ten, or engraven, Complement, but upon the other Two Lines,' which issue from the Perpendicular, and the Base, there is nothing written. -This Upper Plate is to move upon the Under Plate by a Rivet (or such like) through the Centers of both Plates: And so is your Instrument finished.

The Use of the Instrument.

HE Use of the Instrument is principally to give you by

any Cafe.

Example. In the Triangle ABC, suppose there were given Fig. the Perpendicular C A, and the Angle at the Base B, to find the Angle at the Perpendicular C.

Here it is evident, that the Angle at the Base B, is the Middle Part: Turn the Rundle about till you bring the Angle at the Base against Middle Part: Then shall you find that the Perpendicular C A, and the Angle at the Perpendicular C, will stand against Extreams Disjunct; which tells you, that C A and C, are Extreams Disjunt: And now (the Rundle thus resting) you fee that One of the Extreams, as C, is sought; therefore, the other Extream C A must be the first Term in the Proportion, against which you find Co-fine. Wherefore say,

As the Co-fine of the Perpendicular A C,

Is to the Radius:

So is the Middle Part the Angle at the Base B, (against which ftands Sine and Complement, that is Co-fine B.)

To the Angle at the Perpendicular C, against which stands Cofine and Complement; that is Sine; for, Co-fine Complement is the Sine it felf.

And so the Proportion in short is this:

As cs. CA: to Radius :: Socs. B: tos. C.

Another Example.

XLV.

In this Triangle ABC let there be given the Hypotenufe CB, Fig. and the Angle at the Perpendicular C, to find the Base B A.

It is evident that B A is the Middle Part. Bring BA against the Middle Part, then will CB and C stand against Extreams Disjunct: And (because the Middle Part) BA is fought, the Radius must be the First Term in the Proportion, and the Two Extreams the Second and Third, against both which stands Comp. Cofine, (which is Sine); and against B A the Middle

Part

XLVI

XLVI.

Fig.

Part fought, there stands Sine also: So that your Proportion will run thus:

As Radius,

To the Sine of the Angle the Perpendicular C:

So is the Sine of the Hypotenuse C B,

To the Sine of the Base BA.

A Third Example.

In this Triangle ABC, let there be given the Perpendicular

XLVII. CA, and the Angle at the Bafe B, to find the Base B A.

It is here evident that A B is the Middle Part: Turn the Triangle about till the Base A B lye against the Middle Part; then will the Perpendicular C A, and the Angle at the Bafe B, lye against the Extreams Conjunt; and feeing the Middle Part is fought, the Radius therefore must be the First Term in the Proportion: And because the Extreams are Disjunct, the Proportion will be in Sines and Tangents jointly, as by the Instrument appears: And the Proportion will be

As the Radius,

Is to the Tangent of the Perpendicular CA:
So is the Tangent Complement of B,

To the Sine of the Base B A.

But if the fame Things were given, and the Angle at C, the
Perpendicular, had been required, then B had been the
Middle Part, and C A and B, the Extreams Disjunct,
and the Proportion in Sines only. As thus:'

As the Co fine of C A,.י
To Radius:

So is the Co-fine of B,

:

To Co-fine Comp. (that is, to Sine) of C.

And thus, by this Instrument, may the Proportions for the Solution of any Right-angled Spherical Triangle be readily fet down, and the Triangle refolved in any of the XVI. Cafes. On the Back-fide of this Instrument, there is another Contrivance of the forementioned Gentleman's, with Latin Verses for bringing the Rules for the Solution of Oblique-angled Spherical Triangles to Memory.

I shall not here give you any Account of it more than the Figure, and the Latin Verses (or Rules) in English: Referring you, for the farther understanding of it, to what is delivered in the IVth Chapter hereof, Page 68...

Of

:

Of the Solution of Spherical Triangles, by
PLANISPHERE.

A Planisphere is a Projection of the Sphere (or Globe) in
Plano: Of which, there are principally Two; One called Fig.
Stereographical, projected by Circles; The other Orthographical, XLVIII.
projected by Ellipses, and generally known by the Name of
Analemma.

To either of which Planispheres, there belongs proper Indexes, to move upon the Centres of the Planispheres. As to the Stereographical, an Index having Two Legs as a Sector; both which are to be divided, according to the Tangents of half Arks, as the Semidiametre of the Planisphere it self is divided, and must be numbred by 10, 20, 30, &c. to 90 Deg. both Ways, and on both Legs.

To the Analemma, of Orthographical Projection, there must be an Index of the whole Length of the Diametre thereof, to move about upon the Centre of the Projection, which must be divided as the Diametre of the Planisphere is; namely, as a Scale of Natural Sines, and must be numbred both Ways, from the Centre, by 10, 20, 30, &c. to 90 Deg. - Upon this Index, (by help of a Groove made through the former Index) another Index is to be made to move upon, and with, the former, and always keeping at Right Angles with it: And this Index is to be divided as a Scale of Sines, as one half of the other; and fo numbred, by 10, 20, 30, &c. to 90, both Ways: And this second Index (in the Use of this Planisphere) I shall call the Curfor. And thus much for the Descriptions.

