A COMPENDIUM OF FORTIFICATION, BOTH Geometrically and Instrumentally, BY A SCALE, The Making whereof is showed by the Tables, and their Use, both of the Tables and the Scale, for speedy Protracting of any Fort consisting of 8 Bulwarks, whose Bastion-Angles shall not exceed 90 Degrees; and so the like for Bastion-Angles of 12 Bulwarks.  
WRITTEN BY PHILIP STAYNRED Professor and Teacher of the MATHEMATICS in the City of BRISTOL.  JER. XVI. 19 
The Lord is my strength, and my fortress, and my refuge in the day of affliction.  
LONDON, Printed by E. Cotes, Anno Domini 1669.   

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.  

Names of the Sides of a Fort.  
A B the Outward Side of the Polygon, and D E the Inward Side.  
C A the Semidiameter of the Outward Polygon, and C D the Inward.  
I H A G F the Bulwark or Bastion.  
A G, or A H, the Front or Face of the Bastion— 280 Foot.  
A D the Capital or Head Line,  
D I, or D F, the Gorge Line.  
I H, or F G, the Flank; and F M the second Flank.  
D G the Espaule, or Shoulder.  
F K the Curtain— 420 Foot.  
A K the longest Line of Defence— 720 Foot.  
A L the shortest Line of Defence.   

Names of the Angles.  
N A B the Angle of the Polygon, and N A C the half Angle.  
H A G the Flanked Angle of the Bulwark.  
A G F the Angle of the Shoulder.  
F D G the Angle forming the Flank, commonly 40 Degrees.  
G A O the Inward Flanking Angle.  
A P B the Outward Flanking Angle, and A P O half the same.   
Note, That the Bastion or Flanked Angle H A G must never be less than 60 Degrees, neither above 90 Degrees; but as near as you can to an Angle of 90 Degrees: So that it may be defended from the Flank and Curtain on either side.  
The longest Line of Defence K A 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 A B; and upon A, as a Centre, with the Chord of 60 deg. describe an Arch, as C D E; and from C lay down half the Polygon Angle (which in the Table following the Figure you shall find to be) 54 deg. as C D; also the same again from D to E, and draw the Line A E. Now divide the half Polygon Arches C D and D E each into three equal parts, as in F H I G, 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 A B, as K M; and on the Centre 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 K N, which will cut the Semidiameter of the Polygon in the Point O; so shall A O be the Capital Line of the Bastion. Then from O draw a Line parallel to A B, as O P; so shall you have O R for the Gorge Line, and R K the Flank. Now for the Curtain, take half the Front A K, as A T, and lay it down three times from R towards P, which will fall in S; so is R S the Curtain. Then on the Point at Serect a Perpendicular, as S V, equal to R K, which shall be the Flank of another Bastion; and so the Front K A being laid from V, shall cut the first Line A B in B; so drawing V B, you have the Front of the same Bastion.  
Lastly, Divide A B in the middle, as in W, and from W let fall a Perpendicular to A B, which will cut the Semidiameter of the Polygon in the Point D; so is D the Centre of the Polygon. And with the same Semidiameter D A you must describe a whole Circle, of the which A B 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 Polygon 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 ⅔ 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 R S about 420 Foot, and the Front A K 280 Foot: And for the Flank R K, and Gorge R O, their proportion commonly is as 6 to 7: but the Angle K O R forming the Flank is about 40 gr. by which the Proportion is near as 5 to 6.  

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 Angle of 90 deg. in a Dodecagon.  
Of the Works that are in or about Forts of most Importance.  

A B the Breadth or Walk on the Rampire— 40 Feet.  
B C the Breadth of the Parapet of the Rampire, with the Fausse-bray, and Parapet thereof, each 20 Foot; in all— 60  
D E the Breadth of the Moat, Ditch, or Trench— 120  
E F the Coridon, or Covert-way of the Counterscarp— 20  
F G the Argin or Parapet thereof, being— 50 or 60  
H the Ravelines; I the Halfmoons, with their Parapets— 20   
There may be sometimes an occasion in Forts to raise Mounts, Cavaliers, Platforms, 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.  
 
LEt the Fort be an Hexagon, that is, of six Bastions or Bulwarks. First draw the Line A B, and upon A describe an Arch of a Circle, as B D C, 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 A C and A D, Now the ⅓ part of the half Polygon Angle is B G and C F; then draw the Obscure Line A F and A G. Next you shall make choice of the Capital Line, of any sufficient length, which let be A E; and from E draw a Line parallel to A B, as E H, continued; and upon the Point E, as a Centre, describe K I, making it an Angle of 40 deg. as K E I; so shall E I cut out the Front in L, as A L: So from L let fall the Perpendicular L M, which shall be the Flank; and M N 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 Flankers M L and N O unto P and Q; then take the longest Line of Defence A N, and lay it thereon from the shoulders at L and O, unto P and Q; drawing the Line P Q, dividing it into three equal parts; and from those parts 1 and 2, draw parallels unto P L and Q O: also from those Points P and Q draw parallels to the Fronts A L and B O, those will cut the former Parallels 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, Parapet, etc. as in the former Works.  
New 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 Use 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 A B must be divided 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 Directions, 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 Centre 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 A L be 296, and the Flank M L. The Curtain, being once and a half the length of the Front, will be M N 444. Thus you may do for any of the rest. These Tables are useful for Irregular Forts; But first I will show you the Height, Breadth, and Scarpings of the Rampire, Parapet, Ditch, etc. of these Sconces, as they are represented in the Profile, or Section, as followeth.  
Terra Plana  

