Problemata Geometrica varia.

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ßoitld be moved parallcly f'rom Dp to B C the circle upon(\\\ n) Fig. 5(towit rphose diameter is ahvayes ivithin the fides of the cone^ B, C G) Jball be eqnal to the bracelet a e , and this fl>all be1 ° m every point of the radius C K. F'rom bence therefore , itfolloYpeSjthat the circle so increafed , and movedfjall mal^etheßgnre D c K c F G B D eqnal to the cone B C G. But fenee thec °ne B C G is ' of the cylmder D B G 1 , therefore the f gure& c B K c G Fß>all be also ' of the Cylinder DBG F,thereforethe residue D c K e F D ( that is the hemispherium f is * ofthef<tid cylincler.

Corollar. Since therefore a fphere is'ofa cylincler circum-fcr/bccl by the thircl Frop.a Parabola fball be * of a pa-rallelogram circumferibecl.

Ilie delineation of a Varakla.

I N the Schcme(Fig. 6 .No.i .)a f,o d,u w 3 y c 3 l tt 3 m L, areeqnal by ftruBurefeecaufc they are all parallel, & terminatedby the paralleis fai 3 an. Hence if k o be i.t u ßoall be 2 3 b y 3,^ 1 4,11 m 5, ö n 6 : and as k o,t u 3 b y ,&c. are one to another,J° ßoall a o 3 a u 3 a y ,&c. be oneto another. But the plains of d o,w u,u t^of c y,y b ,<&c. are of eqnal altitude 3 whofc alti-tlt des are the eqnal right lines o d 3 11 w 3 y c, Ac. therefore thef a *ne plains are as their bases k o,t u 3 b y 3 Sc 1 3 Ac. Or (as ita Ppeares by whatis abovefaicl J) as a o 3 a u, a l,a y i&*c. Sincetherefore tbe fquares e x, o x, 11 z, y g, 1 p, Ac. are eqnal tothe plains d o k, w 11 t, c y b 3 tt 1 3 c 3 Ac. [becaufe the jicles ofth°fc fquares are mean proportionals between the fides of theNainy .. j therefore the fqnares o x, uz 3 y g, 1 p, Ac. are as theri g>ht a o 3 au 3 ay 3 a 1 3 Ac. in the Scheme Fig 6. No. 1. No. 3. asA rc himedes 3 /;/j' Propofition is,in his Book^dc Quadratura para-res: Propof. 3.

Out of the f e grouncls by the line of lines, and superfetes inf e socior, a Parabole may be des er ibecl thus. *Vpon A 11 as the* a nieter,prick^domvn by the feine of lines,the eqnal parts Ao^ V ) (Fig. 6. Nc. 3.) Ay? AL, A m 3 An, Ac. Andfrom thefe$ oi nts raif e the perpendicnlars o x, v z 5 y g, 1 p 3 m q,n h 3 Ac.^ n d npon the perpendicnlar o x ajfnme the point x , and openfeBor in the line offupcrfcies,fo that o x (being the frflP^pendicularfmay fall imvith the points 1...1 (thefirftof theltie °ffuperfciesyhcn ifyon tafe off from the fame line 2...2,

yon

Fig. 6.