texts/archimedes/xml/stevi_weegc_01_nl_1586.xml - Colored diff

Main History Search Repository tree

Return to stevi_weegc_01_nl_1586.xml CVS log

Up to [CVSROOT] / texts / archimedes / xml

Colored diff for /texts/archimedes/xml/Attic/stevi_weegc_01_nl_1586.xml between version 1.59 and 1.120

version 1.59, 2003/09/09 14:41:09

version 1.120, 2003/09/11 19:21:49

Line 10

<cvs_file>stevi_weegc_01_nl_1586</cvs_file>

<cvs_version/>

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

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

<chap>

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

Line 1208

Line 1210

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

</s>

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

<s>' <emph type="other"/>VOORGAENDE<emph.end/> mach oock verstaen worden van een cloot ligghende op een lini AB als hier neuens, alwaer wy segghen als vooren, gelijck LD tot DO, alsoo M tot P (welverstaende </s>

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

<s>

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

</s>

Line 1235

Line 1230

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

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

<s>

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

</s>

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

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

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

<s>

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

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

Line 1281

Line 1271

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

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

<s>

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

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

Line 1292

Line 1282

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

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

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

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

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

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

Line 1304

Line 1294

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

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

<s>

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

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

<margin.target id="marg144"/>Parallela.

</s>

<s> Dit so ghenomen, laet ons den pilaer der eerste form neerduwen, ofte der tweeder form oplichten, tot dat KF sulcken reden hebbe tot FH, als 3 tot 7, ende alsdan sal G oock euestaltwichtich sijn teghen den pilaer door de voorgaende vervolghen; Inder voughen dat den pilaer hoogher ende leegher verheuen, sal teghen G euestaltwichtich blijuen, t'welck openbaer onmueghelick is, als oock wisconstlick <arrow.to.target n="marg145"/> sal blijcken door t'volghende voorstel. </s><s>KF dan en heeft tot FH gheen ander reden dan I tot G.</s>

<s>

<margin.target id="marg145"/>Mathematicè.

</s>

<s> <emph type="other"/>UYT<emph.end/> dese voorgaende bescrijuen wy een vertooch soodanich.</s>

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

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

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

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

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

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

<s> T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Wesende dan inden as des pilaers een vastpunt, &c. </s><s><emph type="other"/>MERCKT<emph.end/>. </s><s>Soo eenighe der linien als IF, FK, de sijde des pilaers niet en sneen, men sal die sijde voorder trecken tot dat sy ghesneen wort, als inde voorgaende laetste form. </s>

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

<s><emph type="other"/>MERCKT<emph.end/>. </s><s>Soo eenighe der linien als IF, FK, de sijde des pilaers niet en sneen, men sal die sijde voorder trecken tot dat sy ghesneen wort, als inde voorgaende laetste form. </s>

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

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

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

<s> <emph type="other"/>TGHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn, diens as CD, ende vastpunt E, ende roerlick punt F, daer an den pilaer door trechtdaelwicht G in die ghestalt ghehouden wort, daer an oock den pilaer door t'scheefdaelwicht H (welverstaende G gheweert sijnde) in die ghestalt ghehouden wordt, ende de rechtdaellini snie de sijde des pilaers in I, maer de scheefdaellini snie de selue sijde in K. </s><s><emph type="other"/>TBEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck de rechtdaellini IF tot de scheefdaellini FK, alsoo t'rechtdaelwicht G tot het scheefdaelwicht H. </s><s><emph type="other"/>TBEREYTSEL<emph.end/>. </s><s>Laet ons teeckenen t'punt L, alsoo dat EL euen sy an EF, ende voughen an t'punt L t'rechthefwicht M, dat den pilaer in die ghestalt can houden, diens rechtheflini LN: Insghelijcx t'scheefhefwicht O, dat den pilaer oock in die ghestalt can houden, wiens scheefheflini LP euewydich sy met FK. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Ghelijck NL tot LP, also M tot O, duer het voorstel maer de macht van G is anden pilaer euen met de macht van M, ende de macht van H met die van O duer het voorstel, ende IF is euen an LN, ende FK, an LP; Daerom ghelijck de rechtdaellini IF tot de scheefdaellini FK, alsoo t'rechtdaelwicht G tottet scheefdaelwicht H, Sghelijcx sal oock t'bewys sijn van alle d'ander ghestalten als inde formen hier na volghende.</s>

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

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

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

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

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

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

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

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

<s>

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

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

</s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een bekenden pilaer met een vastpunt inden as, &c. </s>

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

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

<s>

<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>

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

openbaer door de <arrow.to.target n="marg149"/>verkeerde ende oueranderde Eueredenheydt, hoe dat elck van d'ander onbekende <arrow.to.target n="marg150"/>palen als Rechthefwycht, Rechtheflini, Scheefheflini, Pilaer, door drie bekende palen altijdt bekent sullen worden, welcker bescriuing wy om de cortheyt achterlaten.</s>

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

<s> <emph type="other"/>TIS<emph.end/> <arrow.to.target n="marg149"/> {Inuersam & alternam proportionem.}<arrow.to.target n="marg150"/> {Terminorum} <arrow.to.target n="marg151"/></s>

<s>

<margin.target id="marg149"/>openbaer door de verkeerde ende oueranderde Eueredenheydt

<margin.target id="marg149"/>Inuersam & alternam proportionem.

<margin.target id="marg150"/>, hoe dat elck van d'ander onbekende palen

<margin.target id="marg150"/>Terminorum

<margin.target id="marg151"/> als Rechthefwycht, Rechtheflini, Scheefheflini, Pilaer, door drie bekende palen altijdt bekent sullen worden, welcker bescriuing wy om de cortheyt achterlaten.

</s>

<s><emph type="other"/>XIIII. VERTOOCH<emph.end/>. <emph type="other"/>XXIII. VOORSTEL<emph.end/>.</s>

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

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

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

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

<s><emph type="other"/>XIIII<emph.end/>. <emph type="other"/>VERTOOCH<emph.end/>. <emph type="other"/>XXIII<emph.end/>. <emph type="other"/>VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>TGHE<emph.end/>. </s><s>Laet AB een pilaer sijn diens as CD, ende vastpunt daer in E, en t'roerlick punt F, an t'welck een scheefhefwicht G sy, dat den pilaer in die ghestalt houde, ende diens scheefheflini FH. </s><s>Laet oock an t'selue punt F gheuoucht wesen een scheefhefwicht I, ouer d'ander sijde, ende met G euewichtich, ende diens scheefheflini FK, den houck KFD euen make anden houck HFC. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat I anden pilaer euen sulcken ghewelt doet als G, te weten dat I (G gheweert sijnde) den pilaer oock in die ghestalt sal houden. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet an t'punt F gheuoucht worden t'rechthefwicht L dat den pilaer oock in die ghestalt can houden, ende sijn rechtheflini sy FM. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Want de linien FH, FK, sijn tusschen de euewydighe <arrow.to.target n="marg152"/> HK, CD, ende dat den houck HFC, euen is (door t'ghegheuen) an den houck KFD, so sijn FH ende FK euen. </s><s>waer uyt volght dat ghelijck MF tot FH, alsoo MF tot FK, Maer ghelijck MF tot FH, alsoo L tot G, daerom oock ghelijck MF tot FK, also L tot G; maer I is euen an G door t'ghestelde, ghelijck dan MF tot FK, alsoo L tot I. </s><s>T'welck so sijnde, I houdt den pilaer in die ghestalt door het voorstel. </s><s>Sghelijcx sal oock t'bewijs sijn in alle anderen voorbeelden. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Euen ghewichten dan der trecklinien van een selfde punt des as, ende op verscheyden sijden met den as euen houcken makende; doen anden pilaer euen ghewelden, t'welck wy bewysen moesten.</s>

<s>

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

</s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Euen ghewichten dan der trecklinien van een selfde punt des as, ende op verscheyden sijden met den as euen houcken makende; doen anden pilaer euen ghewelden, t'welck wy bewysen moesten.</s>

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

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

<s> <emph type="other"/>ALS<emph.end/> des ghewichts trecklini rechthouckich op den as is; Soo doedet anden pilaer ghegeuener ghestalt de grootste ghewelt. </s>

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn diens as CD, ende vastpunt E, ende roerlick punt F, waer an gheuoucht is t'scheefhefwicht G, dat den pilaer in die ghestalt houdt, ende also dat sijn scheefheflini HF rechthouckich op den as CD is; Laet oock an F gheuoucht worden t'scheefhefwicht I, euen an G, ende sijn scheefheflini sy KF. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat G meerder ghewelt doet anden pilaer, dan I, oock gheen meerder ghewelt daer an doen en can. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons an F voughen t'rechthefwicht L dat den pilaer in die ghestalt houden can, diens rechtheflini FM. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s> <emph type="other"/>DES<emph.end/> s <arrow.to.target n="marg153"/> tweede voorstel wort aldus bethoont, T'ghewicht G (t'welck den pilaer in die ghestalt houdt) heeft sulcken reden tot L, als HF tot FM, maer I is euen an G, ende KF is meerder dan FH <arrow.to.target n="marg154"/>, daerom I heeft minder reden tot L, dan KF tot FM, waer duer soo wy bouen gheseyt hebben, t'ghewicht I is te licht om den pilaer in die ghestalt te houden; maer G cander hem in houden, G dan doet anden pilaer meerder ghewelt dan I. </s><s>Maer dat G daer an gheen meerder doen en can, is daer uyt openbaer, dat van F op de sijde des pilaers gheen corter lini en can ghetrocken worden dan FH, anghesien sy daer op rechthouckich is. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Als dan des ghewichts trecklini rechthouckich op den as is, soo doedet an den pilaer ghegheuener ghestalt de grootste ghewelt, t'welck wy bewysen moesten. </s>

<s><emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn diens as CD, ende vastpunt E, ende roerlick punt F, waer an gheuoucht is t'scheefhefwicht G, dat den pilaer in die ghestalt houdt, ende also dat sijn scheefheflini HF rechthouckich op den as CD is; Laet oock an F gheuoucht worden t'scheefhefwicht I, euen an G, ende sijn scheefheflini sy KF. </s>

<s>

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

<margin.target id="marg154"/>

</s>

<s>

<margin.target id="marg155"/>Alle hefwicht dat minder reden heeft tot L, dan siin heflini tot FM, is te licht om den pilaer in die ghestalt te houden, duer het 20 voorstel:

<margin.target id="marg156"/>I is hefwicht dat minder reden heeft tot L, dan siin heflini KF tot FM:

<margin.target id="marg157"/>T'hefwicht I dan is te licht om den pilaer in die ghestalt te houden.

