Elements of the conic sections

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be ordinatet applied to that diameter, o it isnamed simply, an ordinate to that diameter. Also a diameter parallelao a traight line ordinatet applied oranother diameter, is sal tobe ordinalely applied to that other diameter. IX. third proportional to tW conjugate diameter is calle the latus rectum, O the a. rameter of that diameter, hi chris the srs of the three proportional S. X. A straight line hici meet the ellipsisoni in ne potnt, is sal to ue i in that int.

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Becaugem is a pol ni in the Ilipsis, D, H are together equa to the tength of the string ith hicli it is describe es an die causeth potnt like is in the ellipsis, A, E are together equa to the tength of the

string. For a like re ason, EB, DB are together equa to the fame tength: DA, Exare, there fore equa to EB, DB Tahe way the Common part DE, and the re mainder, o Wit,twice AD, ill e qua to the rema indert ice B AD, there re is equa to B.

Ad the common part AE and AD, together illi AE, illae qua to the greater axis: ut AD, together illi AE is qualuo the tength of the string that is, tot toge- the with HE there reminandi are to-gether equat to the greater aXis AB. COR. I. The reater Xis is bisected in thecentre C. For Since des 3. DC is quai toEC, and that Acis equa to EB AC is equal

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to greater, o leSS, than the greater aXis. Coll. 4. The distance of ei ther verte F, or

of the lesser axis Dom ei ther of the foci, is equa totalf the greater axis. Uoin GD, GE: then, ecause C is qua to CE, and GCommon, and the angies at C right angies the triangle DG is quai to the triangle EG there rem is qua to EG but G and EG are together equa to the greater Xis; there re ach of them is quai to the alsos it. COR . . The tesser axis FG is bisected in thecentre Dra stra ight Iines DF, G romthe focus D to the vertices of the lesse aXisathen DF DG by the precedin corollary, illbe quat the angi DF is, there re, qual

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The quare os half the lesser axis h

equa to the rectangle Contained by the segments of the greater avis, intercepte between thevertices of that axis and ei ther of

Fig. i. Proin ither focus a Dio Elther vertex of the esse aXis a G, dra the traight line GE et ob the centre of the ellipsis, and A, B the vertice of the reater axis : the thesquares of GC, CE re together equa to thesquare of GE, that is, to the quare 4. Cor. I. 2. 6 CB, that is, id the rectangle AE toge the wit the Ruare o Ea 5. 2. Elem: takeBWay the Common quare of CE, and the re- maining quare of GC Willi eques to the ream inprectangi AEB.

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PROP. III. THEOR.

Ever diameter of an ellipsis is bi,

sected in the centro.

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est focus half the greater axis isto the distance of this focus Dom

the Centre, a the distance be- tween the Centre and the per pendicular is to the exces of half the greater aXis above the traight line drawn to thi Samo OCUS.

Fig. I. rom H, a potnt in an ellipsis, et Κ ber . . . rawn perpendicular to the greater axis AB, HEiein drawn rom the fame potn to the lacus D then B, the hal of the reater axis, is to CE, the distance of the focus Esrom the centre, as Κ, the distance belweenthe centre C, and the perpendicular HK, is to the excessis CB above HE. Having made L equatuo EH, and drawnHD to the other focus, describe rom the cen-

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hut the alve o magnitudes have the sameratio to one another hicli the wholes have

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Boebe proportionals anda, Κ hein the sameside of the potnim is in an ellipsis given inposition that is, in an ellipsis that has AB rthe reater axis, and the oin E sorine of the foci. Mais A equa to BE and as directed in the sirs definition describe an ellipsis that shallhave A for the greater axis, and the minis

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neares tom, the par BL oeta heu qua to the distance of Hsrom the focus D the quare fili perpendicular Κ, is qualto the XCes of the rectangle AKB contained by the segments into hich the axis is divided in the potnt Κ, bove the rectangle DLE, contained by the segments

into whicli the distance of the sociis divided in the oin L.

For since the strato line CB is cui into any two paris in the potnt L ste quare o BC, CL are together equat T. 2. Elem. t twice

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the rectangle BCL, together illi the quareo BL that is, o tWice preced Prop. and 16. 6. Elem. the rectangle ECΚ, together illithe quare es L, o HE, that is, o twice the rectangle ECΚ, together illi the qua res of ΚΕ, ΚΗ, that is to the T. 2. Elem. Squa res of EC, Κ, and H together the quareS, there fore, o BC, CL are together equa to thesquares of EC, CK, and ΚΗ together: ut thesquare si BC, CL are together equal 5. 2.andi aX. Elem. to the rectangle AKB together illi the quares of CK, CL and thesquare of EC, Κ, and ΚH are together equal 5. 2. andi. X. Elem. to the rectangi DLE, together illi the quare of L, Κ, H; there fore the rectangi AKB, together illithe qua res of CK, CL, are qua to the rectangi DLE together illi the qua res of CL, CK, Id stom these quais ahe way the common quares of Κ, CL, and there ill re- main the rectangi AKB, qua to the rectangi DLE, together illi the quare of ΗΚ:there fore the quare of AHis the Xces of the rectangle AKB ab ove the rectangle DLE.