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the Bafe, and CA for the Cathetus; and to the end of that Fig. Cathetus apply the Index; fo fhall you have the Angle fought XLVIII. for C, upon the Limb.
3. Otherwife, 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 thefe, with the Axis, fhall include a Quadrantal Triangle And the Greater Right Side fhall be on the Limb from the Pole: And the Leffer Right Side fhall be upon the Diameter from the Centre.
Laftly, Two of the Four Parts being had, there will be no Difficulty in finding out of the Hypotenufe.
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 B D; and the Two Oblique Angles B and D, and the Two other Sides DC and BC: The third Angle C, oppofite to B D, is not here enquired, but only the Place of the Angu lar Point C, both for DC and BC.
In the first of the Three Schemes, the inner Angle D is Ob- Fig. tufe, and B Acute: In the fecond 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 firft thing to be done, is to open the Legs of Index to the Widenefs of B D, and thereto fcrew them faft: Alfo I call the Two Legs of the Index, the Leg or Index B, and the Leg or Index D, as they are noted with thofe Letters.
The Angles D and B, of every Triangle propofed, are reckoned upon the Diameter from the Limb, by the Great Cir cles, namely, the Angle B, and the Conjunct of the Angle D, from, but the Angle D, and the Conjund of the Angle B, from : And the Sides DC and B C, are to be reckoned by
Fig. the Parallels; namely, the Sides themfelves, from the Pole;. XLIX. or their Complements and Exceffes, from the Diameter. Note, That of a Triangle, the Conjunt Angle, or the Outer Angle, is all one.
Two Sides of an Oblique-angled Spherical Triangle, wheres 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.
5. The Rule. If the Angles D and B, be reckoned in the Upper Semicircle of the Planifphere, at the Pole N, and the Angle D be Obtufe, or B Acute; the Point C fhall be taken in the former Semicircle towards: But, if the Angle D be Acute, and B Obtufe, the Point C fhall be taken in the latter Semicircle towards, w; and contrariwife, for the lower Pole S. -Then, upon the Diameter, count the Angle given, D or B,. either by it felf, or by its Conjunct, according as was taught in Sect. 3. 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 refpective Side given. DC or BC, for that Point of Concurrence fhall be the Point C, for the given Side, either D C or B C. To this Point C, thus found, apply the Index noted with the proper Letter of the Angle given, D or B; and mark the Diffance of it from the Centre for this Distance being exactly taken on the other Index, fhall in the Planifphere give the true Place of the Point fought: Which being aptly eftimated, will, upon the Diameter, fhew the Angle fought, either B, or the Conjunct D from; or elfe, the Conjunct of B, or the Angle D, from w. And, it will alfo fhew upon the Limb, the Complement of the Side fought.
By what hath been before fhewed, (that the Point C, both of the Sides DC and B C, are evermore equally diftant 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.
In the Upper Semicircle of the Planifphere, if the Angle D be Obtufe: Or if the Side D C be much Leffer than B C, the Point C fhall be taken in the former Quadrant towards: But, if D'be Acute, or DC much Greater than B C; the Point C fhall be taken in the latter Quadrant towards w. And, contrariwife, 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 fought, fhall 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, fo that the Two Points C, may fall within it. And among the Great Circles and Parallels, or both (according as Reafon fhall direct) reckon the Two Circular Parts given, tracing them proportionally, with a Pin in each Hand, until they fhall both exactly meet with their proper Legs of the Index, at an equal Distance from the Centre, and as near to it as poffibly may be: So have you upon the Planifphere, the true Places of both the Points C; and thereby, the Two Circular Parts fought, refpectively, 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 almoft equidiftant, there may be fome Uncertainty in finding the true Points exactly: 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 Ufe of this, as in all other Inftrumental Operations; for the remedying whereof,. Ufus optimus Magifter.
Fig. The Solution of Spherical Triangles, by the OrthographiXLIX. cal Projection, or Analemma.
I. Of Right-angled Spherical Triangles. HE Hypotenufe is, always, reprefented upon the Index. 2. The Right Angle, at the Section of any Meridian, and the Equinoctial.
3. One Leg in the Meridian, and the other in the Aquino
4. One Angle only entering the Queftion, is reprefented at the Centre, between the Equinodial and the Index, and is numbered in the Limb.
5. But the Two Angles entering the Queftion, you must turn the Cafe into an Angle, and its oppofite Leg. For in the Triangle ABC, whofe Right Angle is A; the other Two Angles ABC, Fig. L. and A C B, both entering the Queftion, you mutt lengthen the Hypotenuse to a full Quadrant to E, as alfo, the Legs adjacent to that Extremity of the Hypotenufe, from which it was lengthned. Thus in the Scheme, the Leg A C, adjacent to C, from which Extremity the Hypotenuje was lengthened: And fo, in the Triangle C D E, Right-angled at E: The Angle DCE is equal the Angle BCA; and the Leg DE, equal to he Complement of the other Angle C BA..
II. Of Oblique-angled Spherical Triangles.
1. Three Sides being given; To find an Angle oppofite to any of them.
Reckon the Greatest Leg from the Pole upon the Limb, and where it endeth, apply the Index; upon the Index reckon, from the Limb, the Bafe, (that is, the Side oppofite to the Angle fought:) And to the Point of the Index, where this Bafe endeth; apply (or bring to) the Curfor: Then look where the Curfor cutteth the Parallel of the Leffer Leg, to be reckoned from the Pole, and obferve what Meridian paffeth by this Section of the Cur for and Parallel; for, where this Meridian cutteth the Equinoctial, there you have the Measure of the Angle fought, to be accounted from thence to the Limb.
2. Three Angles given, to find a Side. Turn the Angles into Sides, and deal with them, as with the Sides.
3. The Parts Given and Sought, being altogether Oppofite.
Reckon the Greater of the Two firft Terms upon the Index, and where this Number endeth upon the Index, apply that Point to the leffer of the faid Two firft Terms; which Parallel is to be reckoned from the Equinoctial; then order the Terms, reckoning the First and Third both upon the Index; or both upon the Parallel; and fo likewife do with the Second and Fourth Terms.
4. Two Sides, with an Angle between them Given: To find the Third Side.
Reckon the Greater Side given, from the Pole upon the Limb; and to the end of it, apply the Index. Reckon the other Side Given upon the Parallels, from the Pole: And the Angle given, upon the Equinoctial, from the Limb; and the Meridian it comes to, purfue till it comes to the Paraltel of the Leffer Leg: And to this Section of the faid Meridian and Parallel, apply the Curfor, fo it may juftly lye on the Index and mark what Point of the Index it pointeth out For, this Point of the Index, counted from the Limb is the Third Side required. And after the Third Side is found, you may find either of the Two unknown Angles, by the Rate for Oppofite
5. Two Sides, and an Angle oppofite to the Leffer of them, Given: To find the Third Side.
Reckon the Difference of the given Sides, and the Sum of them (one and the fame Way) from the Pole; and mark the Points where both of them do end. Count alfo, the Angle given, upon the Equinoctial from the Limb, and mark what Meridian it cometh to. Then extending a strait Line (or applying a strait Ruler) between the Two Points marked in the Limb; it will cut the faid Meridian in Two Places: So you are to obferve the Parallels, where the ftrait Line (or Ruler) cutteth the faid Meridian; for, thefe being reckoned from the Pole, will give you the Quantity of the Third Side required: For, this fame Third Side, may be of a Twofold Quantity: The Leffer it is, when the Angle oppofite to the Greater given Side is Obtufe, and the Greater it is, when the faid Angle (oppofite to the Greater given Side) is Acute.,