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version 1.18, 2003/09/04 17:24:06

version 1.36, 2003/09/08 12:33:37

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<cvs_version/>

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

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

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</s>

<s>

<margin.target id="marg2"/>Arithmetica & Geometria.

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

<s> <emph type="other"/>LAETAB<emph.end/>, 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 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"/>ALSE<emph.end/>F, der 8 bepaling wort Handthaef ghenoemt.</s>

<s> <emph type="other"/>ALS<emph.end/> EF, der 8 bepaling wort Handthaef ghenoemt.</s>

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

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

<s> <emph type="other"/>ALSE<emph.end/>, der 8 bepaling wort Vastpunt gheseyt.</s>

<s> <emph type="other"/>ALS<emph.end/> E, der 8 bepaling wort Vastpunt gheseyt.</s>

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

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

<s> <emph type="other"/>ALSA<emph.end/> ende B, inde form der 8 bepaling, tsy haer eyghenwichten euen ofte oneuen sijn, wy noemen die Euestaltwichtighe, ouermidts sy naer de ghestalt euewichtich sijn, want A doet anden balck door tghestelde soo grooten ghewelt als B, ende B als A. </s><s>Dese Euestaltwichticheydt dient nootsaeckelick verstaen, ende onderscheyden vande Eueneyghenwichticheydt, anghesien dit al wat anders is als dat, want om by voorbeelt daer af te spreken, tghewicht ande cortste sijde des onsels hanghende, is somtijts thienmael swaerder als tander, nochtan hebben sy een ghelaet van euewichticheyt, maer ten is niet eyghen, dan alleenlick na de ghestalt. </s>

<s> <emph type="other"/>ALS<emph.end/> A ende B, inde form der 8 bepaling, tsy haer eyghenwichten euen ofte oneuen sijn, wy noemen die Euestaltwichtighe, ouermidts sy naer de ghestalt euewichtich sijn, want A doet anden balck door tghestelde soo grooten ghewelt als B, ende B als A. </s><s>Dese Euestaltwichticheydt dient nootsaeckelick verstaen, ende onderscheyden vande Eueneyghenwichticheydt, anghesien dit al wat anders is als dat, want om by voorbeelt daer af te spreken, tghewicht ande cortste sijde des onsels hanghende, is somtijts thienmael swaerder als tander, nochtan hebben sy een ghelaet van euewichticheyt, maer ten is niet eyghen, dan alleenlick na de ghestalt. </s>

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

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<s> <emph type="other"/>ALS<emph.end/> de Heflini ende Daellini CB der I ende 3 form vande I2 bepaling, om dat sy door t'ghestelde rechthouckich sijn op den sichteinder, wy noemen die Rechtheflini, en dese Rechtdaellini, ende haer ghewichten E Rechthefwicht, Rechtdaelwicht: Maer wesende de Heflini ofte Daellini CB, scheefhouckich opden sichteinder, als inde 2 ende 4 form, dan heeten wy die Scheefheflini, ende dese Scheefdaellini, ende haer ghewichten E Scheefhwicht, Scheefdaelwicht. </s>

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

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

<s> <emph type="other"/>WAER<emph.end/> <arrow.to.target n="marg53"/>Horizon<arrow.to.target n="marg54"/>Finitor<arrow.to.target n="marg55"/>terminator visus<arrow.to.target n="marg56"/>{Astrologia.}<arrow.to.target n="marg57"/></s>

<s>

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<margin.target id="marg61"/> en isser gheen ander, dan dat door des weerelts middelpunt strect.

</s>

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

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

<s> <emph type="other"/>DE<emph.end/> <arrow.to.target n="marg62"/>{Columna.}<arrow.to.target n="marg63"/>{Geometriæ.}<arrow.to.target n="marg64"/></s>

<s>

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<s>Stof Form Daet Grondt Ancleuing Gheslacht Afcomst Bepaling Voorstel Eysch Vertooch Reden Everedenheyt Even Ghelijcke Voorbeelt Swaerheytsmiddelpunt As Middellini Omtreck Euewydeghe Lijckstandighesijden Vlack Plat Pilaer Telconst Meetconst Wisconst Wisconstnaer Wisconstlick. </s>

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

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

<s>

<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.

