126 Opuscula* 
tione in ordiiiata - — = v erit a"' v~ — . u'"^^i 
«2 -{- 1 
qua; ssquatio efl: ad infinitas parabolas , & hyperbolas . 
§. 12. Quaniquam Theorema propofitum maxima gau- 
deat utilitate , inutile tamen a quantitatibus imaginariis ali- 
quando redditur , quum fciiicet eft > i ; tunc enim ordi- 
nata s.i — evadit imaginaria . Ut huic incommodo re- 
medium tutu;n afferam , fequentem methodum adhibeo de- 
raonftrando nimirum , quod fubditur , theorema . Pofita 
dx~sdp defcribatur curva coordinatarum s^pp — 
^ s p' ~~ p X ^ u ajo foit^Jdu^ — dy" ^ s . pp — i -H 
fpdx. 
§. 13. Hoc Theorcma demonftro eodem modo ac Ber- 
noullianum deinoiiftravi . Sumantur enim coordinatarum ele- 
z i 
menta ds . p p — i^--h^spdp.pp~-i z=i dy ^ds . p^ — /7 
^sppdp — sdp-hdxr^du. In his fubftituo dx pro sdp^ 
ut evadant ds . p p — i^-f- ^ p dx . p p — dy y d s . 
-h^ppdxz^du. Elevo ad fecundam poteftatem pp — i. 
z 
ds . p p I ^ p d X ~:z dy^ , pp , d s . p p — i -f- "^p d x 
~ du^ . Quoniam femper eft/7f>;7/7— -i, liquet fore femper 
du^^^dy^; quadratum primum detrahe a fecundo , ut habeas 
ds . pp — 1 ~h p d X du — dy i & extracla radice qua- 
drata ds.pp — 1 -\- ^ p d x ~ dit -—~d y"^ . In prima ^qua- 
tionis parte addatur , & deraatur s .1} . p p — i ^ ut {\\. d s . 
p p — 1 -h s . U p p — I '--^spdp-\-'^pdx-:=.^jdu^~dy^\ 
pro s dp fubftitue dx , & invenies ds . p p - 1 -r- s .Y) p p - i 
-4- p d X da^ — dy^ , & fnfla integratione s . p p — i -4- 
fpdx ~ f^'du' dy\ Q. E. D. 
§. 14. Hoc primum invento, advoco Theoreraa Bernoul- 
lianum ad inveniendam integrationem fo; mulre yVw* - i/jy* per 
rectificationem curvae algebraicje. Ut hoc fiat , pono dy qdu y 
uc 
