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ELEMENTS

OF

Spherical Trigonometry.

DEFINITIONS.

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1.

HE Poles of a Sphere are two Points in the Superficies of the Sphere that are the Extremes of the Axis.

II. The Pole of a Circle in a Sphere, is a Point in the Superficies of the Sphere, from which all Right Lines that are drawn to the Circumference of the Circle, are equal to one

another. III. A great Circle in a Sphere, is that whose

Plane passes thro' the Center of the Sphere, and whoje Center is the same of that of the

Sphere. IV. A spherical Triangle is a Figure comprebend: ed under the Arcs of three great Circles in a

Sphere. V. A Spherical Angle is that which, in the Super

ficies of the Sphere, is contained under two Arcs of great Circles ; and this Angle is equal ta the Inclination of the Planes of the said Circles.

PRO

PROPOSITION I.

Great Circles ACB, A FB, mutually biseEt each

other.

F

VOR since the Circles have the fame Center,

their common Section shall be a Diameter of each Circle, and so will cut them into two equal Parts.

Coroll. Hence the Arcs of two great Circles in the

Superficies of the Sphere, being less than Semicira cles, do not comprehend a Space; for they cannot, unless they meet each other in two opposite Points in a Semicircle

PROPOSITION II.
If from the Pole C of any Circle AFB, be drawn

a Right Line CD to the Center thereof, the
faid Line will be perpendicular to the Plane of
that Circle. Vid. Fig. to Prop. 1.
ET there be drawn any Diameters E F, GH, in

the Circle AFB; then because the Triangles 5 CDF, CDE, the Sides CD, DF, are equal to the

Sides CD, DE, and the Base CF equal to the Base CE; (by Def. 2.) then (by 4. El. 1.) shall the Angle CDF= Angle CDE; and so each of them will be a Right Angle. After the fame Manner we demonstrate that the Angles CDG, CDH, are Right Angles; and so (by 4. El. 11.) CD shall be perpendicular to the Plane of the Circle AFE. W.W.D.

Coroll. 1. A great Circle is distant from its Pole by

the Interval of a Quadrant; for since the Angles CDG, CDF, are Right Angles, the Measures of

them, viz. the Arcs CG, CF, will be Quadrants. 2. Great Circles that pass thro' the Pole of some other

Circle, make Right Angles with it; and contrariwise, if great Circles make Right Angles with some other Circle, they shall pass thro' the Poles of that other Circle, for they must necessarily pafs thro? the Right Line DC,

PRO

PROPOSITION III.
If a great Circle ECF be described about the

Pole A, then the Arc CF intercepted between
AC, AF, is the Measure of the Angle CAF,
or CDF. Vid. Fig. to Prop. 1.
HE Arcs AC, AF, (by Cor. 1. Prop. 2.) are

Quadrants, and consequently the Angles ADC, ADF, are Right Angles. Wherefore (by Def. 6. El. 11.) the Angle CĎF (whose Measure is the Arc CF) is equal to the Inclination of the Planes ACB, AFB, and also equal to the spherical Angle CAF, or CBF. W.W.D.

Coroll. 1. If the Arcs AC, AF, are Quadrants, then

shall A be the Pole of the Circle passing thro' the Points C and F; for AD is at Right Angles to the

Plane FDC (by 14. El, u.) 2. The vertical Angles are equal, for each of them is

equal to the Inclination of the Circles; also the adjoining Angles are equal to two Right Angles.

PROPOSITION IV. Triangles shall be equal and congruous, if they have

two Sides equal to two Sides, and the Angles comprehending the two Sides also equal.

PROPOSITION V. Alfo Triangles shall be equal and congruous, if one Side, together with the adjacent Angles in one Triangle, be equal to one side, and the adjacent Angles of the other Triangle.

PROPOSITION VI. Triangles mutually Equilateral, are also mutually

Equiangular. PROPOSITION VII. InIsoscelesTriangles, the Angles at the Base are equal.

PRO

PROPOSITION VIII. And if the Angles at the Base be equal, then the

Triangle shall be Isosceles. These four last Propositions are demonstrated in the same Manner, as in plain Triangles.

PROPOSITION IX. Any two sides of a Triangle are greater than

the third. FW

OR the Arc of a great Circle is the fhortest

Way, between any two Points in the Superficies of the Sphere.

PROPOSITION X.
A Side of a spherical Triangle is less than a

Semicircle.
ET AC, AB, the sides of the Triangle ABC

be produced till they meet in D; then thall the Arc ACD, which is greater than the Arc A C, be a Semicircle.

PROPOSITION XI. The three sides of a spherical Triangle are less

than a whole Circle. OR BD+DC is greater than BC, (by Prop. 9.)

and adding on each Side B A+AC, DBA+ DCA; that is, a whole Circle will be greater than AB + BC+ AC, which are the three Sides of the spherical Triangle ABC.

PROPOSITION XII. In any Spherical Triangle A BC, the greater An

gle A is subtended by the greater Side. AKE the Angle BAD=Angle B ; then shall

AD=BD (by 8 of this) and so BDC=DA +DC, and thefe Arcs are greater than AC. Wherefore the Side BC, that fubtends the Angle BAC, is greater than the Side AC, that subtends the Angle B.

PRO

F

MAKE

U 3

PROPOSITION XIII, In any spherical Triangle ABC, if the Sum of the

Legs AB and BC be greater, equal, or less, than a Semicircle, then the internal Angle at the Base AC shall be greater, &qual, or less, than the external and opposite Angle BCD, and to the Sum of the Angles A and ACB shall also be greater, equal, or less, than two Right Angles. IRST, let AB BC = Semicircle = AD,

then shall BCEBD, and the Angles BCD and D equal, (by 8 of this) and therefore the Angle BCD shall be = Angle A.

Secondly, Let AB + BC be greater than ABD; then shall B C be greater than BD; and so the Angle D (that is, the Angle A, by 12 of this) shall be greater than the Angle BCD. In like Manner we demonstrate, if AB+B C be together less than a Semicircle, that the Angle A will be less than the Angle BCD. And because the Angles BCD and BCA, are =two Right Angles; if the Angle A be greater than the Angle BCD, then shall A and B CA, be greater than two Right Angles; if the Angie A=BCD, then shall A and BCA be equal to two Right Angles. And if A be less than BCD, then will A and BCA be less than two Right Angles. W.W.D.

PROPOSITION XIV. In any spherical Triangle GHD, the Poles of the

Sides being joined by great Circles; do constitute another Triangie X M N, which is the Supplement of the Triangle GHD, viz. the Sides NX, XM, and NM, shall be Supplements of the Arcs that are the Measures of the Angles D, G, H, to the Semicircles; and the Arcs that are the Measures of the Angles M, X, N, will be the Supplements of the sides GH, G D, and HD, to Semicircles.

ROM the Poles G, H, D, let the great Circles FR XCAM, TMNO, XKBN, be described; then

because

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