THE NEW METHOD OF Fortification. As Practised by Monsieur de VAUBAN, Engineer General of FRANCE: WITH AN EXPLICATION Of all Terms appertaining to that ART: Made English.  
LONDON, Printed for Abel Swall, at the Unicorn at the West-End of St. Paul's Churchyard, 1691.   

To His GRACE, THE DUKE of ORMOND.  
May it please Your Grace,  
THE General Complaint of the want of Books of Fortification in our English Tongue, has encouraged me to Translate ●●at of Monsieur Vauban: To which I ●as invited by the almost Universal Applause 〈◊〉 has received from the most competent ●●dges. Nothing could have emboldened me 〈◊〉 the Presumption of this Address, but the ●●od and Welfare of my Native Country, 〈◊〉 proposing your Grace to the Imitation of our Nobility and Gentry, which seems to be the only Means for this Kingdom to recover its Ancient Glory. I doubt not, but that brave Example which your Grace continually sets to the whole Nation, will inspire the Men of Quality to come up to it, as near as possible: Which, if they had always done, Vauban might, without any great Detriment to us, have been confined to the Territories of his Master. That your Grace may ever prosper, and receive the Honour's due to so great Merits; as it is the greatest Good can be wished you, so it is no less the unfeigned Desire of,  
May it please your Grace, Your Grace's most Obedient and most humble Servant, ABEL SWALL.   

THE PREFACE.  
THis little Tract, which was chiefly designed by the Ingenious Author, for the Use and Benefit of young Gentlemen and others, addicted to the Study of this part of the Mathematics, who have not leisure, or opportunity to read over the vast Systems of Voluminous Writers, has deservedly obtained so great a Repute among the Curious, that it is Translated into several Languages. It contains an exact Explication of all the Terms that are attributed to Mathematical Bodies; and, in general, all that is necessary for the Theory and Practice of a good Geometrician. Moreover, therein is exhibited the true Mensuration of Heights, Lengths, Planes, Solids, and Concaves; as also, the Science called The Transmutation of Bodies; which is the Subject of the first Part. In the Second, are explained the Rules that are observed by the French Engineers at this day, in raising the●r Works; and at the same time, are produced the different Sentiments of the French among themselves, as to this Matter; it is likewise shown wherein their manner of Operation differs from that of the Germans, and other Nations. Lastly, The most clear and accurate Method of Fortification is plainly taught.  
The Word Geometry, is derived from the Greek, 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉, & 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉; importing, to measure the Earth; so that Geometry (with respect to its Etymology,) is nothing else but the Art of measuring Land; yet in a larger, and more proper Sense, it is applied to all sorts of Dimensions. As for the Original of this Science, it is indeed very uncertan; and it is believed by many, that the Egyptians were the ●irst Inventors thereof: For in regard that the River Nile does every Year overflow the Country of Egypt; those Inundations which generally overthrew, and confounded all the Landmarks, obliged the Inhabitants to devise certain Measures, by the means whereof, they might be able to distinguish, and adjust the Limits of their respective Grounds from those of their Neighbours, when the Waters were withdrawn: And this Opinion is not altogether to be rejected; especially if we consider that Moses is generally reputed to have given them the first Rudiments of this Art: However since there were not very many Men who applied themselves to the Study of goo● Literature in those early Ages of the World: The Mathematics, and Geometry in particular, were, as it were, locked up within the Breast of some few Learned and Skilful Artists, until the Time of Thales, who first publicly professed them in Greece; nevertheless the Grecians were not very diligent in promoting and improving these Noble Sciences, and only taught them privately; neither was there any one that undertook to show the Usefulness, and Necessity of this kind of Learning, until Euclid wrote his Elements; which gave occasion to the Publishing of a great Number of excellent Treatises on this Subject, that are now extant.   

An Advertisement to the Book-Binder.  
1. CAre must be taken, not to cut the Blank Paper on which the Plates are wrought off; by reason that the Figures must be so placed as to ●old out of the Book.  
2. There are Eight Figures which must be placed at the End of the Treatise of Geometry, immediately after the Table; they are marked thus: Geom. Pl. 1, 2, 3, 4, 5, 6, 7, 8.  
3. The other Figures belong to the Fortification, and are to be placed at the End of the Book, and set in order as they are marked; A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, V, W, X.   

A TABLE of the Matters contained in this Treatise of FORTIFICATION.  

BOOK I.  


CHAP. I.  


§. I.  
OF the Definition of Military Architecture p. 1    


§. II.  
Of the Original of Fortification. p. 2    


§. III.  
Of the Parts of Fortification p. 3    


§. IV.  
Of the Division of Military Architecture p 4.      


CHAP. II.  
Of the Parts of a Fortress, and its Angel's ib.    


CHAP. III.  
Of the Terms of the Measures p. 6    


CHAP. IV.  
Of the Names and Terms made use of, as well in attacking, as in the Defence of Places, alphabetically set down p. 7     

BOOK II.  


CHAP. I.  
Of the Maxims of Fortification p. 21    


CHAP. II.  
Of the Situation of Places p. 25  


§. I.  
Of the Advantages and Disadvantages of a Fortress situated on a Mountain p. 25, 26    


§. II.  
Of the Advantages and Disadvantages of a Marshy or Fenny Situation p. 27.  


§. III.  
Of the Advantages and Disadvantages of a Place encompassed round with Water ibid.    


§. IV.  
Of the Advantages and Disadvantages of Places situated upon an open Rising Ground p. 28    


§. V.  
Of the Side of a Mountain p. 29    


§. VI  
Of a Valley ibid.    


§. VII.  
Of the Advantages and Disadvantages of a Place situated on the side of a great River ibid.        


CHAP. III.  
Of the Quality of the Earth p. 30    


CHAP. IV.  
Of Provision, and other Necessaries, p. 32     

BOOK III.  


CHAP. I.  


§. I.  
How to describe any Polygon within a given Circle p. 33  


§. II.  
How to make a Scale p. 34        


CHAP. II.  
How to make the Draught of a Square; as also the Orillon and the Hollow Tower p. 35    


CHAP. III.  
Of the Structure of the Body of the Place of the Square p. 38    


CHAP. IV.  
Of the Structure of the Half-Moon before the Curtain of the Square p. 40    


CHAP. V.  


v. 1.  
Of the Structure of the Ravelin before the Curtain ibid.    


§. II.  
To make a Ravelin with Cuneated p. 41    


§. III.  
Of a Ravelin with Counter-guards ibid.    


§. IV.  
To make a Ravelin to be placed at the Entrances into Places p. 42      


CHAP. VI  
To make a Horn-Work before the Curtain p. 43    


CHAP. VII.  
To make a Horn-Work ●efore the Bastion p. 44    


CHAP. VIII.  
To make a Horn-Work with Shoulders p. 45    


CHAP. IX.  
To make a Horn-Work with a Crown ibid.    


CHAP. X.  
To make a Crownwork before the Curtain p. 46    


CHAP. XI.  
To make a Crownwork before the Bastion p. 48    


CHAP. XII.  
To make a Ravelin before the Point of a Bastion ibid.    


CHAP. XIII.  
To make a single Tenail p. 49    


CHAP. XIV.  
To make a double Tenail ibid.    


CHAP. XV.  
To make a Bastion with Counter-guards p. 50    


CHAP. XVI.  
Of the Structure of a Pentagon, and of the Tenail in the Moat ibid.    


CHAP. XVII.  
The Structure of a Hexagon, and other Polygons p. 51    


CHAP. XVIII.  
To make the Profil p. 53    


CHAP. XIX.  
How to trace out the Draught of a Fortress in the Field p. 55    


CHAP. XX.  
To make the Stre●n in a Fortress p. 57     

BOOK IU.  


CHAP. I.  
Of the principal Angles of a Fortress p. 59    


CHAP. II.  
Of the Flank p. 67    


CHAP. III.  
Of the Faces ibid.    


CHAP. IU.  
Of the Orillon and Flank retired, together with the Brisure, or Place where the great Guns are to be planted p. 68    


CHAP. V.  
Of the Curtain p. 69    


CHAP. VI  
Of the Parapet p. 71    


CHAP. VII.  
Of the Banquets p. 72    


CHAP. VIII.  
Of the Rampart p. 73    


CHAP. IX.  
Of the Embrasures and Merlo●s p. 75    


CHAP X.  
Of the Moat in the Body of the Place ibid.    


CHAP. XI.  
Of the Moat belonging to the exterior Works, and the Avant-Moat p. 78    


CHAP. XII.  
Of the Covered Way, and of the Glacis ibid.    


CHAP. XIII.  
Of False Brays p. 79    


CHAP. XIV.  
Of Cavaliers p. 80    


CHAP. XV.  
Of Counterscarps p. 84    


CHAP. XVI.  
Of the Streets, Places of Arms, Corpse de Garde, and Magazines p. 85      

A NEW TREATISE OF Geometry. BOOK I.  

CHAP. I.  
AS an Introduction to our Subject, at our first entry, we shall begin with the Definition of Geometry, and the Explication of those Terms that are hereafter to be made use of.  
Geometry is a Science which treats of Magnitudes, and it is the principal part of the Mathematics.  
1. Magnitude is a continued Quantity, which consists either in Lines or Angles, in Surfaces or Bodies.  
2. A Line is a length without breadth, whose Extremities are called Points.  
3. A Point is something which cannot be divided.  
4. A Line is right, crooked, or mixed: a right Line is that which goes without turnings from one● Point to another. A crooked Line is that which goes off from its bounds, by one or more turnings: and a mixed Line is that which is part right and part crooked.  
5. A Surface is length with breadth: whose bounds are called Lines.  
6. These Lines are either Parallel or Oblique. Parallel Lines are those which are every way equally distant from each other. Oblique Lines are those which meeting at a Point make an Angle.  
7. It is called a right Angle when one of the Lines is perpendicular upon the other. It is called an Oblique Angle when one of the Lines is not perpendicular upon the other. It is called a right-lined Angle when ●t is made up of two right Lines. It is called a Curve-lined Angle, when it is made up of two crooked Lines. It is called a mixed Angle when one of the Lines is right, and the other curved. It is called an Acute Angle when it is less than a right one. And it is called an Obtuse Angle, when it is greater than a right Angle.  
8. A Figure is a Magnitude bounded by one or more Lines and Surfaces, whereof some are plain, some solid.  
9 A Circle is a plain Figure bounded by a Circumference, the middle Point whereof is called the Centre, from which all right Lines drawn to the Circumference are equal.  
10. A Diameter is a right Line, which goes through the Centre, and touches the Circumference in two Points.  
11. A Semicircle is the half of an Entire Circle.  
12. An Oval is a plain Figure bounded by its own Circumference, within which no Point can be taken from which all right Lines drawn to the Circumference can be equal.  
13. A Triangle may be considered either with respect to its Sides, or its Angles: in respect to its Sides, it is either 〈…〉 lateral, which has three equal Sides: or an Isosceles, which has two Sides only equal: whose third is called the Base: or a Scalene, which has three unequal sides: in respect of its Angles, it is either right-angled, which has one right Angle: or Acute-angled, which has three acute Angles: or Obtuse-angled, which has one Obtuse and two Acute Angles. One of the Lines in a right-angled Triangle is called the Base: the second the Perpendicular: and the third the Hypothenusal.  
14. A Parallelogram is a Figure which has four Sides, and four Angles: and it is either equilateral which has four equal sides, or in-equilateral whose sides are unequal.  
15. An equilateral Parallelogram has either right-Angles, as a Square or Oblique Angles, as a Rhombus.  
16. An in-equilateral Parallelogram has either right-Angles as a rectangled Parallelogram: which is likewise called a longsquare; or not, and then it is a Rhomboid: every other square Figure is called a Trapezium.  
17. A Tetraëdron is a solid Figure, or a Body which is bounded by four equal Triangles.  
18. A Pyramid, is a Body bounded by several Triangles, which making one Surface for a Base, meet all in one Point.  
19 A Prism is a Body which has two Bases, equal and parallel each to other.  
20. A Parallelepiped is a Body bounded by six Parallellograms, whereof the two opposite ones are always equal and parallel.  
21. A Trapezium is a Body bounded by plain Surfa●es, which from neither a Prism, nor a Parallelepiped.  
22. A Cone is a Body made by the turning of a right-angled Triangle round a Circle, the angular Point of the right angle being fixed in the Centre.  
23. A Cylinder is a Body made by the turning of a right-angled Parallelogram round two Circles one at the top, the other at the bottom.  
24. A Cube is a Body bounded by six Squares.  
25. An Octaëdron is a Body bounded by eight equal and equilateral Triangles.  
26. A Dodecaëdron is a Body bounded by twelve equal and equilateral Pentagons.  
27. An Icosaëdron is a Body bounded by twenty equal and equilateral Triangles.  
28. A Pentaëdral Prism is a Body bounded by three right-angled Parallellograms, having two equilateral Triangles for its Bases.  
29. A Rhomb is a Body bounded by six equilateral Oblique-angled Squares.  
30. A Rhomboid is a Body bounded by 4 Oblique-angled Parallelograms, having two Oblique-angled equilateral Squares for their Bases.   

CHAP. II. To know how Figures are made, the following Problems must be examined.  
PROBLEMS.  

I. To draw a Line a parallel to a given Line.  
MArk upon the given Line any two points AB where you please, and by them draw two Arches OF, in drawing a right Line by OF you have your Parallel.   

II. From the point G, to draw a parallel to a given Line HI.  
1. Draw an Arch from the point G, which touches the given Line only in K.  
2. With the distance GK, draw another Arch upon the given Line MN.  
3. The Line drawn by GN shall be parallel to the given Line HI.   

III. To raise Perpendicular on a given Line, from a given point.  
1. Take a distance from the given point (what you please) to the right, as AB.  
2. Take the same distance to the jest, as AC.  
3. Take BC and draw two Arches which intersect each other at D.  
4. Join D and A together, and you will have your Perpendicular.   

IV. To raise a Perpendicular at the end of a given Line AB.  
1. Draw a Circle to the point B, from the centre C, which may be taken at pleasure.  
2. From the point D, where the Circle cuts the given Line, draw a right Line through the Centre C, till it touches the Circle on the other side, as at E.  
3. Join E and B together, and you will have your Perpendicular.   

V. From a given point G, to let fall a Perpendicular upon a given Line.  
1. Draw an Arch from the point G, which may cut the given Line in two places, as CD.  
2. From CD draw two Arches under the Line which shal● intersect each other at E.  
3. Join E and G together, and you will have your Perpendicular GF.   

VI To divide a given Line into two equal parts.  
1. From the points AB draw two Arches which shall intersect above the given Line at C, and below it at D.  
2. Join C and D together, and you shall have your Line cut into two equal parts.   

VII. To divide a Line AB into several equal parts.  
1. Draw CD a Line at pleasure, and mark as many parts as the given Line AB ought to have; six for Example, upon this Line as CE.  
2. Take CE and draw two Arches which may intersect in F.  
3. Draw all these six parts in F.  
4. Take the given Line AB, and lay it upon the Lines FE and FC, as GH.  
5. Join GH together, you will have your given Line AB divided into six equal parts.   

VIII. To describe an Angle equal to an Angle given.  
1. Let the given Angle be BAC.  
2. Draw the Arch ADE.  
3. Draw a Line IL, and upon this Line an Arch with the distance AD, as FG.  
4. Set the distance DE upon the Arch FG, as FH.  
5. Draw a right Line from the point I to the point H, which will make an Angle equal to the given Angle.   

IX. To divide an Angle into two equal parts.  
1. Let the given Angle be BAC draw the Arch ADE.  
2. From the points DE draw two Arches which shall intersect each other at F.  
3. The Line drawn from F to A will cut the Angle into two equal parts.   

X. To draw a Circle through three given points ABC, or to find the Centre of a given Circle, or to finish a Circle when there is but one part given.  
1. With the points AB, make two Arches which shall intersect at D above, and at E below the given points.  
2. Draw a right Line from the point D to G.  
3. Do the same thing with BC, so that they may intersect in FG.  
4. Draw another right Line from F to G, and there where this Line intersects the first, as for instance at G, you will find the Centre, by which you may draw the Circle through the three given points.   

XI. To make a Circle two, three, or four times bigger than the given ●ne, etc.  
1. Divide the given Circle into four equal parts ABCD, with the Centre marked at E.  
2. Take the distance AB and put it from E upon the Perpendicular, as OF, and with this distance draw a Circle, as big again as that given.  
3. To make a Circle twice as big, take OF, and put it from E upon the first Perpendicular, as EGLANTINE, and the distance EGLANTINE will be the Semi-diameter of the demanded Circle.  
4. To make a Circle thrice as big, put the distance AGNOSTUS upon the same Perpendicular, as EH, and the distance EH will be the Semi-diameter of the demanded Circle. Thus you may enlarge a Circle for ever.   

XII. To make an Oval.  
1. Draw a Line, and thereupon the Circle BAC.  
2. From C draw another Circle through B, as CBD.  
3. Take care that these two Circles intersect each other, as in EF.  
4. From the point of intersection E, draw right lines through BC, which shall touch the two Circles in GH.  
5. From the point E, draw an Arch from G to the point H.  
6. Keep this distance, and do the same thing from the other point of intersection F, as IK, and so your Oval will be complete.   

XIII. To draw a Spiral Line.  
1. Upon a strait Line, draw the Arch ABC.  
2. Take BC, and fixing one point of the Compass upon B, draw from the point C another Arch, which shall touch the same Line in D.  
3. Take the first Centre A, and draw another Arch from D, that shall touch the line in E.  
4. Return to the point B, and from it as a Centre draw from E another Arch to F.  
5. Take A as a Centre, and draw an Arch from F to G; and so on for ever.   

XIV. To divide a Circle into 360 equal parts.  
1. Divide a Circle into four equal parts, as ABCD, whose Centre is E.  
2. From A through E draw the Arch FG; so also draw from B the Arch IH, and from C draw and the third KL; and at last from D draw the last Arch MN.  
3. Having by these Arches divided the Circle into twelve equal parts, afterwards divide every one of these twelve into three other equal parts, and then you will have thirty six equal parts.  
4. Divide afterwards every one of these thirty six into ten equal parts, and your Circle will be divided into three hundred and sixty equal parts.   

XV● To make an Equilateral Triangle.  
Take a given line AB, and from those two points draw two Arches which shall intersect at C; from C draw two right lines to A and B; these will make an Equilateral Triangle.   

XVI. To make a Triangle equal to a given One.  
1. Let the given Triangle be ABC, draw a line at pleasure, and mark thereupon the length AB, as DE.  
2. Take the distance BC, and from the point E draw an Archippus  
3. Take the distance AC, and from D draw another Arch which shall intersect the first at F.  
4. Join FE and FD together, you will have a Triangle equal to the given one ABC.   

XVII. To divide a Triangle into several equal parts.  
1. Let ABC be the Triangle to be divided into five other Triangles: first divide the longest side BC into five equal parts, so that the fifth part, marked at D may be joined to A.  
2. Divide the longest of the two other sides into four equal parts, and let one of them marked G be joined to D.  
3. Divide the remaining part DC into three equal parts, and join that marked F by a right Line to G.  
4. Divide the remainder GC into two equal parts, and join that marked E by a right Line with F. So you may have five equal Triangles: viz. ABDELLA, ADG, GDF, GFE, and EFC.   

XVIII. To make a Square.  
1. Take the Line AB, and raise a Perpendicular upon B, of the same height as BC.  
2. With the same distance make two Arches from A and C which shall intersect at D.  
3. Join DA, and DC by right Lines: and you will have a Square.   

XIX. To make a Parallelogram.  
1. Draw a long Line AB, and upon B raise a small Perpendicular BC.  
2. Take the distance BC and by the point A draw an Archippus  
3. Once more take the Line AB, and by the Point C make an Arch which shall intersect the other at E; join E to A and C and you will have this Parallelogram.   

XX. To draw a Pentagon of equal Angles without a Circle.  
1. Draw a long Line AB, and mark upon it the distance AC.  
2. Raise a Perpendicular upon C, equal to AC, as CD.  
3. Divide the Line AC into two equal parts, as A, EC.  
4. Set the Point of the Compass upon E; and with the other Point draw an Arch from D, which shall touch the given Line at F.  
5. By the Points A, C, and the distance OF, draw two Arches intersecting at G.  
6. Take only the distance A, C, and by the Points G, A, draw two Arches intersecting each other in ay: so likewise, by the Points G, C, draw two other Arches which shall intersect each other in K.  
7. Join AI, IG, GK, and KC together by right Lines, and you have your Pentagon.   

XXI. To make a right-lined Triangle equal to a Circle given.  
1. Divide the Diameter of AB the given Circle into seven equal parts.  
2. Raise a Perpendicular upon B three times as long as the Diameter AB, and a seventh part over, as BC.  
3. Draw a right Line from C to D the Centre of the given Circle, and that gives you the Triangle.   

XXII. To find the circumference of a Circle, having only the Diameter.  
This Problem must be wrought after the same manner as the foregoing one.   

XXIII. To make a Square equal to a given Circle.  
1. Divide the Diameter AB into seven equal parts.  
2. Double this Diameter, and add a seventh part of itself to it, as AC.  
3. Divide the first Diameter AB into two equal parts, as AD.  
4. Divide the Line DC into two equal parts, and taking the middle Point E, draw from D. to C an arch, as DFC.  
5. Raise a Perpendicular upon E to touch the Circle in F: this Line OF shall be a side of the required Square: the rest is wrought by the eighteenth Problem.   

XXIV. To make one Square equal to two.  
1. Let one of the given Squares be ABCD, and the other EFGH.  
2. Join them both, so that the sides BC, FE, may make one right Line CBE.  
3. Join A to E, and raise a Perpendicular upon E of the same length, as EKE.  
4. Keep the same length, and from the points KA, make two Arches to intersect in I  
5. Join KI and IA, and your Square will be made.   

XXV. To make a Square equal to a Parallelogram.  
Let the given Parallelogram be ABCD.  
1. Lengthen the Base DC, and add to it the other side of the Parallelogram BC, as CE.  
2. Divide the distance DE into two equal parts, and from the middle point F draw an Arch, from D towards E.  
3. Raise a Perpendicular upon C to touch the Arch in G; this will be a side of the required Square; set it upon the first lengthened Line, and you will have another side, as CI.  
4. Keep the same distance, and from the points GI', make two Arches to intersect in H; join IGH, and HI together, and your Square will be made.   

XXVI. To make a Square two, three, or four times greater than it is.  
Let the given Square be ABCD.  
1. Lengthen the side AB, and take the distance BD, and set it upon the lengthened side AB, as A, and it will be a side of a Square double to that given at first.  
2. Afterwards observe what was said above in the Eleventh Problem.   

XXVII. To make an Equilateral Right Angled Square equal to an Oblique Angled Parallelogram ABCD.  
1. From the points AB let fall P●rpendiculars to the lower Line, as A, and join it afterwards to the Perpendicular B, as BF.  
2. And when you have the right angled Parallelogram AEBF, work on by the twenty fifth Problem.   

XXVIII. To make a right Angled Square equal to the Rhomb ABCD.  
1. From the points BC, let fall Perpendiculars upon the Line, and set the distance BE, upon the other point C, as CF.  
2. Join OF together, and the sides BC OF will make a right angled Square equal to a given Rhomb.   

XXIX. To make an Equilateral Right Angled Square equal to the Triangle ABC.  
1. Lengthen the Base CB, and divide the same Base into two equal parts, as CD, DB.  
2. From the point B let fall a Perpendicular equal to BD, as BF.  
3. Lengthen BF upwards, and from the point A draw a Parallel to the Base CB, which shall cut the lengthened Line BF in E.  
4. Divide the distance FE into two equal parts, and from G the middle point draw an Arch from E to F; and from the point H, where this Arch touches the first lengthened Line CB, to B shall be the first side of the Square, which laid upon the Line BE, will give another side BY; afterwards from the points IH, make two Arches which shall intersect each other in K; and join HK and KING together, and your Square will be made.   

XXX. To make a Parallelogram equal to a given Triangle ABC.  
1. Divide the Base AB into two equal parts, as AD, DB.  
2. Divide the side CB into two equal parts, as BE, EC.  
3. Draw a right Line from D to E, what length you please, and take the distance DE, and set it from the point E, to lengthen it, as EF.  
4. Take DF, and from the point B make an Archippus  
5. Take DB, and from F intersect this Arch in G.  
6. Join FG and GB together, and your Parallelogram will be made.   

XXXI. To make a Parallelogram equal to a right Angled Square ABCD.  
1. Lengthen the side of the Square AB, adding double its own length to it, as BF.  
2. Divide the side of the Square BC into two equal parts, as BE, EC.  
3. Take BF, and from the point E make an Archippus  
4. Take BE and intersect this Arch from the point F in G.  
5. Afterwards join EGLANTINE and GF together, and you have the Parallelogram required.   

XXXII. To make a right lined Figure equal to a Figure given.  
Let the given Figure be ABCDE.  
1. Take any one of the points, which you please, A for instance:  
2. From A make as many Triangles as there are opposite Angles in the Figures ABC, ACD, ADE.  
3. Then take a Line at pleasure, of the length of AB, as FG.  
4. From the point G, with the distance BC, make an Archippus  
5. With the distance AC intersect this Arch, from F in H, and join GH together.  
6. From the point H, with the distance CD make another Arch, and intersect it from F in I, with the distance AD, and this must be joined to H.  
7. Take A, and from this point F make the last Arch; intersect it from I in K, with the distance DE, and join it with IF: so the Figure FGHIK, will be equal to the given Figure.    

CHAP. III. Of BODIES.  
PROBLEMS.  

I. To make an equilateral Tetraëdron.  
1. MAke an equilateral Triangle ABC.  
2. Place another equilateral Triangle upon every side, as ABE, BCD, & ACF, and your Tetraëdron is made.   

II. To make a Cube.  
1. Make an equilateral right-angled Square ABCD.  
2. Upon every side place another Square equal to the first, as ABEF, BCGH, DCIK, ADLM.  
3. Join OF, GH, IK, and LM together by right Lines.  
4. Place another Square equal to the rest upon one of these four, as KINO, and the Cube will be made.   

III. To make a Parallelipiped.  
1. Make a Parallelogram ABCD, and lengthen the sides AC, BD, above and below.  
2. Mark any distance above AB you please, as A, BF; and mark the same under CD as CG, DH, then join E, F, and G, H, together.  
3. Above OF, mark the distance AC or BD, at I and K: and join I, K, together.  
4. Lengthen OF and AB; upon the lengthened Lines mark the distance AC, or BD, as FL, BM, EN and AO.  
5. Join L and M, N and O, together, and your Parallelipiped will be made.   

IV. To make a Cylinder.  
1. Make a long Square ABCD.  
2. Divide AB into twenty two equal parts.  
3. Upon what Point you please raise a Perpendicular, of the length of seven of these twenty two parts, as F.G.  
4. Upon the Line DC mark the distance FB, as DH.  
5. Let fall a Perpendicular from H of the length of FG as HI.  
6. Divide FG and HI into two equal parts: and from their middle Points draw Circles: and the Cylinder will be made.   

V. To make a Cone.  
1. From the Point A taken at pleasure draw an Archippus  
2. Divide it into twenty two equal parts; and join the Extremities G, H, with A.  
3. Let fall a Perpendicular where you please from the Arch, equal to seven of those parts as BC.  
4. Divide BC into two equal parts: from its middle point, draw a Circle, which shall be a Base of a Cone made upon the Arch GH.   

VI To make a Pentaëdral Prism.  
1. Make a right-angled Parallelogram ABCD.  
2. Divide AB and CD into three equal parts, as A, OF, FB, and CG, GH, HD.  
3. Join EGLANTINE and FH together by right Lines.  
4. From E, and F make two Arches to intersect in I, and join IE, and IF together.  
5. Upon GH make an equilateral Triangle K, and your Prism will be made.   