</s>

<s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat G meerder ghewelt doet anden pilaer, dan I, oock gheen meerder ghewelt daer an doen en can. </s>

<s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons an F voughen t'rechthefwicht L dat den pilaer in die ghestalt houden can, diens rechtheflini FM. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s>

<s> A. </s><s>Alle hefwicht dat minder reden heeft tot L, dan siin heflini tot FM, is te licht om den pilaer in die ghestalt te houden, duer het 20 voorstel:</s>

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

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

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

<s>

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

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

</s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Als dan des ghewichts trecklini rechthouckich op den as is, soo doedet an den pilaer ghegheuener ghestalt de grootste ghewelt, t'welck wy bewysen moesten. </s>

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

<s> <emph type="other"/>HET<emph.end/> blijft dat hoe de houcken der trecklinien vande ghewichten, op den as den rechthouck naerder sijn, hoe de ghewichten meerder ghewelt doen; Ende ter contrarie hoe sy vanden rechthouck meer verschillen, hoe de ghewichten minder ghewelt doen. </s>

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

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

<s> <emph type="other"/>TWEE<emph.end/> oneuewydighe linien daer een pilaer an hangt beyde oneindelick voortghetrocken, snien malcanderen inde swaerheydts middellini des pilaers. </s>

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn hanghende ande twee oneuewydighe linien CD, EF, welcke voortghetrocken sijn tot G, H, sniende malcanderen in I. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dattet punt I inde swaerheyts middellini is des pilaers AB. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Den houck FEC, ofte IEC, ofte HEC, is al een selfden houck, also oock is DCE, ofte ICE, ofte GCE, daerom wat punten wy inde linien HE, ende CG voor uytersten nemen, den pilaer <pb pagenum="58"/>houdt daer an sijn ghegheuen standt. </s><s>Laet ons nemen I, ghemeen uyterste punt van d'een ende d'ander lini, den pilaer dan houdt daer an sijn ghegeuen standt. </s><s>Maer hanghende den pilaer an t'punt I, so is de hanghende <arrow.to.target n="marg158"/> door I des pilaers swaerheydts middellini inde welcke I is.</s>

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

<s>

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

</s>

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn hanghende ande oneuewydighe linien CD, EF, welcke voortghetrocken sijn tot G, H, sniende malcanderen in I. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dattet punt I, inde swaerheyts middellini is des pilaers AB. </s><s><emph type="other"/>TBEWYS<emph.end/>. </s><s>Laet ons DG ende FH ansien voor stylen ofte stiue linien daer den pilaer op rust, welcke door de begheerte niet en breken noch en buyghen; </s>

<s> der seluer ghewelt is euen ande ghewelt der linien CD, EF, want ghelijck dese den pilaer in sijn ghegheuen standt houden alsoo oock die. </s><s>Ende wat punten wy inde linien DG, FH voor uytersten nemen, den pilaer houdt daer op sijn ghegheuen standt. </s><s> Laet ons nemen I, ghemeen uyterste punt van d'een en d'ander lini; den pilaer dan houdt daer op (Wisconstlick <arrow.to.target n="marg159"/> verstaende) sijn ghegheuen standt, maer rustende den pilaer op t'punt I, soo is de hanghende door I des pilaers swaerheydts middellini, inde welcke I is.</s>

<s>

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

</s>

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn welcke in die standt ghehouden wort door de scheefdaellini CD, ende scheefheflini EF, de selue sijn voortghetrocken tot G, H, sniende malcanderen in I. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat I inde swaerheydts mid<pb pagenum="59"/>dellini is des pilaers AB. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Laet ons GC ansien voor styl, ofte stijue lini ende nemen dat de macht die an D int neertrecken was, nu neerstekende sy in yder punt tusschen C en G daermen haer stelt, ende den pilaer AB, sal alsoo op allen punten diemen tusschen C, G ende E, H voor uytersten neemt, sijn ghegheuen standt houden. </s><s>Laet ons nemen I ghemeen uyterste van d'een en d'ander lini, den pilaer dan houdt daer an sijn ghegheuen standt; maer hanghende den pilaer an t'punt I, de hanghende door I is des pilaers swaerheyts middellini, inde welcke I is. </s>

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn, welcke in die standt ghehouden wort door de scheefdaellini CD, ende de scheefheflini EF, de selue sijn voortghetrocken tot <emph type="other"/>GH<emph.end/>, sniende malcanderen in I. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat I inde swaerheyts middellini is des pilaers AB. </s>

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

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

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

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Twee oneuewydighe linien dan, daer een pilaer an hangt beyde oneyndelick voortghetrocken, snien malcanderen inde swaerheydts middellini des pilaers, t'welck wy bewysen moesten.</s>

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

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

<s> <emph type="other"/>SOO<emph.end/> d'eene der twee linien daer een pilaer an hangt rechthouckich op den sichteinder <arrow.to.target n="marg160"/> is, d'ander salder oock rechthouckich op sijn: Ende sooder d'een scheefhouckich op is, dander salder oock scheefhouckich op wesen: Ende soo dese naer die neigt, die sal naer dese neighen: Maer so dese van die wyckt, die sal oock van dese wycken. </s>

<s>

Line 1483

Line 1446

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn hanghende an twee linien, d'een CD rechthouckich op den sichteinder, d'ander EF (soot mueghelick waer) scheefhouckich, ende GH sy des pilaers swaerheydts middellini. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen t'hinhoudt des voorstels. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet CD ende EF voortghetrocken worden, sniende malcander in I. </s><s> T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Soo den pilaer in die ghestalt blijft hanghende ande linien CD, EF, sy sal op alle vastpunten in die voortghetrocken linien de selue ghestalt houden, ouermidts de houcken ICE, ende IEC, niet en veranderen: </s>

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

<s> Daerom ghenomen I <pb pagenum="60"/>ghemeen vastpunt dier twee linien, den pilaer sal daer an in sijn ghegheuen standt blijuen hanghende, ende IC sal swaerheydts middellini sijn: maer dat is onmueghelick, wanttet GH haer euewydeghe is. </s><s>T'selue sal oock alsoo bethoont worden als de lini EF ouer dander sijde neigt. </s><s>Wesende dan IC rechthouckich op den sichteinder, d'ander lini als EF en cander niet scheefhouckich op sijn; nootsaecklick dan rechthouckich: Ende veruolghens sooder EF scheefhouckich op is, dander moeter oock scheefhouckich op sijn.</s>

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

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Soo d'eene dan der twee linien, &c.</s>

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet AB een pilaer sijn wiens as CD, ende twee punten daer in E, F, welcker scheefhefwichten die hem in die standt houden sijn G, H, ende rechthefwichten I, K, ende scheefheflinien EL, FM, ende rechtheflinien EN, FO. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck LE <pb pagenum="61"/>tot EN, alsoo G tot I, ende ghelijck MF tot FO, alsoo H tot K. </s>

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

<s> T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Laet ons F ansien voor vastpunt, ende E voor t'roerlick, daerom (door het voorstel) ghelijck LE tot EN, alsoo G tot I. </s><s>Laet ons ten tweeden E ansien voor vastpunt, ende F voor t'roerlick; Daerom (door t'voornoemde 20 voorstel) ghelijck MF tot FO, alsoo H tot K. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Hanghende dan een pilaer euestaltwichtich teghen twee scheefhefwichten: Ghelijc scheefheflini tot rechtheflini, alsoo elck scheefhefwicht tot sijn rechthefwicht, t'welck wy bewysen moesten.</s>

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

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

Line 1503

Line 1466

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

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

<s>

<margin.target id="marg161"/>hebben tot veel voorbeelden der voorstellen deses boucx, ghenomen den pilaer, als bequaemste form tot de verclaring des voornemens oock vastpunt ende roerlickpunt ghestelt inden as. Wy sullen nu door dit laetste voorstel, bethoonen de reghelen van dien ghemeen te wesen ouer alle formen der lichamen hodanich sy siin met vastpunt ende roerlickpunt daert valt.

