 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>

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

<s><emph type="other"/>LAET<emph.end/> ons trecken BN, sniende de voortghetrocken AC in N: Insghelijcx DO, sniende de voortghetrocken LI dat is de sijde des pilaers in O, ende also dat den houck IDO, euen sy anden houck CBN. </s><s>Laet ons oock voughen an DO t'scheefhefwicht P, dat den pilaer (de ghewichten M, E gheweert sijnde) in die standthaude. </s><s> Nu anghesien DL, des driehouckx DLI, <arrow.to.target n="marg130"/>lijckstandighe is met BA des driehouckx BAC, ende DI met BC, men besluyt daer uyt aldus: Ghelijck BA tot BC, also t'staltwicht van BA tottet staltwicht van BC (door het 2. vervolg) Ende oock ghelijck DL tot DI also t'staltwicht van DL tot t'staltwicht van DI, dat is alsoo M tot E. </s><s>Maer de lijckstandighe linien van dese ghelijcke driehoucken ABN, LDO, sijn BA met DL, ende BN met DO, Daerom segghen wy als vooren, Ghelijck BA tot BN, alsoo het staltwicht van BA tot het staltwicht van BN (door het 1. vervolgh) Ende oock ghelijck DL tot DO, also het staltwicht van DL tot het staltwicht van DO, dat is also M tot P. </s><s>Maer by aldien de lini BN, ghetrocken waer van B af ouer d'ander sijde van BC, so soude de lini DO, dan oock vallen van D ouer d'ander sijde van DI, dat is, daer DO nu valt onder DI, sy souder dan bouen vallen, ende t'voorgaende bewys soude oock dienen tot sulcke ghestalt, te weten, dat wy noch segghen souden, ghelijck BA tot BN, alsoo t'staltwicht van BA, tottet staltwicht van BN; Ende ghelijck DL tot DO, alsoo t'staltwicht van DL, tottet staltwicht van DO, dat is, also M tot P. </s><s>Inder voughen dat dese <arrow.to.target n="marg131"/>eueredenheydt niet alleen en bestaet inde voorbeelden, alwaer de heflini als DI rechthouckich is op den as, maer op allen houcken. </s>

<s><emph type="other"/>LAET<emph.end/> ons trecken BN, sniende de voortghetrocken AC in N: Insghelijcx DO, <figure/> sniende de voortghetrocken LI dat is de sijde des pilaers in O, ende also dat den houck IDO, euen sy anden houck CBN. </s><s>Laet ons oock voughen an DO t'scheefhefwicht P, dat den pilaer (de ghewichten M, E gheweert sijnde) in die standthaude. </s><s>Nu anghesien DL, des driehouckx DLI, <arrow.to.target n="marg130"/>lijckstandighe is met BA des driehouckx BAC, ende DI met BC, men besluyt daer uyt aldus: Ghelijck BA tot BC, also t'staltwicht van BA tottet staltwicht van BC (door het 2. vervolg) Ende oock ghelijck DL tot DI also t'staltwicht van DL tot t'staltwicht van DI, dat is alsoo M tot E. </s><s>Maer de lijckstandighe linien van dese ghelijcke driehoucken ABN, LDO, sijn BA met DL, ende BN met DO, Daerom segghen wy als vooren, Ghelijck BA tot BN, alsoo het staltwicht van BA tot het staltwicht van BN (door het 1. vervolgh) Ende oock ghelijck DL tot DO, also het staltwicht van DL tot het staltwicht van DO, dat is also M tot P. </s><s>Maer by aldien de lini BN, ghetrocken waer van B af ouer d'ander sijde van BC, so soude de lini DO, dan oock vallen van D ouer d'ander sijde van DI, dat is, daer DO nu valt onder DI, sy souder dan bouen vallen, ende t'voorgaende bewys soude oock dienen tot sulcke ghestalt, te weten, dat wy noch segghen souden, ghelijck BA tot BN, alsoo t'staltwicht van BA, tottet staltwicht van BN; Ende ghelijck DL tot DO, alsoo t'staltwicht van DL, tottet staltwicht van DO, dat is, also M tot P. </s><s>Inder voughen dat dese <arrow.to.target n="marg131"/>eueredenheydt niet alleen en bestaet inde voorbeelden, alwaer de heflini als DI rechthouckich is op den as, maer op allen houcken. </s>

<s>

<margin.target id="marg130"/>Homologa.

<margin.target id="marg131"/>Proportio.

</s>

<s><emph type="other"/>T'VOORGAENDE<emph.end/> mach oock verstaen worden van een cloot ligghende op een lini AB als hier neuens, alwaer wy segghen als vooren, gelijck LD tot DO, alsoo M tot P (welverstaende dat CL <pb pagenum="45"/>rechthouckich ghetrocken is op AB, dat is euewydich met den as GH des cloots D) maer t'ghewicht M is euen an den cloot D, daerom segghen wy ghelijck LD tot DO, also t'ghewicht des cloots, tot P. </s><s>Maer want LD ende DO binnen t'lichaem des cloots niet bequamelick en connen beschreuen worden, so laet ons trecken de hangende CE, ende sullen dan hebben buyten t'lichaem een driehouck CEO, ghelijck anden driehouck LDO, welcker <arrow.to.target n="marg132"/>lijckstandighe sijden sijn LD met CE, ende DO met EO, daerom ghelijck LD tot DO, alsoo CE tot EO, ende veruolghens ghelijck CE tot EO, alsoo t'ghewicht des cloots, tot P. </s>

<s><emph type="other"/>T'VOORGAENDE<emph.end/> mach oock verstaen worden van een cloot ligghende op een lini AB als hier neuens, alwaer wy segghen als vooren, gelijck LD tot DO, alsoo M tot P (welverstaende dat CL <pb pagenum="45"/>rechthouckich ghetrocken is op AB, dat is euewydich met den as <figure/> GH des cloots D) maer t'ghewicht M is euen an den cloot D, daerom segghen wy ghelijck LD tot DO, also t'ghewicht des cloots, tot P. </s><s>Maer want LD ende DO binnen t'lichaem des cloots niet bequamelick en connen beschreuen worden, so laet ons trecken de hangende CE, ende sullen dan hebben buyten t'lichaem een driehouck CEO, ghelijck anden driehouck LDO, welcker <arrow.to.target n="marg132"/>lijckstandighe sijden sijn LD met CE, ende DO met EO, daerom ghelijck LD tot DO, alsoo CE tot EO, ende veruolghens ghelijck CE tot EO, alsoo t'ghewicht des cloots, tot P. </s>

<s>

<margin.target id="marg132"/>Homologa.

</s>

<s><emph type="other"/>LAET<emph.end/> ons nu tot meerder claerheydt dit alleen stellen sonder d'ander linien als hier neuen, alwaer wy segghen ghelijck CE tot EO, also t'ghewicht des cloots D, tot P. </s>

<s><emph type="other"/>LAET<emph.end/> ons nu tot meerder claerheydt <figure/> dit alleen stellen sonder d'ander linien als hier neuen, alwaer wy segghen ghelijck CE tot EO, also t'ghewicht des cloots D, tot P. </s>

<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"/>ENDE<emph.end/> dit niet alleen van clooten <figure/> 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 <arrow.to.target n="marg133"/>sichteinder 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 (<arrow.to.target n="marg134"/>wisconstelick 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 vlieghesterctens 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 <figure/> <arrow.to.target n="marg133"/>sichteinder 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 (<arrow.to.target n="marg134"/>wisconstelick 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. en behouuen gheen vlieghesterctens 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.

</s>

<s><emph type="other"/>MAER<emph.end/> anghesien den driehouck ABN int 6. veruolg, tot dese <arrow.to.target n="marg135"/>eueredenheyt 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 <figure/> driehouck ABN int 6. veruolg, tot dese <arrow.to.target n="marg135"/>eueredenheyt 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>

Line 1060

</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 <arrow.to.target n="marg138"/>euewydich sy van AB, ende QP euewydich van BC. </s><s> Ende om de redenen die int 19. voorstel vande 14 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 <figure/> 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 <arrow.to.target n="marg138"/>euewydich sy van AB, ende QP euewydich van BC. </s><s> Ende om de redenen die int 19. voorstel vande 14 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>

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

<s> <emph type="other"/>LAET<emph.end/> ons nu an I des 7. veruolgs voughen trechthefwicht R euestaltwichtich met den pilaer, diens rechtheflini sy IS, sniende de sijde des pilaers in T, ende IQ snie de sijde des pilaers in V, ende laet an de lini PQ hanghen een ghewicht X, inde plaets vanden pilaer KL, t'welck euen sy anden helft van t'staltwicht des selfden pilaers KL, Laet ons oock weeren den driehouck ABC, ende den pilaer DE doen rusten op t'punt H als hier neuen. </s><s> Ende om de redenen als vooren, ghelijck TI tot IV, alsoo R tot X. </s><s>Ende dit niet alleen als IV rechthouckich is op den as FG, maer cromhouckich soot valt, waeraf men besondet bethooch soude mueghen doen, maer tis openbaer ghenouch door het 6. veruolgh.</s>

<s> <emph type="other"/>LAET<emph.end/> ons nu an I des 7. veruolgs voughen trechthefwicht R euestaltwichtich <figure/> met den pilaer, diens rechtheflini sy IS, sniende de sijde des pilaers in T, ende IQ snie de sijde des pilaers in V, ende laet an de lini PQ hanghen een ghewicht X, inde plaets vanden pilaer KL, t'welck euen sy anden helft van t'staltwicht des selfden pilaers KL, Laet ons oock weeren den driehouck ABC, ende den pilaer DE doen rusten op t'punt H als hier neuen. </s><s> Ende om de redenen als vooren, ghelijck TI tot IV, alsoo R tot X. </s><s>Ende dit niet alleen als IV rechthouckich is op den as FG, maer cromhouckich soot valt, waeraf men besondet bethooch soude mueghen doen, maer tis openbaer ghenouch door het 6. veruolgh.</s>

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

Line 1076

<margin.target id="marg139"/>Proportionem.

