Gravesande, Willem Jacob 's
,
Physices elementa mathematica, experimentis confirmata sive introductio ad philosophiam Newtonianam; Tom. 1
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MATHEMATICA. LIB. I. CAP. IV.
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cunda infinite exigua eſt reſpectu primæ, demonſtramus enim angulum quemcun-
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quein ſecunda ſuperari ab angulo quocunque, id eſt, utcumque exiguo, in prima.</
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<
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xml:space
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">Sit c tertia proportionalis ipſis a & </
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xml:space
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</
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<
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xml:space
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">Multiplicando per c æquationem ax = yy, habemus acx = yyc, id eſt bbx = yyc. </
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in ſecunda curva bbx valet z
<
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; </
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<
s
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xml:space
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= yyc, ſi abſciſſa x fuerit eademiu
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utraque curva.</
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<
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<
s
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xml:space
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">Ex æquatione hac deducimus z, c:</
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s
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xml:space
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<
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: </
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<
s
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xml:space
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">unde patet yy ſuperari à zz, id
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eſt, y minorem eſſe z, quamdiu hæc ac ſuperatur, unde ſequitur curvam ſe-
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cundam dum ex A profluit, antequam z valeat c, inter tangentem & </
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<
s
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xml:space
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primam dari quod univerſaliter obtineri hac demonſtratione conſtat.</
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<
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</
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<
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<
s
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xml:space
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">Ponamus nunc tertiam dari curvam AI, cujus axis etiam eſt AD, & </
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<
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xml:space
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xlink:label
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">42.</
note
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quatio, manentibus iiſdem abſciſſis x, ſit dx = u
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; </
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<
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xml:space
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cunque; </
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">& </
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<
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xml:space
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">d linea determinata; </
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<
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xml:space
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">hanc ſi augeamus, mutamus curvam & </
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nuimus angulum quem curva cum tangente AF efficit; </
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<
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xml:space
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">formaturque hiſce
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curvis tertia claſſis angulorum, qui in infinitum minui poſſunt, & </
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<
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xml:space
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nullus datur angulus, qui non ſuperetur ab angulo quocunque in ſecunda.</
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</
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<
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<
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xml:space
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">Datis b & </
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">d quibuſcunque, ſit bb ad dd; </
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<
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">erit
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ergo bbe = d
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, & </
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<
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xml:space
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">æquatio curvæ bbx = z
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mutabitur in hanc bbex = d
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x
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= z
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e; </
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<
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xml:space
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">ideoque z
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e = u
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, ſi agatur de iiſdem abſciſſis in utraque curva; </
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<
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circo u, e:</
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, u
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; </
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<
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xml:space
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">ergo u ſuperat z, quamdiu e ſuperat u, &</
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<
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">, exeundo ex A,
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curva, cujus abſciſſæ ſunt u, tranſit inter AF & </
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<
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">Curvæ, quarum æquatio eſt f
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x = t
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poſita f quantitate determinatâ in ſin-
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gulis curvis, & </
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<
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">t ordinata quæcunque, dabunt novam claſſem angulorum mino-
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xlink:label
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">43.</
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rum omnibus memoratis, & </
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<
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">eodem modo claſſes in infinitum formari poſſunt,
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ſemperque omnes anguli in claſſe quacunque ſuperantur ab omnibus angulis in claſ-
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ſe præcedenti, & </
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<
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<
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xml:space
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">Inter duas claſſes quaſcunque datur ſeries infinita claſſium; </
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<
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">quæ omnes eandem
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<
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">44.</
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proprietatem babent, ut angulus quicunque unius ſit infinite parvus reſpectu angulo-
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rum claſſis præcedentis, id eſt, ut ab omnibus ſuperetur, & </
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<
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ſpectu claſſis ſequentis, cujus omnes angulos ſuperat.</
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<
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<
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xml:space
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">Curvæ ax = yy & </
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<
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xml:space
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<
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claſſes formant diverſas; </
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<
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xml:space
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menſio z
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id ſecunda unitate ſuperat dimenſionem y
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primæ curvæ; </
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<
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ſtrabimus autem claſſes differre, quantumvis parum hæ dimenſiones differant,
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unde conſtabit propoſitum: </
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<
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<
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cunque, innumeridaripoſſunt, quiinter ſe differunt, quorum nulli, quantumvis
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parum differentes, dari poſſunt, inter quos iterum non alii innumeri dari
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poſſint.</
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</
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<
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<
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">Sit ax = yy & </
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<
s
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xml:space
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">g 1 {1/10} x = s 2 {1/10} id eſt, g {11/10} x = s {21/10}; </
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<
s
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<
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conſtantem lineam, quamdiu curva non mutatur. </
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<
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xml:space
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">Fiat ut a ad g, ita g {1/10} ad
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quartam quantitatem, quæ dicatur b {1/10}; </
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<
s
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xml:space
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">ergo g{11/10} = ab{1/10}; </
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<
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do per b {1/10} æquationem ax = yy datur ab {1/10} x = g {11/10} x = y
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b {1/10} = s {21/10};
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</
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<
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<
s
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xml:space
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<
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xml:space
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<
s
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xml:space
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">Idcirco in viciniis puncti A, ubi s
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neceſſario minor eſt determinatâ b, erit etiam y minor s unde liquet quod de
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angulis dictum.</
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<
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<
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<
s
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xml:space
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">Inter duas claſſes quaſcunque, quantitatum, quæ in infinitum differunt, da-
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<
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xlink:label
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">45.</
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ri in infinitum claſſes intermedias ex conſideratione mediarum proportiona-
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lium etiam deducitur.</
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