</s>

<s><emph type="other"/>XIX. VERTOOCH. XXVIII. VOORSTEL<emph.end/>.</s>

<s><emph type="other"/>XIX. VERTOOCH<emph.end/>. <emph type="other"/>XXVIII. VOORSTEL<emph.end/>.</s>

<s> <emph type="other"/>ALLE<emph.end/> de eueredenheden <arrow.to.target n="marg162"/> , welcke hier vooren beschreuen sijn vanden pilaer tot de ghewichten an hem hanghende, ende dier ghewichten linien: De selue te wesen van yder lichaem tot de ghewichten an hem alsoo hanghende, ende dier ghewichten linien. </s>

<s>

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

</s>

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

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

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

<s>

Line 1531

Line 1489

<s> <emph type="other"/>LAET<emph.end/> nu de form des pilaers (al de stoff bliuende) verandert worden in eenighe ander ongheschickte form, als AB hier neuens, diens swaerheydts middelpunt E sy, ende een rechte lini daer duer CD (welcke vinding des swaerheyts middelpunts ende rechter linien inde verclaert sal worden, werckelick, niet Wisconstelick <arrow.to.target n="marg164"/>) ende alles blijft noch euestaltwichtich, ende ghelijck IG tot GL, also noch H tot K. </s><s><emph type="other"/>LAET<emph.end/> Laet nu t'lichaem AB opghetrocken worden, tot dat FG is inde lini CD, wiens ghestalt dan sy als hier neuens, ende alles blijft noch euestaltwichtich: want het lichaem AB hoogher ofte leegher hanghende, blijft van een selfde ghewicht door de 3 begheerte, ende veruolghens ghelijck IG tot GL, alsoo noch H tot K. </s><s>De eueredenheydt dan des 20 voorstels en is niet alleenelick alsoo met den pilaer, maer met yder lichaem: Ende der ghelijcke salmen oock alsoo bethoonen van al t'ghene hier vooren in alle d'ander voorstellen vanden pilaer gheseyt is. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Alle de eueredenheden dan, welcke hier vooren bescreuen sijn vanden pilaer tot de ghewichten an hem hanghende, ende dier ghewichten linien; de selue sijn van yder lichaem tot de ghewichten an hem alsoo hanghende, ende dier ghewichten linien, t'welck wy bewysen moesten.</s>

<s>

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

</s>

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

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

Line 1549

Line 1511

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

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

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

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

<s> <emph type="other"/>VANDEBEGHINSELEN<emph.end/> <arrow.to.target n="marg166"/></s>

<s>

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

</s>

<s> <emph type="other"/>DERWEEGHCONST<emph.end/>, <emph type="other"/>DWELCKIS<emph.end/></s>

<s> <emph type="other"/>VANDEVINDINGDERSWAER<emph.end/><lb/></s>

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

DER WEEGHCONST, DWELCK IS <lb/>

VANDE VINDING DER SWAER<lb/>

HEYDTS MIDDELPVNTEN, <lb/>

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

<s> <emph type="other"/>HEYDTSMIDDELPVNTEN<emph.end/> ,</s>

<s>

<margin.target id="marg167"/>eschreuen door Simon Steuin.

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

</s>

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

<s> Y hebben in t'eerste bouck tot het beschriuen der wichtighe ghedaenten, ghenomen een pilaer (voldoende aldaer het voornemen) diens swaerheyts middelpunt door ghemeene wetenschap bekent is, maer in veel ander lichamen en ghebueret niet also; wel is waer dattet door een corte ghemeene reghel in allen werckelick te vinden is, so door t'eerste voorstel der <arrow.to.target n="marg168"/> blijcken sal, maer met de Wisconstighe <arrow.to.target n="marg169"/> vinding ist anders ghestelt; Daer af heeft eerst gheschreuen Archimedes in platten, ende naer hem Frederic Commandin in lichamen: Wy sullen tottet een en t'ander (ouermits het een afcoemst <arrow.to.target n="marg170"/> van beghinselen is, byde voorgaende wel dienende, ende tottet volghende, so wel WATERWICHT, als WEEGHDAET, seer noodich) het onse voughen, ende alles naer onse oirden verspreyden, daer af beschrijuende der Beghinselen tweede bouck. </s><s>Wat de bepalinghen <arrow.to.target n="marg171"/> belangt vande Meetconstighe formen, die bygheualle hier yemandt begheeren mocht, wy nemen die door t'ghestelde <arrow.to.target n="marg172"/> voor bekent uyt de Meetconst <arrow.to.target n="marg173"/>; Alleenelick dit daer af segghende, dat wy t'woort <arrow.to.target n="marg174"/>, ofte <arrow.to.target n="marg175"/>, beteeckenen met <arrow.to.target n="marg176"/>: Ende <arrow.to.target n="marg177"/>, met <arrow.to.target n="marg178"/>; Reden, dat dier formen daet <arrow.to.target n="marg179"/> voornamelicxt bestaet int ontsteken ofte branden.</s>

<s>

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

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

<margin.target id="marg170"/>Species.

</s>

<s>Wat de bepalinghen <arrow.to.target n="marg171"/> belangt vande Meetconstighe formen, die bygheualle hier yemandt begheeren mocht, wy nemen die door t'ghestelde <arrow.to.target n="marg172"/> voor bekent uyt de Meetconst <arrow.to.target n="marg173"/>; Alleenelick dit daer af segghende, dat wy t'woort <emph type="italics"/>Parabola<emph.end/>, ofte <emph type="italics"/>Rectanguli coni sectio<emph.end/>, beteeckenen met <emph type="italics"/>Brantsne<emph.end/>: Ende <emph type="italics"/>Conoidale Rectangulum<emph.end/>, met <emph type="italics"/>Brander<emph.end/>; Reden, dat dier formen daet <arrow.to.target n="marg179"/> voornamelicxt bestaet int ontsteken ofte branden.</s>

<s>

<margin.target id="marg171"/>Definitiones.

<margin.target id="marg172"/>Per hypothesin.

<margin.target id="marg173"/>Geometria.

<margin.target id="marg174"/>Parabola

<margin.target id="marg175"/>Rectanguli coni sectio

<margin.target id="marg176"/>Brantsne

<margin.target id="marg177"/>Conoidale Rectangulum

<margin.target id="marg178"/>Brander

<margin.target id="marg179"/>Effectus.

</s>

<s><emph type="other"/>EERST VANDE VINDING DER <lb/>

SWAERHEYTS MIDDELPVNTEN <lb/>

VANDE PLATTEN.<emph.end/> </s><s><arrow.to.target n="marg180"/> </s>

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

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

<s> <emph type="other"/>VANDEPLATTEN<emph.end/>. </s><s><arrow.to.target n="marg180"/> </s>

<s>

<margin.target id="marg180"/>Planis.

</s>

Line 1630

Line 1555

<s><emph type="other"/>BY<emph.end/> aldien de platten eenich ghewicht hadden, ende datmen toeliete die te wesen inde reden haerder grootheden, wy souden eyghentlick mueghen spreken van haer Swaerheydt, Swaerheyts middelpunt, Swaerheyts middellini, &c. Maer <pb pagenum="66"/>anghesien in t'plat gheen ghewicht en is, soo en isser eyghentlick sprekende gheen Swaerheydt, Swaerheydts middelpunt, noch Swaerheyts middellini in; Daerom moetmen dit alles <arrow.to.target n="marg183"/>lijckspreucklick verstaen, ende nemen als door t'ghestelde, dat der platten ghewichten inde reden haerder grootheden sijn, want T'valsche wort toeghelaten, op datmen t'waerachtighe daer duer leere.

<s>

<margin.target id="marg181"/>aldien de platten eenich ghewicht hadden, ende datmen toeliete die te wesen inde reden haerder grootheden, wy souden eyghentlick mueghen spreken van haer Swaerheydt, Swaerheyts middelpunt, Swaerheyts middellini, &c.

</s>

<s><arrow.to.target n="marg182"/>{Metaphoricè.}<arrow.to.target n="marg183"/> T'valsche wort toeghelaten, op datmen t'waerachtighe daer duer leere. </s>

<s>

<margin.target id="marg182"/>Maer anghesien in t'plat gheen ghewicht en is, soo en isser eyghentlick sprekende gheen Swaerheydt, Swaerheydts middelpunt, noch Swaerheyts middellini in; Daerom moetmen dit alles lijckspreucklick

<margin.target id="marg183"/>Metaphoricè.

<margin.target id="marg183"/> verstaen, ende nemen als door t'ghestelde, dat der platten ghewichten inde reden haerder grootheden sijn, want

</s>

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

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

<s> <emph type="other"/>YDER<emph.end/> plats middelpunt der form, is oock sijn swaerheyts middelpunt. </s>

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

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

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een euewydich vierhouck sijn, diens formens middelpunt E. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat E oock het swaerheyts middelpunt is.</s>

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

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

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

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

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

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

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

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

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck sijn van form soot <pb pagenum="68"/>valt, waer in vanden houck A tot in D middel vande sijde BC, ghetrocken is de lini AD. </s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder plats middelpunt der form dan, is oock sijn swaerheyts middelpunt, t'welck wy bewysen moesten.</s>

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

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

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

<s>

<margin.target id="marg184"/>Neuen alle verschillende staltswaerheden, can een swaerheydt ghestelt worden minder dan haer verschil

<margin.target id="marg185"/>Neuen dese staltswaerheden ADC ende ADB, en can gheen swaerheydt ghestelt worden minder dan haer verschil

<margin.target id="marg186"/>Dese staltswaerheden dan ADC ende ADB en verschillen niet

</s>

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

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

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

<s>O. Neuen dese staltswaerheden ADC ende ADB, en can gheen swaerheydt ghestelt worden minder dan haer verschil;</s>

<s>O. Dese staltswaerheden dan ADC ende ADB en verschillen niet</s>

<s> Daerom AD is swaerheyts middellini, ende vervolghens t'swaerheyts middelpunt des driehoucx ABC is in haer. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder driehoucx swaerheydts middelpunt dan is inde lini ghetrocken vanden houck tot int middel der sijde, t'welck wy bewysen moesten. </s>

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

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

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

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

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

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABC een driehouck sijn, ende vanden houck B een lini ghetrocken worden tot D int middel van AC, sghelicx van C tot E int middel van AB, sniende BD in F voor swaerheyts middelpunt des driehoucx ABC. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat CF dobbel is an FE. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Ghetrocken de reden E<emph type="other"/>BI<emph.end/> tot BA 2, vande reden C<emph type="other"/>DI<emph.end/> tot D<emph type="other"/>AI<emph.end/> (dat is <arrow.to.target n="marg187"/> I/2 van <arrow.to.target n="marg188"/> I/I) <arrow.to.target n="marg189"/> daer rest de reden van CF tot FE, maer treckende <arrow.to.target n="marg190"/> I/2 van <arrow.to.target n="marg191"/> I/I daer blijft <arrow.to.target n="marg192"/> 2/I. </s><s>CF dan is tot FE, als van 2 tot I. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Het swaerheyts middelpunt dan eens driehoucx deelt de lini vande houck tot int middel der sijde alsoo, dattet stick naer den houck dobbel is an t'ander. </s><s>t'welck wy bewysen moesten.</s>

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

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

<s>

<margin.target id="marg187"/>Reden

<margin.target id="marg188"/>Reden

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

<margin.target id="marg190"/>Reden

<margin.target id="marg191"/>Reden

<margin.target id="marg192"/>Reden

</s>

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

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

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

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

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

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

<s>

<margin.target id="marg193"/>2. v. 6. B. E.

