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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 those 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, so in every Case there are three Varieties, arising from the three unknown Parts; and consequently there will be thirty Analogies for finding the Quæsita of such a Triangle according to the different Data.
I have annexed a Synopsis thereof; in the first Column of which are the Numbers of the Cases from I to X; in the second Column are the Data, or Parts given; in the third, are the Quæsita, 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 AD E includes the lesser Triangle CDF,) and both these serve to form the Proportions or Theories of each Analogy. In the fourth Column, are the Proportions which are the Reason and Grounds of the Analogies for the Solutions of every Case, and out of which they are severally formed; in the fifth, are the several Theorems, by which the preceeding Proportions were raised and composed. In the sixth and last 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 constant Number ; the intelligent Reader will notwithstanding observe, that even those ten Cases are not absolutely different from one another, and for that Reason, are yet capable of a farther Reduction, viz. to the Number Six. The Reason of which is, because in the Spherical Rightangled Triangle AB C, either of the Legs AB, or AC, may be made the Base or Perpendicular at plea
sure ; and so the Angle B, or C, may be the Angle
For the Case
From hence it appears, that instead of fixteen Cases, only fix are sufficient to solve a Right-angled Spherical Triangle in all its varieties ; however, if we distinguish the Base and Perpendicular, 'tis proper to add the other four similar Cafes, and so make ten; to avoid all Uncertainty to the less skilfull in these Matters.
What I have often wondered at, is, that Authors in general agree there are sixteen Cases of a Right-angled Spherical Triangle, and yet hardly any two of them agree which they are ; some reckoning that a Case, which another thinks is none ; and thus instead of sixteen Cafes we may have twenty-six, or perhaps thirty ( as being all that can be ) if we look on all those as real Cases which different Authors esteem so, and have stated as such. But I have made it evident there are but six Cases really different from each other, whose Analogies are to be deduced from the fix Theorems following, viz. 25, 26, 27, 28, 29, and 30 ; to which the four other similar Cases being added, make in all ten ; and these are really all the Cases that can happen to a Right-angled Spherical Triangle.
The Ambiguities of a Right-angled Spherical Triangle are solved by Theorem 18, 19, 20, and 21 ;
by Ambiguities, I mean the Doubts which occur (in the Solution of the Triangle,) of the Affection or Spe. cies of the Side or Angle found ; that is, whether it be leffer, or Greater than a Quadrant, or a Right-angle ; since the Sine of an Arch is the same with the Sine of that Arch's Complement to 180°, or a Semicircle. Now because, if we have given
1. The Hypothenuse, 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 Cases 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 Case 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 given ; then the opposite Leg and Angle, and consequently, the Hypothenuse are known in Specie, by Theorems 18, 19; and so Case 5, a.d 7, are free,
5. But lastly, if a Leg and an Angle opposite thereto be given, then is the Triangle wholly ambiguous in all its unknown Parts ; because such Data will not determine whether the Parts sought are greater or lesser than 90 Degrees ; therefore the only Cases am. biguous are, the 6, and 8 ; and those in every Part.
Corollary. Hence to avoid the Ambiguity of those Cases, chuse to find the Quasitum by some other Cafe or Data, if it may be ; but if the Data be only fuch as in these dubious Cases, then the best way will be to delineate the Triangle by Spherical Projection, and so shall all Ambiguity vanish.
All the thirty Analogies in the foregoing Synopsis, are composed of Sines and Tangents only ; but if the young Trigonometer please, he may usé Secants also ; and not only fo, but by means of Secants, each of
the faid Analogies may be varied six several ways. For the Manner and Reason of doing this, I shall here reassume the Proportions demonstrated amongst the Geometrical Theorems of Part 1; and they are as follows.
Propor. 3. As the Tangent of an Arch TO,
is to the Tangent of an Arch HD;
Propor. 4. As the Sine of an Arch TO,
is to the Sine of an Arch AH;
By these four Proportions, may Secants be introduced into the Analogy; and every Analogy in the Synopsis may be varied six different ways, whether it confist of Sines only, or Sines and Tangents together. I shall give an Example how this is done, first in an Analogy of Sines only, as Cafe 1, and first Analogy ;
1. viz. R: SC :: SBC : SBA.
In like Manner, may any Analogy, consisting of Sines and Tangents, be varied in a Six-told Difference with the Mixture of Secants : Let an Example be of Case 1, and its third Analogy ; 1. viz. R : csBC :: 10:c1B. By Propor. 1, 3;
ctC : R:: csBC: ctB. 3. se BC : R :: G: 0B: } ing and transpo
and by invertCSBC : ct::R: B.
sing the Terms R: seBC :: C:t B.
of Analogy 6. C: R:: se BC: B.
And after this manner, may any of the other Analogies be varied ; and so an 180 different Canons of Sines, Tangents, and Secants will ensue for finding the Quæfita of a Right-angled Spherical Triangle ; but having given the two Specimens above, I shall leave the other to the Learner's Exercise.
Pursuant to the Order of Part 1, of Plain Triangles, I shall here propose the different and various Methods of solving Right-angled Spherical Triangles ; and set them down in the Order, I design to treat of them in the Sequel of this Book. And they are as follows.