</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, <arrow.to.target n="marg140"/>euewydich 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 <arrow.to.target n="marg141"/>swaerheyts middelpunt 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> Laet AB een pilaer <figure/> 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, <arrow.to.target n="marg140"/>euewydich 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 <arrow.to.target n="marg141"/>swaerheyts middelpunt 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. 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.

Line 1086

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

<s> <emph type="other"/>LAET<emph.end/> ons stellen een form ghelijck an die des 9. veruolghs, alleen daer in verschillende dat dese FH wyct ouer d'ander sijde van FI, ende dat den houck HFC, euen sy anden houck RPO, waer duer G anden pilaer AM euen soo grooten ghewelt doet als Q, ende om de redenen des 9 veruolgs (die wy om cortheyt ouerslaen) G doet euen sulcken ghewelt anden pilaer AB, als Q anden pilaer LM; Nu ghelijck TP tot PR, alsoo S tot Q door het 9 vervolgh, maer IF is euen an TP, ende FH an PR, ende K an S, ende G an Q, daerom ghelijck IF tot FH, also K tot G.</s>

<s> <emph type="other"/>LAET<emph.end/> ons stellen een form ghelijck an die des 9. veruolghs, alleen <figure/> daer in verschillende dat dese FH wyct ouer d'ander sijde van FI, ende dat den houck HFC, euen sy anden houck RPO, waer duer G anden pilaer AM euen soo grooten ghewelt doet als Q, ende om de redenen des 9 veruolgs (die wy om cortheyt ouerslaen) G doet euen sulcken ghewelt anden pilaer AB, als Q anden pilaer LM; Nu ghelijck TP tot PR, alsoo S tot Q door het 9 vervolgh, maer IF is euen an TP, ende FH an PR, ende K an S, ende G an Q, daerom ghelijck IF tot FH, also K tot G.</s>

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

<s><emph type="other"/>LAET<emph.end/> ons stellen een form ghelijck an die des I0 veruolgs, alleen daer in verschillende dat dese PR wyckt ouer d'ander sijde van PT, ende dat PR euewydich sy met FH, waer duer Q anden pilaer AM, euen soo grooten ghewelt doet als G, ende om de redenen des 9 veruolghs, Q doet euen sulcken gheweldt anden pilaer LM, als G anden pilaer AB; Nu ghelijck IF tot FH, also K tot G door het 6 veruolgh: Maer TP is euen an IF, ende PR an FH, ende S an K, ende Q an G, daerom ghelijck TP tot PR, also S tot Q. </s><s>Ende inder seluer voughen salmen vanden anderen ghestalten door haer contrarien altijt dese eueredenheyt bewysen. </s>

<s><emph type="other"/>LAET<emph.end/> ons stellen een form ghelijck an die des 10 veruolgs, alleen <figure/> daer in verschillende dat dese PR wyckt ouer d'ander sijde van PT, ende dat PR euewydich sy met FH, waer duer Q anden pilaer AM, euen soo grooten ghewelt doet als G, ende om de redenen des 9 veruolghs, Q doet euen sulcken gheweldt anden pilaer LM, als G anden pilaer AB; Nu ghelijck IF tot FH, also K tot G door het 6 veruolgh: Maer TP is euen an IF, ende PR an FH, ende S an K, ende Q an G, daerom ghelijck TP tot PR, also S tot Q. </s><s>Ende inder seluer voughen salmen vanden anderen ghestalten door haer contrarien altijt dese eueredenheyt bewysen. </s>

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

<s><emph type="other"/>MAER<emph.end/> dat dese <arrow.to.target n="marg142"/>eueredenheydt oock bestaet inde ghestalt daer den as euewydich is vanden <arrow.to.target n="marg143"/>sichteinder, wort aldus bethoont: Laet AB een pilaer sijn, diens as CD <arrow.to.target n="marg144"/>euewydich 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 1 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 <arrow.to.target n="marg145"/>wisconstlick sal blijcken door t'volghende 22 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 <arrow.to.target n="marg142"/>eueredenheydt oock bestaet inde ghestalt daer <figure/> den as euewydich is vanden <arrow.to.target n="marg143"/>sichteinder, wort aldus bethoont: Laet AB een pilaer sijn, diens as CD <arrow.to.target n="marg144"/>euewydich 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 1 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 <arrow.to.target n="marg145"/>wisconstlick 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.

Line 1110

<s> <emph type="other"/>WESENDE<emph.end/> inden as des pilaers een vastpunt, ende een roerlick, daer an hy door een rechthefwicht ende scheefhefwicht in seker standt ghehouden wort: Ghelijck rechtheflini tot scheefheflini, also rechthefwicht tot scheefhefwicht. </s><s> </s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn diens as CD, ende t'vast<pb pagenum="51"/>punt E, ende roerlick punt F, daeran den pilaer door t'rechthefwicht G in die ghestalt ghehouden wort, daer an oock den pilaer door t'scheefhefwicht H (welverstaende G gheweert sijnde) in die ghestalt ghehouden wort, ende de rechtheflini snie de sijde des pilaers in I, maer de scheefheflini snie de selue sijde in K: Ick seg dat ghelijck de rechtheflini IF, tot de scheefheflini FK, alsoo trechthefwicht G, tot het scheefhefwicht H, waer af t'bewys uyt de voorgaende openbaer is.</s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn diens as CD, ende t'vast<pb pagenum="51"/>punt E, ende roerlick punt F, daeran den pilaer door t'rechthefwicht G <figure/> in die ghestalt ghehouden wort, daer an oock den pilaer door t'scheefhefwicht H (welverstaende G gheweert sijnde) in die ghestalt ghehouden wort, ende de rechtheflini snie de sijde des pilaers in I, maer de scheefheflini snie de selue sijde in K: Ick seg dat ghelijck de rechtheflini IF, tot de scheefheflini FK, alsoo trechthefwicht G, tot het scheefhefwicht H, waer af t'bewys uyt de voorgaende openbaer is.</s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan inden as des pilaers een vastpunt, &c. </s>

Line 1120

<s> <emph type="other"/>WESENDE<emph.end/> inden as des pilaers een vastpunt, ende een roerlick, daer an hy door een rechtdaelwicht ende scheefdaelwicht in seker standt gehouden wort: Ghelijck rechtdaellini tot scheefdaellini, also rechtdaelwicht tot scheefdaelwicht. </s>

<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"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn, diens as CD, ende vastpunt <figure/> 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 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>

Line 1132

<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 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'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, <figure/> 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 14. voorstel, van 4 lb, daer naer salmen ondersoucken wat reden eenighe <arrow.to.target n="marg146"/>hanghende lini als ZÆ, heeft tot Z&OElig;, ick neme als van 2 tot I, daer uyt seg ick 2 gheeft 1, 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 <arrow.to.target n="marg147"/>lijckstandighe linien 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; 1, 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>

Line 1161

<s> <emph type="other"/>EVEN<emph.end/> ghewichten der trecklinien van een selfde punt des as, ende op verscheyden sijden met den as euen houcken makende; doen anden pilaer euen ghewelden. </s>

<s> <emph type="other"/>TGHE<emph.end/>. </s><s>Laet AB een pilaer sijn diens as CD, ende vastpunt daer in E, en t'roerlick punt F, an t'welck een scheefhefwicht G sy, dat den pilaer in die ghestalt houde, ende diens scheefheflini FH. </s><s>Laet oock an t'selue punt F gheuoucht wesen een scheefhefwicht I, ouer d'ander sijde, ende met G euewichtich, ende diens scheefheflini FK, den houck KFD euen make anden houck HFC. </s>

<s> <emph type="other"/>TGHE<emph.end/>. </s><s>Laet AB een pilaer sijn diens as CD, ende vastpunt daer in E, <figure/> en t'roerlick punt F, an t'welck een scheefhefwicht G sy, dat den pilaer in die ghestalt houde, ende diens scheefheflini FH. </s><s>Laet oock an t'selue punt F gheuoucht wesen een scheefhefwicht I, ouer d'ander sijde, ende met G euewichtich, ende diens scheefheflini FK, den houck KFD euen make anden houck HFC. </s>

<s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat I anden pilaer euen sulcken ghewelt doet als G, te weten dat I (G gheweert sijnde) den pilaer oock in die ghestalt sal houden. </s>

Line 1188

<s>I. </s><s>I is hefwicht dat minder reden heeft tot L, dan siin heflini KF tot FM:</s>

<s>I. </s><s>T'hefwicht I dan is te licht om den pilaer in die ghestalt te houden.</s>