VII. To make an Octaëdron.  
1. Upon one Line make three equilaterial Triangles, as ABC, BDE, and DFG.  
2. Lengthen the Line CG; and mark upon the lengthened Line the distance EGLANTINE, as GH.  
3. Join H, F together, and with the same distance from D, F, make 2 Arches to intersect at K, and from C, E, two others to intersect at I.  
4. Join CI and IE together, as also DK and FK, and the Octaëdron will be finished.   

VIII. To make a Dodecaëdron.  
1. Make a regular Pentagon ABCDE: the Centre whereof may be F.  
2. Take the distance OF, and therewith from every side draw two Arches mutually intersecting, as ABG, BCH, CDI, DEK, and EAL.  
3. From the Points G, H, I, K, L, with the distance AB, describe five Circles, and divide each of them into five equal parts: and you will have five other equal Pentagons about your first.  
4. With the distance HIM or HN from M and N draw two Arches to intersect each other in O: from this Point, by the same distance, draw another Circle through N and M, and divide it as you did the other into five equal parts, and as M, N, P, Q, R, and there join them all together.  
5. Take the distance ON, and from P, and Q draw two arches to intersect each other in S: which is a Centre from whence a Circle shall be drawn through PQ, which shall afterwards be divided into five equal parts, QP, PW, WU, UT, and TQ.  
6. Repeat the same operation upon TQ, TU, UW, and WP, and your Dodecaëdron will be finished.   

IX. To make an Icosaëdron.  
1. Make five equilateral Triangles upon the same Line, as ABC, BDE, DFG, FHI, and HKL.  
2. From the Points AC make two arches which shall intersect each other in M.  
3. Join AM and ML together by right Lines.  
4. Upon MC raise the equilateral Triangle N, upon CE, O; upon EGLANTINE, P: upon GI', Q; and lastly upon ILL, R.  
5. Repeat the same operation upon their Bases: for instance, set the equilateral Triangle S upon AB; the Triangle T upon BD; the Triangle U upon DF; W upon FH; and at last the Triangle X upon HK: which will finish the Icosaëdron.   

X. To make a solid Rhomb.  
1. Upon the same Line make four equilateral Triangles. GIM, IKN, KLO, LHP.  
2. Join M, P together by a right Line: as also GM, IN, KN, LO and H.P.  
3. Add to the Line NI its own length beyond it, as IQ.  
4. Add to the Line OK double its own length, as KR, RS.  
5. Join RI and SQ together.  
6● Add to the Line IR it's own length beyond it as RT: do the same to NK, as KU: then join U, T, together, and your solid Rhomb is made.   

XI. To make a solid Rhomboid.  
1. Make an equilateral Triangle ABC, and from BC another D.  
2. Add to the Line CD twice it's own length, as DE, EF.  
3. Add to DB its own length, as BG.  
4. Add to CB thrice its own length, as BH, HI, IK.  
5. With the distance IB from the point G make an arch.  
6. Make another arch with the distance BG from I which shall intersect the first at L, and join G, L and L, I by right Lines.  
7. Add to GH thrice it's own length; HIM, MORE, & OPEN, and join M and I together.  
8. From K, and M draw two Arches to intersect at N, and join KN, NM, MD, OE, and PF together: and the Rhomboid will be finished.   

XII. To make a Polyëdron whose Base may be a Pentagon.  
1. Make a Square ABCD: and divide AB and CD into five equal parts, as CE, OF, FG, GA', HD; as also AI, IK, KL, LM, and MB.  
2. Join EI, and FK, and GL, and HIM, and DB together.  
3. Divide one part of each of these Lines into twenty two equal parts: as FG and KL.  
4. Divide one of those twenty two parts into three other equal parts.  
5. Take eighteen of these twenty two parts, and 2/● of one more, as FN, and KO; and draw two arches from K, L, as also from F, G, which shall intersect in P and Q: from P, and Q as Centres draw two Circles, which being divided into five equal parts by the distances FG and KL, your Figure will be completed.    

CHAP. IU. Of PYRAMIDS.  
PROBLEMS.  

I. To make a Triangular Pyramid.  
1. FRom the point A taken at pleasure draw an arch, which divide into three equal parts, BC, CD and DE.  
2. Join AB, AC, AD and A with right Lines; as also BC, CD, and DE with other right Lines.  
3. From the points C, D, with the distance CD, draw two Arches to intersect in F.  
4. Join CF and DF, and the Pyramid will be completed.   

II. To make a Pyramid with an equilateral Square for its Base.  
1. From the point A draw an Arch: divide it into four equal parts, BC, CD, DE, and OF, which are to be joined together by right Lines, as also AB, OF, AC, AD, and A  
2. From the point C, let fall a Perpendicular equal to CD, as CG.  
3. With the distance CG from G, D, draw two Articles to intersect each other in H.  
4. Join HG and DH together by right lines and you have the required Pyramid.   

III. To make a Pyramid whose Base is a Parallelogram.  
1. From the point A draw a circle, in which first mark a larger arch, as BC, than a smaller one, as CD, then mark a third arch (DE) equal to the first, and a fourth (OF) equal to the second.  
2. Join AB, AC, AD, A, OF, as also BC, CD, DE, OF by right lines.  
3. From C let fall a Perpendicular equal to CD, as CG.  
4. Take the distance CG, and from B draw an arch.  
5. Take the distance BC, and from G draw another arch to intersect the first in H; join BH and HG together, and the Problem is finished.   

IV. To make a Pyramid with an Equilateral Pentagon for its Base.  
1. Draw an arch from A, which divide into five equal parts, BC, CD, DE, OF, FG.  
2. Join them all to A, and one to another, B to C, C to D, D to E, E to F, F to G, by right lines.  
3. Divide one of these five parts, as DE, into twenty two equal parts, and one of them into three.  
4. Take eighteen of the twenty two parts, and two of the three little parts, and from the points DE, make two arches to intersect each other in H, which is the Centre, from whence you are to draw a circle big enough to be divided into five parts, equal to the distance DE. When you have this Pentagon for a Base, the work is done.   

V. To make a Pyramid, which shall have an Equilateral Hexagon for a Base.  
1. Draw an arch from A, and divide it into six equal parts, B, C, D, E, F, G, I  
2. Join them to the centre A, as also B to C, C to D, D to E, E to F, F to G, G to I.  
3. Take one of these six parts (DE) and from D and E make two arches to intersect in H.  
4. From H draw a circle big enough to be divided into six parts equal to DE.  
5. Having this Hexagon as a Base for the Pyramid, the Problem is wrought.   

VI To make a Pyramid with a Heptagon for its Base.  
1. From A draw an arch, and divide into seven equal parts, B, ●, D, E, F, G, H.  
2. Join them to A, and also B to C, C to D, D to E, E to H, H to G, G to F, F to I, by right lines.  
3. Divide one of these parts, FG for instance, into seven equal parts, and then one of these seven into eight more; then take eight of the larger parts, and the half of a little one, and with that distance from the points FG, make two arches to intersect each other in K; for which as a centre draw a circle through FG, which must be divided into seven parts equal to FG, which when joined together, will serve for a Base of the Pyramid required.   

VII. To make a Pyramid with an Octagon for its Base.  
1. From A draw an arch, divide it into eight equal parts, B, C, D, E, H, I, F, G, join these points one to another, and also to A by right lines.  
2. Divide one of these parts, as FG, into seven equal parts, and one of these seven into seven others.  
3. Take nine of the larger, and one and a half of the smaller parts, and with that distance from F and G, make two arches to intersect each other in K.  
4. From K through FG draw a circle, divide it into eight parts equal to FG, and then join them by right lines: so you'll have an Octagon to be a Base to the Pyramid required.   

VIII. To make a Pyramid with an Enneagon for its Base.  
1. From A draw an arch, and divide it into nine equal parts, B, C, D, E, F, G, H, I, K; join these together, and with A by right lines.  
2. Divide any one of these parts, as KM, into seven equal parts, and one of them into seven more.  
3. Take ten of the former, and two of the latter, and from the points KM make two arches to intersect each other in L, with the length of those parts, as a given distance.  
4. From L draw a circle through KM, which must be divided into nine parts equal to K M; join these together by right lines: so you have the Enneagon which was wanted for a Base for the Pyramid required.   

IX. To make a Pyramid with a Decagon for its Base.  
1. From A draw an arch, which divide into ten equal parts, to be joined to each other, and to A, as B, C, D, E, F, G, H, I, K, L.  
2. Divide one of these ten parts into seven, and one of these seven into eight more.  
3. Take eleven of the former, and two and a half of the latter, and with that distance ●rom IK, draw two arches which shall intersect each other in M.  
4. From M draw a circle through IK, which divide into ten parts equal to IK, and there will be a Decagon for a Base to the Pyramid.   

X. To make a Pyramid with an Endecagon for its Base.  
1. From A draw an arch, and divide it into eleven equal parts, to be joined each to other, and all to A, as B, C, D, E, F, G, H, I, K, L, M.  
2. Divide one of them (FG) into seven equal parts, and one of these seven into seven more.  
3. Take twelve of the former, and one and a half of the latter, for a distance, whereby from FG to draw arches which shall intersect each other in N.  
4. From N draw a circle through FG, which may be divided into eleven parts equal to F G● join them together, and you have the Base of the required Pyramid.   

XI. To make a Pyramid with a Dodecagon for its Base.  
1. From A draw an arch, which divide into twelve equal parts, which are to be joined to A, and each to other by right lines, as B, C, D, E, F, G, H, I, K, L, M, N.  
2. Divide any one (FG) into seven equal parts, and one of these seven into seven others.  
3. Take thirteen of the former, and one and a half of the latter, for a distance whereby from FG to draw two arches which shall intersect each other in O.  
4. From O draw a circle through FG, and divide into twelve parts equal to FG, which when joined by right lines, will give a Base of the Pyramid that was required.     

BOOK II.  
AS all Rivers have their particular Branches, and every general Science its Divisions, and its Parts; so Geometry, generally taken, is the Mother of several other Arts and Sciences, whereof we shall treat in this Second Part. These are, Measuring of Heights, of Distances, of Plains, of Solids, of Concaves, and the turning of one Body into another.  

CHAP. I. Of Measuring of Heights.  
MEasuring of Heights is a Science which shows how they are to be measured● But because there are several Instruments for that purpose, as a Quadrant, a Pantometre, an Astrolabe, a Iacob's Staff, etc. We shall at present only make use of the Astrolabe, because it is the surest and the most convenient: and we shall know how it is to be used by the following Problems.  

PROBLEMS.  

I. To take the Vertical Height of any thing, when you can come to the bottom.  
Let the height of a Tower AB be to be measured.  
1. Draw a right Line as you please from the point A, a● AC.  
2. Measure the Line AC with your Scale, or your Chain● of 26 Feet for instance.  
3. Set the Staff, upon which the Astrolabe is Horizontally fastened, upon C, so that by the fixed Sights, one may see the Line CA, from one end to the other, as fas as the Tower, and that by the two other movable sights the point B, which marks the height of the Tower, may also be seen.  
4. When the Instrument is thus fixed, compute whether the Angle CDE be 40 degrees.  
5. Then draw upon Paper a right Line of the length of AC, as FG: But this must be done by a small Scale made upon the Paper, or by a Scale of Wood or Latin: Afterwards erect a Perpendicular upon G.  
6. Take the Reporter [the French word is Rapporteur] and set it directly upon FG, so that the middle of it may answer to the point F: then compute an Angle of 40 Degrees equal to CDE, as FHI.  
7. From F draw a right Line thro● I, ●ill it touches the Perpendicular G in K.  
8. With a small Scale measure the height of KG, and add the height of the Staff to which the Astrolabe is fastened, and the Sum will give you the height of the Tower AB.   

II. To measure a Vertical Height where you cannot come at the bottom.  
Let AB be the height to be measured; the foot inaccessible, because of a River which hinders all approaches.  
1. Draw a right Line upon the Ground, on this side of the River, as CD; let it be 22 Foot long: (Take notice, when you can raise a right-angled Triangle upon this Line, opposite to the height● which you would measure, it is much more convenient, otherwise the Operation is the same.)  
2. Set the Instrument upon C, so that by the immovable ●ights you may see the point D directly, and by the two movable ones you may see the point A beyond the River: When this is done, compute the Angle CEF, which may be of 90 Degrees.  
3. Set the Instrument upon D, so that the fixed sights may make one Line with DC; by the two others you will see the point A: this Angle ought to have 28 Degrees.  
4. Draw upon Paper a Line equal to that drawn in the Field (CD) as GH of 2● Feet.  
5. With the Reporter take an Angle at H equal to the Angle CEF, viz. of 90 Degrees, as IHK, and lengthen the side HK. Afterwards upon the Point G take another Angle of 28 Degrees, equal to the Field-Angle DOP, as GLM, and then lengthen the side GM till it cuts the other HK, in N, and this Point N will mark the foot of the height A.  
6. Having the distance from the Instrument to the foot of the Tower, it will be an easy thing to work the rest by the foregoing Problem.   

III. To measure a Perpendicular Height, where you can come at the bottom, with the shadow of a Staff.  
1. Fasten a Staff a foot long perpendicularly into the ground, as AB, and measure the shade by the Staff, 3 Foot for Instance.  
2. With the shade of the Staff, viz. 3 feet, measure the shade of the House DE, whose height you would know; and as many feet as the shade of the House exceeds the shade of the Staff, so many foot high will the House be: For Instance, if the shade of the House be 24 times longer than that of the S●aff, than we ought to say that the House is 24 Foot high.  
Or thus, 

1. Erect a Stick, divided into 10 equal Parts, Feet, or Inches, perpendicularly in the ground, and at the same time measure the shade caused by this Staff, 20 Foot for Instance.  
2. Measure also the shadow of the House, whose height you would know, for Instance 38 Feet.  
3. By the Rule of Three say, If the shadow of 20 Foot gives a height of 10 Foot, what will the shadow of 38 Foot give? Answ. 19 Foot.     

IV. To measure the Inclination of a Mountain.  
Let the Inclination of the Mountain AB be to be measured.  
1. Draw from the Point A, which stands for the foot of the Mountain, a right Line, 36 Foot long, as AC.  
2. Set the Instrument upon the Point A, and look through the two fixed sights towards the Point C, and through the two movable ones towards the Point B, the top of the Mountain; this Angle, for Instance, may be 110 Degrees.  
3. Then set the instrument upon C, and look through the two fixed sights upon the foot A, and through the two other fixed ones upon the top B: then compute this Angle, which may be for Instance 40 Degrees.  
4. Afterwards, draw a Line upon Paper, 36 Foot long, because it ought to be equal to the Line AC, as DE.  
5. Then by the Reporter set upon D an Angle equal to that of the Figure at A, viz. 110 Degrees, as DFG.  
6. Upon the Point E, set another Angle of 40 Degrees, equal to that at C, as EH●.  
7. Lengthen the sides DF and EI, and where these lengthened Lines intersect one another, as at K, draw from that Point to D a right Line, which will represent the Inclination which was required, provided the height of the Staff upon which the Astrolabe was fastened, be added.   

V. To measure the vertical height of a Mountain.  
This Problem must be wrought as the foregoing one, provided that you let fall a Perpendicular from the Point K, which marks the top of the Mountain, which must touch the Line CD lengthened to L; the Line KL will represent the vertical height of the Mountain.   

VI To measure the height of a Tower built upon a Rock.  
Let the height BD be to be measured.  
1. Seek the inclining height of the Mountain by Prob. 4. and let the Triangle ABC demonstrate its Operation.  
2. Make the Triangle HIK upon Paper equal to the Triangle ABC.  
3. Measure the Field-Angle CBD, for Instance 38 Degrees.  
4. Upon K, which represents the Point B, and so by consequence the top of the Mountain, raise a Perpendicular of what length you please.  
5. With the Reporter raise upon the Point I an Angle equal to the Angle of the Figure ABCD, viz. ILM.  
6. Lengthen the side IN till it touches the Perpendicular raised upon K in N.  
7. Measure the Line NK, it will give you the height of the Tower BD, if you add to it the height of the Staff.   

VII. To measure the depth of a Well.  
1. Measure its Diameter AB, 9 Foot for Instance; and find the Angle BCD of 90 Degrees.  
2. Set the Instrument so upon A, as that by the two fixed ●ights you may see the Point B, and by the two movable ones you may see the bottom of the Pit E, and let this Angle be 60 Degrees, as AHI.  
3. Draw a Line 9 Foot long upon Paper, which shall represent the Diameter AB, as FG.  
4. From F let fall a Perpendicular which may represent the depth BE.  
5. Make an Angle at G equal to the Angle AHI, viz. of 60 Degrees, as GKL.  
6. Lengthen the side GL, and where this lengthened Line touches the Perpendicular F, as in M, measure from M to F, and you will have the depth of the Well.     

CHAP. II. Of Measuring of Distances.  
MEasuring of Distances is a Science which teaches to measure the Distance of one place from another, in an open Field.  

PROBLEMS.  

I. To measure the Distance of one place from another, where they are both accessible.  
Let the Distance from A to B, be to be measured.  
1. From A draw a right Line, what you please, and measure with a Chain or Scale; suppose 22 feet in length, as AC.  
2. Set the Staff upon which the Astrolabe is fixed, upon the Point A, so as that by the two fixed sights, you can only see directly forward towards the Point C, and by the two others the Point B: then count the Angle ADE, suppose 90 Degrees.  
3. Remove the Instrument upon C, and look through the fixed sights towards A, and through the movable ones towards B; then compute this Angle CFG, suppose 22 Degrees.  
4. Draw a Line upon Paper 22 feet long, equal to the Line AC, as HI.  
5. With the Reporter upon the Point H draw an Angle equal to the Angle ADE, that is to say, of 90 Degrees, as HKL: do the same thing at I, where make another Angle equal to the Angle CFG, i. e. of 22 Degrees, as IMN.  
6. Lengthen the sides HK, TM, and from the Point where the●e two lengthened Lines intersect each other, as O, to the Point H, measure the Distance, which shall be equal to the length AB.   

II. To measure the Distance of two places, whereof one is inaccessible.  
Let the Distance from A to B be that which is required, and B inaccessible, because of a River which hinders all approaches.  
1. From A draw a right Line 22 Feet long, as AC.  
2. Set your Instrument upon A, and look through the two fixed sights upon the Point C, and through the two movable ones upon B: then compute the Angle ADE of 93 Degrees.  
3. Carry the Instrument to C, and look through the two fixed sights upon the Point A, and through the two others upon B, and then compute the Angle CFG of 33 Degrees.  
4. Draw upon Paper a Line equal to AC, i. e. of 22 feet, as HI.  
5. Upon H make an Angle equal to the Angle of the Figure ADE, i. e. of 93 Degrees, as HKL: So also upon I make another Angle equal to the Angle CFG, as IMN.  
6. Lengthen the sides HK and TM, and in the place where they intersect, as at O, let fall a Perpendicular upon H, which will give you the Distance AB.   

III. To find the Distance of two places which are both inaccessible.  
Let the Distance AB be to be measured.  
1. Take a fixed Point C, from which draw a right Line CD, which shall be 33 Foot long.  
2. Set the Instrument upon C, and look through the two fixed sights upon the Point D, and through the two others upon the Point A: then compute the Angle CHK for Instance of 90 Degrees.  
3. Leave the Instrument as it is, and remove the movable sights, till through them you may see the Point B, and then compute that Angle, for Instance 36 Degrees, as CHI.  
4. Carry the Instrument to D, and ●it it so, as that through the fixed sights you may see the Point C, and through the two other movable ones the Point B: then compute this Angle, which may be of 100 Degrees, as DEG.  
5. Leave the Instrument as it is, and turn the movable sights till you may see the Point A through them; then compute that Angle, of 30 Degrees, as DEF.  
6. Afterwards draw a Line of 33 foot upon Paper, to represent the Line CD, as LM.  
7. With the Reporter set two Angles at L, one of 90 degrees, and the other of 36 degrees, so that the first LNO may represent the Angle CHK, and the other LNP the Angle CHI.  
8. Lengthen the sides LO and LP.  
9 Add two Angles to the other Point N, one of 100 degrees, as MQR, to represent the Angle of the Figure DEG, and the other of 30 degrees, as MQS, to represent the Angle DEF.  
10. Lengthen the sides MR, MS, and observe where these two lengthened Lines touch the two first, as at TU. For this distance TV will give you the distance AB, which was required.   

IV. To measure the breadth of a River with a Staff.  
1. Fix a Staff perpendicularly upon the bank of a River, as A.  
2. Cleave it at the end, and fasten a cross staff into the cleft, as BC.  
3. Stand behind the small cross-staff B●● and use it as you would the sights of an Astrolabe, i. e. move it downwards, till from BC you can see the other ●ide of the River at PA.  
4. Afterwards letting the small Staff stand as it is, turn the large one perpendicularly, till from BC you can see the whole length of the Bank: Then stand behind B, and observe the last place where from C you can see the Bank on the other side, as at E.  
5. Measure the distance EA, it will give you the breadth of the River AD.   

V. To describe a Figure taken from the Field upon Paper.  
Let the Figure to be described be ABCDE.  
1. Set the Staff to which the Astrolabe is fixed, upon the Point E, and look upon the Point D through the fixed sights, and through the two movable ones upon the Point A. Then compute this single Angle EAD, 60 degrees for Instance.  
2. Measure the Line A of 23 feet.  
3. Measure another Line ED of 24 feet.  
4. Set the Instrument upon D, and look towards C with the fixed sights, and with the movable ones towards E: Then compute this Angle, of 70 degrees for Instance, as DEC.  
5. Let the side DC be of 28 feet.  
6. Seek the Angle Cdb after the aforesaid manner, and let it have 50 degrees.  
7. Measure the Line CB, for Instance, of 13 feet.  
8. The side BASILIUS will be found of itself.  
When this is done in the Field, than 

1. Draw a Line upon Paper 23 foot long, to represent the Figure A, as KF.  
2. With a Reporter make at the Point F and Angle equal to the Angle of the Figure Eda, i. e. of 60 degrees, as Fkg.  
3. Lengthen the side of this Angle Fg, and add its length to it, i. e. 24 feet, as FG.  
4. Upon G make an Angle equal to that of the Figure Dec, i.e. of 70 degrees, as Gfh.  
5. Lengthen the side Gh, and add the side DC, i e. of 28 feet, as GH.  
6. Upon the Point H set another Angle equal to that of the Figure Cdb, of 50 degrees, as Hgi.  
7. Lengthen the side Hi, and add 13 feet to it, as HI, which may represent the side CB.  
8. Join IK together by a right Line, which will give the last side of the Figure BASILIUS, and which at the same time will entirely represent the given Figure upon Paper.     

VI To describe upon Paper a Figure taken from the Field, which is inaccessible.  
Let the Figure ABC be to be described upon Paper from the Point D.  
1. From D draw a right Line DE 32 foot long.  
2. Set the Instrument upon D, and look through the fixed sights towards E, and through the two movable ones towards A: Then compute the Angle Dfg, of 105 degrees.  
3. Leave the Instrument as it is, and remove the movable sights till you can see the Point B, and compute the Angle Dgh of 90 degrees.  
4. Leave the Instrument still as it is; look through the movable sights towards the Point C, and compute the Angle Dgi of 60 degrees.  
5. Set the Instrument upon E, and look through the fixed sights towards the Point D, and through the movable ones towards A, and compute the Angle Ekl, for Instance of 50 degrees.  
6. Leave the Instrument as it is, and look through the movable sights towards B, and compute this Angle Ekm of 70 degrees.  
7. Look through the same sights towards the Point C, and then compute the Angle Ekn, ●or Instance of 90 degrees.  
This being done in the Field, than 

1. Draw a Line of 32 feet upon Paper, as OPEN, equal to the Line DE.  
2. Take an Angle of 105 degrees at O, as Oqr, which may represent the Angle of the Figure Dfg, and lengthen the side Oq.  
3. Leave the Reporter as it is, and compute another Angle of 90 degrees, as Orseolo, to represent the Angle Dgh, and lengthen the side Os.  
4. Seek the last Angle of 60 degrees Ort, which may represent the Angle of the Figure Dgi, and lengthen the side Ot.  
5. Set the Reporter upon the Point P, and compute an Angle of 50 degrees, to represent the Angle Ekl, as Pab; lengthen the side Pb till it meets with the side Oq in C: for this point of Intersection will mark the Point A.  
6. Leave the Reporter as it is, and compute another Angle of 70 degrees, which may represent the Angle Ekm, as Pad; lengthen the side Pd till it cuts the lengthened Line Os in E, which Point will mark the Point B in the Figure.  
7. Compute another Angle of 90 degrees equal to the Angle Ekn as Paf; lengthen the side Pf till it meets with the lengthened Line Ot in G, which answers to the point C in the Field-figure.  
8. Join CE, EGLANTINE, and GC together, and you shall have your Figure ABC represented upon Paper.       

CHAP. III. Of measuring of Plains.  
MEasuring of Plains is a Science which teacheth us to measure all sorts of Surfaces of Things, which Geometricians call Area's.  

PROBLEMS.  

I. To measure the Area of a right-angled Triangle ABC.  
1. Measure the Line AB, 12 foot long for the Base.  
2. Measure the Perpendicular BC, of 8 foot.  
3. Multiply the half of the Base upon the whole perpendicular, or the half of the perpendicular upon the whole Base, and the Sum, (48 foot) will be the Area of the Triangle which was desired.   

II. To measure the Area of an Oblique-angled Triangle DEF.  
1. Let fall a perpendicular from F to its opposite side DE, which may divide the Triangle into 2 right-angled Triangles, DGF, and FGE.  
2. Measure these two Triangles by the foregoing Problem; i. e. multiply the whole Base DG and GE, by the half of the Perpendicular FG, or the whole Perpendicular by the half of the Base DG and GOE  
Thus for instance: Let the Base DG be 12 foot, and the Perpendicular FG 10: Take the half of the Perpendicular, i.e. 5 feet, and multiply it by the Base of 12 feet, and the sum will be 60 feet, which gives the Area of the Triangle DFG.  
3. Let the Base GE of the Triangle FGE be six feet, and the Perpendicular FG stands as it did, i. e. to feet. Multiply 5 by 6, or or 3 by 10; the sum will be 30 feet, which will give the Area of the Triangle. Then add 30 to 60, the sum 90 will be the number of feet in the whole Area of the given Triangle DEF.   

REMARKS.  
Sect. 1. If one of the Lines to be multiplied has only fathoms, and the other fathoms and feet; when you multiply, instead of a fathom take 6 foot. For instance: if you are to multiply 7 fathoms by 8 fathoms and 4 feet, instead of the first sum (seven fathoms) you must put 6 fathoms and 6 feet, because these 6 feet make but one fathom: For since there are no feet expressed, you cannot any otherwise multiply the feet of the second sum: So the number will be 48 fathoms, and 24 feet: for 6 into 8 gives 48: and 6 into 4 gives 24.  
Sect. 2. If one of these lines has only fathoms, and the other has fathoms, inches, and feet; for instance, if you are to multiply 6 fathoms by 8 fathoms, 4 feet, and 6 inches, this is the Method you are to take: 

1. Bring these 8 fathoms, 4 feet, and 6 inches, into one Line.  
2. Instead of 6 fathoms, take 5 fathoms and 6 feet; and put the first cipher 5 under 8, because they both mark fathoms: take one out of the second cipher 6, so as that there may be but 5 feet to put under the feet of the first sum: divide the foot which you substracted into 12 inches, for 12 inches are equal to a foot: then set the inches under the 6 inches of the first sum.  
Sect. 3. Reduce the feet into fathoms, dividing by 6; divide also the inches by 12, and it will give you the feet: for instance, 72 inches gives 6 feet, and 20 feet give 3 fathoms and 2 feet.     

III. To measure the Area of an equilateral right-angled Square ABCD.  
Multiply one side by the other: AB for instance of 12 feet, by BC of 12 feet: because the sides are equal: the product 144 feet will give you the Area which is sought for.   

