his equall CDF the face Dâ— and then in the triangle DCF the lines DF FC FB FP and finally BF 44 93555 which make 〈◊〉 for which reason one must reduce the parts into rodds but this reason may be sett downe more easie in takeing but 7 48926 which make 1000000 parts Now if ye had rather worke it by multiplication without division then ye must take but 100000 parts which make 13352 rodd and so ye shall finde DC 25 38302 CB 9 34672 BA 13 35246 DP 81 01484 BH 33 38115 The angle flancked 79. 33. The 14. Figure the 1. of the tvvo In this Fortresse Hexagonall let EH be 60 rodd and EI BC KH equall the flanke KC 8 rodd it behoues us to finde out the rest BY the help of the rule of Algeber ye shall finde that the angle EIF wil be 69 degrees 4 1 2 minuts then in the triangle IBC the angle I the side IB 8 rodd being giuen ye shall finde BC 20 9229 for the face EI also IC wil be 22 4002 then EC will make 43 3231 the rest is easie without the second flanck The 15. Figure This figure hath for the flanke 10 rodd the rest being like to the former they shal be put in order in the Tables following The 16. Figure 5. Plate The flanke of this Heptagone that is A Fortresse with seuen angles or Bulwarks maketh 10 rodd the distance of the points 72 rodd and the angle flanked 80 degrees how many then will the rest make when as the second flanke is 10 rodd SEing that the angle of the Bulwarke makes 80 degrees and the angle of the Polygone 128 34 17 the difference divided in the halfe wil be the angle flanking interiour 24 17 18 1 2 the flanck CB 10 rodd whence the triangle CBG wil be knowne by adding GH giuen to BG ye shall haue the curtaine 32 1623 which taken from DP 72 rod and then take the halfe of the rest ye shall finde DF and also DC CF for the face DC 21 8525 DG 46 1666 also in the triangle DAE ye shall haue DE consequently EF or AB Moreouer ye shall finde DH in the triangle DKH to be 55 4345. The 17. Figure 5. Plate This Heptagone hath the flanke of 9 rodd the angle flanked 79 degrees 25. 43 DP â—2 rodd and the choice of the second flanke ALB. GIRARD HAuing calculated the angles as in the former OM wil be 21 6436 and HM 19 6836 let us make the second flanke 7 3164 and then the curtaine wil be 26 DF 23 the face OP 25 2901. MP 46 9337. Marolois had giuen here 80 degrees but tooke no more then 79 25 43 for the angle flanked which he did without premeditation The 18. Figure 5. Plate In this present Heptagone the angle flanked is 79 degrees the gorge 12 rodd the curtaine 32 rodd how much will the other lines angles make of this Fortresse TO resolue this question ye must suppute from the triangle CBG the lines CG GB and ye shall haue GB the second flanke now if to BG ye add AB ye shall haue AG then the triangle DAG wil be knowne afterward the triangle DAE and finally the triangles DCF and DHK ALB. GIRARD NOte that in this Question Marolois had sett downe the angle flanked 79 and 2 seuenths yet did not followe this number but 79 degrees the same errour also was committed in his supputation of the former and in the 14 figure One cannot gesse well his Supposition neuerthelesse that needes not to stay the reader for I haue sett downe the question as it seemes he would haue propounded it but the worst is he made 2 figures and one cannot understand well of which of either of them he would speake but we will speake more thereof hereafter howsoeuer those which are most intricate aswell by reason of the faults escaped in the impression of the former Editions as by the errour of his disciples which did calculate them shal be partly omitted and shall giue no impediment but that the rest may giue contentment to the Readers for that which they are desirous to finde out in this booke in the 20 figure following the Letters ye were sent unto were wholly repugnant The 19. Figure In this Heptagone let the angle flanked be 80 degrees the angle from the Capitall line from the imagined DB to wit ADB let be 22 ½ degrees DP 82 rodd the flank CB 10 rodd FRom the angle of the Polygone 128 34 17 take the flanked 80 the halfe of theremainder wil be 24 17 9 for the angle flanking interiour DGA beginn then the triangles CBG DBG whereof the angle in D maketh 17 ½ ye shall finde BG 22 1621 CG 24 3138 BD 30 3118 then GD wil be 49 1096 and DC the face 24 7958 afterward the triangle BDF the line BF wil be found to be 20 1982 and DF 22 6014 whereof the double taken from DP 82 there will remaine for the curtaine BH 36 7972 and consequently for the second flanke GH 14 6350. Now knowing the triangle DEA ye shall finde the Capitall also the gorge AB 12 8744. If in the triangle DKH yee seeke DH then ye shall finde it to be 62 7388. Now the reason of the Demy-diameter or middle line in the side of this Heptagone is as 1000000000 to 867767478. The 20. Figure This present Octogone that is a Fortresse of 8 angles or Bulwarks hath the distance DP divided into 7 equall parts whereof DF FB make each of them 2 BP fichant is 60 rod the angle flanked 82 ½ degrees THe line BF being 2 and FP 5 then the square of BP wil be 29 and seing that BP is 60 rodd its square wil be 3600 rod by which reason ye may knowe one part of the 7 of DP saying if 29 giues me 3600 how many will 1 giue me the square of an other part it will come to 124 1379310345 whereof the square roote substracted wil be 11 14172 and its double 22 28344 for DF or FB the triple is 33 42516 for the curtaine BH and the seventhfould 77 99204 for DP Moreouer the angle flanking interiour or its equall FDC wil be 26 15 so that DC wil be 24 84581 also FC 10 98907 the flanke then wil be 11 29437 for FB was knowne in the triangle right-angle DRG the angle D is also 26. 15 and RC equall to FB then DG the line of defence flanking wil be 50 38218 finally the Capitall wil be found 24 11937 and the gorge AB 13 05342 if ye calculate the triangle DEA The 21. Figure In this Octogone let the face be 24 rod the flanke 12 the curtaine 36 rod the angle flanked right it is required how many the other lines angles wil be The angles ADF 67 ½ ADC 45 will make knowne CDF to be 22 ½ Now DC is 24 rod by consequence DF FC DP FB wil be knowne also seing that
angle G K A wil be equall to the angle D A K and so the figure wil be drawne according to the said proportion The 10. Plate the 51 52 53 54 55 56 57 58 59 Figures SEcondly if the line from angle to angle be giuen AB for example 80 rod ye shall then proportion out the face to the curtaine sesquialtera as 2 to 3. Then in conformitie of the table of angles set downe before wil be made the angle CAB according to its forme to wit in a square of 15 degrees in a Pentagone of 89 ½ degrees in the Exagone 22 ½ degrees and so consequently of the rest then for the forming of the abouesaid proportion of the face to the curtaine which is as we haue said as 2 to 3. aswell in a square Fortresse as in a Pentagone and others following ye shall take upon a scale of a reasonable greatnesse 2. which shal be sett from A to C and 3 from A to D suppose that it be at C D then from the point D wil be made the arch G of the distance of 2 to wit CA from C of the distance of D A 3 wil be made the arches which cutt betweene each other at G by which a right line being drawne to A where the same cutting the line FB as here at F ye haue the face of the Bulwarke and by that the curtaine wil be knowne To knowe also the line of the gorge and the flanke and all the other parts ye must make the angle forming the gorge and the flanke of 40 degrees as is here the angle KIE cutting the diameter P A at I from which point I a right line parallel being drawne from A B as is I N ye shall drawe the line perpendicular to be E K which shal be the flanke and K I the line of the gorge of this Fortresse and then ye haue the thing required In like manner ye shall finde and marke out the other Fortresses according to their seuerall formes takeing heede that as here ye take 2 3 ye must in the others also take 2 3 to place them aswell upon A B as upon A C and the rest being the same as is in the former construction it wil be needelesse to giue yow here any further particular instruction for them The 3 50. Figure and the 9. Plate ALBERT GIRARD THirdly VVhen upon the line of the Polygone interiour AB ye desire to construe or explane a Fortresse hauing 5 termes to vvit the name of the figure the angle of the bastion M The angle forming the flanke GBH and the reason of the face DG to the curtaine HL conceiuing therein that AB is giuen This figure is in stead of the 2 50 being vvorth nothing I am so constrained to make a nevv figure and a nevv explanation Let then NBA be equall to the demy-demy-angle of the Polygone and let the angle forming the flanke be HBG LAK so ye shall haue the intersection or cutting betvveene C then DBF the halfe equall to the demy-demy-angle flanked and finding the point F so that AB to BF is the reason of the curtaine to the face and hauing dravvne CF cutting NB at D and makeing DG a paralell vvith FB meeting vvith BC at G ye haue finally GH the perpendicular vvith BA and doing the like also on the other side ye shall haue all the parts required the demonstration is manifest seing that C is the common point of the like figures FBA DGK then as FB to BA so DG the face vvilbe to GK or HL. The 11. Plate and 60 61 62 63 64 65 66 â—7 â—â— Figures IF there be a question to proportion out the face to the curtaine and the gorge to the flanke when the line EF is giuen ye shall doe as the figures in appearance demonstrate to the eye namely ye shall first proportion out the face to the curtaine by the former rules and make it soe that the curtaine of a square Fortresse be 300 foote and the face HF 250 foote hauing reason as 5 to 6 according to which ye shall finde the point G which is the angle of the shoulder To finde then afterward the gorge the flanke ye shall make the couert line GH which must be diuided into 4 equall parts in foure because ye would haue the reason of the gorge in the flanke as 4 to 3 whereof ye must place three of them from H to I and then drawe I G cutting through the semy-Diameter of the Polygone which is here squared into A and then the line AB being drawne it wil be parallell with EF and so ye haue the side of the Polygone interiour upon which the 2 perpendiculars being drawne GC HD ye haue the 2 flankes and consequently the essentiall parts of a square fortification knowne ye must also obserue the same in the other figures following If the line EF which is the side of the Polygone exteriour or the distance of the angles of the Bulwarks be not giuen but onely the line AB the side of the Polygone interiour ye shall seeke out in the tables a Fortresse of such a forme whereof the face to the curtaine is as 5 to 6 and shall worke it by the rule of proportion or else let it be made after the former manner vvhich is much more easie then all the figures of the 11. Plate If such a side of a Polygone interiour giueth such a side of a Polygone exteriour what will such a side interiour giue ye shall haue the thing required to wit the Polygone exteriour whereby one may easely know the other parts of the Fortresse when the line AB of the Polygone interiour is giuen But if in the tables the proportion be not found make first upon the line AB the triangle ARB whereof the halfe of the basis hath the like reason to the perpendicular as ye will haue proportioned out the gorge of the flanke demonstrated by this exemple as 4 to 3 which wil be done in setting downe upon the said line AB foure equall parts 4 because ye desire to haue proportioned out the gorge to the flanke as 4 to 3 to wit from A to D raising out of the point D a perpendicular and put 3 of the said parts from D to I then doing the like on the other side by this meanes wil be formed the triangle ARB and from the points A B the angles flanking interiour wil be made according to the forme of the Polygone which is a Pentagone conformable to the table of angles described heretofore as appeareth by the letters SBA and TAB which cutt the lines infinite A L. L B at the points S T and the line S T being drawne the angles T S B and S T A wil be equall to the angles S B A and T A B as appeareth by the 28 proposition of the first part of Euclide According to which the reason of the face being putt to the curtaine upon T
because we intend to speake of them hereafter In the meane while it wil be good to note that the said Cats Ravelins and other workes may be applyed to this present Fortification but in this place is omitted for the avoyding of prolixity because we are minded to treate of them briefly in an other place The description of the designe or the plate forme of a Fortresse Hexagonall The 13. Plate and 71. Figure LEt there be giuen a Fortresse Hexagonall to be fortified whereof the face AC maketh 24 rod and the angle flanked 80 degrees according to which the angle flanking interiour will make 20 degrees and the exteriour 140 degrees and let the curtaine be 30 rod which giueth the reason of the face to the curtaine as 3 to 4. To doe this ye shall draw the covert line infinite A B by the helpe of an instrument graduate the other angle CAD of 20 degrees of 20 because that the angle flanking interiour which is alwayes equall to it maketh here 20 degrees by meanes of the line indefinite AC upon which ye make the length of the face 24 rod as from A to C from which point C the perpendicular CD being drawne upon the line A B shal be placed from D the length of the curtaine which is here 32 rod as from D to E finally the distance AD from E to B and the perpendicular EF the distance of CD as from E to F drawing the line FB ye haue the other face so that all the parts of the reason giuen are described and for to finde the curtaine ye must make first the angles GAB GBA of 60 degr seing the whole angle of the Polygone maketh 120 degr by the lines AG GB which cutting through on an other at G is the center of the Polygone and seing the gorge when as ye would make casemates in them or in a bulwarke where Cats or mounts are raised they had neede of a larger extent then otherwise we suppose that it were requisite to make Casemates in them to that end we make the angle HKA which otherwise might be but 40 degrees onely of 35 according to which the gorge in the flanke wil be almost as 4 to 3 or somewhat more by reason of the line H K cutting the line Diagonall AG at H from which point H the line HN being drawne parallell to AB ye shall haue the Polygone interiour upon which the lines CL and FM being drawne in length the lines DC to L and EF to M in so doing all the essentiall parts of the said Fortresse wil be described Now to continew the same draught or platforme in every place ye shall make from the center G a privie circle from the distance GB and shal be sett upon the privie circumference the distance AB which being the 6 part of the said figure Hexagonall the said circumference will contayne justly still fiue parts which finally will come to end at A. In the like manner is the privie circle made out of the same center G and from the distance G N upon the circumference thereof shal be set the line of the Polygone interiour HN afterward ye must onely coppie out the rest Moreouer the parapet wilbet 20 foote which shal be made in the inside of ACL MFB and as the way of the rounds or the falsebray is on the outside of the body of the Fortresse we will make parallells towards the moate of 20 foote broad then 20 foote more for the parapet thereof on the outside of the parapet before ye come to the moate is made an edge or a toe of 6 or 8 foote to keepe the said parapet from falling into the moate which is made of 140 or 150 foote or thereabouts as necessitie and the bottome requires for it must be fitted according to this consideration because the ground lying lowe it will not beare much