Full text: Barrow, Isaac: Lectiones Opticæ & Geometricæ

DEF concurrentes punctis S, T; erit ſemper DT = 2 DS. Quòd
ſi DE ſunt ut cubi ipſarum DF, erit ſemper DT = 3 DS; ac ſi-
mili deinceps modo.

34.1.

Fig. 99.
Fig. 99.

X. Sint rectæ VD, TB concurrentes in T, quas decuſſet poſnio-
ne data recta DB; tranſeant etiam per B lineæ EBE, FBF tales,
ut ductâ quâcunque PG ad DB parallelâ, ſit perpetuò PF eodem or-
dine media Arithmeticè inter PG, PE; tangat autem BR curvam
EBE, opertet lineæ FBF tangentem ad B determinare.

34.1.

Fig. 100

Sumptis NM (ordinum in quibus ſunt PF, PE exponentibus)
fiat N x TD + M \\ - N} x RD. M x TD: : RD. SD; & connecta-
tur BS; hæc curvam FBF continget.

Nam utcunque ducta ſit PG, dictas lineas ſecans ut vides. Eſtque
EG. FG: : M. N. ergò FG x TD. EG x TD: : N x TD. M x TD. Item EF x RD. EG x TD: : M - N x RD. M x
TD. Quapropter (antecedentes conjungendo) erit FG x TD +
EF x RD. EG x TD: : N x TD + M - N x RD. M x TD; (hoc eſt): : RD. SD. Eſt antem LG x TD + KL x RD. KG x TD: : RD. SD. quare FG x TD + EF x RD. EG x
TD: : LG x TD + KL x RD. KG x TD. hinc, cùm ſit EG & gt; KG; erit FG x TD + EF x RD & gt; LG x TD + KL x RD; vel FG. EF + TD. RD & gt; LG. KL + TD. RD; ſeu (dem-
ptâ communi ratione) FG. EF & gt; LG. KL. vel componendo
EG. EF & gt; KG. KL & gt; EG. EL. unde eſt EF & lt; EL. itaque punctum L extra curvam FBF ſitum eſt; adeoque liquet
Propoſitum.

34.1.

11. Lect.
VII.
_Conſtr._
4. Lect.
VII.
_Hyp_
1. Lect.
VII.

XI. Quinetiam, reliquis ſtantibus iiſdem, ſi PF ſupponatur ejuſ-
dem ordinis Geometricè media liquet (planè ſicut in modò præceden-
tibus) eandem BS curvam FBF contingere.

_Exemplnum._ Si PF ſit è ſex mediis tertia, ſeu M = 7; & N = 3; erit 3 TD + 4 RD. 7 MD: : RD. SD; vel SD = {7 MD x RD/3 TD + 4 RD. }

XII. Patet etiam, accepto quolibet in curva FBF puncto (ceu F)
rectam ad hoc tangentem conſimili pacto deſignari. Nempe per F
ducatur recta PG ad DB parallela, ſecans curvam EBE ad E; & per E ducatur ER curvam EBE tangens; fiátque N x TP + M/- N} x RP.

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