</s>

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

<s> <emph type="other"/>NGHESIEN<emph.end/> sommige saken als beghinselen door ghemeene wetenschap bekendt sijn, ende gheen bewijs en behouven, Ander bedectelicker den berispers tot stof souden dienen, om te straffen t'ghene gheen straf en verdient, Wy sullen naer Wisconstenaers ghebruyck <arrow.to.target n="marg67"/>, eer wy tot de voorstellen commen, begheeren dat ons alsulcke toeghelaten worden.</s>

<s> <emph type="other"/>ANGHESIEN<emph.end/> sommige saken als beghinselen door ghemeene wetenschap bekendt sijn, ende gheen bewijs en behouven, Ander bedectelicker den berispers tot stof souden dienen, om te straffen t'ghene gheen straf en verdient, Wy sullen naer Wisconstenaers ghebruyck <arrow.to.target n="marg67"/>, eer wy tot de voorstellen commen, begheeren dat ons alsulcke toeghelaten worden.</s>

<s>

<margin.target id="marg67"/>Mathematicorum more.

</s>

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

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

<s> <emph type="other"/>WY<emph.end/> begheeren datmen toelate euen ghewichten an euen ermen oock euestaltwichtich te sijne. </s>

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

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

<s> <emph type="other"/>ENDE<emph.end/> alle hanghende linien voor evewydighe <arrow.to.target n="marg69"/> ghehouden te worden. </s>

<s>

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

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

VAND<emph type="other"/>EV<emph.end/>OORSTELLEN <lb/>

<emph type="other"/>VANDE VOORSTELLEN.<emph.end/> <lb/>

</s>

<s>I. <emph type="other"/>VERTOOCH<emph.end/>. <arrow.to.target n="marg72"/> I. <emph type="other"/>VOORSTEL<emph.end/>. <arrow.to.target n="marg73"/> </s>

<s><emph type="other"/>I. VERTOOCH<emph.end/>. <arrow.to.target n="marg72"/> <emph type="other"/>I. VOORSTEL<emph.end/>. <arrow.to.target n="marg73"/> </s>

<s>

<margin.target id="marg72"/>Theorema.

<margin.target id="marg73"/>Propositio.

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<s> <emph type="other"/>WESENDE<emph.end/> twee euestaltwichtighe swaerheden, de swaerste heeft sulcken reden tot de lichtste, als den langsten erm tot den cortsten.</s>

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

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

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

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

<s>

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

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

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<margin.target id="marg77"/>Demonstratio.

</s>

<s> <emph type="other"/>MAER<emph.end/> op datmen niet en dencke dit daer also by gheualle ghesciedt te sijne, wy sullender Wisconstich bewys <arrow.to.target n="marg78"/> af doen aldus: </s>

<s> <emph type="other"/>AER<emph.end/> op datmen niet en dencke dit daer also by gheualle ghesciedt te sijne, wy sullender Wisconstich bewys <arrow.to.target n="marg78"/> af doen aldus: </s>

<s>

<margin.target id="marg78"/>Mathematicam demonstrationem.

</s>

<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>

<margin.target id="marg81"/>I. Ghestalt.

</s>

<s> <emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck het lichaem ofte de swaerheydt (twelck hier een selfde is om haer eueredenheydt <arrow.to.target n="marg79"/>, want ghelijck tlichaem EFDA, tot tlichaem EFCB, alsoo diens swaerheyt tot desens, ouermits den pilaer door tghestelde oueral eenuaerdigher swaerheyt is) van EFDA, tot EFCB, alsoo den langsten erm ML, tot den cortsten MK. </s><s><emph type="other"/>TBEWYS<emph.end/>, <emph type="other"/>ILIDT<emph.end/>. </s><s>MH is euen an MG door tghegheuen, laet tot elck doen KM, soo sal dan KH euen sijn an MG met KM; daer naer van d'eene ghetrocken GK, ende van d'ander KI (welcke GK ende KI euen sijn door tghegheuen) soo sal KM met KM euen blijuen an IH; Ende haer helften als KM ende IL sullen oock euen sijn. </s><s><emph type="other"/>IILIDT<emph.end/>. </s><s>Laet tot elck (te weten KM ende IL) doen MI, Ende ML sal euen sijn an IK. </s><s><emph type="other"/>IIILIDT<emph.end/>. </s><s>Ghelijck GI tot haer helft KI, also IH tot haer helft IL, ende door oueranderde eueredenheyt <arrow.to.target n="marg80"/> ghelijck GI tot IH, also KI tot IL, maer KI is euen an ML door het 2 lidt, ende IL an MK door het I lidt, daerom ghelijck GI tot IH, alsoo ML tot MK; Maer ghelijck GI tot IH, also het lichaem ofte de swaerheyt EFDA, tot EFCB. </s><s>Ghelijck dan de swaerste swaerheyt EFDA, tot de lichtste EFCB, also den langsten erm ML, tot den cortsten MK.</s>

