JANUA MATHEMATICA.

SECTION II.

Of Spherical Triangles.

CHAP. Ι.

Definitions and Theorems, necessary to the right underStanding of the Doctrine of the Dimension of Spherical Triangles.

Plain, or Right-lined Triangle) confifteth of Six Parts; namely, of Three Sides, and as many Angles: The Affections whereof shall be demonftrated from these Twenty Theorems following.

A Spherical Triangle (as well as a

1. The Three Sides of every Spherical Triangle are the Arches of Three Great Circles of the Sphere, every one of them being less than a Semicircle, or consisting of fewer Degrees than 180, which is the Measure of a Semicircle. By the 9th of Sect. II.

II. A Great Circle of the Sphere, is such a Circle as divideth the whole Sphere, or Globe, into Two Equal Parts or Hemispheres; and fo is in all Parts diftant from the Poles thereof, by a Qua drant or 90 Deg. of the Great Circle.

III. If one Great Circle of the Sphere, dopass by the Poles of another Great Circle, those Two Great Circles do interfect, or cut, each other at Right Angles: And the contrary.

Thus, Let A EC be a Great Circle of the Sphere; (and let it Fig. XIII.

represent the Horizon of any Place) whose Poles let be B and D, (the Zenith and Nadir of the fame Place, equi-distant from the Horizon AEC 90 Deg.) by which Poles B, and D, let another Great Circle pass; namely, BED, one of the Colures (or any Azimuth, or Vertical Circle). Now, I say, that the Great Circle BED,

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Fig.XIII. BED, cutteth the Great Circle AEC, at Right Angles in the Points E and F. For, if upon the Pole E or F another Great Circle ABCD be described, it is manifest, that A B, BC, CD, and D A, shall be the Measures of the Angles at E and F: (by the 4th of Selt. II.) But the Arches AB, BC, CD, and D A, are Quadrants (by the 3d hereof) and therefore, the Angles at E and F, are Right Angles (by the 13th of Set. II.) Which was to be demonstrated.

Fig.

IV. The Meafure of a Spherical Angle, Is the Arch of a Great Cir-
cle, described upon the Angular Point, and intercepted between
the Two Sides, they being continued out till they be Quadrants:
(by the First hereof.)

So, In the Spherical Triangle A B C, the Measure of the SpheriXIV. cal Angle at A,is not the Arch BC, but the Arch EC, intercepted between the Two Sides AB and AC, continued till they be Quadrants; that is, to the Points E and D; because the Arch B C is not defcribed upon the angular Point A, but the Arch ED is; (by the 1st hereof) and therefore, the Arch BC, cannot be the Measure of the Angle BAC, (by the 4th of Sect. II.)

V. The Sides of a Spherical Triangle, being continued till they meet
together, do make Two Semicircles, and at their Interfection, do
comprehend an Angle, Equal and Opposite to the First Angle.

Fig. XV. Thus, In the Triangle ABC, the Sides A B and AC, of the
Angle BAC, being continued to D, do make the Semicircles
ABD and ACD, which do comprehend the Angle BDC,
equal to the Angle BAC, because the fame Arch G H (being
distant from A and D 90 Deg.) measureth both those Angles,
(by the 4th hereof.)

VI. Every Spherical Triangle, bath from every Angle thereof, ano-
ther Triangle opposite thereunto, whose Bafe and Angle opposite
to the Base, are the fame; and the other Parts of that Triangle,
are the Complements of the Parts of the other Triangle.

Thus, the Triangle ABC, hath another Triangle BD C, oppo-
fite thereunto; whose Bafe BC, and Angle BDC, oppofite there-
unto, are the fame, (by the 5th hereof.) And the Sides B D and
CD, are the Complements of the Sides A B and AC, to a Semi-

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circle: And the Angles DBC, and DCB, are the Complements Fig. XV.
of the Angles A BC, and ACB, to Two Right Angles, or 180
Deg. (by the 18th of Set. II.)

VII. The Sides of a Spherical Triangle may be changed into Angles,
and the contrary; the Complement of the Greatest Side, or
Greatest Angle, to a Semicircle, being taken for the Greatest
Side, or Greatest Angle.

