is known and T 3 is also known or C F and F λ is known therefore C λ or 9 Y is known and so is Y λ known then by the 4. lib. 6. Euclid as I 9 is to 9 Y so is Y λ to λ Z to which if you add one foot the whole base λ Z will be known The Perimeter of this Brestwork or the Lines of it surrounding the whole Fortification whether they be inward or outward must be drawn parallel to the Faces only so that they meet at a point opposite to the middle points of the Courtine and make outward Angles but if Ravelins are built before the Courtines the Brestwork is drawn about them but not about the Horn-works if any should be built IV. Whether it be expedient to make a Ditch about the Out-Brestwork Some affirm it and stand to it But they do not consider when they think to make this Brestwork stronger that they quite over-throw the end it was made for which was that the besieged might safely sally out upon the Enemy and in their return injoy a safe retreat both which will be hindred by a Ditch made about it Insomuch that if the Towns-men do not make it 't would seem fitter for the Enemy to make CHAP. XII An Orthographical Table of Regular Fortifications This Table is collected out of the doctrine of the four foregoing Chapters  Max. Med Min. Rhynland feet The breadth of the base of the Rampar A E. 84 72 60 The inward Talu or line forming the Sloap AB 18 6 14 The outward Talu E F. 9 8 7 The height of the Rampar B L. 18 16 14 The breadth on the top of the Rampar L 3. 57 48 39 The breadth of the base of the Brestwork 3 D 24 18 14 The inward Talu of the Brestwork D T. 1 1 1 The outward Talu of the Brestwork 32. 2 2 2 The inward height of the Brestwork T I. 6 6 6 The outward height of the Brestwork 2 K 4 4 4 The breadth of the top of the Brestwork K O. 21 15 11 The rest of the foregoing Table  Max. Med. Min. Rhynland feet The breadth of the step or Banquet D 4 3 3 3 The height of the step or Banquet 4 G 1 ½ 1 ⅓ 1 ⅕ The Terrepleine or Walk on the Rampar 4 L 30 27 22 The walk of the Fauss-br or chemin des Rondes E 5 21 17 15 The Fauss-bray with its Banquet    The border or bankside Lisier R S 6 6 6 The upper width of the Ditch S V 132 108 84 The outward inward Talu's of the ditch SH V 5 12 12 10 The depth of the Ditch H 8 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 12 12 10 The Width of the bottom of the ditch 87 108 184 64 The Covert way V X 21 17 15 Its step or banquet    The base of the out-brestwork λ Z 79 70 69 Its height 6 6 6 For the base of the Out-brestwork working according to the Rules deliver'd in the 11. Chap. num 3. I find it to be Max. Med. Min. 82. 75 ¼ 69. Therefore Dogen and Fritach are out in their account The first Column shews you the largest and strongest Orthography which is able to sustain the greatest force of the Besiegers The second is able to bear an indifferent Siege The third is made against the least strength which is usually sent against Towns But here we only treat of the Forts themselves I shall hereafter give you the Orthography of Castles and Out-works And again I had no proportion or respect to the diversity of the Figures or Polygons as many Engineers have who for no reason as I can tell give to a Pentagon a different Orthography from that which they give to a Hexagon or a Nonagon For a stronger or weaker Orthography is to be given a Fortification not as it hath more or less Angles or Bulworks but as it ought to resist a greater or less strength of an Enemy Note If the Fortification be made without a Fauss-bray the Out-brestwork will have another Base for it will be a fourth proportional to the three terms I 9 9 Y Y λ but the mean or middle term 9 Y will be less by the space E R if the Fauss-bray be wanting and then the base of the Brestwork λ Z will be max. med 65 50 CHAP. XIII Of the raising of Out-works A Fortification formed according to its essential parts is made stronger if it be surrounded with some Out-works The chief of which is a Raveline a Half-moon a Horn-work a Crown-work and Tongs I shall treat of each of them distinctly in the following propositions PROP. I. I. The definition and form of a Raveline or Target A Raveline is a Bulk of Earth almost like a Bulwork cut off Fig. 10. except that it wants flanques it is surrounded with water and separated from the Fortification by the breadth of the whole Ditch Such an one is F E G H in the 10th Figure its faces are F E G E. It wants as I said for the most part flanques yet it admits of them when it is built before Gates which then will be about 8 or 9 perches Towards the Enemy it is built with a Rampar and Brestwork and lies open towards the Fortification least it might shelter the Enemie when he hath possest it it is rais'd but a little height above the level of the ground that it may be better defended from the main Fortification and the plains the better scowred by it It s Angle must not be less than 60 degr nor more than a right Angle The length of the faces is determin'd in Regular Fortifications numb 3. If they are applyed to the covering of a Courtine that is above its just length observe this that the faces must not be longer than the faces of the Bulworks therefore they may be about 40 50 or 60 paces II. Concerning their place and how they are defended For the most part it is raised before Gates and Courtine but never before the Bulworks The 10th Figure shews the Situation of it as it lyes before a Courtine 't is best to have it of such a breadth as might cover the Courtine only and not the flanques for then 't is defended by the faces and flanques of the Bulworks that it lies betwixt III. The making of it Is various but this is most approved Fig. 10. Raise an infinite perpendicle from the middle point of the Courtine S from this Line on the other side of the Ditch cut off H E equal to ¾ or ⅔ or ½ of the Face then from the point E draw streight Lines either to the ends of the Courtine A B and this will be the best form for the Raveline for the whole Courtine is covered by the Raveline and the Raveline it self not only scowred and defended by the Faces but by the flanques of the Bulworks also or to some other point of the flanque or to the
