JANUA MATHEMATICA.

SECTION II.

Of Spherical Triangles.

CHA P. I.

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

A Spherical Triangle (as well as a Plain, or Right-lined Tri

angle) confifteth of Six Parts; namely, of Three Sides, and as many Angles: The Affections whereof fhall be demonftrated from these Twenty Theorems following.

1. The Three Sides of every Spherical Triangle are the Arches of
Three Great Circles of the Sphere, every one of them being lefs
than a Semicircle, or confifting 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 fuch a Circle as divideth the
whole Sphere, or Globe, into Two Equal Parts or Hemifpheres;
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, do pafs by the Poles of ano-
ther Great Circle, thofe Two Great Circles do interfed, or cut,
each other at Right Angles: And the contrary.

Thus, Let A EC be a Great Circle of the Sphere; (and let it reprefent the Horizon of any Place) whofe Poles let be B and D, (the Zenith and Nadir of the fame Place, equi-diftant from the Horizon AEC 90 Deg.) by which Poles B, and D, let another Great Circle pafs; namely, BED, one of the Colures (or any Azimuth, or Vertical Circle). Now, I fay, that the Great Circle

H 2

BED,

Fig. XIII.

Fig.XIII. BE D, cutteth the Great Circle A EC, at Right Angles in the Points E and F. For, if upon the Pole E or F another Great Circle ABCD be defcribed, it is manifeft, that A B, BC, C D, and DA, fhall be the Measures of the Angles at E and F: (by the 4th of Sea. II.) But the Arches A B, 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 Sect. II.) Which was to be de monftrated.

Fig.

IV. The Measure of a Spherical Angle, Is the Arch of a Great Circle, defcribed 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 A B 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 E D is; (by the 1ft hereof) and therefore, the Arch BC, cannot be the Measure of the Angle BAC, (by the 4th of Sea. II.)

Fig. XV.

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 Oppofite to the First Angle.

Thus, In the Triangle AB C, the Sides A B and A C, of the Angle B AC, being continued to D, do make the Semicircles ABD and AC D, which do comprehend the Angle B D C, equal to the Angle B A C, because the fame Arch G H (being diftant from A and D 90 Deg.) meafureth both thofe Angles, (by the 4th hereof.)

VI. Every Spherical Triangle, hath from every Angle thereof, another Triangle oppofite thereunto, whofe Bafe and Angle oppofite to the Bafe, 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 BDC, oppofite thereunto; whofe Bafe B C, and Angle BDC, oppofite thereunto, are the fame, (by the 5th hereof.) And the Sides B D and CD, are the Complements of the Sides A B and A C, to a Semi

circle:

circle: And the Angles DB C, and DCB, are the Complements Fig. XV.
of the Angles A B C, and AC B, to Two Right Angles, or 180
Deg. (by the 18th of Se&t. II.)

VII. The Sides of a Spherical Triangle may be changed into Angles,
and the contrary; the Complement of the Greateft Side, or
Greateft 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 A B C, obtufe-angled at B: Let D E 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,

K-L

KD)

(LE) are Quadrants, and (L D
LG and FM their common
MH Complement is K H.

KIS

L

SDEZ LM is equal to FG because KM) THIS Therefore, the Sides of the Triangle K L M are equal to the Angles of the Triangle ABC, taking for the Greatest Angle ABC, the Complement thereof F B G.

It may also be demonftrated, That the Sides of the Triangle A B C are equal to the Angles of the Triangle KLM: (by the Converfe of the former.) For, the Side

AB

SOP the Measure (MIK which is the Complement of

BC is equal to FH

of the Angle

DKI

AC)

DI

the Obtufe Angle M KL.

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Therefore, The Sides may be turned into Angles, and the contrary.

Which was to be demonftrated.

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

IX. Suppose the Angles at A and D, viz. BA C and B DC, to Fig. XV. be Right Angles. Then,

SBACZ
CDES

The Triangle BDC hath

One Right Angle, and Two Acute, Two Ob-
tufe Angles, and One Right, One Obtufe,
and Two Acute.

X. A

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

As in the Triangles BA C, and B DC.

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 B A C, 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, Obtufe-angled at B and C, the Sides
D B and DC are greater than Quadrants; and the Side BC leffer,
XIII. ARight-angled Spherical Triangle, with Two Acute Angles,
bath (from the Acute Angle) oppofite to it, a Right-angled
Spherical Triangle, with One Acute, and One Obtufe, 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 Jubtending the Right Angles of a Spherical Triangle, having divers Right Angles, are Quadrants.

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

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XV. A Spherical Triangle, having divers Right Angles, hath either Two or Three Right Angles; and fo 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 A G H fhall have the Three Right Angles at A, G and H;

and