(3.) For the Side LH, which is the Distance,

As the Sine of PHL, 45 Deg. 33 Min.

Is to the Sine of PL, 38 Deg. 30 Min.

So is the Sine of LPH, the Differ. of Longitude, 46 Deg.

To the Sine of L H, 30 Deg. 51 Min.

Which 38 Deg. 31 Min. reduced in Miles, makes 2331 Miles

for the Distance of the Two Places.

3. If the Two Places proposed should be so situate, that one have North, and the other South, Latitude, and under different Meridians. As suppose Constantinople, lying in North Latitude 47 D. at C. and the Cape of Good Hope, lying in 35 Deg. of South Latitude, at V; and differing in Longitude 59 Deg. To find the Di. ftance of these Two Places

Upon the Terrestrial Globe.

Bring one of the Places (as Conftantinople) to the Brass Meridian, and there keep the Globe; then apply the Quadrant of Altirude (or rather a thin Plate of Brass divided as that is) to the Two Places, and you shall find 97 Deg. 42 Min. of the Quadrant (or Brass Plate) to be contained between them, and that is their. Distance.

By Trigonometrical Calculation..

4

The Globe refting in its former Position, you will discover upon it an Oblique-angled Spherical Triangle, composed of P C, an Arch of the Brass Meridian: PV, an Arch of a Meridian, passing through the Place in South Latitude: And of CV, the Quadrant (or Brass Plate) representing the Arch of a Great Circle passing through both the Places: - In which Triangle you have given, (1.) The Side PC, the Complement of the Latitude of Conftantinople, 43 Deg. (2.) The Side PV, the South Latitude of the Cape of Good Hope, with 90 Deg. added, which make 125 Deg. And, (3.) The Angle VPC, the Difference of Longitude, 59 Deg. To find, (1.) The Angles of Position PVC and PCV: And, (2.) The third Side VC, the Distance of the Two Places.

The

Fig. The Canons for Calculations, as in the last, by CASE III. - and IX. of A. O. S. T.

XXVI.

(1.) As the Sine of the Half Sum of the Sides PV and PC, 84 D. Is to the Sine of half their Difference, 41 Deg.

So is the Co-tangent of half the Difference of Longitude, i. e. (half the Angle VPC) 29 Deg. 30 Min.

To the Tangent of 49 Deg. 23 Min.

Which is the Half Difference of the Angles PVC and PCV.. (2.) As the Co-fine of half the Sum of the Sides, PV and P.C, 6 Deg.

Is to the Co-fine of half their Difference, 49 Deg.

So is the Co-tangent of half the Difference of Longitude, 61 Deg.
To the Tangent of 85 Deg. 32 Min.

Which is half the Sum of the Two Angles, PCV and PVC.

(3.) For the Side VC, which is the Distance of the Two Places.

As the Sine of PVC, 36 Deg. 9 Min.

Is to the Sine of PC, 43 Deg.

So is the Sine of V PC, 59 Deg.

To the Sine of 82 Deg. 18 Min.

Whose Complement to 180 Deg. is 97 Deg. 42 Min. for the
Side V C, which is the Distance of the Two Places. Which
in Miles is 5862.

And these are all the Varieties of Positions that any Two
Places upon the Globe can be fituate.

ANCILLA

ANCILLA MATHEMATICA.

VEL,

Trigonometria Practica.

SECTION IV.

OF

ASTRONOMY.

ADVERTISEMENT..

Whereas for the resolving of the following Astronomical Fig.

Problems by Trigonometrical Calculation, it is absolutely XXVII.

necessary, that the true Place (or Longitude) of the Sun in the Ecliptick be first known; I have therefore inserted Astronomical Tables of the Sun's Mean Longitude and Anomaly in Years, Months, Days and Hours; and alfo of Equation, whereby the Sun's true Place in the Ecliptick may be found at any time. It is also to be noted, That the several Problems in this Seltion, being wrought first by the Cæleftial Globe, in order to their Trigonometrical Calculation; I have, therefore, to inform the Fancy, and ease the Memory, of the Reader (in all the Schemes relating to them, (which confift of Great Circles of the Sphere, and represent the Triangle to the Eye, which the Problem is to be resolved by) noted the Circles, Lines and Points, in all the Schemes, with the fame Letters or Charaters; and prefixed Them, and their Significations, next before the Problems.

TABLE