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To find the 'Proportion of the 'Diameter of a Circle its 'Periphery.
Find, by continual Biffecfion, the Sides of the infcrib'd Polygons, till you arrive at a Side fubtending any Arch, howibever fmall : this found, find likewife the Side of a fi- milar circumfcrib'd Polygon ; multiply each by the Number of Sides of the Polygon ; by which you will have the Peri- meter of each Polygon. The Ratio of the Diameter to the Periphery of the Circle, will be greater than of the fame Diameter to the Perimeter of the circumfcrib'd Polygon; but lcfs than that to the infcrib'd Polygon.
The Difference of the two being known, the Ratio of the Diameter to the Periphery, is eafily had in Numbers very nearly true ; tho not juftly lb.
Thus, Wolfilts finds it as iooocooooocoooooo to 3141592- ■$555897932; Archimedes fix'd the Proportion as 7 to 22. Ludolphus a Celllen carries it to a much greater accuracy ; finding, that putting the Diameter for 1, the Periphery is greater than 3 . 141 5y2 tf j 3 589793^3 S40-2S-43 38587950 ; but, lefs than the fame Number, changing the laft Cipher into an Unite. Methls gives us the following Proportion, which is the beft that is cxprefs'd by fmall Numbers : if the Diameter be 113, the Periphery (113. 31415) : 10000; that is, 355, nearly.
To circumfcribe a Circle ahout a given regular 'Polygon : biflcft two of the Angles of the Polygon E and D, (Tab. Geom. Fig. 28.) by the Lines E F and D E ; and on the Point of Concourfe F, as on a Centre, with the Radius EF, defcribe a Circle. See Circumscribing.
•To inferibe any given regular Polygon in a Circle: di- vide 3C0 by the Number of Sides, to 'find the Quantity of the Angle EFD; which being made, in the Centre apply the Chord E D to the Periphery, as often as it will go : Thus is the Figure infcrib'd in the Circle. See Inscribed.
Thro' three given 'Points, not in a right Line, A, B,C, 'to defcribe a Circle. On A and C ftrike Arches interfefling m D and E ; and others, G and H ; from C and B draw the right Lines D E and H G : The Point of Interfcflion, I, is the Centre or the Circle.
Hence, ift, by affuming three Points in the Periphery, or the Arch of any Circle, the Centre may be found, and' the given Arch be pcrfefled. See Centre.
2dly, If three Points of any Periphery agree, or coincide with three Points of another ; the whole Peripheries agree, and the Circles are equal.
3dly, Every Triangle may be infcrib'd in a Circle. See Triangle.
In Opticks, 'tis (hewn, that a Circle never appears truly fuch, unlefs either the Eye be direfled perpendicularly to its Centre; or the dittance of the Eye from the Centre, when direaed obliquely, be equal to the Semidiaineter of the Circle : in every other Cafe the Circle appears oblong ; and to make a Circle that mail appear fuch, it mull be oblong.
Parallel, or Concentric Circles, are fuch as are equally diftant from each other in every Point of their Peripheries ■ or arc dcicrib'd from the fame'Centre : as, on the contrary' thofe (truck from different Centres, are faid to be eccentric. See Concentric, and Eccentric.
The Quadrature of the Circle, or tho manner of making a Square, whofe Surface is perfectly and geometrically equal to that of a Circle, is a Problem that has employ 'd the Geometricians of all Ages. See Quadrature.
Many maintain it to be impoflible ; Des Cartes, in par- ticular, infills on it, that a right Line and a Circle being of different Natures, there can be no Ariel Proportion between 'em : and, in effect, we are likewife at a Loft for the juft Proportion between the Diameter and Circumference of a Circle.
Archimedes is the Pcrfon who has come the neareft to the Quadrature of the Circle : all the reft have made Paralo- gifms.
Charles V. offer'd a Reward of rooooo Crowns to the Perfon who ftiould folve this celebrated Problem ; and the States of Holland have propos'd a Reward for the fame.
Circles of the higher Kinds, are Curves wherein A *» : /M»> : M : pB, (Tab. Analyfis, Fig. 8.) or Ap" : *M>» :: pM* : pB*. ■: r '
Cor. I. Suppofe A p = x, pM =y : AB = a ; then will p B = a — x. Confcquently, x n : y m : : y : a — x. Hence we have an Equation that defines infinite Circles, viz. ym-\- 1 —ax™— xm+i ; and another defining infi- nite other Circles, viz. ym-\- 11= (a — x)nx™.
Cor. II. Um = j, then will y"=ax — x' ; and there- fore a Circle of the firft Kind is contain'd under this Equa- tion alone. If 73 — -, y'—ax'— x'. which Equation de- fines a Circle of the firft Kind.