ends of the flanque C D those parts of these strait lines F E G E cut off from the Bank-side of the Ditch towards E are the Faces of the Bulworks 1. Another way On the Centers A B the extream points of the Courtine with the distance of the same Courtine describe two Arches intersecting one another in E. 2. Produce the marginal lines of the Ditch φ 1 M L till they meet at the point H then from the ends of the flanques C D draw strait lines to the point E that may cut the marginal lines of the Ditch in F G and F E G H shall be the perfect delineation of the Raveline the faces are F E G E the Gorge Lines are F H G H. 1. Another way bisect the Gorge Lines of the Bulwork A R B Q in the points O P then draw strait lines from the points O P by the ends of the flanques C D till theâ—y meet one another in the point E. 2. Then produce the out-lines of the Ditch φ I M L till they meet in H and cut the former lines in F G so shall F E G H be the Ravelin required IV. The Orthographie and Ichnographie or the Profile and Plain This Table following shews the height and breadth of each part The third column shews the Orthographie of the Out-works of Breda The first and fourth shews the Orthographie of the largest The second and fifth of the middle size the sixth shews the least the four last Columns are taken out of Dogen This Table doth not serve only for Ravelins but for all manner of Out-works A Table for the building of Outworks Rhynland feet  Max. Med. Bred. Stab Min. st Temp. The lower breadth of the Ramp 40 36 44 36 24 20 The outward Talu of the Ramp 3 2 6 3 2 2 The inward Talu of the Ramp 6 4 8 6 4 4 The height of the Rampar 6 4 8 6 4 4 The upper thickness of the Ram. 31 30 30 27 18 14 The base of the Brestwork 15 15 16¼ 13 10 8 The outward Talu of the brestw 2 2 3¼ 2 2 2 The inward Talu of the Brestw 1 1 1 1 1 1 The outward height of the Brest 2 2 5 4 4 4 The inward height of the Brestw 6 6 6 6 6 6 The upper thickness of the Brest 15 12 12 10 7 5 The height of the step 1½ 1½ 1½ 1½ 1½ 1½ The breadth of the step 3 3 3 3 3 3 The Walk on the Ramp 12 10 10 1 â— 11 5 3 The rest of the foregoing Table Rhynland feet  Max. Med. Bred. Stab Min. st Temp. The border at the foot of the Ram. 3 3 6 3 3 2 The width of the Ditch 48 30 42 30 24 16 The outward Talu of the Ditch 10 8 7 8 6 4 The inward Talu of the Ditch 10 8 7 8 6 4 The depth of the Ditch 10 8 7 8 6 6 The width of the bottom of the D. 18 14 28 14 12 8 PROP. II. Of the Half-moon or Helmet I. Its definition and place HAlf-moons for the most part do not differ from Ravelins unless it be in bigness perhaps they had this name given them because those which are built before Bulworks are Arch'd in the form of a crescent on that side which lies towards the Bulwork They are placed upon the Covert-way which is beyond the Ditch so that their Capital line produced cuts the Courtine into two equal parts They are built also before the Angle of Bulworks as I said but the greatest use of them is in Irregular Fortification as I shall shew hereafter II. Their Form Let not their Angle be less then 60 degr nor more than 90 degr Let their height be but indifferent and not distant from the Rampar above Musquet-Shot that they may be defended by the Rampar When they are built on the Covert-way their faces must be 25 or 30 paces let the thickness of their Rampar be 15 or 20 feet and they must be so large as to receive 100 or 150 Souldiers III. Their Delineation 1. In the Angle of the Fauss-bray V as in a center with the distance of the breadth of the Ditch V M describe an Arch and produce the Capital line infinitely cutting the Arch in α. 2. On the other side of the Ditch cut off α X from the Capital Line produced which is â…” of the Face of the Bulwork and from the points H and λ where the Gorge lines of the Ravelins intersect one another draw unto λ the lines H X λ X. 3. Produce the faces of the Fauss-bray φ V φ V till they cut the lines H X λ X in 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 and the Arch in 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 or the Fichant lines of the Fauss-bray continued on 2 V 3 V may determine these Intersections So have you a half-moon delineated placed before a Bulwork whose faces are X ζ X â— and its flanques but open are 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 The delineation of other half-moons is like that of Ravelins The Orthographie and Ichnographie is had out of the foregoing proposition Chap. XIII numb 4. IV. Its Vse and Conveniency 'T is the weakest of all the out-works since it cannot entertain a good quantity of Souldiers to defend it by reason of its straits and is also with more difficulty defended from the Fortification Therefore these half-moons seem to be hurtful to the Fortification unless they be arm'd with these cautions to wit that Ravelins be built on both sides and that they consist only of Faces being altogether without flanques otherwise being possest by the Enemie they cannot be hot from the Ravelins and at last let them be every where within Musquet-Shot Yet if all this were perform'd 't will be still doubtful whether they are useful or not Wherefore they seem not to be built without peculiar necessity PROP. III. Of the Horn-work I. Its Definition and Kinds Fig. 10. THat Outwork that runs farthest into the field with two strait sides objecting to the Enemy two half-Bulworks is called a Horn-work The tenth Figure shews an example of it There are three kinds of Horn-works the first hath its sides inclining to one another towards the Field the second towards the Fortification and the third's are parallel II. Its Place and Form They are built opposite to the Courtine or the Angle of the Bulwork beyond the Out-brestwork Yet they are better defended if they cover the Courtine than if the Bulwork especially if the sides are parallel For when they cover the Bulworks with parallel sides they receive no other defence than from the Courtine and that to little purpose since at so great a distance besides after this manner the work would be too narrow Yet if they are to be placed before Bulworks 't is altogether necessary the sides should incline to one another towards the Bulworks that so they may not take in all the faces but exclude some part of them from which they may be defended