IV. To measure the Area of a Parallelogram ABCD.  
Multiply the shortest side BC of six foot, by the longest side AB of 12 foot: the product 72 foot will give you the Area of the Parallelogram which was required.   

V. To find the Area of a Rhomb, ABCD.  
1. Let fall a Perpendicular from A to the Line DC, and A  
2. Let fall another Perpendicular from B, equal to A, as BF; and join CF together. So that instead of a Rhomb you have a Parallelogram ABEF: then work by the foregoing Problem. See Prob. 28. Book. 1. Chap. 2.   

VI To find the Area of a Rhomboid ABCD.  
1. Let fall a Perpendicular from A to the opposite Line DC, as AF.  
2. Let fall another from B, equal to OF, as BE.  
3. Join CE together, and instead of the Rhomboid ABCD you have a Parallelogram ABEF: You will find its Area by the forth Problem of this Chapter.   

VII. To find the Area of any unequilateral Quadrangle.  
Let the given Quadrangle be ABCD.  
1. Divide this Quadrangle into two Triangles ABDELLA, and CDB.  
2. Divide each of these Triangles into two right-angled ones, as ABF, BE, ADF, DEC.  
3. When you have these 4 right-angled Triangles, search their Area's by the two first Problems of this Chapter: then join the Sum of all these Triangles together, and it will give the Area which is required.   

VIII. To find the Area of any regular Polygon.  
Let the regular Pentagon demanded be ABCDE.  
1. Measure one side of the Polygon as AB, for instance, 6 fathoms: multiply this number by the number of the sides of the Polygon, as here by 5, because this Polygon has but 5 sides: the product of this multiplication will give you the length of all the sides.  
2. Let fall a Perpendicular from any side to F the Centre of the Polygon, as FG: then measure this Line, 5 fathoms for instance.  
3. Multiply the sum of all the sides, by the half of the Perpendicular: the product will give the Area which is required.   

IX. To find the Area of any irregular Polygon ABCDE.  
1. Divide the whole Figure into right angled Triangles; as this irregular Pentagon ABCDE into ABG, BCG, ADH, CDH, AEF, and DEF.  
2. When you have only right angled Triangles, work by the first Problem of this Chapter; the sum of all the Triangles reduced to one, gives the Area which is fought for.   

X. To find the Area of a Circle.  
Reduce this Circle to a right angled Triangle, by Prob. 21. Book 1. Chap. 2. and then seek the Area by the Rules of the first Problem of this Chapter.   

XI. To find the Area of an Oval Figure.  
1. Divide the proposed Figure into four equal parts, as ABCD, so as that the line AB may make one Diameter, and CD another.  
2. Seek for a mean proportional between these two Diameters, which you may thus find:  
Set the Diameter AB upon a long line, and upon the same line from the point B, set the other Diameter CD, as BC; then take the half of the distance between A and C, and carry it on to B, as EBB; this shall be a mean proportional; then divide EBB into two equal parts, as OF, FB; then draw a Circle from the point F ●hro the point EBB, which shall be equal to the given Oval Figure.  
3. Having this Circle, find its Area, by the foregoing Problem; i e● raise a Perpendicular upon B, one extreme point of the Diameter, which shall have thrice the length of the Diameter EBB, and a seventh part over, as BG.  
4. Join F the Centre of the Circle to the point G by a right line, which will give a right angled Triangle equal to the given Circle; it's Area may be found by the Rules of the first Problem of this Chapter.  
[This Triangle which he says is equal to the Area of the Circle is not expressed in the Scheme.]   

XII. To measure the Surface of any Equilateral Pyramid.  
1. Let ABCD be an equilateral Square, which shall be the Base of the Pyramid given.  
1. Find the Area of this Base by Prob. 3. of this Chapter, of 48 foot.  
2. Multiply one of its sides, as AB, by the height of the Pyramid OF, that is 12 by 28, whose product will be 336 feet.  
3. Multiply 336 by 4, the number of the sides of this Pyramid, and the product will be 1344 feet.  
4. To this sum add the Area of the Base, 48 feet, and you will have the total sum of the Area of the Pyramid proposed, viz. 1392 feet.   

XIII. To find the Convex Surface of a right lined Cylinder.  
1. Set the height of the Cylinder BD upon a long line.  
2. Set upon the same line, beginning at the point D, the Diameter of the Base of the Cylinder marked CD, as DE.  
3. Divide the line EBB into two equal parts, BF, EF.  
4. With the distance FD draw a Circle, which shall contain the Convex Surface of the given Cylinder: this you may find by Prob. 21. Chap. 2. of the first Book, which teaches you to make a right angled Triangle equal to a given Circle; when you have this Triangle, find its Area by Prob. 1. of this Chapter.  
5. When you have the Area of this right angled Triangle, which represents the Convex Surface of the Cylinder, then find the Area of the two Bases of the Cylinder by Prob. 1 of this Chapter● Add the sum of the two Bases to the Area of that Triangle which gives you the Convex Surface of the Cylinder, without counting the Bases, and you will have the general sum of the whole Surface of the Cylinder, with both its Bases. See Archimedes de Sphaerâ & Cylindro, lib. 1. Prop. XIII.   

XIV. To find the Convex Surface of a regular Cone.  
Let the given Cone be ABDELLA, and its diameter AD.  
1. Set the height of the Cone DB upon a ●ight line.  
2. Set its semi-diameter CD upon the same line, as DE.  
3. Divide EBB into two equal parts, OF, FB.  
4. Take the distance FD, and make a circle, as large as the Cone without the Base.  
5. Reduce this circle to a right angled Triangle by Prob. 21. Chap. 2. of the first Book, and find its Area by Prob. 1. of this Chapter.  
6. By the two forementioned Problems seek the area of the base of your Cone, and add this sum to the former; the sum total will give the whole surface of the given Cone. See Archimed. Lib. cit. Prop. XIV.   

XV. To measure the Convex Surface of a Globe.  
1. Measure the diameter AB, of nine feet.  
2. Find the circumference of the circle, whereof that is the diameter, which is always thrice as long, and a third part over.  
3. Multiply the diameter by the circumference; the product will give you the required surface of the given Globe.   

XVI. To find the Area of any Trapezium, as ABCD.  
You must work by the Rules of Problem 7. of this Chapter.   

XVII. To find the Area of the Base of a round Tower, where you can only come at one part.  
1. Measure the line AB, of fourteen fathoms, and from its middle point draw a perpendicular to the arch, as DC, of four fathoms.  
2. Multiply the distance AD by the remainder DB; i. e. 7 by 7; then divide by 4 the number of the perpendicular DC.  
3. Add to this the length of the perpendicular, i.e. four fathoms, and it will give you the length of the diameter, to complete the base.  
4. When you have found the base, seek for a right angled Triangle, equal to it, by Prob. 21. Chap. 2. Book 1; afterwards you may find the area of that Triangle by Prob. 1. of this Chapter.     

CHAP. IU. Of Measuring of Solids.  
MEasuring of Solids is a Science which shows how one is to find the Solidity of Bodies: Solidity is Magnitude with Breadth, Length, and Thickness.  

PROBLEMS.  

I. To find the Solidity of a Parallelepiped.  
1. Measure the height, length and breadth of the Parallelepiped.  
2. Multiply the breadth AD, of two feet, by the length AB, of four feet, and the sum will give you the area of the Base.  
3. Multiply the number of this Area by the height of the Parallelepiped, which here shall be eight feet; and the product from thence will show you the Solidity required, of sixty four feet.   

II. To find the Solidity of a Prism.  
1. Find the Area of the Base ABC, by Prob. 2. Chap. 3. Book 2. of six feet.  
2. Measure the height CF, here of nine feet.  
3. Multiply the Base by its height, i. e. 6 by 9, and the product 54 will give the Solidity required.   

III. To find the Solidity of a Cylinder.  
1. Find the Area of the Base by Prob. 21. Book 1. Chap. 2. and by Prob. 1. Chap. 3. Book 2. for instance five feet.  
2. Find the height of the Cylinder AB, here of nine feet.  
3. Multiply the Area upon the height, i. e. 5 by 9, and the sum 45 will give the Solidity required.   

IV. To find the Solidity of Pyramids and Cones.  
1. Measure their height FE and AB, for instance, of twelve feet.  
2. Find the Area of their Bases; that of the given Pyramid by Prob. 6. Chap. 3. Book 2. of sixteen feet; and that of the Cone by Prob. 10. Chap. 3. Book 2. of sixteen feet.  
3. Multiply the entire height by the third part of the Base; i. e. twelve ●eet by five feet four inches; or the whole Base by the third part of the height, i. e. sixteen feet by four feet; the sum sixty ●our will give the Solidity of the Pyramid and the Cone; for this is equal to the given Pyramid: and because we are to find its Solidity, that also will be equal to the Solidity of the same Pyramid.   

V. To find the Solidity of a Globe.  
1. Multiply the Semi-diameter AB by the Surface of the Globe, which you shall find by Problem 15. of the foregoing Chapter.  
2. Divide this sum by 3, and the Quotient will give you the required Solidity: for instance, let the Semi-diameter be 28 feet, and the Convex Surface 9856 feet; multiply this by 28, and the product will be 275968; divide that by 3, and your Quotient 91989 1/● will give you the number of feet which make up the Solidity of this Globe.   

VI To find the Solidity of a Tetraëdrum.  
Since a Tetraëdrum is nothing else but a triangular equilateral Pyramid, its Solidity must be sought by Prob. 4. of this Chapter; i. e. by multiplying the Base by a third part of the height, or the height by a third part of the Base; the product will show the Solidity which is required.   

VII. To find the Solidity of an Octaëdrum.  
Since an Octaëdrum is made up of two Pyramids which have one common Base, i e. an equilateral right angled Square.  
1. Find the Solidity of one of these Pyramids by Prob. 4. of this Chapter, for instance of sixteen feet.  
2. Multiply these 16 feet by 2, the product will give the required Solidity of 32 feet.   

VIII. To find the Solidity of a Dodecaëdrum.  
If one draws from the Centre of a Dodecaëdrum a right line to every angular point, one may from twelve equal Pyramids, every one whereof will have a regular Pentagon for its Base; upon which account 

1. Find the Solidity of one of these twelve Pyramids, by the same 4th Problem, and let it be for instance of twelve feet.  
2. Multiply these twelve feet by the the number of the Pyramids, i. e. 12; the product 144 gives the required Solidity.     

IX. To find out the Solidity of an Icosaëdrum.  
This being form of twenty equal Tetraëdra, if one draws right Lines from its Centre to every angular Point, one must, 

1. Find the Solidity of one of these Tetraëdra by the same fourth Problem, for instance of 8 Feet.  
2. Multiply 8 by 20, the number of the Tetraëdra; and the product 160 will show the number of the solid feet in the given Icosaëdrum.     

X. To find the Solidity of a Cube.  
By the third and fourth Problems of the foregoing Chapter, multiply the Area of the Base by the length, and the product gives the Solidity.   

XI. To find the Solidity of a solid Rhomb, and Rhomboid.  
Find the Area of the Base of a Rhomb by Prob. 5. of the foregoing Chapter, and the Area of the Base of the Rhomboid by Prob. 6. of the same: when they are found, multiply them by their respective lengths, as was directed in the foregoing Problem, and the product gives the Solidity required.   

XII. To find the Solidity of a Wall, Rampart, Curtain, etc.  
This is done by Problem 1. of this Chapter.     

CHAP. V. Of Measuring Concave Bodies.  
MEasuring Concave Bodies is a Science which teaches us to find their capacity that so we may know how much they contain: Cylindro-metrical Bodies have need of a Cylindro-metrical Scale: and Cubo-metrical Bodies have need of a Cubo-metrical Scale: So that these two Scales are made use of in the measuring of Concave Bodies.  

PROBLEMS.  

I. To find the Capacity of a Concave Parallelepiped.  
By which I understand all square Bodies; as Chambers, Granaries, Cellars, Towers, Pits, etc. So that if one would know how much they contain, he must, 

1. With his Cubo-metrical Scale measure, the height, length, and depth.  
2. Multiply the length by the breadth, and their product by the height.  
3. Examine how many measures the whole sum takes up upon the Scale, and so the required Capacity may be found.     

II. To find the Capacity of a Ditch.  
1. Measure its breadth at the top, OF, 15 Fathoms and 1 foot.  
2. Measure its breadth at the bottom CD 11 fathoms 5 feet.  
3. Measure its depth CG, 9 fathoms.  
4. Measure its length, 496 fathoms.  
5. Add the 2 breadths together, and multiply the half of the sum by the length of the Ditch; there will be 6438 fathoms.  
6. Multiply these 6438 fathoms by the depth 9 fathoms, and the product will be 57942 fathoms, or 347652 feet, which gives the Capacity required.   

III. To find the Capacity of Columns, Towers, and other Prisms.  
1. With a Cubo-metrical Scale measure the height of the Prism.  
2. Find the Area of the Base.  
3. Multiply that number by the height, and you will find the Capacity.   

IV. To find the Capacity of any regular Pyramid.  
1. Measure the Area of the Base with a Cubo-metrical Scale.  
2. Measure the height likewise, the same way.  
3. Multiply the height upon the Area of the Base.  
4. Divide the Product by 3, and the Quotient will give you the Capacity.   

V. To find the Capacity of a Cone.  
1. Measure the Area of the Base with a Cubo-metrick, or Cylindro-metrick Scale.  
2. Multiply this number by the third of the height.  
3. Divide the Product by 3, and the Quotient will give you its Capacity.   

VI To find the Capacity of a Cylinder.  
1. With a Cylindro-metrical Scale find the Area of its Base.  
2. Multiply that by the height of the Cylinder; the Quotient gives its Capacity.   

VII. To find the Capacity of a Cylinder whose Bases are unequal.  
1. Find the Area of each Base by the Cylindro-metrick Scale.  
2. Add the sums together, and multiply its height by the half of that sum; the Quotient will give its Capacity.   

VIII. To find the Capacity of a Barrel whose Heads are equal.  
We must take notice that the Rod by which a Vessel is to be gauged, is marked with different measures; and so one side of the Gauging●Rod is for the length of a Barrel, and the other for the depth: In answer therefore to the Question proposed, 

1. Measure the Area of the bottom AB.  
2. Measure the thickness with the Gauging-Rod, and by that take the Surface of the Belly of the Barrel.  
3. Add these two sums together, and multiply the half by the length of the Vessel, so the Product gives you its Capacity.     

IX. To find the Capacity of a Barrel whose Heads are unequal.  
1. Measure the Diameter of the Head A, and by this known Diameter, measure its Surface.  
2. Measure the other Diameter OF, and thereby find the Area of that Head.  
3. Add these two sums together.  
4. Find the depth of the Barrel CD, by which you find the Surface of its Circumference; by Prob. 22. Chap. 2. Book 1. and by Prob. 10. Chap. 3. Book 2.  
5. Add the sum of this Surface to the sum of the two Heads; then multiply the half of all these added together, by the length of the Barrel: this will give you the Capacity.     

CHAP. VI Of Transmutation.  
TRansmutation is a Science which teaches us to increase or diminish any given Figures, or to give them another form. We ought to take notice, that what is here said of Figures, may also be understood of Bodies, when one would give them another shape which they had not before: But since we produed somewhat of this matter in Chap. 2. Book. 1. from Prob. 21. to the end of the Chapter, we shall at present only treat of the Transmutation of Bodies.  

PROBLEMS.  

I. To turn a Cylinder into a Parallelepiped of the same height.  
1. Instead of the Base of the Cylinder CBA, make a Square ABCD, by Prob. 23. Chap. 1. Book 1.  
2. Raise upon ABCD four Perpendiculars of the height of AB, which may represent the height of the Cylinder, as OF GH.  
3. Join OF, EH, FG, and GH together by right lines; and you have the Parallelepiped.   

II. To turn a Cone into a Pyramid of the same height.  
1. Make a Triangle, Square, Pentagon, or what other Polygon you please, equal to the Base of the Cone CBD, and let it be the Base of a Pyramid, as ABCD.  
2. Raise from the middle of the Base F a Perpendicular, of the height of the Cone BASILIUS, as FE; then draw from ABCD lines to the point, and the Pyramid is made.   

III. To turn a Parallelepiped into a Cylinder.  
Make a Circle equal to the Base of the Parallelepiped, and raise this Circle to be as high as the Parallelepiped, and the work is done.   

IV. To turn a Pyramid into a Cone.  
1. Make a Circle equal to the Base of the Pyramid.  
2. Raise upon its Centre a Perpendicular as high as the Pyramid, as BASILIUS.  
3. Draw the extremities of the Diameter CD together till they meet at the top of the Perpendicular A, and the Cone will be finished.   

V. To turn a Prism or a Cylinder into a Pyramid or a Cone of the same height, or the contrary.  
1. Enlarge the Base of the Cylinder three times as much as it is, by Prob. 1. Chap. 2. Book 1.  
2. Upon its Centre raise a Perpendicular as high as the Cylinder, and draw the extremities of the Diameter together to the top of the Perpendicular, and the Pyramid will be made: In like manner you may turn a Cylinder or a Prism into a Cone: On the contrary, when you would make a Prism or a Cylinder equal to a Pyramid or a Cone, you must make the Base of the Pyramid or of the Cone three times less, and then upon this Base erect a Cone or Pyramid of the height of the Prism or Cylinder given.   

VI To make a Cube equal ta a Parallepiped.  
1. If the Base of the Parallelepiped be square, find a mean proportional between its height and one side of the Base; this mean proportional will be the true measure of the Cube required.  
2. If the Base be only a Parallelogram, turn it into a Square by Prob. 25. Chap. 2. Book 1. then go on as at first, and your work will be done.   

VII. To make a Cube equal to a Cylinder given.  
1. Make a Parallelepiped equal to a Cylinder given, by Problem 1. of this Chapter.  
2. Afterwards make a Cube equal to that Parallelepiped, by the precedent Problem, and you have the Cube desired.   

VIII. To make a Cube equal to a given Co●e.  
1. Make a Parallelepiped equal to a given Cone, by Problem 5● of this Chapter.  
2. Afterwards make a Cube equal to that Parallelepiped by Problem 6. of this Chapter.   

IX. To make a Cube equal to a Pyramid.  
This Problem must be wrought like the last foregoing.   

X. To make a Cone equal to a Globe.  
1. Take one of the great Circles of your Globe, and make it four times bigger, by Prob. 11. Book 1. Chap. 2.  
2. Raise a Cone upon this Base as high as the Semi-diameter of the Globe AB.  
3. The Cone CDE will be equal to the Globe ABDELLA.   

XI. To make a Cube equal to a given Globe.  
1. Make a Cone equal to a Globe, by the foregoing Problem.  
2. Make a Parallelepiped equal to this Cone, by Prob. 5. of this Chapter.  
3. Lastly, make a Cube equal to this Parallelepiped, by Prob. 6. of the same Chapter.  
Here we conclude our Treatise of Geometry, being persuaded, that the things which we have said, will be enough for those who would make any Progress in Fortification.     
FINIS.   


For Chap: TWO Book: I   

For Chap: 2. Book 1. Geom: ●●: 2.   

For Chap: III Book: 1.   
 

Book: II.   
  

A NEW TREATISE OF Fortification. BOOK I.  

CHAP. I.  

Sect. I. Of the Definition of Military Architecture.  
FOrtification, or Military Architecture, as no other thing than an Art, which teache● Men to fortify themselves with Ramparts, Parapets, Moats, Covert Ways and Glacis', to the end the Enemy may not be able to attack such a part without great loss of his Men; and that th● small Number of Soldiers which defend the Place may be able to hold out for some time.   

Sect. 2. Of the Original of Fortification.  
CErtain it is, that Fortification was Originally very inconsiderable; for in regard that Men had nothing to fear at that time but the Assaults of Wild Beasts, and Robbers they had no other Fortifications than simple Hedges, which continued still in use 328 Years before our Saviour; at what time Alexander the Great found the Hyrcanians and Mardians fortified after that manner, Q. Curtius, l. 6. c. 5. Afterwards they fortified themselves with Walls, those Hedges being easy to be cut down. But Ambition increasing, in regard that they who coveted Dominion over others, soon found a way to leap over these feeble Defences, they added a Moat, and in making this Moat they raised at the same time a Rampart, behind which the Besieged placed himself, to dispierce the Enemy with their Arrows. The Enemy, not able to surmount these Obstacles, resolved to venture some Arrows, to pass the Moat, and lodge himself at the Foot of the Wall, whence the Besieged could not chase him, notwithstanding the great Number of Stones which they threw down to annoy the Assailants, because the Enemies covered themselves with their Bucklers which they held over their Heads. This obliged those within to make Holes in the Walls, by shooting their Arrows at the Enemy to hinder him from attacking them. For which way soever he came, the Arb●lesta annoyed him. Nevertheless the Foot of the Walls were without defence, so that the Enemy lodging himself between the nearest Holes, they bethought themselves of making square Towers with Holes on every side: For they stood at the Hea● of the Walls, to hinder the Enemy from lodging between the Holes in the Wall. True it is, that they soon invented certain Engines which we call at this Day Bartering Ram's, which were strengthened at the Head with a Massy Point o● Iron, and so pushed against the Walls of those square Towers, till they brought them to ruin. After which they made Roundels, which were looked upon as a stronger sort of Fortification, till Powder began to be known in Europe. For we find in History, that in the Y●ar 1380. under the Reign of Wenleslatis, the Son of Charles the IV, a certain Franciscan Monk, whose Name was Berthold Schwarts of Friburg in Brisgau, showed the first use of Powder to the Venetians, then at War with the Genoeses. After which they made Cannon to batter Places, the furious execution of which, ranversed like a Clap of Thunder those feeble Walls; which obliged them that were attacked with these astonishing Engines, to erect a good Rampart behind their Walls, and to make the same Walls much thicker than before.  
This Manner of Fortification would have subsisted a long time, had it not had one defect; which was, that those Round Towers had one part in the form of a Triangle, which could not be seen by those within; and which the Enemy covered to batter, on purpose that he might lodge himself secure from the Shot of the Besieged, who could not see him in that part. To correct this default, they filled that defective part with good Earth, which being environed with a good Wall, formed also two Points which looked toward the Field, as now the Faces of a Bastion, and which covered that part. The Enemy seeing that, began to attack those two Faces before any other thing: So that the Besieged found themselves obliged to add Flanks to those Faces, to plant Cannon upon them, and prevent the Enemy from doing so much Mischief to the Faces as before. And thus the Bastions were invented, as we see them at this Day built upon the Angle of the Gorge.   

Sect. 3. Of the Parts of Fortification.  
THere are three Parts of it; that is to say, Ic●nograp●y, Orthography, and Scenography.  
Ichnography is no other than what we call the Plane, or Representation of the Length and Breadth of a Fo●●ress● of which the Parts are marked out either upon the Ground, or upon Paper.  
Orthography is called the Profil, or the Representation of a Fortress made and raised, so that a Man may see the Length, the Breadth, and Height of its Parts.  
Scenography is the Perspective of the Parts of a Fortress.   

Sect. 4. Of the Division of Military Architecture.  
FOrtification is Regular or Irregular: 

Regular is that where all the Sides, and all the Angles that compose it, are equal one to another.  
Irregular is that where all the Sides and Angles are not all equal nor uniform one to another. And it is either Durable or Temporary.  
Durable, is that which is built to continue a long time.  
Temporary, is that which is erected in case of necessity for a little time: And under this Signification are contained all sort of Works erected for the seizing upon a Passage, or some Hill, or when they make Circumvallations, and Countervallations; that is to say, Redoubts, Trenches and Batteries.      

CHAP. II.  

Of the Parts of a Fortress, and its Angles.  
THE Curtin, see Chap. 4. under C.  
The Flank, see● Chap. 4. under F.  
The Face, see Chap. 4. under F.  
The Face prolonged, is the rest of the Flanking Line, after you have taken away the Face.  
The Flank prolonged, is the Prolonging of the Flank to the Exterior Sides.  
The Demi-Gorge, see Chap. 4. under D.  
The Capital Line, see Chap. 4. under L.  
The Line of Defence is made High and Razing. See Chap. 4. under L.  
The Compliment of the Curtin, see Chap. 4. under C.  
The Distance of the Polygons, see Chap. 4. under D.  
The Half Distance of the Polygons, see Chap. 4. under D.  
The Interior Side of the Polygons, see Chap. 4. under C.  
The Exterior Side of the Polygons, see Chap. 4. under C.  
The Small Diameter, see Chap. 4. under P.  
The Great Diameter, see Chap. 4. under G.  
See the Plan representing all these Parts.   

Of the Angles.  
THE Angle of the Centre, or Angle of the Figure, is that which is made by drawing Right Lines from the Centre to the two meanest Angles of the Polygon.  
The Angle of the Circumference, is the mixed Angle which is made by the Arch, which is drawn from one Gorge to the other.  
The Flanked Angle, is the Angle composed of two Faces.  
The Flanking Angle, is the Angle composed of the Line of Defence, and the Curtin.  
The Angle of Compliment of the Line of Defence, is the Angle which is made of the two Compliments, where they intercut one another.  
The Angle Form-Flank, is the Angle which is composed of one Flank and one Demi-Gorge.  
The Angle Form-Face, is that which is composed by one Flank and one Face.  
The Angle of the Moat, is that which is made before the Curtin, where it cuts itself.  
The Re-entering Angle, is that which retires inward.  
The Salient Angle, is that which advances toward the Field.  
See the Plate A.    

CHAP. III.  

Of the Terms of the Measures.  
A Rod, is the length of 12 Geometrical Feet, and is made use of in Germany and Holland.  
A Fathom, is the length of six Geometrical Feet, of which they make use in France.  
A Geometrical Pace, is the distance of five Geometrical Feet.  
The Simple Pace, is the distance of three Geometrical Feet.  
The Cubit, is the length of one Foot and a half.  
The Geometrical Foot, is the distance of 12 Thumbs breadth.  
A Thumbs breadth, is the length of 12 Lines, of which the one is a Barley-Corn's distance from the other.  
A Germane League, is the space of 4000 Geometrical Paces.  
An Italian League, is the space of 1000 Geometrical Paces.  
A Simple French League, is the space of 1000 Geometrical Paces.  
A Common League, is the space of 2000 Geometrical Paces.  
A Furlong, is the space of 125 Geometrical Paces, of which they make use in Greece.  
A Parasangue, is the distance of 30 Furlongs, or 3750 Geometrical Paces, made use of in Persia.  
The Simple Schoenu● or Cord, is the distance of 30 Furlongs among the Egyptians.  
The Compound Schoenu●, is the distance of 60 Furlongs in Egypt.    

CHAP. IU. Of the Names and Terms made use of as well in Attacking, as in the Defence of Places, Alphabetically set down.  

A.  
WIngs, are the large Sides which terminate and bound to the Right and Left the Exterior Works, as the Horn and Crownwork, etc.  
Wings of Armies, are the Regiments posted upon the Right and Left Hand, when the Army is drawn up in Battalia.  
Antestature, is a small Retrenchment made of Gabions, to possess the remainder of the Grand Attack.  
Approaches, are all sorts of Advantages, by the Assistance of which we advance toward a Place besieged.  
Arriere-Card, is that part of the Army which marches after the Main Battle, to hinder and stop Deserters.  
Arsenal, is a place appointed for the forming and preservation of every thing that is necessary for Defence and Assault.  
Artillery, is all sort of Fire-Arms.  
Assault, is an Attack made upon a Fortress, to win it by main force.  
Attack, is an Effort made by the Besiegers, to approach the Walls of a Fortress.  
False Attack, is an Effort to the same end, but more remiss, to oblige the Besieged to divide their Forces.  
To Attack in Flank, is to attack both sides of the Bastion.   

