Wolff, Christian von: Der Anfangs-Gründe Aller Mathematischen Wiessenschaften. Bd. 4. Halle (Saale), 1710.der Algebrr. ran xr-1 dx = mym-1 dy (§. 398)dx = mym-1dy : ranxr-1 PT = ydx : dy = mym dy : ranxr-1dy = mym : ranxr 1 = manxr : ranxr-1 = mx : r. Setzet z. E. a3x2 = y5 so ist PT = ist/ PT : AP = 5 : 2. Der 5. Zusatz. 417. Jn dem Circul ist ax - xx = yy/ und Der 6. Zusatz. 418. Es sey für unendliche Circul (§. 243) Dem-
der Algebrr. ran xr-1 dx = mym-1 dy (§. 398)dx = mym-1dy : ranxr-1 PT = ydx : dy = mym dy : ranxr-1dy = mym : ranxr 1 = manxr : ranxr-1 = mx : r. Setzet z. E. a3x2 = y5 ſo iſt PT = iſt/ PT : AP = 5 : 2. Der 5. Zuſatz. 417. Jn dem Circul iſt ax - xx = yy/ und Der 6. Zuſatz. 418. Es ſey fuͤr unendliche Circul (§. 243) Dem-
<TEI> <text> <body> <div n="1"> <div n="2"> <div n="3"> <div n="4"> <div n="5"> <p><pb facs="#f0255" n="253"/><fw place="top" type="header"><hi rendition="#b">der Algebrr.</hi></fw><lb/><hi rendition="#et"><hi rendition="#aq"><hi rendition="#u"><hi rendition="#i">ra</hi><hi rendition="#sup"><hi rendition="#i">n</hi></hi><hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">r</hi>-1</hi><hi rendition="#i">dx = my</hi><hi rendition="#sup"><hi rendition="#i">m</hi>-1</hi></hi><hi rendition="#i">dy</hi> (§. 398)<lb/><hi rendition="#u"><hi rendition="#i">dx = my</hi><hi rendition="#sup"><hi rendition="#i">m</hi>-1</hi><hi rendition="#i">dy : ra</hi><hi rendition="#sup"><hi rendition="#i">n</hi></hi><hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">r</hi>-1</hi></hi><lb/> PT = <hi rendition="#i">ydx : dy = my</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi> <hi rendition="#i">dy : ra</hi><hi rendition="#sup"><hi rendition="#i">n</hi></hi><hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">r</hi>-1</hi><hi rendition="#i">dy =<lb/> my</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi> : <hi rendition="#i">ra</hi><hi rendition="#sup"><hi rendition="#i">n</hi></hi><hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">r</hi> 1</hi> = <hi rendition="#i">ma</hi><hi rendition="#sup"><hi rendition="#i">n</hi></hi><hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">r</hi></hi> : <hi rendition="#i">ra</hi><hi rendition="#sup"><hi rendition="#i">n</hi></hi><hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">r</hi>-1</hi> = <hi rendition="#i">mx : r.</hi></hi></hi><lb/> Setzet z. E. <hi rendition="#aq"><hi rendition="#i">a</hi><hi rendition="#sup">3</hi><hi rendition="#i">x</hi><hi rendition="#sup">2</hi> = <hi rendition="#i">y</hi><hi rendition="#sup">5</hi></hi> ſo iſt <hi rendition="#aq">PT = <formula notation="TeX">\frac {5}{2}</formula> <hi rendition="#i">x/</hi></hi> das<lb/> iſt/ <hi rendition="#aq">PT : AP</hi> = 5 : 2.</p> </div><lb/> <div n="5"> <head> <hi rendition="#b">Der 5. Zuſatz.</hi> </head><lb/> <p>417. Jn dem Circul iſt <hi rendition="#aq"><hi rendition="#i">ax - xx = yy/</hi></hi> und<lb/> demnach<lb/><hi rendition="#et"><hi rendition="#aq"><hi rendition="#u"><hi rendition="#i">adx</hi>-2<hi rendition="#i">xdx</hi> = 2<hi rendition="#i">ydy<lb/> dx</hi> = 2<hi rendition="#i">ydy</hi> : (<hi rendition="#i">a</hi>-2<hi rendition="#i">x</hi>)</hi></hi></hi><lb/><hi rendition="#aq">PT = <hi rendition="#i">ydx : dy</hi> = 2y<hi rendition="#sup">2</hi><hi rendition="#i">d</hi>y : (<hi rendition="#i">a</hi>-2<hi rendition="#i">x</hi>) <hi rendition="#i">dy</hi> = 2<hi rendition="#i">y</hi><hi rendition="#sup">2</hi>:</hi><lb/><hi rendition="#et"><hi rendition="#aq">(<hi rendition="#i">a</hi>-2<hi rendition="#i">x</hi>) = (2<hi rendition="#i">ax-xx</hi>) : (<hi rendition="#i">a</hi>-2<hi rendition="#i">x</hi>).