THE ART OF FORTIFICATION, OR ARCHITECTURE MILITAIRE aswell Offensive as Defensive, compiled & set forth, By SAMUEL MAROLOIS Reviewed, Augmented and Corrected by ALBERT GIRARD Mathematician: & Translated out of French into English. by HENRY HEXAM.  
PRINTED At Amsterdam, for M. JOHN JOHNSON. Anno 1638.    

TO The right honourable, & truly noble, Sr. HENRY VANE KNIGHT, MASTER CONTROULER OF THE KING'S HOUSEHOLD, & one of his Majesty's most honourable privy Counsellors of State.  
WHen your honour was last here in the Netherlands Lord Ambassador extraordinary for his Ma.tie of great Brittan●e; with the Lords the State's General of the United Provinces: It pleased your honour, to employ your servant in collecting, and abbreviating some military Abstracts. Since which time, in regard of my long service, I have gained more experience, and studied my profession some what better. To this end, for the good of my Country, and Nation (for every man by a natural inclination, aught to add some thing thereunto) I have of late undertaken, and written a treatise entitled, The Principles of the Art Militaire, practised in the wars of the United Netherlands, represented by figure, the word of Command, an demonstration, digesting the same into three parts, whereof the first, shows the duty, and office of a Captain, his inferior officers, the true form observed amongst us in the exercising of a Company, the Ordering of a Regiment, and our laws, and articles of marshal discipline. The second part treats of the several duties of the officers of the field, and the divers forms of ranging horse and foot in battle-ray, shown by Prince Maurice of famous memory, & his brother his highness the Prince of Orange our General that now is, at sundry times, and in divers places. The third part, handles the place & office of the Master (or General) of the Ordinance and all the officers under his train and Command, together with the Ordinance, Munition, Engines, Instruments, Materials, and all necessary preparations, and Equipage, belonging to an Army & other officers depending on the war. But yet (me thinks) these three parts, are defective and incompleate, unless a fourth be added thereunto, which is the excellent art of Fortification. To this effect, I have translated into English out of French, the works of two famous Mathematicians of our modern times, namely Mr. Samuel Marolois, and Mr. Albert Girard, who have written at large upon this subject. As for exemple, what valour, and this art have done, witness the famous siege of Ostend, for by the means hereof never was town so bravely disputed, never Fortress (in the memory of man) so courageously defended, having to do with so brave an Enemy, who was faint during the time of three years, three months, and odd day's Siege, to get this town (as it were) by ynchmeale: what should I say by ynchmeale? when they were driven to such nonpluses, that they could not get a straw's breadth of us, but held up many days, and weeks (finding such a resistance) their approaches, before they could advance them any further: yea which is more, when they had approached, sapped, and mined into the very bowels of some Bulwarks, blowing them up with the violence of powder, steeple-high into the air, each Enemy than strove at push of the pike, who should become master of that fallen Earth, to turn it up one against an other. And at last being master of the deformed karkeiss of a torn Bulwark or two, found presently a new retrenchment, or cutting of cast up before them; being then forced to sapp forward again, and make new works, and mines, which lasted so long, and was so well disputed, till they got the half of the town, before they had the whole. This men, valour, & fortification (by the help of Almighty God) can do. Again, this was one of Prince Maurice (of blessed memory) his master pieces, for by this art, he was so wary, and careful of the lives of his men, that he choose rather, to spend the States a hundred pound, in casting up a trench, a Sap, and a work, rather than to hazard & lose the life of a man, so good a Menagier was he for the preserving of his men. This Art then, being of such singular use, for the making of all manner of Fortresses, and works both Regular, and irregular, even from a quadrangle which is a fortress with four points or Bulwarks, to a Dodecon, which is a town or a fortress with twelve Bastions, for the preservation of men in a War defensive, comes in most humble wise, to crave your honours gracious patronage, that under your favour, and protection, it may come to the view of the world, an may be helpful to such, as are desirous and ingenious to study this noble art, which if it be acceptable unto your honour, than I shall think my time well spent, and acknowledge myself much bound unto your honour, and shallbe obliged not only to pray unto the Almighty, to preserve you and yours in health; but also to crown you hereafter with everlasting felicity, resting▪  
jour honours humble, and devoted servant HENRY HEXHAM.   

THE FIRST PART OF REGULAR FORTIFICATION, DEMONSTRATED BY THE DECEASED SAMVEL MAROLOIS, And Reviewed, corrected & augmented BY ALBERT GIRARD, SANMIELOIS.  
GOd having found good to create Man, did with all prepare his affection for society, that by this means he might the longer preserve him, and so make him become more strong & able. According to which & to the same end, there were first built houses, and lodgings, to keep men from the injury of the air, as winds, storms, rain, excessive heat, and cold, as the situation of their dwellings, and the commodiousness of those places did afford. And as by divine blessing, and their own industry, they attained unto many Earthly riches, as families, cattle, and other goods: so by the instinct of nature, they sought always to strengthen & defend themselves with the best fortifications they could, that thereby they might not only preserve their bodies against the violence of their Adversaires; but also their goods, kindred, and Associates. To this end (as we may observe as well in ancient histories, as also at this day in some East and West Indian Nations) they drove in piles of timber, and made Impalements close one to an other, that enclossing themselves so narrowly together, they might be the better able to defend and resist the fury of their Enemies. But in process of time, the defects of such enclossure being well marked, they began afterwards to make their Fortifications of a more solid, firm, and durable matter, as stone, brick, earth and such like. And though experience taught them, that these their Impalements of wood, were not able to ●esist fire, & divers other accidents, which might befall them: yet it is certain, they used these and the like fortifications, till they found the harm & conveniency, which they received by them. But necessity, the Inventrix of Arts, found that walls built with stones and bricks were better. These were in use and train, till such time also, as their imperfections were known, and that by the force of man, and Engines they were divers ways attempted; yea many times beaten down & overthrown: which made them seek from time to time to remedy their defects: for perceiving that their walls were heretofore subject to be beaten down (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 down, and to preserve them the better from a breach, and tumbling down, they gave 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 divers manners: first in a round form, 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 closer together, in regard the circle exterior was greater than the interior, and that the Engines could not unfasten, loose and break them but with great difficulty. So on the other side, it was impossible to defend such walls, when approaches was made unto them; because no part of them could be discouvered, or flanked: so that afterward they built them in a square form, with small squares in their angles, for the defence of their Curtains. Then made them also Demycircles with angles interior, and exterior, 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 themselves by discovering every part of the wall, even to the very foundation. And as invention of attempting is grown to the highest degree in these days, by reason of the long experience of the wars in these parts (which is the School of all military actions) and the ability, and capacity of the Assaillants & Defendants: So am I of the opinion, that the Fortifications made in these Low Countries, are the strongest, exactest, & perfectest, which can be invented, and which have been made and practised, not by a simple General, but by one of the greatest Captains of the World, endowed with a singular courage and spirit, an excellent Mathematician, and is not only a Prince of a great house; but also experienced & beaten in all militaire actions, and stratagems; aswell offensive, as defensive: of which Fortification, we have undertaken to treat 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 daily use of arms grown so common: it were in vain for me (in my opinion) to make any further explication thereof: yet to satisfy the ignorant, we will mark out the angles, and sides of a Fortress, by the Letters of the Alphabeth, and opposite to the said letters, ye shall find 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 flank,  
B. N. The Capital line.  
B. C. Q. R. The Moat.  
P. The Raveline, or half moan.  
Q. S. The covert way.  
T. S. The Parapett thereof.  
B. I. The line of defence.  
D. K. The Curtain.  
K. F. The Parapett.  
K. M. The Rampart.  
A. N. The semy-diameter.  
V. C. The flank lengthened.  
C. N. D. The angle forming the flank.  
B. C. D. The angle of the shoulder.    

Orthography, or the Profile.  

2. Figure.  
A. B. The foot or basis of the Rampart.  
G. H. The height 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 foot of the Parapett.  
Z. E. The Parapett itself.  
D. F. The foote-banke.  
F. G. The Terra-plaine or breadth of the Rampart.  
K. A. The way for the round, or the falsebray.  
I. K. The foot bank thereof.  
I. M. The Parapett of the falsebray.  
M. N. The Scharfe.  
P. O. N. M. The moat.  
P. O. The Counterscharfe.  
P. Q. The Covert way.  
R. Q. The foot bank thereof.  
T. S. R. The Parapett of the Covert way.   
The other names, which have need of explanation shallbe declared in their due places.  
Before we come to instruct you particularly in the Art of Fortification; we will briefly treat of the calculation thereof: In which supputation, ye shall have first set down the known terms, and under them the disposition of the Characters or Letters: beginning with a square fortress with four angles, or Bulwarks, and proceed on to a Dodecagone, a fortress with twelve angles, or Bulwarks, making upon every Polygone, 3 or 4 Trials, that afterwards one may choose the best of them: and because the angles will not be much altered by the diversity of the trials: I have thought good to give this general rule for them following.  
It is a thing generally received of all men, that a square fortress with four Bulwarks, is not so good as a Pentagonal with five angles, nor a Pentagonal so strong as an Exagonall with six, and so consequently of the rest. If the cause thereof be sought out; one may observe that this proceeds from the smallness of their angles; as not being able to bear such a body of a Bastion, as the subsequent Polygones: so that a square fortress for this reason, willbe more defective, than the Pentagonal, and this less defensive than the hexagonal, and so well the rest following even to a Dodecagone, which hath the angle of the Bastion right, which is the cause, that constrains one to make the angel's flanked lesser, than the reason of building well doth require, that is, the flanks too little, the Gorge too narrow, and the line of defence too long. To increase then proportionally the angles of Fortresses, according as the angle of their Polygone augmenteth; we will take then the half of their angles, and adding thereunto 15 degrees, the sum willbe the angle of the Bulwark, which we term the angle-flanked, and if the angle-flanked be substracted from the angle of the Polygone, there will remain the double of the angle flanking interior, which double being substracted from 180 degrees, `then will remain the angle flanking exterior, called the Tenaille, and if ye add to the angle flanking interior 90 degrees, than the sum will be the angle of the Shoulder.  
To find out the angle of the Polygone from the number of its subtract, namely 2, the remainder must be multiplied by 2. and the product will be the number of the right angles, which such a Polygone containeth, as ye may see by this exemple following.  
Or thus.  
And by the same rule ye shall find the angles of the Subsequent Polygones, beginning from a square Fortress to a Dodecagone.  
4. 5. 6. 7. 8. 9 10. 11. 12 90. 72. 60. 51 3/7. 45. 40. 36. 32 8/●●. 30 the angl. of the centre. 90. 108. 120. 128 4/7. 135. 140. 144. 147 ●/11. 150 the angle of the Polig. 45. 54. 60. 64 2/7. 67 1/2. 70. 72. 73 7/11. 75 The half. 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. Remains 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 flank interior. 150. 141. 135. 130 5/7. 127 1/2. 125. 123. 121 4/11. 120 The flank exterior. 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 interior. 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 seeing the angle flanked of a Dodecagone is right, and able to resist a battery, which is also made always with right angles to shake the more the face of the Bulwark; one must fortify the Polygones, which are above the Dodecagone, with a right angle, that the line of defence may come the more into the curtain, that one may give 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 shallbe made hereafter.  
Some times we augment, aswell the angles of the Bulwarks, as the Octogone, a Fortress with 8 angles or Bulwarks, with a right angle, and those above are always right, & those under diminishing to the square fortress (which hath the angle of its Bulwark only of 60 degrees.) According to which the Bulwarks are somewhat larger, and the gorges and flank greater than the former, but the second flanks lesser. Now for the finding out of every angle, ye must do this following, where ye may observe, that in the manner above said, the angles flanking interior are the fourth part of the angle flanked, or the 1/● of the angle of the Polygone.  
four 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 centre added thereunto. 150. 144. 140. 137 1/7. 135. the angle Flanking exterior. 30. 36. 40. 42 ●/7. 45. the angle double of the angl. flank. interior. 15. 18. 20. 21 3/7. 22 ●/2. the angle Flanking interior, 90. 90. 90. 90. 90. the angle which is the flank always 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 Fortress of ten Bulwarks, where ye must note also, that before we proceed further, we will make use in the supputation following of the tenths or decinall numbers; which though it gives some imperfection: yet seeing 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 therefore to make use of these following.   

THE I. QUESTION. The 1. Figure, & 1. Plate.  
Let there be made upon a Square a fortification of four Bulwarks, so that the line of the Gorge be 7 parts: DI the curtain 21, and IF, which is the flank of 5, and from the angle of the flank is drawn the line of defence by the angle of the Shoulder, to give it a face. The question is how many the angles will be, and every line of the same fortress, when as the line of defence will take up 600 feet. Now the length of a foot is set down in the 25 model of Geometry, noted by 1, and is divided into 12 ynches, & the inch into 10 equal parts, and is the same foot, whereof 12 makes a rod, which his Excel, useth in all his Fortifications.   

ALBERT GIRARD.  
THe Author hath here above so disposed his calculations, that in stead of explaining them briefly, he confounds them teadiously, as if heretofore there had been no certain rule set down in writing, to calculate lines and angles as is ordinarily done by the Trigometrie of plain Triangles, although there have been a number of Authors, which have treated of them, th' one after one manner, th' other after an other; and the most part of them commixed with long discourses, which moved me not long since, to put into light some tables of sins in a portable volume, with the most succinct method that possibly could, touching the supputation of such plain triangles, reducing them into four divers cases, where I did insert in their places, some of my own inventions to the purpose, known as I suppose to none others heretofore; so that it cannot be, but having 3 known terms, in a triangle, but that one may know the other three, or one of them only, which one desires, as the Reader may know it; the manner & order thereof being much more facile, & easier to conceive, than the reading of our Author in his former editions, being obscure, troublesome, and hard to be attained unto. For this reason the Learners of this science, are required to be forewarned before they come to the reading of this book; & are advertized hereby, that when it is said, a triangle hath three terms, he must understand known, or given, and that when I say that a triangle rightangle hath three terms, that then I am bound but to show two of them; seeing that this word (rightangle) presupposeth that the triangle hath a right angle, to weet of 90 degrees: moreover, that touching this present question, these 7, 21, 5 parts, demonstrate the reason of the lines ND, DIEGO, IF, and not the quantity of the same in feet, as is necessary to find in the manner following.   

To find out the Angleses.  
IN the triangle right angle CDI, the side CD, is to DIEGO, as 5 to 21, (for CD being of 5 parts; then DIEGO willbe the 21 thereof) therefore the angle DIC, willbe found for 13: 24 it's double CLD 26: 48, and its adjunct CLF flanking exterior, willbe of 153: 12. Now in the figure quadrilatere, or of foure-sides, ABLH; the angle exterior BLH is always equal to the 3 interiours B, A & H; now A 90, the angle of the centre, being taken from BLH, 153: 12, there will remain the two Demy-flankes ABL, & LHA 63: 12. for one entire flank ZBC: & seeing that the angle of the Shoulder BCD, is exterior in the triangle CDI, it willbe equal to the two interiours D. 90. and 1. 13; 24, and therefore the Shoulder C, willbe 103. 24.   

To find out the Lengths.  
IN a triangle ambligone BNI, the side BY is 600 feet, & the angles are found, therefore the other terms shall be found, to wit BN the Capital line 196, 64, & NI, 444, 62, but ND is the third of DIEGO (for ND being 7, DIEGO is 21 by the Hypotheses) therefore the fourth part of the found NI, willbe ND, 111, 15 the Gorge, & the rest willbe 333, 46, for DIEGO, the Curtain, now DIEGO to DC is as 21, to 5: therefore one may say, if 21 gives me 5, how many DEVIL 333, 46, give me? facit for CD 79, 40. & so CI willbe 343, 84, which taken from BY 600, there will remain BC 257, 16, for the face.  
Finally, the triangle rightangle VDH hath three terms, DH 600, and the angle H equal to HDI. 13: 24 (because of the parallels DIEGO; VH) than you shall know ud 139, 05, & VH 583, 67, from which side subtract UP (equal to Demy DIEGO) 166, 73, there will remain PH 416, 94, whereof the double willbe for BH 833, 88, and PA willbe 416, 94, for it is equal to PH, and thus must ye do with the rest following.   

The 3. Figure, 1. Plate.  
Let there be a square Fortress, whereof the curtain DIEGO being 4 parts; the flank CD shall have 1 of them, and also the Gorge 1: The defence running from the angle of the flank, formeth the face, & the length BY is of 600 feet, one requires the greatness of the other lines of the same Fortress.   

To find out the Angleses.  
THe triangle rightangle CDI hath three terms, as the reason of the sides CD, 1, to DI, 4, therefore the flancking interior CDI willbe 14 degrees & 2 minutes, whereunto add D 90 degrees, than the Shoulder BCD willbe 104: 2; now if ye double the angle CID, it willbe 28: 4, CMD, and its adjunct CMF, 151: 56 for the angle flancking exterior, from which subtract the angle of the centre A 90: there will remain 61: 56 for the angle flanked entire B, and thus much for the angles.   

For the sides.  
THe triangle BNI, hath three terms, to wit, BY 600 feet, and the angles B, 30: 58. & N, 135. ay, 14. 2, so one shall find the lines BN 205, 76 & NI 436, 59; but in regard that ND to DIEGO is as 1, to 4, according to the Hypotese, than ND will be the fifth of NI, where will follow that ND willbe 87, 32 (and DC as much) and DIEGO the rest 349, 27. Now seeing that in the triangle rightangle CDI, the sides CD, DIEGO are notified, one shall find the Hypotenuse CI of 360, which taken from BY 600. there remains the face BC 240. Finally, in the triangle rightangle VDH, which hath 3 terms DH 600, the angle H, equal to CID 14.2, ye shall have ud 145, 49: also VH 582. 09, from which take UP 174, 64. (equal to the demy-curtaine DIEGO) there will remain PH 407, 45, also PA its equal: then in the triangle rightangle APH, ye shall find AH 576, 22 from whence take BN or HO, the capital line 205, 76, there will remain AO 370, 46, & in takeing ud, from AP, there will remain the perpendicular of the Centre, A upon the midst of the curtain 261, 96. Now BH willbe 114, 90. as being the double of PH.   

4. Figure. & 1. Plate.  
In this square Fortress, the defence BY is 600 feet, 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 Fortress willbe?   

For the angles.  
seeing that the line flanked is 60, then in the triangle BNI the angle B willbe 30, & N 135, (the adjunct of ANY 45:) whereof the angle remaining BEEN willbe 15 degrees: therefore the triangle BDI willbe Isosceles, that is, two sides a like, such that DBI is aswell 15; as BID; The Shoulder C willbe 105, and in the triangle DMI, the angles upon the basis are each of them 15 degrees, the remainder M will be then 150 for the flanking exterior BMH.   

For the sides.  
THe triangle Isoceles BDI hath three terms, the defence BY 600 feet, and the pointed angles every of them 15 degrees. Therefore DIEGO the Curtain willbe 310, 584. Also the triangle CDI, hath three terms, the angle D right; I 15. and the curtain DIEGO, than CD the flank 83, 217, & CI 321, 535, which being taken from BY 600, there will remain BC the face 278, 465. Moreover in the triangle IBN, the angles B 30, I, 15, & the defence 600: so than BN the Capital line willbe 219, 623, & NI 424. 268. from whence DIEGO being taken there will remain the gorge ND, 113. 684. Moreover, the triangle BILL having three terms, L right, B equal to CID 15. and BY 600, then I'll willbe 155, 292, & BL 579, 558, and takeing from it PL 155. 292. (which is the half of the curtain) there will remain BP 424, 266 for AP; also it's double being BH 848, 532, one may also easily know BASILIUS 600. The 5. Figure. & 2. Plate.   

ALBERT GIRARD.  
Heitherto of the defence drawn from the angle of the flank, but in those following, there is a second flank, and the distance of the angle of the flank, even to the angle flanked, which is called Fichant.  
The 5 figure is the design or draught of a square Fortress, whereof the line of defence fichant DH maketh 600 feet, the angle flanked 60 degrees, the line HB is divided into 7 equal parts; whereof the one which is between the Characters 1. 2. is subdivided into 5 equal parts, and from the centre B, is made the Arch, 4 N cutting the capital line at N, from which point is drawn 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 Fortress, will make?   

For the lines in parts indetermined.  
FOrasmuch as BH containeth 35 such parts as BN 9; BV 10, VH 25, ye shall have in the triangle rightangle BTN terms enough to know BT, or TN 6, 364, and TV or, ND 3, 636: Also the triangle rightangle DUH, had three terms, ud equal to TN, 6364, & VH 25, than DII willbe 25, 797 parts or the same DH is 600 foot. We must then calculate according to this reason, the lines above mentioned to bring them into feet, saying, as followeth.   

For the lines in parts determined, to wit, brought into feet.  
THe 25, 797 parts, make 600 foot, how many then will BH 35. come to? There willbe for BH 814, 04 foot, likewise VL, or DZ the curtain may be made so; BH 7 parts make 814, 04 foot, how many VL 3? it will come for the curtain to 348, 87, & for BN the Capital line ye shall say, if 35 gives 814, 04: how many 9? ye shall have for BN 209.32: and so of the other: for TN, or BT, or VD willbe 148. 02, & TV, or the Gorge ND 84 57 and seeing that BH is known, also VL the half of the remainder willbe BV 232, 58. In the triangle rightangle BUC, the angle B willbe 15 degrees: (for PBO is 45, & CBO 30:) and BV being known, therefore BC the face willbe 240, 78, and VC 62, 32, which being taken from VD, there will remain CD the flank 85. 7; and therefore the triangle rightangle CDI, shall have 3 terms, the angle C 75, & CD 85, 7. so DIEGO willbe 319, 84, CI 331, 15, likewise BY flancking 571, 93. & I, Z 29. 03 for the second flank. Now OPEN is known as being the half of BH.  
Now of the Pentagons', or Fortresses with five angles or Bulwarks.   

6. Figure. & 2. Plate.  
In the Pentagone, a Fortress with five Bulwarks, KFBDL, let the line, KL be 63 rods, & divided into 7 equal parts, whereof the Capital line KA is 2 of them, also LE, and from the Character 2 (or G) let GB be the perpendicular. Also the angle flanked is 69 degrees, according to the precedent table, how many then willbe the lines and the angles of such a Fortress?   

ALB. GIRARD.  
Before we come to the Construction hereof, take notice, that KA in this figure is not answerable to the length of KG, yet ye must suppose them to be equal, aswell as the cipher 1, and the letter H, are two differing points, which would have been the better discerned, if the figure had been well made, which may serve as a forewarning for some figures following, upon which one ought not to stand so much as upon the Suppositions, or Hypoteses of the Propositions.  
The triangle rightangle KHA hath three terms, KA 18 rod, the angle AKH 54 degrees, which being of KH willbe 10, 5802, & HA or GB 14, 5623. Now if ye take away KH from KG, there will remain for AB the Gorge, 7. 4198. Item the curtain BD willbe 27 rod, being equal to GM. And because the angle flanked is 69 degrees, if ye add thereunto the angle of the Centre, which is 72, ye shall have the angle flancking exterior, 141 degrees, the half of its adjunct for the interior 19 1/2 degrees. The triangle rightangle KGF, having 3 terms, KG 18, GKF 19 1/2: the face KF willbe 19, 0953, and FG 6. 3741, which being taken from GB 14, 5623. there will remain FB (the flank) 〈…〉, than the triangle rightangle FBI may be known, seeing that FB is found, and the angle 〈…〉 1/2 the compliment of 1; therefore BY willbe 23.1227, which taken from BD 27, there will remain ID for the second flank 3, 8773, also FI willbe found to be 24, 5297, whereunto add KF 19 0953, and ye shall have KING, the defence flancking 43. 6250, finally the triangle rightangle LGB, having three terms, to wit LG 45 rod, and GB 14, 5623; one may easily know BL, and the distance of the Centre of the Fortress by K, because that KP is 31 1/2 rod, and the angle AKP 54 degrees.   

The 7. Figure, & 2. Plate.  
In a Fortress Pentagonal, BOV, let AB the defence flancking be 50 rod, the Flank ED 9 rod, and the angle of the Bulwark 72 degrees. How many then will the other parts of the same make, when the second flank AGNOSTUS makes five rod.  
THe angle of the Polygone is 108, the half is 54 for FBQ, to which add CBF 36 degrees by the Hypoteses, you shall have 90 for CBQ, but CF is parallel to BQ therefore BCA right, the triangle BCA willbe then the rightangle, and BASILIUS of 50, and the angle B of 72 degrees, than ye shall find BC, or QD to be of 15, 451, and CA 47, 553, and the angle A 18 degrees. Also in the triangle rightangle EDA, the angle A is given, and the side ED 9 rod, by the Hypotese, then EA willbe 29, 12463, & DA 27, 69912; now if ye take EA from DA, there will remain BE the face 20, 87537, and if ye subtract DA from AC, there will remain CD, or BQ, 19 854. Also the triangle rightangle BCF, hath 3 terms, BC and the angles; in fine BF the capital line willbe 19, 097 & CF, 11, 226, which being taken from CD, there will remain FD 8, 628, the Gorge; now if to DA, ye add AG 5 rod, the second flank, ye shall have the curtain DG 32. 69912, to which add twice BQ, and ye shall have BV 72, 40711 its half BK 36, 20355, and so the triangle rightangle BKO, having 3 terms, the angle KBO willbe 54 degrees, ye shall find BOY 61, 593099, and KO 49, 82984. Finally VB, BQ being known, than the triangle right angle DQV, having 3 terms given VQ, & QD; ye shall find the fichant DV easily.   

ALB. GIRARD.  
IF one will consider the difference which is between this operation, and that of our Author published in his former Editions, he shall find that this question is defective, for I have added thereunto (that the second flank AGNOSTUS is rod) seeing that in all such questions, where there is a second flank, he ought to have set down 5 known terms & neither more nor less, without the one depending on the other, as in this present, the name of the Fortress, to wit, Pentagonal is a term, in the second place there is the defence; then the flank, the angle flanked, and the second flank, which are five terms: where ye must note, that when there are reasons in the proposition, and that though a reason hath two numbers, nevertheless it is but one term; but where there is no second flank (as in the figures of the first plate, than 4 terms will suffice.) Finally, my Author had so ordained his Supputation, that in stead of making an addition, or a substraction; he made the rules of three very great, so that ye must imagine it was easier for me to change all, then to recorrect it, having no other respect, but to the explication of the figures, and as much as is lawful and possible to show his intention, which I am bound to do: Moreover, if peradventure same one should find this manner of operation strange, which I 〈…〉 heitherto, it is requisite for him to know also (be it spoken under ●●●●ection) that he understandeth not much, if he doth not practice it so himself in other Subjects, yea though I had not set down one only number; which I should have done already heretofore, if I had not had regard to the obscurity, which students might pretend to find, which having cleared heitherto, ye may go on with the more assurance in the rest following: and must also know, that the 8 figure was left without explication by the Author.   

The 8. 9 10 Figure. & 3. Plate.  
In this figure Pentagonal, the angle ELO is divided into 2 equal parts by LD, the flank FB 9 rod, the angle flanked 69 degrees, and the curtain 30 rod, 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 known, also the rest shallbe FKG, or its equal FB, whence follows that the triangle rightangle FBI shall have 3 terms FB, and the angles; therefore the other 3 terms willbe known, also the whole line DB, and the part BY, ergo the rest ID is the second flank. The triangle KBI hath three terms, the angle found, BY & K (the fourth part of the flank by consequence the 3 terms remaining are known, and the difference of the found lines KING, IF willbe for the face KF; which makes that in the triangle rightangle KGF willbe known KG. GF; and if to KG doubled, ye add the curtain, ye shall have KL; also KG to the curtain is for GL, or XB if to GF ye add FB, ye shall have LX, and therefore in the triangle rightangle LB willbe known; seeing that LX, XB are found. Moreover, in the triangle rightangle LXE, the angle E is a demi Polygone, and the line LX being known, ye shall have LE the Capital line, and XE taken from XD (equal to KG) there will remain DE the Gorge.  

BY 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, which is the 8, 9 10. & 3 Plate.  
In the former figure Pentagonal, there are other Hypoteses BL, fichant 60 rod, the angle DLM 36 degrees, and 45 minutes, DM 17 rod, and the angle OLE parted in the midst by LM.  
THe angle MLE is 54: from which take DLM 36, 45; there will remain 17, 15; for DLE its quadruple or fourefould for the angle flanked 69: ye must first calculate the traingles rightangles MDL, QEL, whereby is found DE the Gorge, LE the Capital line, and ML in the triangle rightangle LGB; the line of defence LB 600, and GB 17; therefore ye shall find GL, which added to KG equal to ML, ye shall have 〈…〉 take ML, from it, there will remain GM for the curtain BD, in the triangle MLO, ye shall find the face OL, & MOTHER, which taken from 17, there remains ODD for the flank, & the triangle rightangle ODC may be known, because the angle C, is equal to MLO 19 1/2, then CO & CD being notified, ye shall find CL & BC, and from the Centre of the Fortress willbe known 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  
MOTHER 8, 06179  
ODD 8, 93821  
OC 26, 77657  
CD 25, 24075  
CL 50, 92758  
CB 9, 53481  
A (†) 47, 30000    

The 10. Figure, which is the 8, 9 10.  
In the figure Pentagonal, let LB fichant be 60 rod, and LC the defence flancking 50, 92758 fifth-parts. Also BC the second flank 9, 53481, and the angle OLE parted in the midst by LD.  
THe triangle BCL having the three sides given, than ye shall find the angle C, or the lengthening of the basis BC unto the perpendicular X, as followeth here, the basis BC, 9, 53481 giveth me the Sum of both the other sides, 110, 92758, how much then will their difference 9, 07242 give? it will come to 105, 54816, from which take the basis (because that the perpendicular falls without, which is seen, when the said quotient is more than the basis) there will remain one number, whereof the half is CX 48, 00668, this being done ye must calculate the triangle rightangle CLX, by which ye shall know the angle C, or LMO, which taken from LME, the remainder willbe OWL, 34 1/2, and the angle flanked willbe the double of it 69; also ye shall have LX, or MD, and then shall ye calculate the triangle MDL (for the angle MLD is known) seeing it is equal to MLO 19 1/2, and OLD 17 1/2: Also the triangle LQE, whereby ye shall find the Gorge, the Capital line, & CD, then by the triangle ODC is found the flank, & after that the face: & if to CX above said, ye put BC being given, there will come out BX, or GL, to which add ML by addition, then by substraction ye shall have KL, and the curtain BD, whereof the numbers agree with those of the precedent question.   

The 11. Figure. 3. Plate.  
In this figure Hexagone, that is, a Fortress with six angles or Bulwarks, let the Curtain be sesquialtere, or half as much again to the face, & the face to the flank doubled-sesquialtere, the angle flanked 75 degrees, and DP distant from the points 70 rod. We must find out the rest.  
SEsquialtere, or one and a half, is as 3 in respect of 2; twice sesquialtere is as 5 to 2: so that BH, DC, CB willbe as 15, 10, 4. whereunto if ye add 4 cifphers to each of them, DC willbe 100000 sins of the rightangle. Now ADF is 60, and ADC 37 1/2, than CDF willbe 22 1/2 degrees; therefore CF 38268, and DF 92388 also KP; but BH, or FK is 150000, than DB willbe 334776 parts, which make 70 rod by the Hypoteses, we will bring then these lines into Rodds, according to this reason following; because 334776 parts make 70 Rodd.  

