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.

A COMPENDIUM OF FORTIFICATION.

FIrst, I will describe part of a Tetragon or Square Fort, and explain the Names of the Sides and Angles thereof, as in the Figure following.

[illustration] geometrical diagram

Names of the Sides of a Fort.

• AB the Outward Side of the Polygon, and DE the Inward Side.
• CA the Semidiameter of the Outward Polygon, and CD the Inward.
• IHAGF the Bulwark or Bastion.
• AG, or AH, the Front or Face of the Bastion - 280 Foot.
• AD the Capital or Head Line,
• DI, or DF, the Gorge Line.
• IH, or FG, the Flank; and FM the second Flank.
• DG the Espaule, or Shoulder.
• FK the Curtain - 420 Foot.
• AK the longest Line of Defence - 720 Foot.
• AL the shortest Line of Defence.

Names of the Angles.

• NAB the Angle of the Polygon, and NAC the half Angle.
• HAG the Flanked Angle of the Bulwark.

• AGF the Angle of the Shoulder.
• FDG the Angle forming the Flank, commonly 40 Degrees.
• GAO the Inward Flanking Angle.
• APB the Outward Flanking Angle, and APO half the same.
• Note, That the Bastion or Flanked Angle HAG must never be less than 60 Degrees, neither above 90 Degrees; but as neer as you can to an Angle of 90 De∣grees: So that it may be defended from the Flank and Curtain on either side.
• The longest Line of Defence KA not to be above 12 score Yards, that is, 720 Foot, being within Musket-shot; and the breadth of the Rampire to resist the Battery 100 Foot.

To Describe a Fort of Five Bulwarks, or any other; so that the Bastion, or Flanked Angle of 8 Bastions or Bulwarks exceed not 90 Degrees by the Line of Chords.

FIrst, Draw an Obscure Line, as AB; and upon A, as a Center, with the Chord of 60 deg. describe an Arch, as CDE; and from C lay down half the Polygon Angle (which in the Table following the Figure you shall find to be) 54 deg. as CD; also the same again from D to E, and draw the Line AE. Now divide the half Polygon Arches CD and DE each into three equal parts, as in FHIG, and from two of those parts from D, as F and G, draw Lines unto the Bastion Point at A. Then take any convenient Distance, and lay the same on those Lines from A unto K and L, which shall make the Front or Face of the Bulwark. Next, from the shoulder at K let fall a Perpendicular to AB, as KM; and on the Center at K describe an Arch of 60 deg. from M towards N, and from M lay down on the same Arch 50 deg. or more exact 49 deg. 24 min. and so draw KN, which will cut the Se∣midiameter of the Polygon in the Point O; so shall AO be the Capital Line of the Bastion. Then from O draw a Line parallel to AB, as OP; so shall you have OR for the Gorge Line, and RK the Flank. Now for the Curtain, take half the Front AK, as AT, and lay it down three times from R towards P, which will fall in S; so is RS the Curtain. Then on the Point at S erect a Perpendicular, as SV, equal to RK, which shall be the Flank of another Bastion; and so the Front KA being laid from V, shall cut the first Line AB in B; so drawing VB, you have the Front of the same Bastion.

Lastly, Divide AB in the middle, as in W, and from W let fall a Perpendicular to AB, which will cut the Semidiameter of the Polygon in the Point D; so is D the Center of the Polygon. And with the same Semidiameter DA you must describe a whole Circle, of the which AB is the ⅕ part thereof, which Distance will reach from B unto X, and from X unto Y, and so to Z, and your first Point at A, where you begun your Work.

For the other Bastions, they may be easily transported from the first Bastion. And note, That if your Fort exceed 8 Bulwarks, you must add 15 deg. to half the Poly∣gon Angle, so have you the Bastion Angle; and then work as before. But in the Forts that exceed not 8 Bulwarks, where the Bastion Angle will not be above 90 deg. you must take the 2/3 part of the Angle of the Polygon.