B.  
Bacule, is a Gate that is made like a Pitfall, with a Counterpoise, before the Corpse de Guards advanced near the Gates, and which is supported by two great Stakes● a Portcullis.  
Banquet, a little Footpace at the bottom of the Parapet, upon which the Soldiers get up to fire into the Moat, or upon the covert way.  
Barricadoes, are Trees cut with six Faces, crossed with Battoons as long as a Half-Pike, bound about with Iron at the feet, and set up in Passages or Breaches to keep back as well the Horse as Foot.  
Barriers, are great Stakes planted 10 Foot one from another, about four or five Foot high, with their Transoms or overthwart Rafters, to stop those that would enter with Violence; and where they cry to every Body that show themselves, Whence come ye?  
Bastion, is a Heap of Earth usually raised upon the Angle of the Gorge, composed of two Flanks and two Faces.  
Flat Bastion, is that which is raised upon the Curtin, when it is too long to be defended by the Bastions on each side.  
Battalion, are several Foot-Soldiers ranged in Battalia: It is composed of two Marches of Musqueteers, and a Centre of Pikemen.  
Battery, is a place raised, whereon to plant the Great Guns, and play upon the Enemy.  
Berme, is a little space of four or five Foot, left at the foot of the Rampart without, to prevent the Earth from falling into the Moat.  
Blinds, are pieces of Wood set across upon the Trenches to sustain the Bavins or Hurdles laden with Earth, which are to cover the Trench.  
Red Bullets, are Bullets made red hot in a Forge, with which the Cannon are charged to set Fire upon Houses or Magazines.  
Boyau, is a Moat bordered with a Paraper, which is drawn from one Trench to another for better Communication.  
Breach, is the Ruin which the Cannon or a Mine makes in a Fortification to take it by Assault.  
Brisure, is a Line from four to five Fathom, which is allowed to the Curtin and Orillon to make the hollow Tower, or to cover the concealed Flank.   

C.  
A Bomb-Chest, is a kind of a Wooden Chest wherein are put from three to six Bombs, which is placed under Ground, to blow it up into the Air, in case it be taken.  
Camp Flying, is a small Body of an Army which keeps the Field, to prevent the Incursions of the Enemy, harrass the Country, hinder Convoys, and to throw itself into any besieged Place.  
Caponiere, is a hollow Lodgment● of about four or five Foot, bordered with a little Parapet of above two Foot high, to hold Planks laden with Earth; there are little Embrasures sometimes made in 'em, which are called Murdresses. This Lodgement is wide enough to contain 20 or 30 Soldiers, and is usually placed upon the Extremity of the Counterscarp.  
Carcase, is an Artificial Fire, composed of from three to four Granades, with several Barrels of Pistolets charged, and wrapped up in Tow, dipped in Oily Substances, with a pitched Cloth round about it: All which is put in a kind of Lantern, which has a thin Plate of Iron at each end, held fast by two Branches of Iron, which are crossed by Circles of the same Matter; and in one of these thin Plates there is a hole to fire the Carcase, which is afterwards forced into the Air like a Bomb.  
Cartouche, is a Roll wherein the Charge of the Cannon is put, which besides the Powder consists of several Chain-Rings, ends of Pistols, Bullets, Heads of Cart-Nails, and and other Iron Materials: And the Cannon concealed in the retired Flanks is usually charged with such Cartouches to make the more Havoc.  
Cascans, are certain Wells, some more hollow than others, which are made in the Retrenchment of the Platform near the Wall, to give Air to a Mine.  
Casamatte, is a certain Vault made in a part of the Flank which is next the Curtin, all of Masons Work; it is made to fire upon the Enemy, and defend the face of the opposite Bastion, and the Moat.  
Cavalier, is a mass of Earth raised in a Fortress, to lodge the Cannon, to scour the Field, or oppose a commanding Work.  
Chandeliers, are wooden Parapets covered with Bavins, filled with Earth about a Foot high, made use of in Approaches, Mines and Galleries to cover the Workmen, and hinder the Besieged from constraining them to quit their labour.  
Chauses' Traps or caltrop's, are Irons with four Spikes about four Inches long, so disposed, that which way soever they fall, one Point still lies uppermost; they are made use of to throw into Moats and Breaches, to stop the hasty coming on of the Enemy.  
Covert Way, is a Walk round about the Moat towards the Country, having a Footpace and the Glacis for a Parapet.  
The Chemin de Ronde, is a Walk between the Rampart and the Wall.  
The Shirt or Chemise, is the Solidity of the Wall from the Talus to the Stone Row.  
Cheva●x, or Freisland Horses, so called, because first made use of at Groningen, a City of Freisland, are great pieces of Wood with six Sides, driven through Piles or Stakes of Wood armed with Iron, to hinder the Horse and Foot from going forward.  
Circumvallation, is a Moat bordered with a Parapet, which slanks itself from distance to distance, sunk about the Besiegers Camp, as well to hinder the Relief of the Besieged, as to stop Deserters.  
Clayes, are Twigs interwoven together, having the Figure of a long Square; they are placed upon an Allodgment, and covered with Earth; and sometimes they are laid upon marshy Ground to consolidate it.  
Coff●r, is a hollowed depth from 6 to 7 foot, and broad from 16 to 18, all athwart a dry Moat: Upon the upper part of this they place a Parapet, and a Cross, pieces of Wood, covered with Clayes loaden with Earth.  
Commandment, is a height of 9 foot which one place has above another.  
Compliment of the Curtin, is the remainder of the Curtin, after you have taken away its Flank, to the Angle of the Gorge.  
Compliment of the Line of Defence, is the remainder of the Line of Defence, after you have taken away the Angle of the Flank.  
Counter-Approaches, are Works which the Besieged make to hinder the Approaches of the Enemy.  
Counterscarp, is properly the declining of the Moat on that side next the Country; although by this term is understood, the Covert-way with its Parapet.  
Counterforts, are certain Pillars and Parts of the Walls, distant from 15 to 20 foot one from another, which advance themselves as much as may be in the Ground, and join themselves to the height of the Cordon, by Vaults, to sustain the Chemin de Rondes, and part of the Rampart, to fortify the Wall, and strengthen the Ground.  
Counter-Gards, are Triangular Pieces, in form of a large Parapet, which raise themselves above the Moat before the Faces and the Point of the Bastion to preserve them.  
Countermine, is a Subterraneal Way which the Besieged makes in search of the Enemy's Mine, to take away the Powder, and frustrate the effect of it, though fired by the Assailant.  
Countervallation, is a Moat guarded with a Parapet, which the Besiegers make to secure themselves from the Sallies of the Garrison.  
Corbeills, are a sort of Gabions filled with Earth, placed upon the Parapet to fire upon the Enemy, without being seen by them.  
Cordon, is a row of Stones made round, which is placed where the Wall ends, and the Parapet begins; and it runs round about the place.  
Coridor, is the Covert way above the Counterscarp, round about the place, between the Moat and the Palisadoes.  
Corns or Horns, are Outworks which advance toward the ●●eld, and carry in the fore part two half Bastions in the form of Horns, which they present to the Enemy.  
Crowning, is a Work drawn beyond the Horns to keep the Enemy at a distance.  
Curtin, is the longest straight Line that runs about the Rampart drawn from one Flank to the other, and bordered with a good Parapet 5 foot high, behind which the Soldiers place themselves to fire upon the Covert way, and into the Moat.  
Cu●ette, a little Moat about 4 fathom broad, which is usually sunk in the middle of a great dry Moat, till you meet with Water. It is good to prevent the Besiegers Mining.   

D.  
Defences, are all sorts of Works that flank one another, and which the Enemy desires to ruin, before he strive to pass the Moat.  
De●ilé, is a straight narrow passage, through which there is no marching but in file.  
Dehors, are all sorts of separate Outworks, as Ravelins, Halfmoons, Hornworks, Crownworks, Counter-guards, Tenailles, etc. the better to secure the main Place.  
Demi-distance of Polygons, is the distance between the Exterior Polygons and the Flanks.  
Demi-gorge, is a Line drawn from the Flank to the Angle of the Polygon.  
Demi-lune, is a small flanked Bastion, placed before the Point of a Bastion, when it is too weak; sometimes this Work is placed before the Curtin, when the Moat is a little wider than it ought to be●  
D●scent into the Moat, is a deep digging into the Earth of the Covert way, in form of a Trench, of which the upper part is covered against Artificial Fires, to secure the descent into the Moat.  
Distance of Polygons, is the Line composed from the Flank and its Prolongation, to the Exterior Polygon.  
Donjon, is a Place of Retreat, to Capitulate with more advantage in case of necessity.   

E.  
Embrasure, the Hole through which the Great Guns are discharged.  
Embuscade, the Place where the Soldiers hide themselves to surprise the Enemy.  
Enceinte, the Circumference of a Place, sometimes lined, and composed of Bastions and Curtains, sometimes not.  
Esplanade, is the place void of Houses, between the Citadel and the Town.   

F.  
Face, is the most advanced part of the Bastion toward the Field, and is the first Attacked.  
Fascins, are Bavins of small Wood, two Foot broad, and four long, bound about at the ends and middle, thrown into Moats where there is much Water, to facilitate the passage over to the Wall.  
Falls Braye, is a small Rampart four Fathom wide, bordered with a Parapet, and a Banquet, running round about the Body of the Place. It is made use of to Fire upon the Enemy, when he is already so far advanced, that you cannot force him back from off the Parapet of the Body of the Place; or else, to receive the Ruins which the Cannon make in the Body of the Place.  
Flank, is the part which joins the Curtin to the Face of the Bastion, from which the Face of the next Bastion requires its Defence.  
Flank Covert, is that, the exterior part of which advances to secure the innermost; which advanced part, if it be rounded, is called an Orillon.  
Flank Fichant, that which plays, and fixes its Bullets in a direct Line in the Face of the adjoining Bastion.  
Flank Razant, is that, from the Conjunction of which with the Curtin, the shot razes the Face of the next Bastion; which happens when the Face cannot be discovered but from the Flank alone.  
To Flank a Place, is to play upon a place in such a manner, that there shall be no part of it but what is Defended; and from which you may play upon the Enemy both in Front and Rear, and so oblige him to retire  
Forts of the Field, or Fortins, are all Fortresses, whose Flanked Angles are distant one from another 120 Fathom. They are only for a time to guard some Passage, or dangerous Place; or else they are used in Circumvallation.  
Fougade, is a small Mine made under the Ground, which the Enemy would blow up. It is furnished with some Sacks of Powder, to which the Sawcedge is fixed, which answers to another Post, to the end it may be Fi●'d without danger.  
Fraises, are pieces of Wood from 7 to 8 Foot long, driven in almost half way into the Earth of the Rampart without side, and a little below the Parapet, they present their points somewhat sloping toward the Field; and are made use of to prevent De●erters and Surprises.   

G.  
Gabions, are Baskets equally wide at top and bottom, about four Foot in Diameter; they are filled with Earth, and usually placed upon Batteries and Parapets that have suffered very much, and before other places, to secure them from the Enemy's Shot.  
Gallery, is a covered walk, either of Earth or Turf. The sides of it are made with Planks and Pillars; and they are made use of in the Moat already filled with Faggots and Bavins, to the end the Minor may approach safe to the Bastion.  
Glacis, is the Parapet of the Covert-way, which loses itself insensibly in the Field. The largest are the best.  
Gorge, is the entrance of the Platform of the Bastions and other Works: It never ought to have a Parapet, for fear the Enemy being Ma●ter of it, he should secure himself thereby from the shot of the Garrison.  
Granado, a little hollow Globe filled with Powder, to which there answers a little Fusee of Powder, to set them on Fire; they a●e used to Fire close and narrow Places; or else to disorder the Ranks of the Soldiers.  
Grand Demi Diameter, is the Line composed of the Capital, and the small Demi-Diameter of the Polygon.  
Gazons, are pieces of fresh Earth, covered with Grass, about a Foot long, and half a Foot thick, cut in the form of a Wedge to line the Parapet, and the Traverses of the Galleries.   

H.  
Herisson, a Beam, with a great quantity of Nails having their points outward; it is supported by a Pivot upon which it turns, to stop up any passage instead of a Barrier.  
Hearse, is a Lattice, or Portcullis, of great pieces of sharppointed Wood at the lower end, sustained by a Cord fastened to a Moulinet, which is cut in case of surprise, to the end the Hearse may fall, and stop up the Passage of a Gate, or other Entrance of a Fortress.  
Horse-Shoe, is a Work sometime round, sometime oval, bordered with a Parapet: It is made to secure a Gate, or to relieve an over-tedious Defence.   

L.  
Line Capital, is a Line drawn from the Angle of the Gorge to the Angle of the Bastion.  
Line Cogrital, is the Line drawn from the Angle of the Centre to the Angle of the Bastion.  
Line of Defence, is that which is represented by the Discharge of the small Shot, which uncovers the Face of one Bastion by razing it.  
Line of Defence Fichant, is the Line drawn from the common Point to the Curtin, and from the Flank to the Flanked Angle of the Bastion.  
Line of Defence Razant, is the Line drawn from the Capital Line of a Bastion, to that part of the Curtin where the Defence begins, to discover the Face of the same Bastion.  
Lines of Communication, are the Lines that run from one Work to another, to stop the Fury of the Enemy.  
Lines withinside, are the Moats toward the Place to hinder Sallies.  
Lines withoutside, are the Moats toward the Field to hinder Relief.  
Lodgement of an Attack, is a Work cast up in a dangerous Post, to secure themselves against the Enemies Fire● This Lodgement is made of all the Materials that are capable to make Resistance.  
To Line, is nothing else but to environ a Rampart, Parapet or Moat with a good Wall or good Turf.   

M.  
Madrier, is a thick Plank, sufficient to cover the Mouth of a Petard, and which is applied with the Petard against the Places they would break: This Term is also applied to several Flat Beams which are put at the bottom of a Moat to support the Wall.  
Mantellet, is one or several Planks joined together, and covered with Tin, carried upon two Wheels, which the Pioners in a Siege drive before them to secure themselves against the small Shot.  
Merlon, is that part of the Paparet which is between the two Embrasures.  
Mortier, is a piece of great Cannon very short, which is charged with Bombs, Carcases, or Stones; and is mounted upon a Carriage, the Wheels of which are very low.  
Moulinet, is a cross of Wood, which turns Horizontally upon a wooden Stake, which stands at the side of the Barrier, between the Bars of which the Footmen pass.  
Moyenau, a ●ort of small flat Bastions, placed in the middle of an over long Curtain, of which the Bastions that are at the Extremities, are not well defended from the small shot by reason of their distance. This Work is proper for the placing in it a Body of Musketters, to Fire from all sides.   

N.  
Nailing of Cannon, the driving of a Nail by force into the Touch hole of a great Gun.   

O.  
Orgues, are thick and long pieces of Wood, hung up over a Gate, every one by a Cord, which answers to the Moulinet, that is to be cut in case of a Surprise; and they are to be preferred before Herse's.  
Orillon, is a small Rounding, which joining with the hollow Tower, at the end of the Face of the Bastion, covers one part of the Flank.   

P.  
Pallisadoes, are wooden Stakes from 5 to 7 Foot high, armed with two or three Iron points, which are fixed before Fortresses, Curtains, Ramparts, and Glaces: Some there are which are not armed with Iron.  
Parapet, is an Elevation of Earth upon the Rampart, behind which the Soldiers stand, and where the Cannon is planted for the defence of the Place.  
Park, is a certain Place made choice of in the Camp, out of Canonshot of the place Besieged, to plant the Artillery, and to keep the Artificial Fires, Powder, and warlike Ammunition, which is only guarded by Pike-men.  
Petard, is a hollow Engine made of ●etal, almost in the shape of a Cap, from seven to eight Inches deep, and five Diameter in the Muzzle. It is charged with Powder beaten as small as may be, and then fixed to the Madrier.  
Petty Demi-Diameter, is the Circumference drawn from the Centre through the Gorges.  
Place of Arms, is a Piece of Ground sufficient to draw up the Men to be sent from thence as occasion requires.  
Place of Arms without, is a Place allowed to the Covert Ways, where they make obtuse Angles, whereon to plant some Pieces of Cannon, or Falcons, to force those that advance in their Approaches to retire.  
Place of Arms particular, is a Place near every Bastion, whence the Soldiers sent from the Grand Place to the Quarters assigned them, relieve those that are either upon the Guard, or in Fight.  
Place Irregular, is a Place whose Angles and Sides are unequal.  
Place Regular, whose Angles and Sides are every where equal.  
Plane, the Representation of a Work in its Height and Breadth.  
Platform, is the whole Piece of Fortification raised in a re-entering Angle.  
Polygon Exterior, is the distance of one point of a Bastion, from the point of the other.  
Pont de jonc, is a Bridge made of great Bundles of Bulrushes, that grow in the Marshes, which are afterwards bound together, upon which they lay Planks, and then lay them in muddy and marshy places to secure their footing.  
Pont Flottant, or Floating Bridge, is a Bridge made in form of a Redoubt, composed of two Boats covered with Planks: They ought to be so solidly framed, as to bear Horse and Cannon.  
Potern, is a false Door, often made at the bottom of the Curtin, or near the Orillon, for private Sallies.  
Profil, the Draught of a Work, with all its Lengthy, Breadths and heights.   

Q.  
Quarter of a Siege, is the Incampment upon one of the most principal and important Passages round about the Place besieged, to prevent Relief, and Convoys.   

R.  
Ravelin, is a small Triangular Work without Flanks, and having no more than its two Faces: It is generally raised before the Curtains, where the Moat makes a Salient Angle.  
Redans, Works made in form of the Teeth of a Saw in re-entering and Salient Angles, to the end that one part may defend another: They are usually placed upon the Entrances into Rivers.  
Redoubt, is a small square Fort, having no defence but in Front. It is usually designed to maintain the Trench, for Circumvallations, and Countervallations; and in Watery Places they are often made of Masons Work for the security of the Neighbourhood.  
Reduit, an advantageous piece of Ground Entrenched from the rest of the Place, to retire to in case of surprise.  
Rampart, is the raising of the Earth which runs about a Place, composed of Bastions and Curtains, wherein to put the Soldiers, and plant the Cannon for the defence of the Place.  
Retirade, an Entrenchment formed by two Parapets making a re-entering Angle, which is made in a Work resolved to be disputed Inch by Inch. Sometimes it has a Moat.  
Retrenchment, is a Work made of part of a Rampart, when the Enemy is so far advanced, that he is no longer to be resisted or beaten from the Old Rampart; and to the end that he may not make any New Opposition; it is bordered with a good Parapet, and sometimes with Gabioris and Bavins laden with Earth.  
Ridea●, is a small Elevation of Earth in the plain Field, which extends itself in length; very proper for those that would besiege a Place at a near distance, and to secure those that would approach to the Foot of the Place.  
Rondel, is a Round Tower sometimes raised at the Foot of the Bastions.   

S.  
Sack for Earth, a Sack of Course Cloth to contain a Foot, or a Foot and a half; serviceable upon several Occasions, but especially for the making Retrenchments in haste.  
To Saignor a Moat, to empty the Water by Subterraneal Conveyances, to pass it the more easily, after you have cast Hurdles upon the Mud that remains.  
Sappe, is the deep digging into the Earth of the Covert Way and Glacis in the Form of a Trench. The Earth which is thence digged out, serves for security on the Right and Left, and covers you above against the Fireworks by the help of the Hurdles laden with Earth.  
A Sarrasins, is no more than a Hearse.  
Saucidge, is a piece of long Cloth, of which they sow the sides all along in form of a Gut, which they dip in Pitch, and fill with Powder: The largeness of it ought to be so much as to contain a Tennis-Ball; and then they put one end of it to the place where the Mine is, or a Fougade, or Chest of Bombs, and the other end answers to the place where the Engineer stands that sets it on fire.  
Saucidges, are Faggots made of great Boughs, bound in the middle and both ends; whose use is the same with that of Bavins.  
Shouldering, is nothing but the Orillon.  
Scalade, the mounting of the Wall with Ladders.  
Scarp. The Talus.  
Stars, a Work with several Faces, composed of Parts, of which the one flanks the other.  
Sillon or Envelop, a little Rampart covered with its Parapet raised in a very large Moat, the Draught of which Forms re-entering and Salient Angles, to the end that the Parts may reciprocally flank each other.   

T.  
Talut, is that part which is allowed to Works, whether of Earth or Masonry, the better to suppo●t them; and the worse the Earth is, the more the Sloop ought to be.  
Tenaille, is a Fortification that has in front a re-entering Angle, when it has no Flanks; and this is called a Simple Tenaille: But if it have Flanks, than it is called a Flanked Tenaille.  
Terre-plain, is that part of the Rampart, which is equally level for the Recoil of the Cannon.  
Tower Hollow, is a Rounding made of the Remainder of the two Brisures to join the Curtain to the Orillon; where the Smallshot are placed, that they may not be so much exposed to the view of the Enemy.  
Trenches, are Moats which the Besiegers make to approach the more securely to the Place attacked. The depth of it ought to be from 6 to 7 Foot; the breadth from 8 to 10. They are so to be carried on, as not to be in view of the Enemy.  
Traverse, is a little Moat bordered with a Parapet to the Right and Left, which the Besieger makes quite athwart the Moat of the Place, to pass secure from Flank-Shot, and bring his Miners to the Bastion. This Travers is covered with Hurdles loaden with Earth, for security from the Fire-Works. The only difference between this Work and the Coffer is this, that the one is made by the Besieger, and the other by the Besieged.   

V.  
Vedette, a Sentinel on Horseback.     

A NEW TREATISE OF Fortification● BOOK II.  

CHAP I. Of the Maxims of Fortification.  
SInce it is impossible to build a Fortress, without having considered beforehand what Reasons there are to induce an Engineer to do it after such a manner, that there is no great matter to be objected against it; there are some certain Maxims established, according to which, when a Fortification is raised, there can be but little said against its Advantages to those that are within; as you may easily examine it yourself by the following Observations.  

I.  
All the Parts which are to enclose a space of Ground, aught to be flanked, that is to say, viewed on every side, that so there may be no shelter about the place where the Enemy may lodge himself, but what may be discovered by those within, not only from the front, but from the sides, and even from behind, if it be possible.   

II.  
Whatsoever encloses a durable Fortification, must be either Flank, Face, or Curtain, and built after such a manner, that the first Discharges of the Cannon may not be able to pierce its thickness.   

III.  
Regular Fortification is much to be preferred be●ore that which is Irregular.   

IV.  
It is impossible to fortify a Triangle after the Regular way, because the Angle of the Gorge is always less by 90 Degrees.   

V.  
The straighter the Angle of the Centre is, the Place is by so much the stronger, and consequently better, because gains more Side●.   

VI  
The Angle of the Bastion must never be greate● than 100 Degrees, nor less than 60, for being bigger, the Face cannot be defended as it ought to be, and being less, it is not able to resist the Injury of the Wether, or the Efforts of the Mines and Cannon. See Chap. 1. Sect. 4. of Book IU.   

VII.  
The greatest Angles are best to fortify.   

VIII.  
The Angle of the Curtain ought never to be less than 90 Degrees, nor greater than 110; because, if it be larger, 'tis too much subject to the view of the Enemy. See Chap. 1. Sect. 5. of Book IU.   

IX.  
The great Demi-Gorges are to be preferred before the small ones, as well because there is more space to retrench in, as because one may make retiring Flanks, which adds very much to the strength of the place.   

X.  
Great Flanks are best.   

XI.  
The Face ought to be never less than the half of the Curtain, if it be not in an Irregular Fortification●   

XII.  
The Faces of the Bastion ought to be defended by the small Shot of the opposite Flanks.   

XIII.  
The shortest Faces are best, because they are not so subject to be attacked, as when they come ou● a great way in the Country. See Chap. ●. of Book IU.   

XIV.  
The Curtain ought never to be shorter than 60 Fathoms, which is 360 foot English, nor longer than 100, if it be not in an Irregular Fortification.   

XV.  
The Line of Defence must not be of greater length than 120 or 125 fathoms, because a Musket can carry no farther to do execution: Holdman goes contrary to this Axiom, when he says, that it may be 150 fathoms; and likewise George Pascha, who believes it may be 136 fathoms long.   

XVI.  
The Talu● or Level aught by no means to be forced or streightened, but o●ght to have sloap enough, that the Ground or Earth may settle of itself.   

XVII.  
Large and deep Trenches are to be preferred before those that are narrow and hollow, or large and less deep, because the Enemy will find more difficulty to get over the first sort. See Chap. 10. of Book I.   

XVIII.  
Dry Trenches are to be esteemed before those that are full of Water, especially in great Places, where Sallies, Retreats, and Succours, are necessary.   

XIX.  
Trenches full of Water that cannot easily be drained, aught to be preferred before those that are dry in small Fortresses, where Sallies, Retreats, and Succours, are not so necessary.   

XX.  
The Outworks ought always to be lower than the Body of the Place, and those that are at the greatest distance ought to be the lowest.    

CHAP. II. Of the Situation of Places.  
AS it is very rarely permitted to make choice of that space of Ground which is to be fortified, but on the contrary the Place is almost always determined, whether we design to repair an old Fortification, or surround any Town with Bastions, and other kind of Works; or whether we are to make ourselves Masters of the Passage of a River, of any straight Place, or an Eminence, that is of advantage to command the flat Country, or an Island in a River; or of one or more jetting out of Land into the Sea for the making a Port; and lastly, for several other Uses, the recital whereof would be too tedious; It is necessary upon these Accounts to understand, what kind of Situations are best for the Erecting a strong Hold, and to know the Opinions of others concerning this Matter.  

I.  

What are the Advantages of a Fortress situated upon a Mountain.  
(1.) This Fortification is not easily Ruined by Mines.  
(2.) The Enemy will be at a great deal of difficulty to bring their Machine's to batter down this Fortification.  
(3.) Those of the place may discover the Enemy at a great distance, and so hinder them from making their Approaches nearer.  
(4.) The Horse or Batteries of the Enemy, which are planted either on plain or rising Ground, can no ways do any damage to this Fortification, it being always lower.  
(5.) Upon an Eminence, the Air is always more healthy than in Valleys or lower Places.  
(6.) The Enemy will but very rarely be able to take such a Place by Assault.   

What are the Disadvantages of a Fortress built upon a Rock.  
(1.) They commonly want Water in such a place.  
(2.) It is difficult to bring thither Materials, Provisions, ●nd other Ammunition.  
(3.) These Fortifications are for the most part not very durable, having only a sandy or gravelly Earth for Foundation.  
(4.) Places so situated, are not to be succoured but with great difficulty.  
(5●) Such a Place cannot well be defended, because what is done by the Cannon from top to bottom, is but of little consequence.  
(6.) Sallies are there very dangerous, especially if they be at any distance from the Counterscarps.  
(7.) These Fortifications are almost always Irregular.   

Advertisement.  
It is to be observed, that these kind of Situations are good for Citadels, Castles, and other small Forts, in order to command the Country thereabouts.    

II.  

The Advantages of a Marshy or Fenny Situation.  
(1.) The Enemy cannot come near it without danger of being lost.  
(2.) This Fortification is not easily blown up.  
(3.) These Places may be fortified for little Charge.  
(4.) They have there no need of a great Garrison.   

The Disadvantages of a Marshy Situation.  
(1.) It is almost impossible to relieve it.  
(2.) It must be strengthened with Piles driven in.  
(3.) Sallies thence are commonly unsuccessful.  
(4.) The Air is there always unhealthy, which is the occasion of frequent Distempers, because the Water there is always corrupted.  
(5.) These Places may be attacked without any great loss of Men, provided they stay till the Frosts.    

III.  

The Advantages of a Place encompassed round with Water.  
(1.) It cannot be undermined.  
(2.) There is no occasion for any great Workest  
(3.) The Enemy cannot always discharge their Cannon with any certainty, because the Water abates much of their force.  
(4.) The Besieged may very easily set fire to the Enemy's Shipping.  
(5.) ●he Naval Army is too much exposed to the ●iew of the Artillery of the Place.   

