The mariners magazine, or, Sturmy's mathematical and practical arts containing the description and use of the scale of scales, it being a mathematical ruler, that resolves most mathematical conclusions, and likewise the making and use of the crostaff, quadrant, and the quadrat, nocturnals, and other most useful instruments for all artists and navigators : the art of navigation, resolved geometrically, instrumentally, and by calculation, and by that late excellent invention of logarithms, in the three principal kinds of sailing : with new tables of the longitude and latitude of the most eminent places ... : together with a discourse of the practick part of navigation ..., a new way of surveying land ..., the art of gauging all sorts of vessels ..., the art of dialling by a gnomical scale ... : whereunto is annexed, an abridgment of the penalties and forfeitures, by acts of parliaments appointed, relating to the customs and navigation : also a compendium of fortification, both geometrically and instrumentally / by Capt. Samuel Sturmy.

SECT. III. By the way of your Ship, and any 2 Angles of Position, to find the Distance of any Island, Cape, or Head-Land from you.

YOu have been shewed how to do it with a right-Angle of 45 degr. already; but with a little more trouble, you shall learn to do it by any 2 Angles what∣soever.

As for Example.

Suppose you were Sailing full South from A towards B, and from A should espy Land at C bearing 2 Points from you to the Westward, as S S W, or S W 22 deg. 30 min. and Sailing still upon your Course until you come to B, you observe the Place bears from you just 4 Points, or S W 45 degr. which is the double of the Angle observed at A. If in this manner you double any Angle; that is, let your first Angle be what it will, you must Sail until you have doubled that Number; then you may assure your self that the distance you have Sailed between A and B, is justly equal to the distance between B and C, B being the second Place where you made your last Observation, and C being the Place observed. So that if A B be 12 miles, B C is likewise 12 miles; and this you may do without further trouble or Calculation, and may lay it down by your Plain-Scale, as I have done this following Figure.

In all such Questions remember that the Angles at the second place of Observation, shall be either just the double, if you go nearer to the Place, or else just the half if you go further off than the Angle at the first place.* 1.1 Therefore the first Angle you may take at Random, no matter what it is, so you be careful to observe when you be just upon the double, or the half; so that by Calculation you may resolve it almost with as little trouble as a Right-Angle, which is made plain thus.* 1.2 In the Triangle ABC the acute Angle being the outward at B, being 45 degr. the obtuse or inward-Angle being the Complement thereof to 180 degr. must be 135 degr. and the Angle at A being 22 degrees 30 min. being added to this, makes 157 degr. 30 min. which Substracted from 180 degr. there must needs rest for the Angle at C 22 degr. 30 min. Now this Angle at C being equal to the Angle at A 22 degr. ½; therefore the side A B opposite to the one Angle, must needs be equal to the side B C opposite to the other Angle, as you see by this Case.* 1.3

As the Sine of the Angle A C B 22 degr. 30 min.	958283
to the distance Sailed 12 mile AB	107918
So is the Sine of the Angle CAB 22 degr. 30 min.	958283
to the distance B C 12 miles.	1066201
To find the distance A C	As the Sine of A C B 22 deg. 30 min.	958283	107918
is to A B 12 miles.	107918
So is the Co-sine of 135 d. which is 45 d.	984948
to the distance from the first Place	1092866
of Observation AC 22 miles. 17/199 parts.	134583

Notes

• * 1.1

  See Figure 100 in the follow∣ing Schemes. ☜
• * 1.2

  180	0
  45	0
  135	0
  22	30
  157	30
  180	00
  d. 22	30
• * 1.3

  And by these Rules you may find all the op∣posite sides.