</s>

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

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

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een rechtlinich plat <arrow.to.target n="marg194"/>: Sijn swaerheyts middelpunt te vinden. </s>

<s>

Line 1745

Line 1643

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ongheschict vierhouck wesen. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ongheschict vierhouck wesen. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden.</s>

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

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

<s>

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

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

Line 1760

Line 1658

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCDE een ongheschickt vijfhouck sijn. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden. </s><s>T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal trecken AC, ende vinden t'swaerheyts middelpunt des driehoucx ACB door het 3 voorstel, t'welck F sy, ende vanden vierhouck ACDE duer t'voorgaende I voorbeelt, t'welck G sy, ende de lini FG sal balck wesen, daer naer salmen maken twee euewydighe vierhoucken van een selfde hoochde, als HIKL euen anden vierhouck ACDE, ende HIMN euen anden driehouck ACB, deelende den balck GF in O, alsoo dat den erm OF, sulcken reden hebbe tot den erm OG, als IK tot IM; Ick seg dat O t'begheerde swaerheyts middelpunt is.</s>

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

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

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

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

<s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal trecken AC, ende vinden t'swaerheyts middelpunt des driehoucx ACB door het 3 voorstel, t'welck F sy, ende vanden vierhouck ACDE duer t'voorgaende I voorbeelt, t'welck G sy, ende de lini FG sal balck wesen, daer naer salmen maken twee euewydighe vierhoucken van een selfde hoochde, als HIKL euen anden vierhouck ACDE, ende HIMN euen anden driehouck ACB, deelende den balck GF in O, alsoo dat den erm OF, sulcken reden hebbe tot den erm OG, als IK tot IM; Ick seg dat O t'begheerde swaerheyts middelpunt is.</s>

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

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

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

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

<s> Wy heben hier bouen voorbeelden beschreuen alwaer t'ghegheuen plat verkeert wort in euenhooghe ende euewydighe vierhoucken, wy connen t'selfde oock doen sonder soodanighe verkeering, daer af wy verscheyden voorbeelden beschrijuen sullen als volght.</s>

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

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

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

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

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

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

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

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

<s>

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

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

</s>

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

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

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

<s> 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 4. voorbeelt, t'welck G sy, ende des vierhoucx ADEF, t'welck H sy, ende de lini HG is balck. </s><s>Daernaer ghetrocken BI ende DK rechthouckich op AC, insghelijcx AL ende EM beyde rechthouckich op FD, men sal der drie linien welcker eerste FD, de tweede AC, de derde BI met KD, vinden de vierde euerednighe, welcke NO sy, deelende den balck HG in P, also dat den erm PG, sulcken reden hebbe tot den erm PH, ghelijck AL met EM, tot NO; Ick seg dat P het begheerde swaerheyts middelpunt is.

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

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

</s>

<s>

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

<s> T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Ghelijck int eerste voorbeelt HI tot HL, alsoo den erm NE tot den erm NF, maer <arrow.to.target n="marg199"/> ghelijck HI tot HL, alsoo den vierhouck GHIK, tot den vierhouck GHLM, ghelijck dan GHIK tot GHLM, also NE tot NF, maer GHIK is euen an den driehouck ACD, ende GHLM anden driehouck ACB door t'werck, ghelijck dan den driehouck ACD tot ACB, alsoo den erm NE tot NF. </s><s> Het punt dan N is (door het voorstel des I boucx) des vierhoucx swaerheyts middelpunt. </s><s>Sghelijcx sal oock bewijs sijn des 2 ende 3 voorbeelts. </s><s> T'vierde voorbeelt is openbaer als wy bewesen hebben dat ghelijck DG, tot HB, alsoo den driehouck ACD, tot ACB in deser voughen: Nemende AC voor hoochde, ende DG ende HB voor gronden, soo heeft den rechthouck begrepen onder AC ende DG, sulcken reden tot den rechthouck onder AC ende HB, ghelijck DG tot HB <arrow.to.target n="marg200"/>; Maer ghelijck dien rechthouck tot desen, alsoo den driehouck ACD tot ACB, want elck driehouck is sijn rechthoucx helft <arrow.to.target n="marg201"/>, ghelijck dan DG tot HB, alsoo den driehouck ACD tot ACB. </s><s>Des 5 voorbeelts bewys sal oock claer sijn als wy bewesen hebben, dat ghelijck EK met IC tot LM, alsoo den vierhouck ACDE tot den driehouck ACB, aldus: Anghesien LM vierde euerednighe is der drie AD, AC, HB, de rechthouck begrepen onder AD ende LM, sal euen sijn <arrow.to.target n="marg202"/> an den rechthouck begrepen onder AC ende HB, Laet ons nu EK, IC, LM, ansien voor gronden, wiens ghemeene hoochde AD; </s>

<s>

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

</s>

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

<s>

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

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

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

</s>

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

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

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

<s>[ Figuren ingevoegd: ]</s>

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

<s>

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

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

</s>

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

<s> <emph type="other"/>MY<emph.end/> <arrow.to.target n="marg204"/>{Commentarius in quadraturam paraboles.} <arrow.to.target n="marg205"/></s>

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

</s>

<s>

<margin.target id="marg204"/>is onder het drucken ter handt ghecomen, Fredric Commandins verclaring ouer de viercanting der Brantsne van Archimedes

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

<margin.target id="marg205"/>, alwaer hy onder het 6 voorstel de manier beschrijft, om t'swaerheyts middelpunt te vinden van yder rechtlinich plat, ende dat op een ander wijse als de twee voorgaende. So ymant tottet ouersien der selue begheerich waer, salse daer vinden.

</s>

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

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

<s> <emph type="other"/>HET<emph.end/> swaerheyts middelpunt des vierhoucx met twee euewydighe <arrow.to.target n="marg206"/> sijden, is inde lini tusschen dier sijden middelpunten. </s>

<s>

Line 1854

Line 1732

</s>

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een vierhouck sijn, diens twee euewydighe sijden AB ende DC, ende de lini uyt E middel van AB, tot F middel van DC, sy EF. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat t'swaerheyts middelpunt des vierhoucx ABCD inde lini EF is. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet de drie linien DA, FE, CB, voortghetrocken worden, welcke om de eueredenheyt <arrow.to.target n="marg207"/> der linien AE, EB, DF, FC, vergaren sullen in een selfde punt t'welck G sy. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Laet ons den driehouck GDC ophanghen byde lini GF, ende het deel GFC sal euestaltwichtich sijn, teghen GFD door het voorstel, waer deur oock t'swaerheyts mid<pb pagenum="75"/>delpunt des driehoucx GDC inde lini GF is.</s>

<s>

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

</s>

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

<s> Maer den driehouck GEB, is oock euestaltwichtich teghen den driehouck GEA, daerom van euestaltwychtighe ghetrocken euestaltwichtighe, de resten als de vierhoucken EFDA, EFCB, sullen noch euestaltwichtich blijuen, ende haer swaerheyts middelpunt noch inde lini GF, maer niet uyt de form in EG; Nootsaecklick dan in EF. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Het swaerheydts middelpunt dan des vierhoucx met twee euewydighe sijden, is inde lini tusschen dier sijden middelpunten, t'welck wy bewysen moesten.</s>

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een vierhouck wesen met twee euewydighe sijden AB, DC, ende de lini tusschen haer middelpunten sy EF, ende t'swaerheydts middelpunt sy G. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck tweemael DC met eenmael AB, tot tweemael AB met eenmael DC. </s><s>also GE tot GF. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden DB, ende ghedeelt in drie euen deelen met de punten H, I, ende door de selue ghetrocken worden KL, ende MN, euewydich van DC, sniende EF in O en P. </s><s>Daer naer de lini DE, sniende MI in Q, Ende BF sniende KL in R, Ende ten laetsten RQ. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Anghesien het swaerheydts middelpunt des driehoucx BDC, is in BF, duer het voorstel, ende oock in HL duer het voorstel, soo is R, sijn swaerheyts middelpunt, ende om de selue reden is Q swaerheyts middelpunt des driehoucx ABD, ende QR is dier driehoucken balck, inden welcken haer beyder, dat is des vierhoucx ABCD, swaerheyts <pb pagenum="76"/>middelpunt is, t'selue is oock in EF duer het voorstel, daerom G is t'swaerheyts middelpunt des vierhoucx ABCD. </s><s>Maer want de twee driehoucken CDB ende ABD sijn tusschen twee euewydighe AB ende DC, so sijn sy inde reden van haer gronden <arrow.to.target n="marg208"/>, dat is ghelijck den driehouck CDB tot ABD, alsoo DC tot AB: Maer ghelijck den driehouck CDB tot ADB, also den erm GQ tot GR duer het voorstel des I boucx, ghelijck dan DC tot AB, alsoo GQ tot GR; maer ghelijck GQ tot GR, alsoo PG tot GO (want sy tusschen de euewydeghe MN, KL sijn) ghelijck dan DC tot AB, alsoo GP tot PO [GO], daerom oock ghelijck tweemael DC met eenmael AB, tot tweemael AB met eenmael DC, also tweemael GP met eenmael GO, tot tweemael GO met eenmael GP. </s><s> Maer GE is euen an tweemael GP met eenmael GO, ende GF is euen an tweemael GO met eenmael GP, daerom ghelijck tweemael DC met eenmael AB, tot tweemael AB met eenmael DC, alsoo GE tot GF. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Het swaerheyts middelpunt dan des vierhoucx met twee, &c.</s>

