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either of its poles as a centre with the circles of the sphere. If, however, spherical distance of that pole as a radius. the circles are great circles, the angles For, if this distance be carried round the POA, P'O Al are right angles, and pole, its extremity will lie in the circum- therefore the arcs PĂ, P'A quadference of the circle.
rants: and it is evident that, conv
onversely, Cor. 2. The distances of any circle circles whose polar distances are quadfrom its two poles are together equal to rants pass through the centre of the a semicircumference.
sphere, that is, are great circles of the Cor. 3. A great circle is equally dis- sphere, and are equal to one another. tant from its two poles; but this is not Therefore, &c. the case with a small circle. For if Cor. Circles whose polar distances are A B C be supposed to be a great circle, together equal to a semicircumference the angles POA, P'O A will be right are equal to one another (2. Cor. 2.) angles, and therefore equal to one ano
PROP. 4, ther, so that the polar distances PA, P'A will be likewise equal (III. 12.) ; Any two great circles of the sphere but if ABC be a small circle, the bisecť one another. angles POA, PO A will be, one of For, since the plane of each passes them less, and the other greater than a through the centre of the sphere, which right angle, and therefore the distances is also the centre of each of the great PA, PA will be unequal.
circles, their common section is a dia
meter of each; and circles are bisected PROP. 3.
by their diameters.
Therefore, &c. Equal circles of the sphere have equal
Cor. 1. Any two spherical arcs may polar distances ; and conversely.
be produced to meet one another in two Let A B C and A'B'C' (see the figure points, which are opposite extremities of of prop. 1.) be any two equal circles of
a diameter of the sphere. the sphere; K, K their centres, and P,
Cor. 2. Any number of spherical arcs P' their poles; then, if the radius KA which pass through the same point may is equal to the radius K'A', the polar be produced to pass likewise through distance P A shall be equal to the polar the opposite point. distance PA'; and conversely. For, if O be the centre of the sphere,
PROP. 5. and OK, K P be joined, OK will be perpendicular to the plane A B C (1.), and The spherical arc which is drawn Therefore (def. 3.) O K, K P will lie in from the pole of a great circle to any the same straight line; and in like man- point in its circumference is a quadner OK' will be perpendicular to the rant of a great circle, and is at right plane A' B'C', and OK', K' P' will lie angles to the circumference. in the same straight line. Join O A, Let the point P be the pole of a great PA and OA', PA'. Then, because circle A BC: let any point A be taken the right-angled triangles OKA, OK'A' in the circumference ABC, and let have the hypotenuse o A equal to the hypotenuse O A', and the side KA equal to the side K'A', the angle KOA or PO A is equal to the angle K'O A' or PO A' (I. 13.); and therefore, also, the arc PA (III. 12.) is equal to the arc P'A'. And, conversely, if the arc P A
А be equal to the arc Þ'A', the angle POA will be equal to the angle P'OA PA be joined by the spherical arc (III. 12.); and, therefore, because in the PDA: the arc PDA is a quadrant, right-angled triangles OKA, O K'A', and at right angles to the circumference the hypotenuse OA is equal to the hypo- ABC. tenuse O A', and the angle KO A to the Take O the centre of the sphere, and angle K'O A', the radius K A is equal join OP, O A. Then, because (def. 3.) to the radius K' A' (I. 13.).
OP is at right angles to the plane In the foregoing demonstration it is ABC, the angle PÕ A is a right angle supposed that the points K and K'do (IV. def. 1.); and, therefore, the arc not coincide with the point O, that is, PDA is a quadrant. Again, because that the circles in question are not great OP is at right angles to the plane ABC,
the plane OPDA is at right angles to Q M may be a quadrant, Q R will be the plane A B C (IV. 18.); and, there- equal to MN (I. ax. 3.). And the points fore, the arc P D A is at right angles to Q, R are the poles of A M, A N rethe circumference ABC (def. 7. and spectively, because Q M, Q A, as also def. 8.).
