Barrow, Isaac - Lectiones opticae & geometricae : in quibus phaenomenon opticorum genuinae rationes investigantur, ac exponuntur: et generalia curvarum linearum symptomata declarantur

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9678 ceſcit ex eo, quod earum omnium ad ſe proportiones in eodem ubique
modo fundantur in una ratione I ad R. verbis, & Schematis effin-
gendis parco. Pro ſequentibus hæc adjungo _Lemmatia._

XI. 1. Sint tria quanta A, B, C (quorum maximum A) ſe dein-
ceps æqualiter excedentia; ſint etiam altera totidem M, N, O; &
ſit A. B: : MN; ac B, C: : N, O; dico fore quoque tria M, N,
O in ratione continua _Aritbmetica._ Nam ob A. B: : M. N. erit
diviſim A-B. B: : M-N. N. item ob B. C: : N. O. erit per
rationis converſionem B. B-C: : N. N-O. ergò erit ex æquo
A-B. B-C: : M-N. N-O. itaque cùm ſit ex Hypotheſi
A-B = B-C; erit etiam M-N = N-O: Q. E. D.

XII. 2. In circuli quadrante ZQ trium arcuum ZG, ZH, ZI
Sinus recti F α, F β, F γ æqualiter creſcant (ut nempe ſit αβ = βγ)
dico fore Gα-Hβ& lt; Hβ-Iγ.

11Fig. 115.

Nam ducatur ſubtenſa GI ipſam Hβ ſecans, in X; & ſint XR,
IS ad FQ parallelæ patet ipſas GR, XS æquari hoc eſt fore
Gα-Xβ = Xβ-Iγ; unde liquidum eſt eſſe Gα-Hβ& lt; Hβ
-Iγ: Q. E. D.

XIII. 3. Sunto concentrici bini circulorum quadrantes FZ X,
F ζ ξ; & ad FZ parallela ducatur recta quævis LG γ; circulos inter-
ſecans punctis G, γ; dico fore FZ - LG & gt; Fζ-Lγ.

22Fig. 116.

Nam connexa FG circulum ζ γξ producta ſecet in T; connectan-
túrque ſubtenſæ ZG, ζ T (hæc ipſam L γ ſecans in S) Patétque jam
rectas ZG, ζ T parallelas eſſe; adeóque quadrangulum ZGSζ fore
parallelogrammum; unde GS = Z ζ; adeóque F ζ - LS = FZ
- LG. ergo F ζ- L γ& lt; FZ - LG: Q. E. D.

XIV. Sint jam tres radii paralleli MN, QR, VX, à ſe diſtantes
æqualiter (hoc eſt ut ductis N v, Rρ, Xξ ad axem AC perpendicu-
laribus ſit Xξ-Rρ = Rρ-Nv) & ipſorum refracti cum axe
conveniant punctis K, L, O; erit obliquiorum concurſibus interjectum
33Fig. 117. ſpatium OL majus ſpatio LK, quod à rectiorum occurſibus conti-
netur.

Nam ducantur NC, RC, XC circulo refractario occurrentes
punctis G, H, I; & ad has à refractarii centro F ducantur perpendi-
culares F α, F β, F γ; & quoniam triangula CXξ, CFγ ſimilia
ſunt; erit Xξ. CX: : Fγ. CF. item ſimili de cauſa, eſt CR (CX).

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