digging before ye come to water and therefore in this place one is forced to make the moate to gett earth enough for the rampart but when the ground lies high then ye may digg very deepe before ye come to the water so by this meanes ye may haue asmuch earth as possibly ye can well use seing without it cannot be but hurtfull it wil be better not to make the moate so broad that ye may receiue noe hindrance or hurt by the store of Earth that it affordeth which may be cast upon the outside For to beleeue that the ramparts being raised higher then the dimension giuen in the former figure to wit of 14 or at least 15 foote that they were better as some haue mainteyned yet experience hath showne many times the contrarie for an ennemie hauing once approched to the brinke of the moate the bredth of the parapet hinders one from makeing any defence upon it which notwithstanding is most necessarie because the neerer an ennemie comes with his approches towards us we ought the more to defend our selues and this being committed by our default it is too manifest that one ought to take heede of it and to remedie this errour by the former way namely in makeing the rampart of the heigth abouesaid Some are of the opinion by a simple cōmaund that one ought to heigthen the ramparts aboue the said 14 or 15 foote and not the Bulwarks for which they haue some reason for indeede the Bulwarkes make the greatest defence and when an ennemy is become master of the said bulwarke there were a meanes to cōmaund the said bulwarke more absolutely so that an ennemy could not hinder the use of them if he raised not his works aboue the said rampart to hinder the defences of the besieged which they might make with the said ramparts so raised by a simple commaund which in some manner might hinder the intents approches of the besiegers the more easie would the entrenchements be made and seing the more the bulwarks are raised the more one ought to use diligence about the entrenchment hence would follow this difficulty that the comming to the rampart or the bulwarke would not be so easie as otherwise neither could one make any great defence from such ramparts by reason of their too much heigth in such sort that one bulwarke ought to defend an other in such a case the second flanke would be of noe use after an ennemy is entred into the moate at which time it is then most needfull to giue the greatest resistance that may be seing that when they are gotten ouer the moate and taken in the foot of the bulwarke then the courage of the besiegers begins to encrease the besieged to faile them for it is too apparant that many times resistance failes in such and the like accidents Betweene two bulwarkes are commonly made ravelins or halfe moones which are Quadrangular figures euery face containing 12 15 or sometimes 20 rod which beginn at the brinke of the moate so that its angle interiour or the tenaille O lies just
the midst of the line E F and for asmuch as the angle E is of 112. degrees vvhich comes neere vpon the angle of a Pentagone ye shal be vpon the foresaid angle E describe the angle of a Pentagone and seing that the line E D makes 80 rod vve vvill make the angles flanking interiour of the forme of a Pentagone to the end tha the skirts E L. and D O. be equall as we haue said before for seing the Ennemies force betvveene E D is equall reason requireth that the defense thereof be likewise made equall that by this meanes ye maye take away all occasion from an Ennemie to attempt any further place the most advantagious for him Now in regard that the distance D. C. exceedeth the measure vvhich wee haue spoken of before to wit of 114 ½ rod it vvilbe necessarry to make the Raveline K betweene the said angles D C to supply the defect of the defense the like maye be made betweene C B and the angles of the Bulwarks and B the rest shall be made according to their formes the faces flanks and Curtaines in that forme as vvee haue said aboue euen as this figure Hexagonall 95. demonstrateth The 96. Figure HErein it is requisite to note that vvhen one is bound to fortifie precisely the angles of the figure either interiourly or exteriourly ye meete many times vvith difficulties to vvit here are some angles or sides too small and others too great in so much that this irregularitie maye cause many great defects Which maye greatly be remedied vvhen ye maye haue libertie to change a litle the angles of the figure euen as wee haue done in this 96. figure in such sort that the angle E being but 112. degrees which is the angle of a Pentagone makes the angle of the bulwark S E T too much pointed as when ye would make it on the inside of the angle E. Therefore see that ye make the face E S vpoÌ„ the side ED that by this meanes ye maye haue the angle S E T broader and more open drawing the line E O so that it be equall to ED and that the angle O E D be 20 degrees so that the curtaine Q. P. comes not too farre into the figure and that the line of defense ER maye come a part of it out thereof if it be possible as here to R for the more the said line commeth out of the curtaine it is so much the better which is when the angle O E D is broad and open But this exemple if this had bene observed the Bulwarke L had bene a great distance from the curtaine C D and would haue made it much longer then it is at this present Then vpon the point O shal be made the second face of the Bulwarke W O V so made that O W and O V and ES are of an equall greatnes that the line D X be drawne in such a manner that the angle Y Z 5 be capable to receiue the Bulwark of an Exagone and for the better attayning to such a structure ye shall make first vpon the line C B the flanke noted by 4 5 and the face Z 5 to the end that CB maye serue for a curtaine and so fitted that it maye almost be equall to the curtaine D. C. Then vpon CB ye shall make the Bulwarke I. according to the greatnesse of 4 B. answerable to the Bulwarks G. H. and seing the angles A Bare sharpe ye shall make these two Demy-Bulwarks according to our method mentioned in the 18 Plate and so ye haue finished the thing required Note AS we haue fortified this figure on the inside so ye maye doe the like on the outside in case the ground will afford it but we suppose here that it would be necessaire to make it in such a sort in regard that the lengths of the sides require more the interiour fortification then the exteriour whence appeareth that there are many vvayes to fortifie places irregular yea an infinite number yet bounded with these limits to wit that the angles of the Bulwarks ought not to be noe further assunder then 80 rod at least 60. that the angles flanked must not be lesse then 60. degrees That your line of defense ought not to exceede much aboue 60 rod for by how much the Bulwarkes haue a second flanke by so much they are the better the more spacious and larger the flanks and gorgâ—◠〈◊〉 the Bulwarks are the better for them according to our former Maxims set downe in the end of the first part according to which an expert and skillfull Ingenier will be sure as much as possibly maye be to haue all these advantages abouesaid And for the better facilitating of what is said aboue wee haue made here the figures 97 98 99 100 101 which must be cutt out vpon a pastboord drawne out vpon the same measure as the plott of the place requireth which is to be fortified and to fitt them to the places of the figure with the greater consideration that possible maye bee that he maye follow the rules abouesaid as neere as may be where vnto these figures in my opinion are of very great vse for he may turne and remoue them on what side soeuer he will and after he hath found the most convenient place then he maye joyne fasten them together with a litle waxe that afterwards he maye overcast the advantages and disadvantages which he is to expect How to fortifie an Irregular place lying vpon the side of a river The 22. Plate 102. Figure LEt the forme irregular which ye would haue be the plate of Hardervvijck noted by the numbers 1 2 3 4 5 6 7 8 whereof the extremities or vtmost ends are 1 6. touching the Dike A B and C D which ye desire to fortifie To doe this ye must first overweigh how many bulwarks the circuit of the same place will take vp husbanding them so that ye must make as few as maye be because they are parts of a fortification which will cost much and yet so that you must not place them so farre assunder but that the one bulwark must helpe to defend the other for this reason we haue made the line of defense about 60. rod which is a longer distance then is giuen when one is to defend them with the Muskett or caliver The others that must be defended with the CanoÌ„ maye be 1000 foot distant one from an other or there abouts because at the least it will carry so farr over that ofâ—entimes a Canon will carry much further then to the vtmost end of the line of defense so that they may hinder the batteries which are made to beate downe the flank of the angle from whence the line of defense is drawne also ye make it so that the distance of the said angle vnto the angle of the Bulwark be not so farre as a Canon beares but rather shorter for which reason we
time also as their imperfections were knowne and that by the force of man and Engines they were diuerse wayes attempted yea many times beaten downe overthrowne which made them seeke from time to time to remedie their defects for perceiving that their walls were heretofore subject to be beaten downe by Rames and other ancient Engines they made them upon the right line of the wall Spures and roundles to hinder the beating of them downe and to preserue them the better from a breach and tumbling downe they gaue them the Talude that is a slooping towards the inside of the place that they might be the more able to resist the violence force of these Engines Finally these walls were made after diverse manners first in a round forme which as on the one side it was held the strongest because these Engines beating against the wall crushed the stones made them stick the closser together in regard the circle exteriour was greater then the interiour and that the Engines could not unfasten loose and breake them but with great difficultie So on the other side it was impossible to defend such walls when approches was made unto them because no part of them could be discouvered or flancked so that afterward they built theÌ„ in a square forme with small squares in their angles for the defense of their Curtaines Then made them also Demycircles with angles interiour and exteriour and lastly triangular wise on the one side to resist the force of the furious Canon used at this day and on the other side that they might be the more capable to defend themselues by discovering euery part of the wall euen to the very foundation And as invention of attempting is growne to the highest degree in these dayes by reason of the longe experience of the warrs in these parts which is the Schole of all military actions and the abilitie and capacitie of the Assaillants Defendants So am I of the opinion that the Fortifications made in these Low Countryes are the strongest exactest perfectest which can be invented and which haue bene made and practized not by a simple Generall but by one of the greatest Captaines of the World endowed with a singular courage and spirit an excellent Mathematician and is not onely a Prince of a great howse but also experienced beaten in all militaire actions and stratagems aswell offensiue as defensiue of which Fortification we haue undertaken to treate briefly at this present and as succinctly as possibly may be Of the definitions I. Forasmuch then as the definitions of Fortification are by the dayly use of armes growne so common it were in vaine for me in my opinion to make any further explication thereof yet to satisfie the ignorant we will marke out the angles and sides of a Fortresse by the Letters of the Alphabeth and opposite to the said letters ye shall finde their names and appellations as we may note by the figures 1. and 2 following Icnographie or ground-markeing 1. Figure N. O. The side of the Polygone that is many angles N. D. The line of the gorge D. C. The line of the flanke B. N. The Capitall line B. C. Q. R. The Moate P. The Raveline or halfe mone Q. S. The covert way T. S. The Parapett thereof B. I. The line of defence D. K. The Curtaine K. F. The Parapett K. M. The Rampart A. N. The semy-diameter V. C. The flanke lengthned C. N. D. The angle forming the flank B. C. D. The angle of the shoulder Orthographie or the Profile 2. Figure A. B. The foote or basis of the Rampart G. H. The heigth of the Rampart H. B. The talud or slooping of the infiel of the Rampart A. Y. The Talud on the outside of the Rampart or scharfe Z. D. The foote of the Parapett Z. E. The Parapett itselfe D. F. The foote-banke F. G. The Terra-plaine or bredth of the Rampart K. A. The way for the round or the falsebray I. K. The foote banke thereof I. M. The Parapett of the falsebray M. N. The Scharfe P. O. N. M. The moate P. O. The Counterscharfe P. Q. The Covert way R. Q. The foote banke thereof T. S. R. The Parapett of the Covert way The other names which haue neede of explanation shal be declared in their due places Before we come to instruct yow particularly in the Art of Fortification we will briefely treate of the calculation thereof In which supputation ye shall haue first sett downe the knowne termes and under them the disposition of the Characters or Letters beginning with a square fortresse with foure angles or Bulwarks and proceede on to a Dodecagone a fortresse with twelue angles or Bulwarks makeing upon every Polygone 3 or 4 Trialls that afterwards one may choose the best of them and because the angles will not be much altered by the diversitie of the trials I haue thought good to giue this generall rule for them following It is a thing generally received of all men that a square fortresse with foure Bulwarks is not so good as a Pentagonall with fiue angles nor a Pentagonall so stronge as an Exagonall with sixe and so consequently of the rest If the cause thereof be sought out one may obserue that this proceedes from the smallnesse of their angles as not being able to beare such a body of a Bastion as the subsequent Polygones so that a square fortresse for this reason wil be more defective then the Pentagonall and this lesse defensive then the Hexagonall and so well the rest following even to a Dodecagone which hath the angle of the Bastion right which is the cause that constraines one to make the angles flanked lesser then the reason of building well doth require that is the flanks too litle the Gorge too narrow and the line of defence too longe To encrease then proportionally the angles of Fortresses according as the angle of their Polygone augmenteth we will take then the halfe of their angles and adding thereunto 15 degrees the summe wil be the angle of the Bulwarke which we terme the angle-flanked and if the angle-flanked be substracted from the angle of the Polygone there will remayne the double of the angle flanking interiour which double being substracted from 180 degrees `then will remayne the angle flanking exteriour called the Tenaille and if ye add to the angle flanking interiour 90 degrees then the summe wil be the angle of the Shoulder To finde out the angle of the Polygone from the number of its substract namely 2 the remaynder must be multiplyed by 2. and the product wil be the number of the right angles which such a Polygone containeth as ye maye see by this exemple following Or thus And by the same rule ye shall finde the angles of the Subsequent Polygones beginning from a square Fortresse to a Dodecagone 4. 5. 6. 7. 8. 9. 10. 11. 12 90. 72. 60. 51 3 7. 45. 40. 36. 32 8 â—â— 30 the angl of the center 90.