RN, RA, are quadrants (5. Cor. 2.). Therefore, &c. Cor. 1. If two great circles cut one
PROP. 7. another at right angles, the circumfe
If one triangle be the polar triangle rence of each shall pass through the
of another, the latter shall likewise be poles of the other.
the polar triangle of the first; and Cor. 2. If the spherical distances of a point P in the surface of the sphere from
the sides of either triangle shall be
the supplements * of the arcs which two other points A and C in the same
measure the opposite angles of the other. surface which are not opposite extremities of a diameter be each of them equal
Let A B C be any spherical triangle, to a quadrant, P shall be the pole of the and let A', B', C' be those poles of the great circle which passes through the sides B C, A C, A B, which lie towards points A and C. For, if O be the centre the same parts of the arcs B C, AC, of the sphere, the angles POA and A B, with the opposite angles A, B, C, PO C will be right angles, because the respectively, so that A' and A lie toarcs P A and PC are quadrants; and,
wards the same parts of BC, B' and B therefore, P O is at right angles to the and C towards the same parts of AB:
towards the same parts of A C, and C' plane O AC (IV. 3.); for which reason PO must be the axis, and P the pole of that is, . (def. 11.) let A'B'C' be the the great circle which passes through A polar triangle of ABC: the triangle and C (def. 3.).
À B C shall, likewise, be the polar
triangle 'of A' B'C', and the sides of PROP. 6.
either triangle shall be the supplements Every spherical angle is measured by of the arcs which measure the opposite the sphericul arc which is decribed from angles of the other. the angular point as a pole, and inter- For, in the first place, B' being the cepted between the sides of the angle.
pole of AC, A B' is a Let BAC be any spherical angle, quadrant (5.); and C' and from the point
being the pole of AB, A, as a pole, let a R
A C is likewise a great circle be de
quadrant: therefore scribed cutting the
(5. Cor. 2.) A is the sides AB, AC in
pole of B'C'. Also, B3' the points M, N re
it is upon the same
'c N spectively: the sphe
side of B'C' that Al is : for, because rical angle BAC
A' and A are upon the same side of shall be measured
BC, and that Al is the pole of B C, by the arc MN.
A' A is less than a quadrant; and beTake O the centre of the sphere, and
cause A is the pole of B' C', and that join O A, OM, ON. Then, because A A' is less than a quadrant, A and A' A is the pole of the spherical arc MN, are upon the same side of B'C'. the plane MON is perpendicular to
And, in the same manner, it may 0 A (def. 3.), and M O N O are each shown that B is the pole of A' C', and of them perpendicular to O A. There- B, B' upon the same side of A' C'; and fore the angle MON measures the di- that C is the pole of A'B', and 'C, C' hedral angle MOAN (IV. 17.), or
upon the same side of A' B'. Therefore, which is the same thing, (def. 7.) the the triangle A B C is the polar triangle spherical angle MAN or BAC. There- of A' B'C' (def. 11.). fore, the arc M N which measures the
Next, let the arc B' C' be produced angle MON, measures also the spheri- both ways, if necessary, to meet the arcs cal angle B AC.
AB, AC (produced likewise if necessary) Therefore, &c.
in the points D, E, (4 Cor. 1.). Then, Cor. The angle contained by two because A is the pole of the arc B'C', spherical arcs is measured by the dis- the spherical angle BAC is measured by tance of their poles, which lie towards DE 6.). Again, because B' is the pole the same parts of the arcs. For, if the of A C, B'E is a quadrant; and for the arc N M be produced to R, so that RN
* From this property polar triangles are somemay be a quadrant, and to Q, so that times called supplementary triangles.
like reason CD is also a quadrant: But these angles are respectively meatherefore, the sum of B'E and CD), sured by the arcs A B, AC, and B C. that is, of D E and B' C', is equal to a Therefore AB and AC are together semicircumference, and the side B'C' is greater than BC. And hence, taking the supplement of D E which measures A C from each, A B alone is greater the spherical angle BAC. And, in the than the difference of A C and B°C. same manner, it may be shown that any Therefore, &c. other side of either of the triangles Cor. 1. The three sides of a spherical ABC, A'B'C' the supplement of triangle are together less than the cirthe arc which measures the opposite cumference of a great circle. For, if angle of the other.
A B and A C be produced to meet in D, Therefore, &c.
the arcs ABD, ACD will be semicirScholium.
cumferences; but B C is less than BD angle be each of them equal to a quad- and ACD, that is, less than the cirIf the three sides of a spherical tri- and DC together ; therefore, A B, AC,
and B C are together less than ABD rant, the polar triangle will coincide with be the pole of the side opposite to it. be shown that all the sides of any it; for each of the angular points will cumference of a great circle.
Cor. 2. In the same manner it may (5.Cor. 2.) The surface of the sphere may be divided into eight such triangles, by spherical polygon are together less than dividing the circumference of any great circle into quadrants, and joining the points of division with the poles of the great circle.
SECTION 2.-Of Spherical Triangles.