See the Construction in Dogen pag. 160 161. If they are built before the Courtine let their sides be rather parallel and perpendicular on the Courtine If they cover the whole Courtine as some will have it and as we have express'd in the Figure their defence will be from the faces of those Bulworks that the Courtine lyes betwixt If it does not cover the whole Courtine as others will have it the defence of the sides will be the greater to wit both from the faces of the Bulworks and from part of the Courtine Betwixt the Horn-work and the Courtine there is commonly rais'd a Raveline nay before the work it self betwixt each of its Horns a Raveline or rather an Half-moon may be built To conclude you will add a great deal of strength to this work if you make some Retrenchments But of that afterwards III. Its Delineation 1. Let there be drawn two parallels E I F K for the sides of the Horn-work from the Out-brestwork towards the field at such a distance that if they were produced towards the main Work they might fall in a strait Line with the flanques of the Bulworks or if you desire a less breadth for the Horn-work let them fall within the flanques on the Courtine it self But the ends of these sides must not be above Musquet-shot from the Rampar wherefore they must not run beyond the Rampar above sixty Rhynland Perches Yet these sixty Rhynland Perches use to be counted from the Out-brestwork that so the approaches of the Enemy might be the more infested Joyn E F on which make the Angles F E G E F H twenty five degrees each then bisect one of these F E G with the right Line E L meeting with F H in C then from E G cut off E D equal to F C so will F C E D be the faces of the half Bulworks 3. From the point D and C draw D A and C B equal and parallel to E I and F K for the flanques of the Horns and joyn the Courtine A B the proportion of the flanques D A and C B to the faces will be almost the same as uses to be in Regular Fortifications Also after this manner following the Capital and Gorge lines the Flanques and Courtine will be with more ease determin'd For ⅓ of E F gives the Capitals E N and F M also ⅓ of M N or E F gives the Gorge lines N A M B and there remains for the Courtine A B also ⅓ the right Lines M E N F will determine the length of the Flanques rais'd from A and B and so also the faces E D F C will be found These things being done a Horn-work is delineated such as uses to be stretched before the Courtines in a Regular Fortification the delineation of the rest will be performed almost by the like method having alwayes a respect to the place Note That here is a twofold Practice in building Horn-works 1. That the Courtine might be determin'd by the Faces 2. The faces by the Courtine IV. Its use If Ravelins Half-moons and Horn-works are built about a fortified place the Fortification is accounted most compleat and perfect whose use consists most in this 1. They keep off the Enemy far from the Fortification 2. They are taken with a great deal of difficulty for they are defended from the Courtine from the Bulworks and from the adjacent Works and some Lines from the Out-brestwork it self 3. Being taken and possess'd they can hardly be kept because they lye open towards the Fortification 4. Horn-works are most destructive to the approaches of the Enemy and under their shelter the besieg'd may work counter as occasion shall offer c. PROP. IV. Of the Tonges in French Tenailes I. Their Definition and Kinds THey are Out-works that differ from Horn-works almost only in this that instead of two half Bulworks they have only an external Angle and this sort is called the single one It it called the double one when it hath two outward Angles with one Inward The twelfth Figure shews the single one the thirteenth the double one Now this outward Angle is that which is without the Figure and whose sides incline inwards The inward is that which is within the Figure with its sides running outward II. Their Place The same as that of the Horn-works Yet it will hardly be expedient to lay them before Bulworks by reason of their weakness Of which Num. 4. III. Its Delineation Fig. 12. You must describe a single one after this manner Draw the sides A C B D after the same manner as in the delineation of Horn-works which is already prescrib'd unless these are wont to be shorter viz. than forty or fifty Perches 2. Joyn C D which bisect in F and from F let fall the perpendicular F E equal to ¼ of C D and joyn C E and D E so have you the simple external Angle Fig. 13. Draw the double one after this manner Having drawn the sides A C B D as above joyn C D which being bisected in G from G raise the perpendicular G E equal to ¼ of C D and joyn C E and D E. 2. Produce E G to F till G F be the half of G E and the right Lines C E D E being bisected in K and H joyn F K and F H so will A C K F H D B be the double Tonges or double external Angle IV. Its strength and use They are much inferiour to the Horn-works insomuch that they seem only then to be made use of when some suddain occasion urges Moreover the defect of these Tonges and of all external Angles is this that about its very Angle it affords the Enemy a certain Quadrangular space within which he need not be expos'd to the shot of the Defendants this space is determin'd if the outward sloaping surface of the Brestwork be conceiv'd to be produc'd till it cut the field its capacity is almost equal to twenty three Rhynland Perches which will be easily computed Since then this sort of building is so much against the first Laws of Architecture 't will be almost necessary to raise a Raveline before it The Double Tonges since they have a double external Angle K and H will likewise double the defect already spoken of wherefore they are less used PROP. V. Of the Crown-work I. Its Definition THat work is called a Crown-work that hath on both sides two half Bulworks Fig. 14. and in the middle one or more whole ones Therefore it is the part of some Regular Fortification and seems to have this name given it because it doth as it were incompass part of the Fortification II. Its Place Is the same as that of Horn-works though the Crown-work can cover more of the Fortification than the Horn-work and sometimes they are drawn about Horn-works Their chiefest use is to inclose neighbouring places that might insest the Town as Hills c. and so prevent the Enemy III.