The Disadvantages of a Place surrounded with Water.  
(1.) The En●my may ●asily stop their Provision, and hinder their Succours.  
(2.) Boats alone are sufficient to attack such a place, and so there is no occasion for Horse.  
(3.) These Places are more subject to Diseases than othe●s.   

Advertisement.  
We may affirm, that these Places that are encompassed all round with Water, which cannot be quite drained, or at least with great difficulty, are the best notwithstanding, because they are, as it were, fortified by Nature.    

IV.  

The Advantages of Places situated upon an open Rising Ground.  
(1.) There is great Plenty of good Earth.  
(2.) They are never there in want of Water.  
(3.) Almost all these Places are commonly Regular.  
(4.) They have there Earth enough to make Retrenchments, in case the Enemy make their Attacks with great fury.  
(5.) The Enemy had need to have two or three Armies, if they would intercept the Provision and Ammunition coming to those of the Place from all Parts.   

The Disadvantages of Places erected upon Rising Ground.  
(1.) The good Soil that there is about such a Place, serves only for the Enemy to make Retrenchment 〈◊〉 Batteries, Approaches, Redoubts, and other Works, from whence they may do great Damage to the Besieged.  
(2.) The Enemy may form their Camp, as those of the Place did their Fortification, that is to say, they may give it a Regular Figure, and encompass it with a good Trench and a good Rampart.  
(3.) The Ground is very good to make Mines under such a place.  
(4.) The Enemy may seize all the Fruits growing about these places.   

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These Places however are best, because those of the place will take double the Advantage that the Enemy can have.    

V.  
The side of a Mountain is a very ill place to fortify, and it must be avoided as much as may be, because the top of the Mountain almost always commands the Place within.   

VI  
The Valley is is no good place to fortify, because the Enemy may discover them from the top of the Mountain.   

VII.  

The Advantages of Places situated on the Sides of any Great River.  
(1.) You may bring thither by Water, whatsoever is necessary for Building, and for the Soldiers.  
(2.) You may fortify the Circumference after the Regular manner, or at least very near it: The side towards the Water is fortified with small Charge.  
(3.) It may be easily Relieved.  
(4.) Earth is there very plentiful for repairing th● Works, and for making Retrenchments.  
(5.) You may there make Sluices to put all the Country thereabouts under Water.  
(6.) They never want there any Water.  
(7.) The Enemy must have a great many Troops to ●ay Siege to such a Place, by reason of the distance of the Quarter's.  
(8.) If it be attacked only by Land, nothing hinders but that the Besieged may pass the River at pleasure, and receive Succours by Boats.  
(9) If it be attacked both by Land and by Water, there must be two great Armies, which will be very Chargeable.   

The Disadvantages of Places situated on the Sides of a Great River.  
(1.) The River serves the Enemy for transporting whatsoever is necessary for a Siege.  
(2.) The Enemy may cast up great heaps of Earth to Command the Place.  
(3.) The Enemy may easily shelter themselves from the Firing of the Besieged, having great plenty of Earth.   

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'Tis not to be denied, but that these Places are to be preferred before any of those that we have now mentioned, in case they are Masters of the Pass, by which the Provision and Ammunition is to come in.     

CHAP. III. Of the Quality of the Earth.  
WHen you have once made choice of a Place for the erecting a Fortress, before you Engage yourself in so expensive an undertaking, you must examine the Quality of the Earth, and consider whether it be Good or Bad: For there are very Advantageous Situations, the Earth whereof is good for nothing; and on the contrary, there are Ill Situations that have a very good Soil, but they lie subject to be Commanded after such a rate, that it would be great folly to settle there: As may be evident by the following Observations.  

Sect. I.  
Mountain's commonly have a stony Ground, which is the worst of all, as well because it will not stick together as because the Parapers made thereof are good for nothing: But if you are compelled to fortify in such a place, you must make choice of the best Veins of Earth to make the Parapet, and cause it to be brought from elsewhere. 'Tis true, this sort of Earth is of some advantage, in that the Besiegers will find it difficult to cover themselves in their Approaches for want of good Earth.   

Sect. II.  
THE Gravelly or Sandy Earth is likewise very bad, by reason that it wants a binding Quality, being always apt to tumble down; when you are forced to make use of this, you must mix some good Earth therewith, or some old Dung, and the Ramparts must be well lined with Ston●s or Bricks, and the Parapets with Turfs.   

Sect. III.  
THE Marshy Ground is better than the two former sorts, although generally 'tis not very good, because when it comes to dry, being raised into Ramparts and Parapets, it is subject to fall to pieces; besides, 'tis difficult to find Earth enough about a Marshy place for raising Ramparts, Parapets, and Glacis, of a reasonable height; and moreover, the Foundation of these Works must be always strengthened with Piles. But if there b● a necessity of raising Fortifications in these places, it must be during the Heats, that so the Earth may have the greater Consistence.   

Sect. IU.  
THE best kind of Earth for Fortifying, is that which they call Fat or Fertile Ground, because 'tis pliable; and you are not obliged to strengthen the Foundation with Piles, nor to line the Ramparts, unless you will.    

CHAP. IU. Of Provision, and other Necessaries.  
seeing a Place cannot be able to endure a long Siege, if it has not wherewithal to make resistance, therefore there must be Provision made of whatsoever is necessary for a Soldier; under which Head is to be understood, Money to pay the Troops, Meal, Corn, Hay, Wine, Beer, Brandy, Butter, Salt, Wood, Powder, Led, Match, Bullets, Cannon or Great Guns, Muskets, Pikes, Sulphur, Pitch, Brimstone, Palisadoes, Chevaux de Frise, Hand-Mills, Horsemills, and Windmills, that so, if the Enemy should cut off the Water, they might not be without something to Grind their Corn. A good Commander will take care of all this. An able Physician, Chirurgeon, and Apothecary, are there likewise very necessary. To these there might be added, some other Things requisite, in order to keep the Soldiers in good heart.    

A NEW TREATISE OF Fortification. BOOK III.  

CHAP. I.  

Sect. 1. How to inscribe any Polygon within a Circle giv●n●  
THIS Problem is commonly treated of in Geometry; Nevertheless, because it is impossible to work on a Regular Fortification on Paper, without knowing beforehand how to inscribe Regular Figures in a Circle given or taken at pleasure, we shall at present discourse something thereof.  
(1.) Divide the Diameter of your Circle given, A. B. into as many equal parts, as you would give sides to your Figure: For example, into Five.  
(2.) Draw the Diameter A. B. and make with this distance of points A. B. 〈◊〉 A●●●es to intersect in C.  
(3.) Draw from the Point C. through the second Part which is marked by the Point E. of the said Diameter, a Ri●ht Li●●, t●ll it to●ch the circumference 〈◊〉 be●o●● the Diameter in ●.  
(4.) This distance from the Intersection to the first ●●int A. of the Diameter, will be always one side of the Polygon demanded.  
See the Plate B. Fig. 2.   

Se●t. ●  
How to make a Scale.  
AS every thing is to be done by a Scale, you must know likewise the man●er how it is made, which is as follows.  

(1.) Divide a Line into 10 equal parts, and one of these to show the Feet into 10 others; so one of these last 10 will make the hundredth part of your Line; that is to say, one Fathom of your Scale.  
(2.) To make the Scale well, you must divide also one Fathom into six other equal parts, to show the Feet; for a Fathom is six Geometrical Feet, as have been already said. And by this manner you will have a perfect Scale, for to know all the Breadths, Lengths and Heights.  
(3.) When you would make any thing tha● should be very exact, you must likewise take a Foot, and divide it into 12 other equal parts, and so you will have the Inches, for a Foot contains 12 Inches.     

CHAP. II. How to make the Design of a Square.  
IN France there are three sorts of Fortifications, the Large, the Middlesized, and the Small; but at present they never make use but of the Middlesized: Nevertheless we will first show the Difference, before we speak of the Middlesized.  

The Square.  

The Large Manner.   Fathoms. Feet. (1.) The Semi-diameter 141 00 (2.) The Side or Exterior Polygon 200 00 (3.) The Perpendicular 27 00 (4.) The Face 60 00 (5.) For the Compliment of the Line of Defence take 38 00 (6.) The Flank 22 00 (7.) The Curtin 73 04 (8.) The Line of Defence 141 04  

The Middlesized.   Fathoms. Feet. (1.) The Semi-diameter 127 02 (2.) The Exterior Side 180 00 (3.) The Perpendicl●ar 22 00 (4.) The Face 55 00 (5.) For the Compliment of the Line of Defence take 33 00 (6.) The Flank 18 01 (7.) The Curtin 64 04 (8.) The Line of Defence 1●6 00  

The Small.   Fathoms. Feet. (1.) The Semi-diameter 113 01 (2.) The Exterior Side 160 00 (3.) The Perpendicular 21 00 (4.) The Face 45 00 (5.) For the Compliment of the Line of Defence take 33 00 (6.) The Flank 18 04 (7.) The Curtin 64 00 (8.) The Line of Defence 115 04   

The Explanation of the Middle-sized Table.  
HAving seen the Difference between these three Manners, we shall only speak of the Middlesized, as being the most in use at this Day; and to begin, we will examine the said Middlesized Table.  
(1.) Take 127 Fathoms, and two Foot between the Points of the Compass, and make a Circle.  
(2.) Take 180 Fathoms, and divide them with the Circle into four equal parts: When they are joined the one to the other, they are called Exterior Sides or Polygons; as A, B, C, D.  
(3.) Divide one of these four Sides into two equal parts with 90 Fathoms; as E, F, G, H.  
(4.) Draw from the Points E, F, G, H. straight Lines toward the Centre, which are called Perpendiculars; to which must be allowed 22 Fathoms in length, as I, L, M, N.  
(5.) Set the Rule to the Capital Points A, B, C, D. and draw straight Lines through the end of the Perpendiculars I, L, M, N. at pleasure.  
(6.) Take 55 Fathoms, and place them upon the Lines drawn from the Capital Points, to mark out the length of the Faces, as a, b, c, d, e, f, g, h.  
(7.) Take 35 Fathoms, and put them from the Point where the Perpendicular ends, viz. from the Point I, L, M, N. upon the Lines tha● have gone beyond the said Points, to mark out the length that is to be taken for the Compliment of the Line of Defence, as JO. JP. LQLR. MS. MT. NU. NX.  
(8.) Draw the end of the Faces and these Points OPQRSTUX. which terminate the Compliment of the Defence, and you will have the Flanks.  
(9) Then join together the Feet of the Flanks, and you will have the Curtains.   

To make the Orillon.  
MOnsieur de Vauban does not leave his Flanks, as Monsieur Pagan has done, or as the Dutch and Germans do at this Day; but he conceals one part, the better to cover the Cannon by means of the Orillon: And for that Reason we will show you his manner of raising it, which is not very difficult.  
(1.) Divide the Flank into three equal parts.  
(2.) Divide the first of the three into two other equal parts, to join from this Middle, the two Extremities of the first part, on the side of the Moat; and this little Rounding makes the Orillon.   

To make the Hollow Tower or Flank retired.  
AS you have seen the Reforming of the first of the three parts, placed upon the Flank, now you shall see another of the two remaining parts, the Form of which you must henceforward call the Hollow Tower, the Constructure of which is thus.  

(1.) Draw from the Capital Points of the Bastions a small straight Line within side of the Bastion, through the end of the Orillon.  
(2.) Put five Fathoms upon this Line which you have drawn, as also upon that which is almost parallel to the lower part of the Curtin. This is called the Brisure.  
(3.) Take the Extremities of the Brisure with the Compass, and with that Distance draw two Arches, which intersect one another toward the Moat, in Figure 8.  
(4.) Put the Compass upon that Intersection, and join the said Extremities of the Brisure together by a crooked Line; and thus you will have the Figure which is made at this Day in France in stead of the Flanks, as also the Design of the Entire Square. See under B. Fig. 3.     

CHAP. III. Of the Structure of the Body of the Place of the Square.  
WE are now to see what Largeness or Breadth is to be allowed to the Ramparts, Parapets, Moats, Covert Ways, and Glacis' of Squares, and after what manner they are to be drawn: Observe then what follows.  
 Fathoms. Feet. (1.) The Base of the Rampart 11 00 (2.) The Base of the Parapet 03 00 (3) The first Banquet 00 1 ⅓ (4.) The second Banquet 00 02 (5.) The Moat from 16 to 20 00 (6.) The Covert Way 04 00 (7.) The place of Arms 13 00 (8.) The Glacis 36 00  
In case the Ground will give you leave to take more, you may do it.  

The Explanation of this Table.  
(1.) Take 11 Fathoms, and carry them parallel to the Curtains, Flanks, Faces, Brisures and Hollow Towers, on that side toward the inside of the place, to know after what manner you are to draw the Parallel to the Hollow Tower: Put these 11 Fathoms before; then open your Compasses to the Point from whence you drew your Hollow Tower, and draw from the same Point through the said 11 Fathoms a crooked Line, till it joins the Basis of the Brisure and the Face.  
(2.) Take three Fathoms, and carry them in the same manner parallel to the Design, to the end the Basis of the Rampart may be opposite to it, and parallel. But you ought to make this Parapet, or three Fathoms, parallel also to the Orillon, as far as the Brisure: For there it is, that the Parapet of the Orillon ought to end. As for the Hollow Towers, put as already has been said, three Fathoms, for that is the breadth of the Parapet, before the Hollow Tower, and open the Compass to the point from which you drew the Hollow Tower, and the Base of its Rampart, to draw from the same point through the three Fathoms put before the Hollow Tower a crooked Line joining the Parapet of the lower Brisure, and the prolongation of the upper Brisure together; and there, where I touch the said Prolongation, from thence I draw a small Line to the next Brisure.  
(3.) Take a Foot and a half, and draw with that distance a Parallel round about the Parapet, toward the Base of the Rampart, which will represent to you the first Banquet; round about which you must draw another Parallel of two Foot to mark out the second.  
(4.) To draw the Breadth of the Moat, take from 16● to 20 Fathoms, and draw with that distance a Parallel to the Faces, till they intersect one another before the Curtin. To draw this Parallel round about the point of the Bastion, you must put the Compass of the Overture from 16 to 20 Fathoms upon the said point of the Bastion, and draw a little Rounding from thence to the Parallels of the Faces.  
(5.) To mark out the Covert Way, take from four to five Fathoms, and carry a Parallel to the Moat; but set i● from a point before the Rounding.  
(6.) Where the Covert Way makes a Salient Angle, for example, in G. put there 10 Fathoms to each side, as G. J. and G.H. and afterwards with 15 Fathoms you shall make points where the 10 Fathoms terminate, and draw from the said Points, in the point of the Intersection, straight Lines; and this is that which is called the place of Arms.  
(7.) Take 36 Fathoms, or as much as the Ground will permit, and bring a Parallel round about the Covert Way, and the place of Arms to mark out the Glacis or Parapet of the Covert Way. And thus your Square will be perfectly finished.    

CHAP. IU. Of the Structure of the Half-Moon before the Curtin of the Square.  
FRom the Angle of the Moat, or Counterscarp, raise a Perpendicular as long as the Demi-curtin, with its Brisure for the Capital P. Q. Observe that in Polygons the Capital aught to be 45 Fathoms in length.  
2. From the point Q. draw a straight Line toward the Orillon of the Bastion, terminating on the side of the Moat in S and D.  
3. Take five Fathoms, and put them from S and D. upon the Counterscarp, toward the inside of the Half-Moon, as E. F.  
4. Put the Rule perpendicularly upon the Curtin of the Body of the place, and erect from the points E. F. two small streight● Lines, till they cut the Faces of the Half-Moon, as in VH, and the Lines EV, FH, will make you the Flanks of the Half-Moon.  
5. Draw the Base of the Rampart AB, of 8 Fathoms, parallel to the Faces, and the Flanks of the Half-Moon toward the inside, as also the Parapet AC, of 3 Fathoms, the first Banquet of one Foot and a half, the second of 2 Feet.  
6. Take 12 Fathoms for the wideness of the Moat, which ought to be drawn parallel to the Faces, but not to the Flanks: You must give it also its Rounding before the Capital Point.  
7. The Covert Way, Place of Arms and Glacis are made, as in Chap. 3. L. 3. Sect. 5, 6, 7. See the Plate C.   

CHAP. V. Of the Structure of the Ravelin before the Curtin.  
THis Work is done as the Halfmoons when you allow it no Flanks.  

Sect. 2. To make a Ravelin with Lunettes.  
HAving raised the Ravelin as it ought to be done, 

1. Take two parts of the Flank of the Body of the place, and put them at the Angle of the two Moats; that is, from the Moat of the Body of the place and that of the Ravelin, upon the Counterscarp, to mark out the Demi-gorges of the Lunettes, as AB, and AC.  
2. Take the Flank of the Body of the place entire, and draw from the point BC, two Arches to cut one another in D, and then join BD, and CD, together by a straight Line, to have the Faces of the Lunettes.  
3. You draw the Moat of these Lunettes from the half of that of the Body of the place, allowing it also a Rounding.  
4. The Base of the Rampart, the Parapets, and the Banquets, is made as for the Ravelin. See the Plate D. Fig. V.     

Sect. 3. To make a Ravelin with Counter-guards.  
HAving made the Ravelin alone with its Moat in the same manner as has been set down; 

1. Prolong the Faces of the Ravelin above its Moat, and from the Intersection of that Line and of the Moat put upon the Prolongation 30 Fathoms, as AB.  
2. Put 15 Fathoms upon the Counterscarp of the Moat of the Body of the Place, placing the Compass in the Angle of the two Moats, as CD.  
3. Join DE together by a Right Line, which side, and the other AB, must be fortified with a Rampart, Parapet, Banquets, Moat, Covert Way and Glacis, like the Ravelin itself, the Distances of which are also here to be observed.  
4. Having fixed this Work on the other side of the Ravelin, place a small Reduit at the Head between the two Counter-gards, which are thus to be raised: First, where the Moats of the Counter-gards intersect one another, erect from the same Angle a Perpendicular of 20 fathoms, as OF; then divide the Faces of the Counter-gards into two equal parts, as BG and AGNOSTUS, and from the Capital Point of your Reduit draw a straight Line toward the middle of the Faces, marked by the point G, terminating on the side of the Moat of the Counter-gards in H, and the Line FH will be the Face, which is only to be fortified with a Rampart, Parapet, Moat, Covert-way, and Glacis, if need require. See the Plate C, Fig. VI     

Sect. 4. To make a Ravelin, to be placed at the Entrances into Places.  
1. HAving the place for a Ravelin, you must erect a Perpendicular of 12 fathoms from the Angle of the Counterscarp, as AB.  
2. You must put at the same Angle 6 Fathoms upon the Counterscarp for Demi-gorges, as AC and AD.  
3. Erect upon the points CD Perpendiculars of 4 fathoms for the Flanks.  
4. Joint the Flanks and the Capital together to have the Faces.  
5. Draw a Moat round about this Work of 2 fathoms and a half, parallel only to the Faces, and round before the Point. In this little Work is placed a great Corpse du Guard, the better to guard the Gate and the Bridge, and present the Arms to the Officers.  
6. That which is left of the place designed for the Ravelin ought to be joined together, as if the Corpse du Guard were not there, and fortified with Ramparts, Parapets, etc. as another Ravelin. See the Plate D, Fig. VII.    

CHAP. VI To make a Horn-work before the Curtin.  
1. ERect a Perpendicular from 85 to 88 fathoms at the point of the Ravelin or Half-Moon, as QG.  
2. Draw from the point G to each side an Arch of 60 fathoms.  
3. Take from 114 to 118 fathoms, which is most usual, and resect these Arches in OF, at the Angle of the two Counterscarps DR.  
4. Join ED together by a straight Line, as also FR and FE.  
5. Divide the Lines OF into two equal parts, from the middle of which set down a Perpendicular of 20 Fathoms GH.  
6. Draw the Faces ●s for the body of the place; their length is 38 fathoms.  
7. For the Compliment of the Line of Defence take 21 fathoms 2 feet, which join together after the end of the Face and of the Compliment, to have the Flanks: Draw the Curtin also.  
8. To make the Orillon and the Hollow Tower, go the same way as is set down in Lib. 3. Chap. 2. only that you do not divide the Flank but into two equal parts; Monsieur de Clermont would have allowed to the Orillon of the Square two fifth parts of the Flank.  
9 The Base of the Rampart is to contain 8 fathoms; that of the Parapet, 3; that of the Moat, 12; that of the Covert-way, 5; the first and second Banquet are to have the same wideness, as has been mentioned above.  

The TABLE.   Fathoms. Feet The Capital QG 88 00 The Wing DE 118 00 The Perpendicular GH 20 00 The Face EL 38 00 The Flank LN 15 0● The Compliment, etc. HN, HM. 21 03 The Curtin MN 38 05 The Line of Defence EM 84 04 The Base of the Rampart 08 00 The Base of the Parapet 03 00 The first Banquet 00 01 1/● The second Banquet 00 02 The wideness of the Moat 12 00  

To make a Ravelin before these Horns.  
1. FRom the Angle of the Moat of the Horns, erect a straight Line of 24 fathom for the Capital OPEN.  
2. From the point B draw right Lines toward the Orillon of the two Bastions of the Horns terminating at the Moat, as OQ, OQ.  
3. Draw the Base of the Rampart, and the other distances, as in the working of the Horn-work, only that the Moat ought not to be wider than 8 fathoms, nevertheless it must have its Rounding.  
4. The Covert-way, the places of Arms and Glacis are made, as in the body of the place. See the Plate C, c. 4. l. 3.    

CHAP. VII. To make a Horn-work before the Bastion.  
1. FRom the point of the Bastion draw a straight Line of 86 fathoms, AB.  
2. From the point B draw to each side an Arch of 60 faith.  
3. Take from 118 to 120 fathoms, and resect those two Arches in CD, the Angles where the Moat of the Ravelin throws itself into the Moat of the body of the place, as EF.  
4. Join EC, CD, and DF together, by straight Lines.  
5. Divide the Line CD into two equal parts, from the middle of which draw down a Perpendicular from 18 to 20 fathoms, as BH.  
6. Draw the Faces as usually; their length is to be 38 fathoms, as KING; the Compliment is 21 fathoms 2 feet, as LN; lastly, all the rest is done as in the Horn-work before the Curtin, the parts of which have the same distan●●. See the Plate E, Fig. VIII.   

CHAP. VIII. To make a Horn-work with Shoulders.  
1. PRolong the Flanks towards the Field to the length of the Curtin AB, CD.  
2. From B and D draw a Perpendicular of from 25 to 28 Fathoms, as BE, DF.  
3. From the Points OF raise other Perpendiculars from 118 to 120 Fathoms, as EGLANTINE, HF.  
4. Join GH together by a straight Line, which you shall divide into two equal parts, and from the middle of which you must bring down a Perpendicular of 20 Fathoms, as IK.  
5. Draw the Faces as usually, the length of which must be 35 Fathoms.  
6. The Compliment of the Line of Defence is found, by putting the Compass upon the Extremity of the two Faces, and then keeping one Point of the Compass standing in its place, let the other fall upon the Line of Defence. Which is also to be done on the other side, as LM, NO.  
7. LM and NO are drawn together for the Flanks, as also MORE for the Curtin.  
8. To have the Orillion, take two fifth parts of the Flank of this Work; the rest is made as usually in like manner as the hollow Tower, or Flank retired.  
9 The Base of the Rampart, etc. aught to have the same wideness as the rest of the Horn-works, and to be drawn parallel as far as the Moat of the Body of the place.  
10. The Ravelin before this Work is made, as already has been set down in Chap. 6. Lib. 3. See the Plate F, Fig. IX.   

CHAP. IX. To make a Horn-work with a Crown.  
1. DIvide the Exterior Polygon of the Hownwork into four equal parts, AB, CD.  
2. Take three parts of these four, and describe them with an Arch at the Angle of the Moat of the said Work E.  
3. From the point H erect a Perpendicular as far as the Arch F.  
4. Put the same three parts of the point F upon that Arch● as FG and FH, and join FG and FH together by a Right Line.  
5. From the Line FG and FH, bring down a Perpendicular of 16 Fathoms, as IK, LM; then draw the Faces as usually, of which let the length be 29 Fathoms.  
6. For the Compliment of the Line of Defence take 15 Fathoms, and then draw the Flanks and the Curtin as above.  
7. Prolong the Faces of the Horns to 27 Fathoms, as AN, DO, and join GN and HO together by a Right Line.  
8. Put 24 Fathoms upon the Wing of the Horn, on the side of the Place, at the points AD, as AP and DQ.  
9 From the Points N and O draw straight Lines towards the Points PQ, which terminate on the side of the Moat of the Hornwork in RS.  
10. To make the Orillion, divide the Flank into two equal parts, of which the one serves for the said Orillion, and the other for the Flank retired. You are to observe, that the Flanks which join their Faces to the Moat of the Hownwork, are left without an Orillion ● and a retired Flank, as NR, OS.  
11. The Base of the Rampart, etc. and all the other distances, are form as in the Horn-work. See the Plate F, Fig. X.   

CHAP. X. To make a Crownwork before the Curtin.  
1. FRom the Capital Point of your Ravelin erect a Perpendicular of 100 Fathoms, and if there be no Ravelin, take the Cur●in● of the body of the place with the Brisure to draw two Arches, which cut one the other before the Curtin. From which Point of the Intersection you must do as I have already said concerning the Capital Point of the Ravelin, as AB.  
2. Take the same 100 Fathoms, and make an Arch from the point B to each side.  
3. Take 113 Fathoms, and resect those two Arches as the beginning of the Orillon CD and EF.  
4. Join EBB and BF together by right Lines, and draw straight Lines from the points OF towards the Orillion of the place, terminating upon the side of the Moat GH.  
5. Divide the Line EBB, and the other BF, into two equal parts, from the middle of which bring down Perpendiculars, each from 15 to 16 Fathoms, as IN, LN.  
6. You are to draw the Faces as usually, the length of which must be 31 Fathoms, as OPEN, QR; the Compliment, etc. of 16 Fathoms, as ST, VW: The Flanks and Curtin are made, as already has been frequently showed.  
7. Divide the Flank into two equal parts, of which reserve one for the Orillion, and the other for the Flank retired.  
8. The Base of the Rampart takes up 8 Fathoms, that of the Parapet 3; the first Banquet a Foot and a half, the second 2 Feet; the wideness of the Moats 12 Fathoms; in the Covert-way and Glacis, observe the same breadth as there is round about the body of the place. See the Plate G, Fig. XI.  

The Table for this Work.   Fathoms. Foot. The Capital Line AB 100 00 The Exterior Sides BE, BF. 100 00 The Wings EC and FD, with their Compliments to the Orillions 113 00 The Perpendicular 17 00 The Face 31 00 The Flank 13 03 The Curtin 30 00 The Compliment of the Line of Defence 16 00   

CHAP. XI. To make a Corwn-work before the Bastion.  
1. ERect upon the Capital Line of the Bastion a Perpendicular of 100 Fathoms, as AB.  
2. From the point B draw to each side an Arch of 100 Fathoms.  
3. Take 92 Fathoms, and resect those Arches at the Concourse of the two Moats, viz. That of the Ravelin, and that of the body of the place, in CD, as ED, FC.  
4. Join CB, BD, FC, and ED, together by straight Lines.  
5. Divide the Lines CB and BD into two equal parts, from the middle of which you shall bring down a Perpendicular of 17 Fathoms.  
6. You must draw the Faces as usually, allowing to the length of them 31 Fathoms; to the Compliment, etc. 16 Fathoms; to the Flank 13 Fathoms, 3 Feet; to the Curtin, 29 Fathoms 3 Feet; to the Base of the Rampart, 8 Fathoms; to the Base of the Parapet, 3 Fathoms; and to the Banquets, Coverd-way, and Glacis, as usually. See the Plate H. Fig. XII.   