The longest Line of Defence is from A unto Q, and should not exceed 720 Foot (because of being within Musket-shot) the Curtain RS about 420 Foot, and the Front AK 280 Foot: And for the Flank RK, and Gorge RO, their proportion commonly is as 6 to 7: but the Angle KOR forming the Flank is about 40 gr. by which the Proportion is neer as 5 to 6.

[illustration] geometrical diagram

A Table for 8 Bastions.

Polygons.	½ Angle of the Polygons	½ Angle of the Bastions
	Degrees.	Degrees.
4 Tetragon	45	30
5 Pentagon	54	36
6 Hexagon	60	40
7 Heptagon	64 2/7	42 6/7
8 Octagon	67 ½	45

The ½ Bastion Angle is here found by taking the ⅔ of the ½ Polygon Angle: So the Bastion Angle will be an Angle of 90 degr. in the Octagon. And no more must the Bastion Angle be in any Polygon.

A Table for 12 Bastions.

Polygons.	½ Angle of the Polygon.	½ Angle of the Bastion.
	Degrees.	Degrees.
4 Tetragon	45	30
5 Pentagon	54	34 ½
6 Hexagon	60	37 ½
7 Heptagon	64 2/7	39 9/14
8 Octagon	67 ½	41 ¼
9 Enneagon	70	42 ½
10 Decagon	72	43 ½
11 Undecagon	73 7/11	44 7/12
12 Dodecagon	75	45

The ½ Bastion Angle is here found by adding 15 d. to ½ the Polygon Angle, and take the ½ thereof: So the Bastion Angle will be an An∣gle of 90 deg. in a Dodecagon.

Of the Works that are in or about Forts of most Importance.

[illustration] geometrical diagram

AB the Breadth or Walk on the Rampire	40 Feet.
BC the Breadth of the Parapet of the Rampire, with the Fausse-bray, and Parapet thereof, each 20 Foot; in all	60
DE the Breadth of the Moat, Ditch, or Trench	120
EF the Coridon, or Covert-way of the Counterscarp	20
FG the Argin or Parapet thereof, being	50 or 60
H the Ravelines; I the Half-moons, with their Parapets	20

There may be sometimes an occasion in Forts to raise Mounts, Cavaliers, Plat∣forms, or Batteries, to command all the other Works, and to view the Country about; which may be raised upon the Bastions, if you have room withal to make use of the Flanks: Otherwise let them be raised on the Curtains, a little within the Rampire, so that you may have room left for the Walk.

To Draw the Platform of a Fort, beginning with the Capital (or Head) Line; And also to draw the Horn-works.

[illustration] geometrical diagram

LEt the Fort be an Hexagon, that is, of six Bastions or Bulwarks. First draw the Line AB, and upon A describe an Arch of a Circle, as BDC, whereon lay down half the Polygon-Angle, which in the former Table you shall find to be 60 deg. as from B unto D, and thence to C; and draw AC and AD. Now the ⅓ part of the half Polygon Angle is BC and CF; then draw the Obscure Line AF and AG. Next you shall make choice of the Capital Line, of any sufficient length, which let be AE; and from E draw a Line parallel to AB, as EH, continued; and upon the Point E, as a Center, describe KI, making it an Angle of 40 deg. as KEI; so shall EI cut out the Front in L, as AL: So from L let fall the Perpendi∣cular LM, which shall be the Flank; and MN the Curtain shall be as formerly the whole length of the Front, and a half more. For the rest of the Work, you must proceed as formerly.

For the Horn-works.

YOu must continue the Flunkers ML and NO unto P and Q; then take the longest Line of Defence AN, and lay it thereon from the shoulders at L and O, unto P and Q; drawing the Line PQ, dividing it into three equal parts; and from those parts 1 and 2, draw parallels unto PL and QO: also from those Points P and Q draw parallels to the Fronts AL and BO, those will cut the former Pa∣rallels in R, S, T, and V, which Intersections will limit the Fronts, Flanks, and Curtains, as you may easily perceive; unto which you must make the Rampire, Para∣pet, &c. as in the former Works.

