a

To find the Angle Z CP.

Fig.
This is the Angle of the Sun's Position at the time of the XLIV.
Question, with Reference to the Pole' and Zenith; and may be
thus found.

In regard the Two Sides C Z and C P, comprehending the
enquired Angle Z CP, are both of them less than Quadrants,
we must make use of the alternate and opposite Angle SC N,
· which is equal to Z CP. And, whose Measure is the Arch
of a Great Circle intercepted between the Meridian or Hour-
Circle SCP, and the Azimuth-Circle ZCN, at 90 Deg. Di-
stance from the Angular Point C : And to find the Quantity
hereof, you must,

Lay a Ruler to V, the Pole of the Circle ZCDN, and the Point C, it will cut the Primitive Circle in m, fet 90 Deg. from m, and it will reach ton: A Ruler laid from V to n, will cross the Circle ZDN in the Point X. -Again, Lay a Ru. ler from T, the Pole of the Circle P B S, to C, the Angu. lar Point, and it will cut the Primitive Circle in o, set go Deg. from o top: Then a Ruler laid from T to p, will cross the Circle PBS, in the Point Y. -Lastly, A Ruler laid from C, to X and Y, will cut the Primitive Circle in r and s, so the Distance between r and s, meafured upon the Scale of Chords, will give 49 Deg. 46 Min. for the Quantity of the Anglé Y CX, which is equal to the Angle ZCP enquired. And is the Angle of the Sun's Position at the time of the Question.

The Canon for Calculation.
As s. ZC :
s. ZPC ::

s. ZP s. ZCP.
30 d. 7 m. :

: 49 d. 46 m. To find the Vertical Angle CZ P. This Angle is the Sun's Azimuth from the North Part of the Meridian ZON, whose Measure is the Arch of the Horizon DAV; and to find the Quantity of it, lay a Ruler from Z, the Pole of the Horizon, to D, it will cut the Primitive Circle in d; so the Quantity of the Archd N O measured upon the Scale of Chords, will be found to be uz Deg. 57 Min. And such is the Sun's Azimuth from the Morib Part of the Meridian: The Archd H, 66 Deg. 3 Min. is the Sun's Azimuth from the South; and the Archd N, 23 Deg. 57 Min. is the Sun's Asimuthfrom the East and tl'est

.

30 d.

::

50 d.

Fig. XLIV.

::

The Canon for Calculation.
As s. ZC :
s. ZPC

8. C P : . C ZP.
30 d. 7 m. : 30 d.

66 d. 30 m. 66 d. 3 m.

(Or, 113 d. 57 m. And thus are all the sides and Angles of this Oblique-an

gled Triangle also Measured : And so may any other,

being thus Projected. And, to conclude, This Proje&tive Way will give great

Light to Calculation ; for by the true delineating of your
Triangle upon the Projektion, you shall thereby, discover
whether your Side or Angle be More or Less than a
Quadrant, and so be positive in your Resolution ; which
otherwise you must have Given, or render a double So-
lution: As in this last Angle Ć Z P, whether it were
66 Deg. 3 Min. Or 113 Deg 57 Min. The fame Sing

3
in the Canon answering to both.

JANUA

JANUA MATHEMATICA.

SECTION IV. Of Spherical Trigonometry, Instrumentally

Explained and Performed. THE

HE Explanatory Instrument here described, is for the bet

ter Intormation of the Fancy, by Speculation; and it is deduced from the Catholick, or Universal Proposition, before treated of in Part II. Sec. II. Chap. III. of this Book. Notwithstanding, for the Convenience of the Reader, I shall here, again, infert it.

Proposition Universal.
The Sine of the Middle Part, and the Radius, are Reci-

procally Proportional, with the Tangents of the Extreanı
Parts Conjunct, and with the Có-lines of the Extreams
Disjunct.
In every Right-angled Spherical Triangle, there

are Five Parts, besides the Right Angle, and they are called CIRCULAR PARTS: Of which, those Three which lye most remote from the Right Angle, (as the Hypotenuse, the Angle at the Perpendicular, and the Angle at the Basė) are noted by their COMPLEMENTS.

Of these Five Circular Parts, any Two of them (besides the Right Angle) being given, a Third may

be found. And, Of Three Parts, (Two given, and 'One required) One must (neceffarily) be in the Middle, and must be called the MIDDLE PART.

Of the other Two Extream Parts, they must either Join to the Middle Part, or be Separate from it.

Joined to it,
If they be

Extreams
Separate from it,

Disjun&.

1

3 then are they called S Conjuntt.