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This peculiar Property of the Parabola was first publish'd Anno 1684, by one Mr. Thomas Baker, Rector of Bishop Nympion in Devonshire, in a Treatise intituled, The Geometrical Key: Or, the Gate of Æquations unlock'd; wherein he hath shew'd the Geometrical Construction and Solution of all Cubick and Biquadratick Adfected Æquations by one general Method, which he calls a Central Rule, deduced from this peculiar Property of the Parabola.
Sect. 3. To find the focus of any Parabola. The Focus of every Parabola is that point in its Axis through which the Latus Rectum doth pass. (See Definition 3. Sect. 4. Page 359.) Therefore its Distance from the Vertex of the Parabola may be easily found, either by the Latus Reflum itself, or by any other Ordinate, and its Abscisse.
Thus, suppose the Point at F to be the Focus, S the Vertex, the Ordinate RFR = L the Latus
S. Reftum, and bab any other Or.
R dinate. Then will SFE L.
F oba Or SF= 4 Sa
And 2FR= L; for the Ordinate RFREL as above. 2 m2 30 FR=jOL=;LX;L 1,= 34 SFXL=OL 4 = 5 SF=L, as by Definition 4. Sect. 4. Page 359.
oba Again 6
= L, by the third Step in Page 375.
Oba Conseq. 7 = L, &c. as above,
Q. E, D. 4
Sect. 4. To describe, or draw a Parabola several Ways.
Note, There are two or three Ways of drawing a Parabola inftrumentally at one Motion ; but because those Instruments or Machines are not only too perplex'd for a Learner to manage, but also a little subject to Error, I have therefore chosen to thew how that Figure may be the best) drawn by a convenient Number of Points, viz. Ordinates found, either Numerically or Geometrically, according to the Data ; which, if the Work of the three last Sections be well consider’d, muft needs be very easy.
1. If any Ordinate and its Abscisa are given, there may by them be found as many Ordinates as you please to affign or take Points in the Parabola's Axis ; (by Sect. 4. Page 380) and the Curve of the Parabola may be drawn by the extream Points of those Ordinates, as the Ellipsis was Page 373.
2. If the Latus Re&tum, and either any Ordinate, or its Abfciffe, are given, then any assign'd Number of Ordinates may by them be found (by Seet. 2. Page 381.) either Numerically or Geometrically, &c.
3. If only the Distance of the Focus from the Vertex of the Parabola be given, any assign'd Number of Ordinates may be found by it. For SF=iL the Latus Re&tum, and L=FR as in the last Section ; and it will be, as SF: is to OFR:: so is any other Abscissa, viz. (Sa, or SA, &c.): to the Square of its Semi-ordinate (viz. O ba, or B A) according to the common Property of the Parabola.
Altho' any of these Ways of finding the Ordinates are easy enough, yet that Way which may be deduced from the follow. ing Proposition will be found much more easy and ready in Prase tice.
The Sum of any Abfciffa and focal Distance from Propo&tion.
the Vertex, will be equal to the Distance from the Focus to the extream Point of the Ordinate, wbich
cuts off that Abscissa. For Instance, suppose S to be the Vertex of any Parabola, the Point F to be its Focus, and A B any Semi.ordinate rightly apply'd to its Axis SA: Then I say, where ever the Point A is taken in the Axis, it will be SA+SF = FB. Consequently, if Sf=SF, it will be fA=FB.
B Demonttration. First 11SF= L by the 7th Step, Sect. 3. Ergo 2fA=A+Lby Construction above.
30fA=OFA+ FAXL+ALL Again 4 SA= FA+ L by the supposition and Figure, 4 X L 5 SAXL=FÅ XL till, but SAXL=O AB
Ergo 610 AB= FAXL+ILL 3-6 710 9A-AB=OFA,confe. OfA=FATAB
But 8 OFA+OB=FB, by Theorem I.
