36

. Angles, equal to 180 Deg. there remains 36 D:g. Therefore, the Triang!e E F O is hy of the whole Spherick; that is

Fig. Part: And this most truly, for 20 Pylamides FEOC, fill the XLIII. Solid Place of the Icofaedrum. And so 20 Spherical Bases, (covered over with 20 Triangular Plain Bafes,) compleat the whole Spherick.

FEOC is One of the 20 Pyramids in the Icofaedrum : The Plain Triangle B, is one of the Hedre or Bases : C is the Center of the Body, or Sphere, that circumscribes it.

a

4.77 Icosaedres 4.24 Dodecaedres 9.244 O&taedres 8.000 Cubes

Fill a Solid Place, as will appear out of Precedent Prašice by Triangles, and Spherical Polygons.

That is to say, None of the Five Regular Bodies fill a Solid Place, the Cube only excepted.

Contrary to what Potamon, and from him Ramus, and all

that have followed Ramus; to wit, Snellius, and others,
have delivered.

JANUA MATHEMATICA.

Fig." XLIV.

SECTION IN.
Spherical Trigonometry, Geometrically

Performed.
S the sides of Plain Triangles are Three Right Lines, in-
A ;

ter seating each other upon a Plain: And a Right Line is, The Shortest Extension between any Two Points upon a Plain Site perficies : So the Sides of a Spherical Triangle are Three Arches, of Three Great Circles of the Sphere, inter fedting each other upon the Globe.

And an Arcb of a Great Circle, passing through any Two Points upon a Spherical Superficies, is the neareft Distance between those Two Points..

In Pursuance of the Work in this Chapter intended, I fuppose the Reader to be acquainted with the Circles of the Sphere ; that is, to know their Names and Situations upon the Globe, to what Use each of them ferveth; and also, how to proje&t any of them upon a Plain, answerable to any Position of the Globe: For 10 such as do not, this Se&tion will be but of little Ufe ; and therefore I would advise my Reader, before he enter upon this Geometrical Way of resolving Spherical Triangles, to peruse the Beginnings of the Second and Third Sections of the Third Part hereot; which treat of the Circles of the Sphere, and their several Posi. tions and Affe&tions; in which he may receive very much Satisfa&tion concerning those Particulars. So that it shall suffice, in this Place, that I declare, 1. What a Great Circle is.

a 2. How to Project such a Circle of the Sphere upon a 'Plain,

suitable to any Duy and Time of the Day, at any time of the

Year, and in any Latitude. 3. To discover the Triangle, which is made by the Intersection

of those Great Circles fo projected. 4 To find the Poles of those Great Circles. And, 5. To Measure the Sides and Angles of the Triangle fo laid

down : Which to do, is usually called, The Dodrine of the Dimension of Triangles.

I. Ibat

{

}

Aquino&tial} Colure.

1. What a Great Circle is.

Fig. A Great Circle of the Sphere is such a Circle, as divideth the XLIV. whole Sphere into Two Equal Paris or Hemispheres : Of which, there are generally accounted Six, viz.

Meridian
The Horizon

The
Æquinoctial

Ecliptick,
Besides these Six Principal Great Circles of the Sphere there are
other Circles, which are Great Circles also. As are all Azimuth
or Vertical Circles. Also all intermediate Meridians or Home Cir-
cles. And there are such as we have moft occafion to make use
of in this Place: And here note, That all Great Circles proje{ted
upon a Plain, Steriographically (or Circular,) are either, (1.) Per-
feal Circles as the outward or Primitive Circle is. Or, (2.) Semi-
Great Circles, projected within the Primitive Circle. Or, (3.)

They are Streight Lines, palling through the Centre of the Primi-
tive Circle, as afterwards will appear.
II. How to Project the Circles of the Sphere upon a Plain, and
Swerable 10 a Prefixed Time and Place.

PROBLEM.
L

ET it be required to Projeť such Circles of the Sphere in Plano,

upon the Plain of the Meridian, in the Latitude North, 40 Deg. Upon the oth of June, at the time of the Sun's Rijing or Serring; and also at io in the Morning, or 2 in the Afternoon, the fame Day : The Sun then having 23 Deg. 30 Min. of Norib Declination.

First, With 60 Deg. of a Scale of Chords, upon the Point A, describe the Primitive Circle Z HNO, representing the Meridian of the Place.

Secondly, Draw the Right Line H A O for the Horizon of the Place; and at Right Angles thereto the Line ZAN, for the Æquinoctial Colure, Z being the Zenith, and N the Nadir Points.

Thirdly, Take 40 Deg. (the Latitude given) out of your Scale of Chords, and set them from 0 to P, from Z to Æ, from H to S, and from N to æ ; and draw the Line P A S for the Axis of the

PAS
World, and Hour-Circle of Six; and the Line Æ A &, for the
Aquinoftial,

Fourthly, Because the Sun's Declination at the time given is 23 Deg. 30 Min. take 23 Deg. 30 Min. and set them from # to %, and from a to s : And if you lay a Ruler from A or a

P2

to