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

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

<s>

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

</s>

Line 1878

Line 1751

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

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een rechtlinich plat wesen, diens swaerheyts middelpunt E, ende BDA deel des plats, wiens swaerheyts middelpunt F. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten t'swaerheyts middelpunt vinden des ander deels BDC. </s><s> T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal trecken FE tot in G, alsoo dat FE sulcken reden hebbe tot EG, als t'stick BDC tottet stick BDA: Ick seg dat G t'begheerde swaerheyts middelpunt is des ander deels BDC. </s><s> T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Anghesien t'swaerheyts middelpunt van BDA is F, ende des heels ABCD is E, soo moet t'swaerheyts middelpunt des ander deels BDC sijn in de rechte FE oneindelick voortghetrocken. </s><s>Want soot mueghelick waer, latet daer buyten wesen als H, ende laet ons trecken FH, het swaerheyts middelpunt dan des heels sal in FH sijn, maer dat is teghen t'ghestelde <arrow.to.target n="marg209"/>, wantet E is; Ten is dan niet buyten FE oneindelick voortghetrocken maer daer in. </s>

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

<s>

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

</s>

<s> Latet nu wesen <pb pagenum="77"/>(soot mueghelick waer) tusschen de punten <emph type="other"/>EG<emph.end/> als I; Maer den langsten erm EF sal dan meerder reden hebben tot den cortsten EI, dan de swaerste swaerheyt BDC tot de lichtste BDA, twelck teghen het voorstel des I boucx waer. </s><s>Ten is dan tusschen <emph type="other"/>EG<emph.end/> niet: Sghelijcx salmen oock bethoonen dattet bouen G niet en is. </s><s>Tis dan nootsaecklick G, t'welck wy bewysen moesten.</s>

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een rondt wesen, diens halfmiddellini <arrow.to.target n="marg210"/> EA, ende swaerheyts middelpunt E sy, ende trondt AFGH, deel des rondts ABCD, ende sijn swaerheyts middelpunt I, ende middellini <arrow.to.target n="marg211"/> AG. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten het swaerheyts middelpunt vinden des ander deels, dat is der maen ABCDHGF. </s><s> T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal IE voorttrecken tot in K, also dat IE sulcken reden hebbe tot EK, als de maen ABCDHGF tot het rondt AFGH, ende K sal t'begheerde swaerheydts middelpunt wesen. </s><s>Daer af t'bewys ghelijck sal sijn an tvoorgaende. </s><s>Maer om de reden dier maen tot dat rondt te vinden, men sal trecken CL euen met AG, daernaer AL, vindende <arrow.to.target n="marg212"/> de derde everednighe welcker eerste AL, de tweede LC, ende de derde sy M, Ende AL tot M, sal de reden sijn der maen tot het rondt AFGH. </s><s> Want ouermits ALC rechthouck is (reden dat sy int half rondt staet <arrow.to.target n="marg213"/>) het ront diens middellini AL, sal euen sijn ande maen, ende AL tot M is de ghedobbelde reden <arrow.to.target n="marg214"/> van AL tot LC, dat is van AL tot AG, daerom &c.</s>

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

<s>

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

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

</s>

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

<s>

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

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

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

</s>

<s> Sghelijcx soudemen voortvaren dat int rondt ABCD meer ronden ghebraken; by voorbeelt het rondt NO, wiens middelpunt P. </s><s>Want <pb pagenum="78"/>PK voortghetrocken tot in Q, alsoo dat PK sulcken reden hadde tot KQ, als het restende tot het rondt NO, so soude Q t'begheerde swaerheyts middelpunt sijn. </s><s>Ende alsoo met allen anderen formen welcker deelen reden kennelick is. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen de swaerheyts middelpunten eens plats ende sijns deels, wiens reden an t'ander deel kennelick is: wy hebben het swaerheyts middelpunt gheuonden des ander deels naer den eysch.</s>

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

<s> Sghelijcx soudemen voortvaren dat int rondt ABCD meer ronden ghebraken; by voorbeelt het rondt NO, wiens middelpunt P. </s><s>Want <pb pagenum="78"/>PK voortghetrocken tot in Q, alsoo dat PK sulcken reden hadde tot KQ, als het restende tot het rondt NO, so soude Q t'begheerde swaerheyts middelpunt sijn. </s><s>Ende alsoo met allen anderen formen welcker deelen reden kennelick is. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen de swaerheyts middelpunten eens plats ende sijns deels, wiens reden an t'ander deel kennelick is: wy hebben het swaerheyts middelpunt gheuonden des ander deels naer den eysch.</s>

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

<s> <emph type="other"/>YDER<emph.end/> brantsnees <arrow.to.target n="marg215"/> swaerheyts middelpunt is in haer middellini. </s>

<s>

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

</s>

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een brantsne sijn diens middellini AD. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat t'swaerheyts middelpunt inde lini AD is. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de linien EF, GH, IK, euewydighe van BC, ende sniende AD in L, M, N, daer naer EO, GP, IQ, KR, HS, FT, euewydighe van AD. </s><s> T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Ouermidts EF euewydighe is van BC, ende EO, FT, van LD, soo sal EFTO euewydich vierhouck sijn, wiens EL euen is met LF, oock met OD ende DT, waer duer t'swaerheyts middelpunt van EFTO, in DL is duer het voorstel, Ende om de selue reden sal t'swaerheyts middelpunt des euewydich vierhoucx GHSP in LM wesen, ende van IKRQ in MN, ende veruolghens t'swaerheyts middelpunt der form IKRHSFTOEPGQ, ghemaeckt vande voornoemde drie vierhoucken sal inde lini ND oft AD sijn. </s><s>Maer hoe datter sulcke vierhoucken meer gheschreuen worden, hoe dattet verschil des brandtsnees ABC, ende der binnenschreuen form van die vierhoucken vergaert, minder is, wy connen dan door dat oneindelick naerderen sulck een form binnen de brandtsne stellen, dattet verschil tusschen haer ende de brantsne, minder sy dan eenich ghegheuen plat hoe cleen het sy, waer uyt volght, dat stellende AD als swaerheyts middellini, so sal t'staltwicht des deels ADC, min verschillen van t'staltwicht des deels ADB, </s>

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

<s> dan eenich plat <pb pagenum="79"/>datmen soude connen gheuen, hoe cleen het sy, waer uyt ick aldus : Daerom AD is swaerheyts middellini, ende veruolghens het swaerheyts middelpunt des branders ABC is in haer. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Yder brandtsnees swaerheyts middelpunt dan, is in haer middellini, t'welck wy bewysen moesten.</s>

<s>

<margin.target id="marg216"/>Neuen alle verschillende staltswaerheden, can een swaerheyt ghestelt worden minder dan haer verschil

<margin.target id="marg217"/>Neuen dese staltswaerheden ADC ende ADB, en can gheen swaerheyt ghestelt worden minder dan haer verschil

<margin.target id="marg218"/>Dese staltswaerheden dan ADC ende ADB en verschillen niet

</s>

<s> O. Dese staltswaerheden dan ADC ende ADB en verschillen niet.</s>

<s> Daerom AD is swaerheyts middellini, ende veruolghens het swaerheyts middelpunt des branders ABC is in haer. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder brandtsnees swaerheyts middelpunt dan, is in haer middellini, t'welck wy bewysen moesten.</s>

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

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

<s> <emph type="other"/>ALLER<emph.end/> brantsneens middellinien worden van het swaerheyts middelpunt eueredelick <arrow.to.target n="marg219"/> ghedeelt. </s>

<s>

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

</s>

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD ende <arrow.to.target n="marg220"/> twee onghelijcke brantsneen sijn, diens middellinien AD, ende <arrow.to.target n="marg221"/>, ende swaerheyts middelpunten E, ende <arrow.to.target n="marg222"/>. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat ghelijck AE tot ED, also <arrow.to.target n="marg223"/> tot <arrow.to.target n="marg224"/>. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de linien AB, AC, die deelende in haer middelen F, G, ende trecken FG sniende AD in H, daer naer FI ende GK euewydighe van AD, ende daer naer IA, IB, KA, KC, ende laet ons stellen L in IF, alsoo dat IL dobbel sy an LF, sghelijcx M, alsoo dat KM dobbel sy an MG, ende laet ons trecken LM, sniende AD in N, ende IK, sniende AD in O, ende laet ons stellen P, alsoo dat AP dobbel sy an PD, ende laet ons IF voorttrecken tot Q inden grondt BC. </s><s> Nu anghesien AP dobbel is an PD, so is P t'swaerheyts middelpunt des driehoucx ABC, ende omme de selue reden L, M, <pb pagenum="80"/>swaerheyts middelpunten der twee driehoucken ABI, ende ACK, ende veruolghens, want sy euen sijn, soo is N haer beyde swaerheyts middelpunt. </s>

