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.

CHAP. VI. How to find the Distance of a Fort, or Walls of a City, or Castle, that you dare not approach for fear of Gun-Shot; Or the Breadth of a River or Water, that you cannot pass, or Measure over it, made by 2 Stations, with the Quan∣tity of the Angle at each Station.

SUppose from some private place as at A, you espy a Castle, Fort, Tree, or place whatsoever, that you dare not approach for fear of Gun-Shot, Marsh-Grounds, or a River betwixt you, or some other Impediments, that you cannot make your second Station in any open place, but are forc'd to make it in some other secure Place at B; therefore plant your Instrument or Compass at A, and direct the Sights to C and B, take the Quantity of the Angle C A B 46 degr. 00 min. and go to B, and take the Quantity of the Angle A B C 79 degr. 0 min. then measure the distance of the 2 Stations A and B 350 fathoms.

Then by a Plain Scale, or by the Line of Sines on the Scale of Scales, you may presently resolve the distance, as I do by the Tables.* 1.1

As the Sine of 55 degr. 00 A C B	991336
to 350 Fathoms, A B	354406
So is the Sine of 79 degr. ABC	999194
to the distance	1353600
AC 419:40/100	362264
As the Sine of 55 degr. 00 A C B	991336
is to A B 350 Fathoms.	354406
So is the Sine of 46 degr. 0 C A B	985693
to BC 307 30/100 Fathoms, the	1340099
distance required.	348763

SECT. I. How to take the Breadth of a River.

SUppose you were to take the Breadth of a River, as I have at Crocken-Pill, which runs betwixt Glocester-shire and Somerset-shire, and found the breadth of the Water upon a Spring-Tide 40 Perch or a Furlong; you must do it thus. Being on the River∣side as in the former Figure at E, there set your Compass; Observe some mark on the other side of the Water, as at D; then set a mark at E, and go square-wise either to the right-hand, or to the left from these 2 marks, so far, until you spie the mark D on the other side the Water doth justly make an Angle of 45 degr. with the mark E; and this will be when you come to F; then measure carefully F E, the distance of the 2 Stations, and that shall be equal to the breadth of the River: so that if FE be 10:20:30:40:50: or 100 Poles, or Yards, or Feet, the breadth is the same. The like may be done by any other Angle, as if you go to G, and make an Angle of 26 degr. 30 min. in D; then is the distance GE twice the breadth; but ever if you can get an Angle of 45 degr. for that is the best and readiest Angle to find out such a distance; therefore if you can, use no other.

And the like way of Working you may do at Sea, if you gain the Sight of any Cape, Head-Land, or Island, set it by your Compass when you see it, without altering your Course, make an Angle of 45 degr. And by your Plain-Scale if you have kept a good account of your Way by the same Rules as before, you shall have the true distance of your Ship from the first Place, or Cape, or Head-Land, or Island whatsoever: Or you may get the Slope-side D F or D G if you measure it with your Compasses, and apply it to the same Scale of equal parts by which you put down the distance E F or E G. Thus you may find the distance from the Ship, to any Cape; These are made so plain by the Rules before-going, that it need no further precept.

SECT. II. Being upon the Top of a Hill, Tower, Steeple, or a Ships Top-Mast-Head, there observing the Angle of distance from you, To find the true distance thereof.

YOu may do this by your Quadrant; thus. Let the height of the Hill, Tower, Steeple, or Ships Top-Mast-Head be 40 yards, or any other measure: and from it you see an House, Tree, or Place whatsoever, and you desire the distance from you. You have been shewed already to find the height of a Tree, Tower, Hill, or Steeple; by this Rule we will shew you how to stand upon them, and take the distance from any thing else, viz.

Let the height of the Tower, or Mast, or Hill be 40 yards, and let the Angle of distance taken with your Quadrant be 80 degrees, being 10 degrees under the Line of Level; this is the Rule for all such Questions.

As the Tangent Compl. 80 degr. which is 10 d.	924631
is to the height 40 Yards.	260206
So is the Radius	10
to the distance from the Top of the Tower or Hill 226 9/10	235575 yards.

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.* 1.2 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.* 1.3 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.* 1.4

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 99 in the follow∣ing Schemes. ☜

  46 d.	00 m.
  79	00
  125	00
  180	0
  125	0
  55	0
• * 1.2

  See Figure 100 in the follow∣ing Schemes. ☜
• * 1.3

  180	0
  45	0
  135	0
  22	30
  157	30
  180	00
  d. 22	30
• * 1.4

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