[ 3^9 ] 
onibus compleam, fatis mihi erit invenire squationem 
fpiralis Archimedes relats ad axem, quod fie 
affequor. 
Figura QtJARTA. 
Sit fpiralis Archimedea A/?zM, ejus axis FAF, ab»- 
fciffa AP = x, ordinata PM ad angulum redtum 
— y, eique infinite proxima pm . Dudta mo parallela 
ad axem, erit, mo==zdx, oM.=dy, Sit propterea AM 
radius fpiralis = t , cum quo Am faciat angulum 
infinitefimum MA m, Sc centra A, radio AM, 
defcripto circuli arcu mr> erit M r=dt. Voca arcum 
7nr=ds, Sc eodem centra A, radio quovis conftanti 
— a, defcribe circulum Feb, Sc voca ejus arcum 
infinitefimum cb—du. 
Ex hac prsparatione erit primo AM = AP 
PM\ feu t x — Sc / = unde = 
xdx-\-ydy % z — z — - — z 
—/ ====. Prsterea habebis Mr -4 -rm =M;« feu 
Vf + y* 1 
cif-{-ds 7 ‘~dx z J r dy 1 . Sed ex fimilitudine Sedtorum 
Acb, Amr, eft Ac : cb : : Am : mr ; feu a\ dt : : t : ds. 
& ex squatione fpiralis Archimedes ad focum habes 
cb-=AAr, feu du—dt j unde erit ds~ Ergo fadtis 
opportune fubftitutionibus in altera fuperiori squatione, 
obtinebis df +dy % , feu tandem 
a' + tf + f X 
xdx + ydy 
Vol. LVII. 
x' + f 
: a' X dx'- + dy\ vel potius 
B b b d/Jf 
