Full text: Volumen secundum. Opera geometrica. Opera astronomica. Varia de optica. (2)

DE CIRCULI MAGNIT. INVENTA. goni A C D. Exceſſus igitur perimetrorum eſt H I; cujus
triens I K adjiciatur ipſi G I. Dico totâ G K majorem eſſe
circuli A B circumferentiam. Inſcribatur enim circulo tertium
polygonum æquilaterum A L E M C, quod ſit duplo nu-
mero laterum polygoni A E C B D F. Et ſuper lineis G H,
H I, I K, triangula conſtituantur quorum communis vertex
N, altitudo autem æqualis ſemidiametro circuli A B. Igi-
tur quoniam G H baſis æqualis eſt perimetro polygoni
A C D, erit triangulum G N H æquale polygono, cui bis
totidem ſunt latera, hoc eſt, polygono A E C B D F. Hoc
enim patet, ductis ex centro rectis O A & O E, quarum
hæc ſecet A C in P. Nam triangulum quidem A E O æ-
quale eſt triangulo baſin habenti A P & altitudinem radii
O E. Quanta autem pars eſt triangulum A E O polygo-
ni A E C B D F, eadem eſt recta A P perimetri A C D. Itaque polygonum A E C B D F æquabitur triangulo cu-
jus baſis æqualis perimetro A C D, altitudo autem radio
E O: hoc eſt, triangulo G N H. Eâdem ratione, quo-
niam baſis G I eſt æqualis polygoni A E C B D F
perimetro, & altitudo trianguli G N I æqualis radio circu-
li, erit triangulum G N I æquale polygono A L E M C. Itaque triangulum H N I æquale exceſſui polygoni
A L E M C ſupra polygonum A E C B D F. Trianguli
autem H N I ſubtriplum eſt ex conſtr triangulum I N K. Ergo hoc æquale erit dicti exceſſus trienti. Quare totum tri-
angulum G N K minus erit circulo A B . Altitudo autem trianguli æqualis eſt circuli ſemidiametro. Ergo evidens eſt
rectam G K totâ circuli circumferentiâ minorem eſſe. Quod
erat oſtendendum.

30.1.

TAB. XXXVIII.
Fig. 7.
per 5. huj.

Hinc manifeſtum eſt, ſi à feſquitertio laterum polygoni
circulo inſcripti auferatur triens laterum polygoni alterius
inſcripti ſubduplo laterum numero, reliquum circumferen-
tiâ minus eſſe. Idem enim eſt, ſive perimetro majori adda-
tur {1/3} exceſſus quo ipſa ſuperat perimetrum minorem, ſive
addatur {1/3} perimetri majoris contraque auferatur {1/3} perimetri
minoris. Hinc autem fit ſeſquitertium majoris perimetri mi-

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