<s>

<margin.target id="marg220"/>abcd

<margin.target id="marg221"/>ad

<margin.target id="marg222"/>e

<margin.target id="marg223"/>ae

<margin.target id="marg224"/>ed

</s>

<s> NP dan is balck, de selue ghedeelt in R, alsoo dat den erm NR sy tot RP, als den driehouck ABC tot de twee driehoucken ABI, ACK, dat is, als 4 tot I (want alle brantsne is tot den driehouck als ABC ghelijck 4 tot 3, duer het 24 voorstel der viercanting des brantsnees van Archim. </s><s>daerom, &c.) Laet ons nu derghelijcke linien ende punten oock beschrijuen inde brantsne <arrow.to.target n="marg225"/>. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Ghelijck AD tot AO, also het viercant van DB tottet viercant van OI <arrow.to.target n="marg226"/>; Maer DQ is euen an OI, ende DQ is den helft van DB (want F is t'middel van AB, ende FQ is euewydich van AD) daerom het viercant van DB, is viervoudich an t'viercant van DQ, ofte van OI, ende veruolghens AD is viervoudich tot AO, daerom AO is I/4 van AD, ende OH oock I/4 (want AH is den helft van AD, ouermits FG ghetrocken is uyt de middelen van AB, AC) daerom doet N<emph type="other"/>HI<emph.end/>/I2 van AD, daer toe ghedaen H<emph type="other"/>DI<emph.end/>/2, comt voor ND 7/I2, daer af ghetrocken P<emph type="other"/>DI<emph.end/>/3, rest voor P<emph type="other"/>NI<emph.end/>/4: Maer NR is viervoudich tot RP, daerom RP doet I/20, daer toe P<emph type="other"/>DI<emph.end/>/3, doet voor RD 23/60, daerom RA de reste der lini, doet 37/60, Ghelijck dan 37 tot 23, also AR tot RD, ende met de selue reden is bethoont dat <arrow.to.target n="marg227"/> tot <arrow.to.target n="marg228"/>, oock is als 37 tot 23. </s><s>Dese twee rechtsideghe formen dan ghelijckelick beschreuen in verscheyden brandtsneen, hebben het swaerheyts middelpunt in haer middellinien, also dat de deelen onder malcanderen euerednich <arrow.to.target n="marg229"/> sijn. </s><s>Ende so wy inde brandtsnekens BI, IA, AK, KC, driehoucken beschreuen, soo ghedaen is inde brantsneen ABI, ACK, vindende daer naer t'swaerheyts middelpunt des heels binnescreuen rechtlinich plats, t'welck ick neem dat hier S soude wesen, ende daer <arrow.to.target n="marg230"/>, wy souden inder seluer voughen als vooren bethoonen, dat ghelijck AS tot SR, also <arrow.to.target n="marg231"/> tot <arrow.to.target n="marg232"/>. </s><s>Maer wy connen duer sulck oneindelick inschriuen der rechtlinighe formen oneindelick naerderen nae E, ende <arrow.to.target n="marg233"/>, ende ghelijcksideghe platten sullen altijt der middelliniens AD twee sticken euerednich ghedeelt hebben duer haer swaerheyts middelpunt, ende veruolghens de heele brantsneen ABC, <arrow.to.target n="marg234"/>, sullen die deelen euerednich hebben. </s><s>Want laet (soot mueghelick waer) T t'swaerheyts middelpunt sijn des brantsnees ABC, ende <arrow.to.target n="marg235"/> van <arrow.to.target n="marg236"/>, ende laet ons teeckenen <arrow.to.target n="marg237"/>, dat ghelijck ET tot TS, alsoo <arrow.to.target n="marg238"/> tot <arrow.to.target n="marg239"/>. </s><s>Nu alsmen duer t'inschrijuen veelsidegher formen in <arrow.to.target n="marg240"/>, sal ghecommen sijn tot <arrow.to.target n="marg241"/>, men sal met ghelijcke veelsideghe formen in ABC, ghecomen sijn tot T, daerom T sal t'swaerheyts middelpunt sijn der binneschreuen form, ende oock des heelen brantsneens ABC, t'welck ongheschickt is. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Aller brantsneens middellinien dan, worden van het swaerheyts middelpunt eueredelick ghedeelt, t'welck wy bewysen moesten. </s>

<s>

<margin.target id="marg225"/>abc

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

<margin.target id="marg227"/>ar

<margin.target id="marg228"/>rd

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

<margin.target id="marg230"/>s

<margin.target id="marg231"/>as

<margin.target id="marg232"/>sr

<margin.target id="marg233"/>e

<margin.target id="marg234"/>abc

<margin.target id="marg235"/>e

<margin.target id="marg236"/>abc

<margin.target id="marg237"/>t

<margin.target id="marg238"/>et

<margin.target id="marg239"/>ts

<margin.target id="marg240"/>abc

<margin.target id="marg241"/>t

</s>

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

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

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een brantsne <arrow.to.target n="marg242"/>: Huer swaerheyts middelpunt te vinden </s>

<s> <emph type="other"/>WESENDE<emph.end/> ghegheuen een brantsne <arrow.to.target n="marg242"/>: Huer swaerheyts middelpunt te vinden. </s>

<s>

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

</s>

Line 2008

Line 1825

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brantsne sijn, diens middellini AD. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten haer swaerheyts middelpunt vinden. </s><s>T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal de middellini AD, deelen in E, alsoo dat AE tot ED de reden hebbe van 3 tot 2: Ick seg dat E t'begheerde swaerheyts middelpunt is. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden de rechte linien AB, ende AC, ende de selue ghedeelt in haer middelen, F, G, ende ghetrocken worden FG sniende AD in H, daer naer FI ende GK euewydighe van AD, ende laet ghestelt worden t'punt L in IF, inder voughen dat IL sy tot LF, als AE tot ED: Laet oock ghestelt worden t'punt M in KG, alsoo dat MG euen sy an LF, ende laet ghetrocken worden LM sniende AD in N, ende IK sniende AD in O, ende laet IF voortghetrocken worden tot Q, inden grondt BC, ende laet ghestelt worden t'punt P, alsoo dat AP dobbel sy an PD, ende P sal swaerheyts middelpunt sijn des driehoucx ABC, ende want L, M, als swaerheyts middelpunten ghestelt sijn der brantsnekens ABI, ende ACK, soo sal N swaerheyts middelpunt sijn dier twee brandtsnekens, daerom ghedeelt den balck PN, also dat d'een erm sulcken reden hebbe tot d'ander, als den driehouck ABC tot die twee brantsnekens, wy sullen t'begheerde hebben; maar de heel brantsne heeft sulcken reden tot den driehouck ABC als 4 tot 3 (duer het 24 voorstel vande viercanting der brantsne van Archimed.) daerom den driehouck ABC heeft sulcken reden tot de twee brantsnekens, als 3 tot I; Ghedeelt dan PN alsoo dat het opperste stick, drievoudich sy tot het onderste, wy sullen t'swaerheyts middelpunt des heels hebben. </s><s>Ist dan dat wy bethoonen t'selue, te vallen in E (welcke E duer t'werck soo staet dat AE is tot ED inde reden van 3 tot 2) so is E het ware swaerheyts middelpunt. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>AO ende OH soo wy verclaert hebben int II voorstel, sijn elck I/4 van AD, Maer ghelijck 3 tot 2, alsoo AE tot ED, ende IL tot LF, ende ON tot NH, daerom ghedeelt O<emph type="other"/>HI<emph.end/>/4, in sulcken reden <pb pagenum="82"/>als 3 tot 2, so sal t'stick NH doen I/I0 van AD,</s>

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

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

<s> daer toeghedaen I/2 voor HD, doet voor ND 3/5, daer af ghetrocken P<emph type="other"/>DI<emph.end/>/3 rest voor NP 4/I5, de selue is duer t'bereytsel ghedeelt in E, alsoo dat NE is tot EP, als 3 tot I, daerom EP doet I/I5, daer toe ghedaen P<emph type="other"/>DI<emph.end/>/3, comt voor ED 2/5 van AD: Maer wesende ED 2/5, so sal EA doen 3/5, daerom AE heeft sulcken reden tot ED, als 3 tot 2, ende veruolghens E is t'swaerheyts middelpunt des brantsnees ABC, t'welck wy bewysen moesten. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een brantsne: Wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een brantsne: Wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

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

<s> <emph type="other"/>HET<emph.end/> <arrow.to.target n="marg243"/>{Proportionaliter.} <arrow.to.target n="marg244"/> Te vinden twee punten als N, E, alsoo dat de lini ON sulcken reden hebbe tot NH, als AE tot ED. </s><s><arrow.to.target n="marg245"/> Het sijn twee ghetalen O<emph type="other"/>HI<emph.end/>/4 ende H<emph type="other"/>PI<emph.end/>/6; deelt elck alsoo, dat het minste van OH, met het meeste van HP, drievoudich sy an t'minste van HP, ende dat t'meeste van OH sulcken reden hebbe tot sijn minste, als t'meeste van HP + I/2 tot t'minste van HP + I/3. </s>

<s> <emph type="other"/>HET<emph.end/> schijnt dat Archimedes ter kennis deses voorstels ghecommen is, duer een deser twee manieren: </s><s>D'eerste dat hy lichamelicke brantsneen makende, tot het formen siinder brandtspieghels, ofte om andersins hem daer in te oefnen, beuandt duer de daet, dit deel tot dat te wesen, als 3 tot 2, souckende daer naer de sekerheyt van dien in deser voughen: Anghesien BAI ende BAC beyde brandtsneen siin, soo worden haer middellinien IF ende AD <arrow.to.target n="marg243"/> eueredelick ghedeelt van haer swaerheyts middelpunten (soo int II voorstel bewesen is) daerom moet IL tot LF siin, als AE tot ED, maer ON is euen an IL, ende NH an LF, daerom moet ON sulcken reden hebben tot NH, als AE tot ED. </s><s>Maer als N swaerheyts middelpunt waer der twee brantsnekens, ende P des driehoucx ABC, so moet (ouermits desen driehouck drievoudich is tot die twee brantsnekens) den erm NE drievoudich siin anden erm EP, waer uyt sulcken voorstel rijst: </s><s>Te vinden twee punten als N, E, alsoo dat de lini ON sulcken reden hebbe tot NH, als AE tot ED. </s><s>Stellende daer naer AE te doen 3/5 van AD, ende ED de 2/5 ende versouckende alsoo watter uyt volghen soude, heeft beuonden naer de maniere als bouen, sulcx waerachtelick te ouercommen mettet begheerde. </s><s>Ofte soo hy dit aldus niet ghesocht en heeft al tastende, duer de voornoemde reden van 3 tot 2, maer duer lauter cracht der const, soo schijnet dat hy hem t'voornoemde in ghetalen voorghestelt heeft in deser voughen: </s><s>Het sijn twee ghetalen OH 1/4 ende HP 1/6; deelt elck alsoo, dat het minste van OH, met het meeste van HP, drievoudich sy an t'minste van HP, ende dat t'meeste van OH sulcken reden hebbe tot sijn minste, als t'meeste van HP + 1/2 tot t'minste van HP + 1/3. </s>

<s>

<margin.target id="marg243"/> schijnt dat Archimedes ter kennis deses voorstels ghecommen is, duer een deser twee manieren: D'eerste dat hy lichamelicke brantsneen makende, tot het formen siinder brandtspieghels, ofte om andersins hem daer in te oefnen, beuandt duer de daet, dit deel tot dat te wesen, als 3 tot 2, souckende daer naer de sekerheyt van dien in deser voughen: Anghesien BAI ende BAC beyde brandtsneen siin, soo worden haer middellinien IF ende AD eueredelick

<margin.target id="marg243"/>Proportionaliter.