PROP, 8. The angles which one spherical arc makes with another upon one side of it are either two right angles, or are to the circumference of a great circle. gether equal to two right angles.
This is likewise evident from IV.20. See the Demonstration of Book I.
Scholium. Prop. 2.
By help of this proposition, it may be Cor. 1. If two spherical arcs cut one
shown that the shortest distance of two another, the vertical or opposite angles points on the surface of a sphere, meawill be equal to one another. See the sured over that surface, is the spherical Demonstration of Book I. Prop. 3. Cor. 2. If any number of spherical 10. Scholium.
arc between them. See Book I. prop. arcs meet in the same point, the sum of all the angles about that point will
PROP. 10. be equal to four right angles.
The three angles of a spherical triPROP. 9.
angle are together greater than two Any two sides of a spherical triangle right angles, and less than six right are together greater than the third angles. side; and any side of a spherical tri- For the arcs which measure the three angle is greater than the difference of angles together with the three sides of the the other two.
polar triangle are equal to three semiLet A B C be a spherical triangle; circumferences (7.), or six quadrants : the sides B A and AC shall be to- therefore, the former alone are less than gether greater than BC; and A B alone six quadrants, and consequently the shall be greater than the difference of angles which they measure are less than AC and BC. Take O
six right angles. Again, the sides of the centre of the sphere,
the polar triangle are less than a whole and join O A, O B, O C.
circumference, or four quadrants (9 Cor. Then, because the solid
1.): therefore, the arcs before mentionangle at O is contained
ed are greater than two quadrants, and plane angles
consequently the angles which they AOB, A OC, and BOC,
measure greater than two right angles. the two A O B and A O C
Therefore, &c. are together greater than
Cor. 1. A spherical triangle may have the third BOC (IV. 19.).
two or even three right angles, or two
or even three obtuse angles. For, it is point B to the arc BC, TB is equal evident from the demonstration of the to TC (III. 2. Cor. 3.). Therefore, the proposition, that the sum of the angles triangles SBT and SCT have the depends upon the magnitude of the three sides of the one equal to the three sides of the polar triangle, and since sides of the other, each to each, and the sum of these last may be any what- consequently the angle S BT is equal to ever less than four quadrants, the sum the angle SCT(1.7.). Therefore, also, of the angles of the original triangle the spherical angle ABC is equal to may be any whatever greater than two, the spherical angle ACB. and less than six right angles.
Next, let the angle A B C be equal to Cor. 2. If one side of a spherical tri- the angle A C B: the side A B shall be angle be produced, the exterior angle will equal to the side AC. For, if the polar be less than the sum of the two interior triangle A' B' C' be described, its sides and opposite angles.
A' B' and A' C' which are supplements For the exterior an.
to the measures of the equal angles (7.) gle, together with its
will be equal; and, therefore, by the adjacent interior an- B
former part of the proposition, the sphegle, is only equal to
rical angle at C' is equal to the spherical two right angles (8.) ;
angle at B'. But the sides A B and but the two interior and opposite angles, AC are supplements to the measures together with the same angle, are greater of these angles (7.). Therefore, also, than two right angles.
A B is equal to AC.
PROP. 12. equal to one another, the opposite angles If one angle of a spherical triangle shall be likewise equal ; and conversely. be greater than another, the opposite Let A B C be a spheri
side shall likewise be greater than the cal triangle, having the
side opposite to that other; and conside AB equal to the
versely. side AC; the angle
See the demonstration of Book I. ACB shall likewise be equal to the angle A B C.
