PROBLEM VIII.

To exhibit or point out the general Uses of this · Projection.

The more notable Uses of this Projection are as follows. 1. It manifestly appears, that when the Sun is in

either Aries r, or Libra , it is also in the Equinoctial E R, and consequently that then the Days and Nights are equal all over the World ; that is, the Diurnal Arch Er=g R the Nocturnal Arch; and this happens twice in the Year, viz. March the Tenth, and September the

Twelfth ; as is plain from the Calendar. II. When the Sun is in the Sign Cancer Co, 'tis plain

the Days are then longest, and the Nights the Thortest they can be to us in the Latitude of London ; that is the Semidiurnal Arch, ca, is then the longest, and the Seminocturnal Arch, a T, is then the shortest it ever can be ; and this hap

pens about June the Eleventh. III. It appears that on the longest Day, the Height

of the Sun at Noon is H = 62 Degrees ; its Height when due East or West, Sr= and its Height at the Hour of 6, Morning and Evening, is AK = The Sun's greatest Declination then is E $ = 23° 30' ; its greatest Amplitude is

its greatest ascensional Difference is Aa= it is also evident there is then no dark Night, for the Almacanter Ww, which bounds the Twilight and total Darknefs, doth not touch the Tropic & T, or parallel of the Sun's Motion for that Day.

par

The

IV. When the Sun is in the Sign Capricorn Vs, just

the Contrary will happen to what did when the
Sun was in Cancer ; the Semidiurnal Arch K X is

now the shortest, and the Seminocturnal Arch VOL. II.

E 2

Xvs, is the longest it can be ; the Amplitude X, is equal, but on opposite Points, to ra; so the Declination is now E K = E= 23° 30', but South; the Meridian Altitude is now but HK = 15° 00'; the Length of Twilight is the Arch X7 The ascensional Difference is much about equal to the Twilight in this Case.

PROBLEM IX.

To exhibit in one. View a Synopsis of the Astronomical Affe&tions and Positions of the Heavens, for any particular Time, as May the first, for Example.

PraElice. From the first of May, parallel to the Equinoctial

ER draw the Parallel of Declination f 1, cutting the Ecliptic in o, through which draw the Almacanther g h : Let fall the Perpendiculars fe, om, bi. Suppose the Time of the Day, be Nine a-Clock in

the Morning.
Then draw the Hour Circle of 9 P L QM,
and the Azimuth Circle Zp Na.
Then for May the first, 9 Hour, A M.
The Sun's Place will be, o,
His Longitude in the Ecliptic, ro,
Declination North, E f.
Meridian Altitude, Hf.
Altitude at 9 a-Clock, om.
Altitude at 6 a-Clock, bi.
Altitude when East or West, s V..
Rising and Setting, C.
Right Afcenfion, r L.
Ascensional Difference, bc,

Amplitude of Rising and Setting, rc.
Azimuth at 9 a-Clock, rp.
Semidiurnal Arch, fc.
Seminocturnal Arch, c l.
· The Arch of Twilight, ct.

Note, The Time of the Motion in any of those

Arches is easily known. For since the Sun, or
Earth, moves 360° in 24 Hours, that is at
the Rate of 15° in one Hour, and 15' in one
Minute of Time.

PROBLEM X.

To represent the fixed Stars in the Analemma.

PraEtice. I. Find the Star's Declination from the Equinoctial. II. Also its Latitude from the Ecliptic.

III. Draw the Parallel of the Star's Declination, to the Equinoctial.

IV. Draw the Parallel of its Latitude also to the Ecliptic.

V. Then in the Point where these two Parellels do intersect each other, will the Star be projected.

Thus I suppose the young Learner hath by this time a sufficient Notion of the Nature, Manner, and Use of the Orthographical ProjeЕtion. And I do advise every one that is ftudious this way, to make for his Use a very large one, with the Calendar annexed, and it will serve him as a perpetual Almanack ; and save him the trouble of Calculation in many Aftronomical Problemas; as is evident from what I have already thewn, and will hereafter further appear.

CH A P.

CH A P. IV.

Theorems serving to the Stereographic Pro

jection of the Sphere in Plano.

THEOREM I.

A Point is there projected into a Point in the Plane of the Projection, where a Ray of Light, passing thro' the Eye and the given Point, meets the Plane of the Projection.

Demonstration.
Suppose the com-

mon Section of
the Plane and

Circle be CD.
The Eye placed in

'tis evident the Point A is, by this means, projected in a, and the Point B will be projected in b, which is the Plane infinitely continued. 2. E. D.

THEOREM II.

A straight Line in the remote Hemisphere, or Semi. circle, is projected into a Right Line less than it self ; but in that Hemisphere, or Semicircle, which is con

tiguous