^•,,.yi^„|. De comparaltone arcmtn curvarum irrectificabilium, i59 



10. Problema !• Dalo arcu parabolae Acy in vcrtice A tcrmirtato a puocto quovis /*; laliuin 

 absciiuierc arcum /^, ita ut (liffercntia horum arcuum fg — Ae geometrice assignari queat. ;i ii mi;.) 



Solutlo. Ponatur arcus dati Ae abscissa AE = ^y et abscissa, tcrmino dato f arcus quaesiti 

 fg rcspondens, AF = f; abscissa vero alteri tcrmino g arcus quacsiti rcspondens, AG=g, quae ita 

 accipiatur, ut sit g = fV{i -*-€€) -+-ey{i -^ff); eritque existente parabotae parametro =2, uti 

 constanter assumemus : Arc . fg — Arc . Ae == efg. 



A puncto autem f quoque retrorsum arcus abscindi potest fy, qui superet arcum Ae quantitate 

 algebraica: ob signum radicale V{i -*- ff) cnim ambiguum, capiatur 



AT==y=fy{i^ee)-^eV{i-+-ff). .,. ^ / «„« 



eritquc Arc . fy — Arc . Ae = efy. Q. E. 1. 



-<!r. nuiVf. i\ {>iT,U '»r.;o •i.ir.j o]")fiJ!t! v, ^\'\ 'Hjunuonui» dfirodr.iGq j,i3*ir> olr.ff .$ r.fciolrfot*! .»U 



11. Coroll. 1. Inventis ergo his duobus punctis g et v, erit qnoqne arcuum fg et fy dif- 



lereutia geometiice assignabilis; erit enim 



'•■''•■■ * ■•■ ••;. A ^ A y /iv'"»^""';, .omi!o#! 



Arc. fg — Arc. jy = ef{g — y\. 



At est ^-nr /^26^(1 -*-ff), unde ^ = ^75-^^ Tum vero habemus y-»- y = 2/*l/(l -+-eel^ 

 sive 1/(1 H-ee) =^^ ; unde eliminanda e fit 



^==-tff 4(1"=^)' ^** ^ff{i-^ff)=={9 



Fit ergo 7 = — 5r(lH-2ff)-f-2/'y(l-t-/r)(l 



12. Coroll. 3* Dato ergo arcu quocunque fg, existente AF = fet AG=g., a puncto f 

 retrorsum arcus fy abscindi potest, ita ut arcuum fg et fy differentia fiat gcometrica. Capiatur 

 scilicet AT=y'= — gli-i^^ff^-^-^fVii-^ff^ii-i-gg) eritque ■iAitih.c iJtiia'jKiuii| iu 'JiipJfi 



^^^^^,M9'f9-A.rc.fy = 2f{gy{i-^ff)-fy{i-+gg}yy[i-^ff). 

 Horum ergo aroium diffcrcntia evancsccre neqiiit, nisi sit vel f=0, quo casu fit y = — g, vel 

 g = fy quo casu uterque arcus fg et fy evanescit. minoviiiih luuioJBnimiwai) inuuoin iiio 



13. Copoll. 3. Ut igitur posilii^E = e, AF = f,AG=ig difflerentia arcuum fg et Ae fiat 

 geometrice assiguabilis scilicct Arc .fg — Arc . Ae = efg, oportc.t sit^.,.^^^^^.^^^ ODhiomOD- oupo9i>i 



g = fy[i-^-ee)-^ey{i-^ff), 

 seu ex trium quantitatum e, f, g binis datis tcrtia ita determinatur, ut sit 



vel g=fV{i-^ee)-i-ey{i-t-ff), « 



vel f^.gV{i'-+-ee)'-ey{t-^gg^;' 

 vel e=gy{i-^ff)—fV{i-i-gg). 

 14.. Coroll. 4. Cum sit^f = /*-/(! -i-ee)-H e 1/(1 -n/f), erit 



y( l -f- /jr^r) = ef-i- y (' 1 H- ce) ( 1 H- /f ) , 

 unde coliigitur g -t-V^l -^ gg) = {e -^Vii -t-ee)) {f-t-V^i -i- ff)). 



Ergo ut arcus fg superet arcum Ae quantitate algcbraica efg, oportet ut sil '^ * ^***** * 



\ (ii'»}iir. tiiH 