The Breadth of the Rampire may be 24, 30, or 40 Foot; but here A B is but— 32  
The Inward Scarp A C— 6  
The Height of the Rampire C D— 6  
The Breadth of the Walk of the Rampire D E— 10  
The Breadth of the Bank or Footpace of the Parapet E F— 3  
and the Height of the same Footpace— 1½  
The Inward Scarp of the Parapet F G— 1  
The Inward Height of the Parapet G H— 6  
The Breadth of the Parapet at the Foot F I— 10  
The outward Scarp of the Rampire B K— 3  
The Inward Scarp of the Parapet I L— 2  
The Outward Height of the Parapet L M— 4  
The Thickness of the Parapet at the top M N— 7  
The Brim of the Ditch B O— 3  
The Breadth of the Ditch at the top O P— 32  
The Scarp of the Ditch O Q— 6  
The Depth of the Ditch Q R— 6  
The Breadth of the Ditch at the Bottom R S— 20   
The Profile or Section of a Fort with a Fausse-Bray and Counterscarp; also Subtrenched.  
Terra plana.  
C D is the Fausse-bray, and D M his Parapet: E F G H is the Subtrench, and I K the Coridor, or Covert-way. Lastly, K L is the Argin or Parapet of the Counterscarp. Note, That the Height of the Rampire A B aught 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, than it will command the bottom thereof.   

Of Irregular Fortification.  
IN the seventh Figure following let A B C D E 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 A F, and draw the Bastion-Angle as formerly, being ⅔ of the Polygon-Angle, as A H, and A I continued, being the Sides whereon the Fronts must be laid down.  
Now upon some spare Paper you shall make the half Polygon-Angle G A F, as you may see underneath this seventh Figure, as L K M: Then make choice of the Capital Line, as before; let it be of any convenient length (larger than you think your Bastion will be in the seventh Figure) as underneath K N, and from N the Centre of the Gorge draw a Parallel to K L, continued to O, as N P; and so proceeding as before, you shall find the Point of the second Bastion at O: So have you the Proportion 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 A B the shortest side, and lay it from O unto Q; and from Q draw a Line parallel to the Capital Line K N continued, as Q R. Lastly, drawing a Line from N to O, it shall cut Q R in the Point S; so is Q S 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 Centre of the Gorge. Then for the Front take K V, and lay it on the Capital Line from K to W: so a Ruler being laid from W to O, it shall cut the Line Q R in the Point at X; so is Q X 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 Centres of the Gorge) be parallel to the outward Sides A B, etc. which if they are not, you must correct them; and by your Judgement, 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 near 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.  
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.  
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 easy, 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 C D; from whence draw Parallels to the Sides, and divide C D into 30 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 Polygon, which here I make three Inches; and divide 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 C E, according to the Polygon half Angles, as you may see in the Tables under the Pentagon Fort, being the second Figure: So from the Scale C D you have for the half Angle of a Pentagon 54 Degrees, whereby you may draw the Pentagon parallel F G; 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 Semidiameters 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 Semidiameter of the Inner Polygon, look in the Table of 12 Sides in the second Column 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 H M 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 Centre 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 D A in the same second Figure you shall find to be the Semidiameter of the Outer Polygon, which in the Scale is G F; and taking G I off the Scale for 12 Bastions, or G O on the Scale of 8 Bastions, it will give the Semidiameter of the Inner Polygon, as D E in the same Figure. So likewise G P on the Scale will be equal to the Front A K 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 R S 1 ½ the length of the Front A K.  
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 G A F 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 A B. The rest I leave to your own practice.   

How to Fortify a long Curtain with Bulwarks, or a straight Town Wall.  
LEt the Curtain be A B. First take 200 Foot from the Scale of Fortification, accounting 10 for 100; and lay the same from A unto C, and from C unto D: So shall A D be the breadth of the Neck of the Gorge: and upon the Point C erect a Perpendicular, as C F. 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 E F be the longest Line of Defence, and C F the Capital Line; which Line C F must be laid down from C unto G: and draw G F for the shortest Line of Defence. Lastly, upon D erect a Perpendicular, which will cut the same Line in H: so have you D H for the Flank, and H F the Front. Thus have you finished half the Bastion, 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 without 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.  
These five following Pieces are taken out of Malthus; The Proportions you may find by the Scale, and the Rules before showed.  
Tetragon  
Pentagon  
Hexagon  
Heptagon  
octagon  
FINIS.