CHAP. XII. To make a Ravelin before the Point of a Bastion.  
1. PRolong the Parapet of the body of the place above the Moat, so that it may cut it in QS.  
2. From the point of the Bastion erect a straight and perpendicular Line, upon which put 36 Fathoms, beginning at the Moat, as CV for the Capital.  
3. Upon the prolongation of the Parapet by the points SQ, put 14 Fathoms for the length of the Flanks, as ST, QR.  
4. Join TV, VR, together, and you will have also the Faces.  
5. Round about the Faces of the Flanks you shall draw the Base of the Rampart of 8 Fathoms; that of the Parapet 13 Fathoms, and the two Banquets as usually; and the Moat of 12 Fathoms, parallel to the Faces and the Flanks, with its Rounding, before the Capital Point V, and the Flanks TR. See the Plate I. Fig XIII.   

CHAP. XIII. To make a Single Tenail.  
1. PRolong the Flanks of the Body of the place to 110 Fathoms, as ABCD.  
2. Join BD together by a right Line; divide it into two equal parts, and let fall from the Middle a Perpendicular of 26 Fathoms, as EF. But you must observe, that this Line ought not to be longer than 20 or 22 Feet, when there is a Ravelin or a half Moon before the Curtain of the Body of the place.  
3. Draw the Extremities of the Wings in the point F. as BF. and DF.  
4. Draw the Base of the Rampart 8 Fathoms, that of the Parapet 3, the two Banquets as usually. The Moat 12 Fathoms, the Covert Way and Glacis as for the Body of the place.  

The Ravelin before the Work.  
1. DIvide the Faces of the Tenaille BF and FD. into two equal parts, and put the half of the one upon the Perpendicular erected upon the Angle of the Counterscarp, as G P.  
2. From the point G, toward the Middle of the said Faces of the Tenaille, draw straight Lines for the Faces of the Ravelin, as GL, and GM.  
3. The Distances for the Body of the place of the Ravelin are the same with those of the Tenaille, only that the Moat belonging to it is no wider than 8 Fathoms. See the Plate K. Fig. XV.    

CAAP. XIV. To make a double Tenail.  
1. PRolong the Flanks of the Body of the place to 110 Fathoms, as AB and CD. and join BD together by a right Line.  
2. Divide the Line BD into two equal parts, and let fall from the Middle of it a Perpendicular of 26 Fathoms, as EF.  
3. From the points BD draw straight Lines in the point F for the Faces, each of which must be divided into two equal parts, as BG. DH.  
4. Erect a Perpendicular upon the Angle of the Faces of 45 Fathoms, as FJ.  
5. From the point J. draw the Counter-Faces in the points GH.  
6. To the Wings, to the Faces, and Counter-Faces, draw the Base of the Rampart parallel of 8 Fathoms, and all the other Distances as in the single Tenaille. See the Plate K. Fig. XVI.   

CHAP. XV. To make a Bastion with Counter-Guards.  
FOrmerly they were wont to put these Works before the Bastion, the better to cover them. The Structure is thus.  

1. Draw Lines parallel to the Faces of the Bastion of 15 Fathoms, terminating upon the Counterscarp of the Ravelin or Half-Moon, and forming a point before the point of the Bastion, as ABC.  
2. Draw the Base of the Rampart to these two Faces of 18 Fathoms, as ED. All the other Distances are the same with those of the Ravelin or Half-Moon. See Plate I. Fig. 14. C. XII. Lib. III.    

CHAP. XVI. Of the Structure of a Pentagon.  
 Fathoms. Feet. 1. THe Semi-diameter 153 01 2. The Exterior side AB● CDJ. 180 02 3. The Perpendicular EF. 25 00 4. The Face AF. 50 00 5. For the Compliment of the Line of Defence FG. take 38 03 6. The Line of Defence GB. 131 03 7. The Flank JG. 22 02 8. The Curtain GH. 73 04  

The Explication of this Table.  
SEe Chap. 2. Book 3. There you shall find all that is necessary for the Structure of a Pentagon; provided you take the Distances prescribed here, for those that are mentioned in the same place. See the Plate F. Fig. 16.   

Of the Structure of a Tenaille in the Moat.  
TO justify what has been said above of the Line of Defence, observe this Work, the Structure of which is thus.  

1. Put the Rule upon the advanced Flank of the Body of the place, and draw a Parallel to that Flank of from 5 to 6 Fathoms, which touches the Line of Defence before the Orillon in EBB.  
2. Divide the rest of the Line of Defence from BE. to the nearest Angle C, into two equal parts, as BF. EGLANTINE.  
3. Take the Distance FG, between the points of the Compass, leaving one upon the point F: Carry the other from the point G upon the Compliment of the Line of Defence, mnrk●d F. Then fix one point of the Compass upon the point G. and carry the other from F. upon the other opposite Compliment, marked H. Then join HF and JG together, to form the Flanks: Do the same for the Curtain HJ.  
4. Draw the Base of the Rampart of 8 Fathoms to the Faces, and to the Flanks of the Tenaille; and to the Curtain from 4 to 5 Fathoms. Then the Base of the Parapet of 2 Fathoms 3 Feet round about. You shall make the 2 Banquets as usually. See the Plate M, Fig. 16.     

CHAP. XVII. Of the Structure of a Hexagon, and other Polygons.  
1. THe Semi-diameter takes up 
Fath. 180  2. The Exterior Polygon 
180  3. The Perpendicular 
27  4. The Face 
50  5. For the Compliment of the Line of Defence 
38  6. The Line of Defence 
136  7. The Flank 
25  8. The Curtain 
71   
The other Distances are equal to those of the Pentagon.  

Sect. 2. The Heptagon.   Fathoms. Feet. 1. THe Semi-diameter takes up 207 03 2. The Exterior Side 180 00 3. The Perpendicular 27 00 4. The Face 50 00 5. For the Compliment, etc. 38 00 6. The Line of Defence 135 03 7. The Flank 24 00 8. The Curtain 72 03  

Sect. 3. The Octogon.   Fathoms. Feet. 1. THe Semi-Diameter 238 01 2. The Exterior Polygon 180 00 3. The Perpendicular 28 00 4. The Face 50 00 5. For the Compliment, etc. 28 00 6. The Line of Defence 137 03 7. The Flank 25 04 8. The Curtain 71 04  

Sect. 4● The Enneagon.   Fathoms. Feet. 1. THe Semi-diameter 263 01 2. The Exterior Polygon 180 00 3. The Perpendicular 32 00 4. The Face 50 00 5. For the Compliment, etc. 38 00 6. The Line of Defence 138 00 7. The Flank 29 00 8. The Curtain 70 03  

Sect. 5. The Decagon.   Fathoms. Feet. 1. THe Semi-diameter 291 03 2. The Exterior Polygon 180 00 3. The Perpendicular 36 00 4. The Face 52 00 5. For the Compliment, etc. 38 00 6. The Line of Defence 135 00 7. The Flank 32 00 8. The Curtain 70 00  

Sect 6. The Hendecagon.   Fathoms. Feet. 1. THe Semi-diameter 319 00 2. The Exterior Polygon 180 00 3. The Perpendicular 37 00 4. The Face 152 00 5. For the Compliment, etc. 38 00 6. The Line of Defence 135 00 7. The Flank 32 02 8. The Curtain 70 00  

Sect. 7. The Dodecagon.   Fathoms. Feet. 1. THe Semi-diameter 347 04 2. The Exterior Polygon 180 00 3. The Perpendicular 42 00 4. The Face 52 00 5. For the Compliment, etc. 38 00 6. The Line of Defence 137 00 7. The Flank 37 00 8. The Curtain 69 00   

CHAP. XVIII. To make the Profil.  
HAving now treated of the Breadths and Lengths of the parts of a Place, it is requisite to say something of the Height of the said parts; or rather after what manner to make the Profil of a Place. To which purpose two things are to be observed.  

1. That you erect the Profil of the Exterior Works after the same manner as that of the Body of the principal Place; provided that you apply thereto the heights, of which I shall speak as much as is necessary to be known, whereof the Breadths have been above exhibited.  
2. That 10 Fathoms of the Scale of which I make use in delineating the Body of the Place, make but one, when I would make the Profil of the same Place. So that 100 Fathoms which I have applied to the Body of the Place, for this purpose, afford only a Length of 10 Fathoms.   

The Profil of the Body of the Place.  
FOr Examples sake, let us make the Profil of a Pentagon; and then observe that which follows.  

1. Draw a long straight Line, and there mark 11 Fathoms for the Base of the Rampart, beginning at the Left Hand, as AB.  
2. Put 1 Fathom 4 Feet immediately behind the Point A, upon the same Line as AC. This Triangle marks out the Interior Talus. And to mark out the Exterior Talus, put 1 Fathom immediately before B, as BD. But in the Field you must allow larger Talus to the Rampart, according to the Nature of the Ground.  
3. Erect upon CD two Perpendiculars, each of which contain 3● Fathoms in height, as CE, DF, and then join A, OF, and FB, by a right Line, and that will afford the height of the Rampart with its Talus.  
4. Put 4 Fathoms of the Point E toward the Right Hand, to mark out the Platform of the Rampart, as EGLANTINE.  
5. Erect upon G a small Perpendicular of a Foot and a half for the height of the first Banquet, as GJ. Then draw from the point I a Parallel to the height of the Rampart a Foot and a half broad, as JH; that is, the breadth of the Banquet. Then set another Perpendicular upon the point H, of the same height as HJ; and from the point I draw also another Parallel of two Feet broad, as LM, and then you will have both the Banquets.  
6. Put 4 Feet immediately behind the point G, as GN, and then upon that point erect a Perpendicular of 8 Feet, as NO; which done, join the point M, and the point O together to get the Interior height of the Parapet with its Talus.  
7. Prolong the Exterior Talus BF two Feet farther, as H P, and join O P together to have the breadth of the Parapet with its Exterior Talus.  
8. Put four Feet upon the first Line behind the point B, as BQ; from which point let fall the Perpendicular QR, of 3 Fathoms, and join BR together for the Scarp. Which done, put 20 Fathoms behind the point B, for the breadth of the Moat, as BS, than put two points before the point S, as ST; from whence let fall another Perpendicular of 3 Fathoms, as TV; then join US together to have the Counterscarp of the same RV, and so the Moat will be entirely represented.  
10. After this take 36 Fathoms, so far as the Ground will permit, and put them behind the point W, upon the same first Line, as WA, and then join them together by a straight Line, to represent the Glacis. After this manner you shall have the Body of the Pl●ce according to its Lengths, Breadths, and● heights. See the Plate N. Fig. 17.    

The Profil of Exterior Works.  
THis Profil is made as the preceding in respect of the 7 first points, saving only that the Base of the Rampart be no broader than 8 Fathoms, nor the Rampart higher than 14 Feet. As to the 8 point of the Description of the preceding Profil, the only difference consists in this, that the Moat is not larger here than 12 Fathoms, nor deeper than 2 and a half. All the rest is done as in the preceding. See the Plate N. Fig. 18.    

CHAP. XIX. How to trace out the Draught of a Fortress in the Field.  
AFter you have seen the manner of Fortifying upon Paper, it is requisite for an Engeneer, or any one that would discourse of Fortresses, to understand of what breadth, height, and depth, their Parts ought to be; whence the one or the other part derives its defence, whether it be necessary to fortify one part more than another; where to attack a Place, and after what manner to frustrate the Enemy's design. We shall therefore now show how to trace out a Fortress in the Field: To which purpose it is necessary to know which are the Angles of Exterior Polygons; being these that follow.  

The Angle of the Exterior Polygon.  
Degrees.  The Angle of a Square is of 
90  That of a Pentagon of 
108  Of a Hexagon of 
120  Of a Heptagon of 
128  Of an Octogon of 
135  Of an Enneagon of 
140  Of a Decagon of 
144  Of an Hendecagon of 
147  Of a Dodecagon of 
150   

The Explication of this Table.  
For Example: If you would trace a Polygon of five Bastions, which is called a Pentagon, of which every Exterior Polygon must have the length of the Line given AB, which is 180 fathoms, then proceed thus: 

1. Fix the Ba●oon, upon which you have your Astrolabe, in the hole that is made by the little Peg B, so that you may see through both the Sights of the Astrolabe, from the Point B, or the place where your Instrument is fixed, the Peg A.  
2. Count from the left hand towards the right an Angle of 108 degrees, which is the Angle of the Exterior Polygons, to the Pentagon, (if you make an Hexagon, you take this Angle of 120 degrees, etc.) there fix a little Peg, which you may see through the movable Sights, for Example, C: That done, fix a Cord or a Chain to the Peg B, and draw a straight Line toward the Point C, upon which you shall then put the length of the first Line given AB, as BD.  
3. Put the Instrument upon the Point D, so that through its Diopters or Sights you may see the Peg B. Then seek by the Rule of the Diopters for the Angle of 108 degrees, and there fix another Peg, which you may see through the Diopters, as DE. That done, draw a straight Line from the Point D to the Point E, and then put the length of the Line BD upon the Line DE, as DF.  
4. Do this as many times as is requisite for a Pentagon.  
5. Divide the Flank into three equal parts, and the first also into two others, in the middle of which fix the Peg H. Then fasten the Chain to the Point H, and draw the Rounding, as upon Paper, with the Compass: That done, draw the Brisures of 5 fathoms, and make in their Extremities two Arches to cut each other in I. Then ●ix in the place of the Point I a little Peg, and with the Chain that is fastened to it, you shall draw the Flank retired.  
6. To make the Rounding of the Moat, set the Peg upon the point of the Bastion or Ravelin, and with the Chain draw the Rounding. See the Plate O, Fig. XIX, XX      

CHAP. XX. To make the Streets in a Fortress.  
1. DRaw to the Curtains of the place a Parallel of 10 fathoms for the place of Arms near the Rampart.  
2. Draw all the Points made of that Line before the Bastion of the place; for Example, ABCDE.  
3. Take the Demi-gorge of the place, or which is better, 30 fathoms, and put them from the Centre upon those Lines ABCDE, as abcde: Which distances join together likewise by straight Lines.  
4. Take 3 fathoms, and put them at the Points Aa, towards Bb and Cc, as Ff, Gg; which must also be done afterwards with Bb, Cc, Dd, Ee, etc.  
5. Join Ff and Gg together by Right Lines.  
6. Take 8 Fathoms of the Line Ff, and divide the rest into three equal parts. Then take one part of the three, and put it at the Points F and G, upon the Lines drawn in the Grand Place of Arms, as FG, GL. Which done, put four fathoms upon the same Lines, after the Points HL, as HIM. LN. Lastly, join HL and MN together by right Lines.  
7. Take the second part of the three first, and put them after MN upon the same Lines, as MORE, NP, and join the Line OPEN together. Which done, you shall replace 4 fathoms after the Points OPEN, as QR, and then likewise join QR together.  
8. Divide each Side, FG, fg, into two equal parts, as ST, and join the Lines ST together by a Right Line. Then put 3 fathoms on each side of the Points ST, as Sss, Ttt, and draw ST together; and these will form the Streets of your Fortress. See the Plate P, Fig. XXI.    

A NEW TREATISE OF Fortification, BOOK IU.  

CHAP I. Of the Principal Angles of a Fortress.  
THIS Book contains the Explication of the Parts of a Fortress, whereof we have already spoken: By which are to be understood, the Cavaliers, the Streets, the Places of Arms, the Counter-Guards, the Magazines, Gates, Back-Doors, Draw-Bridges, Bridges, and other Things. But we shall first begin to trea● of the Angles.  

Sect. 1. To find the Angle of the Centre of every Regular Fortress.  
DIvide the Circumference of the Diameter, which contains 360 Degrees, by the number of the Bastions of your Fortress, and the Product of that Division will make the Overture, or Angle of the Centre. For Example: Let us seek the Angle of the Centre of a Pentagon; to which purpose, you must divide 360 Degrees by 5. For in regard our Fortress has no more than five Bastions, a larger Divisor cannot be taken: So that the Product of the Division being 72, we must say, that the Angle of the Centre of a Pentagon has 72 Degrees of Overture.   

Sect. 2. To find the Angle of the Gorge, as it is called af●er Vauban's manner; or the Angle of Exterior Polygons.  
TO find that Angle, there needs no more than to subtract the Angle of the Centre of the same Fortress of 180 Degrees, and that which remains makes the Angl● of the Gorge. For Example: If you would know the Angle of the Gorge of a Pentagon, you must from 180 Degrees subtract 72; for that is the Angle of the Centre of a Pentagon: Then observe what remains, which will be 108 Degrees, and say, That the Angle has 108 Degrees of Overture.   

Sect. 3.  
AS for the Angle of the Bastion, and all the rest, it is impossible to know what their Overtures are, in regard they are not always the same: So that it depends upon the knowledge of the Engineer to make the Flanks, so that they may form a good Angle of the Bastion, according to which almost all the rest take their measures. Consult our Authors upon this matter. Nevertheless we shall here set down the Overtures of the principal Angles after our manner of Fortification.  

In a Square.   Degr. Min. (1.) The Angle of the Centre is 90 00 (2.) The Angle of the Gorge 90 00 (3.) The Angle of the Bastion 63 00 (4.) The Angle of the Courtin 98 30 (5.) The Angle forming the Flank 81 30 (6.) The Angle forming the Front 112 30 (7.) The flanking Angle 67 30  

In a Pentagon.   Degr. Min. (1.) The Angle of the Centre is 72 00 (2.) The Angle of the Gorge 108 00 (3.) The Angle of the Bastion 77 00 (4.) The Angle of the Courtin 101 50 (5.) The Angle forming the Flank 78 10 (6.) The Angle forming the Front 116 00 (7.) The flanking Angle 64 00  

In a Hexagon.   Degr. Min. (1.) The Angle of the Centre is 60 00 (2.) The Angle of the Gorge 120 00 (3.) The Angle of the Bastion 87 00 (4.) The Angle of the Courtin. 106 00 (5.) The Angle to form the Flank 7● 00 (6.) The Angle to form the Front 121 40 (7.) The flanking Angle 58 20  

In a Heptagon.   Degr. Min. (1.) The Angle of the Centre is 52 00 (2.) The Angle of the Gorge 128 00 (3.) The Angle of the Bastion 95 00 (4.) The Angle of the Courtin 102 30 (5.) The Angle to form the Flank 77 30 (6.) The Angle to form the Front 120 00 (7.) The flanking Angle 60 00  

In an Octogon.   Degr. Min. (1.) The Angle of the Centre 45 00 (2.) The Angle of the Gorge 135 00 (3.) The Angle of the Bastion 100 00 (4.) The Angle of the Courtin 103 00 (5.) The Angle to form the Flank 77 00 (6.) The Angle to form the Front 121 00 (7.) The flanking Angle 59 00  

In the Enneagon.   Degr. Min. (1.) The Angle of the Centre 40 00 (2.) The Angle of the Gorge 140 00 (3.) The Angle of the Bastion 100 00 (4.) The Angle of the Courtin 106 00 (5.) The Angle to form the Flank 74 00 (6.) The Angle to form the Front 125 00 (7.) The flanking Angle 55 00  

In a Decagon.   Degr. Min. (1.) The Angle of the Centre 36 00 (2.) The Angle of the Gorge 144 00 (3.) The Angle of the Bastion 100 00 (4.) The Angle of the Courtin 93 00 (5.) The Angle to form the Flank 87 00 (6.) The Angle to form the Front 115 00 (7.) The flanking Angle 65 00  

In an Hendecagon.   Degr. Min. (1.) The Angle of the Centre 33 00 (2.) The Angle of the Gorge 147 00 (3.) The Angle of the Bastion 100 30 (4.) The Angle of the Courtin 101 30 (5.) The Angle to form the Flank 78 30 (6.) The Angle to form the Front 125 00 (7.) The flanking Angle 55 00  

In a Dodecagon.   Degr. Min. (1.) The Angle of the Centre 30 00 (2.) The Angle of the Gorge 150 00 (3.) The Angle of the Bastion 100 00 (4.) The Angle of the Courtin 101 30 (5.) The Angle to form the Flank 78 30 (6.) The Angle to form the Front 128 00 (7.) The flanking Angle 52 00   

Sect. 4. Of the Angle of the Bastions.  
THis Angle has occasioned great Disputes among the most part of the Ancients, who have Treated of Fortification: For some were of opinion, that this Angle was to be always acute; others, that it only aught to be so in the Hexagon, and that it ought to be Right in Figures of more than 6 Bastions; and this was the usual practice of M. Antoine de Ville. Others there are who maintain the contrary, and say, That the acute Angle is never to be allowed of, but in Figures that are under 6 Bastions, and that it always aught to be Obtuse in a Fortess which exceeds that number. True it is, that the greatest number have been for the right Angle; and it were a sort of Ignorance to maintain the contrary, Morsheus●r made choice of this Angle in a Square of 60 degrees, in a Pentagon of 72 deg. in a Hexagon of 80 deg. in a Heptagon of 85 deg. 43 m. in an Octogon to an Hendecagon of 90 deg. Freittag made choice of this Angle from a Tetteragon of 65 deg. to a Pentagon of 74 deg. from a Hexagon of 80 deg. to a Heptagon of 84 deg. 17 min. from an Octogon of 87 deg. 30 min. and from an Enneagon to a Dodecagon of 90. But he soon altered his opinion, maintaining that this A●gle aught to be from a Tetteragon of 60 deg. to a Pentagon of 69; from a Hexagon of 75, to a Heptagon of 79 deg. 17 min. from an Octagon of 82 deg. 30 min. to an Enneagon of 85 deg. to a Decagon of 87 deg. to an Hendecagon of 88 deg. 38 min. to a Dodecagon of 90 deg. Goldman makes use of this Angle for a Square of 65 deg. for a Pentagon of 69 deg. for a Hexagon of 75 deg. for a Heptagon of 79 deg. 17 m. for an Octogon of 82 deg. 30 min. for an Enneagon of 85 deg. for a Decagon of 87 deg. for an Hendecagon of 88 deg. 38 min. for a Dodecagon of 90 deg. Pascha takes it for a Square of 60 fathoms, and augments it still by five, according as the Polygon gains more of the Side. But they who have been accounted most skilful in the Art of Fortification, were always, as I have said, for the Right Angle, though it be alleged, that in making it Acute, there is room for more Firing from the Curtains; or that in making it Obtuse, it is more capable to withstand the Enemy's Batteries, and the Injuries of Time. The reason that kept them so fixed to this Opinion, was, that in their time they made their Breaches with the Cannon, so that this Angle being Right, it had all the perfection that could be desired; whereas the making it Obtuse, diminished the Capaciousness of the Bastion, and of its Flank; besides, that the same Battery could play upon both Fronts: And by making it Acute, the Point of it was so soon broken by the frequent discharges; more especially, if several Guns played at a Time; besides, that bad Wether more easily ruined that Point. But to speak the truth, we ought to believe, that this Angle, whether Right, Acute, or Obtuse, ought never to trouble our Thoughts, provided it be not less than 60 degrees, nor much more than 100: For the Breaches are now made by Mining, wherein there is ●o more trouble to carry the Point of a Bastion with an Obtuse Angle, than with an Angle either Right or Acute. Moreover, there is this in it, that Breaches are rarely made in the Point of a Bastion, but in one of the Fronts, to the end, that in passing the Moat the Assailants may not be exposed to more than one Firing; so that provided the Angle be well able to resist the Injuries of Time, it is enough. However, much more than 100 degrees are not to be exceeded, because that being more open, it would require the Capital Line to be short; in which case, the Bastion would not have room sufficient.   

Sect. 5. Of the Angle of the Curtain.  
ENgineers cannot agree concerning the Overture of the Flank; so that every one orders it according to his Fancy. Erbard makes the Flank perpendicular to the Faces of the Bastions; and his Reason is, because the Flanks, being oblique above the Curtain, that is to say, making an acute Angle with it, are much more exposed to the Enemy, and consequently more difficult to be ruined, as not being battered by the Cannon but very obliquely: But granting that this were so, it is no less a Truth to say, that as they are but little exposed, so neither do they not survey very far; and it is impossible to make good Embrasures upon a Parapet, ordered after that manner. For to play the Cannon above the Parapet, is to make it impossible to preserve it in a good condition; besides that a Flank disposed after that manner, can never defend the Passage of the Moat, nor flank the Breach, nor in the last place, uncover the covert Ways. As for Monsieur le Chevalier de Ville, who makes the Flank perpendicular to ●he Curtain, he has more Reason for what he does, than Erbard; though his Fla●ks are less than those which Monsieur Vauban makes use of, and that they do not batter the Moat sufficiently, nor the Counterscarps, which is the most essential part of the Defence. Besides, it is but too well found by Experience, that when the Counterscarp is gained, the rest holds out but a short time; the Reason of which, if it be demanded, is because the disposing of the Places is so disadvantageous, that one Day sees them reduced to D●st; and the Embrasure● made upon these Parapets, perpendicular to the Curtain, are too acute; especially if it be required that they should clear the covert Ways. I know it may be said, that it is easy to m●ke an Orillon, or Ear, and a retired Flank in a circular fashion: But the Orillon will be too feeble, unless you allow it the half of the Flank, and the remainder will not suffice for its Defence, not having room enough for several Pieces to be placed upon it; besides that, there must be a mixture of great and small Shot, which will cause an inevitable Trouble.  
Monsieur Pagan, a Person experienced to perfection in the Art of War, and one who has justly acquired the Reputation of being one of the most skilful Engineers of our Time, was the first that altered the disposing of Flanks perpendicular to the Curtain, by allowing them much more overture; for he makes them perpendicularly upon the Line of Defence; and he gives very good Reasons for it in his Book of Fortifications; where he says, That his being present at above twenty Sieges, convinced him, that Flanks perpendicular to the Curtains, were of little durance: And he wonders that the Reputation given to the Hollanders, for their Perfection in the Art of Fortification, should be so great, ●ince the Places fortified according to their Method, make so little Resistance; and for that there is not one which is well defended from the Canon: Nevertheless we may presume to say, that Monsieur Pagan'● 〈…〉 altogether without defects; for the Enemy 〈…〉 nk so disposed, before he comes upon the covert 〈…〉 ●efore to make the best of all the Advantages wh 〈…〉: one and the other produces in his defence, Monsieur Vauban allows an Overture to this Angle, from 98 deg. to 106, as may be seen, Book IU. C. ●. Sect. 3. For by this means the Embrasures made in the Parapet in this Flank, perfectly uncover the Moat, and the covert Way. And though Monsieur Vauban should not have made this Flank in a crooked Line, by means of the Orillon, and the retired Flank, however it ought to be preferred before others; because the Ravelin covers the Overture which is allowed it above the Perpendiculars to the Curtain, and consequently it is not too much exposed; b●●ides that it considerably enlarges and diminishes the Faces, which are the weakest Parts of a Fortification.    

CHAP. II. Of the Flank.  
HERE we must repeat what we have already said, That though the largest Flanks are the best; nevertheless it is a Thing disputed among the Engineers. Morsheuser made th● Flank to a Square of 16 Fathoms; to a V, of 18; to a VI, of 20; to a VII, of 22; to an VII, as far as a Dodecagon of 24. Freittag made it to a Square of 12; to a V, of 14; to a VI, of 16; to a VII, of 18; to an VIII, of 20; to IX, of 22; to X, of 24; which Length he retains also in Polygons of 11 and 12 Bastions. Goldman made it a Square of 12; to V, of 16; to VI, of 1●; to a VII, of 20; to an VIII, of 22; to a IX, ●ll he came to Polygons of 12 Bastions of 24 Fathoms. In general, the Hollanders make their Flanks to a Square of 16, Fathoms, 2 Feet; to a V, of 18 Fathom, 2 Feet; to a VI, of 19 Fathom 2 Feet; to VII, 19 Fathoms 5 Feet; to an VIII, 2● Fathoms 1 Feet to a IX, of 22 Fathoms 1 Foot and a half; to a X, 23 Fathoms, 2 Foot; to XI, of 24 Fathoms, 2 Feet. Monsieur the Count of Pagan, allows the Flank in a Square, 19 Fathoms, 1 Foot● to a V, 24, etc. as you may see, L. 3. C. 16, & 17.   

