Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

SECT. IV. Of Spherical Rectangled Triangles.

And here first it will be necessary also to understand those few general Maxims or Rules, that are of special Moment, in the Doctrine of Spherical Triangles.

1. THat a Spherical Triangle is comprehen∣ded and formed, by the Conjunction and Intersection of three Arches of a Circle, described on the Surface of the Sphere or Globe.

2. That those Spherical Triangles, consisteth of six distinct parts, viz. three Sides and three Angles, any of which being known, the other is also found out and known.

3. That the three Sides of a Spherical Trian∣gle, are parts or Arches of three great Circles of a Sphere, mutually intersection each other: and as plain or Right-lined Triangles, are mea∣sured by a Measure, or Scale of equal parts: So these are measured, by a Scale or Arch of equal Deg••ees.

4. That a Great Circle is such a Circle that doth bessect the Sphere, dividing it into two equal parts; as the Equinoctial, the Ecliptick, the Meridians, the Horizon, &c.

5. That in a Right-angled Spherical Triangle, the Side subtending the Right-angle we call the Hypothenuse, the other two containing the Right∣angle we may simply call the Sides, and for distinction either of them may be called the Base or Perpendicular.

6. That the Summ of the Sides of a Spherical Triangle are less than two Semicircles or 360°.

7. That if two Sides of a Spherical Triangle be equal to a Semicircle; then the two Angles at the Base shall be equal to two Right-angles; but if they be less, then the two Angles shall be less; but if greater, then shall the two Angles be greater than a Semicircle.

8. That the Summ of the Angles of a Spheri∣cal Triangle, is greater than two Right-angles.

9. That every spherical Triangle is either a Right, or Oblique-angled Triangle.

10. That the Sines of the Angles, are in pro∣portion, unto the Sines of their opposite Sides; and the Sines of their opposite Sides, are in proportion unto the Sines of their opposite Angles.

11. That in a Right-angled Spherical Triangle, either of the Oblique-angles, is greater than the Complement of the other, but less than the Diffe∣rence of the same Complement unto a Semicircle.

12. That a Perpendicular is part of the Arch of a great Circle, which, being let fall from any Angle of a spherical Triangle, cutteth the oppo∣site Side of the Triangle at Right-angles, and so

divideth the Triangle into two Right-angled Tri∣angles, and these two parts (either of the Sides or Angles) so divided must be sometimes added together, and sometimes substracted from each other, according as the Perpendicular falls with∣in or without the Triangle.

PROP. I. Case 1. A Side and an Angle adjacent thereunto being gi∣ven, to find the other Side.

In the Triangle ABC, there is given the Side AB 27° 54'; and the Angle A 23° 30', and the Side BC is required, to find which this is the Analogy or Proportion.* 1.1* 1.2

PROP. II. Case 2. A Side and an Angle adjacent thereunto being gi∣ven, to find the other Oblique-angle.

In the Triangle ABC, there is given the Side AB 27° 54', and the Angle A 23° 30',* 1.3 and the Angle at C is required, to find which say by this Analogy or Proportion.

As the Radius or S 90° 00',

To Sc. of cr. AB 27, 54.

So is S. V. at A 23, 30,

To Sc. V. at c 69, 22 required.

PROP. III. Case 3. A Side and an Angle adjacent thereunto being given, to find the Hypothenuse.

In the Triangle ABC, there is given the Side* 1.4 AB 27° 54', and the Angle at A 23° 30', and the Hypothenuse AC, is required; which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. of V. at A 23, 30.

So is Tc cr. AB, 27, 54.

To Tc. Hypothenuse AC, 30, 00 required.

PROP. IV. Case 4. A Side and an Angle opposite thereunto being given, to find the other Oblique-angle.

In the Triangle ABC, there is given the Side* 1.5 BC 11° 30', and the Angle A 23° 30', and the Angle C is required, to find which, say by this Analogy or Proportion.

As Sc. cr. BC, 11° 30',

To Radius or S. 90, 00.

So is Sc. V. at A, 23, 30,

To S. V. at C. 69, 22, as required.

PROP. V. Case 5. A Side and the opposite Angle given, to find the Hypothenuse.

In the Triangle ABC, there is given the* 1.6 side BC 11° 30', and the Angle at A 23° 30', and the Hypothenuse AC, is required, which may be found by this Analogy or Proportion.

As S. V. at A 23° 30',

To Radius or S. 90, 00.

So is Ser. BC 11. 30,

To S. Hypothenuse AC 30, 00. as required.

PROP. VI. Case 6. A side and the opposite Angle given, to find the other side.