thence cut off E G E H equal to ⅓ of the side and thence again raise the perpendicle E F equal also to 1 ◠5. Joyn G F H F you have your purpose Fig. 40. There are built also square Forts with two whole Bulworks and on the opposite side the double Tonges See Fig. 40. PROP. VIII To delineate a three-sided Fort with half Bulworks Fig. 41. 1. DEscribe an equilateral Triangle A B C whose sides must be less than those of a Square 2. Cut off from the sides the third part A I B L C K for the Neck-lines 3. From the end of the Neck-lines raise perpendicularly the sixth part of the sides I H L M K G. 4. Add to the sides of the third part B D C E A F and joyn F H D M E G. you have your purpose The four Forts describ'd in the foregoing Propositions are not to be built promiscuously and for varieties sake but with choise and with respect to the place And although they are much weaker than Forts with whole Bulworks nevertheless they are conveniently made use of As to their Profile and Ichnography you may give them the same as to Redoubts and Stars or if they require a greater you may give them that which was used in the siege of Hartogen Bosch in which the base of the Rampar was 27 feet the height 6 the upper breadth of the Rampar 18 the base of the Brestwork 8 the upper breadth of the Brest-work 4 the height of the Brest-work 6 the width of the Ditch 30 feet CHAP. XX. Of Batteries for great Guns and of the Approaches THe Circumvallation being finish'd which is the first act of the Siege deliver'd in the foregoing Chapter you raise batteries for great Guns in certain places and go towards the Out-brestwork cover'd in oblique Trenches Of these therefore in this present Chapter PROP. I. To build an Offensive and Defensive Battery THere is a twofold Battery offensive and defensive Fig. 42. the last is directed towards the enemy without the first towards the besieged You shall build an Offensive one after this manner 1. Multiply the number of Guns that are to be mounted by 12 the product shall give in feet the length of the Battery for each Gun is distant from another 12. feet and the two at the ends are distant from the Brestwork 6 feet each 2. You 'l have the breadth A D if to the length of a Gun mounted in his carriage you add the space A F ten or twelve feet for the recoiling of the peice and the space F D for traversing and passage 3. Let the plat-form of the Battery be made sloaping downwards towards the Enemy that when the Guns are recoil'd they may with more ease be brought back to their places Let its entry behind be I K the way leading to it must not be very steep but gently rising that the Guns may with more ease be got in 4. That part of the Battery that faces the Enemy must be fortified with a Brest-work whose Base you may make 12 15 or 18 feet its height 6 for the sides A D B E a less width will suffice 5. Let there be so many Ports in the Brestwork as there are Guns let their height be three feet their outward width four their inward two the outward width is more than the inward that the Guns may scowr more of the field 6. Behind the Battery you must describe a space D S N E equal and like the Battery in it make a square hole as M whose side must be ten or twelve feet in which the powder must be kept and you must cover the mouth with leather least any sparks should fall in To conclude as well about the Battery it self A E as the space D N you must make a Ditch eight or ten feet wide six feet deep 7. The first Batteries are wont to be raised at a Musquet-shot from the Town afterwads near the very ditches the general rule may be this that the nearer they are the place they do the greater execution 8. The Defensive Batteries are not so full of work their Brestwork if it be made of earth may be six or seven feet thick the height is sufficient if it cover a Gun in its carriage instead of an earthen Brestwork they use commonly great wicker Baskets fil'd with earth PROP. II. To direct the Lines of Approaches to a place Besieged 1. ABout the distance of a thousand feet from the Town open the Trench 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Fig. 43. which you must carry on obliquely towards the place besieged so that it may not be scowred from any part of it which being continued some space you must dig a new one the other way as 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 with the same obliqueness and so by several turnings you proceed to the Out-brestwork it self where at length the Approaches are finish'd drawing two Trenches 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 longer than ordinary and parallel to the place besieged These two last Lines cover the Besiegers like a Brest-work so that being so near at hand they frighten away the Defendants from guarding their Graft and Rampar 2. Although the Approaches ought so to be carried on that they may never be scowred from the Enemies Rampar yet the Engineer shall take good heed he make them no more obliquer than needs to the loss of time and expences I think with two turnings you may alwayes come to the out-brestwork a far shorter way than if more oblique lines had been made For let there be drawn from the point where you began your Approaches the right line 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 which continued may fall a little without the Angle of the brestwork and if another line be produced from β which goes without the Angles of the out-brestwork you will arrive at the out-brestwork in two turnings 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 but why this way of Approaches is less used I think this to be the reason That your long lines of Approach if attempted may be sooner carried and demolish'd by the Enemy than those that are cut in and out with several windings 3. At the end of every Line you must build Redoubts after such a manner that two may be flanqued by one for this is the best situation of Redoubts If one of the Lines be drawn longer than ordinary you may also build Redoubts in the middle for its defence all which the 43 Figure sufficiently expresses 4. The earth which is cast out is thrown towards the Enemy that it may be instead of a Brestwork to the Pioneers Let its Tower width be six feet its upper twelve or fifteen it s least depth must be such that may cover a standing man with the height of its Brestwork joyn'd to it The nearer they advance to the Town tht deeper they must be The width also must be increas'd if there is occasion to bring stuff for the Gallery thorough the Trenches They use frequently to