Thus (in the Figure of Theorem IV. hereof) the Spherical Tri- Fig. XIV. angle ABC, obtufe-angled at B: Let DE be the Measure of the Angle at A, and let F G be the Measure of the Acute Angle at B, (being the Complement of the Obtufe Angle at B, the Greatest Angle in the Triangle :) And let H I be the Measure of the Angle at C: Now,

Therefore, the Sides of the Triangle KL Mare equal to the Angles of the Triangle ABC; taking for the Greatest Angle ABC, the Complement thereof FBG.

It may also be demonstrated, That the Sides of the Triangle ABC are equal to the Angles of the Triangle KLM: (by

the Converse of the former.) For, the Side

Therefore, The Sides may be turned into Angles, and the contrary.
Which was to be demonstrated.

VIII. A Right-angled Spherical Triangle hath One Right Angle,
or more than One.

IX. Suppose the Angles at A and D, viz. BAC and BDC, to Fig. XV. be Right Angles. Then,

Fig. XV. X. A Right-angled Spherical Triangle, hath, from the Right An gle, a Right-angled Triangle opposite thereunto, with Two Obtuse Angles, & contra.

As in the Triangles BAC, and BDC.

XI. The Sides of a Right-angled Spherical Triangle, with Two
Acute Angles, are either of them less than Quadrants.

As the Sides of the Triangle BAC, are all of them, less than
Quadrants.

XII. The Two Sides of. a Right-angled Spherical Triangle, with
Two Obtufe Angles, are greater than Quadrants; and the third
Side, is less than a Quadrant.

As in the Triangle BCD, Obtuse-angled at Band C; the Sides
D B and DC are greater than Quadrants; and the Side BC leffer.

XIII. A Right-angled Spherical Triangle, with Two Acute Angles, bath (from the Acute Angle) opposite to it, a Right-angled Spherical Triangle, with One Acute, and One Obtuse, Angle.

As in the Right-angled Spherical Triangle EDF, having the Angles at E and F Acute, is oppofite to the Right-angled Triangle CD E, with the Acute Angle E CD, and the Obtufe CED.

XIV. The Sides fubtending the Right Angles of a Spherical Triangle, having divers Right Angles, are Quadrants.

As in the Triangle AGH, If the Two Great Circles A Gand A H, do cut the Great Circle G Hat Right Angles in the Points G and H; then A is the Pole of the Great Circle GH (by the third hereof) and AG and A H are Quadrants (by the second hereof.) But if the Angle at A be also a Right Angle, then GH is also a Quadrant, (by the 13th of Selt. II. and by the 4th hereof.)

XV. A Spherical Triangle, having divers Right Angles, bath either Two or Three Right Angles; and so of the Sides, it hath Two or Three of them Quadrants.

As if the Angle at A be put for a Right Angle, the Spherical Triangle AGH shall have the Three Right Angles at A, G and H;

and

and therefore the Three Sides, A G, A H and HG, shall be Fig. XV.
Quadrants. ----But, If you put the Angle at A for an Acute Angle,
then the Spherical Triangle AGH shall have Two Right Angles
at G and H; and thereupon the Two opposite Sides, AG and
AH, Quadrants.

XVI. If the third Angle of a Spherical Triangle, having Two Right
Angles, be Acute; the third Side is less than a Quadrant; but if
Obtuse, then it is greater than a Quadrant.

As in the Spherical Triangle G H I, Acute-angled at G, the third Side HI is less than a Quadrant. --But in the Spherical Triangle AGI, Obtuse-angled at G, the third Side AI is more than a Quadrant.

XVII. An Oblique Spherical Triangle, consisteth fimply of Acute or
Obtuse Angles, or of both of them.

XVIII. A Spherical Triangle, with Two Obtuse Angles, and One
Acute Angle; is opposite to a Spherical Triangle, fimply Acute-
angled, & contra.

As if the Angles at A and D, be supposed Acute; then the Triangle BCD, with Two Obtufe Angles at Band C, and One Acute Angle at D, is oppofite to the fimple Acute-angled Triangle ABC.

XIX. A Spherical Triangle, with Two Acute Angles, and One Obtuse; is opposite to a Spherical Triangle, fimply Obtufe-angled, & contra.

As if the Angles at A and D, be supposed Obtufe; then the Tri angle ABC, with Two Acute Angles at Band C, and One Obtufe Angle at A, is oppofite to the fimply Obtufe-angled Triangle

BDC.

XX. The Three Angles of every Spherical Triangle, are greater. than Two Right Angles.

This is evident in Spherical Triangles, having more Right, or Obtuse Angles, than One: But in Acute-angled Triangles, it may be thus Demonstrated.

DE

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