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version 1.20, 2003/09/04 18:12:57

version 1.51, 2003/09/08 15:32:48

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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> Tis te weten dat de Duytschen in die seer oude tijden vande welcke ter weerelt gheen opentlicke schriften gebleuen en sijn, gheweest hebben een treffelick seermachtich Gheslacht. </s><s>t'welck duer sulcke verderuende oirsaken als meer ander machtighe volcken weeruaren sijn, als oirloghe en dierghelijcke, voorderende uytroeying der wetten, breking van goe oirdens, verwoesting der steden, &c, tot manier van wiltheyt gherocht is, doch niet so volcommen, of den ouden aert der grootmoedicheyt, rechtueerdicheyt, ende ghetrauheyt, daer Tacitus oock af betuycht <arrow.to.target n="marg11"/>, en bleef altijt in hemlien ghewortelt. </s><s>Dese haer woestheyt heeft gheduert tot ontrent de tijden van Iulius Cæsar, welcke daer naer tot beteren staet begon te keeren, soo dat sy eintlick weder ghecommen sijn ter regiering ouer t'eertrijcxdeel Europa, als kenlick is. </s><s>Maer want yemandt van haer voornomde eerste macht twyffelen mocht, ouermits

<pb/>wy daer af, soo weynich als van veel ander volcken diens tijts, gheen opentlicke schriften en hebben, soo sullen wy die aldus beuestighen. </s>

<s>

<margin.target id="marg11"/>Li. de Morib. Germanor.

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

<s>

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

<margin.target id="marg49"/>Pyramides.

<margin.target id="marg50"/>Duer ons ghedacht

<margin.target id="marg51"/>Swaerheydts middelpunt eens lichaems, is door twelck alle plat, tlichaem deelt in euestaltwichtighe deelen.

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<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>

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

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

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<s>

<margin.target id="marg65"/>Materia Forma Effectus Subiectum Aiunctum Genus Species Definitio Propositio Problema Theorema Ratio Proportio Æquales Similes Exemplum Centrumgrauitatis Axis Diameter Circumferentia Parallela Homologalatera Superficies Planum Columna Arithmetica Geometria ArsMathematica Mathematicus Mathematice

</s>

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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.

<margin.target id="marg76"/>Qæsitum.

<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 2/5 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) , ΦΔ alsoo dat SΦ doet 1/5 van SR, ende de swaerheydts middellini vande pilaer is TI, ende TΦ 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>

<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><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>

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

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

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

<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.

</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 , 7, 8, voorstellen Wisconstlick blijckt.<arrow.to.target n="marg101"/></s>

<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.

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

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

</s>

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

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

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

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

<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 947

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 967

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>

Line 979

Line 955

<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"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten het vastpunt vinden daerop sy alle ghestalt houden diemen haer gheeft.</s>

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

Line 1024

Line 988

<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>

Line 1038

Line 1004

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

<s>

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

</s>

Line 1064

Line 1024

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

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

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

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

<s> Wesende dan ghegheuen een bekenden pilaer met sijn vastpunt, &c. </s>

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

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

<s> <emph type="other"/>EEN<emph.end/> daelwicht ende een hefwicht an hem euen, doen met euen houcken an euen ermen euen ghewelden. </s>

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

<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 <pb pagenum="33"/>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 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"/>IIVOORBEELT<emph.end/> met scheefwichten.</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"/>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 hanghen euen an D, diens scheefheflini CI, euewydich <arrow.to.target n="marg115"/> sy van BE, ende CB sy wat voortghetrocken tot in K. </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, euewydich <arrow.to.target n="marg115"/> 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>

<margin.target id="marg115"/>Parallela.

</s>

<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"/>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>

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

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

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

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

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

<s> <emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten een rechthefwicht an V vinden, dat den pilaer in die standt houde. </s><s><emph type="other"/>TWERCK<emph.end/>. </s><s>Men sal de lini QR voorttrecken tot in Y, also dat RY euen sy an RV: Daer naer salmen vinden tghewicht Z an Y, euestaltwichtich met den pilaer, tselue (ghedenckende dat R vastpunt is) sal van 4 lb wesen door het voorstel; Ick seg daerom dat het begheerde rechthefwicht twelck Æ sy, van 4 lb sal wesen. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ouermidts den erm RV des rechthefwichts Æ, euen is anden erm RY des ghewichts Z, ende Æ euen an Z, soo is de ghewelt Æ euen an de ghewelt van Z door het voorstel. </s><s>Maer de ghewelt van Z is (Æ gheweert sijnde) den pilaer in die standt te houden, de ghewelt dan van Æ (Z gheweert sijnde) is oock den pilaer in die standt te houden, twelck wy bewysen moesten. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een pilaer, ende twee punten inden as, t'een vast, t'ander int langste deel verroerlick: Wy hebben gheuonden een rechthefwicht an tverroerlick, dat den pilaer in sijn ghegheuen standt houdt naer den eysch. </s>

