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 Divifions, is wrote Extream Disjunct; and under it the Word, Co-fine. And fuch is the Furniture of each Part or Plate of this Inftrument. And if to these the Practical Stuyoung dent would adapt an Artificial Line of Sines and Tangents, on the Margin of the outmost of these Plates, and contrive them to flide or move by two of the fame Lines fixed on the inner Border of a third Piece (as may Base MIDDLE PART pendicular, omp. may very well be done) then would this little artful Machine anfwer all the Purpofes of Spherical Trigonometry of it felf; for it both ftates the Terms of the Analogy, and gives the Solutions of the Cafe in Numbers. In the Ule of this Inftrument (as has been faid) it is to be confidered, that, as in every Cafe there are two Parts of the Triangle given, and one fought; fo one of these three Parts muft neceffarily lie between 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, muft, by moving the Inner Part about, be fet against the Middle Part wrote on the outer fixed Piece; then shall the Lines, proceeding from the two Extremes, point to the Terms of the Analogy in the outer Parts, and fhew them to be Tangents or Co-fines, as the Extreams are conjunct, or disjoined from the Middle Part. I fhall exemplify the foregoing fix abfolute Cafes of Right-angled Spherical Triangles, in the Ufe of this curious Invention of the noble Scotch Lord aforeaid and afterwards improved in the above defcribed Inftrument by the Right Worshipful Sir Charles Scarborow, M. D. In the following Cafes and Analogies, I fhall (for Perfpicuity's fake) reprefent the Middle Part in a ditferent Character; and ftate every Analogy in the Terms, which arife from, and are peculiar to, this Inftrument; neglecting thofe in the foregoing Synopfis. Given Cafe 1. The Hypothenufe BC, C. 1. To find the Base B A. Here the Part fought is the Middle Part, and the Parts given are Extreams Disjunct; therefore the Analogy begins with Radius, is altogether in Sines, and from the Inftrument, (by fetting the Bafe to the Middle Part) it will be formed thus ; R: SBC: ¿C: SBA. 2. To find the Perpendicular A C. Here the Angle C is the Middle Part, and the Hypothenufe and Perpendicular Extremes Conjunct; whence, (the Inftrument being fet accordingly) the Analogy in Sines and Tangents will be thus; ctBC: CSC:: RtAC. 3. To find the other Angle B.. Now is the Hypothenufe the Middle Part, and the two Oblique Angles, the Extremes Conjunct; one of which being fought, we begin with the other; the Inftrument being ordered, the Analogy in Sines and Tangents will appear thereon as follows, viz. ctC: csBCR ct B. The Method of fetting the Inftrument, and framing the Analogies being thus explained; I fhall only fet down the remaining five Cafes, with the Analogies belonging to each, as they arife from the faid Inftrument. 1. To find the Perpendicular A C. The Analogy, csAB: R::cs BC: cs AC. 2. To find the Angle B. The Analogy, RtAB:: ct BC: cs B. 3. To find the Angle C. The Analogy, s B: R::s BA : s C. 1. To find the Hypothenufe B C.... The Analogy, BA:cs :: R: ct BC. 2. To find the Perpendicular A C. The Analogy, ctB: R:: stAC. 3. To find the Angle C. The Analogy, RSB csBA: csC. Cafe 4, Ambiguous. The Perpendicular Given { and the Oppofite Angle 1. To find the Hypothenufe B C. A C, The Analogy, s B: SAC: Rs BC. 2. To find the Bafe A B. The Analogy, R1 CA:: ct BsBA. 3. To find the Angle C. The Analogy, cs AC: R::cs B: s C. Given Cafe 5. The Bale A B, A C. { and Perpendicular 1. To find the Hypothenufe BC. The Analogy, R: csAB:: cs AC:cs BC. 2. To find the Angle B. The Analogy, AC: R::s A: ct B. 2. To find the Angle C. The Analogy, AB R:: sAC: ct C. Cafe 6. The Angle at Bafe B, Given and the Angle at Perpendicular C. 1. To find the Hypothenufe B C. The Analogy, R: ctB: ct C: cs BC. 2. To find the Base B A. The Analogy, s B: R::cs: csBA. VOL. II. R 2 3. To |