The life and letters of John Locke, with extracts from his journals and common-place booksBy Lord KingNew edWith a general index . mp;c., proportional to thetimes in which they are described, which was to be demorBtrated. LEMMA I. If a right line touch an ellipsis in any point thereof, andparallel to that tangent be drawn another right line from thecentre of the ellipsis which shall intersect a third right linedrawn from the touch point through either focus of theellipsis ; the segment of the last-named right line, lying be-tween the point of intersection and the point of contact, shallbe equa
The life and letters of John Locke, with extracts from his journals and common-place booksBy Lord KingNew edWith a general index . mp;c., proportional to thetimes in which they are described, which was to be demorBtrated. LEMMA I. If a right line touch an ellipsis in any point thereof, andparallel to that tangent be drawn another right line from thecentre of the ellipsis which shall intersect a third right linedrawn from the touch point through either focus of theellipsis ; the segment of the last-named right line, lying be-tween the point of intersection and the point of contact, shallbe equal to half the long axis of the ellipsis. Let A P B Q be the ellipsis, A B its long axis, C itscentre, Yf its foci, P the point of contact, P B the tangent,C D the line parallel to the tangent, and P D the segment ofthe line P E; I say that this segment shall be equal to C B. Eor join P E, and draw/E parallel to C D ; and becauseE/ and E E are bisected in C and D, P D shall be equal (tohalf the sum of P E, and P E, that is, to half the sum of P E,and P/, that is, to half AB, that is) to C B, W. w. to bedemonstrated. LEMMA Every line drawn through either focus of any ellipsis,and terminated at both ends by the ellipsis, is to that di-ameter of the ellipsis, which is parallel to this line, as thesame diameter is to the long axis of the ellipsis. Let A P B Q be the ellipsis, A B its longer axis, E/ itsfoci, C its centre, P Q the line drawn through its focus E,and y C S its diameter parallel to P Q; and P Q shall be to DEMOXSTRATION BY NEWTO?^. 213 VS as YS to AB; for drnw/p parallel to QFP, and cut-ting the ellipsis in p, join P^, cutting Y S in T, and drawP E, wbicli shall touch the ellipsis in P, and cut the diameterin Y S produced in R, and C T will be to C S as C S to C C T is the semi sum of F P and fp, that is, of P P andF Q, and therefore 2 C T is equal to P Q, also ^C S is equalto Y S, and (by the foregoing Lemma) ^C E is equal to A B,wherefore P Q*^ is to
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