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version 1.134, 2003/09/15 11:18:35

version 1.135, 2003/09/15 13:40:52

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<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> <emph type="other"/>ALLER<emph.end/> brantsneens middellinien worden van het swaerheyts middelpunt <arrow.to.target n="marg219"/>eueredelick 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'BEREYTSEL<emph.end/>. </s><s>Laet ons trecken de linien AB, AC, die deelende in haer middelen F, G, ende trecken FG sniende AD in H, daer naer FI ende GK euewydighe van AD, ende daer naer IA, IB, KA, KC, ende laet ons stellen L in IF, alsoo dat IL dobbel sy an LF, sghelijcx M, alsoo dat KM dobbel sy an MG, ende laet ons trecken LM, sniende AD in N, ende IK, sniende AD in O, ende laet ons stellen P, alsoo dat AP dobbel sy an PD, ende laet ons IF voorttrecken tot Q inden grondt BC. </s><s> Nu anghesien AP dobbel is an PD, so is P t'swaerheyts middelpunt des driehoucx ABC, ende omme de selue reden L, M, <pb pagenum="80"/>swaerheyts middelpunten der twee driehoucken ABI, ende ACK, ende veruolghens, want sy euen sijn, soo is N haer beyde swaerheyts middelpunt. </s><s> NP dan is balck, de selue ghedeelt in R, alsoo dat den erm NR sy tot RP, als den driehouck ABC tot de twee driehoucken ABI, ACK, dat is, als 4 tot I (want alle brantsne is tot den driehouck als ABC ghelijck 4 tot 3, duer het 24 voorstel der viercanting des brantsnees van Archim. </s><s>daerom, &c.) Laet ons nu derghelijcke linien ende punten oock beschrijuen inde brantsne <emph type="italics"/>abc<emph.end/>. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Ghelijck AD tot AO, also het viercant van DB tottet viercant van OI <arrow.to.target n="marg226"/>; Maer DQ is euen an OI, ende DQ is den helft van DB (want F is t'middel van AB, ende FQ is euewydich van AD) daerom het viercant van DB, is viervoudich an t'viercant van DQ, ofte van OI, ende veruolghens AD is viervoudich tot AO, daerom AO is 1/4 van AD, ende OH oock 1/4 (want AH is den helft van AD, ouermits FG ghetrocken is uyt de middelen van AB, AC) daerom doet NH 1/12 van AD, daer toe ghedaen HD 1/2, comt voor ND 7/12, daer af ghetrocken PD 2/3, rest voor PN 1/4: Maer NR is viervoudich tot RP, daerom RP doet 1/20, daer toe PD 1/3, doet voor RD 23/60, daerom RA de reste der lini, doet 37/60, Ghelijck dan 37 tot 23, also AR tot RD, ende met de selue reden is bethoont dat <emph type="italics"/>ar<emph.end/> tot <emph type="italics"/>rd<emph.end/>, oock is als 37 tot 23. </s><s>Dese twee rechtsideghe formen dan ghelijckelick beschreuen in verscheyden brandtsneen, hebben het swaerheyts middelpunt in haer middellinien, also dat de deelen onder malcanderen <arrow.to.target n="marg229"/>euerednich sijn. </s><s>Ende so wy inde brandtsnekens BI, IA, AK, KC, driehoucken beschreuen, soo ghedaen is inde brantsneen ABI, ACK, vindende daer naer t'swaerheyts middelpunt des heels binnescreuen rechtlinich plats, t'welck ick neem dat hier S soude wesen, ende daer <emph type="italics"/>s<emph.end/>, wy souden inder seluer voughen als vooren bethoonen, dat ghelijck AS tot SR, also <emph type="italics"/>as<emph.end/> tot <emph type="italics"/>sr<emph.end/>. </s><s>Maer wy connen duer sulck oneindelick inschriuen der rechtlinighe formen oneindelick naerderen nae E, ende <emph type="italics"/>e<emph.end/><arrow.to.target n="marg233"/>, ende ghelijcksideghe platten sullen altijt der middelliniens AD twee sticken euerednich ghedeelt hebben duer haer swaerheyts middelpunt, ende veruolghens de heele brantsneen ABC, <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg234"/>, sullen die deelen euerednich hebben. </s><s>Want laet (soot mueghelick waer) T t'swaerheyts middelpunt sijn des brantsnees ABC, ende <emph type="italics"/>e<emph.end/><arrow.to.target n="marg235"/> van <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg236"/>, ende laet ons teeckenen <emph type="italics"/>t<emph.end/><arrow.to.target n="marg237"/>, dat ghelijck ET tot TS, alsoo <emph type="italics"/>et<emph.end/><arrow.to.target n="marg238"/> tot <emph type="italics"/>ts<emph.end/><arrow.to.target n="marg239"/>. </s><s>Nu alsmen duer t'inschrijuen veelsidegher formen in <emph type="italics"/>abc<emph.end/><arrow.to.target n="marg240"/>, sal ghecommen sijn tot <emph type="italics"/>t<emph.end/><arrow.to.target n="marg241"/>, men sal met ghelijcke veelsideghe formen in ABC, ghecomen sijn tot T, daerom T sal t'swaerheyts middelpunt sijn der binneschreuen form, ende oock des heelen brantsneens ABC, t'welck ongheschickt is. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Aller brantsneens middellinien dan, worden van het swaerheyts middelpunt eueredelick ghedeelt, t'welck wy bewysen moesten. </s>

