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 Conjunt, or
Disjund: For those Three Terms (besides 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 interposed on both Sides :
So are their Circular Parts named, Extreams Conjuntt, or Ex-
treams Disjunct.

VI. But of those Three Terms which may fall in question, we will fubje&t all their Varieties in their Circular Parts, according as every one of them ought (in respect of each other) to be called, The Middle Part; and which, The Extreams Conjunct or Disjunct, as in this following Synopsis is fully demontrated.

If the

Middle Part.

Side

Side

Compl.

Compl.

Compl. B C

Side AC and Comp. B | Com.BCand Com.C
Side A B and Comp. C | Com. B. Cand Com.B
Side A Band Comp. BC | Side A Cand Com. G
Side AC and Comp.BC

Side AB and Com. B

Comp. B and Comp. C | Side AB & Side A C.

Which Synopsis is thus to be Read and Understood: 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 AB and A C, are fupposed 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 Resolution of a Right-angled Spherical Triangle, to know the Mean, and Extream Parts, you must observe, That

1. If One of the Three Terms (which, befides the Right Angle, come in question) doth stand alone by it felf, severed from

the

i

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 interposed) that shall be the Middle Term; and so its Circular Part shall 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 easily shew the Middle Part, and the Extream Terms, the Extream Parts Conjunct.

These 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 Conjunt,
To the Sine of the Middle Part: & contra.

As the Radius,

Then also,

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

II. If

II. If the Extreams (in any Proportion) be Distinct from the Mid. Fig.XXI. dle Part, the Proportion will be performed by Sines only: But if the Extreams be Conjunt to the Middle Part, it must be performed by Sines and Tangents jointly.

The Demonstration of the Universal Propofition is obvious enough: For, where the Extream Parts are Disjunct, the Proportions differ nothing from the common ones: And in the Extreams Conjunct, where it is commonly said,

As Radius, to the Tangent,

We here fay,

As the Co-Tangent, to the Radius.

And likewise Inversly and Contrarily; which is plainly the fame
thing: Because,

The Radius is a Mean Proportional, between the Tangent of
an Arch, and the Tangent Complement of the Same Arch.

Note, That when a Complement in any Proportion doth chance to
concur with a Complement in the Circular Parts, you must then
(always) take the Sine it felf, or the Tangent it felf; inftead
of the Co-fine, or Co-Tangent, in the Circular Parts: Because
the Co-fine of the Co-fine, is the Sine it felf, and the Co-tangent
of the Co-tangent, is the Tangent it self,

As is the Sixth Cafe following; where CB and C A are given,
and the Angle at B is required: Here CA is the Middle Part,
and C B, and B, are the Extreams Disjunct: Wherefore (by the
fecond Part of the foregoing Corollary) the Proportion will be per-
formed by Sines only. - And (by this laft) because the Two
Extreams, CB, and B, fall upon Complements in the Circular
Parts; therefore, inftead of Co-fine BC, and Co-fine B, you
must say Sine BC, and Sine B.

These Things premised, we will exemplifie in the Solution of
Right-angled Spherical Triangles in all the Cafes thereof.

CHAP