108. 120. 128 4 7. 135. 140. 144. 147 â— 11. 150 the angle of the Polig 45. 54. 60. 64 2 7. 67 1 2. 70. 72. 73 7 11. 75 The halfe 15. 15. 15. 15. 15. 15. 15. 15. 15. Sum. 60. 69. 75. 79 2 7. 82 1 2. 85. 87. 88 7 11. 90 The angle flanked Remaines 30. 39. 45. 49 2 7. 52 1 2. 55. 57. 58 7 11. 60 the double of the ang 180. 180. 180. 180. 180. 180. 180. 180 180 The flanke interiour 150. 141. 135. 130 5 7. 127 1 2. 125. 123. 121 4 11. 120 The flank exteriour 15. 19 1 2. 22 1 2. 24 9 14. 26 1 4. 27 1 2. 28 1 2. 29 7 22. 30 The flank interiour 90. 90. 90. 90. 90. 90. 90. 90. 90 105. 109 1 2. 112 1 2. 114 9 14. 116 1 4. 117 1 2. 118 1 2. 119 1 22. 120 the ang of the should And seing the angle flanked of a Dodecagone is right and able to resist a batterie which is also made alwayes with right angles to shake the more the face of the Bulwarke one must fortifie the Polygones which are aboue the Dodecagone with a right angle that the line of defence may come the more into the curtaine that one may giue the more fire upon it but the Polygones which are under a Dodecagone must be fortified according to the precedent Table and the calculation thereof shal be made hereafter Some times we augment aswell the angles of the Bulwarkes as the Octogone a Fortresse with 8 angles or Bulwarks with a right angle and those aboue are alwayes right those under diminishing to the square fortresse which hath the angle of its Bulwarke onely of 60 degrees According to which the Bulwarks are somewhat larger and the gorges and flanke greater then the former but the second flanks lesser Now for the finding out of every angle ye must doe this following where ye may observe that in the manner aboue sayd the angles flanking interiour are the fourth part of the angle flanked or the 1 â— of the angle of the Polygone IIII. V. VI. VII VIII the angle 90. 108. 120. 128 4 7. 135. the angle of the Polygone 60. 72. 80. 85 5 7. 90. the angle Flanked 90. 72. 60. 51 3 7. 45. the angle of the center added thereunto 150. 144. 140. 137 1 7. 135. the angle Flanking exteriour 30. 36. 40. 42 â— 7. 45. the angle double of the angl flank interiour 15. 18. 20. 21 3 7. 22 â— 2. the angle Flanking interiour 90. 90. 90. 90. 90. the angle which is the flanke alwayes 105. 108. 110. 111 3 7. 112 1 2. the angle of the shoulder In the same manner also may be made right the angle flanked of the Decagone a Fortresse of ten Bulwarks where ye must note also that before wee proceede further we will make use in the supputation following of the tenths or decinall numbers which though it gives some imperfection yet seing the things which we omit therein are of no great consequence it were ridiculous to make any further search thereof considering likewise that the tables of Sines tangents and secants are one and the same I thought it good therfore to make use of these following THE I. QUESTION The 1. Figure 1. Plate Let there be made upon a Square a fortification of foure Bulwarkes so that the line of the Gorge be 7 parts DI the curtaine 21 and IF which is the flanke of 5 and from the angle of the flanke is drawne the line of defence by the angle of the Shoulder to giue it a face The question is how many the angles will be and every line of the same fortresse when as the line of defence will take up 600 feete Now the length of a foote is sett downe in the 25 modell of Geometrie noted by 1 and is divided into 12 ynches the ynch into 10 equall parts and is the same foote whereof 12 makes a rodd which his Excell useth in all his Fortifications ALBERT GIRARD THe Authour hath here aboue so disposed his calculations that in stead of explaining them briefly he confounds them teadiously as if heretofore there had bene noe certaine rule sett dovvne in vvriting to calculate lines and angles as is ordinarily done by the Trigometrie of plaine Triangles although there haue bene a number of Authours vvhich haue treated of them th' one after one manner th' other after an other and the most part of them commixed vvith longe discourses vvhich moved me not longe since to putt into light some tables of Sines in a portable volume vvith the most succinct method that possibly could touching the supputation of such plaine triangles reducing them into foure diverse cases vvhere I did insert in their places some of my ovvne inventions to the purpose knovvne as I suppose to none others heretofore so that it cannot be but having 3 knovvne termes in a triangle but that one maye knovve the other three or one of them onely vvhich one desires as the Reader may knovve it the manner order thereof being much more facile easier to conceiue then the reading of our Authour in his former editions being obscure troublesome and hard to be attayned unto For this reason the Learners of this science are required to be forevvarned before they come to the reading of this booke are advertized hereby that vvhen it is sayd a triangle hath three termes he must understand knovvne or given and that vvhen I say that a triangle right-angle hath three termes that then I am bound but to shevv tvvo of them seing that this vvord right-angle presupposeth that the triangle hath a right angle to vveet of 90 degrees moreover that touching this present question these 7 21 5 parts demonstrate the reason of the lines ND DI IF and not the quantitie of the same in feete as is necessary to finde in the manner follovving To finde out the Angles IN the triangle right angle CDI the side CD is to DI as 5 to 21 for CD being of 5 parts then DI wil be the 21 thereof therefore the angle DIC wil be found for 13 24 it s double CLD 26 48 and its adjunct CLF flanking exteriour wil be of 153 12. Now in the figure quadrilatere or of foure-sides ABLH the angle exteriour BLH is alwayes equall to the 3 interiours B A H now A 90 the angle of the center being taken from BLH 153 12 there will remaine the two Demy-flankes ABL LHA 63 12. for one entire flank ZBC seing that the angle of the Shoulder BCD is exteriour in the triangle CDI it wil be equall to the two interiours D. 90 and 1. 13 24 and therefore the Shoulder C wil be 103. 24. To finde out the Lengths IN a triangle ambligone BNI the side BI is 600 feete the angles are found therefore the other termes shall be found to wit BN the Capitall line 196 64 NI 444 62 but ND is the
third of DI for ND being 7 DI is 21 by the Hypotheses therefore the fourth part of the found NI wil be ND 111 15 the Gorge the rest wil be 333 46 for DI the Curtaine now DI to DC is as 21 to 5 therefore one may say if 21 giues me 5 how many DI 333 46 giue me facit for CD 79 40. so CI wil be 343 84 which taken from BI 600 there will remaine BC 257 16 for the face Finally the triangle right-angle VDH hath three termes DH 600 and the angle H equall to HDI. 13 24 because of the parallels DI VH then yow shall know VD 139 05 VH 583 67 from which side substract VP equall to Demy DI 166 73 there will remaine PH 416 94 whereof the double wil be for BH 833 88 and PA wil be 416 94 for it is equall to PH and thus must ye doe with the rest following The 3. Figure 1. Plate Let there be a square Fortresse whereof the curtaine DI being 4 parts the flanck CD shall haue 1 of them and also the Gorge 1 The defence running from the angle of the flanke formeth the face the length BI is of 600 feete one requires the greatnesse of the other lines of the same Fortresse To finde out the Angles THe triangle right-angle CDI hath three termes as the reason of the sides CD 1 to DI 4 therefore the flancking interiour CDI wil be 14 degrees 2 minutes whereunto add D 90 degrees then the Shoulder BCD wil be 104 2 now if ye double the angle CID it wil be 28 4 CMD and its adjunct CMF 151 56 for the angle flancking exteriour from which substract the angle of the center A 90 there will remaine 61 56 for the angle flancked entire B and thus much for the angles For the sides THe triangle BNI hath three termes to witt BI 600 feete and the angles B 30 58. N 135. I 14. 2 so one shall finde the lines BN 205 76 NI 436 59 but in regard that ND to DI is as 1 to 4 according to the Hypotese then ND will be the fifth of NI where will follow that ND wil be 87 32 and DC as much and DI the rest 349 27. Now seing that in the triangle right-angle CDI the sides CD DI are notified one shall finde the Hypotenuse CI of 360 which taken from BI 600. there remaines the face BC 240. Finally in the triangle right-angle VDH which hath 3 termes DH 600 the angle H equall to CID 14.2 ye shall haue VD 145 49 also VH 582. 09 from which take VP 174 64. equall to the demy-curtaine DI there will remaine PH 407 45 also PA its equall then in the triangle right-angle APH ye shall finde AH 576 22 from whence take BN or HO the capitall line 205 76 there will remaine AO 370 46 in takeing VD from AP there will remaine the perpendicular of the Center A upon the middest of the curtaine 261 96. Now BH wil be 114 90. as being the double of PH. 4. Figure 1. Plate In this square Fortresse the defence BI is 600 feete and the angle flanked 60 degrees whereof DBC is the fourth part which is 15 degrees the question is what the quantity of the parts of such a Fortresse wil be For the angles SEing that the line flanked is 60 then in the triangle BNI the angle B wil be 30 N 135 the adjunct of ANI 45 whereof the angle remayning BIN wil be 15 degrees therefore the triangle BDI wil be Isosceles that is two sides a like such that DBI is aswell 15 as BID The Shoulder C wil be 105 and in the triangle DMI the angles upon the basis are each of them 15 degrees the remaynder M will be then 150 for the flanking exteriour BMH. For the sides THe triangle Isoceles BDI hath three termes the defence BI 600 feete and the pointed angles every of them 15 degrees Therefore DI the Curtaine wil be 310 584. Also the triangle CDI hath three termes the angle D right I 15. and the curtaine DI then CD the flanke 83 217 CI 321 535 which being taken from BI 600 there will remaine BC the face 278 465. Moreover in the triangle IBN the angles B 30 I 15 the defence 600 so then BN the Capitall line wil be 219 623 NI 424. 268. from whence DI being taken there will remaine the gorge ND 113. 684. Moreover the triangle BIL having three termes L right B equall to CID 15. and BI 600 then I L wil be 155 292 BL 579 558 and takeing from it PL 155. 292. which is the halfe of the curtaine there will remaine BP 424 266 for AP also it s double being BH 848 532 one may also easely knowe BA 600. The 5. Figure 2. Plate ALBERT GIRARD Heitherto of the defence dravvne from the angle of the flanke but in those follovving there is a second flank and the distance of the angle of the flank euen to the angle flanked vvhich is called Fichant The 5 figure is the dessigne or draught of a square Fortresse whereof the line of defence fichant DH maketh 600 feete the angle flanked 60 degrees the line HB is divided into 7 equall parts whereof the one which is betweene the Characters 1. 2. is subdiuided into 5 equall parts and from the center B is made the Arch 4 N cutting the capitall line at N from which point is drawne NDZ parallel to BH and from the point V which is in the character 2 the perpendicular VD the question is how many these lines and the angles of such a Fortresse will make For the lines in parts indetermined FOrasmuch as BH contayneth 35 such parts as BN 9 BV 10 VH 25 ye shall haue in the triangle right-angle BTN termes enough to knowe BT or TN 6 364 and TV or ND 3 636 Also the triangle right-angle DVH had three termes VD equall to TN 6364 VH 25 then DII wil be 25 797 parts or the same DH is 600 foote We must then calculate according to this reason the lines aboue mentioned to bring them into feete saying as followeth For the lines in parts determined to vvit brought into feete THe 25 797 parts make 600 foote how many then will BH 35. come to There wil be for BH 814 04 foote likewise VL or DZ the curtaine may be made soe BH 7 parts make 814 04 foote how many VL 3 it will come for the curtaine to 348 87 for BN the Capitall line ye shall say if 35 giues 814 04 how many 9 ye shall haue for BN 209.32 and so of the other for TN or BT or VD wil be 148. 02 TV or the Gorge ND 84 57. and seing that BH is knowne also VL the halfe of the remainder wil be BV 232 58. In the triangle right-angle BVC the angle B wil be 15 degrees for PBO is 45 CBO 30 and BV being knowne therefore BC
FB is found or EA ye haue also the triangle DEA by which ye shall finde DA AB shall finde DG in the triangle CBG for hauing CG ye add unto it DC finally DH in the triangle DKH DP 80 34624 DG 55 35756 DH 61 91032 GH 7 02948. AB 13 39837. AI 62 39837. The 22. Figure In this present Octogone the flanke is 11 rod the angle flanked 82 ½ degrees the line DP 76 rod the second flanke to be choosen how many then wil be the other parts of this Fortresse THe angle of the Octogone is knowne and the angle flanked also their halfe ADF ADC therefore the rest and the triangle CBG wil be knowne and BG wil be found 22 3058 suppose the second flanke GH is 9 6942 then BH 32 rod likewise as much for FK but DP is 76 therefore DF makes 22 and after ye haue calculated the triangles DCF DAE ye shall finde DC 24 52978 DG 49 40045 AB 12 94981 and finally in the triangle DKH the fichant DH 58 25283. The 23. Figure Let the angle flanked be 82 ½ of this present Octogone the line of defence DG 50 rod DP 76 and CAB the halfe of the angle flanked AFter ye haue carryed AC which was forgetten if ye count the triangles DAG ACG ye shall finde the Capitall the face and the gorge and BG then the triangles CBG DCF afterward ye shall haue CB CF DF and therfore FK or the curtaine and the second flanke then DH 58 3498. The Capitall 23 93655 The Gorge 12 84323 The Flanke 11 26362 The second flanke 9 15324 The face 24 53340 The Curtaine 31 99346 The 24. Figure Let there be an Enneagone A fortresse with nine Angles or Bulwarks that is with 9 sides whereof the angle flanked is 85 degrees the face 24 rod the flanke 12 and the curtaine 36 rod. THe angle of the Polygone is 140 degrees seeke the triangles DCF CBG DEA DKH and then ye shall finde the lines GH 12 94824 DP 78 57648 AB 12 88709. The 25. Figure As in the tables aboue let the angle flanked of the Enneagone be 85 degrees the fichant 60 rod and DP being 7 parts then let DF be two of them afterward FB the perpendicular 2 for the makeing of the flanke CB wee require the rest DK being 5 parts and KH 2. the square DH wil be 29. which makes 3600 whereof 1 makes 124 13793103 for the square of one part its root substracted wil be 11 14172 for one part whereof the double wil be for DF or FB and the triple for FK or the curtaine 33 42516 BH and after ye haue calculated the triangles DFC CBG DEA ye shall finde the other lines DC 25 1219 DG 48 25855 GH 12 9026 BC wil be 10 68335 the seuen parts of the number aboue sayd wil be for DP 77 99204. The 26. Figure In this present Enneagone let the angle flanked be 85 degrees the defence 50 rod the other 60 the gorge in the flanke as 4 to 3. the rest is required IF ye take away the angle ADC 42 ½ from ADF 70 the angles GDR will remaine and DG makes 50 rod then the triangle DGR wil be knowne to wit DR RG or KH 23. 