Now follow two Tables; the one for 12 Bastions, and the other for Forts of 8 Bastions: Whereby you may trace out any Fort by help of a Line of Equal Parts, which shall divide the Side of the Outer Polygon into 10000 parts.

The First Table for 12 Sides.

Number of Sides	4	5	6	7	8	9	10	11	12
The Side of the Outer Polygon	10000	10000	10000	10000	10000	10000	10000	10000	10000
The Capital Line	2428	2592	2756	2926	3086	3136	3148	3180	3204
The Gorge	1088	1264	1378	1470	1538	1640	1722	1792	1842
The Front	2914	2952	2986	3014	3024	3054	3070	3082	3094
The Flank	970	1128	1246	1360	1516	1526	1536	1542	1546

The Second Table, for 8 Sides, whose Bastion Angle then shall make 90 Degrees.

Number of Sides	4	5	6	7	8
The Side of the Outer Polygon	10000	10000	10000	10000	10000
The Capital Line	2396	2498	2602	2695	2778
The Gorge	1120	1327	1480	1599	1698
The Front	2914	2939	2959	2975	2987
The Flank	940	1113	1242	1342	1423

The Ʋse of these Tables.

LEt it be required to draw the Proportional Dimension of a Regular Fort of 6 Sides: As for Example, in the fourth Figure, whose Side AB must be divi∣ded into 100 equal parts, and each part supposed to be subdivided into 10 parts; so have you 1000 parts, which shall suffice. Now proceeding according to former Di∣rections, until you come to make choice of your Capital Line, you shall here find in the second Table, which is best for the purpose, under the Figure 6, and right against the word Capital in the first Column, 2602, but 260 will serve: Take the

same from the Scale of Equal parts, and lay it from the Bastion Point at A, and it falls in the Point E, which will be the Center of the Bastion. From thence you may lay down the Gorge Line out of the Table, which is 148 unto M: So will the Front AL be 296, and the Flank ML. The Curtain, being once and a half the length of the Front, will be MN 444. Thus you may do for any of the rest. These Ta∣bles are useful for Irregular Forts; But first I will shew you the Height, Breadth, and Scarpings of the Rampire, Parapet, Ditch, &c. of these Sconces, as they are represented in the Profile, or Section, as followeth.

[illustration] geometrical diagram

The Breadth of the Rampire may be 24, 30, or 40 Foot; but here AB is but	32
The Inward Scarp AC	6
The Height of the Rampire CD	6
The Breadth of the Walk of the Rampire DE	10
The Breadth of the Bank or Foot-pace of the Parapet EF	3
and the Height of the same Foot-pace	1 ½
The Inward Scarp of the Parapet FG	1
The Inward Height of the Parapet GH	6
The Breadth of the Parapet at the Foot FI	10
The outward Scarp of the Rampire BK	3
The Inward Scarp of the Parapet IL	2
The Outward Height of the Parapet LM	4
The Thickness of the Parapet at the top MN	7
The Brim of the Ditch BO	3
The Breadth of the Ditch at the top OP	32
The Scarp of the Ditch OQ	6
The Depth of the Ditch QR	6
The Breadth of the Ditch at the Bottom RS	20

The Profile or Section of a Fort with a Fausse-Bray and Counterscarp; also Subtrenched.

[illustration] geometrical diagram

CD is the Fausse-bray, and DM his Parapet: EFGH is the Subtrench, and IK the Coridor, or Covert-way. Lastly, KL is the Argin or Parapet of the

Counterscarp. Note, That the Height of the Rampire AB ought to be raised 15 or 18 Foot above the Terra Plana, although here it is but 12 Foot, which is somewhat too low to command the Trench or Ditch: But if the Trench be made broader, then it will command the bottom thereof.

Of Irregular Fortification.

IN the seventh Figure following let ABCDE be an Irregular Fort, containing 5 Bastions, or Bulwarks. First we will make a Bastion on the Angle at A, which do thus: Divide the Polygon Angle in half with the Line AF, and draw the Bastion-Angle as formerly, being ⅔ of the Polygon-Angle, as AH, and AI con∣tinued, being the Sides whereon the Fronts must be laid down.

