Full text: Volumen primum. Opera mechanica (1)

Quomodo porro ratio O B ad B P, ſive N H ad H L,
non tantum cum A B F parabola eſt, ſed etiam alia quæli-
bet curva geometrica, ſemper inveniri poſſit manifeſtum eſt. Quoniam tantum recta F H ducenda eſt, quæ curvam in
adſumpto puncto F tangat, & F N ipſi F H perpendicu-
laris: unde N H & H L datæ erunt, ac proinde ratio quo-
que earum data.

71.1.

De linea-
RUM CUR-
VARUM
EVOLUTIO-
NE .
TAB. XV.
Fig. 4. & 5.

At non æque liquet quo pacto ratio K L ad M N innoteſcat,
quam tamen ſemper quoque reperiri poſſe ſic oſten-demus.

Sint rectæ K T, L V, perpendiculares ſuper K L, ſit-
que K T æqualis K M, & L V æqualis L N, & ducatur
V X parallela L N, quæ occurrat ipſi K T in X. Quo-
niam ergo ſemper eadem eſt differentia duarum L K, N M,
quæ duarum L N, K M, hoc eſt, quæ duarum L V, K T; eſt autem differentiæ ipſarum L V, K T æqualis X T, & X V ipſi L K; erit proinde N M æqualis duabus ſimul
V X, X T, vel ei quo V X ipſam X T ſuperat. Atque
adeo, ſi data fuerit ratio V X ad X T, data quoque erit
ratio V X ad utramque ſimul V X, X T, vel ad exceſſum V X
ſupra X T, hoc eſt, data erit ratio V X ſive L K ad N M.

Sciendum eſt autem, quoniam K T ipſi K M, & L V
ipſi L N, æquales ſumptæ ſunt, locum punctorum T, V,
fore lineam quandam vel rectam vel curvam datam, ut mox
oſtendetur. Et ſiquidem ſit linea recta; ut contingit ſi A B F
coni ſectio fuerit, & K L axis ejus; conſtat rationem V X
ad X T datam fore, data poſitione ipſius lineæ V T, quæ
locus eſt puuctorum V, T; ſemperque eandem tunc haberi
dictam rationem, qualecunque fuerit intervallum K L.

At ſi locus alia linea curva fuerit, diverſa erit ratio V X
ad X T, prout majus minuſve fuerit intervallum K L. In-
quirendum eſt autem quænam futura ſit iſta ratio, cum K L
infinite parvum imaginamur, quoniam & puncta B, F, pro-
xima invicem poſuimus. Similiter itaque & puncta V, T,
lineæ curvæ minimam particulam intercipere intelligendum
eſt; unde recta V T, cum ea quæ in T curvam contingit,
coincidet. Sit ergo tangens illa T Y; poteſt enim duci quo-

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