How many parts. 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 seeing that PD, DF are known, FP willbe also; and by FC, CB, willbe known FB, or A, 16. 366, now by the 47 pro. 1 of Euclid BP fichant willbe 53, 259: And if one calculates the triangles CBG, DEA, ye shall find DG, 42, 765; and DE, 9, 448, ergo OF, or AB the Gorge willbe 9, 87; the distance from the Centre to D, is equal to DP, 70 rod.   

The 12. Figure. & 4. Plate.  
Let DP be 72 rod, the angles of the Bulwarks are 72 degrees, the curtain BH 32 rod, the flanks of 8, 3638, fourth of rodds, how many then willbe the other parts of this Fortress hexagonal?  
Seeing that DP is 72, & BH, or FK 32, than the half of the rest willbe 20 for DF, and also the angle ABF being 60 degrees, and ADC 37 1/2, than CDF willbe 22 1/2 degrees: whereby ye shall know then the triangle rightangle CDF. Secondly, FC found, with CB given, will make known FB, but PF is 52, than the triangle rightangle BFP hath the sides BF, FP known, & BP willbe known in the triangles rightangles ADE, CBG, their terms will suffice to make the rest to be understood. when ye have DA, its half willbe for DE, which taken from DF, there will remain 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 hexagonal, the second flank is in the flank as 6 to 7, the flank 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 willbe, when as DH makes 60 rod.  
IF ye make GH 6, then HO willbe 7, & HIGH 10, the Curtain willbe 25; Then ye shall calculate the triangles CBG, DAG by which ye have the angle G, or his equal CDF, & the face D● 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 set down more easy, in takeing but 7, 48926, which make 1000000 parts; Now if ye had rather work it by multiplication, without division, than ye must take but 100000 parts, which make 13352 rod, and so ye shall find  

DC 25, 38302  
CB 9, 34672  
BASILIUS 13, 35246  
DP 81, 01484  
BH 33, 38115  
The angle flanked 79. 33.    

The 14. Figure, the 1. of the two.  
In this Fortress hexagonal, let EH be 60 rod, and EI, BC, KH equal, the flank KC 8 rod, it behoves us to find out the rest.  
BY the help of the rule of Algeber, ye shall find that the angle EIF willbe 69 degrees, 4 1/2 minutes; then in the triangle IBC, the angle I, & the side IB 8 rod being given, ye shall find BC 20, 9229 for the face EI; also IC willbe 22, 4002, then EC will make 43, 3231, the rest is easy, without the second flank.   

The 15. Figure.  
This figure hath for the flank 10 rod; the rest being like to the former they shallbe put in order in the Tables following.   

The 16. Figure. & 5. Plate.  
The flank of this Heptagone, that is, A Fortress with seven angles, or Bulwarks, maketh 10 rod, the distance of the points 72 rod, and the angle flanked 80 degrees, how many then will the rest make, when as the second flank is 10 rod.  
seeing that the angle of the Bulwark makes 80 degrees, and the angle of the Polygone 128: 34, 17 the difference divided in the half willbe the angle flanking interior 24, 17, 18 1/2, the flank CB 10 rod, whence the triangle CBG willbe known, & by adding GH given to BG, ye shall have the curtain 32, 1623., which taken from DP 72 rod, and then take the half of the rest, ye shall find DF, and also DC. CF, for the face DC 21, 8525, & DG 46, 1666, also in the triangle DAE; ye shall have DE, & consequently OF, or AB. Moreover ye shall find DH in the triangle DKH, to be 55, 4345.   

The 17. Figure, & 5. Plate.  
This Heptagone hath the flank of 9 rod, the angle flanked 79 degrees 25. 43, DP ●2 rod, and the choice of the second flank.   

ALB. GIRARD.  
Having calculated the angles as in the former. OM willbe 21, 6436, and HIM, 19 6836, let us make the second flank 7, 3164, and then the curtain willbe 26; DF, 23; the face OPEN 25, 2901. MP 46, 9337. Marolois had given here 80 degrees, but took no more than 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 rod, & the curtain 32 rod, how much will the other lines & angles make of this Fortress.  
TO resolve this question, ye must suppute from the triangle CBG, the lines CG, GB, and ye shall have GB the second flank; now if to BG, ye add AB, ye shall have AGNOSTUS, than the triangle DAG willbe known; afterward the triangle DAE, and finally the triangles DCF, and DHK.   

ALB. GIRARD.  
NOte that in this Question Marolois had set down the angle flanked 79 and 2 sevenths, yet did not follow this number, but 79 degrees: the same error also was committed in his supputation of the former, and in the 14 figure. One cannot guess well his Supposition: nevertheless that needs not to stay the reader; for I have set down the question, as it seems he would have propounded it, but the worst is, he made 2 figures, and one cannot understand well of which of either of them he would speak, but we will speak more thereof hereafter; howsoever those which are most intricate (aswell by reason of the faults escaped in the impression of the former Editions, as by the error of his disciples, which did calculate them) shallbe partly omitted, and shall give no impediment, but that the rest may give contentment to the Readers, for that which they are desirous to find out in this book: (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 Capital line, & from the imagined DB (to wit ADB) let be 22 ½ degrees, DP 82 rod, & the flank CB 10 rod.  
FRom the angle of the Polygone 128, 34, 17, take the flanked 80, the half of theremainder willbe 24, 17, 9: for the angle flanking interior DGA: begin then the triangles CBG, DBG. (whereof the angle in D maketh 17 ½) ye shall find BG 22, 1621., CG 24, 3138; & BD 30, 3118, than GD willbe 49, 1096, and DC the face 24, 7958, afterward the triangle BDF, the line BF willbe found to be 20, 1982, and DF 22, 6014, whereof the double taken from DP 82, there will remain for the curtain BH 36, 7972, and consequently for the second flank GH, 14, 6350. Now knowing the triangle DEA, ye shall find the Capital, also the gorge AB 12, 8744. If in the triangle DKH, ye seek DH, than ye shall find 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 Fortress of 8 angles or Bulwarks, hath the distance DP divided into 7 equal 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, than the square of BP willbe 29, and seeing that BP is 60 rod, its square willbe 3600 rod, by which reason ye may know one part of the 7 of DP, saying, if 29 gives me 3600, how many will 1 give me (the square of an other part?) it will come to 124, 1379310345, whereof the square root substracted willbe 11, 14172, and its double 22, 28344 for DF, or FB, the triple is 33, 42516 for the curtain BH, and the seventhfould 77, 99204 for DP. Moreover, the angle flanking interior, or its equal FDC willbe 26, 15; so that DC willbe 24, 84581, also FC 10, 98907, the flank then willbe 11, 29437 (for FB was known) in the triangle rightangle DRG, the angle D is also 26. 15; and RC equal to FB, then DG the line of defence flanking, willbe 50, 38218; finally, the Capital willbe 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 flank 12, & the curtain 36 rod, & the angle flanked, right, it is required how many the other lines & angles willbe?  
The angles ADF 67 ½, & ADC 45, will make known CDF to be 22 ½ Now DC is 24 rod, by consequence DF, FC, DP, FB willbe known; also seeing that FB is found (or EA) ye have also the triangle DEA, by which ye shall find DA, AB, & shall find DG in the triangle CBG: for having 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 flank is 11 rod, the angle flanked 82 ½ degrees, the line DP 76 rod, & the second flank to be chosen, how many then willbe the other parts of this Fortress?  
THe angle of the Octogone is known, and the angle flanked, also their half ADF, ADC, therefore the rest, and the triangle CBG willbe known, and BG willbe found 22, 3058, suppose the second flank GH is 9, 6942, than BH 32 rod, likewise as much for FK; but DP is 76, therefore DF makes 22, and after ye have calculated the triangles DCF, DAE, ye shall find 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 half of the angle flanked.  
AFter ye have carried AC, which was forgotten, if ye count the triangles DAG, ACG, ye shall find the Capital, the face, and the gorge, and BG, than the triangles CBG, DCF, afterward ye shall have CB, CF, DF, and therefore FK, or the curtain, and the second flank, than DH, 58, 3498.  

The Capital 23, 93655  
The Gorge 12, 84323  
The Flank 11, 26362  
The second flank 9, 15324  
The face 24, 53340  
The Curtain 31, 99346    

The 24. Figure.  
Let there be an Enneagone, A fortress with nine Angles or Bulwarks, (that is with 9 sides) whereof the angle flanked is 85 degrees, the face 24 rod, the flank 12, and the curtain 36 rod.  
THe angle of the Polygone is 140 degrees, seek the triangles, DCF, CBG, DEA, DKH, and then ye shall find the lines GH 12, 94824: DP 78, 57648: AB 12, 88709.   

The 25. Figure.  
As in the tables above 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 making of the flank CB, we require the rest.  
DK being 5 parts, and KH 2. the square DH willbe 29. which makes 3600, whereof 1 makes 124, 13793103 for the square of one part, its root substracted willbe 11, 14172 for one part, whereof the double willbe for DF, or FB, and the triple for FK, or the curtain 33, 42516 BH, and after ye have calculated the triangles DFC, CBG, DEA, ye shall find the other lines DC 25, 1219, DG, 48, 25855, GH 12, 9026, BC willbe 10, 68335, the seven parts of the number above said willbe 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 flank as 4 to 3. the rest is required.  
IF ye take away the angle ADC 42 ½ from ADF 70, the angel's GDR will remain, and DG makes 50 rod, than the triangle DGR willbe known, to wit, DR, & RG, or KH 23. 0875, which will make ye know the triangle DKH (for DH is 60) then RK, or GH willbe 11, 0301. Moreover, in the triangle DEA the side EA is equal to KH, by DE & DR, ye shall have AGNOSTUS 35, 94735, then in carrying AC, ye shall seek the angle A from the triangle ABC, setting down AB 4, and BC 3 parts, according to the Hypatese, ye shall find then the angle A of 36 degrees, 52 min. & 12. seconds. Then let us go to the triangle ACG, having AGNOSTUS known, the angle A & G, equal to CDF 27 ½ degrees, to find CG afterward the face, and consequently DF, FK, OF or AB the Gorge, which willbe 14, 72808, therefore if 4 gives 3, how many than AB? ye shall have BC for the flank 11, 04606, & DP 78, 51183, DAMN 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 Fortress non angular?   

ALB. GIRARD.  
THis question is impossible to resolve being exceeding, seeing that there is a condition in it more than one desires, & which is worst, repugnant to the others, for which fault the Author may be excused, seeing that in his time, there were no such advertisements given as we have given thereof in the beginning of the trigonometry, cited in the first question going before, the which though they may seem to be of little consequence to some; yet one must acknowledge, that those that know them shall not fall into the like errors, as these which may be explained thus; There are two reasons given DP to DF, and DF to FB (a reason of equality) two lines of defence the angle flanked, & the name of the figure of nine-side figure, which are six terms, yet one needs but 5 (as ye shall find it noted in the 7 figure, where the question was defective, and of some others afterwards) finally, the proof of this may be seen in making comparaison of this with the 25 figure, where the same question is propounded, and where ye shall find that the defence flanking aught to be 48, 25855, and here he will have it 50, which is absurd, as is said.   

The 28. Figure.  
In this present Decagone, a Fortress with ten angles or Bulwarks, let the angle flanked be 87 degrees, the Gorge in the flank 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, than the imaginall angle BAC willbe 36 degrees, 52 minutes, 12 seconds. Moreover, the angle ADC being 43 ½ then CDF willbe 28 ½, which is an angle of the triangle GDR, and which may be known, seeing that DG is 50 rod, therefore GRACCUS 23, 858, or its equal KH, and for as much as DH is 60, than DK willbe known, and also DR withal, and so ye shall find RK for the second flank: likewise the triangle DEA willbe known, then ED and DR will make known ER, or AGNOSTUS, & consequently ye shall have the triangle ACG for the angle A was found above, and the angle G is 28 ½ degrees, than the face willbe 26, 11334, and having found out DF, ye shall find OF, or the gorge 15, 197: Also FK for the curtain 32, 10821; DP 78, 00619.   

The 29. Figure.  
In this Decagone the angle flanked maked 87 degrees, the defence fichant 60 rod, the flank 12 rod, and the gorge 16 rod, it is required how much the other parts make?  
THe quadrangle ABCD having five terms given, ye shall find the other parts; also the triangle DFC, whereby ye have FB, or KH, & consequently the triangle DKH, and having FD & DK, their sum and difference willbe 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 curtain be 36 rod, and the flank 12, the face 24, and the reason of the Bulwark in the flanking interior, as 58 to 19  
THe half of 58 is 29, than the angles ADG to AGD, or CDF, willbe as 29 to 19, therefore setting down ADF to CDF, it willbe as 48 to 19; but ADF is 72 degrees, than CDF willbe 28 ½ degrees, & then the angle flanked 87 degrees, the rest is easy, for the triangles DCF, CBG will make known 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 flank 12, and the curtain 36, how many will those parts make, when the defence flanking is doubled to the Capital.  
IF DG be set down 2. then DA willbe 1. Now the angle DAG is 108 degrees, than the angle flanking interior G willbe 28. 23. 38, and the angle flanked 87, 12, 44● the rest is easy, & is found in the same manner, as the end of the former, & therefore Marolois leaves its so for this reason.   

The 32. Figure.  
In this present Vndecagone, a fortress with eleven Bulwarks, let the face be 24 rod, the flank 12, the curtain 36, and the capital DA to AG as 5 to 7, the unknown parts are required.  
SEt down DA 5, and AG 7, and the angle DAG is 106 degrees, and 4 elevenths: that is 106, 21, 49, and hence ye shall know the angles remaining ADC, 43, 55, 48, and the other or CDF 29, 42, 23, therefore in the triangle CDF, the angle D is than so, and DC 24 rod, and then DF, DK, DP willbe known, likewise FC, FB, BP, & AD, DE & OF 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 themselves with the figure put here under, which was needless; supposing that they had skill in trigonometry. I will only set down here the reason of the raid (or semy-diameter) in the side of the Vndecagone inscribed in the Circle, is 
as 100000000000000  
to 563●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 curtain be as 2 to 3: & the gorge in the flank 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 find out the other dimensions?  
seeing that ADC, and ADF are known, the rest CDF willbe also, being 29, 19, 6, & setting down DC 2 parts, (than the curtain, or FK willbe 3) the triangle CDF willbe known by the parts, namely DF 1, 74384, as much also is KL, and FK 3, than DP willbe (in parts) 6, 48768, which make by the Hypotese 75 rod, & by this reason, ye shall find out the face, and the curtain, 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 half of the rest willbe for DF, ye shall find then also FC, afterward AB, being set down upon 4. BC willbe 3, than the triangle BAC willbe 36, 52, 12; which taken from DAB, there will remain DAC, & so the triangle DAC shall have 3 terms, ye shall then seek DA, or AC, by which ye shall come to have AB 13, 7969, & BC 10, 34743, then in the triangle CBG, ye shall find BG, afterward DR, or CG being known ye shall have DG 41, 54579, GH 11, 57496, and finally DH the fichant 58, 96636.   

The 34. Figure.  
Moreover for this Vndecagone let the angle flanked be 88 degrees 38, 11, & the face to the curtain as 2 to 3, and the gorge in the flank as 8 to 5. & the fichant 60 rod, how many will all these lines make?  
YE must first find out the Quadrangle ABCD, whereof the angles are known, & set down AB, 8 parts; then BC willbe 5, & DC being found, ye shall find DF, FC, also BH, seeing that DC is to BH as 2 to 3; having DK, and KH ye shall have DH in parts, which make 60 rod, by which reason ye have the lines required, AB, 14, 6, CB 9, 127, DG 42, 374, DC 23, 7363, BH 35, 604, & DP 76, 99675. Upon the 35. Figure.   

ALB. GIRARD.  
THis question is defective, in regard it hath but four terms known, DP 70 rod, & CB 10, the reason of DC to BG is as 10 to 9, and the name of the figure Undecagonall, which are but 4 terms, he comes to the 〈◊〉▪ setting down the face 20 rod, to see if there be not in it repugnancy, as if the question were exceeding, so that he in this search committeth the faulth, which is called petitio principij. Which is spoken not to defame the Author, but to show, how this proceeded from hence, that in this time many thought that Geometry hath attained to her highest degree; though we have had but the A, B, C of it: till that some undertakeing the restauration thereof, having in part made it flourish again, could nevertheless escape the blame of some envious & ignorant men in this divine science, as happens often, have let pass the most difficult of the Analitica, which ought to adorn those, that make profession thereof, who contrariwise setting down the limits of their pretensions from the beginning of their course; by emulating one an other, content themselves to grope together without learning to go forward with a solid, and a firm pace in a fair way.   

The 36. Figure.  
In this Dodecagone, let the face be 24, the flank and the curtain 36 rod, the angle flanked right, how many will the remaining lines make?  
LEt DF be found, also FC, ye shall have DK, & KH for to have DH, afterward in the triangle DGR, having the angles & RG, ye shall find DG. Finally ye shall find DE in the triangle DEA, for to have OF, or the gorge: Now seeing that GCB is 60 degrees, then of necessity GC willbe 24, as being as much again as CB, than DG willbe 48, and GRACCUS, or KH the half is 12, therefore DH willbe 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 flank is as 4 to 3.  
seeing that the angle DCF is 60 degrees. DH 45 willbe the double of GRACCUS 22 ½ rod, DH, HK being known, DK willbe 55, 621, DR 38, 971, than the second flank 16, 65. Also DE willbe found by the triangle DAE, & taken from DR, there will remain AGNOSTUS the basis of the triangle ACG, the angle A thereof willbe 36, 52, 11, because of the reason given, and G 60 degrees, than the said triangle ACG will make known CG 21, 4933, & also the face 23, 5067 (for DG is 45) the half of GC is CB 10, 7466, which multiplied by 4/3, ye shall have BA 14, 3289, DP 75, 979, BH 35, 2637.   

The 38. Figure.  
Let the curtain 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 Fortress Dodecagonall come to?  
THe triangle DAG, afterward ACG willbe known, and also CBG, DCF, DKH, whereby DC willbe found 23, 974, GRACCUS 22 ½ (the half 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 shallbe the double of CF 12; the angle BAC being given, the triangle DAC, willbe known, therefore DA, AC, and the triangles DAE, ACB DGR, DHK willbe known; 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 we agree not, seeing that in the construction, in stead of 37 for the angle BAC, (as he set down) he took 38.  
All questions coming after this are defective even to the discourse, which endeth the order of the same questions; but seeing I know whither it tends, I will explain them, in adding thereunto the things, that were wanting passing over the figures 40, 41, and so we will begin with the 42, as followeth.   

The 42. Figure.  
In a square Fortress with four Bulwarks, let the angle flanked be 60 degrees, the angle forming the flank FID 40 degrees, the face 24 rod, having the reason to the curtain as 4 to 5, the other lines are required.  
Seek the triangles DAL: AID, IFD, DFP, ye shall find also PH, for FH willbe 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 flank ●0 degrees, the curtain to the face as 5 to 4, the face 24 rod; we desire to know the other lengths.  
Ye must do as before, and then there willbe no difference in the operation, Touching the number of the names of the figures in this 9 plate, ye shall find them marked about the angle flanking exterior, the reason of the face to the curtain is marked on the point of the Bastions, the length aswell of the face, as of the curtain next themselves, and must be proposed as the two former, having always the angle forming the flank of 40 degrees: the opening of the flanked angles, is according to the manner of the first table, to wit 15 degrees more than the demy-angle of the Polygone, saving that in the figures 2, 43, & 2, 44, afterward an other time the 45, 46, he taketh for the angles flanked the ⅔ of the angle of the Polygone, according to the second table in the beginning placed before: Of these things ye shall find 2 tables in the end of this book, setting down the faces to the curtains, as 2 to 3, the faces 24 rod, which make the former-flankes 40 degrees, and the flanked according to the two manners abovesaid, where ye must know, that the lines of defence fichant are about 60, 61 rod. The said tables are both calculated 〈◊〉; because of the errors escaped in the other editions. Moreover, after the Dodecagone ye shall have over and above following a Fortification upon a right line, which is called upon a right curtain. Now let us mark what the Author saith.   

Note 1.  
NOw not to take so much pains in remembering the divers proportions of the face to the curtain, whereof the Fortress quadrangular, & Pentagonal Fortress is as 4 to 5, and the Exagonall as 3 to 4, it will not be amiss (seeing that the line of defence will bear 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 do not much exceed 60 rod, which is as far as a musket can well carry, & therefore one ought not to exceed this number; because that always from the flank reciprocally the entrance into the moat must be defended, which is often done, and most commodiously by the Musketeers, 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 carry some 700 feet pointblank, Bulwarks ought to be made no 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 exterior take 80 rod, and the rest according to the former proportion, the parts willbe brought very near together, as appear by the precedent Examples, according to the figures of the 10 plate, marked with the numbers 51. 52. 53. 54. 55. 56. 57 58, 59, whereof we have made no calculation; because they are easily done by the former supputations.   

Note 3.  
THat in all our designs and calculations, we are resolved to use but one measure only, which is in the 25 plate of our Geometry upon the rule of the instrument marked number 1, which is a foot divided into 12 equal parts, whereof 12 foot makes a Rheneland rod in the territoire of Leyden.   

Designs & plat forms of divers Fortifications.  
WE might according to the former supputations give divers constructions of the platforms of Fortifications; but considering the great diversity of rules, (which often causes confusion, and that time will not permit us) we will give but one simple and general 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 given (suppose it to be 24 rod) than ye must do as followeth: Ye shall draw a right line infinite AC from the point A, the angle CAB shallbe made according to the form of the Polygone, upon which A, B shallbe set down the 24 rod abovesaid, as from A to D, and by D is drawn the line, which formeth the angle of the Centre of the Bulwark 40 degrees, then must ye set down upon A, C and A, B, the reason of the curtain to the face that by the same ye may draw the line infinite A, E, making from the point D, a parallel with A, C, cutting the said A, at G, by which point is drawn the line, that formeth the other face G, K, in such sort that the angle G, K, A, willbe equal to the angle D, A, K, and so the figure willbe drawn, 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 given AB (for example 80 rod) ye shall then proportion out the face to the curtain sesquialtera, as 2 to 3. Then in conformity of the table of angles set down before, willbe made the angle CAB, according to its form, 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 abovesaid proportion of the face to the curtain, which is (as we have said) as 2 to 3. aswell in a square Fortress as in a Pentagone, and others following ye shall take upon a scale of a reasonable greatness 2. which shallbe set from A to C, and 3 from A to D, suppose that it be at C & D, then from the point D willbe made the arch G of the distance of 2, (to wit CA) & from C of the distance of D, A, 3, willbe made the arches, which cut between each other at G, by which a right line being drawn to A, where the same cutting the line FB, as here at F ye have the face of the Bulwark, and by that the curtain willbe known. To know also the line of the gorge, and the flank, and all the other parts, ye must make the angle forming the gorge, and the flank 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 drawn from A, B, as is I, N, ye shall draw the line perpendicular to be E, K, which shallbe the flank, and K, I, the line of the gorge of this Fortress, and then ye have the thing required.  
In like manner ye shall find, and mark out the other Fortresses, according to their several forms, takeing heed 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 willbe needless, to give you here any further particular instruction for them. The 3, 50. Figure, and the 9 Plate.   

ALBERT GIRARD.  
THirdly, When upon the line of the Polygone interior AB, ye desire to construe, or explain a Fortress, having 5 terms, to wit, the name of the figure, the angle of the bastion M, The angle forming the flank GBH, and the reason of the face DG, to the curtain HL, conceiving therein that AB is given.  
This figure is in stead of the 2, 50, being worth nothing. I am so constrained to make a new figure, and a new explanation. Let then NBA be equal to the demy-angle of the Polygone, and let the angle forming the flank be HBG, LAK, so ye shall have the intersection or cutting between C, then DBF the half equal to the demy-angle flanked, and finding the point F, so that AB, to BF is the reason of the curtain to the face, and having drawn CF, cutting NB at D, and making DG a parallel with FB, meeting with BC, at G, ye have finally GH the perpendicular with BASILIUS, and doing the like also on the other side, ye shall have all the parts required; the demonstration is manifest, seeing that C is the common point of the like figures FBA, DGK: then as FB, to BASILIUS: so DG the face, will be to GK, or HL.   

The 11. Plate, and 60, 61, 62, 63, 64, 65, 66, ●7, ●●, Figures.  
IF there be a question to proportion out the face to the curtain, and the gorge to the flank, when the line OF is given, ye shall do as the figures in appearance demonstrate to the eye, namely, ye shall first proportion out the face to the curtain by the former rules, and make it so that the curtain of a square Fortress be 300 foot, and the face HF, 250 foot, having reason as 5 to 6, according to which ye shall find the point G, which is the angle of the shoulder. To find then afterward the gorge & the flank, ye shall make the covert line GH, which must be divided into 4 equal parts (in four, because ye would have the reason of the gorge in the flank as 4 to 3) whereof ye must place three of them from H to I, and then draw I, G, cutting through the semy-Diameter of the Polygone, which is here squared into A, and then the line AB being drawn it willbe parallel with OF, and so ye have the side of the Polygone interior, upon which the 2 perpendiculars being drawn GC, HD, ye have the 2 flanks, and consequently the essential parts of a square fortification known: ye must also observe the same in the other figures following.  
If the line OF, which is the side of the Polygone exterior (or the distance of the angles of the Bulwarks) be not given, but only the line AB, the side of the Polygone interior, ye shall seek out in the tables a Fortress of such a form, whereof the face to the curtain is as 5 to 6, and shall work it by the rule of proportion, or else let it be made after the former manner, which is much more easy, than all the figures of the 11. Plate.  
If such a side of a Polygone interior, giveth such a side of a Polygone exterior, what will such a side interior give? & ye shall have the thing required; to wit, the Polygone exterior, whereby one may easily know the other parts of the Fortress, when the line AB of the Polygone interior is given. But if in the tables the proportion be not found, make first upon the line AB the triangle ARB, whereof the half of the basis hath the like reason to the perpendicular, as ye will have proportioned out the gorge of the flank, demonstrated by this exemple as 4 to 3, which willbe done in setting down upon the said line AB four equal parts (4 because ye desire to have proportioned out the gorge to the flank, 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 means willbe form the triangle ARB, and from the points A & B, the angles flanking interior willbe made according to the form of the Polygone, (which is a Pentagone) conformable to the table of angles described heretofore, as appeareth by the letters SBA and TAB, which cut the lines infinite A, L. L, B at the points S, T, and the line S, & T being drawn, the angles T, S, B, and S, T, A willbe equal to the angles S, B, A, and T, A, B, as appear by the 28 proposition of the first part of Euclid.  
According to which the reason of the face being put to the curtain, upon T, S, and T, A, namely, the face upon TA, and the curtain upon T, S; ye have the like distances, and from the points I & X, the arches which cuts between them in a certain place, by which intersection the line T, V, shallbe drawn cutting the line S, B, at V, than the right line V, L. being drawn passing through the centre L, & cutting A, R at G, ye shall have all the parts of this Fortress: for having set down the distance R, G, from R to H, from the same point G is drawn a parallel to S, B, as E, G, cutting the diagonal line L, A, at E, the like ye shall find for F, and consequently all the other parts of the said Fortress Pentagonal. The same must ye also understand of the other figures following, from the figure 60, to the figure 68 Also ye may find the point V, in setting down upon T, A, 2 equal parts, such as ye require, and from the extremity or utmost end, having made it a parallel to S, T, and upon the same, 3 of those parts, and drawn from the extremity the line TV, the said point will consequently be known thereby.  
NOte that the proportion given here, between the face and the curtain is not so much to rely fast upon it, as to show that the general rule set down here before in the 9 and 10 plates, take place also in all other reasons, which may be propounded. For otherwise we are of the opinion; that the former figures would rather be accepted then these here, because that the reason of the curtain to the face is (as we have said above, sesquialtera as well in square Fortress, as of Pentagonal, and others following. Which for the facility, & simplicity together with the goodness of them ought to be preferred before the figures 60, 61, 62, 63, 64, 65, 66, 67, and 68, of the 11 Plate above said. Whereof the face to the curtain is (as the exemples show) of divers reasons; in such sort, that hence forward one ought to resolve, that the reason of the curtain to the face ought in all forms of Fortification to be sesquialtera, and the face 24 rod, each rod containing 12 foot, the length whereof is set down in the 25 plate of our Geometry, noted with the character or the figure 1, & is divided into 12 ynches, which rod is used in the Fortifications of the United Provinces, to the end that the line of defence may not exceed much above 60 rod, which is about as far as a Musket can well bear, the gorge to the flank may be made by the rule given in the 11 plate, according to the reason given, but forasmuch as it is more convenient to make the angle forming the flank GAC (plate the 11) of 40 degrees which giveth the reason almost as 6 to 7, in my opinion one ought to rest thereupon, and so ye shall have a general rule for all manner of Fortifications, whither they be Quadrialtera, Pentagonall, hexagonal, or the others following, as ye may perfectly understand by that which shallbe said hereafter. In the mean while note that I give here the reason of the gorge to the flank 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 designs or Plots of some regular Fortifications.  

The 12. Plate, and the 69. Figure.  
ALthough one may (by that which we have taught) sufficiently now understand the order, and Method requisite to be held in all manner of Fortifications: yet I have thought it good to trace out here some from among the rest, to make the lovers of this Art to conceive the better my intention, and will begin with a regular Pentagone, whereof the angle of the centre maketh 72 degrees, and the angle of the Polygone 108 degrees, and seeing the angle flanked according to the table before mentioned, maketh 69 degrees: It willbe easy to find the angle C, A, D, which is always equal to the angle flanking interior, and shall find the same to be 19 ½ degrees, setting then your graduate Instrument upon 19 ½ degrees, it makes the angle GOD, and drawing first the covert line infinite AB, takeing upon the scale 24 parts, or rod: and set then from A to C, drawing from the point C a perpendicular upon A, B, as is CD afterward having set on DE the length of the curtain, which is here only 34 rod (because the false brays makes the lines of defence too great) & then the distance AD, from B to E, raising from the E the 〈◊〉 cular OF, being equal to D, C, and FB drawn out, the two faces willbe known; for to know the centre of the Bulwark, shallbe made of 2 lines GA', & GB the Demy-diameter of the Polygone, than your instrument being set upon 40 degrees, ye shall find the angle HKA, cutting the said demy-diameter at the same point H: from which point the line HIGH being drawn; ye shall have the side of the Polygone interior upon which out of the points C, and F, shallbe drawn the perpendiculars CL & FM, which will form the flanks, and the gorges of the said Fortress Pentagonal, in the inside of the side of the Polygone interior shallbe drawn a parallel of 5 ½ rod for the thickness of the rampart, as N, O N, R, and RS, and ye must draw for the parapett of the rampart a parallel of 20 feet (also in the inside of the said side of the Polygone HIGH) and on the outside thereof ye shall draw a parallel of 20 foot for the falsebray as LX, and yet more outward 20 other foot for its parapet, & so all the other parts willbe traced out, which are within the moat; for seeing that there is here no Casemates, the falsebray is carried not only about the curtains; but also about the flanks, and the faces of the Bulwarks, in such sort, that the falsebray serveth as a Casemats to the said flanks, the entrance or sallies ought in my opinion to be made in the midst of the curtains, as covertly as possibly may be; the Bulwarks are made either massive, or hollow from Earth; at this present we have drawn them out as being hollow from earth, & so the superficies, or plain in the midst of the bulwark N, R, S, H. is of the same height, as the rest of the enclosure of the Pentagone is, the moat shallbe 10 rod broad, which is traced out as followeth: In the point A, or the angle of the Bulwark, shallbe made an arch of the distance of 10 rod, & in the point V make an other arch of somewhat a lesser extension than the former about 10 or 12 foot, then draw upon the back thereof a covert line infinite; the like must be done on the otherside of the Bulwark, and so from place to place, with covert lines being drawn as abovesaid, the intersections of them will bring forth the angles of the said moat, aswell on the interior as on the exterior parts, ye must make upon the brink of the moat exterior a parallel of 20 or 24 foot; for the breadth of the covert way, and besides somewhat more towards the exterior part, ye shall make a parallel of 50 foot for the thickness of the parapet thereof, which shall go down sloaping being only raised towards the covert way, or the Coridor of 6 foot, and diminishing little by little to the furthest part of the 50 foot, & so ye shall have finished the Icnographia, or ground-draught of this Fortress Pentagonal, the streets between the ramparts and the houses ought to be 30 foot, that with the more convenience they may be entrenched in the time of need, and that the soldiers may stand there in battle-ray. The streets ought to be made of 24 or 30 foot, and for the market place every side shallbe made 12 rod, and of the same form, as the Polygone is, which is fortified, being a Pentagone, in which the town-house is to be made, & other public buildings, & thus we have finished the form thereof.   

