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LCM, MCL, thofe Seftions will be called &e following Sections • And if the Diftance of the primary Vertex ot thofe Hyperbola's from the common Centre C, as C H, or C-j/, be equal to the Semi- tangent K.? or K« ; at the pri- mary Vertex of thefe, thofe Sections are called conjugate Setlnns : And all the Figures together are named the Hyperbolic Syftcm.
'Further, i b the Ordinate to the Axis thro the FOCUS, is
equal to the principal Latin Retlum, or the Parameter of
- the Axis ; and an indeterminate Diameter, which is parallel
to the Ordinates of any determinate Diameter, is called
the conjugate ^Diameter of the fame.
'Properties of the Hyperbola.
1. Any Diameter or right Line paffing thro' the Centre, biffefts all its Ordinates ; that is, all the right Lines termi- nated on both Sides by the Hyperbolic Periphery.
2. The Ordinates of the Axis ate perpendicular to the fame ; but the Ordinates of the reft of the Diameters are oblique to their Diameters : and fo much the more in divers Species, at equal Diftances from the Axis, by how much the Difference of the Angles including the Hyperbola's is the greater : and in the fame Hyperbola, fo much the more oblique, by how much the Diameters are remov'd from the Axis.
5. If any Lines, as H b and Qj, be Semiordinates to any Diameter whatever, as KD ; the Square of the Semi-or- dinatc HZ", is to the Square of the Scmi-ordinate Q_s, as the Reaanglc K H D H is to the Reftangle K Q_D Q_: and fo the Square b n is to [the Square a K., as the Rectangle ibhb is to the Rectangle tab a : And thus every where.
4. The Latin Retlum, or Parameter of every Diameter, is a third Geometrical Proportional, after the Diameter, and the Conjugate thcreof,(or its Tangent, which is equal to it:) That is, if the Latin Retlum of any Diameter, as D K, be y ; then, as the Diameter D K is to its conjugate v, or its equal ». y ; fo that Conjugate- y, or that Tangent m v is to _r. And as the Ordinate to the Axis thro' the Focus is the" principal Latin Retlum, fo it is more than double of the lealt Diftance of the Focus from the Vertex.
5. The Square of any Semi-ordinate, as Qr, is greater than a Rectangle made of the Abfcifs D Q_> dtawn into the Lams Retlum of its own Diameter, asj>: And, in like manner, the Square of the Semi-ordinate bn, is greater than the Rectangle of the Abfcifs i b, into the Latin Rec- tum of the Diameter h i. From which -esrssSWl, or Excefs, this Section hath its Name.
6. If from any Point of the Hyperbola, as B, (Fig. i6\) there be drawn right Lines to both the Foci, as B H, BI, the Difference of thefe Lines will be equal to the Axis DK; as will eafily appear from the Delineation it felf.
7. If the Angle B HI, comprehended by Lines drawn to the Feci, be biffected by the right Line E B, that right Line will be a Tangent to the Hyperbola in the Point B.
8. The right Lines L L and M M, which inclofe the Hy- petbola's, are Afymptotes of the Hyperbola's ; that is, they are fuch, to which, on both fides, the Curve approaches near- er and nearer, but is never able to touch or coincide there- with.
9. TheSpecies of Hyperbola's are various, according to the different Magnitude of the Angle LCM, comprehended by the Afymptotes : but that Angle remaining the fame, the Species of the Hyperbola remains unchang'd ; but accord- ing to the different Magnitudes of the Parallelograms, by which Hyperbola's are defcrib'd, Hypeibola's of divers Mag- nitudes do arife : [If the Angle contained by the Afym- ptotes be a right Angle, the Hyperbola is called Equilate- ral, or Rectangular 5 and the Latin Retlum of all the Dia- meters will (as in a Circle) be equal to their Diameters. And, laftly, if Hyperbola's be defcrib'd about the fame Axis in divers Angles of the Afymptotes, the right Lines per- pendicular ro the Axis will be cut off in a given Proportion by them all ; and the Spaces likewife inclofed by the right Lines, or Ordinates, the produced Axis, and the Curves, will be in the fame given Proportion.
10. If the Diftances from the Centre of the Hypetbola, be taken in a Geometrical Proportion in one of the Afymptotes, fo that CI, CII, GUI, CIV, CV, CVI, be in continued Geometrical Proportion ; and if from thofe Points there be drawn parallel ro the other Afymptote the Lines, I 1, II z, III 5, IV 4, V 5, VI 6, the Spaces I z, II 3, III 4, IV 5, V<S, will be equal among thomfelves. And coniequent- ly, if that Afymptote CM be fuppofed to be divided, ac- cording to the Proportion of Numbers exceeding one ano- ther in a natural Series, thofe Spaces will be proportional to the Logarithms of all thofe Numbers.
Common Properties of all the Conic Sections.
From the whole it may be gather'd, iit, Thar the Conic Setlie/n are in themfelves a Syltem of regular Curves, allied to each other ; and that one is chang'd into another per- petually, when "it is either increas'd, or diminifh'd, in infi- nitum.