Now upon some spare Paper you shall make the half Polygon-Angle GAF, as you may see underneath this seventh Figure, as LKM: Then make choice of the Ca∣pital Line, as before; let it be of any convenient length (larger than you think your Bastion will be in the seventh Figure) as underneath KN, and from N the Center of the Gorge draw a Parallel to KL, continued to O, as NP; and so proceeding as be∣fore, you shall find the Point of the second Bastion at O: So have you the Propor∣tion of your Bastions, whereby you may gain those in the seventh Figure.

Now to reduce it from the Bastion Point at A, you must take AB the shortest side, and lay it from O unto Q; and from Q draw a Line parallel to the Capital Line KN continued, as QR. Lastly, drawing a Line from N to O, it shall cut QR in the Point S; so is QS the length of the Capital Line sought for, which must be laid down on the seventh Figure from A unto T; so is T the Center of the Gorge. Then for the Front take KV, and lay it on the Capital Line from K to W: so a Ru∣ler being laid from W to O, it shall cut the Line QR in the Point at X; so is QX the length of the Front, to be laid down in the seventh Figure from A unto H and I. Thus shall you finish your Bastion, when you have let fallen your Flanks perpendicular on the ends of your Curtains, as you see. The like Method you are to observe for the other Bastions.

And when you have finished your Fort, you must observe whether your Curtain Lines (that is, from the Centers of the Gorge) be parallel to the outward Sides AB, &c. which if they are not, you must correct them; and by your Judgment, by help of the Lines of Defence, you may as you see occasion widen the Necks of the Gorges, and also the Bastion Angles; but not above 90 Degrees: And so let the Flankers be as neer proportional as the Rules (or Ocular Demonstration directeth) which commonly the Gorge Line to the Flank bears proportion as 7 to 6.

Much more could I write of Irregular Fortification: but my purpose at this time is but to make a small Treatise, or an Epitome thereof.

[illustration] geometrical diagram

The seventh Figure, of an Irregular Fort, containing 5 Bastions; being the Platform of the Royal Fort sometimes on St. Michael's Hill, on the North West Side of the City of Bristol.

To make a Scale for Fortification by the Tables.

[illustration] geometrical diagram

THis may be performed Geometrically by observing the former Instructions, whereby you may gain the length of every Line: but it will be sooner done, and more easie, by these Tables following.

First Table, for 12 Sides.

Numb. of Sides	4	5	6	7	8	9	10	11	12
Semi. Out. Pol.	1000	1000	1000	1000	1000	1000	1000	1000	1000
Semi. Inn. Pol.	661	700	731	756	777	795	810	823	834
The Front	412	347	299	261	232	209	190	174	160
The Gorge	158	151	141	132	123	115	108	101	96
The Flank	133	127	119	111	103	97	90	85	80

Second Table for 8 Sides.

Number of Sides	4	5	6	7	8
Semi. Outer Polyg.	1000	1000	1000	1000	1000
Semi. Inner Polyg.	661	706	740	766	787
The Front	412	346	296	259	229
The Gorge	158	156	148	138	130
The Flank	133	131	124	116	109

Make your Scale of a sufficient length, that may hold both Lines, the one for 12 Sides, and the other for 8. Make choice within the breadth of the Scale, between the Borders, any sufficient breadth, as CD; from whence draw Parallels to the Sides, and divide CD into ••0 Equal parts, and begin your Account from C with 45: so shall the end at D be 75 degr.

Now make choice of the length of the outer Po∣lygon, which here I make three Inches; and di∣vide a Line by the Side thereof, equal thereto, in 100 equal parts, and suppose each part into 10; so have you 1000 parts, agreeable to the Tables. The next thing is to draw Parallels to CE, ac∣cording to the Polygon half Angles, as you may see in the Tables under the Pentagon Fort, being the second Figure: So ••••om the Scale CD you have for the half Angle of a Pentagon 54 Degrees, whereby you may draw the Pentagon parallel FG; and so in the lower Scale of 8 Bastions. In the like manner you may do for all the rest.

