CON
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cohere j bat where their Fitnefs of rigure will let them ap- parallel to the Ordinate* thereof, as >, 9, is a W* t0 , u
ptoach rear enough to feel each others ittraflive Power, then Ellipfis ,n that Vertex. A Diameter parallel ,0 the Ordi.
they clofe and hold together. See Cohesion. nates of another Diameter, is term da Conjugate D,a„„„
CONIC SeBion, a curve Line arifing from the Scaion of and the Ordinate to the greater Axis, which pafleth th ro!
a Cone. See Cone, and Section. either of the Foe, as MA ,s termed the prwafat La, m
The Conic Sections are three, viz. the Ellipfis, Hyper- ReBum, or the 'Parameter of the grea.er Axis. See Cen-
Ma, and 'Parabola; befide the Circle, which tho it arifc from the Section of a Cone by a Plane parallel to the Bafe, is not u r ually confider'd in that Capacity. See Circle.
Tho the Equations, Gencfis, and many of the Properties, wi:h the Ratio's, Dimenfions, iSc. of each of the Conic Sec- tions, be feparately given under their refpeclive Articles in this Work, Ellipsis, Hyperbola, and Parabola 5 yet, to make the Doftrine of Conies, which is fo confiderable a Part of the higher Geometry, and of fuch frequent ufe in the new Aftronomy, the Motion _of Projeailes.^c. more
ike, Focos, Axis, Diameter, Ordinate, Parame.
TER, (SC.
Properties of the Ellipfis.
1. The Ordinate? of every Diameter are demonfirated to be parallel to each ether.
2. TheOrdinates of the Diameters or Axes are perpendi- culartothe Axes themfelves; but theOrdinates of the ted of the Diameters are oblique to their Diameters; and in EUipf es of divers Species, fo much the more oblique, at equal Dif.
1 the Axis, by how much the Proportion of the hut in the f ame
tance from
complete, we iliall here put the whole in a new Light, and greater Axis to the letter is the greater
brine, it toaether into one contrafted view. Ellipfis, fo much the more oblique, by how much the m ote
The common Interfeflion, then, of any Plane with a Conic remote the Diameters are from the Axes. Superficies, we obferve, is called a Conic Seaion : And this 3. There are only two Conjugate Diameters, wh,ch are Section varies, and acquires a different Name, according to equal each to other; ws. thole whole Vertices are at equal the different Inclinations of the cutting Plane. For, Diftances from the Vertices of the Axes : Thus the Dia-
ift, If ABC, (Tab. Comes, Fig. 12.) be a Cone any how meter VT is conjugate and equal ,0 that other GM j cut by a Plane A DE, thro' the Vertex; and again by ano- where VF is equal to MF and VD equal to MK.
4. The obtufe Angle VCM of thele two Diameters, which are conjugate and equal, is greater and the acute An- gle V C G is lefs than every other Angle contain'd by the reft of the Diameters that are conjugare to each other.
5. If the Lines (/ P and v B be Semi-ordinatcs to any Dia- meter, as M G, the Square of the Semi-ordinare f< P is to the Square of the Semi-ordinate 11B, as is the Re£hngk
thcr Plane parallel to the former Plane' AD E : then, the Seftion B F G H, made in the Superficies thereof, is called an Hyperbola; the Plane of which being produced to meet the oppofite Superficies, will make the Seaion fbg; which is likewife call'd an Hyperbola : and both thefe, conjunaiy, are called op>p>ofite Sections.
idly, If thro' the Vertex A of a Cone, a Plane DAE, paffes without the Superfices thereof, that is, neither cutting iW> X it G, to the Reflangle M v Xr G; that is, £.r? is t0 nor touching it; and the Cone be again cut by a Plane pa- the Reflangle comprehended under the two Farts, into rallel to the Plane DAE, the Seaion FHG made in the which the Diameter is divided by the Ordinate K.P, as Supetficies thereof, is called an Ellipfis.?B? is to the Reflangle under the Parts of the Diametej
- dly, If a Plane A D E touches the Superficies of a Cone, made by the Ordinate A B.
and the Cone be cut by a Plane, the Seflion is a Parabola. 6. The Parameter, or Zatus ReBunt of any Diameter,
But inftead of considering thefe Curves as arifing by Sec- is a third Proportional to that Diameter and irs conjugate: tion of the Cone it felf, their Defcription, Nature, and Pro- That is, (in Fig. 1.) if the Diameter D K is to its coujuperties are found more eafy of Conception, when confider'd gate Diameter E F, as EF is to Y; tnen I is the Pun- as drawn on a Piane : For which Reafon, after 2)es Cartes meter or Lams RcBum of the Diameter E and moll of the later Writers, we fhall rather chufe to lay
'cm down in this fecond manner.
Gcnefis, or Conflrv.Bion of the Ellipfis. To conceive the Produflion and Nature of an Ellipfis, then, let H and I, (Fig. 15.) be two Points, Nails, or little
A M, an Ordinate to the Axis thro' the Focus is, as above, equal to the principal Parameter, and is a third Proportional after the greater and leffer Axis.