Cor. If one side B C of a spherical Take O the centre of the sphere, and join O A,
triangle A B C be produced to D, the OB, O C. From the point C, in the plane
T À OC, draw CS at right
BS angles to CO (and, therefore (III. 2.), touching the arc C A in C) to meet O A produced in S: at the points B and C exterior angle A C D shall be equal to, draw BT and CT, touching the arc or less than, or greater than, the interior BC, and meeting one another in T, and opposite angle A B C, according as and join BS, ST. Then, because the the sum of the two sides A B, AC is arc AB is equal to AC, the angle AOB equal to, or greater than, or less than, is equal to the angle AOC (III. 12.); the semicircumference of a great circle. and, because the triangles S O B, S O'ú For, if B A and B C be produced to have two sides of the one equal to two meet one another in D, the angles at B sides of the other, each to each, and and D will be equal to one another, the angles SOB, SOC which are in- having for their common measure the cluded by those sides equal to one measure of the same dihedral angle another (1. 4.), the base SB is equal to (def. 7.) ; and B A D will be a semicirthe base SC, and the angle S B O to the cumference. But, by the proposition, angle SCO, that is, to a right angle. the angle A C D is equal to, or less than, Therefore, BS touches the arc A B in or greater than the angle at D, accordB (111. 2.). And, because the spherical ing as A C is equal to, or greater than, angles ABC, ACB are measured by or less than AD. Therefore, the angle the plane angles of the tangents at ACD is equal to, or greater than, or and C (see def. 7. note) they are mea
less than the angle at B, according as sured by the angles SBT, SCTre
A B and A C are together equal to, or spectively. But, because TB and TC greater than, or less than a semicircumare tangents drawn from the same ference,
angle A QP to the angle AlQP (I. 4.); If two spherical triangles have two and in the same manner it may be shown sides of the one equal to two sides of the that A B is equal to A'B, and AC to other, each to each, and likewise the in- A'C. Now, because A P is perpendicluded angles equal; their other angles cular to the plane O B C, and that PQ shall be equal, each to each, viz. those is perpendicular to the line O B in that to which the equal sides are opposite, plane, A Q is likewise perpendicular to and the base, or third side, of the one
OB (IV. 4.); and for the like reason shall be equal to the base, or third side, A'Q is perpendicular to the same O B. of the other.
Therefore, the angles AQ P, A'Q P meaThere are here two cases for consi- sure the dihedral angles formed by the deration; first, that in which the equal planes 0 A B, O B C, and 0 A' B,
OBC sides A B, A Ć and DE, D F lie in the (IV. 17.), or, which is the same thing, same direction; and, secondly, that in def. 7.) the spherical angles A B C and which they lie in opposite directions. A'BC; and because, as has been already The first case may be demonstrated by demonstrated, the angle A Q P is equal superposition, after the same manner as
to the angle A'Q P, the spherical angle Book I. Prop. 4, to which, for brevity's ABC is equal to the spherical angle sake, the reader is referred: the second A' B C. In the same manner, it may be case as follows:
shown that the spherical angles ACB Let ABC, DEF be two spherical and A' C B are equal to one another.
And because the straight line AB is
AC is equal to the arc A'C. ThereB
fore, in the isosceles spherical triangle B A A', the angle B A'A is equal to the angle B A A', and in the isosceles sphe
rical triangle C A A' the angle CAA is A
equal to the angle CAA (11.); and, triangles, which have the two sides AB, consequently, the whole (or, if the points AC equal to the two sides DE, DF, B, C, are on the same side of the arc each to each, viz. A B to D E and A Ć A A', the remaining) angle B A'C is to DF, but DE lying in a direction from equal to the whole or remaining angle DF, which is the reverse of that in which BAC. Therefore, the triangles A' B C, AB lies from AC, and let them likewise ABC have their several sides and angles have the angle B A C equal to the angle equal to one another, but lying in a EDF: their other angles shall be equal,
reverse order. each to each, viz, A B C to DEF; and
Now, because A' B and D E are each ACB to DFE, and the base B C shall of them equal to A B, they are equal to be equal to the base E F.
one another; and, for the like reason, Take O the centre of the sphere: from A'C is equal to DF, and the angle A draw A P perpendicular to the plane BA'C to the angle EDF. Also, the OBC, and produce it to meet the sur- equal parts lie in the same direction face of the sphere in A'. Join PO,PB, from one another in these two triangles, PC, and OB; from P draw P Q per
A'BC, DEF. Therefore, by the first pendicular to OB ; join A Q, AB, AC, case, the base B C is equal to the base A'Q,* A' B,
angles A' BC, A'C B to cal arcs A'B, AC, A'A. Then, be- the angles D E F and D F E respeccause
in the right-angled triangles APO, tively. And, because the angles A B C, A'PO, the hypotenuse A O is equal to ACB are equal to the angles A'BC, the hypotenuse A' O, and the side PO ACB, each to each, the former angles common to both, the remaining sides are likewise equal to D E F and D FE AP and A' P are equal to one another respectively, (I. 13.): and because in the right-angled
Therefore, &c. triangles A PQ, A'PQ, the side A P is
PROP. 14. equal to the side A' P, and the side PQ common to both, the hypotenuse A Q is If two spherical triangles have two equal to the hypotenuse A' Q, and the angles of the one equal to two angles of
the other, each to each, and likewise the * A' Q is omitted in the figure.
interjacent sides equal ; their other sides