<s>

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

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

Line 1834

<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 <arrow.to.target n="marg242"/>brantsne: Huer swaerheyts middelpunt te vinden. </s>

<s>

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

</s>

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<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'GHEGHEVEN<emph.end/>. </s><s>Laet ABC een brantsne sijn, diens middellini AD. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten haer swaerheyts middelpunt vinden. </s><s><emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal de middellini AD, deelen in E, alsoo dat AE tot ED de reden hebbe van 3 tot 2: Ick seg dat E t'begheerde swaerheyts middelpunt is. </s><s><emph type="other"/>T'BEREYTSEL<emph.end/>. </s><s>Laet ghetrocken worden de rechte linien AB, ende AC, ende de selue ghedeelt in haer middelen, F, G, ende ghetrocken worden FG sniende AD in H, daer naer FI ende GK euewydighe van AD, ende laet ghestelt worden t'punt L in IF, inder voughen dat IL sy tot LF, als AE tot ED: Laet oock ghestelt worden t'punt M in KG, alsoo dat MG euen sy an LF, ende laet ghetrocken worden LM sniende AD in N, ende IK sniende AD in O, ende laet IF voortghetrocken worden tot Q, inden grondt BC, ende laet ghestelt worden t'punt P, alsoo dat AP dobbel sy an PD, ende P sal swaerheyts middelpunt sijn des driehoucx ABC, ende want L, M, als swaerheyts middelpunten ghestelt sijn der brantsnekens ABI, ende ACK, soo sal N swaerheyts middelpunt sijn dier twee brandtsnekens, daerom ghedeelt den balck PN, also dat d'een erm sulcken reden hebbe tot d'ander, als den driehouck ABC tot die twee brantsnekens, wy sullen t'begheerde hebben; maar de heel brantsne heeft sulcken reden tot den driehouck ABC als 4 tot 3 (duer het 24 voorstel vande viercanting der brantsne van Archimed.) daerom den driehouck ABC heeft sulcken reden tot de twee brantsnekens, als 3 tot 1; Ghedeelt dan PN alsoo dat het opperste stick, drievoudich sy tot het onderste, wy sullen t'swaerheyts middelpunt des heels hebben. </s><s>Ist dan dat wy bethoonen t'selue, te vallen in E (welcke E duer t'werck soo staet dat AE is tot ED inde reden van 3 tot 2) so is E het ware swaerheyts middelpunt. </s>

<s><emph type="other"/>T'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><emph type="other"/>T'BEWYS<emph.end/>. </s><s>AO ende OH soo wy verclaert hebben int 11. voorstel, sijn elck 1/4 van AD, Maer ghelijck 3 tot 2, alsoo AE tot ED, ende IL tot LF, ende ON tot NH, daerom ghedeelt OH 1/4, in sulcken reden <pb pagenum="82"/>als 3 tot 2, so sal t'stick NH doen 1/10 van AD,</s><s> daer toeghedaen 1/2 voor HD, doet voor ND 3/5, daer af ghetrocken PD 1/3 rest voor NP 4/15, de selue is duer t'bereytsel ghedeelt in E, alsoo dat NE is tot EP, als 3 tot 1, daerom EP doet 1/15, daer toe ghedaen PD 1/3, comt voor ED 2/5 van AD: Maer wesende ED 2/5, so sal EA doen 3/5, daerom AE heeft sulcken reden tot ED, als 3 tot 2, ende veruolghens E is t'swaerheyts middelpunt des brantsnees ABC, t'welck wy bewysen moesten. </s>

<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/> 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> <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 11. 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"/>Proportionaliter.

Line 1875

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

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een 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 1. 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> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet ABCD een <arrow.to.target n="marg246"/>viergrondich wesen, diens formens middelpunt E sy, ende den as van A duer E, tot in F, middelpunt des driehoucx BCD, sy AF. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten bewysen dat E oock is sijn swaerheyts middelpunt. </s><s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>Laet ons t'lichaem ophanghen byde lini AF, maer het viergrondich bestaet uyt vier euen ende ghelijcke naelden een selfder ghestalt, wiens ghemeene sop E, daerom AF is des lichaems swaerheyts middellini, ende om de selue reden sal de lini CE oock des swaerheyts middellini sijn: E dan is oock het swaerheyts <pb pagenum="84"/>middelpunt.</s><s> Sghelijcx sal oock t'bewys sijn van allen lichamen hebbende middelpunten der form, soo wel vermeerde ende ghecorte gheschicte lichamen, als gheschicte, want soomense ophangt byde middellinien deur eenighen lichamelicken houck, ofte duer het middelpunt haerder gronden ende des formens middelpunt, soo hebben al de naelden (wiens ghemeene sop het formens middelpunt, ende gronden de platten des lichaems sijn) tot allen sijden ghelijcke ghestalt, daerom oock duer ghemeene wetenschap, ende duer de 1. begheerte des 1. boucx, alles hangt an die lini euewichtich, ende veruolghens de sne sulcker twee swaerheyts middellinien malcander sniende in des formens middelpunt, is ock het sawerheyts middelpunt. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Yder lichaems formens middelpunt dan, is oock sijn swaerheyts middelpunt.</s>

<s>

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

</s>

Line 1886

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

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

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

<s>

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

</s>

<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><emph type="other"/>T'BEWYS<emph.end/>. </s><s>T'plat ghetrocken duer DE, ende duer haer <arrow.to.target n="marg248"/>lijckstandighe 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="marg248"/>Homologam.