The 70. Figure.  
THe Profile or draught shallbe made in this manner: first a privy line is drawn from the utmost part thereof, than ye shall begin to lay out the breath of the ramparts, moats and ways, & as we have set down in our Icnographia for the breadth of the rampart 66 foot, which make 5 ½ rod, we will take upon the scale joined to the Profile 66 foot, for to place them upon the said covert line as from A to B: then as the way of the rounds, which we otherwise term the falsebray, is counted 20 foot, which one may take upon the said scale, & remove them upon the said covert line, as from B to I, and the same 20 foot from I to M, for the breadth of the parapet, than ye shall make an edge MN of 6 foot, which is called in flemish the Teen, or the toe of the foot of the parapet, the better to prevent the falling of the same into the moat: from whence ye must take the breadth of the moat, which shallbe 120 foot; for so much makes the 10 rod as from N to Q, from Q to R, ye must draw out 20 or 24 foot for the covert way, and from R to T 50 foot for the parapet; and so ye have all the dimensions required. For their heigths above AB ye shall make a parallel of 14 foot, which shallbe the height of the rampart, and underneath a parallel of 10 foot for the depth of the moat, the talud or slooping interior of the rampart A, V, aught to be 14 foot, and the talud exterior XB, must be the half of its height, which is 7 foot, and the trapeze, or unequal sides given A, D, C, B, being the profile of the rampart: for its parapet ye shall take 20 foot from C to E, and from the point E draw a perpendicular of 6 foot, which shallbe the height of the said parapet, and upon the said perpendicular EGLANTINE shallbe laid out 4 foot to draw a parallel to DC, cutting the line B, H, at H, which shallbe the height of the said parapet in this place, and for the superior part ye shall draw the line GH. Your foot bank shallbe made as followeth: from E is drawn the line FE of 3 foot, and 1 ½ foot high, then from the upper part of the footebanke make the line G 2. in such sort, that the said foot-banke shall not be broader in height then 2 foot, and shall have the talud 1 foot, which shallbe the talud interior of the said parapet. The parapet of the falsebray IN shallbe made as is above said; to wit, from the point I, is raised the perpendicular IK of 6 foot, which shallbe the height of the said parapet, and from I towards B draw 3 foot for the footebanke, being a foot and a half high, and so draw out the line K2, that the breadth of the upper part of the footebanke be not (as we have said) above 2 foot, in such sort that the Talud interior (after ye have drawn out the said line K 2) of the said parapet shallbe but a foot: for from the point I, the foot bank is 3 foot broad, but in the point 3 it is found to be but 2 foot broad. For the moat first draw out NO, which is the brink interior thereof, having the talud, as much as is the depth of the moat, to wit, of 10 foot, likewise the line Q, P, is taken, which is the brink exterior of the said moat, the parapet of the covert way Q, P, is made by rising from the point R, the perpendicular line RS of 6 foot, takeing from the said point R towards the left hand 3 foot for the breadth of the footbanke, and to make it a foot and a half high; To give the more firmness to the said parapet, there shallbe made from the upper part of the point S an oblique line upon the said footebanke, which is (as is before said) three foot broad; in such sort, that the said line declining a foot upon the said footebanke, maketh the said footebanke to be no broader than 2 foot. Finally a right line is drawn from S to T, which makes the slooping of the said parapet.   

Note.  
WE have made the rampire no higher than 14 foot, upon the upper part whereof is made a parapet of 6 foot high, all upon a plain field, which is a convenient heith for a place, that hath no hills or heigths about it, but in case there be any hills, that do command the said place, it willbe then necessary to raise the rampart somewhat higher, then is here described, answerable to the height of the said hill, which otherwise would command the Fortress, & for the better blinding of such places, sometimes one makes upon the upper part of the rampart an other parapet about 7 foot broad, or there abouts & 6 foot high, which Parapet serveth so much the better to blind the said place, and to discouver the fields about it, but when an Enemy hath approached near unto such places, than the said parapet can do them no further service, so that necessity will compel them to take it away for their more safeguard, and to lie under the said parapet of 20 foot broad: for Canon shot coming every time to pierce the said parapet being but 7 footebroade; they cannot safely lodge behind it, without extreme danger; and therefore is necessary to be razed, to make use of the other, which is able to resist the force of Canon.  
We have not made here in this 79 figure any Ravelins or half moons, which are loose pieces on the further side of a moat, neither any Cats, or high platforms to plant ordinance upon, which are usually made in divers places within Forts, and upon Bulwarks; because we intent to speak of them hereafter. In the mean while it willbe good to note, that the said Cats, Ravelins, and other works, may be applied to this present Fortification, but in this place is omitted for the avoiding of prolixity, because we are minded to treat of them briefly in an other place.    

The description of the design or the plate form of a Fortress hexagonal.  

The 13. Plate, and 71. Figure.  
LEt there be given a Fortress hexagonal to be fortified, whereof the face AC maketh 24 rod, and the angle flanked 80 degrees, according to which the angle flanking interior will make 20 degrees, and the exterior 140 degrees, and let the curtain be 30 rod, which giveth the reason of the face to the curtain, as 3 to 4. To do this, ye shall draw the covert line infinite, A, B, & by the help of an instrument graduate, the other angle GOD of 20 degrees (of 20, because that the angle flanking interior, which is always equal to it maketh here 20 degrees) by means 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 drawn upon the line A, B, shallbe placed from D, the length of the curtain, which is here 32 rod as from D to E, finally the distance AD from E to B, and the perpendicular OF the distance of CD as from E to F, drawing the line FB, ye have the other face, so that all the parts of the reason given are described; and for to find the curtain, ye must make first the angel's GAB, & GBA of 60 degr. seeing the whole angle of the Polygone maketh 120 degr. by the lines AGNOSTUS, GB, which cutting through on an other at G, is the centre of the Polygone, and seeing the gorge, when as ye would make casemates in them or in a bulwark where Cats or mounts are raised, they had need of a larger extent, than 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 only of 35, according to which the gorge in the flank willbe almost as 4 to 3, or somewhat more by reason of the line H, K, cutting the line diagonal AGNOSTUS at H, from which point H the line HN being drawn, parallel to AB, ye shall have the Polygone interior, upon which the lines CL, and FM, being drawn in length the lines DC to L, and OF to M; in so doing all the essential parts of the said Fortress, willbe described; Now to continue the same draught or platform in every place ye shall make from the centre G a privy circle from the distance GB, and shallbe set upon the privy circumference the distance AB, which being the 6 part of the said figure hexagonal, the said circumference will contain justly still five parts, which finally will come to end at A. In the like manner is the privy circle made out of the same centre G, and from the distance G, N, upon the circumference thereof shallbe set the line of the Polygone interior HN, afterward ye must only copy out the rest. Moreover, the parapet wilbet 20 foot, which shallbe 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 Fortress, we will make parallels towards the moat of 20 foot broad, & then 20 foot more for the parapet thereof, on the outside of the parapet, before ye come to the moat, is made an edge, or a toe of 6 or 8 foot, to keep the said parapet from falling into the moat, which is made of 140, or 150 foot or thereabouts, as necessity, and the bottom requires; for it must be fitted according to this consideration, because the ground lying low, it will not bear much digging before ye come to water, and therefore in this place one is forced to make the moat to get earth enough for the rampart, but when the ground lies high, than ye may dig very deep before ye come to the water, & so by this means ye may have as much earth, as possibly ye can well use, & seeing without it cannot be but hurtful, it willbe better not to make the moat so broad, that ye may receive no hindrance or hurt by the store of Earth that it affordeth; which may be cast upon the outside: For to believe that the ramparts being raised higher than the dimension given in the former figure to wit of 14; or at least 15 foot, that they were better as some have maintained, yet experience hath shown many times the contrary, for an enemy having once approached to the brink of the moat, the breadth of the parapet hinders one from making any defence upon it, which notwithstanding is most necessary: because the nearer an enemy comes with his approaches towards us, we ought the more to defend ourselves, and this being committed by our default, it is too manifest that one ought to take heed of it, and to remedy this error by the former way: namely in making the rampart of the height abovesaid. Some are of the opinion by a simple command that one ought to heighten the ramparts above the said 14 or 15 foot, and not the Bulwarks, for which they have some reason; for indeed the Bulwarks make the greatest defence, and when an enemy is become master of the said bulwark, there were a means to command the said bulwark more absolutely, so that an enemy could not hinder the use of them, if he raised not his works above the said rampart, to hinder the defences of the besieged, which they might make with the said ramparts so raised by a simple command, which in some manner might hinder the intents & approaches of the besiegers, the more easy would the entrenchements be made, and seeing the more the bulwarks are raised, the more one ought to use diligence about the entrenchment; hence would follow this difficulty, that the coming to the rampart or the bulwark, would not be so easy, as otherwise, neither could one make any great defence from such ramparts by reason of their too much height, in such sort that one bulwark ought to defend an other, & in such a case the second flank would be of no use after an enemy is entered into the moat, at which time it is then most needful to give the greatest resistance that may be, seeing that when they are gotten over the moat, and taken in the foot of the bulwark, than the courage of the besiegers begins to increase, & the besieged to fail them, for it is too apparent, that many times resistance fails in such and the like accidents.  
Between two bulwarks are commonly made ravelins or half moons, which are Quadrangular figures, every face containing 12, 15, or sometimes 20 rod, which begin at the brink of the moat: so that its angle interior, or the tenaille O, lies just upon the brink and angle of the moat, the reason why their faces P, Q, and Q, R, are made of a greater distance, is that the defence may be the better: for otherwise their could be no defence made but from one part of the face of the bulwark, which otherwise would not be sufficient to hinder the assaults of an enemy, which ravelins or loose pieces li● so open and are raised above the plain field some four foot, that one may the easier discover the said field, and so also to prevent the inconveniences which might happen, when one would give fire from them upon the besiegers, (if the said ravelin had not been raised) which would have annoyed those which were under the covert way. Upon the said Ravelin being so raised, is made a parapet of 20 foot broad, and 6 foot high, which willbe able to resist Canon, the moat whereof ought to be made 50, or 60 foot, and as deep, as possibly may be about the said moat, ye must make also a cover way of some 20 foot broad, as hath been noted before; Afterward ye must make a parapet of 50 foot broad & six foot high, ending at the end of the 50 foot towards the plain field; as the profile doth show in the 72 figure, and to the intent ye may the better understand my meaning, we have made the said Hexagone in perspective, with the falsebrayes, Ravelins, covert ways, and their parapets apart, as appear by the second 71 figure, but ye must note, that the said Ravelins must be in respect of Ramparts, which have a double height, to see in them the more perfection, also a double height, to make them show the better: for otherwise they would seem too little for the reason abovesaid against our intention.   

The 72. Figure.  
THe Profile shallbe made as in the 12 Plate of the former figure 70, by drawing a privy line infinite, and takeing upon it all the dimensions, as first the breadth of the ramparts, the parapets, the falsebrayes, its parapet, the breadth of the moat, the covert way, and the other parts of the said profile; and seeing we have here ordained Ravelins, we 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 breadth of the Rampart 68 foot from A to C, from C to D 20 foot, from D to E 20 foot, and from E to F 6 foot, the breadth of the moat shallbe 150 foot, as from F to G, which ought to be broader, but seeing the distance falls out too little, it shallbe made but of 150 foot, the Raveline ought to have in this place 180 foot, but seeing the ground fail us, we have made the dimension thereof 180 foot, but this hindrance would not permit us, to make it of that competent greatness as it ought, which I speak to give no stop to the Reader, which Ravelin is raised 4 foot above the plain field from 6 to T, upon which a parapet is made, whereof the basis shallbe 20 foot broad, and 6 foot high, to wit its breadth from Q to R, and the height from Q to S, the footebankes are made of the same breadth & height, & therefore I will make here no mention of them, neither of the taluds or slooping of Ramparts, Parapets, & moats; aswell the interior, as the exterior parts are made according to the nature of the soil; for by how much the ground is lean & sandy, by so much ye ought to give it the talud, & by that means ye shall hinder the falling down of the works, for which reason often times ye must give on the exterior side, being a sandy ground, as much talude as height, and if the said Ravelin be also raised 4 foot above the plain field, me thinks it will not be amiss, to raise the rampart thereof, which is 15 foot, some 6 foot higher to command the better the said Raveline, & leave the Bulwarks of the height of 15 foot, that all inconveniences may be prevented: on the outside of the Raveline is made the edge HIGH of 6 foot, and a moat of 50 foot, than the covert way 20 foot, & the parapet LN 60 foot, with a footebanke of the ordinary breadth of 3 foot in that manner, as the 72 figure demonstrateth.    

The description of the plote of a Fortress Heptagonall.  

The 14. Plate, and 73. Figure.  
LEt there be given to be fortified a Heptagone, whereof the side of the Polygone exterior maketh 63 rod, and the angle flanked 80 degrees. To do this, ye shall first divide the said side AB into 7 equal 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 shallbe drawn being equal to A, D, and E, B, to wit, either of them 2 seventh parts of the line A, B, than ye shall draw from the extremities, or the utmost parts of these perpendiculars, the privy line infinite H, I. which shallbe the side of the Polygone interior, and by consequence N, O the curtain. The angles C, A, B, and C, B, A, shallbe made by the help of a graduate-instrument, as our compass 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, shallbe the centre of the said Fortress Heptagonall. Afterward I, B, 8, of the 40 degr. which is the half of the angle flanked, that ye may have the whole angle of 80 degr. according to the Hypotheses, and where the said lines cut the said perpendiculars D, N, and E, O, in the points 2. & 8, ye shall have 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 Fortress, ye shall do the like as hath been taught in the former figure hexagonal, the faces in this present Raveline shallbe made of 15 rod, the moat of 10 rod, and the moat to the Ravelin of 15 foot, then about it shallbe made a covert way of 20 foot, & a parapet of 60 foot broad descending (as we have said before) sloopingly. According to this form the flank willbe then made of a reasonable greatness, but the gorge so much the lesser, which may be made so, seeing ye intent not to make any flanks in the Casemates, but in case ye would, ye must then make the flank the lesser, and consequently the gorge the greater, to the end ye may have space enough for to make within them the said Casemates. The like is to be understood when ye mean to make a Cat upon the Bulwark, for the gorge being so little (besides other inconveniences) it is impossible to make there such a body, in reserving sufficient space for the flanks, which me thinks in such places, that have such narrow gorges, it were better to make the said Cats upon the Curtain, yet so that they be no 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 near the rampart as possibly may be, the better to discouver, and command the fields about them.   

The 74. 75. Figures.  
THe figure 74 is the profile of the Rampart, Falsebray, its Parapet, moat, covert way, and its parapet, whereof the length, breadth, height 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 drawn out, we have represented but a part of the moat, the form of the Raveline, its height, and the form of its parapet, the breadth and the depth of its side, with their taluds or slooping, 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 have spoken of this present figure 73. we have added hereunto the platform of the Fortress of Coverden in Frizland, whereby ye may the better understand our intention, being accounted the master piece, and the most regular, and royal Fort in the Low-Countries, having a False-braye, Ravelins, or half moons, covertwayes, with which we 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 making then the former.  

The 15. Plate, & 76. Figure.  
LEt the side of the Polygone interior be B, C, upon which ye would have a part of a Fortress Heptagonall made, whereof the face to the curtain is as 3 to 4. and the gorge to the flank as 13 to 9 To do this, ye must do as we have taught you in the 11. Plate, or in the 9 Plate, and Figure 3. 50, where ye shall find the proportion requisite: on the outside of the same face, shallbe made the parallels each of 20 foot, aswell for the parapet, as for the falsebray, the like is to be done in the flanks, and curtains, and on the inside a parallel of 20 foot, with an other parallel of 72 foot for the breadth of the Rampart in the basis, in the same manner as this 76 figure showeth, on the outside ye shall make a parallel of 10 rod for the breadth of the moat, and upon the brink of it to the tenaille, ye shall lay out the Ravelins, as we have said in the former figure 73, in the angles of the Bulwarks ye shall make also Ravelins, such as ye see here marked with ay, K, L, M, which shallbe 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 moats of these Horneworks comes to answer to the Falsebray, which is in the flanks of the Bulwarks: so that their breadth S, T, maketh about 32 rod, of which distance the two demy-Bulwarks V, X, are made (whereof the face, and the curtain are of a like greatness) according to the rule prescribed in the 10. Plate, the flanks willbe found, making from the angle of the shoulder, two lines perpendicular upon the curtain, which is directly opposite to it. even as the figures 77, 78 demonstrate. The utmost angles of the said Horneworks ought not to be further from the curtain of the Fortress, than a Musket can carry pointblank, which is 60 rod or there abouts, and if the Ravelins OF, GH, are not made in the tenailles, than ye may cut the said Hornworke, as is here marked out in the 78 figure: the breadth of the moat shallbe 24 or 25 foot: The Rampart here is no otherwise then the parapet of the same breadth.  
The depth of the moat may be made 6 foot in case the ground lies low; but otherwise the deeper it is made it willbe so much the better, the breadth will not hinder it, though it were made but 36 foot; for according to the same breadth & depth ye may enlarge the Rampart, aswell in height as in breadth, upon which afterward ye may make a parapet as great as it can well bear: but when haste requireth, I would make no Rampart, but only a Parapet of 24 foot thick, as we have made here being 6 foot high, and a footbanke of 3 or 4 foot.  
The figure 79 marketh out a Profile, which is a section or cut drawn through the midst of a curtain passing through the midst of the Ravelines G, and traversing through the Horneworke P, and the Rampart and the moat. The true dimensions thereof are clearly expressed in the said Profile, by the help of the scale; joined to it: as also by the means of the Alphabetical letters, showing the feet to be 12 ynches, whereof the length is 〈…〉 in the 25 Plate of our Geometry.  
The benefit of such works are well known, when they are made in places of advantage for defence, or where no men, victuals, and ammunition of war are wanting, as also where the ground is of a reasonable largeness, 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 give to such works, against the attempt, which may be given by an Army: & as a great assault, cannot be resisted with a few men, but up a small room: so is it manifest that the greatness & largeness of such a place, must be answerable. Therefore in my opinion, it is a thing repugnant to the rule of Fortification, to make such Horneworkes in the angles of Bulwarks, where they are so straightened. For Demy-bulwarkes cannot (by reason of their smallness) be well maintained and defended, and on the other side, their two wings cannot be defended from the main Fortress, but with great disadvantage, as we intent to discuss: thereof more at large hereafter in the plateforme of Gulick.    

How Citadels, or Castles may be joined to Towns, or Towns to Castles,  

The 16. Plate, and the 81. 82. Figures.  
FOr to build a Castle (or a Citadel) to a Town, either to help to defend it the better, or to curb it: ye must find out first the most advantageous place: in case there be a river, than ye shall build your Castle upon the side of it, where it may best command, with all the advantage that possibly may be, also takeing heed that it be built in such a place, that the Castle may receive no disadvantage thereby, but that the Rampart of the Town where the letters A, G are, be the weakest part; to the end that by this means, ye may frustrate the desire of the inhabitans of the said town, from opposing themselves against the said Castle: but if ye fear no such inconvenience, ye may make the places A, G, as strong as the other: saving that the Ramparts in the said angles be no thicker in the angel's A, G (which be angles of imperfect Bulwarks) then in other places, and not massive or full of earth, as the Bulwarks B, & C, D, are. Let there be a town then containing 8 angles, whereof 7 of them are marked out with these Alphabetical letters A, B, C, D, E, F, & G, and that you are desirous to make a Pentagonal Citadel. For the doing whereof ye must first make a map of the said town Octogonall: in case it be not made, according to the method given heretofore in our Geometry, which is here marked out by these letters A, B, C, D, E, F, & G, with the Falsebrays, Parapets, Moats, Covert way, & its Parapet. Then upon a paper of the same scale, whereof the map of the said town is made, being traced out in a Pentagone with all the outworks, such as the figure marked H, I, K, L, M, demonstrats, according to the method given you before in the 10 and 11 Plates, than the same figure Pentagonal must be cut perfectly, and so fitted to the town, removing it so long till you have at length found out the most advantageous place to make there your Castle Pentagone, takeing heed that the tenailles N and O, fall out right, to answer the curtains P, O, and Q, N, to the end that the entrance into the town may be the more commodious, & that the town in the places N, R, S, T, O, be not discovered without, whereof one ought to have a singular care: for there where the said angles of the tenaille, be not well joined to the Ramparts, those places will always be badly defended in the said joinings, as ye may observe in the Castle of Gulick, which platform follows next. It is true; that when ye make these angles in the curtains, that the parts N, R, S, T, O, take up a great deal of the town, which may be prevented in setting the Castle more outward, and making the said curtains meet nearer the faces of the Bulwarks of the Castle. But the side of the town N, R, S, T, O, will in no wise be guarded, as the Platform of the Castle of Gulick showeth: so that necessity will require, for the preventing of such an inconveniency, to make some Ravelins, or some other works without the said place, as the commodiousness of the ground may afford, which would make me resolve rather to approve of the form of the 80 figure, then of the other, were it not for some cause more urgent, then is here alleged, which reason would move me to change my opinion.  
If the town or City A, B, C, D, E, F, G, were greater, than were it unnecessary to take so much care for the place, which the Castle taketh up: for than ye might make the curtains P, O, & Q, N, longer, to wit some 50 rod, that the flanks V, X, and Y, Z, might not come so near the Bulwarks E, M: from whence one may easily discover the Falsebray of the Castle, for the preventing of which inconvenience, it willbe good to raise the Bulwarks of the Castle, which looks into the town, somewhat higher above the Ramparts thereof, but the town being of a small circuit (as we have said) one shallbe constrained to use for it the same figures as are demonstrated by the figures 80. & 81.  
If one would built a town to a Castle, first ye must observe diligently the site of the place, and if it may be made regular, to take care, how many Inhabitants this place will contain, the nature and quality of them, and whether the town may in process of time be enlarged: which ye shallbe able to judge, when ye see the country that lies about it, what ways, avenues and passages there are to come to it, as also to consider the goodness of the air, and of the soil of all which things (as Vitruvius saith) one ought to have a most special care. According to all which one ought to govern themselves: for to enclosse a great piece of ground, where there is none, or little appearance of Inhabitants, this were rather to ruinate, then to edify, to which takeing heed then, ye must resolve of the Circuit, or Capacity, which ye would give to the said town, and suppose ye would make the form thereof Octogonall, and regular: ye must trace out then an Octogonall Fortress regular, according to the former method, with covert ways, and what ye require more upon the same scale, as you made your Castle Pentagonal, than ye must cut the said form Octogonall, and join it to the said Castle, removing it so long, till you have found out & chosen out (with all circumspection requisite) the most convenientest place, and staying there ye may trace out the said town, as the figures 80 and 81 demonstrate, joined to the Castle. But as we have said, it will lose much of its capacity, because the curtains of the town will come to meet with the angles of the tenailles of the Castle in the points marked O, and N.   

The 17. Plate, and the 82. 83. Figure.  
IF ye will not tie yourself so precisely to this form Octogonall, ye may make the town much better by the means which we will now speak of. Let first be drawn the lines D, D, and A, A. etc. right perpendiculars upon the midst of the curtains N, M, and K, Y, of some 60 rod, or there abouts, then let there be made the perpendicular line infinite in the midst, and upon the curtain D, E; in which ye have the centre of the circle, which makes the circle of all the figure, and suppose ye desire to make a form regular of nine Bulwarks, whereof the angles flanked, are distant one from an other 80 rod, from the utmost end Q, shallbe made an arch, in such sort, that within the circuit thereof the distance N, P, is contained 9 times, which shall make the point of the centre C, C. then upon the circumference of the said arch the said 9 points, being marked as they are noted here by the letters Q, R, S, T, V, X, Y, Z, &c ye shall make the angles interior flanking, according to their form, that ye may have the angle flanked right, & that ye may the better make the angle flanking, or the tenaille pointed and small, which is accounted the best, as that skilful, and diligent man M. john Errard of Barleduke, Ingenier to the King of France hath observed well, and to know how much the said angle flanking will make, ye must subtract the half of the angle from the Bulwark, or flanked from the half of the angle of the Polygone, the rest shallbe the angle flanking interior. But if the angle of the Polygone be lesser than 150 degrees, which is the angle of a Dodecagone, ye shall then govern yourself according to the table given before, takeing heed that the angle of the Polygone agreeth most with the angle to be fortified, and in its column is found the angle flanking interior, necessary to the said angle, by which ye may come to the angle of the tenaille, & seeing in this exemple we have set down the distance from angle to angle, as well of the Pentag. as of the town 80 rod, we will set down upon the lines, which make the tenailles 2 equal parts, & from the said 2 parts in the line which formeth the angle flanking, a parallel line being drawn to that of 80 rod, & placed upon the 3 equal parts, a privy line infinite, must be drawn through the said 3 point, & through the angle, and where the same comes to cut the other side of the tenaille in the point O, ye shall have the shoulder, through which the line A, B, being drawn, making the angle forming the flank 40 degrees, ye shall have the centre of the Bulwarks justas we have taught heretofore in the 10 & 11 plates. If the Castle had been made with four angles, the construction would have been the same, and if the lines D, D, Q, and A, A, etc. had been somewhat more than 60 rod, the form would have been better, for though we have made the lines of defence, to be 60 rod in the former Fortresses, there is an other consideration in this present building; in regard the Bulwarks Q, etc. advance themselves towards the angles Q, etc. from whence the said curtains Q, E, E. & B, B etc. may be sufficiently defended, when they do not exceed above 70 rod, the moats, covert ways, falsebrayes and other works, must be made in that form as we have set them down before in the 16. plate.    

How to describe the order which is observed in making of a platform, and of some other appendances.  
THe draught being made upon a paper, ere ye make a platform of it, ye ought first to overcast, aswell the lines, as the body of the Fortress, in that manner as we have shown you before in our Geometry, that ye may know pertinently all the particular dimensions, & so commit no error, in making a plateforme thereof: Before ye let it out, according to our Method given in the conclusion of the practice of Geometry, which being done before the Inginier begins the work, he shall declare the particular conditions, which he would have observed in the making and finishing of this work, as the time when they are to begin, how long it willbe before it be finished, how many men he must ordinarily have to work, what instruments they must use for the driving in of piles, and the laying of the foundation, how many foot, and what edge they are to give to the wall, or the parapet of the falsebray, the thickness of the parapet, and how broad the falsebray ought to be, how much the talud (or slooping) on the inside and on the outside must be▪ and that according to the goodness or weakness of the matter upon which one builds, how many bundles of brush in a sand●e Earth he is to lay in the parapet of the falsebray, and in the Rampart, the height, the Talud or slooping, both on the inside, & on the outside, the breadth and depth of the moat, & its slooping, and generally and specially all things touching the making of such a Castle, whereof he hath made his platform. And some days before they come to the place, or that the form be drawn out, they shall set up billets in the next adjoining towns to give notice to the Work masters, that upon such and such a day, one is resolved to hire and let out such & such works, to him that will take them on upon the least price: and before the day prefixed the said Inginier shall come thither himself with some one of the Deputies, or some other that hath the managing of that work, to draw a draught of the said Fortress, and to prepare all things necessary against that day assigned. Then though day being come, all the Workemasters & Undertakers shall come together, either upon the place where the work is to be made, or else in some other place where the conditions and covenants are read, according as they will have the work hastened: this done, they shall demand, who will take it on for the least money, than one of the Workemasters will say for so much, an other for so much lesser, which is done till no man will take it on at a lesser rate, than underneath they sign unto the articles of the conditions, that I N, N, have undertaken the work upon such and such conditions, for the sum of so much money: sometimes ye shall have two or three Undertakers join together and undertake the whole work, all of them signing to the articles, as also the Commissioners, and Inginier must do on their part: then they begin the work, and commonly the Undertakers are bound by the said articles and Contracts, to deliver all the materials necessary for the said work, which they receive from the Master of the Magazine there or else, where giving him a note under their hands for the restitution of them back again, when the work is finished, and to make good whatsoever they either break or lose.  
Then the Workemaster placeth his men in order, as he knows the quality of the work requireth best, some to dig, and cast up the Earth, others to fill Karrs, and Wheelbarrows, others to level the ground which is pricked out; for in the beginning it willbe necessary to carry the Earth away, with a Horse and a Karre, which is cast up on the outside of the moat upon the basis of the Rampart, & not with Wheelbarrows as they must do at the last, when the work grows high, and that ye are come to the depth of the moat, for than ye cannot so well use Horse and Karrs; by reason of the difficulties known to all men, namely, that the treading of the horses and the Karrs spoil the groundwork, and cannot be laid so well, as when men brings the Earth up in Wheelbarrows upon away ascending with plancies, as hath been found by experience often times by the undertakers of such works.   

Of the Foundations.  