of powder for the Train reaching the powder in K that putting a match to it you may blow up the Mine at the time appointed As for the Attacque it self which is ordered when the Mine is sprung and the breach made I leave that to the Officers and Souldiers to whom these things belong CHAP. XXIV Of defending Towns 1. THe first beginning is to keep the enemy from the Town as far off and as long as you can Therefore whatsoever without the works can put a stop to the Enemy the Besiegers must possess and defend as long as they can 2. They must use all their endeavour to hinder the approaches of the Enemy therefore let them sally frequently but warily least they fall into snares to the irreparable loss of the Town and rout and kill the Pioneers and Souldiers Let them throw down the Lines that are finish'd and if they cannot carry away their Guns they must spike them up by driving Nails in their Touch-holes 3. Those outward works which they can keep no longer must be retrenched see what we said before of Retrenchments but if they are utterly like to be lost they must be blow'n up together with those that possess them 4. The sapping of the out-brestwork must be intercepted by a counter and transverse Sappe 5. The filling of the Ditch and the building of the Gallery must be hindered at a distance by the continual firing of Musquets great Guns hand Granadoes and other fireworks nor is there any other way if the Ditch be full of water But if the Ditch be dry then they must fall upon the builders of the Gallery with handy-stroaks as well as with all that which I declared above And the Gallery it self must either be destroyed by fire or blown up with a Mine 6. But if notwithstanding all this the force of the Enemy prevailing the Gallery is brought over to the Bulwork and the Bulwork it self be undermined Against this plague no remedy remains but to find out the place of the Mine To do this they use several practises Some by the motion of Pease leaping on a Drum-head well braced do conjecture at the place of their digging others boaring a very long Augur into the ground suspected and applying their ear to it think to hear the stroakes of their digging others use other ways to discover it The most certain way is by countermining to search the foundation of the Bulwork The Mine being found the powder must be carried out but if the streightness of time will not permit it must be wetted and a passage opened for the fire 7. The Bulworks being blown up if the Besieged have no inward works remaining the last refuge is that since they can no longer resist the Enemy with wall and Rampar that they stop his passage with arms and hands as he is breaking in at the breach Which since they are rather the parts of Captains and Souldiers than the Engineers I leave the rest to them and put an end to this treatise FINIS A New Exact and most Expeditious METHOD Of Delineating all manner of FORTIFICATIONS Regular and Irregular As well from the INTERIOR as from the EXTERIOR POLIGONE Being comprehended within the Two Faces or Superficies of a MEDALL LONDON Printed in the Year 1672. A New Exact and most Expeditious METHOD Of Delineating all manner of FORTIFICATIONS Regular and Irregular as well from the Interior as from the Exterior Poligon being comprehended within the Two Faces or Superficies of a MEDALL The use of the First Face or Superficies of the MEDALL as represented in Fig. A. THe first Face or Superficies of the Medall serves to delineate all manner of Fortifications from the Base or Exterior Poligon And that 1. From a Pentagon to a Right Line as being the more perfect 2. From a Square to a Pentagon as being the less perfect Figures 1. From a Pentagon to a Right Line Having divided the given Base or Exterior Poligon A A as in Fig. C into two equal parts in the point D from that point D erect a Perpendicular D E of a sufficient length Then take any measure whatsoever be it a Chain Pole Staff or Stake out of a Hedge and mark out 3 or 6 or 9 or 12 or 15 or 18 or 21 or 24 or 27 or 30 Lengths upon the Line A D in the point B from which raise a Perpendicular B C equal to â…“ of A B so is C a visual point through which is to be drawn the Line A F cutting the Perpendicular D E in the point E And E a visual point through which is to be drawn the Line A N. Which done divide the Perpendicular D E into eight equal parts and make either of the Lines E N and E F equal to nine of those parts And then from the points N and F of the Lines A N and A F let fall the Perpendicular N O till it cut the Line A F in the point O. And so likewise the Perpendicular E P till it cut the Line A N in the point P. Lastly joyn the points A O N F P A and so you have A O and A P for the two Faces N O and F P for the two Flancks and N F for the Curtain of that Fortification And observing the same Method from all the Bases the Fortification becomes entire as in Fig. C. 2. From a Square to a Pentagon The Method of delineating from a Square to a Pentagon is the very same with that from a Pentagon to a Right Line Onely instead of marking out three Lengths upon the Half-Base you are to mark out 15 for the Line G H. And instead of 1 you are to allow 4 for the Perpendicular H I. And lastly instead of dividing the Perpendicular K L into eight parts you are to divide it into five and of those parts you are to set off seven both upon the Line L M and upon the Line L R. And this is all the difference as may be clearly seen in Fig. D. The use of the Second Face or Superficies of the MEDALL as represented in Fig. B. THe second Face or Superficies of this Medall directs how to delineate any Fortification by the Interior Poligon And that from a Square to a Right Line For Example The Interior Poligon D F O being given as in Fig. E first draw the Capital Lines A G K T and M Q of a sufficient length Then by the foregoing Directions describe a Fortification inward upon this Interior Poligon as though it were an Exterior Which done continue the Courtains and so you have given you the two Lines A C and K M which you are to divide equally in the points B and L. And from those 2 Points to draw the four Lines B D B F L F and L O As likewise from the two points E and N you are to draw the four Lines E G E H N T and N Q so as E G may be parallel to B D E H to B F