Concerning their Use, I shall only lay down such General Rules as are necessary for the counting of the Quantities of the Sides and Angles of Spherical Triangles upon the several Planispheres and their Indexes; in all Cases, both of Right and Oblique Triangles. Not infisting upon particular Examples, for that throughout all this Book there are such Variety. All which (by these few General Rules here delivered) may be wrought upon either of these Two Projections.

Fig. LXVIII.

I.

T

The Solution of Spherical Triangles, by the
Stereographical Projection.

I. Of Right angled Spherical Triangles.

O retain the Method before observed in the 16 Cafes of Right-angled Spherical Triangles; I will here alfo follow the fame Order, wherein I shall note the Triangle to he resolved by the Letters ABC, setting A at the Right Angle, and B and C at the other Two Acute Angles. So shall the Base be AB; The Cathetus, (or Perpendicular) CA; The Hypotenuse BC. -The Angle at the Bafe B; and the Angle at the Cathetus C..

2. There are therefore, besides the Right Angle, Five Circular Parts; namely, Three Sides, and Two Angles: Of which, Four come into the Account at once, Two of them are given, and the other Two found out.

3. If the Four Parts, which at once come into the Account, be the Three Sides, and One Angle. -Let that Acute Angle be evermore noted with the Letter B; and the Triangle may be refolved thus:

Set the Angle B at the Centre, reckoning it upon the Limb from the Diameter; and the Base B A upon the Diameter from the Centre, and the Cathetus C A upon the Great Circles from the Diameter, by help of the Parallels; and the Hypotenuje BC upon the Index from the Centre.-Note, That if any of those Accounts, fall not just upon fome Line in the Instrument, either of the Great Circles or Parallels, the Excess is to be estimated in Minutes or Parts of a Degree.

Example. The Perpendicular CA, and the Angle at the Base B, given, To find, (1) The Base BA. (2.) The Hypotenuse CB. (3.) The Angle at the Cathetus C.

1. Reckon on the Limb, from the Diameter, the Quantity of the given Angle B; and to the End thereof, fet either of the Legs of the Index.

2. Upon the Limb, from the Diameter, reckon the Catbetus CA; and from the end of that Arch, estimate reasonably, a Parallel Circle, till it meet with the Index: For that Point of Interfection shall shew both the Hypotenuse B C, upon the Index: And the Base BA, upon the Great Circle, meeting also in that Point. -Then, to find the Angle at C: Take CA for

the

the Bafe, and C A for the Cathetus; and to the end of that Fig. Cathetus apply the Index; so shall you have the Angle fought XLVIII. for C, upon the Limb.

3. Otherwise, you may fet the Angle B at the Pole, reckoned upon the Diameter from the Limb; and B A upon the Limb from the Pole; and CA upon the Index from the Limb; and B C upon a Great Circle from the Pole.

4. But if the Four Parts which at once come into the Account, be the Two Oblique Angles, C and B, and the Two Right Sides CA and B A, then the Triangle may be refolved thus.

1. Reckon the Leffer Angle on the Index from the Centre, and the Greater Angle on a Great Circle from the Pole: So both of these, with the Axis, shall include a Quadrantal Triangle: And the Greater Right Side shall be on the Limb from the Pole: And the Leffer Right Side shall be upon the Diameter from the Centre.

Lastly, Two of the Four Parts being had, there will be no Difficulty in finding out of the Hypotenuse.

II. Of Oblique-angled Spherical Triangles.

1. In an Oblique-angled Spherical Triangle BC D, Five of the Circular Parts come into the Account at once; namely, One Side BD; and the Two Oblique Angles B and D, and the Two other Sides DC and BC: The third Angle C, oppofite to BD, is not here enquired, but only the Place of the Angular Point C, both for DC and BC.

In the first of the Three Schemes, the inner Angle D is Ob- Fig. tuse, and B Acute: In the second the inner Angle Dis Acute, XLIX. and B Obtufe: In the third, both the inner Angles D and B are Acute.

2. The Side B D, is evermore understood to be Given: And in every Operation, the first thing to be done, is to open the Legs of Index to the Wideness of BD, and thereto screw them faft: Also I call the Two Legs of the Index, the Leg or Index B, and the Leg or Index D, as they are noted with those Letters.

The Angles D and B, of every Triangle proposed, are reckoned upon the Diameter from the Limb, by the Great Circles; namely, the Angle B, and the Conjunct of the Angle D, from, but the Angle D, and the Conjunt of the Angle B, from: And the Sides DC and BC, are to be reckoned by

R2

the