<s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ouermidts den erm RV des rechthefwichts Æ, euen is anden erm RY des ghewichts Z, ende Æ euen an Z, soo is de ghewelt Æ euen an de ghewelt van Z door het 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>

<margin.target id="marg116"/>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"/>IVERVOLGH<emph.end/>. </s>

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

<s> <emph type="other"/>NGHESIEN<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 , 2 lb rusten. </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 vougde, inde plaets des keghels , als hier neuen, dat sal weghen 2 lb. </s><s> <emph type="other"/>OFTE<emph.end/> soomen an V een keghel vougde, inde plaets des rechthefwichts Æ, als hier neuen, dat op den keghel rusten sal 2 lb, ende op den keghel 4 lb. </s><s> <emph type="other"/>OFTE<emph.end/> soomen den pilaer ophinghe an twee euewidighe <arrow.to.target n="marg117"/> linien R, ende V, als hier neuen, dat ande lini R hanghen sal 2 lb, ende ande lini V 4 lb.</s>

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

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

<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>

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

</s>

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

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

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

<s> <emph type="other"/>WESENDE<emph.end/> twee punten inden as des pilaers, t'een vast t'ander verroerlick: Trechthefwicht an tverroerlic met den pilaer euestaltwichtich, heeft sulcken reden tot den pilaer als het asstick tusschen het swaerheyts middelpunt des pilaers, ende het vastpunt, tot het asstick tusschen tvastpunt ende t'verroerlick punt. </s>

<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 Wisconstelick <arrow.to.target n="marg118"/> 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"/>TBESLVYT<emph.end/>. </s><s>Wesende dan twee punten inden as des pilaers t'een vast t'ander verroerlick, &c. </s>

<s>

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

</s>

<s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan twee punten inden as des pilaers t'een vast t'ander verroerlick, &c. </s>

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

<s><emph type="other"/>VIIIVERTOOCH. XVIVOORSTEL<emph.end/>.</s>

<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 I4 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><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons weeren Æ ende anhanghen Z 4 lb, ende door het voorstel den pilaer sal in die ghestalt bliuen: Maer Æ doet by V soo grooten ghewelt anden pilaer als Z by Y door het I3 voorstel, daerom gheweert Z, ende Æ anghehanghen, soo sal Æ den pilaer oock in die ghestalt houden. </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Wesende dan twee punten inden as des pilaers t'een vast tander verroerlick, trechthefwicht an tverroerlick, dat den pilaer in een ghestalt houdt, sal hem in alle ghestalten houden, twelck wy bewysen moesten.</s>

<s><emph type="other"/>IXVERTOOCH. XVIIVOORSTEL<emph.end/>.</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>

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

<s><emph type="other"/>IX VERTOOCH<emph.end/>. <emph type="other"/>XVII VOORSTEL<emph.end/>.</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 2 lb door I veruolgh des I4 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 .</s>

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, ghedeelt als int I 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 I veruolgh des I4 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 I3 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 I3 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 I 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 eueredenheden <arrow.to.target n="marg119"/> elck van vier palen <arrow.to.target n="marg120"/>, 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.

</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 tot ; maer is euen an tghewicht des pilaers rustende met tpunt V op tpunt Æ, ende tghewicht is euen an t'ghewicht des pilaers rustende met tpunt R op tpunt van , daerom ghelijck TR tot TV, also tghewicht rustende mettet tpunt V op tpunt van Æ, tottet ghewicht rustende mettet punt R op tpunt van . </s><s><emph type="other"/>TBESLVYT<emph.end/>. </s><s>Rustende dan een pilaer op twee punten inden as, &c. </s>

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

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

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

</s>

<s><emph type="other"/>10. VERTOOCH<emph.end/>. <emph type="other"/>XVIII. VOORSTEL<emph.end/>.</s>

