hath a way of some eight foot broad is presently formed that runneth between the battlements of the wall and the Rampier that is called the way of the rounds and in French Le chemin des rondes and this way goeth from one Bastion to another round about the Garrison a very commodious and necessary meanes for the Rounds to go safe in the night and to discover by looking out of the Port holes of the battlements if any appeares neare to their Counterscarp or Corridor that are on the other side of their dike and the Rampier besides the Brest-work upon it is commonly six foot higher then the battlements of the walls and the Brest-worke six foot more so that the splinters of the wall during a battery cannot offend the souldiers that defend the Brest-workes of the Rampier Fourthly the Rampiers are commonly thirty foot high besides the height of the Brest-worke that is six foot high within side for it hath a foot-step of eighteen inches high and two foot broad and the whole height of the Rampier with his Brest-worke is within side thirty six foot and without thirty three foot high because the top of the Brest-work is carried three foot slope because it is alwayes twenty foot broad at top to be of Cannon-proof and the Rampier besides the thicknesse of the Brest-worke and the slope of the two sides defalked is alwayes forty foot broad on both sides on which they plant Elmes or Sicamore Trees that in few yeares make very pleasant and shadie walkes that serve in time of siege for a way for horse men foot and carriages to come to defend and relieve the Rampier Fiftly the distance from the Center of one Bastion to another is commonly from two hundred fifty yards to three hundred yards Sixtly the faces of their Bastions are ordinarily from one hundred yards to one hundred twenty yards besides the turning of their Orillons that is about twenty yards Seventhly the whole gorge of their Bastions are from one hundred yards to an hundred and twenty yards from out to out 8 The brest of their Bastions are from 120. yards to 130. yards 9 And from the center of the Bastion to the point or the utmost Angle of the Bastion from 80 yards to 100 yards 10 The flanks of the Bastions from 42 yards to 50 yards that is divided into three equall parts if they make Orillons one part is allowed for the flank and the other two for the turning and the framing of the Orillons 11 Their curtaines are alwayes betweene 160 yards to 200 yards 12 The Line of Defence to be good for the defence of the musket shot is to be from 220 yards to 250 yards at the most 13 The slope of their Brest-works without side is one foot for every yard if the earth be good Novv let the Reader judge vvhether these costly Fortifications be for our turn and within side a foot in two yards but if it be a sandy ground or a running clay it requires a foot and halfe without side and a foot within for a yard high 14 The foundation of these walls begin from the bottome of the dike and are carried up to the upper water-table of the wall to the Cordeau and from the bottome of the dike on the field-side levell with the ground and all of free stone The middest of the Bastions are filled up with earth to the Cordeau and made slope to the firme ground of the towne and their Rampiers and Brest-works are raised fifteen yards above the Cordeau CHAP. VII Of the superficies contained in the first Plare SInce the Principles of Geometry are the very ground-work of the Art of Fortification I judge it convenient to begin this Abstract by the demonstrations of such superficies as are most commonly used in the practice of this Art Of Superficie A Superficie is properly any kinde of forme demonstrated upon paper or upon the ground inclosed with three lines at the least except it be the circulary forme that hath but one circulary line that begins at one point and ends at the same The point is a small touch of the pen Of the Point that cannot be divided because it hath no parts but is the beginning and end of all lines and the center of all Formes See Figure 1. A line is the continuance of a point that is incapable of division Of Lines but in the length of it it is distinguished by divers termes As by the strait line in Figure 2. by the circulary line in Figure 3. by the perpendiculary line in Figure 7. by the parallel line in Figure 8. by the diagonall line in Figure 11. and by the diametricall line in Figure 12. There are divers other distinctions of lines but they are not usefull in this Art All Angles derive from the conjunction of two lines in one point Of Angles whether they be strait circulary diagonall or mixt with any one of these As Figure 4. is called a plaine Angle because two strait levell lines from one and the same distance meet at one point and a strait perpendicular and a levell line falling in the midst of a strait line produceth an Obtus and a strait Angle as in Figure 7. And two circulary lines meeting at one point maketh a circulary Angle as in Figure 5. And a strait and a circulary line meeting at one point produceth a mixt Angle as in Figure 6. There are also divers other sorts of Angles some of which we shall have occasion to speak of in another place The eighth Figure demonstrates how to make a perpendicularie line cut a strait line given Of the eighth Figure and how you may cut that perpendiculary line in three parts by three demicircles to make up the