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version 1.82, 2003/09/09 21:12:26

version 1.111, 2003/09/11 15:25:35

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<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>

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<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> <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> <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 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><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><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> <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 <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><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>

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

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

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

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

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

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

<s><emph type="other"/>IIIVOORBEELT<emph.end/>.</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"/>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"/>III. VOORBEELT<emph.end/>.</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> <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> <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> <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 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> <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 swaerheydts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal den seshouck deelen in vier driehoucken, met eenighe linien als AC, AD, FD, vindende daer naer het swaerheyts middelpunt des vierhoucx ADCB door het 4 voorbeelt, t'welck G sy, ende des vierhoucx ADEF, t'welck H sy, ende de lini HG is balck. </s><s>Daernaer ghetrocken BI ende DK rechthouckich op AC, insghelijcx AL ende EM beyde rechthouckich op FD, men sal der drie linien welcker eerste FD, de tweede AC, de derde BI met KD, vinden de vierde euerednighe, welcke NO sy, deelende den balck HG in P, also dat den erm PG, sulcken reden hebbe tot den erm PH, ghelijck AL met EM, tot NO; Ick seg dat P het begheerde swaerheyts middelpunt is. </s><s> Ende soo salmen voort mueghen varen met ander veelhouckeghe platten. </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><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een rechthouckich plat: Wy hebben sijn swaerheydts middelpunt gheuonden naer den eysch.</s>

<s>

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

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

</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 1772

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>

<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><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Het swaerheydts middelpunt dan des vierhoucx met twee euewydighe sijden, is inde lini tusschen dier sijden middelpunten, t'welck wy bewysen moesten.</s>

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

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Anghesien het swaerheydts middelpunt des driehoucx BDC, is in BF, duer het 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 1796

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> <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>

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

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

<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. XI. VOORSTEL<emph.end/>. </s>

<s><emph type="other"/>VIII. VERTOOCH<emph.end/>. <emph type="other"/>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> <emph type="other"/>T'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><emph type="other"/>T'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><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>

<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><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 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><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Aller brantsneens middellinien dan, worden van het swaerheyts middelpunt eueredelick ghedeelt, t'welck wy bewysen moesten. </s>

<s>

<margin.target id="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>

Line 1926

Line 1825

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

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