Page:Cyclopaedia, Chambers - Volume 2.djvu/764

 S P H

C 108]

S P H

Of theft, fame are faid to be greater* as the Ecliptic, Meridian, Equator, £gfc. others /t?/}, as the Tropicks, Parallels, $$c. See Greater and Lessee..

Of theft, again, Ibme are moveable* or owe their Origin, to the Motion of the Earth, &c. inch is the Ecliptic, Secon- daries of the Ecliptic, ££<-. See each Circle under its proper Head; as Equator, Ecliptic, Horizon, £5^.

Sphere, in Geography, &c. a certain Dii pofition of the Circles on the Surface of the Earth, with regard to one an- other ; which varies in various Parts thereof. See Earth.

The Circles originally conceived on the Surface of the Sphere of the World, are ahnoil all transferred, by Analogy, to the Surface of the Earth ; where they are conceived to be drawn directly underneath thole in the Sphere* or in the fame Planes therewith; i'a that were the Planes of thofe of the Earth continued to the Sphere, they would coincide with the refpe£Hve Circles thereon.

Thus we have a Horizon, Meridian, Equator, ££?c. on the Earth.

As the Equator in the Heavens divides the Sphere into two equal Parts 5 the one Nortfo and the other South $ 16 does the Equator on the Surface of the Earth, divide the Globe [in the fame Manner. See Equator.

And as the Meridians in the Heavens, pals through the Poles of the Horizon ; fo thofe on the Earth, gjfe. See Meridian.

With regard, then, to the Pofition of Ibme of thefe Circles in refpecl: of others, we have a Right* a Parallel^ and an oblique Sphere.

A right Sphere, is that where the Equator cuts the Horizon of the Place at right Angles : For the particular Phenomena, ££c. whereof; lee Right Sphere.

A parallel Sphere, is where the Equator is parallel to the fenfible Horizon, and in the Plane of the Rational. See Far&i,z,-el Sphere.

An oblique Sphere, is where the Equator cuts the Hori- zon obliqually. See Oblique Sphere.

Armillary* or artificial Sphere, is an agronomical Inflru- ment, reprefenting the leveral Circles of the Sphere* in their natural Order ; ferving to give an Idea of the Office and Pofi- tion of each thereof, and to folve various Problems relating thereto.

'T is thus called, as confifting of a Number of Fafcixor Rims of Brafs, or other Matter, called by the Latins* Armill<e* from their refembling of Bracelets, or Rings for the Arm.

Bythis, 'tis diftinguifhed from the Globe* which, though it have all the Circles of the Sphere on its Surface ; yet is not cut into Armillee or Rings, to reprefent the Circles, limply and alone; but exhibits alio the intermediate Spaces, be- tween the Circles. See Globe.

Armillary Spheres* are of different Fyinds, with regard to the Pofition of the Earth therein ; whence they become diftinguifhed into Ptclomaic and Copermcan Spheres. In the Firtt whereof, the Earth is in the Centre ; and in the latter near the Circumference, according to the Pofition that Planet has in thofe Syflems. See System.

The Ptoloniatc Sphere, is that commonly in Uft, and is reprefented (Tab. Aftronomy, Fig. 21. ) with the Names of the feveral Circles, Lines, i$c. of the Sphere* inferibed thereon. See Ptolomaic

In the Middle, upon the Axis of the Sphere* is a Ball, re- prefenting the Earth ; on whofe Surface are the Circles, &c. of the Earth. The Sphere is made to revolve about the faid Axis, which remains at reft : By which Means, the Sun's Diurnal and Annual Courfe about the Earth, are reprefented according to the 'Ptolemaic Hypothefis : And even by Means hereof, all Problems relating to the Phenomena of the Sun and Earth, are fblved, as upon the Cceleftial Globe; and after the lame Manner 5 which feedefcribed under the Article Globe.

The Copermcan S* here, reprefented (Fig. 22.) is very different from the Ptolemaic* both in its Conflitution and TJft ; and more intricate in both. Indeed the Inftrument is in the Hands of fo few People, and its Ufe fo inconfiderable, except what we have in the other more common Inftruments, particularly the Globe and 'Ptolomaic Sphere* that we fhall be eafily excufed the not filling up Room, with any Defcrip- tion thereof.

SPHERICAL Angle* is the mutual Inclination of Two Planes, whereby a Sphere is cut : Thus the Inclination of the two Planes, C A F and C E F (Tab. Trigon. Fig. 9.) forms the Spherical Angle ACE. See Sphere and Angle.

The Meafure of a Spherical Angle, A C E, is an Arch of a great Circle A E, defcribed from the Vertex C, as from a Pole, and intercepted between the Legs C A and C E.

Hence, i°, Since the Inclination of the Plane C E F, to the Plane C A F, is every where the fame ; the Angles in the oppofite Interferons G and F, arc equal.

2° Hence the Meafure of a Spherical Angle A C E, is de- fcribed with the interval of a Quadrant AC or EC, from the Vertex C between the Legs C A, C E.

