The life of John Locke, with extracts from his correspondence, journals, and common-place booksBy Lord King . raAv PR, which shall touch theellipsis in P, and cut the diameter in VS pro-duced in R, and CT will be to CS as CS toCR. But CT is the semi sum of FP and/p,that is of FP and FQ, and therefore 2 CT isequal to PQ, also CS is equal to VS, and (bythe foregoing Lemma) *C R is equal to A B,wherefore P Q is to V S as V S to AB; W. ic. tobe dem. corol. AB x PQ = VS^^CS^ lem:ma 3. If from either focus of any ellipsis unto anypoint in the perimeter of the ellipsis be drawna riglit line, and anot
The life of John Locke, with extracts from his correspondence, journals, and common-place booksBy Lord King . raAv PR, which shall touch theellipsis in P, and cut the diameter in VS pro-duced in R, and CT will be to CS as CS toCR. But CT is the semi sum of FP and/p,that is of FP and FQ, and therefore 2 CT isequal to PQ, also CS is equal to VS, and (bythe foregoing Lemma) *C R is equal to A B,wherefore P Q is to V S as V S to AB; W. ic. tobe dem. corol. AB x PQ = VS^^CS^ lem:ma 3. If from either focus of any ellipsis unto anypoint in the perimeter of the ellipsis be drawna riglit line, and another right line do touchthe ellipsis in that point, and the angle of con-tact be subtended by any third line drawnparallel to the first line, the rectangle whichthat subtense contains with the same subtenseproduced to the otlier side of the ellipsis, is tothe rectangle which the long axis of the ellipsiscontains with the first line produced to the JOHN LOCKE. 395 other side of the elHpsis, as the square of thedistance between the subtense and the firstline is to the square of the short axis of Let AKBL be the ellipsis, AB its longaxis, KL its short axis, C its centre, F/its foci,P the point in the perimeter, F P the first line,PQ that line produced to the other side of theellipsis, PX the tangent, XY the subtense,XI the same subtense produced to the otherside of the ellipsis, and Y Z the distance betweenthis subtense and the first line, I say, that therectangle YXI is to the rectangle AB x PQ,as YZ^ to KL. For let VS be the diameterof the ellipsis parallel to the first line FP, andGF another diameter parallel to the tangent 396 THE LIFE OF PX, and the rectangle YXI, shall be to PX^the square of the tangent, as the rectangleSCV to the rectangle GCH, that is as SVto GH. This is a property of the ellipsisdemonstrated by all that write of the conicsections, and they have also demonstrated thatall the parallelograms circumscribed about anellipsis are equal. AVhence the r
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