0875 which will make ye knowe the triangle DKH for DH is 60 then RK or GH wil be 11 0301. Moreouer in the triangle DEA the side EA is equall to KH by DE DR ye shall haue AG 35 94735 then in carrying AC ye shall seeke the angle A from the triangle ABC setting downe AB 4 and BC 3 parts according to the Hypatese ye shall finde then the angle A of 36 degrees 52 min. 12. seconds Then let us goe to the triangle ACG hauing AG knowne the angle A G equall to CDF 27 ½ degrees to finde CG afterward the face and consequently DF FK EF or AB the Gorge which wil be 14 72808 therefore if 4 giues 3 how many then AB ye shall haue BC for the flanke 11 04606 DP 78 51183 DA 24 56925. The 27. Figure The distance DP being 7 parts DF FB each of them 2 parts the defences 50 60 rod and the angle flanked 85 degrees how many will the other dimensions be of such a Fortresse non angular ALB. GIRARD THis question is impossible to resolue being exceeding seing that there is a condition in it more then one desires vvhich is vvorst repugnant to the others for vvhich fault the Authour may be excused seing that in his time there vvere no such advertisements giuen as vvee haue giuen thereof in the beginning of the Trigonometrie cited in the first question going before the vvhich though they may seeme to be of litle consequence to some yet one must acknovvledg that those that knovve them shall not fall into the like errours as these vvhich may be explained thus There are tvvo reasons giuen DP to DF and DF to FB a reason of equality tvvo lines of defence the angle flanked the name of the figure of nine-side figure vvhich are sixe termes yet one needes but 5 as ye shall finde it noted in the 7 figure vvhere the question vvas defectiue and of some others aftervvard finally the proofe of this may be seene in makeing comparaison of this vvith the 25 figure vvhere the same question is propounded and vvhere ye shall finde that the defence flanking ought to be 48 25855 and here he vvill haue it 50 vvhich is absurd as is said The 28. Figure In this present Decagone a Fortresse with ten angles or Bulwarks let the angle flanked be 87 degrees the Gorge in the flanke in reason sesquitertia the defences 50 and 60 rod it is required how many the other parts thereof will make THe reason sesquitertia is as 4 to 3 for AB to BC then the imaginall angle BAC wil be 36 degrees 52 minutes 12 seconds Moreouer the angle ADC being 43 ½ then CDF wil be 28 ½ which is an angle of the triangle GDR and which may be knowne seing that DG is 50 rod therefore GR 23 858 or its equall KH and for as much as DH is 60 then DK wil be knowne and also DR withall and so ye shall finde RK for the second flanke likewise the triangle DEA wil be knowne then ED and DR will make knowne ER or AG consequently ye shall haue the triangle ACG for the angle A was found aboue and the angle G is 28 ½ degrees then the face wil be 26 11334 and hauing found out DF ye shall finde EF or the gorge 15 197 Also FK for the curtaine 32 10821 DP 78 00619. The 29. Figure In this Decagone the angle flanked maked 87 degrees the defence fichant 60 rod the flanke 12 rod and the gorge 16 rod it is required how much the other parts make THe quadrangle ABCD hauing fiue termes giuen ye shall finde the other parts also the triangle DFC whereby ye haue FB or KH consequently
the triangle DKH and hauing FD DK their summe and difference wil be for DP 78 65 and BH 30 32633 the face shall be 27 49377 DG 52 64265. The 30. Figure In this Decagone let the curtaine be 36 rod and the flanke 12 the face 24 and the reason of the Bulwarke in the flanking interiour as 58 to 19. THe halfe of 58 is 29 then the angles ADG to AGD or CDF wil be as 29 to 19 therefore setting downe ADF to CDF it wil be as 48 to 19 but ADF is 72 degrees then CDF wil be 28 ½ degrees then the angle flanked 87 degrees the rest is easie for the triangles DCF CBG will make knowne CF FD DG is 49 14888 DH 61 68324 DP 78 18336 AD 24 52336. The 31. Figure Let there be a Decagone whereof the face is 24 rod the flanke 12 and the curtaine 36 how many will those parts make when the defence flanking is doubled to the Capitall IF DG be sett downe 2. then DA wil be 1. Now the angle DAG is 108 degrees then the angle flanking interiour G wil be 28. 23. 38 and the angle flanked 87 12 44â— the rest is easie is found in the same manner as the end of the former therefore Marolois leaues its so for this reason The 32. Figure In this present Vndecagone a fortresse with eleuen Bulwarks let the face be 24 rod the flanke 12 the curtaine 36 and the capitall DA to AG as 5 to 7 the unknowne parts are required SEt downe DA 5 and AG 7 and the angle DAG is 106 degrees and 4 eleuenths that is 106 21 49 and hence ye shall know the angles remayning ADC 43 55 48 and the other or CDF 29 42 23 therefore in the triangle CDF the angle D is then so and DC 24 rod and then DF DK DP wil be knowne likewise FC FB BP AD DE EF for the gorge Here is nothing but that which is ill calculated by the Autheur or rather by his disciples as from the beginning without all doubt seekeing to help themselues with the figure put here under which was needlesse supposing that they had skill in Trigonometrie I will onely set downe here the reason of the raid or semy-diameter in the side of the Vndecagone inscribed in the Circle is as 100000000000000 to 563â—46511368285 So that ye may take here the reason or proportion as precisely as ye will The 33. Figure In this Vndecagone let the face of the curtaine be as 2 to 3 the gorge in the flanke as 4 to 3 the distance of the points of the Bastions 75 rod and the angle flanked 88 degrees 38 min. 11 seconds the question is how ye shall finde out the other dimensions SEing that ADC and ADF are knowne the rest CDF wil be also being 29 19 6 setting downe DC 2 parts then the curtaine or FK wil be 3 the triangle CDF wil be knowne by the parts namely DF 1 74384 as much also is KL and FK 3 then DP wil be in parts 6 48768 which make by the Hypotese 75 rod by this reason ye shall finde out the face and the curtaine saying if 6 48768 parts make 75 rod how many then will make aswell 2 as 3 DC 23 12075 BH 34 68112 as much makes FK which taken from 75 the halfe of the rest wil be for DF ye shall finde then also FC afterward AB being set downe upon 4. BC wil be 3 then the triangle BAC wil be 36 52 12 which taken from DAB there will remaine DAC so the triangle DAC shall haue 3 termes ye shall then seeke DA or AC by which ye shall come to haue AB 13 7969 BC 10 34743 then in the triangle CBG ye shall finde BG afterward DR or CG being knowne ye shall haue DG 41 54579 GH 11 57496 and finally DH the fichant 58 96636. The 34. Figure Moreouer for this Vndecagone let the angle flanked be 88 degrees 38 11 the face to the curtaine as 2 to 3 and the gorge in the flanke as 8 to 5. the fichant 60 rod how many will all these lines make YE must first finde out the Quadrangle ABCD whereof the angles are knowne sett downe AB 8 parts then BC wil be 5 DC being found ye shall finde DF FC also BH seing that DC is to BH as 2 to 3 hauing DK and KH ye shall haue DH in parts which make 60 rod by which reason ye haue the lines required AB 14 6 CB 9 127 DG 42 374 DC 23 7363 BH 35 604 DP 76 99675. Vpon the 35. Figure ALB. GIRARD THis question is defectiue in regard it hath but foure termes knovvne DP 70 rod CB 10 the reason of DC to BG is as 10 to 9 and the name of the figure Undecagonall vvhich are but 4 termes he comes to the 〈◊〉 setting dovvne the face 20 rod to see if there be not in it repugnanciâ— as if the question vvere exceeding so that he in this search committeth the faulth vvhich is called petitio principij VVhich is spoken not to defame the Authour but to shevv hovv this proceeded from hence that in this time many thought that Geometrie hath attained to her highest degree though vvee haue had but the A B C of it till that some undertakeing the restauration thereof haueing in part made it flourish againe could neuerthelesse escape the blame of some envious ignorant men in this divine science as happens often haue let passe the most difficult of the Analitica vvhich ought to adorne those that make profession thereof vvho contraryvvise setting dovvne the limits of their pretensions from the beginning of their course by emulating one an other content themselues to grope together vvithout learning to goe forvvard vvith a solide and a firme pace in a faire vvay The 36. Figure In this Dodecagone let the face be 24 the flanke and the curtaine 36 rod the angle flanked right how many will the remayning lines make LEt DF be found also FC ye shall haue DK KH for to haue DH afterward in the triangle DGR hauing the angles RG ye shall finde DG Finally ye shall finde DE in the triangle DEA for to haue EF or the gorge Now seing that GCB is 60 degrees then of necessity GC wil be 24 as being asmuch againe as CB then DG wil be 48 and GR or KH the halfe is 12 therefore DH wil be 61 648 DP 77 56944. The 37. Figure In this present Dodecagone let the angle flanked be right the defences flanking the fichant 45 60 rod the gorge in the flanke is as 4 to 3. SEing that the angle DCF is 60 degrees DH 45 wil be the double of GR 22 ½ rod DH HK being knowne DK wil be 55 621 DR 38 971 then the second flanke 16 65. Also DE wil be found by the triangle DAE taken from DR there will remaine AG the basis of the
triangle ACG the angle A thereof wil be 36 52 11 because of the reason giuen and G 60 degrees then the sayd triangle ACG will make knowne CG 21 4933 also the face 23 5067 for DG is 45 the halfe of GC is CB 10 7466 which multiplyed by 4 3 ye shall haue BA 14 3289 DP 75 979 BH 35 2637. The 38. Figure Let the curtaine be 36 rod the line of defence flanking 45 the angle BAC 36 45 the flanked right How many will the other parts of this Fortresse Dodecagonall come to THe triangle DAG afterward ACG wil be knowne and also CBG DCF DKH whereby DC wil be found 23 974 GR 22 ½ the halfe of DG 10 5127 DH 61 0593. DP 77 52516 AB 14 733. The 39. Figure In this Dodecagone the angle BAC makes 37 degrees the flanked right the face 24 rod the fichant 62 how many makes then the other lines THe line DC 24 shal be the double of CF 12 the angle BAC being giuen the triangle DAC wil be knowne therefore DA AC and the triangles DAE ACB DGR DHK wil be knowne and consequently the lines required as AB 14 6179. BC 11 01525 BH 36 7851 DP 78 3547. DA 23 82709 He failed herein to calculate them a new together but the reason of this was that wee agree not seing that in the construction in stead of 37 for the angle BAC as he sett downe he tooke 38. All questions comming after this are defectiue euen to the discourse which endeth the order of the same questions but seing I know whither it tends I will explane them in adding thereunto the things that were wanting passing ouer the figures 40 41 and so we will beginn with the 42 as followeth The 42. Figure In a square Fortresse with foure Bulwarks let the angle flanked be 60 degrees the angle forming the flanke FID 40 degrees the face 24 rod hauing the reason to the curtaine as 4 to 5 the other lines are required Seeke the triangles DAL AID IFD DFP ye shall finde also PH for FH wil be 30 IO 48 45532 AK 76 6464 IF 9 22766 FD 7 74298 IA 19 73479 the fichant AH 54 98265. The 43 Figure Is a Pentagone the angle flanked 69 degrees the angle forming the flanke â—0 degrees the curtaine to the face as 5 to 4 the face 24 rod wee desire to know the other lengths Ye must doe as before and then there wil be no difference in the operation Touching the number of the names of the figures in this 9 plate ye shall finde them marked about the angle flanking exteriour the reason of the face to the curtaine is marked on the point of the Bastions the length aswell of the face as of the curtaine next themselues and must be proposed as the two former hauing alwayes the angle forming the flanke of 40 degrees the opening of the flanked angles is according to the manner of the first table to wit 15 degrees more then the demy-angle of the Polygone sauing that in the figures 2 43 2 44 afterward an other time the 45 46 he taketh for the angles flanked the â…” of the angle of the Polygone according to the second table in the beginning placed before Of these things ye shall finde 2 tables in the end of this booke setting downe the faces to the curtaines as 2 to 3 the faces 24 rod which make the former-flankes 40 degrees and the flanked according to the two manners abouesaid where ye must knowe that the lines of defence fichant are about 60 61 rod. The said tables are both calculated 〈◊〉 because of the errours escaped in the other editions Moreouer after the Dodecagone ye shall haue ouer and aboue following a Fortification upon a right line which is called upon a right curtaine Now let us marke what the Authour saith Note 1. NOw not to take so much paines in remembering the diverse proportions of the face to the curtaine whereof the Fortresse quadrangular Pentagonall Fortresse is as 4 to 5 and the Exagonall as 3 to 4 it will not be amisse seing that the line of defence will beare it to make also the said Fortresses of the same proportion as the subsequent to wit in reason sesquialtere as appeareth by the 3 figures of the 9 plate quoted by the numbers 2 42. 2 43. 2 44. where the lines of defence doe not much exceede 60 rod which is as farre as a musket can well carry therefore one ought not to exceede this number because that alwayes from the flanke reciprocally the entrance into the moate must be defended which is often done and most commodiously by the Muskettiers because that Canon cannot so suddenly by reason of the weight be brought thither many good occasions are neglected for which cause they are preferred before Canon and in regard a Musket will but carrie some 700 feete point-blanke Bulwarkes ought to be made noe further one from another for otherwise the line of defence would be too long which should cause imperfection Note 2. IF in stead of takeing the face 24 rod for the line of the Polygone exteriour take 80 rod and the rest according to the former proportion the parts wil be brought very neere together as appeareth by the precedeÌ„t Examples according to the figures of the 10 plate marked with the numbers 51. 52. 53. 54. 55. 56. 57. 58 59 whereof we haue made no calculation because they are easely done by the former supputations Note 3. THat in all our designes and calculations we are resolued to use but one measure onely which is in the 25 plate of our Geometrie upon the rule of the instrument marked number 1 which is a foote deuided into 12 equall parts whereof 12 foote makes a Rheneland rodd in the territoire of Leyden Designes plat formes of diverse Fortifications WE might according to the former supputations giue diuerse constructions of the platformes of Fortifications but considering the great diversity of rules which often causes confusion and that time will not permit us wee will giue but one simple and generall rule for them which is this following The 9 Plate and the 42 43 44 45 46 47 48 49 50 Figures FIrst if the face be giuen suppose it to be 24 rod then ye must doe as followeth Ye shall draw a right line infinite AC from the point A the angle CAB shal be made according to the forme of the Polygone upon which A B shal be sett downe the 24 rod abouesayd as from A to D and by D is drawne the line which formeth the angle of the Center of the Bulwarke 40 degrees then must ye sett downe upon A C and A B the reason of the curtaine to the face that by the same ye may draw the line infinite A E makeing from the point D a parallell with A C cutting the said AE at G by which point is drawne the line that formeth the other face G K in such sort that the