<s> <emph type="other"/>DES<emph.end/> <arrow.to.target n="marg153"/>bewyfredens tweede voorstel wort aldus bethoont, T'ghewicht G (t'welck den pilaer in die ghestalt houdt) heeft sulcken reden tot L, als HF tot FM, maer I is euen an G, ende KF is meerder dan FH <arrow.to.target n="marg154"/>, daerom I heeft minder reden tot L, dan KF tot FM, waer duer soo wy bouen gheseyt hebben, t'ghewicht I is te licht om den pilaer in die ghestalt te houden; maer G cander hem in houden, G dan doet anden pilaer meerder ghewelt dan I. </s><s>Maer dat G daer an gheen meerder doen en can, is daer uyt openbaer, dat van F op de sijde des pilaers gheen corter lini en can ghetrocken worden dan FH, anghesien sy daer op rechthouckich is. </s>

<s> <emph type="other"/>DES<emph.end/> <arrow.to.target n="marg153"/>bewyfredens tweede voorstel wort aldus bethoont, T'ghewicht <figure/> G (t'welck den pilaer in die ghestalt houdt) heeft sulcken reden tot L, als HF tot FM, maer I is euen an G, ende KF is meerder dan FH <arrow.to.target n="marg154"/>, daerom I heeft minder reden tot L, dan KF tot FM, waer duer soo wy bouen gheseyt hebben, t'ghewicht I is te licht om den pilaer in die ghestalt te houden; maer G cander hem in houden, G dan doet anden pilaer meerder ghewelt dan I. </s><s>Maer dat G daer an gheen meerder doen en can, is daer uyt openbaer, dat van F op de sijde des pilaers gheen corter lini en can ghetrocken worden dan FH, anghesien sy daer op rechthouckich is. </s>

<s>

<margin.target id="marg153"/>Syllogismi.

<margin.target id="marg154"/>47.v.i.b. E.

Line 1206

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

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn hanghende ande twee 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>Den houck FEC, ofte IEC, ofte HEC, is al een selfden houck, also oock is DCE, ofte ICE, ofte GCE, daerom wat punten wy inde linien HE, ende CG voor uytersten nemen, den pilaer houdt daer an sijn ghegheuen standt. </s><s>Laet ons nemen I, ghemeen uyterste punt van d'een ende d'ander lini, den pilaer dan houdt daer an sijn ghegeuen standt. </s><s>Maer hanghende den pilaer an t'punt I, so is de <arrow.to.target n="marg158"/>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 twee <figure/> 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>Den houck FEC, ofte IEC, ofte HEC, is al een selfden houck, also oock is DCE, ofte ICE, ofte GCE, daerom wat punten wy inde linien HE, ende CG voor uytersten nemen, den pilaer houdt daer an sijn ghegheuen standt. </s><s>Laet ons nemen I, ghemeen uyterste punt van d'een ende d'ander lini, den pilaer dan houdt daer an sijn ghegeuen standt. </s><s>Maer hanghende den pilaer an t'punt I, so is de <arrow.to.target n="marg158"/>hanghende door I des pilaers swaerheydts middellini inde welcke I is.</s>

<s>

<margin.target id="marg158"/>Perpendicularis.

</s>

<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 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 (<arrow.to.target n="marg159"/>Wisconstlick 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 <figure/> standt. </s><s>Laet ons nemen I, ghemeen uyterste punt van d'een en d'ander lini; den pilaer dan houdt daer op (<arrow.to.target n="marg159"/>Wisconstlick 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>

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

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn welcke in die standt ghehouden wort door de scheefdaellini CD, ende scheefheflini EF, de selue sijn voortghetrocken tot G, H, sniende malcanderen in I. </s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn welcke in die standt <figure/> ghehouden wort door de scheefdaellini CD, ende scheefheflini EF, de selue sijn voortghetrocken tot G, H, sniende malcanderen in I. </s>

<s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat I inde swaerheydts middellini is des pilaers AB. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons GC ansien voor styl, ofte stijue lini ende nemen dat de macht die an D int neertrecken was, nu neerstekende sy in yder punt tusschen C en G daermen haer stelt, ende den pilaer AB, sal alsoo op allen punten diemen tusschen C, G ende E, H voor uytersten neemt, sijn ghegheuen standt houden. </s><s>Laet ons nemen I ghemeen uyterste van d'een en d'ander lini, den pilaer dan houdt daer an sijn ghegheuen standt; maer hanghende den pilaer an t'punt I, de hanghende door I is des pilaers swaerheyts middellini, inde welcke I is. </s>

Line 1228

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn, welcke in die standt ghehouden wort door de scheefdaellini CD, ende de scheefheflini EF, de selue sijn voortghetrocken tot <emph type="other"/>GH<emph.end/>, sniende malcanderen in I. </s>

<s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat I inde swaerheyts mid<pb pagenum="59"/>dellini is des pilaers AB. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons HE ansien voor stijl, ofte stiue lini, ende nemen dat de macht die an E int opheffen was, nu opstekende sy in yder punt tusschen E en H, daermen haer stelt, ende den pilaer AB sal also op allen punten diemen tusschen CG ende EH voor uytersten neemt, sijn ghegheuen standt houden. </s><s>Laet ons nu nemen I ghemeen uyterste punt van d'een en dander lini, den pilaer dan houdt daer op 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'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat I inde swaerheyts mid<pb pagenum="59"/>dellini is des pilaers AB. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons HE ansien voor stijl, <figure/> ofte stiue lini, ende nemen dat de macht die an E int opheffen was, nu opstekende sy in yder punt tusschen E en H, daermen haer stelt, ende den pilaer AB sal also op allen punten diemen tusschen CG ende EH voor uytersten neemt, sijn ghegheuen standt houden. </s><s>Laet ons nu nemen I ghemeen uyterste punt van d'een en dander lini, den pilaer dan houdt daer op 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'BESLVYT<emph.end/>. </s><s>Twee oneuewydighe linien dan, daer een pilaer an hangt beyde oneyndelick voortghetrocken, snien malcanderen inde swaerheydts middellini des pilaers, t'welck wy bewysen moesten.</s>

Line 1239

<margin.target id="marg160"/>Horizontem.

</s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn hanghende an twee linien, d'een CD rechthouckich op den sichteinder, d'ander EF (soot mueghelick waer) scheefhouckich, ende GH sy des pilaers swaerheydts middellini. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen t'hinhoudt des voorstels. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet CD ende EF voortghetrocken worden, sniende malcander in I. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Soo den pilaer in die ghestalt blijft hanghende ande linien CD, EF, sy sal op alle vastpunten in die voortghetrocken linien de selue ghestalt houden, ouermidts de houcken ICE, ende IEC, niet en veranderen: </s><s> Daerom ghenomen I <pb pagenum="60"/>ghemeen vastpunt dier twee linien, den pilaer sal daer an in sijn ghegheuen standt blijuen hanghende, ende IC sal swaerheydts middellini sijn: maer dat is onmueghelick, wanttet GH haer euewydeghe is. </s><s>T'selue sal oock alsoo bethoont worden als de lini EF ouer dander sijde neigt. </s><s>Wesende dan IC rechthouckich op den sichteinder, d'ander lini als EF en cander niet scheefhouckich op sijn; nootsaecklick dan rechthouckich: Ende veruolghens sooder EF scheefhouckich op is, dander moeter oock scheefhouckich op sijn.</s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn hanghende an twee linien, d'een CD rechthouckich op den sichteinder, d'ander EF (soot mueghelick waer) scheefhouckich, ende GH sy des pilaers swaerheydts middellini. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen t'hinhoudt des voorstels. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet CD ende EF voortghetrocken worden, sniende malcander in I. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Soo den pilaer in die ghestalt blijft hanghende ande linien CD, EF, sy sal op alle vastpunten in die voortghetrocken linien de selue ghestalt houden, ouermidts de houcken ICE, ende IEC, niet en veranderen: </s><s> Daerom ghenomen I <pb pagenum="60"/>ghemeen vastpunt dier twee linien, <figure/> den pilaer sal daer an in sijn ghegheuen standt blijuen hanghende, ende IC sal swaerheydts middellini sijn: maer dat is onmueghelick, wanttet GH haer euewydeghe is. </s><s>T'selue sal oock alsoo bethoont worden als de lini EF ouer dander sijde neigt. </s><s>Wesende dan IC rechthouckich op den sichteinder, d'ander lini als EF en cander niet scheefhouckich op sijn; nootsaecklick dan rechthouckich: Ende veruolghens sooder EF scheefhouckich op is, dander moeter oock scheefhouckich op sijn.</s>

<s> <emph type="other"/>VOORDER<emph.end/> , anghesien EF neigt naer de sijde van A, soo sal de lini die den pilaer in die ghestalt houdt moeten neighen naer EF. </s><s>Want laetse (soot mueghelick waer) daer van wycken, als CK, sniende de voortghetrocken EI in K, inder voughen dat de hanghende lini door K, sal om de redenen als bouen swaerheydts middellini wesen des pilaers, t'welck noch ongheschicter is dan doen wy die seyden door I te vallen: D'ander lini dan die den pilaer in de ghestalt can houden, en wyckt van EF niet, sy en is met haer oock gheen euewydighe als bouen bethoont is, ende ter sijden uyt te wijcken is openbaer onmueghelick, sy neigt dan nootsaecklick naer EF. </s><s>Ende soo EF ouer d'ander sijde neigde, men sal insghelijcx bethoonen dat d'ander lini van haer wycken sal. </s>

Line 1249

<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 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'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 <figure/> 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>

Line 1259

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

<s> <emph type="other"/>WY<emph.end/> hebben tot veel voorbeelden der voorstellen deses boucx, ghenomen den pilaer, als bequaemste form tot de verclaring des voornemens oock vastpunt ende roerlickpunt ghestelt inden as. Wy sullen nu door dit laetste voorstel, bethoonen de reghelen van dien ghemeen te wesen ouer alle formen der lichamen hodanich sy siin met vastpunt ende roerlickpunt daert valt.

