Page:Cyclopaedia, Chambers - Volume 1.djvu/461

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j)HB, proportional to the Time; the angular Motion KI B about the other Focus I, will be almotr. proportional to the Time, and confequently without any notable Acceleration or Retardation, and nearly equable ; that is to fay, where the Bilipfis doth not differ much from a Circle. Gene/Is of a Parabola. Let D I be an infinite right Line, and IL another per- pendicular to it 5 (Fig. 15-) Then, taking, in the Line D I, any Point, F, let the Line F I be biffecled in the Point Tj and let there be taken two Threads joined together in the Point T, one T I, the other TF : and let a Pin fixed to the 'Threads in the Point T be moved to the right and left, in fuch a manner, that when the Pin is in any other Pofition, 1 P, the Thread TI which here becomes P L,be always

perpendicular to I L ; or, which is the fame thing, parallel to and thus It is every whe

comprehended by the fame Axis and Ordinate, and the re- fpecVive Curves, will be in the fame given Proportion to one another.

13. Every Parabolic Space comprehended Betwixt the. Curve and the Ordinate, is to the Parallelogram made of the lame Bale and Altitude in a fubfefquialteraj Pioponion 5 that is, as 2 is to 3 ; and to the exrerhal Space in a duplicate Proportion, or as 2 is to 1 : So q / T is to q i I, as 2 is to 3 ; and to /IT as 2 is to r. From whence it becomes eafy to fquare the Parabola. Sec QijatratuRe.

14. The Dillance between the Vertex of the Axis, artl the Point where any Tangent interfeefs it, as J, is equal td the Abfcifs of the Axis which belongs to the Ordinate ap- plied from the Point of Contact : So Tl is equal to T F ;

I) I", but equal to the Thread TF ; which in this cafe becomes P F, ever paffing thro' the Point F.

The Curve thus generated by the Pin, infinitely produced both ways, is a 'Parabola : in which gP/TiRc, is called the 'Periphery ; I D the j£xis t or principal Diameter ; F the Focus f the Point T the principal Vertex.

An Ordinate to the Axis thro' the Pecus, is equal to the principal L.atus Refrum : all right Lines n i, parallel to the Axis, are Diameters, as dividing the Lines P and KT, which are parallel to the Tangents at their Vertices, into two equal Parts ; and they are call'd Diameters belonging to the Vertices in which they terminate, as i. 'Properties of a Parabola. T. Every Diameter, or right Line parallel to the Axis, biffecls all the Lines within the Figure, which are parallel to the Tangent of the Vertical Point : which biffecfed Lines are call'd Ordinate*.

2. The Ordinates of the Axis are perpendicular thereto ; but the Ordinates of the reft of the Diameters are oblique to their Diameters 5 and fo much the more oblique, by how much the Vertex of the Diameter is further remov'd from the principal Vertex of the Parabola.

5. The Latus ReBum, or Parameter to every Diameter,

15. AH Parabolas are like, or of the fame Species $ as are alfo all Circles.

16. If a Diameter be continu'd thro' the Point of Con- courfe of two Tangents ; this Diameter will bifTecT: the Line that joins the Contacts : which Property of the Parabola may Hkewife be underliood of the Ellipfis, and Hyperbola.

Gene/is of an Hyperbola. Suppofe a Staff or Rule of a Sufficient Length, as I Bj (Fig. 16.) let I and H be two central Points, anfwering to the Foci of an Ellipfis, in which let Nails be faften'd 5 tbenj there being tied to one end of the Stick, a Rope or Thread as long again as the Stick, let the other end thereof be bor'd thro', and fo fix'd upon the Kail I ; and fix the other end of the Rope, by a Knot, upon the other Nail H : which done$ laying your Finger on the Point B, where the Rope and Staff are tied together, let your Finger defcend fo long, till you have thereby applied, arid joined the whole Rope to the Staff, or Rule 3 the Staff having been in the meart while, as it needs mult, wheel'd about the Centre I. Thus, with the Point B, the Vertex of the Angle H B I, you will have defcribed a Curve Line XBD, which is part of an Hyperbola ; the whole confifHng of that Curve which will remit from the Curve XBD; which hath added to it the

as turn'd to

5 is a third Geometrical Proportional after any Abfcifs, and its Curve Y D, the Produce of the Rule and Work Semi-ordinate 5 that is, if the Latus ReElum of the Dia- the other Side, meter i w, or that of the Vertex i } bey ; then, as the Ab- Further, transferring the Hole, or Knot of the Rope to fells iq is to the Semi-ordinate q k, fo is that Semi-ordinate the Nail I, and fattening the End of the Staff on the Nail ok to y. H, you will defcribe another Hyperbola, vertically oppofite

4. The principal ZatUS ReBum, or that belonging to the to the former, which is altogether like and equal thereto. Axis, is equal to the Ordinate h i paffing thro' the Focus ; But if, without changing any thing in the Rule and Nails, and quadruple of FT, the leaf! Dillance of the Focus from you only apply a longer Rope 5 you will have an Hyper- bola of a different Species from the former : and if you ilill lengthen the Rope, you will have Hill other forts of Hyper-, bolas ; till at length, making the Rope double the lengtn of the Rule, you will have the Hyperbola ch'arig'd into a right Line.

