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HAving done with plain triangles, we come next to speak of Sphe∣rical.
1 A Spherical triangle is that which is described on the sur-face of the Sphere.
2 The sides of a Spherical Triangle are the arches of three great circles of the Sphere, mutually intersecting each other.
3 The measures of Spherical an∣gles are the arches of great circles de∣scribed from the angular point be∣tween the sides of the angles, those sides being continued to quadrants.
4 Those are said to be great circles which bi-sect the Sphere.
5 Those circles which cut each o∣ther at right angles, the one of them passeth through the poles of the other and the contrary.
6 In every Spherical triangle, each side is less then a semicircle, for if in the triangle ABC, you produce the
or BC is less then a semicircle. In like manner if the sides AB and AC be produced, the side AC may be also proved to be less then a Semicircle.
7 In every Spherical Triang. any two sides are together greater then the third, for otherwise they cannot possi∣bly make a triangle.
8 The sum of the sides of a Sphe∣rical Triangle are less then two se∣micircles. For if any two sides be pro∣duced as suppose AB, BC till they concur in the point D, the arches BAD, BCD shall be each of them a semicircle, but in the train. ADC, the sides AD and CD are together grea∣ter then AC by the last aforegoing, therefore the three sides AB, BC, AC are together less then the two semi∣circles BAD, BCD,
9 If two sides of a Spherical tri∣angle be equal to a semicircle, the two angles at the base shall be equal to two right. If they be less then a se∣micircle, the two angles shall be less
then two right: but if greater then a semicircle, the two angles shall be greater then two right. As in the sphe∣rical Triangle ABC let the sides AB AC be equal to a Semicircle, ther•• are the angles at B & C equal to two right, for the arch BC 〈◊〉〈◊〉 produced to D, the an∣gle ACB shall be equal to B, & seeing that the two angles at C are equal to two right, the two
Again. Let the sides AB and AC be less then a semicircle, seeing that the two angles at C are equal to two right, and the angle B less then the angle ACD, the angles ACB and B are together less then two right.
Lastly, Let the sides AB and AC be more then a semicircle, the angles at C being equal to a semicircle, and the angle at B greater then the an∣gle
ACD, the angles ACB & B shall be greater then two right.
10 The sum of the three angles of a Spherical triangle are greater then two right angles and less then six.
Demonst. In the triangle ABC let the side BC be produced to D, then shall the angle ACD be either more or less or e∣qual to the angle ABC; first, sup∣pose them equal, then the arches AB and AC
Again. Let the angle ACD be less then the angle B, then the sum of the arches AB & AC shall be more then a Semicircle, and therefore the angles ABC and ACB greater also then two
right, and therefore much more are the three angles A.B.C. greater then two right.
Lastly. Let the angle ACD be greater then the angle ABC, and make the angle DCE equal thereto, and the side AB being produced to E that the arch BE and CE may meet, and let the arch CA be produced to F, then shall the arches EB and EC be together equal to a semicircle, and therefore AE and EC are together less then a semicircle, and the angle EAF or BAC is greater then the angle ACE by the ninth hereof, but the angles ACE, ACB and B are equal to two right, therefore the an∣gle ACB, ABC & BAC are grea∣ter then two right.
And because every angle of a Sphe∣rical Triangle is less then two right, the three angles together must needs be less then six, as was to be proved. Therefore,
11 Two angles of any Spherical Triangle are greater then the diffe∣rence
between the third angle and a semicircle also.
12 Any side being continued, the exteriour angle is less then the two interiour opposite ones.
13 In any Spherical Triangle, the difference of the sum of two angles, and a whole circle is greater then the difference of the third angle, and a semicircle.
14 In any Spherical Triangle, one side being produced, if the other two sides be equal to a semicircle, the out∣ward angle shall be equal to the in∣ward opposite angle upon the side produced: if they be less then a se∣micircle, the outward angle shall be greater then the inward opposite an∣gle: if greater then a semicircle, the outward angle shall be less then the inward opposite angle.
In the Spherical Triangle ABC let the side BC be continued to D, and let the sides AB and AC be together equal to a semicircle. I say then that the outward angle ACD is equal to
the inward opposite angle at B, be∣cause
Again, Let the sides AB and AC be less then the semicircle BAD if the common arch AB be taken away, there shall remain the arch AC less then the arch AD and therefore the ang. ACD shall be greater then the angle D, therefore also more then B.
Lastly, If the sides AB and AC be together more then a semicircle, taking away the common arch AB, the remaining arch AC shall be grea∣ter then AD, and the angle ACD lesser then D, and therefore also lesser then B as was to be proved.