</hi> Solcherge-<lb/> ſtalt iſt <hi rendition="#aq"><hi rendition="#i">a</hi>-2<hi rendition="#i">x</hi> : 2<hi rendition="#i">a-x = x</hi> : PT</hi></hi><lb/> Weil <hi rendition="#aq">PT = (2<hi rendition="#i">ax-xx</hi>) : (<hi rendition="#i">a</hi>-2<hi rendition="#i">x</hi>)/</hi> ſo iſt <hi rendition="#aq">AT</hi> =<note place="right"><hi rendition="#aq">Tab. V.<lb/> Fig.</hi> 47.</note><lb/> (2<hi rendition="#aq"><hi rendition="#i">ax-xx</hi>) : (<hi rendition="#i">a-x</hi>) - <hi rendition="#i">x</hi> = (2<hi rendition="#i">ax-xx-ax+xx</hi>) : (<hi rendition="#i">a</hi><lb/> -2<hi rendition="#i">x</hi>) = <hi rendition="#i">ax</hi> : (<hi rendition="#i">a</hi>-2<hi rendition="#i">x</hi>)/</hi> folgends <hi rendition="#aq">BP : 2AP = A<lb/> B : TA.</hi></p> </div><lb/> <div n="5"> <head> <hi rendition="#b">Der 6. Zuſatz.</hi> </head><lb/> <p>418. Es ſey fuͤr unendliche Circul (§. 243)<lb/><hi rendition="#et"><hi rendition="#aq"><hi rendition="#u"><hi rendition="#i">ax</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi> - <hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">m</hi>+1</hi> = <hi rendition="#i">y</hi><hi rendition="#sup"><hi rendition="#i">m</hi>+1</hi></hi></hi></hi><lb/> ſo iſt <hi rendition="#aq"><hi rendition="#u"><hi rendition="#i">max</hi><hi rendition="#sup"><hi rendition="#i">m</hi>-1</hi><hi rendition="#i">dx</hi>- (<hi rendition="#i">m</hi>-1)<hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi><hi rendition="#i">dx</hi> = (<hi rendition="#i">m</hi>+1)<hi rendition="#i">ymdy</hi><lb/><hi rendition="#et"><hi rendition="#i">dx</hi> = (<hi rendition="#i">m</hi>+1) <hi rendition="#i">y</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi> <hi rendition="#i">dy : max</hi><hi rendition="#sup"><hi rendition="#i">m</hi>-1</hi>-(<hi rendition="#i">m</hi>-1)<hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi></hi></hi><lb/> PT = <hi rendition="#i">ydx : dy</hi> = (<hi rendition="#i">m</hi> + 1) <hi rendition="#i">y</hi><hi rendition="#sup"><hi rendition="#i">m</hi>+1</hi> : <hi rendition="#i">max</hi><hi rendition="#sup"><hi rendition="#i">m</hi>-1</hi> - (<hi rendition="#i">m</hi>-1)<lb/><hi rendition="#i">k</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi> = (<hi rendition="#i">m</hi>+1) (<hi rendition="#i">ax</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi>-<hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">m</hi>+1</hi>) :, <hi rendition="#i">max</hi><hi rendition="#sup"><hi rendition="#i">m</hi>-1</hi>-(<hi rendition="#i">m</hi>-1)<hi rendition="#i">x</hi><hi rendition="#sup"><hi rendition="#i">m</hi></hi></hi>.<lb/> <fw place="bottom" type="catch">Dem-</fw><lb/></p> </div> </div> </div> </div> </div> </body> </text> </TEI> [253/0255]
der Algebrr.
ran xr-1 dx = mym-1 dy (§. 398)
dx = mym-1dy : ranxr-1
PT = ydx : dy = mym dy : ranxr-1dy =
mym : ranxr 1 = manxr : ranxr-1 = mx : r.
Setzet z. E. a3x2 = y5 ſo iſt PT = [FORMEL] x/ das
iſt/ PT : AP = 5 : 2.
Der 5. Zuſatz.
417. Jn dem Circul iſt ax - xx = yy/ und
demnach
adx-2xdx = 2ydy
dx = 2ydy : (a-2x)
PT = ydx : dy = 2y2dy : (a-2x) dy = 2y2:
(a-2x) = (2ax-xx) : (a-2x). Solcherge-
ſtalt iſt a-2x : 2a-x = x : PT
Weil PT = (2ax-xx) : (a-2x)/ ſo iſt AT =
(2ax-xx) : (a-x) - x = (2ax-xx-ax+xx) : (a
-2x) = ax : (a-2x)/ folgends BP : 2AP = A
B : TA.
Tab. V.
Fig. 47.
Der 6. Zuſatz.
418. Es ſey fuͤr unendliche Circul (§. 243)
axm - xm+1 = ym+1
ſo iſt maxm-1dx- (m-1)xm dx = (m+1)ymdy
dx = (m+1) ym dy : maxm-1-(m-1)xm
PT = ydx : dy = (m + 1) ym+1 : maxm-1 - (m-1)
km = (m+1) (axm-xm+1) :, maxm-1-(m-1)xm.
Dem-
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Zitationshilfe: | Wolff, Christian von: Der Anfangs-Gründe Aller Mathematischen Wiessenschaften. Bd. 4. Halle (Saale), 1710. , S. 253. In: Deutsches Textarchiv <https://www.deutschestextarchiv.de/wolff_anfangsgruende04_1710/255>, abgerufen am 16.02.2025. |