Fig. XX. Arch A Without. But the Triangle B AE hath the Angles at Band E of different Kinds; and the Perpendicular AC falls Without, and the Quadrantal Arch A Within, the Triangle.

CHA P. III.

Of the Menfuration, or Solution, of Right-angled
Spherical Triangles.

Na Right-angled Spherical Triangle, there are (befides the which are more remote from the Right Angle, the Lord Nepeir changeth into their Complements: -As in this Triangle A BC, Right-angled at A: For the Three Remote Parts, to wit, the Angles B and C, and the Side CB, he takes their Complements: Thefe Three Complements,with the Sides C A and BA, do make Five Parts: Which, by an artificial Term,he calls CIRCULAR.

But the Right Angle at A is fet afide from being any of the
Circular Points.

II. In the Solution of a Right-angled Spherical Triangles, there
are always Two other Parts, or Terms, given (befides the Right
Angle) to find out a Fourth.

III. Thefe Three Terms (namely,
the Third which is Required)
cording to their Circular Parts.

the Two that are Given, and must be first looked upon ac

IV. Of which, One is named the Middle (or Mean) Part; the other Two are called the Extream Parts, borrowing their Appellation from the Scituation of the Terms themfelves: For, of Three Terms, One muft (of neceffity) be in the Middle, and the other Two in the Extreams: Therefore the Circular

Part

Fig.XXI.

Fig. XX

Fig.XXI
Diagram
A.

Part of the Middle Term, is called the Middle Part, and the Fig.XXI.
Circular Parts of the Extream Terms are called the Extream
Parts.

V. But the Extream Parts may be twofold, either Conjunct, or
Disjund: For thofe Three Terms (befides the Right Angle) do
come in question, according as the Two Extreams are from ei-
ther Part immediately joined to a third Mean, or are dif-joined
from the fame, by a Side or Angle interpofed on both Sides:
So are their Circular Parts named, Extreams Conjunct, or Ex-
treams Disjunct.

VI. But of thofe Three Terms which may fall in queftion, we will fubje&t all their Varieties in their Circular Parts, according as every one of them ought (in refpect of each other) to be called, The Middle Part; and which, The Extreams Conjund or Disjunct, as in this following Synopfis is fully demonftrated.

Which Synopfis is thus to be Read and Underflood: Example of the First Line. If the Side A B be the Middle Part, then is the Side A C, and Comp. of the Angle B, the Extreams Conjunct. And the Comp. of the Side B C. and Comp. of the Angle C, the Extreams Disjunct. And fo of all the reft.

And here it is to be noted, That the Sides A B and A C, are fuppofed to be joined together, (as one entire Part,) because the Right Angle at A, is not reckoned amongst the Circular Parts.

VII. Therefore, In the Refolution of a Right-angled Spherical Triangle, to know the Mean, and Extream Parts, you must ob ferve,

That

1. If One of the Three Terms (which, befides the Right Argle, come in queftion) doth ftand alone by it felf, fevered from

the:

Fig.XXI. the other Two on both Sides; (as the Side B C, from the Sides CA and B A, by the Angles B and C interpofed) that shall be the Middle Term; and fo its Circular Part fhall be called the Middle Part; and the other Two Circular Parts are the Extreams Disjunct. But,

2. If the Three Terms do immediately adhere together, the Middle Term doth eafily fhew the Middle Part, and the Extream Terms, the Extream Parts Conjunct.

Thefe Things being all rightly understood, the whole Trigonometry of Sphericals will be abfolved by this One Propofition, which therefore we will call Catholick or Univerfal.

Proposition Universal.

The Sine of the Middle Part and the Radius, are Reciprocally Proportional, with the Tangents of the Extream Parts Conjunct; and with the Co-fines (or Sines Complements) of the Extreams Disjunct. That is,

As the Radius,

To the Tangent of one of the Extreams Conjunct;
So is the Tangent of the other Extream Conjunct,
To the Sine of the Middle Part: & contra.

To the Co-fine of one of the Extreams Disjunct;
So is the Co-fine of the other Extream Disjunct,
To the Sine of the Middle Part: & contra.

COROLLARY

I. If the Middle Part be Sought, the Radius fhall be in the First Place of the Proportion: But if one of the Extream Parts be Sought, then the other Extream fhall be in the First Place. Of the Second and Third Places, it mattereth nothing how they be difpofed.