<margin.target id="marg244"/>ghedeelt van haer swaerheyts middelpunten (soo int II voorstel bewesen is) daerom moet IL tot LF siin, als AE tot ED, maer ON is euen an IL, ende NH an LF, daerom moet ON sulcken reden hebben tot NH, als AE tot ED. Maer als N swaerheyts middelpunt waer der twee brantsnekens, ende P des driehoucx ABC, so moet (ouermits desen driehouck drievoudich is tot die twee brantsnekens) den erm NE drievoudich siin anden erm EP, waer uyt sulcken voorstel rijst:

<margin.target id="marg245"/>Stellende daer naer AE te doen 3/5 van AD, ende ED de 2/5 ende versouckende alsoo watter uyt volghen soude, heeft beuonden naer de maniere als bouen, sulcx waerachtelick te ouercommen mettet begheerde. Ofte soo hy dit aldus niet ghesocht en heeft al tastende, duer de voornoemde reden van 3 tot 2, maer duer lauter cracht der const, soo schijnet dat hy hem t'voornoemde in ghetalen voorghestelt heeft in deser voughen:

</s>

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ghecorte brantsne sijn (welverstaende dat AB euewydighe sy met DC) wiens middellini EF. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten haer swaerheyts middelpunt vinden. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een ghecorte brantsne sijn (welverstaende dat AB euewydighe sy met DC) wiens middellini EF. </s>

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

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

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

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

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

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

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

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een viergrondich <arrow.to.target n="marg246"/> wesen, diens formens middelpunt E sy, ende den as van A duer E, tot in F, middelpunt des driehoucx BCD, sy AF. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat E oock is sijn swaerheyts middelpunt. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Laet ons t'lichaem ophanghen byde lini AF, maer het viergrondich bestaet uyt vier euen ende ghelijcke naelden een selfder ghestalt, wiens ghemeene sop E, daerom AF is des lichaems swaerheyts middellini, ende om de selue reden sal de lini CE oock des swaerheyts middellini sijn: E dan is oock het swaerheyts <pb pagenum="84"/>middelpunt.</s>

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

<s>

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

</s>

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

<s> Sghelijcx sal oock t'bewys sijn van allen lichamen hebbende middelpunten der form, soo wel vermeerde ende ghecorte gheschicte lichamen, als gheschicte, want soomense ophangt byde middellinien deur eenighen lichamelicken houck, ofte duer het middelpunt haerder gronden ende des formens middelpunt, soo hebben al de naelden (wiens ghemeene sop het formens middelpunt, ende gronden de platten des lichaems sijn) tot allen sijden ghelijcke ghestalt, daerom oock duer ghemeene wetenschap, ende duer de begheerte des I boucx, alles hangt an die lini euewichtich, ende veruolghens de sne sulcker twee swaerheyts middellinien malcander sniende in des formens middelpunt, is ock het sawerheyts middelpunt. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Yder lichaems formens middelpunt dan, is oock sijn swaerheyts middelpunt.</s>

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

<s> <emph type="other"/>YDER<emph.end/> pilaers swaerheyts middelpunt is int middel vanden as. </s>

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

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

<s> T' <emph type="other"/>GHEGHE<emph.end/>. </s><s>Laet AB een driehouckich pilaer sijn diens grondt ACD. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat sijn swaerheyts middelpunt int middel vanden as is. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons trecken van D tot E int middel van AC de lini DE: Daer naer FG ende HI euewydighe van AC, sniende DE inde punten K, L, daer naer de linien FM, HN, IO, GP, euewydighe met DE, daer naer van A tot Q int middel der sijde DC, de lini AQ: Laet sghelijcx oock beschreuen worden het decsel, ende laet ons den pilaer doorsnien met een plat <arrow.to.target n="marg247"/> RS euewidich met den grondt ADC, ende S sy int middel van CB. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>T'plat ghetrocken duer DE, ende duer haer lijckstandighe <arrow.to.target n="marg248"/> int decksel, deelt den binneschreuen pilaer uyt die twee vierhouckighe pilaren vergaert, in twee euen ende ghelijcke deelen, ende van ghelijcke ghestalt; het doorsnijt dan dier binneschreuen pilaers swaerheydts middelpunt. </s><s>Maer hoe datter sulcke vierhouckighe pilaren meer beschreuen sijn inde ghegheuen driehouckighen, hoe dat dese min verschilt van die; wy connen dan duer dat oneindelick <pb pagenum="85"/>naerderen sulck een form binnen de ghegheuen pilaer beschrijuen, dat haer verschil vande binnescreuen minder sal wesen dan eenich ghegheuen lichaem hoe cleen het sy, waer uyt volght dat het staltwicht des deels DECB ouer d'een sijde des plats, min verschillen sal van t'staltwicht des deels ouer d'ander sijde des plats, dan eenich lichaem datmen soude connen gheuen hoe cleen het sy, waer uyt ick aldus : Daerom t'plat duer DE ende haer lickstandighe <arrow.to.target n="marg249"/> int decksel, lijt duer t'swaerheyts middelpunt des ghegheuen pilaers, ofte het swaerheyts middelpunt is in dat plat. </s><s>Ende om de selue reden ist oock int plat duer AQ, ende haer lijckstandighe int decksel. </s><s>Maer deser twee platten ghemeene sne is de rechte lini tusschen de swaerheyts middelpunten des grondts ende decsels, welcke lini den as is des ghegheuen pilaers, tswaerheyts middelpunt dan is inden as, het is oock int plat duer RS, want t'selue deelt den pilaer in twee euen, ghelijcke, ende lijckstandighe deelen; Maer dat plat doorsnijt den as in sijn middel, het swaerheyts middelpunt dan is in des as middel.</s>

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

<s>

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

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

<margin.target id="marg249"/>Homologam.

</s>

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

<s>

<margin.target id="marg250"/>Neuen alle verschillende staltswaerheden, can een swaerheyt ghestelt worden minder dan haer verschil

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

<margin.target id="marg251"/>Neuen dese staltswaerheden en can gheen swaerheyt ghestelt worden minder dan haer verschil

<margin.target id="marg252"/>Dese staltswaerheden dan en verschillen niet

</s>

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

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

<s>O. Dese staltswaerheden dan en verschillen niet. </s>

<s> Daerom t'plat duer DE ende haer lickstandighe <arrow.to.target n="marg249"/> int decksel, lijt duer t'swaerheyts middelpunt des ghegheuen pilaers, ofte het swaerheyts middelpunt is in dat plat. </s><s>Ende om de selue reden ist oock int plat duer AQ, ende haer lijckstandighe int decksel. </s><s>Maer deser twee platten ghemeene sne is de rechte lini tusschen de swaerheyts middelpunten des grondts ende decsels, welcke lini den as is des ghegheuen pilaers, tswaerheyts middelpunt dan is inden as, het is oock int plat duer RS, want t'selue deelt den pilaer in twee euen, ghelijcke, ende lijckstandighe deelen; Maer dat plat doorsnijt den as in sijn middel, het swaerheyts middelpunt dan is in des as middel.</s>

<s>

<margin.target id="marg249"/>Homologam.

</s>

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

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

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

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

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

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

<s> <emph type="other"/>YDER<emph.end/> naeldens <arrow.to.target n="marg253"/> swaerheyts middelpunt is inden as. </s>

<s>

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

</s>

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

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

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

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

<s>O. Dese staltswaerheden dan en verschillen niet. </s>

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een naelde sijn, diens grondt den driehouck BCD, wiens swaerheyts middelpunt E, ende den as AE. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat des naeldens swaerheyts middelpunt inden as AE is. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons de naelde snien met een plat FGH euewydich met BCD, ende sniende den as AE in I: Laet oock ghetrocken worden FK, GL, HM, euewydich vanden as AE, alsoo dat de punten K, L, M, int plat sijn des driehoucks BCD, inder voughen dat FGHKLM een pilaer is, wiens grondt IKL euen ende ghelijck is an het decksel FGH, ende ghelijck anden grondt BCD; Daer naer ghelijck de naelde doorsneen was met FGH, laetse noch eenmal alsoo doorsneen sijn met het plat NOP, sniende den as in Q, ende daer uyt oock alsoo ghemaect den pilaer NOPRST, te weten NR, OS, PT, euewydich vanden as AE, ende de punten R, S, T, int plat FGH. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Anghesien de driehoucken NOP, RST, FGH, KLM, alle ghelijck sijn anden driehouck BCD, ende dat haer punten Q, I, E, in haer sulcken stant hebben als E inden driehouck BCD, ende dat E des driehoucx BCD swaerheyts middelpunt is, soo sijn oock die Q, I, E, haer driehouckens swaerheyts middelpunten, waer duer IE as is des pilaers FGHKLM, in wiens middel huer swaerheyts middelpunt is duer het voorstel. </s><s>Sghelijcx is QI oock as des pilaers NOPRST, in wiens middel huer swaerheyts middelpunt is, ende vervolgens het swaerheyts middelpunt des lichaems uyt die twee pilaren vergaert is in QE, daerom oock in AE; Maer hoe datter inde naelde sulcke pilaren meer beschreuen worden, hoe dattet verschil der naelde ende der <pb pagenum="87"/>binneschreuen form van sulcke pilaren vergaert, minder is, </s>