But if you alter the Diftance of the Nails, in the very fame proportion in which you change the Difference be- twixt the Length of the Rope, and that of the Stick ; iri this Cafe you will have Hyperbolas mark'd out, which are altogether of the fame Species, but have their fimilar Parts differing in magnitude,

Laltly, If the length of the Rope arid Rule be equally increas'd, their Difference in the mean while, and the In- terval of the Nails remaining the fame 5 not a different Hy- perbola, cither as to Species or Magnitude, will be defcrib'd^ nor any other than a greater Part of the fame Hyperbola.

It muff be own'd, however, that many Properties of fin Hyperbola are better known from another manner of gene-

thc principal Vertex.

5. The Latus Return belonging to any Vertex or Dia- meter, is alfo quadruple of the diftance of that Vertex from the Focus : Thus, the Latus Rectum of the Vertex s is quadruple F 5, and fo it is every where.

6. The Dillance of any Vertex or Point in the Parabola \vhatcvcr, from the Focus, is equal to the leaf! Diflance of rhc fame from the Line L L, which is perpendicular to the Axis ; and is diftant from the principal Vertex, by a quarter of the principal Latus RcZtltm.

7. The Square of every Semi-ordinate, as q k, is equal to a Rectangle made of the Latus Retlum, of the fame Vertex as Y, and iq the Abfcifs of the Diameter of the Vertex. And from the Equality of the aa^sfoAH, or Compa- nfon in the Figure, betwixt the Rectangle and the Square of the Semi-ordinate, without any Excefs or Defect, the Name of the Section is derived.

8. Since therefore the Latus ReBum in any Diameter

given, the Abfciffes arc as the Squares, or in the duplicate rating the Figure, which Is as follows : Let LL arid M M*

Ratio* of the Semi-ordinates. Thus, TF is to TG as '

i¥q is togG q, and fo likewife is iq to i r, as the Square

of q T is to the Square of r I : and thus every where. From

whence, alfo, when the Abfcifs of the Axis is equal to the

principal Latus Retlum, or fourfold of the Dillance from the

Vertex, it will be equal to its Semi-ordinate.

'9. The Angle, comprehended by any Tangent whatever, and a Line from the Focus, is equal to an Angle compre- hended by the fame Tangent, and any Diameter, or the Axis. Thus, the Angles 1 2 F, and p i n are equal : whence, by the way, all the Rays of Light which fall on the Con- cave part of the Surface, produced by the Convolution of the Parabola about the Axis ; which fall, we fay, on the fame Parallel to the Axis ; will be reflected from a concave

(Fig. 17.) be infinite right Lines interfering each other aE any Angle whatever, in the Point C: from any Point what- ever, as D or e, let DcD^ be drawn parallel to the firlt Lines; orcc,ed; which, with the Lines firft drawn, make the Parallelograms, asDcCi, or e c C d. Now, conceive two fides of the Parallelogram, as Dc D d, or e c e d, to be fo moved, this way and that way, that they always keep 1 the fame Parallelifm ; arid that at the fame time the Areas always remain equal; that is to fay, that Dt and ec re- main always parallel to M M ; and D d or e d always pa- rallel to LL ; and that the Area of every Parallelogram be equal to every other, one Side being increas'd in the fame Proportion wherein the other is diminifh'd : By this means, the Point D or c will defcribe a Curve-Line within the Ari-

paraboloid Figure to the Focus F, and there beget a raoft g!c comprehended by the fir ft Lines 5 which is altogether vehement burning : from which Property, the Point F has the fame as that defcrib'd above. So alfo in the Angle vcr- the Name Focus, and has communicated the fame to the tically oppbfite will be defcrib'd a like and equal Hyperbola j

like Points in the Hyperbola and Ellipfis. See Fotus.

10. A Parabola, like an Hyperbola, does not inclofe a Space, but llretches out in infinitum.

11. A Parabolic Curve always tends more and more, in in- finitum, to a Parallelifm with its Diameters 5 but can never arrive thereat.

[2. If two Parabolas be defcribed, with the fame Axis and Vertex; the Ordinates to the common Axis will be cut off by the Parabola in a given proportion ; and the Areas

if the Parallelogram Cc TLd, equal to the former, be fup- pofed to be moved, in the lame manner as before : which Hyperbolas arc, as was faid before, called eppofite SctlionSi or oppofitc Hyperbola's.

In each Figure, D K is the tranfierfc Axis, or tranfverfi 'Diameter of the Hyperbola of the oppofitc Se&iotis ; the Point C the Centre ; H and I the Foci. In the latter Figure, all the Lines paffing thro' the Cenrre C, as*'/', are Diameters: but if Hyperbola's be defcrib'd in the following Angles, as