Thus, the Circle, the Curvature thereof being never fo
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paffeth into an Ellipfis
its Centre going away^ infinitely, and'Vhe'c
little increas'd or diminifh'd, the Ellipfis,
vaturc being by that means diminifh'd, is titrn'd into a 'P rabola : And when the Curvature of the Parabola is ne/" fo little chang'd, there arifeth the firfl of the Hyperbola',. the Species whereof, which are innumerable, wilfall of them arife orderly by a gradual Diminution of the Curvature ■ u n til the Curvature vanifliing away, the laft Hyperbola 'ends in a right Line perpendicular to the Axis. From whence it is manifeft, that every regular Curvature, like to that of » Circle, from the Circle it felf into a right Line, is a conical Cutvature, and is difiinguifh'd with its peculiar Name ac- cording to the divers Degrees of that Curvature.
idly, That the Latin Rectum of a Circle, is double to the Diftancc from the Vertex : That all the Latera Rrfla of the Ellipfes, are in all Proportions to that diltance be- twixt the Double and Quadruple, according to their different Species : That the Latin Retlum of the Parabola is j u n quadruple to that Diftance ; and, laftly, that the Lcic y& Retla of Hyperbola's are in all Proportions beyond theQ M . druple, according to their various Kinds.
3dly, That all Diameters in a Circle and Ellipfis, inter- fere one another in the Centre of the Figure within the Section : That in the Parabola they are all parallel araonsj themfelves, and to the Axis ; but that in the Hyperbola they interject one another, but this without the Section, i n the common Centre of the oppofitc Sections.
4thly, That the Curvature, with refpect to the Focus, in all thefe Figures, is increas'd or diminifh'd proportionably.
CONICKS, that Part of the higher Geometry, or Geo- metry of Curves, which confiders the Cone, and the feveral Curve Lines arifing from the Setlions thereof. See Geo- metry ; fee alfo Cone, and Conic Setlion.
CONIFEROUS, a Term applied to fuch Trees, Shrubs, or Herbs, as beara fquammous or fcaly Fruit, of a woodySub- ftance, and a Figure approaching to that of a Cone ; in which there are many Seeds, and when they are ripe, the feveral Cells or Partitions in the Cone gape or open, and the Seed drops out : Of this kind are the Firr, the Pine, Beach, and the like. See Plant.
CONJOINT, Conjimtlus, is applied in the antiemMu- fick, in the fame Senfc as Confonant, to two or more Sounds. See Consonance.
Conjoint ^Degrees, are two Notes which immediately follow each other in the Order of the Scale ; as Ut and Re. See Degree.
Conjoint "Tctrachords, are two Tetrachords, where the fame Chord is the higheft of the one, and the loweft of the other. See Chorp.
CONISOR, or COGNISOR, in Law, is ufed in the paf- fing of Fines, for him that acknowledges the Fine. See Fine. He to whom the Fine is acknowledg'd, is called the Ceg- nizee.
CONJUGATE Diameter, or Axis, in Cenicks, or the Sections of the Cone, is a right Line biflecting the tranf- verfe Diameter. See Diameter, and Conic Setlion.
Conjugate ^Diameter, or Axis, of an Ellipfis, is the fhorteft Diameter, or Axis, biffecting the longer, or ttanf- verfe Axis. See Ellipsis.
'Tis demonftrated, ift, That in an Ellipfis, the Conjugate Axis is a Mean Proportional between the Tranfverfe Axis and the Parameter. idly, The Square of the Conjugal Axis, is to that of the Tranfverfe, as the Square of the Scmi-ordinate is to the Rectangle of the Segments of the Axis. 3dly, That a right Line drawn from the Focus to the Extremity of the Half- conjugate Axis, is equal to the tranf- verfe Semi-axis.
Hence, the Conjugate Axes being given, the Focus is ea- fily determin'd. See Focus. And the Ellipfis thence eafiiy defcrib'd.
Conjugate Axis, in an Hyperbola, is a Mean Proportio- nal between the tranfverfe Axis and the Parameter. See Hy- perbola.
It is thus called, becaufe the Conjugate Axis of an Ellip- fis has the fame Ratio.
In an Hyperbola, the Square of the Conjugate Axis, is to the Square of the Tranfverfe, as the Parameter to the tranfverfe Axis. See Parameter.
CONJUGATION, in Grammar, an orderly Diftribufion of the feveral Parts of Verbs ; or, a different Inflection ot Verbs, made according to their different Moods and Tenfes, to diftinguifh. 'em from each other. Sec Verb.
The Latins have four Conjugations, diftinguifh'd by the Terminations of their Infinitive, are, ere, ere, ire ; and molt of the French Grammarians reduce the Conjugations of their Language to the fame Number 3 ending in er, re, ir, ana oir. ,
In Englijh, where the Verbs have fcarce any natural In- flections, but derive all their Variations from additional Par- ticles, Pronouns, gjc. we have fcarce any fuch thing as itrict Conjugations. See Moon, Tense, iSc.
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