<s>I 0. <emph type="other"/>VERTOOCH. XVIII. VOORSTEL<emph.end/>.</s>

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, ende ghedeelt als int voorstel, ende t'vastpunt sy X, ende het roerende punt S, an t'welck gheuoecht sy een onbekent scheefhefwicht Y, met den pilaer euestaltwichtich, ende sijn scheefheflini snie de sijde des pilaers AB in . </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten dat onbekende scheefhefwicht Y bekent maken. </s><s>T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal sien wat rechthefwicht an S den pilaer in die ghestalt soude houden, wort beuonden door voorstel, van 4 lb, daer naer salmen ondersoucken wat reden eenighe hanghende lini <arrow.to.target n="marg146"/> als ZÆ, heeft tot Z, ick neme als van 2 tot I, daer uyt seg ick 2 gheeft I, wat t'rechthefwicht van 4 lb ? comt voor Y 2 lb, t'welck ick seg sijn waer ghewicht te sijne. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de hanghende door S welcke sy AS. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Ghelijck AS tot S, also t'rechthefwicht tottet scheefhefwicht Y door het 20 voorstel, maer den driehouck ZB, is ghelijck anden driehouck SA, welcker lijckstandighe linien <arrow.to.target n="marg147"/> sijn Z met S, ende ZÆ met SA: Daerom ghelijck AS tot S, alsoo ÆZ tot Z, ende vervolghens ghelijck ÆZ 2, tot Z I, also t'rechthefwicht 4 lb tot Y, daerom Y weghende 2 lb is bekent ghemaect, t'welck wy bewysen moesten. </s><s>Ende sghelijcx sal den voortganck sijn in allen anderen voorbeelden. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een bekenden pilaer met een vastpunt inden as, &c. </s>

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een pilaer sijn weghende 6 lb, ende ghedeelt als int voorstel, ende t'vastpunt sy X, ende het roerende punt S, an t'welck gheuoecht sy een onbekent scheefhefwicht Y, met den pilaer euestaltwichtich, ende sijn scheefheflini snie de sijde des pilaers AB in &OElig;. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten dat onbekende scheefhefwicht Y bekent maken. </s><s>T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal sien wat rechthefwicht an S den pilaer in die ghestalt soude houden, wort beuonden door voorstel, van 4 lb, daer naer salmen ondersoucken wat reden eenighe hanghende lini <arrow.to.target n="marg146"/> als ZÆ, heeft tot Z&OElig;, ick neme als van 2 tot I, daer uyt seg ick 2 gheeft I, wat t'rechthefwicht van 4 lb ? comt voor Y 2 lb, t'welck ick seg sijn waer ghewicht te sijne. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de hanghende door S welcke sy AS. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Ghelijck AS tot S&OElig;, also t'rechthefwicht tottet scheefhefwicht Y door het 20 voorstel, maer den driehouck &OElig;ZB, is ghelijck anden driehouck &OElig;SA, welcker lijckstandighe linien <arrow.to.target n="marg147"/> sijn &OElig;Z met &OElig;S, ende ZÆ met SA: Daerom ghelijck AS tot S&OElig;, alsoo ÆZ tot Z&OElig;, ende vervolghens ghelijck ÆZ 2, tot Z&OElig; I, also t'rechthefwicht 4 lb tot Y, daerom Y weghende 2 lb is bekent ghemaect, t'welck wy bewysen moesten. </s><s>Ende sghelijcx sal den voortganck sijn in allen anderen voorbeelden. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een bekenden pilaer met een vastpunt inden as, &c. </s>

<s>

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

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

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<s>

<margin.target id="marg148"/>souden inde wercking hebben mueghen segghen, AS 2, gheeft S I, wat t'rechthefwicht 4 lb ? comt voor Y 2 lb, maer op dat sy lijckformigher soude siin an t'ghene inde daet gheschiet (want men can binnen int lichaem qualick de linien AS, S trecken) wy hebben de hanghende lini ZÆ int voorbeelt uytwendich ghenomen.

<margin.target id="marg148"/>souden inde wercking hebben mueghen segghen, AS 2, gheeft S&OElig; I, wat t'rechthefwicht 4 lb ? comt voor Y 2 lb, maer op dat sy lijckformigher soude siin an t'ghene inde daet gheschiet (want men can binnen int lichaem qualick de linien AS, S&OElig; trecken) wy hebben de hanghende lini ZÆ int voorbeelt uytwendich ghenomen.

</s>

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</s>

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

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

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

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

<s> T' <emph type="other"/>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 4e 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> T' <emph type="other"/>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.

</s>

<gap/>

<s>

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