Now to draw the cross Lines for the Semidiame∣ters of the Inner Polygons, as also the Lines for the Fronts, Gorges, and Flanks, you shall work thus. First, you must note, That the Semidiameter of the Outer Polygon is the Radius or whole Line of 1000

equal parts, and that is drawn at Right Angles, or a cross at F: But for the Semi∣diameter of the Inner Polygon, look in the Table of 12 Sides in the second Co∣lumn under 4, you have 661. Take the same Number off your Scale of Equal Parts, and lay it from E to H: Then in the third Column under 5 you have 700 parts; lay the same down from G to I, and make there a prick or point: Do the like for the Hexagon and Heptagon, as at I and K; proceeding along with all the rest, unto the Dodecagon. And lastly, draw a Line through all those Points: So have you the Arch Line HM for the Semidiameters of the Inner Polygon. In the same manner work for the Front, Gorge, and Flank Lines. The Scale of 8 Sides is the same Method.

I have also inserted on the left side of the Scale a Line of Chords, whose Radius (or Arch of 60 Degrees) is three Inches; and on the left side, a Line for the Sides of the Polygons. The Hexagon, or six Sides, is equal to the Radius: And for the Tetragon, or four Sides, it is equal to the Chord of 90 Degrees. So having described a whole Circle with the Chord of 60 Degrees, you shall find, that if you take from the Center N unto 5, it shall divide the Circle into 5 equal parts, for drawing the Figure of a Pentagon, which in the second Figure of a Pentagon Fort will reach from A to B, and so to X, Y, Z, and A. Now DA in the same se∣cond Figure you shall find to be the Semidiameter of the Outer Polygon, which in the Scale is GF; and taking GI off the Scale for 12 Bastions, or GO on the Scale of 8 Bastions, it will give the Semidiameter of the Inner Polygon, as DE in the same Figure. So likewise GP on the Scale will be equal to the Front AK in the Pentagon Fort. The like you may understand for laying down the Gorge and the Flank. And for the Curtain, as before, you must make RS 1 ½ the length of the Front AK.

This Scale will also be of good Use in Irregular Fortification. As for Instance. In the Irregular Fort, the seventh Figure, you shall find the half Bastion Angle GAF to be 58 Degrees, which falls on the Scale between the Pentagon and Hexagon, from whence you may draw a Bastion on some spare place, and from thence proportionate the same unto the outer side of the seventh Figure AB. The rest I leave to your own practice.

How to Fortifie a long Curtain with Bulwarks, or a strait Town Wall.

[illustration] geometrical diagram

LEt the Curtain be AB. First take 200 Foot from the Scal•• of Fortification, accounting 10 for 100; and lay the same from A unto C, and from C unto D: So shall AD be the breadth of the Neck of the Gorge: and upon the Point C erect a Perpendicular, as CF. Then take 420 Foot off your Scale, and lay the same from D to E, which shall be the length of the Curtain. Next you must take 720 Foot, and lay the same from E, to cut off the Capital Line at F; so shall EF be the longest Line of Defence, and CF the Capital Line; which Line CF must be laid down from C unto G: and draw GF for the shortest Line of Defence. Lastly, upon D erect a Perpendicular, which will cut the same Line in H; so have you DH for the Flank, and HF the Front. Thus have you finished half the Ba∣stion, from whence you may transport all the rest of the Bastions, were they ever so many.

Note, That these Bastions must not exceed 720 Foot, that so it may not be with∣out Musket-shot. But if you will defend the Front with Cannon, you may make the Line of Defence almost twice so much. The like for the Curtain, which may be 800 Foot; and in the middle of the Curtain you may make a Spur, or Point of a Bastion, as at K, which will be necessary for Musket-shot, beside the Cannon; which in the Line drawn about the City of Bristol I have seen many of them.

[illustration] geometrical diagrams

These five following Pieces are taken out of Malthus; The Proportions you may find by the Scale, and the Rules before shewed.

Tetragon

Pentagon

Hexagon

Heptagon

Octagon

FINIS.