CHAP. III. Of the Faces.  
AS the Face is the most dangerous Part of a Fortress; and for that it is extremely exposed to the view of the Enemy, 'tis requisite that it should be as short as may be; for that, by how much it is the more advanced towards the Field, by so much it is the more liable to be attacked: Whence it c●mes to pass tha● the Enemy of ●●●e generally makes choice of that Part, to begin his Breaches. As to the length of the Faces, the Engineers differ as much one among another, as they do about the Flanks. Adrian Matins would have the Face to carry two third parts of the Curtain. Monsieur, the Count of Pagan, allows it in a Square 55 Fathoms, and permits the same Length for the Polygons. Freittag would have the Face to be always 48 fathoms, according to the practice of all the Hollanders. As for the Length of the Faces, after Vauban's Method see Chap. 16. of the Third Book,   

CHAP. IU. Of the Orillon, and Flank retired; together with the Brisure, or Place where the Great Guns are to be planted.  
AS for what concerns the erecting of the Orillon, and Flank retired, according to Monsieur de Vauban's Method, there needs no more than to read what he has written thereupon in the Second Chapter of his Third Book. But here is a new Method; which though it be not better, however it is as good.  
1. After the Out-Lines of the Fortress are drawn, divide the Flank of one of the Bastions into three equal parts, and the Face of the opposite Bastion into two, as, A, ●, B, C.  
2. Draw from the Point B, which makes the middle of the Face, a Line through the third part of the Flank, which is the nearest to the Angle of the Shoulder D, prolonged in the Body of the Bastion of 5 fathoms, as DE.  
3. Divide this same Third part of the Flank, into two equal parts; as FG, and GD● from which Point G, erect a Perpendicular of about two fathoms, as GH. Lastly, Set the Compass upon the Point H, and so draw the Rounding of the Orillon, joining FD, together.  
4 Draw from the re-entering Angle of the Moat P a straight Line through the Angle of the Curtain L, prolonged in the Body of the Place, of an equal bigness to the Line DE; that is to say, of five Fathoms, as, KI; to the end that you may have the Brisure, where you may place the piece of Cannon that points towards the Half Moon, & cannot be battered from the Field; because it is, as it were, covered by the Ravelin. Lastly, To make the Rounding of the Flank retired, take the Distance KE, between the Points of the Compass, and make Sections toward the Moat, in L; from whence afterwards draw the Arch from K to F.  
It cannot be said, but that these orillon's are very good; for these Rounding give less advantage to the Enemy's Cannon, and enlarge the room, as well for placing the Cannon, as for the Musketeers. If you object, That the Distance from that part where the Enemy may pass the Moat, is too great. I answer, That the principal Defence of the Small Shot to hinder the Passage of the Moat, consists in the Ten●ille; where you may set a Body of Musketeers, to fire continually.  
Upon this Method, Monsieur de Clermont observes, That the Orillon of the Square would be too little, if you allow it no more than the Third Part of the Flank, as to the Polygons; and therefore he requires Two Fifth Parts of the same Flank, for the Orillon. See the Plate P. Fig. XXII. at the end of the Third Book.   

CHAP. V. Of the Curtain.  
IN regard that the Curtain ought to cover the greatest part of a Work, it is requisite that it should be of a reasonable Length; that is to say, that it should never be shorter than Sixty Fathoms, and never longer than an Hundred. M●rsh●user allows it 72 Fathoms, and Freittag as much, Goldman would have the Length to be of 96 Fathoms; Usum de Ville 100 Tieu makes it 74 Fathoms in Length; Himsel 84. Monsieur de Clermont allows it Three Fifth Parts of a Polygon. The Count de Pagan 60 Fathoms. Monsieur de Vauban forms it according to the Plurality of the Bastions. As you may see Lib. 3 c. 17.  
As to the Figure of the Curtains there are some Engineers, who will have them ●o be, as is marked at the Letter A; of which the Construction is such. See the Plate P. Fig. XXII. at the end of the third Book.  
1. Divide the Line AB into ● equal parts, as AD, DM, NC, and CB, and take ● for the Demi-Gorge, as BC, AD.  
2. Divide the same Line AB into three equal parts, and take one for the Capital, as BE, A●.  
3. Where these three Parts are marked, as at R and G, bring down from the Points R, G, two Perpendiculars, of which se● each have the third part of the Capital Line for its length, as RH and GI', and then join H and I together by a Right Line.  
4. Make upon the Points D and C, the Angle of the Curtain of 100 degrees, and allow to the ●lank the length of the Demi-Gorge, ●● DP and CQ. This being done, join EQ and DP together, to have the Faces. The defects of the Curtain are, 1. That it is too short. 2. That the ●ortification takes up too much room. 3. That it renders the Bastion too Acute.  
Others there are that form an Angle without side, as that which is marked with the Letter B. The Construction of which is after this manner.  
1. Divide the Line AB into five equal parts, as AC, CD, DE, OF, and FB.  
2. Divide the first Line AB into three equal parts, and allow one for the Capital, as AI, and BK; then join HI and BK together, to gain the Faces.  
3. Divide the Curtain CF into two equal parts, and up●n the middle, marked with the Letter L, erect a Perpendicular; which has the half of the Face GK; as LM.  
4. Draw from CF two straight Lines to the Point M, and your Curtain will be made. See the Plate P. Fig. XXIV.  
●●hers make the Curtain of a Circular Figure withinside, like that which is marked C: The Construction of which is as follows. See the same Plate. Fig. XXV.  
Having observed what has been said touching the three first Points of the Preceding Curtain, take the Extremities of the Curtain BC, and make two Arches to cross or cu● one another underneath; and from this Point of the Intersection draw an Arch, from one end of the Curtain to the other.  
Lastly, There are others that form the Curtain of a Circular Figure without side, like that which is marked D. The Construction of which is like to that which we have already described, only that here you make the Intersection above the Curtain in E: From which Point you afterwards draw the Arch from one end of the Curtain to the other.  
The Breast Curtains are those which are drawn straight outright from one Flank to another. See the same ●lank P● at the end of the third Book. Fig. XXVI.   

CHAP. VI Of the Parapet.  
IN regard that this Work is raised to cover the Cannon and the Soldiers which defend the Place, it must be allowed a reasonable height and breadth. Cellarius allows it six Feet in height, with a Foot of sloping within side● to the end the Soldiers may be able to stand the better when the Parapet is reared after that manner. The height without side from the Cord●n ought not to be above four Feet, because the ●loping which is made by those different Heights, inward and outward, gives the Soldier an easier view of the Covert-ways, and the Moats that are before him. As for the Thickness of the Parapet, Monsieur Clermont allows it for the Basis, the third part of the Basis of the Rampart. The Count de Pagan makes it three Fathoms broad. Of which opinion are several others, as Goldman, Fr●ittag, and Monsieur Vauban himself. So that we may boldly say, That the Thickness of the Parapet is good, when it is not less than 18 Feet, nor broader than 24. For being weaker, the Parapet cannot long resist the force of the great Shot, which would soon levelly it with the Earth● and being thicker, it would hinder the Soldiers from the sight of the Moat and the Covert way.  
If you would line the Parapet, it must be allowed a little ●loaping, that the Soldiers may have the better footing. The best Lining of Parapets is with Turf. As for the Earth or Mould which you are to make use of in erecting a Parapet, it is very requisite to mix it with Withy Twigs, or Brambles, and to sow it with any Weeds that take a deep root, to bind the Earth together, so that the Cannon may not easily crumble it down. Hay of Burgundy is also very good for this purpose, in regard that there is no Plant that puts forth more or deeper Roots.   

CHAP. VII. Of the Banquets.  
AT the foot of the Parapet, upon the Platform, are usually made two little Steps or small Banks, of which the first is about two Feet broad. Goldman makes it four Feet, Fr●ittag 3. the Count of Pagan 2. Monsieur Clermont the same; and one Foot, and one Foot and a half high● wherein they all agree. There are two made, for that the Soldiers are not all of an equal height. So that the low Men are placed upon the upper Step or Bank next the Parapet, and the taller upon the lowermost: so that every one may have a view from the Parapet, and discharge at his ●ase.   

CHAP. VIII. Of the Rampart.  
THIS Work is the Principal Piece of a Fortification; and therefore the Rampart ought to be higher and broader than any of the rest of the Parts. Fournier allows for the Breadth of the Rampart from 15 to 25 Paces; and for the Height, from the Level of the Field from 15 to 25 Feet. Freittag takes for the Base of a Square 9 Fathoms: of a Pentagon, 10. of a Hexagon, 11. of a Heptagon, 12. of an Octagon, 13. of an Enneagon, 14. and observes this Breadth in all other Polygons. The Height of the Rampart he thinks sufficient if it be but 3 Fathoms; and he raises the said Rampart above the level of the Field three Fathoms. Monsieur Clermont would have the fourth part of the Face allowed for the Base of the Rampart, or in the Field from 8 to 12 Fathoms: believing that a larger breadth would not only be useless, but also prejudicial, by reason of the great space of Ground it would take up, and the vast quantity of Earth it would require. Monsieur de Vauban never makes the Basis larger than 11 Fathoms, alleging the same reason that Monsieur Clermont brings. The height of his Rampart never exceeds four Fathoms, whether to cover the Houses of the Place, or to command the Enemy's approaches: For being lower, the Garrison within could never command the Enemy's Works. If they 〈◊〉 me that in some places a higher Rampart is requisite; I answer, That a good Cavalier supplies that defect, without raising the whole Rampart any higher. The upper Breadth of the Rampart ought to be from 4, to 6 Fathoms; for being too narrow, it would require that the Parapet should be made too broad to consume the rest of the Earth, appointed for the Rampart; besides that it would be prejudicial, because too broad a Parapet hinders the Soldiers from the sight of the Moat; and being too large, the remainder of the Earth would not suffice to make a good Parapet, which is also very dangerous. As for the sloaping of the Rampart, it has been said above, That there ought to be allowed ●o the Talu● or sloping of the Rampart, as the Earth tumbling down from the top to the bottom takes up of itself. For every Body knows, th●t, according to the badness of the Earth, the sloaping aught to be more or less, the better to sustain the Rampart. The best Earth is that which is called Grey Clay, next to that the slimy Earth, because that by reason of their clamminess, th●y bett●r resist the inconveniencies of Heat and Rain; besides they bind exactly well together, support themselves with little sloapi●g, and feed the Roots of Weeds and Grass better. To every foot of Earth, where the Rampart is raised, two ●ranches of Willow are to be set no bigger than a Man's Thumb: Besides that, the Earth is to be so hard rammed down, that it may sink four or five Inches, and that there remain not above seven or eight. Lastly, You ought to sow with Hay-Seed and Weeds upon the outside in every Row, to the end the Earth may intermix with the Roots. Nor is there any Herb that casts forth more or deeper Roots than the Spanish Tre●oyle, or winged Clover. Moreover, you ought to erect the Platform so, that the Plane of it may decline somewhat towards the City, to carry off the Wa●er. When you plant Trees upon the Rampart, it is a great Ornament in time of Peace, and a good Provision in time of War. There are some Engineers that do not like this Advices for they say that the Wind makes such a noise, when the Branches hit one against another, that the Men can hardly hear one another: besides that, it is a gr●●● hindrance to the Centinal, whic● is a consideration not altogether to be rejected. Lastly, We cannot pass by in silence the Fancy of some Men, who proportion the thickness of the Rampart to the number of the B●stions of the Fortification, as if a Hexagon ought not to have as good ● Rampart as a Decagon. I would ●ain ask them, whether small places are not attacked with the same A●ms as great onest. If they answer, That there is more Earth taken out of the Moat of a large Place, than a small one; we r●ply, That it may be spent upon the Glacis: ●or that the larger the Glacis●s are, the better they are●   

CHAP. IX. Of the Embrasures and Merlons.  
MANY have proposed several sorts of Embrasure●, but the best are those which are straight and narrow within side, and wide towards the Field; to the end that the great Guns may play with more ease toward the right and left. The M●rlons, to the end that they may be good, aught to be made of Earth, the most easy to be tempered that may be: And this Earth ought also to be mixed with Withy Twigs, or Brambles, provided they take Root, after which they are to be lined with good Turf.   

CHAP. X. Of the Moat in the Body of the Place.  
IN regard that every Fortress ought to be environed with a Moat, we shall briefly examine which sort of Moats are first to be preferred. Some make them broad, others narrow, one draws the Mo●t parallel to the Faces, another draws them narrower toward the Point of the Bastion, than in any other Places. Las●ly, one is for a dry Ditch, another for a Moat that is full of Water. One prefers a Moat very deep, but not so broad; another would have it broad, but not so deep. Let us examine every one of these in order. The Count of Pagan makes his Moat for a Tetragon 16 Fathoms broad; and for Polygons of 20, his depth does not exceed 4 Fathoms. Mons●eur Clermont takes from 16 to 20 Fathoms, for the Breadth of the Moat, without any regard to the Plurality of Bastions; and makes it hollow from 18 to 20 Feet. Fou●nie● would have it broad from 15 to 25 Paces, and deep from 15 to 25 Feet. Cellarius makes his Moat 3 Fathoms deep, and broad at the top about 19, and at the bottom near 13. Monsieur de Vauban prefers Monsieur Clermont's Way before all the rest, as to the Breadth and Depth of the Moat: For as to the way of drawing out the Moat, Monsieur Clermont makes it narrower toward the Point of the Bastion than toward the other Parts; more especially if the Flanks are large; for he says that otherwise, it would be requisite that the Moat should be as large as the Length of the Flank, or else much more narrow; and in the first case there would be a necessity of making it very shallow, which would render it defective, for that the Enemy would easily shoulder himself into it. And supposing it were of a reasonable Depth, there would be Ground which no body would know what to do with, by reason of the extraordinary Wideness. And in the second case, there would be one side of the Flank, that would not uncover the Moat, which would be against the Rules of Fortification. Others answer first, that if the Moat were as large as the Flank, with a moderate Depth, that Wideness would hinder the Enemy from crossing the Moat without great loss of his Men. But if the depth were a little more than ordinary; and it should be asked, what should be done with the Earth taken out of the Moat; they reply, ●twould make the Glacis. The learned may judge as they please; but for my part, I hold with them, that are for making the Moat parallel to the Line of Defence. If they still objects that the rounding of the Moat being as large as in other Places, all the whole Curtain and Face of the opposite Bastion cannot be discovered; we must answer, that if they would uncover those Parts above the rounding, they ought to make no Moat at all, because the Point of the Bastion itself is not proper for that. Add to this, that if the Moat be made narrower toward the Point of the Bastion, the Enemy having once gained the Point, the Earth of which is filled one part of the Moat, he will easily pass it by, throwing B●vins over the Ruins of the Bastions, which will show him, as I may so say, the way to enter directly into the Bastion.  
The next Question is, whether a Moat full of Water is to be preferred before a dry Moat; which a distinction between the great and small Places will quickly decide. As for great Fortresses, it cannot be denied but that dry Moats are to be preferred before those full of Water. Some will object, that a Moat full of Water secures a Place against Scaladoes and Surprises; besides that, is very hard to be filled up; and that it affords great Trouble to the Enemy, that would either pass it, or cover himself, or fight within it: And that on the other side, besides that a dry Ditch is extremely exposed to Surprises, very easy to be filled up, and no less easy to be passed; it is very proper to cover the Enemy in making of Mines, by means of which he may blow up a great Part of the Place. But I answer to this, that as well a Moat full of Water hinders the Sallies of those within, without the help of Boats, which make so great a noise, that 'tis impossible but the Out-Sentinels near the Place besieged, must hear it; and the same that is to be said of the Sallies, is to be said of the Retreats. Beside, if the Water that fills the Moat be not a running Stream, it begets an unwholesome Air, causes frequent Distempers, and renders the Soldiers less courageous, by reason of the noisome Stenches that arise out of it; and on the other side, besides that, a dry Moat is more commodious for Sallies, and safe Retreats; there is no Fear of Sickness proceeding from noisome Water, and unwholesome Air. To say something of the Convenience of Mines, which the Enemy has to make, you must make a Cuvette, or little Dike in the middle of the Moat from 15 to 20 Foot broad, and very deep; for the deeper it is, the better it is: Surround it with a good Parapet with Palisadoes round about; and so you will be secure from Mines and Surprises. As for little Fortresses, a Moat full of Water is better than a dry, because Sallies, Retreats, and Succours, are not so necessary as in great Places. The Moat is also to be rounded before the Point of the Bastion, to the end that it may be of the same Wideness quite through, and that it may not advance too far into the Field; which it would do if it were pointed. You may also set a Corpse du Guard upon this Rounding, the better to defend the Covert way. Lastly, When you would line the Moat, they never allow it any great Talus or Sloaping; only they allow it such as the Earth of the Rampart, and the Moat gives it of itself.   

CHAP. XI. Of the Moat belonging to the Outer Works, and the Ava●t-Moat.  
THe Moats belonging to Outer-Works aught to be neither so great, nor so deep, as that which belongs to the Body of the Place: For a Wideness of 12 Fathoms suffices for these Moats, being hollow from 12 to 15 ●eet.  
In short, there are some Engineers, who make a little Moat about the Glacis, though others practice quite contrary. The first pretend that the Earth taken out of the Moat, serves to fortify the Glacis, and secure the Covert-way. On the other side it is said, that the Avant-Moat serves only to hinder the Sallies and Retreats of the Garrison: and this Reason prevails; for if you have need of Earth, the better to secure your covered Way, you may make the Moat of the Body of the Place deep, and employ the Earth which you take from thence, for that purpose. If you think that this Moat prevents Surprises, and the Approaches of the Enemy; I answer, that a good wide Glacis will do the same, without being at so much Charge for an Avant-Moat Nevertheless sometimes they are constrained to environ the Glacis with such a Moat: They make it five Fathoms wide, and two deep. Such a Moat as this you may see about the Citadel of Strasbur●●.   

CHAP. XII. Of the covered Way, and of the ●lacis.  
A Good Covert way will require at least five Fathoms in Breadth, especially if the Ground will permit it; for one broaden is useless and prejudicial, unless its Parapet be higher than ordinary; otherwise it is easily pierced by the Enemy's Batteries: And one less wide, is defective in this● that there is not room for the great Guns and Men, but one will hinder the other. Besides, that when it is too narrows it is not flanked on so many Parts a● it ought; and it may be said, that this part must be then too much in view of the Enemy, whose chiefest Effort in a Siege it is, to make himself Master of the Covert-way.  
This Covert-way has no other Security than the Glacis, which serves it for a Parapet round about. The Wideness of it ought to be at least from 36 to 40 Fathoms, losing itself insensibly in its Inclination towards the Field. But the better to secure the Covert way, we must say that the largest Glacises are ●he best; for that by this means, the Enemy lies more open; and the less it has of Sloaping, the better they are, because the small Shot is the more secure in the Covert way, and does more Execution. In short, there is nothing to be suffered about the Glacis, that may favour the Enemy in his Approaches: So that if the Glacis be higher than five Foot above the Covert way, you must put one or two Banquets below, as need shall require for the short statu●● Soldiers.   

CHAP. XIII. Of False Brays.  
HEre it is, that some will ask me why the French admit no False Brays in their Fortifications, as being a second Defence, and an admirable Work to dispute the Counterscarp with the Enemy, and to hinder him from crossing the Moa●. Moreover it is certain, that small Shot does more Execution from a low than a high Place. Besides, that the small Shot placed in the False Bray, doubles the Defence of it, as that Work doubles the Defence of the Place. But if you talk of the Defence; Why do you then put the Counterscarp before the Faces, knowing that those Parts defend nothing? You will answer, that this is to dispute the Passage of the Moat. But consider, that the Enemy having broken the Parapet of the Point, may afterwards pierce with his great G●ns the Walk that runs all along the Face: Besides that the ruining of that Work is very advantageous to the 〈◊〉, who would willingly have wherewithal to fill the 〈◊〉. As for the part of the False Bray that is before the Curtain, it is no longer good, because it does not flank the Faces of the Bastions, nor hinder the Passage of the Moat, which is objected always for the principal Reason. Nevertheless the remainder of that Work, which is before the Flank, will be good: But know that this part is easily rendered unserviceable, by the Ruin of the Rampart, and the Parapet of the Flank, which falling within side, fills up its Platform: And if the Rampart be lined, the Splinters of it maim the Soldiers, that guard that Flank of the False Braye. To remedy which Defects, the French now adays erect Tenailles in the Moat before the Curtain, and before the Flanks separated from the Body of the Place, by a little Moat, to the end the fall of the Rampart of the Body of the Place may not kill or wound those that are in it, being lined with Walls There is nothing of all that you talk of False Brays, that can be attributed to these Tenailles; besides that they cast not the half of what those other Works do.   

CHAP. XIV. Of Cavaliers.  
Cavaliers are one of the best Defences of a Place, because they most notably scour the Field, which obliges the Enemy to open his Tre●ches at a great distance, in regard these are as so many second Bastions, which are opposed to the Enemy. And moreover that they defend the Passage of the Moat, command the Covert-way, and above the Glacis; 'tis certain, that being well disposed, they may defend the Head of the Breach, which the Enemy shall make in the Face of the Bastion, wherein those Cavaliers are erected. Now  
A Figure showing all the angles in a Square  
A Draught & Square of Vaubane.  
For § 3: Book: III Chap: V. Fig: VI.  
For § 2. Book: III Chap: V. Fig: V.  
For § 4 Book: III Chap: V. Fig: VII.  
in regard that Opinions are different, as well in respect of the place where ●hey are to be set, as in regard of their Figure. Let us see the Reasons both of the one and the other.  

1.  
There are some that would have the Cavalcers placed upon that part of the Curtains which discovers the Face of the opposite Bastion; that is to say, upon the Lines of the second Flank, as A● the Construction of which is such. See the Plate Q, Figure XXVII.  
1. Bring down a straight Line perpendicularly under the Flank, and upon this Line put two third parts of the Demi-gorge; as A, B.  
2. Divide the Line AB into two equal parts; as, AC, CB; and prolong perpendicularly the Point C, till this Prolongation be equal to the Demi-gorge; as, C, D.  
3. Bring down also a Perpendicular from the Point D, of the Length CB; as, DE; and at length join BE, together by a straight Line.  
4. Having raised the Cavalier to a just Height, they give it a Parapet of th●ee Fathoms, and two Banquets, as is usual.  
1 But it may be objected against this Situation, th●t the Cavalier does not flank, but very obliquely● and by consequence, that the Effects of it are of little consequence.  
2. That the Ruins which the Enemy's Cannon will make, as being opposed to it in Front, falling between the Curtain, and the Cavalier, will hinder the Execution of the Small Shot of th● Second Flank.   

II.  
1 Others lo●ge the Cavalier in the middle of the Curt●in, and give it a Triangular Figure; alleging, That in that part● one and the same Cavalier may flank two Faces of the B●stions. To which it is answered, That a Cavalier cannot be erected after that manner, but in Places which have more than half the Curtain for the sec●nd Flank; which would be very difficult, unless it were in City's where the Angles of the Polygon are of a very large Overture. For to say, we o●ght to make these second Flanks ve●y large, as well to augment the Firing from the Curtain, as to g●in room fo● placing of th● Cavalier, this is to make Bastions in a Place, that serve for nothing; because, that by this means, the Flanks are made so small, that they seem to be only for Ornament.  
2. That the Enemy stands in no fear of the Firing from a Cavalier disposed after that manner, when once he is lodged upon the Counterscarps, because it is too high, and the Cannon cannot plunge low enough.  
The Construction of this Cavalier is such. See the Plate Q. Fig. XXVII, B.  
1. From the Curtain bring down a perpendicular Line● and put there above 14 Fathoms; as, AB.  
2. Take that Distance, and draw from the Point B a Circle.  
3. Divide this Circle into Three equal parts, beginning from the Point A; as, AC, AD, and CD; which join together by right Lines; and this shall be the Base of the Cavalier.  
4. Draw on the Sides AC, and AD, the Base of the Parapet of 3 Fathoms. But it is to be observed, That these Cavaliers are not to be erected but from 9, to 10 Feet above the Platform of the Rampart, unless there be very strong Reasons to raise it higher. In the mean time, 'tis very true, that this Cavalier is not better than the preceding, because the Angles of it are very acute; for when the Angle A is carried away, all the rest is worth nothing; since neither Cannon nor Soldiers can subsist up●n the side AC, and AD, by reason of the Enemy's continual playing upon them.   

III.  
Some raise the Cavaliers upon the Platform of the Bastion, and dispose it in such a manner, that the Faces and Flanks of the Cavalier are parallel to the Faces and Fla●ks of the Bastion; as C. The Advantages that pro●●●d from this Disposition, are, that the Enemy being got u●●n the opposite Counterscarp, will be troubled to lodge his Cannon there, by reason of the Cannon upon the Flank, and upon the Cavalier; which playing without ceasing, will dismount the Enemy's Artillery; hinders him from raising Shouldering to secure himself: Besides that, the Breach of the opposite Face is much better defended, and the Field better scoured. But one would think they might enjoy all these Advantages without possessing the Platform of the Bastion; and by that means deprive themselves of the Advantage of making En●renchments; which favour the Defence of Assaults. Besides that, these Cavaliers are of no use for the Defence of the Breach made in the Bastions where they stand. And there is this moreover, That the Besieger may at the same time, with a Mine carry away a part of this Cavalier with the Face of the Bastion. And lastly, ●Tis for the Defendants to deprive themselves of several Advantages; which we shall observe in the following Method, and which has all the Advantages of the preceding Situations, without being obnoxious to their Imperfections. But before we speak of this Cavalier, observe the Construction of it; of which we have set down the Advantages, and Disadvantages.  

1. If the Bastion is empty, it ought to be filled up, till it come to be as high as the Platform.  
2. Which being done, raise that which was empty upon the Platform, from 9, to 10 Feet.  
3. Having levelled that Height toward the Top, you shall draw the B●se of the Parapet of three Fathoms, which ought not to be raised above 6 Feet; because the Cannon are not to be placed higher; for that being higher, they would require a Battery to be raised on purpose.    

IV.  
Lastly, There are others that place the Cavalier within the Bastion, of ● circular Figure; disposing it thus; 

1. They prolong a Line of Defence in the Body of the Place, till it cut the Radius of the Centre at the Point A.  
2. They divide the Distance between this Point A, and th●t of the Gorge B, into two equal pa●ts; as, AC, CB.  
3. From the Point C, they draw a Circle, which makes the Base of the Cavalier.  
4. They raise this Place from 9, to 10 Feet, above the Platform of the Bastion.  
5. They allow this Cavalier a Parapet Three Fathoms wide, and One high●  
6. The best Lining of this Cavalier, is with Turf; for if it were of Mason's Work, the Bricks and Stones, which will be shattered from it, would annoy those that defend the Rampart.    
It is objected against this, That they do not leave room enough between the Foot of the Cavalier, and the Parapet of the Flank, for the free passage of the Men that are to defend the Breach, and to traverse the Guns, which are upon the Flanks of the Bastion, because their Carriages being long, they will take up a great deal of Ground.  
But it is answered, That this Cavalier being distant, at least 5 Fathoms from the Parapet, this Passage seems to be more than sufficient to communicate between the Rampart, and the Bastion, in respect of the place of the Cannon; and in regard the largeness of the Base of the Cavalier, may be lessened, if it be found to incommode the Passage, we must confess, there is room enough, not only when the Cannon are mounted upon small Carriages, but that when they are mounted upon their usual Carriages, these Cavaliers bear them above all others.    