In the Triangle ABC, there is given the side BC 11° 30', and the Angle at A 23° 30', and the side AB is required, to find which this is the Analogy or Proportion.

As Radius or S 90° 00',* 1.7

To Tc. of V. at A. 23. 30,

So is T. cr. BC 11, 30,

To S. of cr. AB 27. 54 as was required.

PROP. VII. Case 7. The Hypothenuse, and an Oblique Angle given, to find the side adjacent thereunto.

In the Triangle ABC, there is given the

Hypothenuse AC, 30° 00', and the Angle A 23° 30', and the side AB, is required, which is found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. V. at A, 23, 30.* 1.8

So is T. Hypoth. AC, 30, 00,

To T. cr. AB, 27, 54, as was required.

PROP. VIII. Case 8. The Hypothenuse, and an Oblique-angle given, to find the opposite Side.

In the Triangle ABC, there is given the Hypothenuse AC, 30° 00', and the Angle at A 23° 30', and the Side BC, is required, which is found by this Analogy or Proportion.* 1.9

As the Radius or S. 90° 00',

To S. Hypoth. AC, 30, 00.

So is S. V. at A, 23, 30,

To the S. cr. BC, 11, 30. which was required.

PROP. IX. Case 9. The Hypothenuse, and an Oblique-angle given, to find the other Oblique-angle.

In the Triangle ABC, there is given the Hy∣pothenuse AC 30° 00', and the Angle A, 23° 30', now the Angle at C, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To Sc. Hypoth. AC, 30, 00.

So is T. of V. at A, 23, 30,* 1.10

To Tc. of V. at C. 69, 22, as was required.

PROP. X. Case 10. The sides given, to find the Hypothenuse.

In the Triangle ABC, there is given the side AB 27° 54', and the side BC 11° 30', and the Hypothenuse AC is required, to find which say by this Analogy or Proportion.

As the Radius or S. 90° 00',* 1.11

To Sc. cr. BC. 11, 30.

So is Sc. cr. AB 27, 54,

To Sc. Hypothenuse AC 30, 00. required.

PROP. XI. Case 11. The sides given, to find an Angle.

In the Triangle ABC, there is given, the side AB 27° 54', and the side BC 11° 30', and the Angle at A, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',* 1.12

To S. cr. AB. 27, 54.

So is Tc. cr. BC. 11, 30,

To Tc. of V. at A. 23. 30. as required.

PROP. XII. Case 12. The Hypothenuse, and a side given, to find the o∣ther side.

In the Triangle ABC, there is given, the Hy∣pothenuse* 1.13 AC 30° 00', and the side AB 27° 54' and the side BC is required, which may be

found by this Analogy or Proportion.

As Sc. cr. AB. 27° 54',

To Radius or S. 90 00.

So is Sc. Hypothenuse AC. 30° 00',

To Sc. cr. BC. 11° 30' as required.

PROP. XIII. Case 13. The Hypothenuse, and a Side given, to find the contained Angle.

In the Triangle ABC, there is given the Hy∣pothenuse* 1.14 AC 30° 00', and the side AB 27° 54', and the Angle at A is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',

To T. cr. AB. 27° 54'

So is Tc. Hypoth. AC 30° 00',

To Sc. of V. at A, 23° 30', as required.

PROP. XIV. Case 14. The Hypothenuse, and a Side given, to find the oppo∣site Angle.

In the Triangle ABC, there is given the Hy∣pothenuse AC 30° 00', and the side AB 27° 54', ••ow the Angle C, is required, which may be* 1.15 〈…〉〈…〉ound by this Analogy or Proportion.

As the S. Hypoth. C, 30° 00',

To Radius or S. 90° 00'.

So is S. of cr. AB, 27° 54',

To S of V. at C. 69 22, as required.

PROP. XV. Case 15. The Oblique Angles given, to find either Side.

In the Triangle ABC, there is given the An∣gle A 23° 30', and the Angle at C 69° 22', and the side BC, is required, which may be found by this Analogy or Proportion.

As the S. of V. at C, 69° 22',* 1.16

To the Radius or S. 90° 00'.

So is the Sc. of V. at A, 23° 30',

To the Sc. of cr. BC, 11° 30', as required.

PROP. XVI. Case 16. The Oblique-angles given, to find the Hypothenuse.

In the Triangle ABC, there is given the An∣gle A 23° 30', the Angle C, 69° 22', and the Hypothenuse AC, is required, which may be found by this Analogy or Proportion.

As the Radius or S. 90° 00',* 1.17

To Tc. of V. at C. 69°, 22',

So is Tc. of V. at A, 23 30,

To Sc. Hypoth. AC, 30 00, as required.

Notes

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  Fig. 38.