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Of the Solution of Spherical Triangles, by
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. projetted" 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 Planispberes. As to the Ste: reographical, an Index having Two Legs as a Seltor ; 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, or Orthographical Proje&tion, there must be an Index of the whole Length of the Diametre thereof, to move about upon the Centre of the Proje&tion, which must be divided as the Diametre of the
Planisphere is ; namely, as a Scale of Nasural Sines, and must be numbred both ways, from the Cen. tre, by 10, 20, 30, bc. 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 so mumbred, by 10, 20, 30, &c. to 90, both Ways : And this fecond 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 Cafes, both of Right and Oblique Triangles. Not insisting 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 Proje&tions.
The Solution of Spherical Triangles, by the LXVIII.
1. Of Riglitangled' Spherical Triangles. T
"O retain the Method before observed in the 16 Cafes of
Right-angled Spherical Triangles; I will here also fullow the same Order, wherein I shall note the Triangle to he resolved by the Letters A B C, setting A at the Right Angle, and B and C at the other Two Acute Angles. So shall the Base be A B; The Cathetus, (or Perpendicular ) CA; The Hypotenuse BC. — The Angle at the Base B; and the Angle at the Cathetus C.
2. There are therefore, besides the Right Angle, Five Circular Paris; 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 rbé 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 Caihetus CA 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
of those Accounts, fall not just upon some Line in the Instrument, either of the Great Circles or Parallels, the Excess is to be estimated in Minutes or Paris of a Degree. Example. Tie Perpendicular CA, and the Angle at the Base
B, given; To find, (1) The Base B A. (2.) The Hypote
nufe CB. (3.) The Angle at the Cathetus C. 1. Reckon on the Limb, from the Diameter, the Quantiry of the given Angle B; and to the End thereof, set 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 Intersection shall shew both the Hypotenuse B C, upon the Index: And the Base B A, upon the Great Circle, meeting also in that point. -Then, to find the Angle at C: Take CA for
the Base, and C A for the Cathetus; and to the end of that Fig. Cathetis apply the Index; so shall you have the Ang'e fought XLVIII. for C, upon the Limb.
3. Otherwise, you may set the Angle B at the Pole, reckoned upon the Diameter from the Limb; and B A upon the Limb from the Pole; and C A upon the Index from the Limb; and B C upon a Great Circle from the Pole.
4. But if the Four Paris 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 resolved thus.
1. Reckon the Lesser 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 Leser 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 B C 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 D C and BC: The third Angle C, opposite to B D, is not here enquired, but only the Place of the Angular Point C, both for D C and BC.
In the firit of the Three Schemes, the inner Angle D is Ob- Fig. tuse, and B Acute: In the second the inner Angle D.is 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 Ď, as they are noted with those Letters.
The Angles D and B, of every Triangle proposed, are teckoned upon the Diameter from the Limb, by the Great Cir cles; namely, the Angle B, and the Conjun&t of the Angle D, from *, but the Angle D, and the Conjunt of the Angle B, from w: And the sides DC and B €, are to be reckoned by
Fig. the Parallels ; namely, the Sides themselves, from the Polt; XLIX. or their Complements and Excesses, from the Diameter.
Note, That of a Triangle, the Conjunct Angle, or the Outer
Angle, is all one. 4. Two Sides of an Oblique-angled Spherical Triangle, wbere:
of one is B D, with the Angle intercepted, being given : To find out, at one Work, the Third Side; and the other Angle at B D.
There are these Two Varieties s
B. C and B.
to find out
D C and D.
DC and D ?
in Se&. 3.
5. The Rule. If the Angles. D and B, be reckoned in the Upper Semicircle of the Planisphere, at the Pole N, and the Angle D be Obtuse, or B Acute; the Point C shall be taken in the former Semicircle towards a: But, if the Angle D be Acute, and B Obtuse, the Point C shall be taken in the latter Semicircle towards we; and contrariwise, for the lower Pole S. -Then, upon the Diameter, count the Angle given, D or B, either by it felf, or by its Conjunat, according as was taught
And from the end of that reckoning, imagine an Arch of a Great Circle aptly traced, until it meet with the Parallel of the Complement of the respective Side given. D C or B C, for that point of Concurrence shall be the Point C, for the given Side, either D C or BC. To this Point C, thus found, apply the Index noted with the proper Letter of the Angle given, D or B; and mark the Distance of it from the Centre for this Distance being exactly taken on the other Index, shall in the Planisphere give the true Place of the Point fought : Which being aptly estimated, will, upon the Diameter, Thew the Angle fought; either B, or the Conjun&t D from $; or else, the Conjunct of B, or the Angle Ď, from W. And, it will also shew upon the Limb, the Complement of the Side fought.
By what hath been before shewed, (that the Point C, both of the Sides D C and BC, are evermore equally distant from the Centre,) there is opened a Way, whereby Any Two of Five Circular Parts ; together with the Side D B, being given, the other Two may be found at one Operation.
Herein are Four Varieties.
to find out
D and B.
D and BC.
The RULE In the Upper Semicircle of the Planisphere, if the Angle D be Obtufe : Or if the Side DC be much Lesser than B C, the Point C shall be taken in the former Quadrant towards : But, if D'be Acute, or DC much Greater than B C; the Point C shall be taken in the latter Quadrant towards vs. And, contrariwise, in the Under Semicircle. -And in both, if the Angles D and B be Acute, or if the Sides DC and B C, differ but little in length, the true Point sought, shall fall near the Axis, towards w, within a Space, equal to the Index opened at the Wideness of B D.
Set, therefore, the Index thus opened; so that the Two Points C, may fall within it. And among the Great Circles and Parallels, or both (according as Reason shall direct) reckon the Two Circular Parts given, tracing them proportionally, with a Pin in each Hand, until they shall both exactly meet with their proper Legs of the Index, at an equal Distance from the Centre, and as near to it as possibly may be: So have you upon the Planisphere, the true Places of both the Points Ć; and thereby, the Two Circular Parts fought, respectively, to each ? Point C; both of D C, and B C.
And note farther, That if any Point C, falleth, either among : the Parallels, near the Diameter; or among the Great Circles, near the Limb; where they be almost equidistant, there may be fome Uncertainty in finding the true Points exa&tly: The remedying whereof requireth the more Diligence in the Compu. ter ; but this may be remedied, by reducing the Oblique Triangle into Two Right-angled.
Other Inconveniencies may arise in the Use of this, as in all ? other Instrumental Operations ; for the remedying whereof, Uffzus optimus Magifter.