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ten, as is thus evident: For let the five Quantities be a, b, c, d, e, 'tis plain the Combinations of two different Quantities of thofe five, will be as follows; ab, ac, ad, ae; bc, bd, be; cd, ce; de; in all, ten. As there are only ten different Cafes of Right-angled Spherical Triangles, fo in every Cafe there are three Varieties, arifing from the three unknown Parts; and confequently there will be thirty Analogies for finding the Quæfita of fuch a Triangle according to the different Data.

I have annexed a Synopfis thereof; in the first Column of which are the Numbers of the Cafes from I to X; in the second Column are the Data, or Parts given; in the third, are the Quæfita, or Parts fought; in the fourth, are placed the Quadrantal Triangles in the Scheme, each of which includes another leffer Triangle, (as the Quadrantal Triangle ADE includes the leffer Triangle CD F,) and both thefe ferve to form the Proportions or Theories of each Analogy. In the fourth Column, are the Proportions which are the Reafon and Grounds of the Analogies for the Solutions of every Cafe, and out of which they are feverally formed; in the fifth, are the feveral Theorems, by which the preceeding Proportions were raised and compofed. In the fixth and laft Column, are the Analogies adapted to each Variety of every particular Cafe, for the Solution thereof.

But tho' I have reduced the Number of Cafes to a natural Order, and a conftant Number; the intelligent Reader will notwithstanding obferve, that even those ten Cafes are not abfolutely different from one another, and for that Reafon, are yet capable of a farther Reduction, viz. to the Number Six. The Reafon of which is, because in the Spherical Rightangled Triangle AB C, either of the Legs AB, or AC, may be made the Bafe or Perpendicular at pleaVOL. II. fure ;

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fure; and fo the Angle B, or C, may be the Angle at Bafe or Perpendicular indifferently; confequently of the ten Cafes in the Synopfis, four of them will be fimilar to four others; and only two will be abfolutely different and particulnr in their Analogies.

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From hence it appears, that instead of fixteen Cafes, only fix are fufficient to folve a Right-angled Spherical Triangle in all its varieties; however, if we diftinguifh the Bafe and Perpendicular, 'tis proper to add the other four fimilar Cafes, and fo make ten; to avoid all Uncertainty to the lefs skilfull in thefe Matters.

What I have often wondered at, is, that Authors in general agree there are fixteen Cafes of a Right-angled Spherical Triangle, and yet hardly any two of them agree which they are; fome reckoning that a Cafe, which another thinks is none; and thus inftead of fixteen Cafes we may have twenty-fix, or perhaps thirty (as being all that can be) if we look on all thofe as real Cafes which different Authors efteem fo, and have ftated as fuch. But I have made it evident there are but fix Cafes really different from each other, whofe Analogies are to be deduced from the fix Theorems following, viz. 25, 26, 27, 28, 29, and 30; to which the four other fimilar Cafes being added, make in all ten; and thefe are really all the Cafes that can happen to a Right-angled Spherical Triangle.

The Ambiguities of a Right-angled Spherical Triangle are folved by Theorem 18, 19, 20, and 21;

by

by Ambiguities, I mean the Doubts which occur (in the Solution of the Triangle,) of the Affection or Species of the Side or Angle found; that is, whether it be leffer, or Greater than a Quadrant, or a Right-angle; fince the Sine of an Arch is the fame with the Sine of that Arch's Complement to 180°, or a Semicircle. Now because, if we have given

1. The Hypothenufe, and either of the Angles B, C; there can be no Ambiguity, or Doubt about the other Parts, by the faid Theorems: Hence the two first Cafes are free of all Ambiguity.

2. If the Hypothenuse and either Side, AB, AC, are given; there can be no Ambiguity; and thus the third and fourth Cafe are free from doubt.

3. If both the Legs, or both the Angles, are given, as in Cafes 9, 10, there can be no Ambiguity.

4. If a Leg, and an Angle adjacent thereto be gi ven; then the oppofite Leg and Angle, and confequently, the Hypothenufe are known in Specie, by Theorems 18, 19; and fo Cafe 5, ad 7, are free,

5. But laftly, if a Leg and an Angle oppofite thereto be given, then is the Triangle wholly ambiguous in all its unknown Parts; because fuch Data will not determine whether the Parts fought are greater or leffer than 90 Degrees; therefore the only Cafes ambiguous are, the 6, and 8; and thofe in every Part.

Corollary. Hence to avoid the Ambiguity of those Cafes, chufe to find the Quæfitum by fome other Cafe or Data, if it may be; but if the Data be only fuch as in these dubious Cafes, then the best way will be to delineate the Triangle by Spherical Projection, and fo fhall all Ambiguity vanish.

All the thirty Analogies in the foregoing Synopfis, are composed of Sines and Tangents only; but if the young Trigonometer pleafe, he may ufe Secants alfo; and not only fo, but by means of Secants, each of

the

the faid Analogies may be varied fix feveral ways. For the Manner and Reafon of doing this, I shall here reaffume the Proportions demonstrated amongst the Geometrical Theorems of Part I ; and they are as follows.

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Propor. 3. As the Tangent of an Arch TO,
is to the Tangent of an Arch HD;
fo is the Co-tangent of the latter Arch,
to the Co-tangent of the Former.
TS: AH:: BF: FN.

Propor. 4. As the Sine of an Arch TO,
is to the Sine of an Arch AH;

fo is the Co-fecant of the latter Arch,
to the Co-fecant of the Former.
RO DG: BC CN.

By these four Proportions, may Secants be introduced into the Analogy; and every Analogy in the Synopfis may be varied fix different ways, whether it confift of Sines only, or Sines and Tangents together. I fhall give an Example how this is done, firft in an Analogy of Sines only, as Cafe 1, and first Analogy;

1. Viz,

1. viz. R: SC :: sBC: sBA.

:

2. cseC: R:: SBC: sBA, by Propor. 2, inverting. 3. cseBC R: SC: SBA, by Prop. 2, inverting. 4. SBC cseC:: R: cseBA, by Prop. 2, inverting. 5. R: cseBC cseC: cseBA, by Prop. 4, inverting. 6. SC: R:: cse BC: cseBA, by Prop. 4, tranfpofing.

In like Manner, may any Analogy, confifting of Sines and Tangents, be varied in a Six-fold Difference with the Mixture of Secants: Let an Example be of Cafe 1, and its third Analogy;

1. viz. R: csBC :: tC ctB.

2.

3.

4.

5.

6.

ctC: R:: csBC : ctB.
se BC: R:: tC : ctB.
csBC: ct C:: R : tB.
R: seBC: ctC : t B.
tC R se BC: tB.

By Propor. 1, 3; and by inverting and tranfpofing the Terms of Analogy.

And after this manner, may any of the other Analogies be varied; and fo an 180 different Canons of Sines, Tangents, and Secants will enfue for finding the Quæfita of a Right-angled Spherical Triangle; but having given the two Specimens above, I fhall leave the other to the Learner's Exercise.

Pursuant to the Order of Part 1, of Plain Triangles, I fhall here propofe the different and various Methods of folving Right-angled Spherical Triangles; and fet them down in the Order, I defign to treat of them in the Sequel of this Book. And they are as follows.

1. Method

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