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.

PROP. IX. Case 9. Two Sides and their Included Angle being known, to find the third Side.

In the Triangle APZ, there is given the Side* 1.1 ZP 38° 30', the Side PA 70°, and the Angle P, let be 31° 34, and the Side AZ is required.

The Resolution of this Case depends on the Catholike proposition of the Lord of Marchiston, by supposing the Oblique-Triangle to be divided (by a supposed Perpendicular falling either within or without the Triangle) into two Rect∣angulars.

Now in the Triangle AZP, let fall the Per∣perpendicular ZR; so is the Triangle AZP divi∣ded into two Rectangulars ARZ and ZRP. Now the Side AZ may be found at two Opera∣tions thus: say,

As the Radius or S. of 90° 00'

To Sc. of the included V, P. 31 34.

So is T. of the lesser Side PZ. 38 30,

To T. of a fourth Arch. 34 08.

If the contained Angle be less than 90°, take this fourth Arch from the greater Side; but if it be greater than 90°, from its Complement unto 180°, the Remainder is the Residual Arch: Now again say,

As Sc. of the fourth Arch. 34° 08'

To Sc. Residual Arch. 35 52

So Sc. of the lesser Side PZ. 38 30

To Sc. AZ the Side required. 40 00

☞ 1.2 But note that many times the Perpendicu∣lar will fall without the Triangle, as it doth* 1.3 now within; in such case the Sides of the Tri∣angle must be continued, so will there be two Rectangulars, the one included within the o∣ther: as in the Triangle HIK, the Perpendicular let fall is KM, falling on the Side HE, and so the two Rectangulars found thereby will be IM K, and KMH, and so by the directions in the former proposition find out the Side IK, if re∣quired to be found.

Notes

• * 1.1

  Fig. 40.
• ☞ 1.2

  Note that if the Angles at the Base be both of one kind, that then the Perpendicular falls within the Triangle: if of diverse kinds, without the Triangle.
• * 1.3

  Fig. 41.