<s> <emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck het lichaem ofte de swaerheydt (twelck hier een selfde is om haer eueredenheydt <arrow.to.target n="marg79"/>, want ghelijck tlichaem EFDA, tot tlichaem EFCB, alsoo diens swaerheyt tot desens, ouermits den pilaer door tghestelde oueral eenuaerdigher swaerheyt is) van EFDA, tot EFCB, alsoo den langsten erm ML, tot den cortsten MK. </s><s><emph type="other"/>TBEWYS<emph.end/>, <emph type="other"/>I. LIDT<emph.end/>. </s><s>MH is euen an MG door tghegheuen, laet tot elck doen KM, soo sal dan KH euen sijn an MG met KM; daer naer van d'eene ghetrocken GK, ende van d'ander KI (welcke GK ende KI euen sijn door tghegheuen) soo sal KM met KM euen blijuen an IH; Ende haer helften als KM ende IL sullen oock euen sijn. </s><s><emph type="other"/>II. LIDT<emph.end/>. </s><s>Laet tot elck (te weten KM ende IL) doen MI, Ende ML sal euen sijn an IK. </s><s><emph type="other"/>III. LIDT<emph.end/>. </s><s>Ghelijck GI tot haer helft KI, also IH tot haer helft IL, ende door oueranderde eueredenheyt <arrow.to.target n="marg80"/> ghelijck GI tot IH, also KI tot IL, maer KI is euen an ML door het 2 lidt, ende IL an MK door het I lidt, daerom ghelijck GI tot IH, alsoo ML tot MK; Maer ghelijck GI tot IH, also het lichaem ofte de swaerheyt EFDA, tot EFCB. </s><s>Ghelijck dan de swaerste swaerheyt EFDA, tot de lichtste EFCB, also den langsten erm ML, tot den cortsten MK.</s>

<s>

<margin.target id="marg79"/>Proportionem.

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

</s>

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

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

<s><emph type="other"/>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>

<margin.target id="marg81"/>I. Ghestalt.

</s>

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

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

<s> <emph type="other"/>MEN<emph.end/> can tvoorgaende oock bethoonen, blyuende de twee swaerheden hanghende an eenen lichamelicken balck, in deser voughen: Laet den pilaer ABCD ghesneen sijn in twee deelen, met een plat door den as EF, ende den as des ondersten deels EC sy GH, ende EC sy doorsneen met een plat IK euewydich vanden grondt ED, sniende den as GH in L, ende het swaerheydts middelpunt van het deel IKDE sy M int middel van GL, ende van het deel IKCF sy N int middel van LH, ende des heels ABCD sy O in middel van EF, ende OP sy swaerheydts middellini des heels ABCD, ende MQ van IKDE, ende NR van IKCF. </s><s> Dit soo sijnde tis kennelick dat des heels pilaers rechter sijde, euewichtich is teghen haer slincker. </s><s><emph type="other"/>LAET<emph.end/> ons nu het onderste deel EFCD neertrecken, also dat het blyue hanghende ande linien ML [MQ] ende NR, als hier neuens. </s><s>Ende is openbaer dat den lichamelicken balc ABFE noch in haer eerste ghestalt sal blyuen. </s><s>Laet ons nu achten dat het deel IKDE, ghesneen sy van IKCF, ende dat elck deel vallen mach daert wil, maer sy hanghen an haer swaerheyts <pb pagenum="15"/>middelpunten M, N, sy houden dan haer eerste ghegheuen ghestalt door de bepaling, daerom ABFE blijft oock noch in sijn eerste ghedaente. </s>

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

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

<s>

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<s> <emph type="other"/>VYT<emph.end/> het verkeerde des voorgaenden voorstels volcht, dat hebbende de swaerste swaerheydt sulcken reden tot de lichtste, als den langsten erm tot den cortsten, dat die twee swaerheden euestaltwichtich sijn.</s>

<s>I. <emph type="other"/>EYSCH<emph.end/>. <arrow.to.target n="marg86"/> <emph type="other"/>II. VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>I. EYSCH<emph.end/>. <arrow.to.target n="marg86"/> <emph type="other"/>II. VOORSTEL<emph.end/>.</s>

<s>

<margin.target id="marg86"/>Problema.