CHAP. XV. Of Counterscarps.  
IT may be s●id, That Counterscarps cut in the Rock, are the best; for, besides the difficulty of opening them, to descend into the Moat, they need no Talus, or sloping; as being sufficient to support themselves.  
Those that are li●ed, are to be preferred before those that are not, by reason of their durance; though it may be feared, That the Enemy will make use of that Lining, to secure himself, and make a Gallery all along with E●brasures, to play upon those that defend the Moat: But this Reason signifies little; for when a Moat is full of Water● the Counterscarp ought to be lined of necessity; otherwise the Earth being moistened by the Water, tumbles easily down but if the Moat be dry, a Lining is not so absolutely necessary. The Sloaping aught to be allowed after such a manner, That if the Enemy should enter with the Soldiers of the Garrison, upon some hot occasion, he could not easily get out again. That part of the Counterscarp which is before the Point of the Bastion, aught to be rounded, to gain the Space above the Covert-Way; provided you can keep there a Body of your Men, to be in a condition to sustain an Assault.   

CHAP. XVI. Of the Streets, Places of Arms, Corpse de Guard, and Magazines.  
HAving showed, Book 3. Chap. XX. how to make the Streets in a Fortress, we shall first speak of the Places of Arms. The Grand Place of Arms ought to be in the middle of the Fort; the construction of which is contained in Chap● XX. Sect. 3 The lesser Places of Arms ought to be on the side of the Bas●ions; the bigness of which ought to be be proportioned so, as to hold 200 men for every Bastion; or 500, if they are to sustain a Siege: And in regard that every Man marching in Battalia, takes up no more than Three Feet in Rank, and Seven in File, it is easy to know how many Men will serve for every Fortress, and how large the Places of Arms ought to be.  
The most numerous Corpse de Guard ought to be in those Parts, where the most numerous Guard mounts. There must be one or two Chimneys at least; and a little Theatre of Wood all along, about three Feet high from the ground, and seven or eight Feet broad, for the Soldiers to lie upon. The other Corpse de Guard, built after the same manner are to be placed near the Gates, or Passages.  
In regard that there is always Occasion for Ammunition, that is, ●or Powder and ●all, for the Soldiers that are upon the Ramparts, there must be Arsenals near at hand. But great ca●e must be taken to put the Powder where it may not take Wet● much more, where it may be in danger of being Fired.  
The Magazines of Hay and Provender for the Cavalry, aught to be near the Stables where the Horses stand. The others must be built in the most secure Parts of the Garrison.  
In regard the Soldiers, as well Foot, as Horse, are often lodged in Huts; these Huts are to be set up near the Rampart, to the end, that in case of necessity, they may be ready upon the Rampart, to defend the Place. To which purpose, the Town is to furnish them with Beds, Tables, Benches, & Hangings for the Officers Chambers, under whose Government the Soldiers are to serve; their Presence being necessary, to keep them in order, and from being injurious to the People of the Town. Every Hutt ought to be so large as to hold ●our Soldiers. Every Captain ought to have a particular Chamber, and another fo● two Lieutenants. These Huts ought to be 18 Fathoms long, and 7 and a half broad; and four Fathoms high; though there are some that are higher.   

CHAP. XVII. Of the Gates.  
A Gate, if it be well placed, aught to be in the middle of the Curtain; because that in that part it is defended with two Flanks, which lie on both the extremities of it. The Gate cannot be too well secured, because that there Surprises are generally most attempted; whether by Petarding or by Intelligence from the Garrison. Some place the Gates in the Flanks, because, say they, the Sallies are by that means more private, and for that those Gates look a thwart the opposite Flank. But, besides that, being so placed, they very much weaken the Flanks, which are almost the chief strength of the Place, and hinder by that means the opposite Face to that Flank from being well defended; they also hinder the use of the retired Flanks and the Tenailles.  
As for those that are placed in the Faces of the Bastions, they are worse disposed; not only because it is a kind of a Breach already made, but because they are not Flanked, but by a part at a great distance: The Sallies also are too much in view; nor can they be secured by any Work, but such a one as is not Flanked; which is a very great defect.  
The Wideness of the Gates ought to be from 14 to 16 Feet. The Height from 15 to 17. The Thickness like to that of the Rampart of the Wall. They ought to be all vaulted, or else the greatest part of them, and well guarded.  
There must be great S●ones laid at the Extremities of the Gates to hinder the Carts from spoiling the Wall. The Wood of the Gate ought to be of good Oak, joined and fastened with good Nails and strong Bars of Iron.   

CHAP. XVIII. Of Back-Doors, or Sally-Ports, and Draw-Bridges.  
TO sally out of the False Brays, Sally-Ports are made of such a wideness that the Cannon may pass through them: that is to say, ●hey ought to be from 7 to 8 Feet wide; and from 8 to 9 Feet high.  
The Draw-Bridges are made after several Fashions; but the most usual are made with Plyers. Their Length and their Wideness ought to be proportionable to the Borders or Frame of the Gate which are to hold them, being drawn up. The Arms ought to be from 8 to 9 Inches thick, as also the Beam that joins them.  
The Shoulder to which the Arms are annexed, a●d upon which they are to turn, aught to be from 14 to 16 Inches in Diameter. The Extremities ●e●ng bound about with two good Circle's of Iron; you must drive through the Cen●re two Iron Pins, a Foot long, and two or three Inches in Diameter, to the end that it may the more easily move upon a strong Band of crooked Iron, which is to be joined to the Threshold and J●umbs, or Side-Posts, of the Gate. The P●yers a●e to be twice the height of the Ga●e, at least, and one Foot in Diameter. The Inner Square must be traversed with a St. Andrew's Cross, which will serve for a Counter poise. The Chains are to be of Brass, to the end that the Bridge may never be let down without making a noise.   

CHAP. XIX. Of Bridges.  
THE Bridges of a Town are to be from 14 to 15 Feet wide; lower than the Field● and somewhat ●ound. When they are straight and even, they ought to be made a little higher in the middle than toward the Extremities, that the f●ot of the Gate may not be discovered. Though the Supporters of Bridges may be of Stone, nevertheless they will be better, if their Planks and Rails are of Wood, that they may be the more easily cut down upon occasion. When there is no Half Moon b●fore a Gate, it is certain that the Bridge which crosses the Moat ought to be the wider, because there must be a good Corpse de Guard upon it, and a Draw Bridge. B●t i● there be a Half-Moon, some ●u●n aside the way all along the Gorge of the Half-Moon above the Counterscarp, and make a Co●ps de Guard and a Palisado, which hinders the entrance of the Enemy: Others carry the Way through the Extremities of the Face. And a good Corpse de Guard ought to be set at the end of the Bridge with a Portcullis.    

A NEW TREATISE OF Fortification. BOOK V.  

CHAP. I.  
IN regard that Fortification is either Continual, or Transitory, we shall now speak of the latter, as having given full Instructions for the first: Let us begin with a Redoubt.  

Of the Raising a Redoubt.  
1. Make a Rectangle Square, each side of which contains from 12 to 20 Fathoms.  
2. Within this Square draw the Base of the Rampart thr●e Fathoms large; and that of the Parapet from 9 to 10 Feet. The Height of the Rampart above the Level of the Field ought to be t●ree Feet; and that of the Parapet above the Platform of the Rampart, of five, as is usual: or of seven, when there are two Steps or Banquets: though sometimes they raise the Rampart a little higher, as necessity requires.  
3. About the Rampart toward the Field draw another Parallel three Feet wide, which is called the Berme.  
4. About the Berme make a Moat of four fathoms, with a Rounding before the Points of the exterior Sides: the Depth of the Moat ought to be about eight or ten Feet.  
5. The Bridge for entrance into the Redoubt ought to be ten or twelve Feet broad, when you would bring the Cannon into it; otherwise five or six Feet of breadth will suffice. See the Plate Q. Fig. XXVIII. in Chap. 14. Lib. 4.    

CHAP. II. Of the Half Redoubt.  
1. DRaw a Line of 17 Fathoms, as AB.  
2. Make an Isosceles Triangle of 12 Fathoms from the Points AB, as AC, CB, and join AC and CB together.  
3. Prolong the Line AB, and place upon the Prolongation on each side eight Fathoms and three Feet, as AD, BE.  
4. All the rest is done, as in the precedent Work. See the Plate R, Fig. XXIX.   

CHAP. III. Of the Star-Square.  
1. MAke a Square, each side containing 12 Fathoms in length, as ABCD.  
2. From the middle of each exterior side, ●●aw a Perpendicular of two Fathoms, as DEFG; and afterwards draw AB in the Point E, BC in the Point F, DC in the Point G, and DA in the Point H. The rest is done, as in the Redoubt. See Plate R, Fig. XXX.   

CHAP. IU. Of the Star-Pentagon.  
1. DRaw a Circle of 27 Fathoms, and divide it into five equal parts, with 31 Fathoms 5 Feet, as A, B, C, D, E.  
2. Join the Sides together, by straight Lines, and divide each into two equal parts, let fall from the middle of them the Perpendiculars of 5 Fathoms, F, G, H, I, L.  
3. Draw from the Points A, B, C, D, E, straight Lines into the Points F, G, H, I, L.  
4. The Base of the Rampart is 6 Fathoms. That of the Parapet 2. The Berme 3 Feet. And the Breadth of the Moat 5 Fathoms. See Fig. XXXI. of the Plate R.   

CHAP. V. Of the Star-Hexagon.  
1. DRaw a Circle of 12 Fathoms, and divide it into 12 equal parts, as ABCDEF.  
2. Join these Points together by straight Lines, and divide each into two equal parts, then let fall from the middle a Perpendicular of two Fathoms, as GH, IK, MN.  
3. Join these Points with the six first, as in the preceding Works.  
4. Draw the Base of the Rampart of three Fathoms: that of the Parapet nine Feet; the Berme three Feet; and the Moat of five Fathoms. See the Plate R, Fig. XXXII.   

CHAP. VI Of the Triangle of Half Bastions.  
MAke an equilateral Triangle, of which each side contains 24 Fathoms, as ABC.  
2. Prolong the side AB, and place upon the Prolongation 8 Fathoms, as BE. Then also prolong the other side BC, and place upon the Prolongation 8 Fathoms CF. Last, Place upon that of the Line CA the same distance, as AG.  
3. For the Demi-gorges take the same distance, as BH, CI, and ALL, and then raise Perpendiculars upon the Points H, I, L. And for the Flanks a length of four Fathoms HIM, IN, and LO. This done, join the Points ME, OG, and NI together, to gain the Faces: and thus you have the Design of your Triangle.  
4. The Base of the Rampart takes up three Fathoms and a half. The Base of the Parapet 1 Fathom 4 Feet. The Berme 3 Feet. The Moat 5 Fathoms. It ought to be parallel to the Flanks, and round before the Points of the Half Bastions. See the late R, Fig. XXXIII.   

CHAP. VII. Of the Square of the Half Bastions.  
1. MAke a Square, each side containing 24 Fathoms, as A, B, C, D  
2. Prolong the Line, BASILIUS, and upon the Prolongation place 8 Fathoms, as A Then make that of the Line AD, of the same distance as DF. This done, prolong also the Line DC of 8 Fathoms, as CG: Then make the latter CB of the same length as BH.  
3. For the Demi-Gorges, take also 8 Fathoms, B, M, A, I, D, K, and C, L.  
4. Raise upon the Points I, K, L, M, the Flanks perpendicularly, the length of which is 4 Fathoms, as I, O, K, P, L, Q, and M, N.  
5. Draw the Faces, joining together NHQGPF, and OE.  
6. Having the Draught of the Square, let the rest be as in the preceding Triangle. See the Plate S. Figure 34.   

CHAP. VIII. Of the Square of entire Bastions.  
1. MAke a Square, each side containing 31 Fathoms, as A, B, C, D.  
2. From the middle of each side, let fall a Perpendicular of 4 Fathoms, as E, F, G, H  
3. Draw the Faces after Vau●an's manner, and add thereto 10 Fathoms.  
4. Take the Extremities of the two Faces; and from one of them, let the Compass fall upon the Compliment of the other; and in that part where it touches the Compliment, join the same Point to the Extremity of the Face, which is the nearest to it, for the Flank. That done, draw the Curtain.  
5. The Base of the Rampart contains 5 Fathoms, that of the Parapet 2 Fathoms 3 Feet. The Berme 3 Feet, and the Breadth of the Moat 6 Fathoms. See the Plate S. Figure 35.   

CHAP. IX. Of a Pentagon with entire Bastions.  
1. DRAW a Circle of 17 Fathoms, and divide it into five equal parts, with one distance of 31 Fathoms, ● Feet as A, B, C, D, E; and join the Points together by right Lines.  
2. From the middle of each side let fall a Perpendicular of five Fathoms, and draw the Faces as usually, after Vauban's manner; the length of which must be 10 Fathoms.  
3. The Flank is made, by setting the two Points of the Compass upon the ● Extremities of the Faces; of which the one falls upon the Compliment of the Line of Defence, as has been said in the preceding Chapter: This done, join the two Flanks together for the Curtain.  
4. The Base of the Rampart takes up ● Fathoms. The Base of the Parapet 2 Fathoms 3 Feet.  
5. The Berme 3 Feet: The Breadth of the Moat 6 Fathoms. See the Plate S. Figure 36.   

CHAP. X. Of the Half-Hexagon with entire Bastions.  
1. DRAW a Circle of 28 Fathoms, 4 Feet, and divide it with the same distance into 6 equal parts, of which you shall take three for your Temporary Work.  
2. Divide into two equal parts each of the three sides; and from the middle let fall a Perpendicular of 5 Fathoms.  
3. Draw the Faces as usually; of which let the length be ● Fathoms.  
The Flank and Curtain are made, as in the preceding Works.  
5. The Base of the Rampart contains 6 Fathoms. The Base of the Parapet, 2 Fathoms 4 Feet. The Berme 3 Feet. The Breadth of the Moat 6 Fathoms. See the Plate S. Figure 37.  

A Remark.  
IN regard this Fortification is only for present use; and because for that Reason it is sometimes stronger than the other, the Draught and Proportion of it could not be ascertained, as in a continual regular Fortification. Only we may say, that most frequently the Rampart of these sorts of Forts is raised like those of Redoubts. See the first Chapter, Lib. V.    

CHAP. XI. Of the Maxims of Irregular Fortification.  
IRregular Fortification is that, of which all the Sides and Angles are not equal one to another. The Original of this Fortification proceeds from Ancient Towns and Cities● which were built after no regular Figure, and yet required to be encompassed with Walls and Bastions, to defend themselves against the Assaults of their Enemies. But because it was impossible to fortify those irregular Places, and make them, in respect of their Duration, equal to regular Placest there are certain Maxims, according to which they may be environed with a passable Fortification. For Example: 

1. Every irregular Fortification ought to be made according to the Maxims of the regular as near as may be.  
2. The most irregular Forts are the best, provided they be not too ●asie of Access.  
3. Entire B●stions are to be preferred as much as may be before Half Bastions.  
4. No Face ought to exceed 60 Fathoms.  
5. No Flank ought to be less than 16 Fathoms, if it be possible.    
Therefore when you are to make an irregular Fortification, you must consider, 1. The Quality of the Situation. 2. The time allowed to make the Work. 3. The number of Men which are to be put in to defend it. 4. The number of Workmen. 5. The Victuals. After all you must draw the Plane, and fortify it after the following manner.   

CHAP. XII. The manner of Fortifying an irregular Place, where there is nothing but the Field.  
1. HAving the Plane of the irregular Place, divide each exter or side into two equal parts; and from the Middle let fall a Perpendicular●  
2. Upon each exterior side draw an isosceles Triangle of 480 Fathoms.  
The Plan of a Horn work before the Bastion of a Square.  
The Draught of a Horn-work with Shoulders.  
a Horn-work with a Crown before it.  
A draught of a Square with a Crown Work before the Courtin. Fig: 11  
A Pentagone or Work of five Bastions. Fig: XVI.  
A Draught of a Pentagone or five sided Figure with Tenailles within the Ditch. Fig: 16.   

CHAP. XIII. Of the Body of the Placo in an Irregular Fortification.  
THE Body of the Place make thus: 

1. Take 11 Fathoms for the Base of the Rampart, and put them upon each Triangle. The Length across will always give you the true Breadth of the Base of the Rampart.  
2. Take 3 Fathoms, and bring them round about the Place, without putting them upon the Triangles. Draw the Banquets as usually.  
3. Take 20 Fathoms for the Breadth of the Moat, and ●ut them upon all the Triangles, the Distance taken across ●ill be the true Breadth of the Moat for every side. The manner of making this Fortification is the same with that ●f the Regular. See the Plate T. Fig. 38.     

CHAP. XIV. To make a Half-Moon before these Fortifications.  
BE sure to see whether the Length of the curtain exceeds 100 fathoms or no. If it be from 100 to 140 Fathoms, 〈◊〉 50 fathoms for the Capital of the Half-Moon or Rave●●●. But if it be larger than 140 Fathoms, erect before 〈◊〉 Ravelin another Work, whether a Horn or Crownwork. If the Curtain be not longer than 100 Fathoms, 〈◊〉 the half of the Curtain with the Brisure, for the Capi●●● of the Ravelin, and the rest is done as in a regular Fortification. For the Base of the Rampart is 8 Fathoms. That of the Parapet 3. The Breadth of the Moat 12. These Distances are to be put upon the Triangles, before the exterior sides of which you intent to place the Ravelin, and afterwards taken across.   

CHAP. XV. To set other exterior Works before the same Fortifications.  
IF an exterior Polygon exceeds 120 Fathoms, and that you have still room to erect a good exterior Work before it, whether Horn or Crownwork; as you may see in Fig. 39 which is a Crownwork made according to the Rules of Chap. 10. lib. 3. The parts of it ought all to be marked upon the Triangle.  
AB, Is the Basis of the Rampart of the Ravelin, and the Horn-work.  
CD, Is the Basis of the Rampart of the Body of the Place.  
OF, Is the Breadth of the Moat belonging to the Ravelin and Horn-work.  
GH, Is the Perpendicular of the Horn-work, from the end of which are drawn the Faces.  
IK, Is the Breadth of the Moat belonging to the Body of the Place.  
23 is the Perpendicular of the Body of the Place.  
LM, Is the Face of the Horn-work.  
NO, The Compliment of the Line of Defence to th● B●d● of the Place.  
PQ, The Face of the Body of the Place.  
RS, The Capital of the Horn-work.  
TV, The Distance which joins t●e Point of the Capi●●● to the Horn-work.  
WX, Are the Wings of the Horn-work.   

CHAP. XVI. How to fortify an Irregular Place already built.  
TO avoid much Labour, you must fortify the parts of an Old Fortification, and finish them, in the same manner as they were begun; unless they are gone utterly to decay; and than you may make what alterations you please. To the end therefore, that Money may be laid out to good purpose, when the Flanks are repaired, they must allow an Overture of 100 deg. If the Faces are tolerable, let them alone as they are, to avoid the vast expenses in making others. Whence it comes to pass, that there is no ancient place which is fortified after the French manner, because the Engineers have been always forced to labour upon Dutch Fortifications. As for the Rampart and Parapet, they may be better fortified whatever Face they have. But to come to the Thing itself, let us observe the following Rules:  

I●  
The exterior Side, the Length of which is from 60 to 80 fathoms, requires a Demi-Gorge from 14 to 15 fathoms, and a Flank of 15 fathoms.   

II.  
If the exterior Side takes up from 80 to 140 fathoms, it is to be divided into 5 equal parts; of which one is retained for the Demi-Gorge. The Flank changes according to the variety of the Sides. For example, if the exterior Side is of 80 fathoms, the Flank shall be of 16.  

If of 90 the Flank of 17.  
If of 100 the Flank of 18.  
If of 105 the Flank of 19  
If of 110 the Flank of 20.  
If of 115 the Flank of 21.  
If of 120 the Flank of ●2.  
If of 125 the Flank of 23.  
If of 130 the Flank of 24.  
If of 135 the Flank of 25.  
If of 140 the Flank of 25 Fathoms 3 Feet.   
This being well understood, it will not be difficult to reduce it into Practice. Suppose the Pentagon following, of which the Side AB is 130 Fathoms, BC 124, CD 130, DE 113 Fathoms 3 Feet, EA 146 Fathoms, and that you design to Fortify it.  
1. Allow the fifth part of each exterior Side for the Demi-Gorges, as OF, BG, CI, CK, DL, DM, EN, EO, and AP.  
2. Raise upon FG, HI, KL, MN, OPEN, Perpendiculars for the Flanks, in case you cannot allow them an Overture of 100 degrees; the Length of which must be proportionable to the Length of the Sides. For Example: The Flanks FG shall have 24 Fathoms, HI 22, KL 24, MN 20, EA 25 Fathoms and 3 Feet.  
3. In drawing the Line of Defence, you will have the Faces at the same time.  
4. The Base of the Rampart takes up 11 Fathoms. That of the Parapet 3. The Breadth of the Moat 20 Fathoms. The Covert way, and the Glacis, are made as is usual. See the Plate W. Fig. 40.  

Observation.  
The Flank is left after the Holland-manner; that is to say, without either Orillon, or hollow Tower, to avoid overmuch Expense.    

III.  
If the exterior side be from 140, to 170 Fathoms, the Flanks are allowed the following Length;  
If it be 145, allow the Flank 25 fathoms, 3 feet.  
If 150, to the Flank 26 fathoms.  
If 155, to the Flank 26 fathoms, 3 feet.  
If 160, to the Flank 27 fathom.   

IV.  
The exterior Side surpassing the Length of 160 fathoms, to 280, is looked upon then as if it had two sides; that is to say, that besides the Bastions which are placed upon the Angles, there should be a Platform in the middler as you see in Fig. 41, where the Side AB is 200 fathoms, and the Construction of it thus; 

1. Divide the Side AB into two equal parts; as, AC, CB.  
2. Divide either the Distance AC, or the other CB into five equal parts, to give one of them to the Demigorge; as, AD, EBB, CF, CG.  
3. Raise upon DG, FE, the Flanks of an Overture of 100 deg. the Length of which shall be the Demi-gorge.  
4. From the Point C, draw the Capital, which takes up the Length of the entire Gorge; as, CH.  
5. Join the Extremities of the Flanks, and the Point H, together, by a straight Line, to gain the Faces. See the Plate W, Fig 41.    

Observation.  
These sorts of Bastions are perfectly well defended; for the Flanks of the neighbouring Bastions play backwards upon their Faces; besides that, almost all the Curtain serves them instead of a second Flank, and very much augments their Defence.    

V.  
The Exterior Side surpassing 280 Fathoms, two flat Bastions are placed there; holding this for a Maxim that the greatest Side ought to be 140 fathoms.    

CHAP. XVII. How to Fortify an Acute Angle.  
IF you find by chance an Acute Angle with the Circuit of a Place which is to be fortified, since it is impossible to repair this defect, because you are obliged to make use of the old Ramparts; you are only to put upon this Angle a Tenaille-B●●tion; which is thus erected; 

1. Having made the Flanks, draw the Faces as the Line of Defence requires; but in regard they would be excessive long, and would form too acute a Point, that is corrected by putting 40 fathoms from the top of the F●ank above the Face; as, AB, CD.  
2. Join BD together by a right Line, and then divide that Line into two equal Parts, BE, ED; and from the middle of the same Line, let fall a Perpendicular, which has for its Length the third part of the same Line; as, EF.  
3. Draw the Points BD, in the Point F, by right Lines, and your Bastion is made. Oft times they erect little Ravelins before th●se sorts of Bastions: But to say the Truth, this Work is not worth any thing; nor is it ever to be made use of, but upon necessity. But you shall meet with it in some Places; as, at Geneva, and Charlemont. This small Ravelin must not be separated from the Bastion, but by a small Moat of five or six fathoms. Moreover, there is no necessity for its having Ramparts, but only a good Parapet Cannon proof. There are several other Ways of fortifying an acute Angle; but which are of no more value than this. Therefore we must endeavour to avoid them; for notwithstanding all the care a Man can take to fortify this Angle, it will be very defective. See the Plate W. Fig. 42.     

CHAP. XVIII. How to fortify a re-entering Angle.  
THere are few irregular Places, of which the old Enclosure does not form a re-entering Angle; which is almost always secure, as being seen from several parts. But usually it is accompanied with one defect, that it almost always forms the two ●aliant Angles which are next it, too acute. 'Tis true, that if these salient Angles were not too acute, your Firing-place would be very great, from ● Work made upon the re-entering Angle, to defend the Bastions that are in sight of it; nor is it always a Fault to have these re-entering Angles, which are thus fortified.  
If the Angle be not very far sunk in, and very obtuse, as A, it is thus to be fortified;  
1. Place on each Side the Angle great Demi-Gorges, in case the Sides will permit it, and raise perpendicular Flanks, as large as you can.  
2. Join their Extremities together, by a right Line; which you shall divide into two equal parts; and from the middle raise a Perpendicular, containing the Moiety of the said Line.  
3. Draw the Extremities of the Flanks toward the Capital Line of that Perpendicular, and your re-entering Angle will be fortified. See Plate X. Fig. 43.  
If the reentering A●gle be somewhat obtuse, or straight, in this Case, having taken good Demi-Gorges from every side of the Angle, the Flanks are there to be raised; the meeting of which, forms the Salient Angle B; which is capable to defend the neighbouring B●stions. Se● Plate ●o. Fig. 44.   

CHAP. XIX. Of the Artillery.  
HAving given some account of Irregular Fortificaion, reserving to ourselves another Time to speak more of it; we shall only observe what sort of Artillery they have in France. There is 1. The Cannon, which is called the Battering Piece. 2. The Colverin. 3. The Falcon 4. The Saker. 5. The Petard. 6. The Small Petard. The Battering Piece is 10 Feet long, The Bullet about six Inches Diameter, and weighs 33 pounds. requiring two pounds of Powder at least, to charge it; and it carries directly forward 360 Fathoms. At the distance of 100 Fathoms, it pierces two Fathoms of closed Earth. If it be not well settled, it pierces two fathoms and a half; and if the Earth be poor, and hungry, it pierces four fathoms, You may discharge it eighty times in a day. The Dutch Cannon slings a Bullet of 48 pounds, 400 feet distance, and pierces a Rampart of good and well closed Earth 20 feet deep. A half Piece of Battery carries a Bullet of 24 pounds, at 300 feet distance, and enters twelve feet of good Earth. The Falcon carries a twelve pounder 200 feet off, and enters 7 feet into a good Parapet.  
A Small Petard cannot break a double Gate well barred.  
A Great Petard, fixed to a weak Gate, usually makes no more than a Hole; and the greatest Execution it can produce, is to break the fore part of it.  
There is nothing can stop the Execution of Mines, as they make them at present; nevertheless there must be a Proportion between the Petards, the Mines, etc. and the Powder.  
Lastly, A Musket carries 120 Fathoms, and if double charged 150, though it will kill a Man above 300 Paces. Discharged very near it will pierce two Planks two Fingers thick: but discharged at 50 Paces, it enters according to the substance of the Body it meets with. We shall insist no longer on these things at this time; intending to speak more amply of them in another Treatise, where we shall show the manner how to attack a Place.   
THE END.   


A Scale for the 〈◊〉 of a Pentagon according to Vauban's method and of its exterior Works   

The Pentagon. Fig: XIX  
The Hexagon Fig: XX   

These four Draughts or Plains of Courtines belong to CHAP. V. Book IU.   

Fig: 27. Chap: XIV Liv: IV  
Fig: 28 Chap: I BookV.   

CHAP. TWO Book. V.  
CHAP. III Book. V.  
CHAP. X Book. V.  
CHAP. V Book. V.  
CHAP. VI Book. V.   

CHAP. VII. Book. III.  
CHAP. VIII. Book. IV.  
CHAP. IX Book. V.  
CHAP. X. Book. V.   
 

The Scale for these Triangles is in the preceding page.  
The figure of the Triangles of the irregular Pentagon in the preceding Chap XIII XIV & XV. Book V.   

CHAP. XVI. § 2 Book. V  
CHAP. XVI. § 4: Book. V.   

Chap: XVIII Book. v.  
Chap: XVIII Book. v.   

A figure showing all the angles in a Square.