deducted from D F the distance between the second and fourth there remains H F your Divisor which measured I admit 50 Halberds lengths the distance between G E 30 Halberds lengths the space between D F 100 Halberds length now 100 multiplyed in 30 produceth 3000 which divided by 50 leaveth in the Quotient 60. I conclude therefore the distance between A and B to be 60 Pikes lengths This one thing is to be taken notice of especially that whatsoever you mete the space G E withall that you use the same in measuring H F and as for D F it matters not what you measure it withall for your Quotient shall bear the same denomination Preciseness is to be used in placing of your Triangle and in measuring E G and H F otherwise error may ensue especially if D F be but a small distance and the Angle at B very sharp There needeth in this matter no further admonition small Practice will resolve all doubts CHAP. IX To measure the distance between any two Forts Castles or other places howsoever they be scituated though there be Rivers or such like Impediments between that you cannot approach nigh any of them and that without an Instrument also LEt your Angles as before hath been said be prepared of any three Staves c. you shall first at pleasure set up one Staff and applying thereunto your Angle in such sort that the one containing side lye directly to one of the Marks which here for distinction sake I will call the first go backwards too and fro until you find your second Mark precisely covered with your Staff noting what part of the line or side subtending the Angle it cuts by your line visual and there make a fine notch or mark upon that subtending Staff which done you shall go sidewise from the first erected Staff as the other containing side of your Triangle will direct you so far as you list and then set up your second Staff yet pass on from thence in a right line with that containing side of your Angle that riseth from your Staves and cometh somewhat toward the Mark and go so far until you spy your self justly between your third Staff and your first mark there set up your fourth Staff then resort to your Angle again and standing behind the second Staff note whether a right line from the Angle to that notch before made on the subtendent Staff or side of the Triangle will direct you for that way precisely shall you go on until you come in a right line with the second and third Staff and erect there the fifth Staff this done measure the distance between the second and third Staff reserving that for a Divisor then multiply your distance between the first and third Staff by the distance between the fourth and fifth Staff the product divide by your reserved Divisor and it yieldeth in the Quotient the true distance between the two marks Example Let A B be the distance I would know C my first Station where the first Staff is erected I my Triangle made of three Staves and placed at the Station and directed with one of the containing sides to A which is the first mark as you may see in the Figure and with the other side to D and E the second and third Staves H is the notch or mark upon the side subtended to the Angle where the line visual from ☽ passeth to the second mark B my Triangle now I scituate at D as it was before at C the one contained side lying even with the erected Staves the other directed to my fourth Staff F placed in a right line with E the third Staff and A the first mark Again my line visual proceeding from D to H the notch in the subtended side of the Angle is extended to my fifth Staff G scituated exactly between E the third Staff and B the other mark This done I measure the distance between my second and third Staff finding it 20 foot likewise between the fourth and fifth Staff and find it 72 foot finally between the first and third Staff 65 paces so that according to the Rule before given multiplying 65 by 72 I have 4680. which divided by 20 yieldeth in the Quotient 234 and so many paces is there between A and B. I have not set out the Figures in just proportions answering to these numbers for that is not requisite but in such form as may best open and make manifest the scituation of the Staves and Triangle wherein consists all the difficulty of this Practice CHAP. X. How you may readily find out the distance to any Tower Castle Forts c. by help of the former Quadrant LEt the Quadrant be made upon a square Board as is there marked A D B Q. Let D B be divided into 90 Degrees or equal parts and instead of the 12 equal parts or right and contrary shadows g m and h m let the two sides D Q and B Q be divided each into 1200 equal parts or as many as you please and marked from the Center A and have a Ruler or Index to be moved round upon the Center A having two sights upon it set just upon the feducial line of the Index and let it be divided into such equal parts as the Lymb B Q or D Q. Let this Instrument thus fitted be handsomly placed upon its Staff or otherwise lay the feducial of your Index upon the beginning of the Degrees of the Quadrant and turn your whole Instrument the Index not moved till you may espy through the sight your mark then remove your Index to the contrary side of the Quadrant placing the line feducial on the side line where the degrees end and look through the sights and in that very line set up a mark a certain distance the farther the better this done take away your Instrument and set up a Staff there and remove the Instrument to the mark you espyed set your Index on the beginning of the Degrees moving your whole Instrument till you find through the Sights the Staff at the first Station then remove your Index your Quadrant keeping its place till you may again espy through the Sights your mark which done note the Degrees cut by the line feducial and then work thus upon some even smooth Superficies whether it be Board Plate or Paper Draw first a streight line and open your Compasses to some small distance call that space a score and make so many such divisions upon your Line as there is scores between your Stations then upon the end of your line raise a perpendicular and fixing one foot of your Compasses at the other end opening it to what wideness you please draw an Arch rising from the same line that represents your Stationary distance and dividing it into 90 equal parts or Degrees as you was taught in the making your Quadrant extend from the Center to the number of Degrees cut by your feducial line a right line until it concur with
