This peculiar Property of the Parabola was first publish'd Anno 1684, by one Mr. Thomas Baker, Rector of Bishop Nympton in Devonshire, in a Treatife intituled, The Geometrical Key: Or, the Gate of Equations unlock'd; wherein he hath fhew'd the Geometrical Conftruction and Solution of all Cubick and Biquadratick Adfected Equations 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 pafs. (See Definition 3. Sect. 4. Page 359.) Therefore its Distance from the Vertex of the Parabola may be eafily found, either by the Latus Rectum itself, or by any other Ordinate, and its Abfciffa. Thus, fuppofe the Point at F to be the Focus, S the Vertex, the Ordinate RFR L the Latus Rectum, and bab any other Ordinate. Then will SF † L. S F Demonftration. Firft And 202 11S FXL FR. by Sect. 2. Page 375. 2 = 2 FR= L; for the Ordinate R FRL as above. 30 FROL=LX÷L 4 SFXL=L 4 L 5SFL, as by Definition 4. Sect. 4. Page 359. Again 6 Confeq. 7 Oba Sa ☐ ba L, by the third Step in Page 375. =L, &c. as above. Q. E. D. Sect. 4. To defcribe, or draw a Parabola feveral Ways. Note, There are two or three Ways of drawing a Parabola inftrumentally at one Motion; but because those Inftruments or Machines are not only too perplex'd for a Learner to manage, but also a little fubject to Error, I have therefore chosen to fhew how that Figure may be (the beft) 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 laft Sections be well confider'd, muft needs be very eafy. 1. If any Ordinate and its Abfciffa 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 thofe Ordinates, as the Ellipfis was Page 373. 2. If the Latus Rectum, and either any Ordinate, or its Abfciffe, are given, then any affign'd Number of Ordinates may by them be found (by Sect. 2. Page 381.) either Numerically or Geometri cally, &c. 3. If only the Distance of the Focus from the Vertex of the Parabola be given, any affign'd Number of Ordinates may be found by it. For SFL the Latus Rectum, and L = FR as in the laft Section; and it will be, as SF: is to FR:: fo is any other Abfciffa, viz. (Sa, or SA, &c.) to the Square of its Se mi-ordinate (viz. □ba, or □ BA) according to the common Property of the Parabola. Altho' any of thefe Ways of finding the Ordinates are caly enough, yet that Way which may be deduced from the following Propofition will be found much more easy and ready in Practice. Propofition. The Sum of any Abfciffa and focal Distance from the Vertex, will be equal to the Distance from the Focus to the extream Point of the Ordinate, which cuts off that Abfciffa. For Inftance, fuppofe S to be the Vertex of any Parabola, the Point F to be its Focus, and AB any Semi-ordinate rightly apply'd to its Axis SA: Then I fay, where ever the Point A is taken in the Axis, it will be SASF FB. Confequently, if SfS F, it will be fA FB. Demonftration. Si F First 11SFL by the 7th Step, Selt. 3. 2.2 = I 2 B 30ƒAFA+FAXL+ ÷ L L Again 4 SA FA+L by the Suppofition and Figure. 4 X L 5 SAXL=FAXL+LL, but SAXLAB Ergo 6 AB=FAXL+ LL 3-6 70fA-AB-FA,confe. fA=FA+□ AB But 8 FA+□AB=FB, by Theorem 11. Ergo 90 fa=☐ FB Q. E. D. This Propofition being well understood, 'twill be very eafily 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 reprefent the Parabola's Axis, and from its vertical Point, as at S, fet off the Focal Distance both upwards and downwards, viz. make SfSF, the Distance of the given Focus from the Vertex; as in the Scheme: Then by the Propofition 'tis evident, that, if never fo many Lines be drawn Ordinately at Right Angles to the Axis, the true Distance between the Point f out of the Parabola, and any of thofe Lines (or Ordinates) being measur'd or fet off from the Focus F to the fame Line or Ordinate, 'twill affign the true Point in that Line through which the Curve muft pafs; that is, it will fhew 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 Exactnefs 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 cafier and exacter may the Curve of the Parabola be drawn. The fame is to be obferv'd when any other Curve is requir'd to be drawn by Points. Sect. 5. To draw a Tangent 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 muft touch, the Curve) and P the Point where the Tangent will interfect (or meet with) the Parabola's Axis produced: Then if from the Point of Contact В there be drawn the Semi-ordinate BA at a A Right Angles to the Axis SA, wherefoever the Point A falls in the Axis, 'twill be SPS A. Demonstration. Draw the Semi-ordinate b a (as in the Figure) then will the BAP and Aba P be alike. Let yAS the Abfciffa, Ddd and and zSP; put x= A a the Distance between the two Semiordinates, which we fuppofe to be infinitely near each other, as in the Ellipfis, Page 377. Then 1, Or yz:BA::y+z+x: ba, per Theorem 13. 2\y+x:y+z+x:: BA:ba. See Page 192. Again 3y BA::y+x: ba, per Theorem Page 389. 3, Or 4y y+x::DBA: □ ba : 2in's 5{+2yx + x x = y y + 2 x x + 2y x+zz+ sy: y+x::yy +2yz+zz:yy+2yz+ 12yx+xx+2xx + xx ZZX y yy + zyz+yx+xx+2xx+- Z zx y y Suppofe 9o and rejected, as in the Ellipfis, Page 377 Concerning the chief Properties of the Hyperbola. NOTE, any Part of the Axis of an Hyperbola, which is intercepted between its Vertex and any Ordinate (viz. any intercepted Diameter) is call'd an Abfciffa; as in the Parabola. Sect. I. The Plain of every Hyperbola is proportion'd by this general Theorem. As the Sum of the Tranfverfe and any Abfciffa multiply'd into that Abfciffa is to the Square of its SeTheorem.mi-ordinate: fo is the Sum of the Transverse and any other Abfciffa multiply'd into that Abfcifa: to the Square of its Semi-ordinate. That Let the following Figure HVG repreferit a Right Cone cut into two Parts by the Right Line SA; then will the Plain of that Section be an Hyperbola (by Sect. 5. Chap. 1.) in which let SA be its Axis, or intercepted Diameter, bab and BAB Ordinates rightly apply'd (as before in the Parabola) and TS its Tranfverfe Diameter. Again, if the Cone is fuppos'd to be cut by hg, parallel to its Bafe HG, it will alfo be the Diameter of a Circle, &c. as in the Ellipfis and Parabola. Then will the ▲ Sga and ASGA be alike, alfo the A Tah and ATAH will be alike; therefore it will be 'Sa:ag:: SA: AG And 2 Ta: ab:: TA: AH I '.' 3 Sax AG = SAX ag. 24Tax AH=TAX ab S Sax Tax AGX AH= 3X4 5 SAXTA Xag Xah But 6 ag Xah = □ ab SAGXAH=□ AB per Lemma Page 363. h T V And 72 Sax Tax□ AB E H A which give the following 8, Anal. 9 Sax Ta□ab:: SAXTA: □ AB, &c. |