</s>

Line 1900

<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> Daerom t'plat duer DE ende haer <arrow.to.target n="marg249"/>lickstandighe 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.

Line 1909

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

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

<s>

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

</s>

Line 1942

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

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

<s>

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

</s>

Line 1970

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

<s>

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

</s>

Line 1980

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

<s><emph type="other"/>T'BEWYS<emph.end/>. </s><s>L is swaerheyts middelpunt des heels, ende K des deels, ende ghelijck t'onderste deel tottet bouenste, also KL tot LM, Daerom M, door het 1. voorstel des 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"/>T'BEWYS<emph.end/>. </s><s>L is swaerheyts middelpunt des heels, ende K des deels, ende ghelijck t'onderste deel tottet bouenste, also KL tot LM, Daerom M, door het 1. voorstel des 1. boucx is t'begheerde swaerheyts middelpunt, t'welck wy bewysen moesten. </s><s>Sghelijcx sal oock den voortganck sijn in allen anderen ghecorte naelden. </s><s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een ghecorte naelde: Wy hebben huer swaerheyts middelpunt gheuonden, naer den eysch.</s>

<s><emph type="other"/>IX. EYSCH<emph.end/>. <emph type="other"/>XXI. VOORSTEL<emph.end/>.</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"/>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 17. voorstel. </s><s>Daer naer om te vinden t'ghemeene swaerheydts middelpunt van twee naelden, men sal tusschen haer swaerheyts middelpunten een balck trecken, die deelende in sulcken reden als haer twee naelden tot malcanderen sijn, weluerstaende t'cortste deel naer de swaerste naelde. </s><s>Ende inder selue voughen salmen daertoe vergaderen de derde naelde, ende alle d'ander, ende t'punt den balck alsoo ten laetsten deelende, sal t'begheerde swaerheyts middelpunt sijn, waer af t'bewys openbaer is. </s>

<s> <emph type="other"/>T'GHEGHEVEN<emph.end/>. </s><s>Laet A een ongheschickt platgrondich lichaem sijn, dat is omvanghen in platten so veel alst sy. </s><s><emph type="other"/>T'BEGHEERDE<emph.end/>. </s><s>Wy moeten sijn swaerheyts middelpunt vinden. </s><s> <emph type="other"/>T'WERCK<emph.end/>. </s><s>Men sal <pb pagenum="91"/>t'lichaem deelen inde <arrow.to.target n="marg268"/>nalden dieder ten weynichsten ende bequamelicxt uyt vallen willen. </s><s>Ten quaetsten commende men can als duer ghemeene reghel, alle platgrondich lichaem in soo veel naelden deelen alst platten heeft, stellende eenich punt int lichaem voor haer ghemeene sop. </s><s> Dit soo sijnde, men sal yder naeldens swaerheyts middelpunt vinden duer het 17. voorstel. </s><s>Daer naer om te vinden t'ghemeene swaerheydts middelpunt van twee naelden, men sal tusschen haer swaerheyts middelpunten een balck trecken, die deelende in sulcken reden als haer twee naelden tot malcanderen sijn, weluerstaende t'cortste deel naer de swaerste naelde. </s><s>Ende inder selue voughen salmen daertoe vergaderen de derde naelde, ende alle d'ander, ende t'punt den balck alsoo ten laetsten deelende, sal t'begheerde swaerheyts middelpunt sijn, waer af t'bewys openbaer is. </s>

<s><emph type="other"/>T'BESLVYT<emph.end/>. </s><s>Wesende dan ghegheuen een platgrondich lichaem soodanigher form alst valt, Wy hebben sijn swaerheyts middelpunt gheuonden, naer den heysch.</s>

<s>

Line 1995

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

<s> <emph type="other"/>YDER<emph.end/> branders <arrow.to.target n="marg269"/> swaerheyts middelpunt is inden as. </s>

<s> <emph type="other"/>YDER<emph.end/> <arrow.to.target n="marg269"/>branders swaerheyts middelpunt is inden as. </s>

<s>

<margin.target id="marg269"/>Conoidale rectangulum.

</s>

Line 2006

<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><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 <arrow.to.target n="marg270"/>lancronden 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>

<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 strie: </s>

<s> <emph type="other"/>T'BEWYS<emph.end/>. </s><s>Want LD <arrow.to.target n="marg271"/>halfmiddellini 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 strie: </s>

<s>

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

</s>

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