Geometrical trigonometry, or, The explanation of such geometrical problems as are most useful & necessary, either for the construction of the canons of triangles, or for the solution of them together with the proportions themselves suteable unto every case both in plain and spherical triangles ... / by J. Newton ...

Example.

Let DK the sine of DE 27 d. be 45399.04997

And BM the sine of EB 33 d. be 54463.90350

Their sum is BH = AN sine of 87d. 99862.95347

27 The Canon of sines being thus made, a Table of Tangents and Secants may be easily deduced from them, by the following problemes.

28 As the co-sine of an arch, is to the sine thereof, so is Radius, to the Tangent of that arch.

Demonst. In the annexed diagram, the Triangles AEF and AHG are like, because of their right angles at F and G, & their common angle at A. Therefore, AF. FE ∷ AG. GH.

[illustration]

29 The Radius is a meane proportional, be∣tween the tangent and the tangent complement of an arch.

Dem. HG is the tangent of an arch, CK the co-tang. thereof & LH = AG and the triangles, ALH and ACK are like, because of their right an∣gles at L and C, and their common angle at A. Therefore AL = HG. LH ∷ AC. CK.

30 The Radius is a meane proportional, be∣tween the right sine of an arch, and the secant of its complement.

Demonst. In the preceding diagram the triangles AEF & AHG are like, therefore, AF. AE ∷ AG. AH.

31 As the sine of an arch or angle is to Rad. so is the tangent of the same arch, to the secant thereof.

Demonst. In the preceding diagram

the triangles AEF & AGH are like, therefore EF. AE ∷ HG. AH.

32 As Radius, is to the secant of an arch, so is the co-tangent of the same arch, to the co-secant thereof.

Demonst. In the preceding diagram, the triangles ALH and ACK are like, therefore LH, AH ∷ CK. AK.

Other more easie and expedi∣tious wayes of making the Tangents and Secants, you may see in the first Chap. of my Trigonometria Britan∣nica, but the Canons being now al∣ready made, these Rules we deeme sufficient.

The construction of the Artificial Sines and Tangents, we have purpose∣ly omitted, they being nothing els but the Logarithmes of the Natural, of which Logarithmes we have shew∣ed the construction in a former Insti∣tution, by the extraction of roots, and in my Trigonometria Britannica by multiplication: and therefore shall now proceed to the use of the Canon of Sines, Tangents and Secants, in

the solution of all Triangles, whether plaine or Spherical.