</s>

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<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen bekende swaerheden, haer handthaef te vinden.</s>

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

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

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

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

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

<s> E swaerheydts middellini van ABCDZ is ÆW door het 3 voorbeelt, alsoo dat SÆ doet / van SR, ende de swaerheydts middellini van Y is XN, ende ÆX is balck, de selue salmen in tween deelen, alsoo dat de sticken de reden hebben als 30 lb van ABCDZ, tot 6 lb van Y: welverstaende tcortste stick naer de swaerheydts middellini van tswaerste deel, twelck vallen sal in S, inder voughen dat SG wederom de begheerde handthaef is als vooren.</s>

<s><emph type="other"/>DE<emph.end/> swaerheydts middellini van ABCDZ is ÆW door het 3 voorbeelt, alsoo dat SÆ doet / van SR, ende de swaerheydts middellini van Y is XN, ende ÆX is balck, de selue salmen in tween deelen, alsoo dat de sticken de reden hebben als 30 lb van ABCDZ, tot 6 lb van Y: welverstaende tcortste stick naer de swaerheydts middellini van tswaerste deel, twelck vallen sal in S, inder voughen dat SG wederom de begheerde handthaef is als vooren.</s>

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

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

<s> E swaerheydts middellini van YZ, is (door het eerste voorbeelt) , alsoo dat S doet / van SR, ende de swaerheydts middellini vande pilaer is TI, ende T is balck: de selue salmen in tween deelen, also dat de sticken de reden hebben als 30 lb van Y met Z, tot 6 lb vande pilaer, te weten tcortste stick naer de swaerheydts middellini van tswaerste deel, twelck vallen sal in S, inder voughen dat SG wederom de begheerde handthaef is als vooren.</s>

<s><emph type="other"/>DE<emph.end/> swaerheydts middellini van YZ, is (door het eerste voorbeelt) , alsoo dat S doet / van SR, ende de swaerheydts middellini vande pilaer is TI, ende T is balck: de selue salmen in tween deelen, also dat de sticken de reden hebben als 30 lb van Y met Z, tot 6 lb vande pilaer, te weten tcortste stick naer de swaerheydts middellini van tswaerste deel, twelck vallen sal in S, inder voughen dat SG wederom de begheerde handthaef is als vooren.</s>

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

<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"/>TWERCK<emph.end/>. </s><s>De swaerheydts middellini van ABCDYZ is SG door het 4 voorbeelt, ende van Æ is QB, ende SQ is balck: de selue salmen in tween deelen, also dat de sticken de reden hebben als 36 lb vanden pilaer met Y <pb pagenum="18"/>ende Z, tot 12 lb van Æ, te weten tcortste stick naer de swaerheydts middellini van tswaerste deel, twelck vallen sal in T, inder voughen dat TI de begheerde handthaef sal sijn. </s><s>Ende soomen noch hinghe an P 24 lb, d'handthaef soude SG sijn, ende so voorts met allen anderen swaerheden diemen anden pilaer soude mueghen hanghen. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>De swaerste swaerheydt A int eerste voorbeelt, heeft sulcken reden tot de lichtste B, als den langsten erm ED, tot den cortsten EC, daerom EF door de bepaling is d'hanthaef. </s><s>Sghelijcx sal oock tbewys sijn van al dander voorbeelden, twelck wy om de cortheydt achterlaten. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen bekende swaerheden, wy hebben haer handthaef gheuonden naer den eysch. </s>

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

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

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

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

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

<s>

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

</s>

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

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

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen twee euestaltwichtighe swaerheden, d'een bekent dander onbekent, ende d'hanthaef: Die onbekende bekent te maken. </s>

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<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet inde form des 2 voorbeelts van het 2 voorstel den pilaer ABCD voor d'een swaerheyt weghen 6 lb, ende dander onbekende swaerheyt sy tghewicht daer an hanghende Y, ende d'hanthaef sy XN. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten tghewicht van Y bekent maken. </s><s>[ Figuur ingevoegd ] <emph type="other"/>TWERCK<emph.end/>. </s><s>Anghesien TI swaerheydts middellini is des pilaers, ende QB van Y, so sal TQ balck sijn, diens cortsten erm XQ, ende langsten XT; Daerom salmen ondersoucken wat reden den erm XQ, heeft tot XT, wort beuonden neem ick, als van 1 tot 2. </s><s>Ich seg dan, XQ 1, gheeft XT 2, wat den pilaer 6 lb ? comt voor Y 12 lb. </s><s>Der ghelijcke voorbeelden mochten wy hier stellen op dander formen des 2 voorstels, ten waer die door de voorgaende kennelick ghenouch sijn. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet B int eerste voorbeelt, soot mueghelick waer, swaerder sijn dan 1 lb, de swaerste swaerheydt dan en sal niet sulcken reden hebben tot de lichtste, als den langsten erm tot den cortsten; twelck teghen het I voorstel is; B dan en is niet swaerder dan 1 lb. </s><s>Sghelijcx salmen oock bethoonen dat sy niet lichter en is, sy weeght dan effen 1 lb, twelck wy bewysen moesten. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen twee euestaltwichtighe swaerheden, d'een bekent dander onbekent, ende d'hanthaef: Wy hebben die onbekende bekent ghemaect, naer den eysch. </s>