and Engineers And forasmuch as the very same things are to be made use of for Defence it is necessary that the Fortresses to the end they may be convenient for use should be prepared for it long before Now this cannot be practised in small Works where after the springing of one Mine all is lost It is also necessary that the Works should outwardly be defended as well with Canon as with Musquets by a greater number of men and from more places of Defence then can be done by the Enemy conveniently in his Attacques And herein consisteth the vertue of a Fortification For if the Enemy do find place to make a Battery of six or eight Pieces to break the Flanques we ought to have a commodious place to plant Twenty eight or Thirty pieces in Contra-Battery which will soon ruine the Enemies Battery being greater in number and standing already in advantagious places whereas the Enemy must first raise his Battery under the danger of those Canon-shots All which being premised and well considered let us now inquire whether our Modern Fortresses have this conveniency and if not where the difference lies For example see Fig. 2. Num. 2. which is part of a Work of eleven Bastions and Fortified after the best Modern Method and Practice being Fryday's second Way and according to the measure of these following Lines A B Polygone exterior 78.13 Long. F G Polygone interior 64.93 Long. F E Demy-Gorge 14.19 Long. A G Capital-Line 24.40 Long. A C Face 24.00 Long. C D Flanque 11.91 Long. D E Courtine 36.00 Long. A E Line of Defence Fichante 61.68 Long. A H Line of Defence Secant 48.85 Long. Fig 11 The Modern Engineers refer themselves much to the strength of their Out-works whereof some nevertheless are very Murthering-Holes more troublesome then useful as I shall prove in due time but for the present shall onely shew how the faults of Modern Fortifications may be Rectified according to my Method It is notorious to every one who understands Fortifications according to the Maxims above-mentioned that the longer the Defending Lines are the better they are and that according to Axiome 9. the longer Flanques and Gorges are much the stronger always provided they may be made without prejudice to other parts or that they be weakened thereby As also that the great Bulworks are better then the little ones because Retrenchments and Counter-mines may be made within them and because more Pieces of Canon and Musqueteers may be placed there for the defence of the next Bulwork and because it is fitter for all other Warlike actions It is also notorious that the longest Polygones and Regular Fortresses contain most room and therefore ought to be preferred before the shorter in case the Lines of Defence in the Fortification do not exceed the reach of a Musquet that is 60 or 70 Rods for after this manner more space is included with fewer Bulworks and therefore with less charges The other Maxims are also very strictly observed in my Fortifications Now to obtain all these Advantages I describe in Fig. 2. Num. 2. the Ground-Lines of part of an Eleven-angled Work according to Fryday as also part of an Eleven-angled Work according to my own practice both of them Fortified after our second Methods of Fortifying Which is the most commodious and the fittest for Cities of an ordinary extent but to Fortifie very great Towns it will be better to make use of the third Method I shall therefore shew first the difference of Lines in our two Methods and afterwards the manner of applying the whole Work unto practice See Num. 2. Fig. 2. The Length of Lines in an Eleven-angled work Fortified after Fryday's second Method A B Exterior Polygone 78.13 F G Interior Polygone 64.38 .. .. The little Semi-diameter 114.26 F E The Gorge 14.19 A G The Capital-Line 24.40 A C The Face 24.00 C D The Flanque or Shoulder 11.91 D E The Courtine 36.00 A E Line of Defence Fichante 61.68 A H Line of Defence Secant 48.85 E H Second Flanque 14.19 The Length of Lines according to my second Method in a Figure of Eleven Angles A B 90.00 F G 74.03 ... 131.36 F E 24.00 A G 28.32 A C 27.69 C D 16.00 D E 27.00 A E 64.50 A H 54.00 E H 12.00 Here the difference of Lines is clearly seen and that my Figure of eleven Angles containeth in the Circuit of its exterior Polygones 130. 57 ①more than that of Fryday's for his eleven Polygones are 859.43 ①in length and my eleven 990 ⓪ and my Line of Defence is no more than 2.72 ①longer than his which is little or no difference my Flanques Gorges and consequently whole Bulworks are much larger and more convenient My Bulworks-Angle or Angle Flanqued is equal to his as also the Angle of the Polygone and the Angle of the Tenaille and the Faces are as long as the Courtines My Flanques defend the next Bulworks Faces and Moats at right Angles So that all parts do mutually defend themselves with great conveniencie Here it is to be observed That I do include very near as much space with nine of my Bulworks as Fryday doth with eleven of his keeping very near the same Line of Defence which easeth the Charges very considerably besides the great Symetry of the Work And this is all that concerns the Ground-Lines the vertue of this Work will be seen by that which follows I shall proceed now to the Structure of these Fortifications and how the proportions of Lines are to be found I doubt not but there be many Theorici which understand the use of Fortifications no otherwise than upon the Paper who will imagine that this Work cannot be good seeing the Flanques do not according to the profound Error before mentioned stand Perpendicular to the Courtines not considering what strength the Bulworks have by this Obliquity not onely those that are next but also those that are situated on the other side For hereby the Bulworks gain at the entrance upon each Gorge the length D I being 8 ⓪ twice which is 16 Rods gained upon both the Gorges so that the entrance of the Bulwork cometh to be 48 Rods. This affordeth place enough to Lodge upon the Flanques the supposed Artillery and Souldiers which I esteem necessary It affordeth also room for the making of Retrenchments and the Flanques come to be much longer after this manner as is proved by the 47 of the first of Euclid It maketh also the great Line of Defence to keep its due measure not surpassing the length of 64 ⓪ which is the true reach of a Musquet Although my Polygones be considerably longer yet the Lines of Defence are not sensibly longer than those of Fryday's and others I am also of opinion That the Modern Flanques are more for an imaginary beauty than to make good the Defence which notwithstanding is the main end for which they are made and wherein consisteth the strength of Fortresses The Calculation
