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
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
allow them to be of 150. square yards that is a piece of ground of twelve yards and halfe square of all sides by which scantling the 41666. yards being divided there will appeare to be in this continent 277. dwelling houses and as many Inhabitants that may billet foure hundred Foot and two Troopes of Horse and this number is sufficient to maintaine this Fort against an Army of five thousand men three months so it be provided with sixteen piece of Ordnance Balls Ammunition Armes Victualls and all other necessaries fit for a Siege Object Some will wonder why so small a Fort should have sixteen pieces of Ordnance when many of our Garrisons that are of two or three miles Circumference have not so many Ans I answer the more is the pity they are no better provided But if this Method of Fortification were not better stored to what end should Forrainers be at the charges to erect upper and lower flanks and because I have not as yet spoken of these upper and sower flanks I will upon this occasion describe how they are made and for what use they are erected In the Ancient Fortifications by Bastions Of the upper and lovver Flanks of a Bastion there was formerly low Casamates to scour the dikes even with the water for they were built so low that a Cannon being mounted in them and pointed shot upon a levell Line even with the water of the dike and these Casamates were all vaulted with arches of free-stone having two Port-holer and two Demy-Cannons in every one of them their vault being some sixteen foot square and they had a slanting descent from the gorge of the Bastion with a door to come down to them but because they had no other light then the Port-holes nor no other evacuation for the smoke then the doore when the Cannoniers fired their Pieces they were so incumbred with the smoke in the vault that they could not suddenly charge their pieces againe but were inforced to sâ—ay till the evacuation of the smoke were past Whereupon the moderne Enginiers to prevent these defects have invented the upper and lower slanks where they alwaies place two pieces of Ordnance in everie one of them so that according to the Forrain Method every Bastion is to have ten pieces of Ordnance at the least two Demy-Cannons in everie flanke and two long Culverins to defend the faces and the point that is in all ten pieces of Ordnance these Flanks are made after this manner The wall of the flanks is brought up from the bottome of the dike with free-stone of two yards thick to the levell of the water of the dike and within and without laid with Tarris that the water of the dike may not pierce the same and upon this wall and the firme ground after another foundation of stone hath been laid twenty four foot from the brim of the first wall some five foot deep a Brest-work is brought up of earth and turfe of twenty foure foot thicke in the bottome and of twenty foot broad at top and six foot high having three Port-holes and beyond this Brest-worke the ground is digged lower the whole length of the flank that is ordinarily from thirteen yards to sixteen yards in length and in breadth eight yards and in depth five foot and this being laid with tracin and planked is the Platforme of the lower flanke open over-hâ—â—d in which they place two Demy-Cannons and these scouâ—â—â—â—d free the dikes from the assay lants galleries and from the scaling of the Rampiers Now to erect the upper flank they go eight and forty foot wider into the gâ—rge of the Bastion and upon the in ward foundation of the lower flanke wall and the earth of the Bastion they erect another Brest-work of the same height bredth and thicknesse of the former with three Port-holes in it and then they planke another platforme and place two Demy-Cannons more upon the same and these are the manner of their upper and lower flanks that have a slope coming downe from the upper to the lower some ten foot broad which being covered by the Orillons make the Bastions very strong and when these double flanks are taken out on both sides the gorge is not above fiftie yards broad that is narrow enough for the last re-intrenchment Now these flanks being all open over head the smoke of the Ordnance is suddenly evacuated the use of the upper Flank is to scoure the face and the points of the Bastions in the time of a storme and to beat all along the Courtine And so much will suffice once for all concerning the erection and use of these lower and upper Flanks Now I come to shew how you are to set out this Square in the field If men could as soon and as easily set out a Superficie in the field as they may upon paper it were soone done but this last requires a greater labour and care You are then in the first place to make choyse of your Center and there to knock in a stake then you are to stand close to that stake and turne your face full South and then take your Demi-circle and set the sight of it upon the ray of 90 degrees that is the Angle of the Center of the Square then you are to have two men by you one with lines and the other with stakes and he with lines is to have a line of one hundred and * You are to observe that this line of 180 yards is the just demi-diagonall line and distance that is betvveen the Cent r-stake and the Angles of the Square fourescore yards the one end of which line he is to fasten to the Center-stake and when you have taken your Demi-circle and set the sight of it as afore-said you are to take with it having your face turned to the South the right hand ray of the Angle of 90 degrees and when you have it let the man straine the line fastened to the Center-stake along the said ray to the end of the line and when he is just against you and the ray of your Demi-circle let the other man knock in a stake at his feet then turne your selfe and set your Demi-circle to take the left hand ray of the angle of 90 degrees and when you have it let the man with the end of the line come to it and when hee is just against you and the ray of your Demi-circle let the other man with stakes knock in a stake at his feet at the end of the line and these two stakes represent the South-East and the South-West Angles of your Square then turne your selfe fall North standing close to the Center-stake and set out after the same manner the North-East and North-West Angles of your Square and let the man with stakes knock in two stakes as hee did in the South side and these foure utmost stakes will represent the foure Angles of the Square Now to avoyd errour you are to fasten the