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ten, or engraven, Complement; but upon the other Two Lines,' which iffue from the Perpendicular, and the Base, there is nothing written. This Upper Plate is to move upon the Under Plate by a Rivet (or fuch like) through the Centers of both Plates: And fo is your Inftrument finished.
The Ufe of the Inftrument.
THE Ufe of the Inftrument is principally to give you by
Example. In the Triangle A B C, fuppofe there were given the Perpendicular C A, and the Angle at the Bafe B, to find the Angle at the Perpendicular C.
Here it is evident, that the Angle at the Bafe 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 ftand against Extreams Disjunct, which tells you, that C A and C, are Extreams Disjunit: And now (the Rundle thus refting) you fee that One of the Extreams, as C, is fought; therefore, the other Extream G A must be the firft Term in the Proportion, against which you find Co-fine. Wherefore fay,
As the Co-fine of the Perpendicular A C,
Is to the Radius:
So is the Middle Part the Angle at the Bafe B, (against which ftands Sine and Complement, that is Co-fine B.)
To the Angle at the Perpendicular C, against which ftands Cofine and Complement; that is Sine; for, Co-fine Complement is the Sine it self.
And fo the Proportion in fhort is this:
As cs. CA to Radius :: Soc s. B: tos. C.
In this Triangle A B C let there be given the Hypotenufe CB, and the Angle at the Perpendicular C, to find the Bafe BA.
It is evident that B A is the Middle Part. Bring B A against the Middle Part, then will C B and C ftand againft Extreams Disjund: 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, againft both which stands Comp. Cofine, (which is Sine); and against B A the Middle
Part fought, there ftands Sine alfo: So that your Proportion will run thus:
To the Sine of the Angle the Perpendicular C:
So is the Sine of the Hypotenuse C B,
A Third Example.
Fig. In this Triangle ABC, let there be given the Perpendicular XLVII. CA, and the Angle at the Bafe B, to find the Bafe B A.
It is here evident that A B is the Middle Part: Turn the Triangle about till the Bafe A B lye against the Middle Part; then will the Perpendicular C A, and the Angle at the Bafe B, lye against the Extreams Conjunct; 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 Inftrument appears: And the Proportion will be
As the Radius,
Is to the Tangent of the Perpendicular CA:
To the Sine of the Bafe B A.
But if the fame Things were given, and the Angle at C, the
To Co-fine Comp. (that is, to Sine) of C.
And thus, by this Inftrument, 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 Inftrument, there is another Contrivance of the forementioned Gentleman's, with Latin Verfes for bringing the Rules for the Solution of Oblique-angled Spherical Triangles to Memory.
I fhall not here give you any Account of it more than the Figure, and the Latin Verfes (or Rules) in Englifh: Referring you, for the farther underftanding of it, to what is delivered in the IVth Chapter hereof, Page 68...
Of the Solution of Spherical Triangles, by
To either of which Planifpheres, there belongs proper Indexes, to move upon the Centres of the Planifpberes. 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 Planifphere it felf is divided, and must be numbred by 10, 20, 30, c. to 90 Deg. both Ways, and on both Legs.
To the Analemma, or 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 Planifphere 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 fe cond Index (in the Ufe of this Planifphere) I fhall call the Curfor. And thus much for the Defcriptions.
Concerning their Ufe, I fhall only lay down fuch General Rules as are neceffary for the counting of the Quantities of the Sides and Angles of Spherical Triangles upon the feveral Planifpheres and their Indexes; in all Cafes, both of Right and Oblique Triangles. Not infifting upon particular Examples, for that throughout all this Book there are fuch Variety. All which (by these few General Rules here delivered) may be wrought upon either of thefe Two Projections.
The Solution of Spherical Triangles, by the
I. Of Right angled Spherical Triangles.
O retain the Method before obferved in the 16 Cafes of Right-angled Spherical Triangles; I will here alfo follow the fame Order, wherein I fhall note the Triangle to be refolved by the Letters A BC, fetting A at the Right Angle, and B and C at the other Two Acute Angles. So fhall the Bafe be A B, The Cathetus, (or Perpendicular) CA; The Hypotenuse B C.-The Angle at the Bafe B; and the Angle at the Cathetus C..
2. There are therefore, befides 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 Bafe B A upon the Diameter from the Centre; and the Cathetus CA upon the Great Circles from the Diameter, by help of the Parallels; and the Hypotenuse BC upon the Index from the Centre. Note, That if any of thofe Accounts, fall not just upon fome Line in the Inftrument, either of the Great Circles or Parallels, the Excefs is to be eftimated in Minutes or Parts of a Degree..
Example. The Perpendicular C A, and the Angle at the Bafe B, given; To find, (1) The Bafe BA. (2.) The Hypotenufe 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 Cathetus CA; and from the end of that Arch, eftimate reasonably, a Parallel Circle, till it meet with the Index: For that Point of Interfection fhall fhew both the Hypotenufe B C, upon the Index And the Bafe B A, upon the Great Circle, meeting alfo in that Point. Then, to find the Angle at C: Take CA for