S and T A namely the face upon TA and the curtaine upon T S ye haue the like distances and from the points I X the arches which cuts betweene them in a certaine place by which intersection the line T V shal be drawne cutting the line S B at V then the right line V L. being drawne passing through the center L cutting A R at G ye shall haue all the parts of this Fortresse for hauing sett downe the distance R G from R to H from the same point G is drawne a paralell to S B as E G cutting the Diagonall line L A at E the like ye shall finde for F and consequently all the other parts of the said Fortresse Pentagonall The same must ye also understand of the other figures following from the figure 60 to the figure 68. Also ye may finde the point V in setting downe upon T A 2 equall parts such as ye require and from the extremity or utmost end hauing made it a parallell to S T and upon the same 3 of those parts and drawne from the extremity the line TV the said point will consequently be knowne thereby NOte that the proportion giuen here betweene the face and the curtaine is not so much to relye fast upon it as to show that the generall rule sett downe here before in the 9 and 10 plates take place also in all other reasons which may be propounded For otherwise wee are of the opinion that the former figures would rather be accepted then these here because that the reason of the curtaine to the face is as wee haue said aboue sesquialtera as well in square Fortresse as of Pentagonall and others following Which for the facility simplicity together with the goodnesse of them ought to be preferred before the figures 60 61 62 63 64 65 66 67 and 68 of the 11 Plate aboue said Whereof the face to the curtaine is as the exemples shew of diuerse reasons in such sort that hence forward one ought to resolue that the reason of the curtaine to the face ought in all formes of Fortification to be sesquialtera and the face 24 rod each rod containing 12 foote the length whereof is sett downe in the 25 plate of our Geometrie noted with the character or the figure 1 is diuided into 12 ynches which rod is used in the Fortifications of the United Provinces to the end that the line of defence may not exceede much aboue 60 rod which is about as farre as a Musket can well beare the gorge to the flanke may be made by the rule giuen in the 11 plate according to the reason giuen but forasmuch as it is more convenient to make the angle forming the flanke GAC plate the 11 of 40 degrees which giueth the reason almost as 6 to 7 in my opinion one ought to rest thereupon and so ye shall haue a generall rule for all manner of Fortifications whither they be Quadrialtera Pentagonall Hexagonall or the others following as ye may perfectly understand by that which shal be said hereafter In the meane while note that I giue here the reason of the gorge to the flanke as 7 to 6 which ought to be understood in Fortresses without Casemates but if it were my intention to make some of them I would then alter somewhat of the said proportion The manner how to describe succinctly the designes or Plots of some regular Fortifications The 12. Plate and the 69. Figure ALthough one may by that which we haue taught sufficiently now understand the order and Method requisite to be held in all manner of Fortifications yet I haue thought it good to trace out here some from among the rest to make the louers of this Art to conceiue the better my intention and will begin with a regular Pentagone whereof the angle of the center maketh 72 degrees and the angle of the Polygone 108 degrees and seing the angle flanked according to the table before mentioned maketh 69 degrees It wil be easie to finde the angle C A D which is alwayes equall to the angle flanking interiour and shall finde the same to be 19 ½ degrees setting then your graduate Instrument upon 19 ½ degrees it makes the angle CAD and drawing first the covert line infinite AB takeing upon the scale 24 parts or rod and sett then from A to C drawing from the point C a perpendicular upon A B as is CD afterward hauing sett on DE the length of the curtaine which is here onely 34 rod because the false brayes makes the lines of defence too great then the distance AD from B to E raysing from the E the 〈◊〉 cular EF being equall to D C and FB drawne out the two faces wil be knowne for to knowe the center of the Bulwarke shal be made of 2 lines GA GB the Demy-diameter of the Polygone then your instrument being sett upon 40 degrees ye shall finde the angle HKA cutting the said demy-diameter at the same point H from which point the line HI being drawne ye shall haue the side of the Polygone interiour upon which out of the points C and F shal be drawne the perpendiculars CL FM which will forme the flanks and the gorges of the said Fortresse Pentagonall in the inside of the side of the Polygone interiour shal be drawne a parallell of 5 ½ rod for the thicknesse of the rampart as N O N R and RS and ye must draw for the parapett of the rampart a parallell of 20 feete also in the inside of the said side of the Polygone HI and on the outside thereof ye shall drawe a parallell of 20 foote for the falsebray as LX and yet more outward 20 other foote for its parapet so all the other parts wil be traced out which are within the moate for seing that there is here noe Casemates the falsebray is carried not onely about the curtaines but also about the flankes and the faces of the Bulwarks in such sort that the falsebray serueth as a Casemats to the said flankes the entrance or sallies ought in my opinion to be made in the midst of the curtaines as covertly as possibly may be the Bulwarkes are made either massiue or hollow from Earth at this present wee haue drawne them out as being hollow from earth so the superficies or plaine in the midst of the bulwarke N R S H. is of the same heigth as the rest of the enclosure of the Pentagone is the moate shal be 10 rod broad which is traced out as followeth In the point A or the angle of the Bulwarke shal be made an arch of the distance of 10 rod in the point V make an other arch of somewhat a lesser extention then the former about 10 or 12 foote then drawe upon the back thereof a covert line infinite the like must be done on the otherside of the Bulwarke and so from place to place with covert lines being drawne as abouesaid
upon the brinke and angle of the moate the reason why their faces P Q and Q R are made of a greater distance is that the deffence may be the better for otherwise their could be noe defence made but from one part of the face of the bulwarke which otherwise would not be sufficient to hinder the assaults of an ennemie which ravelins or loose peeces li◠so open and are raised aboue the plaine field some foure foote that one may the easier discouer the said field and so also to prevent the inconveniences which might happen when one would giue fire from them upon the besiegers if the said ravelin had not bene raised which would haue annoyed those which were under the covert way Upon the said Ravelin being so raised is made a parapet of 20 foote broad and 6 foote high which wil be able to resist Canon the moate whereof ought to be made 50 or 60 foote and as deepe as possibly may be about the said moate ye must make also a cover way of some 20 foote broade as hath bene noted before Afterward ye must make a parapet of 50 foote broade six foote high ending at the end of the 50 foote towards the plaine field as the profile doth show in the 72 figure and to the intent ye may the better understand my meaning we haue made the said Hexagone in perspectiue with the falsebrayes Ravelins covert wayes and their parapets apart as appeareth by the second 71 figure but ye must note that the said Ravelins must be in respect of Ramparts which haue a double height to see in them the more perfection also a double height to make them show the better for otherwise they would seeme too little for the reason abouesaid against our intention The 72. Figure THe Profile shal be made as in the 12 Plate of the former figure 70 by drawing a privie line infinite and takeing upon it all the dimensions as first the bredth of the ramparts the parapets the falsebrayes its parapet the bredth of the moate the covert way and the other parts of the said profile and seing wee haue here ordained Ravelins wee will make this profile from the midst of the rampart passing also through the midst of the Raveline that ye may the better understand our meaning According to which ye shall take the bredth of the Rampart 68 foote from A to C from C to D 20 foote from D to E 20 foote and from E to F 6 foote the bredth of the moate shal be 150 foote as from F to G which ought to be broader but seing the distance falls out too litle it shal be made but of 150 foote the Raveline ought to haue in this place 180 foote but seing the ground faile us wee haue made the dimension thereof 180 foote but this hindrance would not permit us to make it of that competent greatnesse as it ought which I speake to giue noe stop to the Reader which Ravelin is raised 4 foote aboue the plaine field from 6 to T upon which a parapet is made whereof the basis shal be 20 foote broad and 6 foote high to wit its bredth from Q to R and the height from Q to S the footebankes are made of the same bredth heigth therefore I will make here no mention of them neither of the taluds or sloopings of Ramparts Parapets moates aswell the interiour as the exteriour parts are made according to the nature of the soile for by how much the ground is leane sandie by so much ye ought to giue it the talud by that meanes ye shall hinder the falling downe of the workes for which reason often times ye must giue on the exteriour side being a sandie ground as much talude as heigth and if the said Ravelin be also raised 4 foote aboue the plaine field me thinks it will not be amisse to raise the rampart thereof which is 15 foote some 6 foote higher to commaund the better the said Raueline leaue the Bulwarks of the height of 15 foote that all inconveniences may be prevented on the outside of the Raveline is made the edge HI of 6 foote and a moate of 50 foote then the covert way 20 foote the parapet LN 60 foote with a footebanke of the ordinary bredth of 3 foote in that manner as the 72 figure demonstrateth The description of the plote of a Fortresse Heptagonall The 14. Plate and 73. Figure LEt there be giuen to be fortified a Heptagone whereof the side of the Polygone exteriour maketh 63 rod and the angle flanked 80 degrees To doe this ye shall first divide the said side AB into 7 equall parts as ye see by the points 1. 2. 3. 4. 5. 6. 7. and from the points of the 5 and 2 parts the perpendiculars D N and E O shal be drawne being equall to A D and E B to wit either of them 2 seuenth parts of the line A B then ye shall draw from the extremities or the utmost parts of these perpendiculars the privie line infinite H I. which shal be the side of the Polygone interiour and by consequence N O the curtaine The angles C A B and C B A shal be made by the helpe of a graduate-instrument as our compasse is or an other either of 64 ½ degrees in such sort that the intersection of the lines A C. C B which is C shal be the center of the said Fortresse Heptagonall Afterward I B 8 of the 40 degr which is the halfe of the angle flanked that ye may haue the whole angle of 80 degr according to the Hypotheses and where the said lines cutt the said perpendiculars D N and E O in the points 2. 8 ye shall haue the faces A 2. B 8 the flanks N. 2. O. 8 and also the gorges H N. O I the Parapets Ramparts Falsebrayes for the other parts of this Fortresse ye shall doe the like as hath ben taught in the former figure Hexagonall the faces in this present Raveline shal be made of 15 rod the moate of 10 rod and the moate to the Ravelin of 15 foote then about it shal be made a covert way of 20 foote a parapet of 60 foote broad descending as we haue said before sloopingly According to this forme the flanke wil be then made of a reasonable greatnesse but the gorge so much the lesser which may be made so seing ye intend not to make any flankes in the Casemates but in case ye would ye must then make the flanke the lesser and consequently the gorge the greater to the end ye may haue space enough for to make within them the said Casemates The like is to be understood when ye meane to make a Catt upon the Bulwarke for the gorge being so litle besides other inconveniences it is impossible to make there such a bodie in reserving sufficient space for the flanks which me thinks in such places that haue such narrow