2. The 17. Plate.  
WHen the foundation is bad, it willbe your best course to drive in piles, along the foot of the Rampart, just where the said Rampart beginneth, some piles 25, or 30 foot, or else young Alders-trees, about four or five years' growth along the work about 18 or 20 foot long, close one to an other, and the closer the better, and though this be a great charge, and a long work; yet this is the best foundation ye can choose: But as we have said oftentimes, seeing it happeneth that one cannot stay so long, ye must then take bundles of brush, and bind them up in faggots, & lay them crosswise, driving them into the Earth with stakes to fasten them together, that they may lie firm, and if ye lay the foundation in a ditch or a quagmire, ye must make use of such faggots bound and tied up together with cords, & if it be in a river, than ye must bind brickbats, or stones between two bundles, to make them sink to the bottom: ye may make them as long, or as short as ye please, and so necessity requires, till ye have filled up the water with these sinkers, upon which you must lay your foundation, than ye must also drive in some piles about 18 or 20 foot long, (according to the depth) to make them lie the more firm and solid together. Now these bundles are made in the form of Saucidges, as here followeth.  
First, ye drive into the ground stakes of the height & breadth as ye desire to make these Saucidges, either great or small, the one a foot high, the other a foot and a half, and sometimes two foot, than ye lay between these stakes bulrushes, boughs or small bundles of brush bound fast together, and in the midst of them ye must lay brickbats, when ye sink them in a river, but of Earth if ye will lay them as a foundation in a moat all along the said work, than ye bind the said bundles with stones, or Earth in them, fastening them withwiths in the same manner, as ye bind up a faggot, or a bavin, and in this form are your Saucidges, or sinkers made, which being done, the letters A are bound one to another, as strong as possible may be, as ye may perceive by the letter B in the 2. 17. Plate.  
Then ye sink these down to the bottom, that ye may afterward build your Fortress upon them, according to the former platform, whereof a Plate is made before, which groundworks among the Engineers of the united Provinces are much used, because these parts are maritimate, moorish, and subject to bad foundations, so that in some places the foundations cost more than the rest of the building itself, yea in divers places of these Countries, the Inhabitants are driven to keep their grounds from the violence of the Sea and Rivers, by such and such like means and remedies, to their great and unspeakable charges, for one would have thought it had been impossible for them so many years to do it, since the times that these Provinces were united, had not the superintendants, & Dikegraves' used great care and diligence in continuing the same preservation for the good of the inhabitants, and common weal, whereof we will speak more at large hereafter.  
When the work is raised above the water, and that the Earth in the moat diminisheth, the Worck-master shall then begin to lay his edge, or border upon the line drawn out for the Rampart. Now the line being drawn out upon the ground, as ye see in the line C. by the 2. of the 17. Plate, being five or six ynches broad, and as deep, which is done by the Ingenier himself to take away all excuse from the Workmasters, that he may likewise carry a right line, aswell on the inside as on the outside of the Rampart, as also for the moat, and the other parts of the said Fortress, represented in the platform: Besides if the condition of the undertaker contains, that the outside of the Rampart must be raised with sodds, which are ordinarily 4 or 5 ynches square, and 14 or 15 ynches long diminishing towards the inside, as ye may see here by the figure D, to the end that between the said rows of sodds, one may lay a little earth to settle them the better together to the body of the Rampart. Moreover, they are so laid together that the rows above must lie even in the midst of the jointure of the rows beneath, making them settle together according as the condition bears, and as the articles are which the undertaker hath signed unto. Now for his better ease, he must make use of a triangular Instrument, such a one as the letter F demonstrats, being 〈◊〉 ●●de A, B, about 2, or 3 foot more or less, according to the quality of the work, and the rest so consequently: For as ye desire to have the Talud, either great or little, so ye must enlarge the line A, G, or lessen it: If it be needful to lay bundles of brush to strengthen your works of the Fortress, ye must lay them so that the ends touch the former sodds, to wit the one row half a foot above the other, and between each of them half a foot of Earth, continuing the work so until it be finished: upon the top of the Earth of the Rampart, ye shall raise the parapet with such a Talud and breadth, as the conditions imports, in the same manner as before, raising it with sodds, as we have said. If ye make your Fortress in a fat and a clayie soil, than ye may (in stead of sodds) make a crust of 3, or 4 foot, or more, stamping it well with a stamper made to that end, and of such an inclination as your agreement bears, in which crust ye may set a certain kind of grass, called in Flemish Queeckcruyt, in Latin Gramen, & in English Meadow grass, which root is of such a nature, that it will spread itfelfe over the whole Rampart, and bind the Earth fast together: so that it will make the said crust of a long durance, and almost perpetual: on the outside of the said burst, ye may sow Oats or Hay-seed: they use also to set upon the Bulwark, roots of seven-leaved-grasse, which is likewise very good: but the leaves do not spread themselves over the exterior superficies, or this crust, as the abovesaid grass doth: for this reason some years since his Excellency hath found it good, to repair all the Fortifications of such a crust without sodds; because experience hath taught us, that the said sodds binds not the rest of the Earth, so well as the said crust doth, which being moist by that means joineth so much the better the rest of the wall or Rampart together, which is of very great use, and is quickly done.  
The talud of the Rampart is made somewhat greater, that one may stand an end the better upon it, as ye may see by the profiles here before, aswell of Ramparts as Falsebrayes. This being done, ye must make the parapet of the covert-way of earth digged out of the moat, whereof good store was laid on the outside of the moat, to that purpose: and the said undertaker ought to that end make an exact calculation of the quantity of Earth, which is necessary for that use. But seeing oftentimes the undertakers are not skilful in casting up such accounts: It were good that the Ingenier himself should make an exact calculation, not only of this part, but also of the whole work, which may be done in that manner, as we have shown in our Geometry, or else in this easy manner following.   

The second 17. Plate, and Letters G, H, R.  
LEt the 12 part of the Fortress be Dodecangular G, and AB the face 24 rod; the flank AD 12, and the curtain 36. and for the Rampart H. Let the talud interior be 15 foot, and the height as much, the talud exterior 10 foot, the breadth upon the Rampart 50 foot, which makes that the superficies of the Profile of the said Rampart willbe 933 1/3. Moreover, for the Profile of the parapet, which is K 99 1/4. the height EC 6 foot, HF, 4. AB for the footbanke a foot and a half; AC 3. BD 2. CG 20. HG 2 1/2. the solidity of the Rampart & the parapet is required.   

Of the Rampart.  
THe Icnographie showeth, that CM 10 foot is the talud exterior, and HN 15 foot the interior, the angle B is right in the Dodecagone, and in the Polygones upon it, ye must remember once for all, that if the Profile on all sides hold one & the same dimensions, to wit in the faces, flanks, & 〈◊〉 the angles BADE shallbe cut by halves from the lines HB, EA, KD. Also the angle of the Polygone being 150, & the flanked right; the angle flanking interior willbe 30 degrees, as it is marked, and the shoulder 120 degrees, which makes that the triangles rightangles are as follows: those which have their angles pointed of 45 degrees, shallbe Isosceles, or like angles, BM, MC upon the basis each of them 10 foot, the lines C 3. 3 G, each 50 foot, BX & XH, each 75 foot, GN, NH 15 foot, and the like in the triangles having their Hypotenouses at KD. Moreover in the triangle ORA, in the shoulder RO 10, OA 5, 7735, the line marked 5 A, 43, 30125, therefore takeing the lines BX, and 5 A from BASILIUS 24 rod, or 288 foot, there will remain HF 169, 69875. Likewise AD 144 foot, PD, or RV, willbe 138, 2265, and by adding to it FU. FR will come to 148, 2265, the line A 12 is equal to A 5. taken from AD, there will remain 100, 69857 for T 8, to which add 8 I, 60, TI will come to 160, 69875, and LI, 169, 359, add to the said TI, IZ, 15, ye shall have EKE 175, 69875.  
The demy-curtaine 13 D, makes 216, than 9 F, 226, also 11 K 291, & 10 I, 276: so that if ye add CR 272, 2265, RF 148, 2265, and F 9 which is 226, ye shall have 646, 453 for CFR 9 the circuit exterior.  
Ye shall find also GL 193, 359, & LI, 169, 35904: & 10 I, 276. whereof the sum for GLI 10, willbe 638, 718 the circuit interior, which being added to the exterior abovenamed, will amount to 1285, 171, its half is 642, 5855, which multiplied by G3, the distance of the parallels 50, will come to 32129, 275 for the superficies, between the abovesaid circuits, and then multiply by 15 the heights, ye shall have the solidity of the wall, without the taluds 481939, 125. CRF 9 10. ILGC, which had been too short as here followeth.  
The Author having made this above with so many lines, if from the midst of LR, he had drawn a perpendicular line upon the face, and cut of from it 20, 2071 its double substracted from the sum of the abovesaid perpendicular, and from BA, AD, D 13, which is 683, the remainder willbe 642, 5856, which multiplied by 15, it willbe the same as above.   

Of the Taluds or slooping walls or Ramparts.  
THe Taluds without the Pyramids, which are in the angles, shallbe calculated thus, their Profile is a triangle, which multiplied by the length, or else as Demy-parallelipedes, the Basis multiplied by the height, afterward the half taken, add to it then CR 272, 2265, RV 138, 2265, with D 13, which is 216, the sum 626, 453 multiplied by MC, 10, the superficial basis willbe 6264, 53; likewise HE 169, 69875, EY 160, 69875 & I 10, which is 276, making together 606, 3975, which being multiplied by JY 15, will make 9095, 9625 for the other superficial basis, together with the former will make 15360, 4925 (because they are of a like height) by which height of 15 being multiplied, than the half will come to 115203, 69375 for the taluds, aswell interior, as exterior besides the Pyramids.   

Of Pyramids, and double Pyramids of the Taluds.  
THere are 3 Pyramids in the angles convexe B, A, K, and three double Pyramids in the angles concave, H, E, D, which I say because Marolois making no distinction of them, taketh two things for one: As for exemple, the body upon VFXD is 1000 foot cubical, and he 〈…〉 but 500, and so of the two others.  
The Superficies of the triangle BMC, is 50 from ROAP, is the rightangle RO, OA 57, 735, and YIZK 225, and then the double superficies of GNH 225 (according to that as we have said before touching the Authors fault) ESLT its double Superficies is 259, 8057, & the double superficies VFXC, of 200, whereof the sum is 1071, 5425, which multiplied by the tierce of the common heights, ye shall have for the abovesaid fix bodies 5087, 7125, (whereas Marolois hath but 3375, 69375) to which if ye add, aswell the Taluds 115203, 69375, as the Rampart 481934, 125, the sum willbe 602230, 5312, for the solidity of the Rampart, besides the parapet.  
Those which will not take the pains to have it done so exactly, but are contented to come somewhat near unto it, shall add unto it BADE 13, which is 648, with HEK 11, being 636, 396, and the half of the sum multiplied by the profile of the Rampart 937 1/2 ye shall have 602060, which comes the nearest unto 602230, found out above (being only 170 too little) for Marolois found out but 600518, with much ado, having 1712 too little.   

Of a Parapet.  
WE have before set down the dimensions of a Parapet as ye may see by the letter K, newly cut, with its Rampart H, in this plate, & ye must not think it strange, that the Parapet is made upon a greater measure, then that of the Rampart, the reason is, because there are no more dissections, or cutting of, and that we have not room to make that which we would within the some Plate.  
Now if ye desire to calculate the Profile as the Author hath done, the whole as the brevity of the Rampart shows above mentioned, its situation is termined on the outside by CRF 9 and on the inside by a parallel line, between the abovesaid & GLI 10, distant from the other 20 foot, and for the avoiding of the confusion which the plurality of lines might cause, we will take for the said parallel, the line GLI 10, and therefore 3 G, or 7 S, will make 20, or else the line LT; ye shall find by the triangles, that TR willbe 5, 369. add hereunto C 3, which is 20, and the sum of CR known above, taken from 272, 2265, there will remain for GL 246, 8675, then from RF 148, 2265, substracting the length equal to TR, 5, 359, there will remain 142, 8675, afterward add thereunto 8 V, or its equal 20, ye shall have 162, 8675 for LI; also F 9, being 226, than I 10 willbe 246; then the sum of CRF 9, (added to GLI 10) and the half taken, ye shall have for the middle length of the parapett 651, 094.  
For the superficies of the Profile of the Parapet K, the triangle EIF 17 1/2: and DO 2 1/4, the trapeze, (or unequally side) IFGC 75; and BOCA 4 1/2, the sum is 99 1/4, for the Superficies of the Profile, which multiplied by the middle length above said, ye shall have 64621, 0795 for the solidity of the parapet, which is much too much; for it ought to be as I have calculated it with its dissections 63976, 9475, to which add 602230, 5312 from the Rampart ye shall have 666207, 478, for the whole solidity of the Rampart and Parapet, whereof the double 1332414, 956, willbe the twelfth part of the Circuit, which multiply by 12, ye shall have the total sum 15988979, 472, and divided by 144, ye shall have 111034, 68 plyntides, or as Marolois calls them Chevilles, pegs or pins, for the whole circuit of the Dodecagone.  
Seeing then there are 9252, 88 pins in the 12 part of this Fortress Dodecagonall, the Ingenier may conjecture, for how much he may let out the same work, which is about 16, 20, 25 or 30 pins, according to the situation of the place, the ground and other considerations, and after that the undertakers shall offer to make it according to the calculation, made thereof, they may take on the said work upon condition, that they shall finish it in such a time, & with so many men as is specified in the contract, and so at that instant they begin the work, & finish it according to their bargains and conditions, as the greatness of the work requires, and the time limited, for which also the undertaker is bound to have as many work men upon it, as shall be found needful for the finishing of the said work: & for the better assurance thereof, it is requisite to know how many pins a man may spit, and lay upon a wheelbarrow in a day: for those that are skilled therein, say that a man if he works his best, he may dig & lay up 4 1/2 pins in a day, but ordinarily men suppose that one can do three pins every day, without takeing too much pains, according to which, one may easily make his calculation how many spadmen he must use to make up that work in such a time prefixed, for men which shall do nothing else, but spit and dig up the Earth & the others to carry it away upon wheelbarrows to the Rampart, for ye must have as many men to carry up the Earth in the wheel-barrowes, as that the spiters may be continually kept in work without any cessation or stay, to the intent the Workmaster may finish his work within the time limited.   

Note.  
WE say here, that a man may in case of necessity spit in a day 4 1/2 pins, and commonly three in a day with leisure, but it is to be understood, this must be in a good and fast ground, which willbe bear spiting without falling or breaking; but when the Earth is lean and sandy, it willbe impossible for a man to do so much, & so by consequence one cannot know any certain rule.   

The 1. Maxim.  
ACcording to which plots and calculations all other places aswell regular as irregular may be fortified, and hold this for a maxim, that the place so fortified, as we have taught above, willbe better then the others.   

The 2. Maxim.  
THat the line of defence which comes from the angle of the flank to the angle of the Bulwark, called the line of defence fichant ought not to be much longer than 60 rod, because it is as far as a Musket can well carry, but if ye will defend the face with Canon, than it may be a 100, or 120 foot long.   

The 3. Maxim.  
THat lesser the angle of the tenaille is, it is so much the better, because the faces look more directly one upon another, therefore ye may see that in the plots of the Dodecagone above said: ye must make the angle of the Bulwark only of 90 degrees, that the angle of the tenaille may be shut the closer.   

The 4. Maxim.  
BY how much the flank and the gorge are greater and spacious, by so much they are the fitter for defence.   

The 5. Maxim.  
THe further the defence is made in the curtain, it is the better, because ye may bring the more men to give fire upon it, for the defence of the faces. Therefore in the plots of the Dodecagone abovesaid, ye must see that the angle of the Bulwark be made of 90 degrees, to the end ye may give fire the larger.   

The 6. Maxim.  
That the angle flanked aught to be no less than 60, and not above 90 degrees.   

The 7. Maxim.  
THat the angle of the Polygone, which ye intent to fortify, ought not to be less than 90 degrees, but the greater it is, it willbe so much the better.   

The 8. Maxim.  
THat in all Fortresses we will make the face to be 24 rod, and the curtain 36, according to which the lines of the Polygones exterior will make about 60 rod, and the lines of the Polygones interior 60 rod. In a Heptagone and the Polygones underneath, the Polygones exterior willbe more than 80 rod, and above less than 80. The Polygones interior which are beneath the said Heptagone, are less than 60 rod, and above more than 60, the one increasing, the other decreasing, until that the angles of the Polygones come to be 180 degrees, and then ye have the Polygones exterior, and interior equal, and each of them 70 rod, the line of defence fichant in the whole about 60 rod, & the angle forming the flank of 40 degrees.   

The 9 Maxim.  
THat the angle flanking interior aught to be no less than 15 degrees, but the greater the better they are.   

The 10. Maxim.  
THat the angle of the flank be always right, the angle of the shoulder must be at the least 105 degrees, and the greater the better.   

The 11. Maxim.  
IF it be needful to fortify any Polygone, either regular or irregular, whereof the sides are less than is here spoken of, than ye must proportion out the gorges, flanks and faces, according to the form of the angle, as ye shall see in a table following.  
Herewith we will put an end to our first part of Fortification, and will come now to the fortifying of places irregular.  
FINIS.     

A THE DIMENSIONS OF 〈◊〉 IN TWO MANNERS.  
     The curtain right. The second manner is, When the angle flanked is two thirds of the Polygone. VIII. IX. X. XI XII.  four V. VI VII. VIII. 6. 36. 36. 36. 36. 36. 36. 36. 36. 36. 36. 4. 24. 24. 24. 24. 24. 24. 24. 24. 24. 24. 1.52 21.29. 21.09 20.93 20.78 16.97 23.18 22.82 22.55 22.34 22.17 2.71 13.22 13.65 14.02 14.34 16.97 9.23 10.83 12.00 12.90 13.63 3.48 24.84 25.96 26.90 27.71 33.94 13.05 17.53 20.78 23.26 25.19 ●3. 03 23.60 24.08 24.50 24.88 28.97 19.73 20.40 21.10 21.75 22.32 7.90 28.12 28.30 28.46 28.60 28.97 26.26 26.77 27.10 27.37 27.53. 0.66 11.09 11.45 11.76 12.04 12.00 7.74 9.09 10.07 10.83 11.44 4.37 14.69 14.90 15.05 15.15 24.00 7.10 8.03 8.33 8. 41● 8.39 0.61 11.08 11.45 11.75 12.00 16.97 6.21 7.42 8.21 8.77 9.18 1.33 61.43 61.51 61.59 16.66 60.37 60.80 61.10 61.34 61.54 61.72 8.11 48.02 48.00 48.02 48.07 40.97 53.91 53.41 53.44 53.63 53.89 9.05 78.58 78.18 77.85 77.57 69.94 82.36 81.65 81.10 80.68 80.35 1.42 62.43 63.30 64.04 64.69 69.94 54.45 57.66 60.00 61.81 63.26 ●3. 28 114.87 126.50 138.14 149.85 Infinite 58.24 69.45 81.10 92.55 104.98 0.25 91.27 102.42 113.63 124.97 Infinite 38.50 49.05 60.00 70.80 82.66 1.28 22.17 22.90 23.51 24.04 28.97 13.95 16.51 18.28 19.60 20.62          Angles  35. 140. 144. 147.16 4/11 150. 180. 90. 108. 120. 128.34 ●● 135. ●2. 30 85. 87. 88.38 1/11 90. 90. 60. 72. 80. 85.42 6/7 90. 6.15 27.30 28.30 29.19 1/11 30. 45. 15. 18. 20. 21.25 1/7 22.30 27.30 125. 123. 121.31 9/11 120. 90. 150. 144. 140. 137.8 4/7 135. ●0. 40. 40. 40. 40. 35.16. 40. 40. 40. 40. 40.   

THE SECOND PART, TREATING OF THE FORTIFICATION OF PLACES IRREGULAR. BY SAMUEL MAROLOIS, Revewed, Corrected, & Augmented BY ALBERT GIRARD SAMIELOIS.  
THe Fortification irregular being more variable by an infinite deal then the regular, is therefore more difficult and requireth for this cause much more judgement and discretion, for the effectuating of it well, and in choosing the best way to make it then the former. Now to discuss thereof methodically methinks it will not be amiss to begin with the Fortification of a triangle equilateral, that is, of equal sides; which is well contained among the regular Polygones. And howbeit that their angles fall so little and pointed, that they cannot be duly fortified as they ought: yet notwithstanding, I think it best to begin with them, as being irregular places, fortified no otherwise then with a Wall or a small Moat. But if the ground cann afford a good rampart, and a large moat; then I am of the opinion that sometimes it willbe requisite, yea, and for the most part better, that one fortifies such a place out of the enclossure, than otherwise, as I hope I shall demonstrate unto you hereafter.  

The 18 Plate & 84. Figure.  
LEt then the triangle A, B, C, be Equiangle and Equilateral, which ye would fortify without diminishing of the place in anywise, nor yet enlarging it much. To do this, we will divide the sides of the triangle into 3 equal parts by the points F and D, and from the distance F, D, shallbe made the triangle Equilateral D, F, E: the like must be done with the other triangles upon the two sides A, B, A, C, and so ye shall make a body Hexangular, built as a tenaille, without which ye may make a moat of a competent breadth and capable for a place so fortified; and within the inside, ye may cast up a rampart of some 40, or 50 foot, as necessity shall require, all as appeareth by this present figure. But if then afterward, ye find that the matter requires, that the angles exterior may be made, and reduced into Bulwarks, as than ye must follow our general rule, given for our regular fortification, noted before in the 10, & 11 Plates, and shall draw the utmost angles, to wit privy lines, as from G to A, then having resolved to say out the proportion of the curtain to the face, as 4 to 3. ye must upon G, H, make 3 equal parts, and four of the same parts from G towards A, as from G to A. from which point A from the distance of the 3 parts is made an arch towards I, and from H an other arch of 4 parts towards the same point I. cutting the first at the point I▪ by which the line G. I. being drawn cutting H, A. at R. we have AR. for the face, and by this means the said face willbe in like proportion to the curtain as 3 is to 4: Now for to know the flank, the Capital line and the line of the gorge ye shall make an angle of 40 degrees through the line S. M. passing by the point R, and cutting the line diagonal that is, which extends itself from one corner to the other) at S. and so AS willbe the Capital line, S, T the gorge, T, R the flank of the Bulwark, and T, V, the curtain, which hath the like proportion to A. R, as 4 to 3. And though this plote be not so well proportioned as is to be desired: nevertheless the defence thereof will fall out well, being drawn from the midst of the curtain; but because of the smallness of the angles flanked, which are but 60 degrees, the gorge is very narrow, and the flank little enoug: so that the said bulwark will bear no cutting off; neither can contain men sufficient to endure an assault: so that this manner of fortification, hath many imperfections, and ought not to be built in such a sort, when there is a commodious place to make it otherwise: yea it would be better to alter somewhat the angles by making them more blunt, and open, as in drawing the line Q. P. in such wise, that the angle R. Q. P be 22 1/2 degrees, and the face Q. R. comes not without the fortress; then draw the line P, R which giveth the length of the face Q. R, according to the method abovesaid, & the angle being made 40 degrees, you shall have the capital line Q, S, and R, T the flank S, T the Gorge, & T V, the curtain, which proportion is better than the former; because the angle flanked is more open than the precedent, the Gorge larger, the flank greater and stronger and of sufficient strength to resist a furious battery, so that by this means though the place will bear no great alterations, yet it is made much better. Whence appeareth that by a little change a fortification may be much bettered, without any great charge and labour, which ought not only to be understood of this fortification; but also of all others, as we hope shall appear by the rest.   

How to fortify a square fortress with a Tenaille.  

Figure 85.  
FIrst ye must consider, whither it were more fitting, to fortify the angles of a quadrate, them to fortify it with a tenaille, which in my opinion ought to be done, when as the angles of the square are so far distant, that it is very needful, to make a bulwark in the midst of the curtain, then in stead of building upon the angles of the square pointed Bulwarks, as commonly the quadrate Bulwarks are: I would make the angel's BCD, FGH, KLM, & OPQ, in the midst of the sides of the squares, so that the lines, which form the tenailles be all alike among themselves; as A B C D E F G H &c, and the angles also equal one to another, which is done by drawing the two diagonal lines from the quadrate, to wit, A I, N E, which cuts through each other in the centre R. Then the distance A R being taken; and placed from A to D, from E to B, from E to H, and from I to F, and finally having made the angles DCB Isoceles &c, from the distance D E, ye shall have the figure Octoangular form into a tenaille, whereof all the angles willbe right, which afterwards may be made angles of the Bulwarks, when there is accommodation to make a fort royal thereof, & that the distance from angle to angle, be about 80 rod; for if the right angles should come much nearer than 40 or 50 rod, this would cause an imperfection, and such a fortification cannot be called a fort royal: so that in such a case one ought to take advise, and consider well, whither it were not better, to leave such a fortification in the former estate; then to alter it, because the circuit of such a place, will not be worth the time, labour & expense, in regard the Bulwarks willbe too little, the lines too short, and too weak to work their effects: but if so be, the distance of the angles be capable to bear a perfect fortification them ye may make the plot as followeth. Let the line EGLANTINE, be drawn, and divided into three equal parts at the points 1, 2. Then 2 of those parts shallbe placed upon H, G, as here at S. and from the points S. & I. of the distances I, G, and G, S, shallbe made the arches, which cut through one another at the point 4; by which the line G. 4. being drawn, where the same cutteth the line H, I, at the point 2, willbe I, 2; the face, or the skirt of the Bulwark. For to have its flank, the angle XVI must be made of 40 degrees by the line XV, passing through the point 2, & 5, and cutting the diagonal G, R, in the point X, which shallbe the centre of the Bulwark through which point X a parallel line, being drawn to the line I Gas X X, and from the point 2, drawing a perpendicular line upon it, as Y 2, ye shall have all the essential parts of this fortification, namely, the line I 2. shallbe the face, 2 Y the flank: Y X the line of the gorge: IT the curtain; & ZI the line of defence flanking. According to which the angles G, E, C, A, etc. being placed, ye have the distance I 2. and upon the distance I, X, one may make the plot of such a fortification, which was first built into a tenaille, as appeareth by the two Bulwarks I & G, & nothing altered in all this fortification; but the lines of the tenailles 2, H. 5 placed upon ST, TIE, & 2 IN, which in all are but a little longer than the abovesaid 2, H, & H, 5. Now the Earth which is taken from the space T 5, H 2, Y, is sufficient to supply that defect. The moat which was made before, if it be broad and deep enough, ye may leave it as it was without making any alteration in it. And because that in such places, no falsebrayes cann be made, but with great expense; in regard they are made on the outside of the ramparts: and therefore one must make on the other side of the moat a good covert way, according to our former plots, that your men may give the better resistance to the assaillants. Moreover one may (as appeareth by the figure 87 from 5 to H) make a good parapet, which shall run down slooping about some 20, or 24 foot thick, and between H & 5 some traverses, rising one above an other, to lodge conveniently behind them your musketeers, which may give fire upon the passage to the moat, yea, ye may make these traverses in such a manner that they may easily plant two pieces of ordinance upon them, that with them one may beat upon, and break the Enemies' gallery, when he shall offer to put it into the moat, as also to defend the breach, when the assaulters shall attempt any thing upon one of the Bulwarks, which may be done not only with the said pieces: but also by the continual shooting of the musketeers lodged in the said traverses, which would do great spoil & could hardly be entered, because the breach, having no false bray under the bulwark (as there can be none here) would make it more unentrable, then if there were one. For the Earth of the rampart tumbling down into the moat (which is deep, would cause in my judgement the entrance to be more troublesome, then if there were a false bray; seeing an Enemy should be forced with great difficulty, danger, and loss of time, to dam and fill up the said moat, which is one of the chiefest observations of an Enemy besieged, to gain time, and to hinder and foreslow his Enemies' approaches, whereunto me thinks one ought most diligently to endeavour.  
If one finds this manner of cutting off, good, which nevertheless I will leave to the judgement of Captains experienced in the art militaire: it is manifest, that this fortification with a tenaille will be changed but a very little, by the addition of Bulwarks, and so of less expense: for the Ramparts 5. H. 2. ought to be made low, to the height required, and diminished to the thickness of 20 or 24 foot: so that all may be done without any great loss, & time. But if the moats be so deep, that one cannot from the flank 5 see the bottom of them, being dry, or else cannot discouver the superficies of the water (if there bea●●y) and that by the hindrance of the said Earth, or parapet 5. H, 2. It were in vain to fortify it, seeing reason requires, that it should be slighted: otherwise I see no cause, that it ought to be taken away. For to believe, that the said Earth might give some advantage to the Besiegers, that being passed over the moat, they might have the better access to lodge in the said place: the same may be said aswell of a false bray, which nevertheless is approved of the most expert Captains; It is not likely then, that the said Earth can cause any damage, which might cause this to be left undone.    

To fortify a Quadrilatera irregular in the form of a trapeze.  

Figure. 86.  
LEt the Trapezoide (A fortress of unequal sides) be A B C D, where of the sides A D are 69 rod, D C 45, CB 40, BA, 46: and the angle B 108 degrees (and consequently C. 109: 10. D 71: 52, and A. 70: 50, and that ye would fortify the same into a quadrangle.  
To do this, let there be described two Bulwarks upon the angles B, C, the one being the angle of the Pentagone, and the other C approaching, all in proportioning the gorges, flanks, and faces according to the distance BC, saying by rule of three, if 56, 88 the Polygone interior in the tables, gives the face 24, what will 40 rod give, the side of the Polygone given B, C? the like do with the flank, gorge and second flank, according to which Dimensions afterward the said bulwarks FGHIK, & LMNHE. are described.  
And as the distance AD is two great, to describe two Bulwarks upon the angles A, D, which may defend themselves, because this distance exceedeth our rules given heretofore, and the angles too little: ye shall make or ordain two Bulwarks, as is seen in this figure, to wit, when as the said distance AD is great enough, otherwise, ye must make but one Bulwark, as we will show you hereafter. But in this exemple, the flanks of the abovesaid Bulwarks 1. 2. 3. 4. shallbe made perpendicular lines upon AD. and so far lengthened from the angles AD, that there be means to defend the said flanks 1. 2. 3. 4. so that indeed the two Bulwarks abovesaid, are but in effect as Demy-Bulwarks; whereof their angles 1. 3. make 70. degrees, and are defended from one side of the curtain, as a Pentagone is defended, and from the other sides, are the flanks of the said Curtains 2. A & 4. D defended: so that the flanks 1. 2. & 3. 4. are also sufficient to defend the two lengths A. 2. & 4. D. & forasmuch as it is to be doubted, that the nearer one approaches to the abovesaid flanks of the said angles A & D. that they willbe in the most danger to be ruined: It willbe good, to draw back these flanks of the said angles, at the least some 400. foot, that they may not be beaten down with Canon; on the otherside, ye must take heed that the flanks 1. 2. & 3. 4. be not made too great, nor too far asunder from the said angles A. & D. that the lines of defence coming out of the curtain be made in the same sort, as the defences of the Pentagone are, as is seen upon the curtain B. C. And as the distance A. D. is in this exemple but 69 rod making there in 2 Bulwarks: it followeth necessarily, that the distances 2 A: 4. D. also the flanks, and the faces of the said Bulwarks be much more less, than our former plateformes, will bear: in such sort, that the parts A 2 & 4. D. are but 15 rod, & the flanks 2. 1. & 4 3, only ten rod; because that otherwise the line of defence cannot come out of the curtain F. G. as here from the point O, which of necessity must be made to defend the face K, 3. Afterward the face 3 K. being made 12 rod, the flanks K. G. and V, F, willbe about 7 rod, from the side C. D. ye shall draw a parallel of 10 rod, and the instrument being opened of 70 degrees, it must be moved so much, that the line of defence M. T. may come a little out of the curtain E P: the flank P Q. being made equal to E H. ye have the demi Bulwark P. Q. M N. The like is to be done upon the side A. B. and so this place willbe fortified, according to our intention, and in the same manner, as the figure 86 demonstrateth: and though the Bulwarks be very small: notwithstanding by this means the angles A, & D, willbe well preserved; and the one will defend the other directly. And when as the distances C. D. are but 45 rod: one may in some wise from the flank defend the angle D. But when the flank is no more than 6 9/10 rod, one ought rather to make the demi Bulwark P. Q. M. N, having the flank N M. 10 rod, which is more capable to defend N. D. then is E. H. yet with this caveat, that the distance N D be not too little: for then by reason of the thickness of the parapet, it will be impossible to discouver N D. if the demy-Bulwarke P Q. M. N. be not made lower than the curtain C. D. and in so doing such an inconveniency may be prevented: the like is to be understood of the Bulwarks placed upon the curtain A. D.    

