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

## 82.PROP. XIV. THEOREMA.

Sint duo polygona complicata A, B, nem-
pe A intra circuli vel ellipſeos ſectorem & B extra: continuetur ſeries convergens horum
polygonorum complicatorum ſecundum no-
ſtram methodum ſubduplam deſcriptorum, ita
ut polygona intra circulum ſint A, C, E, & c, & extra cir-
culum B, D, F, & c; dico A + E minorem eſſe quam 2 C: ex prædictis manifeſtæ ſunt ſequentes analogiæ; prima quo-
niam A, C, B, ſunt continue pro-
portionales; & ſecunda quoniam
C, D, B, ſunt harmonice pro-
portionales: & proinde exceſſus
C ſupra A, hoc eſt C — A, eſt ad exceſſum D ſupra C ſeu
D - C in ratione compoſita ex proportione A ad C & ex
proportione A + C ad A, hoc eſt in ratione A + C ad C; at A + C eſt major quam C, & ideo exceſſus C ſupra A eſt
major quam exceſſus D ſupra C, eſt autem D major quam
E, & proinde exceſſus C ſupra A multò major eſt quam
exceſſus E ſupra C; eſt igitur A + E minor quam 2 C; quod demonſtrare oportuit.

### 82.1.

A # B
C # D
E # F
C - A:B - C::A:C
B - C:D - C::A + C:A

## 83.PROP. XV. THEOREMA.

Iiſdem poſitis: dico exceſſum C ſupra A minorem eſſe qua-
druplo exceſſus E ſupra C. ex prædictis manifeſtæ ſunt
ſequentes tres analogiæ, prima quoniam A, C, B, ſunt con-
tinuè proportionales; ſecunda, quoniam C, D, B, ſunt har-
monicè proportionales; & tertia, quoniam C, E, D, ſunt con-
tinuè proportionales; & ideo
exceſſus C ſupra A (hoc eſt)
C - A eſt ad exceſſum E ſu-
pra C ſeu E - C, ut A C + E C
+ AE + CC ad CC; at B ma-

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