V.

The Secant of an Arch is a Right Line drawn from the Centre through the Circumference, by which the Tangent is terminated; wherefore, CT is the Secant of the Arches S A and S D.

VI.

The Sine, Tangent, &c. of the Complement of any Arch is called the Co-Sine, Co-Tangent, &c. of the Arch; Thus HS is the CoSine, and B G the Co-Tangent to the Arch A S.

Note. Sines, Tangents, and Secants, are faid to be fo many Degrees as the Arch contains Parts of 360 Degrees; fo that the Radius, being the Sine of a Quadrant or a Fourth Part of the Circumference, contains 90 Degrees: Thus the Radius is always equal to the Sing of 90 Degrees.

From those Definitions it follows, that the Chord of any Arch and its Supplement to a Circle, is reprefented by the fame Line; wherefore, if the Chord of an Arch less than a Semicircle be given, the Chord of an Arch as much above a Semicircle as the other is deficient, is also given. From Definitions 2d, 4th, and 5th, it follows, that any Arch and its Complement to a Semicircle have one Right Sine, Tangent, and Secant, common to them both. Wherefore, the Right Sine, Tangent, and Secant of an Arch less than a Quadrant, is the fame with the Right Sine, Tangent, and Secant of an Arch as much exceeding a Quadrant as the former is defective; and from Definition the 3d, it follows, that the versed Sine of an Arch greater than a Quadrant, is greater than the Radius; but the verfed Sine of an Arch less than a Quadrant, is less than the Radius ; and the Sum, and Difference of the Radius and Co-Sine of any Arch, fhall be the versed Sine of that Arch, and its Supplement to a Semicircle; for D C + CRDR the verfed Sine of the Arch DS; and A C-CR, is the versed Sine of the Arch A S.

By comparing the fimilar Triangles CR S, and C B G, it appears, that the Co-Sine of an Arch is to the Right Sine, as the Radius is to the Tangent of the fame Arch, and the contrary; CR:RS: AC: CT.

for

And, that the Co-Sine of an Arch is to the Radius, as the Radius is to the Secant of the fame Arch, and the contrary; for R C; SC

AC: CT.

Hence the Radius is a mean Proportional between the Co-Sine and the Secant of the fame Arch, wherefore, the Co-Sines and Secants of Arches are reciprocally proportional.

That the Right Sine of an Arch is to the Radius, as the Tangent of the fame Arch is to its Secant, and the contrary; for RS: SC: AT: T C.

And the Tangent of an Arch is to the Radius, as the Radius is to the Co-Tangent of the fame Arch, and the contrary; for T A: AC::CB: BG.

Hence the Radius is a mean Proportional between the Tangent and Co-Tangent of an Arch; wherefore, the Tangent of Arches and their Co-Tangents are reciprocally proportional.

That the Tangent of an Arch is to the Secant of the same Arch as the Radius is to the Secant of its Complement, and the contrary; for AT: TC::CB: CG.

The Chords, Sines, Tangents, and Secants, or verfed Sines of fimilar Arches, are in Proportion to each other, as the Radii of thofe Circles.

By viewing the Figure, it will appear, that if the longest Side of the Triangle C R S be made the Radius of a Circle, the other Sides will be Sines of the Angles oppofite to them; but if one of the Sides containing the Right Angle be made Radius, the other will be the Tangent of the Angle oppofite to it, and the Hypothenuse, or longest Side, the Secant of the fame Angle.

Having explained the Properties of a Right-angled Triangle at fome Length, it being the Foundation of Plane and Mercator's Sailing, we fhall now proceed to fhew how to determine the feveral Lengths of the Sines, Tangents, and Secants belonging to the feveral Arches, answering to every Degree and Minute of the Quadrant, in Parts of the Radius firft given,, and how to lay them down on the Plane Scale.

THE

THE

FUNDAMENTAL PROJECTION

OF THE LINES OF

Sines, Tangents and Secants, on the Plane Scale.

Ift.

ITH the Radius intend for
you

your Scale defcribe a

W Semicircle A D B C, and upon the Centre C raise the

Perpendicular C D, (which will divide the Semicircle into two Quadrants, A D, B D) continue C D directly to S, and upon B raife the Perpendicular B T, then draw the Right Lines BD and A D.

2dly. Divide the Quadrant B D into 9 equal Parts, then will each of these be 10 Degrees. Again, you may fubdivide each of thefe Parts into fingle Degrees: And thefe again, if your Radius will admit of it, into Minutes, or fome Aliquot Parts of a Degree greater than Minutes.

3dly. Set one Foot of the Compaffes in B, and transfer each of the Divifions in the Quadrant B D to the Right Line BD; then is B D a Line of Chords.

4thly. From the Points .10, 20, 30, &c. in the Quadrant B D, draw Right Lines parallel to CD till they cut the Radius CB, then is the Line CB divided into a Line of Sines, which must be numbered from C towards B.

5thly. If the fame Line of Right Sines be numbered from B towards C, it will become a Line of verfed Sines; which may be continued to 280°, if the fame Divifions be transferred on the other Side of the Centre C.

6thly. From the Centre C, through the feveral Divifions in the Quadrant B D, draw Right Lines till they cut the Tangent B T; fo will the Line BT become a Line of Tangents.

7thly. Setting one Foot of the Compafies in C, extend the other to the feveral Divifions, 10, 20, 30, &c. in the Tangent Line BT, and transfer thefe Extents feverally into the Right Line C S, then. will the Line CS be a Line of Secants.

8thly. Right Lines drawn from A to the feveral Divifions, 10, 20, 30, &c. in the Quadrant B D, will divide the Radius CD into a Line of Semitangents.

9thly. Divide the Quadrant A D into 8 equal Parts, and from A transfer thefe Divifions feverally into the Line AD; then is AD a Line of Rhumbs, each Divifion anfwering to 11° 15' upon the Line of Chords. The Ufe of this Line is for protracting and mea