line given three parallell lines without altering the compasse to any other distance but the first First let a strait line be given then take the just distance of that line with your compasse and set one of the points of it upon the right hand end of the line and with the other point make a small demicircle above then remove the point of your compasse upon the left hand side of the line given and with the other point make another small circle cutting the first and where these two small circles cut one another set your rule and draw a strait line to the line given and it will cut the same with a strait perpendicularie line Now to avoid all errour that might proceed from the mis-placing of your Rule upon the line given make but two other small demicircles below the line as you did above and where these cut one another put one end of your Rule and the other upon the upper cutting of the two small demicircles and draw a strait line and of necessity the perpendicularie line will be strait and without errour Now to cut that perpendicular Line in two equall parts more to make up the Line given three parallel
Lines set your compasse upon the just distance of the length of the perpendicularie Line then set one of the points of the Compasse upon the lower end of the perpendicularie Line and make a demy circle upwards then remove your Compasse upon the upper end of the perpendicularie Line and make a demy-circle downwards then remove the point of your Compasse upon the Center-point where the perpendicularie Line did cut at the first the Line given and make the third demy-circle upwards and this demy-circle will cut the other two in foure places and where they cut set your rule and draw two strait Lines and these will make up the Line given three perfect parallel Lines as it is clearly demonstrated in figure 8. The ninth figure sheweth how to draw as many parallel Lines as you please upon two lines Of the ninth Figure that are the upper and the lower lines in this figure First let two equall lines be given one below another above according to the length you desire to have your parallel lines then divide these two lines with your Compasse in as many parts as you desire to have parallel lines and at everie division make a point and from everie point make a demy-circle above the upper and the lower given lines then set your Rule upon them one after another and draw as many lines as there is demy-circles and these will all be perfect parallel lines Now if you will double these parallel lines it is but to divide with your Compasse the just bredth of them in two parts and at everie division to make a point and to set your Compasse againe upon its first distance and to cut your first demy-circle with it and to draw as many lines and you shall have as many more perfect parallel lines as you had before as it is clearly demonstrated in Figure 9. Of the tenth Figure The tenth Figure sheweth how to make a square and a triangle upon a line given Suppose the Base of the square is the line given now to make a perfect square and foure Scalene triangles of this line you are to set your Compasse upon the just distance of the line and to set one point of it on the right hand end of it and to make a demy-circle and to remove your compasse upon the left end and to make another demy-circle and where they cut one another set your rule and draw a perpendicularie line then turne your rule and draw a strait top line and joyne the base line with this top line by two strait lines and you will have a perfect square and foure perfect square scalene triangles as it is demonstrated in Figure 10. The eleventh Figure sheweth how to make a paralograme out of two circles Of the eleventh Figure First set your compasse upon the demy bredth of the paralograme that you intend to make and then make a circle on the right hand then remove your Compasse-point upon the extreame of the first circle on the left hand upon a strait line and with the other point mark the Center-point of the other circle then turne your compasse round and you will have two circles whose extreams will touch one another then set your rule upon the foure extreames of these two circles and draw foure strait lines and where these lines cut one another there is the foure Angles of the Paralograme as it is demonstrated in Figure 11. The twelfth Figure sheweth how to make a perfect square by the out-side of a circle Of the tvvelfth Figure Make choyse of your Center and set your compasse upon the verie Diameter that you intend to have your square then make a circle and draw foure strait lines upon the extreames of it and it will produce a perfect square as is demonstrated in Figure 12. CHAP. VIII Of the Superficies contained in the second Plate AMong all other Superficies there is none so usefull for the Art of Fortification as the Triangles and yet Mr. Ward in his Animadversions of War makes mention but of three sorts yet there are six principall sorts of Triangles from which all other mixt Triangles are derived and these six sorts may be all set out by the Circle as it is demonstrated in this Plate The equilaterall Triangle deserves the precedencie 1 Of the equilaterall Triangle and the best vvay to set out the same because it is the only Triangle that can be fortified by Bastions for all her sides are equall the best way to set it out is to set your Compasse upon the