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

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

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

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

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

<s><emph type="other"/>IIII. EYSCH<emph.end/>. V. <emph type="other"/>VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>IIII. EYSCH<emph.end/>. <emph type="other"/>V. VOORSTEL<emph.end/>.</s>

<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 middelpunt <arrow.to.target n="marg88"/> 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 /. </s><s> <emph type="other"/>MERCKT<emph.end/>. </s><s>Ghelijck de Meetconstighe ende Telconstighe voorstellen <arrow.to.target n="marg89"/> 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>

wesen, diens as EF, ende haer middelpunt <arrow.to.target n="marg88"/> 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 Meetconstighe ende Telconstighe voorstellen <arrow.to.target n="marg89"/> 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>

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

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

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<s> Ende van het tweede punt I (van het tweede om dat 2 het ander der ghegeuen ghetalen is) salmen den pilaer ophangen byde swaerheyts, middellini IN; Daer naer salmen an tvijfde punt M een ghewicht hanghen als O, euen so swaer dat alles in euestaltwichticheyt sy, twelck soo wesende, ick seg dat tghewicht van O, in sulcken reden is tot tghewicht des pilaers, als 2 tot 3, ofte dat O euen is ande / des pilaers. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>G is swaerheydts middelpunt <arrow.to.target n="marg90"/> des pilaers ABCD, ende MP swaerheyts middellini van O, daerom ghelijck den erm IG tot den erm IM, alsoo O tot den pilaer door het voorstel, maer IG heeft sulcken reden tot IM, als 2 tot 3, daerom O heeft sulcken reden tot den pilaer, als 2 tot 3, twelck wy bewysen moesten. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een pilaer, wy hebben gheuonden een ghewicht in ghestelde reden tot des pilaers ghewicht, naer den eysch. </s><s><emph type="other"/>MERCKT<emph.end/>. </s><s>Wy souden oock mueghen voorbeelden stellen met Redenen van onmetelicke palen <arrow.to.target n="marg91"/>, maer sulcx is openbaer ghenouch door tvoorgaende, metgaders tghene wy vande onmetelicke grootheden elders ghescreuen hebben. </s>

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

<s>

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

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

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<s><emph type="other"/>IIVERTOOCH. VIVOORSTEL<emph.end/>.</s>

<s><emph type="other"/>II VERTOOCH <emph.end/>. <emph type="other"/>VI VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> een hanghende pilaer ghesneen door sijn swaerheydts middelpunt, met een plat euewydich vanden gront, ende wesende tvastpunt in dat plat bouen het swaerheyts middelpunt: Den as des pilaers blijft euewydich vanden sichteinder. </s><s><arrow.to.target n="marg92"/></s>

<s>

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

</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"/>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"/>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, </s>

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

<s> 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>Tis oock te anmercken als voor ghemeenen Weegconstighen Reghel, dat<arrow.to.target n="marg93"/> Maer tswaerheydts middelpunt hier bouen E en is inde tweede form niet in sijn swaerheydts middellini IO, tis dan een onmueghelicke ghestalt. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan een pilaer ghesneen, &c. </s>

<s>Alle swaerheyts middelpunt eens hanghenden lichaems is in siin swaerheydts middellini.</s>

<s>

<margin.target id="marg93"/>Alle swaerheyts middelpunt eens hanghenden lichaems is in siin swaerheydts middellini.

<s>Maer tswaerheydts middelpunt hier bouen E en is inde tweede form niet in sijn swaerheydts middellini IO, tis dan een onmueghelicke ghestalt. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan een pilaer ghesneen, &c. </s>

</s>

<s><emph type="other"/>IIIVERTOOCH. VIIVOORSTEL<emph.end/>.</s>

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

<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"/>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, </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> 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 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"/>IIIIVERTOOCH. VIIIVOORSTEL<emph.end/>.</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 plat <arrow.to.target n="marg94"/> 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 sichteinder <arrow.to.target n="marg95"/> 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 Wisconstelick <arrow.to.target n="marg96"/> ghenomen soo can sy daer op rusten. </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 Wisconstelick <arrow.to.target n="marg96"/> ghenomen soo can sy daer op rusten. </s>

<s>

<margin.target id="marg94"/>Plano.

<margin.target id="marg95"/>Horizonte.

<margin.target id="marg96"/>Mathematice.

</s>

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

<s>A. Al dat ligt moet grondt hebben daert op rust, <lb/>E. Dees pilaer en heeft gheen grondt daer hy op rust, <lb/>E. Dees pilaer dan en can soo niet ligghen.

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

<s>

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

</s>

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

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

<s>

<margin.target id="marg98"/> A. Al dat ligt moet grondt hebben daert op rust, E. Dees pilaer en heeft gheen grondt daer hy op rust, E. Dees pilaer dan en can soo niet ligghen.