the face wil be 240 78 and VC 62 32 which being taken from VD there will remaine CD the flanck 85. 7 and therfore the triangle right-angle CDI shall haue 3 termes the angle C 75 CD 85 7. soe DI wil be 319 84 CI 331 15 likewise BI flancking 571 93. I Z 29. 03 for the second flancke Now OP is knowne as being the halfe of BH Now of the Pentagons or Fortresses with fiue angles or Bulwarks 6. Figure 2. Plate In the Pentagone a Fortresse with fiue Bulwarks KFBDL let the line KL be 63 rods divided into 7 equall parts whereof the Capitall line KA is 2 of them also LE and from the Character 2 or G let GB be the perpendicular Also the angle flancked is 69 degrees according to the precedent table how many then wil be the lines and the angles of such a Fortresse ALB. GIRARD Before vvee come to the Construction hereof take notice that KA in this figure is not ansvverable to the length of KG yet ye must suppose them to be equall asvvell as the cipher 1 and the letter H are tvvo differing points vvhich vvould haue bene the better discerned if the figure had bin vvell made vvhich may serue as a fore-vvarning for some figures follovving upon vvhich one ought not to stand so much as upon the Suppositions or Hypoteses of the Propositions The triangle right-angle KHA hath three termes KA 18 rodd the angle AKH 54 degrees which being of KH wil be 10 5802 HA or GB 14 5623. Now if ye take away KH from KG there will remaine for AB the Gorge 7. 4198. Item the curtaine BD wil be 27 rodd being equall to GM And because the angle flancked is 69 degrees if ye add thereunto the angle of the Center which is 72 ye shall haue the angle flancking exteriour 141 degrees the halfe of its adjunct for the interiour 19 1 2 degrees The triangle right-angle KGF hauing 3 termes KG 18 GKF 19 1 2 the face KF wil be 19 0953 and FG 6. 3741 which being taken from GB 14 5623. there will remayne FB the flanck 〈…〉 then the triangle right-angle FBI may be knowne seing that FB is found and the angle 〈…〉 1 2 the complement of 1 therefore BI wil be 23.1227 which taken from BD 27 there will remayne ID for the second flanck 3 8773 also FI wil be found to be 24 5297 whereunto add KF 19. 0953 and ye shall haue KI the defence flancking 43. 6250 finally the triangle right-angle LGB hauing three termes to wit LG 45 rodd and GB 14 5623 one may easely know BL and the distance of the Center of the Fortresse by K because that KP is 31 1 2 rodd and the angle AKP 54 degrees The 7. Figure 2. Plate In a Fortresse Pentagonall BOV let AB the deffence flancking be 50 rodd the Flanck ED 9 rodd and the angle of the Bulwarke 72 degrees How many then will the other parts of the same make when the second flanck AG makes fiue rodd THe angle of the Polygone is 108 the halfe is 54 for FBQ to which add CBF 36 degrees by the Hypoteses you shall haue 90 for CBQ but CF is paralell to BQ therefore BCA right the triangle BCA wil be then the right-angle and BA of 50 and the angle B of 72 degrees then ye shall finde BC or QD to be of 15 451 and CA 47 553 and the angle A 18 degrees Also in the triangle right-angle EDA the angle A is giuen and the side ED 9 rodd by the Hypotese then EA wil be 29 12463 DA 27 69912 now if ye take EA from DA there will remaine BE the face 20 87537 and if ye substract DA from AC there will remaine CD or BQ 19. 854. Also the triangle right-angle BCF hath 3 termes BC and the angles in fine BF the capitall line wil be 19 097 CF 11 226 which being taken from CD there will remaine FD 8 628 the Gorge now if to DA ye add AG 5 rodd the second flanck ye shall haue the curtaine DG 32. 69912 to which add twise BQ and ye shall haue BV 72 40711 its half BK 36 20355 and so the triangle right-angle BKO hauing 3 termes the angle KBO wil be 54 degrees ye shall finde BO 61 593099 and KO 49 82984. Finally VB BQ being knowne then the triangle right angle DQV hauing 3 termes giuen VQ QD ye shall finde the fichant DV easely ALB. GIRARD IF one vvill consider the difference vvhich is betvveene this operation and that of our Authour published in his former Editions he shall finde that this question is defectiue for I haue added thereunto that the second flanke AG is rodd seing that in all such questions vvhere there is a second flank he ought to haue set dovvne 5 knovvne termes neither more nor lesse vvithout the one depending on the other as in this present the name of the Fortresse to vvit Pentagonall is a terme in the second place there is the defence then the flanck the angle flancked and the second flancke vvhich are fiue termes vvhere ye must note that vvhen there are reasons in the proposition and that though a reason hath tvvo numbers neverthelesse it is but one terme but vvhere there is no second flanck as in the figures of the first plate then 4 termes vvill suffice Finally my Authour had so ordained his Supputation that in stead of makeing an addition or a substraction he made the rules of three very great so that ye must imagine it vvas easier for me to change all then to recorrect it hauing no other respect but to the explication of the figures and as much as is lavvfull and possible to shevv his intention vvhich I am bound to doe Moreouer if peradventure same one should finde this manner of operation strange vvhich I 〈…〉 heitherto it is requisite for him to knovv also be it spoken under â—â—â—â—ection that he understandeth not much if he doth not practize it soe himselfe in other Subjects yea though I had not sett dovvne one onely number vvhich I should haue done already heretofore if I had not had regard to the obscuritie vvhich students might pretend to finde vvhich hauing cleared heitherto ye may goe on vvith the more assurance in the rest follovving and must also knovv that the 8 figure vvas left vvithout explication by the Authour The 8. 9. 10 Figure 3. Plate In this figure Pentagonall the angle ELO is divided into 2 equall parts by LD the flanck FB 9 rodd the angle flancked 69 degrees and the curtaine 30 rodd how many then will the other lines angles be THe angle of the Polygone is parted in the midst by KA and the angle AKF is knowne also the rest shal be FKG or its equall FB whence followes that the triangle right-angle FBI shall haue 3 termes FB and the angles therefore the other 3 termes wil be knowne also the whole line DB and the
gorges it were better to make the said Cats upon the Curtaine yet so that they be noe hindrance to the Rounds going along the Ramparts for which reason they ought to be made more inward and so that they may lie as neere the rampart as possibly may be the better to discouver and commaund the fields about them The 74. 75. Figures THe figure 74 is the profile of the Rampart Falsebray its Parapet moate covert way and its parapet whereof the length bredth heigth and depth are marked out in the said figure 74 which is a section of the Bulwarck as the figure 75 the section of the midst of the Rampart and because all the profile cannot commodiously be drawne out we haue represented but a part of the moate the forme of the Raveline its heigth and the forme of its parapet the bredth and the depth of its side with their taluds or sloopings the covert way with its parapet footebanke as ye may exactly note by the figures hereunto annexed The 2. of the 14. Plate ANd for the better facilitating of that 〈◊〉 we haue spoken of this present figure 73. we haue added hereūto the platforme of the Fortresse of Coverden in Frizland whereby ye may the better understand our intention being accounted the master peece and the most regular and royall Fort in the Low-Countries hauing a False-braye Rauelins or half moones covertwayes with which wee will put an end to the description of our regular Fortresses A succint description of some other works in the said Heptagone which are of an other manner of makeing then the former The 15. Plate 76. Figure LEt the side of the Polygone interiour be B C upon which ye would haue a part of a Fortresse Heptagonall made whereof the face to the curtaine is as 3 to 4. and the gorge to the flanke as 13 to 9. To doe this ye must doe as we haue taught yow in the 11. Plate or in the 9. Plate and Figure 3. 50 where ye shall finde the proportion requisite on the outside of the same face shal be made the parallells each of 20 foote aswell for the parapet as for the falsebray the like is to be done in the flanks and curtaines and on the inside a parallell of 20 foote with an other parallell of 72 foote for the bredth of the Rampart in the basis in the same manner as this 76 figure sheweth on the outside ye shall make a parallell of 10 rod for the bredth of the moate and upon the brinke of it to the tenaille ye shall lay out the Ravelins as we haue said in the former figure 73 in the angles of the Bulwarks ye shall make also Ravelins such as ye see here marked with I K L M which shal be defended by the Ravelins E F G H the better to defend the said ravelins which are in the angles of the Bulwarks ye shall make the other works N O P Q which are called Horne-works which are made in such a manner that the moates of these Horneworks comes to answere to the Falsebray which is in the flanks of the Bulwarks so that their bredth S T maketh about 32 rod of which distance the two demy-Bulwarks V X are made whereof the face and the curtaine are of a like greatnesse according to the rule prescribed in the 10. Plate the flanks wil be found makeing from the angle of the shoulder two lines perpendicular upon the curtaine which is directly opposite to it euen as the figures 77 78 demonstrate The utmost angles of the said Horneworks ought not to be further from the curtaine of the Fortresse then a Musket can carry point-blanke which is 60 rod or there abouts and if the Rauelins EF GH are not made in the tenailles then ye may cut the said Hornworke as is here marked out in the 78 figure the bredth of the moate shal be 24 or 25 foote The Rampart here is noe otherwise then the parapet of the same bredth The depth of the moate may be made 6 foote in case the ground lies low but otherwise the deeper it is made it wil be so much the better the bredth will not hinder it though it were made but 36 foote for according to the same bredth depth ye may enlarge the Rampart aswell in height as in bredth upon which afterward ye may make a parapet as great as it can well beare but when haste requireth I would make noe Rampart but onely a Parapet of 24 foote thick as wee haue made here being 6 foote high and a footbanke of 3 or 4 foote The figure 79 marketh out a Profile which is a section or cutt drawne through the midst of a curtaine passing through the midst of the Ravelines G and traversing through the Horneworke P and the Rampart and the moate The true dimensions thereof are clearely expressed in the said Profile by the helpe of the scale joyned to it as also by the meanes of the Alphabeticall letters shewing the feete to be 12 ynches whereof the length is 〈…〉 in the 25 Plate of our Geometrie The benefit of such workes are well knowne when they are made in places of advantage for deffence or where noe men victualls and amunition of warre are wanting as also where the ground is of a reasonable largenesse to wit at the least 32 rod that the faces of the said horneworkes may be about 12 rod which is the least length that one can giue to such workes against the attempt which may be giuen by an Army as a great assault cannot be resisted with a few men but up a small roome so is it manifest that the greatnesse largenesse of such a place must be answerable Therefore in my opinion it is a thing repugnāt to the rule of Fortification to make such Horneworkes in the angles of Bulwarkes where they are so straightned For Demy-bulwarkes cannot by reason of their smallnesse be vvell mainteyned and defended and on the other side their tvvo vvings cannot be defended from the maine Fortresse but vvith great disadvantage as vvee intend to discusse thereof more at large hereafter in the plateforme of Gulick How Citadels or Castles may be joyned to Townes or Townes to Castles The 16. Plate and the 81. 82. Figures FOr to build a Castle or a Citadell to a Towne either to help to defend it the better or to curbe it ye must finde out first the most advantagious place in case there be a river then ye shall build your Castle upon the side of it where it may best commaund with all the advantage that possibly may be also takeing heede that it be built in such a place that the Castle may receiue noe disadvantage thereby but that the Rampart of the Towne where the letters A G are be the weakest part to the end that by this meanes ye may frustrate the desire of the inhabitans of the said towne from opposing themselues against the said Castle but