just distance of the Base of it and to make two points with your Compasse and to draw two Demy-circles one from the right point and the other from the left hand point and where these Circles cut one another make the third point and draw three strait lines to these three points and it will produce a perfect equilaterall Triangle equall of all sides Some set it out by the Circle Hovv to set out the equilaterall Triangle by a Circle dividing the Diameter of it in foure equall parts and draw a strait line upon the first quarter point as the Base of it and two lines more from the two ends of the Base to the top of the Diametricall Line of the Circle but this way is not so perfect as the other See Figure 13. The Isocele Triangle is also verie usefull in the Art of Fortification 2 Of the setting out of the Isocele Triangle for all the regularie Poligons of many Angles after the Sexagon are composed of Isocele Triangles but the Sexagon is composed of six equilaterall Triangles that have all their sides equall But the Isocele Triangle hath alwayes two sides equall and longer than her Base it is set out after this manner out of the Circle After your Circle is made draw a strait perpendicularie Line Diametricall wise upon the Center of it to the two extreames of the Circle then divide that last Line into eight equall parts and draw a strait line with your rule upon the first point of the eighth part division from one extreame of the Circle to another and that Line represents the Base then draw two strait lines from the two ends of that Base to the top-point of the Diametricall Line of the Circle and these three Lines will produce a perfect Isocele Triangle See Figur 14. The Recteligne Triangle is also usefull in the Art of Fortification to set out Halfe Moones 3 Of the rectiligne Triangle and hovv it is to be set out by the Circle her Base contrarie to the Isocele Triangle is longer then her two sides it is set out after this manner After your Circle is made draw a strait Line from the two side extreamities of the Circle and cut with that Line the Center-point of the Circle and this Line represents the Base then draw two other Lines from the two ends of the Base to the upper point of the Circle and two other Lines from
the two ends of the Base to the lower point of the Circle and it will produce two perfect Recteligne Triangles that makes also a perfect square Rhomb The Scalene Triangle hath all her sides unequall it is very usefull for measuring of land 4 Of the scalene Triangle and hovv it may be set out by the Circle for two of them make an Equilaterall Triangle and foure of them a perfect long Rhomb It is to be set out after this manner After your circle is made divide the side diametricall line of the circle into eight equall parts then cut with a strait line the first eighth point of that division on the left hand point and then divide the diametricall line of the circle that is drawne perpendiculary-wise into foure equall parts and cut with another strait line the first equall point of that division towards the Base or the lower side of the circle then from the two sides of these two lines draw a strait diagonall line and these three lines will produce a perfect scalene Triangle See Figure 21. The acute Angle 5 Of the Oxigon Triangle and hovv it may be set out by the Circle or Oxigon Triangle is onely usefull for measuring of land all her sides are also unequall and all her Angles acute you are to set it out after this manner After the Circle is made draw a Diametricall line perpendiculary-wise that may cut the point of the Center from the top of the Circle to the bottome then from the lower end of this line on the left hand make a point at the fourth part of the Circumference of the Circle and another point on the right hand at the third part of the Circumference of the Circle then draw a line from the left hand point to the right point and two other lines from the right hand and the left hand point to the top point of the Diametricall line and these three lines will produce a perfect Oxigon Triangle see Figure 22. The Obtus Triangle is usefull in Fortification 6 Of the Obtus Triangle and hovv it may be set out by the Circle and for measuring of Land two of her sides are equall and containe but the two third parts of her base her base being longer then her sides by a third part it is to be set out after this manner After the circle is made divide the Diametricall line of it into three equall parts and cut the first point of this division by a strait line and this line represents the base then draw two lines from the two ends of this base to the top of the Diametricall lines and these three lines will produce a perfect Obtus Triangle see Figure 23. Now I come to the 24. Superficie that containeth an Ovall Of the setting out of the Ovall and in that Ovall two long Rhombs a Paralograme two equilaterall Triangles and foure Scalene Triangles But because I have already spoken of all these the Ovall and the two long Rhombs excepted I will begin with the Ovall the most difficult of all other superficies to be rightly set out upon paper or in the field To set it out upon paper you are to set your compasse upon the scale of the third part of the length you intend to have the Ovall and this third part is to be divided againe into two equall parts and your compasse set upon that sixth part then