<s> <emph type="other"/>WY<emph.end/> hebben tot veel voorbeelden der voorstellen deses boucx, ghenomen den pilaer, als bequaemste form tot de verclaring des voornemens oock vastpunt ende roerlickpunt ghestelt inden as. </s><s>Wy sullen nu door dit laetste voorstel, bethoonen de reghelen van dien ghemeen te wesen ouer alle formen der lichamen hodanich sy siin met vastpunt ende roerlickpunt daert valt.

</s>

Line 1270

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

</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"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ons t'voorbeelt nemen der eueredenheydt <figure/> 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 <arrow.to.target n="marg163"/>swaerheyts middelpunt 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> <emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons den <figure/> pilaer AB (blijuende de linien FG ende IL op haer plaetsen) neertrecken, alsoo dat hy bliue hanghende an sijn <arrow.to.target n="marg163"/>swaerheyts middelpunt 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>

<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, <arrow.to.target n="marg164"/>werckelick, niet Wisconstelick) 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 <figure/> (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, <arrow.to.target n="marg164"/>werckelick, niet Wisconstelick) 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>

<s><emph type="other"/>LAET<emph.end/> Laet nu t'lichaem AB opghetrocken worden, tot dat FG is inde lini CD, wiens ghestalt dan sy als hier neuens, ende alles blijft noch euestaltwichtich: want het lichaem AB hoogher ofte leegher hanghende, blijft van een selfde ghewicht door de 3. begheerte, ende veruolghens ghelijck IG tot GL, alsoo noch H tot K. </s><s>De eueredenheydt dan des 20. voorstels en is niet alleenelick alsoo met den pilaer, maer met yder lichaem: Ende der ghelijcke salmen oock alsoo bethoonen van al t'ghene hier vooren in alle d'ander voorstellen vanden pilaer gheseyt is. </s>

<s><emph type="other"/>LAET<emph.end/> Laet nu t'lichaem AB opghetrocken <figure/> worden, tot dat FG is inde lini CD, wiens ghestalt dan sy als hier neuens, ende alles blijft noch euestaltwichtich: want het lichaem AB hoogher ofte leegher hanghende, blijft van een selfde ghewicht door de 3. begheerte, ende veruolghens ghelijck IG tot GL, alsoo noch H tot K. </s><s>De eueredenheydt dan des 20. voorstels en is niet alleenelick alsoo met den pilaer, maer met yder lichaem: Ende der ghelijcke salmen oock alsoo bethoonen van al t'ghene hier vooren in alle d'ander voorstellen vanden pilaer gheseyt is. </s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Alle de eueredenheden dan, welcke hier vooren bescreuen sijn vanden pilaer tot de ghewichten an hem hanghende, ende dier ghewichten linien; de selue sijn van yder lichaem tot de ghewichten an hem alsoo hanghende, ende dier ghewichten linien, t'welck wy bewysen moesten.</s>

Line 1345

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een euesijdich driehouck wesen, diens formens middelpunt sy D. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat D oock het swaerheyts middelpunt is des driehoucx ABC. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden van A tot int middel van BC, de lini AE, sghelijcx van C tot int middel van AB, de lini CF. </s><s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Wesende de driehouck ABC opghehanghen byde lini AE, het deel AEC sal euewichtich hanghen teghen AEB, want sy sijn euen groot, ghelijck, ende van ghelijcker ghestalt; AE dan is swaerheyts middellini des driehoucx ABC, Ende om de selue reden sal FC oock des driehoucx swaerheyts middellini sijn, maer dese snien malcanderen in des formens middelpunt D, ende elck dier linien heeft in haer het swaerheyts middelpunt, tis dan D.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een euesijdich driehouck wesen, <figure/> diens formens middelpunt sy D. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat D oock het swaerheyts middelpunt is des driehoucx ABC. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden van A tot int middel van BC, de lini AE, sghelijcx van C tot int middel van AB, de lini CF. </s><s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Wesende de driehouck ABC opghehanghen byde lini AE, het deel AEC sal euewichtich hanghen teghen AEB, want sy sijn euen groot, ghelijck, ende van ghelijcker ghestalt; AE dan is swaerheyts middellini des driehoucx ABC, Ende om de selue reden sal FC oock des driehoucx swaerheyts middellini sijn, maer dese snien malcanderen in des formens middelpunt D, ende elck dier linien heeft in haer het swaerheyts middelpunt, tis dan D.</s>

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een euewydich vierhouck sijn, diens formens middelpunt E. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat E oock het swaerheyts middelpunt is.</s><s> <emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden FG, tusschen de middelpunten van AD ende BC, insghelijcx HI, tusschen de middelpunten van AB ende DC. </s>

<s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Wesende den vierhouck opghehanghen byde lini HI, Het deel HIDA sal euewichtich hanghen teghen HICB, want sy sijn euegroot ghelijck ende van ghelijcker ghestalt; HI dan is swaer<pb pagenum="67"/>heyts middellini des vierhoucx ABCD, Ende om de selue reden sal FG oock des vierhoucx swaerehyts middellini sijn, maer dese doorsnien malcanderen in E, ende elck dier linien heeft in haer het swaerheyts middelpunt, tis dan E.</s>

<s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Wesende den vierhouck opghehanghen byde lini HI, Het deel HIDA sal euewichtich hanghen teghen HICB, want sy sijn euegroot ghelijck ende van ghelijcker ghestalt; HI dan is swaer<pb pagenum="67"/>heyts middellini des vierhoucx <figure/> ABCD, Ende om de selue reden sal FG oock des vierhoucx swaerehyts middellini sijn, maer dese doorsnien malcanderen in E, ende elck dier linien heeft in haer het swaerheyts middelpunt, tis dan E.</s>

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een gheschickt ofte inschriuelick vijfhouck wesen, diens formens middelpunt F sy. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat F oock het swaerheyts middelpunt is. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een gheschickt ofte inschriuelick <figure/> vijfhouck wesen, diens formens middelpunt F sy. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat F oock het swaerheyts middelpunt is. </s>

<s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden van A tot int middel van DC, de lini AG; sghelijcx van B tot int middel van ED, de lini BH. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Wesende den vijfhouck opghehanghen byde lini AG, het deel AGDE sal euewichtich hanghen teghen het deel AGCB, want sy sijn euegroot, ghelijck, ende van ghelijcker ghestalt: AG dan is swaerheyts middellini des vijfhoucx, ende om de selue reden sal BH oock des selfden vijfhoucx swaerehyts middellini wesen; maer dese doorsnien malcanderen in des formens middelpunt F, ende elck dier linien heeft in haer het swaerheyts middelpunt, tis dan F. </s><s> Sghelijcx sal oock t'bewys sijn in allen anderen hebbende een formens middelpunt als Seshoucken, Ronden, Scheefronden, &c. </s>

Line 1365

<s> <emph type="other"/>YDER<emph.end/> driehoucx swaerheydts middelpunt, is inde lini ghetrocken vanden houck tot int middel der sijde. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck sijn van form soot <pb pagenum="68"/>valt, waer in vanden houck A tot in D middel vande sijde BC, ghetrocken is de lini AD. </s><s> <emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat des driehoucx swaerheyts middelpunt inde lini AD is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken EF, GH, IK, euewydighe van BC, sniende AD in L, M, N, daer naer EO, GP, IQ, KR, HS, FT, euewydighe met AD. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck sijn van form soot <pb pagenum="68"/>valt, waer in vanden houck A tot in D middel vande sijde BC, ghetrocken <figure/> is de lini AD. </s><s> <emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat des driehoucx swaerheyts middelpunt inde lini AD is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken EF, GH, IK, euewydighe van BC, sniende AD in L, M, N, daer naer EO, GP, IQ, KR, HS, FT, euewydighe met AD. </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ouermits EF euewydighe is van BC, ende EO, FT met LD, soo sal EFTO, euewydich vierhouck sijn, wiens EL euen is met LF, oock met OD ende DT, waer duer het swaerheyts middelpunt des vierhoucx EFTO in DL is, duer het I voorstel deses boucx. </s><s>Ende om de selue reden sal het swaerheyts middelpunt des euewydichs vierhoucx GHSP wesen in LM, ende van IKRQ in MN, ende vervolghens het swaerheyts middelpunt der form IKRHSFTOEPGQ ghemaect vande voornoemde drie vierhoucken, sal wesen inde lini ND, ofte AD. </s><s> Nu ghelijck hier in beschreuen sijn drie vierhoucken, also canmender oneindelicke sulcke vierhoucken in beschrijuen, ende des binneschreuens formen swaerheyts middelpunt, sal altijt sijn (om de redenen als vooren) inde lini AD. </s><s>Maer hoe datter sulcke vierhoucken meer sijn, hoe dat den driehouck ABC min verschilt vande binneschreuen form der vierhoucken; want treckende linien euewydich van BC door de middelen van AN, NM, ML, LD, t'verschil des laetsten ghestalts, sal effen den helft sijn van t'verschil des voorgaenden ghestalts. </s><s>Wy connen dan door dat oneindelick naerderen sulck een form binnen den driehouck stellen, dattet verschil tusschen haer ende den driehouck, minder sal wesen dan eenich ghegheuen plat hoe cleen het sy: Waer uyt volght, dat stellende AD als swaerheydts middellini, so sal t'staltwicht des deels ADC, min verschillen van t'staltwicht des deels ADB, dan eenich plat datmen soude connen gheuen hoe cleen het sy, waer uyt ick aldus strie. </s>