An other way.  

The 19 Plate, and 88 Figure.  
NOw in such a fortification, which hath but one simple defence, the angles of the tenaille may lightly be taken in, because that in the figure 86 N M. & N. D. are so short: It were better in my opinion, to fortify the said figure in this form following.  
Let there be made upon the longest side the Bulwark 1, 2, 3, 4, 5, 6, proportioned according to the length A. 6. the half of A. D, and of the nature of the Dodecagone, whereof the gullet (which is the distance from the one flank to the other) maketh 27. 71. and the capital line drawn from the angle thereof through its centre perpendicular upon the said line from flank to flank, called extended, maketh 28. 59 as appeareth by the table here to fore many times repeated, which is the cause of the obliquity of the flanks upon the said curtain, A D, for seeing that the Dodecagone hath the angle in the circumference of 150 degrees, the flanks will fall perpendicularly upon the sides, and consequently the same flanks will make upon the said curtain A, D, an angle of the half of the angle of a Dodecagone, which is 150 degrees, making for every angle of the flank 75 degrees, and as much will the angles A 12, & 45. D make. The rest of the Bulwark in no wise changes the proportion. To know then the Capital line 6, 3. you shall say by the rule of proportion: If the curtain interior and exterior, which are here alike containing 70 rod, in a right curtain giving 28. 59 rod, for the capital line extended, what will 34 1/2 rod, the length A. 6. give? that which it will bring forth, to with 14. 9, is the capital line, 6, 3. for the face ye shall say: if 70 the side of the Polygone interior or exterior in a right curtain (which are equal) giveth for the face 24, what will the side of the Polygone A. 6. 34 1/2 rod, give? ye shall have for the face 11, 83 rod, which being placed from 3 to 4, & from 3 to 2 (first the angel's 6. 3. 4. & 6. 3. 2. being made of 45 degrees, because the whole angle 2. 3. 4 makes 90 degrees by Hypothesis) ye shall set the transporter upon 15 degrees, and shall make the lines 1. 2 & 4. 5 of such an inclination, that shall form the flanks, according to the proportion required, and by the same means shall be known the gullet 1. 5, which may be found likewise by the former rule of proportion, in saying: if 70 giveth 1385 (the half of 27. 71) what will 34 1/2 rod give for A. 6? that which this rule will produce shallbe for 1. 6. or 6. 5. After the same manner must ye make up the Bulwarks, E, F, G, H, I, & S, T, V, W, X. designed in the midst of the sides A, K, & R, P, according to their length. But before they are laid out, the two sides A B, & C, D, aught to be lengthened so far, that one may make the angles B, K, L & C, R, P, of 75 degrees, and that the line of defence O. K. termineth in the curtain M. Q. and to know the length of the faces and flanks, the face shallbe made double in the flank, which is done if ye place upon the perpendicular O N, the half of the line of Defense O K. as from O to N. and from the point K, the privy right line N K, being drawn, cutting through the line C, B, at M, ye shall draw the line M, L, parallel to N, O, or perpendicular upon C, B, and so the flank L, M, will be the half of the face L, K. And thus we have finished the fortification of the abovesaid Quadrilatere irregular, which in my opinion is a better fortification, than the precedent figure 86. because these Bulwarks are more capable to defend the angles A, D, K, R, and the flanks of the said Bulwarks being drawn 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, which may be used against them, and yet make a good defence, as the figure 88 plainly demonstrates.    

An other way to make the fortification of such a place regular.  

The 19 Plate and 89. Figure.  
FIrst of all we will describe the quadrate E F. G H. after such a manner that the line E F is parallel to A. D. & of the greatness of the poligone exterior; in the following table of lengths: drawing the parallel E, F, so, that its distance A, D, be equal to the distance of the Polygons', aswel interior as exterior, to the intent, that the side of the Quadrilatere A, D, may serve for the curtain. Then the angles I, F, E, & I, E, F, being made 15 degrees (because the angle interior flanking is of 15 degrees in the square) ye shall take upon a scale 5. equal parts & place them upon the line F, E, and four of the same parts upon the line F, I. from the extremity (or utmost end) whereof and with the said distances ye shall make two arches, that shall cut through one an other at X, drawing from thence a line to F, cutting through the line, I E, at G. then E, G, willbe the face of the Bulwark, which will have the like proportion to the curtain, as 4 to 5. But we will here after in all the kinds of Polygones, ordain the curtain to the face in proportion as 3 to 2. that is sesquilatera, finding it best, as we have said before. Afterwards, to have your flanks ye must make two perpendicular lines G N. and H K, upon the side A. D. in such manner, that the lines G N and H K, shallbe the flanks, and N K the curtain, which is part of the side A D. The like ye shall do with the three other sides, and so by this means this fortification willbe made regular, and royal, which will not cost much more, than the former irregular fortification abovesaid, the benefit whereof surpasseth the others by far: 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 will permit it: As for the moats, ramparts, and parapets, they must be made as we have 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 have fortified in such sort, that the Bulwarks come again to the angles of the propounded figure; To do this, ye shall first measure the outsides, and the angles, which I supposeye shall find to be as they stand here underneath.  

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 seeing the angle A, is lesser than 90 degrees, reason requireth that the said angle be not fortified; because it would make an angle flanked, lesser than 60 degrees, and flanking greater than 150 degrees, against our former maxims, according to which ye shall make of the same angle A. an angle of the bulwark, making the angle of the Polygone F. so that the right lines FG, and FI, come to cut through the lines lengthened BC, DE, in the points G and I Upon the angles whereof, and according to the proportion of the sides shallbe described the Bulwarks, takeing heed, that the angle of the Polygone show what form of a Bulwark one must built upon, to wit, a Quadrate, a Pentagonal or an exagonall, proportioning out the parts of such a Bulwark, according to the least side of the two, and then the figure willbe described, according to this present form: And seeing the side DE (because it hath been lengthened) is longer, than the proportion can bear of our regular figurs precedent: It willbe necessary, that between the two Bulwarks D & E, a Ravelin be made, which is a loose piece, that may be defended at leastwise from the flanks of the abovesaid two Bulwarks: according to which flank, the angle of the Raveline shallbe made a little more open, or closer as the curtain is either long, or short. The faces whereof shallbe made of 18 or 20 rod, some times a little lesser, as the place and situation of the ground requireth. And to give you to understand more clearly my intention touching the fortification of places irregular, whereof the angles are no less than 90 degrees, which is the angle of a Quadrate, and that the sides do not differ much from those places which are regular: It must be proportioned thus: Suppose that one giveth me the angle C to be fortified, of which the magnitude is III degrees, which comes near the angle of a Pentagone, according to which I take the shortest line of the two BC; & C, D, making therewith the angle B, C, D, which is C. D. containing 46 rod, we will seek 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 giust the face 24, what will then a Polygone of 46 rod give? ye shall have for the face 19 41 rod. The like ye shall have for the flank, and then the Gorge, where by ye shall find the said Bulwark C. as also all the other parts of this fortress Pentagonal: holding this for an infallible rule, that the angles of the Polygone, which you would fortify aught to be at the least right, and in case there be any angle that hath a less opening, than the right, ye must make thereof the angle of a Bulwark or else draw a line (if you cannot lessen the place) which will form an angle competent to build a Bulwark upon, as appeareth by the 90 figure in the angle A. Which line you must so husband, that if it be possible it may be made equal to that, which shall come to cut through it, that so ye may have a new angle, whereupon ye may make likewise an other Bulwark, even as the said figure F, G, C, D, I, demonstrateth, and if the lines exceed by much the length of the Polygones interior, than ye shall take them for sides of the Polygones exterior, in tracing out the said fortresses within them, & that according to the kind of every angle F, G, C, D, I, and in so doing your Pentagone will be fortified.    

An other way.  

The 19 Plate & 91 Figure.  
LEt the Pentagone be A, B, C, D, E, which you desire to fortify after an other manner then above: for the effecting of this, ye shall prepare the angles of the Bulwarks, according to the scale of the figure, that you would have fortified, such as before ye have traced out, all separated one from an other, as they are marked out in the 21 Plate by the numbres 97, 98, 99, 100, 101, which angles shallbe placed upon the angles oft figure to be fortified, observing well by what means the said place may be best made up & with the least expenses, according to our former Maxims, and to that end, if it be so that the place be girt about with a wall or a rampart, and that ye would make as much use of it, as possible might be for the saving of charges the which (though ye help yourself with the old fortification) is always great enough, at the least ye must labour, that the sides of the Pentagone be accommodated between the two Bulwarks, to serve as a curtain, as we have done in the figure 91. where we have placed in the angle A (which is less than 90 degrees) the angle of a Pentagone, or else the Bulwark of a Pentagone, that may be removed so long till ye shall find the buy finesse, and the disposition of the place required drawing from the centre of the said Bulwark F, two lines, as F, G, & F, I, serving for curtains, and meeting with the other two sides of the Pentagone lengthened in the points G, & I. Upon which angles G, C, D, I, ye shall form after our former rule (to wit, according to the angles and the sides) the Bulwarks upon them, and if ye find, that the sides are somewhat longer, than our precedent rules approve of, in the midst thereof we will cast up a good Raveline as we have done here between the two Bulwarks D & H, and take this for a general rule, that your Bulwarks ought not to be any further asunder than some 60 rod, or thereabouts, to wit, that the line of defence fichant, be of this length or near upon it. If it be so, it would not be needful otherwise to make a Ravelin, but when as the said line exceedeth by much the said measure, then is it wholly necessary, that the said Raveline be placed in the midst, between the two Bulwarks abovesaid, to supply thereby the defect of the said line, & the better to blind the flanks which by that great distance would lie very open.    

An other way.  

The 20. Plate & 92. Figure.  
IF the place permits to alter the angles a little, than the fortification willbe the better, as is seen in the Pentagone A, B, C, D, E, which is the same with the former, where the angles of the Bulwarks are changed, and some faces of them unequal, and so by this means the Bulwarks will not be too far distant one from an other, as this present figure 92 demonstrateth. But in such a case ye are constrained that the curtains comes or runs a little out of the sides of the Pentagone, and sometimes cuts through them, as the same figure showeth, and seeing it were hard to trace out this figure so without some direction, ye shall make use (as we have said before in the 19 Plate) of Bulwarks cut out in pastboord, marked in the 21 Plate, according to the space of the angle, which ye would fortify as we have done here; for the angle A being 72 degrees, ye shall make use thereof, then coming to B, and finding it to be the angle of an Exagone; I take the Bulwark of the Exagone, and place it at B, turning and removing it so long, till the line of the tenaille comes to cut through the line AB, as here at O. making A, O, and O, F, equal, that the defence may be made thereby alike: for I am of the opinion, that the two faces, flanks, and lines of defence ought to be equal among themselves, because they must necessarily defend each other alike: seeing the defence cannot be unequal without diminishing the force of the one or of the other, whereunto ye must take special heed, as much as possibly may be, to hinder such an irregularity, that the one receive no prejudice by the other, and so the faces F, H, and AL. willbe equal, but FH, willbe unequal with FG. the second face of the bulwark H, F, G, and yet so, that the said inequality causeth no great hindrance to its form, and force. For first the face H, F, cannot be made no longer; in regard of the distance A, F, which may not be much augmented, unless the Curtain M, N, comes much out of the line B, C. Also the line A, F, cannot be augmented unless consequently the line, or distance F, C, be not more and more augmented so that always the same difficulty remains, to wit, that the Bulwarks F, & I. willbe of a greater distance than the Bulwarks F, & A, and thus the distances differing, will make also differing faces, curtains, and flanks: for according as the space is great between F, I. so ye must have more men to defend it, then if the place were lesser. Now seeing this defence cannot be made more commodionsly, then from the faces of the Bulwarks: It is most certain that the faces ought to be augmented more than the former: Yet never the less without passing the limits of the former plots in our regular fortification, which have their faces about 24 rod, some more some less: but if they be much longer, than the lines of defence will be of such a large extent, that it will be almost impossible to make any use and profit of them. For which reason I have thought fitting to make the augmentation of the faces F, G. & K, I, according to the distance F, I, remaining notwithstanding within the bounds of regular fortification; and within the compass of the maxims grounded thereupon, written before. And ye must note besides, that the interior angle flanking F, A, L, or A, F, H, must never be less than 15 degrees: and observe also, that the more open this angle is, the more closer the angle of the tenaille will be, and consequently the better. But ye ought on the other side to take heed, that the angle F, A, L, being large and open, the line A, F. be not lenghtned to far from the body of the figure to be fortified: so that herein ye must use discretion requisite in such a case, and so accommodate the said bulwarks cut out in pastboord that the said angles be tolerable, and that on the other side the curtains be not too far 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 Bulwarks B, & C, fall upon the side of the Polygone B, C, so that the part G, H, may serve for a curtain; you must do as followeth, the angle B, A, F, is made of 15 degrees, because that in making it greater, the line FAVORINA would run too far from the place A, B, C, D, E: them ye shall make the line parallel, F, G, distant from the line B, C, as far as the Polygone exterior is distant from the Polygone interior, in the Pentagone; because that the angle G, ought to be Pentagonal, and if the line F, G, is shorter, than the Polygone exterior in our table of lengths; ye shall finde out a proportionable distance, in saying: If the side of the Polygone exterior 8125. giveth distance to the Polygones 1677 what distance will FG, give, that which this rule will produce shallbe the distance of the parallels B, C, & F, G, and so consequently the parallel F, G, being made, from that distance ye have the thing required, and before ye begin with the Bulwarks, it is your best way to make the lines GH, HIGH, & IA, and as the line B, C. hath served for a curtain, the line C, D, may serve for the most part thereto, seeing that the line GH, is not ordinarily parallel to the side of the Polygone irregular, C, D, but the side E, D, may be accommodated as before: so that one part of it will serve for a curtain to the two Bulwarks; which shallbe made in the angles I. & H. so that if the angle D, had not been so sharp, and that the line G, H, might have been parallel to E, D, it is evident, that one could have made use of C, D, for the curtain of the two Bulwarks, G & H. Or else if the line F, G, might have been somewhat augmented, the curtain would have fall'n much nearer to C. D: But seeing in this exemple it could be no longer, of necessity the said curtain must fall within the inside of the figure Pentagonal irregular. Then having drawn these your lines A, F, G, H, I with all circumspection requisite, to wit, that the angle flanking interior be at the least 15. degrees: ye shall draw out upon them the Bulwarks, flanks, and curtains, in such sort, that the faces & flanks of the Bulwarks, which are upon the one side of the Polygone, be alike among themselves, as those which are noted in the 92. Figure, & according to our former rules given in our regular fortifications.    

A way how to fortify a right Curtain.  

The 20. Plate & 94. Figure.  
IF it be needful to fortify a right Curtain, whereof the angles of the Bulwarks are right, ye must do as followeth. Let there be taken 70 rod, wanting 6 seconds (that is 69. 64. rod) and put upon the said curtain A, E, as many times as the said curtain will bear it, as appear here by the points A, B, C, D, E, from which points shall rise the privy 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, shallbe placed 16, 97. rod, as from A, to L. and at M, raising the perpendiculars L, N, and M, O, 12 rod, the flanks 6, 11: then the lines NF, and FOE, being drawn, which will make the faces of the Bulwarks, ye shall have that which is necessary for the description of such a Bulwark, whereof the face is 24, rod, the flank 12. the line of defence fichant 60, 37, and the line of the gorge L, A, 16. 97, rod. The moat may be made broader, seeing that the angle of the Tenaille Z hindereth, that the angle of the flank T, cannot discover the angle flanked F: but if the expense were not too much (which happeneth when one makes the moat very deep) one might to that end cut the part X, Y, Z, and so this inconveniency would be remedied. For I find that such Bulwarks are far better, than those that are made upon an angle; because the gorge is very large; the flank, the face, and the curtain of a competent measure, and according to our former rules, to wit, the line of Defense fichant is 60 rod, or thereabouts, the Curtain 36, the flank 12, the face 24, and the gorge well nigh 17, rod, which is much better, and larger, then in the Bulwarks, that are made upon 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 mind, as for to have more place, or otherwise. Now touching the distance of Bulwarks, or a Polygone interior mentioned of 70 rod wanting 6 seconds, these 6 seconds are of no great moment: nevertheless one must observe the dimention as near as possible may be: Otherwise one might say, that the face being 24, the curtain 36, the flank 12, rod, and the angle flanked right, that then the Capital willbe well nigh 29, the gorge 17, the defence flanking 41. the line of the Poligone interior, or exterior 70. rod, the formed-flanked is 35 degrees, and 16 minutes.    

How to fortify a Hexagone irregular.  

The 21. Plate, & 95, 96. Figure.  
LEt the figure hexagonal to be fortified, be A, B, C, D, E, F, whereof the length of every side maketh as many rod, as they are marked out, to wit AB. 70: BC. 132: CD, 114: DE. 80: OF, 124: & FAVORINA, 176 rod. To do this, ye must take notice of the angles thereof, and as their greatness is ye shall order the angles of the Bulwarks, according to their forms. And for as much as the angles A & F, are but 108, & 110½ degrees, which are the angles of a Pentagone, it will be good to make there the angles of demi bulwarks, to have the angles more open, and the angle of the Tenaille more closer, and consequently better. Upon the curtain FAVORINA, shallbe placed two Bulwarks G & H, of a competent greatness to the curtains, proportioning out the Capital lines, the gorge, flanks and faces, according to the greatness of them, & as we have said before in saying: If 70, (which is the distance of each angle, or the centre of the Bulwarks, gives for the Capital line 28, 97. what will the distances of the centres of the bulwarks give? that which this rule will produce will be the Capital line, and in the like manner shall ye find the line of the gorge, the flank, and the face of the Bulwarks H, &, G, the Bulwark I shallbe made in the midst of the curtain M, N. or in the midst of the line E, F, and for as much as the angle E, is of 112. degrees, which comes near upon the angle of a Pentagone ye shallbe upon the foresaid angle E describe the angle of a Pentagone, and seeing that the line E, D maketh 80 rod, we will make the angles flanking interior of the form of a Pentagone, to the end, that the skirts E, L. and D, O. be equal, as we have said before: for seeing the Enemies' force between E & D is equal, reason requireth that the defence thereof be likewise made equal, that by this means, ye may take away all occasion from an Enemy to attempt any further place the most advantageous for him. Now in regard that the distance D. C. exceedeth the measure, which we have spoken of before, to wit, of 114 ½ rod, it will be necessarry to make the Raveline K, between the said angles D, & C, to supply the defect of the defence: the like may be made between C, B, and the angles of the Bulwarks, and B, the rest shall be made according to their forms & the faces, flanks and Curtains in that form, as we have said above, even as this figure hexagonal 95. demonstrateth.   

The 96. Figure.  
HErein it is requisite to note, that when one is bound to fortify precisely the angles of the figure, either interiorly, or exteriorly, ye meet many times with difficulties: to wit, here are some angles or sides too small, and others too great; in so much, that this irregularity may cause many great defects. Which may greatly be remedied, when ye may have liberty, to change a little the angles of the figure, even as we have 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, upon the side ED, that by this means, ye may have the angle S, E, T, broader and more open, drawing the line E O so, that it be equal to ED, and that the angle O, E, D, be 20 degrees, so that the curtain Q. P. comes not too far into the figure, and that the line of defence ER, may come a part of it out thereof, if it be possible, as here to R: for the more the said line cometh out of the curtain, it is so much the better, which is, when the angle O, E, D, is broad and open. But this exemple if this had been observed the Bulwark L, had been a great distance from the curtain C, D, and would have made it much longer than it is at this present; Then upon the point O, shallbe made the second face of the Bulwark, W, O, V, & so made that O, W, and O, V, and ES, are of an equal greatness, & that the line D, X, be drawn in such a manner, that the angle Y, Z, 5, be capable to receive the Bulwark of an Exagone, and for the better attaining to such a structure, ye shall make first upon the line C, B, the flank noted by 4, 5, and the face Z, 5, to the end that CB may serve for a curtain, and so fitted, that it may almost be equal to the curtain D. C. Then upon CB, ye shall make the Bulwark I. according to the greatness of 4, B. answerable to the Bulwarks G. H. and seeing the angles A & Bare sharp: ye shall make these two Demy-Bulwarks, according to our method mentioned in the 18 Plate and so ye have finished the thing required.   

Note.  
AS we have fortified this figure on the inside: so ye may do 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 interior fortification, than the exterior; whence appeareth, that there are many ways to fortify places irregular, yea, an infinite number, yet bounded with these limits, to wit, that the angles of the Bulwarks ought not to be no further asunder then 80 rod, at least 60. that the angles flanked must not be less than 60. degrees. That your line of defence ought not to exceed much above 60 rod for by how much the Bulwarks have a second flank, by so much they are the better; & the more spacious and larger the flanks and gorg●● 〈◊〉, the Bulwarks are the better for them, according to our former Maxims set down in the end of the first part, according to which an expert and skilful Ingenier, will be sure as much as possibly may be, to have all these advantages abovesaid. And for the better facilitating of what is said above, we have made here the figures 97, 98, 99, 100, 101, which must be cut out upon a pastboord & drawn out upon the same measure, as the plot of the place requireth which is to be fortified, and to fit them to the places of the figure, with the greater consideration that possible may be, that he may follow the rules abovesaid as near as may be, where unto these figures (in my opinion) are of very great use: for he may turn, and remove them on what side soever he will, and after he hath found the most convenient place than he may join & fasten them together with a little wax, that afterwards he may overcast the advantages, and disadvantages, which he is to expect.    

How to fortify an Irregular place lying upon the side of a river.  

The 22. Plate & 102. Figure.  
LEt the form irregular, which ye would have▪ be the plate of Harderwijck, noted by the numbers 1, 2, 3, 4, 5, 6, 7, 8, whereof the extremities (or utmost ends) are 1 & 6. touching the Dike A, B, and C, D, which ye desire to fortify. To do this ye must first overweigh, how many bulwarks the circuit, of the same place will take up, husbanding them so, that ye must make as few as may be, because they are parts of a fortification, which will cost much, and yet so, that you must not place them so far asunder, but that the one bulwark must help to defend the other: for this reason we have made the line of defence about 60. rod, which is a longer distance than is given when one is to defend them with the Muskett, or caliver. The others that must be defended with the Canon, may be 1000 foot distant one from an other, or there abouts: because at the least, it will carry so far over, that oftentimes a Canon will carry much further than to the utmost end of the line of defence; so that they may hinder the batteries, which are made to beat down the flank of the angle: from whence the line of defence is drawn, also ye make it so that the distance of the said angle unto the angle of the Bulwark be not so far as a Canon bears, but rather shorter: for which reason we have made it 1000, or a 1200 foot at the most. Now experience hath taught us, of what little use such a defence is, seeing a Canon cannot be used with that agility as a musket, and for some other inconveniences, which happens to ordinance, when they shoot from above downward. It is in no wise advisable, to make the Bulwarks so far from one an other, and it were better to increase the charge, then to let them be so far asunder. For to pretend that when the said Bulwarks lie so far distant, one from an other one hindereth the spoiling of the Flanks, me thinks this reason is too weak to take place, seeing the continual defence made with musketeers hinders much more Besiegers dessignes then that which is made by Canon shot, because that in the interim, while they are a chargeing and making ready, the Besiegers may advance much, and at last get into the moat, the bottom whereof being filled with fagotts and earth cast into it, makes the way accessible, and being ones got over, they come presently to make a mine in the bulwark &, to blow it up: but if the Besieged have good store of ordinance so that they may play continually upon them; than it will foreslow, and hinder them much. But if these places be so built before, that store of Canon cannot be planted upon them & that it is uncertain, whither the time, the lord of the town, and other accidents will permit, to furnish them with so many ordinance, and other necessaries belonging to Bulwarks: which if they were, yet the charge of the said pieces would at last grow so excessive, that the benefit which one receive by it, would finally come to nothing: if the Besiegers should take in that place, considering the loss of their ordinance, would often times be of a greater value, than the place itself; so that this need not be observed; but in cases of great necessity, and where it concerns the conservation, or the ruin of a kingdom, or a country, and that ye are sure to be assaulted by a mighty enemy: which must be beaten off and resisted with an number of ordinance, musketeers, and small shot, as the Great Sultan, who commonly comes into the field with two or three hundred thousand men caring not much for the loss of the lives of this men, in such sort that the space, which is between two Bulwarks, may be easier taken in, when it is small, then being great, for the more the said Bulwarks are distant one from an other, the more men ye must have to man the space which is betwixt them, which would be a disadvantage to the Besiegers: But on the other side, these Bulwarks ought not to be so far asunder, but that the one may defend the other, and in such a case I find it fitting that there be made two, or three Casemates in the flanks, which ye may when as the curtains are so long, that they may be made there great and wide enough, to have the more places of defence which cannot be so soon ruinated by a large space, as by a small.  
According then to these considerations, let us come now to the fortification of such a place irregular, situated (as we have said above) upon the edge of a river, and observe, that commonly, such places are more long, then broad, or deep: because the greatest part of the Inhabitants seek to have the accommodation of the river, and as ordinarily towns are made after a good part of the house are built, by the water side, whence it appeareth manifestly, that such towns are always more long, then large. Which is the reason that the sides 1, 2, 3, 4, 5. 6. are more difficult to be fortified, than the others parts thereof, seeing the angles 2. & 5 are much less, than the others, as ye may conceive by this present plot, and when ye will not go far out of the circuit: It is wholly necessary, that about the angle 5. ye make a bulwark which being pointed, the angle of the Bulwark must be so likewise, as appeareth by the angles of the Bulwarck D, & I. whereof the faces are about 20 rod, and the flanks 8 rod, accommodating them so, that as much as possibly may be the face, or the skirt of them, as O, P, or Q, R, looks directly towards the dike C, D, to the end, that the said face may have the more force against the said dike CD: the like is to be understood also of the Bulwark A, or E: the other Bulwarks shallbe made as the exigency of the cause requireth, as is here noted by the Bulwarks B, C, D, and F, G, H, I, whereof in all this circuit ye find not one Bulwark less than the other; But as on the one side the charge is somewhat less, so on the other side the Bulwarks are so far distant one from an other, that the defence, which is made from the one bulwark to the other, either with musket, or calliver, willbe almost in vain: in such sort, that the discommodity, which one shall reap on the one side, willbe much greater, than the husbanding of it so as to save the fortification of an other Bulwark: Therefore the best way is to make one Bulwarck more, then to deprive it of its best defence, which may be made from the flanks, and so we may say, that the fortification which is accommodated upon the said irregular place, marked with E, F, G, H, & I. is better, then that which is marked with the letters A, B, C, D, because that the lines of defence in this are much more greater: for which one ought to take diligent advice that he gives himself no disadvantage, for an Enemy who will employ all his knowledge & industry thereto will give him enough to do, so that he needs not give himself any: in such sort, that this observation is absolutely necessary. The Bulwarks than ought to be so large and capable, as that they may lodge within them a sufficient number of Musketeers, to defend the space, between the two Bulwarks. For if the said Bulwarks be too far asunder, ye must then defend a large space with as many men, as you would do a little, which notwithstanding is against the order of defence; whereby it appeareth, how requisite it is to observe therein a good proportion, to the end, that the expense, time, and labour may be answerable to the benefit and profit which one ought to receive from it. Moreover the said Bulwark ought to be made so large, that they may the better resist (with a body of pikes and musketeers lodged in the said Bulwark) at that time, when as an Enemy shall give an Assault upon it, and seeks to take it in, and by that means such Bulwarks may be better cut off, than those that are small, as we shall discourse more at large hereafter when we come to treat of cuttingh off: as well in general as in particular and so reserve this discourse to its due place; to the end, we may speak more particularly of the fortification of this place irregular, which I rather choose (to make the Bulwark, I, the stronger) that the side of the Polygone were drawn towards the point C. for the preventing of the smallness of the angle, 5. which causeth the smallness of the angles P & R. for without all question it is better to draw the said line, then to leave the plate, as it is at this present, to the intent, the inconveniency abovesaid may be avoided; but ere ye do this, ye must consider, if the line so drawn, be capable to defend well the face of the Bulwark Q. R. which then would fall back almost upon the angle, 5. so that one might thereby gain a Bulwark, and the defence would be much better than before: for otherwise, it were better that the Bulwark should remain, as little as it is, then to make it large, and open without defence. Moreover if the Dike C, D, have any passable way without it towards the river side, it were dangerous to make the said rampart against it, and much better to leave the brickwall 5, 6, as it was, and to make the fortification according to this draught, then to make it as is abovesaid: unless ye intent to take away the dike, and in stead thereof make a sufficient wall to resist the water, as the brickwall S, T, extending the same from T to D. But seeing experience hath shown, what danger there is to take away an ancient, solid, and a settled dike, almost grown permanent, to build in the place thereof a heap of stones or bricks of sufficient strength to stop the violence of the water, one ought to consider maturely, whither it were not better to fortify the plain, which is here our question, according to our former rules, then to hazard the drowning of all the Country, to the dishonour of the Ingenier, who gave his advice for such a change, to the damage and hurt of the Inhabitants, aswel of the town itself as of the country, yea, by endangering the subversion of many houses, villages, together with the ruin of man and beast. Which an Ingenier of a sound judgement ought well to consider & not lightly give consent to such an alteration, unless it be in a case of extremity, or else in consideration of a very great advantage, which one might receive thereby.  
If the brickwall ST be much raised, as it often happeneth there were the country lies low, and the rivers flow high, that one cannot well defend from the curtain the face R. It were good to make the Bulwarck H, Y, Z which is 9 & 10. 11. A, and the Demy V, X, ●, which would lessen somewhat the circuit of the town; but would hinder much the former inconveniency, and the charge would be lesser: for as then there would be but four Bulwarks, which would have the lines of defence capable, and able to defend the same Bulwarks, and so the Bulwark O, P, or Q. R, would not come so near the Dike C, D, which being wholly commanded by the said Dike, the force will be lesser: for the small distance between Q, and C, or from T, to R, would cause the Besiegers to make that Bulwark continually ineffectual, as well by Canon as Muskets, as by casting in of fireworks into it, which could not be done, if this bulwark stood backward as here H, Y, & Z do, and yet is near enough to defend well the face X, ●, as also the moat: even as this form demonstrateth, marked with the double blind lines, 12, 9, 10, 11, A.  
That which we have spoken of the side 5. 6 may be said also of the side 1, 2. But if the defence of the Bulwarck E, be better than of the Bulwark F, reason requireth, that it be left so, without altering the said Bulwark E, by takeing diligent heed, of the advantages, and disadvantages, which the site of the place may bring to such accidents as may fall out, according to which ye must order & accommodate the Bulwark: for most often the seat is that part, which ought most of all to be had in consideration, not only in this figure, but also in all others whatsoever they be, either regular, or irregular.   

Note.  
THat in maritimate places, such as are situated by a river side, where the banks, or d●kes, come to touch, ye must draw them in such a manner that the skirt S, 6. may be as far distant as possibly may be from the extremity, or utmost end 6, that the said space S. 6, may help to defend the furthest part of the said Dike. D, C.    