make choice of your Center and cut the point of the Center with two strait lines one of them perpendiculary-wise and the other diametricall-wise then set one of the points of your compasse upon the Center point and with the other make a point upon the diametricall line on the right hand and another point upon the same line on the left hand then set the compasse againe upon the third part of the length of the Ovall and set one of the points of it upon the right hand point that represents the End of the third part of the line of the length of the Ovall and with the other point of the compasse make a circle then remove the point of the compasse and set it upon the left hand point of the aforesaid line and make another circle and the two extreams of these two circles represent the length of the Ovall Then remove your compasse and set one point of it upon the lower cutting of these two circles and set it at that distance that having a point upon this lower cutting of these two circles it may conjoyne with a true circularie line the two upper extreames of the two circles that being done remove your compasse without any alteration of the distance and set one of the points of it upon the upper cutting of the two circles and with the other point of it make a true circulary line to joyne the two lower extreames of the two circles and this being traced with a pen will produce a perfect Ovall one third part longer then it is broad some make it halfe as long againe as it is broad but the sides are too flat and the Ovall is by it more deformed and not so seemly and compleat as this is demonstrated in Figure 16. Now to make the most perfect Rhomb that can be made Of the setting out of the Rhomb you are but to draw foure lines dioganall wise from the upper and lower points of the perpendicularie diametricall line that cuts the two broad sides of the Ovall to the two points of the diametricall line of the length of the Ovall and it will produce a perfect Rhomb as in figure 18. And to make the two equilaterall Triangles and the foure scalene Triangles out of the smaller Rhomb Of the setting out of the 6. Triangles you are but to draw four lines diagonall-wise from the two Ends of the first line that did represent the eighth part of the Ovall to the two upper and lower cuttings of the two circles and these lines with the two diametricall lines of the length and breadth of the Ovall already drawne will produce these six fore-said Triangles And to set out the Paralograme Of the setting out of the Paralograme you are to divide the breadth of the Ovall into four equall parts and the length of it into nine equall parts and at every division to make a point and to draw foure strait lines upon the first points of these two divisions and they will produce this Paralograme that is a ninth part longer then twice his breadth CHAP. IX Of the Superficies contained in the 3. Plate THe first Circle of this Plate contains an Equilaterall Triangle Of the setting out of the equilateral triangle out of a square a Square and a Pentagon And because all the Equilateral Triangles that are set out by a Square have their two sides longer then their square by anâ— part I have drawne two demy-circles to shew where the upper Angle of the Triangle should come to be perfect and equall of all her
sides and that is where the two circles cut one another As for the square it is more easie to set it out from without the circle Of the setting out of the square by the inward side of the circle then from within the circle because it is more difficult to reduce the sides of it to that scantling proposed if in case you set it out by the inward side of the circle then it is if you set out by the out-side of the circle for the diametre of the circle is of the just length of the sides but to take out a square out of the inward side of the circle the demy-diametricall line of the circle is to be one third part longer then when you make it by the out-side of the circle But if you are not tyed to a scantling then you may make as perfect a square our of the inward side as well as from the out-side of a circle by dividing the two crosse diametricall lines of the circle into 6. equal parts and by drawing 4. straight lines cutting the foure first points of this division in the foure sides of the circle drawing the lines to the inward extremes of the circle and these lines wil produce a perfect square Concerning the setting out of the Pentagon you are to divide the circumference of the circle into five equall parts and at every division to make a point then you are to draw five straight lines from point to point and these five lines represent the five sides or bases of the Pentagon then you are to draw five other lines from the center point to the end of these five bases diagonal wise and these five lines compleat the five sides of the five triangles of the Pentagon and this being done this superficie is perfected The other three Circles contained in this place are to set out in every one of them a Poligon with different Angles viz. The first a Sexagon the second a Septagon and the third an Octogon having all of them the Basis of their Triangles of an equall length by the encreasing the circumference of the circle of the length of one of the Bases And because this is a point of great concernment to an Enginier that may by the same know at an instant how