Line 1380

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck wesen. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheytds middelpunt vinden. </s><s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal van A tot int middel van BC, trecken de lini AD, sghelijcx van C tot int middel van AB, de lini CE, sniende AD in F: Ick seg dat F t'begheerde swaerheydts middelpunt is. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>T'swaerheyts middelpunt des driehoucx ABC, is inde lini AD, ende oock in CE, duer het 2 voorstel, tis dan F, t'welck wy bewysen moesten. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een driehouck: Wy hebben sijn swaerheydts middelpunt gheuonden naer den eysch.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck wesen. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s> <figure/> <s>Wy moeten sijn swaerheytds middelpunt vinden. </s><s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal van A tot int middel van BC, trecken de lini AD, sghelijcx van C tot int middel van AB, de lini CE, sniende AD in F: Ick seg dat F t'begheerde swaerheydts middelpunt is. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>T'swaerheyts middelpunt des driehoucx ABC, is inde lini AD, ende oock in CE, duer het 2 voorstel, tis dan F, t'welck wy bewysen moesten. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een driehouck: Wy hebben sijn swaerheydts middelpunt gheuonden naer den eysch.</s>

<s><emph type="other"/>III. VERTOOCH<emph.end/>. <emph type="other"/>IIII. VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>HET<emph.end/> swaerheydts middelpunt eens driehoucx deelt de lini vanden houck tot int middel der sijde alsoo, dattet stick naer den houck, dobbel is an t'ander. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck sijn, ende vanden houck B een lini ghetrocken worden tot D int middel van AC, sghelicx van C tot E int middel van AB, sniende BD in F voor swaerheyts middelpunt des driehoucx ABC. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat CF dobbel is an FE. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ghetrocken de reden EB 1 tot BA 2, vande reden CD 1 tot DA 1 (dat is Reden 1/2 van Reden 1/1) <arrow.to.target n="marg189"/>daer rest de reden van CF tot FE, maer treckende Reden 1/2 van Reden 1/1 daer blijft Reden 2/1. </s><s>CF dan is tot FE, als van 2 tot I. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck sijn, ende vanden <figure/> houck B een lini ghetrocken worden tot D int middel van AC, sghelicx van C tot E int middel van AB, sniende BD in F voor swaerheyts middelpunt des driehoucx ABC. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat CF dobbel is an FE. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ghetrocken de reden EB 1 tot BA 2, vande reden CD 1 tot DA 1 (dat is Reden 1/2 van Reden 1/1) <arrow.to.target n="marg189"/>daer rest de reden van CF tot FE, maer treckende Reden 1/2 van Reden 1/1 daer blijft Reden 2/1. </s><s>CF dan is tot FE, als van 2 tot I. </s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Het swaerheyts middelpunt dan eens driehoucx deelt de lini vande houck tot int middel der sijde alsoo, dattet stick naer den houck dobbel is an t'ander. </s><s>t'welck wy bewysen moesten.</s>

<s>

<margin.target id="marg189"/>Door t'verkeerde des I2 cap. I lib. Almag. Ptolem.

<margin.target id="marg189"/>Door t'verkeerde des 12 cap. I lib. Almag. Ptolem.

</s>

Line 1398

<s> <emph type="other"/>WESENDE<emph.end/> twee sijden eens driehoucx elck ghedeelt in drie euen deelen: De lini tusschen de twee punten der deeling naest de derde sijde, streckt door des driehoucx swaerheyts middelpunt. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck wesen, van t'welck yder sijde AB ende BC ghedeelt sy in drie euen deelen, met de punten D, E, F, G, ende tusschen de punten E, G, naest de derde sijde BC, sy ghetrocken de lini EG. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat EG duer des driehoucx ABC swaerheyts middelpunt streckt </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck wesen, van t'welck yder <figure/> sijde AB ende BC ghedeelt sy in drie euen deelen, met de punten D, E, F, G, ende tusschen de punten E, G, naest de derde sijde BC, sy ghetrocken de lini EG. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat EG duer des driehoucx ABC swaerheyts middelpunt streckt </s>

<s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken van A tot int middel van BC, de lini AH, sniende EG in I. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ouermits <arrow.to.target n="marg193"/> AE sulcken reden heeft tot EB, als AG tot GC, soo is EG euewydighe met BC, ende veruolghens EI is euewydighe met BH, daerom ghelijck AE tot EB, alsoo AI tot IH, maer AE is dobbel tot EB door t'ghegheuen, daerom AI is dobbel tot IH, maer wesende AI dobbel tot IH, soo is I t'swaerheyts middelpunt des driehoucx ABC door het 4. voorstel, daerom EG streckt door des ghegheuen driehoucx swaerheyts middelpunt. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan twee sijden eens driehoucx elck ghedeelt in drie euen deelen, de lini tusschen de twee punten der deeling naest de derde sijde, streckt door des driehoucx swaerheyts middelpunt, t'welck wy bewysen moesten. </s>

<s>

Line 1416

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ongheschict vierhouck wesen. </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 vierhouck deelen in twee driehoucken met de lini AC, ende vinden het swaerheyts middelpunt van elck driehouck, duer het 3. voorstel, dat van ACB sy E, ende van ACD sy F, <pb pagenum="71"/>ende de lini EF sal balck wesen. </s><s>Daer naer salmen maken <arrow.to.target n="marg195"/> twee euewydighe vierhoucken van een selfde hoogde, als GHIK, euen anden driehouck ACD, ende GHLM, euen anden driehouck ACB, daer naer deelende den balck <arrow.to.target n="marg196"/> FE in N, alsoo dat den erm NE, sulcken reden hebbe tot den erm NF, als HI tot HL; Ick seg dat N t'begheerde swaerheyts middelpunt is.</s>

<s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal den vierhouck deelen in twee driehoucken met de lini AC, ende vinden het swaerheyts middelpunt van elck driehouck, duer het 3. voorstel, dat van ACB sy E, ende van ACD sy F, <pb pagenum="71"/>ende de lini EF sal balck wesen. </s><s>Daer naer <figure/> salmen maken <arrow.to.target n="marg195"/> twee euewydighe vierhoucken van een selfde hoogde, als GHIK, euen anden driehouck ACD, ende GHLM, euen anden driehouck ACB, daer naer deelende den balck <arrow.to.target n="marg196"/> FE in N, alsoo dat den erm NE, sulcken reden hebbe tot den erm NF, als HI tot HL; Ick seg dat N t'begheerde swaerheyts middelpunt is.</s>

<s>

<margin.target id="marg195"/>Doorhe45.v.I.B.E.

<margin.target id="marg196"/>Door het I0.v.6.B.E.

<margin.target id="marg196"/>Door het 10.v.6.B.E.

</s>

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCDE een ongheschickt vijfhouck sijn. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCDE een ongheschickt vijfhouck sijn.<figure/> </s>

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

Line 1432

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

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCDEF een ongheschickt seshouck <figure/> sijn. </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 trecken AC, ende vinden t'swaerheydts middelpunt des driehoucx ACB duer het 3 voorstel, t'welck G sy, ende vanden vijfhouck ACDEF, duer het voorgaende 2 voorbeelt, t'welck H sy, ende de lini GH sal balck wesen. </s><s>Daer naer salmen maken twee euewydighe vierhoucken van een selfde hoochde, als IKLM, euen anden vijfhouck ACDEF, ende IKNO <pb pagenum="72"/>euen anden driehouck ACB, deelende den balck HG in P, alsoo dat den erm PG, sulcken reden hebbe tot den erm PH, als de lini KM tot KN; Ick seg dat P t'begheerde swaerheyts middelpunt is. </s><s>Welcke maniere van wercking in allen anderen veelsijdeghe platten ghelijck sal sijn ande voorgaende.</s>

Line 1442

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ongheschickt vierhouck wesen. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheydts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal den vierhouck deelen in twee driehoucken, met de lini AC, ende vinden t'swaerheydts middelpunt van elcken driehouck door het 3. voorstel, dat van ACB sy E, ende vanden driehouck ACD sy F, de lini dan EF is balck. </s><s>Daer naer salmen trecken DG ende BH, beyde rechthouckich op AC, deylende den balck FE en I, alsoo dat den erm IE, sulcken reden hebbe tot den erm IF, als DG tot BH; Ick seg dat I t'begheerde swaerheyts middelpunt is.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ongheschickt vierhouck wesen. </s> <figure/> <s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheydts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal den vierhouck deelen in twee driehoucken, met de lini AC, ende vinden t'swaerheydts middelpunt van elcken driehouck door het 3. voorstel, dat van ACB sy E, ende vanden driehouck ACD sy F, de lini dan EF is balck. </s><s>Daer naer salmen trecken DG ende BH, beyde rechthouckich op AC, deylende den balck FE en I, alsoo dat den erm IE, sulcken reden hebbe tot den erm IF, als DG tot BH; Ick seg dat I t'begheerde swaerheyts middelpunt is.</s>