or Perpendicular to G F. There are many other ways which I here omit for brevity sake From a given Point in a Line to raise a Perpendicular THe given point is A and the Line B C in Fig. 5. Set one foot of the Compasses in the point A making with th' other the Ark B C upon these Points set one foot of the Compasses opening th' other at pleasure as to D and make from these Points the intersection of the Arks in D at the distance D C and draw a Line from the intersection by the given point A to the Line B C which is Perpendicular to B G. To make an Equilateral Triangle UPon the Line B H in Fig. 6. set one foot of the Compasses in B and with th' other take the distance B H and with that make an Ark about A as also out of H where these Arks intersect one another there is the third Point of the Triangle and draw from thence the Lines A B and A H and so the Triangle is made To make a Triangle of three right Lines given whatever they be provided that two of them be longer than the third THe Lines are AB CD EF take with the Compasses the length of one of the three which you would have to be the Base which is here A B and mark two Points one in H and th' other in L from the Point L describe the length of the Line C D with an Ark in K and from H the length of the Line E F and cut the former Ark in K and there is the third Point from K draw the Line K H equal to E F K L equal to C D and L H equal to A B c. Three Points being given and not standing in a right Line to draw a Circumference which shall pass through all three THe Points are O P Q make the equal sections on both sides of the Center as from P and also from Q which is done in V as likewise from O which falleth in 5 Perpendicular upon Q P and P O. Where these touch one another there is the Center as is seen in Fig. 7. Numb VIII of the Circular Ark O P Q which being drawn c. The same manner observe in the finding of the Center of a Circular Ark or a whole Circle To divide a given Circle into 360 degrees and to make an instrument from it which Geomemetricians call a Theodolit or Astrolab THis Instrument is of great use in the Mathematicks for by the same all inaccessible distances heights and depths are measured the use of which I shall declare in its own place in the following parts and it is divided thus The Circle A B C D being first divided into four equal parts by the two Diameters A B and C D take with your Compasses the exact length of the half Diameter as A E or E D by which the Circle is divided into six equal parts and after the manner following into 12 with this opening of the Compasses set one foot in C and with th' other make upon the Circumference marks on both sides in F and G likewise from D in H and L c. Thus the Circumference is divided exactly into 12 equal parts The part D A or a quarter of a Circle containeth 90 degrees as a quarter of 360 degrees F C being a sixth part 60 degrees and D H as a twelfth part 30 degrees then divide a twelfth part as D H or D L into two as in O and P and such a part holdeth 15 degrees then divide one of these parts into 3 and one of them will hold 5 degrees of which one being exactly divided into 5 one of these is a degree of which 360 make up the whole Circumference of every Circle whether it be great or small and are always proportional viz. the least Circle that can be made to the greatest Circle of the Sun or any other Astronomical Circle For if you draw several Circles out of one Center and from the same Center several Semidiameters to the utmost Circle the Arks intercepted in the innermost Circles contain as many degrees as those in the outmost c. Each degree containeth 60 Minutes a Minute 60 Seconds a Second 60 Thirds a Third 60 Fourths and so on and they are written thus 4. ° 5. ′ 6. ″ 7. ‴ 8. ' ' ' ' which is as much as 4 Degrees 5 Min. 6 Seconds 7 Thirds 8 Fourths How all kind of Ovals may be drawn by the Compasses is to be seen by the marks 1 2 3 4 5 and needeth in my opinion no further declaration seeing the manner of working appeareth enough by the Figures By 6 is an Oval which can be made by a Cord and 2 or 3 Nails This manner is of much use in Architectura civili to make oblique Vaults By 7 and 8 are seen two sorts of spiral Lines whose structure may also be perceived by the Figures And thus I have shewed the principal Draughts with Compasses serving for Geometrical use END For the Bookbinder To place the Prints of the Copper-Plates Numb after Fol. II.  8 I.  12 III.  22 IV.  28 V.  30 VI.  34 VII  36 VIII  48 TABULA Numerorum QVADRATORVM Decies millium unà cum ipsorum Lateribus ab Unitate incipientibus ordine naturali usque ad 10000 progredientibus 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 Quadratus fit numero aliquo in se multiplicato qui numerus Latus quadrati vocatur DIOPHANTUS Lib. 1. Defin. 1. A TABLE Of Ten thousand SQUARE NUMBERS Namely Of all the Square Numbers between 0 and 100 Millions And of their SIDES or ROOTS Which are all the whole Numbers between 0 and Ten thousand LONDON Printed by Thomas Ratcliffe and Nath. Thompson and are to be sold by Moses Pitt at the White Hart in Little Britain 1672. A Table of whole Numbers and their Squares  0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 00 0 1 0000 4 0000 9 0000 16 0000 25 0000 36 0000 49 0000 64 0000 81 0000 01 1 1 0201 4 0401 9 0601 16 0801 25 1001 36 1201 49 1401 64 1601 81 1801 02 4 1 0404 4 0804 9 1204 16 1604 25 2004 36 2404 49 2804 64 3204 81 3604 03 9 1 0609 4 1209 9 1809 16 2409 25 3009 36 3609 49 4209 64 4809 81 5409 04 16 1 0816 4 1616 9 2416 16 3216 25 4016 36 4816 49 5616 64 6416 81 7216 05 25 1 1025 4 2025 9 3025 16 4025 25 5025 36 6025 49 7025 64 8025 81 9025 06 36 1 1236 4 2436 9 3636 16 4836 25 6036 36 7236 49 8436 64 9636 82 0836 07 49 1 1449 4 2849 9 4249 16 5649 25 7049 36 8449 49 9849 65 1249 82 2649 08 64 1 1664 4 3264 9 4864 16 6464 25 8064 36 9664 50 1264 65 2864 82 4464 09 81 1 1881 4 3681 9 5481 16 7281 25 9081 37 0881 50 2681 65 4481 82 6281 10 100 1