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

</s>

<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> <emph type="other"/>YEMANT<emph.end/> <arrow.to.target n="marg99"/></s>

<s>

<margin.target id="marg99"/>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"/>IMERCK<emph.end/>.</s>

<s> <emph type="other"/>SOO<emph.end/> <arrow.to.target n="marg100"/>{Mathematicè.}<arrow.to.target n="marg101"/></s>

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

<s>

<margin.target id="marg100"/>yemant thinhondt [t inhoudt] der voorgaende drie voorstellen door eenighe eruaring wilde sien, hy mocht nemen een reghel van houdt ofte ander stof eenvaerdigher dickte ende swaerheyt, als ABCD, teeckenende de punten E, F, G, H, inde middelen der linien AB, BC, CD, DA, treckende EG, ende HF, malcander sniende in I,maeckende daer naer een seer cleen gaetken en I, ende daer bouen een gaetken als K, ende onder I een gaetken als L. Ende stekende een naelde door tgatken K, die vrielick daerin drayen mach, d'eruaring sal bethoonen dat HF altijdt euewydich sal blijuen vanden sichteinder. Maer de naelde in I stekende, de reghel sal daerop alle ghestalt houden diemen haer gheeft. Ende de naelde in L ghesteken, alles sal omkeeren ouer <pb pagenum="25"/>die syde daert eerst beghint, tot dat I is in haer swaerheyts middellini, waer af d'oirsaeck inde voornoemde , 7, 8, voorstellen Wisconstlick

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

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

</s>

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

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

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

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

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

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

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet <emph type="other"/>AB<emph.end/> twee swaerheden sijn ende haer middellinien CD, EF, ende haer CE, ende d'hanthaef GH, inder voughen dat CG is tot GE, als de swaerheydt B tot A, Laet IK noch een balck wesen, oneuewydich van CE, ende laet GH oneindelick voortghetrocken worden naer L, sniende den balck IK in M. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat IM ende MK, oock de ermen sijn der swaerheden <emph type="other"/>AB<emph.end/>; 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><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>D'handthaef dan oneindelick voortghetrocken, deelt alle balcken tweer swaerheden in haer ermen, twelck wy bewysen moesten.</s>

<s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>D'handthaef dan oneindelick voortghetrocken, deelt alle balcken tweer swaerheden in haer ermen, twelck wy bewysen moesten.</s>

<s>

<margin.target id="marg102"/>2 v 6 B. E.

<margin.target id="marg103"/>34 v. <emph type="other"/>IB<emph.end/>. E.

<margin.target id="marg103"/>34 v. IB. E.

</s>

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

<s> <emph type="other"/>HIER<emph.end/> uyt blijct datmen om te vinden de swaerheydts middellini tweer swaerheden, niet nootsaeckelick en moet nemen een euewydighe <arrow.to.target n="marg104"/> vanden sichteinder <arrow.to.target n="marg105"/>, maer alsulcke alsmen wil, ende alst best te pas comt.</s>

<s>

Line 948

Line 926

</s>

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

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

<s> <emph type="other"/>ANGHESIEN<emph.end/> alle swaerheydts middelpunt inde swaerheyts middellini is, soo volght dat alle rechte lini begrepen tusschen twee swaerheydts middelpunten, oock dier swaerheden balck is, ende het onderscheydt der ermen diens balcx, oock het swaerheydts middelpunt te wesen der twee swaerheden. </s>

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

<s>5. <emph type="other"/>EYSCH<emph.end/>. I 0. <emph type="other"/>VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>5. EYSCH<emph.end/>. <emph type="other"/>I0. 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 sichteinder <arrow.to.target n="marg106"/> , 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.

Line 968

Line 941

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 4 lb, ende ghesneen door sijn swaerheydts middelpunt E, met een plat FG euewydich vanden grondt AD, ende laet H vastpunt wesen beneden tmiddelpunt E int middel van EG; Ende anden pilaer twee ghewichten hanghen als I, K, elck weghende 4 lb, welcker middellinien vastpunten sijn D, C, ende laet LM den as, ende NO sichteinder wesen. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten vinden of den as LM euewydich sal connen blijuen vanden sichteinder NO; ofte alle gestalt houden diemen haer gheeft; Ofte ommekeeren tot dat haer swaerheydts middelpunt E is inde swaerheyts middellini door H, welcke verscheydenheden vallen connen naer de reden der swaerheyt des pilaers, tot de ghewichten dieder anhangen. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal trecken door E de swaerheyts middellini PQ des pilaers, daer naer door <pb pagenum="27"/>G de swaerheyts middellini RS der ghewichten I, K, ende EG sal balck sijn, daer naer salmen sien door het voorstel waer tvastpunt der hanthaef valt: want commet onder H, soo keert LM tot sy euewydich blijft vanden sichteinder NO; Maer commet in H, sy houdt alle ghestalt die men haer gheeft; Commet bouen H, alles keert om. </s>