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCDE een ongheschickt vijfhouck sijn. </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 vijfhouck deelen in drie driehoucken, met eenighe linien als AD, AC, vindende daer naer het swaerheyts middelpunt des vierhoucx ACDE duer het 4. voorbeelt, t'welck F sy, ende des driehoucx ACB duer het 3. voorstel, t'welck G sy, ende de lini FG, is balck, Daer naer ghetrocken BH rechthouckich op AC; Ende CI met EK rechthouckich op AD, men sal der drie linien AD, AC, HB, vinden de vierde <arrow.to.target n="marg197"/>euerednighe, welcke sy LM, deelende <arrow.to.target n="marg198"/>den balck FG in N, alsoo dat den erm NG sulcken reden hebbe tot den erm NF, ghelijck CI met EK, tot LM; Ick seg dat N het begheerde swaerheydts middelpunt is.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCDE een ongheschickt vijfhouck <figure/> sijn. </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 vijfhouck deelen in drie driehoucken, met eenighe linien als AD, AC, vindende daer naer het swaerheyts middelpunt des vierhoucx ACDE duer het 4. voorbeelt, t'welck F sy, ende des driehoucx ACB duer het 3. voorstel, t'welck G sy, ende de lini FG, is balck, Daer naer ghetrocken BH rechthouckich op AC; Ende CI met EK rechthouckich op AD, men sal der drie linien AD, AC, HB, vinden de vierde <arrow.to.target n="marg197"/>euerednighe, welcke sy LM, deelende <arrow.to.target n="marg198"/>den balck FG in N, alsoo dat den erm NG sulcken reden hebbe tot den erm NF, ghelijck CI met EK, tot LM; Ick seg dat N het begheerde swaerheydts middelpunt is.</s>

<s>

<margin.target id="marg197"/>Proportionalis.

<margin.target id="marg198"/>Door het I2. v. 6. B. E.

<margin.target id="marg198"/>Door het 12. v. 6. B. E.

</s>

<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/>. Laet ABCDEF een ongheschickt seshouck <figure/> sijn. <emph type="other"/>T'BEGHEERDE<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><s>Ende soo salmen voort mueghen varen met ander veelhouckeghe platten.

Line 1464

<s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ghelijck int eerste voorbeelt HI tot HL, alsoo den erm NE tot den erm NF, maer <arrow.to.target n="marg199"/> ghelijck HI tot HL, alsoo den vierhouck GHIK, tot den vierhouck GHLM, ghelijck dan GHIK tot GHLM, also NE tot NF, maer GHIK is euen an den driehouck ACD, ende GHLM anden driehouck ACB door t'werck, ghelijck dan den driehouck ACD tot ACB, alsoo den erm NE tot NF. </s><s> Het punt dan N is (door het 1. voorstel des 1. boucx) des vierhoucx swaerheyts middelpunt. </s><s>Sghelijcx sal oock bewijs sijn des 2. ende 3. voorbeelts. </s>

<s>

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

<margin.target id="marg199"/>1. 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>

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

<margin.target id="marg200"/>1. v. 6. B. E.

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

<margin.target id="marg201"/>41. v. 1. 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>

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

<margin.target id="marg202"/>16. v. 6. B. E.

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

<margin.target id="marg203"/>1. v. 6. B. E.

</s>

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

<s> <emph type="other"/>MY<emph.end/> is onder het drucken ter handt ghecomen, Fredric Commandins <arrow.to.target n="marg205"/>verclaring ouer de viercanting der Brantsne van Archimedes, alwaer hy onder het 6. voorstel de manier beschrijft, om t'swaerheyts middelpunt te vinden van yder rechtlinich plat, ende dat op een ander wijse als de twee voorgaende. So ymant tottet ouersien der selue begheerich waer, salse daer vinden.

<s> <emph type="other"/>MY<emph.end/> is onder het drucken ter handt ghecomen, Fredric Commandins <arrow.to.target n="marg205"/>verclaring ouer de viercanting der Brantsne van Archimedes, alwaer hy onder het 6. voorstel de manier beschrijft, om t'swaerheyts middelpunt te vinden van yder rechtlinich plat, ende dat op een ander wijse als de twee voorgaende. </s><s>So ymant tottet ouersien der selue begheerich waer, salse daer vinden.

</s>

<s>

<margin.target id="marg205"/>Commentarius in quadraturam paraboles.

Line 1494

<margin.target id="marg206"/>Parallelis.

</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 <arrow.to.target n="marg207"/>eueredenheyt 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 2. 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 <figure/> 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 <arrow.to.target n="marg207"/>eueredenheyt 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 2. 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>

Line 1503

<s> <emph type="other"/>HET<emph.end/> swaerheyts middelpunt des vierhoucx met twee euewydighe sijden, deelt de lini tusschen dier euewydighens middelpunten also, dat het stick naer de minste sijde, tot het ander, sulcken reden heeft, als tweemaal de meeste sijde met eenmael de minste, tot tweemael de minste met eenmael de meeste. </s>

<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'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een vierhouck wesen met twee <figure/> 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. 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 2. voorstel, ende oock in HL duer het 5. 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 7. 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 1. voorstel des 1. 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.

<margin.target id="marg208"/>1. v. 6. B. E.

</s>

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

Line 1516

<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 <arrow.to.target n="marg209"/>t'ghestelde , 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 1. voorstel des 1. 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 <figure/> 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 <arrow.to.target n="marg209"/>t'ghestelde , 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 1. voorstel des 1. 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>

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een rondt wesen, diens <arrow.to.target n="marg210"/>halfmiddellini EA, ende swaerheyts middelpunt E sy, ende trondt AFGH, deel des rondts ABCD, ende sijn swaerheyts middelpunt I, ende <arrow.to.target n="marg211"/>middellini AG. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten het swaerheyts middelpunt vinden des ander deels, dat is der maen ABCDHGF. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een rondt wesen, diens <arrow.to.target n="marg210"/>halfmiddellini <figure/> EA, ende swaerheyts middelpunt E sy, ende trondt AFGH, deel des rondts ABCD, ende sijn swaerheyts middelpunt I, ende <arrow.to.target n="marg211"/>middellini AG. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten het swaerheyts middelpunt vinden des ander deels, dat is der maen ABCDHGF. </s>

<s>

<margin.target id="marg210"/>Semidiameter.

<margin.target id="marg211"/>Diameter.

Line 1531

<s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal IE voorttrecken tot in K, also dat IE sulcken reden hebbe tot EK, als de maen ABCDHGF tot het rondt AFGH, ende K sal t'begheerde swaerheydts middelpunt wesen. </s><s>Daer af t'bewys ghelijck sal sijn an tvoorgaende. </s><s>Maer om de reden dier maen tot dat rondt te vinden, men sal trecken CL euen met AG, daernaer AL, vindende <arrow.to.target n="marg212"/> de derde everednighe welcker eerste AL, de tweede LC, ende de derde sy M, Ende AL tot M, sal de reden sijn der maen tot het rondt AFGH. </s><s> Want ouermits ALC rechthouck is (reden dat sy int half rondt staet <arrow.to.target n="marg213"/>) het ront diens middellini AL, sal euen sijn ande maen, ende AL tot M is de <arrow.to.target n="marg214"/>ghedobbelde reden van AL tot LC, dat is van AL tot AG, daerom &c.</s>

<s>

<margin.target id="marg212"/>II. v. 6. B. E,

<margin.target id="marg212"/>11. v. 6. B. E,

<margin.target id="marg213"/>3I. v. 3. B. E,

<margin.target id="marg213"/>31. v. 3. B. E,

<margin.target id="marg214"/>Duplicata ratio.

</s>

Line 1545

<margin.target id="marg215"/>Parabola.

</s>

<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'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een brantsne sijn diens middellini <figure/> 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 1. 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>

Line 1564

<margin.target id="marg219"/>Proportionaliter.

