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In one of which is wrote Middle Part ; and under it, the Word Sine. In the Parts on each side of this, is wrote Extream Conjunct ; and under them, the Word Tangent : In the two other remote Divisions, is wrote Extream Disjunet ; and under it the Word, Co-sine. And such is the Furniture of each Part or Plate of this Instrument. And if to these the Practical young Student would adapt an Artificial Line of Sines and Tangents, on the Margin of the outmost of these Plates, and contrive them to slide or move by two of the same Lines fixed on the inner Border of a chird Piece (as

may

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ne of these I20 Method 3. Of solving R. A. S. Triangles may very well be done ) then would this little artful Machine answer all the purposes of Spherical Trigo. nometry of it felf; for it both states the Terms of the Analogy, and gives the Solutions of the Cafe in Numbers.

vert ácii In the Use of this Instrument ( as has been said ) it is to be considered, that, as in every Case there are two Parts of the Triangle given, and one sought ; the other two, and be the Middle Part, and the other two the Extreams. Now that which is the Middle Part ( whether given or fought) in the Triangle, must, by moving the Inner Part about, be set against the Middle Part wrore on the outer fixed Piece ; then shall the Lines, proceeding from the two Extremes, point to the Terins of the Analogy in the outer Parts, and Thew them to be Tangents or Co-lines, as the Extreams are conjunct, or disjoined from the Middle Part.

I shall exemplify the foregoing fix absolute Cases of Right-angled 'Spherical Triangles, in the Use of this curious Invention of the noble Scotch Lord aforeSails and afterwards improved in the above described Instrument by the Right Worshipful Sir Charles Scarborow, M. D.

In the following Cases and Analogies, I shall (for Perspicuity's fake ) represent the Middle Part in a ditferent Character ; and state every Analogy in the Terms, which arise from, and are peculiar to, this Inftrument ; neglecting those in the foregoing Syn@pfis.

Caje .

Given

s The Hypothenuse

and an Oblique Angle

BC,

C.

BC

1. To find the Base B A. Here the Part sought is the Middle Part, and the Parts given are Extreams Disjunct; therefore the Analogy begins with Radius, is altogether in Sines, and from the Instrument, (by setting the Base to the Middle Part ) it will be formed thus ;

R: BC ::;:50B a.

2. To find the perpendicular A C. Here the Angle C is the Middle Part, and the Hypothenuse and Perpendicular Extremes Conjunct ; whence, ( the Instrument being set accordingly ) the Analogy in Sines and Tangents will be thus ;

ctBC:cs:: R: 1 AC.

3. To find the other Angle B. Now is the Hypothenuse the Middle Part, and the two Oblique Angles, the Extremés Conjunct ; one of which being fought, we begin with the other ; the Instrument being ordered, the Analogy in Sines and Tangents will appear thereon as follows, viz.

CtC: Cs 25 C::R:ct B.

The Method of setting the Instrument, and framing the Analogies being thus explained'; I shall only set down the remaining five Cases, with the Analogies belonging to each, as they arise from the said Instrument. D

Case 2.
Given

The Hypothenuse BC,

and the Base BA, VOL. II.

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1. To find the Perpendicular A C. The Analogy, csAB:R::cs BBC:05 AC.

2. To find the Angle B. The Analogy, R:t AB::ctBC:05 B.

3. To find the Angle C.
The Analogy, sB:R::SWBA : SC.

Case 3.

The Angle Given

? and Angle at Base

BA,
B.

1. To find the Hypothenuse B C., The Analogy, A :cs BB :: R:ct BC.

2. To find the perpendicular A C. The Analogy, ctB:R::S WA : .

3. To find the Angle C. The Analogy, RusB :: CSBA:08C

Case 4, Ambiguous.
The Perpendicular

AC,
Given

{ and the Opposite Angle · B.

1. To find the Hypothenuse B C.
The Analogy, B:59C :: R:s BC.

2. To find the Base A B. The Analogy, R:1CA::01B:s 14.

3. To find the Angle C. The Analogy, cs AC : R:05 15 : s C.

Cafe 5.

The Bale

Given { and Perpendicular

AB,
AC.

1. To find the Hypothenuse B C. The Analogy, R:CSAB ::08 AC:cs B€.

..

2. To find the Angle B. The Analogy, 1 AC:R:: s8B4 : ct B.

2. To find the Angle C. The Analogy, 1 AB:R::sad:ct C.

Cafe 6.

The Angle at Base

B, Given { and the Angle at Perpendicular C.

1. To find tbe Hypothenuse B C. The Analogy, R:ct B::00 C :cs 15 C.

2. To find the Base B A. The Analogy, s B :R::05C : cs B A.

VOL. II.

R2

3. To

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