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ΕHF, and the quare o CD is quai to thesquare of A together With the rectangle ΑΓ a 6 2. Elem; theresere, a the whole quare of CH, is to the whole quare o CD, socis thesquare of CF aken frona the frst, to the quare of A aken fro the second there re theremaining rectangle EH is to the re mainingreClangle Da, a the quare o CD to thesquare of A cor. 19. 5. Elem; and by division the eXces of the rectangle EHWaboveth rectangle Da, is to ADa, a the 6 2. Elem. rectangle AF to the quare of CA: but by the preceding prop. and 4. 3. the Squareo Ginis to the rectangle ADa, a the quareo CB is to that o CA and inversely, thesqua re of A is to the quare o CB, a therectangle ADtis to the quare of D. Coll. The square os traight lines rawn per
pendicular to the transverse axis stom potnis in an hyperbola, o in opposite hyperbolas, are to ne another, a the rectan gles contained by the segments intercepte between hos stra ight Iines and the vertices of the transVerSe Nis , a SH 4hewn in the essimis I. Cor. 6. 2.
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eular to the secon axis the Square of the second axis is tothe quare of the tranSVCTSO, Sthe sum of the squares of half the secon aXis, and of iis segment e tween the perpendiCu lar and the centre, is to thesquare of the perpendicular.
Fig. I. 4. From a potnim o an hyperbola dra GNperpendicular to the second4Xis Bb thesquareo Bb is to the quare of Aa a the sum of the square CB CN, to the square of GN. Because, by the preceding the quare fC is to the quare of CB, a the rectangle ADa is to the quare o GD. heresere, in-
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versely, and by proposition I 2. b. 5. Elem thesquare o Cnis to the quare o CA, a the sumo the quare o CB, Ginis to the quare ofCA, togethe with the rectangi ADa, that is, a the sum of the quare o CB, Ν is to thesquare of CD o GN. Cost Hence, is from two potnis os an hypembola oris oppoSite hyperbolas, perpendicularSbe drawn to the second)Xis, the quare of theone perpendicular is to the quare of the ther, a the sum of the quares of half the secondaxis, and of the distance belween the former perpendicular and the centre, is to the sum os the quares of half the second axis, and of the distanceae tween the lalter and the centre. PROP. IX. THEOR.
A strato line terminate both way by an hyperbola, O OPPO- sit hyperbolas, and parallel toeither axis, is bisected by the
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Fig. 5. First, let the traight lineinbe parallel toth second4Xis Bb, and meet the transverse in F and thus the square o DF is to the quareos EF as the rectangle AFacis to the cor T. 3.)rectangi AFa there re DF, FDare equat. Nexi, et D be parallel to the transverseaXis Aa, and meet the second axis B in and thus the quare of DK is to the quare ofΚG, a the sum of the quare o CB, CK isto the sum of the fame quares cor preced. of CB, C; there re DK, K are equat. PROP. X. THEOR.
A traight line termina ted both way by an hyperbola, O OPPΟ- sit hyperbolas, and bisected byeither agis is parallel to the other
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First, et Et bisected by the tranSverse Fig. 5, axis in R; and drais DK, EL parallel to the Same aXis, and meetin the secon axis in thepoinis , L. then, because DF, FDare quat, , CL are also equat but the square of DK is to the quare o EL, a the quare o CB, C together, to the squares of CB, CL together; there re DK, EL are equat, and the are parallel consequently DE KL are also parallel 33. I. Elem.ὶ Nexi leti Gie bisected by the second axis in theloint , and dra DF, Μ parallel to
lines DF, GM are equat, and the are parallel; consequently G, Fbmare likewis parallel 33. I. Elem.)
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Cost. It is manifest rom the demonstration, that the straight lines DF, 31, hicli are paralleloo ither axis Bb, and ut ostri etweenthe centre and the potnis here the meet theother axis, qua segments FC, C, re also equat. In the fame manner, Κ, EL areequat, hicli are parallel to the Ais Aa, and
cutiss the equat segment CK, L. And the contra i DF, M are equat tueach other, and parallel o Bb, the cutiss equalsegments FC, C. In like manner, i DK, EL be equata cach other, and parallelao Aa, the cutissiquat segments Κ, L. PROP. XI. THEOR.
An straight line perpendicular tolli transverse a Xis, and me elingit elo the vertex, ill me et the hyperbola in Wo potntS.
Fig. 6. 7. I et DC be perpendicular to the transverse axis Aa, and meet it in C, elow the vertex