14S DE LINEJ CELERRIMI DESCENSFS 



vbi efl: s — ^(x-\-c). Hoc igitur cafu erit A B ir ^ I^-^^ 

 et ACzz^l^^-c. 



^. 27. Si autem Thearemate Ht^eniano tanquam ad 



hunc cafum idoneo vfi eflemiis, iktim hanc habuifl*e- 



mus inde aeqnationem 17 — ^^-. Hincque dv~^^^^^ 



a^dy''' ^ a''dr'' 



^gdx- ^^^^^^_, feu c!.adx^ddy—gdxds'--^—^. 



Quae fi<flo dsznpdx abit in ^^—^—gpx^dx 



a^^dxiy-xf 



nr^ ^n—^ " " qU'ie iam per fe eft {eparata, ideoquc 



conftrui poteft:. Si ponatur 72zri, vt brachyftochroua 

 pro medio refiftente in duphcata celeritatiim ratione pro- 

 deat, erit ^ac dx-ddy — cgdx d s^ — a dy- ds- feu lac dx^ 

 dds — cgdxdyds-—ady'ds. Qiiae aequatio, etiamfi 

 lemmate fimphciore nitatur, tamen muko magis cft 

 compofita et perplexa , quam noftra brachyttochrona in- 

 uenta ; id quod per fe fiepe veritatis criteriam eflc fo- 

 let, praecipue fi operofior calculus eo deduxcrit. 



§. 28. Quo autem appareat , qualem figuram bra- 

 chyfl:ochrona noftra in medio fecundum celeritatis qua- 

 drata reflftente habitura fit , aequationem fumamus hanc 



s 



i^ ( c — a c ) ~ s — a X ^ c — a c. Huec, in feriem conuer- 



s 



fo ^c" abit in hanc (c — ac)iji-i---i — ^_j —-x- 



S* 



iTr7~4734-l- etc.) :::r:s—ax-\-c — ac, ex qua repentur 

 pofito —^—k haec aequatio .v — .f - ~^ — -^ rc'~- ..2.,.^.c ' * 

 — etc. Perfpicitur ergo k nccclTirio eflfe debcre numcrum 

 affirmatiuum , ahas enim ficret x^s quod fieri nequit; 



erit 