</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD ende <emph type="italics"/>abcd<emph.end/> twee onghelijcke brantsneen sijn, diens middellinien AD, ende <emph type="italics"/>ad<emph.end/>, ende swaerheyts middelpunten E, ende <emph type="italics"/>e<emph.end/>. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck AE tot ED, also <emph type="italics"/>ae<emph.end/> tot <emph type="italics"/>ed<emph.end/>. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD ende <emph type="italics"/>abcd<emph.end/> twee onghelijcke <figure/> brantsneen sijn, diens middellinien AD, ende <emph type="italics"/>ad<emph.end/>, ende swaerheyts middelpunten E, ende <emph type="italics"/>e<emph.end/>. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck AE tot ED, also <emph type="italics"/>ae<emph.end/> tot <emph type="italics"/>ed<emph.end/>. </s>

<s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de linien AB, AC, die deelende in haer middelen F, G, ende trecken FG sniende AD in H, daer naer FI ende GK euewydighe van AD, ende daer naer IA, IB, KA, KC, ende laet ons stellen L in IF, alsoo dat IL dobbel sy an LF, sghelijcx M, alsoo dat KM dobbel sy an MG, ende laet ons trecken LM, sniende AD in N, ende IK, sniende AD in O, ende laet ons stellen P, alsoo dat AP dobbel sy an PD, ende laet ons IF voorttrecken tot Q inden grondt BC. </s><s> Nu anghesien AP dobbel is an PD, so is P t'swaerheyts middelpunt des driehoucx ABC, ende omme de selue reden L, M, <pb pagenum="80"/>swaerheyts middelpunten der twee driehoucken ABI, ende ACK, ende veruolghens, want sy euen sijn, soo is N haer beyde swaerheyts middelpunt. </s><s> NP dan is balck, de selue ghedeelt in R, alsoo dat den erm NR sy tot RP, als den driehouck ABC tot de twee driehoucken ABI, ACK, dat is, als 4 tot I (want alle brantsne is tot den driehouck als ABC ghelijck 4 tot 3, duer het 24 voorstel der viercanting des brantsnees van Archim. </s><s>daerom, &c.) Laet ons nu derghelijcke linien ende punten oock beschrijuen inde brantsne <emph type="italics"/>abc<emph.end/>. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ghelijck AD tot AO, also het viercant van DB tottet viercant van OI <arrow.to.target n="marg226"/>; Maer DQ is euen an OI, ende DQ is den helft van DB (want F is t'middel van AB, ende FQ is euewydich van AD) daerom het viercant van DB, is viervoudich an t'viercant van DQ, ofte van OI, ende veruolghens AD is viervoudich tot AO, daerom AO is 1/4 van AD, ende OH oock 1/4 (want AH is den helft van AD, ouermits FG ghetrocken is uyt de middelen van AB, AC) daerom doet NH 1/12 van AD, daer toe ghedaen HD 1/2, comt voor ND 7/12, daer af ghetrocken PD 2/3, rest voor PN 1/4: Maer NR is viervoudich tot RP, daerom RP doet 1/20, daer toe PD 1/3, doet voor RD 23/60, daerom RA de reste der lini, doet 37/60, Ghelijck dan 37 tot 23, also AR tot RD, ende met de selue reden is bethoont dat <emph type="italics"/>ar<emph.end/> tot <emph type="italics"/>rd<emph.end/>, oock is als 37 tot 23. </s><s>Dese twee rechtsideghe formen dan ghelijckelick beschreuen in verscheyden brandtsneen, hebben het swaerheyts middelpunt in haer middellinien, also dat de deelen onder malcanderen <arrow.to.target n="marg229"/>euerednich sijn. </s><s>Ende so wy inde brandtsnekens BI, IA, AK, KC, driehoucken beschreuen, soo ghedaen is inde brantsneen ABI, ACK, vindende daer naer t'swaerheyts middelpunt des heels binnescreuen rechtlinich plats, t'welck ick neem dat hier S soude wesen, ende daer <emph type="italics"/>s<emph.end/>, wy souden inder seluer voughen als vooren bethoonen, dat ghelijck AS tot SR, also <emph type="italics"/>as<emph.end/> tot <emph type="italics"/>sr<emph.end/>. </s><s>Maer wy connen duer sulck oneindelick inschriuen der rechtlinighe formen oneindelick naerderen nae E, ende <emph type="italics"/>e<emph.end/><arrow.to.target n="marg233"/>, ende ghelijcksideghe platten sullen altijt der middelliniens AD twee sticken euerednich ghedeelt hebben duer haer swaerheyts middelpunt, ende veruolghens de heele brantsneen ABC, <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg234"/>, sullen die deelen euerednich hebben. </s><s>Want laet (soot mueghelick waer) T t'swaerheyts middelpunt sijn des brantsnees ABC, ende <emph type="italics"/>e<emph.end/><arrow.to.target n="marg235"/> van <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg236"/>, ende laet ons teeckenen <emph type="italics"/>t<emph.end/><arrow.to.target n="marg237"/>, dat ghelijck ET tot TS, alsoo <emph type="italics"/>et<emph.end/><arrow.to.target n="marg238"/> tot <emph type="italics"/>ts<emph.end/><arrow.to.target n="marg239"/>. </s><s>Nu alsmen duer t'inschrijuen veelsidegher formen in <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg240"/>, sal ghecommen sijn tot <emph type="italics"/>t<emph.end/><arrow.to.target n="marg241"/>, men sal met ghelijcke veelsideghe formen in ABC, ghecomen sijn tot T, daerom T sal t'swaerheyts middelpunt sijn der binneschreuen form, ende oock des heelen brantsneens ABC, t'welck ongheschickt is. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Aller brantsneens middellinien dan, worden van het swaerheyts middelpunt eueredelick ghedeelt, t'welck wy bewysen moesten. </s>

<s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de linien AB, AC, die deelende in haer middelen F, G, ende trecken FG sniende AD in H, daer naer FI ende GK euewydighe van AD, ende daer naer IA, IB, KA, KC, ende laet ons stellen L in IF, alsoo dat IL dobbel sy an LF, sghelijcx M, alsoo dat KM dobbel sy an MG, ende laet ons trecken LM, sniende AD in N, ende IK, sniende AD in O, ende laet ons stellen P, alsoo dat AP dobbel sy an PD, ende laet ons IF voorttrecken tot Q inden grondt BC. </s><s>Nu anghesien AP dobbel is an PD, so is P t'swaerheyts middelpunt des driehoucx ABC, ende omme de selue reden L, M, <pb pagenum="80"/>swaerheyts middelpunten der twee driehoucken ABI, ende ACK, ende veruolghens, want sy euen sijn, soo is N haer beyde swaerheyts middelpunt. </s><s>NP dan is balck, de selue ghedeelt in R, alsoo dat den erm NR sy tot RP, als den driehouck ABC tot de twee driehoucken ABI, ACK, dat is, als 4 tot 1 (want alle brantsne is tot den driehouck als ABC ghelijck 4 tot 3, duer het 24 voorstel der viercanting des brantsnees van Archim.daerom, &c.) </s><s>Laet ons nu derghelijcke linien ende punten oock beschrijuen inde brantsne <emph type="italics"/>abc<emph.end/>. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ghelijck AD tot AO, also het viercant van DB tottet viercant van OI <arrow.to.target n="marg226"/>; Maer DQ is euen an OI, ende DQ is den helft van DB (want F is t'middel van AB, ende FQ is euewydich van AD) daerom het viercant van DB, is viervoudich an t'viercant van DQ, ofte van OI, ende veruolghens AD is viervoudich tot AO, daerom AO is 1/4 van AD, ende OH oock 1/4 (want AH is den helft van AD, ouermits FG ghetrocken is uyt de middelen van AB, AC) daerom doet NH 1/12 van AD, daer toe ghedaen HD 1/2, comt voor ND 7/12, daer af ghetrocken PD 2/3, rest voor PN 1/4: Maer NR is viervoudich tot RP, daerom RP doet 1/20, daer toe PD 1/3, doet voor RD 23/60, daerom RA de reste der lini, doet 37/60, Ghelijck dan 37 tot 23, also AR tot RD, ende met de selue reden is bethoont dat <emph type="italics"/>ar<emph.end/> tot <emph type="italics"/>rd<emph.end/>, oock is als 37 tot 23. </s><s>Dese twee rechtsideghe formen dan ghelijckelick beschreuen in verscheyden brandtsneen, hebben het swaerheyts middelpunt in haer middellinien, also dat de deelen onder malcanderen <arrow.to.target n="marg229"/>euerednich sijn. </s><s>Ende so wy inde brandtsnekens BI, IA, AK, KC, driehoucken beschreuen, soo ghedaen is inde brantsneen ABI, ACK, vindende daer naer t'swaerheyts middelpunt des heels binnescreuen rechtlinich plats, t'welck ick neem dat hier S soude wesen, ende daer <emph type="italics"/>s<emph.end/>, wy souden inder seluer voughen als vooren bethoonen, dat ghelijck AS tot SR, also <emph type="italics"/>as<emph.end/> tot <emph type="italics"/>sr<emph.end/>. </s><s>Maer wy connen duer sulck oneindelick inschriuen der rechtlinighe formen oneindelick naerderen nae E, ende <emph type="italics"/>e<emph.end/><arrow.to.target n="marg233"/>, ende ghelijcksideghe platten sullen altijt der middelliniens AD twee sticken euerednich ghedeelt hebben duer haer swaerheyts middelpunt, ende veruolghens de heele brantsneen ABC, <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg234"/>, sullen die deelen euerednich hebben. </s><s>Want laet (soot mueghelick waer) T t'swaerheyts middelpunt sijn des brantsnees ABC, ende <emph type="italics"/>e<emph.end/><arrow.to.target n="marg235"/> van <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg236"/>, ende laet ons teeckenen <emph type="italics"/>t<emph.end/><arrow.to.target n="marg237"/>, dat ghelijck ET tot TS, alsoo <emph type="italics"/>et<emph.end/><arrow.to.target n="marg238"/> tot <emph type="italics"/>ts<emph.end/><arrow.to.target n="marg239"/>. </s><s>Nu alsmen duer t'inschrijuen veelsidegher formen in <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg240"/>, sal ghecommen sijn tot <emph type="italics"/>t<emph.end/><arrow.to.target n="marg241"/>, men sal met ghelijcke veelsideghe formen in ABC, ghecomen sijn tot T, daerom T sal t'swaerheyts middelpunt sijn der binneschreuen form, ende oock des heelen brantsneens ABC, t'welck ongheschickt is. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Aller brantsneens middellinien dan, worden van het swaerheyts middelpunt eueredelick ghedeelt, t'welck wy bewysen moesten. </s>

<s>

<margin.target id="marg226"/>20. v. I. B. Appol.