Circles of the Sphere, are fuch as cut the mundane Sphere, and have their Periphery either on its moveable Sur- face, or in another immoveable, conterminous, and equi- diilant. See Sphere.
Hence arife two Kinds of Circles, moveable, and immove- able.
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The firft, thofe whofe Peripheries are in the moveable Sur- face and which therefore revolve with its diurnal Motion, as the Meridians, &c. See Meridian.
The latter, having their Periphery 'in the immoveable Surface, don t revolve; as the Ecliptic, Equator, and its Parallels, &c. See Ecliptic, &c.
_ If a Sphere be cut in any manner, the Plane of the Sec- tion will be a Circle, whofe Centre is in the Diameter of the Sphere. See Spherics.
Hence, the Diameter of a Circle pairing thro' the Centre, being equal to that of the Circle which generated the Sphere ; and that of a Circle which does not pafs thro' the Centre, being only equal to fome Chord of the generating Circle; the Diameter being the greater! of all Chords, there hence anfes another divifion of the Circles of the Sphere, viz. into great and lefs.
_ A great Circle of the Sphere, is that which divides it into two equal Parts, or Hemifpheres; having its Centre in the Centre thereof. See Great.
Hence all great Circles are equal, and cut each other in- to equal Portions, or Semicircles. SeeSpiiERics.
The great Circles are the Horizon, Meridian, Equator, J&lipic, the Colures, and the Azimuths ; which lee in their Places, Horizon, Meridian, Ecliptic, Igc.
A leffer Circle of a Sphere, is that which divides the Sphere into equal Parts, and has its Centre in the Axis of the Sphere, but not in the Centre thereof. See Lesser.
They arc ufually denominated from the great Circles they arc parallel to, as Parallels of the Equator, &c. See Pa- rallel.
Circles of Altitude, otherwife call'd Almacantars, are Circles parallel to the Horizon, having their common Pole in the Zenith, and ftill diminiftiing as they approach the Zenith. Sec Almacantar.
They have their Names from their Ufe ; which is to ftiew the Altitude of a Star above the Horizon. See Altitude.
Circles of Latitude, or Secondaries of the Ecliptic, arc great Circles parallel to the Plane of the Ecliptic, pairing thro' the Poles thereof, and thro' every Star and Planet. See Secondary.
They are To call'd, becaufe they ferve to meafure the Lati- tude of the Stars, which is nothing but an Arch of one of thefe Circles, intercepted between the Star and the Ecliptic. See Latitude.
Circles of Longitude, are feveral leffer Circles, parallel to the Ecliptic ; ftill diminiftiing, in proportion as they re- cede from it.
On the Arches of thefe Circles, the Longitude of the Stars is reckon'd. See Longitude.
Circles of Declination, are great Circles paffing thro' the Poles of the World. See Declination.
Vertical Circles, or Azimuths. See Vertical, and Azimuth.
Diurnal^ Circles, are immoveable Circles, fuppos'd to be defcrib'd by the feveral Stars, and other Points of the Heavens, in their diurnal Rotation round the Earth • or ra- ther, in the Rotation of the Earth round its Axis. Sec Diurnal.
The 'Diurnal Circles are all unequal : the Equator is the biggeft.
'Polar Circles, are immoveable Circles, parallel to the Equator, and at a diftance from the Poles, equal to the greateft Declination of the Ecliptic. See Polar.
That next the Northern Pole is call'd the next the Southern one the AntarBic. See A
TARCTIC.
e ArtTic ; and that ' rctic, and An-
Circles of F.xcurfion, are Circles parallel to the Ecliptic, and at fuch a diftance from it, as that the Excurfions of the Uanets towards the Poles of the Ecliptic, may be included within it ; which are ufually fix'd at 10 Degrees. See Sphere, and Spherics.
It may be here added, that all the Circles of the Sphere above defcrib'd, are transferr'd from the Heavens to the Earth ; and thence come to have Place in Geography, as well as Aftronomy : all the Points of each Circle being con- ceiv d to be let fall perpendicularly on the Surface of the Terreftrial Globe, and fo to trace out Circles perfeffly fimi- lar to them.
Thus, the Terreftrial Equator is a Line, conceiv'd pre- cifely under the Equinoflrial Line, which is in the Heavens ;, and lo of the reft. See Eojiator, &c.
Horary Circles, in Dialling, are the Lines which ftiew the Hours on Dials ; tho thefe be not drawn circular, but nearly ftrait. Sec Dial.
Circle Equant, in the Ptolemaic Aftronomy, is a Circle defcrib'd on the Centre of the Equant. See Eqjiant.
Its chief Ufe, is to find the Variation of the firft Inequa- lity. Sec Variation.
Circle of perpetual Apparition, one of the leffer Circles, parallel to the Equator; defcrib'd by any Point of the Sphere touching the Northern Point of the Horizon ; and carry'd about with the diurnal Motion.
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