Problemata Geometrica varia.

9

Fig. I

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ea cii v ;per^;ceiB'qucn 1 :,'lindsi

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an y numbers jo handled althokgh (ifyou conficter them iu pro-portio cflhe diameter to the circumferencefhey are mosfalse,dsfor cxample: Lct the diameter be i*6,the circnmference 32;doeft mnltiplyed produce 5 j6for a sphericis, * of which,to rvitI 44,M' the area of the circle. The area mnltiplyed mto the dia-meter fle ali makg 2592 a cy linder put uponthe fphere. Thesphericis 576 multiplyecl by \ ofthe diameter (viz.f 3, jhallproduce 1 ji&the fphere. Non? who kgiows not,that 1 ^28, is to2592, as 2 to 3, notwithfianding the proportion of the diameter*8, to the circnmference 32, are veryfar front truth.

The rcason is this. Becaufe * ofthe sphericis that is to fay,area of the circle, multiplyecl into the whole diameter , andthe ivhole fphericl^Multiplyed by ‘ of the diameter fhall producethe fame number, that is to fay, a cy linder. Bat the whole fphe -r * c \clrawn into * of the diameter, fhall produce a number inpro-P°rtion to that which was front of the diameter, that 4 isto 6,° r 2 to 3. Or thus.

The fcmidiameter 9 clrawn into \, the periphery 16 is i 44,the area ofthe circle,which drawn into the diameter 18 ,mallesa c )Under.

The fcmidiameter 9 drawn into \ the circnmference i 6,is 144,the area of a circle, which fi ili mnltiplyed by 4, and the produBh l of the diameter , that is alwayes byor * of the diameter ,fhall ncceffarily produce the number of the fphere, only * of thatnii »iber wh/ch the whole diameter had beforc produccd that is\ only of a cyhndcr that circumscribes.

The Ratio t her e fore of a fphere to a cyhndcr depends notH P°n the Ratio of the diameter to the circnmference. Which is^nh noting.

Of 2 Sfhcrc , Spheroidcs, and a Cylinder.

l.

S l of a sphericis is to tbc bafe of a cylinder. So is the^ Jyhere to a cylinder, of equal heighth with it.

Demon Tt.Becanfe ' 6 of a sphericis, and the diameter,^al^e the fphere ; and the bafe with the altitude f that is, thefame diameter J makc the cylinder.

11 . As the circle ofthe fphere a c d b, is to the circle of thespheroidcs cd; So is the jphere to the fphcroicles , that is , inf duplicate proportion of the fborter cliamcters.

C Demon-

Fig. 2