<s><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> Maer den pilaer weeght 4 lb, ende I, K, elck 4 lb, tsamen 8 lb door tghegheuen, daerom ghedeelt EG in T, alsoo dat ET, sulcken reden heb tot TG, als 8 tot 4: Ick seg dat LM keeren sal (ouermits T onder H comt) tot sy euewydich is vanden sichteinder. </s><s>Laet nu den pilaer weghen 4 lb, ende I en K elck 2 lb, tsamen 4 lb, daerom ghedeelt EG in H (welcke H tmiddel van EG is door tghegheuen) alsoo dat EH sulcken reden heb tot HG, als 4 tot 4: ick seg dat LM (ouermits het in H viel) alle ghestalt sal houden diemen haer gheeft. </s><s>Laet nu den pilaer weghen 4 lb, ende I, K elck I lb, tsamen 2 lb, daerom ghedeelt EG inV, also dat EV sulcken reden hebbe tot VG, als 2 tot 4, Ick seg dat den pilaer met al de rest omkeeren sal (ouermitds V bouen H comt) tot dat H is in haer swaerheydts middellini. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ten eersten I en K elck 4 lb weghende, dat dan LM keert tot sy euewydich is vanden sichteinder, blijct aldus: De hanghende door T ghelijck TX, is swaerheyts middellini des heels, daerom die latende, ende hanghende tgheheel ande hanghende <arrow.to.target n="marg107"/> door H, als HY (welcke H ons ghegheuen vastpunt is) so sal de sijde naer BCK, swaerder sijn dan naer ADI, daerom oock sal de sijde BCK neerdalen, tot dat H inde swaerheydts middellini is des heels, ende dan sal LM euewydich sijn vanden sichteinder NO. </s><s>Ten tweeden I, K, elck 2 lb weghende, dat dan LM alle ghestalt houdt, wordt aldus bethoont: Laet ons achten dat I ende K opgheschorst sijn, alsoo dat D tswaerheydts middelpunt sy van I, ende C van K, ende door de begheerte sy en sullen anden pilaer gheen oirsaeck van verandering der swaerheydt wesen; Twelck soo sijnde, H is tswaerheyts middelpunt van soodanighen lichaem vergaert uyt den pilaer ende de twee ghewichten <emph type="other"/>IK<emph.end/>, ende door de bepaling tsal daer op alle ghestalt houden diemen hem gheeft, tselfde sal also bewesen worden in alle ghestalten daermen LM in soude connen stellen. </s><s>Ten laetsten I, K, elck I lb weghende, dat dan alles omkeert, wort aldus bethoont: De hanghende door V ghelijck VZ, is swaerheyts middellini des heels, daerom die latende, ende hanghende tgheheel ande hanghende HY door H ghegheuen vastpunt, so sal de sijde naer ADI, swaerder sijn dan naer BCK, daerom oock sal de sijde ADI neerdalen, tot dat H inde swaerheyts middellini is des heels, ende ofmen schoon LM (alles op tvastpunt H draeyende) euewydich stelde vanden sichteinder NO, sy en can so niet blyuen door het voorstel, maer alles sal omkeeren, twelck wy bewysen moesten. </s>

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

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

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

<s>

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

</s>

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<s> <emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een vastpunt des bekenden pilaers, &c.</s>

<s> Uyt het voorgaende is ghenouch blijckelick den ghemeenen voortganck in allen anderen, als van pilaren welcker vastpunt is buyten de lini als FG, ende der ghewichten vastpunten op ander plaetsen dan <emph type="other"/>DC<emph.end/>; Maer ouermits wy hier voornamelick trachten de oirsaecken vande ghedaenten des s grondelick te openbaren (daer af inde Weeghdaet breeder sal gheseyt worden) so en gheuen wy van sulcke ongheschicte ghestaltheden gheen besonder voorbeelden.</s>

<s> Uyt het voorgaende is ghenouch blijckelick den ghemeenen voortganck in allen anderen, als van pilaren welcker vastpunt is buyten de lini als FG, ende der ghewichten vastpunten op ander plaetsen dan DC; Maer ouermits wy hier voornamelick trachten de oirsaecken vande ghedaenten des s grondelick te openbaren (daer af inde Weeghdaet breeder sal gheseyt worden) so en gheuen wy van sulcke ongheschicte ghestaltheden gheen besonder voorbeelden.</s>