<margin.target id="marg226"/>20. v. 1. B. Appol.

<margin.target id="marg229"/>Proportionales.

</s>

Line 1580

<margin.target id="marg242"/>Parabola.

</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, 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 1; 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 <figure/> 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 1; 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 11. 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 1, 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>

Line 1602

<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 ghecorte brantsne volmaken, daer an stellende t'ghebrekende ABG, daer naer salmen teekenen H, alsoo dat GH sy tot HE, als 3 tot 2: Insghelijcx I, alsoo dat GI sy tot IF, als 3 tot 2, daer naer K, alsoo dat IH sulcken reden hebbe tot IK, ghelijck de ghecorte brandtsne ABCD, tot de brantsne ABG; Ick seg dat K t'begheerde swaerheyts middelpunt is. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>I is swaerheyts middelpunt des heels, ende H des deels, ende ghelijck t'ander deel tot dit, alsoo HI tot IK, daerom K, duer het 9. 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 ghecorte brantsne, wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

<s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal de ghecorte brantsne volmaken, daer an stellende <figure/> t'ghebrekende ABG, daer naer salmen teekenen H, alsoo dat GH sy tot HE, als 3 tot 2: Insghelijcx I, alsoo dat GI sy tot IF, als 3 tot 2, daer naer K, alsoo dat IH sulcken reden hebbe tot IK, ghelijck de ghecorte brandtsne ABCD, tot de brantsne ABG; Ick seg dat K t'begheerde swaerheyts middelpunt is. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>I is swaerheyts middelpunt des heels, ende H des deels, ende ghelijck t'ander deel tot dit, alsoo HI tot IK, daerom K, duer het 9. 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 ghecorte brantsne, wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

<s> <emph type="other"/>NV VANDE VINDING DER <lb/>SWAERHEYTS MIDDELPVNTEN <lb/>VANDE LICHAMEN<emph.end/>. </s>

Line 1610

<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 <arrow.to.target n="marg246"/>viergrondich 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 1. begheerte des 1. 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 <arrow.to.target n="marg246"/>viergrondich <figure/> 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 1. begheerte des 1. 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>

Line 1621

<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 <arrow.to.target n="marg247"/>plat 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 <figure/> 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 <arrow.to.target n="marg247"/>plat RS euewidich met den grondt ADC, ende S sy int middel van CB. </s>

<s>

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

</s>

Line 1643

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een vierhouckich pilaer wesen, diens grondt ACDE. </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'BERYTSEL<emph.end/>. </s><s>Laet ons trecken een plat duer AD, ende haer lijckstandighe int decksel, deelende also den ghegheuen pilaer in twee driehouckighe pilaren, welcker yder het swaerheyts middelpunt int middel vanden as heeft duer het 1. voorbeelt, daerom ghetrocken den balck tusschen die twee punten, ende den seluen ghedeelt in sijn ermen, het onderscheydt der ermen sal het swaerheyts middelpunt sijn des ghegheuen pilaers, welck punt valt in t'swaerheyts middelpunt des plats euewydich vanden grondt den pilaer in twee euen stucken deelende, ende t'selue int middel der lini tusschen de swaerheyts middelpunten des gronts ende decksels, dat is int middel vanden as; T'selue salmen oock alsoo bethoonen in yder pilaer. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder pilaers swaerheyts middelpunt dan, is int middel vanden as, t'welck wy bewysen moesten.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een vierhouckich pilaer wesen, diens <figure/> grondt ACDE. </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'BERYTSEL<emph.end/>. </s><s>Laet ons trecken een plat duer AD, ende haer lijckstandighe int decksel, deelende also den ghegheuen pilaer in twee driehouckighe pilaren, welcker yder het swaerheyts middelpunt int middel vanden as heeft duer het 1. voorbeelt, daerom ghetrocken den balck tusschen die twee punten, ende den seluen ghedeelt in sijn ermen, het onderscheydt der ermen sal het swaerheyts middelpunt sijn des ghegheuen pilaers, welck punt valt in t'swaerheyts middelpunt des plats euewydich vanden grondt den pilaer in twee euen stucken deelende, ende t'selue int middel der lini tusschen de swaerheyts middelpunten des gronts ende decksels, dat is int middel vanden as; T'selue salmen oock alsoo bethoonen in yder pilaer. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder pilaers swaerheyts middelpunt dan, is int middel vanden as, t'welck wy bewysen moesten.</s>

<s><emph type="other"/>XI. VERTOOCH<emph.end/>. <emph type="other"/>XVI. VOORSTEL<emph.end/>.</s>

Line 1652

<margin.target id="marg253"/>Pyramidis.

</s>

<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'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een naelde sijn, diens grondt den <figure/> 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 14. 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>

Line 1666

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een naelde wesen, diens grondt sy den driehouck BCD. </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 swaerheyts middelpunten vinden van eenighe twee driehoucken, als E van BCD, ende F van ADC, treckende de linien AE, BF; welcker sne G, ick seg te wesen het begheerde swaerheyts middelpunt. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een naelde wesen, diens grondt sy <figure/> den driehouck BCD. </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 swaerheyts middelpunten vinden van eenighe twee driehoucken, als E van BCD, ende F van ADC, treckende de linien AE, BF; welcker sne G, ick seg te wesen het begheerde swaerheyts middelpunt. </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Des naldens ABCD swaerheyts middelpunt is in AE, ende oock in BF, duer het I6 voorstel, het is dan nootsaeclick G. </s><s> <emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een naelde: Wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Des naldens ABCD swaerheyts middelpunt is in AE, ende oock in BF, duer het 16 voorstel, het is dan nootsaeclick G. </s><s> <emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een naelde: Wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

<s><emph type="other"/>XII. VERTOOCH<emph.end/>. <emph type="other"/>XVIII. VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>HET<emph.end/> swaerheyts middelpunt van yder naelde deelt den as alsoo, dat het stick naer den houck drievoudich is an t'ander. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een driehouckighe naelde wesen, diens sop A, ende grondt BCD, ende den as van B tot int swaerheyts middelpunt E des driehoucx ADC, sy BE, ende van A tot int swaerheyts middelpunt F des driehoucx BCD sy AF, sniende BE in G, voor t'swaerheyts middelpunt der ghegheuen naelde. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat BG drievoudich is an GE. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken van H middel van CD, de linien HA, HB. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een driehouckighe naelde wesen, <figure/> diens sop A, ende grondt BCD, ende den as van B tot int swaerheyts middelpunt E des driehoucx ADC, sy BE, ende van A tot int swaerheyts middelpunt F des driehoucx BCD sy AF, sniende BE in G, voor t'swaerheyts middelpunt der ghegheuen naelde. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat BG drievoudich is an GE. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken van H middel van CD, de linien HA, HB. </s>

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ouermits HA ghetrocken is uyt het middel van DC tot inden houck A, ende dat E t'swaerheyts middelpunt is des driehoucx ACD, soo sal AE dobbel sijn an EH duer het 4. voorstel, ende om de selue reden sal BF dobbel wesen an FH. </s><s>Dit soo sijnde, ghetrocken de reden EA 2, tot AH 3, vande reden BF 2, tot FH 1 (dat is Reden<arrow.to.target n="marg257"/> 3/2 van Reden<arrow.to.target n="marg258"/> 2/1) daer rest <arrow.to.target n="marg259"/> de reden van BG tot GE: Maer treckende Reden<arrow.to.target n="marg260"/> 2/3 van Reden<arrow.to.target n="marg261"/> 2/1 daer blijft Reden<arrow.to.target n="marg262"/> 3/1. </s><s>BG dan is tot GE, als 3 tot 1.</s>

<s>

<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 16. 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 type="other"/>MAER<emph.end/> soo des ghegheuen naeldens grondt een vierhouck waer, <figure/> 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 16. 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>

Line 1700

<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 9. 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 <figure/> 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 9. 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>

Line 1709

<margin.target id="marg267"/>Pyramis truncata.

</s>

<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'GHEGHEVEN<emph.end/>. </s><s>Laet ABCDEF een ghecorte naelde sijn, diens <figure/> 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 1. voorstel des 1. 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>

Line 1717

<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 <arrow.to.target n="marg268"/>nalden 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 17. 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 <arrow.to.target n="marg268"/>nalden dieder ten weynichsten ende bequamelicxt <figure/> 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 17. 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>

Line 1733

<s> Het swaerheyts middelpunt des rechten branders inden as te wesen is duer ghemeene wetenschap openbaer, wy sullen dan alleenelick t'voorbeelt stellen des gheens diens as opden grondt cromhouckich is.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brander wesen diens grondt BC sy, ende den as AD daerop cromhouckich. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet <figure/> ABC een brander wesen diens grondt BC sy, ende den as AD daerop cromhouckich. </s>

<s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dattet swaerheyts middelpunt in AD is. </s>

Line 1757

<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> <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 1, alsoo <pb pagenum="93"/>t'rondt BC tottet rondt IK <figure/> <arrow.to.target n="marg275"/>, soo sal den pilaer BG sulcken reden hebben tot den pilaer IL <figure/> <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>

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

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

</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 2. 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>Laet ons den brander ABC noch eenmael snien duer de middelen <figure/> 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 2. 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>

Line 1788

<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> 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 19. voorstel, is t'begheerde swaerheyts middelpunt, t'welck wy bewysen moesten. </s>

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