layd out the tvvo sides A B C D ought to be lengthened so farre that one maye make the angles B K L C R P of 75 degrees and that the line of defense O. K. termineth in the curtaine M. Q. and to knovv the length of the faces and flanks the face shal be made double in the flanke which is done if ye place vpon the perpendicular O N the halfe of the line of Defense O K. as from O to N. and from the point K the privie right line N K being drawne cutting through the line C B at M ye shall dravv the line M L paralell to N O or perpendicular vpon C B and so the flanke L M vvilbe the halfe of the face L K. And thus vve haue finished the fortification of the abouesaid Quadrilatere irregular vvhich in my opinion is a better fortification then the precedent figure 86. because these Bulvvarks are more capable to defend the angles A D K R and the flanks of the said Bulvvarks being dravvne obliquely as they are here cannot be made into right angles in such sort that they are the better able to resist against the force vvhich maye be vsed against them and yet make a good defense as the figure 88 plainely demonstrates An other way to make the fortification of such a place regular The 19. Plate and 89. Figure FIrst of all vve vvil describe the quadrate E F. G H. after such a manner that the line E F is paralell to A. D. of the greatnesse of the poligone exteriour in the follovving table of lengths dravving the paralell E F so that its distance A D be equall to the distance of the Polygons asvvel interiour as exteriour to the intent that the side of the Quadrilatere A D may serue for the curtaine Then the angles I F E I E F being made 15 degrees because the angle interiour flanking is of 15 degrees in the square ye shal take vpon a scale 5. equall parts place them vpon the line F E and foure of the same parts vpon the line F I. from the extremity or vtmost end vvhereof and with the said distances ye shall make two arches that shall cutt through one an other at X drawing from thence a line to F cutting through the line I E at G. then E G wil be the face of the Bulwarke which will haue the like proportion to the curtaine as 4 to 5. But wee will here after in all the kinds of Polygones ordaine the curtaine to the face in proportion as 3 to 2. that is sesquilatera finding it best as we haue said before Afterwards to haue your flanks ye must make two perpendicular lines G N. and H K vpon the side A. D. in such manner that the lines G N and H K shal be the flanks and N K the curtaine which is part of the side A D. The like ye shall doe with the three other sides and so by this meanes this fortification wil be made regular and royall which will not cost much more then the former irregular fortification abouesaid the benefit whereof surpasseth the others by farre so that in such like accidents I am of the opinion that such places ought to be made regular in case that time and the situation wil permit it As for the moates ramparts and parapets they must be made as wee haue taught in the places of regular fortifications The fortification of a Pentagone irregular The 19. Plate and 90. 91 Figures LEt the Pentagone irregular be A B C D E which one would haue fortified in such sort that the Bulwarks come againe to the angles of the propounded figure To doe this ye shall first measure the outsides and the angles which I supposeye shall finde to be as they stand here vnderneath rod A B 57. B C 50. C D 46. D E 56 degrees A 72. B 135. C 111. D 97. E 125. 540. And seing the angle A is lesser then 90 degrees reason requireth that the said angle be not fortified because it would make an angle flanked lesser then 60 degrees and flanking greater then 150 degrees against our former maximes according to which ye shall make of the same angle A. an angle of the bulwarke makeing the angle of the Polygone F. so that the right lines FG and FI come to cutt through the lines lengthned BC DE in the points G and I. Vpon the angles whereof and according to the proportion of the sides shal be described the Bulwarks takeing heede that the angle of the Polygone show what forme of a Bulwarke one must built vpon to wit a Quadrate a Pentagonall or an exagonall proportioning out the parts of such a Bulwarke according to the least side of the two and then the figure wil be described according to this present forme And seing the side DE because it hath ben lengthned is longer then the proportion can beare of our regular figurs precedent It wil be necessary that betweene the two Bulwarks D E a Ravelin be made which is a loose peece that maye be defended at leastvvise from the flanks of the abouesaid tvvo Bulvvarks according to vvhich flank the angle of the Raveline shal be made a litle more open or closser as the curtaine is either longe or short The faces vvhereof shal be made of 18 or 20 rod some times a litle lesser as the place and situation of the ground requireth And to giue you to vnderstand more clearely my intention touching the fortification of places irregular vvhereof the angles are noe lesse then 90 degrees which is the angle of a Quadrate and that the sides doe not differ much from those places vvhich are regular It must be proportioned thus Suppose that one giueth me the angle C to be fortified of vvhich the magnitude is III degrees vvhich comes neere the angle of a Pentagone according to vvhich I take the shortest line of the tvvo BC C D makeing therevvith the angle B C D vvhich is C. D. contayning 46 rod we wil seeke out then in the table of the lengths of our regular fortifications the dimensions of a Pentagone and will say by the rule of proportion if a Polygone 56 88 giuâ—â— the face 24 vvhat vvill then a Polygone of 46 rod giue ye shall haue for the face 19. 41 rod. The like ye shall haue for the flanke and then the Gorge where by ye shal finde the said Bulvvarke C. as also all the other parts of this fortresse Pentagonall holding this for an infallible rule that the angles of the Polygone which you would fortifie ought to be at the least right and in case there be any angle that hath a lesse opening then the right ye must make thereof the angle of a Bulwarke or else dravve a line if you cannot lessen the place vvhich vvill forme an angle competeÌ„t to builde a Bulwarke vpon as appeareth by the 90 figure in the angle A. Which line you must so husband that if it
fortified so that herein ye must vse discretion requisite in such a case and so accommodate the said bulvvarks cutt out in pastboord that the said angles be tolerable and that on the other side the curtaines be not too farre distant from the sides of the Polygone as B. C. C. D and as this figure represents it to your eye Yet an other way The 20. Plate 93. Figure IF you are desirous that the flanks of the Bulvvarks B C fall vpon the side of the Polygone B C so that the part G H maye serue for a curtaine you must doe as follovveth the angle B A F is made of 15 degrees because that in makeing it greater the line FA vvould runne too farre from the place A B C D E thē ye shall make the line paralell F G distant from the line B C as farre as the Polygone exteriour is distant from the Polygone interiour in the Pentagone because that the angle G ought to be Pentagonall and if the line F G is shorter then the Polygone exteriour in our table of lengths ye shall sinde out a proportionable distance in saying If the side of the Polygone exteriour 8125. giueth distance to the Polygones 1677 vvhat distance vvill FG giue that vvhich this rule wil produce shal be the distance of the paralells B C F G and so consequently the paralell F G being made from that distance ye haue the thing required and before ye beginne vvith the Bulvvarks it is your best vvay to make the lines GH HI IA and as the line B C. hath serued for a curtaine the line C D maye serue for the most part thereto seing that the line GH is not ordinarily paralell to the side of the Polygone irregular C D but the side E D maye be accommodated as before so that one part of it vvill serue for a curtaine to the two Bulwarks which shal be made in the angles I. H. so that if the angle D had not ben so sharpe and that the line G H might haue bene paralell to E D it is evident that one could haue made vse of C D for the curtaine of the tvvo Bulvvarks G H. Or else if the line F G might haue bene somevvhat augmented the curtaine would haue falne much neerer to C. D But seing in this exemple it could be noe longer of necessity the said curtaine must fall vvithin the inside of the figure Pentagonall irregular Then having dravvne these your lines A F G H I vvith all circumspection requisite to vvit that the angle flanking interiour be at the least 15. degrees ye shall draw out vpon them the Bulwarks flanks and curtaines in such sort that the faces flanks of the Bulwarks vvhich are vpon the one side of the Polygone be alike amonge themselues as those vvhich are noted in the 92. Figure according to our former rules giuen in our regular fortifications A way how to fortifie a right Curtaine The 20. Plate 94. Figure IF it be needefull to fortifie a right Curtaine vvhereof the angles of the Bulwarks are right ye must doe as followeth Let there be taken 70 rod wanting 6 seconds that is 69. 64. rod and putt vpon the said curtaine A E as many times as the said curtaine will beare it as appeareth here by the points A B C D E from vvhich points shall rise the privie perpendiculars A F B G C H D I F K the Capitals of 28 97. rod and on each side of the said points A B C D E shal be placed 16 97. rod as from A to L. and at M raysing the perpendiculars L N and M O 12 rod the flanks 6 11 then the lines NF and FO being drawne vvhich vvill make the faces of the Bulwarks ye shall haue that which is necessarie for the description of such a Bulwarke whereof the face is 24 rod the flanke 12. the line of defense fichant 60 37 and the line of the gorge L A 16. 97 rod. The moate maye be made broader seing that the angle of the Tenaille Z hindreth that the angle of the flanke T cannot discover the angle flanked F but if the expence vvere not too much which hapneth vvhen one makes the moate very deepe one might to that end cut the part X Y Z and so this inconveniencie vvould be remedied For I finde that such Bulwarks are farre better then those that are made vpon an angle because the gorge is very large the flank the face and the curtaine of a competent measure and according to our former rules to wit the line of Defense fichant is 60 rod or thereabouts the Curtaine 36 the flank 12 the face 24 and the gorge wel nigh 17 rod which is much better and larger then in the Bulwarks that are made vpon some angle So that such Bulwarks ought to be preferred before the others were it not for some other reason which makes one change his minde as for to haue more place or otherwise Novv touching the distance of Bulwarks or a Polygone interiour mentioned of 70 rod vvanting 6 seconds these 6 seconds are of noe great moment neverthelesse one must obserue the dimention as neere as possible maye be Otherwise one might say that the face being 24 the curtaine 36 the flank 12 rod and the angle flanked right that then the Capitall wil be wel nigh 29 the gorge 17 the defense flanking 41. the line of the Poligone interiour or exteriour 70. rod the formed-flanked is 35 degrees and 16 minutes How to fortifie a Hexagone irregular The 21. Plate 95 96. Figure LEt the figure Hexagonall to be fortified be A B C D E F vvhereof the length of euery side maketh as many rod as they are marked out to vvit AB 70 BC. 132 CD 114 DE. 80 EF 124 FA 176 rod. To doe this ye must take notice of the angles thereof and as their greatnesse is ye shall order the angles of the Bulvvarks according to their formes And for as much as the angles A F are but 108 110½ degrees vvhich are the angles of a Pentagone it vvilbe good to make there the angles of demy bulvvarks to haue the angles more open and the angle of the Tenaille more closser and consequently better Vpon the curtaine FA shal be placed tvvo Bulwarks G H of a competent greatnesse to the curtaines proportioning out the Capitall lines the gorge flanks and faces according to the greatnesse of thē as we haue said before in saying If 70 which is the distāce of each angle or the center of the Bulvvarks giues for the Capitall line 28 97. what will the distances of the centers of the bulvvarks giue that vvhich this rule will produce vvilbe the Capitall line and in the like manner shall ye finde the line of the gorge the flank and the face of the Bulvvarks H G the Bulwarke I shal be made in the midst of the curtaine M N. or in
great Mathematician consumed his time wholly in this point by maintayning that all fortifications aswell great as small ought to be made in a square-forme but seing that the Bulwarks towards the 4 angles by this meanes become lesse forcible then the others as the figures A and B demonstrate in the 23 Plate me thinks this ought to be taken into consideration Whither it were not better to make a Fortresse whereof the Bulwarks and the lines of defense be of a like force then to make them as aboue For it is impossible that one should make a fortresse stronger in one place but ye must diminish the strength thereof in an other place to wit as the common proverb is one ought to cutt his coate according to his cloath But when the situation of the place and the avenues thereof be such that one maye be assured of the resistance which maye be made better here then there reason then requireth that in such a case one must make such an avenue stronger by diminishing the strength of an other which is not so subject to be attempted as the former is For the site of a place sometimes will require this irregularity But when as it falls out in a plaine field it is reason that the strength thereof be also regular so then in such a case one cannot take any advantage in one place more then a◠other without hurting and weakning of an other which one ought maturely to consider and not yeeld easely therevnto vvithout good and pregnant reasons And seing these Countries which lye lowe and are subject to invndations overflowings the rivers are commonly bounded in vvith Banks and Dikes for the preventing of such inconveniencies and the preservation of the Inhabitants from an vtter subversion These dikes coming to touch both the one and the other part of the said tovvnes as here in the points H A. vvhich sometimes are separated from the said towne by a vvall vvhich goes from A to N. and from M to H vvhich in the Figure C is called a Doudan made in the forme of an Asses back narrow in the midst and bending downwards on both sides vvhich is made ouer a moate to stop the vvater vvhich othervvise vvould breake into the land is called by those of these countries a Beer that is a Beare in regard of the strength vvhich makes it almost inviolable Therefore the Basis or foundation of such a worke is layd first with a grate of beames of timber locked one into an other with squared beames bound fast together vpon vvhich the vvall is built these beames and piles vvhich are driuen in and layd in this groundvvorke are some 8 10 or 12 foote long according to the depth of the riuer and about 7 8 9 or 10 ynches thick lying tvvo or three foote distance one from an other vvhich also ought to be in length answerable to the depth of the vvater Sometimes these Beares are made vvholly of timber and are lyned vvith huge thick oaken plancks betvveene them closse together and are much longer then the former because the vpper ends must stand of much aboue the vvater as ye thinke the vvater can rise in winter and on both sides of these piles you shall laye tvvo great beames or bands of timber the one at the endes of the piles and the other in the midst betvveene the bottome and the vpper end of them fastned together vvith yron bolts vvhich are as thick as these piles and passing through these piles and then ye shall line them with good stronge oaken boords in joyning them as closse together as possible maye bee that they maye keepe out the water the better and last the longer Now if ye resolue to make noe such separation but to let the banck or dike stand then it wil be good to cutt and pare it as narrow as you can to keepe an Ennemy from coming vpō it with many men in front or to hinder his approches the better on that side I am of the opinion also that ye ought to make the line C B. F G. about 300 foote longe that one might giue the more fire vpon an Ennemy both at his falling on and going off but when there is noe fortification made on the other side of the riuer it were much better that the dike did not stand against the point A but that it were made neerer to the inside of the towne that