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D E FIN IT ION S.
I. LANE Trigonometry is the Artwhereby, having given any three Parts of a plane
Triangle (except the three Angles) the rest are determined. In order to which, it is not onl
E \ Ż Note, The Degrees, Minutes, Seconds, &c. contained in any Arch, or Angle, are wrote in this Mammer, 50° 18' 35", which signifies that the given Arch, or Angle, confains 52 Degrees, 18 Minutes, and 35 Seconds.
4. The Difference of any Arch from 90° (or a Quadrant) is call’d its Complement; and its Difference from 180° (or a Semicircle) its SuppleIIlent, 5. A Chord, or Subtense, is a Right-line drawn from one Extremity of an Arch to the other: Thus the Right-line BE is the Chord, or Subtense, of the Arch BAE or BDE.
6. The Sine, or Right-sine, of an Arch, is a Right-line drawn from one Extremity of the Arch, perpendicular to the Diameter passing thro’ the other Extremity. Thus BF is the Sine of the Arch AB or DB.
7. The Versed Sine of an Arch is the Part of the Diameter intercepted between the Sine and the Periphery. Thus AF is the Versed Sine of AB; and DF of DB.
* > . 8. The
8. The Co-sine of an Arch is the Part of th Diameter intercepted between the Center and Sine;
and is equal to the Sine of the Complement of
that Arch. Thus C F is the Co-sine of the Arch AB, and is equal to BI, the Sine of its Complement HB.
9. The Tangent of an Arch is a Right-line
touching the Circle in one Extremity of that Arch,
produced from thence till it meets a Right-line
Thus AG is the Tangent of the Arch AB. Io. The Secant of an Arch is a Right-line
reaching, without the Circle, from the Center to the
Extremity of the Tangent. Thus CG is the Se
cant of AB. 11. The Co-tangent, and Co-secant, of an Arch
are the Tangent, and Secant, of the Complement
of that Arch. Thus HK and CK are the Co
tangent and Co-secant of AB. - 12. A Trigonometrical-Canon is a Table exhibiting the Length of the Sine, Tangent, and Secant, to every Degree and Minute of the Quadrant, with respećt to the Radius; which is supposed Unity, and conceived to be divided into 1oooo, or more, Decimal Parts. By the Help of this Table, and the Dočtrine of similar Triangles, the whole Business of Trigonometry is performed; which I shall now proceed to shew. But, first of all, it will be proper to observe, that the Sine of any Arch Ab greater than 90°, is equal to the Sine of
..another Arch AB as much below 90°; and that, its
cant of its Supplement AB; but only are negative,
or fall on contrary Sides of the Points C and A,
from whence they have their Origin: All which is manifest from the Definitions.
R 2 THE
In any right-angled plane Triangle ABC, it will be as the Hypothenuse is to the p. so is the Radias (of the Table) to the Sine of the Angle at the Base.
In any right-angled plane Triangle ABC, it will be, as the Base A B is to the Perpendicular BC, so is the Radius (of the Table) to the Tangent of the Angle at the Base.
For, let AE or AF be the Radius of the Table, or Canon (see the preceding Figure), and FG the
Tangent of the Angle A, or Arch EF (Wid. Def. 3. and 9.); then, by reason of the Similarity of the Triangles ABC, AFG, it will be, AB : BC :: AF : FG. Q. E. D.
... Note, In the Quotations where you meet with two Numbers (as 5. 4.) without any mention of Prop. Theer. &c. Reference is made to the Elements of Geometry published by the same Authur; to which this little Trač is designed as an Appendix. Th - - - - - UIS
In every plane Triangle ABC, it will be, as any one Side is to the Sine of its opposite Angle, so is any other Side to the Sine of its opposite Angle.
For take CF = To
AB, and upon AC
let fall the Perpen- F. diculars B D and FE; which will be the Sines of the . Angles A and C to the equal Radii ATD E C
AB and CF. Now
As the Base of any plane Triangle ABC, is to the Sum of the two Sides, so is the Difference of the Sides to twice the Distance DE of the Perpendicular from the Middle of the Base,