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1. Method by the Canon of Logarithmetic
Sines, &c. 2. Method by the Canon of Natural Sines, Tan
3. Method by Lord Napier's five Circular Parts.
Of these ten Methods, five are common to both Plain and Spherical Triangles ; such are the 1, 2, 4, 5, 6; the other five are peculiar to Spherical Triangles only. And having already applied the five Common Méthods to the Solution of Plain Triangles, and there shewed the Nature and Construction of the Numbers in the Canon, both of Logarithmetic and Natural Sines, &c. and also given a large Description and Manner of making the Lines on the Instruments, I shall refer the Reader to that Volume, to be informed of those Matters ; and shall here only apply them to the Solution of Spherical Right-angled Triangles, with the other five Methods, in the Order I have above exhibited them.
And as there are but fix Cases of Right-angled Spherical Triangles, whose Analogies are really diverse from each other, it will be sufficient to exemplify them in every Method here mentioned ; and thereby the young Trigonometer will be provided with an infallible Clue, to conduct him thro' all the abstruse and meanderous Windings of the Mazy Labyrinth of Spherical Geometry, with safety, ease, and pleasure.
The Six Cafes I here mean are as follows;
1. Given the Hypothenuse, and one Oblique Angle ;
As in Synop. Case 1, and 2. 2. Given the Hypothenuse, and one of the Legs :
As.Case 3, and 4. 3. Given one Leg, and an adjacent Angle ; As
Case 5, and 7.
Cafe 6, and 8.
These Cases, with all the Varieties, I shall shew how to resolve by all the Methods above, as in the following Chapters.
The First Method of Solving Right-Angled
Spherical Triangles, by the Canon of
HE Reader is supposed to have his Eye on
the Schemes and the Proportions and Analom
gies in the foregoing Synopsis, as he proceed in the subsequent Solutions, and thus will he be fure to understand at once both the Theory and Practice of his Art.
The Hypothenuse, Given and an Oblique Angle, as G = 56:57
BC= 44 52
1. To find the Leg BA. The. Analogy R :C:: SBC : SB.A. That is, As Radius
Is to the Sine of the 'Angle C=56 57 = 9:9233405 So the Sine of the Hypoth. BC=44 52 = 9,8484720 To the Sine of the Log. BA=36 15. = 9.7718125
2. To find the other Leg A C. The Analogy R : BC :: csC : t AG. That is, As Radius
ó. To the Tangent of the Hyp. BC=44 52=9.9979787 So is the Co-line of the Arigle C=56 57=9.7366918 To the Tangent of the Leg ACE28-30=9.7346705
3. To find the Angle B: The Analogy, R: CSBC :: C : ct B. That is, As Radius
To the Co-fine of the Hyp: BC=44:52= 9.8504730 So is the Tang, of the Angle C=56.57=10.1865775 To the Co. Tan. of the Ang. B=42 34 =10.0370505
The Hypothenuse - BC = 44 52 and one of the Legs, as BA = 36 15
i. To find the other Leg AC. The Analogy csBA:R :: eBC : csAC. That is, As the Co-fine of the Leg Ba=38 1559.9065745 Is to chę Radius So is the Co-fine of the Hyp. BC=44 52=9.8504730 To the Co-fine of the Leg AC=28 3059.9438985
2. To find the Angle B. The Analogy + BC : RM: BA : CSB. That isa
As the Táng. of the Hypoth. BC=44 52=9.9979787 Is to the Radius
10. So is the Tangeht of tlie Leg,BA=36 1539.8652404 To the Co-line of the Angle B=42 34=9,8672617
3. To find the Angle C. The Analogy s BC : R :: SBA : 3C. That is,
As the Sine of the Hypoth. RC=44-52=9.8484720 is to the Radius fo is the Sine of the Leg BA=36 1$=9.7918150 To the Sitië of clie" Angle ( 56 5739.9233430
BA= 36 15 and an adjacent Angle
B = 42 34
1. To find the Hypothenuse B C. The Analogy R : cB : : ctAB : ctBC. That is, As Radius
Is to the Co-fine of the Angle B=42 34= 9.8672673 so is the Co-tang. of the Leg BA=36 15=10.1347596 To the Co-tang. of the Hy. BC=44 52=10.0020369
2. To find the other Leg A C.
The Analogy R: SBA :: B : t AC. That is, As Radius
Is to the Sine of the Leg BA=36.15=9.7718150 fo is the Tangent of the Angle B 42 3459.9629494 To the Tangent of the Leg AC=28 3039.7347644
3. To find the other Angle C.
The Analogy R :CHBA :: B: CsC. That is, As Radius
Is to the. Co-fine of the Leg BA=36 1539.9065745 fo is the Sine of the Angle B=42, 3459.8302342 To the Co-sine of the Angle C=56 57=9.7368087