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Fig. XLIV.

JANUA MATHEMATICA.

SECTION IV.

Of Spherical Trigonometry, Inftrumentally
Explained and Performed.

THE Explanatory Inftrument here described, is for the bet-
ter Information of the Fancy, by Speculation; and it
is deduced from the Catholick, or Univerfal Propofition, be-
fore treated of in Part II. Set. II. Chap. III. of this Book.
Notwithstanding, for the Convenience of the Reader, I fhall here,
again, infert it.

Propofition Universal.

The Sine of the Middle Part, and the Radius, are Reci-
procally Proportional, with the Tangents of the Extream
Parts Conjunct, and with the Co-fines of the Extreams
Disjunct.

In every Right-angled Spherical Triangle, there are Five Parts, befides the Right Angle, and they are called CIRCULAR PARTS: Of which, thofe Three which lye molt remote from the Right Angle, (as the Hypotenuse, the Angle at the Perpendicular, and the Angle at the Bafe) are noted by their COMPLEMENTS.

Of thefe Five Circular Parts, any Two of them (befides the Right Angle) being given, a Third may be found. And,

Of Three Parts, (Two given, and One required) One muft (neceffarily) be in the Middle, and muft be called the MIDDLE PART,

Of the other Two Extream Parts, they must either Foin to the Middle Part, or be Separate from it.

115

Joined to it,

If they be

Separate from it,

then are they called
Extreams

Conjunct.

Disjuna.

Q 2

If

If the Extreams be Conjunct, the Proportion must be performed by Sines and Tangents jointly. But,

If the Extreams be Disjunct, the Proportion may be performed by Sines only.

And in both Cafes,

If the Part Sought (whether Side or Angle) fall out to be the Middle Part, then the Radius must be the Firft (or leading) Term in the Proportion. But,

If one of the Extreams (whether Conjunct or Disjunct) poffefs the Middle Part, then the other Extream must be the First Term in the Proportion.

Of the Inftrument.

His Inftrument was the Coutrivance (many Years fince) of

The Right Worthipful Sir Charles Scarborote, M. D., and

divers of them have been made in Silver and Brafs about the Bignels of a Crown Piece, with Verfes in Latin about the Rimb, for the better bringing the Rules to be obferved, in the Ufe of it, to Memory, which were to the fame Effect with those which I have even now laid down in English.

The Inftruments confifts of Two Parts, the undermoft is a round Piece, divided into Five Equal Parts, which reprefent the Five Circular Parts before fpoken of, it hath Two Circular Margins, near the outer Edge thereof; in One of the Five Parts is engraven Middle Part, and under it the Word Sine. -On either Side of Middle Part, there is written or engraven Extream Conjunt, and under them, the Word Tangent. In the Two other of the Five Divifions, which are oppofite to Middle Part, are engraven Extream Disjund, and under them, Co-fine. All which Words anfwer directly to the Words of the Univerfal Propofition: And this is all that is upon the Under Plate of the Inftrument.

The Upper Plate of the Inftrument is a Circle, alfo, divided into Five Equal Parts as the former, and within that another Concentrick Circle, between which are Five Lines drawn from the Centre thereof. Within the innermost of these Circles, there is defcribed a Right-angled Spherical Triangle, the Five Circular Parts whereof do lye directly against the Five Lines, drawn from the Centre between the Two Circles: And upon thofe Three Lines which lye against the Hypotenuse, the Angle at the Perpendicular, and the Angle at the Bafe, is writ

ten,

Extre

erit Afterd