How to fortify a place situated by a river side, where it is necessary to make also some fortification on the other side of the water opposite to it.  

The 23. Plate, & 103. Figure.  
LEt the town irregular be A, B, C, D, E, F, G, & H, having many other oblique corns, as well interior as exterior, situated upon the side of a large river, about some 200, or 300, foot broad & that the side M, N, oft said town (as it is ordinary) is in no wise fortified; which place you are desirous to make strong, and capable of resistance. To do this ye must first draw the lines FE, DB. so that they fall not into the old moat, but that there be left space enough to make a rampart thereupon, on the outside whereof ye shall trace out your Bulwarks, according to our precedent rules: so that the lines of defence do not much exceed above 60 rod, which is within a musket short (as we have often times said) to the end, that the one Bulwark may help to defend the other, as the art of defence requireth, without making many angles, or sides, as much as possibly may be, to save charges: because it is certain, that the Bulwarks, which are made upon a right curtain, are stronger than those that are made upon angles: for seeing (as we have said at the first) that the figures of many sides are better to be fortified, than those which have fewer: for as the quantity of the sides diminish, so also the greatness of their angles diminish, which afterwards cause the greatness of the angles of the tenailles, or Flankers: and the more the same angle flanking is open, the less is it capable of flanking; contrariwise, the more the said angle of the tenaille is pointed, it hath always been found the better: so that this is without all dispute. According to which; seeing that the greatness of the angle of the Polygone begetteth the smallness of the angle flanking, & that the, more open the angle is the nearer it comes to a 180. degrees (which make two right angles) it is manifest then, that Bulwarks made upon two right angles (making a continued line) are better, than those which are made upon some angle lesser than the two right angles are, and hereby we have sufficiently proved the goodness of those Bulwarks, which are made upon right lines, which is the cause (as I believe) that a great Mathematician consumed his time wholly in this point, by maintaining that all fortifications, aswell great as small ought to be made in a square-forme: but seeing that the Bulwarks towards the 4 angles by this means become less forcible than the others, as the figures A and B, demonstrate in the 23, Plate: me thinks, this aught to be taken into consideration. Whither it were not better, to make a Fortress, whereof the Bulwarks and the lines of defence be of a like force, then to make them as above. For it is impossible, that one should make a fortress 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 cut his coat according to his cloth; But when the situation of the place, and the avenues thereof be such, that one may be assured of the resistance, which may 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 plain 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 than a● other, without hurting, and weakening of an other, which one ought maturely to consider, and not yield easily thereunto without good and pregnant reasons. And seeing these Countries, which lie low, and are subject to inundations & overflowings, the rivers are commonly bounded in with Banks and Dikes, for the preventing of such inconveniencies and the preservation of the Inhabitants from an utter subversion. These dikes coming to touch both the one and the other part of the said towns (as here in the points H, & A.) which sometimes are separated from the said town by a brickwall, which goes from A, to N. and from M, to H, which in the Figure C is called a Doudan made in the form of an Asses back, narrow in the midst, and bending downwards on both sides, which is made over a moat (to stop the water, which otherwise would break into the land) is called by those of these countries a Beer that is, a Bear; in regard of the strength, which makes it almost inviolable. Therefore the Basis, or foundation of such a work, is laid first with a grate of beams of timber, locked one into an other with squared beams bound fast together; upon which the brickwall is built: these beams and piles which are driven in and laid in this groundwork are some 8, 10, or 12, foot long: according to the depth of the river; and about 7, 8, 9, or 10, ynches thick, lying two or three foot distance one from an other, which also ought to be in length answerable to the depth of the water. Sometimes these Bears are made wholly of timber, and are lined with huge thick oaken planks, between them close together, and are much longer than the former; because the upper ends must stand of much above the water, as ye think the water can rise in winter, and on both sides of these piles, you shall say two great beams or bands of timber, the one at the ends of the piles, and the other in the midst, between the bottom, and the upper end of them, fastened together with iron bolts (which are as thick as these piles) and passing through these piles, and then ye shall line them with good strong oaken boards, in joining them as close together as possible may be, that they may keep out the water the better, and last the longer. Now if ye resolve to make no such separation, but to let the bank or dike stand; than it willbe good to cut and pair it as narrow as you can, to keep an Enemy from coming upon it with many men in front, or to hinder his approaches the better on that side. I am of the opinion also, that ye ought to make the line C, B. & F, G. about 300, foot long, that one might give the more fire upon an Enemy, both at his falling on, and going off; but when there is no fortification made on the other side of the river, it were much better that the dike did not stand against the point A; but that it were made nearer to the inside of the town, that one might the better offend the approaches made on the outside thereof, to wit, towards the river, as is shown in the former Figure and 22 Plate: but if the other side of the riner lying opposite to the town, aught to be fortified, (for the reasons abovesaid) than it matters not greatly: seeing one may sufficiently offend the Approaches on that side: and because the cutting off would be chargeable, if it be made in a circular form (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. may be of the length of one of the sides of an Octogone, or thereabouts, that the Bulwarks, O & P. may be well defended from the curtains, and that the Bulwark also on the other side, may likewise help to defend the curtains: then between O and P. according to their distance may be made the Bulwarks Q. and R, but seeing the distances E, F. and B, D. are too long 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 need requires, to make also therein some special cuttings off, as we shall declare unto you hereafter.  
Note that if the lines D, B, F, & E, be too short to make the Bulwarks upon them, marked S, & T: it is apparent ye may then lengthen the sides so far, that the said Bulwarks may (with conveniency) be made upon them, than ye must draw a line parallel to D, E. but if the distance D, B. and E, F. be so, that the angles B & F may be defended well from the Bulwarks E and D. as then you need not make the said Bulwarks S. and T: seeing that from the others, namely. E & D, they may be sufficiently defended. And for as much as I find these fortifications to be best, which come nearest to the dimensions given before in our regular fortifications, termed Royal, whereof the faces, flanks, defences, and gorges (which are the principal parts of a fortification) are all capable to work well their effects: one ought to have a special care above all things, to fit the sides of places to be fortified, that they may be almost of the length of Polygones, which ye shall find in the table of our dimensions, described hereafter, even as we have done here in lengthening the sides B, D, and F, E. till that X, Z be equal to the lengthened sides B, X. and F, Z. Upon which, and in the very midst of them, ye shall make three Bulwarks and upon the angles X, and Z, the two Bulwarks X, and Z, which are the Bulwarks of an Hexagone, because the angels X, and Z, make each of them 120 degrees, which are (indeed) the angles of an Hexagone as appeareth by the said table, in which are taken all the dimensions belonging to an Hexagone, aswell the gorges, flanks, as the faces: and from them, ye may draw out your Bulwarks X and Z, abovesaid, which willbe more royal, than the two Bulwarks D, and E, yea or of the others likewise: so that this fortification will not cost much more than the former, making the place more spacious, the circuit almost alike, and the Bulwarks much better: and in my opinion, this form of fortification ought to be preferred far above the other, considering what advantages one may get thereby. But if one were tied to the form B, D, E, F, whereof the two sides B, D. and F, E, are longer than 90 rod, which is a distance too far for the Bulwarks D, & E. to help to defend the angles F and B, one should be driven to make the two Bulwarks S. T. lesser than the former, whereof the faces would make but 16, rod, which notwithstanding willbe capable to defend the abovesaid Angles F, and B.  
Note that if the place requires, that ye must lengthen the line O, P. as far as the rive side I, K. lieth, than ye may draw from the furthest end of the Bulwarks O, and P. right lines to I, and K, and then ye may make two Bulwarks upon the said lines, one upon the right curtain, by which ye shall make your fortification the stronger, because the angles of the Bulwarks O, P. willbe so much the larger, and the defence of the other Bulwarks upon the said lines the better.    

How to fortify a place situated by the seaside, and to make thereunto a commodious Haven.  

The 23 & 24. Plates, and the 103. & 104. Figures.  
PLaces lying by the seaside, have oftentimes need of a good haven, aswell to harbour such shipping, as may daily come into them, as also to keep them safe from an Enemy, and from the violence of storms and tempests: And for as much as these places must resist the injuries of wind, weather, and the raging billows of the sea, as is said, they ought to be sound fortified and kept: and because Earth alone will not be sufficient to withstand the beating of the sea upon them in regard of the fragility thereof, which would be washed away, such places ought to be made with freestone, or at least wise with good brick, with driving in on the outside of the wall next the seaside many strong piles, by clasping them together in this manner following:  
FIrst some 300. foot, or thereabouts from the wall A, B. (marked figure 104) ye must drive in a row of piles a foot thick in diameter, and some 8. or 10. foot long, as the 24, Plate, and the 104, & 105, Figures demonstrate. These must be pointed with iron, and sharpened well, that they may enter the better into the ground, and must be driven in with an Engine called in Dutch a Hie, having a block plated with iron of some 1000, or 1200; pound weight, which falls down upon the heads of these piles, and is drawn up with a pulley, with some 40. men, even as one should ring at a Bell, four or five of them guiding the Hie, that the log may fall down perpendicular just upon the head of the pile, that they drive in, and shall drive it in so deep, till the head thereof be but three, or four foot above the ground.  
After ye have driven in the first row, some six or seven foot distance from it, ye must drive in the second row, whereof the upper ends must be above the ground, more than the first row by a foot and a half, or at the most two foot: so that your second row ought to be four foot and a half, or at the most six foot. Then about six foot nearer the town, ye must drive in a third row of piles, the heads being a foot or two foot above the second row, as the site and the necessity of the place shall require, continuing so till you come to the very wall of the town against which you shall say beds made of Bundles of brush (as the figures 106. and 107. shows) whereupon ye shall lay heavy stones 3, or 4, cubical foot thick, to make them lie firm and fast between the rows, which are also driven in with stakes, or sparrs as the figure 105, & 106, demonstrate. All these rows must be so ordered by a masons line, that they may lie even from the one side to the other, and to keep them close together, ye must have long sparrs sawn in two, through which you must boor holes, and drive iron, or wooden pins through the said rows, and clinch them well, and (as is said) remember, that the piles of the second row are longer than the first, and the third row longer than the second, and so consequently the rest must be. Then every ninth foot, ye must say cross sparrs of the same weight as the first, or thereabouts, which ye join to every row by driving great iron bolts through the piles and the other sparrs, which lie cross the other rows, making through them quadrangular chambers, which you lay bundles of brush into in making them lie fast together by diving in of stakes and in laying first a bed of brush, and then a row of flint stones upon it, and your brush must be hedged, and wreathed together, to make it lie the faster and firmer, that it may not be loosened, and driven away, with the violence of the water, as ye may see in the 24. Plate, and the 106. & 107. Figures.  
This being done, than ye shall say the Basis of the wall first by driving in piles into the Earth and foundation, if it be moorish, as the 108. figure showeth. But if your foundation be good and firm, than ye shall make (as it were) but a Gridiron of wood upon it, as the figure 109 noteth, these sparrs wherewith you make it, must be some 8, 10, or 12, foot long (according to the ponderosity of the wall, and as your foundation shall require) and some 10, or 12, ynches thick. Upon this you shall begin to lay your wall about 10, or 12, foot thick, allowing to every foot of t●lude, two foot of height, laying within it spurs of wood from 16, to 20, foot, and four or five foot long, making them lie level with the wall (as the art of Masonry teacheth) and some six foot distant one from an other: then by this means, ye shall bind the vaults one to an other, and the two vaults shallbe comprehended by the third, to make them last the longer, and willbe the better able to resist the waves of the sea, when the wind makes them beat against the brickwall, filling the vaults, which are between the spurs with good Earth, stamped and beaten in well, that there be no holes, or hollowness between them: In doing so ye make your haven as the greatness of the place will bear it, and according to the number of ships, which you mean to harbour in it. The mouth or entrance into the haven being some 20, 24, or 30, foot broad, according to the greatness or smallness of the ships that are to come into it. Ye may make the haven, either of a round form, oval, square, or (a paralellogramma) with a right angle, as the site of the place and the accommodation of the Inhabitants shall require. On the outside of the mouth, ye shall drive in the breath of 100, or 150. foot rows of piles, from one end to the other, as abovesaid, which may serve for the coming in, and going out, to make the entrance of the ships the more easy, as appeareth by the 104, figure: in which the parallel lines AB, CE, OF, FG, GH, HR, are rows of piles filled and stuffed with bundles of brush, and stones upon them, as we have noted in the 106. figure, where ye must observe, that the brickwall of the figure 106, which ye see on the outside, is the same as the figure 110 is, which ye see on the inside. The rest of the town, aswell that part of it which lies next the sea, as the other side may be fortified in that form as the plate of Ostend demonstrates, which is the next following, making your fortification so, that the faces of the Bulwarks may be about 100 foot or more, to hinder the better the approaches, which an Enemy may make on that side, seeing those places willbe the weakest, in regard of the banks, and shelves of sand noted Z, which are commonly in such places: Therefore one is constrained to raise the ramparts higher in such places, than the order of fortifying well requireth, that one may lie the safer under couvert from an enemies battery. If the utmost ends A, and B, be of a rock, ye must cut them off as much as possible may be, both in the height, and in the breadth, as the commodiousness and the site of the place will afford: if it be sandy it will be safer to deepen the moat even up to one's neck, in case the country circumjacent will permit it, as ye may see here in the 25. Plate of Ostend & the Figure 112, in which town hath been practised & found out as many inventions to make it invincible as in any town of our age, and no less inventions tried on the enemies part to gain it, which they did at last after three years, three months and odd day's siege, using all manner of industry, in making their the approaches, galleries, mines, and other inventions, which the art of man could invent them. Neither was there less art used on the defensive part to make them get it by ynches meal as appeared by the general, and particular cuttings off, counter mines, counter-batteries, and other works which were made in such sort, that this place was as it were a school, and a study of evils, to bring men to their ends, where nothing was either omitted, or forgotten, but many strange engines invented, aswell to stop the mouths of the channels and havens, as for the approaches towards the town, as this 25. Plate demonstrateth, where ye shall see the approaches made with gabions, or great Bassketts filled with Earth & woolsacks to stop the water breaches and at other times sand-baggs: for the reason why they made use rather of these things, then to advance their works and by approaches & sapps (whereof we intent to treat hereafter) was because the country lay low, and was sometimes overflown with water, in regard of the ebbing and flowing of the sea, entering in, and going out between the mouths A & B.    

Of some loose Pieces in regular fortresses.  

The 23. Plate & 111. Figure.  
MAny men are of the opinion, that in places of great importance to foreslow and hinder an enemy the more, one ought on the outside of a place regular to make divers works separated from the body of it, by that means to give the Assaillant the more work, that he cannot come to advance his design. And though I dare not approve much of them; in regard of the great expense which they will cost, and the number of men necessary to keep and defend them: yet I will give here some instruction to those, that are desirous to make such works, Let AB then be the side of a Decagone, upon which ye shall make by the help of our former table the Bulwarks Q, T, Y, Z, S, R, than ye shall draw the parallel lines G, I, H, and HP, for the faces of the Bulwarks of the distance of some 10, or 12, rod, for the breadth of the moat, and for to find the angle L, M, N, ye shall draw the angle of the shoulder passing through the angle of the Bulwark R. the line RM, the like ye shall do with the line Q, M, than ye shall make the faces LM, MN, of 24 rod, from whence ye shall draw the flanks LK, & NO, equal to the flanks TIE, or SZ, and finally the lines IK, & OPEN, answering to the points Q, and R. & for the better keeping of them ye shall make the Ravelins VWX, whereof the angle W comes no further from the centre of the fortress than the angle M, drawing from the angle of the flank O, a line unto the point W, and the face V, W, of the length M N. Ye may make likewise the flanks 3, 4, & V 6, but then the defences of the Bulwarks will be of no use, because they blind the faces LM, & MN, on the outside of these loose works, ye may make a good moat with a couvert way, and a parapet, as the figure 111 demonstrateth. These works will cost much, and ye must have a great many men to guard them: so that one ought to be well advised ere he undertakes the making of them: and to see if the means of the Lord of the place, his forces, and time will permit it: more over he ought to consider that the entrances in and comings out of the said loose pieces, are dangerous and difficult, and therefore hardly to be relieved.    

How to make a fortress, where two great rivers are not above 15, or 18, rod asunder one from an other, or thereabouts.  

The 26. Plate & 113. Figure.  
THat you may the better understand my intention and meaning I have thought good to make use of the Plate of S. Andrews Fort made by the Admirant of Arragon, that year as he withdrew his army from the siege of Bommel, which is a Fort situated in Bommelsward, between these two great rivers called the Maze, and the Whale, having no great distance between A & C, then some 80, rod, being the narrowst place of the whole Island, and consequently strong by nature and situation, having but two avenues, or passages to come to it, to wit, one upon the left side from Rossems' field, and the other on the right side from Herwerden situated at the utmost end of Bommelsward, opposite against the Vorne, or Nassaus' Fort, which is a fortress, with five Bulwarks, and well made. Now to ordain the said place, ye must first resolve if ye intent to make a Covert-Way with a parapet round about the place, and with all suppose that the circuit interior (though little enough) is able to lodge in it as many men, as ye mean to leave there in garnison, according to which ye shall first make the angles of the Tenailles A, B, C, and according to the course of the sides of both the rivers ye shall make the two angles of the Tenailles A, I, H, & C, R, D, and finally the other two D, E, F, & F, G, H, etc. which angles, because the sides of the rivers runs somewhat slanting towards one of the sides, they cannot be equal, which causeth the inequality of the sides of the Tenailles, and consequently of this fortress Pentagonal: than ye shall make of the distance of some 6, or 7 rod, parallel lines to the lines A, B, C, R, D, which will form the breadth of the parapet for the covert way: then on the inside ye shall make a parallel of some 18 foot for the breadth of the covert way, the breadth of the moat about some 7 rod, which lines of the moat towards the interior side, make the angles M, N, O, P, Q, which shallbe the angles of the Bulwarks, the faces, flanks, and curtains whereof often ye shall find rules given in our former rules of the 10 Plate, and so ye shall have the thing required. The Heads K, and L, are made to break the course of the water, which otherwise would spoil the fortification and especially the angles D, & H, of the parapet of the covert way, which having a good moat of some 12 rod, as this Plate showeth, it will make a great resistance.   

Note.  
THat this figure for want of takeing heed, is turned the wrong way: so that the angle F ought to be with all that which lies on that side on the left hand; and the angles A, B, C, with all that is on the left hand towards the right hand, the cause is, because they cut the Plate as the Figure was marked.    

How a General shall come to besiege a Town.  
WE have for a time been minded to describe the order, which an Army ought to keep in marching towards the place, which one intends to besiege, where it is necessary to treat of military motions, of the order of the march, which every soldier ought to observe in particular, as also of the Companies, Regiments, and of the whole Army, but seeing I have no leisure to do it at this present, I will reserve that for the next Edition to make this and other things in some places more complete. And now we will begin to treat of quarters, and lodgings of every company, then of every regiment, and finally of the whole army, as also of their entrenchments, and the things which depends thereon, that we may afterward come to the approaches, sapps, mines, galleries and interior cuttings off, as followeth.  

The 27, 28, Plates, and 114, & 115. Figures.  
WHen as an Army is come within 3, or 4, miles of the place, where it shall quarter. The Quartermaster General goes out before with some 50, 80, or 100 horse, that he may the better view the ground & place where the army shall encamp, before they come thither, which ought (if it be possible) to be near unto some river side, nor far from a good wood, and where there is good store of forage, aswell for horse as straw & wood for hutting and making of paillasses. The said Quartermaster General then having viewed all the grounds & chosen the best, shall draw a draught of it, and show it to the General of the Army, which being done and resolved, he shall then go in all diligence with all the other particular Quartermasters of the Regiments, to give them their ground and quarter: which shallbe 300 foot deep, and the breadth according to the number of the Companies and the greatness of the Regiments. After this the particular Quartermasters shall line & draw out their quarters to distribute them according to the number and qualities of the companies, giving to every company, which is about 110 or 120, two rows of Huts, to these which are 150, 3, rows, & to 200, four rows, and all running down in right lines from the front, so that the said rows are all parallels one to another, each row being 200 foot deep, and 8 broad▪ The Sutlars Huts must be 20 foot deep, and 60, foot is allowed for the Captains Hutt, and enbowring, and the street between the Huts, and the said lodging are 100 foot, which with the 200 foot for the soldier's Huts, makes in all 300 foot, which is the depth, that a regiment must have, either for Horse, or foot, & other quarters: and the rows between them, which make the streets & back streets, shall be likewise & must be 8 foot: which rows in their depth shallbe divided into 25 parts▪ so that every of them make 8 foot for the depth of the said Huts, and the breadth being likewise 8 foot, each hutt will contain 64 foot square, for every two soldiers, that they may the better lie together and help each other, as well in making up their Huts as before and afterward. And for the better accommodation of the Army the Quartermaster General ought to take care, that the Quarter (as is said) be near unto some river, wood, and fourrage, and to labour that the said Quarter be provided with all things necessary, and quartered as near the Town besieged, as possibly may be, to the end the trenches and approaches may be relieved, as soon as may be in case of necessity: And seeing oftentimes there happenssome mischief among the Huts, which join close together, when as fire takes them, so that one row, a whole street, yea sometimes the whole Quarter is burnt before any order can be taken, they shall build them I, ●, or 3, foot asunder one from another, to the end one may pull them down quickly without endangering the other, and so by this means may be kept from fyring. From the front of the Huts, ye shall Kitspit 10, foot, where ye shall se●t up your crutches for your piks and arms, and beyond them some 16, or 10, foot further is the distance between the Captain's lodging and the arms being 28, 30, or 32, foot deep, and 24, or 28, broad in the front. But when the streets between the Huts, and the Captain's lodging are but 20 foot, than the said lodging is to be 40 foot deep. Beyond the Captain's Huts there is a place of 200, or 250, foot or thereabouts called the Alarm place to draw in arms, at the end whereof the line of entrenchment for the Camp runneth, to wit, having a ditch some 6, 7, or 8, foot broad, 5, or 6, foot deep, and the parapet of the same breadth and height, to wit, the basis of 8, foot, and 4, or 5, foot high: but when it is 6 foot high, than ye must make a foot bank which must be a foot high, and 2, or 3, foot broad, that ye may the better discouver the fields about ye. Behind the Regiment are the sutlars Huts or tents, leaving a street between them and the soldier's Huts of 20 foot and the breadth of their huts are likewise 20, foot. Between the sutlars and the other Regiments there is a street, or a separation made between regiment and regiment about 40, 50, yea 100, foot as the ground will afford it, and the greatness of the said Regiments. And the better to accommodate every Regiment, the Quartermaster General ought to know the greatness, and the number of the companies of them. And so, as is said, ye have the depth of a regiment, which is 300, foot, in which are contained the Captains, and the sutlars lodgings with their streets, and the distances between them and the soldiers.  
The Colonel is lodged in the midst of his regiment leaving a street of 80 foot, as the Figures 114 & 115 demonstrate, to lodge therein his train and the officers of his Regiment, as the Quartermaster, the Marshal, Chirurgeon, & others. In the midst of the quarter the General of the Army is lodged some 600 foot, or more from the entrenchment, according to the greatness of the Camp, and the accommodation of the place, having a front of some 700 foot more or less, as his train is either great or small, being 300 foot deep (as all the other quarters are) having in the midst, where they set his pavilions, an overture of 400 foot broad, for to discouver by sight the place where they draw up in arms. Behind him the master of the ordinance is quartered, having his quarter 300 foot deep, and 700 foot in front more or less according as his train is great. For in case that the carrages and victuals must be lodged within the compass of the Camp, it is very necessary the said place be larger, that it may contain them all, and as than ye may separate the train of victuals from the train of the ordinance. Upon the right hand are lodged sometimes the other Officers belonging to the army, leaving between them and the rest, a street of some 40, 50, or 60, foot, that they may freely pass through it with the ammunition of war, to the place where they are to be used: and then round about it are the foot lodged, according to the former order, being as is said) 300 foot deep with the Captain's tents, and the sutlars huts, and the breadth of the front according to the greatness of the regiment, every company of 100 men takeing up 32 foot with the streets, which are between the huts, and before them is the place of Arms 200, or 250 foot, to draw up by companies into their divisions and battalions, to wit, every company directly before his Captain's lodging & their to draw up in parado & to receive the commands of the General of the Army. In the front of the huts the Lieutenants and Ensigns are lodged, and the Sergeants & Gentlemen of the arms in the rear next the 〈◊〉, they may presently prevent any disorders which might fall out. And for the more easy understanding of what is said above in the 27 Plate, & 114 Figure, we have here represented the quarter of an English Regiment consisting of 20 Companies, as they were quartered (as I remember) before Gulick whereof the Plate K, L, M, N, O, P, is the Colonel's lodging, Q, R, S, T, his Kitchen & Stable, H, I, the Captain's lodgings 24 foot broad, and 30, or 32, foot deep. G, the crutches or the forks against which the Pikes and Muskets stand. CA, the soldier's Huts, whereof every file containeth 25, and 8 foot in depth and breadth, the goings out are between the two files; but the four utmost huts of every company have their goings out towards the streets, CH, & A, which is the Sutlars street, E F, are the Sutlars Huts, which are made either large or strait, as every one's occasion shall require, leaving a little space between them, for their more freedom, and for the danger of fyring. The scale which is hereunto annexed will show you the true measures.   

The 28, Plate, and 115. Figure.  
THe 28, Plate is the draught of a Colonel's lodging, in a larger form, that one may the better understant the order, which is here to be observed. 9, is his tent, & his gallery to go into his tent, 5, 6, is his sleeping tent, where it may be so ordered, which would not be amiss, that there be a gallery from 5, to 6, to go in and out prively from one tent to an other, 7, is a hutt for the Colonel's servants: 11 his Kitchen, 10 & 12 are also Huts, 13 in his Stable, 14 and 15 are the officers Huts of the Regiment, and all other officers, which are not ranged under any Company, are lodged in this street, behind the Colonel. 2, & 3, are the Captain's tents, 16, & 17, the places where they draw up in arms, 4, 4, 4, 4. are the soldier's Huts, the side KL, from the colonels lodging is a right line with the Captain's lodgings, not to hinder the sight of the front of the Huts. But as for me I think it were more convenient that the front of the Captain's lodgings, were made in a right line even with the front of the Colonel, to wit, that 2 & 3, and M, & N, may be in a right line, whereby the alarm place should be of an equal breadth.   

The 29. Plate & 121. Figure.  
IF necessity requires to quarter the Horse with the foot within the circuit of one line, which is usually done when one fears an Enemy to fall on upon many sides, and that there is no other accommodation for the horse without the camp, otherwise it were much better to lodge the Horse out of the foot quarter, for the avoiding of many inconveniences, which the foot might receive by them, and which the Horsemen might receive from the foot, leaving between the Sutlars Huts of the foot and them, a street of 40, 50, yea sometimes of 100 foot according as the place will afford it, having their front towards the place of arms, & every regiment quartered together, being commonly 8 troops in a regiment allowing to every hutt 10 foot in length: between them and their Horse there is a space, or a street given them of 5, or 6, foot broad, to lay in their fourrage: then between the same 5 foot, & the great streets (which are ordinarily 20 foot) is made a space of 10 foot for their horse, allowing to every horse 4 foot in breadth. The great streets are made as is said of 20 foot, the narrow of 8, or 12, foot, that they may the better come one to an other, and to join together in case of necessity, and for other considerations, in such wise that in one row, which is 200, foot in depth, there are three such streets of 8 or 12 foot: For otherwise the rows, because of the great spaces, which such Huts will take up, makes the entrance into them uneasy, and thus the lodging of a troop of horse of 70 foot, to wit, the lodging of the Horsemen 110 foot & 2 small streets 5 foot, the street between the Horse 20 foot, and twice 10 foot for their horses, which makes in all 70 foot: and when as the streets between the Horseheads and the Huts shallbe six foot, the front willbe 72 foot, of which breadth the Captain's lodging is made, leaving between him and his soldier's Huts a street of 18 or 20 foot, and the depth 40, or 42, foot, and seeing the depth of the Horseman's Huts is 200 foot, as the Huts of the foot are and 40 the Captains, with the street of 20 foot, makes 60, foot and 40 for the Sutlars street, it is manifest then that the quarter for the Horse (as is said) willbe 300 foot deep. Then between the companies there is a street of 20 foot, which if necessity requires may be made narrower. And seeing the Captain which commandeth a Regiment of 4 or eight troops, hath no greater train than an other horse Captain hath, he is lodged no otherwise then they are; neither takes up any more ground, only the honourablest place is given him, because he commandeth for that time: but the army being in garnison, hath no command over them.  
In the front of these Huts are lodged the Lieutenants and Corners, and next unto the sutlars street are the Huts of the Corporals and Quartermaster, that they may upon any occasion be ready to prevent any disorders that may happen. Beseides the ordinary Sutlars, which use to follow the Regiments, there are a great number of other Sutlars and Tradesmen, which follow the Army, as Drapers, Merchants and others, which are all lodged in a quarter by themselves a part; to the end that those which have need of any thing, may go presently thither and buy it. And this is the reason why all of one profession are lodged in a street, or two rows, leaving a large street between the principal Tradesmen of some 200 foot or there abouts, which shall serve as a Markett-place, where all things necessary aswell for the sustentation of man as other wares are daily sold, which are brought thither by the Country men, and others. The other rows which are of an other profession, have their streets more narrow as 15, 18, or 20, foot. Those which are of a contrary profession have a street of some 30, or 40, foot, as the place will afford it, and the quality of the merchants. The Butcher's quarter is ordinarily made in one of the furthest corners of the Camp, because of the garbages, and paunches of the Beasts they kill, which they are toburie without the quarter in a hole or a ditch digged for that purpose some 200 foot without the entrenchment, that no evil sent or filthy stink may cause the infection of the air. If the Camp be quartered by a riverside (which one must seek to do if it be possible) it will not only be able to furnish the soldiers with water for their accommodation, and to preserve them on that side from an enemy, but also willbe good for the shipping, which may bring them all manner of victuals and commodities as experience hath shown in divers sieges, which if it so falls out then the market place is made by the river side in the same place where they should draw in arms being 300 foot broad or there abouts, without having any Huts in all this plain, that they may not hinder the entrances in, and going out of the ships, from whence they fetch all things necessary, aswell for ammunition, as for victuals, and the Merchants, Sutlars, and Mercers and others following the army are lodged on this side, and separated by streets some 12, 15, or 18, foot, according to the site and commodiousness of the place, as we have said above.  
In the quarter of the Master of the ordinance is 〈◊〉 lodged the Commissary of the Victuals, and his Conductours, as we have said before, and that when as his magazine is not very great in respect of the place, as also all the other officers, over which the said master of the ordinance commandeth, as the Commissary of the ordinance, his Conductours, the Master gunner, and Gentlemen of the ordinance and the ordinary Gunners. The Commissary of the Ammunition▪ with his train, as Carpenters, smith's, Watermen, Pioniers, Miners, and others, all of them being lodged by themselves, that one may find their lodgings the better. The Master of the ordinance hath commonly the gentlemen of the ordinance lodged within his circuit, that they may be ready upon any occasion to receive his commands, and to execute them as the General of the Army shall think good, as appeareth by the Figure 122: In which A is the place ordained for the master of the ordinance: B, the quarter of the Engineers, the Commis and his conductours, C, the quarter of the Carpenters, belonging to the Army. D, The quarter of the Carpenters which attends upon the ordinance: E, the quarter of the saylours' following the train of the ordinance. F, G. the Pioniers quarter. ay, is the master Gunners having 100, or 150. Canoniers under him, and the Provost marshal of ordinance. H, The Miners quarter. L, the Magazine for the ammunition, and his Commis: E, the Magazine for the ordinance; with his Commis, M, N, the Magazine for the Victuals, and O, a place where the wagons stand: but when they fear no enemy then this place is appointed without the Camp some 200, 300, or 400, footewithout the entrenchment. But when there is any doubt, or that the wagons may give any hindrance being set without, they appoint them ordinarily to stand on that side by the master of the ordinance his quarter, where also the Wagon master is lodged, with his lieutenant and their conductours, the better to keep all things in good order, and that they may presently be employed upon any occasion, when the waggonmaster shall call for them, and for the better understanding hereof we have made here a draught of it, as appear by the Figure 122.   