one might the better offend the approches made on the outside thereof to wit towards the river as is showne in the former Figure and 22 Plate but if the other side of the riner lying opposite to the towne ought to be fortified for the reasons abouesaid then it matters not greatly seing one maye sufficiently offend the Approches on that side and because the cutting off would be chargeable if it be made in a circular forme me thinks the best course is to draw the right line O P and the two others O Y and P. K so that O Y and P. K. maye be of the length of one of the sides of an Octogone or thereabouts that the Bulwarks O P. maye be well defended from the curtaines and that the Bulwarke also on the other side maye likewise helpe to defend the curtaines then betweene O and P. according to their distance maye be made the Bulwarks Q. and R but seing the distances E F. and B D. are too longe to be defended from the Bulwarks D E. ye must make the two Bulwarks S T. whereof the faces are 16 rod the flanks 8 rod the faces of the other Bulwarks are each of them 20 rod or thereabouts which distance is capable to lodge men enough in it to defend it and if neede requires to make also therein some speciall cuttings off as wee shall declare vnto you hereafter Note that if the lines D B F E be too short to make the Bulwarks vpon them marked S T it is apparant ye maye then lengthen the sides so farre that the said Bulwarks maye with conveniency be made vpon them then ye must drawe a line paralell to D E. but if the distance D B. and E F. be so that the angles B F maye be defended well from the Bulwarks E and D. as then you neede not make the said Bulwarks S. and T seing that from the others namely E D they maye be sufficiently defended And for asmuch as I finde these fortifications to be best which come neerest to the demensions giuen before in our regular fortifications termed Royall whereof the faces flankes defences and gorges which are the principall parts of a fortification are all capable to worke well their effects one ought to haue a speciall care aboue all things to fitt the sides of places to be fortified that they maye be almost of the length of Polygones which ye shall finde in the table of our demensions described hereafter euen as wee haue done here in lengthning the sides
if the cutting off be in the tenaille of the angles t. l. Figure 156 b 159 of an easie accesse being raised as litle as possibly may be whereof the one maye serue for an entrance the other for a comming out choosing the one or the other for the most commodious according to the situation of the place And seing the cuttings off as we haue said before are esteemed the best which haue two angles flanking ye must at the first make the cutting off h k l m n i Figure 159 to be raised euen with the height of the rampart or according to the height of the Bulwarks if they be a litle lower then the curtaine if that the batteries doe not commaund them For in such a case you must raise it much higher that from thence with the more vivacity and courage ye maye repulse the Assaillants But seing this cutting off is of a better defense and is much more labour then the cutting off a b c ye must consider well it time will giue you leaue to doe it if not ye maye make vse of the cutting off a b c for oftentimes necessity hath noe lawe And seing experience hath taught vs too much what difficulties one shall meete with all in such cutting off the onely way is in my minde to hinder as much as possibly maye be the Ennemies descent into the moate which besides other inventions that are in vse maye be done by the meanes of the cuttings off at the lines a b c d Figure 159 which are made here right opposite to the angles of the Bulwarkes from one part to an other besett with small muskett baskets filled with Earth as we haue said before Of Casemates The 37 Plate 157 158 Figures FOr asmuch as we haue seene the difficulties which Casemates haue caused to the besieged and the small benefit they haue receiued by them which not withstanding haue bene made with all the industrie that possibly might be thereby to hinder an Ennemie from putting ouer a moate and makeing his batteries vpon the brinke of the moate to beate downe the flanks and to dismount the peeces of ordinance planted in the said Casemates to wit that besides the expence the gorges are made by this meanes lesser the Orillon or pillow being noe more then the 2 3 of the flanck and is of litle resistance and on the other side giues but litle advantage being soone stopt as we haue seene in time past I was minded not to haue spoken of them at all though I esteeme them good if they were made in such a sort that there mouths might not be stopped vp and the peeces within them dismounted which hath not bene done hitherto to my remembrance Forâ—fâ—hese Casemates be made of brick batts when the Besiegers shall play vpon them with there ordinance the bricks flying into the port-holes will doe more hurt to the Canoniers and other men then the Ennemies bullets themselues and by this meanes the portholes wil be easely filled and stopped vp as wee haue seene in diverse places If your Casemate be made of Earth ye must giue it a great Talude that is much slooping which maketh the gorge so narrow namely in those Fortresses which are vnder an Hexagone that oftentimes there wil be hardly any entrance into the bulwark which we call the gullet the Orillon and the flanke very litle and consequently wil be of litle resistance wherein they finde so many difficulties that many great Captaines haue resolued wholly to leaue them vnmade If one could not preserue them otherwise then they haue done to this present I should approue of them But seing I cannot resolue of a thing which I dare not wholly approue off because experience and many men slaine in the warres haue not found it good this is my opinion also In the Figures 157 158 of the Plate 37 a b is the vtmost end of the shoulder the double of p a Figure 157 the mouth or port-hole of the Casemate and as b p maketh 150 foote so p a will make 50 foote p t equall to p a wil be likewise 50 foote t v is 36 foote from a ye shall draw a line to u that ye maye the better discouver the exteriour brinke of the moate and the said t u shall containe three portholes for three peeces of Canon which shal be vauted ouer from d e to t u with steps as the Figure 158 demonstrats marked betweene t v and e d in such sort that the first Vault on the side of e d is closse by the superficies of the water enlarging or raising the said Vaults more and more vntill that the last vault towards t v be raised aboue the superficies l f g k which is the plateforme of the Casemate some 3 foote or thereabouts and seing that t e d v are about 20 foote ye shall advance as farre as possibly ye can the parapet t f and g v as much as the canon and the place will permit you to giue the better soliditie firmenesse to the parapet of the Casemates Then vpon the topp of the basis f g e d the said parapet shal be so raised that the inside f g shal be lined with a wall that it maye prevent the falling downe of such a heigth so that the ennemie maye not discover the vpper part of your vault h i k l and firmely joyned asvvel to the wall as to the Orillon and on the outside with hard Earth as strong as possibly maye going downe sloopingly that it maye not be subject to tumble downe into the moate and this will hinder the Ennemie greatly from entring into the moate and putting his gallerie ouer For by this meanes they must be driuen first to beate downe the shoulder and make it to fall into the moate at the space p a e d which for this reason must be made as deepe as possibly ye can to the end that the portholes 3 4 5 be not easely stopt the place l k f g is about 20 foote vncovered and the vault i h l k also 20 foote broade The line h i is about some 54 foote the Colomne is made in the midst of l k to make the vaults crosswise because the distance betweene l k is too great to make there a single vault which maye serue to make vpon it the parapet of the superiour place to gayne more place for the gorge and to lodge the Canoniers and there amunition dry The entrance into the Casemate must be in that place where m n is vnder the rampart and must be vaulted from m to i from n to o being some 10 or 12 foote broad or thereabouts that ye maye the better draw in your ordinance all what ye haue neede of into the Casemate and make it as high as necessity requires The orillon a v y b is al together massie that it maye giue the greater resistance And because ye maye the
part BI ergo the rest ID is the second flanke The triangle KBI hath three termes the angle found BI K the fourth part of the flanck by consequence the 3 termes remayning are knowne and the difference of the found lines KI IF wil be for the face KF which makes that in the triangle right-angle KGF wil be knowne KG GF and if to KG doubled ye add the curtaine ye shall haue KL also KG to the curtaine is for GL or XB if to GF ye add FB ye shall haue LX and therefore in the triangle right-angle LB wil be knowne seing that LX XB are found Moreouer in the triangle right-angle LXE the angle E is a demy Polygone and the line LX being knowne ye shall haue LE the Capitall line and XE taken from XD equall to KG there will remayne DE the Gorge BI 25 4151 FI 26 9617 ID 4 5848 KI 51 2795 KF 24 3178 KG 22 9229 GF 8 1175 KL 75 8459 XB 52 9233 LX 17 1175 LB 55 6227 LE 21 1583 XE 12 4366 DE 10 4863 The 9. Figure vvhich is the 8 9. 10. 3 Plate In the former figure Pentagonall there are other Hypoteses BL fichant 60 rodd the angle DLM 36 degrees and 45 minutes DM 17 rodd and the angle OLE parted in the midst by LM THe angle MLE is 54 from which take DLM 36 45 there will remaine 17 15 for DLE its quadruple or fourefould for the angle flancked 69 ye must first calculate the traingles right-angles MDL QEL whereby is found DE the Gorge LE the Capitall line and ML in the triangle right-angle LGB the line of defence LB 600 and GB 17 therefore ye shall finde GL which added to KG equall to ML ye shall haue 〈…〉 take ML from it there will remaine GM for the curtaine BD in the triangle MLO ye shall find the face OL MO which taken from 17 there remaines OD for the flanck the triangle right-angle ODC may be knowne because the angle C is equall to MLO 19 1 2 then CO CD being notified ye shall find CL BC and from the Center of the Fortresse wil be knowne the distances towards K P A. DE 10 41454 LE 21 01319 ML 22 76572 GL 57 54128 KL 80 30700 BD 34 77556 OL 24 15101 MO 8 06179 OD 8 93821 OC 26 77657 CD 25 24075 CL 50 92758 CB 9 53481 A †47 30000 The 10. Figure vvhich is the 8 9. 10. In the figure Pentagonall let LB fichant be 60 rodd and LC the defence flancking 50 92758 fifth-parts Also BC the second flancke 9 53481 and the angle OLE parted in the midst by LD THe triangle BCL hauing the three sides giuen then ye shall finde the angle C or the lengthning of the basis BC unto the perpendicular X as followeth here the basis BC 9 53481 giueth me the Summe of both the other sides 110 92758 how much then will their difference 9 07242 giue it will come to 105 54816 from which take the basis because that the perpendicular falls without which is seene when the sayd quotient is more then the basis there will remaine one number whereof the halfe is CX 48 00668 this being done ye must calculate the triangle right-angle CLX by which ye shall know the angle C or LMO which taken from LME the remainder wil be OLE 34 1 2 and the angle flancked wil be the double of it 69 also ye shall haue LX or MD and then shall ye calculate the triangle MDL for the angle MLD is knowne seing it is equall to MLO 19 1 2 and OLD 17 1 2 Also the triangle LQE whereby ye shall finde the Gorge the Capitall line CD then by the triangle ODC is found the flanke after that the face if to CX aboue sayd ye put BC being giuen there will come out BX or GL to which add ML by addition then by substraction ye shall haue KL and the curtaine BD whereof the numbers agree with those of the precedent question The 11. Figure 3. Plate In this figure Hexagone that is a Fortresse with sixe angles or Bulwarks let the Curtaine be sesquialtere or halfe as much againe to the face the face to the flancke doubled-sesquialtere the angle flancked 75 degrees and DP distant from the points 70 rodd Wee must finde out the rest SEsquialtere or one and a halfe is as 3 in respect of 2 twise sesquialtere is as 5 to 2 so that BH DC CB wil be as 15 10 4. whereunto if yee add 4 cifphers to each of them DC wil be 100000 sines of the right-angle Now ADF is 60 and ADC 37 1 2 then CDF wil be 22 1 2 degrees therefore CF 38268 and DF 92388 also KP but BH or FK is 150000 then DB wil be 334776 parts which make 70 rodd by the Hypoteses wee will bring then these lines into Rodds according to this reason following because 334776 parts make 70 Rodd How many partes they will come to 150000 31 364 BH 100000 20 909 DC 40000 8 364 CB 38268 8 002 CE 92388 19 318 DF And seing that PD DF are knowne FP wil be also and by FC CB wil be knowne FB or AE 16. 366 now by the 47 pro. 1 of Euclide BP fichant wil be 53 259 And if one calculates the triangles CBG DEA ye shall finde DG 42 765 and DE 9 448 ergo EF or AB the Gorge wil be 9 87 the distance from the Center to D is equall to DP 70 rodd The 12. Figure 4. Plate Let DP be 72 rod the angles of the Bulwarks are 72 degrees the curtaine BH 32 rod the flancks of 8 3638 fourth of rodds how many then wil be the other parts of this Fortresse Hexagonall Seing that DP is 72 BH or FK 32 then the halfe of the rest wil be 20 for DF and also the angle ABF being 60 degrees and ADC 37 1 2 then CDF wil be 22 1 2 degrees whereby ye shall know then the triangle right-angle CDF Secondly FC found with CB giuen will make knowne FB but PF is 52 then the triangle right-angle BFP hath the sides BF FP knowne BP wil be knowne in the triangles right-angles ADE CBG their termes will suffice to make the rest to be understood when ye haue DA its halfe wil be for DE which taken from DF there will remaine AB DG 21 64780 FB 16 64800 BP 54 60000 AD 19 22344 DE 9 61172 AB 10 38828 GH 11 80804 DG 43 50349 The 13. Figure In this Figure Hexagonall the second flanke is in the flanke as 6 to 7 the flanke hath its gorge as 7 to 10 the Gorge to the line of the Polygone as 2 to 9. the question is how many they the other dimensions wil be when as DH makes 60 rodd IF ye make GH 6 then HO wil be 7 HI 10 the Curtaine wil be 25 Then ye shall calculate the triangles CBG DAG by which ye haue the angle G or