<s> Sghelijcx sal oock t'bewys sijn val naelden wiens gronden sijn Vierhoucken, Veelhoucken, Ronden &c. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder naeldens swaerheyts middelpunt dan is inden as.</s>

<s> blijuende nochtan het swaerheyts middelpunt der binneschreuen form altijt inden as AE; Wy connen dan duer dat oneindelick naerderen sulcken form binnen de naelde stellen, dattet verschil tusschen haer ende de naelde, minder sal wesen dan eenich ghegheuen lichaem hoe cleen het sy, waer uyt volght dat stellende AE voor swaerheyts middellini, der naelde, soo sal het staltwicht van d'een sijde tot d'ander, min verschillen dan eenighe swaerheyt diemen soude connen gheuen, waer uyt ick aldus : Sghelijcx sal oock t'bewys sijn val naelden wiens gronden sijn Vierhoucken, Veelhoucken, Ronden &c. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Yder naeldens swaerheyts middelpunt dan is inden as.</s>

<s><emph type="other"/>VI. EYSCH<emph.end/>. <emph type="other"/>XVII. VOORSTEL<emph.end/>.</s>

<s>

<margin.target id="marg254"/>Neuen alle verschillende staltswaerheden, can een swaerheyt ghestelt worden minder dan haer verschil

<margin.target id="marg255"/>Neuen dese staltswaerheden en can gheen swaerheyt ghestelt worden minder dan haer verschil

<margin.target id="marg256"/>Dese staltswaerheden dan en verschillen niet

</s>

<s><emph type="other"/>VI. EYSCH. XVII. VOORSTEL<emph.end/>.</s>

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een naelde wesen, diens grondt sy den driehouck BCD. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten haer swaerheyts middelpunt vinden. </s><s>T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal de swaerheyts middelpunten vinden van eenighe twee driehoucken, als E van BCD, ende F van ADC, treckende de linien AE, BF; welcker sne G, ick seg te wesen het begheerde swaerheyts middelpunt. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Des naldens ABCD swaerheyts middelpunt is in AE, ende oock in BF, duer het I6 voorstel, het is dan nootsaeclick G. </s><s> T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een naelde: Wy hebben huer swaerheyts middelpunt gheuonden naer den eysch.</s>

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

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

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een driehouckighe naelde wesen, diens sop A, ende grondt BCD, ende den as van B tot int swaerheyts middelpunt E des driehoucx ADC, sy BE, ende van A tot int swaerheyts middelpunt F des driehoucx BCD sy AF, sniende BE in G, voor t'swaerheyts middelpunt der ghegheuen naelde. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat BG drievoudich is an GE. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons trecken van H middel van CD, de linien HA, HB. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Ouermits HA ghetrocken is uyt het middel van DC tot inden houck A, ende dat E t'swaerheyts middelpunt is des driehoucx ACD, soo sal AE dobbel sijn an EH duer het voorstel, ende om de selue reden sal BF dobbel wesen an FH. </s><s>Dit soo sijnde, ghetrocken de reden EA 2, tot AH 3, vande reden BF 2, tot F<emph type="other"/>HI<emph.end/> (dat is <arrow.to.target n="marg257"/> 3/2 van <arrow.to.target n="marg258"/> 2/I) daer rest <arrow.to.target n="marg259"/> de reden van BG tot GE: Maer treckende <arrow.to.target n="marg260"/> 2/3 van <arrow.to.target n="marg261"/> 2/I daer blijft <arrow.to.target n="marg262"/> 3/I. </s><s>BG dan is tot GE, als 3 tot I.</s>

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

<s>

<margin.target id="marg257"/>Reden

<margin.target id="marg258"/>Reden

<margin.target id="marg259"/>Door t'verkeerde des I2 cap. I. lib. Almag. Ptol.

<margin.target id="marg260"/>Reden

<margin.target id="marg261"/>Reden

<margin.target id="marg262"/>Reden

</s>

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

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

<s>

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

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

</s>

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

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

<s>

<margin.target id="marg264"/>Neuen alle twee punten in verscheyden plaetsen staende, connen twee punten ghestelt worden die malcander naerder siin

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

<margin.target id="marg265"/>Neuen dese twee punten en connen gheen twee punten ghestelt worden die malcander naerder siin

<margin.target id="marg266"/>Dese twee punten dan en staen in gheen verscheyden plaetsen

</s>

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

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

<s>O. Neuen dese twee punten en connen gheen twee punten ghestelt worden die malcander naerder siin; </s>

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

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

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

<s><emph type="other"/>VIIEYSCH. XIXVOORSTEL<emph.end/>.</s>

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

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

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

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

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

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

<s>

Line 2190

Line 1961

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABCDEF een ghecorte naelde sijn, diens decksel ABC, ende grondt DEF. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten huer swaerheyts middelpunt vinden. </s><s>T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal de ghecorte naelde volmaken, daer an stellende het ghebrekende ABCG, vindende H swaerheyts middelpunt des driehoucx DEF, treckende den as GH, wiens punt inden driehouck ABC, sy I, daernaer salmen teeckenen K, alsoo dat GK drievoudich sy an KI: Insghelijcx L, also dat GL drievoudich sy an LH, teeckenende M, alsoo dat KL sulcken reden hebbe tot LM, ghelijck de ghecorte naelde ABCDEF, tot de naelde ABCG, Ick seg dat M t'begheerde swaerheyts middelpunt is. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>L is swaerheyts middelpunt des heels, ende K des deels, ende ghelijck t'onderste deel tottet bouenste, also KL tot LM, Daerom M, door het voorstel des I boucx is t'begheerde swaerheyts middelpunt, t'welck wy bewysen moesten. </s><s>Sghelijcx sal oock den voortganck sijn in allen anderen ghecorte naelden. </s><s>T' <emph type="other"/>BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een ghecorte naelde: Wy hebben huer swaerheyts middelpunt gheuonden, naer den eysch.</s>

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

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

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

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

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet A een ongheschickt platgrondich lichaem sijn, dat is omvanghen in platten so veel alst sy. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden. </s>

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

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

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

<s>

<margin.target id="marg268"/>Pyramids.

</s>

Line 2218

Line 1989

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brander wesen diens grondt BC sy, ende den as AD daerop cromhouckich. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dattet swaerheyts middelpunt in AD is. </s><s>T' <emph type="other"/>BEREYTSEL<emph.end/>. </s><s>Laet ons den brander snien met twee platten EF, GH euewydich vanden grondt BC, welcker ghemeene sneen met den as AD, sijn I, K; Ende laet ons trecken de linien EL, FM, GN, HO: <pb pagenum="92"/>ende LM, NO, GH, sullen lancronden <arrow.to.target n="marg270"/> wesen ghelijck an t'lanckront BC; ende laet EM met GO pilaren sijn uyt de selue beschreuen.</s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brander wesen diens grondt BC sy, ende den as AD daerop cromhouckich. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dattet swaerheyts middelpunt in AD is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ons den brander snien met twee platten EF, GH euewydich vanden grondt BC, welcker ghemeene sneen met den as AD, sijn I, K; Ende laet ons trecken de linien EL, FM, GN, HO: <pb pagenum="92"/>ende LM, NO, GH, sullen lancronden <arrow.to.target n="marg270"/> wesen ghelijck an t'lanckront BC; ende laet EM met GO pilaren sijn uyt de selue beschreuen.</s>

<s>

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

</s>

Line 2226

Line 1997

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

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

<s>

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

</s>

Line 2252

Line 2023

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

<s> T' <emph type="other"/>GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brander wesen diens sop A, ende as AD sy. </s><s>T' <emph type="other"/>BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden. </s><s>T' <emph type="other"/>WERCK<emph.end/>. </s><s>Men sal den as AD in E deelen, alsoo dat AE dobbel sy an ED, ende E sal t'begheerde swaerheyts middelpunt sijn; T'welck bewesen is duer Frederick Commandin int 29 voorstel, waer af den sin verclaert naer onse manier soodanich is. </s><s>T' <emph type="other"/>BEWYS<emph.end/>. </s><s>Laet den brander doorsneen worden met een plat FG, euewydich vanden grondt BC, ende duer t'middel des as H, ende sniende de uytersten des branders in I, K, ende laet BCGF ende IKLM twee pilaren sijn, beschreuen omme den brander, wiens middelpunten N, O, ende IKPQ een pilaer binnen den brander, wiens swaerheyts middelpunt oock O sijn sal. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brander wesen diens sop A, ende as AD sy. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal den as AD in E deelen, alsoo dat AE dobbel sy an ED, ende E sal t'begheerde swaerheyts middelpunt sijn; T'welck bewesen is duer Frederick Commandin int 29 voorstel, waer af den sin verclaert naer onse manier soodanich is. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet den brander doorsneen worden met een plat FG, euewydich vanden grondt BC, ende duer t'middel des as H, ende sniende de uytersten des branders in I, K, ende laet BCGF ende IKLM twee pilaren sijn, beschreuen omme den brander, wiens middelpunten N, O, ende IKPQ een pilaer binnen den brander, wiens swaerheyts middelpunt oock O sijn sal. </s>

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

<s>

Line 2268

Line 2039

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

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

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

Line 2278

Line 2049

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

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

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

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

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

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

Legend:

Removed from v.1.59
changed lines
	Added in v.1.120