Ergo 90 fa=OFB 9 we? 101fA=FB
Q. E. D.
This Proposition being well understood, 'twill be very easily apply'd to Practice, fuppofing the Focal Distance given, or any other Data by which it may be found. Thus draw any Right Line to represent the Parabola's Axis, and from its vertical Point, as at S, set off the Focal Dijlance both upwards and downwards, viz. make
Sif =S F, the Distance of the given focus from the l'ertex; as in the Scheme : Then by the Proposition 'tis evident, that, if never so many Lines be drawn Ordinately at Right Angles to the Axis, the true Distance between the Point f out of the Parabola, and any those Lines (or Ordinates) being measur'd or fet off from the Focus F to the same Line or Ordinate, 'twill assign the true Point in that Line through which the Curve must pass; that is, it will Thew the true Limits or Length of that Ordinate ; as at B in the laft Scheme.
Proceeding on in the very fame Manner from Ordinate to Ordinate, you may with great Expedition and Exactness find as many Ordinates (or rather their points only, like B) as may be thought convenient, which, being all join'd together with an even Hand, will form the Parabola requir’d.
N. B. The more Ordinates (or their Points) there are found, and the nearer they are to one another, the casier and exaéter may the Curve of the Parabola be drawn. The same is to be observ'd when any other Curve is requir’d to be drawn by Points.
Sect. 5. To draw a Langent to any given Point in the
Curve of a Parabola. Tangents are very eafily drawn to the Curve of any Parabola ; For, fuppofing S to be its Vertex, B the Point of Contact (viz. the Point where the Tangent must touch the Curve) and P the Point where the Tangent will interfeet (or meet with)
P the Parabola's Axis produced: Then if from the Point of Contaxt B there be drawn the Semi-ordinate B A at Right Angles to the Axis S A, wheresoever the Point A falls in the Axis, 'twill be SP-SA.
and z =SP; put x = Aa the Distance between the two Semiordinates, which we suppose to be infinitely near each other, as ia the Ellipsis, Page 377. Then
11y+z:BA::y+z+x: ba, per Theorem 13. 1, Or 2y+z:y+z+x:: BA :ba. See Page 192.
Again 3ly : OBA::y+x:oba, per Theorem Page 380 3, Or 4ly: y +*::0 BA: Oba 2 in O’s 5
22x+xx :: OBA: Oba
sy:y+x::yy + 2 y 2+xz:yy + 2yz + 4, 5
12 yx+z2+2zx + x x 6
yy +2yz + yx+zz +22*+7.
у yy + 2yz + 2yx+zz+ 2zx+xx. That is, 8 = y* * * *, consequently -=y+*
Q. E. D.
CH A P. IV.
Concerning the chief Properties of the Hyperbola. NOTE, any Part of the Axis of an Hyperbola, which is inter
cepted between its Vertex and any Ordinate (viz. any intercepted Diameter) is callid an Abscisa; as in the Parabela.
The Plain of every Hyperbola is proportion'd by this general
As the Sum of the Transverse and any Abfcifa mul
tiply'd into that Abscissa : is to the Square of its SeLycorem. mi-ordinate :: fo is the Sum of the Transverse and
any other Abscissa multiply'd into that Abscisa : ta the Square of its Semi-ordinate.
That is, if TS be the Transverse Diameter,
Sa, SA Abscissæ.
ba, B A Semi-ordinates, Then is
Ta = TS+ Sa
And it will be
Let the following Figure HVG represent a Right Cone cut into two Parts by the Right Line SA; then will the Plain of that Section be an Hyperbola (by Seet. 5. Chap. 1.) in which let S A be its Axis, or intercepted Diameter, bab and BAB Ordinates rightly apply'd (as before in the Parabola) and TS its Transverse Diameter. Again, if the Cone is suppos'd to be cut by hg, parallel to its Base HG, it will also be the Diameter of a Circle, &c. as in the Ellipsis and Parabola. Then will the A Sga and ASGA be alike, also the A Tah and A TAH
T will be alike ; therefore it
will be i'Sa:ag::SA: AG
And 2 Ta: ah::TA: A H I: 3 Sax AG = SAX a'g. 2 ... 4 Tax AHTAXah
ŞSax Tax AGXAH= 3 X 4 5 Ś AXTAX ag X ah But 6 ag X ah = o ab
l which give the following 8, Anal. 9 Sa X Tarab::SAXTA: 0 AB, &c.
Q. E. D.