15 A Spherical Triangle is either right or angled or oblique.
16 A right angled Spherical Tri∣angle is that which hath on right an∣gle at the least.
17 The legs of a right angled sphe∣rical Triangle are of the same affection with their opposite angles.
In the spherical Triangle ABC right angled at A let the side AB be a quadrant, I say then that the an∣gle ACB shall be a quadrant also, be∣cause B is then the pole of the arch AC, and the arch BC perpendicu∣lar
the angle ACE shall be also less then a quadrant, it being less then the right angle ACB.
18 In a right angled spherical Tri∣angle, if either leg be a quadrant, the Hypotenusa shall be also a quadrant; but if both the legs shall be of the same affection, the Hypotenuse shall be less then a quadrant; if of different, then greater, and the contrary.
In the right angled triangle ABC right angled at A let the side AB be a quadrant, I say then that the Hypo∣tenuse BC is also a quadrant, because the angle ACB is right, by the last a∣fore-going, and the arches AB and BC which are perpendicular to the arch AC doe meet in the pole B.
Again, Let the sides AB and AC be continued to their opposite pole at F, then shall the Triangle FBC be equal to the triangle▪ ABC, but the arch GH being drawn by the points G and H, the base GH will be common to the right angled triangle
AHG, whose legs AG and AH are greater then the quadrants AB & AC and also to the other right angled tri∣angle FGH whose legs FG and FH are less then the quadrants FB & FC, and GH is less then the quadrant BC, which is the measure of the right an∣gles
Lastly, In the triangle DAH right angled at A, the leg AD is less then the quadrant AB, and the leg HA
is greater then the quadrant AC, therefore the Hypotenusa DH is also greater then a quadrant, for AC and DC are each of them quadrants by the work, if therefore upon the pole D you describe the arch CI it will cut the Hypotenuse DH in the point I, and therefore DI is a qua∣drant and DH more then a quadrant, as was to be proved.
19 In a right angled spherical tri∣angle, if either of the angles at the Hypotenusa be a right angle, the hy∣potenusa shall be also a quadrant, but if both shall be of the same affection it shall be less, if of different then greater and the contrary.
In the triangle ABC right angled at C if either of the angles at A or B be right, the side opposite thereto shal be also right by the 17 hereof, and the Hypotenusa AB shall be a quadrant by the last aforegoing, but if the an∣gles at A aud B be both acute or ob∣tuse the sides AC and CB shall be both acute or obtuse also by the 17th
hereof, and the Hypotenuse AB less then the quadrant by the last afore∣going: but if either of the angles at A & B be acute, and the other obtuse, one of the legs shall be less, the other more then a quadrant, by the 17th hereof, and the Hypotenuse AB more then a quadrant by the last afore∣going, as was to be proved. Therefore
20 In a right angled spherical Tri∣angle either of the Oblique angles is greater then the complement of the other, but less then the difference of the same complement to a semicircle.
21 An Oblique angled spherical Triangle, is either acute or obtuse.
22 An acute angled spherical Tri∣angle hath all its angles acute.
23 An obtuse angled spherical Triangle hath all its angles either ob∣tuse or mixt, viz. acute and obtuse.
24 In any Spherical Triangle, whose angles are all acute, each side is less then a quadrant.
In the Triangle ABC let all the angles be acute, and A the greatest
angle, then is BC greater then either of the other arches AB, AC, and the arch A E being drawn at right angles to the arch AB and made equal to AC, the arch BE is less then a quadrant, because the legs of the right angled Triangle ABE, viz. AE
25 In any oblique angled spheri∣cal Triangle, if the angles at the base be of the same affection, the perpen∣dicular
drawn from the vertical an∣gle shall fall within, if of different without.
In the oblique angled Triangle ABC whose angles at B & C are acute, the perpendicular AD shall fall within the triangle, for if it fall not within, it must be the same with one of the sides, or els fall without the Triangle if it be the same with either side, the angle at B or C must be right, which is contrary to the proposition, if it fall without the triangle, suppose at E the angle AEB shall be right,
and therefore the side AE is greater then a quadrant, by the 17th thereof, and the angle ACE being acute AE shall be also less then a quadrant by the same Theorem, but that the same side should be both more and less then a quadrant is absurd in this case there∣fore the perpendicular shall fall with∣in the triangle.
But in the Triangle AEB, obtuse-angled at B, acute at E, the perpendi∣cular AD shall fall without the tri∣angle, upon the side EB continued, or if otherwise, it must be the same with one of the sides, or fall within the triangle, it cannot be the same with either of the sides, for then the angle at B or E should be a right angle, and cannot fall within the triangle be∣cause then the angles at B and E must either be both acute or both obtuse as hath been already proved, if therefore the angles at the base be of different affection, the perpendicular shall fall without, as was to be proved.