many Bastions the place will require if he doth but know the true circumference of it I will expresse my selfe more clearly Suppose then that the six Bases of the six equilaterall Triangles of a Sexagon contain 250. yards a piece that is in all 1500. yards circumference And that you would set out a Septagon with seven Bases that should contain every one of them 250. yards a piece you are then to adde to the demy-diametricall line of the Sexagon a sixt part more of the extent that it had before and by this meanes the circumference of the circle will be able to afford you seven Bases of 250. yards a piece that is in all 1750. yards The reason of it is that the circumference of a circle containes three diametricall lines so that you are of necessity to adde to the former demy-diametre line of the Sexagon a sixt part more then it had before because a demy-diametre is but the sixt part of the circumference of a circle And by this rule you may make all the Bases of any Foligon from six Angles to twenty foure Angles nay to eight and forty Angles if you please of two hundred and fifty yards a piece adding alwayes to the demy-diametre as foloweth To make a Septagon you are to adde the â…™ part to make an Octogon you are to adde a seventh part to make a ninth Angle Poligon you are to adde the eighth part and so as you goe on the ninth the tenth the eleventh or twelfth part And as I have said you are to observe that course till you come to set out a Poligon of eight and forty Angles That I conceive would be sufficient to fortifie the City of London on both sides the water with as large a Line of Communication as it hath at this present having at every two hundred and fifty yards distance an Angle to forme a strong complete and defensible Bastion A farre more beseeming a Fortification for so famous a City then such sleight winding Angles and ill flanked Redouts wherewith it is now fortified CHAP. X. Of the five Redouts contained in Plate 4. BEfore I come to speake of the Superficies of these five Redoubts it will not be amisse to inform the Reader of their extent continent and dimensions And first of their extent Of the extent of Redouts All square or circullary redouts are to bee of one hundred yards from out to out in their diametricall line otherwise their inward continent will be too small But a Triangle redout is to have her three sides of two hundred yards a piece because two equilaterall Triangles containe no more then a square Secondly their continent is to be so large Of the continent of redouts as it may lodge a competent number of Souldiers for their defence that is two hundred men at least and every two men cannot have a smaller piece of ground allowed them for their lodging then a piece of eight yards square that makes in all sixty four square yards Now a redout of one hundred yards diametre reduced in the forme of a square will contain but 10000. square yards out of which you are to defalke a third part for the Rampiers the place and the streets and then there will remaine but 6667. yards for the lodgings and this being divided by sixty four yards it will containe but two hundred and eight Soldiers lodgings if two of them be lodged together in a chamber of 24. foot square And by this you may judge whether the greater part of the Redouts about the City of London are of a proportionable extent seeing most of them have not fifty yards of diametre Thirdly for their dimensions Of the bredth and depth of their ditches Their ditches are to be tenne yards broad and five yards deep and the slope of the sides of these ditches are to be but one foot slope in three that the bottome of the ditch may remain to be twenty foot broad when it is fifteen foot deep for the reasons that will be shewn when I come to speak of the contre escarpe and the Rampiars their Breast-works are to be proportionable to this ditch Of the height and thicknesse of their Rampiers The height of it within side is to be twelve foot high and nine foot without the slope of it within side is to be but one foot in six and one foot in three without side there are to be five footsteps of two foot broad a piece Of their slope within and without and of eighteene inches high a piece that the breastwork may not be above foure foot and an halfe high that is of a convenient height for tall and middle siz'd Soldiers to
brest-works broad and eighteen inches high The slope * the slope of brest-work walls of the in-side wall is to be but a foot in six foot because the foot-steps serve instead of buttresses to that inward side but the step of the out-side wall is to be a foot in three and a water table is to be left of eight inches broad if the earth be good between the brim of the ditch and the turfe of the out-side wall if the ground be sandy or a tunning clay it is to bee eighteen inches broad notwithstanding the chat of ignorant men The bottome or foundation of the brest-work is to be thirty five foot in the bottome with the five foot-steps and twenty foot broad at top that it may be of Cannon proofe and it is to have a slope towards the ditch of three foot and for that purpose Of the water table of the brest-works Of the slope of the top of the brest works and of their thicknesse the out-side wall is but nine foot and the inward twelve foot that the slope may be from twelve to nine And observe once for all that all the Angles of this