Figure 123.  
HAving described the particular lodgings, and quarters of every company of a Regiment as well of foot as of Horse and other officers, it will be requisite to represent them together in a General quarter. And because exemples are of great use: I will take first this Figure 123, which was his Excel▪ quarter before Gulick in the year 1611, G is then the quarter and tents of the General: I the quarter of the Master of the; ordinance K the officers of the Army; L the market place, where all the tradesmen and merchants stand, which follows the Army, as we have said before. F & H are the French Regiments, namely F, Monsieur Chastillon his Regiment, H, Monsieur Bethunes. A, the Regiment of Count Ernest of Nassou: B English companies commanded by Monsieur Medkercke Lieut. Colonel to General Horatius Vere. Care 8, companies of Frizes. E, be 6 Germany companies. D the 4 companies of the guards, to which quarters are added the number, which show the length, and the breadth of every quarter in particular, also the length of the streets, & the greatness of the whole quarter in General.   

Figure 124. & 125.  
ANd because commonly the Captains, after their soldiers are hutted, build Huts in the place, where their tents stood, being warmer, & cooler against the sun, and more durable. I thought it not amiss to represent the form of them: A, then is the Captains Hutt, B his Kitchin. C, his stable, D. the place where his fourrage lies. E, is the street between the Captain's lodging, which is 8 foot. the scale noted by the numbers 124 and 125 will show the depth and breadth of the said lodgings, and the largeness of the streets.   

Figure 126.  
THe Figure 126, is a quarter of Horse and foot, whereof A, is the General's lodging. ay, the lodging of the master of the ordinance. G, L, F, H, are four Regiments of Horse. B, C, D, E, M, N, O, P, are 8 regiments of foot. K the Market place for the Sutlars; the Half squares 1. 2. 3. 4. 5. 6. 7. 8. 9 & 10. are the places of defence, to hinder the giving on of an Enemy, whereof every face is about 100 foot (as also in the Figure 123. are the exterior angles 3, & 4) to the intent that one may lodge more men in them to defend the skirt AB, the Overtures 13, 14, 15, & 16, are the avenues, passages and entrances into the general quarter: whence it is necessary, that a Quartermaster General ought to have true notice of the number and quality of the Regiments & companies, that he may dispose of them according as the place and the nature of the ground will afford, and for the better effectuating of what is said before, it would be good, to draw out a draught of the greatness of every Regiment in paper, and above it to put the colonels names, to alter and change them according to the accommodation of the place & the number and greatness of the Regiments, setting them down in two, three or four rows, or in length if the Entrenchment be by a riverside, lodging the Nations which cannot agree well together as far asunder as possibly may be one from an other, to prevent mischiefs, quarrels, and other disorders, which may happen between them. If he quarter's the Horse with the foot, the General of them, aught to have his quarter in the same front as the General of the Camp hath, or between the troops of Horse, being not far distant from the said General, and his front, and there must be, if it be possible, an overture made of 300. or 400, foot, that one may have the perfect sight of the General's quarter towards the place of arms, for to draw in and out. The sutlars and merchants, which follow the Army ought to be quartered behind the General's quarter, on that side where the master of the ordinance lies: but the Butcher's shambles, as is said before, in one of the corners of the camp, or if they be within, to be strictly commanded to carry out their offald and filth out of the Camp, or else to dig up great holes to cast them in to them, that by such noisome smells the air may not be infected.   

The 29 Plate, Figures 116, 117, 118, 119, & 120.  
AT the end of the said 200, or 250, foot, the entrenchment is kilspitted made by the soldiers for nothing, every company against his own quarter, if it so falls out, which entrenchements are (as we have said some 6, 7, or 8, foot more or less, as necessity requires. For when one fears an Enemy, it is needful that the entrenchements be made stronger, and greater, them when there is no appearance to be attempted: in the midst of these entrenchments there may be made spurs, & redoubts on every side of some 4, 5, or 6, rod, for the better defending of the said trenches: for from them they may bed scouvered all along, and so by night or otherwise they may hinder an unexpected attempt, being distant one from an other some 40, or 50 rod, according to the greatness of the skirts of the trenches. The reason why one maketh these square redoubts is, because others cannot be so soon made, otherwise I should be of the opinion that one ought to make great flanks, and large skirts, to draw the more men to flancker than, and not to entrench the inside (to wit in the entrenchments of the camp) as appeareth by the entrenchment A B C, Figure 116, which would make a better defence, if one should assault the squares D, E, F, G. It is true, that when one hath passed the entrenchment, that the space A & C, being open, would be sooner taken in then the square: but I leave this to your consideration, if when ye have lost your trenches, the square redoubts may be kept, in regard of their smallness and the fewness of men that man's them, to wit, some 25, 30, or 40, men at the most: now for the better preventing of such a danger, ye may make the lines, and entrenchments D, F, leaving only a gap open of some 3, or 4, foot, for an entrance in and out with a drawbridge, and so this would be as difficult to be taken in as the whole squares; so that I should rather resolve to make the entrenchment of a Camp with skirts of demysquares: so much the greater on the outside (as is said) without making whole squares, or redoubts. For in so doing the defence would be much better, and the coming to defend it more commodious and easy. Those which find the other manner better may make use of it, for my own part I rather approve of these Demysquares. The Entrenchments which are made without the enclossure of the Camp, as those which enclosse a town, or for to cut of the passage of an Enemy, to hinder their Victuals & such like things ought to be made with strongh redoubts as Prince Maurice did at the siege of the Grave, where the Enemies' army lay close by his camp, not daring once to attempt the passage: because of the strongh entrenchment, which he had made by the said square-redoubts, being not further distant one from an other than some 50, or 60, rod, and about as much also upon the Entrenchment, which were made (as I have said) in the form of Bulwarks, and within them were the said squares, which in my opinion are exceeding good in such entrenchments, which are made without the circuit of the quarter, especially, upon the Avenues that they may lodge in them some 25, or 30 men to guard them, & to hinder an Enemies' passage, but when one is not resolved to make the said entrenchments like Bulwarks, there ought to be made at the least such square works as the Figures 116, & 117, demonstrate, because they are wholly enclossed, having but one entrance as narrow as possibly may be, being advantageous aswell without as within. For as long as the Approaches are not much advanced before, and about the town, they may send men unto them, and occasion many hindrances, if they be not well entrenched: The Figure 117, will give us clearly to understand, that which we come to speak off, the square D, E, F, G, being the same of the squares of the Figure 116, which is an entrenchment running round about the town, which is besieged, and is as far distant from it at the least as a muskett can well bear, beginning and ending at the entrenchment of the quarter; DE, is the outside of the Redoubt containing 4, rod, or 48, foot; the space marked I, is the talud of the parapet; 2, the superficies thereof; 13, the foot-banck; 5, the ditch eight foot broad, and 6 foot deep, as ye may more plainly understand by the Figure 118, which is the Profile of the said entrenchment, whereof the height, and the breadth are noted here, the bottom of the ditch being 2, foot wide, and consequently every side of the talud is the half of the height, or depth, the foot bank in the basis 3 foot, a foot high, and above 2● foot. And there ought to be one the outside of the parapet an edge of some 2, foot, which was forgotten in this Figure 118, but ye may see it marked in the 120, A, T. And seeing the said squares or Redoubts are chiefly made upon the Advenues and passages, to hinder thereby the attempts of an Enemy, which secks as much as possibly may be to keep a free passage, and entrance into the town besieged: It is sometime (for ones better assurance) necessary to use the greatest care & perfection to hinder him; so that those squares be made like little Bulwarks, as appeareth by the Figure 119, and then the sides of these square works are greater than those above, according to the quality of the passage, their curtains being at the least of those provisional Forts some 4, rod, according to which the other parts being proportioned as the table of our dimensions shows, the faces F, C, & D, E, 3 2/5 rod, the flank C, A, 103, rod, and the line of the gorge 1. 22, rod, the Rampartas almost all other provisional works are only made of 6, foot high at the least, and at the highest 9, or 10 foot, making the basis of 14, or 15, foot, and 3, or 4, foot of height with the ordinary taluds, upon which is made a parapett of 7, or 8, foot broad, & 5, or 6, foot high, with a footebanke some 3 foot with the taluds, according to the quality of the ground, to wit, sometimes but a foot high, and as much Talude, and otherwhiles but half a foot only, than one supputes how wide the ditch will be according to the said Rampart, being everywhere 6 foot deep or thereabouts: And when one raises the rampart above 6 foot, which is usually made in provisional fortresses, whereof the curtain is 5, rod or more; I think it were not amiss, to make in the ditch a foot bank afect the fashion of a counterscharfe some; foot, that if need be the musketeers may run along it, raising the parapet thereof above the plain field, as much as the said rampart exceedeth 6 foot in height, or somewhat less, as the draught shows, which we have made thereof marked with the number 120, where we have raised the parapet 2 foot higher than the field, as likewise the edge of the ditch, and cutting the Talud of the ditch towards the same place some 3 foot, making together 5, foot, with the allowance of a little Talud, as commonly one gives to parapets, to serve as a parapet to the ditch, and there to be under covert, and seeing there rests yet three foot, before ye come to the bottom of the ditch, ye may make a second foot-banke, a foot and a half high, that ye may the better descend from the other to charge & discharge your muskett, while the other are a giving fire, and for to know the greatness of the parapets, and how wide the ditch must be answerable to it, we will make here this calculation thereof following.  
Before we come to the supputation of the said profile (Figure 120) we must set down 6, in the place of a 5, for the line S, R. To mark out 10, for M, K, & an O omitted at the interfection of F, H, and of a parallel through E, to B, C.    

A B C D.  
A D, making 14. foot, BC, 9 1/2, the sum being multiplied by C, G, 3, the half will be 35 1/4 for the superficies A B C D.   

B E F I  
F, H, 5, being multiplied by ay, H, 1/● the half of the product will be 1 1/4 for the triangle F, H, ●, then F, O, I, multiplying the half of O, E, ye shall have 2, for the triangle F, O, E. Finally, the sum of O, E, 4. & H, B, 6, multiplied by 2, the half of O, H, ye shall have 20, to which add 3 1/4 for the two triangles abovesaid, ye shall have 23 1/4 for the superficies of B E F I   

L M K, and all the rasing.  
The half of 1/4 LM, being multiplied by M, K, 10, ye shall have 10, for the triangle LM, K: to which add A B C D 35 1/4 & B, E, F, I, 23 1/4 there will come for all the rasing 68 1/2, which ought to be equal to that which is void, M N O P Q R S T, but to the end we may resolve this by the Trapeze M, T, S, R, we will subtract from the Sum abovesaid the two Triangles M N O, 4 1/2, & O P Q, 1 1/●, which make 5 1/●, and will remain 62 7/● for the trapeze M, T, S, R, which divided by the depth which is 6, it will give 10 23/4● the Talud exterior 6, with the interior 3, maketh 9, the half is 4 1/2, which added to 10 23/48 ye shall have 14 47/48 for T, M, which is very near 15 foot, and likewise 4 1/2 substracted from 10 48/23 ye shall have 5 47/48 for S, R, (making very near 6) the proof is that the superficies of M N O P Q R S T, is 68 1/2 equal to the superficies of the whole rasing, the like ye may do with all provisional fortifications.  

The 33 Plate, and 140. Figure.  
ABout the entrenchment of a Camp, there are made some gapps, avenues, and passages for men, and wagons to go out, and in, which have no ditches, or ramparts, some being 6, 7, or 8, foot wide, as occasion serves, shut in with a wooden gate, or Turnepikes made of sparrs some two ynches and a half thick in diameter, and about five or six foot high, plated with iron heads at the points, and having two great Iron nails driven through them eight or nine ynches long, and blund headed on the other side of the thickness of an inch, half an inch, or thereabouts; they are pointed to drive them in the better into the ground in two places, to stop up the passage, as the 33 Plate, and the 140 Figure showeth, which pikes three or four rows of them, must be driven in close one to another as high as a man's girdle, which is about 3 foot, or three foot and a half, the first and utmost row must be driven into the ground deeper than the other, and so the one above the other row, that they may not be pulled up. These are of good use also to be driven in upon the top of a breach (as Sr. Francis Vere did in Ostend) when an Enemy is ready to give an assault, and that one hath not time enough, to cast up a breast of Earth upon it. These and an other Instrument called in Dutch a freeze Ruyter, and by us a turnepike (as also your quadrangular tanternailes cast down upon a breach) are of singular use, to barrocadoe, and stop up places▪ A Turnpike is made in this manner following:   

The 33 Plate & 141 Figure.  
TO make a turnpike, ye must take a round sparre some 12 or 15 foot long, and about 5, or 6, ynches in diameter, and boor holes through it in many places: so that these holes meet not one with an other, being boored right one under another, about an inch in a sexangular form, that the circuit may be divided into three equal parts, and in every part, ye must boor a hole as is said one under an other, that the demypikes of six foot length, well sharpened with iron heads at both ends, about an inch or an inch and a half, may be put crosswise fast through the holes of the spars 3 or 4, ynches one from an other, even as the Figure 141 demonstrateth.  
These turnepikes are of a very good defence against horsemen, to stop a passage into the Camp, to stand at the entrance into a work, or upon the gap of a work: They may be made so that ye may run them upon wheels, to remove them from one place to an other and join their axeltrees together with pins of iron, so that on a sudden ye may shut in the passages unto an army, and being so joined ye cannot remove the one without the other, so that these turnepikes joined together gives the longer and better resistance. There are other inventions to keep out an Enemy, but having no leisure now at this time to treat of them, we will pass them over, and come to handle of the Approaches.    

OF APPROACHES.  

The 31, & 32 Plates, and 127. 128. 129. 130. 131. 132. 133. 134. Figures.  
FOr as much as in approaches (whereof we intent anon to speak) one must of necessity use showels, spades, fire-rakes, pickaxses, mattocks, hatches, bills, and axses, to give the more lustre to our business I have thought good to describe them here. And seeing Monsieur Doncker in his life time Controller of the fortifications in the united provinces, and besides at the siege of Gulick Controller of the princes of Brandenburg and of Nieuburgh their ordinance, a man being very skilful in making these materials, gave me the model of them in that true form as ye shall see them represented here to your view in the 31, and 32 Plates, by the numbers 127, 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. and 139.  
The Figure 127 is an ammunition spade, whereof the part A B, is a foot high and A, C, two foot, being in all three foot long: the point A B, is plaited with a sharp edge of iron some six ynches broad, that ye may spit a foot into the Earth with it.  
The 128 Figure is a showell, made more hollow, and some what greater, than a spade, which cannot be used with that nimbleness, as the former: because of the great quantity of Earth, which it will take: so that for ordinary use I should rather advise one to use the spade, for those that are not able, and strong men.  
The showel 129, hath a plate of iron only, being fit to level, and plain the ground, and for this reason (in my opinion) ought to be made longer, than the other, because it must be used with as much strength as the other.  
The Figure 130 is a pickaxe, well known to workmen, wherewith ye dig up a stony & hard ground, into which a spade or a showell will not enter.  
The 131 Figure is a Mattock, the head whereof is like an axe made bending, which is also well known, and used to dig up hard and stony grounds.  
The 132 Figure is a hatchet, to be used with one hand.  
The 133 Figure is an Axe, to be used with both hand:  
The 134 is a less kind of axe, used with both hands not usual in these parts.  
The Figure 135 demonstrated in the 32 Plate, is an ammunition wheelbarrow, which will hold a foot of Earth deep, and if it be well loaded a foot and a half, which wheelbarrow, as the Figure 136 showeth, is made of an i●nographick form, containing 16 ynches in length, and 14, or 15, ynches in breadth, and 7, ynches high, the boards being made for lightness of fir wood; saving the axletree, and the handles, which are of other wood some 17, ynches long, but the shorter they are, the better they are, and that for divers reasons.  
The Figure 137 is a bill, to cut brush and branches with all, and useful for the making of Bavins, and bundles of brush, Gabions, and for other necessary occasions, aswell in hutting, as to be used in trenches & works.  
The 138 Figure is a kind of a crooked bill, serving for the same use.  
The Figure 139 is a long showel, or a rake for the casting up of Earth out of galleries, to fill up a moat, and in other dangerous places. And the form A, in the 31 Plate, is good also to plain the Earth which is cast upon a gallery, to keep it from fyring, which otherwise might happen to a gallery.   

The 33 Plate & 144. 145. Figures.  
BEfore we come to speak of Approaches, it is very necessary, that one should get an exact draught, as well of the interior, as of the exterior part of that town, of fortress, which one resolves to approach unto, with all the marks, and observations, as namely, of hedges, ways, hills, valleys, and the like, that ye may know the better how to order and run your approaches. Ye ought not to break ground no nearer at the first than within a muskett shot, or at the furthest as far as a harquebusse a Crock can reach, and (as I say) ye ought to begin your trenches, somewhat further off from the town; if there be a commodious access from the quarter to the trenches, that when need requires ye may the sooner relieve, and second your men in the approaches, if the Enemy shall sally, or fall out upon them. For, when the place where you first break ground is a great way from the quarter, it will be disadvantageous unto you, because ye must always keep in them a strong guard, to repulse an Enemy, and to beat them back when they shall fa●lout, and while your men are a making ready, and coming from the quarter, many good occasions may be lost, before they cann be at hand. Therefore you must have a great care, and circumspection, where your trenches are to begin, and where your quarter must be made, as near to and as covertly from the town as possibly may be, considering well, whither it were not better to quarter sometimes the more incommodiously, that ye may begin your approaches in a place of most advantage, near unto the town, or place, which ye are resolved to take in: then to choose a place more commodious for your quarter, that would be a hindrance to your design, by reason of the great distance between that and the approaches, or which may lie too open to Canon shot, all which in my judgement ought to be maturely considered, that your design may not be hindered or fore-slowed.  
The next thing is for you, to know the weakest part, and side of the town, or place, where you may begin your approaches, and which place you intent to gain. For, it is a great help, especially in places regular, to choose out the most advantageous place. If you should make a breach in a curtain, between two bulwarks, ye shall find it to be the retyredst place, the best to be defended, soon cut off, and the hardest for you to approach unto, therefore would advise you, not to begin there. And if it hath a tenaille, than ye have less reason to make any attempt upon that place, having a retired angle, and so consequently the cutting of willbe according to their wish. But to approach to the angles of a Bulwark, which are angles shooting out, being the exterior parts of a town, or fortress willbe the easiest for you to come unto, and the hardest for them within to cut off: seeing the place is little, and that the ramparts themselves may give some hindrance there unto.  
The place then being resolved on, a certain number of soldiers (or Commanded men) are choosen out of every Regiment, and Company to the number of 4, or 500 men, more or less, to go down in the night to break ground, and to begin the approaches. Also there must be placed a competent number of horse, and foot, to help to guard and defend these workmen, if an enemy should sally out upon them. These commanded men have each of them ten stivers a night, or more as the danger of the work shall require: they having men (as is said) in arms lying round about them, to defend them as appeareth by the numbers 20. 20: in the 145 figure. Then the Ingenier, who hath the ordering of the approaches, sets these men out in a right (or a crooked) line placing three or four men upon every rod, and all in a file, in as right a line as possibly may be, takeing special care, that his men may stand as much out of danger of shot as is possible, which is done also by the help of these gates, which lies about them in keeping their marches close, that they may not be discouvered by those of the town. More over, they must say out Sentinel perdues, before and round about them, which if they hear, or see any thing, may give the alarm silently before an Enemy can fall upon the workmen: For the alarm being once given, these guards are to draw up, and help to defend the Workmen. But if they see that the Enemy sallies out to strong for them, these guards than may draw into these Corpse de garde made purposely for them, and withal the commanded men must quit their work for that time, and bring off their arms, spades, showells, and mattocks; but is haste and necessity constrains them, to defend themselves; then they must betake themselves to their arms, and cast away their materials, retiring softly (with the guards abovesaid) if they cannot make the place good, to the Corpse de garde made first for their retreat. These works aught to be strong, and able to repulse and keep out an enemy, yea it often happeneth, that they upon a disorderly retreat, into the town may be followed, and beaten back with the loss of many men, even to their very moat side, and that the Besiegers may discouver the strength, or weakness of their works, and spy out some advantages for them. The Enemy being beaten back, the commanded men may fall instantly to their work again, and being got some 3 foot deep, and casting up the Earth towards the town, they may work with the more safety, and out of danger. And for this reason every soldier will make as much haste as possibly he can to get into the ground for his own preservation, and the Earth being cast up thus out of holes, they may afterwards repair it, and make thereof a large trench, and ditch, namely, (as is said) 3 foot broad, and 3 foot high, casting the Earth upon the edge of the trench, and the next day may make it 6 foot broad, and six foot high: for the larger the trench is, the higher ought the parapet to be; because otherwise, they might be discouvered by those of the town, and so your men may be the better under covert, and lie safer in the said approaches. And because this cannot be done so exactly by night, fresh commanded men are sent down in the morning from the quarter to repair, and enlarge the trenches, and if the trenches be so large, and so high, that one cannot well discover the fields round about them, than ye must make a foot-banke or two, for the musketeers to come to the top of the trench to give fire over them through muskett baskets, which is done mosttimes when a sap is begunn: At the first entrance into the trenches, or approaches, ye make a square work, or two, called Redoubts, or a Corpse de garde being distant one from an other some 40, or 50, rod, that in them ye may keep a strong guard the day following, and the Ingenier, who hath the managing of the approaches; may so order & encourage his men, that the said Corpse de garde may be in defence before day, to the end, that if the Enemy should sally out upon the trenches, that from the same Corpse de garde they may be beaten back: the breadth of every side ought to be some 4, or 5, or at the most 6 rod, and the ditch broad & deep as necessity shall require. They ought to be made in my opinion in such a manner, that the two opposite angles do enfile the said thrench as appeareth by the Figure 117, Plate 29, and by the Corpse de garde D E F, Figure 145. For then the said trenches will not only lie open unto it, but also one may discouver the fields round about it. But if it be not found good to make them so, than the best way in my judgement were to make them out of the Trenches a rod and a half, or at most two rods distant from them: so that the said trenches ought to be between the town, and the said Corpse de garde: one of the sides must be parallel to the said trench that ye may march by it in the night, both with men and wagons, and to draw up the ordinance between the said Corpse de garde, and the Trench, and so the said Corpse de garde will flank the said trenches on that side where they lie most open as ye may note by the Corpse de garde in the 4, Figure, 145 & 33 Plate.  
After your commanded men have wrought almost to the break of the day, than they must be drawn off (without beating of a drum) by a Sergeant or the Quartermaster of a Regiment, and being returned to their quarter the Sergeant shall deliver the note of the number of his men to the Quartermaster of his regiment, and the Quartermaster to the controller of the works, who to that end comes to v●site the said work and to see if there were as many men as was appointed, lest there should be any fraud committed therein. Then the Controller is to give the Quartermaster his attestation for the work men, or in his absence the Ingenier gives the said Certificate, and the Quartermaster brings it to the States, who gives an ordinance upon it to the Treasurer, and so the Quartermaster receives money for the workmen of every company, to wit, ten stivers, gives it to the Sergeants & the Sergeants to the soldiers of every company.  
The repairing of the trenches are made by the undertakers (as we have said) for so much, or so much the rod, according as one desires to have them large and spacious, as some times 6, 8, foot, or more, as necessity requires. For oftentimes they must be so broad, that a waggon loaden with fagotts, Brush, or Gabions may go in them. The wages of every soldier is then augmented some times to 15, 20, or 30. stivers a day, when there is evident danger, as there is commonly, at last before one begins to sapp: for the musketeers playing continually upon the points of the Trenches, where they imagine that the approaches will run, kills many men: yea so that sometimes they are forced to give over the work, because many men are not able to work in it, and being too hit, they put in a resolute man or two, into the work, promising to give them more than ordinary if they will hasten the work, and this is considerable, whither this extraordinary expense be not profitable for the advancing of the approaches, and gaining of time, then to spare a little money and foreslow them: For experience hath many times taught us: that expedition in such cases hath wrought many great effects.  
When you are approached so near your Enemy, that by reason of their continual shooting upon your points ye cannot advance your trenches any further; than you must begin your sap, which ye shall run, if it be possible, directly upon the point or side of the Bulwark, which ye intent to take in, as the point, I, & K, Figure 145, demonstrate. And because two men at the most can but work in them, these sapps are ordinarily taken on by some resolute soldiers for some 7, 8, yea or 12, gilders a rod, as the danger requireth: they first make the sap some 3 foot broad, and about 3. 4. 5. or 6 foot deep according as the ground is, either high or low, than these undertakers or others are to repair the Sapps, and receive for their pains 6. 7. or 8, gilders a rod, and so make it 6 foot broad, or thereabouts casting up the Earth on both sides, that they may be the better under covert, and the safer from the Enemies' shot.  
According to which we have made the Figure 144, whereof P, Q, are two Bulwarks. A is the beginning of the Approaches, A, B, the first trench or line where the workmen break ground, carried in such a manner towards the angle R. (in case the same line be lengthened and comes out of the Corpse of the Fortress) in the angle B is made a Corpse de garde for the assurance of those, which guard the trenches, as appear by the Figure 145 to the letter B. At C is made a line towards the angle S, which must be drawn in such a sort, that if it be lengthened, it will fall on the outside of the Corpse of the fortress Q. which is continued (as the said Figure demonstrated) to I, the parapet of the covert way, where one begins a mine to blow up the counterscharfe, that ye may come to the brink of the moat. From D, is drawn the line E, K, to assure one the better of the enclossure K, E, I, between K, E, & I, E (before ye come to pierce through with your sap) ye ought to make the lines L, O, & F, M: that from these lines you may give fire upon the Enemies' Musketeers, setting along upon these lines muskett basketts that ye may play continually upon the besieged, that under the favour thereof, you may advance your sapps towards the faces R, T, & V, S, as the said Figures 144, & 145. shows. If the Corpse de garde be in some eminent places as they ought to be (if it be possibly) after ye are advanced to them, ye make a battery upon them, but if there be one piece of ground higher than another, that is more commodious, which ye may choose for your batteries, as we have done here in the Figures G, H, P, O, X, Z, Y, Figure 145.   

Figure 142. 143.  
WHen ye begin your trenches of Approach, ye commonly make some battery, that under the favour thereof your men may work forward with the more safety, and hinder the Enemies from falling out, which would much foreslow your works. Your Batteries then are made first in such a manner that they may beat upon the Parapet of the ramparts & Bulwarks of the town, to dismount the enemies ordinance, and for this reason you must raise your Batteries high according to the height of the ramparts, so that your Canon may play freely about two foot lower, than the top of the parapet, according to which, and in consideration of the distance, ye must raise your said batteries, takeing heed that your Canon be planted upon a plain superficies, and elevated some 13, degrees, when the distance is far off: ye must not raise them so high as when the batteries are near unto the place: so that ye must either raise, or sink them, as need shall require. Ye make your batteries, and platforms according to the greatness, and number of your pieces: for a Demy-Canon, being shorter than a whole, of necessity the platform of the one, must be longer, and deeper than the other, and seeing a Canon being mounted upon its Carriage, is some 16 or 18 foot long; it is evident, that the batteries ought to be made for recoiling at the least 10, or 12 foot longer making together 28, or 30 foot: 12, or 15 of the first foot towards the parapet, must be underlaid with thick & strong oaken planks and the other with hurdles, when ye have not planks enough: upon the said batteries ye make a parapet 12, 16, or 20 foot thick or thereabouts, with portholes for your canon as the Figure 143 demontrateth. Sometimes ye set up Gabions 6 foot high and 3 foot broad filled with Earth, for your ordinance to play out of, leaving a little space between them, to put out the mouth of your Canon, which space so soon as the Canon is discharged is presently stopped and blinded with a bundle of brush full of leaves, that the enemy may not discouver the portholes, but when you make your batteries upon the Counterscharfe, or upon the brink of the moat, than the portholes as soon as your Cannon is sho●● off, are shut with doors of thick oaken planks musket proof, that the said portholes may not be seen. Sometimes ye set only Gabions upon the parapets of Batteries which contain 7 foot in Diameter, and 10 in height, yea oftentimes set 3 rows of them upon he said parapet, which make in all 21 foot thick as appeareth by the Figure 149, & the 34, Plate, There are usually made three sorts of Gabions, to wit, the ordinary of 6 foot high & 3 foot wide: the middle sized 7 foot high, and 5 wide, and the double Gabions 10 foot high, and seven foot wide in the mouths, as appeareth by the Figures 146. 147. & 148. the platform or bed, is a foot higher behind then before, or by the parapet to the end the pieces may not recoil too far, as also that with the more ease they may be drawn up towards their portholes, and seeing every piece is broad at the Axletree some 7 foot, ye must allow 5, or 6 foot for a space between the Axeltrees of each piece one both sides & so having 4, 5, or 7. foot distance between them one may the better come to the mouths of the pieces to lad them, for it is necessary to have room enough unto both sides to come to them readily, that being discharged ye may draw them up again into their places.  
If the sides lie open, they must be blinded with a parapet of Earth or with Gabions; at the furthest end of the battery, ye must make a cellar or a place for your powder bullets & match and other necessaries, for the conductor, which shall give out the powder, the bullets, with a gentleman of the ordinance, who shall give order to the Canoniers, when and how often they shall shoot, and without whose knowledge, and Command they shall do nothing: the ammunition which is there must be covered, with horse skins and hair cloth to prevent the danger of fyring, and to that end this hole is made in the Earth. The circuit of the battery is oftentimes trenched about as the others in the approaches are, but otherwise, when an enemy is not to be feared, only is compassed about with match bound up with stakes, that no man unwares or without leave may come within the compass thereof. The entrance into the said battery is made slooping, that one may the better come up, and go down, and chiefly that ye may with the more ease draw up the Canon into them. The Batteries must not be far from the trenches, but that upon all occasions they may be seconded.  
According to which directions ye shall easily make all sorts of batteries, as for exemple: The General would have you to make a battery for four pieces of Canon, ye must take for every piece at least 12 foot, in length: so that for the four pieces ye must have 48 foot, and then for the two utmost pieces each 5 foot makes 10 foot, which with the 48 make in all 58 foot, which shallbe the breadth of the said battery for 4 pieces, excepting the Parapets & Taluds, which oftentimes have as much Talud, as heith, and sometimes the half, and for the depth, ye must take 28, or 30 foot, or thereabouts besides the Parapets and Taluds, the first 12, or 15 foot are laid with Oaken planks, and the other with hurdles, for whole Canon, (as we have said) which carry a bullet of 48 pound weight, but some two foot less for the half Canon, and for other pieces accordingly; The said planks must be laid both at length and sidelings, and the hurdles along the pieces, as appeareth by the Figure 143, in which H is the ditch, G the edge, F the Talud, E the parapet, B the bedding of the planks, C the Talud. The Figure 142 is the profile of the said battery 8. 12 is the cellar, where the powder stands, which ought to be somewhat nearer the battery than is marked here, and was made in the same time as the battery was, and the Earth which is cast up out of it, is used to raze the said battery: when ye have not earth enough to make up the battery with all, than ye must dig round about it, as we have said in this exemple 143. But for a battery, which is made upon the brink of a moat, it is not commonly so high, nor taketh up such a great space, as this present battery doth; because the ground will not afford it, and that they cannot so well cover such places: so that one is constrained to entrench them more narrowly, within the place to be the more secure under convert from the short of the besieged, which forbears not to offend, ye aswell by hand granadoes as otherwise.    