If a Circle of the Sphere, A E B F (Fig, 8/ ) cut anothe- C E D F, the adjacent Angles, A E C and AED are equ a [ tQ two Right ones ; and the vertical Angles A E C and D £ g equal to one another. The former likewile, holds of leveral Angles fornfd on the fame Arch C ED, at the lame Point E.

Hence, any Number of Spherical Angles* as AEC, AED DEB, B E C, g?c. made on the fame Point E, are eq ua [ to four right Angles. SeeSpHERicAL Triangle.

Spherical Triangle* a Triangle comprehended between three Arches of great Circles of a Sphere* interfering each orher in the Surface thereof. See Triangle.

'Properties of the Spherical Triangles,

1. If in two Spherical Triangles, (Tab. Trigon. Fig. 10.) ABC and a be A— a, B A=£^and C A=ca: Then will B and the Sides,) including the Angles, be relpeftively equal, the whole Triangles are equal: That is,BC=^c, B=£and C=c.

Again, if in two Spherical Triangles A=a } C=c and AC— ac; thenwiIlB=£, AB=ab and£c=BC. Laftly, if in two Spherical Triangles AB=ab* AG=ac* and B C=k, then will A=#; B=£andC=£; the Demon- strations whereof, coincide with thole of the like Properties in plain Triangles. The Theorems of the Congruency of rectilinear Triangles, extending to all other curvilinear, circular, parabolical, &c. provided their Sides be Similar. See Triangle.

2. In an equilateral Triangle ABC (Fig. n.) the Angles at the Bafe, B and C, are equal ; and if in any Triangle, the Angles B and C, at the Bale B C, are equal ; the Tri- angle is equilateral,

3. In every spherical triangle* each Side is lels than a Semicircle : Any two Sides taken together are greater than the Third ; and all the three Sides together are lels than the Periphery of a great Circle : And a greater Side is always oppofed to a greater Angle, and a lefs Side to a lels Angle.

4. If in a Spherical Triangle B A C (Fig. 1 2.) two Legs AB and B C taken together, be equal to a Semi-circle ; the Bafe AC being continued to D; the external Angle BCD will be equal to the internal oppofite one B A C.

If the two Legs together, be lels than a Semi-circle, the external Angle B C D, will be greater than the internal op- pofite one A : And if the Legs be greater than a Semi-circle, the external Angle BCD, will be lefs than the internal op- pofite one A ; and the Converfe of all thefe holds, *»«. If the Angle B C D be equal to, greater, or lefTer than A ; the Sides AB and BC arc equal to, greater, or lefler than a Semi- circle.

$. If in a Spherical Triangle ABC, two Sides A B and B C, be equal to a Semi-circle ; the Angles at the Bafe A and C, are equal to two Right ones : If the Sides be greater than a Semi-circle, the Angles are greater than two Right ones; and if lels, lels. And, converftly.

6. In every Spherical Triangle* each Angle is lefs than two Right ones ; and the Three together, lefs than Six right Angles, and greater than two.

7. If in a. Spherical Triangle BAC (Fig. 13. J the Sides A B and A C be Quadrants ; the Angles at the Bafe, B and C, will be right Angles. And, converftly, If the interfered Angle A be a right Angle, B C will be a Quadrant : It A be obtufe, B C will be greater than a Quadrant ; and if acute, lefs. And, converftly,

8. If a Spherical rectangular Triangle, the Side BC (Fig. 14 J adjacent to the right Angle B, be a Quadrant; the Angle A will be a right Angle ; it B E be greater than a Quadrant, the Angle A will be obtuft ; and if B D be lefs than a Qua- drant, the Angle A will be acute. And, converftly.

9. If in a Spherical rectangular Triangle, each Leg be either greater or lefTer than a Quadrant ; the Hypothenufe will be lefs than a Quadrant. And, converftly.

10. If in a Spherical Triangle ABC (Fig. iy.) re&angular only at B, one Side C B be greater than a Quadrant, and the other Side A B lefs; the Hypothenufe AC will be greater than a Quadrant. And, converftly.

11. If in a ^/^rWobliquangular Triangle ACB (Fig- 16. ) both Angles at the Bale, A and B, be either obtufe or acute ; the Perpendicular C D let fall from the Third Angle C to the oppofite Side A B, falls within the Triangle s if one of them. A, be obtuft ; and the other, B, acute ; the Pe^ pendicular falls without the Triangle.

12. If in a Spherical Triangle ACB, all the Angles A, B and C be acute ; the Sides are each lefs than a Quadrant. Hence, if inan obliquangular Spherical Triangle, one Side be greater than a Quadrant, one Angle is obtuft, viz. that op- pofite to this Side. .

13. If in a Spherical Triangle ACB, two Angles A and B, be obtufe, and the third *C acute ; the Sides A C and C B oppofite to the obtuft Sides, are greater than a Qua- drant ; and that oppofite to the acute Side A B, left tnan a

I Quadrant-