method where you erect Platformes are to have another brest-work fix foot higher then the rest in which you make your Port-holes and that this brest-work is to have a * Of the frize of the Platform frise when it is twelve foot from the ground made of wooden pikes sharpe at one and of six foot long and foure inches broad and three inches thick that are to be placed eight inches distant one from another jetting three foot over the ditch and running a yard into the brast-works These high brest-works are at every foure course of turfe to be laid with brush or bavin wood well rammed with the carth if the crectors intend they should be of continuance Having so clearly expressed the dimensions of this method of Fortification I shall not need hereafter to speak any thing at all of them because I have observed in these five Superficies following these proportions related in their Sides Courtines Flanks Brests Faces Gorges and Lines of defence and so I come to the method it selfe CHAP. XXVI Of the Fortifications of the Octogon fortified by small Flankers demonstrated in Plate 19. THe eight sides of this Octogon containes two hundred and fifty yards apiece and by consequence the circumference of it is of two thousand yards that is an English mile and the â…• part of a mile The Angle of her Center is of 45 degrees it is to be set out in the field as the Sexagon by the Demi-cirle and Lines as it is described in Plate twelve and Chapter 15. I will therefore only give you some directions how to set out the small flankers because it is a new method of Fortification that wee have not as yet given you any directions for When the Superficie of this Octogon hath been set out and fully traced after the manner you were directed to set out the Sexagon you are to divide all the sides of it in two equall parts and to knock in a stake * From these middle stakes is the line of defence drawn that is in this figure of 150. yards and the Courtine 200 yards in every one of their divisions then you are to fasten a line of twenty five yards to every one of the eight Angle stakes one after another and to strain that line first on the right hand and then at the left hand upon a strait line upon the sides traced on the left and right hand of that stake and at the end of the line you are to knock a stake and these two stakes will represent the two Demi-Gorges * The whole Gorge of the flankers is to be 50 yards of that flanker then remove your line to another Angle stake and doe as you did before till you have set out all the demi-gorges after that manner Then set by that line and fasten another line of fiftie * The Line of fiftie yards represent the distance from the center to the point of the flanker yards to every angle-stake of the sides and straine the same into the field upon a strait diagonall line standing at the center-stake with your demi-circle to guide the same because the demi-diagonall line from the center-stake to the utmost point of the flank is of 380 yards in this Figure and therefore too long to be strained conveniently and at the end of this line of fiftie yards drive in a stake and this stake will represent the utmost angle or point of the said flanker and as you have done this set out the other seven angles of the flankers after this verie manner then you are to fasten a Line of 28 * 28 yards is the extent of the slope flank yards to everie one of the sixteene stakes one after another that represent the Demi-Gorges of the Flankers and to straine it three yards more slope then upon a strait perpendicularie Line towards the other Flanker of that side of the Octogon and at the end of th Line you are to drive in a stake The reason why you straine this Line three yards more slope then strait is to preserve the Flankes from the Enemies batterie that are otherwise too much exposed to be ruinated by it when the Line of the Flank is drawne upon a strait perpendicularie Line you are also to observe that the strait perpendicularie Line of the Flank is but five and twentie yards and that these three yards are added to it that the Flanks may be three yards more slope then strait because a slope line in five and twentie yards extent comes to be three yards longer then a strait And as you have set out this Flank set out the other sixteen after this manner and drive in stakes in them all that being done you are to fasten a Line of 150 yards to the middle stake of every side and straine it first on the right hand and then on the left and as you straine it make it touch the Flank-stake and fasten it to the stakes that represent the vtmost point of your Flankers and this Line doth represent the Line of defence and the * The face of the flanker is to be 25 yards besides the three yards added to thicken the Angle of the shoulder face of that Flanker and before you remove the Line have the face traced by the Pioniers and this face extends it selfe from the Flank-stake to the stake that represents the vtmost angle of the Flanker And as you have set out this face-stake set out all the fifteene other and they being set out and traced your eight Flankers will be perfected Then you are to set out all your dikes after the dimensions described in Chap. 25. And these directions for the setting out of the Flankers of all the other ensuing Superficies shall suffice once for all to avoyd repetitions This Octogon thus set out and