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3. To find the Angle C.

The Analogy SBC: R::s BA : s C. In Numbers 70545: 100000 :: 59130: 83819; Now 83819 is the Natural Sine of 56° 57'=C.

Cafe 3.

Given

One Leg, as
and an adjacent Angle

BA= 36 15
B42 34

1. To find the Hypothenufe BC.

The Analogy Rcs B: ct AB: ct BC. In Numbers 100000: 73649 :: 136382: 100291 ; Now 100291 is the Tangent of the Complement of 44° 52′ B C.

2. To find the other Leg A C.

The Analogy R: SBA::t Bt AC.

In Numbers 100000: 59130 :: 91847 54295; fo is 54295 the Natural Tangent of 289 30'-AC.

3. To find the other Angle C.

The Analogy Rcs BA SB: cs C.

In Numbers 100000 80644: 67644 54537 $ fo fhall 54537 be the Co-fine of 56° 57′ = C.

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1. To find the Hypothenufe B C.

The Analogy B: SAC: R: SBC.

I:

::

In Numbers 67644 47715 100000 70545; then is 70545 the Sine of 44° 52′ = B ̊C.

2. To find the other Leg B A.

The Analogy B: t AC :: R: s B A.

In Numbers 91847: 54295: 100000: 59130; thus is 59130 the Sine of 36° 15′ = BA.

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3. To find the other Angle C.

The Analogy esCA: R::cs B: s C. In Numbers 87881: 100000 :: 73649: 83819; fo is 83819 the Natural Sine of 56° 57' = C.

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1. To find the Hypothenufe B C.

The Analogy Rcs CA cs BA : cs BC. In Numbers 100000: 8788180644 70875; Thus fhall 70875 be the Co-fine of 44° 52′ = BC.

2. To find the Angle B.

The Analogy s BAR::t CA: t B.

In Numbers 59130: 100000 54295 918473 fo is 91847 the Tangent of the Angle B 42° 34'.

VOL. II.

Q 2

3. To

To find the Angle C.

3. The Analogy s CA: R::t BA: t C. In Numbers 47715: 100000 :: 73323: 153692'; thus fhall 153692 be the Tangent of 56° 57′ C.

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1. To find the Hypothenufe B C."

The Analogy C: ctB : Rcs BC;

In Numbers 153692: 108876: 100000

thus

70857; you have 70857 the Co-fine of 44° 52′ = BC.

2. To find the Leg A B.

The Analogy s B: cs C:: R: csBA. In Numbers 67644: 54537 :: 100000: 80644; fo that 80644 is the Co-fine of 369 15′ = BA.

3. To find the Leg A C.

The Analogy s Ccs B

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T

:

Res AC;

In Numbers 83819: 73649: 100000: 87881; and thus we have 87881 the Co-fine of 28° 30' AC.

Thus we have the Manner of working all the Cafes of a Right-angled Spherical Triangle by Natural Sines and Tangents.

С НА Р.

CHAP. XI.

The third Method of folving Right-angled Spherical Triangles, by the Lord Napier's Five Circular Parts.

HAT thefe Five Circular Parts are, I have

W already faid in the Definitions; and I have

also demonstrated in Theorem 40, and 41, their Properties, wherein the Ufe and Excellency of this Invention of that noble Lord doth wholly confift; and that is, by fhewing at once the Proportion for refolving any Cafe of a Right-angled Spherical Triangle, by one Univerfal Propofition or Catholic Canon; viz.

The Radius and Sine of the Middle Part, are reciprocally proportional to the Tangents of the Ex Stream Parts conjunct, and to the Co-fines of the Extreams disjunct.

Now in order to fet forth the Advantage of this Univerfal Canon, there is contrived an Inftrument which fhews by Infpection the Order of the Terms, in any Proportion or Analogy, for the Solution of any Cafe, by having the feveral Parts of this Canon, and the Five Circular Parts, fo aptly placed thereon, as to correfpond to each other; and thereby conftitute the Analogy at once.

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But

But before I come to defcribe this Inftrument, and fhew its ufe; I muft defire the young Learner to confider well the following Particulars.

1. That if the Extreme Parts be conjunct, then the Analogy must be performed by Sines and Tangents together.

2. But if the Extremes be Disjunct, then the Analogy will confist of Sines only.

3. If the Side or Angle fought chance to be the Middle Part, then muft Radius begin the Analogy; or be the First Term.

4. But if one of the Extreams be the Part fought, then the other Extream must be the first Term of the Analogy.

5. That in ufing the Inftrument, if it chance that Co-fine and Complement fhould correfpond, then you muft understand the Sine of the Part thereby. For the Co-fine of the Complement of an Arch, is the Sine itfelf of that Arch.

The Inftrument now to be defcribed (a Scheme of which I have here delineated) may be made of Paper, Paftboard, Wood, Brafs, Silver, &c. and confifteth of two Parts, of which the leffer and inmost moveth on the other, by fome kind of Rivet, on the Center, in order to be turned round as Occafion requires. On this inmoft moveable Piece is defcribed a Right-angled Spherical Triangle ABC, whofe five Circular Parts are, by five ftraight Lines referred to the outmoft fixed Piece, which is divided into two Circular Margins, both which are divided into five equal Parts.

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