<s>6. <emph type="other"/>EYSCH. II. VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>6. EYSCH<emph.end/>. <emph type="other"/>11. VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een bekende pilaer, ende bekende swaerheden daer an hanghende: Te vinden het vastpunt daer op hy alle ghestalt houdt diemen hem gheeft. </s>

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

<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>

<margin.target id="marg108"/>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>

<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 <arrow.to.target n="marg109"/>an haer ghewichten hinghe euerednich

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

met de ermen, dat E noch altijt tvastpunt sal blijuen daer sy alle ghestalt op houden diemen haer gheeft.</s>

<s> <emph type="other"/>TIS<emph.end/> <arrow.to.target n="marg109"/>{Proportionales.}<arrow.to.target n="marg110"/></s>

<s>

<margin.target id="marg109"/>openbaer dat wesende de twee vastpunten der ghewichten als <emph type="other"/>CD<emph.end/>, ende tvastpunt des handthaefs als E, alle drie in een rechte lini als hier bouen, ende an <emph type="other"/>CD<emph.end/> 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 <emph type="other"/>CED<emph.end/> in een rechte lini wesende C ende D niet euewyt en waren van E, ende datmen an haer ghewichten hinghe euerednich

<margin.target id="marg109"/>Proportionales.</s>

<margin.target id="marg110"/> met de ermen, dat E noch altijt tvastpunt sal blijuen daer sy alle ghestalt op houden diemen haer gheeft.

</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 voorstel, dat is ghelijck FI lb, tot H 4 lb, also den erm KI, tot KG, daer naer salmen trecken EK balck des pilaers ter eender, ende der ghewichten FH ter ander sijden, de selue EK ghedeelt in L, also dat den erm EL sulcken reden hebbe tot LK, als 5 lb van FH, tot I0 lb des pilaers, L sal tbegheerde punt sijn op twelck sy alle ghestalt sullen houden diemen haer gheeft, waer af tbewys openbaer is door het voorstel.</s>

<s><emph type="other"/>VII. EYSCH<emph.end/>. <emph type="other"/>XII. VOORSTEL<emph.end/>.</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 <emph type="other"/>FH<emph.end/>, daer naer salmen vinden haer ermen door het voorstel, dat is ghelijck <emph type="other"/>FI<emph.end/> lb, tot H 4 lb, also den erm KI, tot KG, daer naer salmen trecken EK balck des pilaers ter eender, ende der ghewichten <emph type="other"/>FH<emph.end/> ter ander sijden, de selue EK ghedeelt in L, also dat den erm EL sulcken reden hebbe tot LK, als 5 lb van <emph type="other"/>FH<emph.end/>, tot I0 lb des pilaers, L sal tbegheerde punt sijn op twelck sy alle ghestalt sullen houden diemen haer gheeft, waer af tbewys openbaer is door het voorstel.</s>

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

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een bekende pilaer, met sijn vastpunt ende bekende ghewichten daer an hanghende die den as euewydich houden vanden sichteinder:<arrow.to.target n="marg111"/> Te vinden een ghewicht hanghende ter begheerder plaets des pilaers, dat den as in ghegheuen ghestalt houde. </s>

<s>

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

<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> 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><pb pagenum="31"/><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal vinden door het voorstel, tvastpunt daer op den as alle ghestalt houde diemen haer gheeft twelck N sy: Daer naer salmen trecken DN, ende de hanghende <arrow.to.target n="marg112"/> EO, sniende ND in O, daer naer salmen sien wat reden NO heeft tot OD, ick neme als van I tot 2, daerom hanghe ick an D een ghewicht P van 6 lb, te weten in sulcken reden tot den pilaer met de twee ghewichten G, I, al tsamen I2 lb, als van I tot 2; Ick seg P 6 lb, te wesen het begheerde ghewicht. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Tswaerste ghewicht I2 lb des erms ON, heeft sulcken reden tot het lichtste 6 lb des erms OD, ghelijck den langsten erm OD, tot den cortsten ON; Daerom hanghet al euestaltwichtich ande handthaef EF door het voorstel. </s><s>Ende veruolghens den as IK blijft in haer ghegheuen ghestalt.</s>

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

<s>

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

</s>

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<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 sichteinder <arrow.to.target n="marg113"/> NO, ende P sy een punt inden pilaer voor de begheerde plaets. </s>

<s>

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

</s>

<s> Daer naer werdt den as LM (alles draeyende op E) verheuen, als in de tweede form.</s>

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

<s> Wy moeten een ghewicht an P vinden, dat den as LM in die ghestalt houde.</s>

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

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

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

<s>

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