§. 2. Ponatur ergo ad vniformitatem obtinendam 



— -^^ . vt fiat p V a —v X [r -{- p p) ^ vnde colligitur 



p=yl^- et y(i_|-pp)zz^2-Y)' """c ob dy-pdx 

 pro curua quaefita nancifcimur hanc aequationem differen- 

 tialem : dy — dx ^^-^^-^ = 7^7*4^:^) ' P'^» cuius integrali 

 inuenicndo capiatur in axc interuallum A B zz ar et fuper 

 AB tamquam diametro defcribatur femicirculus ANB, 

 in quo crit applicata PNzzV^tfjf — ^jf), hinc 



JTJXT \adx-xdx 

 y [a X — X X) 

 €t differentiale arcus 



^- a d X 



^. A N = -^^— — . , 

 y [a X — X x) 



atque hinc collieimus </. A N — </. P N z= — -— — r — dv, 



vnde patet efle P M —y = A N — P N ; ex quo manife- 



ftum efl , curuam inuentam effe cycloidem, prouolutione 



circuh, cuius diametcr efl «, fub reda horizontali A G 



natam. 



§. 3. Deinde etiam patet arcum curuae quaefitae 



fore K-U.-fdx^/[i-\-pp)~f^±^^~^a-:L-V{aa-ax), 



lam vero quum fit B P zz a — .v, duda chorda BN me- 



dia proportionalis inter B P et B A erit haec chorda 



B N=y («« — <2jr), ficque fit arcus AM= 2 AB— 2 BN. 



Promoueatur pundum P ad B vsque , quo pado curua 



A M porrigetur vsque in E eiitque E pundum cycloidis 



imum, ibique appHcata B E zz arcui A N B, tum vero ar- 



cus AMEzzaAB; hinc ergo prodibit arcus EM32BN. 



Dehinc vero continuetur curua A M E vltra E, donec ad 



horizontalem redeat in D, eritque A D zz 2 A N B. 



§. 4.. Quoniam vero porro variatio continet mem- 

 brum ^;(~^)P, vbi P elt quantitas conflans, euidens eft 



lianc 