How to pierce through a Counterscharfe, and to make a Gallery over a moat.  

The 34 Plate, and 150. 151. 152. & 153, Figures.  
WHen you are come with your sap to the parapet of the covert way, than you must make batteries to beat down the flanks, and the other places of defence of the Fortress or town, and withal ye begin then to pierce into their counterscharfe: and for the better effectuating thereof, if it be high ye must make an entrance into it through a mine, so that your descent into the moat may be made level with the Superficies of the water of the moat, which descent must go down slanting as F, L, & the Figure 150, G, H, I, K, is the furthest end of the mine, or the descent into the moat F, G, the height of the mine, being 6 foot or above, and some 5 foot broad, or some what more, that the Earth which is taken out of it may the more commodiously be carried away, and that the more men may march in front in it. Before ye get in the ground at the entrance ye must underprop it with posts as the former F, G, H, shows, until such time as ye are got so deep, that ye are under the Earth, and then for keeping the earth from falling down, ye must drive in posts on both sides, and lay oaken planks (which are not very broad cross upon them, continuing so as you advance your mine in keeping the Earth from tumbling down: these posts and planks must stand and join close one to another as ye see represented by the Figures 150, & 153, Plate 34. In which F, G, H, is the entrance into the mine to descend into the moat. The 3 first posts (because ye are not yet entered much into the ground) are made in that manner as the Figure 152 demonstrates, the rest are set in that form, as the Figure 150 shows, in such sort that I, K, is the entrance into the Counterscharfe. If the ground lie low, and that ye cannot get into the counterscharfe after the manner abovesaid, than ye must continue your sap, to the brink of the moat, and that ye may keep it from fyring, ye must cover it over with fir planks and cast earth upon it, that your men may work with the more safety and that ye may hinder the besieged from casting in of granadoes, and other fire works, ye must place musketeers round about the approaches, which if they perceive the enemy cast any fire works from the rampart upon it, they may give fire presently upon them.  
Being gotten through the Counterscharfe and come to the edge of the moat: ye presently cast abundance of faggots, brush & earth into it, to fill it up, and one or two of your resolutest men leaps into the moat to lay them right upon the face of the Bulwark, that as you fill up the moat to come to the skirt of the Bulwark so ye may advance your gallery till you are over in setting your posts forward, and laying planks over it and casting Earth upon the gallery, which must be 7 or 8 foot high, and 6, 7, or 8 foot broad for the larger it is so much the better and the more men may march in Front in it: the posts, or supporters may be set some 5, or 6 foot distance one from an other, which may be plancked on the in side, and on the outside with oaken planks, and the posts being some 6 ynches thick, the gapps, or spaces between then, are filled with Earth, to resist the force or violence of canon, and above the gall●rie, as is said, ye cast upon it a foot or half a foot of earth thick, and then spread it abroad with a fire-rake to keep it from fyring, which fire-rake is described in the 31 Plate, by the Figure A.   

The 151 Figure.  
WHen you have put over your gallery (as we have said) than you must begin to mine, as the place, and as the Assaillants shall find it best, either upon your right, or left hand, high or low, if the water hinder you not mining & working in this manner following: The Earth which you dig out of the mine, must be carried away in wheelebarrows through the gallery whither you will. If you please you may cast it into the water towards the angle of the Bulwark, and so fill up the moat with it, if it be not incommodious for you to carry it through the gallery: But on the other side, ye must mark well the turnings of your mine, which may occasion the Besieged to countermine, and so to hinder the design of the Besiegers. For if they have once discouvered, or met with the mine of the Assailants, they must be forced to stop it up & to abandon it, and so begin another. The Countermines which are made in ramparts, or Bulwarks when a fortress is new made, being some 5, or 6 foot high, and 3, or 4 foot broad, are of singular use, which do encircle the place, and from them ye may hear the least noise that is made on the outside, and which way one works, and by that means may hinder them the better from myning.  
These mines are commonly made in the form of a paralellograme or a long square, to wit, the chamber in which the powder is laid, must be 4 or 5 foot high, and 3 or 4 foot broad, and in length answerable to the ponderosity or weight of the rampart, and according to the breach, which ye intent to make. If ye would blow up the rampart on the inside, you must pierce into it some 6, or 8 foot, and then ye must make your chamber, but only 4 foot high, some 3 or 4 foot broad, and some 6 foot long, according to the number of the barrels of powder, which you mean to say in it, and the greatness of the wall, which you intent to blow up. The reason why the chamber is made here but only 4 foot high, is to the end that the exhalation finding the least resistance towards the inside, forces it with the more violence and makes therein a greater shaking and breach to the terror, and hurt of the besieged: when ye would blow up the upper part of the rampart, them you make your mine some what ascending upwards: if the moat be very deep, that your design be not frustrate by reason of the water, for your better assurance, ye must make your chamber some 5, 6, or 7 foot high, to the intent that the exhalation may break upwards according to the intention of the Master-miner. But the entrance into the said chamber, as we have said above, must be only 4 foot high and 3 broad, to the end one may the better stop it, and hinder the exhalation from breaking out backward, towards the gallery, which one ought diligently to prevent, in shunning those inconveniences and mischiefs, which oftentimes by such casualties have happened heretofore.  
Then having chambered your powder, and noted well, that those within have not discouvered it: you must stop, and shut up your mine exceeding firm that it may take the better effect. To do this, the best way willbe to stop it at E, with two huge planks, just at the said entrance at E with great sparts of timber, and driving them into the Earth, as firmly as possibly may be. The chamber F G H I is 4 foot broad, and F G, & F, I, is six foot in length: Oftentimes the breadth is but 3 foot, that one may be the better assured of the resistance of the posts and planks. In one of the planks, which stoppeth up the entrance E into the mine, ye make a hole in it, to put your train through which runs from E, to B, that through it ye may give fire to the powder in the said chamber: After these planks ye dam up your mine with good Earth, from E, to B: the turnings C D E, are made to delude the besieged, that they may not find out your mine by counter myning; but most often it runs right forward, or somewhat winding. The length of the chamber F, I, (which is here but six foot) is made as long as necessity requireth, but the breadth F G, is ordinarily no more than 3, or 4, foot at the most: and to weaken the more the place which you intent to blow up; me thinks it would be good before ye spring your mine to dig some Holes in some corners that the exhalation may have the better vent, and easy bursting out. Some are of the opinion that a barrel of powder will blow up 12 foot of Earth, according to which ye may make the said chamber, and lay in as many barrils of powder as you please, to the end you may make your breach the larger & more spacious. But seeing this is yet unresolved, I will leave the judgement thereof to those which have more experience therein then myself.    

How one must be prepared against a Siege.  

The 35 Plate and 154 & 155. Figures.  
HAving succinctly spoken of Approaches, Sapps, the descent into a moat, galleries and mines: me thinks it will not be amiss now to treat of the preparations which must be made against a siege.  
If than one should be suddenly surprised: so that there is no time to make any outworks, half moons, Horn works or other pieces of fortifications as well loose, as joining to the wall, or place: one ought in my opinion (besides the care that ought to be taken for materials, ammunition, and victuals (whereof I do not intend here to speak) one is to take special heed upon what side of the town or Fortress the Enemy will begin and run his approaches, towards what Bulwark he makes them, on which side I would make some works to hinder an enemy, as we shall declare in the 38 Plate & Figure 159. But if one be advertized of the Siege some time before, or that ye imagine, it willbe good to provide such necessaries and to fortify those places with a more capable defence, which we intent to speak of hereafter (according as ye shall find the weakness of the place to require, & as time will permit you to do it before ye be besieged) as well without the town as within, to make good entrenchments & to provide munition, victuals, and men sufficient to defend that place, without the Town, you must make Horneworks invented of late years, Halfmoones, traverses, and other works, all tending to hinder the Besiegers from getting into your moat, and to prolong the siege with hopes of relief, seeing experience hath taught us, that when an Enemy comes once to enter your moat, and to put over his gallery, that town or place cannot long hold out: if you have not made strong works, entrenchments, and cuttings off without (to give an Enemy his handful, and to make them gain your work● inch by inch) which notwithstanding are not of so good a resistance, as is your settled, firm, and solid rampart. These works aught to be drawn out according to the greatness of the place, and the men which ye have to man them. For, if ye have many of those works, they will require a great many men to defend them so that if ye have not men enough to keep them, that labour and expense will be in vain, and so an Enemy may soon become master of them. And seeing in all observations, exemples are of great efficacy I have thought good, and profitable to make you understand my intention by representing unto you the Plate of Gulich prepared against that siege, whereof the Bulwarks of the town are marked by the numbers 2, 3, 4, 5, the Castle, which is quadrangular by the cyphers 6, 7, 8, 9 And seeing the town stands upon a low ground, and the Castle upon the side 6, 9, 8, upon a higher ground: in all likelihood the approaches was to be begun on this side, as the Besiegers did in the year 1611, for which reason, and to hinder the Approaches on that side, they made the angel's 9 8. and between them the Horneworks c, d, f, the Ravelins c, g, h, b, being not further asunder one from an other, but that from the curtain one might defend the utmost horns c, d, f. But seeing c, and f, are very narrow, to wit, only some 24 rod, and shoot out far from the body of the Fortress: in my opinion they ought rather and it had been much better to have made the angles in the form of the Raveline c, & f, and so have quitted the said horne-works, because (as we have said) they were too strait, and narrow, and to have made them at g, or upon the curtain 6, 9, the Horneworks 1, and (also whither the Raveline b, could not have defended the Raveline c, in the place of the angle of the Bulwark 8) and to have made the hornworke q. upon the curtain ●, 8, and so might have saved the Raveline a. And between the two Bulwarks of the town 4, & 5, the Raveline r, & o, between 3, & 4, and so have saved the entrenchment n, which runs out far from the angle 3, and is of no great defence. The Raveline k, is made between the Bulwarks 2, & 3, to take away the entrenchment m, made in a Tenaille, but the sides being of a very small distance, & for this consideration of little use, yea many times hurtful as experience hath shown oftentimes. The reason why I make of the angles 8, and 9, the Ravelins, c, and f, in stead of the Horneworks, is, that the Horneworkes d, & i, might be the better defended, which willbe reciprocaly defended from the said Horneworks, and will not be so far distant from the body of the said fortress. Those which are of an other opinion, may follow their own intention and experience, against whom I will not contend acknowledging that experience strikes a great stroke in all occurrence, saving that I am only permitted to give advise thereupon.  
The said Horn works are made in breadth, and height as necessity requires and as they may resist the forces of an Enemy, and according to the time they have to make them. For the first one makes a Rampart, or a parapet, some 12, 16, or 20, foot broad, 6 foot high, and a ditch of the same breadth of 12, 16, or 20 foot, and some 6 foot deep. If one cares not for the expense and hath time, ye may add to the former breadth 6, 8, 10, or 12 foot more, in enlarging also to so much the breadth of the ditch, and then the Rampart is made some 3. 4. 5. or 6 foot high, and upon it also ye shall make a parapett of 6 foot high, with a foot-banck some 3 foot broad, and a foot high, the ditch is enlarged and deepned, according to the greatness of the Rampart, and as we have taught in the Plates of our regular fortification before: and the higher the Ramparts be, the entrenchments are the better covered: one may make them the further from the place, to wit within a musket shot, so that there willbe a better means for entrenching, and cuttings off, and to foreslow and hinder an Enemy from advancing his approaches.   

The 38 Plate & 159 Figure.  
IF one hath not time to make these works abovesaid, as when an Enemy is nearer at hand, than one did expect: After ye have observed well the place towards which the Assaillant makes his account to begin his approaches, as towards these two Bulwarks, one might make little ditches without them as A B C D, which are in the extension of the Capitals beginning in the Angles A, C, of the length of 600 foot, or thereabouts (to wit, that the utmost end B, D, be not out of muskett shot) of the breadth of some 6 foot at the most, and 5 foot deep without the parapet, the Earth being plained from the one part to the other, to the end that you may not hinder the sight of the musketeers, which you shall lodge in them, and in stead thereof upon the edge of these shallbe set small muskett basketts filled with Earth, easy to be removed from one place to an other: It willbe good for one's better security to hold there a Corpse de garde, that if the enemy seek to beat them from thence, they may be the more able to discharge their duties, whereby (in my opinion) the approaches of the Besiegers may be much hindered, & being so puzzled they willbe constrained every time to alter their designs and so to begin their approaches again a good way further off, and more crookedly, as towards B, D, to which being come they can get no advantage, seeing they are enfiled with the angles of the Bulwarks: If they had time to make chests in them, or to make them in such a sort, that the Besiegers could not drive them out of them, it would be much better.    

Of Cuttings off, aswell general, as particular.  

The 36 Plate & 156 Figure.  
IF an Enemy be got into the skirt of the Bulwark g, h, and that one hath not time to cut it off royally, & yet you are resolved to make him gain the place foot by foot, you must make the cuttings off f, l, m, whereof n, o, p, is the ditch, making the angle of the Tenaille l, as much pointed as possible may be, that the lines f, l, and l, m, may be the better seen from one an other, carrying the outward edge of the ditch n, o, & p, as near the skirt g, h as possibly may be, that ye may have the ditch as large as may be, to hinder the taking away of the tenaille l. If the whole face be not ruined, but only the angle of the Bulwark r, one shall make the angle t, from the Tenaille s, t, u, (which in regard of the smallness of it willbe better than the former) so that on the inside of the cutting off x, y, z, one may have space there to make the ditch as great and as deep as may be: the entrance ought to be in the angle of the tenaille t. But seeing that fortification is better, which hath two flanked, and flanking angles, ye must make the entrances in and out also double, to have the same the more easy, and if time & occasion would permit, they may make in the angle t, two issues, which in my opinion might be made upon the two sides s, t, and t, u, as near the angle t, as one can: and ye must note by the way, that the Bulwark against our intention is made massy, and not hollow.   

The 38 Plate & 159 Figure.  
IF the two faces be wholly spoilt, as the Figure 159. demonstrateth, than I should think it good, to make the angel's a, b, c, containing the angle of the tenaille b, as little as may be to gain the better place of defence in making the ditch, as large and as deep as is possible, as it is here marked out by the lines d, g, & e.  
When the Bulwark hath a Cat noted B, and that its angle h, is ruined, than one might so order the cuttings off l, m, n, o, & the ditch p. q. r. s▪ 't, that the besiegers should be compelled to undermine the Cat, and to pass the ditch, which the lines p, q, r, s, t, represents upon the edge exterior.  
But if the whole Bulwark be ruined, you shall draw from the midst of the curtain of the Bulwark ruined a privy line marked by the letters h, I, upon the inside whereof, ye shall make the two skirts of the bulwarks, according to our general method given in our regular fortifications, and the other cuttings off h, k, l, m, n, i, shall have: the other Bulwark B, being also ruined, one may draw the privy line s, h, & make within it two other skirts, as h, oh, p, q, r, s, and so o, h, k, p, l, willbe a perfect bulwark, having flanks, gorges, faces and curtains of the same proportion, as the two Bulwarks had which were ruined, and by this means every place aswell regular as irregular may be cut off. Many other sorts of cuttings off might be described, according to the situation of the place, and the manner of attempting, by the industry of the lngenier, which hath the ordering of such cuttings off: But seeing the like inventions were abundantly practised in the Town of Ostend, the Lovers thereof may peruse the 25 Plate of this bocke, and see what cuttings off was used there, the enemy having possessed half the town, before he gained the whole.    

How one ought to carry himself upon an Assault.  

The 36, & 38 Plates, & 156, & 159, Figures.  
IF the enemy prepares to give you an assault, and that his breach be great enough, those within must labour by all means and seek to stop it & to defend themselves in the best manner they are able, and above all if it be possible to drive in some strong piles upon the top of the breach, which we before have called palisadoes, for an assault, described in the 33 Plate and Figure 140, each palissadoe having 2 iron pins some 10 ynches long, driven through them, as we have said that by this means one may keep them from coming up and entering the rampart or breach. Now seeing the breach ought to be defended by able men, furnished with arms fitting to such an end, ye must make choice of your best ablest, and most courageous men, which must stand under the breach to second those which helps to defend it, and shall fall on, when they see their fellow-soldiers repulsed back. On the inside of the cutting off you shall place some other troops of a stronger body than the former, to the end, that if those men, which defends the breach, should be forced to retreat into the ditch, that the others at that very instant may show themselves upon the top of the rampart of the new cutting off, and if there be any means to plant a piece of ordinance or two upon it (which may be blinded till the Besiegers falls on) it will greatly offend them. The entrances, and sallies to the said breach, aught in my opinion (if the cutting off be in the tenaille of the angles t. l. Figure 156, & b, 159) of an easy access, being raised as little as possibly may be, whereof the one may serve for an entrance, & the other for a coming out, choosing the one or the other for the most commodious, according to the situation of the place. And seeing the cuttings off (as we have said before) are esteemed the best, which have two angles flanking, ye must at the first make the cutting off, h, k, l, m, n, i: Figure 159, to be raised even with the height of the rampart, or according to the height of the Bulwarks, if they be a little lower, than the curtain, if that the batteries do not command them. For in such a case, you must raise it much higher, that from thence (with the more vivacity, and courage) ye may repulse the Assaillants. But seeing this cutting off is of a better defence, and is much more labour, than the cutting off, a, b, c, ye must consider well, it time will give you leave to do it, if not ye may make use of the cutting off a, b, c, for oftentimes necessity hath no law.  
And seeing experience hath taught us too much what difficulties, one shall meet with all in such cutting off: the only way is in my mind to hinder (as much as possibly may be) the Enemies' descent into the moat: which besides other inventions that are in use, may be done by the means of the cuttings off, at the lines a, b, & c, d, Figure 159, which are made here right opposite to the angles of the Bulwarks from one part to an other, besett with small muskett baskets & filled with Earth as we have said before.    

Of Casemates.  

The 37 Plate & 157, & 158, Figures.  
FOr as much as we have seen the difficulties, which Casemates have caused to the besieged, and the small benefit they have received by them, which not withstanding have been made with all the industry that possibly might be, thereby to hinder an Enemy from putting over a moat and making his batteries upon the brink of the moat, to beat down the flanks, and to dismount the pieces of ordinance, planted in the said Casemates, to wit, that besides the expense, the gorges are made by this means lesser, the Orillon or pillow being no more than the 2/3 of the flank, and is of little resistance, and on the other side gives but little advantage, being soon stopped, as we have seen in time past: I was minded not to have spoken of them at all, though I esteem them good, if they were made in such a sort, that there mouths might not be stopped up, and the pieces within them dismounted: which hath not been done hitherto to my remembrance. For●f●hese Casemates be made of brick bats, when the Besiegers shall play upon them with there ordinance, the bricks flying into the portholes will do more hurt to the Canoniers, and other men, than the Enemies' bullets themselves, and by this means the portholes willbe easily filled, and stopped up, as we have seen in divers places. If your Casemate be made of Earth, ye must give it a great Talude (that is much slooping) which maketh the gorge so narrow, namely in those Fortresses, which are under an Hexagone, that oftentimes there willbe hardly any entrance into the bulwark, which we call the gullet, the Orillon, and the flank very little, and consequently willbe of little resistance, wherein they find so many difficulties, that many great Captains have resolved wholly, to leave them unmade. If one could not preserve them otherwise then they have done to this present: I should approve of them: But seeing I cannot resolve of a thing, which I dare not wholly approve off, because experience, and many men slain in the wars have not found it good, this is my opinion also. In the Figures 157, & 158, of the Plate 37, a, b, is the utmost end of the shoulder, the double of p, a, Figure 157, the mouth, or porthole of the Casemate, and as b, p, maketh 150, foot, so p, a, will make 50 foot, p, t, equal to p, a, willbe likewise 50 foot, t, v, is 36 foot, from a ye shall draw a line to u, that ye may the better discouver the exterior brink of the moat, and the said 't, u, shall contain three portholes for three pieces of Canon, which shallbe vauted over from d, e, to, t, u, with steps as the Figure 158 demonstrats, marked between 't, v, and e, d, in such sort that the first Vault on the side of e, d, is close by the superficies of the water enlarging or raising the said Vaults more and more, until that the last vault towards 't, v, be raised above the superficies l, f, g, k, which is the plateforme of the Casemate, some 3 foot, or thereabouts, and seeing that 't, e, & d, v, are about 20 foot, ye shall advance as far as possibly ye can the parapet t, f, and g, v, as much as the canon, and the place will permit you, to give the better solidity & firmness to the parapet of the Casemates. Then upon the top of the basis f, g, e, d, the said parapet shallbe so raised that the inside f, g, shallbe lined with a wall, that it may prevent the falling down of such a height, & so that the enemy may not discover the upper part of your vault, h, i, k, l, and firmly joined aswel to the wall, as to the Orillon and on the outside with hard Earth, as strong as possibly may, going down sloopingly, that it may not be subject to tumble down into the moat, and this will hinder the Enemy greatly from entering into the moat and putting his gallery over. For by this means they must be driven first to beat down the shoulder, and make it to fall into the moat at the space p, a, e, d, which for this reason must be made as deep, as possibly ye can, to the end, that the portholes 3, 4, & 5, be not easily stopped, the place l, k, f, g, is about 20 foot uncovered, and the vault i, h, l, k, also 20 foot broad. The line h, i, is about some 54, foot: the Colomne is made in the midst of l, k, to make the vaults crosswise: because the distance between l, k, is too great, to make there a single vault, which may serve to make upon it the parapet of the superior place, to gain more place for the gorge, and to lodge the Canoniers, and there ammunition dry.  
The entrance into the Casemate must be in that place, where m, n, is, under the rampart, and must be vaulted from m, to i, & from n, to o, being some 10 or 12 foot broad, or thereabouts, that ye may the better draw in your ordinance & all what ye have need of into the Casemate, and make it as high as necessity requires: The orillon a, v, y, b, is all together massy, that it may give the greater resistance. And because ye may the better understand our intention, we have added hereunto the Figure 158, which raised work is represented in perspective: in which ye may see the parapet of the false bray, the portholes of the Casemate, and other things, which we have thought necessary to make known.   

The 38, Plate & 160, Figure.  
A Fortress being thus provided with good falsebrayes, aswell under the Bulwarks, as along the curtains, broad according to our former plots: I could wish that a Casemate were made in that form which we shall now describe, and that the curtains, were drawn in as the far as the breadth of the falsebray with the parapet contains, to wit, that the false brays under the Bulwarks might end on the outside of the foundamentall lines a, b, and the like: and contrariwise, that in the curtains the said falsebrayes may end within the said fundamental line c, d, so that the space i, f, g, may be accommodated, as the Figure 160, representeth, to plant therein two pieces of ordinance, which will work no small effect: & by this means the Casemates, willbe better preserved then otherwise, because ye need not fear the flying of brick-batts about your ears, seeing that the flank e, will serve as a stay and a covert thereunto: so that in stead of one shoulder ye shall have two, to wit, b, p, and c, e.  
The descent into the falsebray, which goes round about the Bulwark ought to be made at m, and to come out at y, and seeing one makes sometimes some privy sallies, between A and p, one might also make the entrance into the Casemate in this place, making a vault from m, to y, to wit, in the underpart of the Casemate: The entrance into the false bray from the curtain ought to be made at w, from one & the other part. The letters q, r, s, t, v, represents the place where I would makes Cats, if one resolves to make them: but seeing I am not resolved about this point, I will rather speak nothing thereof at this present.   

Figure 2. 160. & 38. Plate.  
IF the Bulwark be not massy, but hollow, ye may make your entrance into the Casemate, as here towards o. r. p. l. z. and the entrance into it should be at the point, l. all the space l. z. p. r. o. being open.    

Of Brass ordinance.  

The 39 & 40. Plates and 161. 162. 163. 164. Figures.  
WE were minded to have treated of divers other dependences belonging to fortification, as of bridges, gates, ports, foundations etc. how towns & public places ought to be fortified; but the Printer not willing to stay any longer for the finishing of this book, it is not possible for me to effect my intention, so that my project also touching the casting of brass ordinance is for this reason also frustrated, having been minded to have described the proportion, aswell of their carriages as of their charges, and other dependences about them. To which end I had ordained these 4 brass pieces which are usually cast in the united provinces, according to the scale hereunto annexed: whereof the least carries a bullet of 6 pound weight with the Figure noted 161: the second is the Figure 162 & carries a bullet of 12 pound weight, the third 163 carries a bullet of 24 pound, and the fourth marked with the Figure 164. carries a bullet of 48 pound, of the two last, the first is a half Canon, and the last a whole canon, or pieces for battery: the two others are field-pieces, bearing a bullet of 6, and 12 pound. In the mean while, he that desires to know the proportions of them may make use of this scale, till we write more at large thereof.    

ALBERT GIRARD.  
BEfore we come to describe the two tables, whereof we have made mention before, supputed of a new: we will explain some terms, which we have introduced, because they were nameless before. To that end we will take one of the Figures of the eleventh Plate excepting the last which is the 68, and the line N C being drawn, which we will term the gullet (to distinguish it from N A, or A C, the gorges) which comes to cut through the Capital, being drawn further within the Figure, in a point, where the letter Z, is made, then E, Z shallbe called (the extension) that is the lengthened capital, for the Bulwarks extends so far outward. Then the angle G, A, C, (the forme-flanck). Also in the 60, or 61, Figure E L, shallbe called the Raid majour, and L A, the Raid minour, finally, in the sixth Plate, the line D, F, or K, L, is called the surface as being placed before the face (for surface is a word not much used, and as superficies signifies all one thing: ye must not think that the second table is calculated amiss; because the progression of the second flanks do increase, even to that of the Heptagone, and then diminisheth: for if one should continue the table from the one end to the other, it would follow, that the angles of a Polygone, should be equidistant 130, degrees (which well nigh contains the second flank majour 8. 412) & so have the second flanks equal. we understand equidistants the one so much more than 130, as the oth●● less. Likewise the defence flancking, diminisheth even to a Pentagone & then augmenteth. The least that we can observe is, where the angle of the Polygone is 113, degrees, to wit, of 53, 366, & the Equidistant are equal. One may say the like of the first table, which decreaseth to 48 rod, and afterwards increaseth. The fortification of a right line holding not with the one or the other table, is placed between them both. Those that will make use of them proportionally, for the building of irregular Figures commits no small errors, though Marolois was of this opinion, and so much the more, because the angles of the Polygone are about the right angles, for the other Polygones differ always less one from an other, the further they fall from the quadrate: so that we to this end, if we were willing to make use of them, the tables whereof the angles of the Polygones are progressive of 3, or 4, degrees of space, would be of a better choice, & should here take place if time would permit us. Let there be here some small exemple thereof, and suppose, that one would fortify an angle of 99, degrees, whereof the two sides are each of them 36 rod, and 3 primes, & take a Pentagone, or a quadrate to imitate, and let us take a quadrate by which means, he shall find the lines as they are hereunto annexed: which done, let us take the 7, Figure of the second Plate: to speak more intelligibly thereof let us demand of him, how much the angle C B Ewilbe. 

BF Capital 13, 15  
BE The face 16, 00  
ED The flank 5, 16  
DF The gorge 6, 15  
BA Defen. Flanc. 33. 94  
Angles required  
FBE 32 deg. 15. min.  
BFD 130. 30  
EDF 90.  
Angles of a quadrate  
30 degrees  
135.  
90.   He will say, that to the half of 99 he hath added 15, and the sum is for the said angle C, B, E, and that the angle B, F, D, is adjunct of the half of 99, to wit, 130, 30, afterwards he cannot deny, but the number of lines, which he hath found may receive the angles of the square-bulwarks halued as there annexed: for the lines are drawn proportionally from thence: as then the consequence will be absurd, seeing that the quadrangle, having the four lines known & D the right, which are 5 common terms of both sorts of a quadrangle abovesaid: The unknown angles shall receive each of them a certain number of degrees, as appeareth by our trigonometry, and not two sorts of numbers for each angle, so that if F, comes to be 135 degrees as it is, it can never make 130, and 30, and so of the rest. Now of one should take the angles as he requireth it, and the two sides F D, D E, the other lines will be otherwise, than he hath calculated them, for in stead of 16, 00, the face, one shall have 51, 04; and in stead of its flanking 35, 94, ye shall find only 32, 44, and so of the rest, which is an error of 3● rod, upon 32, here then are small lines & great faults, what would those be then in great and solid fortifications. To conclude, we will propound one question aswell to exercise those which are addicted thereunto, as for to end this discourse: A square fortress regular hath the face 24 rod, & the curtain 36, with an angle forming the flank of 40 degrees: without a second flank: how many then will the other angles and lines be? The manner to resolve this, and the solution ye shall see hereafter, when we have made some treaty upon that subject, with hope shortly to bring to light, the restauration of Geometry, touching the parts thereof, which have been lost hitherto, whereof the Authors are Euclid, Aristeus the elder, Eratosthenes, and Apolonius Pergeus, with some other parts of the mathematics all according to our weak power, by the help and grace of God.  
FINIS.    

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


1. De oude polder.  
2. Cassemat.  
3. Het nieuwe west ravelyn.  
4. De west poort.  
5. Nieu porcespic.  
6. Helmont.  
7. De oude Sluis.  
8. Santhil.  
9 Halve maen so hooch als de wall.  
10. Sluyseerst gemaeckt omt'water inde graft te setten.  
11. Trencement.  
12. Bolwerck.  
13. Contrescherp.  
14. Halve maen.  
15. E'en sluyseerst gemaeckt.  
16. Sorte nae de polder halve maen.  
17. Eee sort.  
18. Nieu polder.  
19 Sorte.  
20. Nieu helmont.  
21. Sorte.  
22. Nieu west poort.  
23. Vlammenburch.  
24. Peeckels bolwerck.  
25. Nieu Polder.  
26. Spaens bolwerck.  
27. Koestal. oft Suyt host bolwerck.  
28. Suyt bolwerck oft Treurenburch.  
29. De urygaerden.  
30. Polder halve maen.  
31. Baterie.  
32. Cat.  
33. Higher leggen de schepen.  
34. Nieu Santhil daer de oude kerck plachte sta●●●  
35. Nieurve haven.  
36. Noortwest bolwerck.  
37. De host poort en het ravelyn.  
38. host ravelyn.  
39 Spaens halve maen.  
40. Slimmers halve maen.  
41. Elcks verdriet.  
42. Suyt host ravelyn.  
43. Suyt ravelyn.