Huygens, Christiaan - Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

NOn opus eſt ut hic demonſtrem majorem duarum me-
diarum arithmeticè continuè proportionalium inter duas
inæquales quantitates majorem eſſe quam major duarum me-
diarum Geometricè continuè proportionalium inter eaſdem,
& igitur hujus propoſitionis approximationem præcedentis
eſſe exactiorem, quod etſi fiat; præcedente tamen ob facilita-
tem potius utimur.

PROP. XXIII. THEOREMA.

Sint duo polygona complicata
11
A B # A
C D # C
E F # G
K L # H
Z # X
A, B, nempè A extra hyperbolæ
ſectorem, B intra. continuetur ſe-
ries convergens horum polygono-
rum complicatorum ſecundum me-
thodum noſtram ſubduplam de-
ſcriptorum, ita ut polygona extra
hyperbolam ſint A, C, E, K, & c, & intra hyperbolam B,
D, F, L, & c; Sitque ſeriei convergentis terminatio ſeu hy-
perbolæ ſector Z. dico Z majorem eſſe quam C dempto tri-
ente exceſſus A ſupra C. ſit exceſſus C ſupra G quarta pars
exceſſus A ſupra C, & exceſſus G ſupra H quarta pars ex-
ceſſus C ſupra G, continueturque hæc ſeries in infinitum ut
ejus terminatio ſit X. exceſſus A ſupra C major eſt quadru-
plo exceſſus C ſupra E, & ideo exceſſus C ſupra E minor
eſt exceſſus C ſupra G, eſt ergo E major quam G. Deinde
exceſſus C ſupra E major eſt quadruplo exceſſus E ſupra K,
& ideo exceſſus C ſupra G multò major eſt quadruplo ex-
ceſſu E ſupra K, & igitur exceſſus G ſupra H major eſt
exceſſu E ſupra K; cumque E major ſit quam G, ma-
nifeſtum eſt K etiam majorem eſſe quam H: eodem pror-
ſus modo demonſtratur in omni ſerierum A, C, E, K; A,
C, G, H, continuatione, terminum quemcumque ſeriei A,
C, E, majorem eſſe quam idem numero terminus ſeriei

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