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whicli solve his case, a re determined is a parabola be found haying ABD a diameter, and the vertex of AB, and whicli passes through the potia re and is the talus rectum ofBAie suci, that a segmenti a circle describe tapon it, 'hen placed above A, and in the directior of AB, a contain an angi equat OBAD and that a traight in drawn throughD, S a tot parallel o AB, may ouch the Circumferen ce of that segment. . Suppos Whatis require done : Iet AG be the parameter of the diameter B an dispo AG et the segmenti a circle containing an angi equa to
the Circumferen ce of that segmen iram, and
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dat. ygium in position the mini H too salven where AH meetsi give in position and the angi AH Gis iven: ence HG is given 29 dat . in positiona and there fore the potnt G is givem henc the traight line AG is givenin magnitude since then, AG in the parabola i ichiasse through the o in D, is the latus rectun of the diameter AB, of whicli Acis the
Vertex be cause a traight line touching the parabola in the vertex of the diameter AB, meeis,asmath been proved the diameter drawn through D, in the potnt here this diameter meet tho ctrcumference of the circle, the segmen ofwhich, describe dipon the latus rectum o the diameter, passing through A, Contains an angle equat o ADH an Since, in the pre Sent ca Se, the diameter D me et the circumferen celsiliis segmen in H; therei rem touches thoe
parabola in A and AB, AH ein give inposition, and AG give in magnitude, the parabola, ac cord in to the I 8th proposition canbe described. The compositio of this case is ac Ilows :
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and et a parabola be describe which may have AB for a diameter, and AG for the latus
rectum os AB and whicli AH may I8. I. touch in Aci his parabola illias through D, and DH ill ouch the circle described about AH G. Dra D parallel o AH .and sincethe trian gles AH, H a recisosceles andequiangular, H, HA, AG and consequently ΚΑ, D, AG are proportionals the quareo D is, heresere, qua to the rectangle ΚΛG and DK is parallel to the tangent AH: henc the oin D is in the I cor I 3. I. parabola: and ecause the angi AHinis equalto AGH in the opposite segment, DII tolaches Conv. 32 3. Elem. the circle in H.
I rem ain tot enquired, hether the parameter AG be reater o les than the para- meter of the diameter B in an other parabola, avin AB, or a diameter, and A so the verte of AB, and hicli passe through D. Let there e an other parabola admittin of
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these conditions and let AE touch i in A, and me et the diameter passing through D in the poliati, and the circle GH in L hau in joined LG, dra E parallel o D dra also DF parallelao EA ; DF, there fore, is ordinatet applied to the diameter AB and be- cause the angi ADE is qua to the angle AH o ALG, that is, o AEC, and that theangle DEA is qua to EAC, the triangle EDAis equi angula to the triangle AEC: heres ore the traight lines DE, EA, AC, that is, AF, FD, AC are proportionalsa the quare fiFis, therefore, qua to the rectangi FAC: and for this reason, AC is the parameter of the diameter B in his parabola. And ecause DE Ouches the circle in H, AL is es than AE and therei ore A is les than ACci there- fore A is the leas of at the possibi parameter of the diameter AB, in parabolas whichliave AB for a diameter, and A for the vertex
of AB, and whichias through D.
Aster the fame manne Hi may be hewn, that in an parabola hateuer, hicli an Swer theconditions of the proposition the latu rectum
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of the diameter AB is greater o leSS, ACcord- inyas the tangent drawn through A, and situ at-ed on ei ther fide of the tangent AH, is more
remo te from, Ormeare to the Same AH.
To proce e to the compositionis hat was analysed in therars case the propoSed parameteri equat o AG found in the mannei abOVementioned the parabola, achath been Shewn, a b described, and willae the onlyone that cata fulsit halcis require in the r blem. Ic neXt, the propoSed parameter beles than AG it is impossibi to construct theproblem or i the proposed parameter, foreXample AC, beareater than AG pon AC describe the segmen o a circle Containing an angle equat o ADH, o DAB and since DHtouches the circle AHG, it must ut the segmen described ponis in two potnis: et Eae ne of them; and oin AE and et a parabola 18. I. be described, aving AB fora diameter, and A for the parameter of AB, and o whicli the straight line AE may be a tangent and dra DF paralleloo AE then it ma be haWn, as bove, tha DE, EA, AC,
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that is that AF, FD, AC, re proportionalsa and there re the square of DF is equat to therectangi FAC, containe by the abscissa FAand the parameter ACci, and that, ConSequently the parabola passes through the oin D. The sam thing may be demonstrated with re-gard to the ther parabola, hici has for a tangent the straight line Oining A, and theother potnt of intersectionis Anyas in the investigatio of the problem, it has been prov-ed that the angies AH, HAD are equat; the angi AK is, heresere, qua to ADΚ and of con Sequence, AK is equat o AD aut AG is a third proportional to AK, KD, o toAD DK; that is, the eas parameter is athir proportional to the traight line whiclijoin the vertex of the diameter give in position and the give potnt, and the straight linewhicli is drawn Do the fame potnt to the diameter oras to cui offfro the diameter a segment equat to the first proportionat.
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Ap. Def. 1. VIII. 1 a straight line Oining an potiatand the circumference of a circle no in the Same plane illi the oint, e produced stomthe poliat in the opposite direction, and then, while the pol ni rem alias Xed, b carried round in the directionis that circumserenc tili it re- turn to the place from hence the motion commence&; by the revolutionis that strat glit line,
a Urisce, Calle the conica surface, and whicli Consist of two furfaces connecte together atthe fixe poliat, ill e describe d. The two Connecte suri aces may each of themae inmnitet increased i the straight line illi,hichthe are describedie produced both way to an infinite distance. 2. IX. the Xedioin is called the verter of the conical furface.
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S. X. The strat glit line dra n through thepolat and the centre of the circle, is called the aiis. 4. I. The figure contained by the circle, and the surisce hicli is intercepte heime enthe vertex and the circumference of the circle, is called the corie. XII. The fame fiXedio int v hichris theverte of the Suriace of the cone, is name the
XIII. The traight line drawn from thevertecto the Centre of the circle, is called the of the CUNE T. XIV. And the circle iself is named thebas of the CONe. 8. U. Cones Which have thei axis at rightangles to the base, are called right-angled cones. 9. XVI. An cones hicli ave no their axis a right an gles to the base, are calle Sca-