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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
# Folio 31, page id 1041163592:0065
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II
PRENOIDLA MATEEMAETCA.
cancurrentes in C; dein accedant puncta D, B, G, 2d puHCLal b, C.
scque F incerfectio lincarum BC, AC ultimo Facta ubi puncta L, 
accedune usque ad . Manifestum est quod distantia C f mIIIBL
esse POteC quam assignata quaeris. Fs autem (exnatura CIrCuIOrum
per puncta ABC, Abg transcuncium) AB quad.
aequale AG X BD, et Ab quad. ae quale Ag X b d
adcoque ratio AB quad. ad Ab quad. compo
nitur ex rationibus AC 2d A∞ ec B D ad b
Sed quoniam C Y astumi pocest minor longitu
dinc quaris assignata, heri potest ut ratio AC
ad A minus disterat a ratione aequalitatis quam
Dro difterentia quaris astignata, adeoque ut ra
tio AB quad. 2d Ab quad. minus difterat a ra-
tione BD ad b a quam pro disterentia quavis
alsignata. Est ergo, per Lemma 1, ratio ultima
AB quad. 2d Ab quad. aequalis rationi ultima
BD ad bd. d. F. D.
Ca5. 2. Inclinetur jam B D ad AD in angulo
quoris datos e eadem semper erit ratio ultima B D ad b a quxe
Deius, adeoque eadem ac AB quad. ad Ab quad. d. V. D.
Ca5. 3. EF quamvis angulus D non detur, sed recta B D ad da-
cum punctum convergente, vel alia quacunque lese constituaturs
tamen anguli D, a communi lege constituri ad aequalitatem semper
vergene ec propius accedent ad inricem quam pro difterentia qua-
vis assignata, adeoque ultimo equales crunt, per Lem. L, ee prop
cerea linex B D, bd. Tune in cadem ratione ad inricem ac prius.
Q. F. D.
Corol. 2. Unde cum tangentes AD, A d, aecus AB, Ab, et c0.
eum SinUS BC, bC Fane ultimo chordis AB, Ab equaleS; CrUNE
etiam illorum quadrata ulcimo ue subcensae B D, bd.
Corol. 2. Forundem quadrara sune etiam uleimo ue sune arcuum
Pagitex quae chordas bisceant sc ad dacum punctum COnverGUINE.
Nam Cagitex illae Tune uE CuBtenCae B D, bd.
Corol. 3. Ideoque Sagitta est in duplicata ratione temporis quo
corpus data velocirare deseribit areum.
Corol. 4. Friangula rectilinea AD B, A d b Tunc ulcimo in cripli-
cata rarione lacerum AD, Ad inque sesquiplicara lacerum D B,
db; vexore in composita ratione laterum AD, _ DB, Ad et db
existencia. Sic ee triangula ABC, Abc sunc ultimo in eriplicata
ratione lacerum BC, bc. Rationem vero Sesquiplicatam voco eri-
plicatae sudduplicatam, quae nempe ex simplici ee subduplicata com-
Donitur, quamque alias Sesquialteram dicunt.
Corob.
ETBEK
PRIMVSS