7. The Square of every Scmi-ordinate, as M I, (Fig. 1:.) ....,..„ is lefs than the Reflangle made of any Abfcifs whir- Pegs', about which put a Thread B H I, then putting your ever; as I K drawn into the Laws ReBum of its own Dia- Finper to the Thread, and keeping the fame always in an meter, or than IKxjp. And in the other Figure, P t ta is equal Tenfion, move the Finger round from the Point B, lefs than the Reflangle made or the Abfcifs M p, and the till you return to the fame Point B again. Latus ReBum of M G : From which Deject, or ewaJjf,
By this Revolution of the Point B, is defctib'd the curve this Seflion hath its Name. Line, called the Ellipfis; which differs from the Delineation 8. If from any Point, as B m the firli Figure, you draw of a Circle only in this, That a Circle hath only one Centre, the tight Lines B H and B I to the Foci, the Sum of them but the Ellipfis two; which, if the Points H and I fhould will be equal to the grearer Axis, as was fhew'd above come together, into one, the elliptic Curve would become perfeflly circular.
But by how much greater the diftance is betwixt thofe
And if the Angle I B H, comprehended by thofe Lines, be bifefled by the right Line ba, the Line A is perpendicular to the Tangent V B in the Point B; that is, to the Curve in the Point of Contaa.
9. The Diftance of a Body turn'd round in an Ellipfis, about the Focus H, from the fame Focus, is the grearelt of all in the Point K; lead of all in the Point D; and mean i:
Points, the fame length" of the Thread flill remaining; fo
much the farther is this Figure removed from the Citcular.
So that according to the diverfe Proportion of the Diftance
HI to the Thread BHI, or to the Line D K, which is
equal to the fame Thread, divers Species of Ellipfes will be the Points E and F; and that mean Diftance HF is equa
defcrib'd. to the greater Half-Axis D C or C K; as is manifeft from
But then, if the Length of the Thread be increas'd or the Ptoduaion of the Ellipfis. diminifli'd, in the fame Proportion as the Diftance of the 10. The vanifhing Subtenfe of the Angle of Connfl, pa- Points FI and I is increafed, or diminifli'd, there will in- rallel to the Diftance from the Focus, at an equal perpeedt- decd be defmb'd divers Ellipfes, but all of the fame Spe- cular Interval from that^ diftance, always remains given^and
whence it appears, that Ellipfes are not only innume- rable in Magnitude, but in Species alfo; and reach from a Circle to a right Line : For, as when the Points H and I meet together, the Ellipfis becomes a Circle; fo, when they are removed from each other half the length of the Thread, it becomes a right Line, both fides meeting together.
Whence alfo ir appears, that every Species of Ellipfes is no lefs different from any other, than the Extremes of them are different on this fide from a Circle, and on that from
unvaried in the fame Ellipfis, yea, and in the fame Para- bola, and Hyperbola too. Thus if d Z be always given,Jf<< alfo will always remain given in a diftance infinitely fmall.
11. The Area of the Ellipfis is to the Area of the Circle circumfcrib'd, as the leffer Axis is to the greater; and fo are all correfpondent Parts whatfoever among themfelves, as M I K, »1K: and the Ordinates to the greater Axis, as m I, are divided by the elliptick Periphery always in the fame Proportion, fo that MI is to ml always in the faros Proportion; to wit, that of the leffer Axis to the greater.
a right Line. It alfo appears from this Delineation, that if
from a Point raken at pleafure in the Elliptick Periphery, as And we are to reafon in the fame manner concerning
the Point B, you draw two Lines to the two central Points; cle inferib'd in an Ellipfis.
thefe two Lines BH and B I, taken together, will be equal 12. All Parallelograms defcrib'd about the conjugate Dia-
to the gteateft Diameter DK; and confequently, that the meters of the Ellipfis, and comprehending the Ellipfis, are
Sum of them is always given. equal. Thus, the Parallelogram a £ y J 1 is equal to the
In the Ellipfis DFKR, (Fig.14.) thePointC is call'dthe other e £ n * : And thus it is every where. ] .
Centre, the Points H and I thcFoci, D K the gre aler Axis, or 13. If a right Line always pairing thro' one of the Fm
tranfverfe Axis, or the principal 'Diameter, or Latus tranf- be fo moved, that the Elliptic Area defcrib'd by the lame,
verfnm; and FR is the leffer Axis : All the right Lines is proportional to the Time; the angular Motion of a rignt
pairing through the Centre C are Diameters; and all right Line drawn from the other Focus to the former Line, wi
Lines terminated at the Periphery, and divided into two equal Parrs by any Diameter, are called Ordinates. That Parr of every Diameter intercepted betwixt the Vertex thereof, and the Ordinate as M (/, is called the Abfcifs thereof. A Line drawn from the Vertex of the Diameter,
be almoft equable : Thus, in the former Figure, if the an- gular Motion of the Line HB be fo attemper'd, that tje fame being according to the reciprocal proportion of the Diftance accelerated or retarded, doth dektibe the Area
/ DHBi
